ACTA SCIENTIARUM MATHEMATICARUM

Transcription

ACTA UNIVERSITATIS SZEGEDIENSIS
ACTA
SCIENTIARUM
MATHEMATICARUM
ADIUVANTIBUS
B. CSÁKÁNY
S. CSÖRGŐ
E. DURSZT
F. GÉCSEG
L. HATVANI
A. HUHN
L. KÉRCHY
L. MEGYESI
F. MÓRICZ
P. T. NAGY
J. NÉMETH
L. PINTÉR
G. POLLÁK
L. SZABÓ
I. SZALAY
Á. SZENDREI
B. SZ.-NAGY
K. TANDORI
V. TOTIK
REDIGIT
L. L E I N D L E R
TOMUS 50
FASC. 3—4
S Z E G E D , 1986
INSTITUTUM BOLYAIANUM UNIVERSITATIS SZEGEDIENSIS
A JÓZSEF ATTILA TUDOMÁNYEGYETEM KÖZLEMÉNYEI
ACTA
SCIENTIARUM
MATHEMATICARUM
CSÁKÁNY BÉLA
CSÖRGŐ SÁNDOR
DURSZT ENDRE
GÉCSEG FERENC
HATVANI LÁSZLÓ
HUHN ANDRAS
KÉR.CHY LÁSZLÓ
MEGYESI LÁSZLÓ
MÓRICZ FERENC
NAGY PÉTER
NÉMETH JÓZSEF
PINTÉR LAJOS
POLLÁK GYÖRGY
SZABÓ LÁSZLÓ
SZALAY ISTVÁN
SZENDREI ÁGNES
SZŐKEFALVI-NAGY BÉLA
TANDORI KÁROLY
TOTIK VILMOS
KÖZREMŰKÖDÉSÉVEL SZERKESZTI
LEINDLER LÁSZLÓ
50. KÖTET
FASC. 3—4
S Z E G E D , 1986
J Ó Z S E F A T T I L A T U D O M Á N Y E G Y E T E M BOLYAI I N T É Z E T E
Acta Sei. Math., 50 (1986), 253—286
Abstract Galois theory and endotheory. I
MARC KRASNER
Introduction
Let us consider the situation in the classical Galois theory. Then we have a
field extension K/k which is either normal and algebraic or algebraically closed.
A permutation a: K—K is called an automorphism of K/k if, for each a,b£K
and agfc, we have
(1)
o-(a-\-b) =
o-a+a-b,
(2)
and
(3)
a • ab = (<7 • a)(<7 • b),
a • a = a.
Observe that the assumption "bijective" can be replaced by the weaker one "surjective", i.e., the automorphisms of K/k are just the surjective mappings a: K-+K
satisfying (1), (2) and (3). This observation follows readily from the fact that fields
have no non-trivial ideals.
The automorphisms of K/k are known to form a group under the composition of mappings. This group, denoted by G(K/k), is called the Galois group of
K/k. Let A be a subset of ^ a n d let G(K/k; A) denote {o£G(K/k); (\/a£A)(<r • a=a)},
which is a subgroup of G{K/k). Then Ak={a£K; (y<r£G(K/k; A))(<r-a=a)}, the
set of all elements in .fiT preserved by each a£G(K/k\ A), is called the Galois closure
of A. The classical Galois theory asks and answers the following two questions:
(a) How to characterize Ak in terms of A and the field extension structure
(K; x+y, xy, k) of K/kl Answer: Ak is the closure of A\Jk with respect to the
operations x+y, xy (defined on KxK),
JC-1 (defined on AT\{0}) and, if the
characteristic p of k is not zero, \x
(defined on K"= {ap; a£K}).
Received January 6, 1982- The original version of the paper was revised by the referee. The
author could not be asked to agree to the.modifications as he passed away in 1985.
l
M. Krasner
254
(b) Which subgroups of G(K/k) are "Galoisian", i.e. of the form G(K/k; A)
for some AQK? When the degree [K:k] is finite then the answer is: all subgroups.
Conditions (1), (2) and (3) in defining automorphisms seem heterogeneous at
the first sight since (1) and (2) concern the preservation of some binary operations
while (3) concerns the preservation of some elements. Yet, these conditions turn out
to be of the same nature when formulated in terms of relations. There are two manners of defining the automorphisms of K/k in this way:
I. An automorphism of K/k is a permutation a of K such that
(a) a + b = c oa
(fi) ab = co{a-
• a + a • b = a • c,
a)((x -b) = a-c,
and
(y) for each a£fc, a = a o a • a = a.
We can formulate Conditions (a), (fi) and (y) by saying that a preserves the relations x+y=z, xy—z and, for each a£k, x=a, where a is said to preserve
a, say ternary, relation Q if for any triple ( a , b , c ) £ K 3 we have (a, b,
C)£QO
«=>(<7 • a, a • b, a • c)6 g.
II. An automorphism of K/k is a self-surjection n of K such that
(a') a + b = c => (i- a + fi-b = /i-c,
(f}') ab = c =>({i-a)(ji-b) = ¡i- c, and
(y') for each a£fc, a — a=>n-a — a.
If we drop the condition that ¡i is surjective and consider self-mappings of K satisfying (a')> (/O and (y') then we obtain the notion of endomorphisms of K/k. It is
not hard to show that all the endomorphisms of K/k are automorphisms, provided
K/k is normal and algebraic. But in the general case the endomorphisms of K/k
form only a monoid, i.e. a unitary semigroup, D(K/k) with respect to the composition of mappings. This monoid always contains 1 K , the identical mapping of K.
As previously, D{K/k, A) will denote the set {5£D(K/k); (\/a£A)(8-a=a)},
which
is a submonoid of D(K/k). Further, it can be proved that {a£K;
(\/8£D(K/k))
( < 5 i s the closure of AUk with respect to the operations x +y, xy, x~x
p
and, if p9±0, Yx, as before.
We can formulate Conditions (a'), (/O and (y') by saying that 5 stabilizes the
relations x+y—z, xy=z and, for each a(Lk, x—a, i.e. 5 transforms every system
of values satisfying any of these relations into a system of values satisfying the same
relation, but nothing is required for systems of values not satisfying these relations.
In both of these manners we have a particular case of the following general
situation: given a relational system, i.e. a base set A (here A=K) and some rela-
M.Krasner:Abstract Galois theory and endotheory. I
tions on it (here x+y=z, xy=z and x=a for a£k). (In the sequel relational
systems will be referred to as structures though the term "structure" has a more
general meaning in the literature.) We consider the group G of all permutations of
the base set preserving the given relations and the monoid D of self-mappings of
this set stabilizing these relations. Then in Question (a) we asked what was the set
of relations of a certain form (in our case x=a for a^K) preserved by all <r£G
and stabilized by all SdDl The answer was that it is the set of relations x=a where
a belongs to the closure of A(Jk with respect to some operations arising from the
relations x+y=z and xy=z.
The above considerations naturally raise the idea of considering any first order
structure E/R where J? is a set of (not necessarily finitary) relations on a base set
E, the automorphism group alias Galois group G of E/R (consisting of all permutations of E that preserve each r £-/?), and the endomorphism monoid alias stability
monoid of E/R (consisting of all self-mappings of E stabilizing each r£R). Then
the question analogous to (a) is how to characterize the relations on E that are
preserved by all a£G or that are stabilized by each <5£D, respectively. Of course,
we want the answer be somewhat similar to that 'in the classical Galois theory.
Therefore the answer should be (and, in fact, will be) that they are the relations
belonging to the closure of R with respect to some appropriate operations. But,
first of all, all such relations do not form a set because any set occurs among their
argument sets. So, at least in the first study, we have to limit the argument sets of
the considered relations so that we fix a set X° (of sufficiently large cardinality)
and consider the relations whose argument sets are subsets of X°. On the other
hand, we cannot hope in this general situation that the sets
R(X°) = {r; rQ Ex for some XQX°
and each <r£G preserves r}
and
R(X°) = {r; rQ Ex for some X Q X° and each S£D stabilizes r}
are the closures of R with respect to some operations on the base set E arising from
the relations r£R. A priori, it may be hoped only that R(X°) and R(X°) are
closures of R with respect to some set theoretical operations on relations and these
operations do not depend on the particular choice of R. Such is the case, indeed:
there exists a family of such operations, called fundamental operations, so that R(X°)
is the closure (within the set of relations with argument sets included in X°) of R
with respect to these operations while R(X°) is the closure of R with respect to a
part of this family, called the family of direct fundamental operations. The theory
concerning the preservation of relations by permutations of the base set and the
Galois group is called abstract Galois theory while that dealing with the stability of
relations by arbitrary self-mappings of E and with the stability monoid is called
abstract Galois endotheory. Although Question (a) has the same answer for both
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M. Krasner
theories in the classical Galois theory, it has quite different answers in the general
case. Question (b) in the general case asks: which permutation groups on E are
the Galois groups of appropriate structures E/R and which monoids of self-mappings
of E are endomorphism monoids of some structures E/R1 The answer is that any
group and any monoid on E are such.
Once this study with a fixed X° has been done we can introduce the classes R
and R of all relations preserved by each o£G and stabilized by each d£D, respectively, and we can study them in a non-axiomatic way. In the present paper we will
do it roughly within the frame of Bernays—Godel axiomatic set theory albeit this
is not the only possibility. On the deep analogy of the terminology of classical Galois
theory, the classes R and R will be called abstract fields and abstract endofields,
respectively. With some precaution, it is possible to consider and to study certain
mappings between them. In particular, a bijection of an abstract field or endofield
onto another one (with a different base set in general) which commutes with all funda-:
mental operations is called an isomorphism. It will be proved that any isomorphism
between abstract fields or, under certain conditions, between abstract endofields
is of a special form called transportation of structures. After these so-called isomorphism theorems the notion of abstract endofield homomorphisms is also intror
duced and a much more difficult homomorphism theorem is proved in order to
characterize these homomorphisms.
Let k be an abstract endofield (which may be, in particular, an abstract field),
and let A be a subset of the corresponding base set E. Denote by k(A) the endoextension of k by the set of relations {x=a; a£A}, i.e., the abstract endofield
generated by k\J{x—a; a£A}. Then k(A) does not depend on the particular choice
of x. Such extensions k{A) of k are called its set extensions, and their study is called
abstract Galois set theory. A theorem is proved, which describes the relations in
k(A) in terms of A and the relations in k. As a consequence of this theorem, a family
of partial operations on E is defined from k such that Ak={e~, (x=e)£k(A)},
the
so-called rationality domain of k(A), is just the closure of A with respect to these
partial operations. This result can be considered as the first step of deducing the
answer to (a) in the classical Galois theory from that given in the general Galois
theory. The second step is the theory of "eliminating structures'-, which will not be
exposed in this paper. The third step starts from the main result of the second one
and allows us to understand and to foreseee the deep reasons why Ak is the closure
of AUk with respect to x+y, xy, x~x and \x in the classical Galois theory.
While dealing with the theory concerning R(X°) and R(X°) then X° is of sufficiently large cardinality means that card X° ^card E. In particular, when the base
set E is finite, we can-take a finite X°. Then the fundamental operations become the
realisations of operations in the (first order) predicate calculus with equality for
O
X as the set of object variables on the model E/R. Similarly, the direct fundamental operations are just the realizations of a "strongly positive" part of this
calculus, which is generated by \J, &, 3x for x£X°, adjunctions of variables belonging to Xo, equalities x¡=xj (x,, XjZX0, x^x/),
and the identity. So in the general
case the fundaméntal operations can be considered as infinitary generalizations of
the previous finitary operations.
Except for certain points our approach is independent from the axiom of choice,
but this is not the cáse for the afore-mentioned theory of "eliminating structures".
In Section 1 we give the precise notions of points, relations, structures^ etc.,
define the action of mappings of the base set on them, and introduce the Galois
group, the stability monoid and their "invariants". The fundamental operations
are defined and studied in Section 2. In Section 3 we study the interaction between
mappings (and, in particular, self-mappings) of base sets and fundamental operations. In Section 4 we prove the main theorems, i.e. the equivalence and existence
theorems, of the abstract Galois theory and endotheory, but considering only relations whose argument sets are included in some fixed set Xo such that card Xo ^
Scard E. In Section 5 we introduce the abstract fields and endofields, study their
mappings and, in particular, prove isomorphism and homomorphism theorems.
Finally, Section 6 is devoted to the abstract Galois set theory.
Historical remarks. I found the abstract Galois theory during the summer
vacation of 1935, submitted it to the jubelee volume of Journal des mathématiques
purés at appliquées dedicated to J. Hadamard in 1936, and this first exposition
[1] of the theory appeared in 1938. It is the following question that was the intuitive
origin of this research: Let zlt..., z„ be the roots of a polynomial f ( X ) of degree
n over some base field k, and let G be the Galois group o f f ( X ) \ how can the system
of equations satisfied by the «-tuples (cr - z ^ ...,A-Z„) (<t6G), and only by these
«-tuples, be obtained from the rational relations among the roots zlt ..., z„? (This
situation is somewhat obscured in the usual treatments of the classical Galois theory
because of the use of the so-called "Galois resolvent", i.e. the replacement of the
«-tuple (z l5 ..., z„) by a convenient linear combination
of z„
and, in more modern treatments, because of the emphasis put on the field structure
and the normal case.) This question led me to the fundamental operations and gave
me the key idea to the proof of the equivalence theorem, i.e., the idea of considering
the relation r * (cf. Section 4, later).
This theory, as elaborated in a set theoretical frame, had certainly no precursors.
Yet, it might be connected with some vague ideas or projects expressed before in
much narrower contexts. A rather enigmatic phrase in the last letter by Galois
to his friend Auguste Chevalier ("Tu sais, mon cher Auguste que ces sujets ne sont
pas les seuls que j'aie explores. Mes principales méditations depuis quelque temps
258
M. Krasner
étaient dirigées sur l'application à l'analyse transcendante de la théorie de l'ambiguïté. Il s'agissait de voir à priori dans une relation entre les quantités ou fonctions transcendantes quelles échanges on pouvait faire, quelles quantités on pouvait
substituer aux quantités données sans que la relation pût cesser d'avoir lieu. Cela
fait reconnaître tout de suite l'impossibilité de beaucoup d'expressions que l'on
pourrait chercher. Mais je n'ai pais le temps et mes idées ne sont pas bien développées
sur ce terrain, qui est immense") suggests that he had a vague feeling that his theory
for roots of polynomial equations is, maybe, only a particular case of a much wider
theory, where the polynomial relations among roots are replaced by some more
or less arbitrary relations on convenient domains. Clearly, these domains and relations could be only as general as conceivable for him and his contemporaries. However, the mathematics of that time was not set theory but a science of magnitudes,
i.e., of real and complex numbers and sufficiently smooth functions (in the quoted
phrase Galois speaks, among others, of the "quantités ou fonctions transcendantes"),
and the relations used in this mathematics were differential, functional, etc. equations and systems of equations. Some people see, in this phrase, an allusion to
what he knew about abelian functions. But if it had been so, he would not have
spoken of his "méditations" and he certainly would not have written "cela fait
reconnaître tout de suite l'impossibilité de beaucoup d'expressions ..."; in fact, he
would have spoken rather of the results known by him.
Cayley and Silvester's theory of invariants may be considered as a vague precursor of the abstract Galois theory in some very particular case. Indeed, if A is
an invariant of some "generic" form (i.e., its coefficients are independent variables),
a is a value of A, and the set of all points (in some field) of all projective varieties
for which a is the value of A is assigned to a, then this set of points is preserved
by all elements of the projective group or some of its subgroups. So, an invariant
A can be interpreted by the set {rAta} consisting of relations of the form rA a which
are invariant with respect to some (very particular) groups. Remarkably enough,
Cayley raised the problem of determining all the invariants. Yet, he could not consider the definition of these groups as that of Galois groups with respect to a system
of invariants.
The "Galois type theories" which appeared for differential equations at the
end of the nineteenth century (S. Lie, Picard—Vessiot, etc.) can hardly be considered as the beginning of the abstract Galois theory because they were developed,
like the classical Galois theory, by using quite particular methods, specific for each
case and without any more general spirit. Nevertheless they created a vague feeling
that Galois' ideas were, in some unknown manner, applicable in a wider framework.
In some very faint sense the abstract Galois theory for arbitrary (first order)
structures can be considered analogous to F. Klein's "Erlangener Programm" for
geometries (1872), though the former goes far beyond this program. Klein's main
idea is that a geometry is a system of "all invariants" of an "arbitrary" permutation group of a "Mannigfaltigkeit" ("variety"). As no set theory existed that time
(the first results on abstract sets were obtained by Cantor in 1873), Klein could not
express his ideas in a precise manner. In particular, he did not give any precise
meaning of "all invariants" of "Mannigfaltigkeit" (which seems to be a riemannian
variety or something stronger), and his "arbitrary" groups are far from being arbitrary as their elements must be automorphisms of the structure of "Mannigfaltigkeit". Anyhow, though the abstract Galois theory deals with the duality between
permutation groups and classes of their invariant relations, it is much more than
a simple idea of this duality.
In my first paper [1] on abstract Galois theory the fundamental operations
are defined almost in the same way as in this paper, but in a rather misleading manner: instead of arbitrary argument sets canonical argument sets, indexed by ordinals,
are used, whence the axiom of choice is widely used without real necessity. In later
publications ([2] and other papers, which I do note quote) I only formulated the
results of the theory in a clearer form by using canonical identifications (cf. Section 3 later), but without giving new proofs because of those in [1] being easily
adaptable to the new manner. There the essential part of the theory is developed
for relations with a fixed, sufficiently large (i.e., of cardinality not smaller than
card E) argument set X. The defect of this approach is that certain rather complicated
and not very intuitive fundamental operations (called "mutations") have to be
considered as well. That is why I return to the first manner, but without its past
imperfections.
I found the abstract Galois endotheory, if my memory is good, in 1964, and
formulated it in printed form first in the paper [3] of the International Congress of
Mathematicians, Moscow, 1966. I found the first but not-completely true version
of the homomorphism theorem of this theory in 1966, before the above-mentioned
congress. The gap in its proof was remarked by some my 3rd year students in 1973,
After having tried to fill this gap unsuccessfully I found the necessary modification
which made the theorem correct, i.e., I replaced norms by regular pseudo-norms,
and published it in [5]. Yet, until the present paper, the abstract Galois endotheory
(even without its homomorphism theorem) has never been published with complete
proofs. Only some rough ideas of certain proofs were indicated in [4].
The idea of abstract Galois set theory (in relation with eliminative structures
and the passage from the abstract Galois theory to the classical one) goes back as
early as to the end of thirties. However, the proof of its main theorem was put in
a clear form only in the first years after the second world was (1946? 1947?). This
theorem and the characterization of rationality domains (in case of abstract fields)
were formulated in several papers, but never with proofs. I exposed the abstract
M. Krasner
260
Galois theory and, after it had come to existence, the abstract Galois endotheory
in my Seminar (1953—1959) and in my courses for 3rd year students (Clermont-F d ,
1960—1965, and Paris, 1965—1980).
Terminology and notations. We shall use the ordinary notations of set theory
and mathematical logic. The set difference {x£A; x$B} will be denoted (in Russian
manner) by A\B, though personally I prefer the notations A- -B or A~\B to
A\B as they cannot interfere with algebraic notations. When a£A, AQA and
<p: A—B is a mapping, the <p-image of a and A will be denoted by <p • a and <p • A,
respectively. This will be the only mathematical use of the dot " •". The product
of two objects, say x and y, will be denoted by xy. Similarly, the composition of
mappings [// and (p will be denoted by cpij/ or sometimes, following Bourbaki's notation, by cpoij/. The mappings i/r C— D and cp: A—B are considered composable
iff q>-A<^C; in this case \p(p (or \J/oq>) is the mapping A—D such that \j/(p-a=
=[j/ -(q> -a) for every a£A. If <p\ A—B is a mapping, a£A and b=q>-a, then
we write <p: a—b if a is an arbitrary element of A while cp: a<-*b is reserved for
the case of fixed a. The equivalence relation {(x,y); cp-x=(p-y} on A, referred
to as the kernel of <p in the literature, will be called the type of cp and will be denoted
by T((p). If (p: A —B is a mapping and C is an equivalence relation oh A then cp
is said to be compatible with C iff T(<p) is thicker than C, i.e., iff x=y (mod C )
always implies x=y (mod T(<p)). Similarly, a mapping y: A—C is called compatible with a mapping /?: A—B iff it is compatible with
i.e., if for every
a£A the image y • a depends only on /? • a.
Let F and G be two families of subsets of A and B, respectively, and suppose
F and G are complete semilattices with respect to the union U • Consider a mapping
F—G. This mapping is said to be additive if <P- (J X— IJ <P-X holds for
every subfamily F of F, i.e., if commutes with the operation U. If
F—G is
additive and, in addition, F is the family of all subsets of some subset A of A, then
4> is called A-hyperpunctual. Then we have -x instead of
If
F—G is an /4-hyperpunctual mapping
and the image 0 • {*} of any singleton in F is a singleton (denoted by { is injective, 4> is called an injectively punctual mapping. Then <p
induces a bijection (p° of A onto <p •A'ZB, and if
is the mapping of the family
P(A) of all subsets of A onto P{<p • A) which prolongs (p° (i.e., induced by <P),
then
clearly commutes with all boolean operations. So, if P is a subfamily of
P{A), we have
(J X=
IJ X= U
U *
X€P
X€P
XZP
XIP
XIP
0 . N X = $°XTP
XIP
N
XIP
0--3T
and, for
XQA,
<P-(A\X) = <P° • (A\X)
= <p-A\q>-X = <P-A\<P-X.
Sometimes, in order to define certain notions or to formulate certain results
with elegance and well, we shall have to deal not only with sets but with classes
as well. This will be done only for the sake of convenience and better understanding,
but not for raising any question of Foundations. In fact, all we do could be done
in the language of sets, though in a longer and more complicated way. In principle,
the word "class" will be understood in Bernays' sense*) but in a freer and more
naive manner of speaking. However, as it can be shown, this manner of speaking
is only an "abuse of the language" from the point of view of Bernays' theory, since
the existence of classes and their mappings occurring in this paper can be proved
based on Bernaiys—Godel axioms.
1. Relations, structures and mappings
Let E and Xbe two sets called base set and argument set, respectively. Generally,
for the sake of some of the proofs, we assume that E consists of at least two elements, though our results are trivially valid for a one-element base set E, too. The
elements of Ex are called X-points while subsets of Ex are called X-relations. I.e.,
Appoints are mappings of X into E (denoted by, e.g., P: X-+E or XZ-+E) and
.Y-relations are sets of such points. When there is no danger of ambiguity, we often
do not indicate the argument set X. An (ordered) pair S=(E, R) is called a (first
order) structure on E (or with base set E) provided R is a non-empty set of relations
on E. (The argument set Xr of r£R may depend on r.) In particular, when all the
relations in R have the same argument set, say X, then S—(E, R) is called an X-structure. By the arity of an Z-point, X-relation or .^-structure we mean the cardinal
card X of X. We say that a structure S=(E, R) is under a set X° if XrQX° for
all r£R.
Let d: E-*E' be a mapping between the sets E and E'. This mapping can be
extended to points, relations, sets of relations and structures on E in the following
evident way: put d-P=doP
(Bourbaki's notation!), d-r={d-P;
P£r},
d-R=
= {d-r\ /•£/?}, d-(E, R)=(E',d-R).
In particular, when 8 is a self-mapping of
E (i.e. E'=E), we say that S stabilizes a relation r on E (or, in other words, r is
stable by 5) iff S-r^r.
We say that a permutation a of E preserves r (in other
*) But this does not mean that I consider the foundation of mathematics based on the Bernays—Godel axioms (and, generally, based on any predicative calculus formalism) as an
adequate one.
M. Krasner
262
words, r is preserved by a or invariant by a) iff<r-r=r. More generally, a self-mapping
5 of E is said to preserve r iff it stabilizes both r and its complement Ex-\r.
We say
that 5 is stabilizing or preserving on a set R of relations if it stabilizes or preserves
•each r£R, respectively. For a structure S=(E, R) the set of all self-mappings of
E that are stabilizing on R constitutes a monoid with respect to the composition of
mappings. This monoid is called the stability monoid or endomorphism monoid of
E/S (or S), and it is denoted by D(E/S) or, sometimes, by D(E/R). The endomorphism monoid is never empty for it always contains the identical mapping of E.
The set of all permutations of E that are preserving on R is called the Galois group
o r automorphism group of E/S (or S) and is denoted by G(E/S) or G(E/R).
R e m a r k 1. G(E/S) is the largest permutation group contained in
Indeed, a permutation a belongs to G(E/S) iff a and <r -1 belong to
whence the assertion follows.
D(E/S).
D(E/S),
R e m a r k 2. Suppose R is a set of relations on E such that r£R implies 1 r =
=Ex-\r£R.
Then G(E/S) consists of the permutations that belong to D(E/S).
This remark is a straightforward consequence of the definitions.
Particular relations. Firstly, we mention the empty relation 0, which is the
only relation without a unique argument set and base set. I.e., 0 can be considered
an Z-relation on E for any X and E. The X-identity on E is Ex and is also denoted
by I(X, E). Let C be an equivalence relation on X. Then
IC{E) = {P£EX; (Vx£X)(\/x'£X)(x
= x' (mod C)^Px
=
P-*')}
is called the C-multidiagonal on E. It consists of all Appoints that are compatible
with C. In particular, if XQX is a C-class and all C-classes but X are singletons then
DX(E) — IC(E) = {P• Ex; ( V x x ' d X ) ( . P • x = P• xty
is called the Z-diagonal of E. When
X={x,y),
DXti{E) = D{x,y){E) = {P£EX- P x = P y}
is called the {x, ^-diagonal on E, and such diagonals are called simple. The relation
1»C{E) = {Pac{E)\
T(P) = C}
is called the strict C-multidiagonal on E, while EX\IC(E)
is referred to as the
C-antidiagonal on E.
A relation r will be called semi-regular iff there exists an equivalence relation
C on its argument set X such that rQIc(E) and C=T(P) for some P£r. When
r is semi-regular then this C is unique, is denoted by T(r), and is called the type
of r. Further, t(r)={P£r; T(P)=T(r)} is called the head of the semi-regular rela-
tion r. If
for
some C, i.e. r is semi-regular and r=t(r), then r is said
to be regular.
An A'-point P: X—E is said to be surjective, injective and bijective if it is
such as a mapping, while a relation r is said to be such if every P g r is such. In
particular, r is injective iff it is regular and T(r) is the discrete equivalence relation
on X.
Incase X=0 there is only one 0-point Pa(E): &—E (indeed, no two mappings
can differ at any argument belonging to 0). Hence there are only two 0-relations:
0 and /(0, E)={Pe(E)}.
Let D(E) denote the monoid consisting of all self-mapping of E, called the
symmetric monoid of E, and let S(E) stand for the (full) symmetric group of E,
consisting of all permutations of E. For a subset A of D(E) the class of all relations
on E that are stabilized resp. preserved by each
A will be denoted by s-Inv A
resp. p-Inv A, and will be called the stability invariant resp. preservation invariant
of A. (Note that these classes are never sets.) When the context shows clearly what
kind(s) of invariants is considered, the letters s or p before Inv may be omitted.
Clearly, the mappings 4i-»-s-Inv .d and A*-*-p-ln\ A are decreasing. Further, if
© is a family of subsets of D(E) and Inv stands for any of our two invariants, we
have Inv( U
D Inv^l. For a set X" let R(E; X°) denote the set of all
¿ee
¿e.0
relations on E under X°. Now the sets
s-Inv(*°> A = s-Inv A D R ( E ; X°)
and
p-Inv(*°> A = p-Inv AC\R(E; X°)
are called the stability and preservation invariants of A under X°, respectively.
If R is a set of relations on E then R=s-lnv D(E/R) and
p-Inv G(E/R)
are called the stability and preservation closures of R, respectively, while
j?(*°> = s-Imtx°) D(E/R)
and
= p-Inv^°> G(E/R)
are called the stability and preservation closures of R under X", respectively. The
main problem of the next two paragraphs is to characterize these closures in terms
of R but without any intervention of self-mappings of E.
2. Fundamental operations
In order to characterize the above-mentioned closures of R in terms of relations we have to introduce certain operations acting on relations. Some of these
operations act on sets of relations while others on single relations, but any of these
operations results in single relations. Some of these operations are only partial,
i.e. they are defined for (sets of) relations satisfying some prescribed conditions.
264
M. Krasner
While defining our fundamental operations in the sequel, all relations are assumed
to have a fixed base set E.
I. Infinitary boolean operations:
la. Infinitary union, lb. Infinitary intersection. Both of these operations act on
non-empty sets R of relations, and are defined iff all r£R have the same argument
set, say X. These operations are denoted by U • R= U r and fl • R= f ) r> and
r£R
r£R
their results are relations with the same argument set X.
Ic. Negation. This fundamental operation acts on any single relation r, and
is denoted by 1 . If X denotes the argument set of r then
• r=Ex\r,
the complex
ment of r in E , has the same argument set X.
R e m a r k 1. The above three fundamental operations are not independent.
Indeed, if "1 R denotes {"1 -r; r£R}, we have U R= "1(0 • 1/?) and fl R=
= ~|(U • 1-R)- However, U and fl are independent.
R e m a r k 2. Two well-known properties of these operations, namieiy
/•fl("l • r ) = 0 and rU(~l -r)=Ex,
where X is the argument set of r, will be of
relevance later.
R e m a r k 3. When E and X are finite then there are only a finite number of
X-relations, whence the infinitary boolean operations are in fact the ordinary
(Unitary) ones.
R e m a r k 4. We define the following preorder for sets R and R' of relations.
Put RsR'
iff there exists a surjective mapping <p: R'—R such that for every
r'ZR' we have r'^cp-r'.
A (possibly partial) operation <o, acting on sets of relations, will be said increasing if for arbitrary sets R and R' of relations RsR'
implies
co(R)Q03(/?')> provided that to is defined for R and R'. Further, an operation a>
(possibly partial) that acts on relations is said to be increasing if for any two relations r and r' belonging to the domain of ft), rQr' implies co-rQm-r'.
It is easy
to see that the infinitary union and intersection are increasing, while the negation is not.
II. Projective operations, which act on relations:
Ha. Projections (or restrictions) p r x . This operation is defined for a relation
r iff the argument set X of r contains J a s a subset. For an Appoint P: X-*E let
(P|Af) denote the restriction of P onto X^X,
and define p r x - r as {(P|X); P£r).
This relation will also be denoted by pr£ • r, r% and, abusing the scripture, even by
(r|Z). This fundamental operation transforms a relation with argument set X ^ X
into a relation with argument set X.
lib. Antiprojections (or extensions) ext x .. This operation is defined for relations r with argument set XQX', and ext x .-r is the cartesian product rXEx s<x.
With the usual identification in cartesian products, ext x . • r is the set of all points
(P, P') such that P(ir and P' is an arbitrary (Z'\A r )-point. This relation will also
be denoted by e x t a n d
R e m a r k 5. We can extend the operations prx and extx- to any relation r
with an arbitrary argument set X via defining p r x - r as p r ^ n x - r and ext x . -r as
ext* u x ,-r. Then we have pr x p r x , = p r x n x , and ext x . ext x .=ext x , u x ..
R e m a r k 6. We have p r 0 - r = { P 0 } for
and p r o - r = 0 for r = 0 .
R e m a r k 7. Both projections and extensions are increasing.
R e m a r k 8. When relations are considered as sets of points, extensions are
hyperpunctual mappings and projections are even punctual, if relations with a fixed
argument set are considered. Therefore, the projections commute with U, and it
is easy to see that the extensions commute with all boolean operations.
R e m a r k 9. While p r J ext*,-r=r is always true, r is only a subset of
ext* pr£ • r. If ext^prjf •/•=/• then r is said to be identical on X\X
or outside
X, and the arguments belonging to X\X
are called fictitious. The operation cx=
=ext* pr^, which preserves the argument set, is called the X-cylindrification.
For XQX' we have Ex'=extx.-Ex
and, in particular, I(X, E)=Ex=extx
Ea=
=ext x -{/>„}.
Canonical identification. Let r and r' be relations on E with argument sets X
and X'. Let r ~ r ' mean that there exists a set X"^XUX'
such that ext^- • r=
=extx„ • r'. If this equality holds for some set X"^XUX'
then it holds for every
X"^XUX'.
Consider ~ as a relation on the class of all relations on E; then ~
is easily seen to be an equivalence relations, i.e., ~ is reflexive, symmetric and
transitive. As infinitary boolean operations commute with extensions, they are compatible with this equivalence. For
we have pr£=pr|(pr£*ext*.)=
=(pix p r f ) ext£.=pr*' ext*., whence pr y • r=pr x (ext x - • r). Hence it is easy to
conclude that if an JT-relation r and an Z'-relation r' are equivalent modulo ~ and
X ^ X D X ' then pr x • r=prx • r'. So projections are also compatible with this
equivalence. Further, we have e x t - r ~ r . It is also obvious that if r and r ' have
the same argument set X and r ~ r ' then r and r' must coincide. Therefore, for
any X-relation r and X', there is at most one ¿"-relation equivalent to r, and there
is certainly such an ¿"-relation if X'
X.
Relations on E can be considered modulo
i.e., we may identify relations
that are equivalent. This means that if a relation can be obtained from another
one via omitting and adding some fictitious arguments then these two relations
are considered the same. This identification will be called canonical. Since infinitary
boolean operations and projections are compatible with
they are meaningful
M. Krasner
266
after canonical identification. It is trivial that ext x . becomes the identical operation
when relations are canonically identified. Further, the infinite union and intersection become defined for every set R of relations in this case. Indeed, take a sufficiently large set X that includes the argument set of any r£R. Then for each r£R
there is exactly one ¿"-relation r' such that r ~ r ' , so we can and must define U • R
and fl • R as (J r' and H r', respectively.
III. Contractive operations.
Ilia. Contraction {cp). Given a surjection (p: X-* Y and an X-point P: X-»E
which is compatible with <p, we can define a Y-point Q: Y-*E by the condition
Q-y=P-x
where xdtp-1 • y. The mapping that sends P to Q and maps the multidiagonal IT(<,)(E) onto EY=I(Y, E) will be denoted by (cp). It is easy to check
that (cp) is injective. An .^-relation r is said to be compatible with q> iff every P£r
(as a mapping of X into E) is compatible with <p. In this case ((p) • r= {(q>) • P; P(Lr)
is a Y-relation. This mapping (<p), which maps the set of ^-relations compatible
with )• (/ r ( „)(£) 0(1/-))=(<?) • {ITM(E)\r)=
l((cp) • r). Clearly, cp is
increasing. If cp: X— Y and cp': Y->-Z are surjections then (cp') • ((<p) • P) is defined
iff P is compatible with (p'cp, and in this case ((p'<p) • P=(q>') • ((<p) • P). The same
formula holds for relations.
Illb. Dilatations [up]. Let ip: Y-*X be a surjection. Then for any X-point
P: X-+E the mapping [ip]-P=P'\p: y^-P-ty -y) is a Y-point compatible with
ip. The mapping P—lip] • P is injective and maps Ex onto the multidiagonal ITm(E).
For an ^-relation r let [ip] • r stand for {[ip] -P; P£r}. Then [ip], called the dilatation
[\p], is a mapping of P(EX), the set of all ^-relations, onto P{ITm(E)).
Further,
this mapping is punctual and injective. Hence it commutes with the infinitary intersection and union, and we have [ip]-(~\r)=ITm(E)\Jip]-r.
If ip: Y-*X and
ip'\ Z—Y are surjections then
• P=[i/>'] • (OA] • P ) and bp'>p]-r=[ip']-([ip]-r).
Obviously, (ip)[ip] • r = r for every ^-relation r, and [\p](ip)-r=r for every ^-relation r compatible with \p.
R e m a r k 10. Every multidiagonal can be obtained by an appropriate dilatation
from some identity.
Note that any multidiagonal can also be obtained as an intersection of (extensions of) simple diagonals. Indeed, IC{E)= f ) e x t x • Dx JE) or, up to canonical
identification, IC(E)=
H
x=J>(C)
x = S'(C)
Ac v(-£)- The strict multidiagonal lr{E) can be obtained
by an intersection of simple diagonals and antidiagonals.
Contractive operations and canonical identification. Let cp: X-+ Y be a surjection and let r be an ^-relation compatible with (p. For x£X the 7X<p)-class of x
is a singleton, provided JC is a fictitious argument of r. If y=(p • x,
F = y\{j>} and <p=((p\X), then is injective on X\X, (p=(<p\X}
maps X onto F = Y\cp -(X\X), ( r | Z ) is compatible with cp, (^)-(r|Z)=(((p)-/-|F}
and (q>)-r is identical on Y\Y.
So,
(<p)-(r\X)~((p)-r.
Two relations, say r and r' with respective argument sets X and X', are equivalent if and only if ( i r \ x f ] x ' ) = ( r ' \ x n x ' ) and both are identical outside
XilX'.
Indeed, r ~ r ' iff ext x u x , - r = e x t x u x , - r ' , and we can use the equalities {r\XC\X')=
= ( e x t x u x , • r \XC\X') and {r'\XC\X')={extxux,-r'\XC\X:').
Furthermore, if r and
r' are equivalent, r=ext x -(r|XflA").
Let r and r' be relations with respective argument sets X and X', and let
<p: X—Y and <p': X' — Y be mappings. The pairs (r, cp) and (r',<p') will be
said equivalent iff r ~ r ' and, furthermore, there exists a subset X of XC\X' such
that (<p\X)=((p'\X) and r is identical on X\X.
(Note that in this case ( r | Z ) =
= ( r ' | X ) and r' is identical on X'\X.)
Let (ir,(p)~(r',q>') denote that (r, cp)
and (/•', <p') are equivalent. It is easy to see that this binary relation is in fact an
equivalence.
Similarly, let r be an ¿'-relation, r' be an ¿"-relation, further let \ji: Y—X
and
: Y—X' be surjections. Then the pairs (r, \jj) and (r', ij/') are said to be
equivalent iff r ~ r ' and there exists a subset X of XC\X' such that (a)
-X=
(this set will be denoted by F) and (^|F)=(iA'|F), (fi) (¡l/\Y\Y): Y \ F —
-X\X
and (ij/'\Y'\Y):
Y'\Y-X'\X
are bijections, and (y) r and r' are
identical outside X. Note that if (r, \j/) and (r\ \j/') are equivalent then [ij/]-r~
Floatage. Floating equivalences. Free and semi-free intersections. For a bijection
cp: X-+ Y, every ¿f-relation r is compatible with (p. Then the contraction (<p), which
coincides with the dilatation [<p-1], is called a flotage (of arguments). A subset X
of ¿"such that (<p\X) is the identity mapping is called an anchor set of the floatage
{(p), while the maximal anchor set of (cp) will be called its anchor and is denoted by
A(q>)= {x£X; <p • x=x}. The elements of A(<p) are referred to as anchor arguments.
When considering a floatage with an anchor set X, we say that we let the arguments
outside X float.
Two relations, say r and r' with respective argument sets X and X', are called
floatingly equivalent (in notation ryr') if there exists a bijection q>: X-+Y such
that (cp) • r ~ r ' . It is easy to verify that this binary relation is really an equivalence,
called floating equivalence. The existence of a floatage (cp) such that (cp)-r=r' is
called restricted floating equivalence and is denoted by
When we allow floatages with a fixed anchor set X only then we obtain the analogous notions of semi-
268
M. Krasner
floating and restricted sermfioating equivalences of anchor X (in notation
r~fr'
and
resp.).
Let r be an ¿"-relation and let X Q X . It is clear that the X-projection of r
does not change when we let the "dumb" arguments x(LX\X float. We have
also seen that (r\X) is invariant under canonical equivalence. Therefore p r x • r =
= p r x - r ' , provided
r'. In particular, it is without changing p r y • r that
we can let the "dumb" arguments
float
so that their images avoid some prescribed set
If it is so then the image of г by this floatage is said to be regularized for X.
Let R be a set of relations on E, and let Xr denote the argument set of r£R.
For each r£R take a floatage (<pr) so that the sets Yr=q>r-Xr (r^R) be pairwise
disjoint. If У is a set including U Yr then П ext y (<p,) • r does not depend,
r£R
r£R
modulo floating equivalence, on the choice of floatages (<pr) and of Y. Hence
n exty ((pr) • r can be called the free intersection of r£R; it is determined up to
гея
floating equivalence. When we take Y= U Yr then П exty (q>r) • r is clearly the
cartesian product
r€R
JJ (<pr) • r, which is equivalent to ]J r. So the free intersec-
r(R
r€R
tion is, up to floating equivalence, the (complete) cartesian product. The free intersection of all r£R will be denoted by П f-R.
Given a set X and a set R of relations r on E and with argument sets Xr, the
semi-free intersection of anchor X of R is defined as follows. Put Xr=XC)Xr
and
choose floatages (<pr) with anchor sets Xr so that XC\(pr=0 and the sets
(pr • {Xr\Xr), r£R, be pairwise disjoint. Further, take a set У including all <pr- Xr.
Then f | e x t r (<Pr) • r> called the semi-free intersection of anchor X of R and denoted
r?R
_
by i l ® . R, does not depend, up to semi-floating equivalence of anchor X, on the
choice of У and that of (pr (г£Л).
L e m m a 1. Let R be a set of relations r with argument sets X„ let
and put X= (J Xr. Then
'€Я
XrQXr,
П e x t x p r x r . r = p r x . ( n Jf ) - i ? ) .
r€R
P r o o f . Put Q= П e x t z p r x .f and д*=ртх-(Г\{/)•
R).
Let
П(/> be re-
presented by П e x t x u r (<pr) r where <pr: Xr-»X,U Yr are bijections subject
reR
to the previous conditions. I.e., the Yr are pairwise disjoint and Y'= (J Yr is
_
_
_ _
r£R
disjoint from X. An AT-point P: X—E is in Q iff for every r£R (P|A"r) is a point
such that there exists an (A,\J r )-point P'r with ((P\Xr), P r ') belonging to r. But
this is equivalent to (<pr) • ((P\Xr), P't)£{<pr) • r and, as (<pr\Xr) is the identity map, also
to the existence of a y r -point P"-((pr\Xr\IR )-P^
such that (P,P r ")eext X u r p (<?,)• r.
Since the sets Y are pairwise disjoint, for any set {P": Y — E ; r£R} of points
there exists exactly one point P": Y'-E
such that (P"\Y )=P*
holds for each
r£R. Clearly, (P, P " ) e e x t x u r ( q > r ) - r is equivalent to
R
R
R
(P, P'0€ext X U y ext x u r£<Pr) •r = ext Z ur(?r) * r.
Thus the simultaneous existence of all P"\ Y -*E
with (P, P") belonging to
e x t X u r (cpr)• r is equivalent to the existence of a single P": Y'—E such that
(P, P ^ e e x t f y j r , (<p,)-r for every r£R, i.e., ( F , P " ) belongs to f ) e x t x u r , (<pr)-r—
T
= H ext r (q>r) • r, which is equivalent to P€Q*. I.e., P£Q
rgR
is equivalent to P^Q*,
which completes the proof.
The operations la, lb, Ic (U, fl, ~|), Ha, lib (pr x ,ext x ,)> Etta and Illb
((<p), [i/r]) are called fundamental operations. The increasing fundamental operations, i.e. all but the negation 1 , are called direct fundamental operations. Two
miliary operations, namely IVa: (adding to any set of relation) the empty relation 0
and IVb: (adding) the 0-identity 7(0, E ) = { P } , are also considered as direct fundamental operations. These two miliary operations are combinations of the rest of
the fundamental operations when we start from a nonempty set R of relations.
Indeed, take an r£R, then 0 = r n ( l r ) and 7(0, £')=pr 0 -(rU("lr)). Yet, they
are not combinations of direct fundamental operations in general.
If r ~ r ' or ryr' (or, in particular, r ~ r ' , rx~fr' or r~fr')
then each
of r and r' can be obtained from the other by a suitable combination of direct fundamental operations. On the other hand if we identify the canonically equivalent
relations, we can drop all extensions from (direct) fundamental operations. When
passing from relations to floatingly (and even restricted floatingly) equivalent ones is
permitted, we may fix a representative set X(c) of cardinality c for each cardinal
c, e.g., we may put X(c)= {a; a, is an ordinal and a<<»(c)} where ©(c) denotes
the smallest ordinal with cardinality c. (I follow Cantor's point of view rather than
that of von Neumann. In fact, the second point of view has been adopted in my
first paper [1] on abstract Galois theory, while the first one in all of my other
papers.)
When the axiom of choice is admitted, contractions become combinations of
projections and floatages, while dilatations become combinations of extensions,
floatages and intersections with simple diagonals. Indeed, if q>: X—Y is a surjection, for each y£Y we can choose an x(y)£X such that cp-x(y)=y.
Put
X= {*(;>); y£Y}. As {(p\%)'. X—Y is a bijection, ((<p|Z)) is a floatage, and
(<jj). r=(cp\X) pr x • r, provided r is compatible with (p. Similarly, if ^ : y—Xis a surjection, ^-«-^(x) is a mapping of X into 7 such that ij/-y(x)=x, and F={j>(x); x£X)
then ¡¡?=(t/r|F): j(;t)—* is a bijection of F onto X. Hence [ i i s a floatage,
E
M. Krasner
270
and for any ^-relation r we have
W-r
= extY&]-rn(n
fl
D-y,y). ,
ye T ysXTM)
When the image of <p or
is finite, the axiom of choice is not necessary for the
above results.
-°
We say that fundamental or direct fundamental operations are used below X°,
if these operations are used only for" relations with argument sets included in X°
and only when these operations result in relations whose argument sets are also
included in J f 0 . I.e., in case of p r x , ext x ., (cp: X-~ Y) and [ip: Y-*X] the inclusions
X'QX0
and YQX° are-also required. In particular, if .E is finite and
these operations are used below some finite X° then they are equivalent to the realizations of operations of the predicate calculus with equality on the finite model E,
the set of object variables being X°. Indeed, any A'-relation r on E (where XQ X°)
can be considered as the realization of some predicate Pr=Pr(X) on E. Further,
P,ur.=P,\/Pr.,
Prnr.=Pr&Pr.,
Plr=~\Pr,
Ppr y .r=(Bx 1 )...(Bx s )P r
where
fo,...,
Pextx,.r=Pr(X')
where Pr(X') is the predicate obtained from
Pr=Pr{X) by adding the set A " \ Z o f fictitious variables, if <p: X-~ 7is a bijection and
X={xl7 ...,x„} then Piq>yr=Pr9) is a predicate such that P^\(p-xlt...,
(p-xn)=
= P r ( x 1 , ..., x„), and we have PBx ^={x=y). The direct fundamental operations
are equivalent to the part of predicate calculus generated by \J, &, existential quantifiers, addition of fictitious variables, substitutions of object variables, the inequality
xt7^x{ and equalities x—Xj (in particular, x—xj). In the general case we may
consider the fundamental operations as realizations on models of an (unlimited)
infinitary generalization of predicate calculus, and direct fundamental operations as
that of certain "positive" part of it.
L e m m a 2. For any relation r there exists a set Rr of relations with the same
argument set such that
(1) every relation in Rr can be obtained from r by a combination of direct fundamental operations;
(2) all the relations in Rr are semi-regular, and for each point P of an arbitrary
relation s£Rr there exists a relation s£Rr such that sQs and Pgi(s); and
(3) r = U -R r .
P r o o f . Firstly, every multidiagonal is obtained by a successive use of direct
fundamental operations (starting from the empty set!). Indeed, it is obtained by
dilatation from some identity I(X, E)=extx • 1(9, E). Let P be a point of r with
type T(P) and let rP=rC\IT{P)(E).
Put jR r ={r P ; P€r}. Then (1) is clearly satisfied. It is easy to see that rP is semiregular and of type T(P), and P€t(rp). For
PZsZR, we have s=raQr where Q is a point of r such that P is compatible with
271
T(Q). Then T(P) is thicker than T(Q), so ITlP)(E)QInQ)(E)
s=rP, and (2) is clearly satisfied. Finally, P£rPQr implies
r={P;
P€r}= U ^ i U
Pír
and rPQra=s.
Put
rPQr,
Pír
U rt—r-, i-e- (3) holds.
Pír
The set Rr constructed in the previous proof is called the semi-regular decomposition of r.
which yields U
Formalism of fundamental operations. If we apply a (generally infinite) combination of fundamental operations to relations, we obtain an operation, which
can be represented, by a (generally infinite) formula of the "fundamental operation
calculus". The best way of denoting these formulas is to use (generally infinite)
trees with finite branches. By a tree we mean an unoriented, connected graph without loops, without circles (i.e. closed paths), and with a distinguished vertex, called
its root. Let T be a tree with root r. Then for any vertex v in T (in notation, v£ T)
there is exactly one path in T that connects v and r, provided v^r. Let w be the
vertex next to v onthis path. Now w is called the father o f f , while v is called the son
of w. A vertex v^r has one and only one father, while the set S(v) of sons of v
can be of arbitrary cardinality. Vertices without sons are called extremities of T.
We shall write
if v^v' and v is on the path connecting r and «/• For a vertex
v the set {v'; v^v'} of vertices spans a subgraph, called the subtree of T of root v.
The number of edges in the path connecting v and r is denoted by h(v) and is called
the height of v. In particular, h(r)=0. Let h(T)—sup h(v) be called the height
viT
of T, which is a non-negative integer or
T is called of finite or infinite height
according to h(T)7± <» or h(T)= «>.
A branch of T is a maximal linearly ordered set of vertices together with the
edges connecting them. We shall deal only with trees without infinite branches.
I.e., we always assume that our trees have no infinite, linearly ordered sequence of
vertices v ^ v ^ v ^ . . . . For such trees there is another invariant, the so-called
depth, which is more important than the height. Given a tree T, a mapping v from
its vertex set into the class of ordinals is called a depth function of T if for any vertex
v£T we have v(v)= sup (v(«)+l) (in particular, v(v)=0 for any extremity
«€S(B)
of T). If a tree has a depth function then it cannot have infinite branches. Indeed,
if r = V 0 < V 1 C V 2 < c , < . . . were an infinite branch then v(Ü0)>V(«J>V(w2)>V(V3 )>...
would be an infinite decreasing sequence of ordinals, which is impossible. As to
the converse, admitting the denumerable axiom of choice, we can prove the following
.
L e m m a 3. Any tree without infinite branches has one and only one depth function.
2•
272
M. Krasner
P r o o f . Assume that a vertex v £ T is irregular in the sense that the lemma is
not true for the subtree TB. I.e., T0 has no depth function or lias more than one
depth functions. Then v must have at least one irregular son u. Really, if for all
S(v) the subtree T„ has a unique depth function v„, then the mapping v„ defined by
V0(H>)=VU(W) if w£Tu and « € S ( v ) and, further, vB(v)= sup (*.(«)+ 1) is a depth
function of Tv. As the restiction of this v„ to any Tu, u£S(v), "is unique by the
assumption^ v„ is the only depth, function of Tv, which is a contradiction. Thus
we have seen that any irregular vertex has an irregular son. Now, if the lemma is
not true for a tree T, then its root r is irregular as T=Tr. Hence an irregular son
of r, then an irregular son r 2 of rlt etc. cán be chosen. I.e., the dénumerable axiom
of choice yields the existence of a denumerable sequence r=r 0 , r x , r 2 , . . . of irregular vertices such that each r¡, / >0, is a son of r t - v But then r 0 < r i < r 2 < . . . contradicts the fact that T has no infinite branch.
When T is a tree without infinite branches and v is its unique depth function,
d(T)=v(r)
is called the depth of T. Any ordinal can be the depth of some tree:
The depth of T is finite iff h(T) is finite; in this case
d(T)=h(T).
A formula is a mapping -Ffrom the vertex set of a tree T without infinite branches
such that
(1) F(v) is a fundamental operation provided v is not an extremity, and F(v)
is either U or fl if S(v) is not a singleton;
(2) for any extremity v with father w, F(v) is a relation set variable denoted by
X(v) (with capital X) provided
{w} and F(w) is U or D , a n d F(v) is a relation variable x(v) in all other cases.
Note that F(u) and F(v) may coincide even for distinct extremities u and v.
Let E be a base set. A map F' from T is called an E-formula if it is obtained
from some formula F via replacing certain relation set variables X(v) and certain
relation variables x(v) at all of their occurrences by some relation sets R(v) and some
relations r(v), resp., on E. Clearly, X(u)=X(v) or x(u)=x(v) must imply R(u)—
= R(v) or r(u)=r(v), respectively.
Given a base set E and a formula F, let <P(F)= {F(v); v\ is an extremity of T}.
A mapping Q of 4>(F) is called the system of values of variables of F if Q • F(v) is
a set of relations on is with the same argument set when F(v)=X(v) is a relation
set variable while Q • F(v) is a single relation on E when F{V)=x{v) is a relation
variable." A mapping t(g) from the vertex set of T, í(Q): v-»t(Q; v), will be called
á coherent valuation of F for Q if:
(1) t(g\ V)=Q • F(v) for every extremity v of T;
(2) if v is not an extremity then t(g;v) is a relation on E;
(3) if F(v) is U or D,
v' is an extremity, and F(v')=X(v')
then
e.F(v)=F(v).(e.F(v'));
(4) in any other case when F(v) is U or D, all the Q • F(v'), v'd S(v), have the
same argument set and Q • F(v)— U (Q • F(v')) or Q-F(V)=
f ) (Q • F(V')\
O'€S(R)
»-ESI»)
respectively;
(5) if F(v) is "1, p r | , ext*., (cp: X—Y) or [\j/: Y—X] then S(v) is a singleton
{u} and Q • F(u) is an ¿"-relation such that, in case F(v)=(cp), Q • F(u) is compatible
with cp, and, in all cases, Q • F(v)=F(v) • (Q • F(u)).
Given Q, an easy induction on v(v),
T, shows that there is at most one such
coherent valuation t(e).
It is possible to define the fundamental operations for formulas. Let U be a
set of formulas (or ^-formulas), and let rF denote the root of the formula F£ U.
We construct a new formula F ° = U U or F°= D U, resp., via taking a new root r°,
adding a new edge (r°,rF) to every F€ U, and putting F°(r°)= U or i r 0 ( r ° ) = n ,
respectively. Note that S(r°)={rF; F£U} and each F becomes F^. If J7 is a formula or ^-formula and co is one of the operations "1, p r f , ext*., (<p: X—Y), and
[\Jf: Y—X] then the formula (or ^-formula) a> • F is constructed so that we join
a new root r° with the root r of F by a new edge (then r becomes the only son of r°)
and we put (co • F)r=F and (co • F)(r°)=co. It is easy to verify that if QF is the
value of $(F), FZ U, and Q is the value of <P(U £/) or 0(0 U), respectively,
such that we have (el^(F))=e J ? for every F£U, then (UC/)(e) resp. (DC/)(e)
is defined iff all the F(Qf), F £ U , are defined and U resp. D is applicable to
{F(Q )I
F ^ U } . If this is so then we have ( U i / ) ( e ) = U ( 6 F ) and
(fli/)(e)=
FIU
= H F(6F)- If co is one of the operations ~l, p r i , ext x -, (<p), and
and Q is a value
F€l7
of 0(F) then (co • F)(Q) is defined iff F(Q) is defined and co is applicable to it. In
this case we have (co • F)(Q)=CO
• F(Q).
The formalism can be defined modulo canonical identification, too. Then ext
disappears and the results of the rest of fundamental operations are always defined
for arbitrary relations and, for U and D, for arbitrary sets of relations. Indeed, we
only have to consider relations and operations modulo canonical identification and
then to replace (cp) • r, where cp: X—Y is a surjection, by (<p)(prx ext*r • rf)IT(^(E))
where X, is the argument set of r and X' includes XrUX.
F
F
3. Fundamental operations and mappings
£
a
a
d
Let eo(...) be one of the considered fundamental operations. We denote by
an arbitrary value of its argument, which may be a set of relations (for la, lb) or
relation (for 13, Ila, lib, Ilia, Illb) or nothing (for IVa, IVb). Let d: E-E'
be
mapping from the base set E into another set E'. We say that co commutes with
if for any £ (with base set E) to is defined for d-£ provided it is defined for <?,
M. Krasner
274
and d• (ü(£)=(ü(d• £). We say that co sefrd-commutes with d if m(d- £) is defined
jvhen tű((J) is,; and d• co(^)^(o(d• £).
P r o p o s i t i o n 1. (1) Every fundamental operation commutes with any bijection.
(2) Every direct fundamental operation semi-commutes with any mapping.
(3) More precisely, the infinitary union, projections, contractionsdilatations, and
the addition of 0 and Ia(E) commute with all mappings, the infinitary intersection
commutes only with injections, and extensions commute only with surjections.
• P r o o f . As every mapping d of the base set preserves the argument sets of
relations, it is clear that if any of the operations la, lb, Ic, Ila, lib, M b , IVa, and
IVb is defined for some value £ of its argument then it is also defined for d •
If
cp: X^~Y is a surjection and P: X—E is compatible with q> then d • P is also
compatible with <p, where d: E-+E' is an arbitrary mapping from E. Indeed, for
x£X, (d- P) • x=d • (P • x) depends only on P • x, which depends only on cp • x.
Therefore, if an ^-relation r is compatible with (p then so is d • r, i.e., (cp) • r being
defined implies that ((p)-(d-r) is also defined. So the preliminary condition on
commutation and semi-commutation is always fulfilled.
,
Let us compute:
d-(UR)
= d-{P-,{3r£R)(Per)}
= {d-P; (3r
= {Q; (3sed-R)(Q£s)}
For XQX
;
=0
£*)(>€/•)}=
(d-R).
and an ¿"-relation r we have
d-(r\X)
= d-{(P\X);
Per}=
= {(Q\Xy,Qed.r}
{ d - № ) ; P€r} = {(d-P\Xy,
= (d-r\X),
i.e.,
P£r} =
d-prx r = pix(d • r).
If P: X-+E is an Appoint compatible with the surjection q>: X-*-Y then (q>)-P
is the mapping YÊ, y-»P-x where (p-x—y. So, d• ((<p) • P) is the mapping
y-~d-((((p)-P)-y)=d-(P-x)=(d-P)-x,
where (p-x=y.
Thus d-((<p) P ) =
<=(<p)-(d- P), and, if r is an ¿"-relation compatible with cp, we have
d • ((<?) • r) = d • {(<p) • P ; P£r} = {(<p) • (d • P); P€r} =
= №
Q; Q£d r} =
(cp).(d-r).
When \Jt: Y—X is a surjection and r is an ¿'-relation,
d • № - r ) = d . W - P - , P £ r ) = {do(PoiP); P g r } = {(doP)óiJ/; P g r } =
={Q°<l>-,Q€d.r} = [<l,].(d.r].
It is clear that 0 and /(0, E) depend on no argument, d • 0=0, and d • /(0, E)=
=7(0, E).
If jR is a set of AT-relations, we have
d-(r\R)
= d-{P-, (VreÄ)(Per)} = { d . P ; (Vr€Ä)(P6r)}.
Since P g r implies d-P£d-r,
Q£d-(C\R)
implies Q£d-r, for each r£R, i.e.,
-(d-R). The converse implication d-P£d-r^>P£r
holds for all rQEx
iff d is injective. Therefore the equality d • (C\R)=
- (d • R) holds for any set R
of Z-relations only if d is injective.
Let r be an ^-relation and let I ' d I , i.e.,
Then
d-extx.r=
=d-(rXEx'^x)=(d-r)X(d-E)x^xQ(d-r)X(E')x,^x=extx.
(d-r) and we have the
equality d-extx. r=extx. (d-r) (even for only one arbitrary r ^ 0 ) iff d is surjective. This proves (3). Now (1) and (2), except the case of
are consequences of (3).
But if s: E—E' is a bijection then it commutes with all the Boolean operations,
so, in particular, with the negation
: r—Ex\r.
The proof of Proposition 1 is
complete.
Applying Proposition 1 to the particular case E=E'
we obtain
C o r o l l a r y 1. (1) Every fundamental operation commutes.with any permutation
of the base set.
(2) Every direct fundamental operation semi-commutes with any self-mapping of E.
(3) The infinitary union, projections, contractions, dilatations, and the addition
of 0 and Ie(E) commute with all self-mappings of E, the infinitary intersection commutes only with its self-injections, while the extensions commute only with its selfsurjections.
P r o p o s i t i o n 2. Let a be a permutation of E, let R be a set of relations on E,
and assume that a fundamental operation co is applied to a subset or element I; of R.
(It is a subset when co= U or co= D, and it is an element otherwise.) If a is preserving on R then a preserves co(£).
P r o o f . As <T
=a>(£), indeed.
and co commutes with a, we have a• co(^)=co(a • £)=
P r o p o s i t i o n 3. If 5 is a self-mapping of E stabilizing on a set R of relations
and if 03 is an increasing fundamental operation semi-commuting with 8 which is applicable to a subset or element £ of R then 8 stabilizes co (<!;). P r o o f . Indeed, we have 8 • a>(£,)Qco(8 • £). Further, if £ is a set of relations,
<p: r—S-r (r£%) is a surjection of £ onto <5 • £ such that cp • r=8-r^r
for all
rg£. Hence
When £ is a single relation, S-£QTherefore,
as co is
increasing, we have co(8 • £ ) Q a ( £ ) and 8 • co(£)Qco(S • £), whence <5 • co(£)Qco(£).
276
M. Krasner
C o r o l l a r y 2. Let m be a fundamental operation which is applicable to a subset
or element £ of a set R of relations on E. If a self-mapping 5 of E is stabilizing on R
then it stabilizes co(0It follows from the preceding results that for any set AQD(E) of self-mappings
of E, s-Inv A is closed with respect to all direct fundamental operations. I.e., if a
direct fundamental operation a> is applied to an element or a subset £ of s-Inv A
then a>(£) belongs to s-Inv A. Similarly, s-Inv(XO) A is closed with respect to these
operations below X°. In particular, the same closedness is true for R and
Rm,
where J? is a set of relations on E. If AQ S(E) is a set of permutations of E then
p-Inv A is closed with respect to all fundamental operations, and so is the preservation closure R of a set R of relations on E. Similarly, p-Inv(*0) A and
are
closed with respect to all fundamental operations below X°.
4. Equivalence and existence theorems of abstract Galois theory and endotheory
Let X° be a set and let J? be a set of relations under X°. I.e., the argument sets
of relations in R are subsets of X°. The set R is said to be logically resp. directly
closed below X° if it is closed with respect to all fundamental resp. all direct fundamental operations below X°. If F is a logically or directly closed family of sets of
relations on E then the intersection p| R of this family is also logically or directly
closed, respectively. If R is a non-empty set of relations on E then the family of all
relation sets that include R, are under X° and are logically resp. directly closed is
not empty as it contains Rm, the set of all relations on E under X°. The intersection Rf0) resp. R f f i of this family is called the logical resp. direct logical closure
of R below X°. Rf0) and RfP are the smallest relation sets (on E) under X° that
are logically and directly closed, respectively. Let S=(E, R) and S'=(E, R') be
two structures on E so that both R and R' be under X°. (In this case S and S" are
said to be structures under X°.) We say that S and S" are equivalent resp. directly
equivalent below X° if Rf0)=R'fm
resp. Rffi=R'Jf\
Generally, these equivalences depend on X°. Yet, as it will be shown, they do not depend on X° when
card Z ° ^ c a r d E is assumed. Indeed, our main purpose in this paragraph is to
prove the following four theorems, in which card Z°Scard E is always supposed.
E q u i v a l e n c e t h e o r e m of a b s t r a c t G a l o i s e n d o t h e o r y . Let S and S'
be structures under X° and assume that card X° s c a r d E, where E is the common
base set of these structures. Then S and S' are directly equivalent i f f DE/S=DE/S..
E q u i v a l e n c e t h e o r e m of a b s t r a c t G a l o i s t h e o r y . Lets and S' be two
structures unde:r X0 on E where card Jf 0 Scard E. Then S and S' are equivalent i f f
Gejs—GEJS'-
E x i s t e n c e t h e o r e m of a b s t r a c t G a l o i s e n d o t h e o r y . For any semigroup
D of self-mappings of E, if D contains the identical mapping \B and X° is a set with
card Af'Scard E then there exists a structure S under X° on E such that
D=Dm.
E x i s t e n c e t h e o r y of a b s t r a c t G a l o i s t h e o r y . Let G be an arbitrary
permutation group on E and let X° be a set with card Af°^card E. Then there exists
a structure S under X° on E such that
=GS/B.
P r o o f of equivalence
and existence theorems of abstract Galois endotheory.
M
(a) Consider R =s-Inv(*0)
DE/S . This set is directly closed below X°, so it contains
Hence D
¡ 2 D
where Q stands for RFFI. As Q ^ R , we also have
D e i s = D e / r ^ D e / q , i.e. D
= D
. Therefore if S and S' are directly equivalent
below X°, i.e. R?P=R'}p,
then D
=D
..
(b) Let D be a submonoid of D(E). For an X-point P on E the set D P=
= {8-P; 5£D) is called the D-orbit of P. Every Z>-orbit is stabilized by all 8eD,
as S-(D-P)=SD-PQD-P.
It is also clear that any d£D stabilizes every union
of £>-orbits. Conversely, assume that each ő£D stabilizes a relation r. Then, f o r
any P £ r , we have
E / Q
E / S
E / Q
E / S
E / S
E / S
{P} = {l £ .P}i£>.P={<5.P; <5€i>}= (J {á-PjE U S-rQ U r = r,
SÍD
i.e., {P}^D-PQr.
SÍD
SÍD
By forming unions we infer that
r= U W i U í - í i U r = r,
Pír
Pír
Pír
i.e., r= (J D-P. Hence every relation stabilized by D is a union of certain Z)-orbits.
Pír
The set of all relations under X0 that are stabilized by D will be denoted by
0
•RjJ *. It is clear that the endomorphism monoid of E/Rfg0* includes D.
If P : X-*E
is a surjective point then the mapping D ( E ) — D ( E ) , <5—(5 • P
is injective. If card X°^card E then there exist surjective Appoints P with
If 8 stabilizes the D-orbit of such a point P then 8 must belong to D. Indeed,
8 - ( D ' P ) = 8 D - P , 8-P=S-(1 -P)£S-(D-P),
SO S$D
would imply
D-P§D-P,
i.e., 5 would not stabilize D - P . This means that, denoting R ^ by Q , Ő $ D
,
i.e. D / Q = D , which proves the existence theorem of abstract Galois endotheory.
(c) As card X° ^ c a r d E, there is a bijective point P: %—E
under X°, i.e.
