B U L L E T I N

Transcription

B U L L E T I N

B U L L E T I N
DE LA SOCIÉTÉ DES SCIENCES
ET DES LETTRES DE £ÓD
SÉRIE:
RECHERCHES SUR LES DÉFORMATIONS
Volume LX, no. 1
Rédacteur en chef et de la Série: JULIAN £AWRYNOWICZ
Comité de Rédaction de la Série
P. DOLBEAULT (Paris), H. GRAUERT (Göttingen),
O. MARTIO (Helsinki), W.A. RODRIGUES, Jr. (Campinas, SP), B. SENDOV (Sofia),
C. SURRY (Font Romeu), P.M. TAMRAZOV (Kyiv), E. VESENTINI (Torino),
L. WOJTCZAK (£ódŸ), Ilona ZASADA (£ódŸ)
Secrétaire de la Série:
JERZY RUTKOWSKI
N
£ÓD 2010
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BULLETIN DE LA SOCIÉTÉ DES SCIENCES ET DES LETTRES DE L
ÓDŹ
2010
Vol. LX
Recherches sur les déformations
no. 1, 174 pp.
FROM COMPLEX ANALYSIS
TO ASYMMETRIC RANDERS-INGARDEN
STRUCTURES III
Editors of the Volume
Julian L
awrynowicz, Dariusz Partyka
and Józef Zaja̧c
Julian L
awrynowicz
Institute of Physics, University of L
ódź
Pomorska 149/153, PL-90-236 L
ódź, Poland
Institute of Mathematics, Polish Academy of Sciences
L
ódź Branch, Banacha 22, PL-90-238 L
ódź, Poland
Dariusz Partyka
Faculty of Mathematics and Natural Sciences
The John Paul II Catholic University of Lublin
Al. Raclawickie 14, P.O. Box 129, PL-20-950 Lublin, Poland
State University of Applied Science in Chelm, Pocztowa 54
PL-22-100 Chelm, Poland, e-mail: [email protected]
Józef Zaja̧c
State University of Applied Science in Chelm
Pocztowa 54, PL-22-100 Chelm, Poland
Chair of Applied Mathematics
The John Paul II Catholic University of Lublin
Al. Raclawickie 14, P.O. Box 129, PL-20-950 Lublin, Poland
e-mail: [email protected]
Note. Since this year the numbering of volumes of the series Recherches coincides
with that of the Bulletin. This permits to simplify the citations: Bull. Soc. Sci.
Lettres L
ódź Sér. Rech. Déform. 60, no. 1 (2010) etc.
4
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ÓDŹ, POLOGNE
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References
[1]
Affiliation/Address
[5]
TABLE DES MATIÈRES
1. P. Dolbeault, Complex Plateau problem: old and new results
and prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11–31
2. D. Mierzejewski, Spheres in sets of solutions of quadratic
quaternionic equations of some types . . . . . . . . . . . . . . . . . . . . . . . . .
33–43
3. A. K. Kwaśniewski, Cobweb posets and KoDAG digraphs are
representing natural join of relations, their di-bigraphs and the
corresponding adjacency matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45–65
4. C. Boloşteanu, The Riemann-Hilbert problem with isolated
points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67–75
5. D. Partyka and J. Zaja̧c, Generalized problem of regression .
77–94
6. J. Dziok, Extremal problems in a generalized class of uniformly
convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95–108
7. B. Bochorishvili and H. M. Polatoglou, Electronic properties
of quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109–121
8. D. Georgakaki, Ch. Mitsas, and H. M. Polatoglou, Time
series analysis of the response of measurement instruments . . . .
123–135
9. R. S. Ingarden and J. L
awrynowicz, Model of magnetic electron microscope including the scanning microscope III. Variational approach and calculation of the focal length in a Randerstype geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
137–154
awrynowicz, Finsler-geometrical model of quantum electrodynamics I. External field vs. Finsler geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
155–174
Left to right (sitting): Julian L
awrynowicz (Lódź), Bogdan Bojarski (Warszawa), Ilpo
Laine (Joensuu), and Luis Manuel Tovar Sánchez (México, D.F.) during opening ceremony
of the XV International Conference on Analytic Functions in Chelm (Poland), July 2009.
Behind, in the centre (standing): Józef Zaja̧c (Chelm/Lublin)
9
PL ISSN 0459-6854
BULLETIN
DE
LA SOCIÉTÉ DES SCIENCES ET
2010
DES
LETTRES DE L
ÓDŹ
Vol. LX
no. 1
pp. 11–31
Pierre Dolbeault
COMPLEX PLATEAU PROBLEM: OLD AND NEW RESULTS
AND PROSPECTS
Summary
The Plateau problem is the research of a surface of minimal area, in the 3-dimensional
Euclidean space, whose boundary is a given continuous closed curve. The complex Plateau
problem is analogous in a Hermitian complex manifold: it is a geometrical problem of
extension of a closed real curve or manifold into a complex analytic subvariety, or into a
Levi-flat subvariety. Wirtinger’s inequality in Cn is recalled. Minimality of complex analytic
subvarieties and analogous properties of Levi-flat subvarieties, in Kähler manifolds, are
given. Known results in Cn and CP n are recalled. Extensions to real parametric problems
are solved or proposed, leading to the construction of Levi-flat hypersurfaces with prescribed
boundary in some complex manifolds.
1. Introduction
Given a Hermitian manifold X, the complex Plateau problem is the research of an
even dimensional subvariety with negligible singularities, with given boundary, and
of minimal volume in X. We will call mixed Plateau problem the research of a real
hypersurface with given boundary, and of minimal volume in X. More briefly, both
problems will be called complex Plateau problem.
First we shall recall or show that complex analytic subvarieties, resp. Levi-flat
hypersurfaces are solutions of the Plateau problem when X is Kähler (Sect. 2).
Then we will consider the complex Plateau problem as the research of the extension of an odd dimensional, compact, oriented, connected submanifold into a
complex analytic subvariety, and recall known solutions (Sects. 3, 4).
To solve the mixed Plateau problem as the research of the extension of an oriented, compact, connected, 2-codimensional submanifold into a Levi-flat hypersurface, we will need solutions of the complex Plateau problem with real parameter, in
Cn and CPn ; in CPn , it is an open problem to explicit satisfactory conditions on
12
P. Dolbeault
the boundary. In this way, we get very peculiar solutions of mixed Plateau problems
(Sect. 5).
Finally, the mixed Plateau problem is solved in Cn , in particular cases, as a
projection of a Levi-flat variety, and set up in CPn (Sects. 6, 7): known solutions
are recalled in Cn when the complex points of the boundary are elliptic; special
elliptic and hyperbolic points of the boundary are defined, and a solution when the
boundary is a ”horned sphere” is described; this will be the opportunity to precise
and complete results announced in ([D 08], Sect. 4). Problems when the boundary
has general hyperbolic points are still open.
Proofs of the results in Sects. 6 and 7 will appear in detail elsewhere [D 09].
2. Volume minimality of complex analytic subvarieties and of
Levi-flat hypersurfaces in Kähler manifolds
2.1. Wirtinger’s inequality (1936) [H 77]
In Cn ,with complex coordinates (z1 , . . . , zn ), we have the Hermitian metric
H=
n
dzj ⊗ dz j
j=1
and the exterior form (standard Kähler)
n
n
i
ω=
dzj ∧ dz j =
dxj ∧ dyj .
2 j=1
j=1
From the real vector space IR2n ∼
= Cn , we consider the real vector space Λ2p IR2n
of the 2p-vectors with the associated norm |.|; every decomposable vector (exterior
product of elements of IR2n ) defines a real 2p-plane of Cn i.e. an element of the
Grassmannian G2p
2n . We define the norm
|| ζ ||= inf
| ζj | where ζ =
ζj , ζj
j
j
is decomposable.
N
Let Ppp = {
λj ζj ; ζj decomposable defining a complex p-plane of Cn ; λj ≥
0; N ∈ IN ∗ }.
j=1
2.1.1. Theorem. For every ζ ∈ Λ2p Cn , we have:
equality uniquely for ζ ∈ Ppp
1 p
ω (ζ) ≤|| ζ ||;
p!
[W 36].
2.1.2. Corollary. Let V be a smooth real oriented 2p-dimensional submanifold of a
Hermitian manifold X = (X, ω) of complex dimension n. Then
Complex Plateau problem: old and new results and prospects
V
13
ω p /p! ≤ vol2p (V )
with equality iff V is complex.
2.2. Currents with measure coefficients [H 77]
2.2.1. Comass of an r-form; mass of a current with measure coefficients.
Let ϕ ∈ Λr IR2n , the comass of ϕ is defined as
|| ϕ ||∗ = sup{ϕ(ζ) : ζ ∈ Gr2n ⊂ Λ2p IR2n }.
Let Ω be an open subset of Cn , for every differential form ϕ of degree r on Ω, let
|| ϕ ||∗ = sup{|| ϕ(z) ||∗ : z ∈ Ω}
where || ϕ(z) ||∗ is the comass of ϕ(z).
Let T be a current with measure coefficients on Ω, K be any compact subset of
Ω and χK the characteristic function of K,
MK (T ) = sup |χK T (ϕ)|
||ϕ||∗≤1
is, by definition, the mass of T on K.
The measure which assigns the number MK (T ) to each compact set K ⊂ Ω is
called the mass or volume measure of T and denoted || T ||, so that MK (T ) =|| T ||
(K).
2.3. Complex Plateau problem
2.3.1. [H 77] On Ω ⊂ Cn , or more generally, on a Hermitian manifold (X, ω), let
B be a d-closed current of dimension 2p − 1 with compact support, and let T be
a (2p)-current with compact support and measure coefficients such that dT = B.
The complex Plateau problem is to find such a T with minimal mass, i.e. for every
compactly supported current S, with measure coefficients such that dS = B, to have
M (T ) ≤ M (S), or equivalently, for every compactly supported, d-closed (2p)-current
with measure coefficients R,
M (T ) ≤ M (T + R).
Such a T is said absolutely volume minimizing on X.
Let T be a d-closed (2p)-current with measure coefficients on X. If, for each
compact subset K of X,
MK (T ) ≤ M (χK T + R)
for all compactly supported d-closed (2p)-current R with measure coefficients on X,
then T is said to be absolutely volume minimizing on X.
14
P. Dolbeault
2.3.2. Theorem. [H 77] Let T be a 2p-current with measure coefficients on a Hermitian manifold (X, ω) and K be a compact subset of X. Then
(χK T )(ω p /p!) ≤ MK (T )
and equality holds iff χK T is strongly positive.
2.3.3. [H 77] Volume minimality of complex analytic sets in a Kähler manifold.
2.3.4. Corollary to Theorem 2.3.2. Assume that X = (X, ω) is a Käkler manifold and does not contain compact p-dimensional complex subvarieties. Let V be a
p-dimensional complex subvatiety, and T = [V ], then T is absolutely volume minimizing on X.
Proof. T is strongly positive. Let K be a compact subset of X and R be a compactly
supported d-closed (2p)-current with measure coefficients. From Theorem 2.3.2,
MK (T ) = (χK T )(ω p /p!).
But locally ω = ddc ψ, then ω p = ω p−1 ∧ddc ψ = d(ω p−1 ∧dc ψ), so in the neighborhood
of any point of X, R(ω p ) = R(d(ω p−1 ∧ dc ψ)). Let (αj )j∈J be a partition C ∞ of
unity subordinate to a locally finite open covering (Uj )j∈J of X such that for every
j, ω|Uj = ddc ψj . Then
αj R(d(ω p−1 ∧ dc ψj )) = ±
d(αj R)(ω p−1 ∧ dc ψj ) = 0,
R(ω p ) =
j
because:
j
j
d(αj R) =
j
dαj ∧ R +
αj ∧ dR = 0
j
and, as in the proof of ([H 77], Corollary 1.25), in an open set of the Hermitian Cn ,
MK (T ) = (χK T )(ω p /p!) = (χK T + R)(ω p /p!) ≤ MK (T + R).
2.3.5. Remark. If X contains a compact p-dimensional complex subvariety W ,
d[V ] = 0, but MK ([V ] > 0; then T is relatively volume minimizing on X.
2.4. Volume minimality of Levi-flat hypersurfaces in Kähler manifolds
We suppose to be in the category of currents with measure coefficients.
Recall the definition: A Levi-flat subvariety (with negligible singularities), of odd
dimension, is, outside of the singularities, a submanifold with Levi form ≡ 0, or,
equivalently, is foliated by complex analytic hypersurfaces.
Let M be a C ∞ Levi-flat hypersurface of a C ∞ Kähler manifold X = (X, ω)
bearing a foliation L by complex hypersurfaces Ml and let L be the space of the
foliation L assumed to be a C ∞ real curve.
Let M be a C ∞ hypersurface of X bearing a foliation L with the same space
L; the leaves of L being C ∞ subvarieties with negligible singularities.
15
Let S be a C ∞ compact submanifold of codimension 2 of X. We denote by the
same notation the hypersurfaces and submanifold and the integration currents they
define.
2.5. Mixed Plateau problem
Given S to find a C ∞ hypersurface in X \ S whose boundary is S in the category
H of foliated hypersurfaces with the same space of foliation, a real curve. If M is
such a hypersurface
whose space of foliation is L and the leaves (Ml , l ∈ L), then
vol(M ) = L vol(Ml )dl. From Sect. 2, for every l ∈ L, vol(Ml ) ≥ vol(Ml ) then M
is relatively volume minimizing in the category H and, by definition, M is solution
of the mixed Plateau problem.
2.6. Research of solutions of the complex Plateau problem
The present method of resolution consists in finding complex analytic, resp. Levi-flat
subvarieties, in X \ S, whose boundary S (in the sense of currents) is a submanifold
of X with convenient properties.
3. Possible origin: holomorphic extension; polynomial envelope
of a real curve
3.1.
The extension theorem of Hartogs, obtained at the beginning of the 20th century,
has been completely proved by Bochner and Martinelli, independently, in 1943. The
simplest version is:
Let Ω be a bounded open set of Cn , n ≥ 2. Suppose that ∂Ω be of class C k
(1 ≤ k ≤ ∞) or of class C ω (i.e. real analytic). Let f be a function in C l (∂Ω),
1 ≤ l ≤ k.
Then the two conditions are equivalent:
(i) f is a CR function, i.e. the differential of f restricted to the complex subspaces
of the tangent space to ∂Ω, at every point, is C-linear;
(ii) there exists F ∈ C l (Ω) ∩ O(Ω) such that F |∂Ω = f .
Then the graph of f is the boundary of the complex analytic submanifold defined
by the graph of F in Cn+1 .
3.2.
Let M be a compact submanifold of dimension 1 of Cn , we call polynomial envelope
of M , the compact set {z ∈ Cn ; | P (z) |≤ max | P (ζ) |; P ∈ C[z], the polynomial
ζ∈M
ring with complex coefficients }.
Then (J. Wermer (1958)), the polynomial envelope of M is either M , or the union
of M with the support of a complex analytic variety T , of complex dimension 1,
whose boundary is M [We 58].
16
P. Dolbeault
4. Solutions of the complex Plateau problem (or boundary
problem) in different spaces
4.1.
The first result has been obtained in 1958, by J. Wermer, in Cn , for p = 1 and M
holomorphic image of the unit circle in C [We 58]; this result has been generalized to
the case where M is a union of C 1 real connected curves by Bishop and Stolzenberg
(1966), looking for the polynomial envelope of M according to Sect. 3.2.
In Cn , after preliminary results by Rothstein (1959) [Rs 59], the boundary problem has been solved by Harvey and Lawson (1975), for p ≥ 2, under the necessary
and sufficient condition: M is compact, maximally complex and, for p = 1, under
the moment condition: M ϕ = 0, for every holomorphic 1-form ϕ on Cn [ HL 75].
For n = p + 1, the method, inspired by the Hartogs’ theorem consists in building T
as the divisor of a meromorphic function the defining function R; this function itself
is constructed, step by step, from solutions of ∂-problems with compact support.
T can also be viewed as graph (with multiplicities on the irreducible components)
of an analytic function with a finite number of determinations. For any p, we come
back to the particular case using projections.
In CPn \ CPn−r , 1 ≤ r ≤ n, for compact M , the problem has a une solution if
and only if, for p ≥ r + 1, M is maximally complex and if, for p = r, M satisfies
the moment condition: M ϕ = 0, for every ∂-closed (p, p − 1)-forme ϕ. The method
consists in solving the boundary problem, in Cn+1 \ Cn−r+1 , for the inverse image
of M by the canonical projection [HL 77].
In both cases, the solution is unique.
Harvey et Lawson assume the given M to be, except for a closed set of Hausdorff
(2p − 1)-dimensional measure zero, an oriented manifold of class C 1 ; we will say: M
is a variety C 1 with negligible singularities.
The boundary problem in CPn has been set up, for the first time, by J. King
[Ki 79]; uniqueness of the solution is no more possible, since two solutions differ by
an algebraic p-chain.
4.2.
In CPn , a solution of the boundary problem has been obtained by P. Dolbeault et
G. Henkin for p = 1, (1994), then for every p (1997) and more generally, in a qlinearly concave domain X of CPn , i.e. a union of projective subspaces of dimension
q [DH 97].
The necessary and sufficient condition for the existence of a solution is an extension of the moment condition: it uses a Cauchy residue formula in one variable and
a non linear differential condition which appears in many questions of Geometry or
Mathematical Physics. In the simplest case: p = 1, n = 2, this is the shock wave
equation for a local holomorphic function in 2 variables ξ, η, f ∂f /∂ξ = ∂f /∂η.
From a local condition, the above relation allows to construct, by extension
17
ot the coefficients, a meromorphic function playing, in Cn , the same part as the
Harvey-Lawson defining function described above; it defines a holomorphic p-chain
extendable to CPn using the classical Bishop-Stoll theorem.
4.2.1. The conditions of regularity of M have been weakened, first in Cn , and for
p = 1, to a condition, a little stronger than the rectifiability, by H. Alexander [Al 88]
who, moreover, has given an essential counter-example [Al 87], then by Lawrence
[Lce 95] and finally, and for any p, in Cn and CPn , by T. C. Dinh [Di 98]: M is
a rectifiable current whose tangent cone is a vector subspace almost everywhere.
Moreover, Dinh has obtained the reduction of the boundary problem in CPn to
the case p = 1, with weaker conditions than above and by an elementary analytic
procedure [Di 98].
All the previous results are obtained as corollaries. New progress by Harvey and
Lawson [HL 04].
5. Extension to real parametric problems
5.1.
∼ R × Cn−1 , and k : R × Cn−1 → R be the projection. Let N ⊂ E be a
5.1.1. Let E =
compact, (oriented) CR subvariety of Cn of real dimension 2n − 4 and CR dimension
n − 3, (n ≥ 4), of class C ∞ , with negligible singularities (i.e. there exists a closed
subset τ ⊂ N of (2n − 4)-dimensional Hausdorff measure 0 such that N \ τ is a CR
submanifold). Let τ be the set of all points z ∈ N such that either z ∈ τ or z ∈ N \ τ
and N is not transversal to the complex hyperplane k −1 (k(z)) at z. Assume that N ,
as a current of integration, is d-closed and satisfies:
(H) There exists a closed subset L ⊂ Rx1 with H 1 (L) = 0 such that for every
x ∈ k(N ) \ L, the fiber k −1 (x) ∩ N is connected and does not intersect τ .
5.1.2. Theorem [DTZ 09] (see also [DTZ 05]). Let N satisfy (H) with L chosen
accordingly. Then, there exists, in E = E \ k −1 (L), a unique C ∞ Levi-flat (2n − 3)subvariety M with negligible singularities in E \ N , foliated by complex (n − 2)subvarieties, with the properties that M simply (or trivially) extends to E as a
(2n − 3)-current (still denoted M ) such that dM = N in E . The leaves are the
sections by the hyperplanes Ex01 , x01 ∈ k(N )\L, and are the solutions of the “HarveyLawson problem” for finding a holomorphic subvariety in Ex01 ∼
= Cn−1 with prescribed
boundary N ∩ Ex01 .
5.2. In a real hyperplane of CPn+1
5.2.1. The simplest significant case is the boundary problem in CP 3 . For the boundary problem with real parameter in C3 , we considered a boundary problem in IR×C3 ,
i.e. in the subspace of C4 , in which the first coordinate is real. In the same way, we
will consider in CP 4 , with homogeneous coordinates (w0 , w1 , . . . , w4 ), a boundary
problem in the subspace E defined by w1 = λw0 , with λ ∈ IR. Then, for personal
18
P. Dolbeault
convenience, we will follow, step by step, the known construction in CP 3 in the
oldest version [DH 97].
Particularly, the coefficients Cm of the defining function R of the solution are
estimated as for the problem in CP 3 .
The end of the proof of the main theorem seems analogous to the known case in
IR × C3 .
5.2.2. The projective space CP 3 has homogeneous coordinates w = (w0 , w2 , . . . , w4 );
denote Q = {w0 = 0} the hyperplane at infinity of CP 3 .
For w0 = 0, let k be the projection: E → IRλ , (w0 , w1 = λw0 , w2 , w3 , w4 ) → λ;
π : E → CP 3 ,
for w0 = 0, λ is indeterminate.. We also have the projection:
(w0 , λw0 , w2 , w3 , w4 ) → (w0 , w2 , w3 , w4 ). In the same way, (E \ {w0 = 0}) ∼
= IR × C3 .
5.2.3. Let N ⊂ E ⊂ CP 4 be a submanifold of class C ∞ , CR, oriented, compact of E,
of dimension 4, of CR dimension 1, with negligible singularities. N being compact
in E, k(N ) is compact in IR, i.e. in N , the parameter λ varies in a closed, bounded
interval Λ of IR.
Assume that N satisfies the same properties as in Sect. 5.1.1.
5.2.4. Consider the complex hyperplanes of CP 4 , whose equation is h̃(w) = w4 −
ξ2 w0 − η2 w1 − η2 w2 = 0 and, in E, the subspaces Pνλ whose equation is
h̃1 (w , λ) = w4 − ξ2 w0 − η2 λw0 − η2 w2 = w4 − (ξ2 + η2 λ)w0 − η2 w2 = 0,
of real dimension 5. Their restrictions to (E \ IR × Q) ∼
= IR × C3 are real affine
subspaces of dimension 5.
We note νλ the 1 × 2-matrix (ξ2 + η2 λ) η2 .
Generically, Γνλ = N ∩ Pνλ is of dimension 2.
/ L, N ∩ Eλ is of dimension 3 and
For z ∈ N, λ = k(z). let Eλ = k −1 k(z); for λ ∈
is contained in Eλ ∼
= CP 3 .
Consider the linear forms
h(w) = w3 − ξ1 w0 − η1 w1 − η1 w2 ,
h̃0 (w , λ) = w3 − ξ1 w0 − η1 λw0 − η1 w2 = w3 − (ξ1 + η1 λ)w0 − η1 w2 .
Denote by νλ = (ξλ , η) the 2 × 2-matrix
(ξ1 + η1 λ)
(ξ2 + η2 λ)
η1
η2
.
For fixed λ, νλ is a coordinate system of a chart of the Grassmannian GC (2, 4), i.e.
νλ is a coordinate system of a chart of GC (2, 4) × IR. and we identify νλ with the
point of GC (2,4) × IR having these coordinates.
Let ξλ =t (ξ1 + η1 λ) (ξ2 + η2 λ) ; η =t (η1 η2 ). Remark that ξλ depends on
(ξ1 , ξ2 , η1 , η2 ); to get effective dependance on the parameter λ, it suffices to fix η1 = 0,
.η2 = 0.
Recall: ξλ =t (ξλ1 ξλ2 ), ξλl = ξl + ηl λ, l = 1, 2, η =t (η1 η2 ).
19
Let zj = wj /w0 , j = 2, 3, 4, be the non homogeneous coordinates; h̃0 defines the
affine function:
h = z3 − (ξ1 + η1 λ) − η1 z2 .
The two forms h̃0 et h̃1 are linearly independent, then the set of their common
zeros Dνλ is of real dimension 3, is contained in Pνλ ; in general, Dνλ ∩N is a finite set
Zνλ ; then, for general enough fixed λ and νλ , Zνλ = ∅. For every fixed real number
λ∈
/ L, the situation in Eλ is the classical situation in CP 3 .
5.2.5. Boundary problem. Given N , find a complex analytic subvariety M depending
on the real parameter λ such that dM = N in the sense of currents, under a necessary
and sufficient condition on N .
To do this, we can check, step by step, the solution of the boundary problem in
CP 3 [HL 97], introducing the parameter λ.
For λ ∈
/ L, γνλ = N ∩ Pνλ ∩ Eλ is of dimension 1. Under the notations of the
Sect. 5.2.4, consider the function
1
dh
z2 .
(1)
G(νλ ) =
2πi γν h
λ
5.2.6. Tentative statement. The following two conditions are equivalent:
(i) There exists, in E = E \ k −1 L, a C ∞ Levi-flat subvariety M , (with negligible
singularities), of dimension 5, foliated by complex analytic subvarieties Mλ of complex dimension 2, such that M extends simply (or trivially) to E as a current of
dimension 5 (still denoted M ) such that dM = N in E . The leaves are the sections
by the subspaces Eλ , λ ∈ k(N ) \ L, and are the solutions of the boundary problem for
finding complex analytic subvarieties in Eλ ∼
= CP 3 with given boundary N ∩ Eλ .
(ii) N is a submanifold CR, oriented, of CR dimension 1 outside a closed set of
4-dimensional Hausdorff measure 0.
There exists a matrix νλ∗∗ in the neighborhood of which
Dξ2λ G(νλ ) = Dξ2λ
N
fj (νλ )
j=1
where fj , j = 1, . . . , N , is a holomorphic function in νλ , C ∞ en λ, and satisfies the
system of P.D.E.
(2)
fj
∂fj
∂fj
=
, l = 2, 3.
∂ξλl
∂ηl
5.2.7. Remark. This result is not satisfactory because the relation of the analytic
conditions with the geometry of the submanifold N is not explicit.
20
P. Dolbeault
5.3. Boundary problem in a real hyperplane of Cn+1 or CP n+1
Cn+1 and CP n+1 are both Kähler. The solutions of the above boundary problems
are both Levi flat, hence, from a plain extension of Sect. 2.5, volume minimal, i.e.
solution, of codimension 3, of mixed Plateau problems.
6. Levi-flat hypersurfaces with prescribed boundary:
preliminaries
6.1. Introduction
Let S ⊂ Cn be a compact connected 2-codimensional submanifold. Find a Levi-flat
hypersurface M ⊂ Cn \ S such that dM = S (i.e. whose boundary is S, possibly as
a current).
For n = 2, near an elliptic complex point p ∈ S, S \ {p} is foliated by smooth
compact real curves which bound analytic discs (Bishop [Bi 65]). The family of these
discs fills a smooth Levi-flat hypersurface.
In 1983, Bedford-Gaveau considered the case of a particular sphere with two
elliptic complex points. If S is contained in the boundary of a strictly pseudoconvex
bounded domain, then the families of analytic discs in the neighborhood of each
elliptic point extend to a global family filling a 3-dimensional ball M bounded by
S. In 1991, Bedford-Klingenberg [BeK 91] and Kruzhilin extended the result when
there exist hyperbolic complex points on S with the same global condition.
Results of increasing generality have been obtained by Chirka, Shcherbina, Slodowski, G. Tomassini until 1999. The global sufficient condition of embedding of S
in the boundary of a strictly pseudoconvex domain is still required in these papers.
A first result for n ≥ 3 (in the sense of currents), and for elliptic points only,
has been obtained four years ago ([DTZ 05] and [DTZ 09] in detailed form); we
got new results when S is homeomorphic to a sphere, with three elliptic and one
hyperbolic special points (see [D 08] for a first draft), or a torus, with two elliptic
and two hyperbolic special points and, more generally, a manifold which is obtained
by gluing together elementary models.
A local condition is required because, in general, S is not locally the boundary of
a Levi-flat hypersurface. The proof uses the construction of a foliation of S by CR
orbits, Thurston’s stability theorem for foliations on S, and a parametric version of
the Harvey-Lawson theorem on boundaries of complex analytic varieties. There is
no global condition.
6.2. Preliminaries and definitions
6.2.1. A smooth, connected, CR submanifold M ⊂ Cn is called minimal at a point
p if there does not exist a submanifold N of M of lower dimension through p such
that HN = HM |N . By a theorem of Sussman, all possible submanifolds N such
21
that HN = HM |N contain, at p, one of the minimal possible dimension, called a
CR orbit of p in M whose germ at p is uniquely determined.
6.2.2. S is said to be a locally flat boundary at a point p if it locally bounds a Leviflat hypersurface near p. Assume that S is CR in a small enough neighborhood U of
p ∈ S. If all CR orbits of S are 1-codimensional (which will appear as a necessary
condition for our problem), the following two conditions are equivalent [DTZ 05]:
(i) S is a locally flat boundary on U ;
(ii) S is nowhere minimal on U .
6.2.3. Complex points of S [DTZ 05].
At such a point p ∈ S, Tp S is a complex hyperplane in Tp Cn . In suitable holomorphic coordinates (z, w) ∈ Cn−1 × C vanishing at p, S satisfies
(1)
w = Q(z) + O(|z|3 ), Q(z) =
(aij zi zj + bij zi z j + cij z i z j )
1≤i,j≤n−1
S is said flat at a complex point p ∈ S if
bij zi z j ∈ λR, λ ∈ C. We also say that
p is flat.
Let S ⊂ Cn be a locally flat boundary with a complex point p. Then p is flat.
By making the change of coordinates (z, w) → (z, λ−1 w), we make bij zi z j ∈ IR
for all z. By a change of coordinates (z, w) → (z, w + aij zi zj ) we can choose the
holomorphic term in (1) to be the conjugate of the antiholomorphic one and so make
the whole form Q real-valued.
We say that S is in a flat normal form at p if the coordinates (z, w) as in (1) are
chosen such that Q(z) ∈ R for all z ∈ Cn−1 .
6.2.4. Properties of Q.
Assume that S is in a flat normal form; then, the quadratic form Q is real valued.
Only holomorphic linear changes of coordinates are allowed. If Q is positive definite
or negative definite, the point p ∈ S is said to be elliptic; if the point p ∈ S is is not
elliptic, and if Q is non degenerate, p is said to be hyperbolic. From Sect. 6.4, we will
only consider particular cases of the quadratic form Q.