Let us fix such a point P arbitrarily. Let P : X—E be another arbitrary
point under X°, and put EP=PX
and ÍP=P~1-EP.
Then P^P:
X-+1 is
a mapping, which induces a surjection ePt p: X-~%P. Since P is bijective, the type
T(sPip) of this mapping is equal to that of P. Hence P is compatible with eP P .
For any x£X and x=sPiP-x£%P
we have
E
E / Q
E
((e P ,].)-P).jc = P . * = (PP~v)Px
= P(P~lPx)
= P{ePtP-x)
=
Px.
278
M. Krasner
Note that (sP,p)-P denotes the image of P by the contraction (sP P ) induced by
the surjection sPtp: X—£P,
not the composite eP PoP, which even does not
exist in general. So, we have (ePP) • P=(P\2P) and, conversely, P=[e P j p] • (P\%P).
As every self-mapping 5 of E commutes with projections, contractions and dilatations, we also have 5-(P\XP)=(ePiP)-(S-P)
and 5-P=[EPiP]-(S-P\2p).
Further,
if P is compatible with some surjection q>: X— Y then, clearly, so is 5 • P. Finally,
if D is a semigroup of self-mappings of E, we have D -(P\%P)=(ePtp)
(D-P)
and D • P=[eP P] (D • P\%P). If, in particular, D is a monoid containing 1 E , then
the £>-orbit D-P of an arbitrary point P : X-+E can be obtained from the D-orbit
of the fixed bijective point P by a combination of direct fundamental operations.
If P is under X° then these operations are below X°. Therefore R(jpQ {D - P f j p :
Indeed, every r d R ^ , which is a-union of D-orbits by (b), is obtainable from
D-P by means of direct fundamental operations (more precisely, by infinitary
union, projections, and dilatations) below X". Suppose D=DE/S. Then, if S=(E, R),
we have RQR^p, whence R f f > g ( R ^ p = O n the other hand, {D • Pyjp Q
QR^
is trivial and {D • P f f p ^ R ^ has already been proved, whence
{D • Py*p=R(g°\ Therefore if we prove D-pQRfp
then we also have R(P=
= { D - P f j p Q R V p and R ^ J p ^ R ^ , from which the equivalence theorem follows. Indeed, if S=(E, R) and S'=(E, R') are two structures under X° with the
same stability monoid D=DE/S=DE/S.
then RiJp=R^0)=R'dfa)
would mean the
equivalence of S and S' below X°.
(d) Now we prove D-PzR^jp
via obtaining this orbit explicitly from R by
direct fundamental operations. Firstly, we replace every r^R by the set Rr of semiregular relations having the same argument sets as r, which has been
defined in Lemma 2, Section 2. Then R is replaced by the set of relations
k= (J Rr, which is under X° provided so is R, and which has the following properties implied by the quoted lemma:
(1) every
can be derived from some r£R by direct fundamental operations;
(2) for any P £ f £ k there exists an r ' ^ k such that r ' Q r and P g i ( r ' ) ; and
(3) every r£R is the union U • k' of some subset k' of k.
These three properties show that kQRfp
and RQk$p,
whence R<ff>=
(
=k?p
and it is sufficient to prove D • P£k Jp. Therefore it suffices to prove
D-p£R$p
only for sets R of semi-regular relations under X° that have property (2).
v
Let R be such a set of semi-regular relations and consider the relation
r* = n
f l ext*(e P ,p)-r.
r£8f{l(r)
Clearly, r* is obtained from R by direct fundamental operations, whence it belongs
to Rjp- We shall, prove that r* is precisely D • P where
D=DE/S.
First, as P£/(r), (e P t P )-r is defined and (P\JtP)=(ePtP) -P£(ep,p) -r. Thus
p£extg-((ePiP)-r)=extx(ePip)-r
and P(Lr*. But, as r* is obtained from R by a
combination of direct fundamental operations and all 8£D are stabilizing on R,
every 8£D stabilizes r* and
D-PQr*.
For an arbitrary i'-point Q we have Q = Q O 1\ = Q O ( P - 1 O P ) = { Q O P ~ 1 ) O P ,
for P is injective. Hence Q=8 -P where 8 — QOP~
is a self-mapping of E. Therefore every J?-point of E is the transform of P by some self-mapping <5 of E.
Now assume that 5$D, and let*us prove that <5• P$r*. Since
8$D=DE/R,
there exists some r£R not stabilized by 8, i.e. 5-r%r. Therefore there is a point
P£r such that 8-P$r. Since (2) holds for R, there exists an r'^R such that P£t(r')
and r'Qr. Then <5 -P^r'. As the contraction (sPP), which is defined also for 6 • P,
is injective (for points), we have
X
(,§ •P\XP) = 5- {P\XP) = 8 • ((ep.p) • P) = (sP,p) • (8 • PMsP,P)
• r'
and that {<5-.P|J?p}X.E'iNA' is disjoint from ( ( e P , P ) - r ' ) x E * \ S P = e x t s (e P _ P )-r'.
Thus 5 • P£(8 • P \ X P ) X E * \ x p does not belong to ext* ( e P t P ) - r ' ^ r * and 8-P$r*.
Therefore, 5 • Pdr* iff 8£D. Thus the equivalence theorem of of abstract Galois
endotheory is proved.
P r o o f of the equivalence and existence theorems of abstract Galois theory.
Since Rix0) is closed with respect to all fundamental operations below X°, an argument analogous to (a) shows that if S=(E,R)
and S'=(E,R')
are equivalent
below X° then GE/S=GE/S,.
We have already seen (cf. Remark 1 in Section 1) that
if a is a permutation and both a and a~x stabilize a relation r then a preserves r.
Consequently, if a monoid G consisting of some self-mappings of E happens to be
a group, i.e., a permutation group on E, then every er£G even preserves and not only
stabilizes all reR^°\
Therefore R(ax0)=j-Inv^
G=p-lnvm
G and, if card
A^Scard E, G is the stability monoid and also the preservation monoid of E/R
Hence for any permutation group G on E there exists a structure S—(E, R) such that
G=GB/S, which proves the existence theorem of abstract Galois theory.
Considering the particular case D=G and keeping the notations of (c) of the
preceding proof we have G • (P\%p)=(ep p) • (G • P) and G • P=[ep p]-(G • P\% ).
If G=GB/S then RQR^0* and, as Rix0) is closed with respect to all fundamental
operations, R f ^ ^ R ^ . Since G • P and R^
are equivalent (and even directly
(x<f)
)
equivalent), to prove R =R^°
it is sufficient to show G • PZ R(x°\
Let S=(E, R) be a structure under X° with card A b o a r d E and let G=GE/S.
Consider S*=(E,RD~[R)=(E,R*)
where
r€.R}. Then S and S*. are
equivalent below X° (really, RQR* and R*<^Rf)) and GE/S* = GE/S. Thus it suffices
P
M. Krasner
280
X 0 )
to prove G - P £ K F
, i.e., to prove G-P€RF°>
under the hypothesis R = ~ \ R . In
this case, by Remark 2 of Section 1, we have G
— D
C ] S ( E ) . AS P is bijective,
S-8-P
is injective, implying G - P = D P F ) S ( E ) - P , where D = D
. If 8 is an
injective, surjective or bijective self-mapping of E t h e n 8-Pis injective, surjective or
bijective as well, respectively. I. e., S (E)>P is the set of all bijective ^-points while
G'P
is the set of all bijective points of D - P . On the other hand, R * = D - P ^ R J P Q
QR1*0* has already been proved. Hence it is sufficient to show that the set of all
bijective points of r*=D- P can be obtained from this relation via a combination of
fundamental operations.
First, an v?-point Q is not injective iff there exist x,
x^y, such that
Q,'X=Q-y, i.e., iff ggextjf • jD^J,. Therefore the set of injective points of r is
E / S
E / S
B
/
S
L
r** = r*n(-|. u Dx„) = r + n( n (l-Ac.,))x.yit
x^y
x.yiX
x*y
As we do not want to use the axiom of choice, two cases have to be handled even
if E is infinite.
(1) There exists no bijection from E(and also of T) onto any of its proper subsets.
Then every injective ^-point of E is surjective and r** is the set of all bijective points
of r*. Now r ** is obtained from r* and from simple diagonals via infinitary boolean
operations, whence (cf. Remark 10 of Section 2 and the discussion of operations
IV. 1-2 in the same section) r** can be obtained from r* via a combination of fundamental operations; which was to be proved.1*
(2) There exist bijections from E onto some of its proper subsets. Then a set
$
Z°) and a mapping P can be chosen so that Jt^X0. Let y be an element of
X°\% and put X^-i'LIf}'}. Let Q: %—E be an injective point and consider an
arbitrary
{Q} XEM. Now, if Q is bijective (i.e., surjective) then Q' -y belongs to
E=Q- Z=(Q'\X)- X. Hence there exists an x^X such that Q'-x=Q'-y.
I.e., Q'
cannot be injective. Conversely, if Q is not surjective then there are an e£E and a Q'
such that e$QQ' -y—e and Q' is injective. So when Q' ranges over the set of all
injective points of ext x , • r** then (Q' ]%) ranges over the set of all non-bijective points
of r **. The set of injective points of ext x . • r** is, visibly,
est*- • r**fl(~l • (J ext x .
Dx,y).
*) Originally I proved the equivalence theorem of abstract Galois theory without using the
axiom of choice under the assumption card X 0 Scard E+1. It was B. Poizat who found the present
proof with card E instead of card E+1 for the case (1). P. June proved that card E can be replaced
even by card E—L when E is finite.
Therefore the set of bijective points of r** is
G. P = r*** = r**n(~| • p r , • (extjf- • r* + n(~l • U e x t X - D x , y ) ) \
xtX
which is obtained from r** and also from r*=D • P via a combination of fundamental operations. This completes the proof.
R e m a r k 1. If card Z°Scard E then we have Rm=Rp
and
Km=R?f\
Indeed, we have seen that D El ^x ! >)=D E/R and GEiR(x")=GE/R. Thus the
equivalence theorems together with the fact that R (x0) resp. RixV) is closed with
respect to direct resp. to all fundamental operations yield
= ( i F ^ f = jp°>
and
R(p = (R«yp = R«\
This means that a relation under X° belongs to R f f i iff it is stabilized by all selfmappings of is that stabilize every r£R, and, analogously, it is in Rp iff it is preserved by all permutations of E that preserve every rdR.
R e m a r k 2. Let S and S' be structures under X°, where card X°^card E. It
follows from the equivalence theorems that it does not depend on the particular choice
of X° whether S and S" are (directly) equivalent below X° or not. Therefore the notion
of equivalence and that of direct equivalence can be defined without any reference
to a particular X° in the following way: Two structures, S and S', are said to be
equivalent resp. directly equivalent (in notation S~S' resp. S~S') if there exists
a set A-0 such that card Z°Scard E, S and S' are under X°, and S and S' are
equivalent resp. directly equivalent below X°.
In particular, if R and R' are under X° and X0' 5 X° then we have
R<,pnR<Ex0)=Xtf0),
RT} =
R^HR?0)
and
= R<X°\
Rjxo-> = (R}x°>)tf°->.
Thus a set of relations under X0', which is closed with respect to direct resp. all fundamental operations and its part below X°, which is also closed with respect the same
operations below X°, mutually determine each other. More generally, if card X° s
== card E and card X0' =?card Eand R is under x ° n x 0 ' then R(xp and R f p mutually determine each other (because any of them is characterized by
and
so do R<p and R f ' \
R e m a r k 3. We are going to define two preorders, the thin (alias direct) Galois
preorder ^ and the thick Galois preorder 3 . For structures S=(E, R) and S"=
<J
=(E, R') we write S^S'
resp. S^S'
if there is a set X° such that c a r d Z ° S
Q
is card E, S and S' are under X and RpQR'Jf^
resp. RpQR'}x\
Note that
f and ~ are the equivalence hulls of
and S , respectively. For a structure S under
M. Krasner
282
X° let C S f ^ S ) and C< x0) (S) denote the ~-class and ~-class of 5, respectively.
Then, by the equivalence theorems, the mappings C < £ 0) (S)—D E/s and C ^ i S ) —
- * G e / s are injective, are decreasing with respect to thé orders ^ and
and do not
depend on the choice of X°. (Here the orders are induced by the similarly denoted
preorders.) These mappings map
and R^which
do not depend on X°,
onto the set of semigroups of self-mappings of E that contain 1 B and onto the set of
permutation groups on E, respectively. (This is an easy consequence of the existence
theorems.)
R e m a r k 4. Let S—(E, R) be a structure under X°, where card A^&card E,
put D=Dm,
let J? be a subset of X° with card
card E, and let P: X—E be a
bijective point. We have seen that any relation in R!ff)=R<g°)= R^
can be obtained from D • P via a combination of direct fundamental operations. (More precisely, first projections, then dilatations and finally an infinitary union have to be
applied.)
Case of finite base set. In case E has only a finite number of elements, say m
elements, it suffices to take a finite set X°={x1, ..., xn} with « ë m in the equivalence and existence theorems. If S=(E, R) is a structure under this X" then R is a
finite set of relations. Further, the infinitary boolean operations are, in fact, the ordinary ones. So every structure can be considered as a model (with base set E ) of some
finite system Pi(Xj), ..., PS(XS) of predicates (with no axioms). (Here each P^Xi)
depends on a set XtQX° of object variables.) By a model (with base set E) of the
previous system of predicates we mean a mapping r: Pi(X^)—r(Pi) ( / = 1 , ..., j )
where r(P¡) is an ^-relation on E. This mapping will be extended, in the following
way, to a mapping F-*r(F) from the set of all formulas F=F(P1, ..., Ps) obtained
from Plt..., P, via the operations of the predicate calculus with equality, used below
X°, in to the set of relations under X°. Firstly, these operations are generated by the
following ones via superposition:
(1) disjunction, i.e.,
(P,Q)-P\jQ\
(2) conjunction, i.e., (P,
(3) negation, i.e.,
Q)-*PbQ\
P—~\P;
(4) existential quantification, i.e., ? ( I ) - ( 3 x ) / > ( I ) where
(5) adjunction of a set of fictitious variables, i.e., P(X)—Px\X')
QX'QX0
and X^X
is the set of fictitious variables;
xeXQX0;
where X<^
(6) floatages, i.e., P(X)—Pl(X-X)
where A is a bijection of
X—{x,(l), xl(S), ...,xi(l)}
( l s / ( l ) < / ( 2 ) < . . . < / ( r ) ^ n ) onto a subset X-X
and
of
X°
(7) adjunction of equality predicates x=y (JC, y£X°), which may be proper
equality predicates if x and y are distinct or the x-identity predicates x=x.
It is not hard to see that any fundamental operation is a superposition of some
of these seven kinds of operations. Really, U and D are iterations of disjunctions and
conjunctions, pr y is a suitable iteration of (4), contractions can be composed from
projections and (6), and any dilatation [t/r] is a superposition of a floatage, of an extension, and of an intersection with some simple diagonals (see the discussion on the
axiom of choice after the definition of direct fundamental operations). The above considerations allow us to extend r to all formulas F(PX,..., Ps) below X° in the following obvious way, via induction: put
(1)
r(PV0 = r(P)Ur(0,
(2)
r(P&Q) =
(3)
r (IP)
(4)
r((3Z)P(Z)) = p r x X w r ( P ( X ) ) ,
(5)
r(Px') = ext x , r(P),
(6) r(Px) =
(7)
r(P)C\r(Q),
=lr(P),
(X)-r(P),
r(x = j>) = DXty if x and y are distinct while
r(x — x) = 7({JC}; E) =
Now it is clear that the set of all r(F(P1(X1), ..., PS(XS))), where
F=
= F(Tl(X1), ..., T,(XS)) ranges over all formulas of the predicate calculus with
equality sign that depend on.the predicate variables Tx(A^), ..., TS(X^, coincides
with the logical closure R(p of R={r(Pj),...,
r(Ps)} below X°. On the other hand,
it is easy to see that the realizations (i.e., r-images) of the operations TV T', Tb T',
(3x)r(A-) for x£X, T(X)-*Tx'(X')
for XQX',
the adjunction of x=y
and also of "1 (x=x) are direct fundamental operations. Conversely, every direct
fundamental operation is the realization of an appropriate superposition of these
operations. Let us call the part of predicate calculus (below X°) generated by these
operations strictly positive predicate calculus (below X°). Then BSjfp is the set of
r(F(Pl, P 2 , ..., PJ) where F(Tlt ...,T,) ranges over the formulas of strictly positive
predicate calculus (below X°) that depend on the predicate variables
T^T^XD
(i— 1,2, ..., j). So we have
• E q u i v a l e n c e t h e o r e m s f o r a f i n i t e b a s e set E. Let M=(E\ Pt(X{)-*
— riQExi (i=l, ..., J)) be a model of a system of predicates {P£X,); i = l , 2, ..., j}.
Then a relation r on E is of the form r(F(P1, ..., P,)) for some formula
M. Krasner
284
F(7'1(A'1), ..., Ta(Xs)) of the predicative resp. strictly predicative calculus below X° if
«and only if r is preserved resp. stabilized by all erg S(E) resp. d£D(E) that preserve
resp. stabilize all rt (/=1, ...,s). (The set X° is supposed to contain all Xt and to
have at least as many elements as E.)
Examples of some classical structures
1. The structure of the classical Galois theory. Let E/k be a commutative field
•extension which is normal algebraic or algebraically closed. Consider two {x, y, z}relations on E, +(x,y,z)
and X(x,y, z) such that P£ +(x,y, z) iff
P-x+P-y=
=P-z and P£X(x,y,z)
iff (P-x)(P-y)=P-z.
For e£E let (x; e) denote the
{x}-relation on E with the property P g ( x ; e ) i f f P x=e. Put RX)={+(x,y, z),
X(x, y, z)}U{(x; a); a£k}, and let A be a subset of E. We can consider the struct u r e S=S0(A)=(E,
a); a£A}). Then GE/S coincides with the ordinary
•Galois group of the field extension E/k(A) while DE/S is the monoid of all isomorphisms of E/k (A) into E. Note that GE/S and DE/S are the same when E/k (A) is
.algebraic.
For f ( x l t ...,xn)€k[xlt
...,*„] let ( / = 0 ) denote the { x j , . . . , ^ - r e l a t i o n
on Esuch that an {xi, . xj-point P belongs to ( / = 0) iff f(P-xx,
...,P-xn)=0.
Then
+(x,y,z)
= (x+y-z
= 0),
X(x, y, z) = (xy-z
= 0)
and
(x;e) = (x-e
= 0).
Let
R¿ = { ( / = 0); / ( № , x2,...] = k[{xn; n€N*}]}
•(here N* stands for the set of positive natural numbers).
We claim that R0 and R'0 are deducible from one another by means of direct fundamental operations. Really, a standard argument shows that the same self-mappings of
E stabilize RQ as R'0, whence the equivalence theorem of abstract Galois endotheory
yields this assertion.
The aim of the classical Galois theory, as we have seen it in the introduction, is to
•determine the set A of all a £ E that are preserved by each a £ G E / k W . (Note that A
coincides with the set A of all elements in E that are preserved by each 5gD £ / t ( i 4 ) .)
It is clear that <7-5=5 is equivalent to <r-(x; 5) = (JC; 5). The abstract Galois theory
answers this problem by describing A as follows:
a€A
iff (*;a)€(^,U{(*;a); a € ¿ } ) / = W { ( x ; a ) ; a<LA})f.
On the other hand, the classical Galois theory says that A is the closure of k U A
with respect to addition x+j>, multiplication xy (both defined on EXE), inversion
j c - 1 (defined on £*=JET\{0}) and, when the characteristic p of k is not 0, forming of
285
p
pth roots f x (defined on Ep={xp;
The second result can be deduced from
the first by means of "abstract Galois set theory", to be exposed in Section 6, and of
the theory of "eliminative structures", to be exposed elsewhere. Although this deduction is quite complicated, it reveals deep reasons why the operations x+y, xy, JC-1
p
and \ x , and only these operations, occur in the above-mentioned result of the classical Galois theory. Moreover, it can be shown that {GE/k(A); AQE) is the set of all
subgroups of G m that are closed with respect to the finite topology ("Krull topology")
on D(E), provided Ejk is algebraic.
2. A slightly different structure is obtained if we replace (x; e)=(x—e=0)
by
(x,y; e)=(y—ex=0).
I.e., R0 is replaced by
and R0(A)
by
-RJ = {+ (*» * z)> X(*. y>z))u
R*(A) = {(x,y;a);
Let S*(A) stand for (£,1%(A)).
~S0(A).
Indeed,
and
(y-ex
(x-e
y;a);aek}
adA)\JB$.
Then we have
= 0) = pr{Xir} •((y-zx
= 0) = pr {x} • ((x-ey
•((yz-t
{(*>
= 0)D l(y+z-t
= 0)f)(z-e
i.e.,
S*(A)~
= 0))
= 0)flpr {;CiJ ,j ext{x,„,Zft} •
= 0)DD { y , z , t } (E))).
Similarly, DEisZ(A)=DEjSo(A) U {0}, where 0 denotes 2?—{0}, the zero homomorphism of E.
3. Linear Galois theory. Let fc be a not necessarily commutative field, and let E/k
be a field extension. We consider
and
K L ) = {+(*,;>, z)}U {(*,;>; a); a€fc}
SlL>(A) = (E,
U {(x, y, a); a£A}).
The stability monoid DBIS^-\A) of E/S(0L)(A) is the semigroup AE/k(A) of all linear
transformations k: E—E of the left vector space E over the field k(A) (the field generated by kUA), while the Galois group GE/S^HA) of E/S^L)(A) is the group of bijective linear transformations of the same vector space, i.e., it is the general linear group
GLk(A)(E) of E over k(A). The two main questions of this theory in classical algebra
are the following: how to determine the set A of all a£E such that every S^DE/S^Â)
stabilizes (x,y;a);
and which submonoids of DEIS^=AB/k
are of the form
DEISU-\A) for some AQE. It can be shown, in an elementary, way, that 2,—k(A).
As regards the second question, Jacobson's density theorem yields the following
3
M. Krasner: Abstract Galois theory and endotheory. I
answer. For a given e£E let (e) denote the linear transformation x—xe of the left
vector space E/k, and put (E)={(e); e£E). Then (E) is a field with respect to the
addition and multiplication in Am, and (E) is anti-isomorphic to E. Now, a submonoid A of AB/k is of the form DE/S^ for some AQE if and only if A is a subring
of AB/k containing (E) and closed with respect to the finite topology on D(E), the set
of self-mappings of E.
4. Homogeneous Galois theory. Let E/k be the same as in the first example; we
put
Kh) = {+(*, y, z), n(x, y, z, t) = (xy-zt
and
R&h)(A) — jR£h)U{(x;a);
= 0)}U{(x; a); ccÇk}
a£A}.
It is easy to show that Gew»\a) and DmRw(A) are the group generated by the ordinary Galois group G ( together with the group
E / K
A )
(£*) = {(e): * - xe; e(LE* = £ \ { 0 } }
of multiplications by the non-zero elements of E, and the semigroup generated by the
ordinary stability monoid DElkiA) together with (E*). In order to describe the set A
of all elements 5Ç.E that are preserved by every oÇ.GEiRu>\a), which is the same as
the set of elements preserved by all
let k(B) denote the perfect closure
(in E) of the field k{B) generated by B (where BQE).
Then
A=ak(a~1A)=
A}) for some (moreover, for any) non-zero element a in A, and {0} =
= {0}. Finally, note that R^ can be replaced, up to direct equivalence, by the set
{ ( / = 0 ) ; f€k[x!,x2,...]
and / is homogeneous}.
References
[1] M.
KRASNER,
Une generalisation de la notion de corps, J. Math. Pures Appt.,
17
(1938), 367—
385.
Generalisation abstraite de la théorie de Galois, in : Algebra and Number Theory
(24th International Colloquium of CNRS, Paris, 1949), 1950; pp. 163—168.
[3] M . KRASNER, Endothéorie de Galois abstraite, in: Proceedings of the International Math. Congress
(Moscow, 1966), Abstracts 2 (Algebra), p. 61.
[4] M . KRASNER, Endothéorie de Galois abstraite, P. Dubreil Seminar (Algebra and Number Theory),
22/1 (1968—1969), no. 6.
[5] M. KRASNER, Endothéorie de Galois abstraite et son théoreme d'homomorphie; C- R. Acad.
Sci. Paris, 232 (1976), 683—686.
[61 B. POIZAT, Third cycle thesis, 1967.
[7] P . LECOMTE, Doctoral thesis.
[2] M . KRASNER,
Acta Sei. Math., 50 (1986), 287—298
Rédei-Fnnküonen und ihre Anwendung in der Kryptographie
RUPERT NÖBAUER 1 )
'
1. Einleitung. In [8] hat L. RÉDEI eine Klasse von rationalen Funktionen über
einem endlichen Körper K ungerader Ordnung untersucht, die Permutationen von K
induzieren.
In der vorliegenden Arbeit erfolgt nun u.a. eine Übertragung der von L. Rédei
gefundenen Ergebnisse von endlichen Körpern K auf Restklassenringe Z/(m) des
Rings der ganzen rationalen Zahlen modulo ungerader natürlicher Zahlen m. In
Abschnitt 2 werden rationale Funktionen über Z/(m) definiert, die den von L. Rédei
untersuchten Funktionen entsprechen und die daher als Rédei-Funktionen bezeichnet
werden. Es wird gezeigt, daß bestimmte Mengen von Permutationen von Z/(m), die
durch solche Rédei-Funktionen induziert werden, bezüglich der Komposition Gruppen bilden. In Abschnitt 3 wird die Anzahl der Fixpunkte der durch Rédei-Funktionen
induzierten Permutationen von Z/(m) berechnet. In Abschnitt 4 wird die Struktur
von durch Rédei-Funktionen induzierten Permutationsgruppen von Z/(/w) ermittelt,
und es werden alle ungeraden m bestimmt, für die diese Gruppen zyklisch sind. In
Abschnitt 5 wird u.a. gezeigt, daß es für jedes ungerade wiS3 Permutationen von
Z/(m) gibt, die durch Rédei-Funktionen induziert werden und die nur einen Fixpunkt aufweisen, und es wird eine Aussage über die Anzahl derartiger Permutationen
hergeleitet.
In Abschnitt 6 erfolgt die Beschreibung eines Public-Key Kryptosystems — also
eines Verschlüsselungssystems mit öffentlich bekanntem Schlüssel (vgl. [1]) — auf
der, Basis von Rédei-Funktionen. Das Verfahren kann als Variante des RSA-Schemas (siehe etwa [9]) angesehen werden: Beim RSA-Schema erfolgt die Verschlüsselung der Nachrichten mit Hilfe solcher Permutationeri von Z/(m), die durch Potenzen yp induziert werden, beim vorliegenden System erfolgt sie mit Hilfe von durch
Received March 21,1984 and in revised form July 17,1984.
l
) Der Autor dankt dem Institut für Mathematik der Universität Klagenfurt für die Ermöglichung dieser Arbeit. Weiters dankt der Autor dem Referenten für wertvolle Verbesserungsyorschläge.
3*
R. Nöbauer
288
Redei-Funktionen induzierten Permutationen. Das vorliegende System ist eine Verallgemeinerung des von R. LIDL und W. B. MÜLLER in [3] vorgeschlagenen Systems,
und zwar von Z/(m), m quadratfrei, auf Zl(m), m beliebig.
2. Einige grundlegende Tatsachen. Wir wollen zunächst einige allgemeine Bemerkungen über rationale Funktionen machen, die Permutationen von Z/(m) induzieren.
Sei m eine beliebige natürliche Zahl und sei f(x)=g(x)/h(x)
Quotient von ganzzahligen Polynomen, welche teilerfremd in Z[x] sind. Man nennt / ( * ) Permutationsfunktion von Z/(m), wenn h(u) für jedes ganze u eine prime Restklasse mod m ist
und wenn die Abbildung n:
mod m von Z/(m) in sich eine Permutation ist (vgl. [6]). Ist speziell h(x)—l und f(x)=g(x)/l
eine Permutationsfunktion
von Z/(m), so nennt mao/(x) Permutationspolynom von Z/(m). Ist m=ab mit 2)
(a, b)—l, so ist f ( x ) genau dann Permutationsfunktion von Zfcab), wenn es Permutationsfunktion von Z/(o) und von Z/(b) ist. Weiters ist f ( x ) genau dann Permutationsfunktion von Z/(p e ), e > l , wenn gilt: f ( x ) ist Permutationsfunktion von
Z/(j>), und / ' ( m ) ^ 0 mod p für jedes ganze u (vgl. [6]).
Man nennt zwei Permutationsfunktionen /!(*), f2(x) von Z/(m) äquivalent,
wenn es ein lineares Polynom p(x)=cx+d,
(c, m)= 1, gibt, sodaß gilt
ft(x) =
p-1(x)of1(x)op(x),
wobeip~*(x) das bezüglich der Komposition o zu p(x) inverse Polynom c~xx—c~xd
bezeichnet.
Sei n eine ungerade natürliche Zahl, und sei a eine ganze Zähl mit a Nichtquadratelement in Z, also mit a Nichtquadratelement in Q, dem Körper der rationalen Zahlen. Seien gn(x), k„(x)€Z[x] definiert durch
(x+Y*)a
(2.1)
= gn(x) + hn(x))fc.
Eine explizite Darstellung von g„(x) und hn{x) ist gegeben durch®)
&(*) = 2
("i)
*.(*) = 2
(2i +
lj«'X"-2-1.
Es gilt hn(x)9i0 in Q(fa)(x), dem rationalen Funktionenkörper über Q{fä),
und somit h„(x)y^0 in Z[x], Denn wäre h„(x)=0, so würde folgen ( x + y a ) " =
=g„(x), also (^ct)"€Z, und dies ergäbe einen Widerspruch.
Die Polynome g„(x), h„(x) sind teilerfremd in Z[x], denn für d„=(gn(x), h„(x))
erhält man aus (2.1) durch Herausheben von d„(x) unter Beachtung der Tatsache,
') (fli,..., a„) bezeichne den größten gemeinsamen Teiler, [ a l t . . . , a j das kleinste gemeinsame Vielfache der Zahlen a x
a^.
a
) [a] bezeichne die nächstkleinere ganze Zahl von a.
Rldei-Funktionen
289
daß Q{fä)[*J ein ZPE-Ring ist: i i n ( x ) = ( x + / ä ) s mit O^s-zn.
s=0, also d„(x)= 1.
Daraus ergibt sich
Wir setzen nun f„(x)=g„(x)jh„(x). Nach den obigen beiden Bemerkungen ist
der Nenner h„(x) der rationalen Funktion/, (x) von 0 verschieden, und die Darstellung
g„(x)/h„(x) ist bereits gekürzt. In Q(fä)(x) gilt
(22)
fx+^T
/„(*) + /«
U-j/aJ
/.(*)-/«'
Daraus folgt
ÄJx)+Vä
A(/„(x))+A
=
/ * „ ( * ) A ( L ( x ) ) - y ä '
und man erhält
(2.3)
Mx)ofn(x)
=fkn(x)
für beliebige natürliche Zahlen k und n.
Für eine ungerade Zahl m—pe^...pe/ soll die Menge aller Permutationen von
Z/(m) untersucht werden, die durch bestimmte Permutationsfunktionen f„(x) induziert werden. Daher wird als erstes nach Bedingungen gefragt, under denen fn(x)
Permutationsfunktion von Z/(m) ist.
Es sei m=pe (p ungerade Primzahl,
ein quadratischer Nichtrest m o d p
und n eine natürliche Zahl mit {n,pe~1(p+\))=\,
dann ist fn(x) Permutationsfunktion von Z/(pe). Für e—1 folgt dies aus der Arbeit von L. R I D E I [8]. Für e=>l
erhält man aus (2.2) durch beiderseitiges Differenzieren
J x + f t ) " ' 1 —2 ~\fä.
ix-fc)
{x-fcy
also
f'M
-2<f;jx)YZ
(fn(x)-l/^y
'
_
\(x)-j*y
_
(x-yäy^
-
J n W
=
n(xl-a)n-1(fn(x)-f^)2
(x-i^r
=
=
«(x2-«)"-1^^)-^")2
(gn(x)-K(x)Väf
=
«(x2-«)"-1
K(xY
'
und daraus folgt f ' n ( m o d p für alle ganzen u. Wir nennen derartige Permutationsfunktionen Redei-Funktionen. Es sei Pp„ die Menge aller jener Permutationen
von Z/(pe), die durch Redei-Funktionen mit einem festen a induziert werden. Wegen
(2.3) bildet Pp„ bezüglich der Komposition eine Halbgruppe, die als Unterhalbgruppe
der vollen Permutationsgruppe von Z/(pe) regulär und endlich, also sogar eine Gruppe
ist. (Im Spezialfall e=l folgt dies ebenfalls aus [8]).
290
R. Nöbauer
Betrachtet man den allgemeinen Fall m=p\i...pe/, so folgt aus dem Vorherigen,
daß /„(x) eine Permutationsfunktion von Z/(m) ist, wenn (n, pf'~1(pi+l))=l
gilt
und a ein quadratischer Nichtrest mod pt, / = 1 , ..., r, ist. Ist Pm die Menge aller
durch Redei-Funktionen mit gegebenem a induzierten Permutationen von Z/(m),
dann bildet — wiederum wegen (2.3) — Pm eine kommutative Halbgruppe, die regulär und endlich, also sogar eine Gruppe ist.
.
Die Struktur dieser Gruppe ist unabhängig von der speziellen Wahl von a :
Ersetzt man a durch aß2, ßÔ m o d p t , i= 1, ..., r, so gehen die Permutationsfunktionen /„(x) über in die äquivalenten Permutationsfunktionen ßf„(x/ß), wie man
mit Hilfe von Gleichung (2.1) leicht erkennt. Bezeichnet x die durch ßx induzierte
Permutation von Z/(m), so geht also Pm über in die isomorphe Gruppe zPmx~\
Die Klärung der Struktur der Gruppe Pm wird in Abschnitt 4 vorgenommen;
dabei wird als wesentliches Hilfsresultat die in Abschnitt 3 berechnete Anzahl der
Fixpunkte von Permutationen n„£Pm herangezogen.
3. Fixpunkte. Zur Berechnung der Fixpunktanzahl von Permutationen n„€P m
werden Kenntnisse über einen speziellen Erweiterungsring von Z/(/>e) benötigt. Sei p
eine ungerade Primzahl, sei a eine ganze Zahl mit a quadratischer Nichtrest mod p,
und sei e s l . Man bilde den Faktorring Z[x]/(x2—oc,pe) des Polynomrings Z[x].
Da
u(x)(x2—a)-\-v(x)pe = a mit a£Z
dann und nur dann gilt, wenn <2=0 mod pe, ist durch
rj(a mod pe) = a mod
(x2—a,pe)
ein Monomorphismus von Z/(pc) in Z[x]/(x2—a,pe)
definiert, also kann man
Z/(p°) in Z[x]/(x2—a,pe) einbetten. Bezeichnet man die Restklasse von x mit ]fä, dann
gilt ( / a ) 2 = a . Somit läßt sich jedes Element von Z[x]/(x2—<x,pe) darstellen in der
Form a 1 f ü + b mit a, b£Z/(pe). Da aus
ax+b = u(x)(xz—a) + v(x)pe
folgt a = 6 = 0 mod/?®, ist diese Darstellung eindeutig. Wir setzen im folgenden
Z ( p e , a)=Z[x]/(x 2 —a, pe).
Es gilt: Ist ^ der kanonische Epimorphismus von Z/(pe) auf Z/(p), so ist durch
ö(a]fa +b)=(£a)fix +£b ein Epimorphismus von Z(pe, a) auf Z(p, a) definiert, der
ebenfalls als der kanonische Epimorphismus bezeichnet werden soll. Wir zeigen nun
Lemma 1. Die invertierbaren Elemente von Z(pe,a) sind genau die Elemente
a~fü+b, für die nicht gleichzeitig a=0 m o d p und 6 = 0 mod p gilt.
Beweis. Ist ß£Z(pe, a) invertierbar, dann ist auch öß£Z(p, a) invertierbar,
also gilt nicht gleichzeitig a = 0 mod p und 6 = 0 mod p. Gilt umgekehrt nicht
291
Rldei-Funktionen
gleichzeitig a=0
mod p und b = 0 mod p, dann gilt wegen der Wahl von a
\b a
= b2—aa2 ^ Omod/7,
laa b
über Z/(pe) invertierbar, also ist das lineare Gleichungs- .
also ist die Matrix ^
system
bu+av = 0 m o d p e ,
txau+bv = 1 modê
lösbar, und für die Lösung des Systems gilt (a ]fü+b){u fä+v)=\.
invertierbar.
Somit ist a
fä+b
Es werde im folgenden mit G p . die Gruppe der invertierbaren Elemente von
Z(p , a) bezeichnet. Wegen Lemma 1 gilt | Gp„ | = ( p e f - (p*-1)2=(pe~ T O 2 - 1 )• In
Gp. bilden die invertierbaren Elemente von Z/(pe) eine Untergruppe Z p „ der Ordnung
pe~\p-\).
Somit gilt \Gp*/Zp.\ =pe~1(p +1).
Klarerweise ist 5 ein Epimorphismus von Gp„ auf Gp. Grundlegend für das folgende ist
e
L e m m a 2. Die Gruppe Gp„!Zpe ist zyklisch.
Beweis. Da Z(p, a) ein Körper ist, ist (^zyklisch. Daher ist auch GJZp zyklisch.
Sei gZp erzeugendes Element von Gp/Zp und sei ß ein Urbild von g bei 8. Sei o die
Ordnung von ßZp., dann gilt ß°£Zp.. Es folgt g°iZp, also o=(p+\)r.
Die
r
r
Ordnung des Elements (ßZp„) =ß Zp. beträgt somit p +1, und wir haben gezeigt:
In Gp„/Zp. gibt es ein Element der Ordnung p +1.
Wir betrachten nun das Element y = ( l +p]fc)eZ(pe,a).
Es gilt / ° = 1 +
1
2
e
+p fä +p r0 mit r 0 €Z(p , a). Es sei für ein/'mit 0î<e—l
schon gezeigt, daß
yp*
= l + p-'+i]/«" + pt+2rt
mit
r£Z(pe,
x).
Dann gilt
yp'*'
= (yp')p
=
(l+p'+i
=
l+pi+2
Daraus folgt, daß die Ordnung von yZp. den Wert pe~1 hat, und damit haben wir
gezeigt: In Gp„/Zp„ gibt es ein Element der Ordnung pe~x.
In abelschen Gruppen gilt: Haben zwei Elemente xlt x2 die teilerfremden Ordnungen o1 und o 2 , dann hat das Element x1x2 die Ordnung o1o2. Somit hat in Gp./Zpa
das Element yßrZp0 die Ordnung pe~1(p+1), und Lemma 2 ist bewiesen.
L e m m a 3. In der Gruppe Gp./Zp. bilden die Elemente (r/a-f-s)Z p . mit r = 0
mod p eine zyklische Untergruppe der Ordnung pe_1.
292
R. Nöbauer
Beweis. Die Menge der r/ä+s£Gp. mit r = 0 m o d p ist die Urbildmenge von
Z p bei <5, also selbst eine Untergruppe Up.czGp. mit Zp.<zUp.. Somit ist Up.IZp.
Untergruppe von G p ./Z p „, die als Untergruppe einer zyklischen Gruppe zyklisch ist.
Wegen \Up.\=pe~1(p—\)pe~1
folgt |t/ p „/Z p .|=p e - 1 , womit Lemma 3 bewiesen ist.
Es soll nun die Anzahl der Fixpunkte der durch f„(x), (n,pe~1(p +1))=1,
induzierten Permutationen n„ von Z/(^ e ) bestimmt werden. Dazu zunächst eine
Vorbemerkung:In Q (/a)[x] bildet Z [ / ö ] [x]einen Unterring; es sei q der kanonische
Epimorphismus von Z []/<*] [x] auf Z(pe, a)[jc]. Dann erkennt man durch Anwendung von Q auf (2.1), daß
(x + lfcY =
gn(x)+h„(x)fc
auch in Z(pe, a ) M gilt- Zum Abzählen der Fixpunkte von n„ benötigen wir folgendes
L e m m a 4. Die Anzahl der Fixpunkte von n„ ist gleich der Anzahl der Elemente
u£Z/(pe), für welche
gilt.
Beweis. Sei u Fixpunkt von n„. Dann gilt /„(«) = « m o d p e , also g„(u)=u • h„(ü)
mod pe, und somit gilt in Z(pe, a)
(M + }^)" =
Daw+Va" regulär ist, folgt (u+Y^f^^K
h„(u)(u+fc).
(u)£Zp,, und daher gilt ((w+/cT)Z p<I ) n - 1 =
Sei umgekehrt ((u+iföi)Zp.y-1=Zp.,
also (u+fä.)n~1=yZZpc.
Dann gilt
n
(u+fü) —yu+yia
=gn(u)+h„(u)fä,
und es folgt gn(u) = uh„(w) modp e , d.h.
e
f„(u) = u m o d p . Damit ist Lemma 4 bewiesen.
Da die Elemente von UpC/Zp. kein Element der Gestalt ^öT+w, die übrigen
Elemente von G p „/Z p . aber genau ein Element dieser Gestalt enthalten, ist die Anzahl
der Fixpunkte von n„ gleich der Anzahl der Lösungen der Gleichung £ n - 1 = 1 in der
Gruppe Gp,/Zp. minus der Anzahl der Lösungen in der Gruppe l/ p ,/Z p „. Da
\Gp./Zp.\=pe-\p+l),
gilt
in Gp./Zp. genau dann, wenn ^-I.P-'CP+I)) = I.
Die Anzahl der Lösungen dieser Gleichung in der zyklischen Gruppe
Gp./Zp.
beträgt (n — l,pe~1(p + l)). Ferner gilt wegen \Up4Zp.\=pe~x
in Up./Zp. genau
dann
wenn ¿; ( "~ 1 ' P °" I) =1; die Anzahl der Lösungen dieser Gleichung in
der zyklischen Gruppe t/ p ./Z p . beträgt (n — l,pe~1). Wegen (n—l,pe~1(p + l))=
=(n — l,pe~1)(n — l,p + l) gilt also folgendes
L e m m a 5. Sei (n, pe~1(p + l))=l. Dann ist fix (pe,n), die Anzahl der Fixpunkte der durch f„(x) dargestellten Permutation von Zj(j>e), gegeben durch
43
Rldei-Funktionen
S a t z 1. Sei m eine ungerade natürliche Zahl mit der Primfaktorzerlegung
m=
e
=Pil-P%
i— 1» und sei ol eine ganze Zahl mit a. quadratischer Nichtrest m o d p l y
/ = 1, ..., r. Sei v=[pe^~1(j>1+\), .•.,pe/~1(j>r+1)],
sei n eine natürliche Zahl mit
(n,v)—l
und sei d=(n—l, v). Dann ist fix (m,ri), die Anzahl der Fixpunkte der
durch f„(x) dargestellten Permutation von Z/(m), gegeben durch
fix (m, n)=
n (d, Pp-1)^,
Pi+l)~
i=i
1).
r
Beweis. Aus dem Chinesischen Restsatz ergibt sich ßx(m, n ) = ] / ß x (pf', n)~
¡=i
Gemäß Lemma 5 gilt fix (pf>, n ) = ( n — l , p f ' ~ l ) ( ( n — l , p , + l ) — l ) . Aus pf'~ 1 lv folgt
(rt-l,p?-1)=(n-l,v,tf-1)=((n-l,v),rt-l)=(d,tf-1),
und aus
(p,+l)|t>
folgt ( n — l , p t + l ) — ( n — l , v , p f + l ) = ( ( n — l, v ) , p t + l ) = ( d , p { + l ) .
Somit erhält
man fix (Pf', n ) = ( d , p f ' ~ 1 ) ( ( d , p i + l ) — l), und daraus ergibt sich die Behauptung.
4. Die Gruppenstruktur. Es seien m, a, v, n und d wie in Satz l,und es bezeichne
7in die d u r c h i n d u z i e r t e Permutation von Z/(m).
L e m m a 6. Genau dann induziert f„(x) die Einheitspermutation e von Zl(m), wenn
gilt « = 1 m o d v .
Beweis. Genau dann gilt n„=e, wenn jedes Element von Z/(/w) Fixpunkt
bezüglich 7in ist, und dies ist gleichbedeutend mit fix (m, n)=m. Nach Satz 1 gilt
Gx{]Jpp,n)=
¿Pf'
i=i
i=i
genau dann, wenn gilt (d,peil~1)=peii~1
und {d,pl+\)=pi+\
für / ' = 1 , . . . , ry
also dann und nur dann, wenn
\ d und (/»¡+1)1«/, / = 1 , ..., r, und dies ist gleichbedeutend mit v\d. Da nach Definition von d stest gilt d\v, ist somit 7i„=e äquivalent zu v=d, und damit ist die Behauptung bewiesen.
L e m m a 7. Genau dann induzieren f i x ) und f„{x) dieselbe Permutation
Z!(m), wenn gilt k = n mod v.
Beweis. Sei nk=n„.
von
Man wähle /=>0, so, daß In = 1 mod v. Gemäß Lemma 6
gilt
8 =
n
ln =
=
=
nlk.
Aus s—nlk folgt — wiederum aus Lemma 6 — lk= 1 mod v, also Ik = In mod v,.
und daraus erhält man k = n mod v.
Sei nun andererseits k=n mod v. Wieder wähle man / > 0 so, daß /« = 1 mod vy
es gilt dann auch lk= 1 mod v. Mit Hilfe von Lemma 6 erhält man
7lk = UkO 7t,„ = 7ik0 7tl0 7in = TtklOn„ = 71„.
R . Nöbauer
294
S a t z 2. Die Gruppe Pm aller jener Permutationen von Z/(m), die durch eine Funktion fn(x) mit gegebenem a und («, v)= 1 induziert werden, ist isomorph zu Zv, der
primen Restklassengruppe von Z/(v).
Beweis. Nach Lemma 7 ist durch i{/(rtn)=n mod v eine wohldefinierte und
bijektive Abbildung von Pm nach Z„ gegeben. Diese ist wegen (2.3) mit o verträglich,
also sogar ein Isomorphismus.
Es sollen nun alle ungeraden m ermittelt werden, für die Pm zyklisch ist. Entsprechende Resultate für andere durch Permutationsfunktionen dargestellte Permutationsgruppen findet man z.B. in [2], [5] und in [7].
S a t z 3. Die Gruppe Pm ist genau dann zyklisch, wenn einer der beiden folgenden
Fälle vorliegt:
(a) m=3,
(b) m >3, m quadratfrei, und es gibt eine ungerade Primzahl q, sodaß sämtliche
Primteiler pt von m darstellbar sind in der Gestalt
pt = 2qki-l,
k ^ l .
Beweis. Nach bekannten Sätzen der Zahlentheorie ist Zw genau dann zyklisch,
wenn w gleich einem der folgenden Werte ist:
(3.1)
1,2,4, qe, 2qe
(q ungerade Primzahl, e s 1).
Nach Satz 2 ist Pm isomorph zu Z 0(m) . Wir haben also alle ungeraden natürlichen
Zahlen m^3 zu bestimmen, für die v(m) einen der Werte (3.1) annimmt. (Da es
keine quadratischen Nichtreste modulo 1 gibt, existiert P1 nicht.)
Es gilt für alle ungeraden m mit m s 3 , daß v(m)^4, also nimmt v(m) für
kein in Frage kommendes m die Werte 1 oder 2 an. Weiters gilt v(m)=4 genau
dann, wenn m—3.
Sei im folgenden q eine feste ungerade Primzahl, und sei e S l .
Angenommen, es gibt ein ungerades wiS3 mit v(m)=qe. Sei pt ein Primteiler
von m, dann gilt ( / J j + l j H w ) , also (Pi+l)\qe. Daraus folgt 2\qe, und dies ergibt
einen Widerspruch. Es gibt also keine ungeraden Zahlen m mit v(m)—qe.
Sei mm m^3 eine ungerade Zahl mit v(m)=2qe. Sei pt ein Primteiler von m,
und sei v (m) die Vielfachheit, mit der pt in der Faktorzerlegung von m vorkommt.
Wäre V P ( (/M)>1, dann gälte Pi(pt+l)\v(m),
also pi(j>i+\)\2qe,
und dies ergäbe
einen Widerspruch zu pt ungerade. Also gilt vp (m)=l. Wegen (/> f +l)|«(m) gilt
(j?i + \)\2qe, und daraus folgt pi+\=2qs
mit l W s S e , also p =2qs-\.
DieZahl
m ist also von der Gestalt (b).
Sei andererseits m von der Gestalt (b), und sei r die Anzahl der Primfaktoren von
m. Dann gilt
v(m) = [2qki, ..., 2qkr] = 2qmax^v -K),
Rldei-Funktionen
295
d.h. v(m) ist von der Gestalt 2q e , und Pm ist zyklisch. Damit ist Satz 3 vollständig
bewiesen.
Als unmittelbare Folgerung erhält man: Mit Ausnahme von 3 gibt es keine
natürlichen Zahlen m mit m= 3 mod 4, für die Pm zyklisch ist.
5. Eigenschaften der Fixpunktanzahl. Es sollen nun einige mit der Fixpunktanzahl fix (m, n) in Zusammenhang stehende Fragen erörtert werden. Wie Satz 1 zeigt,
ist fix (m, n) bei gegebenem m durch d eindeutig bestimmt. Bezeichnet man für einen
festen Teiler d von v die Anzahl der Permutationen n„£Pm, für die gilt
(n—l,v)=d,
mit o(d, v), dann gilt
s
a
Satz 4. Sind v= JJ q f , £/— 1, und d= JJqV,
j=i
j=i
legungen der Zahlen v und d, so gilt
a(d, v) = (vld) #(1 -Ej/qj)
mit Ej =
Oshj^gj,
die Primfdktorzer-
' 2 für h j = 0,
1 für 0 <= hj < gj,
0 für hj = gj.
Beweis. (Vgl. [4], wo ein analoges Resultat für die Gruppe der durch die Potenzen yf induzierten Permutationen von Z/(m) hergeleitet wird.) Klarerweise ist a(d, v)
die Anzahl der Restklassen a mod v mit (a,v)=d und ( a + l , w ) = l . Nach dem
Chinesischen Restsatz kann man die Restklassen a mod v bijektiv zuordnen den
j-Tupeln (at, ..., a,), in denen Oj jeweils alle Restklassen modulo qf' durchläuft. Mit
Hilfe der Gleichungen a=aj+kjqgji, j=\, ..., s, erkennt man
(a, v) = ]J(a, q f j ) = ¿(aj+kjqfj,
j=i
i
qf>) = ¿(üj,
J=i
qfj).
Da der Restklasse fl + 1 das i-Tupel ( ^ - f 1, ..., a s +1) entspricht, ist
(a+1, v)= 1 gleichbedeutend mit (a ; + l, qp)=l für alle j und dies wiederum ist
gleichbedeutend mit a^ — 1 mod q}, j=\, ...,s. Ferner ist (ßj,qp)=qp
gleichbedeutend damit, daß aj=bqmit
{b, qj)—1. Die Anzahl der verschiedenen
Restklassen ay- mod qp, die diese und die vorhergehende Bedingung erfüllen, ist also
gegeben durch
q?J-hJ-HqjSj)
=
qfj-hj(l~£jlqj),
woraus die Behauptung folgt.
L e m m a 8. Sei dein Teiler von v. Es gilt a(d, v)=0 genau dann, wenn d ungerade
ist.
Beweis. Da m ungerade ist, hat m einen ungeraden Primteiler pt, und es folgt
(,Pi+l)K also 2|v. Sei o.B.d.A. q1=2. Aufgrund von Satz 4 gilt o(d,v)=0 genau
dann, wenn für ein ./€{1,
gilt l - e y / ^ = 0 , d.h. wenn für ein j gilt e j = q j .
296
R. Nöbauer
Wegen
gilt dies höchstens für q}=2, also für 7 = 1 . Es gilt £ 1 =q r i, also
e 1 = 2 genau dann, wenn ^ = 0 , also wenn ¿/ungerade ist. Damit ist das Lemma bewiesen.
Für kryptographische Anwendungen sind speziell die beiden folgenden Problemstellungen von Interesse: (i) Welches ist die kleinste Zahl u>min, die als Fixpunktanzahl einer Permutation nn£Pm auftreten kann? (ü) Wieviele Permutationen nndPm
gibt es, die genau wmin Fixpunkte aufweisen?
Aufgrund von Satz 1 und Lemma 8 treten als Fixpunktanzahlen von Permutationen n n ^P m genau die Zahlen
r
i(d, m) = 77 (d,
Pi+l)~
1),
d gerader Teiler von v
auf. Setzt man d= 2, so erhält man t(2, m ) = l . Es gibt also Permutationen nn£Pm
mit nur einem Fixpunkt. Um die Anzahl aller n„£Pm mit nur einem Fixpunkt
bestimmen zu können, benötigt man
Satz 5. Die Funktion x(d, m) ist streng monoton auf dem Verband der geraden
Teiler von v.
Beweis. Es ist zu zeigen, daß T(C, m)<t(d, m), wenn c ein echter Teiler von
d ist. Da dann (c,
und (c,
1)1(^,^+1) für alle /, also
( c , / ^ - 1 ) ^ , / ? ? ' - 1 ) und ( 0 , ^ + 1 ) ^ ( ^ , ^ + 1 ) , folgt jedenfalls t ( c , v ) ^ x { d , v ) .
Sei qj eine Primzahl, die in c mit der Vielfachheit vqj(c) und in d mit der Vielfachheit
vq (d)>vq (c) vorkommt. Wegen d\v gilt vq (d)Svq (v) = max {vq (pf , - 1 (i?i + l))}.
J
1
j
J
lî^r
J
Also gibt es ein / mit q ^ ^ l p ^ i p t + l ) - Ist qj=Pi, dann gilt (c, pei'~1)^(d,
p^'1).
Ist aber q j ^ P i , dann gilt (c, +
/>¡ + 1). Daher gilt jedenfalls
(c,PF'~ 1 )((C,/' i +l)-l)<(i/,/'F'- 1 )((i/,A+l)-l)> und daraus folgt T ( C , m ) ^ z ( d , m ) .
F o l g e r u n g 1. Die Anzahl der Permutationen nn£Pm mit nur einem
beträgt
1 für v ^ 0 mod 4
(«,/25)77(1-2/?;)
mit ö =
2 für v = 0 mod 4.
Fixpunkt
Beweis. Sei d ein gerader, von 2 verschiedener Teiler von v. Dann ist 2 echter
Teiler von d, und aus Satz 5 folgt 1—t(2, m)-<-x(d, m). Somit ist die Anzahl der
Permutationen n„ von Z/(w) mit nur einem Fixpunkt gleich <r(2, v). Aufgrund von
Satz 4 folgt daraus unmittelbar die Behauptung.
6. Kryptographische Anwendung. Es soll nun die Beschreibung eines Pubüc-Key
Kryptosystems auf der Basis der Redei-Funktionen erfolgen.
297
Rldei-Funktionen
Jeder potentielle Kommunikationsteilnehmer C wählt eine Faktorenanzahl r und
r große Primfaktoren plt...,pr
mit zugehörigen Vielfachheiten ex,..., er, bildet
r
das Produkt m=JJ p\l und berechnet die gemäß Satz 1 definierte Größe v. Weiters
wählt C eine ganze Zahl a mit a Nichtrest mod pt, i= 1, ..., r — etwa durch sukzessives Testen zufällig gewählter Testzahlen a mit Hilfe des Eulerschen Kriteriums-.
Schließlich wählt C einen Chiffrierschlüssel « > 0 mit
(6.1)
und
(n,v)= 1
(6.2)
(« —1, p) = 2
und berechnet zu diesem n durch Lösen der linearen Kongruenz
(6.3)
nk = 1 mod v
einen zugehörigen Dechiffrierschlüssel
0. Im öffentlich zugänglichen Schlüsselverzeichnis veröffentlicht C die Größen mc=m, a c = a und nc=n, hält jedoch die
Faktorzerlegung von m sowie kc=k und vc=v geheim.
Angenommen, A möchte an B die Nachricht <p übersenden. A sucht aus dem
öffentlich zugänglichen Schlüsselverzeichnis die Schlüsselparameter von B, also m B , a B
und nB, stellt die Nachricht cp als Folge von Blöcken xt, xfcZUjn^, dar, chiffriert
die Xi mittels
(6.4)
xt •* yt = f„B(.xd mod mB
und übersendet die yt an B. Der Empfänger berechnet mit Hilfe des nur ihm bekannten Dechiffrierexponenten kB aus den yt die Nachrichtenblöcke xt:
fkB(yd =fkB°fnB(xi)
=fkBnB(.xt) =fi+tVB(Xi)
= Xi mod mB.
Dabei steht t für eine geeignete natürliche Zahl, und die letzte Gleichung gilt aufgrund von Lemma 6.
Da zur Berechnung von vB die Faktorzerlegung von mB benötigt wird, jedoch
bis heute keine schnellen Algorithmen zur Faktorisierung großer Zahlen bekannt
sind, ist es nicht möglich, aufgrund der öffentlich bekannten Informationen mB,
aB und nB das zur Berechnung des Dechiffrierschlüssels kB benötigte vB zu ermitteln.
Forderimg (6.2) garantiert, daß die durch f„(x) induzierte Chiffrierfunktion nur einen
Fixpunkt aufweist.
R. Nöbauer: R6dei-Funktionen
298
Literatur
und M . HELLMAN, New directions in cryptography, IEEE Trans. Information Theory.
IT—22 (1976), 644—654.
H . HULE und W . B . MÜLLER, Grupos ciclicos de permutaciones inducidas por polinomios
sobre campos de Galois, An. Acad. Brasil- Cienc., 45 (1973), 63—67.
R. LIDL und W. B. MÜLLER, Permutation polynomials in RSA-cryptosystems, in: Proceedings
Crypto 83, University California, Santa Barbara, 1983.
W . B . MÜLLER und W . NÖBAUER, Über die Fixpunkte der Potenzpermutationen, Österr. Akad.
Wiss. Math. Naturwiss. Kl. Sitzungsber., II (1983), 93—97.
W. NÖBAUER, Über eine Gruppe der Zahlentheorie, Monatsh. Math., 58 (1954), 181—192W . NÖBAUER, Über Permutationspolynome und Permutationsfunktionen für Primzahlpotenzen,
Monatsh. Math., 69 (1965), 230—238.
W . NÖBAUER, Über Gruppen von Dickson-Polynomfunktionen und einige damit zusammenhängende zahlentheoretische Fragen, Monatsh. Math., 77 (1973), 330—344.
L . R£DEI, Über eindeutig umkehrbare Polynome in endlichen. Körpern, Acta Sei. Math-, 1 1
(1946), 85—92.
R . RIVEST, A . SHAMIR und L. ADLEMAN, A method for obtaining digital signatures and publickey cryptosystems, Comm. ACM, 21 (1978), 120—126.
[1] W . DIFFIE
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
UBW KLAGENFURT
UNIVERSITÄTSSTRASSE 67
9010 KLAGENFURT, AUSTRIA
Acta Sci. Math., SO (1986), 299—330
Some sufficient conditions for hereditarily finitely
based varieties of semigroups
G. POLLÂK
Introduction
The proofs of most theorems saying that one or another variety is hereditarily
finitely based are very similar to each other (in so far as syntactic proofs are concerned).
The general scheme of such proofs has been described in [6] (see also Theorem 2.5
of the present paper); however, this description does not help much more in future
proofs as finger-posts do in alpinism. The essential difficulty usually lies in proving
that the objects and relations in the scheme are what they ought to be (sometimes it is
not even easy to construct.them). We think therefore that every unification which
renders possible to claim that a more or less broad class of varieties is h.f.b. is of
interest.
In the present paper we give sufficient conditions of the following two types:
a) if J is a fully invariant ideal of the countably generated free semigroup F, and a
certain quasi-ordered set (connected with F\J) is well-quasi-ordered then the variety SG (J) defined by all identities u=v, u, v£J is h.f.b.; b) if V is a variety, MczF
is a standard form for elements of J (i.e. every w £ / equals to some w* iM), and M
itself, as well as the "process of standardization" are subject to certain conditions,
then every variety in the lattice interval [V, V(~)SG(J)] is finitely based over V.
Furthermore, we show that certain concrete subsets of F satisfy these conditions. As
an application, we find all h.f.b. identities in one of the four classes of "candidates"
to such identities (see [3]; class (d)). This result accomplishes, in a certain sense, the
investigations concerning such identities; namely, classes (c) and (d), as well as balanced h.f.b. equations are completely described now (see [1], [4], [6]), homotypical and
some other equations of class (b) are settled in [5], and it looks likely that presently
known syntactic methods cannot help us much further.
Received October 3, 1983.
300
G. Pollik
Part L General sufficient conditions
1. Preliminary. We need rather a lot of not generally known concepts and
of notations running through this paper; so we have collected most of them here.
By the free semigroup F we always mean the countably generated free semigroup
F(X) on the set of generators X— {jq, x2, ...}• As we need also the free monoid F°,
we shall call the elements of F terms, the elements of F° words (i.e. a word can be
empty, a term cannot). The coincidence of words will be denoted by u=v; the formula u=v is an identity which holds in some subvariety V of the variety of all semigroups SG. The empty word, as well as the empty set, we denote by 0; this will not
lead to confusion.
The set of all variables (letters) which occur in u will be denoted by X(u). Furthermore, \u] denotes the length of u, and \u\t the number of occurrences of xt in u.
The words w(i), t/° are the prefix and the suffix of length / of the word u (of length
S / ) , resp.:
(1.1)
|«(I)| = |«(,)| = /.
u = ua)u' = u"u"\
A third kind of denoting equality is defined by
def
—r^
(1.2)
u =! utu2 o u = uxu2 and X(u{) Pi X(u^ = 0.
Notethat u=! uxu% implies \X(u)\ = ^(MJ)! + ^(u^.
I f a word has no decomposition of the form (1.2), it is said to be irreducible. Every word has a unique irreducible decomposition:
r
drf
u =!! JJ utou=l
1=1
(1.3)
r
JJ u,,
t=I
u{ irreducible for i = 1,..., r.
We call the components w, the irreducible factors of u. The word u is said to be
semiirreducible if |«,| > 1 for every / (in particular, 0 is semiirreducible). The decomposition
s
u = ! w0 i7*c(i) w i>
]=i
(1.4)
Wj semiirreducible
will be called the semiirreducible factorization of u, and w>„,..., M>s its semiirreducible
factors. A word is said to be simple if its semiirreducible factors are empty (i.e. if no
letter occurs in it more than once). Besides (1.4), we shall also make use of the reduced
semiirreducible factorization
(1.5)
*
w = ! w 0 IJaiWi>
i=1
semiirreducible, a, simple, wt, a{ £ 0 for i = 1, ...,s— 1 .
Clearly, both (1.4) and (1.5) are unique.
Hereditarily finitely based varieties of semigroups
301.
By F (n) we denote the set of all words having only semiirreducible factors of length
In particular, F ( 0 ) =F ( 1 ) consists of all simple words. Obviously, F (N) C.F (n+1)
for every n £ l , and F°\F(n)=J(n)
is a fully invariant ideal in F. Let Mc.F°. Set
(1.6)
= {u: u = a0 ]J «¡a,, |a0...a,| S m, fi =!
¡=i
M ( m ) = {«; u£Mlm\
(1.7)
X(a0...a,)nX(G)
G^.M^M},
= 0}.
It is easy to see that
L e m m a 1.1. F$QF$n+1»
if n^ 1.
Indeed, if «€ F j f f , then at most m factors of the factorization (1.4) of u can contain some element of X(a0...at) (as \X(a0...a^\=in),
and every such factor is of
length ^ n . Put
f
¡=i
where every a\ is equal to a factor of (1.4) which contains a letter of a0...a,.
\a'0...a't-\ ^ ( n + l ) m ,
and
x(a'0...a't.)nx(u[...t4.)
Clearly,
= 0.
Two words u, u' are said to be of the same type if there is an automorphism
agAut F° which maps u into u': ua=u'. The set of all words of the same type
(an orbit of Aut F°) is called a type. E.g. X and 0 = {0} are types. The type of u will be
denoted by T(u). If u is irreducible, simple etc., the same is said about T(u).
An endomorphism <p€End F" is said to be disjoint if X(xt <p) fl X(xj <p)=0
provided 1V7. The endomorphism 1 for at most finitely many
Xi's. The number
y(<p)=
2(1*^1-1)
i=1
is called the growth of <p. The set of all disjoint endomorphisms will be denoted by
Dend F°, that of all finite disjoint endomorphisms by Fde F°.
The proof of the following facts is straightforward (see also [4]).
L e m m a 1.2. If u^X is (semi)irreducible and <p£Dend F°, then uq> is (semi)irreducible.
m
L e m m a 1.3. If M=! JJwiand
i=l
Let r(u) ( = r ) , s(u)(—s)
tively.
m
<p£Dend F0 then uq> = \ IJwt<p.
1=1
be the number of factors in (1.3) and (1.4), respec-
L e m m a 1.4. If <p£Fde F° then r(u)Sr(u(p)Sr(u)+y((p),
s(u)^s(u<p)S
^s(u)+y(q>), and the image of an irreducible factor
X of u is an irreducible factor
of u<p.
302
G. Pollik
We have to deal with several order and quasi-order relations. The most important
order relation will be the lexicographical order of words defined by
def
u < I e x vov
= uv,
v £F
or
u = wx.u ,
v = wXjV ,
.
.
.
i<;.
The lexicographical order is not a well-order on F°, however, it is a well-order on
F°\Fn (the set of words of length <«).
Let F b e a variety of semigroups, and F(V) the free semigroup in F on the infinite set of generators {^v, x 2 v, ...}, where v: F—F(V) is the canonical homomorphism. A fully invariant ideal J is said to be a V-ideal i f / i s the full inverse image
of Jv. In particular,
/(V)={u£F: there is a v£F such that vû, F t = « = » }
is a F-ideal.
Let a be a set of identities. By F(CT) we denote the subvariety of F consisting of
those algebras which satisfy a. If / is a fully invariant ideal then
V(J) = V(T)
where
x ={u =
v:u,v£J}
(i.e. V(J) is generated by the algebra' F(V)/J).
Following Petrich, we term an identity u=v homotypical if X(u)=X(v)
and
heterotypical else. The ideal
J0(V)={u£F:
there is a v£F such that X(u)^X(v),
Vt=u=v}
is a F-ideal, too. Obviously, J0(V)QJ(V),
and J0(V)v is the kernel of F(V).
Two systems of identities
cr2 are said to be V-equivalent if
V N
<=> as.
Similarly, a system a is V-finite, V-independent etc., if it is F-equivalent to a finite
system, not F-equivalent to any proper subsystem of itself etc. Furthermore, V'( Q V)
is said to be finitely based over V if V = V(a), a finite (or, equivalently, F-finite).
The interval [V, V) of the lattice of varieties is finitely based if every element of [V, V)
is finitely based over V. If F i s finitely based, too, then we say that [V, F] is finitely
based.
We say that the system of identities a (or, also, the identity u=v in the case
a— {u=v}) is hereditarily finitely based (h.f.b. for short) if the variety SG(o) is, where
SG is the variety of all semigroups (which, by definition, means that every subvariety
of SG(o) is finitely based).
Let J be a fully invariant ideal in F. A subset M<ZF is termed a standard form
for Vin J (or, for short, M is standard for V in J) if for every u£J there is a u*£M
such that Ft= u—u* (neither uniqueness nor the existence of an algorithm for finding it* is demanded). Clearly, if M is a standard form, then so is every M'
M.
Moreover, if / is a F-ideal, u£J and u=u* imply u*iJ. Thus, JC\M is also a
Hereditarily finitely-based varieties of semigroups
303
standard form for V in J. However, sometimes it is more convenient to work with
larger standard forms. If J=F we simply say that M is a standard form for V.
Usually we state oiir theorems for standard forms in an arbitrary J because the more
elegant special case of standard forms for V does not suffice in the applications.
Finally, if o={us=vs} then by c we denote the system c U {ws=«s}.
2. Well-qnasiôrders and h.f.b. varieties. A quasi-order relation «< is said to
be a well-quasi-order (wqo for short) if there are neither infinite (strictly) descending
-<-chains nor infinite <-antichains or, equivalently, if every infinite sequence contains
an infinite (not necessarily strictly) ascending subsequence ([2]). The following quasiorderings on subsets MQF° will occur:
wok'
u<iyu'
iff
=
uq>- w2 for some
iff V N u' =
iff u' = uip for some
u
iff
u l 3 u2£F°,
0
• u(p • w2 for some
u<ihu'
u'
(¡»Ênd-F0,
cpdEndF ,
wl9 M2€F°,
<p£End F°,
u' = ucp for some
(p£FdeF°.
Note that in the last case it is sufficient to find a <p'£Dend F° such that u'=ucp':
this can be always modified so as to obtain a <p€Fde F°.
Let, furthermore, P denote the set of positive integers and I the symmetric group
on P. For every. k£P we define two quasi-orders, one on Mxl
and one on M2XZ
as follows:
(u, JI) <Kk («', 7zr) iff 3<p£Fde F°(I«p = u\ xin v';n)
iff 3<p€Fde F°
(u<p = u', xincp = xin,
for 1 ^ i S k, v<p£M).
The remark made above about <p is valid here, too. Also, it is worth noting that in the
definition of <sc2 the word v' does not play any role, and that « is a wqo if <sck is.
In proving varieties to be h.f.b., often it is crucial to know that one or the other
of these quasi-orders is a wqo for some standard form M. As for the second one, this
is even indispensable:
L e m m a 2.1. If Vis h.fb., then
visa
wqo on F°.
^
P r o o f . Obviously, no infinite descending
-chain can exist. If «i,« 2 , ...
were an infinite <aK-antichain, then consider the system o r ={«2t-i :=: "2t : k = l , 2, ...}.
We are going to show that V(a) is not finitely based, moreover, if
ak=o\{u2k_1—u2k}
then V(<rk)¥=u2k-1=u2k. Indeed, suppose there exist (u2k-i = )vi^ •••ivi( = u2k) such
that vt=v\ • Wi<Pi -v",
• w'î-v" for / = 1,...,/— 1, where <?>,•€ End F,
v\, v"€F° and either FNw—w- or w—w'^^.
If it is always the first instance
that prevails then F N u2k =u 2 k , which is not the case. Let i be the minimal index such
4*
304
G. Pollik
that V&v~vi+1,
and, say, wû^^,
w\ = u2r, r^k.
Then
Vt=u2t_1=vi=
=*>' • u2r_lq>i • v", contrary to the assumption.
Clearly, if <sct is a wqo (for some fixed M) then <sz, for 0
is a wqo, too.
However, already the condition that « 1 is a wqo on some MXE is rather restrictive
as the following lemma shows.
L e m m a 2.2. If <scL is a wqo on MX Z then there is a natural number p such that
\u\^p for every u£M and /6P.
P r o o f . Put p(u)=max \u\h and suppose p(u1)~zp(u2)<...,
\Uj\kU)=p(uj).
Then we obtain an infinite «vantichain ( U j , n j ) , j= 1 , 2 , . . . by putting itj=
=(1 k(J)): if cp is disjoint and xk(J)<p=xm,
jî,
then
whence UjCpÛi.
Let G n>fc ={(a 1 ,..., iJr)£(F°)r: \av..ar\^.n,
\X(ai...ar)\^k}.
The following
proposition enables a more flexible handling of
P r o p o s i t i o n 2.3. (MXE, <zk) is wqo i f f MxGn
(2.1)
(u;
...,«,)-<(«';
ai, ...,a'r.)
3(p£Fde F°(u<p = «',
k
is wqo for every n under
i f f r = r'
and
= a't for i = 1, ..., r).
P r o o f . The sufficiency is obvious because (u, n)<s:k(u', n') is equivalent to
(u; xu, ...,***)-<(«'; xw, ...,x kx .). Now let <ck be a wqo, and let x ^ , ...,xs(q) be
the variables of av..ar (in the order of their first occurrence). Set
and put (u;ax, . . . , a r X 0 ( « ' ; a i , ...,0^) iff r=r',
T(al...ar)=*T(£T(a,)=
= T(a'i) for i = l , ...,r, (u, it)<s:q(u', n'). Now -<0 is a wqo because q^k and r,
T(ax...ar), T(as) can take only a finite number of different "values". Furthermore,
(u; alt ..., ar)<0(u'; a[, ..., a',) implies (u; ax,..., a r )-<(«'; a[, ...,ar) because the
very endomorphism <p which satisfies u<p=u\ xsu)(p=xs'u)
maps a, on a\. Hence
-< is a wqo.
The following proposition shows that the conditions that <sck is a wqo for different numbers k are not very far from each other.
P r o p o s i t i o n 2.4. If M<zF°, and there is a natural number q such that uv£M
implies \X(u)C\X(v)\ =q, furthermore, <s:29+1 is a wqo on M, then <sct is a wqo on Mfor
every k.
P r o o f . For k^2q+\
(in particular, for k= 1) the assertion is obvious. So
suppose it holds for some k. For (u, n)£MXZ consider the decomposition
r
" = "o nxun"i,
f=i
h
k, X(ut)N{*!«,...,
= 0,
305.
and put
r
« = 77*/,=»
1=1
« = «o-"r>
"i = «0—«i-i.
^(«) = (z(iii)nz(«i))\{x№+1)lt}, J(«)= u
*,(u),
i=1
«i =
X(ud
=x(u)nx(ui).
It is easy to see that r^kp where p is the bound from Lemma 2.2, |X(u)|^rg, and
X(«¡)^Xj(u)iJXl+1
(u), whence qi=\X(u^)\^2q.
Choose permutations
n0,...,nr
such that ljr,=(Ar+l)jr, x 2 ) t j ,..., * (9(+1)it| are the different elements of X(u t )
(e.g. in the order of their first occurrence in «,), and define
(«, it) -< («', n") iff («, n)
(u\ n'),
(2.2)
(«¡,«29+!(",',0
for
i = i,...,r,
and
r ®,+l
r
«i+l
T(77 77 x №( ) = T(N 77 *<*;)•
i=0 t=l
i=o r=i
As (w, n)<ak(u', 7t') implies T(u)=T(u'),
r is the same for («, 7t) and for (u\ n'),
and the definition makes sense. Furthermore, (2.2) implies that qi=q'i for 0 ^ / S r ,
because l7t ( =(fc+l)jr, IN'I=(K+1)TC'
for every /; in particular, x ( i + 1 ) „ a n d jc(fc+1)n.
are the first letters of the corresponding products, and this fixes the length of the inner
products.
From the assumptions it follows that -< is a wqo on M X I , so it is sufficient to
show that -< is weaker than < £ t + 1 , i.e. if (2.2) holds then there is a q> €Fde F° such
that u<p=u', x,„(p=x,„. for t^k+l.
By (2.2), there are disjoint endomorphisms
<p0, ...,<pr such that
mj/ = u',
u,(Pi =
Put
xtv\p = xtK,
x,„t(pi = x,n\
v
m =
for
for
1 s t s k,
I S i S qt+1,
\x*Vt
i= 1, ..., r .
xs£X(Ui),
if
x£X(u).
Then <p is well-defined: if xseX(ui)DX(uj),
i<j, then s=ini=t'itj
for some
i ^ q t + 1 , t ' ^ q j + l, however, then, the third condition in (2.2) guarantees that
tn\=t'n'j, i.e. xs(pi = x8(pj. Also, it is not difficult to see that <p is disjoint. Furthermore, uij/=u' whence u'=nxt{g,
(in general, the sequence
..., tT depends on
(«, 7i)). Thus, u(p = u', xtK<p=x,n. for t=l,...,
k+l.
In [6], the generally used syntactic method of proving varieties to be h.f.b. is
formulated in Proposition 2.1. Here we give a slightly different version. Let MczF0,
V a variety, and J a fully invariant ideal of F. We say that M is a good standard form
for V in J (or a good standard form for V if J— F) if there exist a linear order relation
G. Pollik
306
< on M and a quasi-order relation -< on M*={(u,v):
following conditions are fulfilled:
w > » } c M 2 such that the
C) For every u£J there is a u*£M such that Vt=u=u*;
O) (M, < ) is a well-ordered set;
Q) (M*, < ) is a wqo set;
A) If.(«, v), («', v')£M*, (u, v)-<(u', v') then there is a wÇA/ such that w-=u'
and
Vt=(ii=v=>u'=w).
If one replaces Fr by / in the first part of the proof of Proposition 2.1 in [6], .oneobtains
•
P r o p o s i t i o n 2.5. If there is a good standard form for V in J then the interval
[V(J), V) is finitely based.
It is easy to see that if M is a good standard form for Via J then it is a good standard form in the minimal F-ideal JY which contains J. Thus, JVC\ M is a standard
form for F in / (even in Jv).
In order to obtain a sufficient condition for F to be h.f.b., we need the following
P r o p o s i t i o n 2.6. Let V be a variety and J a V-ideal. If the interval [V(J), V)
is finitely based and V(J) is h.f. b. over V then every subvariety of V is finitely based
over V.
The proof is based on
L e m m a 2.7. Let V be a variety, J a fully invariant ideal in F, and F ' = V(J).
Let, furthermore, a be a system of identities and u£F such that V(a)Y=u=v for any
v£J. Then V'(a)\=u=u'
implies (u'$J and)
V(a)(=«=«'.
P r o o f . Let ( « = ) v1,v2,...,vl
( = « ' ) be a sequence of terms such that Vj=
=vj-wj(pj-vj,.vj+1=v'j-wj(pj-wj
for 7 = 1 ,
/ - 1 , where v), v'JeF0, q>j£End F,
and either Wj, w'j£J or V\= Wj=Wj or (Wj=Wj)Ç.â. Suppose/ 0 is the least index for
which wlo, w'i^J- Then V(o)t= u=v,jcJ, contrary to the assumption. Hence
vfiJ ( y = l , . . . , / ) and v1, ...,vl yields a proof of u=u' in F(cr).
P r o o f of P r o p o s i t i o n 2.6. A system of identities a can be supposed, without
loss of generality, to consist of three parts o}={{u=u')£a:
u, u'ÇJ}, a'j=
= {(u—u')da: u, u'^J) and <r0={(u=u')£a: u€ J, u'$J}. Using Lemma 2.1, we can
replace all but finitely many members of <r0 by identities of type o f . if («=«'),
(V=V')£G
and u'<I V'
then V(J)T=V'=u1 • u'<p • u2 . (uu u2£F°, <p£End F);
however, v'fyJ whence V\f= v'=ux- u'(p • w2 and therefore {u=u\ v=v'} is F-equiyalent to {«=«', v=u1- u'cp • w2}. Thus, one can assume that <t0 is finite. The same
holds for aj because V(J)(a'j) is finitely based by assumption, whence o'j is F ( / ) equivalent to some finite system o*, but then Lemma 2.7 implies also V(o'j)= F(CT+).
0
VW
307.
Finally, V(<jj)£[V(J), V] and therefore is finitely based over V. This completes the
proof.
Putting together Propositions 2.5 and 2.6, we obtain a condition which can be
used in proving varieties to be h.f.b. In these applications, the following lemma will
be referred to several times.
L e m m a 2.8. Let Vbe a variety, J a fully invariant ideal in F, and M a standard
form for V in J subject to the condition
(i) for every u£j
there is a u*£M such that V\=u=u*
and |X(H*)|^|X(M)|.
If u,v£ J, X{u) ^ X{v) then for every <p£Dend F with \X(u(p)\?£\ there is a w£M
such that \X(w)\<\X(uq>)\ and V)=u—v=>u=w.
P r o o f . Choose an arbitrary y£X(v(p)C\X(u(p)
y = xx else. Define i/Ênd F by
, =fx)r\X(u<p)^0 and put
x£X(u),
*<№).
Then wi/ = uq>, and X(v\jj)=X{vcp)C\X(u<p) if X(vq>)C[X(ucp)9i0, X(v\]/)={y}
else. Now X(v)C\X(u)c:X(u);
hence X(v<p) fl X(uq>) a X(ucp) because q> is disjoint.
Thus, \X(v\j/)\<\X(uq>)\, and, in virtue of (i), the term w=(v\p)* meets the requirements.
Now we give a sufficient condition, which may seem rather sophisticated at first
glance, however, can be applied to reasonable classes of varieties. By (u, v) we denote
the greatest common prefix of u and v, i.e. the longest subword w such that u=wu,
V = wv.
•
T h e o r e m 2.9. Let V be a variety of semigroups, J a fully invariant ideal in F,
lifiQF0 for i=l, ...,/, and let
M = {u = Mj...«;: u^Mi,
and u = uu=> X(U)flX(u) < g}
be a standardform for V in J with some natural number g. Suppose, moreover, that the
following conditions are fulfilled with some natural numbers n and k^n +(l— l)g+2:;
conditional)from Lemma2.7, furthermore,
(ii) (AfjXl, « J is a wqo set for / = 1 , . . . , / ;
(iii) i/" » = v ^ v ^ . - v j ^ v j , Vj=VjVj, (p£Fde F°, and vq> is a
prefix of some v'..vx^M
such that v'i=vt for /<j, furthermore,
(n)
\v cp\ =n, then (i) 'is satisfied for vcp with some (vq>)* s. th. either (vq>)* =
= vt(pfor i= 1, . . . , j f - l , \(vj(p, (vcp)*)\^\vj(p\-n or, for some h^j, (vcp)* =
= Vi(p for /</?, (vq>)l is a proper prefix of vhq> (of Vj(p, if h=j).
Then [V(J), V) is finitely based.
G. Pollik
308
P r o o f . Fix a factorization u=ux...ui,
M by
v< u
or
ufcM^
iff either
|A"(c)| = |JT(«)i,
for every u£M.
Define < on
< \X(u)\,
v, = u, for
i = 1
j-1,
Vj<.,tlUj.
Furthermore, for (u,v)£M* set ut=v, if /<=/', Uj^vj, and
U = WXeU, W = ^...Uj^Vj,
Vj = (Uj, Vj>,
{ Vj
or
_ (wv
or
—
VjXjVVj, \v\ = n or |£| < tl, Vj = 0,
1 wxdvv.
If Vj=Vj, we put d—e, ¿3=0. Let u0 denote the product of all different variables of
U
i<h
n*(«„))= (J
-",)nz(wi+1..M,)) (which shows that
»=1
|w0|s(/-l)g)
in the order of their first occurrence in u, say. Suppose that x c ( 1 ) , . . . , x c ( r ) is the
sequence of all different letters of a=xdxe(wxdv)wu0;
clearly, r s | a | s ( / — l ) g +
+n+2=k. Choose n£Z such that tn=c(t) for t^r, and for (u, v), (u,v) (¿M* put
(u,v)<(u',v')
iff (u„7t)<£k(u'„ 1ir) for
r = r\
\w\e=\w'\e„
j=j\
|AT(m)| = |ir(®)| -o- \X(if)\ = \X(o%
T(xdxev)
w = vow'
i= 1
I,
= T(xd.xe.v 0,
= v',
vj = vj *>v'j = v'}
(letters with' denote objects which belong to («', «'))•
Lemma 2.2 implies that (M, < ) is a well-ordered set, because
\u\^p\X(u)\
with some constant p. Furthermore, as r,j, \w\e and \xdxev\ are bounded, and the
equivalences decompose M* in eight -<-independent classes, the qo-set (M*, -<) is
wqo in virtue of (ii). Thus, it remains to prove that (A) is satisfied.
If («,»)-<(«',»') then there are disjoint endomorphisms (pi, ...,(pl such that
«,<¡»1=«^, xelt)q>i=xc.it) for t=\,...,r
and / = 1 , . . . , / ; in particular, (pi and cph
coincide on XiuÔXiu,,). Hence the endomorphism <p given by
___ = fxs<pt if x s €X(u/),
-\xs(Pl
if
x£X(u)
Xs(p
is well-defined and disjoint. We have ut(p=u't, ucp=u'. Hence V$=u'=(v(p)*, and
all we have to show is (wp)*<t/'. Clearly, we can suppose |xs<p| = l for xs$X(u);
hence, by (i),
| j r ( « 0 H * ( ( ^ ) * ) | ^ \X(u<p)\-\X(v<p)\ S |Z(«)|-|Z(t>)|
S 0.
This proves the assertion for the case \X(v)\< |A^(M)| (in virtue of Lemma 2.8, even for
X(v)*X(u) if \X(u')\*\).
So let \X{v)\ = \X{u)\,
=
Then we have
also \X(u')\=\X(v')\. Next note that fw| e =jvf'| e . and xecp=xe. (as e=2n,
e'=2n')
guarantee w<p=w'. If, moreover, Vj=Vj then v'j=v'j, i.e. vt(p=v\ for the first /
309.
components of
Hence, according to (iii), either (vcp)*=v[, .••,(v<p)^_1=v'h_lr
(vcp)% is a proper prefix of v'h for some h^j or (vcp)f=v', for / = 1 , ...,j. In the first
case (vcp)*<v'<u', in the second one (vcp)*<u\ too, because
u'j. If, on the
other hand, vj=vjxdvvj, then vj=v'jxi.v'vj,
d<e, d'<e' (because v<u, «'<«')»
and T(xd xe v)=T(xd. xe.v) implies, in particular, \v\ — \i>'\. Now either |t3|<«, then
we have vj=vjxd,v'=(vjxdv)cp=vjcp,
and the same argument as for the case Vj=vj
prevails (only V'J<UJ follows now from <f'<e'); or |<5|=n, and, again by (iii),
either there is an h^j such that (vq>)*=.vi(p=v,i for I</J, (vcp)l is a proper prefix of
v'h (of v'jXfV' if h=j) whence (vcp)*<v\ or (vcp)*=v't for / = 1 , ...,j—l,
\{v'jy(vcp)*)\^\vjxd.v'\,
i.e. (vcp)*=v'jXd.Wj<lexu'j, which yields the proof for
this case, the last one.
In the special case / = 1 Condition (iii) reads somewhat simpler:
C o r o l l a r y 2.10. Let V be a variety of semigroups, J a fully invariant ideal in F,
and Mc. F° standardfor VinJ. Suppose that (M, <sck) is a wqo setfor some k^ 2, (i)
holds, and for some n^k—2 we have
(iii') if v=vv£ M, <p£Fde F°, vcp is a prefix of some v'dM, and \v{n)cp\=n,
then (i) satisfied for some standard form (vcp)* of vcp such that either (vcp)*
is a prefix of vcp or \(vcp, (vcp)*) | ^\vcp\ —n.
Then [V(J), V) is finitely based.
R e m a r k 1. If Vis homotypical (i) is fulfilled.
R e m a r k 2. It is not difficult to distil from the proof that (iii) can be weakened in
the following manner. Instead of v£M we consider pairs (v, Q ) € M X Z , and we require (iii) only for those <p£Fde F° such that 1x^1 = 1 for 1 î^k—n—2.
This
{n)
enables us to dispose not only of the elements of X(xdxe(wxd€) u0)
but of some
more variables, too, provided k is sufficiently large. It is precisely in this form that
Theorem 2.9 will be applied at the end of the paper.
The next two theorems are devoted to special cases where the conditions can be
weakened.
T h e o r e m 2.11. Let I be a fully invariant ideal of F. If ((F\I)XE,
wqo set then SG(I) is h.fb.
<K2) is a
P r o o f . Choose an a£l, and set M = ( F \ / ) U {a}. Define v<u iff either
v=a, u£F\I
or v,u£F\I
and «<,exw. By Lemma 2.2, < is a well-order.
For (u,v)£M* put
G. Pollik
310
if v ^ w . Define
(u, v) -< («', t/)
iff either
or
u <szu\
v = a,
(M, it) <c2 («', TIO,
or
»' ^ a,
u <sc u',
v = w,
W.
Clearly, -< is a wqo relation on M*. Note that u, u'â and if v=w then also vâ.
Anyway, there is a <p£Fde F such that u'=uq>. If |Ar(«)|<|A'(«)| then we can
suppose that \X(v<p)\<\X(uq>)\ whence vqxucp. Now let \X(v)\=\X(u)\. If u«u',
v=a then u'=ucp=(v(p)*=a<u'.
If u<zu', v=w, then u'=uq>=wq> • (xeu)(p
>w<p=v(p. If v, v'${a, W}, («, 7T)<S:2(M', n'), then u'=u<p, xd.=xd(p, xe.=xe<p, d'<
< e ' . Hence u(p=wcp- xe. • ucp, v(p=wcp- xd. • v<p, and either v<p£l, (vcp)*=a<u',
or v(p£F\I, v(p<lexu' whence vqxu'. This completes the proof.
T h e o r e m 2.12. If V, J, Mare as in Corollary 2.10, (i) holds, and
is wqo, then[V(J),V) is finitely based.
2
<M XS,
2
<< >
P r o o f . Define •< by
v •<• u iff either
|Z(i;)| < |Z(«)|
or
|X(t>)| = |JT(«)|, V < l e x u.
For (u,v)£M* set w=(u,v), u = wxeu, v=w or v=wxdv,
+max {/: x£X(u)\JX(v)},
v=& if v=w. Let
dê,
and put d= 1 +
and define
(u,v)<(u',v')
iff (u,v,7i)«l(u',
X(v) = X(u) <=> X(v') = X(u'),
v';n),
\w\e = \w'\e.,
v = wov'
= w'.
Obviously, < is a well-order on M and -< is a wqo on M*. If X(v)^X(u)
and
\X(u')\7i\ then (A) follows from Lemma 2.8. If \X(u')\ = \ then |Z(u)| = l by w « u '
and \X(v')\ = \X(v)\ = \ by / < « ' , «<«. If, besides, X(v)^X(u)
then
X(v')^
7±X(u'); hence
v=x%, u'=x^, v'=x%,, d<e, d'<e', m\m', and putting
xe(p=x£lm, xdcp=xd., we have u'=u<p, vq>=(v<p)*<u'. Finally, let
X(v)=X(u).
Then X(v')—X(u'), and there is a <p£Fde F° such that uq>=u', xdq>=xd., xe<p=.
=xe>, vcpdM. Now \w\e=\w'\e- implies iwpHw', and either vsw, vq> = wq>=
= w'=v'cu'
or v = wxdv, vq> = w'xd- -vqxu' because d'<e'. This completes the
proof.
2
R e m a r k . Of course, (M XL,
is wqo if and only if {M*XI,
<K2) is.
2
Hence we could have defined in advance and replaced M by M* in the text of the
theorem. This will be our way in the next section (Lemma 3.7).
311.
3. Standard forms and h.f.b. varieties. In this section we show some (comparatively large) particular subsets of F° to be good standard forms whenever they are
standard and part of the conditions (i)—(iii) (or (iii')) is satisfied.
T h e o r e m 3.1. If
is standard for V in J and (i), (iii') hold (with some
then [V(J), V) is finitely based.
nÔ),
|
P r o o f . By Corollary 2.10, it is sufficient to show that ( F $ X E , <szk) is a wqo
set for every k. The proof will be accomplished through a succession of lemmas.
L e m m a 3.2. F (n) is wqo tinder <sc.
P r o o f . Denote by T„ the set of all semiireducible types of length SM, and define
the quasi-order relation -< on P x T „ by
(k,T)<(k',Tr)
iff
k < k',
T = T'.
As Tn is finite, PXT„ is a wqo set under -<, and, according to [2], so is the set V of
vectors (<*!,..., a(), a(CPxT„, of arbitrary length under the relation
(«,, ..., a,) -< ( f t , . . . , /?m) if there is a sequence
such that
i( 1) «=...< i(i) = m
ocj -< f}iU)
(the additional condition i(l)=m accepted here obviously does not change the situation). Assign to
F (n) the vector a(w)=(a 1? ..., a s ) where a,=(|a,|, T(wt))
(see (1.5)), and put
u<u'
iff
a(u) < a («')
and
T(>v0) = T(w'0).
This relation is a wqo, too, because
FXT„
T(iv0))) is a strict homomorphism of (F£n), -<) onto a wqo set. On the other hand, if «-<«' then we can
construct a <p€Fde F° such that uq> = u': there exist endomorphisms (pjdFde F°,
j=0,...,s,
with (tfj-vvj)cpj = a'i(J)w.(J) (where /(0)=0, ao = a'o = 0). Denote the first
letter of a j by xrU), and put
xtq>j
x,cp
if
xt£X(ajWj),
t ^ r(j),
>U)-i
(
II
a'kw'k)-xrU)(pj
k=iU-1)+1
xt+N
if
x,iX(u),
if
t = r(j),
where N= max i. It is easy to see that (p fits for our aim.
*i€X(u')
L e m m a 3.3. If M is closed for subwords, « is a wqo on M, and the length of
the irreducible elements of M is bounded then <sk is a wqo on MxS for every kÔ.
P r o o f . Let (u,n)£MXZ
and suppose that Hf(1), ..., wi(l) (/(1)<...</(/))
are those irreducible factors of u which contain some xm, s^k. Put.
un=uuiy..
G. Pollik
312
...u, (l) and suppose that xm occurs in ux for the first time at the m 4 -th place (i.e.
u, = u„x„uB,
x„$X(un),
| u j = m s - l ) ; if x„$X(u)
put /ws==°. d e a r l y ,
i
u=! v0 JJ uiU)Vj. Assign to (u, n) the vector
b(u, n) = (mj,..., mk; I; T(u l ( 1 ) ),..., T(ui(l));
v0,...,
vt)
and set
b(K)n)<b(«',70
vj^v'j
for
iff
ms = m's,
lSs^k,
l = V,
T(ui(0) =
1 S r ^ /,
T(u'iin),
0 S j ^ Z.
Then (u,n)-<(u',n')ob(u,n)-<.b(u',7i')
defines a wqo on MXl. Indeed, (u, 7t)i—>-»-b(t/, ;t) is then a strict homomorphism between quasi-ordered sets, and the set
of the vectors b is wqo under -< because if N is an upper bound of |«| for irreducible
u£M then l^k, msS\u„\^kN,
and T(ul(r)) can take also only a finite number o f
different values and the assertion follows from Lemma 3.2. Furthermore, if (w, 7r)-<
«<(«', n') then we can define <p€Fde F° so that vj(p = v'j, ui(r)<p = u'm, and then
\uK\=\i/ms=m's
guarantees also xsx(p = xm. for s = l , ..., k, i.e. (u, k)<szk(u', n').
As an immediate consequence of Lemmas 3.2 and 3.3 we get
C o r o l l a r y 3.4. F(n)XZ is wqo under <zkfor every ¿ > 0 .
Note, however, that Lemma 3.3 is only seemingly more general than Corollary
3.4 because it is not difficult to see that if the conditions of the lemma are fulfilled
then M Q F ( n ) for some«.
Finally we prove
L e m m a 3.5. F$XZ
is a wqo set under <nk for every
0.
P r o o f . For ( u , n ) £ F $ X Z let s ^ . - . ^ s ^ k be those indices for which x SJt €
£X(a0...at) (see (1.6)), and suppose that
occurs in a0...at for the first time at t h e
mrth place. Assign to (M, n) the vector
c{ju,n) = Q(l,f,s1,
ntx,..., m,; T(a0), ...,T(at),
T(a0...at);
...,Gt)
and put
c(u, n) -< c(u',nr)
if the first 21+t+4
vectors coincide and
components of both
(Gj, TC) <sct (fij, n')
for j = 1 , . . . , t.
Define (w, 7I)-<(M', n')oc(u, n)-<c(u', n'). As in the proof of Lemma 3.4, we can see
that F $ X Z is a wqo set under -<. Furthermore, if (u, TZ)<(U', 7t') then we can define <p0£Fde F° such that (a0...at)<p0=a'0...a't which guarantees also
x^tp0=
=xs^, ar<p0=a'r for / = 1 , . . . , / ; r=0,...,t,
and <py6Fde F° such that Gj(pj = Gjy
x^cpj^x^..
Putting together <p0,..., <pj (which is possible in virtue of (1.6)—(1.7)),
313.
we obtain a <p£Fde F° with u=x„.
lemma and also the theorem.
for s= 1, ..., k. This proves the
In some special cases one can omit condition (iii'). We give here one theorem of
this kind.
T h e o r e m 3.6. If F$ is standard for Vin J and (i) holds then [V(J), V) is finitely based.
In virtue of Theorem 2.12, it suffices to prove
L e m m a 3.7. Fffi*XE is a wqo set under «j* for every
k^l.
P r o o f . Let (u, v; n)£F$*XZ,
v=b0 TT vfii be the decomposition of v indii=i
cated in (1.6). By Lemma 3.5 and Proposition 2.3, F$XG 2 p + k t k is w ( l 0 under -<
defined in (2.1). Put
/
\
/ / /\
(m, V; %) < (w , v ; % ) *>
def
•<=>• (u\ Xin)...,
xkn, ao,..., a,, b0,...,
t = f,
r = r\
br)
(u'\ x-w>..., xkn>, a0,...,
|a,-| = \a[\,
,
av, b0,...,
br.),
\bj\ = \b'j\.
Clearly -< is a wqo on F$*XI,
and there is a <p€Fde F° such that u is simple for every xt£X
(for xt£X(u) this holds automatically, as ùcp = û', and |x,ç>| = l for .xl€X(a1...ar)).
Thus, v(p=b'0 TT vt(p-bj£F$,
lemma.
i=l
T h e o r e m 3.8. Let
M={u
=
because Vi<p€.Fa), I/7A'I—/>•
¿=0 1
for i=l,...,l.
u£M„
This proves the
If
|X(M,.)nZ(Mj)| S q for i * j )
is standard for V in J and (i), (iii) hold (with some nÔ), then \V(J), F) is finitely
based.
P r o o f . It is easy to see that if UÛÇ.M then | A ' ( f i ) P i Z ( H ) | p ) / 2 + P q / 4 .
Now the assertion follows from Lemma 3.5 and Theorem 2.9.
We mention two more special cases.
T h e o r e m 3.9. If F^n) is a standardform for Vand (i) holds then Vis h.fb.
P r o o f . The theorem becomes a special case of Theorem 3.1 (with J=F)
show that (iii') holds (with « + 1 instead of n). So let v=vv€F(n),
v=v0
if we
s
]J xcmvt
314
G.Pollák:Hereditarilyfinitelybasedvarietiesofsemigroups
its semiirreducible factorization, ¡«¡I S/z,
1-1
a
V =! v0 ¡1 xcWvt • xcWV,, v = \v IJ Xc(i)Vi,
/=1
V,V, = V,
i=l+l
and <p£FdeF°,
r
v(p=\v'0
IIxi0)v'j,
.7=1
\v'j\^n (i.e. v(p£Fw), |» { B + 1 Vl=» + L As
we have |xc(i)<p| = l. Furthermore, by Lemma 1.3, v(p—\w-xcm(p-w.
Hence w • xc(l)cp •
since it can be
obviously achieved that X(w• xcm<p)f)X(iv*)=0.
This proves the assertion, as
\v<p\ = \v\^n.
P r o p o s i t i o n 3.10. Let J—
. The variety SG(J) is h.f.b.
P r o o f . Follows from Theorem 2.11 and Lemma 3.5.
Part II. Application: a class of h.f.b. identities
The aim of this part is to prove the following
T h e o r e m A. A non-balanced identity of the form
(*)
u = x1...xixi+1xixi+i...xn
=
= v (n€Z, e(j) s 2)
is h.f.b. if and only if v is not of the form
(* *)
v = x1...xi^1x2nx2i+1)nxi+2---X„
(7t=(i i +1) or identical, n > 2).
The assertion will be broken up into several propositions. From know on V
denotes SG(*).
4. Two special cases. To start with, we settle the negative part of the assertion.
Of course, here one cannot utilize the results of Part I.
P r o p o s i t i o n 4.1. The identity
(T)
XiXi+1XiXi+2...xn
=
x1...xi-ixfx2+1xi+2...x„
is h.f.b. i f f n=2.
P r o o f . The fact that xyx=x2y2 is h.f.b. has been proved in [5]. So let n > 2
and, say, i > 1 . Consider the identity
(O
tyztx\...x\m
=
tyztx\...x\m^x\mx\m_x
315.
and the infinite systems (a) = {am: m = 1, ...}, (<r*)= {<rm: m ^ k ) . We claim that the
system <TU {T} is independent. Clearly T does not follow from a, so we have to prove
that A * U { T } ( / «"IT• To see this, we show that if <T*U (T}I-tyztx\...x% k =w then
W =! (tyzt) • ( t Pj),
j=i
(4-1)
Pi = X2k
OI
" Pi — X2k-lX2kX2k-ll
Pfi T(xyx) U T(x*),
i
U Pj) = {•*!, • •
J=i
whence w^tyztx%...x%k_sx%kz%k_1. Indeed, let w9(=tyzt x\...x\^,
..., wr( = w)
be a sequence of terms such that, for every s=\,...,r,
there exist M/, w'^F0,
cps£End F, and ( M S = V S ) € C T * U {T} which satisfy ws_1 = w's-us(ps-w",
VVS=H^ •
•vscps- w", Suppose, furthermore, that w s _ 1 is of the form (4.1) for some i S r (this
certainly is the case for s= 1). First let us=vs£ok; by symmetry, we can assume that
us = tyztxl...xlm.
Sure enough, m<k because tyztx\...x^,-={a ws_ t for m>k. Moreover, the only subword of w s _ 1 which is an endomorphic image of tyzt is tyzt
itself, and the only subwords which are squares are those of the form x j . Hence
us<p£T(us) and
i
i
i
yfs-!=\us(ps
¡1 Pj = tyztx}n...x}2m)K
IJ Pj, ws = ! vs(ps- J[
Pj,
j=2m+l
y=2m+l
/'=2m+l
n£X 2m , / s 2 w + 2 , whence also ws is of the form (4.1). If
u
s
— X1...Xi-1XiJ$+iXi+2---X„,
Vs = Xi...XiXi+1XiXi
+
2...Xn
then (x?xf+1)<ps£T(x2y2) by the same reason as above, i.e. (x?x*+1)<Ps=Pj-iPj for
some y's/, and ws differs from
only in these factors which are replaced by
some p'j£T(xyx); thus, ws again is of the form (4.1) because if j=l (which, by the
way, can occur only if / + l=w) then x^-i-xL is replaced by x2k-1x2kx2k-1.
Finally, if us = x1...xixi+1xixi+2...xn
then (XiXi+1Xi)<ps€T(xyx) because the only
subwords of w s _ 1 which are endomorphic images of xyx are those of the form xqxrxq
( = p j for some j) and tyzt, but this latter one is out of consideration because if
(XiXi+jXj)cps£tyzt then us 1 , which is impossible.
Therefore this case is similar to the previous one.
If / = 1 then n >/'+1 and we can consider the identities
(ff'J
x\.. .x\m tyzt = xlx\x%.. ,x\m tyzt
instead of <rm, and dualize the above reasoning (with the only — unessential — difference that here Pi=x\ or x1x2x1 which is not dual to pt=x2k_ix^x^-i).
This
completes the proof.
Next we deal with a special case.
316
L e m m a 4.2. The identity
<4.2)
Js h.f.b.
(u =) xl...xixi+lxi...xB
=
v) (TI£Z)
P r o o f . First suppose that the symmetrical difference X(u)X(v)=
{ x j . By
repeated applications of (4.2), any word w of Fn+1 can be brought to the form w* =
= w a-i) x $iy"*e№) w *° , ~' _1) » d(J)—2; besides, the number of variables either does
not change or decreases by 1 at every step. Furthermore, applying (4.2) twice, using
the endomorphisms
xi+J
if
if j > i+2,
f*2i
if j = h
i _ x2i+J
Xa,-...^;-!
if ./ = ¿ + 1,
+
+ 2
if J =
'+2,
respectively, we obtain
2
_—
* l " - * 2 i - l * 2 i * 2 i + lit " • * 2 i + (ll-l)!l*2I + ll + l •••*2n + f - l
2
2
x
X
= XJ ...Xzi-lXzi • V - l * 2 i
— Xx ... X2i-lX2iX2i + l • • • X3iîX2iX3i
+
•
•*2i+n + l " - * 2 n + i - l
~
lX2iX3i + 2 . . . X2„ + i-l =
= XI ...Xi • Ulj/ •X2i+„+i---X2n + i-i — Xx-.-Xi • V\jl • X2;+N+L • • • X^FI+J—I = x1...xiwx2iw'x2iw"x2i+n+1...x2n+i-1,
w' £ 0,
and a third application of (4.2) relieves us from x 2 i . Hence F|=w*=w** =
= ^ i - i ) * / ( i ) " x f(r)w** (2n-,-2) €i^y" 1 " i-31
and
Z(w^)CZ(w + )gA-(vv).
Thus,
2n+i
by Lemma 1.1 and Theorem 3.6 [V(F ~% V) is finitely based, whence the assertion
follows obviously.
If (4.2) is heterotypical, and X(u) + X(v)?i {x^, we can assume that h > n for
some k ^ m (i.e. x ^ X(u)). Indeed, if this is not the case, then there is a y'Sn, j ^ i ,
such that
(i.e. x y does not occur on the right side). First let
1, say
7 > / ' + l , and let j be maximal. Executing in (4.2) the substitution Xji-—XjXn+1 on
the one hand, and xJ+1y-*xB+1, xJ+2>—Xj+1, ..., x„>-»x„_1 on the other, we obtain
U = XI-..XJX I + 1 XI...X N = X 1 ...X J X; + IX I ...(X;X I I + I)XJ + 1 ...X I I = *I E ...X(,, + X) C
where g=n-(n+l
n...j+l),
•/+1${1, ..., m) then
and x n + 1 occurs on the right side. Furthermore, if
and by means of the substitution x ^ i - o r ^ x ^ xt>-*xi+3, xi+1»->-x,xj+2, xt+t>—
*-*xi+t+2 (2^t^n)
we can bring about that x i + 2 did not figure on the right side
-which is the previous case.
Now if xk„$X(ii),
'
317
kSLm then (4.2) implies
Xx • • • Xm = Xi... Xk-i(XfcJCjtO • • - xm
and every term is equal to one of length S m in the variety
V=SG(u=xlK...xmn)
whence V is h.f.b.
Now let (4.2) be homotypical. Next we show that (4.2) implies either
(4.3)
x1...xkxk+1xk...
Xi = Xi-.-Xi
for some kî, l^n or the dual of (4.3) — the latter if 7t=(j"/+l). In this case, as
well as if n = i (identical), the assertion is obvious. In the opposite case there is an
rSn,
{/', / + 1}, m^r. Let e.g. r < i and minimal. Then
(4.4)
x0...x„ = Xq ... xr-1xrK-i...
=
Xia-iX(i+1)„-iXi„-i... xm-i =
X0...Xr-2Xr...(Xr-1Xnt-i)...Xn.
Thus, (4.3) implies a permutative identity
Xi---X„+2 = x1...xr-1xre...xsexs+1...x„+i,
rg
r,
SQ ^ s,
and, according to [7], for sufficiently large / we have
XI-.-X/
=
X1 ...XR -1 XR<T ...X(S+ I-„-1 )<T XS+L -N ...XI
for every permutation a of the symbols r,..., s+l—n — l whence, in particular,
(4.3) follows.
Using (4.3), an arbitrary word w can be easily transformed to the form w =
= w (k - 1) w'w (f ~ k ~ 1> , x y x ^ i w ' , i.e. the irreducible factors of w' are contained in
XU T(x2). However, (4.3) implies also
X1 ...XK XK+1 XK+2 ...XI
=
= X1 ...XK XK+1 XK XK+1 ...XT
X1 . . . XK XK
+ 1
X K X K + 2 ...XT =
=
XX.-.XI
whence w=w ( t ) >v'V- f c - 1 ) , w " £ F m . Thus, w* = w ( k ) w"w ( l - k - 1 >eI$ ) -VgF$- 2 \
and (i) holds as (4.4) is homotypical. Hence the assertion of the lemma follows by
Theorem 3.6 (here J=F).
5. Some auxiliary identities. From now on we can suppose, in virtue of
Lemma 4.2 and the results of [6], that
(5.1)
e(k) = 2
for some
k^m,
kn~x ^ i.
We proceed by some identities which follow from (*) and (5.1). Note that
(5.2)
5
( * ) I- x"+ a =
318
as (*) is supposed to be non-balanced. Hence (x®")v is an idempotent in
Moreover, if (*) is homotypical then
F(V).
( * ) 1- x" + 2 = x"+ 3
(5.3)
and already (x" +2 )v is idempotent.
L e m m a 5.1. Suppose that (5.1) holds. If (5.3) holds, too, then
(5.4)
V z " + 2 = JC"+2J2"+2 for
( * ) b-
q fe 1.
If(*) is heterotypical (in particular, if (5.3) does not hold) then
(5.5) ( * ) 1- jC" + 2_jj24+1z" + 2 = xn+2yzn+z,
x"+:2/«zn+2
= JC" + V
+2
for
0.
P r o o f . If (5.3) holds and (*) is heterotypical then [5], Lemma 2 yields even
x"+2yz"+2=x"+2z"+2.
So let (*) be homotypical, and first suppose nî,
n^
(ii+1). There is a k${i, / + 1} such that kn^k;
let e.g. kfc;
e(s)
.
lye{s>
Xt* = Y
2
if
t = ln,
else>
2
if i = fc,
= \y2
,
l x,q> else.
We have in virtue of (*)
x?+2yzT+2 = X"+2 • uq> • z"+2 = JC"+2 • vip • z"+2 =
_
_ j^+2^+2 . ^
. ¿1 + 2 _
_ ^+2^+2 . ^ . ^ +2 _ ^ + 2 ^ +2^(5)^ +2 _
= JC"+2 • vcp' • f+2
which implies (5.4). If, on the other hand,
x,X = \y
z"+2
=
• U(p' • z"+2 =
xr+2y2z"+2,
then the substitution of
if
if
if
/</,
' =
/>Z,
in (*) yields (5.4) immediately.
If (*) is heterotypical, we can confine ourselves, by the remark made above, to the
case where (5.3) does not hold. According to [5], Lemma 2, it suffices to prove
However, in our case there is a k<m with e(k)=2, kn>n. The
x2ny2nx2n_x2n
319
substitution
- . J / '
l* 2 "
^ ' =
else
in (*) yields the required identity.
Define
mi = min {j: jn ¿¿j},
m2 = min { j : e(j) = 2};
if Wj^min (m, n ^ m a x (m, n), we put n=(m+l
>max(m,ri), let n = i , m ^ «>.
m+2), » i r = m + l , and if mx>
L e m m a 5.2. Jf ( * ) | - x " + 2 = x " + 3 and either m1on2
= w 2 = / + l then
( * ) ^ xx...xn-xyn+*
(5-6)
=
xx...xn-2xl-xy»+\
P r o o f . First consider the case w 1 < w 2 . Choose
x,9 =
if
x,'
" t' <
^ "ml 1l
:
if t = mxn,
xu.
™U p k p .
2
y+
else;
or w2(J {/,/+1} or mx=
i^£End F such that
f x2mi
xtil/ = {
W
if t = mxn,
,
else.
In virtue of (*) and Lemma 5.1 we have
8
=
2 if m^ = i, \
0 if m x i i > n , I.
1 else
/
Hence (5.6) follows.
Next let m 2 Smin (i — 1, mx), and put
y+
2
if
if
Then
XX"-Xmsy ,n+ 2 _=
)//n
+2
t ^ m2,
i > m2.
_ ij" + 2 _
— nra . v " + 2 — I X 1--- X ">2-1 ^maJ'"
*i-*n,2-iJ'n+1*^)}''i+2
if m 2 < m 1 ;
if
In the second case we define i^gEnd F by
, _J*i,
lx,
5»
if t = m2
else
MI = M
L
= STT.
G.PoMk
320
and obtain
= x1...xmil-1yn+2x%(s)yn+i
*!••
2
= ^ . y»+ = uty.y«** =
=
B+t
X1...Xm^1^miy
.
In both cases (5.6) follows.
If i+
we get, by obvious substitutions in (*),
xi...xi+1yn+2
= x1.'..x,xl+1x,y"+2
Finally, if m1=m2=i+l,
= X1...XiXn.1XiXi+1y"+2
= ...
put
if
if i > i + l ;
if
Applying (*) and Lemma 5.1, we obtain
x1...xiyn+2
M+
+
= w<p.;y"+2 = x1...xiyn+2xiyn+2
=
+a
+2 n+2
= x1...x,y '*l *y"
= t4-x!l y
n+%
= JCi... *|_1 • tlx • y
= Xi...Xi-iOe?+V+2)2
=
+i
= X!...
=
• VX • y"
+!
2
= X!... x i _ 1 x ? V " = xt... xi^xf
=
yn+*.
This completes the proof.
Lemma 5.3. If (*) is heterotypical and either m^m^
( * ) 1- xl...xa-lJr+i
(5.7)
=
or m2î,
then
x1...xn.1x2nyn+i.
Proof. If m 1 S/n 2 then set
v t ( p=
X
if
-\ f
y ,/,
X,w
if
= /**! * t =
- \xtq> else.
m n
i >-
By (*) and Lemma 5.1, we have (taking in account that «(wJÔ as
xi...xm.1yn+2
= vcp-yn+2 = U<p-yn+2 = i4 • y^x^l^
= vxlf.y"+!t =
>
yn+i
m^m^Sm)
=
x1...xmi-1x2miyn+\
which implies (5.7).
If iVmgC/Mi then (*)h-x n + 1 =x" + 2 and in consequence of Lemma 5.2 and
Lemma 5.1 we have even
x1...x„-1yn+1
= x1...xn.2x2-1yn+1
= *1...xn_2*S±i*S+V+1
= x1...x„-2X£\y"+1
= x1...xn.1^n+1y+1
=
Clearly, both (5.6) and (5.7) imply
(5.8)
n
xi...xn-iy
+2
= x 1 ...x n _ 2 x®_ 1 /'+ 2 .
=
+1
xi...xaf
.
321.
Furthermore, it is easy to see that
(5.9)
= x1...xa.ix2n.1xny''+2
(5.6) H ^...x.y**
n+2
(5.10) (5.7) 1- x1...x„y
=
+t
xl...xn-ix%+1xny
3
(5.11) (5.8) y- xl-..:xmjT
=
xl...x„-t3Ztlxmy'+\
n+z
= x 1 ...x n _ 2 x n _ 1 x n j
+
n+2
(5.12) (5.6)A(5.7) hXl...xHy» *
2
= x^.x^y"*
,
n+2
=
xi...xn_2x^xny
=
,
x1...xn^x„y^.
R e m a r k . The only cases when neither the conditions of the Lemmas 5.2, 5.3,
nor those of their duals are fulfilled, are given by (**) and
(5.13)
L e m m a 5.4.
v=
x1...x^1xUi+i-X„.
2a 2
2n 2n
(*)y-(x "y ) =x y .
2
P r o o f . If (*) is heterotypical the assertion follows from Lemma 5.1. So let (*)
be homotypical. We indicate the substitutions in (*) which yield the required identity.
If n = i , substitute
fx 2 " if i s i,
X ,
~ \ y2» if i > i .
If kn^k
and set
for some
{/,/+1} (suppose e.g. k<i), choose A: to be minimal
fx 2 " if
W - X y 2 » if
t ^ k,
i>fc.
The remaining case n = ( i / + 1 ) is dual to n = i .
L e m m a 5.5. If nî,
i + 1 ) then
. ( * ) h- Xi+1yzj£+* =
P r o o f . If (*) is heterotypical and ( * ) | - x n + 2 = x n + s then xn1+2yzxn2+2=x^+2x^+2=
n +2
=x 1 zyxl+2. If ((*) is heterotypical and) (*)\-xn+1=xn+s
then, by (5.10),
xi+V*3+2
= xn1+2(yz)2n+1xn2+2 = xr1+z^y2"(yz)2n+1^+2
+2 2n 2n+1
= jtr1 z y
Put
+2
z^
z
x,(p = \y2n+1
z2n
Then
=
+2
n 2n+1 2n+1
Xx z* y
z
+2
xZ .
if t = i,
if t = i +1,
else.
xj+ 2 z 2n j 2 " +1 z 2n+1 JcS +2 = xZ+zz2n+1 • uq> • z^x3+2 =
= ¿I+228«+1. ^ . 23»^+2 = xn+2z2n+ly2n+lz2nxn+2
and, by symmetry,
x?+2z1*+1y*n+1z2"x"2+!1=xl+*zyx"2+!!.
=
322
If (*) is homotypical then k n ^ k for some
xj+2
y
x, m x ;
{/, /+1}; say,=
JCJ+2
if
Set
t <mi,
yxS + s if t = mu
x,<l> =
z*5 +a if t = piiit,
x%+2 else.
We obtain (using also (5.4) and Lemma 5.4 if m1n=i)
xZ+iV*S+2
=
• uq> - x£+a = x!+a • v'*S+V(s)*a+a =
=
;
= .
= x? + 2 -¡4 • *s+2 = x? +2 • # • xs + 2 = * i + 2 ^ + 2 ) e ( m . ) 0 ' j c S + 2 ) e ( s ) = Jtf +a zx2 +8 j>*s +8 ,
where sn=m1
By symmetry,
and
{21
if w^rt = i,
else.
xni+2zxZ+2yxZ+2=x"i+2zyxn2+*.
C o r o l l a r y 5.6. Either (*)<r-xn1+2yzyxn2+2=xn1+!1y2zxn2+2
=xZ zy2x"2+2.
or:(*)\-^l+2yzyx"2+2^
+2
P r o o f . If JIÎ, n?£(ii+l) then both identities hold by Lemma 4.8. If
then xn1+2yzyxn2+2=xn1+2y^zx"2+2=xn1+2y2zxn2+2
by Lemma 5.1. The case
= ( / / + 1 ) is dual.
n=i
n=
(5.9), (5.10) and Corollary 5.6 imply
C o r o l l a r y 5.7. If either the assumptions of Lemma 5.2 or those of Lemma 5.3 are
fulfilled then either
(5.14)
or
( * ) (— x1...xn-1ztzx„ytt+2
(5.140
( * ) H xx:..xn-iztzxny^2
=
=
xi...x„-iz2txny"+t
xx...xn_itz2xnyn±2.
Furthermore, applying (5.9), (5.10), (5.14), (5.14'), we obtain
C o r o l l a r y 5.8. If either the assumptions of Lemma 5.2 or those ofLemma 5.3 are
fulfilled then
( * ) h w ^ + a = w (n) x c(1) ...x c(0 y" +a ,
(5-15)
c(j\ 7* c(k), if j * k,
for every
w£F".
{x c ( 1 ) ,..., x c ( 0 } g X(w)
323.
6. Standard forms. Now we are able to construct standard forms for V in the
F-ideal J defined by
J= {w: there are arbitrarily long terms which equal to w in V},
and for V(J). Although the considerations below could be performed in the same
generality as till now, some special cases, in particular .the one where (*) is homotypical and e(k)=2 only if k=(i+l)n~1,
would demand a separate consideration.
Therefore we continue the investigation of (*) accepting in what follows the further
restriction
(T) e(k)=2
for at least two different values of k.
This suffices for the proof of Theorem A, since the case when e(k)=2
k has been settled in [6].
L e m m a 6.1. If (*) is homotypical, (T) holds, and 1|wj|
then
w=hû^h^/.
—n for j—1,2, 3
P r o o f . There is a k such that e(k)=2, knî. let e.g. kn>i.
that if \uj\ S n for j— 1,2,3 then there exist vx, v2, vaZF such that
u0 = u1ulus = v14vs~v0,
Kls>l"ols
for x£X(vJ,
u2 = u'2v2,
\vj\^\uj\-n+2
for exactly one
First we show
• '
'
for j = 1,2,3.
Indeed, put ux="i.u2=xct,)xc(i+1y..xc(kîi,
w 8 =x c(tat+1 )...x c(ll) w3.
Then
"o = uixc(l)---xc(i)(xc(i + l)--xc(kii)fi)xc(i)xc(i+l)--- •
(*)
v • Xc(kn-1) i.xc(kn)ft)Xc(kx+1) • • • Xc(n) u3 = U1U1 (Xc(kx) US u3
with some ux,u^£F°,
and
v2=xc(k„p
v%=un%u's meet all requirements of
(6.1).
If-w is as stated then (6.1) can be applied n times. The term w=iv1wîv3 thus obtained contains every variable of w2 at least n + 1 times. Now suppose some w£F
B+l
contains a letter xs at least « + 1 times. Then w is of the form w=u0 TT xsu,
j=I
where U £ F ° for j=0, ...,n + l. Put
xt(p =
Ut—lxs
if
xs
if
UiXaUi
xsut
+ 1
if
if
/ <
i,
t=i,
t =
i+1,
*>i+l.
Then w>=w<p, and
|f^| s >|vf| s , i.e. we can obtain from u> a longer word
of the same form, and therefore w£J. The same holds, then, for H> and hence also
for w. .
324
G.Poltek
L e m m a 6.2. Suppose (T) holds. If (*) is heterotypical then
(6.2)
J = {w: u < w or p < w } ,
If (*) is homotypicaland e(k)=2 for some k^{iK~\
subset
L= {w: w = w(ai..n)vw(to'-',),
If (*) is homotypical, e(k)=2
contains the subset
(/ + l)n-1}
then J contains the
u -ci w}.
i f f kn£ {/, / + 1 } , 6M/ V is not of the form (4.1), then J
L' = {w: w = wln*+n)ww<n!+n\
wt$T(xyx)UT(x*)\JX
for some irreducible factor w, of w}U
U{w: w = w'xexdxew"xrxsxrwm;
|w'|,
s n 2 +2«, |w"| ^
n-2}.
P r o o f . The first assertion is trivial. If (*) is homotypical, e(k)=2, kn${i, / + 1 } ,
say, kn>i+l,
and w£L then w=w (2/l! _ n) w'• ucp• w"w(2n'~"'> with some q>£End F,
and we can modify the mapping (p in such a way that xt(p'=x,'| = 1 if t^kn, and w=w(2n,_n)w' • ucp' • wmw^nl~n). However, then
w = W(2n»-n)H'' • vcp' • wmwini~"y
= ^ - „ y i X l ^ - ^ ) i)Cp' • (Xkn9y
=
• ( ^ ' . . . ^ V ' ^ lV(»,-»>,
and the assumptions of Lemma 6.1 are fulfilled.
Now let ¿7t=/, k'n=i+l,
e(k)=e(k')=2,
e(J)^ 1 if j^k,
k'. Suppose first
that w is contained in
the first component of L'.
a) If » ^ ( „ . . „ J W " ' " ' 1 , |H'| c >2 for some c £ P (in particular, if | # | e > 2 ) ,
then n consecutive applications of (*) with substituting each time <pj:x,t—xc,
(pj\ xl+1>—bj, |6,| e >0 gives us a word w* with |w*| c >n + l, and the second part of
the proof of Lemma 6.1 verifies the assertion.
b) If w=w(nt)K>H'(nt), xyxy<iw
(in particular, if xyxyow),
then w =
~vj'xcax,aw",
and choosing <p8€End F such that x,<p=a, xi+1<p=xc,
wcp=
=w (n ,_ n) iv• ucp • ww'" 1- ' 0 , we have \w• vq> • w\c>2, and this case can be reduced
to a).
c) If xyzx«aw, then w=w'xrbxrw",
| 6 | s 2 . Putting x,cp=xr,
xi+1<p=b,
2
w=w-ucp • iv with |w|, |w|&« , we have (*)(-w-ixp-w, which gives us case b )
as bz is a subword of v<p.
It is easy to see that all possibilities are exhausted by a)—c). Finally, let w£L'v By
assumption, J t ^ i , iir*(ii+1),
whence there is an lî, / + 1 such that In ¿¿I;
let e.g. / > / + 1 and maximal (then, clearly, ln<l). Set w=w x -wj/ -w2- ux < wa>
325.
K l , \щ\^пг+п,
х,ф=хе,
x 9 =
'
х1+1ф=хл,
x,x=xr,
xl+1x=xs,
and
х,ф
if
• w2 • (*i-*i+i)x
(*i*i+2-*i+ik
x,x
if t = '»
if t = l+l,
if i = - / + l .
t < /,
We have
w=
- uB • ws =
because М ^ и ^ ^ + л ,
tes the proof.
-
w3 =
axrbxra'€L{,
\a'\^\ws\^n2+n,
This comple-
Lemma 6.2 and Proposition 3.10 immediately yield.
P r o p o s i t i o n 6.3. Tf(T) holds and v is not of the form (**) then SG(J) is h.fb
P r o o f . Set T="|J 1 T(x\...x?),
1=0
and let J(T) be the set of all w£F°
every
semiirreducible factor of which is contained in T. Put
= {w: w w ww< n \ vt>€/(T)}U{w: |w| < 2л},
(6.30
(6.3a)
(6.38)
Мг = {w: w ^ j i i w « ,
=
?sn,w
/(!")} U (w: |w| < 4л2},
w (n i + „ ) wwww ( " ,+n) , |w>| =L
= wyw£ I(J) (y i A'(wîv))} U {w : |w| < 2л 2 +2л}.
Lemma 6.2 implies that F\JQMj
( j = 1, 2, 3), if either (*) is heterotypical or (*>
is homotypical and e(k)=2 for some
{in-1, ( / + 1 ) я - 1 } or (*) is homotypical,.
e(k)=2 iff kn£{i, /+1}, and nî, 7гИ(//+1), respectively. Indeed, if w $ M j then,
w is long enough to be written in the form given in the first bracket of (6.3y) but with
w having a semiirreducible factor н>Д/(T). Then either
or х 2 ...х 2 <]иг
and, consequently, if w' is a suffix of length n of w (n) , w(2n!), or w (n , +n) , resp., and,,
similarly, w" a prefix of length n of w(B), w12"^, or w{n'+"\ we have u<\w'ww" or
î w w ' m " (even v<iw). As the latter case can be reduced to the first one, both imply
V/Ç.J by Lemma 6.2. Hence the assertion of the proposition follows by Proposition!
3.10, since M ^ F ™ ,
In order to obtain a standard form in J, we prove first
L e m m a 6.4. If (Г) holds and w£J then there is a w£J such that
(*) 1- w = w = Wj*;+2W2,
Xc£X.
P r o o f . If (*) is heterotypical the assertion is obvious; so let (*) be homotypical.
There is а кшп which satisfies e(k)=2, knî+l.
We are going to show that f o r
326
•every triple r, s, I of natural numbers there is an xc£X and a term w such that w=
s
= w=w 0 JJx? c Wj,
f o r / = 0 , . . . , J; this is somewhat more than stated. First
•consider the case r = 1. Let \X(w)\=L As w=w' implies now X(w)=X(w'), we
only have to take a sufficiently long word w, say, |w| >2l+l(s—1)(/+1),
in order to
obtain a factorization of the required form. Now suppose the assertion holds for
some r > 0 and arbitrary s, I. In the same way as above, one can find a w'=w such
that the same subword x ^ p c j occurs in w' a sufficiently large number of times and at
a sufficiently large distance from each other (this is necessary in order to cover the
•cases where kn=i—\ or kn=i+2;
c,d,f need not be different). Say, w' =
as
=w0 flxjx'cxjwj, \wj\^n+l. One can define endomorphisms <plt..., <ps such that
s
xkn(Pj=xrc for 7 = 1 , . . . , s and w'=w'0 [[U(Pj-wp
sV
w'=w'0 IJ j' wj=wo
II
»
K I s / . Applying (*), we have
with so" 16 wj> \w"j\ — \w'jV This completes the proof.
P r o p o s i t i o n 6.5. If (T) holds and v is not of the form (4.1) then [V(J), V\ is
finitely based.
P r o o f . Let w£j; we can suppose that w=w'^ , c +2 w" by Lemma 6.4. First let the
assumptions of both Lemma 5.2 and its dual (or those of Lemma 5.3 and its dual) be
fulfilled. In virtue of (5.9) (or (5.11)) and its dual, and by Corollary 5.8, we have
(6.4)
w = w (n _ 2) x3 +2 x c(1) ...x c(i) ^ +2 w("- 2 > = w*
(c(j) * C(k) if j * k).
Thus, viêig 4 , and the assertion follows by Theorem 3.6, because (i) automatically
holds if (*) is homotypical, and if it is not, then, starting from an arbitrary w ' = w
with a minimal number of variables in X(w'), we can transform it to the form (6.4) by
steps of the following two types: 1) first, the insertion of the (n +2) nd power of some
variable — however, doing so it is not necessary to introduce new variables, and 2)
a sequence of applications of (5.10), (5.11) and (5.14) or (5.14'); besides, (5.10) is
always used for reducing the power of the elements, and the other three transformations do not change the set of the variables involved.
For the rest we can suppose, according to the Remark made on p. 321, that the
assumptions dual to those of Lemma 5.2 or 5.3 hold, but the assumptions of these
lemmata do not, i.e., a) v=x1...x,-.1xfyf
if (*) is heterotypical, and ft) v=xt...
...xîXf^v' or y) v^xx-.x^^v'
if (*) is homotypical. Let w=M'(i_1)wjc^)2jcc(X)...
...X C(I) M» ( " -2) , and write w in the form
327.
Suppose that w is chosen in such a way (from the class of all w'=w of the same
form), that the vector v=v(w)=(|X(ïv)|, p, I) is minimal in the lexicographical
ordering. We claim that
(6.5)
ВДП{хс(0),
Indeed, let d(q)=c(r),
...,*,(,_!)} = 0.
and set
If m2=i, put
'*J|8
if t = u
if t = i + 1 ,
if 7 > i + 1 ,
X, • Xd(q)w4 = щ(w2^0 )w3) v2
if in = i +1,
WlXd(q)V3
if in > i + 1
1
with some w, vr, v2, va£ F", and in the first case by iteration also
w = ЩхХ) v4,
We have a contradiction with the choice of w in all cases.
If m2=i+l, set
/
\xt(p
if
+1,
X,i+1.
As in this case (*) l- x" + 2 =x" + 3 , we can make use of the dual of (5.9). Hence
W = w^SfêJ w2x"c(+0? w3x$> *S(+r2+1) x c ( r + 2 y. .x c(l) w<"-2> =
= U(p-^r2+1)xcir+2)...xcil)w^-z)
=
in
= V(p-j$lrl1)xcir+2y..xmw -v
(6.7)
щУ^сЫг) ХФ+2У--ХС(1) w<"-2> =
= Wi *2fêj
=
=
•
•
W3
•
•^ X c i r + D - X c O ^ - V
=
328
if /->0 and
(6.8)
w = w^îjgw.affitfw« = w ^ l f â w l x » ^ = . . . = w1xl{$w»a+*xZ$wi
if r = 0 . In the first case we have reduced / by 1; in the second,
and
we can create the (n+2) n d power of a variable in this part of the word by Lemma 5.10.
This proves (6.5).
Furthermore, w ^ ^ w ^ J by the same lemma and (6.5). Let vf(i_i)W' be the longest one of all words it equals to. Then ««âH^-îv (else we could replace its subword
of thé form wj/ by the longer word vty), and
x(n+I...xm+II_,_1*JaM'(i_1)M'. Indeed,
if e(k)=2, kn>i+1 for some k then let
Wj
for j < kn,
Xj(p = Wj = yjWjZj, Xjlp = Wj = wkxykx+1
for j = kn,
9jzjyj+!
for j =- kn
( y j = z j if |wy| = l). We have even
v<P-ym+1 = (ijWjin)ym+i
y=i
= u<P-ym+1 = w1...wiwi+1wl...w„ym+1
=
= wi...w'iwU1w'l...w'n = uxli = vxl> = wie„V...w'melm>,
and 101^1=lwp|+2>|i><p-j'ni.(.1|. If, on the other hand, e(fc)=2 iff kn£{i, / + 1 }
(this occurs only in case /?) with in£_{i, /+1}), then nî, n?±(i i+1), whence
ln+l?*(l+l)n
for some / t t > / + 1 . Put
wt
if t£{i,
i+l,In},
W/, w?, +1 if t = ln + l,
w?
else.
We have
wi«...w®,*1...
s wl/'xl...xn-t-i
= wf ...w?_1wiwi+1wiw?+2...wlx1...xn-i-1
= w}'-x1...xa-,-1
=
=
wi„:.wuww,„...wl„x1...x„-i-1.
Applying (*) to this latter word, we obtain some word w* which contains w2 and all
factors w* which do not enter iv, as well as x x , . . . , x , , - , ^ . Hence |W*|>|H'21I...
•••^«^l-^n-i-llAs in the proof of Lemma 4.14, we can conclude now that
w = wwl"-'-»,
w=\w0
(6.9)
W* s w l i - 1 ) ww i ''- 1 \
w ='.
]]xfU)Wj,
j=i
w / u
T(xi...x?),
t=o
(5 = 1 or 2).
329.