From [Bi 65], for n = 2, in suitable holomorphic coordinates, Q(z) = (zz +
λRe z 2 ), λ ≥ 0, under the notations of [BeK 91]; for 0 ≤ λ < 1, p is said to be
elliptic, and for 1 < λ, it is said to be hyperbolic. The parabolic case λ = 1, not
generic, is omitted [BeK 91]. When n ≥ 3, the Bishop’s result is not valid in general.
6.3. Elliptic points
6.3.2. Proposition ([DTZ 05], [DTZ 09]). Assume that S ⊂ Cn , (n ≥ 3) is nowhere
minimal at all its CR points and has an elliptic flat complex point p. Then there
exists a neighborhood V of p such that V \ {p} is foliated by compact real (2n − 3)dimensional CR orbits diffeomorphic to the sphere S2n−3 and there exists a smooth
function ν, having the CR orbits as the level surfaces.
22
P. Dolbeault
Sketch of Proof (see [DTZ 09]).
In the case of a quadric S0 (w = Q(z)), the CR orbits are defined by w0 = Q(z),
where w0 is constant. Using (1), we approximate the tangent space to S by the
tangent space to S0 at a point with the same coordinate z; the same is done for the
tangent spaces to the CR orbits on S and S0 ; then we construct the global CR orbit
on S through any given point close enough to p.
6.4. Special flat complex points
We say that the flat complex point p ∈ S is special if in convenient holomorphic
coordinates,
(2)
Q(z) =
n−1
(zj z j + λj Re zj2 ), λj ≥ 0.
j=1
Let zj = xj + iyj , xj , yj real, j = 1, . . . , n − 1, then:
n−1 (3) Q(z) = l=1 (1 + λl )x2l + (1 − λl )yl2 + O(|z|3 ).
A flat point p ∈ S is said to be special elliptic if 0 ≤ λj < 1 for any j.
A flat point p ∈ S is said to be special k-hyperbolic if 1 < λj for j ∈ J ⊂
{1, . . . , n − 1} and 0 ≤ λj < 1 for j ∈ {1, . . . , n − 1} \ J = ∅, where k denotes the
number of elements of J.
Special elliptic (resp. k-hyperbolic) points are elliptic (resp. hyperbolic).
6.5. Special hyperbolic points
6.5.1. We will not consider special parabolic points (one λj = 1 at least) which don’t
appear generically.
S being given by (1), let S0 be the quadric of equation w = Q(z). Suppose that S0
is flat at 0 and that 0 is a special k-hyperbolic point. Then, in a neighborhood of 0,
and with the above local coordinates, it is CR and nowhere minimal outside 0, and the
CR orbits of S0 are the (2n − 3)-dimensional submanifolds given by w = const. = 0.
The section w = 0 of S0 is a real quadratic cone Σ0 in R2n whose vertex is 0
and, outside 0, it is a CR orbit Σ0 in the neighborhood of 0.
6.6. Foliation by CR-orbits in the neighborhood of a special 1-hyperbolic
point
We mimic the begining of the proof of 2.4.2. in ([DTZ 05], [DTZ 09]).
6.6.1. Local 2-codimensional submanifolds.
In C3 , consider the 4-dimensional submanifold S locally defined by the equation
(1)
w = ϕ(z) = Q(z) + O(|z|3 )
and the 4-dimensional submanifold S0 of equation
(4)
w = Q(z)
23
with
Q = (λ1 + 1)x21 − (λ1 − 1)y12 + (1 + λ2 )x22 + (1 − λ2 )y22
having a special 1-hyperbolic point at 0, (λ1 > 1, 0 ≤ λ2 < 1), and the cone Σ0 whose
equation is: Q = 0. On S0 , a CR orbit is the 3-dimensional submanifold Kw0 whose
equation is w0 = Q(z). If w0 > 0, Kw0 does not cut the line L = {x1 = x2 = y2 = 0};
if w0 < 0, Kw0 cuts L at two points.
6.6.2. Remark. Σ0 = Σ0 \ 0 has two connected components in a neighborhood of 0.
Proof. The equation of Σ0 ∩ {y1 = 0} is
(λ1 + 1)x21 + (1 + λ2 )x22 + (1 − λ2 )y22 = 0 whose only zero , in the neighborhood of
0, is {0}: the connected components are obtained for y1 > 0 and y1 < 0 respectively.
6.6.3. Behaviour of local CR orbits.
Under the notations of [DTZ 09], follow the construction of the complex tangent
space E(z, ϕ(z)) to the CR orbit at z; compare with E0 (z, Q(z)). We know the
integral manifold, the orbit of E0 (z, Q(z)); deduce an evaluation of the integral
manifold of E(z, ϕ(z)).
6.6.4. Lemma. Under the above hypotheses, if k = 1, the local orbit Σ corresponding
to Σ0 has two connected components in the neighborhood of 0.
Proof. Use Remark 6.6.2 and the adaptation of the technique of [DTZ 09].
6.7. CR-orbits near a subvariety containing a special 1-hyperbolic point
6.7.2. Proposition. Assume that S ⊂ Cn (n ≥ 3), is a locally closed (2n − 2)submanifold, nowhere minimal at all its CR points, which has a unique spcial 1hyperbolic flat complex point p, and such that:
(i) the orbit Σ whose closure Σ contains p is compact;
(ii) Σ has two connected components σ1 , σ2 , whose closures are homeomorphic
to spheres of dimension 2n − 3.
Then, there exists a neighborhood V of Σ such that V \Σ is foliated by compact
real (2n − 3)-dimensional CR orbits whose equation, in a neighborhood of p is (3),
and, the w(= xn )-axis being assumed to be vertical, each orbit being diffeomorphic
to
the sphere S2n−3 above Σ ,
the union of two spheres S2n−3 under Σ ,
and there exists a smooth function ν, having the CR orbits as the level surfaces.
6.8. Geometry of the complex points of S
6.8.1. Let G be the manifold of the oriented real linear (2n − 2)-subspaces of Cn .
The submanifold S of Cn has a given orientation which defines an orientation of the
24
P. Dolbeault
tangent space to S at any point p ∈ S. By mapping each point of S into its oriented
tangent space, we get a smooth Gauss map
t : S → G.
6.8.2. Dimension of G. dim G = 2(2n − 2).
6.8.3. Proposition. For n ≥ 2, in general, S has isolated complex points.
Proof. Let π ∈ G be a complex hyperplane of Cn whose orientation is induced by
C
= CPn−1∗ ⊂ G, as real
its complex structure; the set of such π is H = Gn−1,n
submanifold. If p is a complex point of S, then t(p) ∈ H or −t(p) ∈ H. The set
of complex points of S is the inverse image by t of the intersections t(S) ∩ H and
−t(S) ∩ H in G. Since dim t(S) = 2n − 2, dim H = 2(n − 1), dim G = 2(2n − 2),
the intersection is 0-dimensional, in general.
6.8.4. Homology of G. (cf [P 08]). G has the structure of a complex quadric; let S1 , S2
be generators of H2n−2 (G, ZZ); we assume that S1 and S2 are fundamental cycles of
complex projective subspaces of complex dimension (n−1) of G. Then, denoting also
S, the fundamental cycle of the submanifold S and t∗ the homomorphism defined
by t, we have:
t∗ (S) ∼ u1 S1 + u2 S2
where ∼ means homologous to.
6.8.5. Lemma (proved for n = 2 in [CS 51]). With the notations of 6.8.1, we have:
u1 = u2 ; u1 + u2 = χ(S), Euler-Poincaré characteristic of S.
The proof for n = 2 works for any n ≥ 3.
6.8.6. Local intersection numbers of H and t(S) when all complex points are flat.
Proposition (known for n = 2 [Bi 65], here for n ≥ 3). Let S be a smooth, oriented,
compact, 2-codimensional, real submanifold of Cn whose all complex points are flat
and special. Then, on S, (special elliptic points) + (special k-hyperbolic points,
with k even) - (special k-hyperbolic points, with k odd) = χ(S). If S is a sphere,
this number is 2.
7. Levi-flat hypersurfaces with prescribed boundary:
particular cases
7.1.
To solve the boundary problem by Levi-fllat hypersurfaces, S has to satisfy necessary
and sufficient local conditions. A way to prove that these conditions can occur is to
construct an example for which the solution is obvious.
25
7.2. Sphere with elliptic points
7.2.1. Example. In C3 , Let S be defined by the equations:
z1 z 1 + z2 z 2 + z3 z 3 = 1
(S)
= z3
z3
We have CR-dim S = 1 except at the points z1 = z2 = 0; z3 = ±1 where
CR-dim S = 2. S is the unit sphere in C2 × IR; it bounds the unit ball M in C2 × IR,
which is foliated by the complex balls C2 × {x3 } ∩ M . The leaves are relatively
compact of real dimension 4 and are bounded by compact leaves (3-spheres) of a
foliation of M .
7.2.2. Theorem [DTZ 05]. Let S ⊂ Cn , n ≥ 3, be a compact connected smooth real
2-codimensional submanifold satisfying the conditions
(i) S is nonminimal at every CR point;
(ii)every complex point of S is flat and elliptic and there exists at least one such
point;
(iii) S does not contain complex manifold of dimension (n − 2).
Then S is a topological sphere, and there exists a Levi-flat (2n − 1)-subvariety
M̃ ⊂ C × Cn with boundary S̃ (in the sense of currents) such that the natural
projection π : C × Cn → Cn restricts to a bijection which is a CR diffeomorphism
between S̃ and S outside the complex points of S.
7.3. Sphere with one special 1-hyperbolic point (sphere with two horns)
7.3.1. Example. In C3 , let (zj ), j = 1, 2, 3, be the complex coordinates and zj = xj +
iyj . In R6 ∼
= C3 , consider the 4-dimensional subvariety (with negligible singularities)
S defined by: y3 = 0,
x3 (x21 + y12 + x22 + y22 + x23 − 1) + (1 − x3 )(x41 + y14 + x42 + y24 + 4x21
0 ≤ x3 ≤ 1;
−2y12 + x22 + y22 ) = 0,
−1 ≤ x3 ≤ 0; x3 = x41 + y14 + x42 + y24 + 4x21 − 2y12 + x22 + y22 .
The singular set of S is the 3-dimensional section x3 = 0 along which the tangent
space is not everywhere (uniquely) defined.
S being in the real hyperplane {y3 = 0}, the complex tangent spaces to S are
{x3 = x0 } for convenient x0 .
The set S will be smoothed along the complement of 0 (origin of C3 ) in its
section by the hyperplane {x3 = 0} by a small deformation leaving h unchanged. In
the following S will denote this smooth submanifold.
From elementary analytic geometry, complex points of S are defined by their
coordinates:
e3 : xj = 0, yj = 0 (j = 1, 2), x3 = 1;
h: xj = 0, yj = 0 (j = 1, 2), x3 = 0;
e1 , e2 : x1 = 0, y1 = ±1, x2 = 0, y2 = 0, x3 = −1.
26
P. Dolbeault
Lemma. The complex points are flat and special. The points e1 , e2 , e3 are special
elliptic; the point h is special 1-hyperbolic.
Remark that the numbers of special elliptic and special hyperbolic points satisfy
the conclusion of Proposition 6.8.6.
7.3.1’. Shape of Σ = S ∩ {x3 = 0} in the neighborhood of the origin 0 of C3 .
Lemma. Under the above hypotheses and notations:
(i) Σ = Σ \ 0 has two connected components σ1 , σ2 ;
(ii) The closures of the three connected components of S \ Σ are submanifolds
with boundaries and corners.
Proof. (i) The only singular point of Σ is 0. We work in the ball B(0, A) of C2
(x1 , y1 , x2 , y2 ) for small A and in the 3-space
πλ = {y2 = λx2 },
λ ∈ IR.
For λ fixed, πλ ∼
= IR3 (x1 , y1 , x2 ), and Σ ∩ πλ is the cone of equation
4x21 − 2y12 + (1 + λ2 )x22 + O(|z|3 ) = 0
with vertex 0 and basis in the plane x2 = x02 the hyperboloid Hλ of equation
3
4x21 − 2y12 + (1 + λ2 )x02
2 + O(|z| ) = 0;
the curves Hλ have no common point outside 0. So, when λ varies, the surfaces
Σ ∩ πλ are disjoint outside 0. The set Σ is clearly connected;
Σ ∩ {y1 = 0} = {0},
the origin of C3 ; from above:
σ1 = Σ ∩ {y1 > 0};
σ2 = Σ ∩ {y1 < 0}.
(ii) The three connected components of S \ Σ are the components which contain,
respectively e1 , e2 , e3 and whose boundaries are σ 1 , σ 2 , σ 1 ∪ σ 2 ; these boundaries
have corners as shown in the first part of the proof.
The connected component of C2 ×IR\S containing the point (0, 0, 0, 0, 1/2) is the
Levi-flat solution, the complex leaves being the sections by the hyperplanes x3 = x03 ,
−1 < x03 < 1.
The sections by the hyperplanes x3 = x03 are diffeomorphic to a 3-sphere for
0 < x03 < 1 and to the union of two disjoint 3-spheres for −1 < x03 < 0, as can be
shown intersecting S by lines through the origin in the hyperplane x3 = x03 ; Σ is
homeomorphic to the union of two 3-spheres with a common point.
7.3.2. Proposition (cf [D 08], Proposition 2.6.1). Let S ⊂ Cn be a compact connected
real 2-codimensional manifold such that the following holds:
(i) S is a topological sphere; S is nonminimal at every CR point;
27
(ii) every complex point of S is flat; there exist three special elliptic points ej , j =
1, 2, 3 and one special 1-hyperbolic point h;
(iii) S does not contain complex manifolds of dimension (n − 2);
(iv) the singular CR orbit Σ through h on S is compact and Σ \ {h} has two
connected components σ1 and σ2 whose closures are homeomorphic to spheres of
dimension 2n − 3;
(v) the closures S1 , S2 , S3 of the three connected components S1 , S2 , S3 of S \ Σ
are submanifolds with (singular) boundary.
Then each Sj \ {ej ∪ Σ }, j = 1, 2, 3 carries a foliation Fj of class C ∞ with
1-codimensional CR orbits as compact leaves.
Proof. From conditions (i) and (ii), S satisfying the hypotheses of Proposition
6.3.2, near any elliptic flat point ej , and of Proposition 6.7.2 near Σ , all CR orbits
are diffeomorphic to the sphere S2n−3 . The assumption (iii) guarantees that all CR
orbits in S must be of real dimension 2n − 3. Hence, by removing small connected
open saturated neighborhoods of all special elliptic points, and of Σ , we obtain,
from S \ Σ , three compact manifolds Sj ”, j = 1, 2, 3, with boundary and with the
foliation Fj of codimension 1 given by its CR orbits whose first cohomology group
with values in R is 0, near ej . It is easy to show that this foliation is transversely
oriented.
7.3.2’. Recall the Thurston’s Stability Theorem ([CaC], Theorem 6.2.1). Let (M, F )
be a compact, connected, transversely-orientable, foliated manifold with boundary or
corners, of codimension 1, of class C 1 . If there is a compact leaf L with H 1 (L, R) =
0, then every leaf is homeomorphic to L and M is homeomorphic to L×[0, 1], foliated
as a product,
Then, from the above theorem, Sj ” is homeomorphic to S2n−3 × [0, 1] with CR
orbits being of the form S2n−3 × {x} for x ∈ [0, 1]. Then the full manifold Sj is
homeomorphic to a half-sphere supported by S2n−2 and Fj extends to Sj ; S3 having
its boundary pinched at the point h.
7.3.3. Theorem. Let S ⊂ Cn , n ≥ 3, be a compact connected smooth real 2codimensional submanifold satisfying the conditions (i) to (v) of Proposition 7.3.2.
Then there exists a Levi-flat (2n − 1)-subvariety M̃ ⊂ C × Cn with boundary S̃ (in
the sense of currents) such that the natural projection π : C × Cn → Cn restricts
to a bijection which is a CR diffeomorphism between S̃ and S outside the complex
points of S.
Proof. By Proposition 6.3.2 , for every ej , a continuous function νj , C ∞ outside ej ,
can be constructed in a neighborhood Uj of ej , j = 1, 2, 3, and by Proposition 6.7.2,
we have an analogous result in a neighborhood of Σ .
Furthermore, from Sect. 7.3.2’, a smooth function ν”j whose level sets are the
leaves of Fj can be obtained globally on Sj \ {ej ∪ Σ }. With the functions νj and
ν”j , and analogous functions near Σ , then using a partition of unity, we obtain a
28
P. Dolbeault
global smooth function νj : Sj → R without critical points away from the complex
points ej and from Σ .
Let σ1 , resp. σ2 the two connected, relatively compact components of Σ \ {h},
according to condition (iv); σ 1 , resp. σ 2 are the boundary of S1 , resp. S2 , and σ 1 ∪σ 2
the boundary of S3 . We can assume that the three functions νj are finite valued and
get the same values on σ 1 and σ 2 . Hence a function ν : S → R.
The submanifold S being, locally, a boundary of a Levi-flat hypersurface, is orientable. We now set
S̃ = N = gr ν = {(ν(z), z) : z ∈ S}.
Let
Ss = {e1 , e2 , e3 , σ1 ∪ σ2 }.
λ : S → S̃ z → ν((z), z) is bicontinuous; λ|S\Ss is a diffeomorphism; moreover λ
is a CR map. Choose an orientation on S. Then N is an (oriented) CR subvariety
with the negligible set of singularities τ = λ(Ss ).
At every point of S \ Ss , dx1 ν = 0, then condition (H) (Sect. 5.1.1) is satisfied at
every point of N \ τ .
Then all the assumptions of Theorem 5.1.2 being satisfied by N = S̃, in a particular case, we conclude that N is the boundary of a Levi-flat (2n − 2)-variety (with
negligible singularities) M̃ in R × Cn .
Taking π : C × Cn → Cn to be the standard projection, we obtain the conclusion.
7.4. Case of a torus
7.4.1. Euler-Poincaré characteristic of a torus is χ(Tk ) = 0.
7.4.2. Example. In C 3 , let (zj ), j = 1, 2, 3, be the complex coordinates and zj = xj +
iyj . In R6 ∼
= C3 , consider the 4-dimensional subvariety (with negligible singularities)
S defined by:
y3 = 0
0 ≤ x3 ≤ 1;
x3 (x21 + y12 + x22 + y22 + x23 − 1) + (1 − x3)(x41 + y14 + x42 + y24 + 4x21 − 2y12 + x22 + y22 ) = 0
− 12 ≤ x3 ≤ 0;
x3 = x41 + y14 + x42 + y24 + 4x21 − 2y12 + x22 + y22 ,
glue it with the symmetric with respect to the real hyperplane x3 = − 12 , and and
smooth along {x3 = 0}, {x3 = ± 12 }. The complex points are flat and special.
7.4.3. Theorem. Let S ⊂ Cn , n ≥ 3, be a compact connected smooth real 2codimensional submanifold satisfying the following conditions:
(i) S is a topological torus; S is nonminimal at every CR point;
(ii) every complex point of S is flat; there exist two special elliptic points e1 , e2
and two special 1-hyperbolic points h1 , h2 ;
29
(iii) S does not contain complex manifolds of dimension (n − 2);
(iv) the singular CR orbits Σ1 , Σ2 through h1 and h2 on S are compact and, for
j = 1, 2, Σj \ {hj } have two connected components σj1 and σj2 ;
(v) the closures S1 , S2 , S3 , S4 of the four connected components S1 , S2 , S3 , S4 of
S \ Σ1 ∪ Σ2 are submanifolds with (singular) boundary.
Then there exists a Levi-flat (2n − 1)-subvariety M̃ ⊂ C × Cn with boundary S̃
(in the sense of currents) such that the natural projection π : C × Cn → Cn restricts
to a bijection which is a CR diffeomorphism between S̃ and S outside the complex
points.
7.5. Generalizations
7.5.1. Elementary models and their gluing. The examples and the proofs of the theorems when S is homeomorphic to a sphere (Sect. 7.3) or a torus (Sect. 7.4) suggest
the following definitions.
7.5.2. Definitions. Let T be a smooth, locally closed (i.e. closed in an open set),
connected submanifold of Cn , n ≥ 3. We assume that T has the following properties:
(i) T is relatively compact, non necessarily compact, and of codimension 2;
(ii) T is nonminimal at every CR point;
(iii) T has exactly 2 complex points which are flat and either special elliptic or
special 1-hyperbolic;
(iv) If p ∈ T is 1-hyperbolic, the singular orbit Σ through p is compact, Σ \ p
has two connected components σ1 , σ2 , whose closures are homeomorphic to spheres
of dimension 2n − 3;
(v) If p ∈ T is 1-hyperbolic, in the neighborhood of p, with convenient coordinates, the equation of T , up to third order terms is
zn =
n−1
(zj z j + λj Re zj2 );
λ1 > 1;
0 ≤ λj < 1
for j = 1
j=1
or in real coordinates xj , yj with zj = xj + iyj ,
n−1
(1 + λj )x2j + (1 − λj )yj2 + O(|z|3 ).
xn = (λ1 + 1)x21 − (λ1 − 1)y12 +
j=2
Other configurations are easily imagined.
up- and down- 1-hyperbolic points. Let T be the (2n− 2)-submanifold with (singular)
boundary contained into T such that either σ 1 (resp. σ 2 ) is the boundary of T near
p, or Σ is the boundary of T near p. In the first case, we say that p is 1-up, (resp.
2-up), in the second that p is down. Such a T will be called an elementary model.
For instance, T is 1-up and has one special elliptic point, we solve the boundary
problem as in S1 in the proof of Theorem 7.3.3.
7.5.3. The gluing (to be precised) happens between two compatible elementary models along boundaries, for instance down and 1-up.
30
P. Dolbeault
7.6. Other possible generalizations
The mixed Plateau problem can be set up in projective space CPn and in subspaces
of CPn on which the complex Plateau problem can be solved, using Statement 5.2.6,
its gemeralisation to any n ≥ 3 and a better geometric condition on the given
boundary.
Acknowledgments
I thank G. Tomassini and D. Zaitsev for discussions, corrections and remarks about
several parts of this paper.
References
[BeK 91] E. Bedford & W. Klingenberg, On the envelopes of holomorphy of a 2-sphere in
C2 , J. Amer. Math. Soc. 4 (1991), 623-646.
[Bi 65] E. Bishop, Differentiable manifolds in complex Euclidean space, Duke Math. J. 32
(1965), 1-22.
[CaC] A. Candel & L. Conlon, Foliations. I. Graduate Studies in Mathematics, 23. American Mathematical Society, Providence, RI, 2000.
[CS 51] S.S. Chern and E. Spanier, A theorem on orientable surfaces in four-dimensional
space, Com. Math. Helv., 25 (1951), 205-209.
[Di 98] T.C. Dinh, Enveloppe polynomiale d’un compact de longueur finie et chaı̂nes holomorphes à bord rectifiable, Acta Math. 180 (1998), 31-67.
[D 08] P. Dolbeault, On Levi-flat hypersurfaces with given boundary in Cn , Science in
China, Series A: Mathematics 51, no. 4 (2008), 551–562.
[D 09] P. Dolbeault, On Levi-flat hypersurfaces with given boundary: special hyperbolic
points, in preparation.
[DTZ 05] P. Dolbeault, G. Tomassini, D. Zaitsev, On boundaries of Levi-flat hypersurfaces
in Cn , C. R. Acad. Sci. Paris, Ser. I 341 (2005), 343–348.
[DTZ 09] P. Dolbeault, G. Tomassini, D. Zaitsev, On Levi-flat hypersurfaces with prescribed
boundary, Pure and Applied Math. Quarterly 6, N. 3 (Special Issue: In honor of
Joseph J. Kohn), 725–753, (2010) arXiv:0904.0481
[DH 97] P. Dolbeault et G. Henkin, Chaines holomorphes de bord donné dans CP n , Bull.
Soc. Math. France 125, 383–445.
[H 77] R. Harvey, Holomorphic chains and their boundaries, Proc. Symp. Pure Math. 30,
Part I, Amer. Math. Soc. (1977), 309–382.
[HL 75] R. Harvey and B. Lawson, On boundaries of complex analytic varieties, I, Ann. of
Math. 102, (1975), 233–290.
[HL 77] R. Harvey and B. Lawson, On boundaries of complex analytic varieties, II, Ann. of
Math. 106, (1977), 213–238.
[HL 04] F. R. Harvey and H. B. Lawson Jr., Boundaries of varieties in projective manifolds,
J. Geom. Anal. 14, no. 4, (2004), 673–695.
[Ki 79] J. King, Open problems in geometric function theory, Proceedings of the fifth international symposium of Math. p. 4, The Taniguchi foundation, 1978.
[Lce 95] M. G. Lawrence, Polynomial hulls of rectifiable curves, Amer. J. Math. 117 (1995),
405–417.
31
[P 08] P. Polo, Grassmanniennes orientées réelles, e-mail personnelle, 21 fév. 2008
[Rs 59] W. Rothstein, Bemerkungen zur Theorie komplexer Räume, Math. Ann. 137
(1959), 304–315.
[We 58] J. Wermer, The hull of a curve in Cn , Ann. Math. 68 (1958), 550–561.
Université Pierre et Marie Curie – Paris 6
UMR 7586, I.M.J.
F-75005 Paris
France
Presented by Wlodzimierz Waliszewski at the Session of the Mathematical-Physical
Commission of the L
ódź Society of Sciences and Arts on March 2, 2010
ZESPOLONY PROBLEM PLATEAU: STARE I NOWE WYNIKI
ORAZ PERSPEKTYWY
Streszczenie
Problem Plateau polega na badaniu powierzchni o minimalnym polu w 3-wymiarowej
przestrzeni euklidesowej, przy czym brzeg powierzchni jest dana̧ cia̧gla̧ krzywa̧ zamkniȩta̧.
Zespolony problem Plateau jest analogiczny na hermitowskiej rozmaitości zespolonej: jest
to geometryczny problem uogólnienia zamkniȩtej krzywej lub rozmaitości rzeczywistej na
analityczna̧ podrozmaitość zespolona̧, lub podrozmaitość plaska̧ w sensie Leviego. Przypominamy nierówność Wirtingera w przestrzeni Cn . Uzyskujemy minimalność analitycznych podrozmaitości zespolonych i analogiczne wlasności podrozmaitości plaskich w sensie
Leviego na rozmaitościach Kählera. Przypominamy znane wyniki w przestrzeni Cn i przestrzeni rzutowej CP n . Rozwia̧zujemy lub proponujemy rozszerzenia do rzeczywistych zagadnień parametrycznych, co prowadzi do konstrukcji hiperpowierzchni plaskich w sensie
Leviego o danym z góry brzegu w przypadku pewnych rozmaitości zespolonych.
PL ISSN 0459-6854
BULLETIN
DE
DES
LETTRES DE L
ÓDŹ
2010
Vol. LX
no. 1
pp. 33–43
Dmytro Mierzejewski
SPHERES IN SETS OF SOLUTIONS OF QUADRATIC
QUATERNIONIC EQUATIONS OF SOME TYPES
Summary
We study sets of solutions of quadratic quaternionic equations of some types by the
method of sections by hyperplanes perpendicular to the real axis. Namely, we look for
spheres in such sections. We get necessary and sufficient conditions for such section to
be spherical for a quaternionic equation of the form ax2 + x2 b = c and also (as a simple
corollary) of the form
ax2 + x2 b + apx + xpb + axp + pxb = q.
We prove that for any quaternionic equation of the form
ax2 + x2 b + xcx +
m
p() xq () = d
=1
with c ∈ R any such section is not spherical.
1. Notations and terminology of the paper
We use H as the standard notation for the set of all (real) quaternions. (We do
not deal with so-called complex quaternions whose components are complex.) The
notation R has its usual sense (the set of real numbers).
We use the standard notations i, j, k for the quaternionic imaginary units; recall
that
i2 = j 2 = k 2 = −1,
ij = −ji = k,
jk = −kj = i,
ki = −ik = j.
We always use subindices to denote the components of a quaternion as follows:
ξ = ξ0 + ξ1 i + ξ2 j + ξ3 k,
34
D. Mierzejewski
where
ξ0 , ξ1 , ξ2 , ξ3 ∈ R.
We often treat sets of some quaternions geometrically using the well-known interpretation of any quaternion ξ0 + ξ1 i + ξ2 j + ξ3 k as the point (ξ0 , ξ1 , ξ2 , ξ3 ) of the
four-dimensional space (here ξ0 , ξ1 , ξ2 , ξ3 ∈ R).
We name a quaternionic equation any one in which every known parameter is a
quaternion; as for solutions of such equations, we always consider solutions being
quaternions. Analogously, real equations are ones with real parameters and real solutions being considered. In every quaternionic equation of this paper the letter x
denotes the unknown and other letters denote given parameters, if there is no other
explanation.
2. Introduction
Investigations of solutions of polynomial quaternionic equations (or, by other words,
zeros of quaternionic polynomials) were performed in many works during 20th and
especially 21st century. Some examples of these works are in References.
The topic turns out to be much more difficult than similar investigations of real or
complex polynomials. For readers that are not very familiar with the topic we would
like to note that due to the absence of commutativity in the system of quaternions
even an arbitrary quaternionic monomial has comparatively complicated general
form, namely
a(1) xa(2) x . . . a(n) xa(n+1) ;
moreover a quaternionic polynomial written the simplest way can contain any number of terms of the same degree. For example, the general form of an arbitrary linear
quaternionic polynomial is
m
a() xb() + c,
=1
and the general form of quadratic one is
m
=1
a
()
xb
()
xc
()
+
q
d(p) xf (p) + g.
p=1
Note that if in the last expression there is such for which b() = 1 then the
corresponding term can be rewritten as a() x2 c() ; in this case we say that here is
a non-split square, while any term of the form a() xb() xc() is referred to as one
with a split square. In fact if b() is a real number then it is easy to make the square
non-split because every real number commutes with every quaternion.
It turns out that, in contrast to real and complex polynomials, it occurs often
that a quaternionic polynomial has infinitely many zeros. In particular, the set of
the zeros of a quaternionic polynomial or a part of this set may constitute a sphere
(see [7], [5], [3]). This paper is devoted to just this shape. We look for spheres in
Spheres in sets of solutions of quadratic quaternionic equations of some types
35
the sets of the solutions of polynomial (namely, quadratic) quaternionic equations of
several certain types. More precisely, we consider sections of the set of the solutions
of an equation by (three-dimensional) hyperplanes perpendicular to the real axis;
and we look for the sections being spheres.
As for method of investigations, we pass from a quaternionic equation to a system
of four real equations with four unknowns, that is, we equate coefficients at the
same quaternionic units after fulfilment arithmetical operations with the unknown
and all parameters written in the form ξ0 + ξ1 i + ξ2 j + ξ3 k. This method may be
called very simple and simultaneously it often leads to very complicated expressions.
Nevertheless it was successfully used in [8], [3], [9], [4]. By the way, consideration of
a section of the set of the solutions by a hyperplane perpendicular to the real axis
now means a fixed x0 in a system of equations, so that only x1 , x2 , x3 are considered
as unknowns; just this point of view was used in [3] and [4].
In Section 3 we investigate an arbitrary quaternionic equation of the form
ax2 + x2 b = c.
This is a particular case of a quadratic quaternionic equation with only non-split
squares. It was shown in [4] that for every such equation every above-mentioned
section is of one of the following shapes: a linear manifold, a sphere, a circle, a set of
two points, or the empty set. Now, according to the task of this paper, we look only
for spheres, and in the main theorem of Section 3 we get necessary and sufficient
conditions for such spherical section. Moreover a remark and a corollary at the end
of the section provide some generalizations of the main theorem.
In Section 4 we investigate some quaternionic quadratic equations with one split
square, namely:
m
p() xq () = d,
ax2 + x2 b + xcx +
=1
where c ∈ R (to ensure that the split square is “really” split). In particular, here
c = 0, while it is possible that a or b (or even both a and b) equals 0. Thus this type
of equations is very wide. But by some simple reasonings performed in Section 4 we
conclude that any equation of this type has no sphere in any from the considered
sections.
3. Some quadratic equations with two non-split squares
Theorem 1. Let a quaternionic equation of the following form be given:
(1)
ax2 + x2 b = c.
For each real number ξ0 consider the set Sξ0 of such solutions x of (1) that x0 = ξ0 .
Then Sξ0 can be a sphere only in the case ξ0 = 0. Namely, S0 is a sphere if and only
if either
c0
c1
c2
c3
(2)
=
=
=
< 0,
a0 + b 0
a1 + b 1
a2 + b 2
a3 + b 3
36
D. Mierzejewski
or these proportions and inequality hold true in wide sense, namely: for at most three
values of ∈ {0, 1, 2, 3} it is allowed
c = a + b = 0,
and (2) must get true after deleting every fraction with zeros in both the numerator
and the denominator. The centre of the corresponding sphere is always in the origin,
and its radius equals −c /(a + b ) with any ∈ {0, 1, 2, 3} for which this fraction is
well-defined.
Proof. Direct calculations give the following system of real equations equivalent
to (1):
⎧
(a0 + b0 )(x21 + x22 + x23 ) + 2(a1 + b1 )x0 x1 +
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2(a2 + b2 )x0 x2 + 2(a3 + b3 )x0 x3 = (a0 + b0 )x20 − c0 ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(a1 + b1 )(x21 + x22 + x23 ) − 2(a0 + b0 )x0 x1 +
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ 2(a3 − b3 )x0 x2 + 2(b2 − a2 )x0 x3 = (a1 + b1 )x20 − c1 ,
⎨
(3)
⎪
⎪
⎪
(a2 + b2 )(x21 + x22 + x23 ) + 2(b3 − a3 )x0 x1 −
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2
⎪
⎪
⎪ 2(a0 + b0 )x0 x2 + 2(a1 − b1 )x0 x3 = (a2 + b2 )x0 − c2 ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(a3 + b3 )(x21 + x22 + x23 ) + 2(a2 − b2 )x0 x1 +
⎪
⎪
⎪
⎪
⎪
⎪
⎩
2(b1 − a1 )x0 x2 − 2(a0 + b0 )x0 x3 = (a3 + b3 )x20 − c3 .
We have to investigate this system treating x0 as a fixed number, so that the unknowns are x1 , x2 , x3 and we consider shapes in the tree-dimensional space of points
(x1 , x2 , x3 ). The equations from (3) have similar structure. About each of them
it is essential to know whether its coefficient at the expression x21 + x22 + x23 equals
0. If yes then the equation generates, as a rule, a plane, but sometimes the whole
three-dimensional space (hyperplane) or the empty set; if no then the equation generates a sphere, a point, or the empty set.
We are interesting in the cases where the intersection of the four figures generated
by the equations is a sphere. Obviously, it occurs if and only if one of the following
four situations takes place:
1) every equation generates the same sphere;
2) one equation generates the hyperplane, and each from other three equations
generates the same sphere;
3) two equations generate the hyperplane, and each from other two equations
generates the same sphere;
4) three equations generate the hyperplane, and other equation generates a
sphere.
37
We will investigate every situation. But firstly we will rewrite every equation from
(3) in such a way that it will be convenient to determine the centres and the radii
of the spheres:
⎧ 2 2 2
a1 + b 1
a2 + b 2
a3 + b 3
⎪
⎪
x
+
x
+
x
+
x
+
x
+
x
=
⎪
1
0
2
0
3
0
⎪
⎪
a0 + b 0
a0 + b 0
a0 + b 0
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(a0 + b0 )2 + (a1 + b1 )2 + (a2 + b2 )2 + (a3 + b3 )2 2
c0
⎪
⎪
x0 −
,
⎪
2
⎪
(a
+
b
)
a
⎪
0
0
0 + b0
⎪
⎪
⎪
⎪
⎪
2 2 2
⎪
⎪
⎪
a0 + b 0
a3 − b 3
b 2 − a2
⎪
⎪
−
x
+
x
+
x
+
x
+
x
=
x
1
0
2
0
3
0
⎪
⎪
a1 + b 1
a1 + b 1
a1 + b 1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ (a0 + b0 )2 + (a1 + b1 )2 + (a2 − b2 )2 + (a3 − b3 )2 2
c1
⎪
⎪
x0 −
,
⎪
⎪
⎨
(a1 + b1 )2
a1 + b 1
(4)
⎪
2 2 2
⎪
⎪
b 3 − a3
a0 + b 0
a1 − b 1
⎪
⎪
+
x
+
x
−
x
+
x
+
x
=
x
⎪
1
0
2
0
3
0
⎪
⎪
a2 + b 2
a2 + b 2
a2 + b 2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(a0 + b0 )2 + (a1 − b1 )2 + (a2 + b2 )2 + (a3 − b3 )2 2
c2
⎪
⎪
⎪
x0 −
,
⎪
2
⎪
(a
+
b
)
a
2
2
2 + b2
⎪
⎪
⎪
⎪
⎪
⎪ 2 2 2
⎪
⎪
a2 − b 2
b 1 − a1
a0 + b 0
⎪
⎪
⎪
+
x
+
x
+
x
+
x
−
x
=
x
1
0
2
0
3
0
⎪
⎪
a3 + b 3
a3 + b 3
a3 + b 3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2
2
2
2
⎪
⎪
⎩ (a0 + b0 ) + (a1 − b1 ) + (a2 − b2 ) + (a3 + b3 ) x20 − c3 .
(a3 + b3 )2
a3 + b 3
Of course, each equation from (4) is well-defined only if any denominator in it does
not equal 0, but it is just a condition under which the equation can generate a sphere
and cannot generate any hyperplane. Thus we have to use an equation from (4) when
it generates a sphere, but one from (3) when it generates a hyperplane.
Let us investigate firstly the case where all the four equations generate spheres.
We have to look for the situation where these spheres are identical. It means that
they have the same centre and the same radius. The coordinates of the centre are
subtracted from x1 , x2 , x3 in the brackets in the left-hand side of an equation from
(4), and the radius equals the square root of the right-hand side. By the way, the
right-hand side has to be positive for the case of the sphere (if it is negative then
the equation generates the empty set, and if it equals 0 then the equation generates
a point). So, the following conditions arise:
(5a)
a0 + b 0
b 3 − a3
a2 − b 2
a1 + b 1
x0 = −
x0 =
x0 =
x0 ,
a0 + b 0
a1 + b 1
a2 + b 2
a3 + b 3
(5b)
a3 − b 3
a0 + b 0
b 1 − a1
a2 + b 2
x0 =
x0 = −
x0 =
x0 ,
a0 + b 0
a1 + b 1
a2 + b 2
a3 + b 3
38
D. Mierzejewski
a3 + b 3
b 2 − a2
a1 − b 1
a0 + b 0
x0 =
x0 =
x0 = −
x0 ,
a0 + b 0
a1 + b 1
a2 + b 2
a3 + b 3
(5c)
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
(5d)
(a0 + b0 )2 + (a1 + b1 )2 + (a2 + b2 )2 + (a3 + b3 )2 2
c0
x0 −
(a0 + b0 )2
a0 + b 0
=
(a0 + b0 )2 + (a1 + b1 )2 + (a2 − b2 )2 + (a3 − b3 )2 2
c1
x0 −
2
(a1 + b1 )
a1 + b 1
⎪
⎪
(a0 + b0 )2 + (a1 − b1 )2 + (a2 + b2 )2 + (a3 − b3 )2 2
c2
⎪
⎪
⎪
=
x0 −
⎪
2
⎪
(a2 + b2 )
a2 + b 2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(a + b0 )2 + (a1 − b1 )2 + (a2 − b2 )2 + (a3 + b3 )2 2
c3
⎪
⎩ = 0
x0 −
> 0.
(a3 + b3 )2
a3 + b 3
Note that the condition ∀ ∈ {0, 1, 2, 3} a + b = 0 (necessary for the case of four
spheres) follows from (5a-d) because otherwise (5a-d) has no sense.
Supposing that x0 = 0 we at once conclude from the first equation of (5a-d) that
a1 + b 1
a0 + b 0
=−
,
a0 + b 0
a1 + b 1
but it is impossible for real numbers. Therefore x0 = 0. Then (5a-d) can be sufficiently reduced as follows:
⎧
⎪
⎨ x0 = 0,
⎪
⎩
c1
c2
c3
c0
=
=
=
< 0.
a0 + b 0
a1 + b 1
a2 + b 2
a3 + b 3
This is just what is claimed in the theorem, and moreover it is easy to see that now
the centre and the radius of the sphere are just as claimed in the theorem.
Now we pass to the case where three equations generate spheres and one generates
the hyperplane. Firstly let the first equation of (3) generate the hyperplane. It means
that a0 + b0 = 0 and the equation can be rewritten as
2x0 ((a1 + b1 )x1 + (a2 + b2 )x2 + (a3 + b3 )x3 ) = −c0 .
Moreover the case of the hyperplane occurs if and only if every coefficient of this
linear equation equals 0, that is,
x0 (a1 + b1 ) = x0 (a2 + b2 ) = x0 (a3 + b3 ) = c0 = 0.
If x0 = 0 then a1 + b1 = a2 + b2 = a3 + b3 = 0, but this is in contradiction with
the fact that the last three equations of (3) generate spheres. Thus we conclude that
x0 = 0.
Then, analogously to the previous case, taking into attention that the spheres
have to be identical we get:
c2
c3
c1
=
=
< 0,
a1 + b 1
a2 + b 2
a3 + b 3
39
that, along with the obtained in the previous paragraph equalities x0 = a0 + b0 =
c0 = 0, means that the theorem again holds true in this case.
The cases where the only second, third, or fourth equation of (3) generates the
hyperplane are analogous. The most essential difference is the fact that supposing
x0 = 0 one refutes the case of a sphere for only the first equation, not for all the
three. But the result is the same.
Passing to the cases of two spheres and two hyperplanes we again obtain analogous pictures. The condition x0 = 0 arises every time due to the presence of a1 + b1 ,
a2 + b2 , and a3 + b3 in the first equation and of a0 + b0 in every other equation.
Other conditions are of the form
a + b = am + bm = c = cm = 0,
cn
cp
=
<0
an + b n
ap + b p
and show that the theorem holds true.
And at last the reader can easily verify that the cases of one sphere and three
hyperplanes are also analogous. In these cases the conditions look as follows:
x0 = a + b = am + bm = an + bn = c = cm = cn = 0,
cp
< 0.
ap + b p
By the way, here is already no need to equate the radii (since here is only one sphere)
and the last inequality may be rewritten also as
cp (ap + bp ) < 0.
Remark 1. In fact Theorem 1 provides also information about every quaternionic
equation of the form
(6)
αx2 β + γx2 δ = λ,
because (6) can be easily reduced to the form (1) by multiplication by β −1 on
the right and by γ −1 on the left (and if β or γ equals 0 then the corresponding
multiplication is simply unnecessary).
Corollary 1. Let a quaternionic equation of the following form be given:
(7)
For each real number ξ0 consider the set Sξ0 of such solutions x of (7) that x0 = ξ0 .
Then Sξ0 can be a sphere only in the case ξ0 = −p0 . Moreover putting
c := q + ap2 + p2 b
one obtains that S−p0 is a sphere if and only if either (2) holds true, or those
proportions and inequality hold true in wide sense, namely: for at most three values of
∈ {0, 1, 2, 3} it is allowed c = a +b = 0, and (2) must get true after deleting every
40
D. Mierzejewski
fraction with zeros in both the numerator and the denominator. The centre of the
corresponding sphere is always in the point −p, and its radius equals −c /(a + b )
with any ∈ {0, 1, 2, 3} for which this fraction is well-defined.
Proof. The point is that if one changes x by x + p in (1) then one gets just (7) with
q = c − ap2 − p2 b.
So, the set of the solutions of (7) can be obtained by the corresponding shifting of
the set of the solutions of (1), and that is all. 4. Some quadratic equations with one split square
The main result of this section is Theorem 2 situated at the end. Every other proposition of this section gives information about a particular case of an equation considered in Theorem 2. There is no formal reason to prove other propositions before
the main theorem, but we do this for convenience of the reader; namely, it is easier
firstly to understand a proof with more or less simple expressions and then to think
over how it changes after a certain complicating of the expressions.
Proposition 1. Let a quaternionic equation of the following form be given:
(8)
xax = b,
where a ∈ R. For each real number ξ0 consider the set Sξ0 of such solutions x of (8)
that x0 = ξ0 . Then for any ξ0 ∈ R Sξ0 is not a sphere.
Proof. Direct calculations give the following system of real equations equivalent
to (8):
⎧
−a0 x21 − a0 x22 − a0 x23 − 2a1 x0 x1 − 2a2 x0 x2 − 2a3 x0 x3 = b0 − a0 x20 ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ −a x2 + a x2 + a x2 + 2a x x − 2a x x − 2a x x = b − a x2 ,
⎪
⎨
1 1
1 2
1 3
0 0 1
2 1 2
3 1 3
1
1 0
(9)
⎪
⎪
⎪
a2 x21 − a2 x22 + a2 x23 + 2a0 x0 x2 − 2a1 x1 x2 − 2a3 x2 x3 = b2 − a2 x20 ,
⎪
⎪
⎪
⎪
⎪
⎪
⎩
a3 x21 + a3 x22 − a3 x23 + 2a0 x0 x3 − 2a1 x1 x3 − 2a2 x2 x3 = b3 − a3 x20 .
Obviously, with any fixed x0 each equation of (9) generates a surface of at most
second degree (in particular, it may be the empty set) or the whole hyperplane (in
very special cases). Obviously, a sphere in the intersection of such figures can arise
only if at least one of these figures is a sphere; moreover for this aim each equation
has to generate either a sphere, or the hyperplane.
From simple geometrical considerations we see that among these for equations
only the first one can generate a sphere. Therefore in order to obtain a sphere in the
intersection it is necessary to obtain the hyperplane by every equation excepting the
41
first one. But it can occur only if every coefficient from the equations equals 0. In
particular it means that
a1 = a2 = a3 = 0,
but it is in contradiction with the fact that a ∈ R. (Note that here it is impossible to
get a0 = 0 because a0 appears only along with x0 , which is considered as a constant
now.) So, any sphere in the intersection is impossible, and the proposition is proved.
Proposition 2. Let a quaternionic equation of the following form be given:
(10)
xax +
m
p() xq () = b,
=1
where a ∈ R. For each real number ξ0 consider the set Sξ0 of such solutions x of
(10) that x0 = ξ0 . Then for any ξ0 ∈ R Sξ0 is not a sphere.
Proof. Constituting the system of real equations equivalent to (10) one gets a system
differing from (9) by only presence of some additional terms being linear with respect
to x0 , x1 , x2 , x3 . It is easy to verify that under the circumstances all considerations
from the proof of Proposition 1 work as well for the current proof. Proposition 3. Let a quaternionic equation of the following form be given:
(11)
ax2 + x2 b + xcx = d,
where c ∈ R. For each real number ξ0 consider the set Sξ0 of such solutions x of (11)
Proof. Now the corresponding system of real equations is the following:
⎧
−(a0 + b0 + c0 )(x21 + x22 + x23 ) − 2(a1 + b1 + c1 )x0 x1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
− 2(a2 + b2 + c2 )x0 x2 − 2(a3 + b3 + c3 )x0 x3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
= d0 − (a0 + b0 + c0 )x20 ,
⎨
(12a)
⎪
2
2
2
⎪
⎪
⎪ −(a1 + b1 + c1 )x1 − (a1 + b1 − c1 )(x2 + x3 ) − 2c2 x1 x2 − 2c3 x1 x3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
+ 2(a0 + b0 + c0 )x0 x1 + 2(b3 − a3 )x0 x2 + 2(a2 − b2 )x0 x3
⎪
⎪
⎪
⎪
⎪
⎪
⎩
= d1 − (a1 + b1 + c1 )x20 ,
42
D. Mierzejewski
⎧
−(a2 + b2 + c2 )x22 − (a2 + b2 − c2 )(x21 + x23 ) − 2c1 x1 x2 − 2c3 x2 x3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
+ 2(a3 − b3 )x0 x1 + 2(a0 + b0 + c0 )x0 x2 + 2(b1 − a1 )x0 x3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
= d2 − (a2 + b2 + c2 )x20 ,
⎨
(12b)
⎪
⎪
⎪
−(a3 + b3 + c3 )x23 − (a3 + b3 − c3 )(x21 + x22 ) − 2c1 x1 x3 − 2c2 x2 x3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
+ 2(b2 − a2 )x0 x1 + 2(a1 − b1 )x0 x2 + 2(a0 + b0 + c0 )x0 x3
⎪
⎪
⎪
⎪
⎪
⎪
⎩
= d3 − (a3 + b3 + c3 )x20 .
Analogously to the proof of Proposition 1 we can see that for the presence of a
sphere in the section each equation of (12a, b) (with a fixed x0 ) has to generate either
a sphere, or the whole hyperplane. Supposing that the second equation generates a
sphere we obtain:
a 1 + b 1 + c1 = a 1 + b 1 − c1 ,
c2 = c3 = 0,
but it is the same that
c1 = c2 = c3 = 0,
so that c ∈ R, a contradiction with the condition of the proposition. Analogously it
is easy to show that the third and fourth equations cannot generate any sphere. So,
the only possibility to get a sphere in the intersection is to provide for all the last
three equations to generate the hyperplane. But for this aim all coefficients of the
corresponding polynomials (with respect to x1 , x2 , x3 ) have to equal 0; in particular,
we obtain again the impossible equalities
c1 = c2 = c3 = 0.
So, it is impossible to obtain any sphere in the intersection, and the proposition is
proved. Theorem 2. Let a quaternionic equation of the following form be given:
(13)
ax2 + x2 b + xcx +
m
p() xq () = d,
=1
where c ∈ R. For each real number ξ0 consider the set Sξ0 of such solutions x of (13)
Proof. Analogously to the proof of Proposition 2, it is easy to understand that this
theorem can be proved by practically the same considerations as in the proof of
Proposition 3 (that is, additional linear terms do not disturb). Spheres in sets of solutions of quadratic quaternionic equations of some types
43
References
[1] S. Eilenberg, I. Niven, The “fundamental theorem of algebra” for quaternions, Bull.
Amer. Math. Soc. 50, no. 4 (1944), 246–248.
[2] D. Janovská, G. Opfer, Linear equations in quaternionic variables, Mitt. Math. Ges.
Hamburg 27, (2008), 223–234.
[3] D. Mierzejewski, Investigation of quaternionic quadratic equations I. Factorization
and passing to a system of real equations, Bull. Soc. Sci. Lettres L
ódź. 58 Sér. Rech.
Déform. 56 (2008), 17–26.
[4] D. Mierzejewski, Quasi-spherical and multi-quasi-spherical polynomial quaternionic
equations: introduction of the notions and some examples, Advances in Applied Clifford Algebras, submitted.
[5] D. Mierzejewski, V. Szpakowski, On solutions of some types of quaternionic quadratic
equations, Bull. Soc. Sci. Lettres L
ódź. 58 Sér. Rech. Déform. 55 (2008), 49–58.
[6] I. Niven, The roots of a quaternion, Amer. Math. Monthly 49 (1942), 386–388.
[7] A. Pogorui, M. Shapiro, On the structure of the set of the zeros of quaternionic polynomials, Complex Variables and Elliptic Equations 49, no. 6 (2004), 379–389.
[8] V. Szpakowski, Solution of general linear quaternionic equations, The XI Kravchuk
International Scientific Conference, Kyiv (Kiev), Ukraine, 2006, p. 661 [in Ukrainian].
[9] V. Szpakowski, Solution of quadratic quaternionic equations, Bull. Soc. Sci. Lettres
L
ódź. Sér. Rech. Déform. 60 (2010) – to appear.
Teatralna Street, 5-b, flat 6
UA-10-014 Zhytomyr
Ukraine
Chair od Mathematics
Kielce University of Technology
(Politechnika Świȩtokrzyska)
Al. Tysia̧clecia Państwa Polskiego 7, bud. C
PL-25-314 Kielce, Poland
Presented by Julian L
awrynowicz at the Session of the Mathematical-Physical Commission of the L
SFERY W ZBIORACH ROZWIA̧ZAŃ ROWNAŃ KWADRATOWYCH
KWATERNIONOWYCH NIEKTORYCH TYPÓW
Streszczenie
Badamy zbiory rozwia̧zań równań kwadratowych kwaternionowych niektórych typów
metoda̧ przekrojów hiperplaszczyznami prostopadlymi do osi rzeczywistej. Mianowicie,
szukamy sfer w takich przekrojach. Otrzymujemy warunki konieczne i wystarczaja̧ce dla
sferyczności takich przekrojów dla równania kwaternionowego postaci ax2 + x2 b = c oraz
(jako prosty wniosek) postaci
Udowadniamy, że dla jakiegokolwiek równania kwaternionowego postaci
ax2 + x2 b + xcx +
m
=1
z c ∈ R żaden taki przekrój nie jest sferyczny.
p() xq () = d
PL ISSN 0459-6854
BULLETIN
DE
2010
DES
LETTRES DE L
ÓDŹ
Vol. LX
no. 1
pp. 45–65
Andrzej Krzysztof Kwaśniewski
COBWEB POSETS AND KoDAG DIGRAPHS ARE REPRESENTING
NATURAL JOIN OF RELATIONS, THEIR DI-BIGRAPHS
AND THE CORRESPONDING ADJACENCY MATRICES
Summary
Natural join of di-bigraphs (directed bi-parted graphs) and their corresponding adjacency matrices is defined and then applied to investigate the so-called cobweb posets and
their Hasse digraphs called KoDAGs. KoDAGs are special orderable Directed Acyclic
Graphs which are cover relation digraphs of cobweb posets introduced by the author few
years ago. KoDAGs appear to be distinguished family of F errers digraphs which are natural join of a corresponding ordering chain of one direction directed cliques called di-bicliques.
These digraphs serve to represent faithfully corresponding relations of arbitrary arity so
that all relations of arbitrary arity are their subrelations. Being this chain − way complete
(compare with Kompletne, Kuratowski Kn,m bipartite graphs) their DAG denotation is
accompanied with the letter K in front of descriptive abbreviation oDAG.
The way to join bipartite digraphs of binary into multi-ary relations is the natural join
operation either on relations or their digraph representatives. This natural join operation
is denoted here by ⊕→ symbol deliberately referring – in a reminiscent manner – to the
direct sum ⊕ of adjacency matrices as it becomes the case for disjoint di-bigraphs.
1. Introduction to cobweb posets
1.1. Notation
One may identify and interpret some classes of digraphs in terms of their associated
posets. (see [1] Interpretations in terms of posets Section 9).
Definition 1 (see [1]). Let D = (Φ, ≺) be a digraph. w, v ∈ Φ are said to be equivalent
iff there exists a directed path containing both w and v vertices. We then write: v ∼ w
for such pairs and denote by [v] the ∼ equivalence class of v ∈ Φ.
46
A. K. Kwaśniewski
Definition 2 (see [1]). The poset P (D) associated to D = (Φ, ≺) is the poset P (D) =
(Φ/ ∼, ≤) where [v] ≤ [w] iff there exists a directed path from a vertex x ∈ [v] to a
vertex y ∈ [w].
The graded digraphs case:
If D = (Φ, ≺) is graded digraph then D = (Φ, ≺) is necessarily acyclic. Then
no two elements of D = (Φ, ≺) are ∼ equivalent and thereby P (D) = (Φ/ ∼, ≤)
associated to D = (V, ≺) is equivalent to: P (D) ≡ (Φ, ≤) = transitive, reflexive
closure of D = (Φ, ≺).
The cobweb posets where introduced in several paper (see [2–6] and references
therein) in terms of their poset [Hasse] diagrams. Here we deliver their equivalent
definition preceded by preliminary notation and nomenclature.
Notation: nomenclature, di-bicliques and natural join
In order to proceed proficiently we adopt the following.
Definition 3. A digraph D = (Φ, ≺·) is transitive irreducible iff its transitive reduction
(D) equals D.
Definition 4. A poset P (D) = (Φ, ≤) is associated to a graded digraph D = (Φ, ≺)
iff P (D) is the transitive, reflexive closure of D = (Φ, ≺).
Obvious
D = (Φ, ≺ ·) is transitive irreducible iff its transitive reduction (D) = D iff D =
(Φ, ≺·) is Hasse diagram of the poset P (D) = (Φ, ≤) associated to D ≡ D = (Φ, ≺·)
is cover relation ≺· digraph ≡ D = (Φ, ≺·) is P (D) = (Φ, ≤) poset diagram.
1.2. Further on we adopt also the following nomenclature
We shall use until stated otherwise the convention: N = {1, 2, ..., k, ...}. n ∈ N ∪{∞}.
The Cartesian product Φ1 × ... × Φk of pairwise disjoint sets Φ1 , ..., Φk is a k-ary
relation, called sometimes the universal relation and here now on Kompletna relation
or K-relation, (in Polish this means complete). The purpose of introducing the letter
K is to distinguish in what follows [for k = 2] from complete digraphs notions
established content.
Convention 1. [identification]. The binary relation E ⊆ X × Y is being here identified with its bipartite digraph representation B = (X ∪ Y, E).
→
Notation Km,n ≡ B = (X ∪ Y, E) if |X| = m, |Y | = n. Colligate with Kuratowski
and Km,n .
Comment 1.
Complete n-vertex graphs for which all pairs of vertices are adjacent are denoted by
Kn , The letter K had been chosen in honor of Professor Kazimierz Kuratowski, a
Cobweb posets and KoDAG digraphs are representing natural join of relations
47
distinguished pioneer in graph theory. The corresponding two widely used concepts
for digraphs are called complete digraphs or complete symmetric digraph in which
every two different vertices are joined by an arc and complete oriented graphs i.e.
tournament graphs.
The binary K-relation E = X × Y equivalent to bipartite digraph B = (X ∪
→
Y, E) ≡Km,n is called from now on a di-biclique following [6].
Example of di-bicliques obtained from bicliques: See Fig. 1.
If you imagine arrows → left to the right - you would see two examples of dibicliques
Fig. 1: Examples of di-bicliques if edges are replaced by arrows of join direction
if you imagine arrows ← right to the left, you would see another examples of dibicliques.
Convention 2. [recall] The binary relation E ⊆ X ×Y is identified with its bipartite
digraph B = (X ∪ Y, E) unless otherwise denoted distinctively deliberately.
The natural join
The natural join operation is a binary operation like Θ operator in computer
science denoted here by ⊕→ symbol deliberately referring – in a quite reminiscent
manner – to direct sum ⊕ of adjacency Boolean matrices and – as a matter of fact
and in effect – to direct the sum ⊕ of corresponding biadjacency [reduced] matrices
of digraphs under natural join.
⊕→ is a natural operator for sequences construction. ⊕→ operates on multi-ary
relations according to the scheme: (n + k)ary ⊕→ (k + m)ary = (n + k + m)ary .
For example: (1 + 1)ary ⊕→ (1 + 1)ary = (1 + 1 + 1)ary , binary ⊕→ binary = ternary.
Accordingly an action of ⊕→ on these multi-ary relations’ digraphs adjacency matrices is to be designed soon in what follows.
Domain-Codomain F -sequence condition dom(Rk+1 ) = ran(Rk ), k = 0, 1, 2, ...
Consider any natural number valued sequence F = {Fn }n≥0 . Consider then any
chain of binary relations defined on pairwise disjoint finite sets with cardinalities
appointed by F -sequence elements values. For that to start we specify at first a
relations’ domain-co-domain F -sequence.
48
A. K. Kwaśniewski
Domain-Codomain F -sequence (|Φn | = Fn )
Φ0 , Φ1 , ...Φi , ... Φk ∩ Φn = ∅ for k = n, |Φn | = Fn ; i, k, n = 0, 1, 2, ...
n
Let Φ = k=0 Φk be the corresponding ordered partition [anticipating-Φ is the vertex
set of D = (Φ, ≺· ) and its transitive, reflexive closure (Φ, ≤)]. Impose dom(Rk+1 ) =
ran(Rk ) condition, k ∈ N ∪ {∞}. What we get is binary relations chain.
Definition 5 [Relation‘s chain]. Let Φ = nk=0 Φk , Φk ∩ Φn = ∅ for k = n be the
ordered partition of the set Φ. Let a sequence of binary relations be given such that
R0 , R1 , ..., Ri , ..., Ri+n , ..., Rk ⊆ Φk × Φk+1 , dom(Rk+1 ) = ran(Rk ).
Then the sequence Rk k≥0 is called natural join (binary) relation’s chain. Extension to varying arity relations’ natural join chains is straightforward.
As necessarily dom(Rk+1 ) = ran(Rk ) for relations’ natural join chain any given
binary relation’s chain is not just a sequence; therefore we use “link to link” notation
for k, i, n = 1, 2, 3, ... ready for relational data basis applications:
R0 ⊕→ R1 ⊕→ ...⊕→ Ri ⊕→ ...⊕→ Ri+n , ...is an F − chain of binary relations
where ⊕ → denotes natural join of relations as well as both natural join of their
bipartite digraphs and the natural join of their representative adjacency matrices
(see Section 3).