Let M be the set of all words of the same form as w*:
M l
M2 = {w: w=lw0
= {щ:
ПX/u)wJ>
j=1
M 3 = {w3: |w,| = n - i - l } ,
M5 = Fm,
M = { w i . . . w , : vt£Mu
= i-1},
*A>
f=o
T{x\...xf)},
М 4 = Г(х п + 2 )иГ(У+ 1 ),
M6 = (we: |we| = n - l } ,
*(и>()ГЩи>у) = 0 if i,j£{2,4,
5}, i
We have seen that M i s a standard form for Vin J. Thus, by Theorem 2.9 and Lemma
3.5, the proof will be accomplished if we show that conditions (i) of Lemma 2.8 and
(iii) of Theorem 2.9 are fulfilled. Now (i) holds — if (*) is homotypical, by Remark 1
made after Corollary 2.10, and if (*) is heterotypical, by the fact that in assuming
that x"c+2 is a subword of w, we could choose xc as well from the variables of the original word which was to be transformed, and while transforming w to w* we never
needed to introduce a new variable. Furthermore, let w=ww£M, <p£ End F°, wq>
a prefix of some w'£M, and |й ( 2 ш - 1 ) ф|=2/и —1. By Remark 2 (after Corollary
2.10), we can confine ourselves for such q> that
\x,(p\ = 1 for
x ( €Z(w 1 w 3 w 4 w e ).
If
1, (iii) holds obviously. If w ^ w ^ v , then к с р — и ^ щ щ • w', u ( £M t ,
and we obtain (wcp)* by applying (several times) the dual of either 5.9 or 5.10, and that
of 5.14 or of 5.14', not changing thereby w<p. Finally, if vt>sw((_1)w2t;, w2v a prefix of
ív, and either v = x f a ) x 2 i m . . . _ уХод,
|wy|s2i; then, by assumption,
\xfU)(p\ = \xda)(p\ = ... = \xd(t)(p\ = l, and X(w2<p)ПX(w<p)QX(w&cp). Thus, we can
transform (w (i _ 1) w a x /( y ) )?» ( ' -:l) (xj (1) ...x d(() )v) to a standard form iv without changing
either w0 or w e ; thereby we "standardize" the whole word w<p, without changing
(wit-^WzXfy^tp. Or #2=0, \v\=2m—\, and the assertion is obvious once more.
This completes the proof.
Theorem A follows now from [6], Proposition 4.1, Lemma 4.2, Propositions 6.3,
6.5 and 2.5.
References
[1] А. Я. Айзенштат, О перестановочных тождествах, Современная Алгебра, 3 (1975),
3—12.
[2] G. HIGMAN, Ordering by divisibility in abstract algebras, Proc. London Math. Soc. (3), 2 (1952),
326—336.
[3] G . POLLAK, On identities which define hereditarily finitely based varieties of semigroups, in:
Algebraic Theory of Semigroups (Proc. Conf. Szeged, 1976), Coll. Math. Soc. János Bolyai
20, North-Holland (Amsterdam, 1979); pp. 447—452.
330
G. Pollák: Hereditarily finitely based varieties of semigroups
[4]
, A class of hereditarily finitely based varieties of semigroups, in: Algebraic Theory
of Semigroups (Proc. Conf. Szeged, 1976), Coll. Math. Soc. János Bolyai, 20, North-Holland
(Amsterdam, 1979); pp. 433—445.
[5]
, On two classes of hereditarily finitely based semigroup identities, Semigroup Forum, 25
(1982), 9—33.
[6] G . POLLÁK—M. VOLKOV, On almost simple semigroup identities, in: Semigroups (Proc. Conf.
Szeged, 1981), Coll. Math. Soc. János Bolyai 39, North-Holland (Amsterdam—OxfordNew York, 1985); pp. 287—324.
[7] M . S. PUTCHA—A. YAQUB, Semigroups satisfying permutation identities, Semigroup Forum, 3
(1971), 68—73.
SOMOGYI B. U. 7
6720 SZEGED, H U N G A R Y
Acta Sei. Math., 50 (1986), 331—33*
The arity of minimal clones
P. P. PÂLFY
Clones play a central role in universal algebra and in multiple-valued logics
A set of finitary operations is a clone of operations if it is closed under composition
and contains all projections. The clones on a fixed set form a complete lattice with
respect to inclusion. If the set is finite, then the lattice of clones is atomic and coatomic.
The coatoms, i.e. the maximal clones, were classified in Ivo ROSENBERG'S profound
paper [4]. On the contrary, quite little is known about the minimal clones. Recently
B£LA CSAKANY [1], [2] determined all minimal clones on the three-element set.
By definition, the arity of a minimal clone of operations is the minimum of
arities of the nontrivial operations in the clone. Identifying any two variables in
such an operation turns this operation into a projection. As observed by SWIERCZKOWSKI [5], if such an operation has at least four variables then it is a semiprojection,.
i.e. there is an index / (1 î^k) such t h a t / ( % , ..., ak)=al whenever K ^ , . . . ,
Since the arity of any nontrivial semiprojection does not exceed the cardinality of the
underlying set, it follows that the arity A: of a minimal clone of operations on an
«-element set must satisfy
(see [3,4.4.7]). It is easy to find unary, binary and ternary minimal clones, for examplethose generated by a constant function, a semilattice operation, or a median operation
of a lattice, respectively (see [3, pp. 114—115]). If « > 2 , then any minimal clone contained in the clone generated by an arbitrary nontrivial w-ary semiprojection has arity
n. But for
the existence of a k-ary minimal clone of operations on an «-element set was not known.
T h e o r e m . There exists a k-ary minimal clone of operations on an n-element set
(«S3) i f , and only i f ,
Isk^n.
Received May 17,1984.
P. P. Pdlfy
332
P r o o f . By the preceding remarks it is enough to point out a k-axy minimal clone
for
Fix different elements bx, ...,bk,bk+1£A,
where \A\=n. Define a
.fc-ary nontrivial operation / b y
•Я6*»
\alt
otherwise.
Since/is a semiprojection, the clone [ / ] generated by/ i s &-ary. We are going to prove
that [ / ] is a minimal clone. For any term t representing a function in [ / ] (in short,
*£[/]) we denote the first (from the left) variable of t by a(t), for example
<K/(/(*2. xx, x3), xx, x1)) = x2.
First we show that/satisfies the following identities:
(1) / ( / . к , • • •, ' * ) = / for any fc-ary i*
*»€[/].
(2) / ( * ! , t2, ...,tk)=f
if the fc-ary terms t2,..., /*€[/] are such that <r(ti)=;сг
<i=2,..., к),
(3) Дхх, t2,..., tk)=Xl if t2, ..., /*€[/] and for
some /, 2
<т(/,-)=л,1,
(4) / ( x l 5 ? 2 ,..., / Jl )=x 1 if /2> • • > ' * € [ / ] and for some i and j, 2
•o(td=a(tj).
Indeed, substitute a x , a 2 ,
A for the variables
, x 2 , . . . - Observe that, by
the definition o f / , if / € [ / ] and <j(t)=xt, then at the given valuation t takes on
or bk+x and the latter can occur only if щ—Ь^ Hence if axj^bx then (1)—(4) obviously hold. Suppose ax=bx. In (3) takes on bx or bk+1, in both cases the left hand
side is bx. Similarly, in (4) either tt and tj have equal values or one of them takes on
bk+1, again forcing the left hand side to be bx. If {a2,..., ak}={b2,...,
then/
takes on bk+1 and we have equality in (1), moreover, in (2) the set of values of t2, ..., tk
is also {b2,..., bk} hence we have equality here as well. If {a2, ..., ak}?± {b2, ...,
then some (2 s/sA:) cannot occur as the value of any £>ary term hence in (1) and (2)
both sides take on bx.
Now the minimality of [ / ] will be derived from the identities (1)—(4). We prove
by induction on the length that any term is either equal to a projection or it turns into
fby suitable identification and permutation of variables. Take a term t=f{tx, ..., tk).
By the inductive hypothesis, if tx is not a projection then suitable identification and
permutation of variables yields a term < ' = / ( / , t2,..., /£), where t'2, ...,t'k are jfc-ary.
By (1) we have t'=f. Now let tx be a projection. Without loss of generality we may
assume that tx=xx. If there is a t{ ( 2 s / s f c ) with a(t^)=xx or there are tt,
tj (2si<jsk)
such that a(t^=a(tj) then t=xx by (3) or (4), respectively. Otherwise, by permuting the variables we may assume that a(ti)=xi for / = 2 , . . . , к.
Then identifying
... with xx we obtain / by (2). Hence for any nontrivial
•*€[/] we have /€[/], therefore [ / ] is a minimal clone. Thus the proof is complete.
The arity of minimal clones
333
Note that by our proof any nontrivial Är-ary operation satisfying the identities
(1)—(4) generates a minimal clone. Indeed, the minimality of a clone is an inner
property, hence it can be advantageous to consider clones abstractly. Following
W. TAYLOR [6, pp. 360—361], an abstract clone T is a heterogeneous algebra on a
series of base sets T l5 T2, ... equipped with composition operations Crm:
—Tm (m,r= 1,2,...) and constants (that correspond to the projections) e^£T„
(/= 1, ..., n; n= 1, 2, ...) satisfying the identities
C;(f, Ci(ult
= Ci(c:(t,
C£(e?,
vlf...,
v„),..., CZ(ur, vlt...,
vn)) =
uu ..., ur), vx,..., v„) (m, n, r = 1, 2,...),
..., O = t, (m, n = 1,2,...; i = 1,..., n),
C;(f,eJ, ...,«» = * ( » = 1 , 2 , . . . ) .
Subclones, homomorphisms, etc. are defined in the natural manner. An abstract
clone is minimal if it is generated by any of its nontrivial members. Any homomorphism of a minimal abstract clone onto a nontrivial clone of operations on a set
yields a minimal clone of operations. We will pursue this line of research in a forthcoming paper.
Acknowledgements. I am deeply indebted to Béla Csákány: his pioneering work
in this area inspired several ideas of the present paper, and our correspondence and
discussions helped me to develop these ideals. I am also grateful to Ágnes Szendrei for
her advice and to László Szabó for calling my attention to this problem.
/
References
[1] B.
CSÁKÁNY,
Three-element groupoids with minimal clones, Acta Sei. Math., 45 (1983), 111—
117.
All minimal clones on the three-element set, Acta Cybernet., 6 ( 1 9 8 3 ) , 2 2 7 — 2 3 8 .
und L . A . KALUZNIN, Funktionen- und Relationenalgebren, D V W (Berlin, 1979).
[4] I . G . ROSENBERG, Über die funktionale Vollständigkeit in den mehrwertigen Logiken, Rozpr.
CSAVÄada Mat. Prir. Véd, 8 0 , 4 (1970), 3 — 9 3 .
[5] S. SWIERCZKOWSKI, Algebras which are independently generated by every n elements, Fund.
Math., 49 (1960), 93—104.
[6] W. TAYLOR, Characterizing Mal'cev conditions, Algebra Universalis, 3 (1973), 351—397.
[2] B . CSÁKÁNY,
[3] R . PÖSCHEL
MATHEMATICAL INSTITUTE O F THE
HUNGARIAN ACADEMY OF SCIENCES
REÁLTANODA UTCA 13—15
1053 BUDAPEST, HUNGARY
6
Acta Sei. Math., 50 (1986), 335—33*
A note on minimal clones
z s . LENGVARSZKY
For
the existence of jfc-ary minimal clones of operations on the «-element set was proved by P. P. PALFY [1] who exhibited concrete examples. Here we
give a very simple nonconstructive proof of this theorem.
First we prove the following claim:
Let (A,
be a partially, ordered set and k a natural number such that the cardinality of an arbitrary antichain in (A,
is at most k— 1. Then the arity of any
nontrivial monotone semiprojection on (A,
does not exceed k.
Indeed, let / be a monotone semiprojection on (A,
with arity
m^k+l.
Without loss of generality we may assume that / i s a semiprojection to the first variable
Let ax,..., am be arbitrary elements of A. By assumptions for some i and j,
lî^j^m,
fljÔj
holds. For l S / S m we define the elements b, and c, by
,
=
'
f a , if 7 = i,j,
l a , if
1
(dj if
l a , if
I = i,j,
Since / is monotone,
a1 = b1=f(b1,...,bm)^f(a1,...,am)^f(c1,...,cm)
= c1 = a1,
i.e. / is trivial.
In order to prove P&lfy's theorem, let k and n be natural numbers such that
3 ^ k s n . Suppose we have an «-element poset (A,
such that the cardinality of
an arbitrary antichain is at most k—1 a n d / i s a nontrivial, monotone, A>ary semiprojection on (A, ^ ) . Since A is finite, there exists a minimal clone C contained in
the clone generated by / . Let g be a nontrivial function of minimal arity from C.
Then g is a semiprojection (see [2]) with arity m. Clearly m^k; on the other hand g is
also monotone, hence m S i by our previous claim, i.e. m=k.
We will be done if for all k and «,
we give an «-element poset (A,
and a nontrivial, monotone, k-ary semiprojection / on it. For this aim let A be an
Received August 29, 1985.
336
Zs. Lengvárszky: Minimal clones
n-element set and u3,...,uk
distinct fixed elements in A, L=A\{u3,...,
ut}. Let ^
be a parial order on A such that the distinct elements u and v are comparable if and
only if u, v£L (see the diagram).
0
1
o
:
O
O-.-O
| «8 »4 "fc
O
L
Then any antichain of (A,
has at most k—1 elements. Define a k-aiy operation/
by
,,
faz if alta2£L
and a3 = w3, ...,ak — uk
f(ai,...,ak) = l a i otherwise
Since \L\ ^ 2 , / i s a nontrivial semiprojection to the first variable. Suppose that a^
^ ¿ x , . . . , ak^bk. If aâ^L
and a3=u3,..., ak=uk, then bt, b2£L and
b3=u3,...
...,bk—uk, hence f(at, ..., ak)=a2^b2=/(b1,
...,bk). If either at, a^L or a3=
=u3,..., ak=uk does not hold then either
b2aL or b3=u3, ..., bk=uk does not
hold. Thus, / ( « ! , . . . , ak)=a1^b1=f(b1,
..., bk), i.e. / is monotone, completing the
proof.
References
[1] P . P . PÁLFY, The arity of minimal clones, Acta Sci. Math., 5 0 (1986), 0 0 0 — 0 0 0 .
[2] I- G. ROSENBERG, Minimal clones I: The five types, in: Lectures in Universal Algebra (Proc.
Conf. Szeged, 1983), Coll. Math. Soc. János Bolyai, Vol. 43, North-Holland (Amsterdam,
1986); pp. 405—427.
BOLYAI INSTITUTE
ARADI VÉRTANÚK TERE 1
6720 SZEGED, HUNGARY
Acta Sei. Math., 50 (1986), 337—33*
Intervallfallende Folgen and volladditive Funktionen
Z. DARÖCZY, A. JÄRAI und I. KÄTAI
1. Einführung
CO
Es sei A n >A n + 1 >0 («€N) und L'.= 2 Ki^- 00 eine Folge mit der folgenden
n= l
Eigenschaft: für beliebiges
L] gibt es eine Folge e„£{0,1} (n£N) derart, dass
oo
x= 2! ¿n^n »st. Die Folge {4} wird dann intervallfüllend genannt. Z. B. ist {1/2"}
n=i
intervallfüllend mit L= 1.
Ist {A„} eine intervallfüllende Folge, so nennen wir die unbekannte Funktion
F: [0, Z ] - R volladditiv, falls
/1=1
»1 = 1
für jede Folge e„€{D,1} (w£N) gilt. Unser Hauptziel in dieser Arbeit ist es, für geeignete intervallfüllende Folgen {!„} die volladditive Funktionen zu bestimmen.
Ähnliche Untersuchungen in dieser Richtung kann man in der Arbeit [1] finden.
Z. B. ist in [1] der Fall {^=1/2"} betrachtet.
2. Intervallfüllende Folgen
Es bezeichne A die Menge der reellen Folgen {A„} mit A n >A n + 1 >0 (n€N) und es
oo
sei L : = 2 !
n= l
D e f i n i t i o n 1. Wir nennen die Folge {A„}6 A intervallfüllend, falls es für beliebiges jc€[0, L] eine Folge £„€{0,1} (n€N) gibt, für welche
(2.1)
Eingegangen am 24 Mai, 1984.
x=2*nK
»=1
Z. Daröczy, A. Järai und I. Kätai
338
ist.
Satz 2.1. Die Folge {X„}€A ist genau dorm intervallfüllend, falls
(2.2)
für jedes n€N
2 A;
i=n+l
gilt.
Beweis, (i) Nehmen wir an, dass A„> 2
für irgendein ra£N, und es sei
(=0 + 1 •
co
lt'—l'
2
Wir zeigen, dass dann die Zahl x:= 2 A.+.yfP), L] sich nicht
1=1
•= H+ 1
in der Form (2.1) darstellen lässt, d.h. dass die Bedingung (2.2) notwendig ist. Gilt
£,=0 für irgendein i ^ n , so ist
¡=1
i=1
i=n+1
gilt hingegen £¡=1 für alle / S n , so ist
i=1
-r
2M >
1=1
••
d.h. x kann nicht in der Form (2.1) dargestellt werden.
n-1
(ii) Es sei x£[0, Z,] und wir setzen induktiv e„=l für 2 i A H - *
und
e„=0 andernfalls. Wir zeigen, dass 2 Z t h ^ x unmöglich ist, d.h. dass x in der
Gestalt (2.1) darstellbar ist. Gilt £„=0 für unendlich viele Werte n, dann ist für diese
Werte 71
CO
n—1
- • •
,
'•
x
2eî
~ 2
<
b,
i=l
i=l
OO
woraus wegen A„-»0 die Darstellung x = 2
folgt-
Gih e„=0 nur für endlich
i= l
viele Werte n, so sei jVder grösste dieser Werte. Wir haben
woraus
x-2
¡=1
2
i=N+1
x< 2
i=l
A,.=
'
i=W+l
j
• '
•••••••
i-
folgt, ein Widerspruch.
D e f i n i t i o n 2. Es sei {1„}ZA eine intervallfüllende Folge. Wir nennen die Zahl
x€[0, L] eindeutig, falls es genau eine Folge e„€{0,1} (n€N) gibt, für welche (2.1)
gilt.
.-"-MN1' •>-•• •••..•
Intervallfüllende Folgen und volladditive Funktionen
339
Bemerkung. Offenbar sind 0 und L eindeutige Zahlen.
D e f i n i t i o n 3. Wir nennen die intervallfüllende Folge
beliebiges N£ N die Restfolge {!„} — {A^} intervallfüllend ist.
locker, falls für
Bemerkung. Im Falle einer lockeren Folge sei
LN :=
B=1
für beliebiges
=
N£N.
n^N
Dann ist jedes JC€[0, Z,^] darstellbar in der Form
oo
x = 2*t*t
j=I
mit
£¡€{0,1}.
D e f i n i t i o n 4. Wir nennen die intervallfüllende Folge
jede Zahl JC£]0, Z,[ nichteindeutig ist.
Satz 2.2. Ist die intervallfüllende Folge
A ergiebig, falls
locker, so ist sie auch ergiebig.
Beweis. Es sei x€[0, L]; Dann gibt es eine Folge £„£{0,1} (n€N) derart, dass
(2.3)
x=2^K-
(i) Ist in (2.3) e„=l nur endlich viele n erfüllt, so sei N der grösste dieser Werte.
Da die Folge intervallfüllend ist, gilt
~
, X = 2ENK=
/1=1
JV-l
JV-l
~
2 ENK + *N= 2 EN*N +
2
n=1
n=1
n—N+1
mit 5„€{0, 1} (n^N+l).
Es ist also JC nicht eindeutig.
(ii) Ist in (2.3) e„=l für unendlich viele n erfüllt, so gibt es ein JV(E N mit
e w = l und X<L—1N =Ln.
Darum gibt es eine Folge 5„6{0,1} (n£N, n ^ N ) für
welche
x =
2
• "=i
HfiN
gilt, d.h. x ist nicht eindeutig.
'
Es sei 1 < ? < 2 und A n : = W («<=N). Dann ist {1 /q"}£A und L:= 2 W =
. . • •.
"=i
= l/(q—1)>1. Es sei k£N festgewählt imd 1
2 d i e Wurzel der Gleichung
(2.4)
L-1
= 1 /qk.
Satz 2.3. Für 1 < 9 < 2 ist die Folge {l/g"}€/i intervällfullend. Für
ist {l!<f} locker, und folglich ergiebig.
l<qSq(l)
340
Beweis, (i) Im Falle l < g < 2 gilt für jedes n
l/i"^(l/i*)(l/fo-l)) =
2
W,
so dass wegen Satz 2.1. die Folge {1 /q"} intervallfüllend ist.
(ii) Für l<q^q(l)
hat man L—l^l/q.
Dann ist bei beliebigem festgewählten
N für die Restfolge {l/q"}-{l/qN} (2.2) trivialerweise erfüllt, falls nur
n^N+l
gilt. Ist hingegen n < N , so folgt aus
q"/qN ^ l/q S
die Ungleichung
L-l
1/9-=£ 2 1 Iq'-l/q".
i=n+1
d.h. es gilt (2.2). Somit ist die Folge {1/V} locker, und wegen Satz 2.2. auch ergiebig.
Satz 2.4. Für q(l)<q<2
also auch nicht locker.
ist die intervallfüllende Folge {\/qn}£A nicht ergiebig,
Beweis. Der Beweis ergibt sich aus dem folgenden.
Lemma. Für
ist die Zahl
(2.5)
*:=
¿l/^-^JO.Lt
13 = 1
eindeutig.
Beweis des Lemmas. Nehmen wir an, dass die mittels (2.5) definierte Zahl x
auch noch eine andere Darstellung der Form (2.1) hat, wobei A„ := 1 /q" ist. Es sei N
die erste natürliche Zahl für welche sN^8N gilt; hierbei ist <5„=1 für ungerades n,
und S„=0 für gerades«. Nun gilt für SN=0 (gerades N) es=l, d.h. wegen
x = 2 W
>1=1
= N2W+
n = l
sjqn = "2 W+ViN+
2
/1 = 1
n=N+l
2
n=N+l
haben wir
2
2
n=N+1
n=N+l
W-
Daraus folgt
d / ?
W
) ( ? / ( ?
2
- l ) ) 2
Bjq'^l/q",
n=N+1
d.h. L—l^l/q,
dies ist aber unmöglich, da im Falle <7(1)<<7<2 die Ungleichung
L—l<l/q
erfüllt ist. Für SN = l (Nungerade) und für eN=0 verläuft der Beweis
ähnlich.
341
3. Über volladditive Funktionen
Es sei {!„}£ A eine festgewählte intervallfüllende Folge.
D e f i n i t i o n 5. Wir nennen die Funktion F: [0, ZJ-«-R volladditiv, falls für jede
Folge e„€{0, 1} («£N) die Beziehung
(3.1)
«ei
gilt.
n=1
S a t z 3.1. Es sei
eine intervallfüllende imd ergiebige Folge. Ist F: [0,1,]-•R
volladditiv, so gibt es ein c£R derart, dass
(3.2)
für alle *€[0,L].
F(x) = cx
Beweis. Wir setzen
(3.3)
F(x):=F(x)-F(L)xIL
(x€[0,IJ).
Dann ist F volladditiv und es gilt / ( 0 ) = / ( L ) = 0 .
:= {n|«6N, a„>0}. Es sei noch
(3-4)
Es sei P(k^=\an
und
P:=
{ := 2 V
ntP
Für ^ 6]0, L[ gibt es eine Folge e„€{0,1} derart, dass
(3.5)
n=i
gilt, und die Darstellung (3.5) von der Darstellung (3.4) verschieden ist. Darum existiert ein n0£P mit e„ o =0, woraus wegen (3.1)
n£P
> 2
11 = 1
niP
n£P
= 2 ZnHK) = F{2
n=l
U=1
= HZ)
folgt, was unmöglich ist. Somit haben wir £ = 0 oder £=L, d.h. P=0 o d e r P = N .
Der letztere Fall ist unmöglich, weil dann P(g)= P(L)>0 wäre. Ist aber P=0, so
gilt F(x)^0 für jedes ;c€[0,L]. Da — P ebenfalls volladditiv ist, ergibt sich durch
eine Widerholung des vorigen Gedankenganges — / ( ; c ) s 0 , woraus P(x)=0 für
jedes x€[0, L] folgt. Aus (3.3) folgt jetzt mit c:= F(L)/L die Behauptung.
B e m e r k u n g . In Satz 3.1. kann man die Bedingung der Ergiebigkeit durch diejenige der Lockerheit ersetzen.
342
K o r o l l a r 3.1. Falls 1
die Bedingung
ist und die Funktion F: [0,
(L:=
l/(g-l))
F( 2 ejq") = 2 e„ F(\!qn)
/1=1
n=1
(3.6)
für jede Folge £„€{0,1} (w(EN) erfüllt, so gibt es ein c£R derart, dass
für jedes x€[0, L\ gilt.
F(x)=cx
Beweis. Wegen Satz 2.3. ist jetzt {\lqn}£A intervallfüllend und locker (d.h.
ergiebig), so dass sich Satz 3.1. anwenden lässt.
Die Beweisidee des Satzes 3.1. ermöglicht es uns, das folgende Resultat auszusprechen :
S a t z 3.2. Es sei {)•„}£ A eine intervallföllende Folge, und F: [0, Z,]—R eine
volladditive Funktion. Ferner sei anF(X„)
(n£N). Für P:= {«|«€N, a„>0} und
2 K ist im Falle / V 0 und i V N die Grösse
L[ eindeutig.
n€P
Beweis. Wegen der Bedingung i V 0 und i V N gilt £ €]0, L[. Ist £ nicht
eindeutig, so gibt es eine Folge-en€{0, l} und n0£P derart, dass £ ^ = 0 und
f =
2*nK
n=i
ist. Daraus folgt auf Grund der Volladditivät
F(0 = F(2
n£P
K) = 2
niP'
<
> 2 e„an = F( 2 ejn)
/1 = 1
/1 = 1
= F(g),
was einen Widerspruch bedeutet. Folglich ist £ eindeutig.
D e f i n i t i o n 5. Es sei {X„}£A eine intervallfüllende Folge. Für
induktiv nach n
(3.7)
"¿Bitoli+Xn^x,
0, für
"2 S i t o X i + l ^ x .
i=1
L] sei
£„(*):=
Sodann gilt wegen Satz 2.1.
(3.8)
1, für
1
¿e B (x)A„.
n=i
Die Darstellung (3.8) nennen wir die reguläre Darstellung von x.
Satz 3.3. Es sei {A„}£ A eine intervallfüllende Folge. Bei beliebigem x£[0, L]
gilt £„(.*)=0 für unendlich viele Werte von n.
343
Beweis. Für x = 0 ist die Behauptung trivial. Es sei 0 < x < Z , . Dann gibt es in
(3.8) ein n mit en(x)=0. Gilt entgegen unserer Behauptung e„(x)=0 in (3.8) nur
für endlich viele n, so sei iVder grösste dieser Werte. Wegen (3.7) ist nun
"Z
> x = "2et(x)Xi+
i=1
2
h
i=N+l
i=l
in Widerspruch zu (2.2).
S a t z 3.4. Ist {X„}£A eine intervallfüllende Folge und F: [0, Z,]—R volladditiv,
so giltfür an := F(X„) (n£N) die Beziehung
2 \an\
n=l
(3.9)
<0
°-
Beweis. Es sei P:= {w|»£N, ß„>0} und M:=N—P.
wegen der Volladditivität
m
und
F(L-Q
woraus
. . . . . .
= F(2
n€P
I := 2
n(P
K
K) = 2 F(K) = 2
ntP
n£P
kl
= F( 2 4 ) = 2 F(k) = ~ 2
M
n£M
niM
n£M
2 kl =
n=i
folgt.
Dann gilt für
F(0-F(L-0
S a t z 3.5. Ist {Ä„}£A eine intervallfüllende Folge und F: [0, Z,]—R volladditiv, so
ist Ffür beliebiges x€ [0, L[ rechtsstetig.
Beweis. Für beliebiges e > 0 gibt es wegen Satz 3.4. ein N derart, dass
(3.10)
2
ITOIê/2.
i=N+1
Es sei
2
«>N+1
CO s
K~
2
.
n=N+1
x
wobei x= 2 n( )K eine reguläre Darstellung ist. Wegen Satz 3.3. ist <5^ >0. Falls
n=i
N
e
x -c y < x+<5 w ~ 2 *(x)K+
• * n=1
~
2
n=N+1
^
344
so gibt es wegen y—
dass
N
eo
У
n=l
л=7Г+1
Л> eine Folge e*€{0,1} ( n ^ N + l ) derart,
n=l
n=N+l
ist, und daraus folgt wegen der Volladditivität von F u n d wegen (3.10)
\F(y)-F(x)\
= | 2
2
л=JV+1
/i=N+l
e„(*)F(An)| =i 2
2
n=JV+l
Satz 3.6. Ist {;.„}£ A eine intervallfüllende Folge und F: [0, i ] - R
so ist F im abgeschlossenen Intervall [0, L] stetig.
volladditiv,
Beweis. Für beliebiges JCC[0, L] hat man wegen der Volladditivität von F
(3.11)
F(x)+F(L—x)
— F(L).
Wegen Satz 3.5. ist F für alle x£[0, L[ rechtsstetig, also auf Grund von (3.11) f ü r
alle j>£]0, L] linksstetig, und somit stetig auf [0, L\.
4. Volladditive Funktionen im Falle spezieller intervallffillender Folgen
Es sei l < g < 2 und {\/cf}eA eine intervallfüllende Folge. Ist F: [0, Z,]—R
( Z , : = l / ( ? - l ) ) volladditiv (d.h. gilt (3.1) für die Folge A„:=l/?"), so gilt nach Korollar 3.1. für
die Beziehung F(x)=cx mit c konstant. Es ist natürlich
danach zu fragen, was sich im Falle q(\)<q<2
sagen lässt, wenn also wegen Satz
2.4. die Folge {1 /q"} nicht ergiebig (nicht locker) ist.
S a t z 4.1. Für q=q(k) (&€N) und volladditives Fgibt es ein c£R derart, dass
F(x)=cx für alle x€[0, 1/(?(A:)-1)] gilt.
Beweis. Es sei Jt€N festgewählt und F: [0, L] — R (L:= 1 /(q(k) — 1) volladditiv. Dann ist auch die Funktion F(x) := F(x) — F(L)x/L (x£[0, L]) volladditiv und
es gilt F(0)=F(L)=0.
Es sei a„:=F(l/qn) (n€N) mit q:=q(k). Wegen der Volladitivität gilt
(4.1)
0 = F(L) = F(2
1 ¡q") = 2 HVq")
n=l
n- 1
2
am 1
o«4
2
71 = 1
Auf Grund der Voraussetzung ist
1 =L-l/qk=
=
Vi"
woraus für beliebiges N£ N
(4.2)
345
+n
i¡q» = 2
Vq"
n=i
n^k
folgt. Aus (4.2) ergibt sich der Volladditivität von F
oo
(4.3)
aN = 2
a
n=i
N+»
und
eo
(4.4).
a N + 1 = 2°s+n+1
n=i
n&k
für alle NeN.
Aus (4.3) und (4.4) folgt nun aN-aN+1=aN+1+aff+k+1-aN+k
(4.5)
aN+k+1—aN+k+2aN+1—aN
für alle N£ N.
Wegen (4.1) ist die Potenzreihe
(4.6)
d.h.
= 0
№:=2a„z»
n=i
konvergent für | z | < l und für z= 1. Indem wir jetzt die Gleichung (4.5) mit der
Zahl zN+k+1 multiplizieren und für N summieren, erhalten wir unter Berücksichtigung
von (4.6) für |z| < 1 und für z=1
f(z)~
*+l
2 a,z'-2[/(z)/=1
k
2 aizc]+2izk[f(z)-a1z]-zk+1m
¡=1
=0
und daraus folgt
/(z)[zk+1—2zk+z—l]
= z(z-l)
2a zi-1
1=1i
+
zk+1(-ak+1-2a1).
Aus der letzteren Gleichung erhalten wir für z = 1 und unter Berücksichtigung von'
/ ( 1 ) = 0 die Beziehung —ak+1—2a1—0, d.h.
(4.7)
/ ( z ) [ z * + 1 - 2 z * + z - l ] = Z(Z-1) ¿ « . Z » - 1 = 2 ( 2 - 1 ) 0 ^ ( 2 )
1=1
für alle | z | < 1, wobei Qk-i(z) ein Polynom von z höchstens (k—l)-ten Grades ist.
Wir betrachten jetzt das Polynom
(4.8)
P(z) := Pk+1(z) :=
zk+1-2z*+z-l
für welches l<z:=q(k)<2
wegen l/(q(k)—l)—l = l/q(k)k eine reelle Wurzel ist.
Es sei
(4.9)
A(z):=zk+1-hz
und 5 ( z ) : = 2 z * + l .
346
Es ist A(z)=P(z)+B(z)
|z| ^ 1 sind, ferner ist
(4.10)
wobei P, B holomorph im abgeschlossenen Einheitskreis
\A(z)\ = \P{z)+B(z)\ < \P(z)\ + \B(z)\
für |z| = l. Die Ungleichung (4.10) ergibt sich daraus, dass für |z| = l
\A(z)\ = | z | | z * + l | = |z*+l| < \2z*+l\ = \B(z)\
gilt. Darum haben nach dem Satz von Rouchi P und B mit Multiplizität gerechnet
dieselbe Anzahl von Wurzeln im ofiFenen Einheitskreis |z|< 1. Da für die Wurzeln
k
von B die Ungleichung |z| =
1 /2|< 1 gilt, haben sowohl B als auch P im offenen
Einheitskreis genau k Wurzeln. Da z = 0 und z = 1 keine Wurzeln von P sind, folgt
auf Grund von (4.7), dass sämtliche sich im offenen Einheitskreis befindenden Wurzeln
von P auch Wurzeln von ß*_i(z) sind, so dass Qk-i ebenfalls k Wurzeln hat. Dies ist
oo
nur möglich falls ß*_i(z)=0 für alle z, und so folgt aus (4.7) / ( z ) = ^ a n z " = 0 für
| z | < l . Somit ergibt sich ak=0
oo
x= 2
(ngN). Da nun jedes
n=l
L] eine Darstellung
e
11 = 1
J<ln (en€{0, 1}) zulässt, erhalten wir auf Grund der Volladditivität von P
F(x) = F(2*nll")=
11=1
woraus F(x)=cx
(cF(L)/L)
Aal
= 0,
folgt.
B e m e r k u n g . Die Behauptung des Satzes gilt auch für q=q( 1). Dies bedeutet
einen neuen, von der Beweismethode des Korollars 3.1. verschiedenen Beweis für das
vorige q=q( 1).
5. Eindeutige Zahlen
Gemäss Satz 2.4. gibt es eindeutige Zahlen für die intervallfüllende Folge
{l/i"}€/l, falls ? ( 1 ) < ? < 2 .
Satz 5.1. Es sei q(l)<q<2.
Falls x€]0, 1[ für die intervallfüllende Folge
eine eindeutige Zahl ist, gilt .
(5.1)
jc€ U I W , 1 fq-1).
n=1
Beweis. Für *6]0, 1[ gibt es ein n derart, dass x£A„-.=[\lqn, 1 ¡q"'1) ist. Da für
^ ( l ) < g < 2 die Ungleichungen l l q " < L l q n < l / q n ~ 1 gelten, zeigen wir, dass im Falle
x€B„:=[l/q", Llq")c.A„ die Zahl x nicht eindeutig ist, also für eindeutiges x die
Beziehung x£[L/qn, l/q"'1) gilt, d.h. (5.1) erfüllt ist.
347
Dann lässt sich nämlich im Falle JE£B„ wegen
1/ ? " <
i
1
l=n+1
/q'^L/q-
jedes Element von [0, L/gf[, also auch x, in der Gestalt
x=Zei/q"+1
¡=i
(5.2)
schreiben, mit £¡€{0,1} (/€N). Anderseits hat x die Gestalt x= l/q"+9,
,+i
wegen 0 s = 3 = ; c - l / q ' ^ L / q " auch 3 in der Form 9=ß^8i/q' ,
mit
wobei
<5,6{0, 1}
(i£N) geschrieben werden kann. Darum gilt
(5.3)
x =
W + J < W H
t=i
Offenbar ist wegen (5.2) und (5.3) x nicht eindeutig.
S a t z 5.2. Gilt q(l)<qSq(2)
und ist x€]0,1[ eindeutig in Bezug auf die intervallfüllende Folge {1/(7"} £/1, so gibt es ein
N derart, dass
(5.4)
X = (1 /q»-1)
gültig ist.
i
*=i
l/q*-1
Beweis, (i) Es sei x£[l/q, 1) eindeutig. Dann gilt wegen Satz 5.1. x£[L/q, 1). Da
x=l/q+Tx/q
gilt, ist
]0, 1[ eindeutig, weil andernfalls x nicht eindeutig wäre.
Anderseits haben wir Tx£[L—1, q—1), woraus wegen
l/q2^L—l<q—l^L/q
(diese Ungleichungen sind für q ( \ ) < q ^ q ( 2 ) richtig) unter Berücksichtigung von
Satz 5.1.
Tx$XL\q\ 1 lq)
folgt. Daraus ergibt sich, dass
Sx:= qTxZ[Liq,
1) c [1 \q, 1)
und Sx eindeutig ist, weil Tx eindeutig war. Nun folgt
(5.5)
X = 1 Iq+Sx/q*,
wobei Sx£[L/q, l)c[l/$r, 1) eindeutigist. Wir zeigen mittels Induktion nach«, dass
(5.6)
2/1—1
x = 2 1
2
Iq^+S-x/q "
348
gilt, wobei
1) eindeutig ist. Für n=1 ist (5.6) wegen (5.5)
•erfüllt. Gilt (5.6), so haben wir wegen (5.5)
Snx = 1
wobei S"+1x£[l/q,
<5.7)
/q+Sa+1x/q2,
1) eindeutig ist, also gilt
l/q 2 k - 1 +Qlq+S n + 1 xlq s )lq* n =
x =
i=l
2n—l
«:=i
•d.h. (5.6) ist auch für n + 1 richtig. Aus (5.6) folgt wegen lim S"x/q2"=0
fl-»-oo
x = 2 1 \q№~x
<5.8)
k=l
und zwar ist auf Grund des im Beweis von Satz 2.4. auftretenden Lemmas diese
Darstellung tatsächlich eindeutig.
(ii) Falls x in ]0, 1[ enthalten und eindeutig ist, so gibt es ein N£ N derart, dass
x£[llqN, l/q11'1) gilt. Daraus ergibt sich, dass qN~1x£[l/q, 1) eindeutig ist, d.h. es
gilt wegen (i) notwendigerweise (5.8). Also ist x in der Tat von der Gestalt (5.4).
B e m e r k u n g . Auf Grund der Definition sieht man leicht ein, dass für eindeutiges
x€]0, L[ auch L—x eindeutig ist.
6. Volladditive Funktionen im Falle q ( l ) < q ^ q ( 2 )
Satz 6.1. Ist q(l)<q^q(2)
und F: [0, X]—R volladditiv bezüglich der intervallfüllenden Folge {\/qn}£A {L:=\l(q—Vf),so gibt es ein cgR derart,dass
F(x)=cx
für alle x€[0, L\ gilt.
Beweis. Setzen wir
<6.1)
/ ( * ) : = F(x)-(F(L)IL)x
so ist P volladditiv und es gilt F(0)= F(L)=0.
(x£[0, L]),
Es sei an:=F(l/q") (n£N) und
P := {/iln^N, a„ => 0}.
Ferner sei
{ := 2
nf.P
1/?".
Wir zeigen dass
ist. Andernfalls ist PT^H und folglich
4 = L wegen der Volladditivität von F folgen würde, dass
L[, weil aus
0 ist, also
349
ein Widerspruch. Somit gilt Z V 0 und P ^ N , also ist auf Grund von Satz 3.2. £
eindeutig. Ist nun x£]0, L[ beliebig, so hat man wegen der Volladditivität von P
(6.2)
/ ( * ) + f { L - x ) = P(L) = 0,
woraus P(L/2)=0 folgt. Daraus ergibt sich ^ L / 2 , d.h. £<E]0, Lß[V]Lß,
L[.
Auf Grund der Identität (6.2) kann man ohne Beschränkung der Allgemeinheit
£€]0, Lß [voraussetzen. Wegen q>q( 1) gilt £ / 2 < l , d.h. £<=]0, L/2[c]0, 1[, und
auf Grund der Eindeutigkeit von £ folgt aus Satz 5.2.
1
1
i = i / q » - 2 Iii*"- = 2 i/g w + 2 ( l - 1 >
fc=i
*=i
für irgendein N£ N. Es sei
(6.3)
(6.4)
oc:=
und
(6.5)
Dann gilt
(6.6)
*=i
22lqN+ilk~1)
ß := 2 2
k=l
und
a+ß = 2f.
L e m m a . f(a)=2F(<xß),
P(ß)=2F{ßß)t
Beweis d e s L e m m a s . Es sei
«,:= 2 V q N + i ( k - 1 )
t=/
(6.7)
(«6N)-
Wegen q ^ q ( 2 ) < q hat man
q3-q2-q-1
woraus 2qa/(q*— \)<L
(6.8)
< ? 3 - 2 ? 8 + ? - l SO,
folgt. Aus (6.7) ergibt sich nun
a , = 2q*lqN+«'-»-1(q*-l)^LIqri+«i-1>-1
(i<EN).
Definitionsgemäss gilt
(6.9)
« l = " ? / 9 w . + i ( | - 1 ) + « i + i (i€N)
und wegen (6.8)
«f := l / ? w + 4 ( i - 1 ) + a ; + 1 = a,—l/<7" + 4 ( i - 1 ) <
< Llq*+*Q-»-i-llq*+W-*>
= (¿0-1)/$»+«'-« = 1/}»+«'-«
woraus wir wegen Satz 5.1. die Darstellung
«f =
2
<*/?*
*=N+4(J-1)+1
(^€{0,1})
350 Z. Daróczy, A. Járai und L Kátai: Intervallfüllende Folgen und volladditive Funktionen
entnehmen, d.h. wegen der Volladditivität von / u n d wiederum wegen Satz 5.1. gilt
P(aü = Pil/qH+W-V+at)
= F(l/qN+iil-»)+F(<xf)
= f(l/q»+4(l-»)+F(l/q!i+4U-iy+<xi+1)
= 2f(l/qN+«'-»)
=
+ F(ai+J.
Aus der vorigen Gleichung erhalten wir
(6.10)
,
P(a,) = 2fl J V + 4 ( l _ 1 ) +/(a i + i )
(<6N)..
Aus (6.10) erhalten wir mittels Induktion (ai=a) für beliebiges ATgN
F(a) = 2 2
+
i=1
woraus für K—«o wegen der Stetigkeit von F im Nullpunkt
F(a) = 2 2 «iY+4(;-i) = 2/(a/2)
i=l
folgt. Für ß verläuft der Beweis ganz ähnlich.
F o r t s e t z u n g des Beweises des Satzes. Mit Rücksicht auf das Lemma
folgt aus (6.6)
F(a)+F(ß) = 2F(ccl2)+2F(ßl2) = 2F(£).
Anderseits gelten nach der Definition von % die Ungleichungen / ( a ) < P(£) und
P(ß)< P(£), womit wir auf einen Widerspruch gestossen sind. Es gilt also P— 0, und
darausfolgt # ( x ) s 0 für alle x£[0, £,]. Indem wir diesen Gedankengang für —F
wiederholen, erhalten wir — P ( x ) ^ 0 und somit gilt F(x)=0 für jedes x€[0, L],
Hieraus folgt nun wegen (6.1) die Behauptung des Satzes.
Literátor
[1] Z . DARÓCZY, A . JÁRAI, I . KÁTAI,
On functions defined by digits of real numbers, Acta Math.
Hung., in print.
(Z. D . und A. J.)
MATHEMATISCHES INSTITUT
L. KOSSUTH UNIVERSITÄT
PF. 12
4010 DEBRECEN, U N G A R N
a. K.)
MATHEMATISCHES INSTITUT
L. EÖTVÖS UNIVERSITÁT
MÜZEUM KRT. 6—8
,
1088 BUDAPEST, U N G Á R N
>
Acta Sci. Math., 50 (1986), 351—380
3
Canonical number systems in Q{]/2)
S. KÖRMENDI
1. Let us given an algebraic number field Q(Y) defined as a simple extension of
the rational number field determined by y. Let
denote the ring of the integers in
Q(y)We shall say that an algebraic integer Q£ 5[y] is the base of a full radix representation in
if every aÇ.
can be written in the form
a = J«,<?k,
(1.1)
k=0
where the digits.a t are nonnegative integers such that 0 ^ a k < N = |Norm (g)|.
The largest set that we could hope to represent in the form (1.1) is the ring Z[Q],
i.e. the polynomials in g with rational integer coefficients. The reason that the norm
N yields the correct number of digits is due to the fact that the quotient ring Z[Q]/Q
is isomorphic to ZN by the map which takes a polynomial in Q to its constant term
modulo N.
Any such radix representation is unique. Let P(X) denote the minimum polynomial of Q. Since Q is an integer in iS[y], therefore the coefficients of P(X) are rational
integers, the constant term of P ( X ) is ± N . Suppose A(X), B{X)£Z[X]
are polynomials whpse coefficients are integers in the range from 0 to N— 1. If A(G) and B(Q)
represent the same element of Z[Q], then A ( X ) — B ( X ) is in the ideal generated by
P(X)
in Z[X].
Since the coefficients of A ( X ) - B ( X ) are in the interval
[—JV-f-1, N— 1] and the constant term of P ( X ) is ±N, therefore A ( X ) - B ( X )
must be the zero polynomial, i.e. A(X) and B(X) have the same coefficients.
I. KÁTAI and J. SZABÓ [1] proved that the only, numbers which are suitable
basés fór all thé Caüssian integers, using 0 , 1 , . . . , N— 1 as digits, are —n±i where
n is a positive integer, N=n2+l is the norm of —n±i. Their work was generalized
by I. KÁTAI and B. KOVÁCS [2], [3], namely they determined all the bases for quad-
Received April 20,1984.
7*
352
S. Körmendi
ratic number fields, using natural numbers as digits. Similar results have been achieved by W. GILBERT [4], independently.
B. KOVÁCS [5] gave a necessary and sufficient condition for the existence of
number base in algebraic number fields. Namely he proved: If Q(y) is an extension of
degree n of Q, then there exists a number base in S[y] if and only if there exists a
SeSfy] such that {1, 9, ..., 3" - 1 } is an integer-base in S[y].
However, the determination of all the number bases in algebraic number fields
seems to be a quite hard problem. Our purpose in this paper is to determine all the
3
number bases in Q\]f2\ This is the simplest case that has not been considered until
now. We hope to extend our investigation for all cubic fields.
3
.
•
2. Let a = f l , and let K(X)=X3—2
use some lemmas.
•
.
....
• •
be the minimum polynomial of a. We shall
P L e m m a 1. Let <x=a+bo+co2 with a,b,c£Q,
and let
3a, E2=
=3(a2—2bc), E3= -(a3+2b3+4c3-6abc).
Then a is a root of the polynomial
P r o o f . Let í = e x p (27T//3) be one of the cubic roots of unity, and let a 1 =ir,
cc2=a+bâ+c^2a2,
az=a+b£,2a+c(^2<J)2 be the conjugates of a. Expanding the
product (X—a1)(X—a2)(,X—ix3) we get immediately that this is T(X).
L e m m a 2. (1, a, a2} is an integer base, i.e. cc=a+ba+ca2
if and only if a, b, c are rational integers.
is an integer in Q(&)
P r o o f . This is well known.
L e m m a 3. Let aGSfir] {1, a, a 2 } is an integer basis if and only .if a = M ± « r , or
a=M±(a+a2)
with a rational integer M.
P r o o f . Let a = a + b a + c o 2 . Then a 2 =(a 2 +Abe) +(2aA + 2c*)o+(2ac + b2)^.
The matrix A of the basis transformation [1, a, <r2]-»[l, a, a2] has the form
A =
1
a
a2+4bc
0
b
2ab+2c*
.
0
c
2ac+b*
det A=±V if and only 6s—2c3 = ± 1. It is well known, see e.g. [6], that, all the solur
tions of this Diophantine equation are:
(2.1)
(b, c) = (1,0), ( - 1 , 0 ) , (1,1), ( - 1 , 1 ) . .
;
Let B denote the set of the number bases in Q(o).
L e m m a 4. If a£B, then ( l , a , a 2 } is an integer base.
.'
-
3 __
353
Canonical number systems in Q 0 2 )
P r o o f . Obvious.
L e m m a 5. Let $€S[er] be such that {1, 9, 5 2 } is an integer basis. Let the minimum
polynomial T(X)=X3+E1X2+E2X+E3
of 9 satisfy the conditions
E3^2. Then 9£B.
P r o o f . See [4].
Lemma6.7rSs-l,
SgSM
then 9$B.
P r o o f . Let H a denote the set of those numbers a that can be written in the form
a = <10+0!$+...+«¿9*
with suitable digits a,€[0, |JV(3)|-1]. If 9 SO, then J / 9 g [ 0 ,
and so - 1 cannot be represented. If S = - l , then |JV(9)| = 1, a j = 0 , and so #¡>={0}. If |S|<1
and a € i / s , then
|a| S a 0 + f l l |9| + . . . + a k
(|
- 1 ) |3|/(|9| - 1 ) ,
consequently Hs is a bounded subset of the real numbers. Since Z[9] is not bounded,
the proof is finished.
L e m m a 7. Let T(X)=Xa+E1XiE2X+E3
be the minimum polinomial of a,
,
,
2
andlet y = ( £ 2 + £ 1 + l)+(£ 1 + l ) + a . Then (1 - a ) y = T ( l ) . Consequently, if |r(7)| <
< 1 , then y or —7 cannot be represented in the form r 0 + r 1 a + ... +rkak, /-^{O, 1 , . . . ,
. . . , | W ( a ) | - l } , i.e. a$B.
P r o o f . The assertion 7 , (1)=(1—a)y is obvious. Let c=sgn T(l). Then
cy = cr(l)+(cy)a,
cT(l)6{0,..., |AT(a)|-l}.
Let us assume in contrary that c has a representation in the form
cy = r 0 +r 1 a+...+/•»<**.
Then r0=cT( 1), c y = r 1 + r 2 a + . . . + r t a k _ 1 . Repeating this procedure we get that
cT(\)=rQ=r1...=rk,
< 7 = 0 , which does not hold.
3. From Lemmas 3 and 4 it follows that if a£B, then a = M ± < r or a =
=M±(a+a2).
Let T(X)=X3+E1X2+E2X+E3
be the minimum polynomial of a.
Let us consider the table below.
a
Ei
Et
Et
M+a
M-a
M+a+o*
M-0-0'
-3M
-3M
- 3M
-3M
3 M*
3 M*
M»+2
3(M*—2)
3(M»-2)
M'-2
M*-6M+6
M*—6M—6
Conditions of Lemma 5 are
satisfied if
MS-3
Msor
4
M = -1
M S -5
M s -4
The numbers a satisfying the conditions stated in the last column belong to B.
S. Körmendi
354
From Lemma 7 we get that a i f
a s — 1, i.e. if
a = M+a
and
Ms-2,
a = M—<r
and
a = M+o+o*
and
Ms-3,
a = M—o—o*
and
Ms2.
AT s i ,
It remains to consider the following set of integers
...,a B , the minimum
polynomials of which are denoted by T^X),...,
TB(X), resp.
a
:.
Jf8 + 9JT2+21X+ 25
— 3+tr
1
2
3
4
5
6
7
8
• 9
.Y3 + 6-X"2 + 12JT+10
*» + 2
jr» + 12Jr 2 +42Jr+34
—2—ff
—a
—4+tr+o -2
— 3—<r—<r2
-2-ff-cr2
-1-ff-tr2
X3 + 6X* — 6X+2
X» + 3X*-3X+1
X*-6X+6
—a—d1
X3-3^-3X4-11
l-O—(T2
v;
N(oc)
25 ,.
10
2
34
. 15
2
1
6
11
L e m m a 8. We have a 6 , a 7 , a 8 , ot9$B.
P r o o f . The conditions of Lemma 7 hold for a 6 , a 8 , a 9 . a 7 is a unit, the set of the
digits contains only one element, the zero, so a 7 $B.
L e m m a 9. a 3 = —a^B.
•\ • •
P r o o f . The set of the allowable digits are (0, f}. Let a 3 = a . First we observe
that —1 = 1+a 3 , 2 = a 3 + a 6 . The general form of the integers id Q(a) is Z =
=X0+X1a+Xza2,
X£Z. By the relation —1 = 1+a 3 , each'Z can be written in the'
form
(3.1) .
. .
Z = r»+na-l-...,+rsa5
s . ..
" •.
:
with nonnegative integers K0, ..., Y5.
Let now Z ° # 0 be an arbitrary integer, written in the form (3.1). We shall define
the following algorithm:
tÇZ°):=>Y0+Y1+J,+Ys',-.h
V
\
I = F 0 -2[r„/2}€{0,1}.
Z ( J ) = y x +Y 2 a + ( 7 3 + h ) a 2 + - Y ^ + r 5 a 4 + h a 5 :
Then Z i 0 >=/+aZ ( 1 ) , furthermore
(3.2)
= [Yj2],
=
-
'
=
-
s_
Canonical number systems in Q(lf 2)
355
Let us continue this procedure with Z (1) instead of Z (0) , and so on. We get a sequence
Z (1) , Z ( 2 ) ,.... We say that the procedure terminates if Z (JV) =0 for a suitable N. It is
obvious that a £B, if the procedure terminates for every Z. Let us assume in contrary
that there exists a Z for which it does not terminate. Since the sequence ¿(Z^) the
values of the members of which are positive integers, is monotonically decreasing,
we get that *(Z< N >)=i(Z (N+1 >)=...=m>0. From (3.2) we get that ocZ<"+>+1>=
=Z(N+n
O'=0,1, 2,...), i.e. a* divides Z w for every positive integer k, which
implies that Z ( W ) =0, contrary to our assumption.
4. It remains to consider the cases a l 5 ct^, a 4 , a 6 . We shall prove that the question whether they belong to B can be decided by a finite amount of computations.
Let a£Q[o], a.=a+bo+co2, A={0,1,...,
|AT(a)|-l}. For y£Q[o] the algorithm
(4.1)
y ^ a y i + i + n , r£A, y0 = y
is well defined. Let
r
0
£(0
0
'
e
=
Let A denote the matrix that describes the multiplication by a in the basé 1, tr, a 2 ,
i.e. for which
*
(4.2)
holds.
From (4.2) we get that
(4.3)
RT =
ARi+1+rte
rl+1 = A - ^ - r t A ^ e
(i = 0,1, 2,...),
where A~l has the following explicit form :
(4.4)
a2—2bc 2b2—2ac
1
2c2—ab a2—2bc
A~ =
N(ot)
b2—ac 2c2—ab
1
4c2-2ab
2b2-2ac
a2-2bc
The algorithm
terminates
if 7 w = 0 f o r a suitable N, i,e. if TF,—0 in (4.3).
Let || • || be a vector norm for which, with the corresponding matrix norm,
(4.5)
|| A-i\\ = x
1
is satisfied. From (4.3) we get that
(4.6)
ri+N = (A-YTi-2
N-l
k=0
rÂ-y-îA-ie),
S. Körmendi
356
and hence that
(4.7)
||ri+JJ 3
|r,|
- 1 ) ( « / ( 1 "*))•
From (4.7) we get immediately that the sequence r 0 , r x , . . . is bounded for every f 0 .
Let us assume that there exists a y which cannot be represented in the base a.
Then (4.3) does not terminate. Since any bounded domain contains only a finite
number of vectors with integer entries, we get that (4.3) is cyclic. From (4.7) we get
that
(4.8)
limsup \\rN\\ s ( | A T ( a ) | - ( x / ( 1 - * ) ) .
Furthermore, the integer yN corresponding to TN cannot be represented in the base a.
So we have proved the following assertion. Let e>0, and let S e be the set of
those y for which
\\r\\ s (|tf(«)|-
ljBii-^BCx/Cl-^
+e
=:L+e,
r = T ( y ) holds. If a t h e n there exists a y w h i c h cannot be written in the
base oc.
Furthermore, if ||r,||sZ,/(l — x), then ||r t + 1 ||s£./(l—x), which is an obvious
consequence of (4.7). This implies that the number of arithmetical operations that
needs to be executed to determine the whole periodic sequence F 0 , r l t . . . is finite.
By using the spectral norm for the matrices At corresponding to ah we get by an
easy computation that
Ur%
« 0,63,
\\At% « 0,75,
\\A^\\S * 0,97,
¡¡A^h
« 0,75,
i.e. the condition (4.5) holds.
5. So we have proved the following
T h e o r e m . The question whether the integers a x = — 3<7, (3^= —2—a, a 4 = —4+
+<r+<72, a6 = — 3—a—<t2 do or do not belong to B can be decided by executing a finite
number of arithmetical operations. All the remaining elements of B are the following
integers:
(a) a = M+a,
Ms-4,
(b) a = M-a,
Afs-3
(c)
M s - 5,
a = M+<x+o\
(d) a = M—a—a*,
Ms-4.
or M = - I
or
Af=0,
3
357
Canonical number systems in ß ( / 2 )
References
[1] I . KÁTAI, J . SZABÓ,
Canonical number systems for complex integers, Acta Sei. Math.,
3 7 (1975),
255—260.
B. KOVÁCS, Canonical number systems in imaginary quadratic fields, Acta Math.
Hung., 3 7 ( 1 9 8 1 ) , 1 5 9 — 1 6 4 .
[3] I. KÁTAI, B. KOVÁCS, Kanonische Zahlensysteme in der Theorie der quadratischen algebrischen Zahlen, Acta Sei. Math., 4 2 ( 1 9 8 0 ) , 9 9 — 1 0 7 .
[4] B . KOVÁCS, Canonical number systems in algebraic number fields, Acta Math. Hung., 3 7 ( 1 9 8 1 ) ,
[2] I . KÁTAI,
405—407.
[5] W . GILBERT,
Radix representations of quadraticfields,J. Math. Anal. Appl.,
Elementary theory of numbers (Warsawa, 1 9 6 4 ) .
[6] W . SIERPINSKI,
D E P A R T M E N T O F COMPUTER SCIENCE
EÖTVÖS L O R Á N D UNIVERSITY
B O G D Á N F Y Ü T 10/B
1117 BUDAPEST. H U N G A R Y
8 3 (1981), 264—274.
Acta Sei. Math., 50 (1986), 359—33*
On measurable functions with a finite number of negative squares
z . SASVARI
To Professor Heinz Longer on the occassion of his 50th birthday
1. Introduction
The complex-valued function / defined on the interval (—2a, 2a),
0 i s
called Hermitian if f(—x)=f(x)
for every x£(—2a, 2a). Let x be a nonnegative
integer. The Hermitian function / i s said to have x negative squares if the matrix
(1)
(/(*<-*;))?,;=!
has at most % negative eigenvalues for any choice of n and x 1} ..., xnZ(—a, a), and
for some choice of n and xlt..., xn the matrix (1) has exactly x negative eigenvalues.
Denote by
0P™;a) the class of all Hermitian functions defined on (—2a, 2a) which
are continuous (measurable, respectively) and which have x negative squares.
In [1] the question was raised if /€^3™ „ implies that / is locally bounded on
(—2a, 2a). The aim of this note is to answer this question in the affirmative (Theorem
1). Therefore in [1] in the definition of
a the condition of local boundednes of/ can
be dropped. We mention that an arbitrary positive definite function / (that is x=0)
is bounded. If, however, %>0, then there exist nonmeasurable functions with x
negative sguares which are unbounded on each subinterval of (—ia, 2a), see [1].
As a consequence of Theorem 1, of the decomposition result in [1] and of [3,
Theorem 2] we show that an arbitrary function
has at least one continuation
in
(Theorem 3). An extension of Theorem 1 to locally compact groups is given
in Section 3. A survey and a bibliography on functions with a finite number of negative squares can be found in [4].
Received April 28, 1984.
360
Z . Sasviri
2. The main result
T h e o r e m 1. Jf /€^3™ „ then f is locally bounded on ( - 2 a , 2a).
P r o o f . We show that / is bouiided on every interval It=[—2a+5, 2a—5],
0<<5<a. For c > 0 the function / + c has at most x negative squares. Therefore we
can suppose that / ( 0 ) > 0 . If K > 0, define
5*/=
1/(01 < K}.
The sets SKf are measurable and we have
lim k(SKf)
J-.«»
= 4a
where A denotes the Lebesgue measure. I f / i s not bounded on I s then there exists a
sequence {/„}~c/ a with | / ( O I — °°- Let K > 0 be such that
X(SKf) > 4a—¿/x 3 and
(2)
0€S K /.
We show that for every n = 1, 2,... there exist elements jc^, ..., x ^ j with the following properties:
xln)e[-5/2,S/2]
(3)
(4)
= Js
for
i=l,...,x+l
x ^ - x < f > e S K f for
(5)
x^-x^
+ tn£SKf
for
i,j=
and
jc{n) = 0,
l,...,x+l,
i,j = 1, . . . , x + l ,
i jt'j.
Let x("^=0, and suppose that
l s l < x + l , have been found such that
(3), (4) and (5)hold with A:+l replaced by/. For each i=\,...,l
we define
Mi = (SKf+xnnJs,
( 0
i.j.m^l
{(Ji\Mt)l>(Js\Nj)U(Ji\Qm)})^l3S/^s5
A( n
Let
Qt = (SKf+xl*
+ t„)nJs.
| ^ n ) ± i „ | < 2 a - 5 / 2 and (2) it follows that .
From
and so
Nt = (SKf+xln)-t„)DJi,
= X(Ji)
(M(n^nej)>o.
be an arbitrary element of the set
f|
(Af ( nA^ng m ). Then (3), (4) and
t.j,m=l
(5) hold with x + 1 replaced by / + 1 . Therefore, for every « = 1 , 2 , . . . we can
choose x*f\ ...,
so that (3), (4) and (5) are satisfied. Now let
..., z£>+a be
defined by
Measurable functions with a finite number of negative squares
361
We consider the matrix
A(n) = (a <;>)2*ti
= (/(zi-> -Zj">))?.y=
The relations (3), (4) and (5) imply K l i . a / N K l i w - i l = 1/(01, / = 1 ,
+l
and \etff\ < K for the other entries of Aw. Using these facts it is easy to see that by
setting Z><n)=det (e//j\J=v
r=l, ...,2x+2,
we have
lim ^fiVl/WI"
= (~ l)r» r = 1,
lim D£\J\f{tB)\*
= (-1/,
...,*+1,
r = 1,..., x,
Di">> 0.
It follows that for it sufficiently large, the signs in the sequence
,Dl°\Din\...,Di$+i
1
change exactly x + l times. Consequently, by Frobenius's rule the matrix A(n) has
x+l negative eigenvalues. This is a contradiction to the assumption that / h a s exactly
x negative squares. The theorem is proved.
Combining Theorem 1 and the decomposition theorem in [1] we have the following
T h e o r e m 2. Every function /(E^P™. a admits a unique decomposition
(7)
№ = f
such that /c€î0,
c
( f ) + m (-2a<f<2a)
and /(0=0
a.e. on (-2a, 2a).
C o r o l l a r y 1. If a. is finite then fZ^.îs
bounded on (-2a, 2a).
Indeed, in the decomposition (7) the function fc is bounded on (—2a, 2a) according to [2] a n d / is bounded as it is a positive definite function.
T h e o r e m 3. Lei
where 0 < a < ° ° and x is a nonnegative integer. Then
there exists a function /€^P™;oo such that
f(t) =f(t)
(-2a<i<2a).
P r o o f . By Theorem 2 the function/admits a decomposition
f ( t ) = m + m
(~2a^t^2a)
such that
f s ^ o - a and / , ( 0 = 0 a.e. on (~2a,2a). In [2] it was shown that
fe has an extension
and by Theorem 2 in [3]/, has an extension
If we set f=fc+j,
we have
and f(t)=?(t)
(-2a<f<2a).
Z. Sasvari
362
3. Some remarks
1. Let x again be a nonnegative integer. The definition of a function with x negative squares can be formulated on a group G as follows. Suppose that / i s a complexvalued function defined on a symmetric set VczG which contains the unit of G and
has the property that f(g~y)=f(g)
for every g£ V. The function/is said to have x
negative squares if the matrix
(/(ifcg/ 1 ))?,/-!
(8)
has at most x negative eigenvalues for any choice of « and glt..., g„€ V for which
gigJ^ V ( / , 7 = 1 , . . . , « ) , and if for some choice of n and gi,..,g„
the matrix (8)
has exactly x negative eigenvalues. We denote by
the set of functions on F which
have x negative squares. If G is a locally compact group with Haar measure A and Vis
A-measurable we set
K = [fZtyx-.v- f is A-measurable on V}.
It is not hard to see that the arguments in the proof of Theorem 1 can be used in order
to show the following result.
T h e o r e m V. Let G be a locally compact group and /£ ^J™. v where V is an open
symmetric subset of G. Then f is locally bounded, that is,f is bounded on every compact
set
KcV.
2. It follows immediately from Theorem 2 that a function
with s e a l
cannot vanish almost everywhere on (—2a, 2a). This fact remains valid for functions on a locally compact group. In order to see this we need the following.
L e m m a 1. Let G be an arbitrary group and
with finitely many translates of the support of f .
If
then V can be covered
P r o o f . Let glt ...,g„€V be such that the matrix A=(f(gigJ1))"J=1
has x
negative eigenvalues. Suppose that V cannot be covered with finitely many translates
of S u p p / = {g€ V:f(g)?iO}. Then there exists a g£ V for which
(Supp f ) g j \
/ , 7 = 1 , . . . , « . Let tx, ...,t2n be defined by
Then we have
h = gi>
t„ = gn,
tnAri = gtg,...,
Thus B has 2x negative eigenvalues, a contradiction.
t^ = g„g.
Measurable functions with a finite number of negative squares
C o r o l l a r y 2. Let G be a locally compact group and let /€^P"
If x^ 1 then f cannot vanish almost everywhere on V.
363
v where A ( F ) > 0 .
T h e a u t h o r is indebted t o Professor Heinz Langer f o r suggesting t h e p r o b l e m discussed in this p a p e r a n d f o r several helpful discussions.
References
LANGER, On measurable Hermitian indefinite functions with a finite number of negative
squares, Acta Sci. Math., 45 (1983), 281—292.
[2] V . I . PLUS&VA, The integral representation of continuous Hermitian-indefinite kernels, Dokl.
Akad. Nauk SSSR, 145 (1962), 534—537. (Russian)
[3] Z. SASVARI, The extension problem for measurable positive definite functions, Math. Z-, 191
(1986), 475—478.
[4] J. STEWART, Positive definite functions and generalizations, an historical survey, Rocky Mountain J. Math., 6 (1976), 409-434.
[1]
H.
SEKTION MATHEMATIK
TECHNISCHE UNIVERSITAT
MOMMSENSTR. 13
8027 DRESDEN, G.D.R.
Ааа ¿с/. Мшк-, 50 (1986), 365—366
Представление полиномов Лагерра
о. в. висков
Пусть И=¿¡с1х — оператор дифференцирования и
(1)
Ц,(х) = (1/|»0е я *~"Л"[е~**" + 1.
я = 0,1, 2.
— обобщенные полиномы Лагерра (см. [1], 10.12). В настоящей заметке устанавливается справедливость следующего представления для полиномов Ь%(х):
(2)
Ц,(х) =
(1/п1)х—ех(х2П+ах+х)п[е-х].
Это соотношение является в известном смысле двойственным к формуле
Родрига (1) и установленному в [2] представлению
(3)
Ц(х) = ((-
1)п/п\)ех(хП2+аВ+0)п[е-х].
Отметим, что представление (3) в частном случае а = О было доказано
Л. Б. Р е д е й [3].
Доказательство (2) можно было бы провести, опираясь на установленную
в [2] операционную лемму и почти дословно следуя использованному там
ходу рассуждений (с заменой оператора /) оператором умножения на независимую переменную, и наоборот). Однако в настоящей заметке предлагается
иное доказательство, основанное на единственности решения задачи Коши,
для подходящего уравнения в частных производных.
В самом деле, хорошо известно [1], что
(4)
2 «(*)<" = (1 - О ^ - ' е х р { * * / ( * - 1)}.
Кроме того, ясно, что функция
"('»х)= 2
Поступило 5-ого февраля 1984.
8
№)(х2£+ссс+х)п[е-*]
366
О. В. Висков: Представление полиномов Лагерра
дает решение задачи Копш
(5)
— =
\)хи,
«|,=о =
С другой стороны, легко проверяется, что функция
u(t, х) = (1 -xt)-*-1
exp {xftxt—1)}
тоже является решением задачи Копш (5). Поэтому
(6)
2 (taln\)éx(^D+oLX+Xy[e~x]
BS о
= (1 -xt)-'-1
exp
{*»//(*/-1)}.
Сравнение (6) и (4) с учетом (2) завершает доказательство.
Литература
[1]
Higher Transcendental Functions, vol. 2, McGraw-Hill (New
York, 1953).
[2] О. В. Висков, О тождестве JI. Б. Редей для полиномов Лагерра, Acta Sei• Math., 39
(1977), 27—28.
[3] L . В . RÉDEI, An identity for Laguerre polynomials, Acta Sei. Math., 3 7 ( 1 9 7 5 ) , 1 1 5 — 1 1 6 .
BATEMAN MANUSCRIPT PROJECT,
М А Т Е М А Т И Ч Е С К И Й И Н С Т И Т У Т И М . В. А. СТЕКЛОВА А Н С С С Р
У Л . ВАВИЛОВА 42
МОСКВА 117333, С С С Р
Acta Sci. Math., 50 (1986), 367—380
Orthonormal systems of polynomials in the divergence theorems
for double orthogonal series
F. MÓRICZ*
1. Introduction. Let (X, !F, fx) be a positive measure space and
/, k=
= 0 , 1 , . . . } an orthonormal system (in abbreviation: ONS) defined on X. We will
consider the double orthogonal series
(1.1)
2 2 a <PiÁx)
i=0k=0 lk
where {ajt} is a double sequence of real numbers for which
(1.2)
¿¿a?*<~.
i=0»=0
The rectangular partial sums and (C, a, /?)-means of series (1.1) are defined by
m
n
*»»(*)= 2
2aik<pik(x)
i=0k=0
and
<Ax)
= (1 IA'mAS) 2 2
i=o*=o
'" = {mma)
{«^-hP^-U
respectively, where
A
A'-JiAttsxix),
m,n = 0,l,
...).
2. Preliminary results: Convergence theorems. The extension of the Rademacher—MenSov theorem proved by a number of authors (see, e.g. [1], [5, Corollary 2],
etc.) reads as follows.
This research was conducted while the author was a visiting professor at the Indiana University,
Bloomington, Indiana, U.S.A., during the academic year 1983/84, on leave from the University of
Szeged, Hungary.
7*
F. Móricz
368
Theorem A .If
2 2 flatDog 0+2)] 2 [log (k+2)]2 <
i=ot=o
(2.1)
then series (1.1) regularly converges a.e.
In this paper the logarithms are to the base 2.
The convergence behavior improves when considering <f%,(x) with a s O and
/?s0 instead of Jmn(x). The following two extensions of the MenSov—Kaczmarz
(a= 1) and Zygmund (a>0) theorems were proved in [7].
T h e o r e m B. If a > 0 and
(2.2)
2
ia? t [loglog(i+4)] 2 [log(fe+2)] 2 <oo,
i=0k=0
then series (1.1) is regularly (C, a, 0)-sammable a.e.
T h e o r e m C. If a > 0 , )5>0, and
2 ¿a?*[loglog(i+4)] 2 [loglog(fc+4)] 2 <«>,
(=0*=0
(2.3)
then series (1.1) is regularly (C, a, fi)-summable a.e.
and
The next three theorems give information on the order of magnitude of ^ ( x )
respectively, in the more general setting of (1.2).
T h e o r e m D [6, Corollary 2]. If condition (1.2) is satisfied, then
SmnC*) = ^ { ^ ( » 1 + 2 ) l o g ( / i + 2 ) }
(2.4)
a.e. as max(m,ri) —
T h e o r e m E [9, Theorem 1]. If a > 0 and condition (1.2) is satisfied, then
(2.5)
<T^n(x) = ox {log log (m+4) log («+2)} a.e. as
max (m, n) —
T h e o r e m F [9, Theorem 2]. If a > 0 , ^ > 0 , and condition (1.2) is satisfied, then
(2.6)
(x) = ox {log log (m+4) log log (n+4)}
a.e. as max (m, h) —
3. Preliminary results: Divergence theorems. The conditions (2.1)—(2.3) and
statements (2.4)—(2.6) are the best possible.
To see this, from here on let (X, OF, ¡i) be the unit square X=[0,1]X[0,1]
in
2
5
R , J " the ff-algebra of the Borel measurable subsets, and n the Lebesgue measure,
in the sequel, the unit interval [0,1] will be denoted by./, the unit square / X / by S,
and the plane Lebesgue measure by | • |.
Orthonormal systems of polynomials in divergence theorems
369
T h e o r e m A' [10, Theorem 4]. If {aik} is a double sequence of numbers for which
(3.1)
|a t t | S max { k + 1 > t | , |a i > k + 1 |} (i, fe = 0, 1, ...)
and
oo
(3.2)
oo
<Vog(i+2)f\log{k+2)]12
2 2
i=0k=0
then there exists a uniformly bounded double ONS {<pik(x1, x 2 )} of step functions on S
such that
(3.3)
lim sup |s mn (x l5
a.e. as
min (m, n) —
.
A function <p is said to be a step function on S if S can be represented as a finite
union of disjoint rectangles with sides parallel to the coordinate axes, and <p is constant on each of these rectangles.
R e m a r k 1. As a matter of fact, (3.3) was proved in [10] under (3.1) and the
stronger condition that for all pairs of i0 and k0
(3.20
2 2 <4[log(i+2)] 2 [log(fc+2)] 2 = «,.
i=i0k=k0
Assume (3.2') is not satisfied for a certain pair of /„ and k0, but (3.2) is. Then
either
¿'S1«&Dog(i+2)]»=«»
1=0k=0
or
S1i«iDog(fc+2)]»=co.
i=o t=0
Now, it is a routine to construct an ONS {«¡»»(xj, x 2 )} such that
(3.3')
lim sups )mi (jc 1 , x^)
a.e. as
and
n — k0—1,
or
m = ¿o — 1 and n—<=°.
On the other hand, since (3.2') is not satisfied, by Theorem A the truncated series
00
(1.10
00
2 2 a <Pik(.x1, *a)
i=i0*=fc0 ik
regularly converges a.e. Consequently, the divergence expressed in (3.3') cannot be
spoilt as min (/«,«)->• 00 and this is (3.3) to be shown.
R e m a r k 2. In particular, it follows from Theorem A' that log («+2) cannot
be replaced in condition (2.1) by any sequence g(n) tending to °° slower than
l o g ( n + 2 ) as n—co. Similar observation pertains to Theorems B' and C ' below.
370
F. Móricz
T h e o r e m B' [11, Theorem 2]. Set
A-2,k = M ,
If
and
A-lik
Apk = { V£ alfl*
I=2"+1
= |au|,
4Pk = max {A„+lik,APik+1}
2
(p, k = 0,1, ...).
(p = - 2 , - 1 , 0, ...; k = 0, 1,...)
2 a?*[loglog(i+4)] 2 [log(fc+2)] 2 = ~ ,
j=0k=0
then there exists a double ONS {cpik(xA'2)} of step functions on S such that
lim sup Iff^C*!, x^l
a.e. as
min (m, n) —
'
Making the convention that for p— —2 and — 1 by 2" we mean — 1 and 0, respectively, the definition of the Apk can be unified as
= { *Z
<4}1/2
/=2»+l
(P = - 2 » - • • • ;
k
=
•••)•
T h e o r e m C' [12, Theorem 1]. Set
A*pq =
{
*2
<4}1/2 0», 9 = -2, -1,0, ...)•
i=2* + lfc=2«+l
If
and
A*pk^maK{A*p+1,k,A*p,k+1}
2
i=0
(p = - 2 , - 1 , 0 , . . . ; fc = 0, 1, ...)
2 «fJ'og log (i +4)]»Qog log (/c+4)]2 =
k=0
then there exists a double ONS {(p^^,
x2)} of step functions on S such that
lim sup \o%n(?cl, x2)| = °° a.e. as
min (m, n) — 00.
The divergence theorems corresponding to Theorems D, E and F will be stated
in Section 5.
4. Main results. Following the arguments due to ME№>ov[4]andLEiNDLER[3], we
can conclude an approximation theorem for double ONS of L 2 -functions by double
orthonormal systems of polynomials (in abbreviation: ONSP) in xt and xz. This
theorem can be considered an extension of [3, Theorem 1] from single to double ONS.
T h e o r e m 1. Let
x 2 ): '\k=0, 1, ...} be a double ONS on S,
r, s= 1, 2, ...} a double sequence of positive numbers, {Mr: r= 1,2,...} and
{Ns: s= 1, 2, ...} two strictly increasing sequences of nonnegative integers. Then there
exist a double ONSP { P ^ ^ , x 2 ): i, k=Q, 1, ...} on S and a double sequence
371
{E„: r, J = 1 , 2, ...} of measurable subsets of S such that the following properties are
satisfied:
(i) | £ „ | s e „ ( r , j = l , 2 , . . . ) ;
(ii) For every (xl5 x2)€S\Ers,
andfor every M,.1<i§Mr
and
(4.1)
| * 2 ) - ( -
L Y - ^ A T F R I , XJ\ =§ £„
(r,s = 1 , 2 , . . . ; M0 = N0 = - l ) .
where _/„(xl5 x2) equals 0 or 1 depending on r, s,
and x2, but not depending on i
and k\
(iii) If the functions <p,k are (not necessarily uniformly) bounded on S, then
max li^Ct!, xj\ ^ 2{ sup l ^ f o , x^l +1}
(i, k = 0,1, ...).
R e m a r k 3. If the (pik are bounded, in particular, step functions on S, then it
suffices to require that the functions q>ik are orthonormal only in each block A/ r _ 1 <
< / s ¥ , and
(r, j = l , 2, ...), but not altogether.
R e m a r k 4. If in each block
represented in a product form, i.e.
(4.2)
<pik(Xl, x,) =
and
the cpA can be
(xd<pi2-s) (x2)
where both {<p?' r \xj: M r _ x < i ^ M r ) and {(pf' s) (x 2 ): N ^ ^ k ^ N , } are bounded
orthonormal functions on I (r, s— 1, 2, ...), then the resulting Pik can also be taken in
the product form
(4.3)
Pik(xlf x2) = P i 1 - ' ) ( x J P j ? ^ (x2)
in each block M r _ 1 < i ^ M r and N ^ ^ k ^ N , , where both {P f ( 1 , r ) (^:
^ M , } and {Pf ,s) (x 2 ):
are orthonormal polynomials on I (r,s=
= 1,2,...).
The above approximation theorem enables us to strengthen Theorems A', B',
and C ' in the same sense as it was done by LEINDLER [3, Theorems A and G ] in the case
of single ONS. Namely, if there exists a double ONS for which such and such a series
or sequence diverges a.e., then there exists a double ONSP which exhibits this divergence phenomenon.
T h e o r e m 2. In each of Theorems A', B' and C' the double ONS {(pik(xi,x2)} can
be replaced by a double ONSP {P tt (*i, x2)} of the form (4.3).
5. Immediate consequences of Leindler's results. Here we cite the main lemma
of LEINDLER [3, Lemma 3] in the form of the following
T h e o r e m G. Let {er: r = l , 2, ...} be a sequence of positive numbers,
{Mr: r= 1,2,...} a strictly increasing sequence of nonnegative integers, and
F. Móricz
372
{(pi (xj: /=0, 1,...} a system of boundedfunctions such that the (p{ are orthonormal on
I in each block
( r = l , 2 , . . . ; A f 0 = - 1 ) . Then there exist an ONSP
{P,(xi): / = 0 , 1, ...} on I and a sequence {Er : r= 1, 2, ...} of measurable subsets of I
such that the following properties are satisfied:
(i) \Er \êr (r—1,2,...;
here | • | means the linear Lebesgue measure);
(ii) For every x1 £l\Er
andfor every
\<p,(?cd--(-ly^îî)!
Mr _1 <i^Mr ,
Si,
(r = 1,2,...),
where jr (xequals 0 or 1 depending on r andxx , but not on i;
(Hi)
max|J»,(*,)| s 2{sup |<z>;(*i)l + 1 } (i = 0 , 1 , . . . ) .
Assume we have two sequences {e®: r = 1 , 2 , . . . } and {e<2): J = 1 , 2, ...} of
positive numbers, two strictly increasing sequences {Mr: r= 1 , 2 , . . . } and
{Ns : s=l, 2, ...} of nonnegative integers, and two systems { ^ ( x j : / = 0 , 1, ...}
and {(pf\x2 ): k=0,1, ...} of bounded functions on I with bounds B ( p and
respectively, such that the (pf^ are orthonormal on I in each block
(/•=1,2, ...; M0=— 1) and the <p£2) are orthonormal on / in each block iV s _i<
cArSiV, ( J = 1 , 2 , . . . ; N0 = — 1). Applying Theorem G separately to both cases
yields two ONSP { P ^ f o ) : / = 0 , 1, ...} and {Pf'(x 2 ): k=0, 1, ...}, two sequences
{/¿r(1): r = l , 2, ...} and {£j 2) : s= 1, 2, ...} of measurable subsets of / s o that properties (i)—(iii) are satisfied, respectively.
It is not hard to verify that the product ONSP given by
x2 ) = PVixJPPixz)
provides an approximation to the product
i,k=0, 1, ...} with the following properties:
(T) Setting
Ers =(E^XI)U(lXE^),
(5.1)
(i, k = 0, 1, ...)
ONS
i)
{i>№(x1, x2 )=(pj (x1 )(pf\x2 ):
( r , s = 1,2,...);
(ii) For every (x l 5 x2 )£S\Ers ,
(5.2)
| ( X M
2
and for every M r _ 1 < i S A f r and
)
(*.) • - ( - lyP'WPWpf»(Xj)Pi2)
- (N# max ^
(xj\ S
B i ^ + s ^ a™
(r, s = 1, 2, ...),
where both j r (1) (î) and j®\x2 ) equal 0 or 1;
(iii)
max
S 4{sup
IPp'ixOPrix^s
(Xj)| + 1 } {sup \<pfp(Xa)| + 1 }
(i, fc = 0,1, ...).
373
Relation (5.2) immediately follows via the identity
(pWfpW-^ !),<«>+7"»pU)p(2) = ç,(2)|yi)_(_ 1)/"p(l)] + ç,(l)[ç)(2)_(_ iy<»/X*>] _
The main trouble is that the right-hand sides in (5.1) and (5.2) are of
O {e^+fi®} and thus do not tend to 0 as max (r,
In spite of this disadvantage, the approximation result just obtained is enough to state, for instance, that the
double ONS can be replaced by double ONSP in the divergence theorems showing the
exactness of Theorems D, E, and F. This is due to the fact that in these cases r and s
can be chosen so as to depend on a single parameter /, say: r = r , and s=st, while
both rt—00 and s ^ c o as /-*However, we can proceed another way. Starting with the strengthened versions
of [3, Theorems D and E], the following three theorems can be deduced simply by
forming the product system of two appropriate single ONSP as well as the product
system {aik=cfpc^:
i, k=0, 1, ...} of the corresponding single sequences {a^}
and {Û£2)} of coefficients.
T h e o r e m s D'. If (g(w): w=0,1, ...} is a nondecreasing sequence of positive
numbers for which
(5.3)
Q(n) = o{log (n+2)} as
then there exist a double ONSP {Pik(x1 ,x2)=P[i:>(xJP^(x2):
i, k=0,1, ...} on S
and a double sequence {aik} of coefficients such that condition (1.2) is satisfied and
(5.4)
where
lim sup I S ^ f e , X^)\/Q(m) e (") =
m
S M N F A , X2 )
00
a.e. as
n
= 2 2 <*ikPik(X1,
i=0k=0
min (m, m)
(m, n = 0,1, ...).
X2 )
Using a double ONS
, x2)=(pf\x1)(pf\x2)}
this theorem was proved in [11, Theorem 2].
in the counterexample,,
T h e o r e m E'. If {g(/j): n = 0 , 1, ...} and {r(m): m=0, 1,...} are two nondecreasing sequences of positive numbers, Q satisfying (5.3) and r satisfying
(5.5)
T (m) = o {log log (m+4)}
as
m
then there exist a double ONSP {/^(xj, xJ^P^Xx^P^Xxz)}
on S and a double
sequence {a tt } of coefficients such that (1.2) is satisfied andfor every a > 0
lim sup \Zm„ (x 1 , x^l/z (m) q (m) = °° a.e. as min (m, n)
where
1, *2> = (MAI) 2
2 AÏT-iSidx!,
X2)
(m, n =
0,1,...).
F. Móricz
374
T h e o r e m F'. If [x(m)} is a nondecreasing sequence of positive numbers satisfying (5.5), then there exist a double ONSP {P^x^ x 2 )=P I i 1) (x 1 )Pi 2) (x 2 )} and a double
sequence {a tt } of coefficients such that (1.2) is satisfied and for every a > 0 and /?>0
{5.6)
where
lim sup
x2)\/x(m)x(n) = <=o a.e. as
*2) = MUT)
2 2
1=0 (¡=0
min (m, ri)
X,)
(m, n =
0,l,...).
Using double ONS { p ^ f o , x 2 ) = y ^ f a ) ( p < k \ x 2 ) } in the counterexamples,
Theorems E' and F ' were included in [8, Section 5].
R e m a r k 5. Examining the structure of the counterexamples in Theorems D '
and F', the following slightly sharper result can be concluded: Estimates (5.4) and
(5.6) remain true if g(m)g(«) and x(m)x(n) in the denominators are replaced by
e 2 (min (m, n)) and x2(min (m, n)), respectively, provided
6(2n) S Cg(n)
(n = 0, 1, ...)
and x(m2) S Cx(m)
(m = 0, 1, ...),
where C is a positive constant.
R e m a r k 6. A couple of other divergence theorems can be strengthened by inserting double ONSP in the above way. For example, the corresponding two-dimensional versions of [3, Theorems B, C, and F | hold also true.
Proof of Theorem 1. It relies on the basic ideas of MENSOV [4] and LEINDLER
[3]. Here we reformulate four lemmas of their papers for the two-dimensional case.
These lemmas were stated and proved in the one-dimensional case. However, the
proofs of the two-dimensional reformulations closely follow the original proofs.
Where is needed, we indicate the necessary modifications.
6.
L e m m a 1. Let { f ( x x , x2): lî^N}
be continuousfunctions, while
{gj(x1, x2): I s j ' s W } step functions on S and let eÔ. Then there exist functions
{Gy(xls x 2 ): 1 ^j^N'}
and a measurable subset E of S with the following properties:
(a) The functions Gj are continuous on S;
(b)
(c) For every (x l5 x2)dS\E
andfor every
l^j^N',
Gj(xls x2) = ( - l y t ' W g / X i , x2)
where j(x1,x2)
(d)
equals 0 or 1 depending on (x l5 x2) but not on j;
max |GJ(X 15 x^l S
(*I.*T)€S
(e)
J
max
k ^ f o , x 2 )|
( l s j s
N');
C*I.*S)€S
1 1
| / / f (*i,Xa)Gj(x,, xjdXldx2\ê
o o
(1 S i == N, 1 S j
N%
375F.Móricz:Orthonormal systems of polynomials in divergence theorems
This auxiliary result is the restatement of [4, Lemma 2, pp. 30—32] with a similar proof. The only thing to explain is that if g is a step function on S, then for every
£ > 0 there exists a continuous function G on S such that
(6.1)
and
(6.2)
| { f o , Xa): G Ox, Xa) ^ g(x1, Xa)>| S e
max |G(x l5 xj\ == max |g(x l t x2)\.
(Xj.XsKS
(xl,xi)€S
This is clear if g (x x , x2)=gm (xjg™ (x2) even with some G (x x , x 2 )=G (1) (x^G^ (x2).
In the general case, it is enough to consider the characteristic function g(x l 5 x 2 )=
= X r ( X I > X2) of a rectangle R inside S given by R=(a1, a2> X (fix, /?2>, where <al5 a 2 )
denotes one of the intervals (a1 ,a 2 ), [a l5 a2), (a l5 a2], [a ls a j and </?i/?2) has a
similar meaning. Setting
8 = min {e/4, («,-«0/2, 03 2 -ft)/2}
we define G(x l5 x2) to be G 0 ^ ) G(2H*2) where
G(1)(x!) =
1
for ax+5 S jq ^ a 2 —<5,
0
for 0 S Xi ^ a x and a 2 X l
1,
linear for
<
< ax+8 and a2—£> < xx < a 2
in such a way that G(1) is continuous on I; and G(2) is defined in an analogous manner.
It is easy to check that this G meets the conditions (6.1) and (6.2) .
R e m a r k 7. If N'=N^N2
and
and the functions ft and g} are given in the forms
f,(x1, *2) =/i (1) (*i)/i (2) (*2)
gjiixi, x2) = gfHxdgîxJ
(1 ^ i ^ N)
( l s j g N{, l s / s
Ni),
then the functions Gj can be also represented in the form
(6.3)
G j ^ , x2) = Gj" ( Xl ) G/2) (x2) (1 —j = Ni, 1 = 1^ N£).
In order to see this, apply the original Mensov's lemma [4, Lemma 2] separately
to the following two systems {/,(1)(xi), ^'(x,.): l ^ i ^ N , lSy'SiV^} and
{f?\x2), g?\x2): lî^N,
l=s/=§JVa'} with £ ( 1 W 2 ) = e / 2 instead of £>0. As a
result, we obtain two systems {Gf\Xl): l ^ N ; } and {G?\x2)\ l^l^N'J
with
corresponding sets Em and is(2), and corresponding exponents
c2) and y'(2)(x2).
Letting (6.3),
£ = (£' 1 »X./)U(/X£ ( a ) ),
and j(xl, x j = j ^ W + j ^ W ,
properties (a), (b), (c), and (e) are obviously satisfied (in case (e) provided e ^ 1).
To verify the fulfillment of (d), we have to take into account that the extreme values of
F. Móricz
376
the step functions g^'and gf> are not altered during the linearization process. Thus,
for every (x l5 x2)6 S there exists a pair (xj, x£)6 S such that
IGÔcOl s |Gj»(xOI = |*j»(*D|
and
|G,(2)(*2)I ^ \Gi2Hxd\ =
|gP(x0|.
Consequently, (d) is also satisfied.
L e m m a 2. Let 0 < r < r ' , {77^^, jca): l S / S r } and {Qk(Xl, x2):
be nonidentically vanishing polynomials in xx and x2,
v
=
max №(*i> *«)l., max ie*(*i> *i)l}.
IX
j< = m T n { J J lPi(x1,x^)dx1dx2,
0
a = max{f
r<k^r'}
i
0
f Iliix^ x^Qk(xlt
x 2 ) dx^ dx2,
f
0
0
Qk(x1,x2)dx1dx2},
1 1
f f Qk(x^ x^Q^,
0 0
y = max {4r's v, 1/x}, and
If the polynomials {Il^Xi, x2): lS/Sr}
exist polynomials {nk(xlt x2):
ties are satisfied:
i
f
x^ dxx
dx2},
A=
are orthogonal on Sand < T A < 1 , then there
in xx and x2 such that the following proper-
(a) The polynomials {77 i (x 1 ,x 2 ): l S / S r ' } are orthogonal on S;
(b)
max \Qk(x1} x^-n^x^
x^l ^ ok
(r < k S r').
The proof is essentially a repetition of the proof of the corresponding result of
MENSOV [ 4 , Lemma 3 , pp. 3 2 — 3 6 ] .
R e m a r k 8. If r'^r'^
and
nt(xlt
and the polynomials J7f and Qk are given in the forms
X2)
=
(xJIlW (X2)
(1 S i S r )
Qu(x1, X2) = QP (Xl) g / » ( x j ( r < / c s r i , r
r'2),
then the polynomials n k can also be represented in the form
(6.4)
n k l ( X l , x,) = I i p ( x j n p ( x j ( r < f c S r i , r
Indeed, apply the original Mensov's lemma [4, Lemma 3] separately to the systems
W f o ) , fif}(*i): l^ter^teiQ
and {n™(x2), Qf\x2):
with
the corresponding notations v(1), x(1),
y(1), A(1) and v(2), ..., A(2). If
and u (2) A (2) <l, then we can obtain polynomials IJ^ 3 for r < k ^ r [ and 77j2) for
such that the systems { i l f f e ) :
orthonormal on I, respectively, and
and {nf\x2):
l s / s r ^ } are
(r < fc ^ ri)
and
Now letting (6.4), the fulfillment of (a) is clear, while a slightly modified version of (b) follows via the elementary estimate
max
|QPCxJQPCxJ-iiPMnloCxJl*
=5 v<2> max \Qp ( x j - l i p (xJI+v (1 > max [&(2) (xj-ll™
(x2)\ S
V(2)<T(1)A(1) + V(1)<7(2)A(2>.
This form is still enough during the proof of Lemma 4 below (cf. [3, p. 26, formula
(3.4)]).
L e m m a 3. Let {?>a(xi, x2): i, k=0, 1, ...} be a double ONS on S,
{£„: r, s=l, 2, ...} a double sequence of positive numbers, {Mr:r= 1,2, ...} and
{Ns: s= 1,2, ...} two strictly increasing sequences of nonnegative integers. Then there
exist a double system
x2): i, k=0,1,...}
of bounded functions on S and a
double sequence {Ers: r,s= 1,2,...} of measurable subsets of S such that the following
properties are satisfied:
(a) The functions \J/ik are orthonormal on S in each block
and
N.-^teN.
(r, j=l,2,...;
M0=N0=-l);
(P) \EJ^srs
(r,j=l,2,...);
(y) For every (x l5 x2)£ S\Ers,
and for every Mr_1î^Mr
and
I<Pik(Xi, xj-tikix,.,
s= fi„ (r, s = 1,2,...).
This lemma is a straightforward. extension of a lemma due to LEINDLER [3,
Lemma 4, pp. 33—36]. The original proof works in the two-dimensional setting, since
the blocks {(pik(Xl, x2): Mr_1î^Mr
and
and the corresponding e rs
(r, s~ 1,2,...) can be treated in an arrangement similar to the Cantor diagonal process, i.e. in the following succession: e u , s 12 , e21, s^, s22, e14, ... and the blocks are
taken accordingly.
L e m m a 4. Let (e„: r,s=1,2,...}
be a double sequence of positive numbers,
{ M r : r = l , 2, ...} and {Ns: J = 1 , 2, ...} two strictly increasing sequences of nonnegative integers, and {t/iit(xl5 x2): i, k=0, 1, ...} a double system of bounded functions
such that the \¡/a are orthonormal on S in each block
and
(r, s= l j 2,...; M0=N0= -1). Then there exist a double ONSP {¿V*!» x2): i, k=
378
F. Móricz
= 0 , 1 , . . . } on S and a double sequence {Ers: r, s= 1 , 2 , . . . } of measurable subsets of
S with the following properties:
(ot) |£„|S£„ (r,s= 1,2, ...);
(/?) For every (x l 5 x2)£S\E„,
and for every Mr_x<i^MT
x j - i - iy-<'"»»>P tt (xi, *2)l S e„
where
(?)
and J V j . ^ i ^ J V , ,
(r, s = 1 , 2 , . . . )
x 2 ) equals 0 or 1;
, max I P t t f o , * ^ s 2 { sup
IM*i,*2)l + l}
(*, fc = 0 , 1 , . . . ) .
R e m a r k 9. If in each block M r _ 1 < i ^ M r and N ^ ^ k ^ N ,
lpit can be represented in a product form analogous to (4.2):
the functions
where both {ipf-r)(x,): M^î^M,}
and
tyf's)(x2):
Ns^k^Ns)
are
bounded orthonormal functions on / (r, i = l , 2,...), then the Pik can be
also represented in the product form (4.3) where both
and
are orthonormal polynomials on / (r, s= 1,2,...).
The proof of Lemma 4 can be modelled on that of Leindler's basic lemma
[3, Lemma 3, pp. 26—33]. We only note that both the Egorov theorem and the Weierstrass approximation theorem are valid in two-dimensional setting, as well.
The former one states that if tK*i> x 2 ) is a bounded measurable function on S,
then for every e > 0 there exist a measurable step function <p(xlt x 2 ) on S and a
measurable subset E of S such that
(1)
\E\T~T;
(2) № (*!, Xjj)-(a)(Xj|).
Concerning the two-dimensional version of the Weierstrass theorem, we refer
to [2, pp. 89—90]. Choosing T= S and 91 to be the set of the polynomials P(xlt x 2 )
in xx and x 2 , properties (i)—(iv) in the cited paper are obviously satisfied, thereby
ensuring the uniform approximation of continuous functions on S by the elements
of 21.
Finally, the validity of Remark 9 can be verified by means of Remarks 7 and 8.
7. Proof of Theorem 2. We will present only the proof of the sharpening of
Theorem A'. The sharpenings of Theorem B' and C' can be proved similarly.
So, assume the fulfillment of (3.1) and (3.2). In the proof of [10, Theorem 4]
a double ONS {<PTT (JCx , x2)- i, k=0,l,...} of step functions on S and a double sequence {H„: r, s= —1,0,1, ...} of measurable subsets of S were constructed with the
following properties:
(i) |//] = 1 where / f = l i m sup Hrs as max (r, s)->-°°;
(ii) For every
, x2)€
(7.1)
r
max
r
max
1
m
l
S - > < n S ! 2*- -<nS2* / =
Z
2 r
,
n
+l
Z aikVikixt, xj s C^mM(r>s) (r,s = - 1 , 0 , 1 , . . . )
fc=2'- +l
,
where C is a positive constant and {tjr: r= — 1, 0,1, ...} is an increasing sequence of
positive numbers tending to oo as
(iii) In each block 2 r - 1 < / s 2 r and 2 s - 1 < & s 2 s the functions (p^ can be represented in the product form (4.2).
Given any 5>0, on the basis of Theorem 1 we can construct a double ONSP
{Ptkixi, x2)} and a double sequence {fl„} of measurable subsets of S such that
(i) |H| = 1 where J7=lim sup Hrs as max (r, .?)—«=;
(ii) For every (x l 5 x 2 )€ff„
(7.2)
max
max
m
Z
n
Z,
a<*A*(*i,*2) ^ (C-5)f/max(r,s) (r,s = - 1 , 0 , 1 , . . . )
(cf. [3, Theorem 2, p. 21 and its proof on pp. 38—39]);
(Hi) In each block 2 r ~ 1 < i ^ 2 r and 2 S " 1 <&;§2 S the polynomials
represented in the product form (4.3).
Relation (7.2) implies the a.e. divergence of the double series
eo
can be
00
Z Z a*Pik(x!, Xg)
i=0
k=0
in the same way as (7.1) implies the a. e. divergence of the double series (1.1).
References
[1] P . R . AGNEW,
On double orthogonal series, Proc. London Math. Soc., Ser. 2 ,
3 3 (1932), 4 2 0 —
434.
and F . DEUTSCH, An elementary proof of the Stone—Weierstrass theorem,
Proc. Amer. Math. Soc., 8 1 ( 1 9 8 1 ) , 8 9 — 9 2 .
[2] B . BROSOWSKI
380
F. Móricz: Orthonormal systems of polynomials in divergence theorems
Über die orthogonalen Polynomsysteme, Acta Sei. Math-, 2 1 ( 1 9 6 0 ) , 1 9 — 4 6 .
Sur les multiplicateurs de convergence pour les séries de polynômes orthogonaux, Recueil Math. Moscou, 6 (48) (1939), 27—52.
[5] F. MÓRICZ, On the convergence in a restricted sense of multiple series, Analysis Math-, 5
(1979), 135—147.
I [6] F. MÓRICZ, On the growth order of the rectangular partial sums of multiple non-orthogonal
series, Analysis Math., 6 (1980), 327—341.
[7] F . MÓRICZ, On the ( C , asO, /?sO)-summability of double orthogonal series, Studia Math.,
81 (1985), 79—94.
[8] F. MÓRICZ, The order of magnitude of the arithmetic means of double orthogonal series,
Analysis Math., 11 (1985), 125—137.
[9] F. MÓRICZ, The order of magnitude of the (C, a SO, /?ë0)-means of double orthogonal series,
Rocky Mountain J. Math., 1 6 ( 1 9 8 6 ) , 3 2 3 — 3 3 4 .
![10] F . MÓRICZ and K . TANDORI, On the divergence of multiple orthogonal series, Acta Sei. Math-,
[3] L . LHNDLER,
[4]
D . MENCHOFF,
4 2 (1980), 133—142.
MÓRICZ and K . TANDORI, On the a. e. divergence of the arithmatic means of double orthogonal series, Studia Math., 82 (1985), 271—294.
[12] K . TANDORI, Über die Cesàrosche Summierbarkeit von mehrfachen Orthogonalreihen, Acta
Sei. Math., 49 (1985), 17£—210.
!
[11]
F.
UNIVERSITY O F SZEGED
BOLYAI INSTITUTE
ARADI VÉRTANÚK TERE I
«720 SZEGED, HUNGARY
Ada Sci. Math., 50 (1986), 381—390
Birkhoff quadrature formulas based on Tchebycheff nodes
A. K. VARMA
1. In 1974 P. Turin [6] raised the following problems on Birkhoff quadrature.
If in the n ,h row of a matrix A, there are n interpolation points l £ x i , > ^ 2 l l > . . . >
^~x„n s — 1 then A is called "very good" if for an arbitrary set of numbers j>to and y£,
there is a uniquely determined polynomial D„(f; A)=D„(f)
of degree at most
2n — 1 for which
Dn(f;A)^Xkn
= ykn=f(xkn),
A))x
x
=yL, fc = 1,2,..., ».
In that case D„(f\ A) can be uniquely written as
D„(f ; A) =2
Kxin)yin(x\
¡=1
A)+ 2 y?n8,n(xi A)
i=i
where yin(x; A), Qin(x; A) are fundamental functions of the first and second kind,
respectively.
P r o b l e m XXXVI. What is the best class of functions for which the integfals of
the polynomials
n
2
¡=i
f(Xin)ytn(x,n)
(n
even)
i
tend to J f(x) dxl
-i
Here the ntb row of the II matrix is referred to the zeros of nn(x)=
where P„(x) is the Legendre polynomial of degree n.
P r o b l e m XXXVIII. Does there exist a matrix A satisfying
i
/ yin(x; A) dxl* 0,
i = l,2
Received June 21, 1984, and in revised form June 4, 1985.
9
n0?
X
J
-1
Pn-X(t)dt
A. K . Varma
382
P r o b l e m XXXIX. Determine the "good" matrices for which
¿ | f yin(x;
A)dx|
is minimal.
In 1982 the author [7] was able to answer the above problems. This can be summarized by the following quadrature formula exact for polynomials of degree S i n — 1 :
-
m n + f i - m
n(2n — l)
J j w a x
,
1
n(n—l)(n—2)(2n
,
+
2(2«—3)
n(n-2)(2n-l)
— l) j f ,
y
/(*J
,e2Pan.i(xin)
(1 -xl)f"(xin)
PU(Xi„)
,+
•
-
In the above formula xin's are chosen to be the zeros of IIB(x).
In 1961, the author [9] extended the results of SAXENA and SHARMA [6] of (0,1, 3)
interpolation to Tchebycheff abscissas. We proved that if n is even, then for preassigned values j i 0 , ya, yi3 (/=1, 2, ..., n) there exists a uniquely determined polynomial f„(x) of degree S3n—1 such that
(1.1)
/„(*,„) = ym,
/ n '(x ; „) = ya,
/„'"(*;„) = ya,
i = 1,2, ..., n
where xin's are the zeros of Tn(x).
The object of this paper is to obtain new quadrature formulas based on / (xin),
f'(xin), fm(X[n) where xin's are the zeros of Tn(x). We now state the main theorems
of this paper.
(1.2)
T h e o r e m I. Let
.
1 >Xln
be the zeros of T„(x)=cosn6,
S3n—1. Then we have
(1-3)
/
_i
}
f(x\
/ —% / 2 dx=
U X)'
r
>...>x„„ > - 1
cos 8=x.
Let f ( x ) be any polynomial of degree
n
2 f(xin)Ain+
,-=i
"
2 f\xin)Bin+
,=i
n
2
,=i
"
f"'(xin)Cin
where
(1
-Xl)2
dx-
( l
i f n
- x „3
iT'(x^
]2
Xi
l - "x f j) '
C
(I Si
(1.5)
B — n (1
1 I (3 + 2 ( 2 / 2 2 + 1 X 1 }
l' (x)dx
Bin - — — (1 — xtn)T„ (XIN) + ^
—yirf.
;
J J (!in
12fj2
jf
Vin(X)
(1_x2)i/2
i(i.V
l4)
Birkhoff quadrature formulas
and
/1
U.b)
i
(i
A _ » f , .
¿ « . - - [ H - 3(1-xf„)
A
l
2nm-x№
383
,
+
XiT^(xt)
4
, r.'fej]
2rP J '
T h e o r e m 2. Le/ / 0 ( x ) = l — x 2 then
(1-7)
f
dx— ^f 0 (Xi„)A i n =
-
y
An interesting consequence of Theorem 2 is the following:
C o r o l l a r y 1. For
f0(x)=\—x%
lim 2Mxin)uin(x)^l-x2
at some x<E[-l, 1],
i=l
where uin(x) are the fundamental polynomials of the first kind (0,1, 3) interpolation
based on Tchebycheff nodes. The explicit representation of uin(x) is given in the next
section.
(1.8)
T h e o r e m 3. There exist positive constants cx and c2 independent of n such that
CiTllnn
<
n
2 Mini - c 2 nlnn.
i=1
Theorems 1, 2, 3 reveal an important fact that the quadrature formula obtained
by integrating the Birkhoff interpolation polynomials of (0, 1, 3) interpolation based
on Tchebycheff nodes is essentially very different from those obtained by integrating
Lagrange or Hermite interpolation.
(1.9)
2. Explicit representation of the interpolatory polynomials; (0,1,3) case. In an
earlier work [9] we obtained the explicit form of the polynomial R„(x) (n even positive integer) of degree S 3n—1 satisfying
(2.1)
Unix*») = fkn,
Kb*,)
= g*„, K
' (xkn) = hn>
k = 1, 2,..., n.
It is given by
(2.2)
Ra(x) = 2 fi»"in(x)+ 2 ginVin(x)+ 2 Kwin(.x),
i=1
i=l
/=1
where the polynomials « ta (x), vin(x), u>in(x) are uniquely determined by the following
conditions:
(2.3)
u'in(xkn) = uZ(xk„) = 0,
w1B(xta) = { J
(2.4)
vin(Xlm) = i f i i x j = 0,
v'in(xkn) = j j
(2.5)
w,a(Xka) = wUxJ
vC(xk„) = { j
9*
= 0,
^
*
k=l
k =
>2>
h2
'
jj ' .fc = 1 , 2 , . . . , « .
384
A. K. Varma
The explicit form of fundamental polynomials is given by
(¡0
where <7„-i,;(x) is a polynomial of degree S n — 1. It is given by
(2.7)
qn.Ux)
=
f J ^ d t + f
J ^ d t + c * ]
where ain, cin are chosen so that q„~iti(x) is a polynomial of degree Sn —1.
„ in = _ _ L
rJMD-jt
nn _{ ( l - / 2 ) 1 ' 2 '
(b)
* = 0/2) _
/
(
(2 8)
-
l
t;M>
/
+
(i-oô+o^
dt
>
r.'Ocj
where J„_1i ¡(x) is a polynomial of degree S n — 1. It is expressed by the formula
(2.9)
1 ^ / J ^ d t + f j f ^ d t2 + y , ,
;,_„(*)
= (1 -x*yi*[«
f n J ^^ ^^J /d! "t ' +
S„_!
~ , • ln—
(1 _ ¿2)3/2*" J (I—i«)»/
1Vv»7
where
(2.10)
2
2
Fin(t) = ( l - i ) ^ ( 0 - ^ ( 0 + ( « + l ) i i n ( 0
2 r.'CxJ
nin
(2.H)
«
f
«,„ - — J
n\7\
R 2v
1
_
{
tF
!»«)+PMt) dt
( T T ^
>
in2~7
/AXO+^C)
(l-i2)1/2
1
j f
(2.13)
_ f
y
- U - *))+f t o (0
(i-01/2(i+08'2
o .
385-
(c)
(2.14)
+ 3 ( ^ 2 ) Vin(x) +
¿in(*)
where
A,. =
x,,
3(1-JC
The above representation of uin(x)
.
r
A
is new and very useful in obtaining
(I-*2)1'2
3. Preliminaries. Here we shall prove the following lemmas.
L e m m a 3.1. For k=\, 2, ..., n we have
( 3.2)
(I-x2)-/2dx=* A _
/
_{
1 -xl„
4n I
P r o o f . According to a theorem of
nomial of degree s 4 n —1 then
MICHELLI
and
I
/ g(x)(l-xa)-1/2Jx =
(3.5)
1
= ^ [1
Id
where xin's are the zeros of T„(x). First let
g(x)
T„(x)ll(x)
. T;(Xkn)
RIVLIN [5]
x,
n2(l-^)J
if g(x) is a poly-
A. K. Varma
386
and note that
g(*,„) = 0,
g'(xtn) = {l
J
J
= k
J
k
and
8K
,
f 3 x , J ( l - x L ) if i = k
10
if i
k,
in)
on applying (3.5) we obtain (3.1). The proofs of (3.2) and (3.3) are similar, so we omit
the details. The proof of (3.4) is on the following lines. Since
t ( I - * 2 ) ( x - x j 1%,(x)Ij,(x) . _
u
J
(T^D
1 / (1 — x2)1 (x—xtn)
= J_{
(1-xL)
_ _ 1 f ljn(x) Lj
~
3 J (l-x2kn)K(
- _ I
f ll(-x)
~
3 J (1-xD
= - j
v2\1/2
X )
2
}
1/a
.
d ,3 . , , .
=•
x(x Xfa,) ^
(1 - x 2 ) 1 ' 2 /
,
_
f2(l-x2)-(l-xxtn)^
I
(l-x2)1/2
)ax ~
f j r ^ B ~ l L ( x ) ( 1 - x 2 ) ~ 1 / 2 dX+J
( (1~-xl)
(
^ W - ^ - ^ d x .
Now applying (3.2) and (3.3) we obtain (3.4). This proves the lemma.
L e m m a 3.2. For k= 1,2, ..., n we have
(3.6)
f i ^ t x i - v - v u t ^ ^ i x j - - ^ ] ,
and
ftlL№-t2)-1,3dt
=
(3.7)
P r o o f . It is well known that
(3.8)
Zta(x) =
n
n ,=1
Tr(Xkn)Tr(x)
.
387-
and
(3.9)
K(x)
= 4r Z
i=1
(*),
K.^x)
= ( 2 r - 1 ) [ 1 + 2 2 Г.«(*)]1=1
Therefore
2ж "/22 ( 2 г - Ц Т ъ - М .
= —
/}
(3.10)
But a simple computation shows that
[и/а]
2 2 ( 2 ^ 1 ) 7 ^ 0 0 = Г.'ОО-*ьУ(1 -xL).
r=l
(3.11)
From (3.10), (3.11) we obtain (3.6). For the proof of (3.7) we first note from (3.8),
(3-9)
}
2 n>2
}
г а к т d t
= =2 t*,-i(xj
f
tK-1m~t2ri/2dt+
n
-i
9
л/2
}
f tT£(t)(l-t2)~1/2
_1
+ -2Пг(хк„)
" r=1
dt.
But
J tT£(t)(l —12)~112 dt = 0,
-i
and
2 f tTZ.1W-W-v»dt
-l
= (ji2r-iy-i)
f
-i
T ^ m - t r ^ d t ^
= ((2r — l) 3 —(2r — 1))7T.
Therefore
1
(3.12)
TT л/2
f tl'Lm-tT^dt
=- 2
^-i(xj((2r-l)3-(2r-l)).
But a simple computation shows that
n/2
2(2r-l)37'2r_1(xta) =
(3.13)
~
l-x£,+
в'Г.'бО
2
3n(l+xL)
/
(I-*« J
Ш
{\-t2?'*at-
From (3.10), (3.17), (3.13) we obtain (3.7). This proves Lemma 3.2.
388
A. K. Varma
4. Proof of Theorem 1. First we will show that
fwtoW-W-Wdt
=
(4.1)
M
flL(t)(l-n-1/2dt
-i
=
Since
T*(x) = {\+Tin{x))l2,
(4-2)
it follows from (2.6) and orthogonal properties of Tchebycheff polynomials
fwkn(t)(l —t2)~1/2 dt =
(4.3)
= ((1 -xl)/l2n*) f (1 •+T2n(t))qn_lik(t)(\ -/2)"1/2 dt =
—l
= ((1 -xLW2rfi) f qn.Uk(t)(l--/2)-1/2dt.
-l
On differentiating (2.7) twice it follows that
(4.4)
(\-x^q'Utk(x)-xqUAx)+qn-i,kix)
=
akX(x)+Vkri(x).
Next, we note that
/
((i -x2)
tf-uW-*?;-!^))
(i -x2)-1'2
dx =
(4.5)
= / (d(( 1 -xnîxWdx)
dx = 0.
Therefore on using (4.4) and (4.5) we obtain
/
J . - u W d - / ) -
1
^ ^
(4.6)
= akn f T:(x)(l-x2)~1'2dx+ JlL(x)(l-x2)-y2dx.
We also note that n is an even positive integer. Therefore
of JC, and it follows that
i
(4.7)
T'n{x) is an odd
polynomial
f T^x)(i-x2)-1'2dx = 0.
On using (4.6), (4.7) and (4.3) we obtain (4.1). We also obtain from (3.6) the second
389-
part of (4.1). Next we will prove that
fvin(x)(l-x2)-ll2dx
=
(4.8)
-
" ( 11
4n
- f ) r f a ) + , f 3 + 2 0 ^ + 12X 1 - x D ^
{
12/i
J J
f j U 201 _2
(1 - f ) '
'
On using (2.8), (3.1) and (4.2) we obtain
fvin(x)(l-x*)-ll2dx
1
(4 9)
=
J L X . +
2naXm+2T;{xin)
=
=
/(1+^(0)5^(0
(l-t2)1'2
J
1
J L * +
rJsz2VL.it
Next, differentiating twice we obtain from (2.9)
(4.10)
(1
=
On using (4.5), (4.7) we obtain
/a i n
(
' }
r
J
s
- i ( * ) j.. _
(1 -x) 1 / 2 *
P
Pitt
r
J
l
'in(x)
(l-x2)1/2
^ ,
+
/
J
^fa)
(l -x 2 ) 1 / 2
X
'
where pin is given by (2.12) and Fin(x) by (2.10). From (2.10) we obtain
J
r
KM
(1-x 2 ) 1 ' 2
J
f1
/(1-^)^(0-^(0
~ J
2Tn'(xin)(l-t2)1/2
2r„'(xJ(l-i2)1/2
a t +
/ w ^ x o -/ 2 ) i / 2 /;;(o)
J
2T;(xin)
/ —2//fa(Q+H2l'in(Q ^
J
27,n/(xin)(l — z2)1/2
/
-2?c(o+»2/f;(o
2r„'(x in )(l—i 2 ) 1/2
/ —2tl"n(t) + n2l'in(t)
~ J
2TJ (xin) (1 — i 2 ) 1 ' 2
Therefore, on using (4.9), (4.11), (2.12) together with the above statement it follows
that
J
1
r,.(x)
dx
ax
= —x-~
2n3
( l
~ x2l )
f
2n
J (l-x2)1/2
(4.12)
X,
"-'l4+4n2l
3
1-xfJJJ
(l-tyi*
dx ++
390
A. K. Varma: Birkhoff quadrature formulas
F r o m (3.7) we h a v e
N o w , on using (4.12) a n d (4.13) we obtain (4.8). Lastly, f r o m (2.14), (3.2), (3.4), (4.1),
(4.8) a n d (3.7) after simplyfying we obtain
P
<4.14)
F r o m (4.1), (4.8) a n d (4.14) o n e can p r o v e T h e o r e m 1.
T h e p r o o f s of T h e o r e m 2 a n d Theorem 3 follow easily f r o m T h e o r e m 1, s o w e
omit t h e details.
References
and P . TÚRÁN, Notes on interpolation. I I , Acta Math. Acad. Sci. Hungar., 8 ( 1 9 5 7 ) ,
201—215.
J . BALÁZS and P . TÚRÁN, Notes on interpolation, ÜL, Acta Math. Acad. Sci. Hungar., 9 ( 1 9 5 8 ) ,
195—213.
G. D. BIRKHOFF, General mean value and remainder theorems with application to mechanical
differentiation and integration, Trans. Amer. Math. Soc., 7 (1906), 107—136.
G . G . LORENTZ, K . JETTER, S . D . RIEMENSCHNEIDER, Birkhoff Interpolation, Encyclopedia of
Mathematics and its applications, Vol. 19, Addison-Wesley Pub. Co. (Reading, Mass.,
1983).
C . A . MICEHELLI and T. J . RIVLIN, The Turán's formula and highest precision quadrature rules
for Chebyshev coefficients, I.B.M. J. Res. Devebp., 16 (1972), 373—379.
R. B. SAXENA and A. SHARMA, Convergence of interpolatory polynomials (0,1, 3) interpolation, Acta Math. Acad. Sci. Hungar., 10 (1959), 157—175.
P . TÚRÁN, On some open problems of approximation theory, J. Approx. Theory, 2 9 ( 1 9 8 0 ) ,
23—85.
A. K. VARMA, On some open problems of P. Túrán concerning Birkhoff Interpolation, Trans.
Amer. Math. Soc., 274 (1982), 797—808.
A. K. VARMA, Some interpolatory properties of Tchebycheff polynomials; (0,1,3) case, Duke
Math. J., 28 (1961), 449—462.
[1] J . BALÁZS
[2]
[3]
[4]
{5]
{6]
|7]
[8]
£9]
D E P A R T M E N T O F MATHEMATICS
UNIVERSITY O F FLORIDA
201 WALKER HALL
GAINESVILLE, FLORIDA 32611, U.S.A.
Acta Sei. Math., 50 (1986), 391—396
Remarks on the strong summability of numerical
and orthogonal series by some methods of Abel type
L. REMPULSKA
In this paper we shall prove that Leindler's theorems on the strong summability
of orthogonal series, given in [2], are true for the methods (A; £„) of Abel type.
1. Let C " ( 0 , 1 ) be the class of real functions defined in (0, 1) and having derivatives of all orders in (0, 1). Denote by B„={bk(r; n)}k=0 a sequence of functions of
the class C°°(0, 1) and such that
(!)
'
P
p
tl:
for k,p—0, 1, ..., n. As in [3] we write
Rk(r\Bn)=
£ bp{r-,
and ARk(r; B„)=Rk(r; B„)-Rk+1(r; B„) for k=0, 1, ... and r£(0, 1). In [3] the
following definition is given: A real numerical series
(2)
£ uk
t=o
(Sk = u0+... + uk)
is summable to s by the method (A; Bn) if the series 2
fc=0
and if the function L(Bn),
(3)
£ ( r ; B„) = 2
k=0
R
k(r\ B„)uk = 2 ARk(r\
(rC(0,1)) satisfies the condition ^lim L(r;Bn)=s.
fc=0
is convergent in (0, 1)
B„)Sk
The classical Abel method, i.e.
(A; B0) method, will be denoted by (A). We shall write L(r) for L(r; B0).
L. Rempulska
392
In [3] and [4] there were given fundamental properties of the methods (A; B„) of
summability of numerical and orthogonal series, and some applications of those
methods to the Dirichlet problem for some equations of Laplace type. In [3] and [4]
it was proved that
T h e o r e m A. Series (2) is summable to s by a method (A; B„) if and only if
l i m
r-i-
(
l - r y * pL
dr
( r ) s s
\'
10
* ? = <>>
if p =
l,...,n.
If the sequence B„ is defined as follows :
b0(r; n) = 1, b„(r; n) = (r-r 2 )"/«!,
(4)
bk(r\ n) = bk(r;
for k=l,
n-l)+-^bk(r;n-l)}
..., n — 1 and « = 0 , 1, ..., then
L(r;Bn) = ( l - r r ^ t Z ( k l " ) r * S k
for 7J=0, 1, ... and r£(0, 1) ([3], [4]).
Moreover in [3] it was proved that
(5)
ARk(r- BB) = ZWp(r;
B„)
r*
for rÇ(0, 1), k=0, 1, ... and n è 0 , if Wp(B„) are some functions of the class
0 ( 0 , 1 > with
(6)
Wp(r; BS] ^ = 0
for p, q = 0, 1, ..., n.
From (5)—(6) and from the Taylor formula for Wp(Bn) we obtain
L e m m a 1. Let r 0 €(0, 1). Then, for every sequence B„, there exist positive constants Mi(r0) and M2(r0) depending on r0 such that
M 1 (r 0 )(l-r)"+ 1 (fc+l) n r* S \ARk(r;Bn)\
M2(r0)(l-r)"+1(k+ 1)V
for k=0, 1, ... and rÇ(r 0 , 1).
2. We shall say that series (2) is strongly (A; -8„)-summable to s with exponent
0 if the function H(B„, q),
(7)
H(r-, B„, q) = J MJ?k(r; 2?„)|\S k -s\*
k=0
(r€(0, 1)) satisfies the condition
(8)
r lim
H(rx Bn, q) = 0.
393
Strong summability of numerical and orthogonal series
By (7), (8) and Lemma 1, we obtain
L e m m a 2. Series (2) is strongly (A; B„)-summable to s with exponent
only if
0 if and
lim (1 - r ) B + 1 2 ( k + l y » * ! ^ - s|* = 0.
i*=o
(9)
Lemma 2 implies
C o r o l l a r y 1. Forafixedn, the methods (A; B„),for the strong summability of numerical series (2) are equivalent, i. e. if B„ and B* are two sequences having properties (1),
then r lim H{r\ B„, q)=0 if and only if \\im H(r; B*, q)=0.
C o r o l l a r y 2. If series (2) is strongly (A; B„)-summable to s with exponent
then it is strongly (A; B„)-summable to s with every exponent 0<q<-ql.
0,
The next statement is obvious:
L e m m a 3. If series (2) is strongly (C, 1 )-summable to s with exponent q> 0, i.e.
lim (l/(» + l ) ) l to-sl« = 0,
then
for every p> 1.
lim (l/(n+l) p ) 2 { k + i y - l \ S k - s \ " = 0
k=0
Applying Lemma 3, we shall prove
L e m m a 4. If series (2) is strongly (C, Y)-swnmable to s with exponent q> 0, then
condition (9) is satisfied for n=0,1, ... .
P r o o f . If (2) is strongly (C, l)-summable with exponent q>0, then |5fc—s|®=
Hence the series 2 (Ar+l),,rk|5rit— i f is convergent in (0,1) and for n=
k=o
= 0 , 1, .... Applying the Abel transformation, we get
=o(k).
(l-r)n+1 ¿(fc+lFÎSit-sl« = (l-r)B+2 ¿ ( k + l ) " + V / t
k=0
k-0
for r€(0,1) and n = 0 , 1, ..., where
lk = (l/(fc+l) B + 1 )
p=o
By Lemma 3 and by Toeplitz's theorem ([1], p. 14), we obtain (9) for « = 0 , 1 , . . . .
Thus the proof is complete.
394
L. Rempulska
L e m m a 5. Suppose that for series (2) condition (9) holds with n=p+\
(p£N)
and with some exponent q>0. Then (9) also holds with n=p and with exponent q.
P r o o f . Let
u.(r\
for r€<0,1). Hence
q) = ( i - r ) " + 1 ¿ ( f c + i y ^ i s i - s i «
k=0
rUp(r; q) = ( 1 - r ^ 1 f\l-t)->-*Up+1(t',
•
.
o
for r6<0,1). But,
f (l-t)7*-*Up+1(t;
o
q)dt = oidl-r)-'-1)
q)dt
(as r -*- '1 —)
if lim Up+1(r; q)=0. This proves our statement.
Lemmas 2 and 5 prove
T h e o r e m 1. For a fixed integer « s 0, the following conditions are equivalent for
every numerical series (2), for every q>0 and every sequence B„: (here H(r; q)=
= H(r;B0,q)).
(a)
Bn,q)
= 0,
(b)
lim ( l - r ) B + 1 2 (k+iyt*\Sk-s\>
(c)
limJl-ry-^Hir;
fc=0
q) = 0 for
= 0,
m = 0, 1, ..., n.
3. Let {<P*(.K)}£°=o be a real and orthonormal system on the interval <0, 1). We
shall consider the strong summability of orthogonal series
(10)
2ckcpk(x)
k=0
with
k=0
by the methods (A;B„). The strong summability of (10) by the methods (A; B„),
defined by the sequence'(4), was examined by L . L E I N D L E R [ 2 ] .
k
Let Sk(x)= 2 cp<Pp(x) and let / b e the function given by the Riesz—Fischer
p=o
theorem, having expansion (10). By L(r, x; Bn) and H(r,x;B„,q)
with
s=f(x)
(r€(0,1), x€(0,1)) we denote the functions as in (3) and (7) but for series (10). As
usual ([1]) we say that two methods of summability are equivalent in L* if the summability of series (10) in a set E of positive measure by one of those methods implies the
summability of (10) to the same sum almost everywhere in E by the other method.
Strong summability of numerical and orthogonal series
395-
In [3] we proved the following
T h e o r e m B. The methods (A; B„), n=0, 1, ..., and the Cesàro (C, 1) method
of summability for orthogonal series (10) are equivalent in L,2.
In [2] L .
LEINDLER
proved
T h e o r e m C. If orthogonal series (10) is Abel-summable to f ( x ) in (0, 1) almost
everywhere, then
lim (1 -r)n+i 2 Pt 1 r*|S*(*)-/(*)l4
'-1t=oV ' /
=0
for any nÇN and q>0 in(0, 1) almost everywhere.
Applying Theorem C and Corollary 1, or arguing as in [2], we obtain
T h e o r e m 2. If orthogonal series (10) is (A)-summable to f (x) in (0,1) almost
everywhere, then it is strongly (A; B^-summable to f ( x ) in {0, 1) almost everywhere
with every exponent q>0 and every sequence B„(n=0, 1, ...)•
Applying Lemma 1 and arguing as in the proof of Theorem 5 given in [2], we can
prove
T h e o r e m 3. Suppose that a and q are two positive numbers and B„ («S0) is a
sequence having properties (1). If the coefficients ofseries (10) satisfy the condition
k=l
then
H(r,x-Bn,q)
=
ox((l~rY)
if qa.< 1 ; and
H(r,x;
in the case qa^l
B„, q) =
ox((l-r)*)
ox((l-r)*|log(l-r)|W)
O x ((l-r) ( B + 1 ) /«)
if
if
if
n+1 >qa,
n+1 = qa,
n+l<qa
but 0 < g s 2 , almost everywhere in <0, 1) as r->-l —.
.396
L. Rempulska: Strong summability of numerical and orthogonal series
References
and H . STEINHAUS, Theory of orthogonal series, Gosudarstv. Izdat. Fiz.-Mat.
Lit. (Moscow, 1 9 5 8 ) . (Russian)
LEINDLER, On the strong and very strong summability and approximation of orthogonal series
by generalized Abel method, Studio Sei. Math. Hungar., 16 ( 1 9 8 1 ) , 3 5 — 4 3 .
REMPULSKA, On some summability methods of the Abel type, Comment. Math. Prace Mat.,
in print.
REMPULSKA, Some properties and applications of summability methods of the Abel type,
Funct. Approximatio Comment. Math-, 14 ( 1 9 8 3 ) , 1 7 — 2 2 .
![1] S . KACZMARZ
[2] L .
[3] L .
'[4] L .
T E C H N I C A L UNIVERSITY O F P O Z N A Î J
INSTITUTE O F MATHEMATICS
3 A P I O T R O W O STREET
•60—965 P O Z N A N . P O L A N D
/
Acta Sci. Math., 50 (1986), 397—403
Behavior of the extended Cauchy representation of distributions
DRAGlSA MITROVI6
1. Introduction. There is an interesting correspondence between the spaces
®'a=@'a( R) of distributions and the class of functions that are analytic in the complex
plane C with a boundary on R, except for a set of points lying in C — R, and vanish
at the point of infinity. In the present paper we make a study of this correspondence.
As we shall see it depends essentially on the support of distributions, order relation
of included functions, and their distributional boundary value in either half plane
A+ = {z£C: Im(z) > 0}, A~ = {z£C: Im(z) < 0}, z =
x+iy.
The problem under study is motivated by some facts from the theory of integrals
of the Cauchy type. Namely, to every function it: R—C Holder continuous with
compact support equal K there corresponds the function
If v(z) = u(z), then: (1) v is an analytic function in C—K; (2) v(z) has the boundary
values
and v~(x) in C; (3) «(z)=0(l/|z|) as |z[ —
Conversely, given a
function v which satisfies the conditions (1)—(3), then it is the Cauchy integral of
some u with supp u=K.
If T is a distribution, then the notation Tt is used to indicate that the testing
functions on which Tis defined have t as their variable. The pairing between a testing
function space and its dual is denoted by (T, q>). The space of C°°=C°°(R) functions
having compact support is denoted by
its dual 2>'=3>'(\l) is the space
of Schwartz distributions on R. As regards the general properties of the spaces <5a and
<S'a we refer to [1].
2. Definitions. In order to describe the correspondence in question in a condensed form we introduce some classes of functions.
Received April 10, 198410
D. Mitrovic
398
Let AT be a closed subset of R and let {alf a2,..., a„} (n£N) be a finite set of
distinct complex points lying in A+UA~. A function / i s said to belong to the class
{M} if
( m . l ) / i s analytic in the domain C—(ATU^, a2, ..., a„}), ak being a pole of
order <xk ( ¿ = 1 , 2 , . . . , « ) ;
(m.2) / ( z ) converges (weakly) in either half plane to a ^'-boundary value;
(m.3)
/ ( z ) = 0(l/|z|)
as
|z|
We use the notation {Mc} for the class of functions in {M} that satisfy (m.l)
when AT is a compact set. Also, a function/is said to belong to the class {M0} if it satisfies the conditions (m.l), (m.2) and the condition
(m.4)
/(«») = lim / ( z ) = 0.
Z-* OO
Thus we have the inclusions {M c }cr{Ai}c{M 0 }. Further, the class of functions that
satisfy the conditions (m.l)—(m.3) when the set of poles is empty is denoted by {^4}.
The class {Ac} is the subclass of {A} relative to compact set K in (m.l). If here the
condition (m.3) is replaced by (m.4) we have the class {A0}.
R e m a r k . The arbitrary sets involved in (m.l) are not necessarily the same for
all functions in a class defined above.
Now denote by R(z) a meromorphic function, vanishing at the point z = <»,
with prescribed poles a1,ai,...,an
(in A + {JA~) and their principal parts. Let
Te0a' ( a s - 1 ) . The function Ffrom C - ( s u p p TU{a1,a2, ..., an}) to C defined by
(1)
P(z) = (1/2*0(7;,
l/(t-z))+R(z)
will be referred to as the extended Cauchy representation of T.
Let us observe that every function / i n {A0} ({M0}) is sectionally analytic in C
with a boundary on R (except for the poles), that is, it can be decomposed into two
independent functions/ + (z) and f~(z) such that / ( z ) = / + ( z ) for z£A+,
f(z)=
= / ~ ( z ) for z£A~ (the half planes being punctured at the points of poles).
3. Main result. We need the following
L e m m a [5]. I f f + ( z ) is a function analytic in A + with / + ( z ) = 0 ( l / | z | ) as |z| —°o
in A
and if f+(x+ie)
converges to 3>'-boundary value f+ as e— +0, then: 1) f*
belongs to for all a < 0 ; 2 ) / + ( x + i e ) converges to 0'a-boundary value f+ as e—
- • + 0 ( a < 0 ) ; 3)
generates the Cauchy representation
a,
%
Extended Cauchy representation of distributions
399
For a function f ~(z) analytic in A ~ and satisfying here the conditions similar to
ones of / + (z), we have
~(z) for z£A~
(3)
- ( i / 2 « o o ; - . !/(<-«» =
for zÇA+.
The distributional version of the previous discussion concerning the integral of
the Cauchy type leads to the following
T h e o r e m . Let T£@'x ( a s - l ) with supp T=K and let {a±, a2, ...,a„} (n£N)
be a set of distinct complex points located in A + \JA~. If f(z)= F(z), then / 6 {M0}.
Conversely, given an /6{M}, then it is the extended Cauchy representation of some
for all a€[— 1,0) with supp T=K.