Relation’s F -chain naturally represented by [identified with] the chain of their
bipartite digraphs
R0 ⊕→ R1 ⊕→ ...⊕→ Ri ⊕→ ...⊕→ Ri+n , ... ⇔
⇔ B0 ⊕→ B1 ⊕→ ...⊕→ Bi ⊕→ ...⊕→ Bi+n , ...
results in F -partial ordered set Φ, ≤ with its Hasse digraph representation
looking-like specific “cobweb” image [see figures below].
1.3. Partial order ≤
The partial order relation ≤ in the set of all points-vertices is determined uniquely
by the above equivalent F -chains. Let x, y ∈ Φ = nk=0 Φk and let k, i = 0, 1, 2, ... .
Then
(1)
c R
c i+k−1 )y
x ≤ y ⇔ ∀x∈Φ : x ≤ x ∨ Φi x < y ∈ Φi+k if f x(Ri ...
c stays for [Boolean] composition of binary relations.
where “”
Relation (≤) defined equivalently:
x ≤ y in (Φ, ≤) iff either x = y or there exist a directed path from x to y; x, y ∈ Φ.
Let now Rk = Φk × Φk+1 , k ∈ N ∪ {0}. For “historical” reasons [2–6] we shall
call such partial ordered set Π = Φ, ≤ the cobweb poset as theirs Hasse digraph
representation looks like specific “cobweb” image (imagine and/or draw also their
transitive and reflexive cover digraph Φ, ≤. Cobweb? Super-cobweb!... – with fog
droplets loops?).
49
1.4. Cobweb posets (Π = Φ, ≤)
Convention 3. [recall]. The binary relation E ⊆ X×Y is identified with its bipartite
→
digraph B = (X ∪ Y, E) ≡Km,n where |X| = m, |Y | = n.
Definition 6 [cobweb poset]. Let D = (Φ, ≺·) be a transitive irreducible digraph. Let
n ∈ N ∪ {∞}. Let D be a natural join D = ⊕→nk=0 Bk of di-bicliques
Bk = (Φk ∪ Φk+1 , Φk × Φk+1 ), n ∈ N ∪ {∞}.
Hence the digraph D = (Φ, ≺·) is graded. The poset Π(D) associated to this graded
digraph D = (Φ, ≺·) is called a cobweb poset.
Convention 4. In a case we want to underline that we deal with finite cobweb poset
(a subposet of appropriate – for example infinite F -cobweb poset Π(D)) we shall use
a subscript and write Pn .
See: [2–6], [10], [13], [18].
Comment 2.
Graded graph is a natural join of bipartite graphs that form a chain of consecutive
levels [i.e. graded graphs’ antichains].
Graded digraph is a natural join of bipartite digraphs that form a chain of
consecutive levels [i.e. graded digraphs’ antichains].
Comment 3. (Definition 6. Recapitulation in brief.)
Cobweb poset is the poset
Π = Φ, ≤,
where
n
Φ=
and
k=0
≺· = ⊕→n−1
k=0 Φk × Φk+1 , n ∈ N ∪ {∞}.
Cobweb poset is the poset
Π = Φ, ≤,
where Φ =
n
k=0
and
→
≺· = ⊕→n−1
k=0 Kk,k+1 ,
n ∈ N ∪ {∞},
where ≤ is the transitive, reflexive cover of ≺·.
Comment 4. (F -partial ordered set)
Cobweb poset Π = Φ, ≤ is naturally graded and sequence F – denominated thereby
we call it sometimes F -partial ordered set Φ, ≤.
2. Dimension of cobweb posets – revisited
2.1. oDAG [7]
Observation 1. [cobwebs are oDAGs]. In [2] it was observed that cobweb posets’
Hasse diagrams are the members of the so-called oDAGs family i.e. cobweb posets’
50
A. K. Kwaśniewski
Hasse diagrams are orderable Directed Acyclic Graphs which is equivalent to say that
the associated poset P (D) = (Φ, ≤) of D = (Φ, ≺·) of is of dimension 2.
Recall: DAGs – hence graded digraphs with minimal elements always might be
considered – up to digraphs isomorphism – as natural digraphs [8] i.e. digraphs with
natural labeling (i.e. xi < xj ⇒ i < j).
Definition 7 [Plotnikov – see [7], [2] and then below]. A digraph D = (Φ, ≺) is called
the orderable digraph (oDAG) if there exists a dim 2 poset such that its Hasse
diagram coincides with the digraph G .
The statement from [2] may be now restated as follows:
Observation 2. [oDAG]. Cobweb P (D) = (Φ, ≤) posets’ Hasse diagrams D =
(Φ, ≺·) are oDAGs.
Proof. Obvious. Cobweb posets are posets with minimal elements set Φ0 . Cobweb
posets Hasse diagrams are DAGs. Cobweb posets representing the natural join of
are then dim 2 posets as their Hasse digraphs are intersection of a natural labeling
linear order L1 and its “dual” L2 denominated correspondingly in a standard way
by:
L1 = natural labeling: chose for the topological ordering L1 the labeling of minimal elements set Φ0 with labels 1, 2, ..., from the left to the right (see Fig. 2) then
proceed up to the next level Φ1 and continue the labeling “→” from the left to the
right [Φ1 is now treated as the set o minimal elements if Φ0 is removed] and so
on. Apply the procedure of subsequent removal of minimal elements i.e. removal of
subsequent labeled levels Fk – labeling the vertices along the levels from the left to
the right.
L2 = “dual” natural labeling: chose for the topological ordering L2 the labeling of
minimal elements set F0 with labels 1, 2, ..., from the right to the left to (see Fig. 1)
then proceed up to the next level F1 and continue the labeling “←” from the right
to the left [Φ1 is now treated as the set o minimal elements if Φ0 is removed] and
so on. Apply the procedure of subsequent removal of minimal elements i.e. removal
of subsequent labeled levels Φk – labeling now the vertices along the levels from the
right to the left q.e.d.
2.2. Brief history of the short oDAG’s name life
On the history of oDAG nomenclature with David Halitsky and others input one is
expected to see more in [15]. See also the December 2008 subject of The Internet
Gian Carlo Rota Polish Seminar (http://ii.uwb.edu.pl/akk/sem/sem rota.htm). Here
we present its sub-history leading the author to note that cobweb posets are oDAGs.
According to Anatoly Plotnikov the concept and the name of oDAG was introduced by David Halitsky from Cumulative Inquiry in 2004.
51
oDAG-2004 (Plotnikov)
Quote 1. A digraph G ∈ Dn will be called orderable (oDAG) if there exists are dim
2 poset such that its Hasse diagram coincide with the digraph G.
The Quote 1 comes from [9] in [2] i.e. from A. D. Plotnikov, A formal approach to
the oDAG/POSET problem (2004) html://www.cumulativeinquiry.com/Problems/
solut2.pdf (submitted to publication – March 2005).
The quote of the Quote 1 is to be found in [9]:
oDAG-2005 [2]
Quote 2. A digraph G is called the orderable digraph (oDAG) if there exists a dim
2 poset such that its Hasse diagram coincides with the digraph G [2].
oDAG-2006 [7]
Quote 3. A digraph G is called the orderable if there exists a dim 2 poset such
that its Hasse diagram coincides with the digraph G [7].
For further use of oDAG nomenclature see [6], and references therein. For further
references and recent results on cobweb posets see [10] and [11].
Definition 8 [KoDAG]. The transitive and reflexive reduction of cobweb poset Π =
Φ, ≤ i.e. posets’ Π cover relation digraph [Hasse diagram] D = (Φ, ≺·) is called
KoDAG; see [11–14].
Comment 5. Apply Comment 1.
Why do we stick to call KoDAGs graded digraphs with associated poset
Π = Φ, ≤ the orderable DAGs on their own independently of the nomenclature
quoted?
Let D = (Φ, ≺·) denote any transitive irreducible DAG [for example any graded
digraph including KoDAG digraph for example as above]. Let a poset P (D) = (Φ, ≤)
be associated to D = (Φ, ≺·).
Definition 9 [Ferrers dimension]. We say that the poset P (D) = (Φ, ≤) is of Ferrers
dimension k iff it is associated to D = (Φ, ≺·) of Ferrers dimension k.
Observation 3. [Ferrers dimension]. Cobweb posets are posets of Ferrers dimension
equal to one.
Proof. Apply any of many characterizations of Ferrers digraphs to see that cobweb
posets are posets’ cover relation digraphs [Hasse diagrams] are Ferrers digraphs. For
example consult Section 3 and see that biadjacency matrix does not contain any of
two 2 × 2 permutation matrices.
Comment 6. Any KoDAG digraph D = (Φ, ≺ ·) is the digraph stable under
the transitive and reflexive reduction i.e. [“irreducible”] Hasse portrait of Ferrers
relation ≺·. The positions of 1’s in biadjacecy [reduced adjacency] matrix display
the support of Ferrers relation ≺·. D = (Φ, ≺·) is then interval order relation digraph. The digraph (Φ, ≤) of the cobweb poset P (D) = (Φ, ≤) associated to KoDAG
52
A. K. Kwaśniewski
digraph D = (Φ, ≤) is the portrait of Ferrers relation ≤. The positions of 1’s in biadjacecy [reduced adjacency] matrix display the support of Ferrers relation ≤. Note:
for F -denominated cobweb posets the nomenclature identifies: biajacency [reduced
adjacency] matrix ≡ zeta matrix i.e. the incidence matrix ζF of the F - poset (see:
Fig. ζN and Fig. ζF ). Recall that this F -partial ordered set Φ, ≤ is a natural join of
F -chain of binary K-relations (complete or universal relations as called sometimes).
→
These relations are represented by di-bicliques Kk,k+1 which are on their own
the Ferrers dimension one digraphs. As for the other – not necessarily K-relations’
chains we may end up with Ferrers or not digraphs in corresponding di-bigraphs’
chain. See below, then Section 4 and more in [15].
3. The natural join ⊕→ operation
We define here the adjacency matrices representation of the natural join ⊕→ operation.
3.1. Recall
Let D(R) = (V (R) ∪ W (R), E(R)) ≡ (V ∪ W, E); V ∩ W = ∅, E(R) ⊆ V × W . Let
D(R) denotes here down the bipartite digraph of binary relation R with dom(R) = V
and rang(R) = W . Colligate with the anticipated examples
→
R = Rk ⊆ Φk × Φk+1 ≡Kk,k+1 , V (R) ∪ W (R) = Φk ∪ Φk+1 .
3.2. The adjacency matrices and their natural join
The adjacency matrix A of a bipartite graph with biadjacency (reduced adjacency
[16]) matrix B is given by
0 B
A=
.
BT 0
Definition 10. The adjacency matrix A[D] of a bipartite digraph D(R) = (P ∪L, E ⊆
P × L) with biadjacency matrix B is given by
0k,k B(k × m)
.
A[D] =
0m,m
0m,k
where k = |P |, m = |L|.
c = composition of binary relations S and
Convention 5. S R
c S
= BR B
R ⇔ BRS
c
where
(|V | = k, |W | = m) BR (k × m) ≡ BR
is the (k × m).
biadjacency [or another name: reduced adjacency] matrix of the bipartite relac apart from relations composition denotes also Boolean
tions’ R digraph B(R) and Cobweb posets and KoDAG digraphs are representing natural join of relations
53
multiplication of these rectangular biadjacency Boolean matrices BR , BS . What is
their form? The answer is in the block structure of the standard square (n × n) adjacency matrix A[D(R)]; n = k + m. The form of standard square adjacency matrix
A[G(R)] of bipartite digraph D(R) has the following apparently recognizable block
reduced structure [Os×s stays for (k × m) zero matrix]:
Ok×k AR (k × m)
A[D(R)] =
.
Om×k Om×m
Let D(S) = (W (S) ∪ T (S), E(S)); W ∩ T = ∅, E(S) ⊆ W × T ; (|W | = m, |T | = s);
hence
Om×m AS (m × s)
A[D(S)] =
.
Os×m Os×s
Definition 11 [natural join condition]. The ordered pair of matrices A1 , A2 is said
to satisfy the natural join condition iff these matrices have the block structure of
A[D(R)] and A[D(S)] as above i.e. iff they might be identified accordingly: A1 =
A[D(R)] and A2 = A[D(S)].
Correspondingly, if two given digraphs G1 and G2 are such that their adjacency
matrices A1 = A[G1 ] and A2 = A[G2 ] do satisfy the natural join condition, we
shall say that G1 and G2 satisfy the natural join condition. For matrices satisfying
the natural join condition one may define what follows.
c →
First we define the Boolean reduced or natural join composition and secondly the natural join ⊕→ of adjacent matrices satisfying the natural join
condition.
c
Definition 12 (→
composition).
c
c
A[D(RS)]
=: A[D(R)]→
A[D(S)] =
Ok×k
Os×k
ARS
c (k × s)
Os×s
c S (m × s).
where ARS
c (k × s) = AR (k × m)A
According to the scheme:
c [(m + s) × (m + s)] = [(k + s) × (k + s)].
[(k + m) × (k + m)]→
Comment 7. The adequate projection makes out the intermediate, joint in common
dom(S) = rang(R) = W , |W | = m.
c
The above Boolean reduced composition →
of adjacent matrices technically
reduces then to the calculation of just Boolean product of the reduced rectangular
adjacency matrices of the bipartite relations‘ graphs.
We are however now in need of the Boolean natural join product ⊕→ of adjacent
matrices already announced at the beginning of this presentation. Let us now define
it.
54
A. K. Kwaśniewski
As for the natural join notion we aim at the morphism correspondence:
S⊕→ R ⇔ MS⊕→R = MR ⊕→ MS ,
where S⊕→ R = natural join of binary relations S and R while MS⊕→R = MR ⊕→
MS = natural join of standard square adjacency matrices (with customary convention: M [G(R)] ≡ MR adapted). Attention: recall here that the natural join of the
above binary relations R⊕→ S is the ternary relation – and on one results in k-ary
relations if with more factors undergo the ⊕→ product. As a matter of fact ⊕→
operates on multi-ary relations according to the scheme:
(n + k)ary ⊕→ (k + m)ary = (n + k + m)ary .
For example: (1 + 1)ary ⊕→ (1 + 1)ary = (1 + 1 + 1)ary , binary⊕→ binary = ternary.
Technically – the natural join of the k-ary and n-ary relations is defined accordingly the same way via ⊕→ natural join product of adjacency matrices – the
adjacency matrices of these relations’ Hasse digraphs.
With the notation established above we finally define the natural join ⊕→ of two
adjacency matrices as follows:
Definition 13. Natural join ⊕→ of biadjacency matrices:
=
A[D(R⊕→ S)] =: A[D(R)]⊕→ A[D(S)] =
Om×m AS (m × s)
Ok×k AR (k × m)
⊕→
=
Om×k Om×m
Os×m Os×s
⎤
⎡
Ok×k AR (k × m) Ok×s
= ⎣ Om×k Om×m
AS (m × s) ⎦ .
Os×s
Os×k Os×m
Comment 8. The adequate projection used in natural join operation lefts one copy
of the joint in common “intermediate” submatrix Om×m and consequently lefts one
copy of “intermediate” joint in common m according to the scheme:
[(k + m) × (k + m)]⊕→ [(m + s) × (m + s)] = [(k + m + s) × (k + m + s)].
3.3. The biadjacency matrices of the natural join of adjacency matrices
Denote by B(A) the biadjacency matrix of the adjacency matrix A.
Let A(G) denote the adjacency matrix of the digraph G, for example a di-biclique
relation digraph. Let A(Gk ), k = 0, 1, 2, ... be the sequence adjacency matrices of the
sequence Gk , k = 0, 1, 2, ... of digraphs. Let us identify B(A) ≡ B(G) as a convention.
Definition 14 [digraphs natural join]. Let digraphs G1 and G2 satisfy the natural
join condition. Let us make then the identification A(G1 ⊕→ G2 ) ≡ A1 ⊕→ A2 as
definition. The digraph G1 ⊕→ G2 is called the digraphs natural join of digraphs G1
and G2 . Note that the order is essential.
We observe at once what follows.
55
Observation 4.
B(G1 ⊕→ G2 ) ≡ B(A1 ⊕→ A2 ) = B(A1 ) ⊕ B(A2 ) ≡ B(G1 ) ⊕ B(G2 ).
Comment 9. Observation 4 justifies the notation ⊕→ for the natural join of relations digraphs and equivalently for the natural join of their adjacency matrices and
equivalently for the natural join of relations that these are faithful representatives
of.
As a consequence we have
Observation 5.
B (⊕→ni=1 ) ≡ B[⊕→ni=1 A(Gi )] = ⊕ni=1 B[A(Gi )] ≡ diag(B1 , B2 , ..., Bn ) =
⎤
⎡
B1
⎥
⎢
B2
⎥
⎢
⎥ , n ∈ N ∪ {∞}.
⎢
B3
=⎢
⎥
⎦
⎣
... ... ...
Bn
3.4. Applications
Once any positive integer-valued sequence F = {Fn }n≥1 is being chosen its KoDAG
digraph is identified with Hasse cover relation digraph. Its adjacency matrix AF is
sometimes called Hasse matrix and is given in a plausible form and impressively
straightforward way. Just use the fact that the Hasse digraph which is displaying
cover relation ≺· is an F -chain of coined bipartite digraphs – coined each preceding with a subsequent one by natural join operator ⊕ → [resemblance of ⊕ → to
direct matrix sum is not naive - compare “natural join” of disjoint digraphs with no
common set of marked nodes (“attributes”) ].
Note: I(s×k) stays for (s×k) matrix of ones i.e. [I(s×k)]ij = 1; 1 ≤ i ≤ s, 1 ≤ j ≤ k.
Let us start first with F = {Fn }n≥1 = N ; see Fig. 2. Then its associated F partial ordered set Φ, ≤ has the following Hasse digraph displaying cover relation
of the ≤ partial order.
The Hasse matrix AN i.e. adjacency matrix of cover relation digraph i.e. adjacency matrix of the Hasse diagram of the N -denominated cobweb poset Φ, ≤ is
given by upper triangular matrix AN of the form
⎡
AN
⎢
⎢
=⎢
⎢
⎣
O1×1
O2×1
O3×1
O4×1
...etc.
I(1 × 2)
O2×2
O3×2
O4×2
...
O1×∞
I(2 × 3)
O3×3
O4×3
and
⎤
O2×∞
I(3 × 4) O3×∞
O4×4
I(4 × 5) O4×∞
so
on...
⎥
⎥
⎥.
⎥
⎦
One may see that the zeta function matrix of the F = N choice is geometrical series
in AN i.e. the geometrical series in the poset Φ, ≤ Hasse matrix AN :
56
A. K. Kwaśniewski
Fig. 2: Display of a finite subposet Π6 of the N natural numbers cobweb poset
c
ζ = (1 − AN )−1
.
Explicitly:
c
c
≡ I∞ + AN + A2
ζ = (1 − AN )−1
N + ... =
⎡
⎢
⎢
=⎢
⎢
⎣
I1×1
O2×1
O3×1
O4×1
...etc
I(1 × ∞)
I2×2
O3×2
O4×2
...
⎤
I(2 × ∞)
I3×3
I(3 × ∞)
O4×3
I4×4
and
so
⎥
⎥
⎥,
⎥
I(4 × ∞) ⎦
on...
c
. Indeed, let AN = A with Akij being the number of maximal
ζ = (1 − AN )−1
k-chains (k > 0) from the x0 ∈ Φi to xk ∈ Φj , i.e. here
0
k = j − i
1 k = j − i,
c
=
Akij =
, and hence Ak
ij
j!/i! k = j − k
0 k=
j − k,
c
c
and Am
are disjoint for k = m.
and the supports (nonzero matrices blocks) of Ak
Indeed: the entry in row i and column j of the inverse (I − A)−1 gives the number
of directed paths from vertex xi to vertex xj . This can be seen from geometric series
with adjacency matrix as an argument
(I − A)−1 = I + A + A2 + A3 + ...
taking care of the fact that the number of paths from i to j equals the number of
paths of length 0 plus the number of paths of length 1 plus the number of paths of
length 2, etc.
c
gives the
Therefore the entry in row i and column j of the inverse (I − A)−1
answer whether there exist directed paths from a vertex i to vertex j (Boolean value
57
1) or not (Boolean value 0), i.e. whether these vertices are comparable, i.e. whether
xi < xj , or not.
Remark: In the cases – Boolean poset 2N and the “Ferrand-Zeckendorf” poset of
finite subsets of N without two consecutive elements considered in [17] one has
c
c
≡ I∞×∞ + A + A2
+ ...
ζ = exp[A] = (1 − A)−1
because in those cases
1 k
0 k = j − i
1 k =j−i
c
Aij = Ak
, and hence
=
.
Akij =
ij
k! k = j − k
0 k = j − k
k!
How does it go in our F -case? Just see
For example:
⎡
O1×1 O1×2 I(1 × 3)
⎢ O2×1 O2×2 O2×3
⎢
c
⎢
A2
O3×2 O3×3
N = ⎢ O3×1
⎣ O4×1 O4×2 O4×3
...etc. ...
and
c
c
c
c
0
1
2
A2
N and then add AN ∨ AN ∨ AN ∨ ... .
O1×∞
I(2 × 4)
O3×4
O4×4
so
⎤
O2×∞
I(3 × 5) O3×∞
O4×5
I(4 × 6) O4×∞
on...
⎥
⎥
⎥,
⎥
⎦
Consequently we arrive at the incidence matrix ζ = exp[AN ] for the positive integers
cobweb poset displayed by Fig. 3. Note that incidence matrix ζ representing uniquely
its corresponding cobweb poset exhibits (see below) a staircase structure of zeros
above the diagonal which is characteristic to Hasse diagrams of all cobweb posets.
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
.
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
.
1
1
1
0
1
0
0
0
0
0
0
0
0
0
0
0
.
1
1
1
0
0
1
0
0
0
0
0
0
0
0
0
0
.
1
1
1
1
1
0
1
0
0
0
0
0
0
0
0
0
.
1
1
1
1
1
0
0
1
0
0
0
0
0
0
0
0
.
1
1
1
1
1
1
0
0
1
0
0
0
0
0
0
0
.
1
1
1
1
1
1
0
0
0
1
0
0
0
0
0
0
.
1
1
1
1
1
1
1
1
0
1
1
0
0
0
0
0
.
1
1
1
1
1
1
1
1
0
0
0
1
0
0
0
0
.
1
1
1
1
1
1
1
1
0
0
0
0
1
0
0
0
.
1
1
1
1
1
1
1
1
1
1
0
0
0
1
0
0
.
1
1
1
1
1
1
1
1
1
1
0
0
0
0
1
0
.
1 ···
1 ···
1 ···
1 ···
1 ···
1 ···
1 ···
1 ···
1 ···
1 ···
1 ···
1 ···
1 ···
1 ···
1 ···
1 ···
. .···
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Matrix ζN : The incidence matrix ζ for the natural numbers i.e. the N-cobweb poset.
58
A. K. Kwaśniewski
Fig. 3: Display of the F -Fibonacci numbers cobweb poset
Comment 9. The given F -denominated staircase zeros structure above on the
diagonal of zeta matrix zeta is the unique characteristic of its corresponding
F -KoDAG Hasse digraphs.
For example see Fig. ζF . below (from [6]).
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
.
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
.
1
1
1
0
1
0
0
0
0
0
0
0
0
0
0
0
.
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
.
1
1
1
1
1
0
1
0
0
0
0
0
0
0
0
0
.
1
1
1
1
1
0
0
1
0
0
0
0
0
0
0
0
.
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
.
1
1
1
1
1
1
1
1
0
1
0
0
0
0
0
0
.
1
1
1
1
1
1
1
1
0
0
1
0
0
0
0
0
.
1
1
1
1
1
1
1
1
0
0
0
1
0
0
0
0
.
1
1
1
1
1
1
1
1
0
0
0
0
1
0
0
0
.
1
1
1
1
1
1
1
1
1
1
0
1
1
1
0
0
.
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
0
.
1 ···
1 ···
1 ···
1 ···
1 ···
1 ···
1 ···
1 ···
1 ···
1 ···
1 ···
1 ···
1 ···
0 ···
0 ···
1 ···
. .···
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Matrix ζF : The incidence matrix ζ for the Fibonacci cobweb poset associated to
F -KoDAG Hasse digraph.
59
The zeta matrix, i.e. the incidence matrix ζF for the Fibonacci numbers cobweb
poset [F – KoDAG], determines completely its incidence algebra and corresponds
to the poset with Hasse diagram displayed by the Fig. 3.
The explicit expression for zeta matrix ζF via known blocks of zeros and ones
for arbitrary natural numbers valued F -sequence is readily found due to brilliant
mnemonic efficiency of the authors up-side-down notation (see Appendix in [13]).
With this notation inspired by Gauss and the reasoning just repeated with “kF ”
numbers replacing k – positive integers one gets, in the spirit of Knuth [18], the
clean result:
⎡
⎢
⎢
AF = ⎢
⎢
⎣
01F ×1F
02F ×1F
03F ×1F
04F ×1F
...
I(1F × 2F )
02F ×2F
03F ×2F
04F ×2F
etc.
01F ×∞
I(2F × 3F )
03F ×3F
04F ×3F
...
⎤
02F ×∞
I(3F × 4F ) 03F ×∞
04F ×4F
I(4F × 5F ) 04F ×∞
and so on
...
⎥
⎥
⎥
⎥
⎦
and
c
c
−1
ζF = exp
≡ I∞×∞ + AF + A2
c [AF ] ≡ (1 − AF )
F + ... =
⎡
⎢
⎢
=⎢
⎢
⎣
I1F ×1F
O2F ×1F
O3F ×1F
O4F ×1F
...
I(1F × ∞)
I2F ×2F
O3F ×2F
O4F ×2F
etc.
⎤
I(2F × ∞)
I3F ×3F
I(3F × ∞)
O4F ×3F
I4F ×4F
I(4F × ∞)
...
and so on ...
⎥
⎥
⎥.
⎥
⎦
Comment 10. (ad “upside down notation”)
Concerning Gauss and Knuth – see remarks in [18] on Gaussian binomial coefficients.
Observation 6. Let us denote by Φk → Φk+1 (see the papers quoted) the dibicliques denominated by subsequent levels Φk , Φk+1 of the graded F -poset P (D) =
(Φ, ≤) i.e. levels Φk , Φk+1 of its cover relation graded digraph D = (Φ, ≺·) [Hasse
diagram]. Then
B (⊕→nk=1 Φk → Φk+1 ) = diag(I1 , I2 , ..., In ) =
⎡
⎢
⎢
=⎢
⎢
⎣
I(1F × 2F )
⎤
I(2F × 3F )
⎥
⎥
⎥,
⎥
⎦
I(3F × 4F )
...
I(nF × (n + 1)F )
where Ik ≡ I(kF × (k + 1)F ), k = 1, ..., n and where – recall – I(s × k) stays for the
(s × k)-matrix with [I(s × k)]ij = 1; 1 ≤ i ≤ s, 1 ≤ j ≤ k. and n ∈ N ∪ {∞}.
60
A. K. Kwaśniewski
Observation 7. Consider bigraphs chain obtained from the above di-bicliques chain
via deleting or no arcs making thus (if deleting arcs) some or all of the di-bicliques
Φk → Φk+1 not di-bicliques; denote them by Gk . Let Bk = B(Gk ) denote their biadjacency matrices correspondingly. Then for any such F -denominated chain (hence
any chain) of bipartite digraphs Gk the general formula reads:
B (⊕→ni=1 Gi ) ≡ B[⊕→ni=1 A(Gi )] = ⊕ni=1 B[A(Gi )] ≡ diag(B1 , B2 , ..., Bn ) =
⎡
⎢
⎢
=⎢
⎢
⎣
⎤
B1
⎥
⎥
⎥,
⎥
⎦
B2
B3
...
n ∈ N ∪ {∞}.
Bn
Observation 8. The F -poset P (G) = (Φ, ≤), i.e. its cover relation graded digraph
m
G = (Φ, ≺·) =⊕→ Gk
k=0
is of Ferrers dimension one iff in the process of deleting arcs from the cobweb poset
Hasse diagram
n
D = (Φ, ≺·) =⊕→ Φk → Φk+1 ,
k=0
does not produce 2 × 2 permutation submatrices in any bigraphs Gk biadjacency
matrix Bk = B(Gk ).
Examples (finite subposets of cobweb posets)
Fig. 4 and Fig. 5 display a Hasse diagram portraits of finite subposets of cobweb
posets. In view of Observation 2 these subposets are naturally Ferrers digraphs
i.e. of Ferrers dimension equal one.
Fig. 4: Display of the subposet P5 of the F = Fibonacci sequence F -cobweb poset and σP5
subposet of the σ permuted Fibonacci F -cobweb poset.
61
Fig. 5: Display of the subposet P4 of the F = Gaussian integers sequence (q = 2) F -cobweb
poset and σP4 subposet of the σ permuted Gaussian (q = 2) F -cobweb poset.
4. Summary
4.1. Principal – natural identifications
Any KoDAG is a di-bicliques chain ⇔ Any KoDAG is a natural join of complete
bipartite graphs [di-bicliques] =
Φk ,
Ek ) ≡ D(Φ, E),
(Φ0 ∪ Φ1 ∪ ... ∪ Φn ∪ ..., E0 ∪ E1 ∪ ... ∪ En ∪ ...) ≡ D(
k≥0
k≥0
→
Φk+1 ≡Kk,k+1
and E = k≥0 Ek .
where Ek = Φk ×
Naturally, as indicated earlier, any graded posets’ Hasse diagram with finite width
including KoDAGs is of the form
Φk ,
Ek ) ⇔ Φ, ≤,
D(Φ, E) ≡ D(
k≥0
k≥0
→
Φk+1 ≡Kk,k+1
where Ek ⊆ Φk ×
and the definition of ≤ from 1.3. is applied.
In front of all the above presentation the following is clear.
Observation 9. “Many” graded digraphs with finite width including KoDAGs D =
(V, ≺·) encode bijectively their correspondent n-ary relation (n ∈ N ∪ {∞} as seen
from its following definition:
→
Ek ⊆ Φk × Φk+1 ≡Kk,k+1
where (n-ary relation)
n−1
n
k=0
k=0
E(n) =⊕→ Ek ⊂ × Φk
i.e.
E(n)(n ∈ N {∞})
i.e. identified with the graded poset Vn , E obtained via, i.e. E(n) is an from natural
join obtained (n + 1)-ary relation E which is a subset of the Cartesian product
achieved by approapriate deleting (Observation 8) the universal (n + 1)-ary relation
identified with cobweb poset digraph Vn , ≺·); V∞ ≡ V .
Which are those “many”? The characterization is arrived at with au rebour point
of view. Any n-ary relation (n ∈ N ∪ {∞}) determines uniquely (may be identified
62
A. K. Kwaśniewski
Fig. 6: Display of the example ternary = Binary1 ⊕→ Binary2 .
with) its correspondent graded digraph with minimal elements set Φ0 given by the
(n-ary rel.) formula
n
E = ⊕→n−1
k=0 Ek ⊂ ×k=0 Φk ,
→
where the sequence of binary relations Ek ⊆ Φk × Φk+1 ≡Kk,k+1 is denominated by
the source n-ary relation as the following example shows.
Example (ternary = Binary1 ⊕ → Binary2 ). Let T ⊂ X × Z × Y where X =
{x1 , x2 , x3 }, Z = {z1 , z2 , z3 , z4 }, Y = {y1 , y2 }, and
T = {x1 , z1 , y1 , x1 , z2 , y1 , x1 , z4 , y2 , x2 , z3 , y2 , x3 , z3 , y2 }.
Let
X × Z ⊃ E1 = {x1 , z1 , x1 , z2 , x1 , z4 , x2 , z3 , x3 , z3 },
and
Z × Y ⊃ E2 = {z1 , y1 , z2 , y1 , z3 , y1 , z4 , y2 }.
Then T = E1 ⊕→ E2 .
More on that – see [15] and see references to the recent papers therein.
Comment 11. As a comment to Observation 9 and Observation 3 consider Fig. 7
which was the source of inspiration for cobweb posets birth [2–6] and here serves
as Hasse diagram DFib ≡ (Φ, ≺·Fib ) of the poset P (DFib ) = (Φ, ≤Fib ) associated to
DFib . Obviuosly, P (DFib ) is a subposet of the Fibonacci cobweb poset P (D) and
DFib is a subgraph of the Fibonacci cobweb poset P (D) Hasse diagram D ≡ (Φ, ≺·).
The Ferrers dimension of DF ib is obviously not equal one.
Exercise. Find the Ferrers dimension of DFib . What is the dimension of the poset
P (DF ib ) = (Φ, ≤Fib )? (Compare with Observation 2). Find the chain Ek ⊂ Φk ×
Φk+1 , k = 0, 1, 2, ... of binary relations such that DFib,n = ⊕→nk=0 Ek , n ∈ N ∪ {∞}.
Find the Ferrers dimension of DFib,n .
63
Fig. 7: Display of Hasse diagram of the form of the Fibonacci tree.
Ad Bibliography Remark
On the history of oDAG nomenclature with David Halitsky and others input one
is expected to see more in [15]. See also the December 2008 subject of the Internet
Gian Carlo Rota Polish Seminar (http : //ii.uwb.edu.pl/akk/sem/sem rota.htm).
Recommended readings on Ferrers digraphs of immediate use here are [19–25]. For
example see pp. 61 and 85 in [19], see page 2 in [20]. The J. Riguet paper [21] is the
source paper including also equivalent characterizations of Ferrers digraphs as well
as the other paper [22–24]. The now classic reference on interval orders and interval
graphs is [25].
Acknowledgments
Thanks are expressed here to the student of Gdańsk University Maciej Dziemiańczuk
for applying his skillful TeX-nology with respect to the present work as well as for
his general assistance and cooperation on KoDAGs investigation.
References
[1] J. Jonsson, Simplicial Complexes of Graphs Series, Lecture Notes in Mathematics,
1928, XIV (2008), 382 (see [JJ] Interpretations in terms of posets Section 9).
[2] A. K. Kwaśniewski, Cobweb posets as noncommutative prefabs, Adv. Stud. Contemp.
Math. 14 (1) (2007), 37–47. arXiv:math/0503286v4, [v1] Tue, 15 Mar 2005 04:26:45
GMT.
[3] A. K. Kwaśniewski, The logarithmic Fib-binomial formula, Adv. Stud. Contemp.
Math. 9, no.1 (2004), 19–26 arXiv:math/0406258v1 [v1] Sun, 13 Jun 2004 17:24:54
GMT.
64
A. K. Kwaśniewski
[4] A. K. Kwaśniewski, Fibonomial cumulative connection constants, Bulletin of the ICA
44 (2005), 81–92, see ArXiv:math/0406006v2 [v6] Fri, 20 Feb 2009 02:26:21 GMT,
upgrade of Bulletin of the ICA 44 (2005), 81–92.
[5] A. K. Kwaśniewski, First Observations on Prefab Posets Whitney Numbers, Advances
in Applied Clifford Algebras 18, no. 1 (2008), 57–73, arXiv:0802.1696v1, [v1] Tue,
12 Feb 2008 19:47:18 GMT.
[6] A. K. Kwaśniewski, On cobweb posets and their combinatorially admissible sequences,
Adv. Studies Contemp. Math. 18 (1), (2009), – in press (upgraded version of
arXiv:math/0512578v4 [v4] Sun, 21 Oct 2007 15:02:09 GMT).
[7] A. D. Plotnikov, About presentation of a digraph by dim 2 poset., Adv. Stud. Contemp.
Math., Kyungshang 12, no. 1 (2006), 55–60).
[8] R. P. Stanley, A matrix for counting paths in acyclic digraphs, J. Combinatorial Theory (A) 74 (1996), 169–172.
[9] Ewa Krot, Characterization of the Fibonacci Cobweb Poset as oDAG,
arXiv:math/0503295v1 Tue, 15 Mar 2005 11:52:45 GMT.
[10] A. K. Kwaśniewski and M. Dziemiańczuk, Cobweb posets - Recent Results, Adv. Stud.
Contemp. Math. 16 (2) (2008), 197–218; arXiv: math. /0801.3985 Fri, 25 Jan 2008
17:01:28 GMT.
[11] Ewa Krot-Sieniawska, Reduced Incidence algebras description of cobweb posets and
KoDAGs, arXiv:0802.4293 Fri, 29 Feb 2008.
[12] Ewa Krot-Sieniawska, Characterization of Cobweb Posets as KoDAGs,
arXiv:0802.2980v1 [v1] Thu, 21 Feb 2008 05:32:10 GMT.
[13] A. K. Kwaśniewski and M. Dziemiańczuk, On cobweb posets’ most relevant codings,
arXiv:0804.1728v1 [v1] Thu, 10 Apr 2008 15:09:26 GMT.
[14] http://www.faces-of-nature.art.pl/cobwebposets.html
[15] A. K. Kwaśniewski, On natural join of posets properties and first applications
arXiv:0908.1375v1 [v2] Sat, 22 Aug 2009 10:42:44 GMT.
[16] V. E. Levit and E. Mandrescu, Matrices and ?-Stable Bipartite Graphs, Journal of
Universal Computer Science, 13, no. 11 (2007), 1692–1706
[17] E. Ferrand, An analogue of the Thue-Morse sequence, The Electronic Journal of Combinatorics 14 (2007), #R 30.
[18] D. E. Knuth, Two notes on notation, American Mathematical Monthly 99, (5) (1992),
403–422.
[19] T. A. McKee and F. R. McMorris, Topics in intersection graph theory, [SIAM Monographs on Discrete Mathematics and Applications #2] Philadelphia 1999.
[20] S. Chatterjee and S. Ghosh, Ferrers Dimension and Boxicity, arXiv:0811.1882v1 [v1]
Wed, 12 Nov 2008 12:32:12 GMT.
[21] J. Riguet, Les Relations des Ferrers, C. R. Acad. Sci. Paris 232 (1951), 1729.
[22] O. Cogis, A characterization of digraphs with Ferrers dimension 2, Rapport de
Recherche, 19, G. R. CNRS no. 22, Paris, 1979.
[23] M. Sen, S. Das, A. B. Roy, and D. B. West, Interval Digraphs: An Analogue of Interval
Graphs, J. Graph Theory 13 (1989), 189–202.
[24] I.-J. Lin, M. K. Sen, and D. B. West, Classes of interval digraphs and 0, 1-matrices
(with). Proc. 28th SE Conf., Congressus Numer. 125 (1997), 201–209.
[25] P. C. Fishburn, Interval Orders and Interval Graphs: A Study of Partialy Ordered
Sets, John Wiley & Sons, New York, 1985.
65
Institute of Combinatorics and its Applications
High School of Mathematics and Applied Informatics
Kamienna 17, PL-15-021 Bialystok
Poland
ódź Society of Sciences and Arts on July 16, 2010
SKIEROWANE GRAFY COBWEB POSETÓW
OKREŚLAJA̧ ZLA̧CZENIE NATURALNE DI-GRAFÓW RELACJI
CZYLI RÓWNOWAŻNIE OPERACJȨ ZLA̧CZENIA NATURALNEGO
MACIERZY SA̧SIEDZTWA
Streszczenie
Zdefiniowano zla̧czenie naturalne (“natural join”) skierowanych grafów dwudzielnych
oraz odpowiadaja̧ce im zla̧czenie naturalne ich macierzy sa̧siedztwa. Operacjȩ natural join
zastosowano do wprowadzonych tutaj szczególnych czȩściowo uporza̧dkowanych zbiorów
ze stopniowaniem zwanych już wcześniej “cobweb posets”. Operacjȩ “natural join” zastosowano zatem i do ich di-grafów tworza̧cych lańcuch tzw. “bi-di-klik”. Stanowia̧ one
w zla̧czaniu naturalnym cia̧gi Kompletnych Grafów dwudzielnych – uporza̧dkowanych (ordered) oraz skierowanych i acyklicznych (DAG’s). Na cześć Profesora Kazimierza Kuratowskiego – wspóltwórcy wspólczesnej teorii grafów – grafy Hassego nazwano tu KoDAGs.
Jak wykazano – KoDAGs stanowia̧ wyróżniona̧ rodzin di-grafów Ferrersa. Mianowicie ich
wymiar Ferrersa wynosi jeden, przy czym podano warunki dostateczne i wystarczaja̧ce na
to, by “natural join” lańcucha di-grafów dwudzielnych tworzyl di-graf o wymiarze Ferrersa
równym jeden.
Wyróżnione di-grafy w zla̧czaniu naturalnym tworza̧ cia̧gi di-grafów określonych dla
odpowiednio licznej rodziny relacji k-narnych – w tym np. binarnych. Operacjȩ “natural
join” świadomie i z uzasadnieniem oznaczono symbolem nawizuja̧cym do symbolu operacji
sumy prostej.
PL ISSN 0459-6854
BULLETIN
DE
DES
LETTRES DE L
ÓDŹ
2010
Vol. LX
no. 1
pp. 67–75
Carmen Boloşteanu
THE RIEMANN-HILBERT PROBLEM WITH ISOLATED POLES
Summary
Let L be a simple smooth closed contour dividing the plane Ĉ = C ∪ {∞} into the
interior domain D+ and the exterior domain D− and a non-vanished function of position
on the contour, G(t) which satisfies the Hölder condition. We have to find a meromorphic
function Φ which has given isolated poles in D+ and D− respectively, which satisfies on L
the relation
Φ+ (t) = G(t)Φ− (t).
The same problem is solved on the real projective plane, which is the factor manifold of Ĉ
with respect to the group {1Ĉ , k}, where k(z) = −1/z̄.
1. Introduction
The Riemann-Hilbert problem for dianalytic functions on the Möbius strip was
solved in [6] using the point of view introduced by I. Bârză and D. Ghişa in [1], [2], [3]
and the method used by Gakhov [7] and Muskelishvili [8].
In this paper we solve the same problem, but for meromorphic functions on the
complex plane and then for N-meromorphic functions on the real projective plane.
Some definitions introduced in other specified papers are used.
2. The case of complex plane
Let D+ ⊂ Ĉ be a relative compact domain, 0 ∈
/ D+ . Suppose that ∂D+ = L, where
−
+
L is a smooth curve. Denote by D = Ĉ \ {D ∪ L}. On L a Hölder function G(t) is
defined which does not vanish. We have to find a meromorphic function Φ such that
(1)
Φ+ (t) = G(t)Φ− (t)
on L,
where Φ+ (z) = Φ(z) for z ∈ D+ and Φ− (z) = Φ(z) for z ∈ D− .
68
C. Boloşteanu
Suppose that
Φ+ (z) =
Φ1 (z)
(z − z1 )k1
Φ− (z) =
Φ2 (z)
,
(z − z2 )k2
and
where Φ1 and Φ2 are holomorphic functions in D+ and D− , respectively, and z1 , z2
are given poles of orders k1 , k2 > 0, with z1 ∈ D+ and z2 ∈ D− . The relation (1)
becomes
Φ2 (t)
Φ1 (t)
= G(t)
.
(t − z1 )k1
(t − z2 )k2
We have
(2)
Φ1 (t) = G(t)
(t − z1 )k1
Φ2 (t).
(t − z2 )k2
Denote by
G1 (t) = G(t)
(t − z1 )k1
.
(t − z2 )k2
If the index of G with respect to the curve L is n, then the index of G1 is n + k1 − k2 .
Let
Π(z) = (z − z1 )n+k1 −k2 .
Multiplying the relation (2) by Π−1 (t), we obtain
Φ1 (t)Π−1 (t) = G(t)
and
Φ1 (t) = G(t)
(t − z1 )k2 −n
Φ2 (t)
(t − z2 )k2
(t − z1 )k2 −n
Φ2 (t)Π(t).
(t − z2 )k2
Now we define a new unknown holomorphic function Ψ(z), where
⎧ +
if z ∈ D+ ,
⎨ Ψ (z) = Φ1 (z)
Ψ(z) =
⎩ −
Ψ (z) = Φ2 (z)Π(z) if z ∈ D− .
The problem is reduced to a classic Riemann-Hilbert problem. We have to find the
sectional holomorphic function Ψ such that
(3)
Ψ+ (t) = G0 (t)Ψ− (t),
where
G0 (t) = G(t)
(t − z1 )k2 −n
(t − z2 )k2
and its index is zero. Obviously, G0 is a Hölder function and it does not vanish on L.
By taking logarithms, the relation (3) becomes
log Ψ+ (t) − log Ψ− (t) = log G0 (t)
The Riemann-Hilbert problem with isolated poles
69
and we can apply the Sokhotski-Plemelj formulae. We obtain
log G0 (ζ)
1
dζ
log Ψ(z) =
2πi
ζ −z
L
=
1
2πi
L
log G(ζ) + (k2 − n) log(ζ − z1 ) − k2 log(ζ − z2 )
dζ.
ζ −z
Γ(z)
, where
1
log G(ζ) + (k2 − n) log(ζ − z1 ) − k2 log(ζ − z2 )
Γ(z) =
dζ.
2πi
ζ −z
Then Ψ(z) = e
L
+
For z ∈ D , we have
+
Φ1 (z) = Ψ+ (z) = eΓ
and for z ∈ D− we have
(z)
−
eΓ (z)
Ψ− (z)
=
.
Φ2 (z) =
Π(z)
Π(z)
The solution to the problem (1) is the meromorphic function Φ(z), where
⎧
+
eΓ (z)
⎪
⎪
,
if z ∈ D+
⎨
(z − z1 )k1
Φ(z) =
(4)
−
⎪
eΓ (z)
⎪
⎩
, if z ∈ D− .
(z − z1 )n+k1 −k2 (z − z2 )k2
According with the Sokhotski-Plemelj formulae, we obtain
1
Γ+ (t) = (log G(t) + (k2 − n) log(t − z1 ) − k2 log(t − z2 ))
2
⎛
⎞
1
log
G(ζ)
+
(k
−
n)
log(ζ
−
z
)
−
k
log(ζ
−
z
)
2
1
2
2
+
CP V ⎝
dζ ⎠ ;
2πi
ζ −t
L
1
Γ− (t) = − (log G(t) + (k2 − n) log(t − z1 ) − k2 log(t − z2 ))
2
⎛
⎞
1
log G(ζ) + (k2 − n) log(ζ − z1 ) − k2 log(ζ − z2 ) ⎠
+
CP V ⎝
dζ ,
2πi
ζ −t
L
where CP V is the principal value of the Cauchy integral from the brackets.
Making simple calculus, one can see that Φ+ , Φ− and G satisfies the relation (1).
3. The case of real projective plane
The real
plane P2 appears as a factor manifold of Ĉ with respect to the
projective
group 1Ĉ , k , where k(z) = −1/z̄. P2 is a nonorientable Riemann surface, Ĉ is an
70
C. Boloşteanu
orientable Riemann surface and the double cover of P2 , and k is an antianalytic
involution without fixed points:
P2 =
[z] | z ∈ Ĉ , where [z] = {z, −1/z̄} .
Let the function p : Ĉ → P2 be the canonical projection with p(z) = [z]:
In this paper we use the symmetry in the sense of Klein. A set D ⊂ Ĉ is called
symmetric with respect to k if k(D) = D [1].
Let D ⊂ Ĉ be a symmetric set and f : D → Ĉ a function. f is called symmetric
function (with respect to k) if f (z) = f (k(z)) for every z ∈ D [1].
Let D̃ be a subset of P2 and let f˜ : D̃ → Ĉ be a function. One can define a
symmetric function f : p−1 (D̃) → Ĉ by
f (z) = f (k(z)) := f˜([z])
for every [z] = {z; k(z)} ∈ D̃.
If one considers only the functions f˜ : D̃ → C, then it is obvious that the
algebra of these functions is canonically isomorphic with the algebra of the symmetric
functions f : p−1 (D̃) → C. The study of functions on P2 is canonically reduced to
the study of functions on Ĉ.
Definition 3.1. [4], [5] A function ϕ̃ fulfills the Hölder condition on a smooth curve
L̃ ∈ P2 if
|ϕ̃([t1 ]) − ϕ̃([t2 ])| ≤ C δ([t1 ], [t2 ])α
for all [t1 ], [t2 ] ∈ L̃ and 0 < α ≤ 1.
Here δ([z1 ], [z2 ]) is the distance between two points [z1 ] and [z2 ], which was
introduced in [5]. It is given by
1
,
δ([z1 ], [z2 ]) = min d(z1 , z2 ), d z1 , −
z¯2
where
1
1
1
d(z1 , z2 ) = (|z1 − z2 | + |k(z1 ) − k(z2 )|) = |z1 − z2 | 1 +
.
2
2
|z1 z2 |
71
Let g : Ĉ → Ĉ be a meromorphic function. Define f = g + g ◦ k and F : P2 → Ĉ
with
(5)
F ([z]) := f (z).
Definition 3.2. [1] The function F is called N-meromorphic function if for every
[z] ∈ P2 its value is given by a function f as in the relation (5).
Let D̃+ ⊂ P2 be a relative compact domain bounded by a smooth contour and
[0] ∈
/ D̃+ . Suppose that on ∂ D̃+ = L̃ a non-vanishing function G̃ is defined which
satisfies the Hölder condition. We have to find an N-meromorphic function Φ̃ with
given poles [z1 ] ∈ D̃+ and [z2 ] ∈ D̃− = P2 \ (D̃+ ∪ L̃) such that
(6)
Φ̃+ ([t]) = G̃([t])Φ̃− ([t]) on L̃,
where Φ̃+ ([z]) = Φ̃([z]) for [z] ∈ D̃+ and Φ̃− ([z]) = Φ̃([z]) for [z] ∈ D̃− .
Let us denote:
D+ = p−1 (D̃+ ),
D− = p−1 (D̃− ),
L = p−1 (L̃).
Obviously, D+ , D− , and L are symmetric sets. We can define the function G = G̃ ◦ p
on L.
Theorem 3.1. The function G fulfils the Hölder condition on L. Besides, G is symmetric with respect to k (i.e. G(t) = G(k(t)), for t ∈ L).
Proof. See in [5]. The conclusion is
G̃([t]) = G(k(t)) = G(t).
Now the problem is transferred to the complex plane, where we have a Hölder
function G(t) defined on the boundary of the set D+ . Solving the problem for the
symmetric set D+ and the function G on L, we obtain a meromorphic function Φ
which fulfils
(7)
Φ+ (t) = G(t)Φ− (t) for t ∈ L.
Theorem 3.2. If Φ is the solution to the problem (7), then the N-meromorphic
function
1
Φ̃([z]) = {Φ(z) + Φ(k(z))}
2
is the solution to the problem (6).
Proof (cf. [6]). For [z] ∈ D̃, we have
Φ̃+ ([z]) =
1 +
{Φ (z) + Φ+ (k(z))}.
2
72
C. Boloşteanu
Taking the limit [z] → [t] ∈ L̃ in the above relation and using the relation (7)
together with the symmetry of the function G with respect to k, we obtain:
1
1
Φ̃+ ([t]) = {Φ+ (t) + Φ+ (k(t))} = {G(t)Φ− (t) + G(k(t))Φ− (k(t))}
2
2
1 −
= G(t) Φ (t) + Φ− (k(t)) = G̃([t])Φ̃− ([t]),
2
as desired.
3.1. Explicit solution
We have to solve the problem on Ĉ for the set D+ and the non vanishing function
G(t) defined on ∂D+ , which satisfies a Hölder condition.
3.1.1. Solution for the complex plane
Suppose that D+ = D1 ∪ D2 , where D̄1 ∩ D̄2 = ∅ and D1 = k(D2 ). Let Li = ∂Di ,
i = 1, 2. Obviously, D− = k(D− ), D+ = k(D+ ) and L1 = k(L2 ). Moreover,
p−1 ([z1 ]) = {z1 , k(z1 )} and p−1 ([z2 ]) = {z2 , k(z2 )}.
We have to find an N-meromorphic function Φ(z) with given poles z1 , k(z1 ) ∈ D+
and z2 , k(z2 ) ∈ D− which fulfils the condition
Φ+ (t) = G(t)Φ− (t) on L1 ∪ k(L1 ).
(8)
We can write
Φ+ (z) =
Φ1 (z)
(z − z1 )(z − k(z1 ))
Φ− (z) =
Φ2 (z)
,
(z − z2 )(z − k(z2 ))
and
where Φ1 and Φ2 are holomorphic functions on D+ and D− , respectively. It was
proved in [6] that the index of a function with respect to a curve L is equal to the
index of the same function with respect to k(L).
Let n be the index of G on L1 . Then the index of G(t) given on L1 ∪ k(L1 ) is 2n.
The relation (8) becomes
Φ1 (t) = G(t)
(t − z1 )(t − k(z1 ))
Φ2 (t).
(t − z2 )(t − k(z2 ))
Introduce the function
Π(z) = (z − z1 )n (z − k(z1 ))n .
Then the argument of
(9)
G0 (t) = Π−1 (t)G(t)
(t − z1 )1−n (t − k(z1 ))1−n
(t − z1 )(t − k(z1 ))
= G(t)
(t − z2 )(t − k(z2 ))
(t − z2 )(t − k(z2 ))
73
will return to its initial value after any circuit of the contours L1 , k(L1 ) and hence
log G0 (t) is a definite function, one-valued, continuous and satisfying the Hölder
condition on L1 ∪ k(L1 ).
Introduce a new unknown holomorphic function Ψ(z) with
+
if z ∈ D+
Ψ (z) = Φ1 (z)
Ψ(z) =
Ψ− (z) = Φ2 (z)Π(z) if z ∈ D− .
Using the same method as in Section 2, we obtain the solution of a new problem.
Theorem 3.3. The solution of (8) is the meromorphic function
⎧
+
eΓ (z)
⎪
⎪
if z ∈ D+
⎨
(z − z1 )(z − k(z1 ))
Φ(z) =
(10)
−
⎪
eΓ (z)
⎪
⎩
if z ∈ D− ,
Π(z)(z − z2 )(z − k(z2 ))
where
(11)
Γ(z) =
1
2πi
log G(t) + log
L1 ∪k(L1 )
(t − z1 )1−n (t − k(z1 ))1−n
(t − z2 )(t − k(z2 ))
dt.
t−z
3.1.2. Solution for the real projective plane
Theorem 3.4. If the function Φ(z) is the solution on the complex plane, then the
function
+
Φ ([z]) if [z] ∈ D̃+
Φ̃([z]) =
Φ̃− ([z]) if [z] ∈ D̃−
with
1
Φ+ ([z]) =
2
and
1
Φ̃− ([z]) =
2
+
+
eΓ (z)
eΓ (k(z))
+
(z − z1 )(z − k(z1 )) (k(z) − z1 )(k(z) − k(z1 ))
−
−
eΓ (z)
eΓ (k(z))
+
Π(z)(z − z2 )(z − k(z2 )) Π(k(z))(k(z) − z2 )(k(z) − k(z2 ))
is the solution of the problem on P2 , where Γ(z) is given by (11).
Proof. Obviously, [z] ∈ D̃+ ⇐⇒ z ∈ D+ and [z] ∈ D̃− ⇐⇒ z ∈ D− . By Theorem
3.2 we can write the formula for
1
Φ̃([z]) = {Φ(z) + Φ(k(z))}.
2
74
C. Boloşteanu
Taking limiting values in the formula (11) and using the Sokhotski-Plemelj formulae for multiply-connected domains, we find
1
(t − z1 )1−n (t − k(z1 ))1−n
Γ+ (t) =
log G(t) + log
+ Γ(t),
2
(t − z2 )(t − k(z2 ))
1
(t − z1 )1−n (t − k(z1 ))1−n
Γ− (t) = −
log G(t) + log
+ Γ(t)
2
(t − z2 )(t − k(z2 ))
where Γ(t) is the Cauchy principal value of the integral (11). For Γ+ (k(t)) and
Γ− (k(t)) the formulae are analogous. Then we obtain
Φ̃+ ([t]) = lim Φ̃+ ([z]), Φ̃− ([t]) = lim Φ̃− ([z]).
[z]→[t]
[z]→[t]
Using the symmetry of G(t) (Theorem 3.1) and making a simple calculus, one can
check that Φ̃+ ([t]), Φ̃− ([t]), and G̃([t]) fulfil the relation
Φ̃+ ([t]) = G̃([t])Φ̃− ([t]).
Acknowledgements
The author would like to express her gratitude to Professor Bogdan Bojarski for his
constant kind help and to Professor Jaroslav Zemanek for the financial support in
the frame of the European Community project TODEQ (MTKD-CT-2005-030042).
References
[1] I. Bârză, Calculus on Nonorientable Riemann Surfaces, Libertas Mathematica 15
(1995), 1–45.
[2] I. Bârză, Integration on Nonorientable Riemann Surfaces. In: Almost Complex Structures. (Eds. K. Sekigawa and S. Dimiev). World Scientific, Singapore-New JerseyLondon-Hong Kong, 1995, 63–97.
[3] I. Bârză and D. Ghişa, Boundary value problems on non orientable surfaces, Rev.
Roumaine Math. Pures Appl. 43 (1998), 1–2, 67–79.
[4] C. Boloşteanu, The Sokhotski-Plemelj formulae on the Möbius strip, Complex Variables and Elliptic Equations 53, no. 7 (2008), 657–666.
[5] C. Boloşteanu and M. Boloşteanu, Distances and Hölder functions on fonorientable
Riemann surfaces, Buletin Ştiinţific – Universitatea din Piteşti 13 (2007), 15–26.
[6] C. Boloşteanu, The Riemann-Hilbert problem on the Möbius strip, Complex Variables
and Elliptic Equations, to appear.
[7] F. D. Gakhov, Boundary Value Problems. Moscow, 1963 (Russian); English translation, Pergamon Press, Oxford 1966.
[8] N. I. Muskhelishvili, Singular Integral Equations. Moscow, 1946 (Russian); English
translation, Noordhoff, Groningen 1953.
Faculty of Accounting and Finance
“Spiru Haret” University Bucharest
R-115100 Câmpulung Muscel, 223 Traian Street
Romania
75
Presented by Leon Mikolajczyk at the Session of the Mathematical-Physical Commission of the L
ZAGADNIENIE RIEMANNA-HILBERTA Z IZOLOWANYMI
BIEGUNAMI
Streszczenie
Niech L bȩdzie prostym gladkim konturem zamkniȩtym dziela̧cym plaszczyznȩ Ĉ =
C∪{∞} na obszar wewnȩtrzny D+ i obszar zewnȩtrzny D− , G(t) zaś – nieznikaja̧ca̧ funkcja̧
polożenia na konturze spelniaja̧ca̧ warunek Höldera. Poszukujemy funkcji meromorficznej
Φ posiadaja̧cej dane bieguny w obszarach D+ i D− , spelniaja̧cej na konturze L relacjȩ
Φ+ (t) = G(t)Φ− (t).
Takie samo zagadnienie jest rozwia̧zane na rzeczywistej plaszczyźnie rzutowej, która jest
czynnikiem plaszczyzny Ĉ ze wzglȩdu na grupȩ {1Ĉ , k}, gdzie k(z) = −1/z̄.
PL ISSN 0459-6854
BULLETIN
DE
DES
2010
LETTRES DE L
ÓDŹ
Vol. LX
no. 1
pp. 77–94
Dedicated to Professor Julian L
awrynowicz
on the occasion of his 70-th birthday
Dariusz Partyka and Józef Zaja̧c
GENERALIZED PROBLEM OF REGRESSION
Summary
The authors present a generalization of the classical regression idea by studying and
then solving certain extremal problem, well defined on the ground of finite or infinite
dimensional Hilbert space. Given empiric data, discrete or continuous, the class of solutions
is determined and uniquely expressed in a new form of the regression functions sequences
(RFS). A large variety of observed phenomena, in different areas of practical and theoretical
sciences, can be described and researched, with certain precision and a help of the RFS
technique.
Introduction
Contemporary world is characterized by the increasing influx of information. The
observation of even simple processes requires numerical data methods of analysis
which take individual parameters, distinguished by these processes. Some of them
are possible to achieve in applied selected moments of time, whereas others can be
observed in continuous time system. In the avalanche of pouring information one has
to know how to find interesting dependence of both relational and functional type.
The latter are often too complicated to capture and describe by means of simple
mathematical expression.
Approximate functional dependences which would describe interesting phenomena with assigned accuracy should be sought. Studying of appropriately constructed
approaching functions can lead to detection of so far unknown dependences as well
as assess separate and combined effects caused by several observed variables. It has
78
D. Partyka and J. Zaja̧c
a huge significance, especially in situation when dependence, expressed in physical,
chemical and biological laws, between observed parameters is unknown. Particular,
although simplified, example of solution to such set scientific problem is a method of
linear regression with its various modifications, formulated on the basis of probability
calculus; cf. [13], [11], [9], [1] and [7]. The method found series of implementations
and experienced numerous theoretical modifications, crucial due to seriousness of
implementation problem; cf. [12], [11], [3], [8], [4] and [2].
The generalized regression problem considered in this paper has a form of solution of properly formulated extremal problem well stated in the Hilbert space
environment, both finite and infinite dimensional.