P r o o f . Consider the direct part of the theorem. To prove the statement (m.l)
it suffices to note that the Cauchy representation f ( z ) of Tis an analytic function in
the domain C—K ([1, p. 56]). The statement (m.2) follows directly from [4, Theorem 2]:
/+ = Fx+ = Tx/2—(l/2ni)(Tx * vp 1 / * ) + * ( * ) .
/ T - Fx — — TJ2—(l/27ii)(Tx*vp
l/x)+R(x).
Observe that the rational function R(x) is a regular distribution (in G'a for all a<0).
As regards the statement (m.4) it is a simple consequence of the hypothesis _R(°°)=0
and the fact that every sequence of functions q>„(t)= l/(t—z„) converges to zero in
( a ^ — 1) as z„-»«=(n—
Conversely, suppose given an /6{M}. Then in view of Lemma the assertions
(m.2) and (m.3) together imply
f ~ t K for all a < 0 . Now define Tx=
= / + - / - . Since ( / + —f~)eO'x for all a < 0 and the Cauchy kernel belongs to <S'a
for all a S — 1, we can associate to Tthe Cauchy representation
t(z) = 0/2ni)(Tt,
l/(t-z))
= (l/2^/)<(/, + ~ f r ) , !/('-*)>
for all a€[—1, 0). Clearly, f is analytic in C—supp T and vanishes at the point
z = ° ° ; moreover, it is easy to show that in this situation t(z)=0(\l\z\)
as |z| —
To prove that f is the extended Cauchy representation of T first we shall show that
the function H from C—(supp J U {ax, a2, ..., a„}) to C defined by
(4)
H(z)=f(z)-f(z)
is meromorphic in C. In fact, after a simple computation we have
((Hi -H-),
for all
(5)
10*
<p) = <(/+ -/-),
Since T x = f + - f
x
<p) = ( ( f t -*£), <P)
it follows
(Hi, <p) = (Hx, cp)
D. Mitrovic
400
for all (pgSf. Further,let A = {z£ C: - < / < I m ( z ) < d } be the strip of the half height
¿ = Min{|Im(a 1 )|,|Iiii(fl 1 )|, ...,|Im(a n )|}.
-
+
Let (a, b) be an arbitrary finite open interval in R, E and E~ two open rectangles
contained in A which have (a, b) as a common edge. Evidently, His an analytic function in A except the boundary on R consisting of the set supp TU K. Applying the
distributional analytic continuation principle ([6], [3, p. 244]) the' equality (5) implies
that the function H is analytic in i s + U (a, b)K)E~, and consequently, in all of A.
Thus H is analytic everywhere in C except for the poles ak of / , and as a meromorphic
function which vanishes at the point of infinity it may be written uniquely in the form
(6)
H(z)=
2
t=ip=I
2BkJ(z-aky,
where the coefficients Bk p must be determined (by means o f / ) . Since the function T
is analytic in G — supp T, using Theorem on the partial fraction expansion of rational
functions ([2]) from (4) we get
(7)
BkiCCk-m = (1/m!) lim
dm[(z-aky,f(z)]/dzm
(w=0, 1, 2, ..., ak—1). Returning to equality (4) with (6) and (7) it follows the representation
(8)
/ ( z ) = (l/27ii)(Tt, l / ( i - z ) ) + 2 2
*=ip=I
BkJ(z-aky.
So we have established that the given / 6 {M} is the extended Cauchy representation
of Tx=(f+ -f~)£0'a for all a e [ - l , 0 ) . Next we have to prove that supp T=K.
First let AT be a closed proper subset of R. Since the function/is analytic on the open
set R—K, it follows that
( f x , (P) = ejim </+(*+is), (p) = lim (f~(x-ie),
q>) = ( f ~ , <p)
for all cp with support disjoint from K (<p<i@(R-K)). Thus <(/+ - f ~ ) , <p)=
={TX, <p)=0 for all such (p. Hence we may conclude that supp T=K. The assumption that supp Tc. K properly leads to the conclusion that there exists an open inters
val (a, ¿)c(K—supp T) on which (//" - / " ) is zero of 3>' ((a, b)). Therefore (by
the analytic continuation principle)/would be analytic on (R—AT)U(a, b) contrary
to the hypothesis. For the same reason supp T= R in the case K~ R. Finally suppose that there exists a distribution
— 1) distinct from T and such that
f(z)=$(z)+H(z)
for z1C-(K[J{a1,
a2, ...,<*„}). According to [4, Theorem 2] we
have / + - f ~ - § ~ = SX. Hence TX=SX on Si and this implies TX=SX on <Sa
(since S> is dense in 0a for all a^R). But this contradicts the hypothesis on S. Thus,:
the distribution T is unique. The proof is complete.
401
In particular, if all poles % of the function / are simple (&=1,2,..., ri), then in
the representation (8) instead of double sum we have
n
2 res [/(z), ak]l(z—ak).
k=l
4. Consequences. First assume that the set of poles of the function F is empty.
In this case f is reduced to the Cauchy representation t of T. Thus we have at once
C o r o l l a r y 1. Let
( a S r - 1 ) with supp T=K. If / ( z ) = f ( z ) , then
fE{Ao}. Conversely, given an f£{A}, then it is the Cauchy representation of some
T£G'a for all a € [ - l , 0) with supp T=K.
Nevertheless we can prove the second part of this Corollary directly, that is,
without intervention of the meromorphy. In fact, since the distributions f* and f ~
belong to <J£ ( a < 0 ) we may define Tx=f+ —f~. As Tx is a linear continuous functional on @a generating the Cauchy integral t(z) we have for all a€[—1,0)
f(z) = (l/27rO</,M/(i-z)>-(l/27tO</f,
W-Z)).
Using the formulas (2) and (3) we get at once the required result
f M =
KJ
i / + ( z ) for
I f - ( z ) for
zZA+,
z£A~,
that is, / ( z ) = f ( z ) . So we have proved by Lemma that the given /€{^4} is the Cauchy
representation of some T£&'a.
Denote in Schwartz's notation by $'=$'(¥£) the space of distributions on R
with compact support (recall that £ ' c <S'a for all agR, but an T£<9'a with compact
support belongs to <?'). From Theorem we derive
with supp T=K. If f(z)=F(z),
then /<E{MC}. Conversely, given an /€{Af c }, then it is the extended Cauchy representation of some T^S'
with supp T=K,
We have to comment only the assertion (m.3). The function F around the point
z = 0 has the Laurent expansion of the form
t
P(z) =
C0+C1/Z+C2/Z2+...
which converges.uniformly and absolutely outside the smallest disk containing K and
allpoles ak (&=1, 2 , . . . , n). The fact that P{z) vanishes as z—°° impliesthat c 0 =0,
and the required result follows at once.
Consequently,: to every pair (T, {a1, a2, ..., a„}) with T££", nCN, there corresponds an f(L{Mc} and to every f€{Mc) there corresponds a pair (T, {a1,a2, ...,a„})
with
T£#',n£N.
D. Mitrovic
402
versely, given an f(z{Ac},
supp T=K.
with supp T=K. If f ( z ) = f(z), then f£{Ac}.
then it is the Cauchy representation of some T^S'
Conwith
Thus one can place distributions in &' into a one-to-one correspondence with
functions in {Ac}.
It may happen that f=F£{M}
(any given /£{Af} is the extended Cauchy
representation of some T£Q'a ( a s —1) with supp T=K). For example, the function
f defined by •
/ ( z ) = F(z) = (1/2TT0(VP 1//, l / ( I - z ) ) + l / ( z - I )
belongs to {M} with K=R.
This follows from
/+(z) = l/2z+l/(z-i),
/"(z)=-l/2z+l/(z-l),
z£A
+
— {i],
z€d-
,
with f+ = l/2(x+i0) + l/(x-i),
f - = l/2(x-i0) + l/(x-i).
Conversely, given
/ + ( z ) and / _ ( z ) we reconstruct / ( z ) starting with fx
—/T=VP
In addition, by means of the second part of Theorem we' sblve the following
boundary value
P r o b l e m 1. Let T be a given distribution in 0'a ( a s —1). Find a function
f€{M) whose 3l'-boundary values fx and f~ satisfy the condition fx—f~ — Tx on R.
The general solution is given by (8), where Bk p are arbitrary real or complex
coefficients.
It is of interest to sketch the following results: if in Corollaries 2—3 we replace
(via condition (m.2)) the convergence in the 3>' topology by one in 0'a for a given
a£[—1, 0), we get new corollaries 2.1—3.1 respectively.
F a c t 1. Corollaries 2—3 are equivalent to Corollaries 2.1—'3.1.
To prove this first observe that /£{M} remains in {M} if we substitute the convergence in Q>' for one in 0'a (a£ R). Next, we use the representation (8) or Lemma.
Also, if in Problem 1 we replace
( a s - 1 ) by T£G'X ( - l ^ a c O ) and the
convergence in 2/' by one in &a, we come to
P r o b l e m 1.1. Let T be a given distribution in 0'a (— 1 ^ a < 0 ) . Find a function
/€{M} whose O'a-boundary values fx and f~ satisfy the condition fx —fx = TX
on R.
F a c t 2."Problem 1 with T£<9'x ( - 1 ^ a < 0 ) is equivalent to Problem 1.1.
Similarly, substituting under previous Conditions the class1 {M} for {Mc} and
{Ac} we come to an equivalent Problem 1.2 and Problem 1.3, respectively.
403
References
Distributions, complex variables, and Fourier transforms, Addison-Wesley
(New York, 1965).
J. W. DETTMAN, Applied complex variables, Macmillan (New York, 1965).
A . MARTINEAU, Distributions et valeurs au bord des fonctions holomorphes, in: Proceedings
of an International Summer Institute held in Lisbon (1964).
D. MiTROVié, Tbe Plemelj distributional formulas, Bull. Amer. Math. Soc., 71 (1971), 562—563.
D. MrrRovié, A distributional representation of strip analytic functions, Internat. J. Math.
Math. Sci., 5 (1982), no. 1,1—9.
W . R U D I N , Lectures on the edge-of-the wedge theorem. Amer. Math. Soc. (Providence, R . I . ,
1971).
[1] H . J - BREMERMANN,
[2]
[3]
[4]
[5]
[6]
UNIVERSITY O F ZAGREB
FACULTY OF TECHNOLOGY
PIEROTTIJEVA 6
ZAGREB, YUGOSLAVIA
Acta Sci. Math., 50 (1986), 405—410
On Stieltjes transform of distributions behaving
as regularly varying functions
V. MARlC, M. SKENDZlC, A. TAKACI
1. Introduction. The asymptotics of the Stieltjes transform of distributions J" with
support in [0,
belonging to a subset, specified below, of the space of (Schwartz)
distributions, whose behavior (at zero) is of the type T~x* log-'x+ in the Lojasiewicz sense, or (at
in the Sebastiao E Silva sense, is studied by LAvoiNEandMisRA
[1], [2]. The aim of this paper is to extend their results to the cases when
T~xyL0(x)+,
v
JC—0+ and T~x L(x),
Here L0 and L belong to the class of slowly varying
functions (s.v.f.) introduced by KARAMATA [ 3 ] in 1 9 3 0 .
A real valued function L(x) is slowly varying at infinity, if it is positive, measurable on [a, for some
0, and such that for each
(1.1)
lim L(Xx)fL(x)
= 1.
A function JLoC*) is s - v - a t z e r o if Z, 0 (l/x) is s.v. at infinity. E.g. all positive functions
tending to positive constants are s.v. at infinity, products of powers of iterated logaX
rithms, the function (1/x) J dt/In t, are such, etc.
Slowly varying functions , are of frequent occurrence in various branches of analysis (Fourier analysis, number theory, differential equations, Tauberian theorems)
and of stochastic processes, whenever more information than the mere fact of convergence is needed. Here we also emphasize the use of s.v.f. in the theory of distributions.
Also, the function i?(x)=x v I,(x) is called regularly varying at infinity with
index v.
1.1. Following [2] we denote by J'(r), Re r > — 1, the space of distributions T
with support in [0,
admitting the decomposition
(1.2)
T =
Received April 28, 1984.
B+&f(x)
V. Marié, M. Skendzié, A.TakaCi:Stieltjestransformofdistributions
406
where A: is a non-negative integer, EP is the distributional differentiation operator of
order k,f(x) is a locally integrable function with support in [a,
for some a s 0
and such that
OO
(1.3)
/
|Wx-'-^dx^co,
and B is a distribution with support in [0, a]. Notice that T is a tempered distribution.
Further, the Stieltjes transform of T£J'(r) is defined by
F(s) = ys{T} = (Tx, ( x + s ) " ' " 1 )
(1.4)
= (lx,(x+s)-'-1)+r^+1> f
-
f{x)(x+s)-'-k-*dx,
s € C \ ( - c o , 0].
Here the first term on the right-hand side in the second line of (1.4) exists since
( x + j , ) _ r ~ 1 coincides on the support of B with some infinitely differentiable function.
Throughout the paper we assume that s > 0 .
The following result of
then
LAVOINE
and
MISRA
[2] is needed in the sequel:
L e m m a 1.1. Let B be a distribution with support in [a, b] where 0 < a < ¿ > <
S f M - (B, x - ' - 1 ) ,
(1.5)
if a=0 then
(1.6)
f+1Sfs{B}
s — 0+;
- (Bx, 1>,
s-oo.
#
1.2. We next give the basic properties of the slowly varying functions ([3], [4])
needed in the paper.
(i) The limit (1.1) holds uniformly in any finite interval [a, b], a >0.
(ii) For any p > 0 there holds
xpL(x)
x-p£(x)-0,
x-°°.
The next property is a Theorem of A U A N É I Ó , BOJANIÓ and TOMIÓ [ 5 ] , and represents
our main tool for the proofs.
(iii) Let g(x) be a Lebesgue integrable function on an interval /. Then
J g(x)L(Ax)dx~L(A) J g(x)dx,
i
i
A—
provided that one of the following conditions is satisfied:
A) / = [ 0 , b],
and the integrals
(1.7)
/
g(x)L(Xx)dx,
0+
converge; the latter for some /?>0.
/x-'|g(x)|dx
0+
Stieltjes transform of distributions
407
B) I=[a, oo), a > 0 , and the integral
(1.8)
/
x*|g(x)|dx
a
converges for some />>0.
C) /=[0, oo) and (1.7) and (1.8) both hold.
1.3. Among the various definitions of the behaviour (at zero and at infinity)
of generalized functions, we use the following two:
Definition
as x v Z, 0 (x) + , i.e.
1.1
(cf.
LOJASIEWICZ [ 6 ] ) .
T~xvL0(x)+,
The distribution
x-0+,
behaves at zero
Re v > — 1,
if there exist a > 0 and a distribution R with support in [0, a] such that T=xvL0(x)+
for x£[0, a] and
(1.9)
i^-1Lo1(t)+(R„<p(xJt))^0, t ^ 0+
for each function cp infinitely differentiable on some neighborhood of [0,
subscript " + " means that the functions bearing it are equal to zero for x s O .
Definition
infinity as xvL(x)
(cf.
i.e.
1.2
SEBASTIAO E SILVA [ 7 ] ) .
T~xvL(x),
if for some
The
The distribution T£3>'+ behaves at
x-oo
1 and x€[a, •»), there holds Tx=Dkf(x)
L-H*)*-'"^*)^-^,
(where (v + l)*=(v + l)(v+2)...(v+fc),
+R
and
* —
and (v + l ) 0 = l ) .
2. Results. We prove the following two theorems giving the behavior of the
Stieltjes transform^ of the distribution T£J'(r) at zero and at infinity respectively.
Notice that
= F(s) is a (holomorphic) function and the asymptotics has to be
understood accordingly.
T h e o r e m 2.1. Let R e r > - 1 , Re v > - 1 , R e ( r - v ) > 0 , further let L0(x) be
slowly varying at zero and T£J'(r). If
(2.1)
then
T~xvL0(x)+,
ys{T}~B(v+1,
x —0+
r-v)s'-rL0(s),
s - 0+.
408
V. Marié, M. Skendzié, A.TakaCi:Stieltjestransformofdistributions
T h e o r e m 2.2. Let R e r > - 1 , Re v > - l , R e ( r - v ) > 0 , further let L(x) be
slowly varying at °o and T£J'(r). If
T~xvL(x),
(2.2)
then
x -*oo
y,{T}~B(y+\ir-v)sv-,L(s),
s —oo.
By taking in Theorem 2.1 L0(x)= 1 and L0(x)=In1 x, j£N, one obtains respectively Theorems 3.2.1 in [1] and 2.1 in [2] for Re v > — 1 of Lavoine and Misra.
Similarly, by taking in Theorem 2.2, k=0, L(x)=1, L(x)—kiJx one obtains their
Theorem 5.1. m in [1] and Theorem 3.1 in [2].
3. Proofs. P r o o f of T h e o r e m 2.1. By virtue of (1.2) and (2.1) one has
(3.1)
T =
x"L0(x)++R+B+Dkf(xy,
here the supports of R, B, f (x) are in [0, a], [a, b], [b,
respectively, and R satisfies
(1.9); L0(x) may be chosen conveniently in [a, «>). We apply the Stieltjes transform
to the distributions at both sides of (3.1), multiply by (sy~rL0(s))~1, and then estimate each summand of the right-hand side.
To treat the first term, put in the occurring integral s= I/A, x=s/u, and to the
obtained integral
oo
A ' " / H r - v - 1 (l + ")- r - 1 L(Au)dM
o
with L(x) = L0(\]x)
apply the result 1.2 (iii) C). Thus for the first term one obtains
($v-'Z,0(s))-i/^^--*B(v+l,r-v),
s
0+.
To complete the proof one has to show that the remaining three terms tend to
zero with s. The second term is such due to (1.9) and to the property 1.2 (ii) (with
L0(s)=L(l/s)).
The third term is such due to (1.5) of Lemma 1.1, and to thé property 1.2 (ii) as above. The fourth term is
(s*-'L0(s))->Sf,{D*f(x)}
= (s'-'Us))-1
^
^
^
f
AxXx+sy-^dx.
The occurring integral is bounded by an absolute constant and (j v "" r Z 0 (i)) _1 —0,
as before, which completes the proof.
P r o o f of T h e o r e m 2.2. Because of (2.2) the function f ( x ) in (1.2) is of the
form
f(x) = c*+vL(x){ l + o ( x ) ) ,
Stieltjes transform of distributions
409
where co(x) is a locally integrable function with support in [a, » ) and such that
©(*)—0, je — oc, and c _ 1 = ( v + l)fc. Hence
(3.2)
T =
cÜt{^+vL(x)+)+Bí+Dk{^vL(x)ú3(x)}
where B1 is with support in [0, a] \ L(x) may be chosen conveniently in [0, a]. Now we
proceed as in the proof of Theorem 2.1, i.e. apply the Stieltjes transform to both sides
of (3.2), multiply by (i v - r Z,(j)) _ 1 and estimate each term on the right-hand side
separately.
We apply to the integral occurring in the first term the result 1.2 (iii) C), yielding
í^{i>tjck+vL(x)}~5(v+1,
r—v)s*~rL(s),
s -co.
Now we see that the second term tends to zero for
this time we have to use
1.2 (ii) and (1.6) of Lemma 1.1. The third term 73, say, is estimated as follows
|/ 3 | S (L(s))- 1 /
s^+kL(sy) dy+iLis^-'Mis)
0
f y^-^~1L(sy)dy
«life
where M ( j ) = s u p 1^(^)1 for yâs^ 1 1 2 . Hence | / 3 | ^ e ^ E . , for
since
¡©(¿jOI is bounded, M(j)—0, j—
and by applying to the occurring integrals (1.2),
(iii) A) and (1.2), (iii), B) respectively.
R e m a r k . By altering slightly the method of proof used above we can obtain
similar results when the distributions behave as some functions more general than the
regularly varying ones. Thus the following result holds:
T h e o r e m 3.1. Let Re r > — 1, T£J'(r),
CO
/?>0,
f
a
\h(x)xp+v\dx converges (a>0).
T~xyh(x),
then ^{Tj^ms-'-1,
and let h(x) be such that for some
If
JC-CO
where m is a constant that can be calculated.
References
[1]
J . LAVOINE
{2]
J.
[3] J .
et O . P . MISRA, Théorèmes abéliens pour la transformation de Stieltjes des distributions, C.R. Acad. Sci. Paris. Série A, 279 (1974), 99—102.
LAVOINE and O . P . MISRA, Abelian theorems for the distributional Stieltjes transformation,
Math. Proc. Cambridge, Philos. Soc., 8 6 ( 1 9 7 9 ) , 2 8 7 — 2 9 3 .
KARAMATA, Sur un mode de croissance régulière des fonctions, Mathematica (Cluj), 4 ( 1 9 3 0 ) ,
38—53.
410
[4]
[5]
[6]
[7]
V. Marié, M. Skendzié, A. TakaCi: Stieltjes transform of distributions
Regulary varying functions, Lecture Notes in Math- 508, Springer-Verlag (Berlin—Heidelberg—New York, 1976).
S . AuANCié, R . BOJANIC, M . ToMié, Sur la valeur asymptotique d'une classe des intégrales
définies, Pubi. de l'Inst. Math. Beograd, V N ( 1 9 5 4 ) , 8 1 — 9 4 .
S. LOJASIEWICZ, Sur la valeur d'une distribution en un point, Studia Math., 16 ( 1 9 5 7 ) , 1 — 3 6 .
J- SEBASTIAO E SILVA, Integrals and order of growth of distributions, Inst. Gulbenkian de Ciencia
Lisboa, 35 (1964), 7124.
E . SENETA,
INSTITUTE O F MATHEMATICS
UNIVERSITY O F NOVI SAD
NOVI SAD, YUGOSLAVIA
Acta Sei. Math., 50 (1986), 411—396
Normal approximation for sums of non-identically distributed
random variables in Hilbert spaces
V. V. ULYANOV
The problem of estimation of the speed of convergence in the central limit theorem in Hilbert spaces has a history of nearly twenty years. In most papers the speed is
studied on the class of balls with a fixed center. This is an important and natural class
of sets, but it is not very rich. However, in contrast to the finite dimensional case, one
has shown that in Hilbert spaces (in general) it is impossible to construct estimates
which are uniform with respect to rich classes, since not even on the class of all balls
the speed of convergence in the central limit theorem in Hilbert spaces is uniform
(see e.g. [1], p. 70).
The first estimates of order n~1/2 on balls with a fixed center under the assumption
of finiteness of some moments and without any other additional restrictions (such as
independence of coordinates) were obtained by GÖTZE [ 2 ] . Following Götze's paper a
series of papers appeared, mostly based on Götze approach, which improved and
extended his results (see e.g. [3], [4], [5]).
The estimates for the case of independent not necessarily identically distributed
random variables (i.non-i.d.r.v.) were obtained by BENTKUS [5] (see Theorem 3.3
in [5] or the condition (5) and the estimate (6) below). The previous results for the
case of i.non-i.d.r.v. were obtained by BERNOTAS, PAULAUSKAS [6], ULYANOV [ 7 ] , [8].
However, corollaries from results [6]—[8] for i.i.d.r.v. give estimates of the order n _1/G .
In the present paper we construct estimates on some classes of Borel sets, in
9
particularly on balls with a fixed center, for the case of i.non-i.d.r.v. Our results
improve the corresponding estimates of BENTKUS [5], have the "natural" form and
at the same time they are obtained under somewhat different conditions. One of the
main features of our estimates is that they require minimal moment conditions and
their dependence on the (truncated) moments has an explicit form. We shall use the
methods of [2], [3], [5] and some ideas from [7].
Received July 2, 1984 and in revised form, April 5, 1985.
412
V. V. Ulyanov
In what follows, H is a real separable Hilbert space with inner product (x, y),
x,y£H, and norm |x|=(x, x) 1/2 , D : H-»H is a bounded symmetric operator
\D\ = sup | ( D x , x)|/|x| 2 ,
xÔ
h: H-+R1 is a linear continuous functional,
W(x) = (Dx,x) + h(x),
Aa,r = {x£H: W(x+a)
< r},
N„ =
Let A: H-*H
r S 0,
a^H;
{\,2,...,n}.
be a bounded symmetric operator with eigenvalues
oo
=
. We
write A£G(P, k), JS>0, if and only if <rk^p. Denote tr A= 2 <*i- Let Z be a ranj=I
dom variable with values in H. Denote by % the symmetrization of X, i.e. %= X1—X2,
where Xx and X2 are independent copies of X. For <5 > 0 put
where IB is the indicator of the set BczH. By c (resp. c( •)) with or without indices,
we denote constants (resp. constants depending only on quantities in the parentheses); the same symbol may stand for different constants. Let Aj, i£N n , be any numbers and 0<zN„. Put /1(0)= 2 V
ae
Our main result is the following theorem.
T h e o r e m . Let Xt, i£N„, be i.non-i.d.r.v. with values in H with zero means and
covariance operators Aiy i£Nn, respectively. Assume that there exists an operator A0
such that
(1)
Ai = liA0,
¿¡SO,
i£Nn,
2 k = i
i=1
and
jttrAi=
1=1
1, (DA0feG(P,
13), for some
0 > 0.
n
Put Sn= 2
and let Z be a Gaussian (0, A0) r.v. with values in H,
¡=i
Then for all h S 1
(2)
A = sup \P{SniAa.r)-P{ZiAa>r)\
rEO
where At= ± E\Xki\\
¡=i
L3= % E\Xl\\
»=1
^
Cl(l
+ |a|3)(/l2+L3),
c ^ c M , №1, P)-
'
Random variables in Hilbert spaces
413
C o r o l l a r y 1. Assume that A0€G(fi, 13)for some /?>0. Then for all n=1
(3)
Ax = sup |P(|5„+a| < r)-P(\Z+a\
< r)| S c(/?)(l+|a|»)(/l 2 +L 8 ).
r SO
C o r o l l a r y 2. Assume that A1=A2=...=A„,
E\Xi\asL,i^Na.
Then for all «si
(4)
A ^ C W D
+
A^G0?,
W^LN-
1
13), for some /J>0,
'*.
R e m a r k s . 1. Our theorem improves estimate (3.14) of Theorem 3.3 in [5];
For completeness we recall the corresponding result proved by BENTKUS [5] (Theorem 3.3).
Let Xt, i€N„, be i.non-i. d.r.v. with values in H with zero means and covarictnce
n
operators At,i£N„, respectively. Assume that ^tTAi=
1 and there exists a notmegative operator A0: H-+H such that (5)
A{ S A0ln,
i£Nn,
and (DA0)2€G(fi, k), for some
&>0. Let q,e,B be any numbers,
2, 0-c
< e S l , i ? > 0 . Then there exists a constant c=c(e) such that if feSc then for all
»Si
(6)
A ^ c,(l +1a s |)(^2+^s+(«/cr|) E - 1 + (n max £|Z ( 1 | 2 ) B ),
where c2=c2(fi, e,q,\D\, \h\, B), o= max
(nÊ\X}\^-%\
Thus our theorem shows that the last two terms on the right hand side of (6)
may be omitted. At the same time we replace condition (5) by condition (1).
2 . Estimate ( 4 ) was obtained earlier by YURINSKI! [ 3 ] .
The proof of the theorem is based on a series of lemmas.
L e m m a 1. Let A(, i£N„, be nonnegative numbers such that
max A, s 1/3.
Then there exist sets
/=1,2.
@l5 0 2 such that
0 i n© 2 =0, 0 1 U0 2 =A' n , A ( 0 j ) S l / 3 ,
<0
<0+1
P r o o f . It is easy to see that there exists an/„such that ^ A , s l / 3 , ^
Put 0 ! = { 1 , 2 , . . . , / 0 + l } . The above construction implies Lemma 1.,
L e m m a 2. Let A(,/£JV„, be nonnegative numbers such that
¿ A , = 1,
it
max A, m . R s 1/3.
A, >1/3.
414
V. V. Ulyanov
Let Mt, i€N„, be any nonnegative numbers. Then there exists. 0<zNn
R^X(0)s3R,
such
that
M
M(0)/A(0)g f
i1=1
P r o o f . As in Lemma 1 it is easy to see that there exist 0 l 5 0 2 , . . . , &„, such that
0 , n 0 j = 0 , i V j , IJ 0,=N„
and R^X(0t)s3R,
/ = 1 , 2 , ...,m.
Assume that the
ratios M(0,)/A(0,) are arranged in the following way
M(eh)
^ M(0j „
A(0(l) - A(0it)
^
M ( 0 j
X(0im) •
This implies
M(0il)/X(0ll)^
2
/=i
Put 0 = 0 , . Lemma 2 is proved.
L e m m a 3. Let X be a r.v. with values in Hand EX—0, E\X\i<
z£H, 5>0
<*>. Then for any
zf S 2 E ( X , z) 2 —4|z| 2 (£|A r 1 | 2 +2.E|.Jf 1 | 8 /i),
(7)
where
P r o o f . We have
(8)
E({2y,zf
(9)
As in (8) we get
(10)
Since EX=0,
= E(X\ z ) 2 - £ ( ( ^ ) 3 , z)2 S E(X\
E(X\ z)2 = 2 E { X \ zf-2{E(X\
z))2.
z)2 S £ ( * , z ) 2 - | z | 2 £ | * i l 2 .
(11)
Moreover
(12)
zY-\z\*E\(X\\\
E(X\z)
=
-E(Xi,z).
¡E(Xi,z)\ ^ IzKÎÎ 2 ) 1 / 2 .
From (9)—(12) it follows that
(13)
Furthermore
(14)
£(^1,z)2s2£(Z,z)2-4|z|2£|A'1|2.
£ KJ?1),!2 S E |£T/<5 ^ 8 £ |.Y l | 3 /5.
Using (8), (13) and (14) we get (7). Lemma 3 is proved.
L e m m a 4. Lei A, B be any bounded linear operators in H, AÔ. Then the sets
of non-zero eigenvalues of the operators AB, BA and A1,2BA112 are the same (taking
into account the multiplicity of the eigenvalues).
R a n d o m variables in Hilbert spaces
P r o o f . See
VAKHANIA [9],
p.
84,
415
or Lemma 2.3 in [5].
L e m m a 5. Let X,(Yt), i£N„, be i.non-i.d.r.v. (independent Gaussian r.v.) with
values in H with zero means and covariance operators A„ i£N„, respectively. Let R be
any positive number, Rs 1/6. Assume that
A, — XtA0,
(15)
2 trJi,= 2
f=J
i=l
AJ £ 0,
= l> (DA0)KG(fi, 13) for some
Let 6 > i n 0 2 = 0 J 01\J02=N„.
V, =($}Y
/? > 0,
jR»/(200|£)|2).
max X, =SR, A2
i simn
(16)
i£N„,
Put A= 2 cov Vt, B= 2 cov Fj, where
itOx
i£0s
3 = 400\D\ 2 (A 2 +L 3 )lf}.
or Vt —
Then there exists 0^ such that
(DAD)llzB(DAD)ll2£G(Rp/2,
(17)
P r o o f . Let x£H.
13), tiA S 12R.
z=(DAD)1/2x,
Put
A2(Q)=
2E
¡(.0
Note that
\Xa\\
E(?i,x)2
(18)
1^(0) = 2
E\Xl\\
i£0
=
2E(Yi,x)2.
From (18) and Lemma 3 it follows that
(19)
(Bz,
z ) S 2(A„z,
Furthermore
(20)
z)X(02 )
\Z\2(A2 (02)+2L3 (02 )/S).
- 4
\z\*^4\D\2
\x\2A(0J.
By Lemma 4 the sets of the non-zero eigenvalues of the operators (DAD)ViA^DAD)1,%
and All2DADAlla are the same. Put y=DA]j2x. As in (19) we have
(21)
Moreover
(22)
(Ay,y) S 2(A0y,y)A(0J-4
\y\2(A2(0d+2L3(01)IS).
\y\2^4\D\2\x\2X(0J^4\D\2\x\2.
Since under the assumptions of Lemma 5 (DA0)2£G(/?, 13), by Lemma 4 we have
(23)
All2DAoDAl'aeG(0,
By Lemma 2 there exists a
such that
(24)
n*
13).
R S A ( 0 J ) ^ 3 R , LS (01 )/X(0L )
Z*.
V. V. Ulyanov
416
Now from (15), (21), (22) and (24) we have for any x£H
2
2
2
2
•V:
2
(All DADAl'*x, x) S 2(Ay DA0DAy x, x)R-l6\D\ \x\ {A&+.2L3(01)IS)
(25).
.
S 2(Ay2DA0DAl'2x,
x)R-pR\x\s.
3*
From (23), (25) and the results of §4, Ch. Zin [10] it follows that
Ay*DADAy2<iG(Rp, 13).
(26)
Similarly from (15), (16), (19), (20), (24), (26) and from the simple inequality
(27)
tr A s= 4-1(0!)
we get (17). Lemma 5 is proved.
P r o o f of T h e o r e m . Put
+
F(r)^P(S n Â a w r ), F ^ r ) ^
=P(3*eAar),
b(r)=P(Z€Aar).
Let f ( t ) and g(t) be the Fourier—Stieltjes transforms of F x and b respectively. Using the inequality
|fr'(r)|=§ /
g(t)dt,
— oo
the symmetrization inequality (see Lemma 2.1 in [3]) and Lemhia 2.4 in [5] we get
(28)
It is easy to see that
(29)
\b'(r)\^c(\D\,m
+ \a?).
I F O O - F . w i s 2 n \ x > \ s l) -s A 2 . ••••••••
i=l
By Theorem 2, §1, Ch. 5 in [11], from (28), (29) we have for any T > 0
(30)
]P(SnZAa,r)-P(Z£Alhr)\Â2+c(l
;
+ \a\*)IT+ /
U n / i O - g i O l dt.
' '
5
-T
Now we estimate \f(t)—g(t)\.
(31)
where
and
By Theorem 4.6 in [12] we havd
1 / ( 0 - «(01 ^ c(|/| + 1/1^(1 + \a\*)(A2+Ls)x(t),
;
any subset of Nn containing not more than one element*
F„), V = X F or VI=?I,
i£N„, Y Y ,.... Y are independent
Gaussian r.v.'s with zero means and covariance operators A , A ,..., A„, respectively.
V = { V
is
. .;
M
X
,
V2 ,...,
T
1 ?
2
N
T
2
167
Denote 4=A+-Z-3- The left hand side of (2) is not greater than 1. Hence we
may assume without loss of generality that
(32)
Amc3,
where c 3 is a constant small enough. In fact, if (32) is not true then (2) is obvious with
c t ^ l / c 3 . We may also assume that
max A,- S A 2 ' 2S .
ISiSn
(33)
In fact, if there exists an z0 such that Af >/1 2/25 then tr A, >/l 2 / 2 5 . Since tr A, =
we have E ^ f ^ A 2 ' ™ . Moreover,
E\Xk\2 =
E\Xiai\2+E\X>\\
Now we consider two possible cases.
Case 1. E\X,Ê\Xfj2.
Then E\X^>A^jl.
Since A ^ E t f ^
2 №
25/23
have A>A l /2, that is /l>(l/2)
. This contradicts assumption (32).
Case 2. E\Xff>E\XioL\\
(£|A? 0 | 2 ) 3/2
Then E\Xlf=-A2'25/2.
2s'2
we
We have
(E|A?0|2)3/2 ~
'
75/44
Hence /1>(l/2)
. This also contradicts assumption (32).
Furthermore, from the definition of x(t), (33) and Lemma 1 it follows that if we
denote
(34)
S{ = inf Eexp { 2 * 7 { 2 DV}, £ Vj)}, 81 > 0,
where @ i n ® 2 = 0
(35)
and
2
then
(36)
Note that for any symmetric r.v. X,z£H,
8>0 we have
£ e x p {i(z, X)} ^ £ e x p {i(z, Xs)}.
Therefore
(37)
<5fS inf £ e x p {2 ii ( Z DV'}, 2 Vj)}>
where V;=Vf if Vj=X} and V'j=Vj if V j = f j t
[5] we get for all t and M that
U 0 2 . By Lemma 2.5 in
4|D|«5 e |i| s 1, 4 \D\ MS max {y, e} ^ 1,
(38)
di^c
inf (exp ( - 0 2 /(tr ¿ + 5 g ) ) +exp ( - y2/tr B) + respectively. From (1.4) in [3] it follows that
<p(s) = ]J (1 +s 2 /^/4) _ 1 / 2 ,
j=i
(39)
where Pj,j= 1, 2, ... are the eigenvalues of
(DAD)1/2B(DAD)112.
Now we estimate the right hand side of (38) for different values of t.
Case I. U|sSc/(/lln(l//l)). Put y=c ln x / 2 (l//l),
= c/(A In (1//1)). From (1), (27) we get
(40)
tx
ö=cln(l//4),
ő=cA,
M=
A+trBÂ.
Hence for the above values of y, q, S, M we have
exp (— g 2 /(tr A+ŐQ))+exp
(41)
(— y 2 /tr B) S. cA°.
By Lemma 5 and (32), (33), (35) there exist 6>x and a constant c such that
(42)
(DADf/2B(DADyi\G(cP,
From (39), (42) we have
(43)
( K s ^ c O + i 1 8 )- 1 .
13).
Case2. c/(A In (\IA))^\t\^clA.
Put y = c l n ^ O / z i ) , Q = C , M = C / ( A In
S=cA. By Lemma 5 and (32), (33), (35) there exist © i and a constant c such that
{DAD)1'2 B {DAD)1'2 6 G (cA2'25 p, 13), trA^
(44)
(I/A)),
cA2'25.
From (39), (40), (44) we have
(45)
Si
ciA'+il+A^KA
Furthermore, put T=c/A,
(45) we get
(46)
f
In (1/4)))- 13 )
T^cftA
W-^M-gWdt^cil
c(Ac+A312'25
In13 (I I A)).
In (1/A)). From (31), (34), (36), (37), (41), (43),
+W A
f
-T
(1 +
t2)x(t)dtm
-T
^ c(l + |a|3)/l /
(\ + t2)5ídt^c(l
— T
+ \a\s)A{
f +
— TX
f + /^(1
Tt
OO
^ c(l+|a|3)/l(l+ /
^ d r s
—T
*
fiil+t^l^dt+A^lnîl/A^^cil
Estimates" (30) and (46) imply (2). The theorem is proved.
+ laÂ.
419
P r o o f of C o r o l l a r y 1. This follows from the theorem. To this end it is enough
to put h=0, D=T, the identity operator, and to note that if A0£G (/5,13), then
A*£G(P, 13).
The proof of Corollary 2 is obvious.
I am grateful to professor J. Mogyoródi for his attention to my work.
References
Normal approximation — some recent advances, Lecture Notes in Math.,
Springer-Verlag (Berlin, 1 9 8 1 ) .
[2] F . GOTZE, Asymptotic expansions for bivariate von Mises functionals, Z. Wahrsch. Verw.
Gebiete, 50 (1979), 333—355.
[3] V. V. YuRiNSKil, On the accuracy of Gaussian approximation for the probability of hitting a
ball, Teor. Verojatnost. i Primenen., 27 (1982), 270—278.
[4] B . A . ZALESSKII, Estimates of the accuracy of normal approximation in a Hilbert space, Teor.
Verojatnost. i Primenen., 27 (1982), 279—285.
[5] V . BENTKUS, Asymptotic expansions for distributions of sums of independent random elements in a Hilbert space, Litovsk. Mat. Sb., 24 (1984), No. 4, 29—48.
[6] V . BERNOTAS, V . PAULAUSKAS, Non-uniform estimate in the central limit theorem in some
Banach spaces, Litovsk. Mat. Sb., 19 (1979), No. 2, 23—43.
[7] V. V. ULYANOV, On the rate of convergence in the central limit theorem in a real separable
Hilbert space, Mat. Zametki, 29 (1981), No. 1,145—153.
[8] V . V . ULYANOV, On the accuracy of closeness estimates for two distributions of sums of
random variables with values in some Banach spaces, Vestnik Moskov. Univ. Ser. 15, 1981,
No. 1, 50—57.
[9] N . N . VAKHANIA, Probability distributions in linear spaces, Mezniereba (Tbilisi, 1 9 7 1 ) .
[10] N. DUNFORD, J . T . SCHWARTZ, Linear operators. II, Wiley (New York, 1 9 6 2 ) .
[11] V. V. PETROV, Sums of independent random variables, Springer-Verlag (Berlin, 1 9 7 5 ) .
[12] V . BENTKUS, Asymptotic expansions in the central limit theorem in a Hilbert space, Litovsk.
Mat. Sb., 24 (1984), No. 3, 29—50.
[1] V . V . SAZONOV,
vol.
879,
MOSCOW STATE UNIVERSITY
FACULTY O F APPLIED MATHEMATICS A N D CYBERNETICS
117234 MOSCOW V—234, USSR
Acta Sci. Math., 50 (1986), 421—410
A connection between the nnitary dilation and the normal extension
of a subnormal contraction
C. R. PUTNAM
1. Introduction and theorem. Let H be an infinite dimensional, separable, complex Hilbert space and let T be a bounded (linear) operator on H. If T is a contraction (|| I'll S L ) on H, then, by a well-known result of B. S Z . - N A G Y [7] there exists aHilbert space Kzj H and a unitary operator U on K for which
(1.1)
Tn = PU"\H
and
T*" = PU*"\H,
n=0,1,2,...,
where P is the orthogonal projection P:K-*H. If K is the least subspace of K which
reduces U and contains H (as will be supposed) then U is the (unique) minimal unitary dilation of T. See HALMOS [3], S Z . - N A G Y and FOIA§ [8].
Next, let T be any subnormal operator, so that there exists a Hilbert space
K'^H and a normal operator N on K' for which NHczH and T=N\H. Thus, N
is a normal extension of T and, if P' denotes the orthogonal projection P': K'-*-H,
(1.2)
T" = Nn\H and T*n = P'N*a\H,
n = 0, 1, 2, ....
In case Kf is the least subspace of K' which reduces N and contains H (as will be
supposed) then N is the (unique) minimal normal extension of T. (See HALMOS [3]
and, for an extensive treatment of subnormal operators, C O N W A Y [2].) The operator T
is said to be a pure subnormal operator if it has no normal part.
Henceforth, it will be supposed that T is both a contraction and a pure subnormal operator. Let the associated operators U and N defined above have the corresponding spectral resolutions
(1.3)
U = f zdGz
C
on K and
N =
f zdEz
on K',
D-
This work was supported by a National Science Foundation research grant.
Received July 4,1984.
422
C. R. Putnam
where D is the unit disk D= {z: |z|< 1} with closure D~ and C = { z : |z| = l}. The
main object of this note is to point out the following explicit connection between the
operators U and N:
T h e o r e m . Let The a pure subnormal contraction on H with the minimal unitary
dilation UonK and the minimal normal extension N on K' of .3). Then, for any Borel
set p on C andfor any vector x in H,
<1.4)
WP)x\\*+
fhp(z)djEzxr
=
\\G(fi)xr,
D
where
(1.5)
hp(z) = (l/2n) fRe(Pz(t))dt,
Po
with
(e*+z)!(eu-z)
Pz(t) =
ea6j?}.
•and 0o={t:OiSt^2n,
The function Re (Pz(.t)) is the Poisson kernel and, as is well known, is positive
for all t£C, z£D, while hfi(z) is harmonic in D.
The formula (1.4) is contained "between the lines" in [6], pp. 333—334, but
apparently has not appeared explicitly in the literature. For completeness, the argument
-will be given below.
As above, let P and P' denote the orthogonal projections P: K-+H and P' : K'-*
—H, so that, by (1.1) and (1.2), for x in H and « = 0 , 1, 2, ..., one has Tnx=
=PU"x=Nnx
and Tnx=PU'"x=P'N*nx.
Hence, if / = / ( z ) is analytic in D and
•continuous in D~ then
f fdEzx
= P / fdGzx
D-
and
P' f fdEzx
C
•Consequently, if h=h(z)
D~, then
= P f
D~
JdGzx.
C
is any real harmonic function in D which is continuous on
P' fhdEzx
= P JhdGzx,
D-
x£H.
C
Next, let P be any closed subset of C and let q> be a real-valued continuous function on C satisfying <p=1 on/? and O^qxl
on C— p. Then there exists a function
•h(z), given by the Poisson integral, which is harmonic in D, continuous in D~, and
satisfies h=cp on C. Consequently,
P'(fhdEz+
D
fq>dEz)x
= P fq>dGzx,
C
xiH.
C
On forming inner products with x one obtains
>(1.6)
f<pd\\Ezxl*+
fhd\\Ezxr=
f<pdiGzx\\*.
Unitary dilation a n d normal extension
423
On replacing (p by <p„=<pn and h by the corresponding hn, n=1,2,...,
one sees that
( p ^ c p ^ . . . and the sequence {<?„} converges to the characteristic function of /?. Similarly, O s h „ s l , h ^ h ^ . . . , and {A„} converges to h f (z) of (1.5). Clearly, one has
the relation ( 1 . 4 ) , when /? is closed, and the extension of ( 1 . 4 ) to arbitrary Borel sets /?
readily follows.
It is known that ||G z x|| 2 is, for each x in H, xÔ, equivalent to arc length
Lebesgue measure on C ; see SZ.-NAGY and FOIA§ [8], p. 8 4 . The absolute continuity
of Ez on C follows from ( 1 . 4 ) . (That is, E{fi)—0 whenever /? is a Borel set on C having
arc length measure zero.) Other proofs of this last result are given in CONWAY and
OLIN [2], p. 3 5 , OLIN [4] and PUTNAM [5] (see also [6]).
2. U as the sweep of N. It may be noted that (1.4) of the Theorem or, equivalently, ( 1 . 6 ) , in which cp is now any continuous function on C and h=h(z)=0(z)
is
its harmonic extension to D~ (via the Poisson integral), can be interpreted in terms of
the sweep of a measure. (For the concept of "sweep" see CONWAY [2], p. 3 3 4 . ) Thus,
one has
f Qdfi — J <p dfl,
Dc
where dn=dlEzx\\*
and d(l=d\\Gzx\\2,
so that (I on C is the sweep of p on D~.
3. Two corollaries.
C o r o l l a r y 1. Under the hypotheses of the Theorem, suppose that, in addition,
(3.1)
Then
(3.2)
X = E(C)x
for some
x£H,
Xr*0.
Ez on C is equivalent to arc length measure on C,
that is, E(fi)=0 for a Borel set /? of C if and only if f} has arc length measure zero.
P r o o f . In view of the remarks at the end of section 1, it is sufficient to show that
P has arc length measure zero whenever E(fi)=0. It follows from (3.1) and (1.4) that
£ ( | z | < l ) x = 0 , so that ||.E t x|| i! =||G z x|| 2 , z on C. However, as noted above, ||0.x|| 2 is
equivalent to arc length measure on C and the proof is complete.
For use below, let a t ={e , s : O s j s / } ,
G ^ G i c O . I f A=[a,b]c[0,2n],
let AE=E„-Ea
C o r o l l a r y 2. Suppose that for some x£H,
(3.3)
where A is any subinterval of
holds.
(3.2),
and
and put Et=E(at)
AG=Gb-Ga.
and
xÔ,
2
inf{MGx|| /M|} = 0,
a
[0,
In] and
\A\>0
is its length. Then
(3.1),
hence also
C. R. Putnam
424
P r o o f . By (1.4) and (1.5),
(3.4)
№ D 7 M I + (1/2k) / ( M \ A \ ) ( ¡Re{Pz{t))dt)d\Ezx\*
D
=
\\AGxfl\A\.
A
By (3.3), there exists a sequence of intervals A„, |/4„|>0, « = 1 , 2 , . . . , where An=
=[a„, b„\ and a„,bn-*c for some c in [0, 2n]. It follows from (3.4) and Fatou's lemma that
J R e ( P z ( c f j d \ E z x Y = 0.
D
Consequently, E(D)x=0,
that is, (3.1), and hence (3.2).
4. Remarks. It follows from Corollary 2 that if Ez on C is not equivalent to arc
length measure on C (so that there exists a Borel set j? on C of positive arc length
measure for which EQ?)=0), then, for every x in H, xÔ, there exists a constant
kx for which ÂGx\\2l\A\^kx^0
for all subintervals A of [0, 2n]. This result can be
regarded as a refinement of the relation
(4.1)
log(.d\\Gtx\\*ldt)£L(0,2n),
which is valid whether or not E, on C is equivalent to arc length measure on C. In
fact, (4.1) holds for arbitrary completely nonunitary (not necessarily subnormal)
contractions T; see S Z . - N A G Y and FOIA§ [8], p. 8 4 .
Since for each x£H, both ||is(x||2 and ||ir t x|| 2 are absolutely continuous on
[0,2n], it is seen from (3.4) with A=[t, t+At] that
(4.2)
d\EtxVldt
+ (1/2TI) J^{Pz(t))d\\E:xr
=
d\\Gtx\m
D
holds a.e. on
In fact, the relation corresponding to (4.2) but with the
equality replaced by " S " follows from Fatou's lemma. That, in fact, equality holds
(a.e.) follows from an integration and the fact that
x=E(D~)x=G(C)x.
We are indebted to K. F. Clancey and to J. B. Conway for very helpful discussions, especially in connection with the "sweep" interpretation of (1.4) as given in
section 2.
References
[1] J. B. CONWAY, Subnormal operators, Research Notes in Mathematics, 51 Pitman Advanced
Publishing Program, 1981.
[2] J. B . CONWAY and R . F . OUN, A functional calculus for subnormal operators. I I , Mem. Amer.
Math. Soc., 10, No. 184 (1977).
Unitary dilation and normal extension
A Hilbert space problem book, Second edition, Graduate Texts in Mathematics,
vol. 19, Springer-Verlag (Berlin—Heidelberg—New York, 1982).
R. F. OUN, Functional relationships between a subnormal operator and its minimal normal extension, Thesis, Indiana Univ., 1975.
C. R. PUTNAM, Peak sets and subnormal operators, Illinois J. Math., 2 1 (1977), 388—394.
C . R . PUTNAM, Absolute, continuity and hyponormal operators, Internat. J. Math, and Math.
Sei., 4 (1981), 321—335.
B . SZ.-NAGY, Sur les contractions de l'espace de Hilbert, Acta Sei. Math., 1 5 (1953), 87—92.
B . SZ.-NAGY and C. FOIA§, Harmonie analysis of operators on Hilbert space, North-Holland
Publishing Co., American Elsevier Pub. Co., Akadémiai Kiadó (1970).
[3] P . R . HALMOS,
[4]
[5]
[6]
(71
[8]
425
P U R D U E UNIVERSITY
WEST LAFAYETTE, I N 47907, U.S.A.
Acta Sci. Math., 50 (1986), 427—410
C*-nonns defined by positive linear forms
JÁNOS KRISTÓF
It is well known that the norm of a C*-algebra A is uniquely defined by a set of
its positive linear forms, namely
M = sup/7(* 5 *)
(1)
/€S
for all x£A, where S is the set of positive linear forms f on A such that | | / | | ^ 1
(cf. [1]). It is worth mentioning that the set S in (1) can be defined by an entirely algebraic way; the linear f o r m / o n A is in S if and only if / ( A ) í R + and | / ( x ) | 2 S
Sf(x*x) for all x€A. This means that in the right side of (1) only such entities enter
which are related to the algebraic structure of A.
The latter observation leads to the result that defining a C*-norm on a #-algebra
is equivalent to indicating a certain set of positive linear forms on it. So the question
arises naturally; how to choose a set S of positive linear forms on a given *-algebra
so as to obtain a C*-norm on it by the definition (1)?
The aim of this paper is to examine this problem. Due to the elementary character of our further investigations we shall constantly refer to the basic works [1]
and [2].
I. C*-seminorms defined by positive linear forms
In the present study the vector space of the linear forms on the »-algebra A is
denoted by A* and co (P) stands for the a (A*, v4)-cIosed convex hull of the subset
P of A*.
If A is a »-algebra with unity and P is a set of positive linear forms on A then the
set {/<E/>|/(l)Sl} is denoted by P(l), where 1 is the unit element of A. Further,
assuming that P ( l ) is a (A*, ^-bounded, || • ||P denotes the mapping from A into R +
Received August 23,1984.
-428
J. Kristóf
-defined by the equation
<!)'
M l r : = sup
/e«i)
ffi?*)
for all x£A. It is obvious that || • ||P is a seminorm on A. The dual seminoma of
|| • Hp is denoted by || • ||p.
Given a »-algebra A and a linear form / on it, for all x£A we define the linear
forms x.f and f.x on A as the mappings y>-*f{xy) and y*-+f (yx), respectively. Clearly, x.(f.y)=(x.f).y
thus one may use the simple notation x.f.y instead of
x.(f.y) or (x.f).y.
Our first result concerns a sufficient condition on a set of positive linear forms
providing C*-seminorms by the aid of (1)'.
T h e o r e m 1 .Let Abe a *-algebra with unity and P a nonvoid set of positive linear
forms on A satisfying
(I) P(l) is o(A*, Aybounded.
(II) R + P<=P and x*.P.xczco(P) for all x^ A.
Then || • ||p is a C*-seminorm on A and every f£co (P) is || • \\P-continuous and
| | / | | p = / ( l ) . Moreover, i f P separates the points of A then || • ||P is a norm on A and
the involution of A is proper.
P r o o f . Since co (P) is nonvoid and a (A*, ^-bounded, we may form the seminorm || • ||Co(p) besides || • || P . We claim that these seminomas are equal. Of course,
|| • ||p51| • II^CP)order to prove the reverse inequality it suffices to show that
_/(x*x)S||x|| 2 for all x<=A and / e ( c o ( P ) ) ( l ) , / # 0 . If /E(co (P))(l) then there is
generalized sequence (fdiii in co (P) such that
/ pointwise on A. In particular,
/ ( 1 ) - / ( 1 ) and / ( 1 ) > 0 , thus we may suppose that f,( 1 ) > 0 for all /£/. Then
take f{:=(f(l)/fi(i))fi
(/€/). Clearly, //<E(co(P))(l) with regard to the first part
of (II) and it is easy to see that co (P(l))=(co (P))(l). On the other hand, f,'-*f
pointwise on A. From this we infer that given an element x of A we have (X*JC) —
-*f(x*x) and / / ( X * X ) ^ | | J C | | 2 , since / ' 6 c o ( P ( l ) ) , thus /(x*x)S||x|| 2 . Now fix an
• element/in (co(P))(l). Then
\f(xW ^f(i)f(x*x)
& Ml CP) = 1*111
for all x£A, i . e . / i s || • || P -continuous and | | / | | p S l . The first part of (II) now
yields that R + c o ( P ) c c o ( P ) , hence/is || • || P -continuous and | | / | | p = / ( l ) for all
_/€co (P).
We show next that
(2)
f(y*x*xy)
\\x\\2Pf(y*y)
for all / € c o (P) and x,y£A. If f(y*y)=0
then choose an arbitrary positive real
number e to obtain
f(y*x*xy)/e =
((ylfi)*.f(ylfc))(x*x).
C*-norms defined by positive linear forms
429
Since by (II) the linear form standing on the right hand side belongs to (co (P))(l),
we now have f(y*x*xy)ê\\x\\%
for all e>0, i.e. the equality
f(y*x*xy)=Q=
= \\x\\%f(y*y) holds. If f(y*y) >0, then
f(y* x* xy)ff(y*y)
=
((y/yfU^)*.f.(yfyJU^y)))(x*x)
and the linear form on the right hand side belongs to (co (P))(l), hence we obtain
the desired inequality.
From the inequality (2) it follows that ||x.y||pS||x||p||j>||p for all x,y£A. The
only thing that remained to be proved is the inequality ||x||pS||x*x||p for all x£A.
If x€A and f£P(l) t h e n S ||/||;i|x*x||p=/(l)||^||pS[|x*xIlp thus || • ||, is
a C*-seminorm on A. The last assertion of the theorem is an immediate consequence
of our previous considerations.
II. C*-norms defined by positive linear forms
The assumptions (I) and (II) introduced in Theorem 1 are not sufficient for P to
ensure that || • ||P be a C*-norm on A. Our next aim is to impose further conditions
on P in order to have possibility of proving the completeness of A with respect to the
uniform structure defined by || • || P .
' We need two auxiliary lemmas. Before formulate them we agree that given a
»-algebra A with unity and a nonvoid set P of positive linear forms on A satisfying
(I), the linear subspace and the || • Hp-closed linear subspace of A* spanned by P will
be denoted by sp (P) and sp (P), respectively.
Lemma 1. Let Abe a *-algebra with unity and P a separating set ofpositive linear
forms on A satisfying (I). Then the a (A, sp (P)) and g(A, sp (P)) topologies coincide
in each || • \\P-bounded subset of A.
Proof. Let (x,) ier be a generalized sequence in A such that xt—x in the
o(A, sp (P))-topology and suppose that we have M:=sup ||x;|| P <
We claim
—
—
¡e/
that Xi—X in the cr(A, sp (P))-topology. If f{sp (P), / £ s p (P) and /'€/ then
\F(x,)-m\
^ \f(xd-f(xd\
+ l/W-f(x)\
+
\f(x)-J(x)\
s II/—/||p(M+ ||x||p)+1fixd-/(x)|.
Since / € i p (P) and /(x,)—/(x) for all / € s p (P), we easily deduce that /(x{)-<-fix).
It is obvious that a (A, sp (P)) is a stronger topology on A than <r(A, sp (P)),
however, Lemma 1 indicates a certain strict intrinsic relation between them.
12
430
J. Kristóf
L e m m a 2. Let A and P be as in Lemma 1 and suppose
(III) x.P<=ip (P) for all x£A.
Then the multiplication in the algebra A is || • ||P-boundedly left ctnd right continuous in the a (A, sp (P))-topology.
P r o o f . Consider a || • || F -bounded generalized sequence (x()iei in A such that
Xi—X in the a (A, sp (P))-topology. Then combine Lemma 1 and (HI) to obtain that
f(yxt) = {y.f)(xd
- 0'•/)(*)
= f(yx)
for all / € P and y£A, i.e. yx^yx
in the a (A, sp (P))-topology.
Due to the obvious equality fy=(y*f)*
and the trivial fact that the set sp (P)
is closed with respect to the natural adjunction of A*, we deduce that xty-*xy in the
same topology.
We are now ready to formulate our main result on C*-norms defined by positive
linear forms.
T h e o r e m 2. Let Abe a *-algebra with unity and P a separating set of positive
linear forms on A satisfying (I), (II), (III) and
(IV) A is sequentally complete with respect to the uniform structure defined by
the <t(A, sp (P))-topology.
Then A is a C*-algebra whose (unique) C*-norm equals || • ||P.
P r o o f . With regard to Theorem 1, A, equipped with || • ||p is a pre -C*-algebra.
Let A denote the envelopping C*-algebra of this pre-C*-algebra. For every || • Hpcontinuous linear form f on A l e t / denote its unique norm continuous extension to
A. We shall prove that A=A.
First we show that there exists a unique mapping n from A onto A with the property fon=f for all / € P . The uniqueness of n is an immediate consequence of the
assumption that P separates the points of A. To prove the existence of it let us consider a fixed element x of A. There is a sequence (x„)ngN in A such that )|jc„—x||P-»-0,
where the C*-norm of A is also denoted by || • || P . Then ( x n ) „ e N is a. j] • || P -Cauchy
sequence in A, hence a Cauchy sequence in the weaker uniform structure defined by
the a(A, sp (P))-topology. Now, condition (IV) implies the existence of an element x
in A with the property that x„ — x in the a (A, sp (P))-topology. We define it(x) to be
equal to x. Then, for all / € P , by virtue of Theorem 1 we have
/(*(*)) = f(x) = lim
n /(*„) = lim
n / C O =/(*),
thus the existence of n is verified.
We claim that fon=f holds for all / £ s p ( P ) . Indeed, if / € s p P and (/ n )„ e N
is a sequence in s p ( P ) such that \\ f„—f\\p-*Q then we also have ||/„—/||p-^0,
431
where || • \\'P denotes the dual norm of the C*-norm of A. Thus /„-+/ pointwise on
A, hence
f(it(x)) = lim f„(n(x)) = Mm}n(x) =f(x)
n
n
for all x£A, thus providing the desired equality.
Obviously, it is linear and it is continuous in the topology defined by the C*-norm
on A and a(A, sp (P)) on A, respectively. It is easy to see that n preserves the involution of A and A, respectively. On the other hand, non=n, since f(x)=f
(x)=
=/(n(x)) for all / g P and x£A, i.e. we have x=n(x) (x£A).
We show that the map n is multiplicative, thus TT, in fact, is a »-algebra morphism
between A and A. Let x£A and y£A. Applying (III) we infer x / £ s p (P) for every
f£P, hence
f(n(xy))=f(xy)
= (x.f)(y)
= ( £ f ) ( y ) = (x.f)(n(y))
=f(xn(y))
i.e. 7t(xy)=xn(y). Next, let x£A and y£A. Then there is a sequence (x„)„€N in
A such that ||jc„—i||P—0. The above established continuity property of n now yields
xn=7i(x„)~*7t(Jc) in the a(A, sp (P))-topology (moreover, in <r(A, sp (P))). Since
n ( x j ) = x „ n ( y ) (n€N) and the sequence (x n ) n6N is || • || P -bounded in A, Lemma 2
implies that n(x„y)—Ti(x)Ti(y) in the a (A, sp (P))-topology. Furthermore, we obviously have x„y->-icy in the C*-algebra A, thus applying the continuity of n again we
obtain n(x„y)-+ n(xy) in the a (A, sp (P))-topology, i.e. 7i(x)n(y)=7t(xy).
To finish the proof we refer to the well known fact that a *-algebra morphism
from a C*-algebra into a pre-C*-algebra is necessarily norm continuous (cf. [1] or
[2]). Thus 71 is norm continuous and A=Ker (id^—7t) is dense and closed in the
C*-algebra A, i.e. A=A.
III. Examples for C*-norms defined by positive linear forms
This last section contains two important examples for C*-norms defined by positive linear forms. Then it becomes clear that it is not difficult to construct in certain
»-algebras with unity such sets of positive linear forms which satisfy the conditions
(I)—(IV) formulated in sections I and II.
E x a m p l e 1. Let A be a von Neumann algebra in the Hilbert space H and for
all z£H let f . denote the positive linear form on A defined by the formula fz(T):=
:= (Tz\z) for all T£A. Then the set P:= {fz\z£H} satisfies the axioms (I)-(IV).
The proof of this assertion is provided by the standard application of the Banach—Steinhaus theorem; we omit the details. In this case a (A, sp(P)) and
a (A, sp(P)) are the weak and ultraweak operator topologies on A, respectively. Of
course, the norm || • ||P coincides with the usual operator norm on A.
12*
432
J.Kristóf: C*-norms defined by positive linear forms
E x a m p l e 2. Let .s/denote a <7-algebraofsubsetsintheset T and A be the »-algebra of complex valued, bounded, s/—i8(C) measurable functions (where âS(C)
denotes the Borel c-algebra of C). Let us define P as the set of integrals on A arising
from <r-additive finite positive measures defined on sé. Then P satisfies the axioms
(I)—(IV).
In this case sp ( P ) = s p (P) and a sequence (<?>„)„€N in A converges to the function (p£A in the a(A, sp (P))-topology if and only if <p„—<p pointwise on T and the
sequence is uniformly bounded. This proposition can be verified by the standard use
of the theorem of Lebesgue and that of Banach—Steinhaus. Of course, the norm
|| -||p coincides with the sup-norm.
These examples make clear that the class of C*-algebras with unity whose norms
are defined by positive linear forms satisfying the axioms (I)—(TV) contains strictly
the class of von Neumann algebras. It is easy to show that there are C*-algebras with
unity (even commutative ones) that do not belong to this class. However, these C*algebras have certain properties very close to those of von Neumann algebras. As a
matter of fact, further investigations on these C*-algebras yield an interesting version
of the spectral theorem for normal elements, analogous to the spectral theorem for
normal operators in Hilbert spaces.
References
[1] J . DIXMIER,
Les C*-algébres et Leurs Représentations, Gauthier Villars Éditeurs
Théories spectrales, Hermann (Paris, 1 9 6 7 ) .
[2] N . BOURBAKÍ,
D E P A R T M E N T O F ANALYSIS
EÖTVÖS L O R Á N D UNIVERSITY
M Ú Z E U M K R T . 6—8
1088 BUDAPEST, H U N G A R Y
(1968).
Bibliographie
M. Barr—C. Wells, Toposes, Triples and Theories (Grundlehren der mathematischen Wissenschaften, Band 278), XIII+345 pages, Springer-Verlag, Berlin—Heidelberg—New York—
Tokyo, 1985.
This book provides an introduction to the three concepts of topos, triple and theory, and describes
the connections between them. Among the three topics, topos theory is central in the book. That
reflects the current state of development and the importance of topos theory as compared to the
other two.
A topos is a special kind of a category defined by axioms saying roughly that certain constructions one can make with sets can be done in the category. However, the notion of topos originated
as an abstraction of the properties of the category of sheaves of sets on a topological space. Later,
mathematicians developed the idea that a theory in the sense of mathematical logic can be regarded
as a topos. In this book a topos is regarded as being defined by its elementary axioms, saying nothing
about set theory in which its models live. One reason for this attitude is that many people regard
topos theory as a possible new foundation for mathematics.
Chapter 1 is an introduction to category theory which develops the basic constructions in
categories needed for the rest of the book. A reader familiar with the elements of category theory
including adjoint functors can skip nearly all of these. Chapters 2, 3 and 4 introduce the three
topics of the title, respectively, and develop them independently up to a certain point. Each of
them can be read without the other two chapters. Chapter 5 develops the theory of toposes further,
making use of the theory of triples. Chapter 6 covers various fundamental constructions which
give toposes, with emphasis on the original idea that toposes are abstract sheaf categories. Chapter 7
provides the basic representation theorems for toposes. Theories are then carried further in Chapter 8, making use the representation theorems. Chapter 9 develops further topics in triple theory,
and may be read independently after Chapter 3.
The book provides a fairly thorough introduction to topos theory covering topologies and
representation theorems but omitting the connection with algebraic geometry and logic. Each
chapter ends with a list of exercies. The exercises provide examples or develop the theory further.
The book can be read by graduate students with a familiarity with elementary algebra.
Gy. Horvdth (Szeged)
Kenneth L. Bowles—Stephen D. Franklin—Dennis J. Volper, Problem Solving Using UCSD
Pascal. Second Edition with 106 illustrations, XI+340 pages, Springer-Verlag, New York—Berlin—Heidelberg—Tokyo, 1984.
This book is designed both for an introductory course in computer problem solving and for
individual self study. This is a revised and extended edition of one of the most successful introductions to Pascal, "Microcomputer Problem Solving Using Pascal", Springer-Verlag, 1977.