A short presentation of the classical approach to the regression problem can be
formulated as follows.
For any p, q ∈ R set Zp,q := {k ∈ Z : p ≤ k ≤ q}. Let F be the family of all
functions R t → at + b, where a, b ∈ R, and let x, y : Z0,n → R be arbitrarily given
sequences. It is well known that if x is not a constant sequence, then there exists
the unique f0 ∈ F satisfying the following condition
n
n
(0.1)
(f (xk ) − yk )2 ≥
(f0 (xk ) − yk )2 , f ∈ F .
k=0
k=0
In fact, the function f0 is of the form f0 (t) = a0 t + b0 as t ∈ R, where
n
n
n
n
n
xk yk −
xk
yk
y k − a0
xk
(n + 1)
k=0
k=0
k=0
k=0
k=0
a0 :=
(0.2)
;
and
b
:=
0
n
n
n+1
(n + 1)
x2k − (
xk )2
k=0
k=0
cf. e.g. [13] and [2]. The function f0 is usually said to be the regression line for the
empiric sequences x, y : Z0,n → R. In view of (0.1), the function f0 has a natural
interpretation as an optimal function with the smallest quadratic deviation from
the empiric observations {(xk , yk ) : k ∈ Z0,n }. The function f0 plays an essential
role in different areas of applied mathematics; cf. e.g. [4] and [2]. It shows that
the mentioned above extremal problem, can be considerably qeneralized and solved,
which is a subject of this paper. To this end we introduce the regression structures
R := (A, B, δ; x, y), where:
I.1. A, B are nonempty sets;
I.2. x : Ω1 → A and y : Ω2 → B for some nonempty sets Ω1 and Ω2 ;
I.3. δ : (Ω1 → B) × (Ω2 → B) → R.
A set F is called the functional model of R if F ⊂ (A → B), where A → B
means the class of all functions acting from A to B.
According to the extremal problem (0.1), the components δ, x and y as well as a
functional model F of R have the following interpretations:
– F is a theoretic functional model of the considered phenomena, i.e. F consists
of all functions describing theoretically the considered phenomena;
Generalized problem of regression
79
– x : Ω1 → A and y : Ω2 → B are empirical functions derived from an experiment
or observation, called the empirical data functions in the sequel;
– δ is a deviation criterion of theoretic functions from empirical ones.
Given a regression structure R and a functional model F of R we seek the
optimal theoretic functions f0 ∈ F which are the best fitted to the empirical data –
represented by the empirical data functions x and y – with respect to the criterion
δ. To be more precise, we consider the extremal problem of determining all functions
f0 ∈ F minimizing the functional
(0.3)
F f → F (f ) := δ(f ◦ x, y) ∈ R ,
i.e. all functions f0 ∈ F satisfying the following inequality
(0.4)
F (f ) ≥ F (f0 ) ,
f ∈F .
The set of all f0 ∈ F satisfying the inequality (0.4) will be denoted by M(F , R).
Each function f0 ∈ M(F , R) is said to be the regression function in F with respect
to R. The problem of describing all regression functions in F with respect to R we
shall call the regression problem for F and R.
Example 0.1. Consider an electric circuit with direct current. According to Ohm’s
law the voltage V depends on the intensity I by the equality V = RI, where the
multiplier R is the resistance of the circuit. We want to determine the parameter R by
means of measurements samples of intensity and voltage represented by a sequence
Z0,n k → (ik , vk ). To this end we consider the regression structure R, where
A := R, B := R, the empiric data functions are defined by Z0,n k → x(k) := ik
and Z0,n k → y(k) := vk , and, as a criterion of deviation δ, we take the smallest
squares method, i.e.
n
(0.5)
(f (k) − g(k))2 , f, g : Z0,n → R .
δ(f, g) :=
k=0
The theoretic functional model F is represented by linear functions R t → rt for
n
r ∈ R. Calculating the critical point of the function R r → k=0 (rik − vk )2 we
obtain
n
n
(0.6)
ik vk
i2k .
R=
k=0
k=0
In what follows we shall study the regression problem for a wide range of theoretic functional models F and regression structures R involving a generalized variant of quadratic deviation applied in (0.1). We aim at showing the main idea of
our approach. Therefore we confine ourselves to the basic concepts. In particular
we focus our attention on the case where the theoretic functional model F is a
finite-dimensional linear set with respect to the standard operations of adding and
multiplying complex-valued functions. The complete version of this article will be
published elsewhere.
80
Most of results in this paper were presented by the first named author during
the seminar “Hypercomplex Seminar 2009: From Schauder Basis to Hypercomplex,
Randers-Ingarden and Fractal Structures, and Nanostructures”, Mathematical Conference Center at Bȩdlewo (Poland), July 24 – July 31, 2009.
1. The generalized quadratic deviation
From now on we shall study the family of regression structures R := (A, B, δ; x, y)
satisfying additional assumptions:
II.1. B = R or B = C;
II.2. There exist a σ-field B of subsets of the cartesian product Ω1 × Ω2 and a
measure μ : B → [0; +∞] such that the function δ satisfies the following equality
(1.1)
|u(t1 ) − v(t2 )|2 d μ(t1 , t2 ) ,
δ(u, v) =
Ω1 ×Ω2
provided both the functions
Ω1 × Ω2 (t1 , t2 ) → u(t1 ) and Ω1 × Ω2 (t1 , t2 ) → v(t2 )
are B-measureable, and δ(u, v) = +∞ otherwise;
II.3. The function Ω1 × Ω2 (t1 , t2 ) → y(t2 ) is B-measureable.
Example 1.1. Consider a regression structure R defined as follows. Given n, m ∈ N
let Ω1 := Z0,n and Ω2 := Z0,m . Let B be the set of all subsets of the cartesian
product Ω1 × Ω2 . Obviously, the set B is a σ-field of subsets of Ω1 × Ω2 , and hence
we can define a unique measure μ : B → [0; +∞) satisfying the condition
(1.2)
μ({(k, l)}) = ρk,l ,
k ∈ Ω1 , l ∈ Ω2 ,
where Ω1 × Ω2 (k, l) → ρk,l ∈ R is a given non-negative function. Then, for any
functions Ω1 t → u(t) ∈ B and Ω2 t → v(t) ∈ B, we derive from (1.1),
δ(u, v) =
(1.3)
|u(t1 ) − v(t2 )|2 d μ(t1 , t2 )
Ω1 ×Ω2
=
|u(t1 ) − v(t2 )|2 d μ(t1 , t2 )
(k,l)∈Ω1 ×Ω2{(k,l)}
=
m
n ρk,l |u(k) − v(l)|2 .
k=0 l=0
In particular, assuming m = n and setting
1 as k = l ,
(1.4)
ρk,l :=
0 as k = l ,
81
we conclude from (1.3) that
δ(u, v) =
(1.5)
n
|u(k) − v(k)|2 ,
k=0
which is exactly the classical square deviation used in (0.1). Namely, combining (0.3)
with (1.3) and (1.4) we obtain
(1.6)
F (f ) =
n
2
|f ◦ x(k) − y(k)| =
k=0
n
|f (xk ) − yk |2 ,
f ∈F ,
k=0
for given empirical data functions Z0,n k → xk and Z0,n k → yk , where F is the
set of all functions R t → at + b as a, b ∈ R.
From now on we may confine ourselves to the case where B = C, which naturally embrace the case where B = R. Fix a regression structure R satisfying the
properties II.1–II.3. We consider the set L1 (R) of all functions f : A → B such that
Ω1 × Ω2 (t1 , t2 ) → f ◦ x(t1 ) is a B-measureable function and
(1.7)
|f ◦ x(t1 )|2 d μ(t1 , t2 ) < +∞ .
Ω1 ×Ω2
We shall also consider the set L2 (R) of all functions g : Ω2 → B such that Ω1 × Ω2 (t1 , t2 ) → g(t2 ) is a B-measureable function and
(1.8)
|g(t2 )|2 d μ(t1 , t2 ) < +∞ .
Ω1 ×Ω2
From (1.7) and the inequality
(1.9)
|zw| ≤
1
(|z|2 + |w|2 ) ,
2
it follows that the functional
(1.10)
z, w ∈ C ,
L1 (R) × L1 (R) (u, v) → u|v :=
u ◦ x(t1 )v ◦ x(t1 ) d μ(t1 , t2 )
Ω1 ×Ω2
is well defined. Hence, u|u ≥ 0 as u ∈ L1 (R), and so the functional
⎛
⎞1/2
L1 (R) u → u := u|u = ⎝
(1.11)
|u ◦ x(t1 )|2 d μ(t1 , t2 )⎠
,
Ω1 ×Ω2
is also well defined. It can be shown, in much the standard way, that the structure
H(R) := (L1 (R), +, ·, · | · ) is a complex (resp. real in case B = R) pseudo-Hilbert
space. Here the symbols “+” and “·” denote the standard operations of adding and
multiplying functions. This means that the structure (L1 (R), +, ·) is a linear space,
the following properties:
82
αu + βv|w = αu|w + βv|w ;
u|v = v|u ;
(1.12)
u|u ≥ 0 ,
hold for all α, β ∈ B and u, v, w ∈ L1 (R) as well as every Cauchy sequence from
L1 (R) is convergent to certain function in L1 (R) with respect to the pseudo-norm
· . A more delicate treatment needs the proof of its completeness.
Remark 1.2. The properties (1.12) yield the well known Schwarz inequality
(1.13)
|u|v| ≤ uv ,
u, v ∈ L1 (R) ;
cf. e.g. [6] and [10].
By (1.8) and (1.9) we see that for each y ∈ L2 (R) the functional
∗
(1.14)
u ◦ x(t1 )y(t2 ) d μ(t1 , t2 )
L1 (R) u → y (u) :=
Ω1 ×Ω2
is well defined. It is clear that the structure (L2 (R), +, ·) is a complex (resp. real in
case B = R) linear space. Moreover, from the algebraic properties of the Lebesgue
integral and Schwarz’s integral inequality it follows that for each y ∈ L2 (R) the
functional y ∗ is linear and bounded on H(R) and the supremum norm of y ∗ satisfies
the following inequality
⎛
⎞1/2
(1.15) sup{|y ∗ (f )| : f ∈ L1 (R) and f ≤ 1} ≤ ⎝
|y(t2 )|2 d μ(t1 , t2 )⎠
.
Ω1 ×Ω2
2. Solution of the regression problem
From now on we shall study the regression problem for F and R, where R is a given
regression structure satisfying the assumptions II.1-II.3 and F is a linear functional
model of R with respect to the standard operations of adding and multiplying functions, i.e. f + g ∈ F and λf ∈ F for f, g ∈ F and λ ∈ B. If additionally F ⊂ L1 (R),
then the regression problem means the extremal problem determining all functions
f0 ∈ F which are minimizing the functional F . It satisfies, according to (0.3) and
(1.1), the following equality
F (f ) =
(2.1)
|f ◦ x(t1 ) − y(t2 )|2 d μ(t1 , t2 ) , f ∈ F .
Ω1 ×Ω2
We shall start with the following basic characterization of the regression functions.
Lemma 2.1. If F = ∅ is a linear set in H(R) and y ∈ L2 (R), then for every f ∈ F
the following property holds:
83
(2.2)
f ∈ M(F , R) ⇔ h|f = y ∗ (h) ,
h∈F .
Proof. Fix f, h ∈ F and λ ∈ C. By the definitions of the functions F and δ we have
F (f + λh) = δ((f + λh) ◦ x, y)
|(f + λh) ◦ x(t1 ) − y(t2 )|2 d μ(t1 , t2 )
=
Ω1 ×Ω2
|f ◦ x(t1 ) − y(t2 ) + λh ◦ x(t1 )|2 d μ(t1 , t2 )
=
Ω1 ×Ω2
|f ◦ x(t1 ) − y(t2 )|2 d μ(t1 , t2 )
=
Ω1 ×Ω2
Re(f ◦ x(t1 ) − y(t2 ))λh ◦ x(t1 ) d μ(t1 , t2 )
+2
Ω1 ×Ω2
+ |λ|2
|h ◦ x(t1 )|2 d μ(t1 , t2 ) .
Ω1 ×Ω2
Hence, and by (2.1), (1.14) as well as by (1.11) we get
F (f + λh) = F (f ) + |λ|2 h2 − 2 Re λ y ∗ (h)
⎤
⎡
f ◦ x(t1 )h ◦ x(t1 ) d μ(t1 , t2 )⎦
+ 2 Re ⎣λ
Ω1 ×Ω2
= F (f ) + |λ|2 h2 − 2 Re λ y ∗ (h) + 2 Re λf |h .
Therefore
(2.3) F (f + λh) − F (f ) = 2 Re [λ(h|f − y ∗ (h))] + |λ|2 h2 ,
f, h ∈ F , λ ∈ C .
Assume now that f ∈ F satisfies the right hand side condition in (2.2). Then
setting λ := 1 we deduce from (2.3) that F (f + h) − F (f ) = h2 ≥ 0, and so
f ∈ M(F , R). Conversely, suppose that f ∈ M(F , R). Then (2.3) yields
(2.4)
2 Re[λ(h|f − y ∗ (h))] + |λ|2 h2 ≥ 0 ,
h∈F,λ∈C.
Replacing h by (−h) in (2.4) we get
(2.5)
−2 Re[λ(h|f − y ∗ (h))] + |λ|2 h2 ≥ 0 ,
h∈F,λ∈C.
Combining (2.4) and (2.5) we have
1
1
− |λ|2 h2 ≤ Re[λ(h|f − y ∗ (h))] ≤ |λ|2 h2 ,
2
2
h∈F,λ∈C,
84
and consequently,
1
1
(2.6) − |λ|h2 ≤ Re[ei α(λ) (h|f − y ∗ (h))] ≤ |λ|h2 ,
2
2
h ∈ F , λ ∈ C \ {0} ,
where α(λ) ∈ [0; 2π) is a unique number satisfying the equality λ = |λ| ei α(λ) . Thus
(2.6) yields, in the limiting case as |λ| → 0, the following equality
Re[ei α (h|f − y ∗ (h))] = 0 ,
h∈F,α∈R.
Choosing appropriately α we see that h|f − y ∗ (h) = 0 for h ∈ F, which completes
the proof.
By the properties of a pseudo-norm we see that the set Θ := {h ∈ L1 (R) : h =
0} is linear. We call it the null set of H(R). As a matter of fact Θ is the closed
ball with radius 0 and center at the zero function θ, defined by θ(t) := 0 for t ∈ A.
We may extend the standard operations of adding and multiplying functions by a
constant to any sets F1 , F2 ∈ (A → B) as follows:
F1 + F2 := {f1 + f2 : f1 ∈ F1 , f2 ∈ F2 } ;
λ · F1 := {λf1 : f1 ∈ F1 } ,
λ∈C;
f + F1 := {f } + F1 and F1 + f := F1 + {f } ,
f ∈ (A → B) .
Corollary 2.2. If F = ∅ is a linear set in H(R) and y ∈ L2 (R), then
M(F , R) = F ∩ M(Θ + F , R) .
(2.7)
If additionally F ⊂ Θ, then M(F , R) = F .
Proof. Fix f, h ∈ L1 (R). If h = 0, then from Schwarz’s inequality (1.13) and (1.15)
it follows that
⎛
|h|f | ≤ hf = 0
and |y ∗ (h)| ≤ ⎝
⎞1/2
|y(t2 )|2 d μ(t1 , t2 )⎠
h = 0 .
Ω1 ×Ω2
Hence
(2.8)
h|f = 0 = y ∗ (h) ,
f ∈ L1 (R) , h ∈ Θ .
Assume that f ∈ M(F , R) and h ∈ Θ + F . Then h = h0 + h1 for some h0 ∈ Θ and
h1 ∈ F. Applying now Lemma 2.1 and (2.8) we can see that
h|f = h0 |f + h1 |f = 0 + y ∗ (h1 ) = y ∗ (h0 ) + y ∗ (h1 ) = y ∗ (h) ,
h∈Θ+F .
By definition, f ∈ F ⊂ Θ + F . Thus, applying Lemma 2.1 once more, we get
f ∈ F ∩ M(Θ + F , R), and so
(2.9)
M(F , R) ⊂ F ∩ M(Θ + F , R) .
85
Conversely, assume now that f ∈ F ∩ M(Θ + F , R) and h ∈ F. Since h ∈ Θ + F ,
we conclude from Lemma 2.1 that
h|f = y ∗ (h) ,
h∈F .
Thus applying Lemma 2.1 once more we get f ∈ M(F , R), and so F ∩M(Θ+F , R) ⊂
M(F , R). Combining this inclusion with the inclusion (2.9) we derive the equality
(2.7).
Since Θ ⊂ L1 (R), the equalities in (2.8) hold for all f, h ∈ Θ. Then Lemma 2.1
yields M(Θ, R) = Θ. If now F ⊂ Θ, then the equality (2.7) takes the form
M(F , R) = F , which proves the theorem.
Given a nonempty set S ⊂ L1 (R) we denote by lin(S) the set of all linear
combinations nk=1 λk vk where n ∈ N, Z1,n k → λk ∈ C and Z1,n k → vk ∈ S.
It is easy to check that lin(S) is the smallest linear subset of L1 (R) containing S.
By S ⊥ we denote the orthogonal complement of S in the space H(R), i.e.
S ⊥ := {f ∈ L1 (R) : f |h = 0 for h ∈ S} .
The following theorem is motivated by Lemma 2.1 and the well known representation
of a linear and continuous functional in a Hilbert space by Riesz and Fréchet; cf.
e.g. [5] and [6].
Theorem 2.3. If F is a closed and linear set in H(R) and y ∈ L2 (R), then
M(F , R) = ∅ and M(F , R) = Θ + f for each f ∈ M(F , R). Moreover, if F ⊂
S := (y ∗ )−1 (0), then M(F , R) = Θ, and otherwise (F ∩ S)⊥ ∩ F \ Θ = ∅ and
M(F , R) = Θ +
(2.10)
y ∗ (h)
h,
h2
h ∈ (F ∩ S)⊥ ∩ F \ Θ .
Proof. Assume that M(F , R) = ∅ and choose arbitrarily f ∈ M(F , R) and f ∈
L1 (R). If f ∈ M(F , R), then, by Lemma 2.1,
g|f = y ∗ (g) ,
(2.11)
g∈F ,
∗
and g|f = y (g) for g ∈ F. Hence, setting h := f − f we conclude from (2.11)
that
h2 = h|f − f = h|f − h|f = y ∗ (h) − y ∗ (h) = 0 .
Thus f ∈ Θ + f for f ∈ M(F , R), and so M(F , R) ⊂ Θ + f . Conversely, suppose
that f ∈ Θ + f . Then by Schwarz’s inequality (1.13) we see that for every h ∈ F,
|h|f − h|f | = |h|f − f | ≤ hf − f = 0 .
Hence, and by (2.11), we get h|f = h|f = y ∗ (h) for h ∈ F. Applying Lemma 2.1
once again we see that f ∈ M(F , R) for f ∈ Θ + f , and so Θ + f ⊂ M(F , R). Both
the inclusions yield the equality M(F , R) = Θ + f , provided that M(F , R) = ∅,
and so we obtain the following implication
86
M(F , R) = ∅ =⇒ M(F , R) = Θ + f .
(2.12)
Assume now that F ⊂ S. Then
h|θ = 0 = y ∗ (h) ,
h∈F ,
which shows, by Lemma 2.1, that θ ∈ M(F , R). Hence, and by (2.12), we see that
M(F , R) = Θ + θ = Θ.
It remains to consider the case where the inclusion F ⊂ S does not hold. Then
F ∩ S ⊂ F = F ∩ S. By the assumption F is a closed set in H(R). Since y ∈ L2 (R),
y ∗ is a continuous functional on H(R), and so S is also closed set in H(R). Therefore
F ∩ S is a closed set in H(R), and consequently
Θ ⊂ F ∩ S = F .
(2.13)
Hence F \ (F ∩ S) = ∅. Since F ∩ S is a closed set in H(R), it follows that each
h ∈ F \ (F ∩ S) has an orthogonal projection hS on F ∩ S, i.e.
hS ∈ F ∩ S
(2.14)
and h − hS |g = 0 ,
g ∈F ∩S ;
⊥
cf. [6]. Hence h − hS ∈ (F ∩ S) ∩ F. If h − hS ∈ Θ, then from (2.13) and (2.14) it
follows that h = hS + (h − hS ) ∈ F ∩ S + Θ = F ∩ S, which is impossible. Therefore
/ Θ, and so h − hS ∈ (F ∩ S)⊥ ∩ F \ Θ. Thus (F ∩ S)⊥ ∩ F \ Θ = ∅. Given
h − hS ∈
h ∈ (F ∩ S)⊥ ∩ F \ Θ we see that h = 0, and so y ∗ (h) = 0. Hence, for each g ∈ F,
(2.15)
Since
(2.16)
gS := g −
y ∗ (h)
h2 h
y ∗ (g)
h∈F ∩S
y ∗ (h)
and g − gS =
y ∗ (g)
h ∈ (F ∩ S)⊥ ∩ F .
y ∗ (h)
∈ (F ∩ S)⊥ , we conclude from (2.15) that
y ∗ (h) y ∗ (g) y ∗ (h) y ∗ (h) g
h
h
=
g
−
g
h =
h
S
h2
h2
y ∗ (h) h2
y ∗ (g) y ∗ (h)
= ∗
h|h = y ∗ (g) , g ∈ F .
y (h) h2
Applying now Lemma 2.1, we see that
(2.17)
f :=
y ∗ (h)
h ∈ M(F , R) ,
h2
h ∈ (F ∩ S)⊥ ∩ F \ Θ .
Hence M(F , R) = ∅, and, combining (2.17) with (2.12), we derive the equality (2.10)
provided the inclusion F ⊂ S does not hold.
In the both cases M(F , R) = ∅, which completes the proof.
3. Calculating regression functions
Assume that F is arbitrarily chosen linear and closed set in the space H(R) and
y ∈ L2 (R). Then Theorem 2.3 yields M(F , R) = ∅. Moreover, Theorem 2.3 enables
us to find regression functions in F with respect to R provided we can determine
87
the linear set (F ∩ S)⊥ ∩ F. This is rather difficult task in general. However in the
case where the quotient linear space F /F ∩ Θ is finite dimensional we can effectively
calculate all the regression functions in F with respect to R in terms of any base of
this space. Obviously, this case is the most essential from practical point of view.
Given f, g ∈ L1 (R) we will write f ⊥ g iff f |g = 0. Given p, q ∈ Z, p ≤ q, and
a sequence Zp,q k → Fk of nonempty sets in the space H(R), we write
q
Fk for the set of all
k=p
Obviously,
2
k=1
q
fk where Zp,q k → fk ∈ Fk .
k=p
Fk = F1 + F2 . We have
Theorem 3.1. Given p ∈ N let Z1,p k → hk ∈ F \ Θ be a sequence satisfying the
following two conditions
(3.1)
lin({hk : k ∈ Z1,p }) = F
as well as
(3.2)
hk ⊥ hl ,
k, l ∈ Z1,p , k = l .
If y ∈ L2 (R), then
(3.3)
M(F , R) = (Θ ∩ F) +
p
y ∗ (hk )
k=1
hk 2
hk .
Proof. Fix p ∈ N and a sequence Z1,p k → hk ∈ F satisfying the assumptions.
From (3.1) and (3.2) it follows that F0 := Θ + F is a closed set in H(R). Therefore
M(F0 , R) = ∅ by the assumption y ∈ L2 (R) and Theorem 2.3. If y ∗ (hk ) = 0 for
k ∈ Z1,p , then by (3.1), F0 ⊂ S := (y ∗ )−1 (0). From Theorem 2.3 it follows that
M(F0 , R) = Θ. Hence, and by Corollary 2.2, we obtain M(F , R) = Θ ∩ F, and so
the equality (3.3) obviously holds.
Assume in contrary, that y ∗ (hk ) = 0 for some k ∈ Z1,p . Then F0 \ S = ∅ and
applying again Theorem 2.3 we can see that (F0 ∩ S)⊥ ∩ F0 \ Θ = ∅ as well as that
the equality (2.10) holds. Thus we have to find an element h ∈ (F0 ∩ S)⊥ ∩ F0 \ Θ.
If p = 1, then y ∗ (h1 ) = 0. Hence, and by (3.1), (F0 ∩ S)⊥ ∩ F0 = Θ⊥ ∩ F0 = F0 ,
and so h1 ∈ (F0 ∩ S)⊥ ∩ F0 \ Θ. Then Theorem 2.3 leads to
(3.4)
M(F0 , R) = Θ +
y ∗ (h1 )
h1 .
h1 2
It remains to consider the case where p > 1. Without lost of generality we may
assume that y ∗ (h1 ) = 0. Then by (3.1) there exists a sequence Z1,p k → λk ∈ C
such that
p
(3.5)
λk hk .
h=
k=1
88
Since
hk −
y ∗ (hk )
h1 ∈ S
y ∗ (h1 )
for k ∈ Z1,p
and h ∈ (F0 ∩ S)⊥ we have
h ⊥ hk −
(3.6)
y ∗ (hk )
h1 ,
y ∗ (h1 )
k ∈ Z1,p .
Combining this with (3.2) and (3.5) we see that for each l ∈ Z1,p ,
y ∗ (h ) y ∗ (hl ) l
0 = hhl − ∗
h1 = h|hl − h ∗
h1
y (h1 )
y (h1 )
p
p
y ∗ (h ) l
λk hk hl − ∗
λk hk h1
=
y (h1 )
k=1
=
p
k=1
k=1
p
l)
λk hk |hl − ∗
λk hk |h1 y (h1 )
y ∗ (h
k=1
y ∗ (hl )
y ∗ (hl )
h1 |h1 = λl hl 2 − λ1 ∗
h1 2 .
= λl hl |hl − λ1 ∗
y (h1 )
y (h1 )
Using this we can see that
λl =
(3.7)
λ1 y ∗ (hl )
h1 2 .
hl 2 y ∗ (h1 )
Thus
h=
p
λk hk = λ1 h1 +
k=1
p
p
λ1 y ∗ (hk )
y ∗ (hk )
λ1
2
2
h
h
=
hk ,
h
1
k
1
2
∗
hk y (h1 )
hk 2
y ∗ (h1 )
k=2
k=1
and, consequently, λ1 = 0 and
(3.8)
h̃ :=
p
y ∗ (hk )
k=1
hk
2
hk =
y ∗ (h1 )
h ∈ (F0 ∩ S)⊥ ∩ F0 .
λ1 h1 2
Applying now (2.10) and (3.2) we obtain
p
y∗ (hk )
∗
h
y
p
hk 2 k
y ∗ (h̃)
y ∗ (hk )
·
hk
M(F0 , R) = Θ +
h̃ = Θ + k=1
2
2
p
hk 2
y∗ (hk ) h̃
k=1
hk 2 hk p
=Θ+
p
k=1
k=1
k=1
∗
|y (hk )|
hk 2
2
|y ∗ (hk )|2
2
hk 4 hk ·
p
p
y ∗ (hk )
y ∗ (hk )
h
=
Θ
+
hk .
k
2
hk hk 2
k=1
Hence, and by (3.4) we see that for each p ∈ N,
(3.9)
M(F0 , R) = Θ + f ,
k=1
89
where, in view of (3.1),
(3.10)
f :=
p
y ∗ (hk )
hk ∈ lin({hk : k ∈ Z1,p }) = F .
hk 2
k=1
From Corollary 2.2, (3.9) and (3.10) it follows that
M(F , R) = F ∩ M(F0 , R) = F ∩ (Θ + f ) = (Θ ∩ F) + f .
Combining this with (3.10) we derive the equality (3.3), which completes the proof.
As far as applications are concerned we will study theoretic models F = lin({hk :
k ∈ Z1,p }) spanned by sequences Z1,p k → hk which are not orthogonal in the
space H(R) in general, because the pseudo-inner product · | · depends on the
empirical data function x : Ω1 → A and measure μ. Therefore we will not apply
Theorem 3.1 directly. However, in such a case we shall ortogonalize such sequences.
To this end we recall that for a given p ∈ N, a sequence Z1,p k → hk ∈ L1 (R) is
said to be an orthogonalization of a sequence Z1,p k → hk ∈ L1 (R), provided
⊥
\Θ ,
hk ∈ Hk ∩ Hk−1
(3.11)
k ∈ Z1,p ,
where H0 := Θ and Hk := lin({h1 , h2 , . . . hk }), k ∈ Z1,p . Every linearly independent
sequence Z1,p k → hk ∈ L1 (R) \ Θ has a sequence being its ortogonalization
result. A sequence Z1,p k → hk may be constructed by using the Gramm-Schmidt
recursive method, i.e.
(3.12)
h1 := h1
and hn := hn −
n−1
k=1
hn |hk h ,
hk 2 k
n ∈ Z2,p .
Corollary 3.2. Given p ∈ N let Z1,p k → hk ∈ F \ Θ be a linearly independent
sequence satisfying the condition (3.1) and let a sequence Z1,p k → hk ∈ L1 (R) be
its orthogonalization result. If y ∈ L2 (R), then
(3.13)
M(F , R) = (Θ ∩ F) +
p
y ∗ (hk ) h .
hk 2 k
k=1
In particular, the above equality holds for the sequence Z1,p k → hk , defined by the
formulas (3.12).
Proof. Given p ∈ N fix sequences Z1,p k → hk ∈ F \ Θ and Z1,p k → hk ∈ L1 (R)
satisfying the assumptions. From the property (3.11) it follows that hk = 0 for
k ∈ Z1,p and hk ⊥ hl for k, l ∈ Z1,p such that k = l. Moreover, by the condition
(3.1) we obtain
lin({h1 , h2 , . . . hp }) = lin({h1 , h2 , . . . hp }) = F .
Thus, applying Theorem 3.1 for the sequence Z1,p k → hk replaced by its ortogonalized associate Z1,p k → hk we derive the equality (3.13), which is our
claim.
90
4. Complementary remarks
In this section we present comments and examples which complete and illustrate
our consideration from previous sections. In the following remark we gather a few
simple observations from Corollary 3.2.
Remark 4.1. Under the assumptions of Corollary 3.2, we have θ ∈ Θ ∩ F, and so the
equality (3.13) yields
(4.1)
f :=
p
y ∗ (h )
k
k=1
hk 2
hk ∈ M(F , R) and M(F , R) = (Θ ∩ F) + f .