434
Bibliographie
In practice, it seems best to learn programming first using Pascal, and then to shift over to
one of the other languages. Pascal is clearly the best language now in widespread use for teaching
the concepts of structured programming at the introductory level. Structured programming is a
method designed to minimize the effort that the programmer has to spend on finding and correcting
logical errors in programs. Put more positively, it is a method designed to allow a program to
produce correct results with a minimum amount of effort on the part of the programmer. The key
to structuring in Pascal is the procedure. Therefore, this book introduces and emphasizes procedures
from the very start. Procedures for graphics are introduced in the first chapter and the first programming assignment requires the student to use them. By the second chapter, students are writing
and using their own procedures. The early and continuing emphasis on procedures is the major
organizational difference between this book and most others which, in contrast, often do not introduce procedures until a third or even half way through their coverage of Pascal.
The subject matter has been chosen to be understandable to all students at about the college
freshman level, with almost no dependence on a background in high school mathematics beyond
simple algebra. At an introductory level, the basic methods of programming and problem solving
differ very little between applications in engineering and science and applications in business, arts ór
humanities. It is possible to motivate and teach students across this entire spectrum using problem
examples with a non-numerical orientation: the manipulation of text strings and graphics.
Chapter 1 of the book introduces the students to the use of the UCSD software system for
Pascal. Chapter 2 through 7 present basic tools for programming and for expressing algorithms.
Chapter 8 through 12 add tools for working with data transmitted to the computer from external
devices and for working with complex data. Chapters 13—15 provide illustrations of complex
problems of types that are frequently encountered by virtually all programmers. The appendices
at the end of the book are included for reference purposes and survey some additional features óf
the Pascal language. Each chapter starts with a statement of the objectives that the students should
attain before proceeding to the next chapter.
If you have a personal computer, the UCSD software system for Pascal, and this book in
addition, you can easily learn to write computer programs and will master the bases of computer
:
problem solving. The book is equally suitable as a textbook of a course.
' J
. K. Dévényi (Szeged)
Differential Geometric Methods in Mathematical Physics, Proceedings, Clausthal, August 30-September 2, 1983. Edited by H. D. Doebner and J. D. Hennig (Lecture Notes in Mathematics,
1139), VI+337 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1986.
The aim of this series of conferences is to promote the application of geometrical, analytical
and algebraic methods and their interplay for the modelling of complex physical systems. This
volume contains 20 papers submitted by participants óf the conference (the 12th in the series) organized by H. D. Doebner, S. I. Andersson (Clausthal) and G. Denardo (Trieste) at the Institute for
Theoretical Physics A, Technical University of Clausthal, F.R.G. The main topics treated in the
proceedings are described by the following key words which are also the titles of the chapters:
Momentum Mappings and Invariants — Aspects of Quantizations — Structure of Gauge Theor i e s — Non-Linear Systems, Integrability and Foliations — Geometrical Modelling of Special
Systems.
'•• ~ .
,
.
•
The volume is dedicated to the memory of the outstanding mathematician and mathematical
physicist Steven M. Paneitz, who died in a tragic accident while attending this conference at the
age of 28.
. . .Piter T.. Nagy (Szeged)
Bibliographie
•435
Dynamical Systems and Bifurcations (Proceedings, Groningen 1984). Edited by B. L. J. Braaksma, H. W. Broer and F. Takens (Lecture Notes in. Mathematics, 1125), 129 pages, Springer-Verlag,
Berlin—Heidelberg—New York—Tokyo, 1985.
Readers paying attention regularly to the review section of these Acta have certainly realized
that nowadays more and more monographs, texts and conferences are devoted to the modern
theory of differential equations and especially to bifurcation theory. This is of course due to the
interest in the topic, both in pure and applied mathematics. The present volume is the proceedings
of the International Workshop on Dynamical Systems and Bifurcations, organized by the Department of Mathematics of Groningen University, April 16—20, 1984.
Almost all of the articles are concerned with the geometric theory of dynamical systems (papers
by M. Chaperon, R. Dumortier, A. Floer and E. Zehdner, J. Palis and R. Roussavie, F. Takens,
G. Vegter). One paper (written by R. Dieckerhoff and E. Zehdner) deals with the boundedness of
the solutions of a second order timedependent nonlinear differential equation, and another one
(by J. A. Sanders and R. Cushman) is devoted to global Hopf bifurcations.
Li Hatvani (Szeged)
E. Grosswald, Representations of Integers as Sums of Squares, XI+251 pages, Springer-Verlag,
New York—Berlin—Heidelberg—Tokyo, 1985.
. There are a number of branches of mathematics which seem to have no use in practical applications and in which the problems are merely investigated for their own beauty. The first of these is
clearly the theory of numbers, called "The Queen of Mathematics" by Gauss. Everybody knows something about integers. Here many questions or problems can be formulated within the capacity of
the man of the street. These questions are sometimes so interesting as the best puzzles having surprising results.
One of the oldest problems is the finding of Pythagorean triangles, right triangles whose sides
are integers. The next question is which of the integers can be represented as a sum of two or more
squares? Is it true that every integer can be written as a sum of four squares? These are curious
questions in themselves and surprisingly they do have applications, for example in lattice point
problems, in crystallography and in certain problems of mechanics. In the last few decades it was
in fact practice that has raised several questions which can be answered by the aid of number-theoretic results.
This book contains the enlarged versions of lectures given by the author in 1980—1981 for
an audience consisting of particularly gifted listeners with widely varying background. Therefore
the theme of this work, starting at the most elementary level but pushing ahead as far as possible
in some directions, seems to be ideal. Besides the elementary — but by no means simple — methods
-there appear various other notions from other branches of mathematics too, e.g. quadratic residues,
elliptic and theta functions, complex integration and residues, algebraic number theory etc.
The list of mathematicians who have obtained results in the study of the problems in this
book includes so illustrious names as Diophantus, Fermat, Lagrange, Gauss, Artin, Hardy, Littlewood, Ramanujan, Siegel and Pfister for instance.
The author made a serious effort to make the book accessible to a wide circle of readers. He
follows roughly the historical development of the subject matter. Therefore, in the early chapters;
the prerequisities are minimal. More advanced tools are necessary in the later chapters.
A number of interested problems is proposed at the end of many chapters, ranging from
•fairly routine to difficult ones. Also, open questions are mentioned.
•
436
Bibliographie
Finally we cite the last sentence of the author's "Introduction": "The success of this book
will be measured, up to a point, by the number of readers who enjoy it, but perhaps more by the
number who are sufficiently stimulated by it to become actively engaged in the solution of the numerous problems that are still open."
L. Pintér (Szeged)
Erich Hecke, Lectures on the Theory of Algebraic Numbers (Graduate Texts in Mathematics,
Vol. 77), Springer-Verlag, New York—Heidelberg—Berlin, 1981.
This is a translation of Erich Hecke's classic book, originally published in German in 1923.
It is a welcome event that, after sixty years of its first edition, this excellent book became available
to the English-reading public.
The first three chapters are of preparatory character: Chapter I contains the elements of the
theory of rational integers, Chapter II summarizes the basic facts on Abelian groups that are needed
later on, and Chapter III deals with the structure of the group 5R(n) of the residue classes mod n,
relatively prime to n. The theory of algebraic numbers starts in Chapter IV with a short introduction into the algebra of number fields (Galois theory is not discussed). Ideal theory, the core of
algebraic number theory, is developed in Chapter V. Next, in Chapter VI, analytic methods are
applied to the problem of the class number and to the problem of the distribution of prime ideals.
Quadratic number fields are treated in detail in Chapter VII; in particular, the Quadratic Reciprocity
Law is derived, and the class number of k ( Í d ) is determined with, as well as without, the use of
the zeta-function. Finally, Chapter VIII is devoted to the proof of the Quadratic Reciprocity Law
in arbitrary algebraic number fields.
After that a book, like this one, had been in continuous use for sixty years, there is no doubt
it will serve new generations of students and mathematicians who want to get acquainted with
algebraic number theory. There are, however, two things which the present-day reader will miss: a
subject index, and a short survey of what has happened in the field since 1923.
Unfortunately, in certain places the translation is not quite correct.
Agnes Szendrei (Szeged)
Dan Henry, Geometric Theory of Semilinear Parabolic Equations (Lecture Notes in Mathematics, 840), IV+348 pages, Springer-Verlag, Berlin—Heidelberg—New York, 1981.
Nowadays the geometric theory of differential equations is an intensively developing branch
of mathematics. The present book also deals with such general questions as existence, uniqueness,
continuous dependence of solutions on the right-hand side, the semigroup properties of solutions,
stability questions for nonlinear semigroups, Lyapunov functions, invariance principle, existence
of equilibria and periodic solutions and the behaviour of solutions in their neighbourhood, questions of bifurcation, the application of the adjoint system, invariant manifolds and the behaviour
of solutions in their small neighbourhoods, exponential dichotomies, orbital stability, perturbations of right-hand side, etc. The feature of the book is that the main results are formulated for
general differential equations written in Banach spaces. The conditions of the theorems can be met
by semilinear parabolic differential equations. Typically, the linear part of the right-hand side of
the equation is a sectorial operator.
The questions studied are well-prepared by examples. The results are illustrated by interesting
applications concerning nonlinear parabolic equations that arose in physical, biological and engineering problems. One of the chapters gathers the examples that accure in the book. The titles of the
sections of this chapter show a wide range of the applications: Nonlinear heat equation. Flow of electrons and holes in a semiconductor, Hodgekin—Huxley equations for nerve axon, Chemical reac-
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437
tíons in a catalyst pellet, Population genetics, Nuclear reactor dynamics, Navier—Stokes and related
equations.
There are sections concerned with special problems of semilinear parabolic equations, e.g.
the existence of travelling waves. Some exercises make this book interesting. They can be well comprehended after the study of the material, but the author also gives hints in more difficult cases.
This book is of interest to experts of the geometric and qualitative theories of differential
equations. I feel it essential for researchers of parabolic partial differential equations. It gives a
well-structured theoretical background and shows the directions of interesting applications.
/ . Terjéki (Szeged)
E. Horowitz, Fundamentals of Programming Languages, Second Edition, XV+446 pages,
Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1984.
Traditional books on programming languages are like abbreviated language manuals. This
book takes a fundamentally different point of view by focusing on a few essential concepts. These
concepts include such topics as variables, expressions, statements, typing, procedures, data abstractions, exception handling and concurrency. By understanding what these concepts are and how
they are realized in different programming languages, the reader arrives at a level of comprehension
far higher than what can be achieved by writing programs in various languages. Moreover, the
study of these concepts provides a better understanding of future language designs.
Chapter 1 is devoted to the study of the evolution of programming languages. Twelve criteria
by which a programming language design can be judged are presented in Chapter 2. Subsequent
chapters develop the concepts mentioned above. Much work has been done on imperative programming languages. The last three chapters cover non-imperative features such as functional programming, data flow programming and object oriented programming. A large number of exercises
enriches the text.
This book is warmly recommended as a good text for a graduate course whose objective is
to survey the fundamental features of current programming languages.
Gy. Horváth (Szeged)
O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics (Applied Mathematical Sciences, 49), XXX+322 pages, Springer-Verlag, New York—Berlin—Heidelberg—
Tokyo, 1985.
If an instantaneous state of a physical system is described by a function (e.g. string, membrane,
temperature of a body, fluid stream) then the mathematical model of the system is often a second
order partial differential equation. The purpose of mathematical physics is to study these equations
using pure mathematical methods. The basic problems are the following: Under which conditions
on the domain and on the functions involved in the equation does the equation have a solution
with prescribed values on the boundary? Is the solution unique? How can the solutions be approximated by solutions of simpler problems?
The book is concerned with these problems for linear equations. The investigations for the
existence and uniqueness of the solutions of boundary value problems are governed by a very natural principle. If a boundary value problem admits more solutions then it cannot be a good model
of a uniquely determined physical process. So the first step is to find conditions for the uniqueness
of the solutions. By experience, in case of linear problems the existence of the solutions is a consequence of their uniqueness (e.g. for systems of n linear algebraic equations in n unknowns). The
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Bibliographie 438
author tries to establish this principle also for boundary value problems introducing various classes
of generalized solutions.
There are other features of the book. The author considers elliptic, parabolic and hyperbolic
•equations alike. She studies also the smoothness of the generalized solutions and the connection
between the generalized and classic solutions. To illuminate this, let us have a look at Chapter 2
devoted to the elliptic equation. First the solvability of the boundary value problems is established
in the space W\(Q) consisting of all the elements in 1^(0) that have generalized first derivatives in
£j(i3). The problems are reduced to equations with completely continuous operators and their
Fredholm solvability is proved. Examples show that these solutions may not have second order
derivatives (not even generalized derivatives either), and they salitisfy the conditions of the problem
in some generalized sense. It is proved that under a minor improvement of the functions in the
equation and a certain increase in the smoothness of the boundary, all generalized solutions in
Wi (Q) belong to Wl (Q) and satisfy the equation in the ordinary sense for almost all x€S2.
The lengthy final chapter is devoted to the Method of Finite Differences. Introducing grids
in the basic domain, the method reduces various problems for differential equations to systems of
algebraic equations in which unknowns are the values of grid functions at the vertices of the grids:
The limit process obtained when the lengths of the sides of the cells in the grid tend to zero is also
examined here. The author is one of the pioneers of proving convergence in Z^QHiorm in case
of initial-boundary value problems for hyperbolic equations.
The original Russian edition is supplied in the present translation by "Supplements and Problems" located at the end of each chapter. They are useful to awaken the reader's creativity by providing topics for independent work.
This excellent monograph is highly recommended to everyone interested in the theory of
partial differential equations and its applications.
L. Hatvani (Szeged)
S. Lang, Complex Analysis, VII+367 pages, Springer-Verlag, New York—Berlin—Heidelberg—Tokyo, 1985.
The first edition of this book was published in 1977. The author has rewritten many sections,
added some new material and made a number of corrections.
The book consists of two parts. Part I (Basic Theory) is intended as an introduction to complex analysis for use by advanced undergraduates and beginning graduate students. Part II (Various Analytic Topics) contains more advanced topics, the chapters here are titled as follows: Applications of the maximum modulus principle; Entire and meromorphic functions; Elliptic functions;
Differentiating under an integral; Analytic continuation; The Riemann mapping theorem.
The author has made the necessary material on preliminaries as short as possible to get quickly
to power series expansions and Cauchy's theorem. In every book on complex functions, one of
the most interesting questions is the presentation of Cauchy's theorem. Perhaps the author's words
about this problem enlighten the basic attitude of the book: "I have no fixed idea about the manner
in which Cauchy's theorem is to be treated. In less advanced classes, or if time is lacking, the usual
band waving about simple closed curves and interiors is not entirely inappropriate. Perhaps better
would be to state precisely the homological version and omit the formal proof. For those who want
a more through understanding, I include the relevant material." He has included both Artin's
proof and the more recent proof of Dixon for the theorem.
Several solved examples illustrate the results making easier the understanding and carefully
selected exercises are proposed.
The chapters of Part II are logically independent and can be read in any order. Here we mention only one of the most striking applications, generally omitted from standard courses, the pro-
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439
blem of transcendence: Given some analytic function / , describe those points z such that /(z) is an
algebraic number.
The most attractive feature of the book for the reveiwer is the successful, composition of the
classical and modern methods and results.
L. Pintér (Szeged)
László Leindler, Strong Approximation by Fourier Series, 210 pages, Akadémiai Kiadó, Budapest, 1985.
Shortly after L. Fejér had proved his summability theorem, Hardy and Littlewood noticed
that.it was true in a stronger sense. This was the first instance when strong means were considered
but, except for a few feeble attempts, no one had analysed the question further until the late 1930's
when J. Marcinkiewicz and A. Zygmund verified that (with the usual terminology)
(*)
—
n+ 1f k=0M f , x) -f(x)|» - 0 (p > 0)
almost everywhere. It had taken another quarter of a century before G. Alexits and D. Králik
turned to the analogous approximation problem. What they found was quite surprising: for Lip a
( 0 < a < l ) functions the left-hand side of ( * ) with p—1 has order {«"'}, i.e. the classical estimate
of S. N. Bernstein holds true in the strong sense, as well. This result inevitably brought up the question about the mechanism behind these — literally — strong results and a rapid development in
this field was predictable. At this crucial stage appeared L. Leindler on the scene and began to
systematically study various kinds of strong means. His papers in the period 1965—1980 exerted
dominating influence on the course of events and his one is the lion's share in the fact that today
we have the theory of strong approximation at all. By the end of the 1970's the results of Leindler
and several other authors began to take the shape of a more or less complete theory and the time
was ripe for a monograph collecting and organizing the various results.
The book under review is an up-to-date summary of results about strong means. It can and
certainly will serve as a reference book. Its content can be recommended not only to the experts
of the field but to anyone interested in trigonometric approximation. Leindler's work may also
be useful for graduate students for it contains many important techniques of mathematical analysis
together with intricate counterexamples which have already proven to be useful in various other
branches of analysis. The organization of the material is clear, the whole book is well got up. It was
a good choice to omit the proof of certain theorems when they would have required repetition of
earlier used techniques. At the same time the reader can learn about the latest and most general
results — the author paid special attention at least to announce them. The only criticism I make
is that with a little more effort and some 20—30 extra pages the parallel theory of strong summability could have been incorporated into the book, giving thereby a complete account of results
about strong means. (The seemingly analogous theory of strong summability of general orthogonal
series requires distinctly different arguments.)
The book consists of four chapters. Chapter I deals with the order of strong approximation,
i.e., with so-called direct theorems. One of the most general estimate of this type was proved in
Leindler's very first paper on the subject and for every classical mean the estimate he obtained
turned out to be sharp in many function classes. However, an inequality being sharp does not mean
that it can be reversed, so the question of what can be said in the opposite direction, i.e. when we
inquire about properties of functions provided the order of some kind of strong mean is known,
is very interesting. Chapter II is devoted to this problem. Chapter III establishes various imbedding
relations of function spaces yielding in many cases the identification of several of these spaces. In
the last chapter some miscellaneous results in three new directions are treated.
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Bibliographie 440
Finally, it is appropriate to note that in spite of the fact that the theory presented in the book
seems to be complete (e.g. without embarrassing gaps), some very exciting questions in connection
with strong means have not or barely been touched upon until now (one of them is the saturation
problem). It is the reviewer's sincere hope that László Leindler's work will stimulate interest in
some of these problems.
V. Totik (Szeged)
Model-Theoretic Logic, Edited by J. Bairwise and S. Feferman (Perspectives in Mathematical
Logic), XVIII+ 893 pages, Springer-Verlag, New York—Berlin—Heidelberg—Tokyo, 1986.
This is the second monograph in the Perspectives series where Bairwise acts as an editor,
between the two he was the editor of the successful Handbook of Mathematical Logic. He and
S. Feferman are among those mathematicians who considerably contributed to the flourishing of
abstract model theory in the 1970's. The present monograph is research-oriented, the expositions
in it reflect intensive present-day investigations. "The aim ... would be to give an entry into the field
for anyone sufficiently equipped in general model theory and set theory, and thereby to bring
them closer to the frontiers of research". Accordingly, several open problems are mentioned and a
bibliography of over a thousand items can serve as a reference for the experts and to-be-experts in
the field. The book is divided into six parts:
1. Introduction, Basic Theory and Examples (written by Bairwise, Ebbinghaus, Flum).
2. Finitary Languages with Additional Quantifiers (Kaufmann, Schmerl, Mundici, Baudisch, Seese,
Tuschik, Weese).
3. Infinitary Languages (Nadel, Dickmann, Kolaitis, Eklof).
4. Second-Order Logic (Baldwin, Gurevich).
5. Logics of Topology and Analysis (Keisler, Ziegler, Steinhorn), and
6. Advanced Topics in Abstract Model Theory (Váánánen, Makowsky, Mundici).
V. Totik (Szeged)
Bernt Oksendal, Stochastic Differential Equations. An Introduction with Applications (Universitext), XIII+205 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985.
"Unfortunately most of the literature about stochastic differential equations seems to place
so much emphasis on rigor and completeness that it scares the nonexperts away. These notes are
an attempt to approach the subject from the nonexpert point of new: Not knowing anything ...
about a subject to start with, what would I like to know first of all? My answer would be: 1) In
what situations does the subject arise? 2) What are its essential features? 3) What are the applications and the connections to other fields?"
The author, a lucid mind with a fine pedagogical instinct, has written a splendid text that
achieves his aims set forward above. He starts out by stating six problems in the introduction in
which stochastic differential equations play an essential role in the solution. Then, while developing
stochastic calculus, he frequently returns to these problems and variants thereof and to many other
problems to show how the theory works and to motivate the next step in the theoretical development. Needless to say, he restricts himself to stochastic integration with respect to Brownian motion.
He is not hesitant to give some basic results without proof in order to leave room for "some more
basic applications". The chapter headings are the following: Introduction, Some mathematical
preliminaries, Itö integrals, Stochastic integrals and the Itö formula, Stochastic differential equations, The filtering problem, Diffusions, Applications to partial differential equations, Application
to optimal stopping, Application to stochastic control. There are two appendices on the normal
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441
distribution and on conditional expectations, a bibliography of 71 items, a list of notation, and a
subject index.
It can be an ideal text for a graduate course, but it is also recommended to analysts (in particular, those working in differential equations and deterministic dynamical systems and control) who
wish to learn quickly what stochastic differential equations are all about.
Sándor Csörgő (Szeged)
Ordinary and Partial Differential Equations (Proceedings, Dundee 1984), Edited by B. D. Sleeman and R. J. Jarvis (Lecture Notes in Mathematics, 1151), XIV+357 pages, Springer-Verlag,
Berlin—Heidelberg—New York—Tokyo, 1985.
These proceedings contain one half of the lectures delivered at the eighth International Conference on Ordinary and Partial Differential Equations which was held at the University of Dundee,
Scotland, June 25—29, 1984.
The 36 lectures show a very wide spectrum. However, it is common in the articles that the
authors investigate nonlinear differential equations, some of which are in fact concrete models of
systems or processeses in the real world. The following main directions can be recognized among
the themes. Many articles investigate the asymptotic behaviour (stability, oscillation etc.) of solutions of nonlinear second, third and n-th order ordinary differential equations. Differential equations with delays are treated, too. Special attention is paid to periodic problems both for ordinary
and partial differential equations (e.g. the probleme of the existence of a periodic solution of prescribed period for Hamiltonian systems is studied). Stability and bifurcation problems are also
considered for equations of several types.
The applications give a very valuable part of the proceedings. The reader can find interesting
(mostly biological) models such as a hydrodynamical model of the sea hare's propulsive mechanism,
models for a myelinated nerve axon, vector models for infectious diseases and multidimensional
reaction-convection diffusion equations.
The collections can be recommended to those wishing to get a snapshot of the theory and
applications of non-linear differential equations.
L. Hatvani (Szeged)
Richard S. Pierce, Associative Algebras (Graduate Texts in Mathematics, Vol. 88) SpringerVerlag, New York—Heidelberg—Berlin, 1982.
The study of associative algebras has long been, and still is, an area of active research which
draws inspiration from and applies tools of many other branches of mathematics such as group
theory, field theory, algebraic number theory, algebraic geometry, homological algebra, and category theory. The purpose of this book is twofold: to treat the classical results on associative algebras
more deeply than most student-oriented books do, and at the same time to bring the reader to the
frontier of active research by discussing some important new advances. The emphasis is put on
algebras that are finite dimensional over a field.
The highlights of the classical part include the characterization of semisimple modules, the
Wedderburn(—Artin) Structure Theorem for semisimple algebras, Maschke's Theorem, the Jacobson radical, the Krull—Schmidt Theorem, the structure and classification of projective modules
over Artinian algebras (Chapters 1 through 6), the Wedderburn—Malcev Principal Theorem,
Jacobson's Density Theorem, the Jacobson—Bourbaki Theorem (Chapters 11—12), and, within a
detailed, systematic study of central simple algebras and the Brauer group of a field (Chapters 13
through 20), the Cartan—Brauer—Hua Theorem, Wedderburn's theorem that all division algebras
of degree 3 are cyclic, the classification of central simple algebras over algebraic number fields,
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Bibliographie 442
and Tsen's Theorem. Among the more recent developments of the subject presented in the book
are the theory of representations of algebras, including a proof of the first Brauer—Thrall Conjecture and an exposition of the representation theory of quivers (Chapters 7 and 8), and Amitsur's
Theorem on the existence of finite dimensional central simple algebras that are not crossed products
(Chapter 20). Together with these results the reader is acquainted with the most powerful tools of
the theory of associative algebras: tensor product, homological methods, cohomology of algebras,
Galois cohomology, valuation theory, and polynomial identities.
Each of the twenty chapters ends with a note containing historical remarks and suggestions
for further reading. In addition, every section is followed by a set of exercises; some of them are
intended to help understanding the material, others call for working out details of proofs omitted
in the text, and there are also exercises presenting important results which otherwise couldn't be
included.
This excellent book is warmly recommended to students as well as established mathematicians
who are interested in associative algebras. For those wishing to read only parts of the text, the
subject index and the list of symbols are of great help.
Ágnes Szendrei (Szeged)
D, Pollard, Convergence of Stochastic Processes. XIV+215 pages, Springer-Verlag, New York—
Berlin—Heidelberg—Tokyo, 1984.
Properties of empirical processes play an important role in mathematical statistics. Earlier
results on this field, especially on the weak convergence of empirical measures can be found in
the books of 1.1. Gikhman and A. V. Skorokhod (Introduction to the Theory of Random Processes,
Saunders, 1969 (in Russian 1965)), K. R. Parthasarathy (Probability Measures on Metric Spaces,
Academic Press, 1967) and P. Billingsley (Convergence of Probability Measures, Wiley, 1968).
These books have become classics and have generated a lot of work in the area. Recently R. M. Dudley (A Course on Empirical Processes, Lecture Notes in Math. vol. 1097, Springer-Verlag, 1985)
and P. Gaenssler (Empirical Processes, IMS Lecture Notes vol. 3, IMS, 1983) have summarized
the newest results on this topic and the book under review is another addendum to the literature.
These three monographs cover nearly the same area of the theory of stochastic processes.
In Chapter I Pollard introduces notations of stochastic processes and empirical measures.
Chapter II contains generalizations of the classical Glivenko—Cantelli theorem. He proves uniform
convergence of empirical measures over classes of sets and classes of functions. The author gives
a review on the weak convergence of probability measures on Euclidean and metric spaces in Chapters III and IV. Chapters V and VI deal with the weak convergence of empirical processes, properties of the limit processes and also contain some applications of the obtained results in mathematical statistics. Martingale central limit theorems are proven in Chapter VII and an application
to the Kaplan—Meier (product-limit) estimator is sketched.
Pollard writes in his Preface: "A more accurate title for this book might be: An Exposition
of Selected Points of Empirical Process Theory, With Related Interesting Facts About Weak Convergence, and Applications to Mathematical Statistics." It is certainly true that it is nearly impossible
to write a book on empirical processes which would cover all the interesting and important theorems on these processes and also their applications. However, I think a research monograph must
contain at least an up-to-date and rich enough list of references, and this cannot be replaced by
saying "according to the statistical folklore..." (cf., for example, page 99). This is completely useless
for a reader and hence I doubt that Pollard's monograph can be used as a reference book. On the
other hand, the interested reader can certainly find some material collected together which was
mainly published only in research papers earlier.
Lajos Horváth (Ottawa)
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443
Probability in Banaeh Spaces V, Proceedings of the International Conference Held in Medford, USA, July 1984. Edited by A. Beck, R. Dudley, M. Hahn, J. Kuelbs and M. Marcus (Lecture
Notes in Mathematics, 1153), VI+457 pages, Springer-Verlag, Berlin—Heidelberg—New York—
Tokyo, 1985.
The proceedings of the first four conferences on probability in Banach spaces have been published as Volumes 526,709,860, and 990in these Lecture Notes series. The present collection containes
25 articles, almost all of which are very high level research papers. 15 papers deal with limit theorems (the central limit theorem, empirical processes, large deviations, ratio limit theorems for
sojourns and other weak theorems; the law of large numbers and of the iterated logarithm, almost
sure convergence of martingale-related sequences and other strong theorems), the rest address
various problems in sample path behaviour, representation questions, moment and other bounds,
stable measures and infinite divisibility. The illustrious list of the authors may give an impression
of the contents: de Acosta; Alexander; Austin, Bellow and Bouzar; Berman; Borell; Bourgain;
Czado and Taqqu; Dudley; Fernique; Frangos and Sucheston; Giné and Hahn; Heinkel; Hoffman—J0rgensen; Jain; Jurek; Klass and Kuelbs; LePage and Schreiber; Marcus and Pisier;
McConell; Rosinski and Woyczynski; Samur; Vatan; Weiner; Weron and Weron; and Zinn.
Murray H. Protter—Charles B. Morrey, Jr., Intermediate Calculus (Undergraduate Texts
in Mathematics), X+648 pages, Springer-Verlag, New York—Berlin—Heidelberg—Tokyo, 1985.
If you teach a course on analytic geometry and calculus at a college or university, your first
task is to choose a text-book. This is not easy at all. The course usually consists of three semesters.
Typically, the first two semesters cover plane analytic geometry and the calculus of functions of
one variable, almost independently of the majors in the audience. The decision on the subjectmatter of the third semester (and of the fourth one, if any) requires more care. One must take
into account the further study of the students: do they go on in mathematics or do they need the
calculus only for applications? This book has been written to help the instructors in this decision
by providing a high degree of flexibility in structuring the course.
The first five chapters present the material of a standard third-semester course in calculus
(the knowledge of plane analytic geometry and one-variable calculus are a prerequisite): 1. Analytic
Geometry in Three Dimensions; 2. Vectors; 3. Infinite Series; 4. Partial Derivatives. Applications;
5. Multiple Integration. The further chapters give an opportunity for the instructor to include less
traditional topics in the third semester and to design a fourth semester of analysis depending upon
special needs: 6. Fourier Series; 7. Implicit Function Theorems. Jacobians; 8. Differentiation
under the Integral Sign. Improper Integrals. The Gamma Function; 9. Vector Field Theory; 10.
Green's and Stokes' Theorems.
The main feature of the book is its flexibility. Of course, this can be meant in several ways.
On the one hand, the chapters are independent of each other. On the other hand, within each chapter
readers can find material of different levels. E.g. in Chapter 10 Green's theorem is established first
for a simple domain (this results is adequate for most applications), but it is followed by Green's
and Stokes' theorems for the general cases proved by using orientable surfaces and a partition of
unity.
'
,
.
.
*
The book is concluded with three appendices on matrices, determinants, on vectors in three
dimensions, and on methods of integration. The sections contain many interesting problems
(answers to the odd-numbered ones are available at the end of the book).
This excellent text-book will be very useful both for instructors and students.
L: Hatvani (Szeged)
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Bibliographie 444
Representations of Lie Groups and Lie Algebras, Proceedings, Budapest 1971. Edited by
A. A. Kirillov, 225 pages, Akadémiai Kiadó, Budapest, 1985.
The present book contains the second part of the Proceedings of the Summer School on representation theory which was held at Budapest in 1971. The first part of the Proceedings was published by the Akadémiai Kiadó (edited by I. M. Gelfand) in 1975. The scientific program of the
school was divided into two sections: advanced and beginner. Most of the lectures published here
were read at the latter section but some of them are now rewritten or completed by new authors.
A. A. Kirillov's introductory lecture gives a summary of the basic notions, results and methods
in the representation theory of finite and compact groups. The article by B. L. Feigin and A. V. Zelevinsky makes the reader acquainted with the theory of contragradient Lie algebras and their representations. In his paper D. P. Zhelobenko dissusses the constructive description of Gelfand—Zetlin
bases for classical Lie algebras gl(n, C), o(n, C), sp (n, C). The method presented here is very
effective in obtaining explicit formulas of the representation theory. The lecture by S. Tanaka is
devoted to an instructive example; explicit description of all irreducible representations of the
group SI (2, F) where Fis a finite field. The survey by I. M. Gelfand, M. I. Graev and A. M. Vershik
"Models of representations of current groups" deals with a class of infinite dimensional Lie groups
which is of great importance for physical applications. The authors describe in detail several classical
as well as new models of the representations of current groups. G. J. Olshansky's article "Unitary
representations of the infinite symmetric group: a semigroup approach" is devoted to the representation theory of another type of the "big" groups. The last lecture by G. W. Mackey is addressed
first of all to physicists. It explains how the imprimitivity theorem and the concept of induced representations play an important role in quantum mechanics.
Most of the content of this book is accessible for beginners, but experts will also find some
new and useful information in the articles published here.
L. Gy. Fehér (Szeged)
Representation Theory II, Proceedings, Ottawa, Carleton University, 1979, edited by V. Dlab
and P. Gabriel (Lecture Notes in Mathematics, 832), XIV+673 pages, Springer-Verlag, Berlin—
Heidelberg—New York, 1980.
The first volume of these proceedings in two volumes contains reports from the "Workshop
on the Present Trends in Representation Theory" held at Carleton University, 13—18 August,
1979. The present volume contains the majority of the lectures delivered at the second part of the
meeting "The Second International Conference on Representations of Algebras" (13—25 August,
1979), but some papers which were not reported at the conference are also published here. These
two volumes together (the first volume contains also an extensive bibliography of the publications
in the field covering the period 1969—1979) provide the interested reader with a comprehensive
survey on the modern results and research directions of the representation theory of algebras. The
list of the authors contributed to this volume is: M. Auslander—I. Reiten, M. Auslander—
S. O. Smalo, R. Bautista, K. Bongartz, S. Brenner—M. C. R. Butler, H. Brune, C. W. Curtis,
E. C. Dade, V. Dlab—C. M. Ringel, P. Dowbor—C. M. Ringel—D. Simson, Ju. A. Drozd,
E. L. Green, D. Happel—U. Preiser—C. M. Ringel, Y. Iwanaga—T. Wakamatsu, C. U. Jensen—
H. Lenzing, V. C. KaC, H. Kupisch—E. Scherzler, P. Landrock, N. Marmaridis, R. Martinez-Villa,
F. Okoh, W. Plesken, C. Riedtmann, K. W. Roggenkamp, E. Scherzler—J. Waschbüsch, D. Simson,
H. Tachikawa, G. Todorov, J. Waschbüsch, K. Yamagata.
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445
Rings and Geometry, Proceedings of the NATO Advanced Study Institute held at Istambul,
Turkey, September 2—14, 1984. Edited by R. Kaya, P. Plaumann and K. Strambach (NATO ASI
Series C: Mathematical and Physical Sciences, Vol. 160), XI+567 pages, D. Reidel Publishing
Company, Dordrecht—Boston—Lancaster—Tokyo, 1985.
This volume contains 11 lectures given at the summer school on Rings and Geometry.
Until quite recently, when looking for applications of ring theory in geometry, we could
mainly think of abstract algebraic geometry which can be formulated in the language of commutative algebra. Of course the connections between ring theory and geometry cannot be circumscribed
only by the subject of algebraic geometry. There are some old and new areas in mathematics, highly
developed in the last decades, which are the results of the interaction of ring theory and geometry.
The editors write in the Preface: "It is the aim of these proceedings to give a unifying presentation
of those geometrical applications of ring theory outside of algebraic geometry, and to show that
they offer, a considerable wealth of beautiful ideas, too. Furthermore it becomes apparent that
there are natural connections to many branches of modern mathematics, e.g. to the theory of (algebraic) groups and of Jordan algebras, and to combinatorics".
The book consists of four parts. In the first one the non-commutative analogue of the function field of an algebraic variety and the generalization of classical objects of algebraic geometry
to the case of non-commutative coordinate field are studied. (P. M. Cohn, Principles of non-commutative algebraic geometry; H. Havlicek, Application of results on generalized polynomial identities in Desarguesian projective spaces.) The second part is devoted to the study of topological,
incidence-theoretical, algebraic and combinatorial properties of Hjelmslev planes. (J. W. Lorimer,
A topological characterization of Hjelmslev's classical geometries, D. A. Drake—D. Jungnickel,
Finite Hjelmslev planes and Klingenberg epimorphisms.) The third part contains the treatment
of the theory of projective planes over rings "of stable rank 2". (J. R. Faulkner—J. C. Ferrar,
Generalizing the Moufang plane; F. D. Veldkamp, Projective ring planes end their homomorphisms.)
The authors of the papers in the fourth part study the possibility of the extension of the classical
relations between projective geometry and linear algebra, investigate the structure of linear, orthogonal, symplectic and unitary groups and matrix rings over large classes of rings, the geometric
structure of the units of alternative quadratic algebras and the coordinatization of lattice-geometry.
(C. Bartolone—F. Bartolozzi, Topics in geometric algebra over rings; B. R. McDonald, Metric
geometry over localglobal commutative rings, Linear mappings of matrix rings preserving invariants;
H. Karzel—G. Kist, Kinematic algebras and their geometries; U. Brehm, Coordinatization of
lattices.) An appendix describes some concepts of geometry indispensable for mathematics. (C. Arf,
The advantage of geometric concepts in mathematics.)
This book is warmly recommended to anybody interested in the interaction of algebra and
geometry. It can be used as an up-to-date reference book "in that area of mathematics where the
ring theory accuring outside of algebraic geometry is harmonically unified with geometry".
Peter T. Nagy (Szeged)
Murray Rosenblatt, Stationary Sequences and Random Fields, 258 pages, Birkhauser, Boston—
Basel—Stuttgart, 1985.
The first systematic book on time series analysis was "Statistical Analysis of Stationary Time
Series" by Ulf Grenander and Murray Rosenblatt published in 1957 by Wiley. Since then a number of excellent monographs of the field have appeared reflecting the fast growth of knowledge.
Now, 28 years after his first book, here is a second look of one of the foremost researchers of the
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Bibliographie 446
topic. As he writes in the preface: "This book has a dual purpose. One of these is to present material
which selectively will be appropriate for a quarter or semester course in time series analysis and
which will cover both the finite parameter and the spectral approach. The second object is the
presentation of topics of current research interest and some open questions".
Chapter I presents basic results on the Fourier representation of the covariance function of
a weakly stationary process and the harmonic analysis of the process itself, that is the Herglotz
and Cramdr theorems. Chapter II formulates the problem of linear prediction, or linear mean
square approximation, gives the basic relations between moments and cumulants, fully discusses
the linear prediction problem for autoregressive and moving average (ARMA) processes, describes
the Kalman—Bucy filter, and investigates the identifiability of the phase function of a non-Gaussian
linear process (one of the several novelties of the book, to be fully taken up in Chapter VIII). Of
particular interest to a probabilist is Chapter III. It first gives Gordin's device to obtain central
limit theorems for partial sums of strictly stationary sequences from those for a martingale difference sequence, then this is used to prove asymptotic normality for general covariance estimators
and quadratic forms including the periodogram under cumulant summability conditions. A nice
discussion of strong mixing, one of the many important spiritual children of the author, is given
together with a basic central limit theorem of the author for strongly mixing sequences. Exotic
behaviour under long-range dependence is also discussed here. Chapter IV gives the estimation
theory for ARMA schemes with new results, based partly on those in Chapter III, on asymptotic
normality in which, besides strong mixing, only the finiteness of a few moments are assumed. Chapters V and VI deal with second and higher order cumulant spectral density estimation, respectively,
assuming strong mixing and requiring again only the finiteness of a few moments in the asymptotic
normality results. The brief Chapter VII is on kernel density and regression function estimates
under "short range" dependence. The final Chapter VIII is on various questions of estimating the
phase or transfer functions and moments of non-Gaussian linear processes and fields with the associated deconvolution problems. A short appendix gethers some facts of measure theory,' of Hilbert and Banach spaces and Banach algebras. Author and subject indices help the reader.
The last sections of the chapters discuss how the results for sequences extend for fields, what
are the difficulties, limitations, etc. There are Monte Carlo simulations for the comparison of the
deconvolution procedures and various concrete applications involving turbolence and energy
transfer are scattered through the book beginning with the fourth chapter. Occasional figures illustrate the text and the arising computational questions receive sensitive discussions. Each chapter
ends with a se' of problems that can be given to students, followed by a set of notes providing further
examples, explanations, references to and descriptions of the literature and many open problems
for further research are discussed at some length.
The text goes very smoothly, it is thoughtful, sympathetic and (sometimes overly) modest.
You feel as if you would be inventing what you read and forget that you hear" everything from
the horse's mouth. Both the mathematics and the discussion concentrates on the essence of the
problem at hand, heuristics constitute an integral part of the matter, and unnecessary details are
elegantly sidestepped. The main achievment of the book is, I think, that it can be read enjoyably
and profitably on several different levels of understanding, with various backgrounds. Thus the
author has considerably surpassed his "dual purpose".
The reviewer was fortunate enough to have the opportunity of sitting in a course "Time Series
Analysis" that Professor Rosenblatt gave for upper undergraduate and early graduate students
in the spring quarter of 1985, at the University of California, San Diego, using the galley proofs
of the present book. He covered most of the material in Chapters I, II, IV, and V, motivating the
necessary results from Chapter III and from the elements of probability theory and Fourier analysis
on heuristic grounds. (This is what he advises to do in his preface.) The course, attended besides
Bibliographie
447
the 15—20 students by other senior guests with the Department of Mathematics and a few research *
workers from some applied departments of UCSD, was a real success.
The book will be inevitable for students, instructors, regular users, occasional appliers, and
researchers of time series analysis as well.
James G. Simmonds, A Brief on Tensor Analysis, VIII+92 pages, Under-Graduate Texts in
Mathematics, Springer-Verlag, New York—Heidelberg—Berlin, 1982.
The book is divided into four chapters. The first chapter introduces the concept of vectors
and defines the dot product and cross product of vectors. A second order tensor is defined as a
linear operator that sends vectors into vectors. Chapter 2 is devoted to the description of tensors
by using general bases and their dual bases and covariant and contravariant coordinates of vectors.
Chapter 3 deals with the Newton law of motion, introduces moving frames and the Christoffel
symbols. The last chapter presents a short glimpse into vector and tensor analysis, introducing
the notions of a gradient operator, divergence and covariant derivatives. All the chapters end with
a set of exercises. The book may serve as a text for first or second-year undergraduates.
L. Gehir (Szeged)
Gary A. Sod, Numerical Methods in Fluid Dynamics: Initial and Initial Boundary-Value Problems,
IX+446 pages, Cambridge University Press, Cambridge—London—New York—New Rochelle—
Melbourne—Sydney, 1985.
In fact there is practically very little about fluid dynamics in this book, intended to provide
the foundations for Volume II, which — according to the author's preface — will deal exclusively
with the equations governing fluid motion. This volume is rather a text on numerical methods
for solving partial differential equations and it is directed at graduate students.
The book deals with the finite difference method, the other, not less powerful method of finite
elements is not discussed. Nevertheless it is very useful for everyone, wishing to actually apply ¿he
results of numerical analysis of partial differential equations in any field.
The book consists of the following chapters: I. Introduction, II. Parabolic Equations,
III. Hyperbolic Equations, IV. Hyperbolic Conservation Laws, V. Stability in the Presence of
Boundaries. It is written from the physicist's and engineer's point of view, which does not mean
that it would lack mathematical rigor. It is simply detailed enough to be comprehensible for the
mathematically less trained reader. It emphasises the concepts and contains all the necessary definitions and theorems about convergence and stability, indispensable for those too who want only
good recipes. Several methods for solving parabolic and hyperbolic equations are introduced and
analysed, and the problem of boundaries is dealt thoroughly. The more sophisticated mathematical
theorems and proofs are avoided, some of them are left to the 4 appendices.
Reading the book, all the natural questions arising in the reader are answered by the author,
except for one: Why this title? It certainly calls the attention of practitioners in the field of fluid
dynamics, but one is afraid that the set of potential readers will unnecessarily be restricted. One
can warmly recommend this book to anybody who just wants to find a good introduction to the
finite difference method for solving parabolic and hyperbolic equations.
M. G. Benedict (Szeged)
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Bibliographie 448
Edwin H. Spanier, Algebraic Topology, XTV+528 pages, Springer-Verlag,- New York—Heidelberg—Berlin, 1982.
This is the second edition of the original text first published in 1966 by McGraw-Hill. It starts
with an introductory part summarising the basic concepts of set theory, general topology and alge^
bra. The material is divided into three main parts each of which consists of three chapters. The
first three chapters introduce the concepts of homotopy and the fundamental group and apply
these in the study of covering spaces and polyhedras. The second three chapters are devoted to
homology and cohomology theory. The general cohomology theory and duality in topological
manifolds are studie here, too. For each new concept, applications are presented to illustrate its
utility. In the last three chapters homotopy theory is studied; basic facts about homotopy groups
are considered, some applications to obstruction theory are presented and in the last chapter some
computations of homotopy groups of spheres can be found.
No prior knowledge of algebraic topology is assumed but some faimiliarity of the reader
in general topology and algebra is preassumed.
L.Gehér (Szeged)
Stability Problems for Stochastic Models, Proceedings of the 8th International Seminar Held
in Uzhgorod, September 1984. Edited by V. V. Kalashnikov and V. M. Zolotarev (Lectures Notes
in Mathematics, 1155), VI+447 pages, Springer-Verlag, Berlin—Heidelberg—New York—
Tokyo, 1985.
Although the 23 papers collected in this volume represent somewhat more diverse research
activities than the 22 papers in the predecessor collection under the same title (Lecture Notes in
Mathematics, 982), almost everything that I wrote about that one (these Acta, 47 (1984), page 260)
is valid for the present proceedings. Characterization problems and related questions still constitute a strong topic in the volume, but there is a noticable shift towards limit theorems and convergence-rate problems.
Stochastic Space-Time Models and Limit Theorems. Edited by L. Arnold and P. Kotelenez
(Mathematics and its Applications), XI+266 pages, D. Reidel Publishing Company, D o r d r e c h t Boston—Lancaster, 1985.
This is the first volume of a new series launched by the D. Reidel Company. In his preface
Series Editor M. Hazewinkel emphasizes "the unreasonable effectiveness of mathematics in science''
(to quote one of his six mottoes, this one due to E. Wigner) and outlines the aims of this new programme.
.. ,
.
The book presents the 13 invited papers presented at a workshop held at the University of
Bremen, FRG, in November 1983, describing the state of the art of the mathematics of spacetime phenomena, a specific combination of the modern theory of stochastic processes and func?
tional analysis. These widely applicable models describe motions which change with time : and
are randomly distributed in space, and are usually given by stochastic differential equations. The
first seven surveys (by Albaverio, Hoegh—Krohnand Holden; Da Prato; Dettweiler; Ichikawa;
Kotelenez; Krée; and Ustunel) are devoted to the existence, uniqueness and, regularity problems
of solutions of stochastic partial differential equations, and in general to stochastic analysis am)
Markov processes in infinite dimensions. The other six papers (by-Van den Broeck; Grigelioni$
and Mikuleviöius; Metivier; Pardoux; Rost; and Zessin) deal with various limit theorems where
stochastic processes describing a space-time model emerge as limits of sequences of stochastic
processes on lattices or of positions of finitely many particles with several kinds of interaction.
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449
An intelligent unifying introduction by Kotelenez puts the whole contents in a broad perspective
and a subject index helps orientation.
If the editors will be able to maintain the high level set by this very carefully compiled first
volume, this series will no doubt be a success.
Symposium on Anomalies, Geometry and Topology, Proceedings, Chicago, United States,
1985. Edited by W. A. Bardeen and A. R. White, XVIII+ 558 pages, World Scientific, 1985.
This book contains the proceedings of the Symposium on Anomalies, Geometry and Topology,
which took place at the Argonne National Laboratory of the US and at the University of Chicago
on March 28—30, 1985.
The 56 papers published here report on the recent progress made in field theory and particle
physics mostly by means of the application of sophisticated geometrical-topological methods and
results. One of the main concerns pursued in the lectures is the new superstring physics—hopefully
the right candidate to become a real "Theory of Everything". The other subject which dominated
the talks is the mathematical structure of anomalies. Anomalies, originally appeared in perturbative
calculations, are now related to properties of Dirac operators and to the connected index theorems
of Atiyah and Singer, as well as to homotopy, cohomology and complex structure of manifolds
in four and higher dimensions. The investigation of anomalies is an important part of the consistency analysis of quantum field theories. For example, anomaly cancellations played a crucial
role in the recent breakthrough achieved in superstring theory. In addition to the main topics of
the Symposium, anofnalies and superstrings, there were special sessions devoted tó charge fractionalization and compáctification issues and to the method of effective Lagrangians in various
models. Beyond the mentioned ones, the book also contains articles concerning several subjects
somehow related to the above. Just to give examples, there are papers addressed to the quantum
mechanics of black holes, or dealing with the global structure of supermanifolds. Amongst the
authors are M. F. Atiyah, M. Green, D. Gross, R. Jackiw, V. Kac, J. R. Schrieffer, J. Stasheff,
G.'t Hooft, E. Witten, S. T. Yau, B. Zuminö and many other outstanding scientists. This list
clearly proves how close is the connection between pure mathematics and theoretical physics
these days.
In conclusion, this collection of high level papers is warmly recommended to everybody interested in the exciting developments reported in it.
Universal Algebra and Lattice Theory (Proceedings, Charleston 1984). Edited by S. D. Comer
(Lecture Notes in Mathematics 1149), VI+282 pages, Springer-Verlag, Berlin—Heidelberg—New
York—Tokyo,. 1985. .
This volume contains research papers on universal algebra and lattice theory, mostly based
on lectures presented at the conference in Charleston, 1984. "In keeping with the tradition set by
its origin, it is notable that there are a number of papers that deal with connections between lattice
theory and universal algebra and other areas of mathematics such as geometry, graph theory, group
theory and logic." The list of the papers is as follows.
M. E. Adams and D. M. Clark: Universal terms for pseudo-complemented distributive lattices
and Hey ting algebras; H. Andréka, S. D. Comer and I. Németi: Clones of operations on relations;
M. K. Bennett: Separation conditions on convexity lattices; A. Day: Some independence results
in the co-ordinization of Arguesian lattices; Ph. Dwinger: Unary operations on completely distributive complete lattices; R. Freese: Connected components of the covering relation in free lat-
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Bibliographie 450
tices; B. GanterandT. Ihringer: Varieties of linear subalgebra geometries; O. C. Garcia and W. Taylor: Generalized commutativity; A. M. W. Glass: The word and isomorphism problems in universal
algebra; M. Haiman: Linear lattice proof theory: an overview; D. Higgs: Interpolation antichains
in lattices; J. Jezek: Subdirectly irreducible and simple Boolean algebras with endomorphisms;
E. W. Kiss: A note on varieties of graph algebras; G. F. McNuIty: How to construct finite algebras which are not finitely based; R. Maddux: Finite integral relation algebras; J. B. Nation: Some
varieties of semidistributive lattices; D. Pigozzi and J. Sichler: Homomorphisms of partial and of
complete Steiner triple systems and quasigroups; A. F. Pixley: Principal congruence formulas in
arithmetical varieties; A. B. Romanowska and J. D. H. Smith: From affine to projective geometry
via convexity; S. T. Tschantz: More conditions equivalent to congruence modularity.
The book gives a good account on several directions of the current research, and it is warmly
recommended to research workers interested in the subject.
Gábor Czédli (Szeged)
Vertex Operators in Mathematics and Physics, Proceedings, Berkeley, 1983. Edited by
J. Lepowsky, S. Mandelstam and I. M. Singer (Mathematical Sciences Research Institute Publications, 3), XIV+482 pages, Springer-Verlag, New York—Berlin, Heidelberg—Tokyo, 1985.
This volume includes proceedings from the Conference on Vertex Operators in Mathematics
and Physics, held at the Mathematical Sciences Research Institute, Berkeley, November 10—17,1983.
Among the subjects of highest current interest in physics and mathematics are the superstring,
supergravity theories on the one hand and the theory of infinite dimensional Lie algebras on the
other hand. In the last few years interesting connections have been found between the dual-string
theory and the affine Kac—Moody algebras, through the use of vertex operators that originally
appeared in the interaction Hamiltonians of the string models. The central role of the mentioned
theories in contemporary mathematics and physics made it possible to explore interrelations between
seemingly so remote topics as string models, number theory, sporadic groups, integrable systems,
the Virasoro algebra and so on. To explore further connections — this was the purpose of the Conference. Most informatory for an expert can be the table of contents of the book under review, so
we give it below.
Section 1 — String models. S. Mandelstam: Introduction to string models and vertex operators; O. Alvarez: An introduction to Polyakov's string model; T. Curtright: Conformally invariant
field theories in two dimensions.
Section 2 — Lie algebra representations. P. Goddard and D. Ovive: Algebras, lattices and
strings; J. Lepowsky and R. L. Wilson: Z-algebras and the Rogers—Ramanujan identities; J. Lepowski and M. Prime: Structure of the standard modules for the affine Lie algebra Af1* in the homogeneous picture; K. C. Misra: Standard representations of some affine Lie algebras; A. J. Feingold:
Some applications of vertex operators to Kac—Moody algebras; M. Jimbo and T. Miwa: On a
duality of branching coefficients.
Section 3—The Monster. R. L. Griess: A brief introduction to finite simple groups; I. B. Frenkel, J. Lepowsky and A. Meurman: A Moonshine Module for the Monster.
Section 4 — Integrable systems. M. Jimbo and T. Miwa: Monodromy, solitons and infinite
dimensional Lie algebras; K. Ueno: The Riemann—Hilbert decomposition and the KP hierarchy;
Ling-Lie Chau: Supersymmetric Yang—Mills fields as an integrable system and connections with
other non-linear systems; Yong-Shi Wu and Mo-Lin Ge: Lax pairs, Riemann—Hilbert transforms
and affine algebras for hidden symmetries in certain nonlinear field theories; L. Dolan: Massive
Kaluza—Klein theories and bound states in Yang—Mills; I. Bars: Local charge algebras in quantum
chiral models and gauge theories; B. Julia: Supergeometry and Kac—Moody algebras.
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451
Section 5 — The Virasoro algebra. C. B. Thorn: A proof of the no-ghost theorem using the
Kac determinant; D. Friedan, Zongan Qiu and S. Shenker: Conformal invariance, unitarity and
two dimensional critical exponents; A. Rocha—Caridi: Vacuum vector representations of the
Virasoro algebra; N. R. Wallach: Classical invariant theory and the Virasoro algebra.
J. Lepowsky writes in the Introduction: "The excitement of discovering surprising connections between different disciplines has become a rule in the subject of this volume". There is no
doubt that the present interaction between mathematicians and physicists continues and that this
book will be very useful for all experts involved in it.
M. Zamansky, Approximation des Fonctions (Travaux en Cours), V+91 pp., Hermann,
Paris, 1985.
A more appropriate title would be "Approximation of periodic functions" because Zamansky's
small treatise is entirely devoted to periodic functions of a single or several variables. The author
belongs to that prominent group of mathematicians who nursed the theory of harmonic approximation through its infancy and in this book he twisted again on the usual course of events, namely
he provided us with a rather unconventional treatment of the subject. Today it is uncommon, though
it may be refreshing, to put multipliers into the heart of the subject. The notations are also peculiar
which may cause some headache for the expert who wants to use the book as a reference. Nevertheless, the topics covered are familiar and they provide an independent introduction to harmonic
approximation in LP and C spaces. Both the book's strength and its weakness lies in its concise
form. On the one hand this enables the author to consider several spaces and approximation methods
simultaneously, on the other hand too general statements blur the necessary distinction between
important and unimportant (methods, for example) and hide the differences between certain spaces
(e.g. L1 and LP, p=-l) that play(d) so important role in mathematics.
Very briefly the contents: Chapter 1: Approximation of functions of a single variable (Fourier
series, moduli of smoothness, generalizations of Jackson's estimate and its converse, conjugate
series, derivatives of functions, saturation with many examples); Chapter 2: Convolutions (equivalence of convolution processes, saturation, direct and converse theorems); Chapter 3: Multidimensional case. A 9 page historical account closes the book.
V. Totik (Szeged)
Livres reçus par la rédaction
W. Ballmann—M. Gromov—V. Schroeder, Manifolds on nonpositive curvature (Progress in Mathematics, Vol. 61), IV+263 pages, Birkhäuser Verlag, Boston—Basel—Stuttgart, 1985.
M. Barr—C. Wells, Toposes, triples and theories (Grundlehren der mathematischen Wissenschaften,.
Bd. 278), XIII+345 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985.
DM 138,—.
M. J. Beeson, Foundations of constructive mathematics. Metamathematical studies (Ergebnisse der
Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 6), XXIII+466 pages, Springer-Verlag,
Berlin—Heidelberg—New York—Tokyo, 1985. — DM 168,—.
C. Camacho—A. Lins Neto, Geometrie theory of foliations, V+205 pages, Birkhäuser Verlag,
Boston—Basel—Stuttgart, 1985.
J. B. Conway, A course in functional analysis (Graduate Texts in Mathematics, Vol. 96), XIV+
404 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985. — DM 118,—.
Differential Geometric Methods in Mathematical Physics. Proceedings of an International Conference held at the Technical University of Clausthal, FRG, August 30—September 2, 1983.
Edited by H.-D. Doebner, J.-D. Hennig (Lecture Notes in Mathematics, Vol. 1139), V I +
337 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985. — DM 51,50.
B. A. Dubrovin—A. T. Fomenko—S. P. Novikov, Modern geometry — Methods and applications,
Part 2: The geometry and topology of manifolds. Translated by R. G. Burns (Graduate Texts
in Mathematics, Vol. 104), XV+430 pages, Springer-Verlag, Berlin—Heidelberg—New York—
Tokyo, 1985. — DM 158,—.
Dynamical Systems and Bifurcations. Proceedings of a Workshop held in Groningen, The Netherlands, April 16—20, 1984. Edited by B. L. J. Braaksma, H. W. Broer, F. Takens, V+129
pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985. — DM 21,50.
E. Grosswald, Representations of integers as sums of squares, XI+251 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985. — DM 148,—.
W. R. Knorr, The ancient tradition of geometric problems, IX+411 pages, Birkhäuser Verlag, Boston—Basel—Stuttgart, 1986.
H. König, Eigenvalue distribution of compact operators (Operator Theory: Advances and Applications, Vol. 16), 262 pages, Birkhäuser Verlag, Boston—Basel—Stuttgart, 1986.
O. A. Ladyzhenskaya, The boundary value problems of mathematical physics. Translated from the
Russian by J. Lohwater (Applied Mathematical Sciences, Vol. 49), XXX+322 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985. — DM 198,—.
S. Lang, Complex analysis. 2nd edition (Graduate Texts in Mathematics, Vol. 103), XIV+367
pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985. — DM 128,—.
L. Leindler, Strong approximation by Fourier series, 210 pages, Akadémiai Kiadó, Budapest, 1985.
J. Marsden—A. Weinstein, Calculus I, 2nd edition (Undergraduate Texts in Mathematics), X V +
385 pages, Springer-Verlag, Berlin—Heidelberg—Néw York—Tokyo, 1985. — DM 69,—.
_ , Calculus II, 2nd edition (Undergraduate Texts in Mathematics), XV+345 pages,
Springer-Verlag, Berlin—Heidelberg—New York—Tokyo," 1985. — DM 69,—.
454
, Calculus III, 2nd edition (Undergraduate Texts in Mathematics), XV+341 pages,
Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985. — DM 69,—.
Model-Theoretic Logics. Edited by J. Barwise, S. Feferman (Perspectives in Mathematical Logic),
XVIII+895 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985. — D M
480,—.
Nonlinear Analysis and Optimization. Proceedings of the International Conference held in Bologna,
Italy, May 3—7, 1982. Edited by C. Vinti (Lecture Notes in Mathematics, Vol. 1107), V +
214 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1984. — D M 31,50.
B. Oksendal, Stochastic differential equations. An introduction with applications (Universitext),
XIII+205 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985. — D M
42,-:
Orders and their Applications. Proceedings of a Conference held in Oberwolfach, West Germany,
June 3—9, 1984. Edited by I. Reiner, K. W. Roggenkamp (Lecture Notes in Mathematics,
Vol. 1142), X+306 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985.
— DM 45,—.
Ordinary and Partial Differential Equations. Proceedings of the Eighth Conference held at Dundee,
Scotland, June 25—29, 1984. Edited by B. D. Sleeman, R. J. Jarvis (Lecture Notes in Mathematics, Vol. 1151), XIV+357 pages, Springer-Verlag, Berlin—Heidelberg—New York—
Tokyo, 1985. — DM 51,50.
Probability in Banach Spaces V. Edited by A. Beck, R. Dudley, M. Hahn, J. Knelbs and M. Marcus
(Lecture Notes in Mathematics, Vol. 1153), VI+460 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985. — DM 64,—.
M. H. Protter—C. B. Morrey, Intermediate calculus. 2nd edition (Undergraduate Texts in Mathematics), X+650 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985. —
DM 128,—.
Recursion Theory Week. Proceedings of a Conference held in Oberwolfach, West Germany, April
15—21, 1984. Edited by H.-D. Ebbinghaus, G. H. Müller, G. E. Sacks (Lecture Notes in
Mathematics, Vol. 1141), IX+418 pages, Springer-Verlag, Berlin—Heidelberg—New Y o r k Tokyo, 1985. — DM 57,—.
Representations of Lie Groups and Lie Algebras. Edited by A. A. Kirillov, 225 pages, Akadémiai
Kiadó, Budapest, 1985.
M. Rosenblatt, Stationary sequences and random fields, 258 pages, Birkhäuser Verlag, Boston—
Basel—Stuttgart, 1985.
N. Z. Shor, Minimization methods for non-differentiable functions. Translated from the Russian
by K. C. Kiwiel, A. Ruszczynski (Springer Series in Computational Mathematics, Vol. 3),
VIII +162 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985. — D M
84,-.
C. Smorynski, Self-reference and modal logic (Universitext), XII+333 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985. — DM 88,—.
G. A. Sod, Numerical methods in fluid dynamics, IX+446 pages, Cambridge University Press,
Cambridge—London—New York, 1985. — £ 30.00.
F. H. Soon, Student's guide to calculus by J. Marsden and A. Weinstein, Vol. 1, XIV+312 pages,
Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985. — DM 48,—.
Stability Problems for Stochastic Models. Edited by V. V. Kalashnikov and V. M. Zolotarev (Lecture
Notes in Mathematics, Vol. 1155), VI+450 pages, Springer-Verlag, Berlin—Heidelberg—
New York—Tokyo, 1985. — DM 64,—.
Stochastic Analysis and Applications. Proceedings of the International Conference held at Swansea,
April 11—15, 1983. Edited by A. Truman, D. Williams (Lecture Notes in Mathematics, Vol.
455 Livres reçus par la rédaction
1095), V+199 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1984. —
DM 31,50.
A. Terras, Harmonie analysis on symmetric spaces and applications I, XV4-341 pages, SpringerVerlag, Berlin—Heidelberg—New York—Tokyo, 1985. — DM 138,—.
Theoretical Approaches to Turbulence. Edited by D. L. Dwoyer, M. Y. Hussaini, R. G. Voigt
(Applied Mathematical Sciences, Vol. 58), XII+373 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985. — DM 118,—.
A. N. Tikhonov—A. B. Vasileva—A G. Sveshnikov, Differential equations. Translated from the
Russian by A. B. Sossinskij (Springer Series in Soviet Mathematics), VIII+238 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985. — DM 98,—.
Topics in the Theoretical Bases and Applications of Computer Science. Proceedings of the 4th Hungarian Computer Science Conference Gyôr, Hungary, July 8—10, 1985. Edited by M. Aratô,
I. Kâtai, L. Varga, X + 514 pages, Akadémiai Kiadô, Budapest, 1986.
Universal Algebra and Lattice Theory. Edited by S. D. Corner (Lecture Notes in Mathematics,
Vol. 1149), VI+286 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985.
— DM 38,50.
R. L. Vaught, Set theory, X+141 pages, Birkhäuser Verlag, Boston—Basel—Stuttgart, 1985.
Vertex Operators in Mathematics and Physics. Edited by J. Lepowski, S. Mandelstam, I. M. Singer (Mathematical Sciences Research Institute Publications, Vol. 3), XIV+482 pages, Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1985. — DM 98,—.
M. Zamansky, Approximation des fonctions, VII+93 pages, Hermann, Paris, 1985. — 160 F.
TOMUS L—1986—50. KÖTET
Apikian, S. V., On cell complexes generated by geodesies in the non-Euclidean elliptic
plane
Atzmon, A., Characterization of operators of class C0 and a formula for their minimal
function
.
Bosznay, A. P.—Garay, B. M., On norms of projections
Daróczy, Z.—Járai, A.—Kátai, I., Intervallfüllende Folgen und volladditive Funktionen .
Davidson, K. R., The distance between unitary orbits of normal operators
Dhompongsa, S., Uniform laws of the iterated logarithm for Lipschitz classes of functions..
Douglas, R, G.—Paulsen, V. I., Completely bounded maps and hypo-Dirichlet algebras...
Dudek, J., Polynomial characterization of some idempotent algebras
Garay, B. M., cf. Bosznay, A. P.,
Ginovian, M. S., On the asymptotic estimate of the maximum likelihood of parameters of
the spectral density having zeros
Huruya, T., Decompositions of completely bounded maps
Járai, A., cf. Daróczy, Z.—Kátai, I.,
Kátai, 1., c f . Daróczy, Z.—Járai, A.,
59—66
191—211
87—92
337—350
213—223
105—124
143—157
39—49
87—92
169—182
183—189
337—350
337—350
Körmendi, S., Canonical number systems in Q ( / T )
351—357
Krasner, M., Abstract Galois theory and endotheory. I
253—286
Kristóf, J., C*-norms defined by positive linear forms
427—432
Leindler, L.—Tandori, K., On absolute summability of orthogonal series
99—104
Lengvárszky, Zs., A note on minimal clones
335—336
de León, M.—Salgado, M., Diagonal lifts of tensor fields to the frame bundle of second
order
67—86
Loi, N. V.—Wiegandt, R., Involution algebras and the Anderson—Divinsky—Suliñsky
property
5—14
Lamiste, Ü., Bäcklund's theorem and transformation for surfaces Ka in E„
51—57
Malkowsky, E., Toeplitz-Kriterien für Klassen von Matrixabbildungen zwischen Räumen
stark limitierbarer Folgen
125—131
Manó, V.—Skendzié, M.—Takaíi, A., On Stieltjes transform of distributions behaving
as regularly varying functions
405—410
Mitrovié, D., Behavior of the extended Cauchy representation of distributions
_
397—403
Móricz, F., Orthonormal systems of polynomials in the divergence theorems for double
orthogonal series
:
367—380
Nöbaner, R., Rédei-Funktionen und ihre Anwendung in der Kryptographie
287—298
de Pagter, B., A note on integral operators
225—230
Pálfy, P. P., The arity of minimal clones
331—333
Pap, E., A generalization of a theorem óf Dieudonné for ¿-triangular set functions
159—167
Paulsen, V. I., cf. Douglas, R. G
143—157
Pollák, G., Some sufficient conditions for hereditarily finitely based varieties of semigroups
299—330
РгбЫе, P., Does a given subfield of characteristic zero imply any restriction to the endomorphism monoids of
fields?