Since Θ ∩F is a linear set, the second equality in (4.1) shows that the class M(F , R)
forms an affine variety in the space H(R). Moreover from (4.1) we easily deduce that
the following properties are pairwise equivalent:
(i) f is a unique regression function in F with respect to R;
(ii) Θ ∩ F = {θ};
(iii) h > 0 for every h ∈ F \ {θ};
(iv) h = 0 ⇒ h = θ for every h ∈ F.
If additionally the sequence Z1,p k → hk ∈ F \ Θ satisfies the orthogonality
condition (3.2), then the formulas (3.12) yield hk = hk , k ∈ Z1,p , and consequently
the property (4.1) remains valid after replacing hk by hk as k ∈ Z1,p .
According to (4.1) the class M(F , R) is determined by the sequence Z1,p k →
y ∗ (hk )hk −2 hk . We will call it the regression functions sequence (RFS) generated
by a linearly independent sequence Z1,p k → hk ∈ F \ Θ satisfying the condition
(3.1).
It is worth noting that our approach to the regression theory is very flexible.
We provide an universal and simple theory covering classical cases of regressions
where the theoretic functional model F is spanned by polynomials, trigonometric
polynomials and other specific functions; cf. e.g. [13] and [2]. Moreover, we study
regression functions with respect to the wide range of regression structures R involving the generalized quadratic deviation (1.1) by means of certain measures μ. This
simplifies much theoretical considerations on the ground of Hilbert spaces. On the
other hand side we gain the possibility of using the modified smallest square method
which can be more adequate in specific situations.
In Example 0.1 the classical smallest square method was used. According to the
equality (1.3) in Example 1 this is a special case of the criterion δ with the measure
μ satisfying (1.2) and (1.4). In what follows we present an example which motivate
using a more sophisticated measure μ.
Example 4.2. Following Example 0.1 we wish this time to determine the electric
circuit resistance R by means of measurements samples of intensity and voltage
91
represented by two sequences Z0,n k → ik and Z0,m k → vk for some n, m ∈ N.
Assume that all measurements were made independently. Given a precision rate let
ρk be the probability that the intensity sample ik satisfies the precision rate for
k ∈ Ω1 := Z0,n and let ρl be the probability that the voltage sample vl satisfies
the precision rate for l ∈ Ω2 := Z0,m . As in Example 0.1 we consider the regression
structure R, where A := R, B := R, the empiric data functions are defined by
Z0,n k → x(k) := ik and Z0,n k → y(k) := vk , and as the deviation criterion
δ we take the generalized quadratic deviation given by (1.1). Following Example 1
we define the measure μ as a unique measure satisfying the equalities (1.2). We now
need to define the numbers ρk,l for k ∈ Ω1 and l ∈ Ω2 . We can do it in many ways.
In our case all measurements were made independently, so it seems to be natural to
set
(4.2)
ρk,l := ρk · ρl ,
k ∈ Ω1 , l ∈ Ω2 .
Then each coefficient ρk,l is equal to the probability of the event that both the measurement samples ik and vl satisfy simultaneously the prescribed precision rate. As
a matter of fact the coefficient ρk,l reflects accuracy of the measurement samples ik
and vl , and thereby reflects accuracy of the measurements devices used for getting
these samples. If the coefficient ρk,l is closer to the value 1 then intuitively the corresponding pair (ik , vl ) of samples is more valuable for us. Therefore the generalized
quadratic deviation criterion δ defined by (4.2) seems to be more natural in this
case as compared to the classical least squares method, where all samples (ik , vk )
are treated equivalently and samples of the form (ik , vl ), k = l, are not considered
at all.
As in the Example 0.1, we consider the theoretic functional model F represented
by linear functions R t → rt for r ∈ R. Then F = lin({h1 }) where h1 is the
identity mapping on R, i.e. h1 (t) = t for t ∈ R. Thus we can apply our theory from
previous sections in order to determine all regression functions in F with respect to
R. The condition (1.7) obviously holds for every function f : A → B, which means
that L1 (R) = (R → R). From (1.11) we have
m
n 2
h1 =
(4.3)
|(h1 ◦ x)(t1 )|2 d μ(t1 , t2 ) =
i2k ρk,l .
Ω1 ×Ω2
k=0 l=0
It is also easily seen that each function g : Z0,m → B satisfies the condition (1.8),
and so L2 (R) = (Z0,m → R). From (1.14) it follows that
n m
y ∗ (h1 ) =
(4.4)
(h1 ◦ x)(t1 )y(t2 ) d μ(t1 , t2 ) =
ik vl ρk,l .
Ω1 ×Ω2
k=0 l=0
Assume that h1 = 0. Then F ⊂ Θ, which implies, by Corollary 2.2, that
M(F , R) = F . Suppose for simplicity that ρk,l > 0 for k ∈ Ω1 and l ∈ Ω2 . From
(4.3) we see that h1 = 0 iff ik = 0 for k ∈ Ω1 . Thus the equality h1 = 0 means
that the current intensity vanishes (current does not flow) or the current intensity
92
is below the sensitivity of intensity measurements devices. In both the cases we are
not able to determine the resistance R. This provides a natural interpretation of the
equality M(F , R) = F .
Assume now that h1 = 0. Then h1 ∈ F \ Θ, and so Θ ∩ F = {θ}. Theorem 3.1
now leads to
∗
y ∗ (h1 )
y (h1 )
M(F , R) = (Θ ∩ F) +
h1 =
h1 ,
(4.5)
h1 2
h1 2
which means that the set M(F , R) consists of the unique regression function
Rt→
y ∗ (h1 )
y ∗ (h1 )
h1 (t) =
t.
2
h1 h1 2
Hence, and by (4.3) and (4.4), we can uniquely determine the resistance
n m
n m
y ∗ (h1 ) R=
(4.6)
=
i
v
ρ
i2k ρk,l .
k l k,l
h1 2
k=0 l=0
k=0 l=0
Note that if n = m and the coefficients ρk,l are defined by (1.4), then (4.6) yields
(0.6). Such a situation naturally corresponds to the sequence Z0,n k → (ik , vk ) of
n + 1 simultaneous measurements of current intension and voltage with the same
precision.
The following example illustrates the usage of Corollary 3.2 in the case where
the theoretic functional model F is spanned by two functions.
Example 4.2. Given a regression structure R with y ∈ L2 (R) consider the case where
the functional model F is spanned by two arbitrary and linearly independent fixed
functions h1 , h2 ∈ L1 (R) \ Θ, i.e. F = lin({h1 , h2 }). Applying Corollary 3.2 we can
see that
2
y ∗ (hk ) M(F , R) = (Θ ∩ F) +
(4.7)
h ,
hk 2 k
k=1
where according to (3.12),
(4.8)
h1 := h1
and h2 := h2 −
h2 |h1 h2 |h1 h = h2 −
h1 .
h1 2 1
h1 2
Hence, h2 ⊥ h1 , and consequently
2
h2 |h1 2
2
2
= h2 2 − |h2 |h1 | .
h2 = h2 − (4.9)
h
1
h1 2
h1 2
Setting
(4.10)
a2 :=
y ∗ (h2 )
h2 2
and a1 :=
y ∗ (h1 )
h2 |h1 −
a2
h1 2
h1 2
we conclude from (4.7) and (4.8) that
(4.11)
M(F , R) = (Θ ∩ F) + a2 h2 + a1 h1 .
93
Combining (4.10) with (4.8) and (4.9) we obtain
(4.12)
a2 =
y ∗ (h2 )h1 2 − y ∗ (h1 )h2 |h1 h2 2 h1 2 − |h2 |h1 |2
and
a1 =
y ∗ (h1 ) − h2 |h1 a2
.
h1 2
In particular, if h2 (t) = t and h1 (t) = 1 as t ∈ R, and the regression structure R is
defined as in Example 1 under the assumption that m = n and the coefficients ρk,l
satisfy (1.4), then the equalities in (4.12) yield a0 = a2 and b0 = a1 where a0 and b0
are defined in (0.2).
Acknowledgement
Presented results were obtained during the Monday Mathematical Seminar, gained
in the State University of Applied Science in Chelm by both the authors.
References
[1] T. W. Anderson, Last squares and best unbiased estimates, Ann. Math. Statist. 33
(1982), 266–272.
[2] N. R. Draper and H. Smith, Applied Regression Analysis, John Wiley and Sons, Inc.,
New York - London - Sydney, 1966.
[3] I. Durbin, A note on regression when there is extraneous information about one of
the coefficients, IASA 48 (1953), 799–808.
[4] N. I. Fisher and A. I. Lee, Regression models for an angular response, Biometrics 48
(1992), no. 3, 665–677.
[5] P. R. Halmos, Introduction to Hilbert Space and the Theory of Spectral Multiplicity,
Chelsea, New York, 1957.
[6] W. Mlak, Hilbert Spaces and Operator Theory, Mathematics and Its Applications,
Kluwer Academic Publishers and PWN-Polish Scientific Publishers, Dordrecht,
Boston, London and Warsaw, 1991.
[7] T. Naes and I. S. Helland, Relevant components in regression, Scand. Journal of
Statistics 20 (1993), no. 3, 239–250.
[8] P. C. B. Philips and Y. Sun, Regression with an evaporating logarithmic trend, Econometric Theory 19 (2003), no. 4, 692–701.
[9] R. E. Quandt, The estimation of the parameter of a linear regression system obeying
two separate regimes, IASA 53 (1968), 873–890.
[10] W. Rudin, Functional Analysis, McGraw-Hill, New York 1991.
[11] L. G. Teser, Iterative estimation of a set linear regression equations, IASA 59 (1964),
845–862.
[12] W. B. Whiston, Sequential Selection of Variables in Multiple Regression, University
of Cincinnati, 1964.
[13] E. I. Wiliams, Regression Analysis, John Wiley and Sons, New York, 1959.
94
Faculty of Mathematics
and Natural Sciences
The John Paul II Catholic
University of Lublin
Al. Raclawickie 14, P.O. Box 129
PL-20-950 Lublin, Poland
State University of Applied Science in Chelm
PL-22-100 Chelm, Pocztowa 54, Poland
State University
of Applied Science in Chelm
PL-22-100 Chelm, Pocztowa 54, Poland
Chair of Applied Mathematics
The John Paul II Catholic
University of Lublin
Al. Raclawickie 14, P.O. Box 129
PL-20-950 Lublin, Poland
Presented by Adam Paszkiewicz at the Session of the Mathematical-Physical Commission of the L
UOGÓLNIONE ZAGADNIENIE REGRESJI
Streszczenie
Autorzy przedstawiaja̧ uogólnienie klasycznej idei regresji przez studium i rozwia̧zanie
pewnych zagdnień ekstremalnych, określonych poprawnie w ramach skończenie- lub nieskończenie-wymiarowej przestrzeni Hilberta. Przy danych empirycznych, dyskretnych lub
cia̧glych, określamy i jednoznacznie wyrażamy klasȩ wszystkich rozwia̧zań zagdnienia w nowej postaci cia̧gów funkcji regresji (RFS). Znacza̧ca rozmaitość obserwowanych zjawisk,
w różnych dziedzinach nauk praktycznych i teoretycznych, może być przy użyciu techniki
RFS opisana i zbadana z istotna̧ precyzja̧.
PL ISSN 0459-6854
BULLETIN
DE
DES
LETTRES DE L
ÓDŹ
2010
Vol. LX
no. 1
pp. 95–108
Jacek Dziok
EXTREMAL PROBLEMS IN A GENERALIZED CLASS
OF UNIFORMLY CONVEX FUNCTIONS
Summary
The object of the present paper is to investigate classes of analytic functions with
varying argument of coefficients defined by subordination. The classes generalize the wellknown class of uniformly convex functions. By using the extreme points theory we obtain
coefficient inequalities and distortion theorems in the classes of functions with varying
coefficients. Some integral mean inequalities are also pointed out.
1. Introduction
Let B denote the class of functions f : U → C, where
U= {z ∈ C : |z| < 1},
and by A we denote the class of functions f ∈ B which are analytic in U.
Let F be a subclass of the class A.
A function f ∈ F is called an extreme point of F if the condition
f = γg + (1 − γ) h
(g, h ∈ F, 0 < γ < 1)
implies g = h. We shall use the notation EF to denote the set of all extreme points
of F . It is clear that EF ⊂ F.
We say that F is locally uniformly bounded if for each r, 0 < r < 1, there is a
real constant M so that
|f (z)| ≤ M
(f ∈ F, |z| ≤ r) .
We say that a class F is convex if
γf + (1 − γ)g ∈ F
(f, g ∈ F, 0 ≤ γ ≤ 1).
Moreover, we define the closed convex hull of F as the intersection of all closed
convex subsets of A that contain F . We denote the closed convex hull of F by HF .
96
J. Dziok
If
F = {fn ∈ A : n ∈ N = {1, 2, ...}} ,
then
(1)
HF =
∞
γn fn :
n=1
∞
γn = 1, γn ≥ 0 (n ∈ N) .
n=1
A real-valued functional J : A → R is called continuous on F if for any sequence
{fn } in F which locally uniformly converges to f the sequence {J (fn )} converges
to J (f ) .
Furthermore, a real-valued functional J : A → R is called convex on a convex
class F ⊂ A if
J (γf + (1 − γ) g) ≤ γJ (f ) + (1 − γ) J (g)
(f, g ∈ F, 0 ≤ γ ≤ 1) .
For each fixed value of m, n ∈ N, z ∈ U, the following real-valued functionals are
continuous and convex on A:
(n)
(2) J (f ) = f (0)/n! , J (f ) = |f (z)| , J (f ) = f (m) (z)
z ∈ U, f ∈ A .
Moreover, for λ > 0, 0 < r < 1, the real-valued functional
⎛
⎞1/λ
2π
λ
1
(n)
J (f ) = ⎝
(3)
f ∈ A
reiθ dθ⎠
f
2π
0
For λ ≥ 1, by Minkowski’s inequality it is also convex on A.
is continuous on A.
The extreme points theory for analytic functions was intensively investigated by
Hallenbeck and MacGregor [1] (see also [2] and [3]).
We say that a function f ∈ B is subordinate to a function F ∈ B, and write
f (z) ≺ F (z) (or simply f ≺ F ), if and only if there exists a function ω ∈ B, |ω(z)| ≤
|z| (z ∈ U) , such that
f (z) = F (ω(z))
(z ∈ U ) .
In particular, if F, F ∈ B, is univalent in U, in analogy to the case of analytic
functions, we have the following equivalence:
f (z) ≺ F (z) ⇐⇒ f (0) = F (0) and f (U) ⊂ F (U).
For functions f, g ∈ A of the form
∞
∞
f (z) =
an z n and g(z) =
bn z n ,
n=0
n=0
by f ∗ g we denote the Hadamard product (or convolution) of f and g, defined by
(f ∗ g) (z) =
∞
n=0
an b n z n
(z ∈ U) .
Extremal problems in a generalized class of uniformly convex functions
97
We denote by A the class of functions f ∈ B of the form
∞
f (z) = z +
(4)
an z n (z ∈ U).
n=2
Also, by Tη (η ∈ R) we denote the class of functions f ∈ A of the form (4) for which
arg(an ) = π + (1 − n)η
(5)
(n = 2, 3, ...).
For η = 0 we obtain the class T0 of functions with negative coefficients.
Moreover, we define
T :=
(6)
Tη .
η∈R
The class T was introduced by Silverman [4] (see also [5]). It is called the class
of functions with varying argument of coefficients.
Let k, A, B be real parameters, k ≥ 0, 0 ≤ B ≤ 1, −1 ≤ A < B, and let
ϕ, φ ∈ A.
By W (φ, ϕ; A, B; k) we denote the class of functions f ∈ A such that (ϕ ∗ f ) (z) =
0 (z ∈ U {0}) and
(φ ∗ f ) (z)
1 + Az
(φ ∗ f ) (z)
(7)
− k − 1 ≺
.
(ϕ ∗ f ) (z)
(ϕ ∗ f ) (z)
1 + Bz
Here we use, of course, the definition of subordination in the class B.
If −1 < B < 1, then the condition (4) is equivalent to the following:
(φ ∗ f ) (z)
1 − AB (φ ∗ f ) (z)
B−A
(8)
(ϕ ∗ f ) (z) − k (ϕ ∗ f ) (z) − 1 − 1 − B 2 < 1 − B 2 (z ∈ U),
and if B = 1, then we have
1+A
(φ ∗ f ) (z)
(φ ∗ f ) (z)
(9)
− 1 >
Re
− k (ϕ ∗ f ) (z)
(ϕ ∗ f ) (z)
2
(z ∈ U) .
In relation to the classes T and Tη , we define the following two classes:
T W (φ, ϕ; A, B; k)
:= T ∩ W (φ, ϕ; A, B; k) ,
T W η (φ, ϕ; A, B; k)
:= Tη ∩ W (φ, ϕ; A, B; k) .
For the present investigations we assume that ϕ, φ are functions of the form
∞
∞
n
(10)
αn z , φ(z) = z +
βn z n (z ∈ U),
ϕ(z) = z +
n=2
n=2
where the sequences {αn } , {βn } are real, and
0 ≤ αn < βn
(n = 2, 3, . . . ) .
Moreover, let us set
(11)
dn := (k + 1) (1 + B) βn − (kB + A + k + 1) αn
(n = 2, 3, . . . ) .
The families W (φ, ϕ; A, B; k) and Wη (φ, ϕ; A, B; k) unify various new and wellknown classes of analytic functions. We list a few of them in the last section.
98
J. Dziok
The object of the present paper is to investigate extreme points of the class
Wη (φ, ϕ; A, B; k) . By using the extreme points theory we obtain coefficients inequalities and distortion theorems in the classes of functions. Some integral means
inequalities are also pointed out.
2. Extreme points
Since A is a complete metric space Montel’s theorem [6] implies following lemma:
Lemma 1. A class F contained in A is compact if and only if F is closed and
locally uniformly bounded.
First we mention a sufficient condition for the function to belong to the class
W (φ, ϕ; A, B; k).
Theorem 1. Let {dn } be defined by (11), 0 ≤ B ≤ 1, −1 ≤ A < B. If a function
f of the form (4), (ϕ ∗ f ) (z) = 0(z ∈ U {0}), satisfies the condition
∞
(12)
dn |an | ≤ B − A,
n=2
then f belongs to the class W (φ, ϕ; A, B; k).
Proof. If 0 ≤ B < 1, then for a function f of the form (4) we have
1−
∞
αn |an ||z|n−1 > 0
n=2
and
≤
≤
(φ ∗ f ) (z)
1 − AB (φ ∗ f ) (z)
(ϕ ∗ f ) (z) − k (ϕ ∗ f ) (z) − 1 − 1 − B 2 (φ ∗ f ) (z)
B (B − A)
(k + 1) − 1 +
(ϕ ∗ f ) (z)
1 − B2
∞
(βn − αn ) |an ||z|n−1
B (B − A)
n=2
(k + 1)
+
∞
1 − B2
1−
αn |an ||z|n−1
(z ∈ U ).
n=2
Thus, by (11) and (12), we obtain (8) and, consequently, f ∈ W (φ, ϕ; A, B; k) .
Let now B = 1. Then simple calculations give
(φ ∗ f ) (z)
(φ ∗ f ) (z) 1 + A
− 1 − Re
−
k (ϕ ∗ f ) (z)
(ϕ ∗ f ) (z)
2
∞
(βn − αn ) |an ||z|n−1
(φ ∗ f ) (z)
n=2
− 1 ≤ (k + 1)
≤ (k + 1) ∞
(ϕ ∗ f ) (z)
1−
αn |an ||z|n−1
n=2
(z ∈ U ),
99
and, by (11) and (12) again, we obtain (9). Hence f ∈ W (φ, ϕ; A, B; k) and the
proof is complete.
The next theorem shows that the condition (12) is necessary as well for functions
of the form (4), with (5) to belong to the class T W η (φ, ϕ; A, B; k).
Theorem 2. Let f be a function of the form (4), with (5). Then f belongs to the
class T W η (φ, ϕ; A, B; k) if and only if the condition (12) holds true.
Proof. In view of Theorem 1 we need only show that each function f from the class
T W η (φ, ϕ; γ, k) satisfies the coefficient inequality (12). Let f be a function of the
form (4), satisfying the condition (5) and belonging to the class T W η (φ, ϕ; γ, k).
Then, putting z = riη in the conditions (8) and (9), we obtain
∞
(βn − αn ) |an |rn−1
B−A
n=2
.
(k + 1)
<
∞
1+B
1−
αn |an |rn−1
n=2
Thus we have
∞
[(k + 1) (1 + B) βn − (k (1 + B) + 1 + A) αn ] |an |rn−1 < B − A,
n=2
which, upon letting r → 1
−
, readily yields the assertion (12).
Since the condition (12) is independent of η, Theorem 2 yields the following
theorem.
Theorem 3. Let f be a function of the form (4), with (5). Then f ∈ T W (φ, ϕ;
A, B; k) if and only if the condition (12) holds true.
Theorem 4. The class T W η (φ, ϕ; A, B; k) is a convex and compact subclass of A.
Proof. Let functions f, g belong to the class T W η (φ, ϕ; A, B; k) , 0 ≤ γ ≤ 1. Since
∞
∞
n
n
an z
bn z
(13) γf (z) + (1 − γ)g (z) = γ z +
+ (1 − γ) z +
=
z+
∞
n=2
n=2
(γan + (1 − γ) bn ) z n ,
n=2
by Theorem 2 we have
∞
dn |γan + (1 − γ) bn | ≤
n=2
γ
∞
n=2
≤
dn |an | + (1 − γ)
∞
dn |bn |
n=2
γ (B − A) + (1 − γ) (B − A) = B − A,
and consequently the function h = γf + (1 − γ)g belongs to the class T W η (φ, ϕ;
A, B; k). Hence the class is convex.
100
J. Dziok
Furthermore, for f ∈ T W η (φ, ϕ; A, B; k) , |z| ≤ r < 1, we have
|f (z)| ≤ r +
(14)
∞
dn an
n=2
Since
∞
rn
rn
≤ r + (B − A)
.
dn
d
n=2 n
n1
1
= r lim sup (dn )− n ≤ r < 1,
lim sup rn d−1
n
(15)
n→∞
the power series
∞
n=2
n→∞
rn d−1
converges. Thus, by (14) we conclude that the class
n
T W η (φ, ϕ; A, B; k) is locally uniformly bounded. By Lemma 1, we only need to
show that it is closed i.e. if fm ∈ T W η (φ, ϕ; A, B; k) (m ∈ N) and fm → f, then
f ∈ T W η (φ, ϕ; A, B; k) . Suppose that
fm (z) = z +
∞
an,m z n (m ∈ N; z ∈ U)
n=2
and f is given by (4). Using Theorem 2 we have
∞
dn |an,m | ≤ B − A (m ∈ N) .
n=2
Since fm → f, we conclude that an,m → an as m → ∞ (n ∈ N). This gives the
condition (12), and, in consequence, f ∈ T W η (φ, ϕ; A, B; k) , which completes the
proof.
Theorem 5. Let f1 (z) = z and
(16)
fn (z) = fn,η (z) = z −
B − A i(1−n)η n
e
z
dn
(n = 2, 3, ...; z ∈ U) ,
where {dn } is defined by (11). Then
ET W (φ, ϕ; A, B; k) = {fn ; n ∈ N} .
Proof. By using (12) we verify easily, that the functions of the form (16) are extreme
points of the class T W η (φ, ϕ; A, B; k). Now suppose that a function f belongs to
the set ET W η (φ, ϕ; A, B; k) and f is not of the form (16). If
f (z) = z − γ
(B − A) n
z
dn
(0 < γ < 1, n = 2, 3, ...; z ∈ U) ,
then
f (z) = (1 − γ)f1 (z) + γfn (z), z ∈ U
and f is not the extreme point of T W η (φ, ϕ; A, B; k) . In other way there exist
m, l ∈ N, m = l, so that the coefficients am and al do not vanish in the power series
(4). Setting
g(z) = f (z) − al z l +
al m
am l
dm am
z , h(z) = f (z) − am z m +
z, γ=
,
dm
dl
dm am + dl al
101
we have
g, h ∈ T W (φ, ϕ; A, B; k) , g = h, 0 < γ < 1 and f = γg + (1 − γ) h.
It follows that f ∈
/ ET W η (φ, ϕ; A, B; k) , and the proof is complete.
3. Applications
The following lemmas will be useful later on.
If f ≺ g, then
Lemma 2. (cf. [7]). Let f, g ∈ A.
2π
0
f (reiθ )λ dθ ≤
2π
g(reiθ )λ dθ
(0 < r < 1, λ > 0) .
0
Lemma 3. (The Krein-Milman theorem). If F is a compact convex subclass of the
then HEF = F .
class A,
The Krein-Milman theorem (see [8] and [9]) is fundamental in the theory of
extreme points. In particular, it implies the following lemma:
Lemma 4. (cf. [1]). Let F be a compact convex subclass of the class A and J : A →
R be a real-valued, continuous and convex functional on F . Then
max {J (f ) : f ∈ HF } = max {J (f ) : f ∈ F } = max {J (f ) : f ∈ EHF } .
Using the extreme points of the class T W η (φ, ϕ; A, B; k) we obtain results listed
below. By (1) the Krein-Milman theorem gives
Corollary 1. Let f1 (z) = z and fn,η be defined by (16). Then
∞
∞
γn fn,η ;
γn = 1, γn ≥ 0 ( n ∈ N) .
T W η (φ, ϕ; A, B; k) =
n=1
n=1
Combining (2) with Lemma 4 we get
Corollary 2. If a function f of the form (4) belongs to the class T W η (φ, ϕ; A, B;
k), then
(17)
|an | ≤
B−A
dn
(n = 2, 3, . . . ),
where dn is defined by (11). The result is sharp. The functions fn,η of the form (16)
are the extremal functions.
102
J. Dziok
For the extreme points fn,η of the form (16) we have
(18)
fn,η
(z) = 1 −
(19)
(l)
(z) = −
fn,η
(l)
(z) = 0
fn,η
(B − A) n i(1−n)η n−1
e
z
,
dn
(B − A) n! i(1−n)η n−l
e
z
(l = 2, 3, ..., n),
(n − l)!dn
(l > n) .
(l)
Let l ∈ N0 , 0 < r < 1, and the sequence δn be defined by
δn(l) =
(20)
(B − A) n! n−l
r
(n − l)!dn
(n ≥ max {l, 2}) .
Applying (15) we obtain
lim sup δn(l) = 0 (l ∈ N0 ) .
n→∞
Thus there exist nl ∈ N (l ∈ N0 ) , such that
δn(l)l = max δn(l) : n ≥ max {l, 2}
(21)
(l ∈ N0 ) .
Therefore, by Lemma 3 we have
Corollary 3. If a function f belongs to the class T W η (φ, ϕ; A, B; k) , |z| = r < 1,
then
B − A n0
B − A n0
(22)
r
≤ |f (z)| ≤ r +
r ,
r−
dn0
dn0
(23)
1−
(B − A) n1 n1
r
dn1
(l) f (z)
(24)
≤ |f (z)| ≤ 1 +
≤
(B − A) n1 n1
r ,
dn1
(B − A) (nl )! nl −l
r
(nl − l)!dnl
(l ≥ 2) ,
where nl is defined by (21). The result is sharp. The functions fnl ,η of the form (16)
From Corollary 3 we get
Corollary 4. Let a function
f belong to the class T W η (φ, ϕ; A, B; k) , |z| = r <
(l)
(l ∈ N0 ) defined by (20) is nonincreasing with respect to
1. If the sequence δn
n, then
B−A 2
B−A 2
(25)
r ≤ |f (z)| ≤ r +
r (l = 0) ,
r−
d2
d2
(26)
1−
2 (B − A) 2
2 (B − A) 2
r ≤ |f (z)| ≤ 1 +
r
d2
d2
(l = 1) ,
103
(l) (B − A) (l)!
f (z) ≤
dl
(27)
(l ≥ 2) ,
where nl is defined by (21). The result is sharp. The functions fnl ,η of the form (16)
Now, we consider some integral means inequalities.
Corollary 5. Let 0 < r < 1, λ ≥ 1, l ∈ N0 and assume that the sequence
(l)
δn
defined by (20) is nonincreasing with respect to n. If f ∈ T W η (φ, ϕ; A, B, k) , then
(28)
2π
(l) iθ λ
f (re ) dθ
≤
0
2π
(29)
2π
(l) iθ λ
f2,η (re ) dθ
(l = 0, 1) ,
0
(l) iθ λ
f (re ) dθ
≤
0
2π
(l) iθ λ
fl,η (re ) dθ
(l = 2, 3, . . . ) ,
0
where fl,η are the functions defined by (16).
Proof. Since
f2,η
fn,η
≺
and fn,η
≺ f2,η
(n ∈ N) ,
z
z
then using Lemma 2 we have
⎧ 2π
⎫
⎨ λ
⎬ 2π
λ
(l)
(l)
iθ max
fn,η (re ) dθ : n ∈ N = f2,η (reiθ ) dθ
⎩
⎭
0
(l = 0, 1) .
0
Thus, Lemma 4 yields (28). The inequality (29) is an immediate consequence of (27)
and (19).
Making use of (6) and Corollaries 2, 3, 4, and 5, we obtain the corollaries listed
below.
Corollary 6. If a function f of the form (4) belongs to the class T W (φ, ϕ; A, B; k),
then the coefficients estimates (17) hold true. The result is sharp. The functions fn,η
(η ∈ R) of the form (16) are the extremal functions.
Corollary 7. If a function f of the form (4) belongs to the class T W (φ, ϕ; A, B; k),
|z| = r < 1, then the bounds (22), (23), and (24), hold true. The results are sharp,
the functions fnl ,η (η ∈ R) of the form (16) are the extremal functions.
Corollary 8. Let a function f of the
form (4) belong to the class T W (φ, ϕ; A, B; k),
|z| = r < 1. If the sequence
(l)
δn
(l ∈ N0 ) defined by (20) is nonincreasing with
104
J. Dziok
respect to n, then the inequalities (25), (26), and (27) hold true. The result is sharp,
the functions fn,η (η ∈ R) of the form (16) are the extremal functions.
l ∈ N0 and assume that the sequence
Corollary 9. Let 0 < r < 1, λ ≥ 1,
(l)
δn
defined by (20) is nonincreasing with respect to n. If a function f belongs to the class
T W (φ, ϕ; A, B; k), then the inequalities (28) and (29), hold true.