Pntnam, C. R., A connection between the unitary dilation and the normal extension of
a subnormal contraction
Rempulska, L., Remarks on the strong summability of numerical and orthogonal series by
some methods of Abel type
Salgado, M., cf. de León, M
Sallam, S., Error bounds for certain classes of quintic splines
Sasvári, Z., On measurable function with a finite number of negative squares
Skendiié, M., cf. Marié, V,—Takaíi, A
Такай, A., cf. Marié, V,—Skendiié, M.,
Tandon, K., cf. Leindler, L
Totik, V.—Vincze, I., On relations of coefficient conditions
Ulyanov, V. V., Normal approximation for sums of nonidentically distributed random
variables in Hilbert s p a c e s . . . :
Varma, A. К., Birkhoff quadrature formulas based on Tchebycheff nodes
Vincze, I., cf. Totik, V
Висков, О. В., Представление полиномов Лагерра
Wiegandt, R., cf. Loi, N. V.,
15—38
421—425
391—396
67—86
133—142
359—363
405—410
405—410
99—104
93—98
411—419
381—390
93—98
365—366
5—14
Bibliographie
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Analysis II — Surveys and New Trends. — J. P. Aubin—A. Celldma, Differential
Inclusions. — M. J. Beeson, Foundations of Constructive Mathematics, Mathematical Studies. — G. W. Bluman, Problem Book for First Year Calculus. —
K- W. Chang—F. A. Howes, Nonlinear Singular Perturbation Phenomena:
Theory and Applications. — Convex Analysis and Optimization. — J. B. Conway, A Course in Functional Analysis. — J. B. Conway, Functions of One
Complex Variable. — J. Diestel, Sequences and Series in Banach Spaces
Differential Geometry and Complex Analysis. — H. E. Edwards, Galois Theory. —
L. Euler, Elements of A l g e b r a — L. R. Foulds, Combinatorial Optimization for
Undergraduates. — D . S. Freed—K. K. Uhlenbeck, Instantons and FourManifolds. — J. K . Hale—L. T. Magalhaes—W. M. Oliva, An Introduction
to Infinite Dimensional Dynamical Systems: Geometric -Theory. — InfiniteDimensional Systems. — D . Kletzing, Structure and Representation of Q-Groups.
— M. A. Krasnosel'skiI—P. P. ZabreIko, Geometrical Methods of Nonlinear
Analysis. — Lie Group Representations 111. — Linear and Complex Analysis
Problem Book. — D . H. Luecking—L. A. Rubel, Complex Analysis: A Functional Analysis Approach—B. Malgrange, Lectures on the Theory of Functions of
Several Complex Variables. — J. Marsden—A. Weinstein, Calculus I—II—M.
— T. Matolcsi, A Concept of Mathematical Physics: Models for Space-Time
V. A. Morozov, Methods for Solving Incorrectly Posed Problems
J. D . Murray, Asymptotic Analysis— Nonlinear Analysis and Optimization— Y. Okuyama,
Absolute Summability of Fourier Series and Orthogonal S e r i e s — D . P. Parent, •
Exercises in Number Theory- — Probability Theory on Vector Spaces HI. —
R. Remmert, Funktiontheorie I — R. A. Rosenbaum—G. P. Johnson, Calculus:
Basic Concepts and Applications. — Yu. A. Rozanov, Markov Random Fields.
— N. Z. Shor, Minimization Methods for Non-Differentiable Functions. —
E. Solomon, Games Programming. — D . W. Stroock, An Introduction to the
Theory of Large Deviations. — N. M. Swerdlow—O. Neugebauer, Mathematical Astronomy in Copernicus's De Revolutionibus. — A. N. Tikhonov—
A. B. Vasil'eva—A.G. Sveshnikov, Differential Equations. — V. S. Varadarajan, Lie Groups, Lie Algebras and their Representations. — S. Watanabe,
Lectures on Stochastic Differential Equations and Malliavin Calculus— A. Weil,
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231—252
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D . J. Volper, Problem Solving Using UCSD Pascal. — Differential Geometric
Methods in Mathematical Physics. — Dynamical Systems and Bifurcations. —
E. Grosswald, Representations of Integers as Sums of Squares. — E. Hecke,
Lectures on the Theory of Algebraic Numbers. — D . Henry, Geometric Theory
of Semilinear Parabolic Equations. — E. Horowitz, Fundamentals of Programming Languages. — O. A. Ladyzhenskaya, The Boundary Value Problems of
Mathematical Physics. — S. Lang, Complex Analysis. — L. Leindler, Strong
Approximation by Fourier Series. — Model-Theoretic Logic. — B. Oksendal,
Stochastic Differential Equations. — Ordinary and Partial Differential Equations.
— R . S. Pierce, Associative Algebras. — D . P o l l a r d , Convergence of Stochastic
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Probability in Banach Spaces V
M. H. Procter—C. B. Morrey,
Jr., Intermediate Calculus. — Representations of Lie Groups and Lie Algebras.
— Representation Theory II. — Rings and Geometry. — M. Rosenblatt, Stationary Sequences and Random Fields. — J. G. Simmonds, A Brief on Tensor
Analysis
G. A. Sod, Numerical Methods in Fluid D y n a m i c s — E. H. Spanier,
Algebraic Topology. — Stability Problems for Stochastic Models. — Stochastic
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433—451
453—455
LIST OF PAPERS IN THE ALPHABETICAL ORDER
OF THE NAMES OF AUTHORS
TOM. 1 (1922)—TOM. 50 (1986)
A
Tom. Year
Page
Abdalla, S. M. and Szűcs, J., On an ergodic type theorem for von Neumann algebras
36 (1974),
167—172
Aczél, J., Zur Charakterisierung nomographisch einfach darstellbarer
Funktionen durch Differential- und Funktionalgleichungen
12 A (1950), 73—80
On some sequences defined by recurrence
13 (1949/50), 134—139
Einige aus Funktionalgleichungen zweier Veränderlichen ableitbare Differentialgleichungen
13 (1949/50), 179—189
Über die Gleichheit der Polynomfunktionen auf Ringen
21 (1960),
105—107
Aczél, J. und Fenyő, I., Über die Theorie der Mittelwerte
11 (1946/48), 239—245
Ádám, A., A theorem on algebraic operators in the most general sense . . . 18 (1957),
205—206
Zur Theorie der Wahrheitsfunktionen
21(1960),
47—52
Über die monotone Superposition der Wahrheitsfunktionen
23 (1962),
18—37
Ádám, A. and P o l l á k , G., On the congruences of finitely generated free
semigroups
31(1970),
259—278
Adams, M. E. and Sichler, J., Disjoint sublattices of lattices
46(1983),
77—83
Adeniran, T. M., Some absolute topological properties under monotone
unions
32(1971),
221—222
Adhikary, D. N., see Kishore, N.
Aghdam, A. M., On the strong nilstufe of rank two torsion free groups . . 49 (1985),
53—61
Agmon, S., see Mandelbrojt, S.
Ahmad, K., see Jain, P. K .
Ahmad, K., Jain, P. K. and Maskey, S. M., A note on boundedly complete decomposition of a Banach space
44 (1982),
365—-369
Alexits, G., Über den Grad der Annäherung durch die Cesàroschen Mittel
der Fourierreihe
3 (1927),
32—37
Sur la convergence des séries lacunaires
11 (1946/48), 251—253
Sur la convergence et la sommabilité presque partout des séries
de polynomes orthogonaux
12 B (1950), 223—225
Sur la convergence des séries orthonormales lacunaires
13 (1949/50), 14—17
Sur la convergence d'une classe de séries orthonormales
13(1949/50), 18—20
Eine Bemerkung zur starken Summierbarkeit der Orthogonalreihen 16 (1955),
127—129
Sur la convergence et la sommabilité des séries orthogonales
lacunaires
18(1957),
179—188
Une contribution à la théorie constructive des fonctions
19 (1958),
149—157
Über die Konvergenz fast überall der Orthogonalreihen bei jeder
Anordnung ihrer Glieder
19(1958),
158—161
Einige Beiträge zur Approximationstheorie
26(1965),
211—224
1
Alexits, G., On the characterization of classes of functions by their best
linear approximation
On the convergence of function series
Alexits, G. and Knapowski, S., A note on the best approximation by
linear forms of functions
Alexits, G. und Krâldc, D., Über die Bedeutung der strukturellen Eigenschaften einer Funktion für die Konvergenz ihrer Orthogonalentwicklungen
Über die Approximation im starken Sinne
Alextts, G. and Sharma, A., The influence of Lebesgue functions on the
convergence and summability of function series
Alexits, G- and Sunouchi, Gen-Ichiro, Characterization of classes of
functions by polynomial approximation
Aumov, S. A. and Joô, I., On the Riesz-summability of eigenfunction
expansions
On the convergence of eigenfunction expansions in Я ' - n o r m
Amemya, I. and Ando, T., Convergence of random products of contractions in Hilbert space
Amemya, I. and Halperin, I., Complemented modular lattices derived
from nonassociative rings
Anderson, J., Bunce, J. W, Deddens, J. A. and Williams, J. P., C*-algebras and derivation ranges
.^nderson, W. N. and Schreiber, M., The infimum of two projections . . .
Ando, T., On pair of commutative contractions
—
Matrices of normal extensions of subnormal operators
Truncated moment problems
Operators with a norm condition
Structure of operators with numerical radius one
Lebesgue-type decomposition of positive operators
An inequality between symmetric function means of positive operators
see Amemiya, I.
Ando, T., Ceauçescu, Z. and Foiaç, С., On intertwining dilations. П.
Apikian, S. V., On cell complexes generated by geodesies in non-Euclidean
elliptic planes
Apostol, С., Products of contractions in Hilbert space
Apostol, С., Foia§, C. and Voiculescu, D., Strongly reductive operators
are normal
Arany, D., Note sur «Le troisième problème de jeu»
Intégration de deux équations aux différences finies linéaires à
deux variables
Aratô, M., Round-off error propagation in the integration of ordinary
differential equations by one-step methods
Parameter estimation and Kaiman filtering in noisy background..
Arestov, V. V., see Арестов, В. В.
Arov, D. Z; see Аров, Д. 3.
Arsene, Gr. et Zsidô, L., Une propriété de type de Darboux dans les
algèbres de von Neumann
2
29 (1968),
34 (1973),
107—114
1—9
27 (1966),
49—54
18 (1957),
26(1965),
131—139
93—101
33 (1972),
1—10
31 (1970),
1—7
45 (1983),
48 (1985),
5—18
5—12
26 (1965),
239—244
20 (1959),
181—201
40(1978),
33 (1972),
24 (1963),
24 (1963),
31 (1970),
33(1972),
34(1973),
38 (1976),
211—227
165—168
88—90
91—96
319—334
169—178
11—15
253—260
45(1983),
19—22
39(1977),
3—14
50(1986),
33 (1972),
59—66
91—94
38(1976),
261—263
2 (1924/26), 39—42
10(1941/43),
42—47
45 (1983),
48(1985),
23—31
13—23
30(1969),
195—198
Asatiani, V. O. and Chanturia, Z. A., The modulus of variation of a
function and the Banach indicatrix
Askey, R., Smoothness conditions for Fourier series with monotone coefficients
ATKINSON, F. V., On relatively regular operators
Atzmon, A., Characterization of operators of class C 0 and a formula for
their minimal function
Aumann, G., Über eine Ungleichung der Wahrscheinlichkeitsrechnung . . .
Aupetit, B. and Zemánek, J., On zeros of analytic multivalued functions
Azoff, E. A. and Davis, Ch., On distances between unitary orbits of selfadjoint operators
45 (1983),
51—66
28 (1967),
167—172
15(1953/54), 38—56
50(1986),
191—211
13 (1949/50), 163—168
46 (1983),
311—316
47 (1984),
419-439
B
Baker, I. N., On some results of A. Rényi and C. Rényi concerning
periodic entire functions
27 (1966),
The value distribution of composite entire functions
32 (1971),
Baker, I. N. and Liverpool, L. S. O., The value distribution of entire
functions of order at most one
41 (1979),
Baker, J. A., D'Alembert's functional equation in Banach algebras
32 (1971),
Bakhshetsyan, A. V., see EaxmeiwH, A. B.
Baldwin, J. T. and Berman, J., Definable principal congruence relations:
Kith and kin
44 (1982),
Bandelt, H. J., Toleranzrelationen als Galoisverbindungen
46 (1983),
Bansal, R. and Yadav, B. S., On systems of TV-variable weighted shifts . . 47 (1984),
Barbu, V., On local properties of pseudo-differential operators
30 (1969),
Barnes, B. A., On thin operators relative to an ideal in a von Neumann
algebra....
38(1976),
Barria, J. and Bruzual, R., On approximation by unitary operators
43 (1981),
Bauer, M., Über ein Problem von Dedekind
1 (1922/23),
Über die Erweiterung eines algebraischen Zahlkörpers durch Henselsche Grenzwerte
1 (1922/23),
Bemerkungen zur Theorie der Differente
1 (1922/23),
Zur Theorie der algebraischen Körper
2 (1924/26),
Bemerkung zur Algebra
4 (1928/29),
Elementare Bemerkungen über identische Kongruenzen
• 6 (1932/34),
Über die alternierende Gruppe
6 (1932/34),
Bemerkungen zum Hensel—Oreschen Hauptsatze
8 (1936/37),
Zur Theorie der Kreiskörper
9 (1938/40),
Über zusammengesetzte relativ Galoissche Zahlkörper
9 (1938/40),
Über die Zusammensetzung algebraischer Zahlkörper
9 (1938/40),
Bauer, M. und Tschebotaröw, N., /j-adi scher Beweis des zweiten Hauptsatzes von Herrn Ore
4 (1928/29),
Baumgartner, K., Über charakteristische Eigenschaften der Divisionsringe
27(1966),
Baumslag, G., Kovács, L. G. and Neumann, B. H., On products of normal subgroups
26(1965),
Beaumont, R. A. and Pierce, R. S., Subrings of Algebraic Number Fields 22 (1961),
M*
197—200
87—90
3—14
225—234
255—270
55—58
93—99
263—270
265—274
239—242
14—17
74—79
195—198
69—71
244—245
46—48
222—223
64—67
110—112
206—211
212—217
56—57
229—231
145—148
202—216
3
Beaumont, R. A. and Wisner, R. J., Rings with additive group which is a
tor sion-free group of rank two
20 (1959),
Bekes, R. A., A convolution theorem and a remark on uniformly closed
Fourier algebras
38 (1976),
Benado, M., Sur une propriété d'interpolation remarquable dans la théorie des ensembles partiellement ordonnés
22(1961),
Berberian, S. K., A note on operators whose spectrum is a spectral set 27 (1966),
Produit de convolution des mesures opératorielles
40 (1978),
Bercovici, H., Jordan model for some operators
38 (1976),
Co-Fredholm operators. I
41 (1979),
Co-Fredholm operators. II
42 (1980),
On the Jordan model of C 0 operators. II
42 (1980),
Bercovici, H-, Foiaç, C , Kérchy, L. and Sz.-Nagy, B., Compléments à
l'étude des opérateurs de classe C0. IV
41 (1979),
Bercovici, H., Foiaç, C. et Sz.-Nagy, B., Compléments à l'étude des
opérateurs de classe C 0 .
ffl
37 (1975),
Reflexive and hyper-reflexive operators of class C 0
43 (1981),
Bercovici, H-, Foia?, C., Pearcy, C. M. and Sz.-Nagy, B., Functional
models and extended spectral dominance
43 (1981),
Factoring compact operator-valued functions
48 (1985),
Bercovici, H. and Kérchy, L., Quasisimilarity and properties of the commutant of Cu contractions
45 (1983),
Bercovici, H. and Voiculescu, D., Tensor operations on characteristic
functions of C0 contractions
39 (1977),
Berens, H. and Westphal, U., A Cauchy problem for a generalized wave
equation
29(1968),
Berger, C- A. and Stampfli, J. G., Norm relations and skew dilations . . . 28 (1967),
Berman, J., see Baldwin, J. T.
BernshteIn, S. N., see BepHnrrefiH, C. H.
Berwald, L., Elementare Sätze über die Abgrenzung der Wurzeln einer
algebraischen Gleichung
6 (1932/34),
B k y a l i s , A. P., see Bhksjihc, A. II.
Birkenhead, R., Sauer, N. and Stone, M. G., The algebraic representation of semigroups and lattices; representing lattice extensions... 43 (1981),
Birkhof?, G. D., A remark on the dynamical rôle of Poincaré's last geometric theorem
4(1928/29),
Blair, W. D., Quotient rings of algebras which are module finite and
projective
36(1974),
Blum, J. R., Ergodic sequences of integers
46 (1983),
Blum, J. R., Eisenberg, B. and Hahn, L.-S., Ergodic theory and the
measure of sets in the Bohr group
34(1973),
Boas, R. P., Fourier series with a sequence of positive coefficients
12 B (1950),
Bochner, S., Über Faktorenfolgen für Fouriersche Reihen
4 (1928/29),
Bochner, S. and Chandrasekharan, K., Lattice points and Fourier
expansions
12 B (1950),
Bogdanovió, S-, Bands of power joined semigroups
44 (1982),
Bognár, J., Operator conjugation with respect to symmetric and skewsymmetric forms
; . . 31 (1970),
4
105—116
3—6
1—5
201—203
3—8
275—279
15—27
3—42
43—56
29—31
313—322
5—13
243—254
25—3667—74
205—231
93—106
191—196
209—221
255—261
6—11
265—266
243—247
17—24
35—37
125—129
1—15
3—4
69—73
Bognár, J., Star-algebras induced by non-degenerate inner products
A proof of the spectral theorem for /-positive operators
Bognár, J. and Krámli, A., Operators of the form C*C in indefinite
inner product spaces
Bohr, H., Über einen Satz von J. Pál
On limit periodic functions of infinitely many variables
Bojadziev, Hr., A note on a paper of S. Watanabe
Borsuk, К., On the imbedding of л-dimensional sets in 2/i-dimensional
absolute retracts
Bosznay, A. P. and Garay, В. M., ON norms of projections
Bouldin, Я-, The convex structure of the set of positive approximants for
a given operator
—
Essential spectrum for a Banach space operator
Compact approximants
Bouldin, R. H. and Rogers, D . D., Normal dilations and operator
approximations
Boyd, D. W., The spectrum of the Cesàro operator
Boyd, P. V-, Spectra of convolution operators
Brehmer, S. Über vertauschbare Kontraktionen des Hilbertschen Raumes
Brelot, M., Familles de Perron et problème de Dirichlet
Remarque sur le prolongement fonctionnel linéaire et le problème
de Dirichlet
BrodskiI, M. S., see Бродский, M. Ш.
Brodskiî, S. D . and Kogalovskű, S. R., see Бродский, С. Д. и Когаловский, С. Д.
Brodski!, V. M., see Бродский, В. M.
Brown, A. and Douglas, R. G., On maximum theorems for analytic
operator functions
Brown, A., Halmos, P. R . and Shields, A. L., Cesàro operators
Brown, A. and Pearcy, C., Compact restrictions of operators
Compact restrictions of operators. II
Bruck, R. H., Sums of normal endomorphisms. П
Bruun, de N. G-, On unitary equivalence of unitary dilations of contractions in Hilbert space
Bruzual, R., see Barria, J.
Bunce, J. W., see Anderson, J.
Burgess, J. P., On a set-mapping problem of Hajnal and Máté
Burnham, J. T. and Goldberg, R. R-, The convolution theorems of Dieudonné
Butzer, P. L. und Pawelke, S-, Ableitungen von trigonometrischen
Approximationsprozessen
Butzer, P. L., Scherer, K. and Westphal, U., On the Banach—Steinhaus theorem and approximation in locally convex spaces
Butzer, P. L. and S c h u l z , D., The random martingale central limit theorem and weak law of large numbers with o-rates
An extension of the Lindeberg—Trotter operator-theoretic approach to limit theorems for dependent random variables. I
39 (1977),
45 (1983),
29(1968),
7 (1934/35),
12 В (1950),
46 (1983),
15—26
75—80
19—30
129—135
145—149
303—304
12 A (1950), 112—116
50 (1986),
87—92
37 (1975),
41 (1979),
44(1982),
177—190
33—37
5—11
39 (1977),
29 (1968),
35 (1973),
22 (1961),
9 (1938/40),
233—243
31—34
31—37
106—111
133—153
12 В (1950), 150—152
26(1965),
26 (1965),
32 (1971),
33 (1972),
22 (1961),
325—328
125—138
271—282
161—164
6—30
23(1962),
100—105
41 (1979),
283—288
36 (1974),
1—3
28 (1967),
173—184
34 (1973),
25—34
45 (1983),
81—94
48 (1985),
37—54
5
c
Campbell, S. L., Subnormal operators with nontrivial quasinormal extensions
Carathéodory, C., Über die Variationsrechnung bei mehrfachen Integralen
G a r l i t z , L., A note on exponential sums
Permutations in
finite
fields
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6
37 (1975),
4 (1928/29),
21 (1960),
24(1963),
6 (1932/34),
191—193
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135—143
196—203
85—104
12 A (1950),
81—100
21(1960),
38 (1976),
111—124
281—290
40 (1978),
41(1979),
9—32
457—459
48 (1985),
26 (1965),
42 (1980),
44(1982),
46 (1983),
55—63
283—288
229—232
13—16
35—40
39(1977),
245—245
44 (1982),
17—21
14 (1951/52), 188—196
39 (1977),
29—30
41 (1979),
289—293
48 (1985),
65—70
45 (1983),
39 (1977),
95—109
31—37
36(1974),
47 (1984),
36 (1974),
22 (1961),
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15 (1953/54),
Sur une généralisation de la notion de dérivée
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45 (1983),
Csernyák, L. and Leindler, L., Über die Divergenz der Partialsummen
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27 (1966),
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35—52
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71—74
71—76
77—82
165—171
187—189
3—10
25—31
33—36
7—11
111—117
75—84
57—65
17—23
157—160
119—129
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48—50
140—142
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36(1974),
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48(1985),
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44 (1982),
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45 (1983),
48 (1985),
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41(1979),
39-45
43(1981),
44 (1982),
267—272
35—42
46 (1983),
41—54
49 (1985),
143—149
46 (1983),
381—388
22 (1961),
31—41
50(1986),
50 (1986),
29 (1968),
19 (1958),
29 (1968),
31 (1970),
31 (1970),
337—350
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225—246
172—187
69—86
75—86
301—318
32(1971),
127—139
27 (1966),
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28 (1967),
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40 (1978),
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47 (1984),
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24 (1963),
236—243
48 (1985)
103—128
12 A (1950),
14 (1951/52),
15 (1953/54),
16 (1955),
22(1961),
213—227
145—156
29—30
246—250
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45 (1983),
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25 (1964),
177—178
11 (1946/48), 17—18
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30 (1969),
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27 (1966),
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37 (1975),
39 (1977),
45(1983),
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299—304
87—89
195—199
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161—163
129—134
12 B (1950),
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33 (1972),
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15 (1953/54), 99—103
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15 (1953/54), 211—222
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35 (1973),
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194—198
113—115
11—17
57—63
255—259
17 (1956),
18 (1957),
250—260
243—260
20 (1959),
21 (I960),
81—90
154—163
34(1973),
69—79
30 (1969),
313—324
41 (1979),
295—305
8 (1936/37),241—255
40 (1978),
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42 (1980),
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44 (1982),
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12 A (1950),
28 (1967),
49(1985),
36 (1974),
36 (1974),
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329—334
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117—124
258—260
101—104
267—270
151—155
27—28
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38 (1976),
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21
KALMÁR, L . , E i n e B e m e r k u n g z u r E n t s c h e i d u n g s t h e o r i e
Über die mittlere Anzahl der Produktdarstellungen der Zahlen.
(Erste Mitteilung.)
Ein Beitrag zum Entscheidungsproblem
Ein Beweis des Ruffini—Abelschen Satzes
Über einen Löwenheimschen Satz
Über die Axiomatisierbarkeit des Aussagenkalküls
On the possibility of definition by recursion
Another proof of the Gödel—Rosser incompletability theorem.
4 (1928/29), 2 4 8 — 2 5 2
5 (1930/32),
5 (1930/32),
6 (1932/34),
7 (1934/35),
7 (1934/35),
9 (1938/40),
12 A (1950),
95—107
222—236
59—60
112—121
222—243
227—232
39—43
41 (1979),
33 (1972),
317--325
307--314
30 (1969),
27 (1966),
277--284
1--16
41 (1979),
29 (1968),
29 (1968),
127--132
199--206
207--212
29 (1968),
30 (1969),
30 (1969),
33 (1972),
39 (1977),
44 (1982),
45 (1983),
47 (1984),
271--282
99--106'
305--311
81—89
319--328
299--305
253--260
85--92
KÁTAI, I. and KOVÁCS, B., Kanonische Zahlensysteme in der Theorie der
42 (1980),
quadratischen algebraischen Zahlen
Multiplicative functions with nearly integer values
48 (1985),
KÁTAI, I. and SZABÓ, J., Canonical number systems for complex integers . 37 (1975),
99--107
221--233
255--260
KALOUJNINE, L . , see KRASNER, M .
KAN, CHARN-HUEN, On Fong and Sucheston's mixing property of operators
in a Hilbert space
KAPP, K. M., Bi-ideals in associative rings and semi-groups
KAPPE, W. P., Über gruppentheoretische Eigenschaften, die sich auf ¿-Produkte übertragen
KARLJN, S., Generalized Bernstein inequalities
KÁSZONYI, L., A characterization of binary geometries of types K(3) and
K(4)
KÁTAI, I-, O n t h e s u m £
<#(/(«))
On a classification of primes
On oscillation of the number of primes in an arithmetical progression
Some algorithms for the representation of natural numbers
Some results and problems in the theory of additive functions
On random multiplicative function
Change of the sum of digits by multiplication
Additive functions with regularity properties
On arithmetic functions with regularity properties
On additive functions satisfying a congruence
s e e CORRADI, K . A .
see DARÓCZY, Z .
see ERDŐS, P.
KATO, Y . , see IZUMDMO, S.
KAUFMAN, R., A type of extension of Banach spaces
A problem on lacunary series
KELLER, O.-H., Darstellungen von Restklassen (mod ri) als Summen von
zwei Quadraten
KELLOGG, O. D., Some notes on the notion of capacity in potential theory
27 (1966),
29 (1968),
163--166
313--316
191--192
25(1964),
4 (1928/29),, 1-- 5
KEPKA, T . , s e e JEZEK, J .
KÉRCHY, L., On C'0-operators with property (P)
On p-weak contractions
On the commutant of C u -contractions
On invariant subspace lattices of Cii-contractions
Injection-similar isometries
22
42 (1980),
42 (1980),
43(1981),
43 (1981),
44 (1982),
109--116
281--297
15--26
281--293
157--163
KÉRCHY, L., Approximation and quasisimilarity
48 (1985),
227—242
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KERÉKJÁRTÓ, B., AZ analysis és geometria topologiai alapjairól. (Sur les
fondements topologiques de l'Analyse et de la Géométrie.)
Remarques sur des propriétés topologiques
Sur les familles de surfaces et de courbes
Involutions et surfaces continues (Première communication)
Involutions et surfaces continues (Deuxième communication)
The plane translation theorem of Brouwer and the last geometric
theorem of Poincaré
Démonstration élémentaire du théorème de Jordan sur les courbes
planes
Über die fixpunktfreien Abbildungen der Ebene
Topologische Charakterisierung der linearen Abbildungen
Ergänzung zu meinem Aufsatz: Topologische Charakterisierung
der linearen Abbildungen
Über reguläre Abbildungen von Flächen auf sich
Über die regulären Abbildungen des Torus
Démonstration nouvelle d'un théorème de Klein et Poincaré
Sur l'indice des transformations analytiques
Bemerkung über reguläre Abbildungen von Flächen
Sur les groupes transitifs de la droite
7 (1934/35), 58—59
7(1934/35), 65—75
7(1934/35), 76—84
7 (1934/35), 160—162
7 (1934/35), 163—172
7 (1934/35), 206—206
10 (1941/43), 21—35
KERLIN, E . a n d LAMBERT, A . , Strictly cyclic s h i f t s o n I*
3 5 (1973),
KERTÉSZ, A., On a theorem of Kulikov and Dieudonné
Systems of equations over modules
Correction to my paper "Systems of equations over modules"...
:
On independent sets of elements in algebra
Noethersche Ringe, die artinsch sind
Eine neue Charakterisierung der halbeinfachen Ringe
KERTÉSZ, A., FUCHS, L. and SZELE, T., On abelian groups whose subgroups
are endomorphic images
KERTÉSZ, A. and SZELE, T., On abelian groups every multiple of which is a
direct summand
Abelian groups every finitely generated subgroup of which is an
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KHAN, R. S., On power series with positive coefficients
15 (1953/54), 61—69
18(1957),
207—234
19 (1958),
251—252
21 (1960),
260—269
31 (1970),
219—221
34 (1973),
169—170
1 (1922/23),
2 (1924/26),
2 (1924/26),
3 (1927),
3 (1927),
46—54
157—161
162—166
49—67
242—249
4(1928/29), 86—102
5(1930/32), 56—59
6 (1932/34), 226—234
6 (1932/34), 235—262
16 (1955),
87—94
77—88
14 (1951/52), 157—166
15 (1953/54), 70—76
30 (1969),
255—257
KHARAZOV, D . F . , see X a p a 3 o a , J \ . <b.
KHAZAL, R. R., Multiplicative periodicity in rings
41 (1979),
Classes of universal algebras, their non-factors and periodic rings 43 (1981),
KIM, Kl HANG and ROUSH, F. W., Counting additive spaces of sets
40 (1978),
Quasi-varieties of binary relations
40 (1978),
133—136
295—299
81—87
89—91
KISHORE, N . a n d ADHIKARY, D . N . , I n d e p e n d e n c e of t h e c o n d i t i o n s of
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33 (1972),
Dependence of associativity conditions for ternary operations . . . 40 (1978),
KISHORE, N . a n d HOTTA, G . G , O n | # , / > „ | s u m m a b i l i t y f a c t o r s
KISS, E. W., Term functions and subalgebras
KTTTANEH, F., Some characterizations of self-adjoint operators •
KLEE, V., The generation of affine hulls
3 1 (1970),
: . 47 (1984),
47(1984),
. 2 4 (1963),
275—284
285—290
9—12
303—306
441—444
60—81
23
KLUG, L., Einige Sätze über Kegelschnitte
1 (1922/23),
Über konjugierte Kegelschnitt-tripel
2 (1924/26),
KLUKOVITS, L., On commutative universal algebras
34 (1973),
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37(1975),
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KNAPOWSKI, S., s e e ALEXITS, G .
KNOEBEL, A., The finite interpolation property for small sets of classical
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44(1982),
KOCHENDÖRFFER, R., Hallgruppen mit ausgezeichnetem Repräsentatensystem
21 (1960),
KOEHLER, J., Some torsion free rank two groups
25 (1964),
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KOGALOVSKÍ, S . R . , see BRODSKII, S . D .
KOKSMA, J. F. and SALEM, R., Uniform distribution and Lebesgue integration
12 В (1950), 87—96
KOLLÁR, J., Automorphism groups of subalgebras; a concrete characterization
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KOMLÓS, J . a n d RÉVÉSZ, P . , R e m a r k t o a p a p e r of G a p o s h k i n
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KOMLÓSI, S., Mean ergodicity in G-semifinite von Neumann algebras
KOMORNIK, V., On the Korovkin closure
Lower estimates for the eigenfunctions of the Schrödinger operator
:
Upper estimates for the eigenfunctions of higher order of a linear
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Local upper estimates for the eigenfunctions of a linear differential
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Singular perturbations of singular systems
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KÖNIG, D., Sur les rapports topologiques d'un problème d'analyse combinatoire
Über eine Schlussweise aus dem Endlichen ins Unendliche
Über trennende Knotenpunkte in Graphen (nebst Anwendungen
auf Determinanten und Matrizen)
KORÁNYI, Ä., Note on the theory of monotone operator functions
On a theorem of Löwner and its connection with resolvents of
selfadjoint transformations
Я'-spaces on compact nilmanifolds
41 (1979),
43 (1981),
327—334
41—43
44(1982),
95—98
45 (1983),
261—271
48(1985),
49 (1985),
243—256
345—352
3 _
2 (1924/26), 32—38
3 (1927),
121—130
5 (1930/32), 155—179
16 (1955),
241—245
17 (1956),
63—70
34 (1973), . 175—190
KÖRMENDI, S., Canonical number systems in ß(K2)
50(1986),
KORVIN, G-, On a theorem of L. Rédei about complete oriented graphs 27 (1966),
351—357
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KÓSA, A., On the homotopy type of some spaces occurring in the calculus
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45(1983),
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47 (1984),
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Subalgebra lattices, simplicity and
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24
rigidity
KOVÁCS, I., Un complément à la théorie de 1'« intégration non commutative»
Sur certains automorphismes des algèbres hilbertiennes
Ergodic theorems for gages
Dilation theory and one-parameter semigroups of contractions . .
21 (1960),
22 (1961),
24(1963),
45 (1983),
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234—242
103—118
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48 (1985),
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3 8 (1976),
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KOVÁCS, I . a n d MOCANU, G H . , U n i t a r y d i l a t i o n s a n d C * - a I g e b r a s
KOVÁCS, I. and Szűcs, J., Ergodic type theorems in von Neumann algebras
27 (1966),
A note on invariant linear forms on von Neumann algebras
30(1969),
KOVÁCS, L. G., On the existence of Baur-soluble groups of arbitrary height 26 (1965),
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KOVARIK, Z. V., Similarity and interpolation between projectors
KŐVÁRY, T., Asymptotic values of entire functions of finite order with density conditions
KRABBE, G. L., Integration with respect to operator-valued functions
26(1965),
39 (1977),
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26 (1965),
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50 (1986),
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KRASNER, M. et KALOUJNINE, L., Produit complet des groupes de permutations et problème d'extension de groupes. I
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13 (1949/50), 208—230'
14(1951/52), 39—66
14 (1951/52), 69—82
KRAUSS, P . H . , see CLARK, D . M .
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KREÏN, M. G. und LANGER, H., Über die ß-Funktion eines ji-hermitischen
Operators im Räume 17«
Analytic matrix functions related to operators in the space IIx
KRISTÓF, J., Ortholattis linéarisables
KRISZTIN, T., On the convergence of solutions of functional differential
equations
Convergence of solutions of a nonlinear integrodifferential equation arising in compartmental systems
KROÓ, A., On the unicity of best Chebyshev approximation of differentiable
functions
KROTOV, V. G., Note on the convergence of Fourier series in the spaces AZ
KROTOV, V. G. and LEINDLER, L., On the strong summability of Fourier
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fl"
KÜHNE, R., Minimaxprinzipe für stark gedämpfte Scharen
KULBACKA, M., Sur l'ensemble des points de l'asymétrie approximative
34(1973),
43(1981),
49 (1985),
50 (1986),
191—230»
181—205
387—395
427—432
43 (1981),
45—54
47 (1984),
471—485
47(1984),
41 (1979),
377—389
335—338-
40(1978),
40 (1978),
21 (1960),
93—98
93—98
90—95
'
KUMAR, A . , see GUPTA, D . K -
25
KUMAR, R . D . CH., s e e SINGH, R . K .
KÜMMERER, B . a n d NAGEL, R . , M e a n e r g o d i c s e m i g r o u p s o n W * - a l g e b r a s 4 1 (1979),
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KUREPA, S., A cosine functional equation in Banach algebras
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KÜRSCHAK, J., Über das identische Verschwinden der Variation
Eine Verallgemeinerung von Moivres Problem in der Wahrscheinlichkeitsrechnung
Rösselsprung auf dem unendlichen Schachbrette
Eine Ergänzung zur Abhandlung von Herrn L. La Paz über
Variationsprobleme, für welche die Extremalflächen Minimalflächen sind
Die Irreduzibilität einer Determinante der analytischen Geometrie
44 (1982),
23(1962),
1 (1922/23),
151—159
345—357
255—267
6—13
1 (1922/23), 139—143
4 (1928/29), 12—13
5 (1930/32), 246—246
6 (1932/34), 21—26
KUZHEL', A . B . , s e e KyaceJU., A . B .
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L
LAITERER, YU., see r o x 6 e p r , H . I J .
LAJOS, S., Generalized ideals in semigroups
A criterion for Neumann regularity of normal semigroups
A note on completely regular semigroups
On semigroups which are semilattices of groups
On (m, n)-ideals in regular duo semigroups
Characterization of completely regular inverse semigroups
A note on semilattices of groups
Characterizations of completely regular elements in semigroups . .
Contributions to the ideal theory of semigroups
LAJOS, S. and SzAsz, F., Some characterizations of two-sided regular rings
Bi-ideal in associative
rings
22 (1961),
217—222
25 (1964),
172—173
28(1967),
261—265
30(1969),
133—136
31 (1970),
179—180
31 (1970),
229—231
33 (1972),
315—317
40 (1978), • 297—300
42(1980),
117—120
31 (1970),
223—228
32 (1971),
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LAKSER, H . , s e e CHANDRAN, V . R LAKSER, H . , see GRÄTZER, G .
LALLEMENT, G- and PETRICH, M-, A generalization of the Rees theorem in
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2 (1924/26),
LANGER, H., Über eine Klasse nichtlinearer Eigenwertprobleme
35 (1973),
Factorization of operator pencils
38 (1976),
On measurable Hermitian indefinite functions with a finite number
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45(1983),
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73—86
83—96
281—292
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LANGER, H. und KREIN, M. G., Über die Q-Funktion eines s-hermitischen
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LANGER, H. and TEXTORIUS, B., Generalized resolvents of contractions . . . 44 (1982),
26
191—230
125—131
LA PAZ, L., Variation problems of which the extremals are minimal
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LEE, A. and TÔTH, G., On variation of spaces of harmonic maps into spheres 46 (1983),
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32 (1971),
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37 (1975),
VAN LEEUWEN, L. C , Über die zulässigen Ideale in Szépschen Ringerweiterungen
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28 (1967),
Square extensions of finite rings
29(1968),
Remarks on endomorphism rings of torsion free abelian groups 32 (1971),
:
— On p-pure subgroups of torsion-free cotorsion groups
35(1973),
Rings with e as a radical element
38 (1976),
Complements of radicals in the class of hereditarily artinian rings 39(1977),
On questions of hereditariness of radicals
47(1984),
LEIBOWITZ, G. M., Spectra of finite range Cesàro operators
35 (1973),
LEINDLER, L., Ü b e r die orthogonalen Polynomsysteme
Zur Frage der Approximation durch orthonormierte Polynomsysteme
Über die absolute Summierbarkeit der Orthogonalreihen
Über die starke Summierbarkeit der Orthogonalreihen
Abschätzungen für die Partialsummen und für die (R, Â(n), l)Mittel allgemeiner Orthogonalreihen
Über die Rieszschen Mittel allgemeiner Orthogonalreihen
Über verschiedene Konvergenzarten trigonometrischer Reihen...
Über Konvergenz- und Summationseigenschaften von Haarschen
Reihen
Über verschiedene Konvergenzarten der trigonometrischen Reihen.
n (Die Entwicklung der Ableitungen)
Über die Unverschärfbarkeit eines Satzes von Orlicz bezüglich
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Über verschiedene Konvergenzarten trigonometrischer Reihen. IH
On the strong summability of orthogonal series
On the absolute summability factors of Fourier series
Corrigendum to the paper: On the strong summability of orthogonal series
On summability of Fourier series
Note on power series with positive coefficients
On embedding of classes H "
Generalization of inequalities of Hardy and Littlewood
On the strong approximation of orthogonal series
On some inequalities concerning series of positive terms
On a certain converse of Holder's inequality. II
83—86
149—158
21—76
247—260
345—350
95—106
97—102
313—318
307—319
27—29
21 (1960),
19—46
22(1961),
22 (1961),
23(1962),
129—132
243—268
82—91
23 (1962),
24 (1963),
25(1964),
227—236
129—138
233—249
26(1965),
19—30
26(1965),
117—124
26(1965),
27 (1966),
27 (1966),
28(1967),
245—248
205—216
217—228
323—336
28 (1967),
29(1968),
30 (1969),
31 (1970),
31 (1970),
32 (1971),
33(1972),
33(1972),
337—338
147—162
259—261
13—31
279—285
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11—14
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27
LEINDLER, L., On a generalization of Bernoulli's inequality
On imbedding theorems
Remarks on inequalities of series of positive terms
On the strong approximation of orthogonal series
Note on integral inequalities
An integrability theorem for power series
On the strong approximation of Fourier series
Generalization of a converse of Holder's inequality
Imbedding theorems and appstrongroximation
Strong approximation and generalized Zygmund class
On the absolute Riesz summability of orthogonal series
Limit cases in the strong approximation of orthogonal series
33 (1972),
34(1973),
35 (1973),
37 (1975),
37(1975),
38 (1976),
38(1976),
38(1976),
43 (1981),
43 (1981),
46 (1983),
48 (1985),
225—230
231—244
107—110
87—94
261—262
103—105
317—324
107—112
55—64
301—309
203—209
269—284
47(1984),
39 (1977),
371—375
95—102
24 (1963),
228—230
43(1981),
311—319
45(1983),
293—304
50 (1986),
99^104
32(1971),
101—107
50 (1986),
335—336
see CSERNYÁK, L .
see KROTOV, V . G .
LEINDLER, L. and MEIR, A., Embedding theorems and strong approximation
LEINDLER, L. and NÉMETH, J., An integrability theorem for power series
LEINDLER, L. und PÁL, L. G-, Über die Konvergenz von Partialsummen der
Orthogonalreihen
LEINDLER, L. and SCHWINN, H., Über die absolute Summierbarkeit der
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On the strong and extra strong approximation of orthogonal
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LEINDLER, L. and TANDORI, К., On absolute summability of orthogonal
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LENARD, A., Probabilistic version of Trotter's exponential product formula
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LÄNGER, H . , see DORNINGER, D .
LENGVÁRSZKY, Zs., A note on minimal clones
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48 (1985),
DE LEON, M. and SALGADO, M., Diagonal lifts of tensor fields to the frame
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50(1986),
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LEONT'EV, A . F., see Леонтьев, А . Ф.
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12 В (1950),
LŒBERMAU, A., Finite type representation of finite symmetric groups
38 (1976),
Entropy of states of a gage space
40(1978),
LINDEN, H., Zur Symmetrisierung gewisser rationaler Eigenwertaufgaben 49 (1985),
LINDNER, CH. С., Extending mutually orthogonal partial latin squares . . . 32 (1971),
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113—114
99—105
357—364
283—285
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LIPKA, ST., Über die Abgrenzung der Wurzeln von algebraischen Gleichungen 3(1927),
73—80
Über asymptotische Entwicklungen der Mittag-Lefflerschen
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3(1927),
211—223
Zur Theorie der algebraischen Gleichungen mit positiven Koeffizienten
5 (1930/32), 69—77
28
LIPKA, ST., ZU den Verallgemeinerungen des Rolleschen Satzes
Über die Descartessche Zeichenregel
Über den Zusammenhang zweier Sätze der Funktionentheorie . . .
6 (1932/43), 180—183
7 (1934/35), 177—185
8 (1936/37), 160—165
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Compactification, Baire functions, and Daniell integration
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LORCH, E . R . and HING TONG, Compactness, metrizability, and Baire iso-
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LORENTZ, R. A. H. and REJTÖ, P. A., Some integral operators of trace
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LOSERT, V-, Uniformly distributed sequences on compact, separable, nonmetrizable groups
LOSONCZI, L., Bestimmung aller nichtkonstanten Lösungen von linearen
Funktionalgleichungen
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41 (1979),
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25 (1964),
34 (1973),
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50(1986),
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47(1984),
2 (1924/26),
12 B (1950),
14 (1951/52),
18 (1957),
32 (1971),
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129—133
199—203
144—144
23—28
335—343
34(1973),
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26(1965),
30 (1969),
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285—288
50 (1986),
17 (1956),
38(1976),
44 (1982),
15 (1953/54),
15 (1953/54),
16 (1955),
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190—194
149—152
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17 (1956),
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19(1958),
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1 (1922/23),
2 (1924/26),
2 (1924/26),
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18—31
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2 (1924/26), 209—225
6 (1932/34), 136—142
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7 (1934/35), 85—94
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40 (1978),
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STACHÖ, T., Operationskalkül von Heaviside und Laplacesche Transformation
35 (1973),
26 (1965),
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273—282
38(1976),
365—374
39 (1977),
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39 (1977),
42(1980),
44 (1982),
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23 (1962),
24 (1963),
28(1967),
31(1970),
32 (1971),
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203—218
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Über die regulären duo-Elemente in Gruppoid Verbänden
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38 (1976),
—
On derivations in generalized matroid lattices
44 (1982),
SUNGL, V., Eine Kennzeichnung der endlichen einfachen Gruppe der Ordnung 604 800
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STRÄTILÄ, S., O n the tensor product of weights o n W'*-algebras
41 (1979),
STRÄHLÄ, S. and ZSIDÖ, L., A spectral characterization of the maximal
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STRATTON, A. E. and WEBB, M. C , Type sets and nilpotent multiplications
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45
STROMMER, J., Ein elementarer Beweis der Kreisaxiome der hyperbolischen
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STROTH, G., A characterization of 3
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49 (1985),
34 (1973),
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Classification and construction of complete hypersurfaces satisfying
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47(1984),
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SZÁSZ, F., Über Gruppen, deren sämtliche nicht-triviale Potenzen zyklische Untergruppen der Gruppe sind
Die Ringe mit lauter isomorphen nichttrivialen endlich erzeugbaren Unterringen
Bemerkung zu meiner Arbeit „Über Gruppen, deren sämtliche
nichttriviale Potenzen zyklische Untergruppen der Gruppe sind"
Die Halbgruppen, deren endlich erzeugte echte Teilhalbgruppen
Hauptrechtsideale sind
Einige Kriterien f ü r die Existenz des Einselementes in einem Ring
Simultane Lösung eines halbgruppentheoretischen und eines ringtheoretischen Problems
46
17 (1956),
83—84
22 (1961),
196—201
23 (1962),
64—66
25(1964),
28 (1967),
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31^-38
30 (1969),
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SZÁSZ, F., Reduktion eines Problems bezüglich der Brown—McCoyschen
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;
Ringe in welchen jedes Element ein Linksmultiplikator ist
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38 (1976),
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SZÁSZ, G., On the structure of semi-modular lattices of infinite length
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Generalization of a theorem of Birkhoff concerning maximal
chains of a certain type of lattices
On weakly complemented lattices
Correction to my paper „Generalization of a theorem of Birkhoff
Die Translationen der Halbverbände
On relatively complemented lattices
—
On complemented lattices
Note on complemented modular lattices of finite length
Remarks to the theory of semi-modular lattices
Über Halbgruppen, die ihre Ideale reproduzieren
Derivations of lattices
Decomposable elements and ideals in semigroups
Unitary subsemigroups in commutative semigroups
SZÁSZ, G. und SZENDREI, J., Über die Translationen der Halbverbände
SZÁSZ, O., Über die Cesäroschen Mittel Fourierscher Reihen
On the Gibbs' phenomenon for Euler means
SZÁSZ, P., Neue Herleitung der hyperbolischen Trigonometrie in der Ebene
Verwendung einer klassischen Konfiguration Johann Bolyai's bei
der Herleitung der hyperbolischen Trigonometrie in der Ebene
Neue Bestimmung des Parallelwinkels in der hyperbolischen Ebene
mit den klassischen Hilfsmitteln
Beweis der Hauptformel der hyperbolischen Trigonometrie unabhängig von der Stetigkeit
Herleitung der hyperbolischen Trigonometrie in der Poincaréschen
Halbebene
Elementargeometrische Herstellung des Klein—Hilbertschen Kugelmodells des hyperbolischen Raumes
On a mean-value theorem of Schwarz—Stieltjes
On a theorem of L. Fejér concerning trigonometric interpolation
SZEGŐ, G., Über die Tschebyscheffschen Polynome
14 (1951/52), 239—245'
15 (1953/54), 20—28
15 (1953/54), 130—142
16(1955),
16 (1955),
89—91
122—126
16 (1955),
17 (1956),
18(1957),
19(1958),
19 (1958),
21 (1960),
27(1966),
37(1975),
38 (1976),
38(1976),
18 (1957),
3 (1927),
12 B (1950),
12 A (1950),
270—270
166—169
48—51
77—81
224—228.
319—323
141—145
149—154
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14 (1951/52), 174—17»
14 (1951/52), 247—251.
15(1953/54), 57—601.
15 (1953/54), 126—129
16(1955),
19 (1958),
21 (1960),
2 (1924/26),
1—8
46—50
164—165
65—73
Zur Summation der Fourierschen Reihe
3 (1927),
17—24
Über die Laplacesche Reihe
4 (1928/29), 25—37
Zwei Extremalaufgaben über Abbildungen, die durch Polynome
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47
SZÉKELY, L. A., Holiday numbers: sequences resembling to the Stirling
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48(1985),
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SZÉKELY, S., s e e GÉCSEG, F .
SZÉKELYHÍDI, L., Almost periodic functions and functional equations
The stability of d'Alembert-type functional equations
SZEKERES, G., Ein Problem über mehrere ebene Bereiche
The asymptotic behaviour of the coefficients of certain power
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On finite metabelian /»-groups with two generators
42 (1980),
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44(1982),
313—320
6 (1932/34), 247—252
12 B (1950), 187—198
21 (1960),
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SZEKERES, G. und TÚRÁN, P., Über das zweite Hauptproblem der „Factorisatio Numerorum"
6 (1932/34), 143—154
SZELE, T., Ein Satz über die Struktur der endlichen Ringe
11 (1946/48), 246—250
Die Abelschen Gruppen ohne eigentliche Endomorphismen
13(1949/50), 54—56
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SZELE, T. and FUCHS, L., On Artinian
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SZELE, T. und SZÉLPÁL, I., Über drei wichtige Gruppen
SZÉLPÁL, I., Die Abelschen Gruppen ohne eigentliche Homomorphismen
Über Ringerweiterungen
17 (1956),
13 (1949/50),
13 (1949/50),
14 (1951/52),
30—40
192—194
51—53
113—114
38 (1976),
171—182
13 (1949/50),
19 (1958),
21 (1960),
34 (1973),
231—234
63—76
166—172
377—381
39 (1977),
44 (1982),
31(1970),
33 (1972),
36 (1974),
38 (1976),
367—389
185—213
141—156
137—151
323—344
383—398
43 (1981),
353—367
45 (1983),
49(1985),
33 (1972),
381—388
107—117
231—235
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SZENDREI, Á., Idempotent reducts of abelian groups
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:SZENDREI, J., On the extension of rings without divisors of zero
Über eine allgemeine Ringkonstruktion durch schiefes Produkt . .
Über die Szépschen Ringerweiterungen
Über die Verallgemeinerung einer Erweiterung von Ringen
s e e SZÁSZ, G .
:SZENDREI, M. B., On an extension of semigroups
Strong subband-parcelling extensions of orthodox semigroups . . .
:SZENTHE, J., On immersion of locally bounded curvature
A metric characterization of homogeneous Riemannian manifolds
On the topological characterization of transitive Lie group actions
Sur la connexion naturelle ä torsion nulle
On the orbit structure of orthogonal actions with isotropy subgroups of maximal rank
On symplectic actions of compact Lie groups with isotropy subgroups of maximal rank
A reduction in case of compact Hamiltonian actions
SZÉP, A., The non-orthogonal Menchoff—Rademacher theorem
SZÉP, J., On the structure of groups which can be represented as the
product of two sub-groups
On factorisable, not simple groups
On factorisable simple groups
48
I
12 A (1950), 57—61
13 (1949/50), 239—241
14 (1951/52), 22—22
SZÉP, J., Zur Theorie der endlichen einfachen Gruppen
14 (1951/52),
—:
Zur Theorie der einfachen Gruppen
14 (1951/52),
Zur Theorie der faktorisierbaren Gruppen
16(1955),
Über endliche Gruppen, die nur einen echten Normalteiler besitzen 17 (195©,
Über eine neue Erweiterung von Ringen. I
19 (1958),
Über eine neue Erweiterung von Ringen. II
20 (1959),
•
Über die Nichteinfachheit von faktorisierbaren Gruppen
21 (1960),
see ITÔ, N.
111—112
246—246
54—57
45—48
51—62
202—214
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see MIGLIORTNI, F .
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SZÉP, J., u n d ITÔ, N., Über die Faktorisation von G r u p p e n
16 (1955),
SZÉP, J . u n d RÉDEI, L . , O n f a c t o r i s a b l e g r o u p s
1 3 (1949/50), 2 3 5 — 2 3 8
Eine Verallgemeinerung der Remakschen Zerlegung
SZÉP, J. und ZAPPA, G., Über dreifaktorisierbare Gruppen. I
SZIGETI, F., Multivariable composition of Sobol'ev functions
SZILÁGYI, T., Generalization of the implicit function theorem and of
Banach's open mapping theorem
SZ.-NAGY, B., Über die Gesamtheit der charakteristischen Funktionen im
Hilbertschen Funktionenraum
Über ein geometrisches Extremalproblem
Über Integralungleichungen zwischen einer Funktion und ihrer
Ableitung
Approximation der Funktionen durch die arithmetischen Mittel
ihrer Fourierschen Reihen
On uniformly bounded linear transformations in Hilbert space
Une caractérisation affine de l'ensemble des fonctions positives
dans l'espace £ a
.'
Méthodes de sommation des séries de Fourier. I
—:
Séries et intégrales de Fourier des fonctions monotones non
bornées
Méthodes de sommation des séries de Fourier. III
Perturbations des transformations linéaires fermées
Approximation properties of orthogonal expansions
Sur les contractions de l'espace de Hilbert
Transformations de l'espace de Hilbert, fonctions de type positif
sur un groupe
Ein Satz über Parallelverschiebungen konvexer Körper
Remarks to the preceding paper of A. Korányi
Sur les contractions de l'espace de Hilbert. H
Note on sums of almost orthogonal operators
Über Parallelmengen nichtkonvexer ebener Bereiche
Bemerkungen zur vorstehenden Arbeit des Herrn S. Brehmer...
Positive definite kernels generated by operator-valued analytic
functions
On a property of operators of class C0
Diagonalization of matrices over H°°
IS (1953/54), 85—86
31 (1970),
173—178
48 (1985),
469—476
39 (1977),
229—231
391—396
8 (1936/37), 166—176
9 (1938/40), 253—257
10(1941/43), 64—74
11 (1946/48), 71—84
11 (1946/48), 152—157
12 A (1950), 228—238
12 B (1950), 204—210
13(1949/50),
13 (1949/50),
14 (1951/52),
15 (1953/54),
15 (1953/54),
118—133
247—251
125—137
31—37
87—92
15 (1953/54),
15 (1953/54),
17 (1956),
18 (1957),
18 (1957),
20 (1959),
22(1961),
104—114
169—177
71—75
1—14
189—191
36—47
112—114
26(1965),
36 (1974),
38 (1976),
191—192
219—220
223—238
see BERCOVICI, H .
s e e FOIAÇ, C .
17
49
SZ.-NAGY, B., s e e RIESZ, F .
SZ.-NAGY, B. et FOIAJ, C „ SUT les contractions de l'espace de Hilbert, M
Une relation parmi ies vecteurs propres d'un opérateur de l'espace
de Hilbert et de l'opérateur adjoint
Sur les contractions de l'espace de Hilbert. IV
Sur les contractions de l'espace de Hilbert. V. Translations bilatérales
Sur les contractions de l'espace de Hilbert. VI. Calcul fonctionnel
Remark to the preceding paper of J.Feldman
Sur les contractions de l'espace de Hilbert. VII. Triangulations
canoniques. Fonction minimum
Sur les contractions de l'espace de Hilbert. VHI. Fonctions caractéristiques. Modèles fonctionnels
Sur les contractions de l'espace de Hilbert. IX. Factorisations
de la fonction caractéristique. Sous-espaces invariants
Sur les contractions de l'espace de Hilbert. X. Contractions similaires à des transformations unitaires
Corrections et compléments aux Contractions IX
Sur les contractions de l'espace de Hilbert. XI. Transformations
uniceUulaires
On certain classes of power-bounded operators in Hilbert space
Sur les contractions de l'espace de Hilbert. XII. Fonctions intérieures admettant des facteurs extérieurs
Correction à la Note « Sur les contractions de l'espace de Hilbert. XI. Transformations unicellulaires »
Forme triangulaire d'une contraction et factorisation de la fonction caractéristique
Echelles continues de sous-espaces invariants
Commutants de certains opérateurs
Opérateurs sans multiplicité
Modèle de Jordan pour une classe d'opérateurs de l'espace de
Hilbert
:
Complément à l'étude des opérateurs de classe C0
Vecteurs cycliques et commutativité des commutants
Compléments à l'étude des opérateurs de classe Q . H
Accretive operators: Corrections
Echelles continues de sous-espaces invariants. II
On the structure of intertwining operators
Jordan model for contractions of class C 0
An application of dilation theory to hyponormal operators
Commutants and bicommutants of operators of class C0
Vecteurs cycliques et commutativité des commutants. II
On injections, intertwining operators of class C0
The functional model of a contraction and the space L l IHà
SZ.-NAGY, GY„ Die Lage der ^-Stellen eines Polynoms bezüglich seiner
Nullstellen
Über die allgemeinen Lemniskaten
50
19 (1958),
26—45
20 (1959),
21 (1960),
91—96
251—259
23(1962),
23 (1962),
23(1962),
106—129
130—167
272—273
25 (1964),
12—37
25 (1964),
38—71
25 (1964),
283—316
26 (1965),
26 (1965),
79—92
193—196
26 (1965),
27 (1966),
301—345
17—26
27 (1966),
27—34
27 (1966),
265—265
28(1967),
28 (1967),
29(1968),
30(1969),
201—212
213—220
1—18
1—18
31 (1970),
31 (1970),
32(1971),
33 (1972),
33(1972),
33(1972),
35(1973),
36(1974),
37(1975),
38 (1976),
39(1977),
40 (1978),
41 (1979),
91—115
287—296
177—183
113—116
117—118
355—356
225—254
305—321
155—159
311—315
169—174
163—167
403—410
11 (1946/48), 147—151
11 (1946/48), 207—224
SZ.-NAGY, G., Über die Lage der Doppelgeraden von gewissen Flächen
gegebener geometrischer Ordnung
i
Totalreelle rationale Funktionen
Merkwürdige Punktgruppen bei allgemeinen Lemniskaten
Verallgemeinerung der Derivierten in der Geometrie der Polynome
Über die Lage der kritischen Punkte rationaler Funktionen
11 (1946/48), 234—238
12 A (1950),
1—10
13 (1949/50), 1—13
13 (1949/50), 169—178
14 (1951/52), 179—185
see SZ.-NAGY, J. ( J u l i u s = G y u l a )
SZ.-NAGY, J., Über die Lage der Wurzeln von linearen Verknüpfungen
algebraischer Gleichungen
Über einen v. Staudt-schen Satz
Über die irreduziblen ebenen Kurven von Maximalindex
Über einen topologisçhen Satz
Zur Topologie der geschlossenen Kurven auf der Sphäre
Einige Sätze über ebene Elementarkurven
Über die Lage der Nullstellen gewisser Maximal- und Extremalpolynome
Über die Ovaloidschalen der Flächen vom Maximalindex
Über die reellen Nullstellen des Derivierten eines Polynoms mit
reellen Koeffizienten
Über eine Zerlegung der ebenen Kurven vom Maximalindex
Über die Zirkulation der ebenen Kurven vom Maximalindex
Über die reellen Nullstellen gewisser Polynome mit Parametern
Über konvexe Kurven und einschließende Kreisringe
Szßcs, A., Sur la variation des intégrales triples et le théorème de Stokes . . .
Sur les équations définissant une matrice en fonction algébrique
d'une autre
Szöcs, J., Diagonalization theorems for matrices over certain domains...
On non-localizable measure spaces
—
Some weak-star ergodic theorems
On G-finite »'»-algebras
T
1 (1922/23),
2(1924/26),
3 (1927),
4 (1928/29),
5 (1930/32),
5(1930/32),
127—138
65—68
96—106
246—247
1—12
83—89
6(1932/34), 49—58
7 (1934/35), 244—248
8(1936/37),
8 (1936/37),
8 (1936/37),
10 (1941/43),
10 (1941/43),
3 (1927),
42—52
136—146
147—148
36—41
174—184
81—95
7(1934/35),
36 (1974),
37 (1975),
45 (1983),
48 (1985),
48—50
193—202
293—295
389—394
477—481
s e e ABDAIXA, S. M .
see KovAcs, I.
Szöcs, J., MÁTÉ, E. and FOIA§, C., Non-existence of unicellular operators
on certain nonseparable Banach-spaces
31 (1970),
297—300
S z O s z , P . , see VINCZE, ST.
«
T
TAFEL, W., VOIGT, J. and WEIDMANN, J., The singular sequence problem 40 (1978),
383—388
TAKACI, A . , see MARIÓ, V .
TAKXCS, L„ On a linear transformation in the theory of probability
•I—
On an identity of Shih-Chieh Chu . . . '
-r-r:
On some reccurrence equations in a Banach algebra
—
Random walk on a finite group
TAKAHASHI, K., On the converse of the Fuglede—Putnam theorem
TAMURA, T., Basic study of general products and homogeneous homomor;
phisms. I
17»
33 (1972),
34(1973),
38(1976),
40(1983),
43 (1981),
15—24
383—391
399—416
395—408
123—125
33 (1972),
31-44
51
TANDORI, K., Über die Cesärosche Summierbarkeit der orthogonalen
Reihen
14 (1951/52),
Über die Konvergenz singulärer Integrale
15 (1953/54),
Über die Divergenz der Fourierreihen
15 (1953/54),
Über die starke Summation von Fourierreihen
16 (1955),
Über orthogonale Reihen
:
16 (1955),
Über die orthogonalen Funktionen. I
18(1957),
Über die orthogonalen Funktionen. II (Summation)
18 (1957),
Über die orthogonalen Funktionen. lH (Lebesguesche Funktionen) 18 (1957),
Über die orthogonalen Funktionen. IV
19 (1958),
Über die orthogonalen Funktionen. V (Genaue Weyische Multiplikatorfolgen)
20 (1959),
Über die orthogonalen Funktionen. VI (Eine genaue Bedingung
für die starke Summation)
20(1959),
Über die orthogonalen Funktionen. VII (Approximationssätze).. 20 (1959),
Bemerkung zur Divergenz der trigonometrischen Reihen
20 (1959),
Über die orthogonalen Funktionen. VIII (Eine notwendige Bedingung für die Konvergenz)
20 (1959),
Bemerkung zu einem Satz von G. Alexits
21 (1960),
Ein Summationssatz für Orthogonalreihen mit monotoner Koeffizientenfolge
21 (1960),
Über die orthogonalen Funktionen. IX (Absolute Summation)... 21 (1960),
Bemerkung zu einem Satz von A. N. Kolmogoroff
22 (1961),
Über die orthogonalen Funktionen. X
23 (1962),
Über die Konvergenz der Orthogonalreihen
24(1963),
Über die Konvergenz der Orthogonalreihen. II
25 (1964),
Über ein Problem von S. B. Stetschkin
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OBITÜARÍES
Fodor, G. (1927—1977)
Haar, А. (1885—1933)
Huhn, А. Р. (1947—1985)
Kalmár, L. (1905—1976)
Kerékjártó, В. (1898—1946)
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61
86-2044 — Szegedi Nyomda — Feleifis vezető: Surányi Tibor igazgató
INDEX
M. Krasner, Abstract Galois theory and endotheory. I
R. Nöbauer, Rédei-Funktionen und ihre Anwendung in der Kryptographie
.
G. Pollák, Some sufficient conditions for hereditarily finitely based varieties of semigroups
P. P. Pálfy, The arity of minimal clones
Z$. Lengvárszky, A note on minimal clones
Z. Daróczy, A. Járat und I. Kátai, Intervallfüllende Folgen und volladditive Funktionen
253
287
299
331
335
337
IS. Körmendi, Canonical number systems in Q(Í2)
Z. Sasvári, On measurable functions with a finite number of negative squares
О. В. Висков, Представление полиномов JIareppa
F. Móricz, Orthonormal systems of polynomials in the divergence theorems for double orthogonal series
A. K. Varma, Birkhoff quadrature formulas based on Tchebycheff nodes
L. Rempulska, Remarks on the strong summability of numerical and orthogonal series by some
methods of Abel type
D. Mitrovic, Behavior of the extended Cauchy representation of distributions
V. Marie, M. Skendzic, A. Takaci, On Stieltjes transform of distributions behaving as regularly
varying functions
V. V. Ulyanov, Normal approximation for sums of non-identically distributed random variables
in Hilbert spaces
C. R. Putnam, A connection between the unitary dilation and the normal extension of a subnormal contraction
J. Kristóf, C*-norms defined by positive linear forms
Bibliographie
INDEX 1—50 (1922—1986)
351
359
365 '
8_
367
381
391
397
405
411
421
427
433
SZEGED (HUNGARIA), ARADI VÉRTANÚK TERE 1
On peut s'abonner à l'entreprise de commerce des livres et journaux
„Kultúra" (1061 Budapest, I., Fő utca 32)
ISSN 0324-6523 Acta Univ. Szeged
ISSN 0001 6969 Acta Sei. Math.
INDEX: 26024
86-2044 —Szegedi Nyomda—F. v.: S lírányi Tibor igazgató
Feleifis szerkesztő és kiadó: Leindler László
A kézirat a nyomdába érkezett: 1986. április 29.
Megjelenés: 1987.
Példányszám: 1000. Terjedelem: 18,2 (A/5) 1т
Készült monószedéssel, íves magasnyomással,
az MSZ 5601-24 és az MSZ 5602-55 szabvány szerint

ACTA SCIENTIARUM MATHEMATICARUM

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