4. Concluding remarks
We conclude this paper by observing that, in view of the subordination relation
(7) and so also (8), (9), choosing the functions φ and ϕ, we can consider new and
well-known classes of functions. Let
n−1
k
ϕ x z ; A, B; k ,
Wn (ϕ; A, B; k) := W zϕ (z) ,
k=0
where n ∈ N, xn = 1. In particular, the class
Wn (ϕ; A, B) := Wn (ϕ; A, B; 0) ,
contains functions f ∈ A, which satisfy the condition
z (ϕ ∗ f ) (z)
n−1
k=0
(ϕ ∗
f ) (xk z)
≺
1 + Az
.
1 + Bz
It is related to the class of starlike functions with respect to n-symmetric points.
Moreover putting n = 1, we obtain the class
W (ϕ; A, B) := W1 (ϕ; A, B)
defined by
1 + Az
z (ϕ ∗ f ) (z)
≺
.
(ϕ ∗ f ) (z)
1 + Bz
The class is related to the class of starlike function. Analogously, the class
Hn (ϕ; γ, k) := Wn (ϕ; 2γ − 1, 1; k)
(0 ≤ γ < 1)
contains functions f ∈ A, which satisfy the condition
z (ϕ ∗ f ) (z)
z (ϕ ∗ f ) (z)
Re n−1
− γ > k n−1
− 1
k
k
k=0 (ϕ ∗ f ) (x z)
k=0 (ϕ ∗ f ) (x z)
(z ∈ U) .
It is related to the class of k-uniformly convex function of order γ with respect to
n-symmetric points. Moreover setting n = 1, we obtain the class
H (ϕ; γ, k) := H1 (ϕ; γ, k)
defined by
Re
z (ϕ ∗ f ) (z)
−γ
(ϕ ∗ f ) (z)
z (ϕ ∗ f ) (z)
> k − 1
(ϕ ∗ f ) (z)
(z ∈ U) .
105
The class is related to the class of k-uniformly convex function of order γ. The classes
z
; γ, k ,
U ST (γ, k) := H
1−z
z
U CV (γ, k) := H
2 ; γ, k ,
(1 − z)
are the well-known classes of of k-starlike function of order γ and k-uniformly convex
function of order γ, respectively. In particular, the classes U CV := U CV (1, 0) ,
k − U CV := U CV (k, 0) were introduced by Goodman [10] (see also [11, 12]), and
Kanas and Wiśniowska [13] (see also [14]), respectively.
We note that the class HT (ϕ; γ, k) := T0 ∩H (ϕ; γ, k) was introduced and studied
by Raina and Bansal [15].
If we set
h(α1 , z) := z q Fs (α1 , . . . , αq ; β1 , . . . , βs ; z),
where q Fs is the generalized hypergeometric function (for details see [16] and [17]),
then we obtain the class
UH (q, s, λ, γ, k) := HT (λh(α1 + 1, z) + (1 − λ) h(α1 , z); γ, k)
(0 ≤ λ ≤ 1)
defined by Srivastava et al. [18].
Let λ be a convex parameter. A function f ∈ A belongs to the class
ϕ (z)
+ (1 − λ) ϕ (z) , z; A, B; 0
Vλ (ϕ; A, B) := W λ
z
if it satisfies the condition
(ϕ ∗ f ) (z)
1 + Az
λ
+ (1 − λ) (ϕ ∗ f ) (z) ≺
.
z
1 + Bz
Moreover, a function f ∈ A belongs to the class
ϕ (z)
+ (1 − λ) ϕ (z) ; A, B; 0
Uλ (ϕ; A, B) := W λ
z
if it satisfies the condition
z (ϕ ∗ f ) (z) + (1 − λ) z 2 (ϕ ∗ f ) (z)
1 + Az
.
≺
1 + Bz
λ (ϕ ∗ f ) (z) + (1 − λ) z (ϕ ∗ f ) (z)
The classes
Wn (ϕ; A, B) ,
Hn (ϕ; γ, k) ,
Uλ (ϕ; A, B)
and Vλ (ϕ; A, B)
generalize well-known important classes, which were investigated in earlier works,
see for example [19–36].
If we apply the results presented in this paper to the classes discussed above,
we can obtain several additional new results. Some of these were obtained in earlier
works, see for example [31–36].
106
J. Dziok
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Institute of Mathematics
University of Rzeszów
Rejtana 16A, PL-35-310 Rzeszów
Poland
Presented by Zbigniew Jakubowski at the Session of the Mathematical-Physical
Commission of the L
108
J. Dziok
ZADANIA EKSTREMALNE W UOGÓLNIONEJ KLASIE FUNKCJI
JEDNOSTAJNIE WYPUKLYCH
Streszczenie
Zadaniem obecnej pracy jest badanie funkcji analitycznych o zmieniaja̧cym siȩ argumencie wspólczynników określonych przez podporza̧dkowanie. Klasy te uogólniaja̧ dobrze
znane klasy funkcji jednostajnie wypuklych. Przez zastosowanie teorii punktów krańcowych
uzyskujemy nierówności miȩdzy wspólczynnikami i twierdzenia o dystorsji w klasach funkcji
o zmieniaja̧cych siȩ wspólczynnikach. Wskazujemy też na pewne nierówności dotycza̧ce
średnich calkowych.
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PL ISSN 0459-6854
BULLETIN
DE
DES
LETTRES DE L
ÓDŹ
2010
Vol. LX
no. 1
pp. 137–154
Roman Stanislaw Ingarden and Julian L
awrynowicz
MODEL OF MAGNETIC ELECTRON MICROSCOPE INCLUDING
THE SCANNING MICROSCOPE III
VARIATIONAL APPROACH AND CALCULATION OF THE FOCAL LENGTH
IN A RANDERS-TYPE GEOMETRY
Summary
In the first part of the paper we have recalled general properties of electromagnetic
“lenses” and some phenomenological and approximate formulae for the length as a preparation for constructing the electromagnetic space. In the second part we have constructed
a Randers-type electromagnetic space. In this final, third part we present a variational approach and calculation of the focal length in the constructed here Randers-type geometry.
We conclude in comparison of the potentials generating functions dependence of the focal
length calculated and of its immersion electromagnetic “lenses” dependence.
9. Torsion-depending deformations within the electromagnetic
spaces
Following the general idea of Randers [R], from the relations (49) and (50) in [IL2]
we deduce
e2
2e
(54)
m2 c2 γμν + 2 Aμ Aν dxμ dxν − Aμ dxμ ds = 0.
c
c
With the notation
e2
Aμ Aν ,
c2
the relation (54) becomes
Gμν = m2 c2 γμν +
(55)
e
Gμ5 = G5μ = − Aμ ,
c
GAB dxA dxB = 0
G55 = 1,
dx5 = ds,
for A, B = 1, 2, . . . , 5.
This trick of Randers leads us to a sort of Kaluża-Klein theory [LR] in which we consider a five-dimensional pseudoriemannian metric with the additional condition (55).
138
R. S. Ingarden and J. L
awrynowicz
Randers did not observe that the condition (55) is of a non-holonomic character.
This follows from looking at a non-holonomic geometry of Wagner [W] which implies
that the motion of test bodies does not follow the shortest line in four and in five
dimensions as well. By this fact the initial point of the geometrization programme is
violated. On the other side the variational principle of the Kaluża-Klein theory in five
dimensions cannot be regarded an intrinsic geometrical principle in the Randers-type
space [R, I] related to (49).
A more direct solution of the problem was proposed in [SK], where a variational
principle
(56)
1
μν √
−g dτ = 0
δ
R + Fμν F
2
with Fμν = 2A[μ|ν] , dτ = dx1 dx2 dx3 dx4
was postulated. The disadvantage of that approach is caused by the fact that in the
√
Randers and even Finsler space the expression −g dτ has no invariant meaning
μ
because, in general, it depends on x or dxμ and hence the integral in (56) is not a
functional of field functions.
Let us begin with the static case: where all field functions are independent of
time. Then it is possible to diminish the number of co-ordinates by one: to eliminate
the time co-ordinate by means of the energy conservation law:
1
2
(57)
1 − 2 δjk ẋj ẋk + eϕ = E with j, k = 1, 2, 3,
mc
c
where E is a (constant) total energy of the electron. For eliminating the time, we
rearrange the Lagrangian (43) in [IL2] in the form
1
e
i
j
2
L(x , x ) = −mc 1 − 2 δjk ẋj ẋk + Aj (xj )ẋj − eϕ(xj ) + E,
c
c
(where an additive constant is unessential). Hence by (57) we get
2
2 2
L = (E − eϕ) − (mc )
(E − eϕ) + (e/c)Aj ẋj .
With the notation
d =
δjk ẋj ẋk dt,
the expression (57) becomes
dt = (E − eϕ)d
x̂j = (d/d) xj ,
c
ẋj = x̂j (d/dt) (E − eϕ)2 − (mc2 )2 .
In consequence the initial Hamilton principle yields Maupertuis’ principle or Fermat’s principle in the relativistic form
⎡
⎤
2
E
−
eϕ
e
δ Ldt = δ ⎣
− (mc)2 + Aj x̂j ⎦ d = 0.
c
c
Model of magnetic electron microscope including the scanning microscope III
139
In an arbitrary parametrization we have
2
E − eϕ(xj )
e
j
j
j
=
− (mc)2 δjk xj xk + Aj (xj )x .
L x ,x
c
c
With the notation
j
ψ(x ) =
(58)
E − eϕ(xj )
c
2
− (mc)2
we finally get
e
ds = ψ(xj ) δjk dxj dxk + Aj (xj )dxj
c
and in arbitrary curvilinear co-ordinates:
e
ds = ψ(xj ) γjk (xj )dxj dxk + Aj (xj )dxj
c
with γ as in (54).
Now, coming back to the definition of the unit directional vector (Pμ ) in [IL2],
Section 6, we have
k
(59)
δkl x k x + (e/c)Aj
Pj = ψδjk x
or, with the notation
pj = ψδjk x
k
δkl x k x ,
we get
Pj = pj + (e/c) Aj .
Yet, δ jk pj pk = ψ 2 , so with
1 2
μ
μ
ν
g μν xμ , x =
∂ /∂x ∂x L2
2
we obtain
(60) gjk
pj pk
Pj Pk
− 2 ,
= M δjk +
M
ψ
e
where M = δ jk Pj pk = ψ 2 + δ jk Aj pk .
c
Therefore, by Definition 1 in [IL2], Section 6, in the static case it is natural to accept
Definition 2. In the electromagnetic space (M, ds, N c ) in the static case the torsion
tensor may be presented in the form
1
M
M
M
(61) Tjkl = M Pj − 2 pj δkl + Pk − 2 pk δij + P − 2 p δjk
2
ψ
ψ
ψ
1
M
− 2 (Pj pk p + pj Pk p + pj pk P ) + 3 4 pj pk p .
ψ
ψ
140
awrynowicz
We may still calculate matrices reciprocal to αjk = ajk + λbj bk if the reciprocal
to ajk is known: aj ak = δkj . We try to find a solution in the form
αjk = ajk + κbj bk ,
(62)
where
bj = ajk bk
and κ is the coefficient to be determined. Clearly,
αj αk = δ jk + κ + λ + κλb b bm bm = δkj ,
and hence
κ = −λ/(1 + λb2 ),
(63)
where
b2 = b b .
According formulae (62) and (63) twice to (61), we conclude that
1
1 jm km N
jk
pm pn − Pm pn − pm Pn ,
g =
(64)
δjk + δ δ
M
M
M
where
N = M + δ jk Pj Pk = δ jk Pj (pk + Pk ).
With the help of (61) and (64) we can calculate the torsion vector
3
M
(65)
Tj =
Pj − 2 pj .
2
ψ
This is an elegant extension of the formula (10) in [IL1]. By suitable changes we can
adapt the formulae (61) and (65) to the non-static case. We have
ν
ν
Pμ = imcδμν x
δρσ x ρ x σ + (e/c)Aμ , pμ = imcδμν x
δρσ x ρ x σ ,
(66)
(67)
gμν
Pμ = pμ + (e/c)Aμ , δ μν pμ pν = −(mc)2 ,
pμ pν
Pμ Pν
+ 2 2 , M = δ μν Pμ pν = −(mc)2 + δ μν Aμ pν ,
= M δμν +
M
m c
1
1 μρ νσ N
μν
μν
g =
δ δ
pρ pσ − Pρ pσ − pρ Pσ ,
δ +
M
M
M
where
N = M + δ μν Pμ Pν = δ μν (pν + Pν ).
Finally we arrive at
Theorem 2. With the previous notation, as the natural extension of the torsion
tensor in the electromagnetic space (M, ds, N c ) in the static case, the torsion tensor
in the non-static case may be presented in the form (cf. [I]):
1
M
M
M
Tμν = M Pμ + 2 2 pμ δνρ + Pν + 2 2 pν δρμ + Pρ + 2 2 pρ δμν
2
m c
m c
m c
1
3M
+ 2 2 (Pμ pν pρ + pμ Pν pρ + pμ pν Pρ ) +
pμ pν pρ .
(68)
m c
(m2 c2 )2
141
In consequence the non-static torsion vector reads:
M
(69)
Tmu = 2 Pμ + 2 2 pμ .
m c
10. Application to an electromagnetic microscope
In order to characterize an electromagnetic microscope in terms of the corresponding
electromagnetic space, we introduce the principal curvature tensor of Varga [V]:
(70)
ν
ν
ν
ν
= Rμ·ρσ
− Tμ·σ
R0·ρσ
Hp·ρσ
and the Berwald affine curvature tensor [B]:
τ
τ
τ
(71) Kμνρσ = Rμνσ − Tμ·ν
R0τ ρσ + Tμ·ρ
Tντ σ|0 − Tμ·σ|0
Tντ ρ|0 + Tμνρ|0|σ − Tμνσ|0|ρ
or, equivalently,
∂
∂
ν
τ
τ
Kμ·ρσ
=
Gν −
G ν + Gμ·ρ
Gτν·σ − Gμ·σ
Gτν·ρ + Gτρ Gτν·μσ − Gτσ Gτν·μσ ,
∂xσ μ·σ ∂xρ μ·σ
where
∂
∂ 2 Gν
∂ 3 Gν
ν
ν
ν
Gνμ =
Gμ·ρ
=
Gμ·ρσ
=
.
μG ,
μ
ν,
μ
∂x
∂x ∂x
∂x ∂x ρ ∂x σ
Here Gν is as defined at the beginning of Section 6 in [IL2] and Gμ ν ·ρ is as in the
formula (35) in [IL2]. Formula (70) may be, by (70) and the symmetry of Tμνρ
rewritten as
τ
τ
Tντ σ|0 − Tμ·σ|0
+ Tμνρ|0|σ − Tμνσ|0|ρ ,
Kμνρσ = Hμνρσ + Tμ·ρ|0
so the Berwald and Varga curvature tensors are closely related.
μ
The Berwald curvature measure in the line element (xμ , x ) with respect to two
μ
μ
linearly independent vectors (x ) and (η ) is given by the formula
μ
μ
μ ρ
μ ρ
(72) R(xμ , x , η μ ) = Kμνρσ (xμ , x )x x η ν η ρ (gμρ gνσ − gμσ gρν )x x η ν η σ .
A Randers space is said to have the scalar curvature if (72) does not depend on η μ ,
μ
i.e. on two linearly independent vectors (x ) and (η μ ). If (72) is constant – does not
μ
μ
μ
depend on x , x and η , the Randers space is said to have a constant curvature. It
μ
appears [B] that if (72) does not depend on η μ and x , then it is also independent of
xμ , that is the Randers space possesses a constant curvature. Besides, for a Randers
space with constant curvature
(73)
Kμνρσ = R (gμρ gνσ − gμσ gρν ) ,
where R is a constant. This form of Kμνρσ is characteristic for spaces of constant
curvature.
Since in the case of the general electromagnetic microscope (73) appears to be
quite complicated we apply Berwald’s result [B] that in a Randers space of dimension
n and scalar curvature we have
1
(n + 1)Rμ + RTμ + Tμ|0|0 = 0.
3
142
awrynowicz
If the space is of constant curvature, then
(74)
Rμ = 0 and, by (73), RTμ + Tμ|0|0 = 0.
For and electromagnetic space, n = 3, we use Latin indices. By (65) with Pj and
M given by (59) and (60), respectively, we obtain
Theorem 3. With the previous notation, in the electromagnetic space (M, ds, N c )
corresponding to an electromagnetic microscope, the corresponding torsion vector
reads
k
3e
(δjk x )(A x )
Aj −
.
Tj =
2c
δmn x m x n
Corollary 2. In the case of an electromagnetic microscope, the torsion vector (Tj )
is independent of ψ [as defined by (58)] and thus also of the electric (scalar) potential
ϕ. The torsion vector vanishes with the magnetic (vector) potential (Aj ).
By the second equation in (74), we have
m n
m k
s
δmn x x δjk − δjm x x
L2 RAk + Ak||s x x = 0
which is a differential equation with respect to xj and thus Aj (xj ), but algebraic
j
with respect to x . We can rearrange it as
m n
m k
s
(75)
L2 RAk + Ak||s x x = 0.
δmn x x δjk − δjm x x
We may look at (75) as to a system of homogeneous linear equations for
(76)
qk = L2 RAk + Ak||s x x
with the discriminant
(x 2 )2 + (x 3 )3
2 1
D=
−x x
3 1
−x x
1
2
−x x
1
3 2
(x ) + (x )2
3
2
−x x
s
1
3
−x x
2 3
−x x
1
2
(x )2 + (x )2
= 0.
Therefore the system (75) has infinitely many solutions (76) which may be expressed
in the form
m n
qk = aδkl x δmn x x ,
where a is an arbitrary constant. Taking into account the expression for ds2 in
Section 9, we arrive at
m n
Rψ 2 Ak − aδk x δmn + R(e/c)2 Ak Am An + Ak|m|n x x
(77)
+2(e/c)RψAk A x
δmn x m x n = 0.
j
j
j
Next we take in (77) subsequently x = δ j1 , x = δ j2 , x = δ j3 , add the
equations obtained and apply the Maxwell equations in the form [IL2], (52) with
γ ρσ = δ ρσ :
3ψ 2 + (e/c)2 δ mn Am An + 2(e/c)ψ(A1 + A2 + A3 Ak = λ,
143
λ = 3a/R.
With the gauge transformation Ak ⇒ Ak + (∂/∂xk )χ we may always assume that
A1 + A2 + A3 = 0, so finally the condition discussed takes a simpler form
2
3ψ + (e/c)2 δ mn Am An Ak = λ.
The expression in the square brackets is positive as the sum of squares of real quantities, of which ψ does not vanish by (58). Hence A1 = A2 = A3 , so the vector (Ak )
has constant direction in the space, and therefore the magnetic field vanishes:
Hj = εjk (∂/∂xk )A = 0,
where εjk stands for the Ricci antisymmetric pseudo-tensor.
In such a way the general problem is reduced to an analogous problem for the
electrostatic microscope [I].
11. Deformation of potentials with the help of generating
functions
Let us consider again an electromagnetic field with the rotational symmetry along
the axis x3 = z. We set x1 = ρ cos α, x2 = ρ sin α and express the electric (scalar)
potential ϕ by a generating function f whose existence follows from the theory of
harmonic functions [Sch, Lu]:
1
ϕ(ρ, z) =
2π
(78)
2π
f (z + iρ cos ψ) dψ,
ϕ(0, z) = f (z).
0
In analogy we express the magnetic (vector) potential (Aj ) by a generation function g:
(79)
(Aj (x)) = −x2 A(ρ, z), x1 A(ρ, z), 0
1
A(ρ, z) =
2π
(80)
2π
g (z + iρ cos ψ) sin2 ψdψ,
A(0, z) =
0
1
g (z).
2
Condition (79) has to assure that div(Aj ) = 0.
In spite of the imaginary unit appearing in the integrals in (78) and (80) the
potentials ϕ and A are real. Indeed, if we expand f and g into the Taylor series in z
and integrate over ψ, all the integrals containing i, which correspond to odd powers
of cos ψ, are zero:
1
2π
2π
2ν+1
cos
0
ψdψ = 0,
1
2π
2π
0
cos2ν+1 ψ sin2 ψdψ = 0,
ν = 0, 1, . . .
144
awrynowicz
Since
1
2π
then
(81)
2π
2ν
cos
0
(2ν)!
ψdψ = 2ν
,
2 (ν!)2
1
2π
2π
cos2ν ψ sin2 ψdψ =
0
(2ν)!
,
22ν+1 ν!(ν + 1)!
2ν
∞
(−1)ν 1
ϕ(ρ, z) =
ρ
f (2ν) (z)
2
(ν!)
2
ν=0
1
1
= f (z) − ρ2 f (z) + ρ4 f (4) (z) + . . . ,
4
64
∞
(82)
A(ρ, z) =
1 (−1)ν
2 ν=0 ν!(ν + 1)!
1
ρ
2
2ν
g(2ν) (z)
1
1 4 (4)
1
ρ g (z) + . . .
= g (z) − ρ2 g (z) +
2
16
192
If we construct the potentials ϕ, A with the help of formulae (78), (80), it is natural to
assume that f, g are arbitrary ρ-dependent real-analytic functions of one variable z.
We notice that ϕ and A satisfy the differential equations
2π
d i
1
1
2
(sin ψ)f (z + i cos ψ) dψ ≡ 0
∇ ϕ ≡ ϕρρ + ϕρ + ϕzz =
ρ
2π
dψ ρ
0
(since the function in square brackets is periodic with period 2π) and
Aρρ + (3/ρ) Aρ + Azz = 0.
We also recall that ϕ, A are not directly observable quantities in contrast to the
field vectors E, H and the Lorentz force F acting on an electron; they are directly
observable and unique. The choice of the constant 12 in the second formula in (80)
is due to the relations
H = ∇ × (Aj ) = −x1 Az , −x2 Az , 2A + ρAρ ,
H(0, z) = (0, 0, 2A(0, z)) = (0, 0, g(z)).
Take now the principle of energy conservation
ẋj ∂/∂ ẋj L − L = C,
where C is a constant and the Lagrangian L does not depend explicitly on time. In
order to eliminate time, let us consider the problem of variation
1 L xj , xj /ρ + p(s)(ρ − t + C) t ds = 0,
δ
0
where p(s) is a Lagrangian multiplier. Variations with respect to ρ and s give
(83)
L(xj xj /ρ) − (xj /ρ)(∂/∂ ẋj )L(xj , xj /ρ) + p(s) = 0
145
and
p (s) = 0,
(84)
p(0) = p(1) = C,
respectively. Variation with respect to p(s) gives ρ = t and from (84) we deduce
that p(s) = C for any s ∈ [0; 1]. Taking into account (83), we arrive at
L xj , xj /ρ + C − xj /ρ ∂/∂ ẋj L xj , xj /ρ
= 0.
= (∂/∂ρ) ρ C + L xj , xj /ρ
By ρ = t and the latter relation, we may reformulate our problem as the Fermat
principle:
1 (85)
δ F xj , xj ds = 0,
0
where
F(xj , xj ) = G xj , xj , ρ(xj , xj ) = ρ C + L(xj , xj /ρ)
and ρ = ρ(xj , xj ) is determined by the equation (∂/∂ρ)G(xj , xj , ρ) = 0. Hence we
get
(86)
F = (1/ρ)gjk xj xk + (C − U ),
−(1/ρ2 )gjk xj xk + (C − U ) = 0,
U being the potential energy: U = U (xj ) = T − L, where T = gjk ẋj ẋk denotes the
kinetic energy.
Eliminating ρ from the system (85) we finally arrive at the Maupertuis principle:
1
(87)
δ
F (xj , xj )ds = 0,
0
where
j
F (x , x ) = 2 C − U (x ) gjk xj xk .
j
j
We can see that now F contains the constant C assumed as given. Since U is fixed
up to an arbitrary constant, the latter constant is eliminated in (87), because it
is contained both in C and in U . Thus F has a unique physical meaning. Yet, F
does not contain derivatives with respect to time, but derivatives with respect to
an arbitrary parameter s. Therefore all the conditions of the Fermat principle are
satisfied and we can see that F has the dimension of energy as the Lagrangian L
has, provided that s has the dimension of time; [AIM], pp. 173–177.
The role of generating functions f, g and the integral representations of potentials
(78) and (80) will appear when we take into account that the image of electron beams
is rotated in the electric microscope by the magnetic field around the optical axis
by an angle ω expressible by f and g; [Lu], pp. 386–398. In other words, we need
geometry with torsion ([IL1], Sect. 3) and torsion-depending deformations ([IL2],
Sect. 8). We shall study this problem in the next two sections.
146
awrynowicz
12. “Lens”-thickness depending deformations in relation
with the scanning microscope
Let us consider the Lagrangian (in the previous notation) in the form
L(x, ẋ) = mc2 1 − 1 − ẋ2 /c2 − eϕ(x) + (e/c) Aj (x)ẋj .
Then from (85) we deduce
1
G/mc = cρ 1 − 1 − x 2 /ρ2 c2 + a · x + ρcΦ,
2
where
Φ = 2 (C − eϕ) /mc2 ,
(88)
a = e/mc2 A
are dimensionless potentials [Lu].
Extremals of the variational problem (85) are not influenced by a multiplicative
constant of F, so we may modify it as
F = (1/mc) G xj , xj , ρ ,
(89)
(∂/∂ρ) (1/mc)G xj , xj , ρ = 0
and therefore
1
Φ + Φ2 x 2 + a · x .
4
In absence of the magnetic field we have a dimensionless index of refraction
1
n = Φ + Φ2
4
j
j
F(x , x ) =
and the expression for F is 1(p)-homogeneous in xj .
Indeed, by (85) we have
xj Gxj = G − ρGρ
and
so
Fxj = (1/mc) Gxj + Gρ ∂/∂xj ρ = (1/mc)Gxj ,
xj Fxj = (1/mc) (xj Gxj ) = (1/mc) (G − ρGρ ).
Yet, Gρ = 0 and G = mcF, and we obtain the Euler equation of the required
homogeneity xj Fxj = F. Since from the latter relation it follows the initial condition
for xj = 0, the equation provides also a sufficient condition for homogeneity.
It is clear that F may be treated as a Randers metric [I]:
1
F(x, y) = Φ(x) + Φ(x)2 (y 1 )2 + (y 2 )2 + (y 3 )2 + aj (x)y j .
(90)
4
Because of an analogy of F with the index of refraction, an electromagnetic field can
thus be interpreted as an optical medium whose properties vary from point to point.
147
Then to every point there is an associated ray surface given by the equation
1
Φ + Φ2 (y 1 )2 + (y 2 )2 + (y 3 )2 + aj y i = 1.
4
For slow electrons Φ2 (x) ≈ 0, so the equation approximatively represents an ellipsoid
symmetric with respect to the vector a with centre offset in the same direction. If
a = 0, the ellipsoid becomes a sphere. The expression discussed is the indicatrix
of the Randers space constructed in the case of nonrelativistic electrons and in the
general case as well. We conclude with
Theorem 4. In the electromagnetic space corresponding to a rotationally symmetric
electron microscope we can insert into the Randers metric function (90) the general
solution for the fields (78), (80) or (81), (82) using (88). The case of a scanning
microscope is included.
Remark. The system of equations involved in Theorem 4 does not include time which
is estimated. Precisely, we may rewrite (90) in the form
1
F(x, y, z, ẋ, ẏ) = Φ(x, y, z) + Φ(x, y, z)2 1 + ẋ2 + ẏ 2 + a1 ẋ + a2 ẏ + a3 ,
4
where
ẋ = (d/dz)x and ẏ = (d/dz)y.
13. Explicit formula for the focal length and some numerical
results
Following the Remark, we calculate the Hamilton function
(91)
H(z, u, v, w) = ẋp + ẏq − F(x, y) = − n2 (u, z) − a2 u + aw − v,
where
u = x2 + y 2 ,
v = p2 + q 2 ,
w = 2(xq − yp),
p = (∂/∂ ẋ)F,
q = (∂/∂ ẏ)F,
and
n(u, z) = n(x, y, z)
is the index of refraction. In terms of (91) we have the Hamilton equations
ẋ = Hp ,
ṗ = −Hx ,
q̇ = −Hy .
As we have already mentioned at the end of Sect. 11, the image of electron beams
is rotated in the electric microscope by the magnetic field around the optical axis
by an angle ω expressible by f and g:
z 1
1
2
α(ζ) + α(ζ) dζ,
β(ζ)
ω(z) = −
2
4
z0
148
awrynowicz
with
2
e
[C − ef(ζ)] , β(ζ) =
g(ζ);
mc2
mc2
where f and g the generating functions for the potentials (78)–(80), whereas ζ = z0
and ζ = z are optically conjugated planes corresponding to the object and image,
respectively. Finally one may calculate ([Lu], p. 398) that the focal length of an
electromagnetic “lens” of thickness d is given by the formula
α(ζ) =
1
1
=
f
4
(92)
d
0
α + β 2 + 12 αα
dζ.
α + 14 α2
In order to have an idea how the curves
(93)
f = f (d) = f (d; f, g, C)
look like, we take
f1
f2
f3
f4
f5
f6
f7
f8
1√
= f d; (1 + z 2 )
g · cm,
s
1√
= f d; (1 + z 2 )
g · cm,
s
1√
= f d; (1 − z 2 )
g · cm,
s
1√
= f d; (1 − z 2 )
g · cm,
s
1√
= f d; (1 − z 2 )
g · cm,
s
1√
= f d; (1 − z 2 )
g · cm,
s
1√
= f d; (1 + z 2 )
g · cm,
s
1√
= f d; (1 + z 2 )
g · cm,
s
1√
1
3
(1 − z )
g · cm, −2
g · cm ,
s
s
1√
1
(1 + z 2 )
g · cm, −4
g · cm3 ,
s
s
1
2 1√
3
(1 − z )
g · cm, 6
g · cm ,
s
s
1√
1
(1 + z 2 )
g · cm, 8
g · cm3 ,
s
s
1√
1
(1 + z 2 )
g · cm, 2
g · cm3 ,
s
s
1
2 1√
3
(1 − z )
g · cm, 4
g · cm ,
s
s
1√
1
(1 + z 2 )
g · cm, −6
g · cm3 ,
s
s
1
2 1√
3
(1 − z )
g · cm, −8
g · cm .
s
s
2
The curves f1 , . . . , f4 are shown on Fig. 7, accompanied by Tab. 1 with the corresponding focal length values. The curves f5 , . . . , f8 are shown on Fig. 8, accompanied
by Tab. 2 correspondingly.
149
Fig. 7: Four cases f1 , ..., f4 of the focal length f dependence on the “lens” thickness d.
Table 1. Focal length f values corresponding to f1 , ..., f4 .
d [cm]
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
f1 [cm]
16.305
8.403
5.663
4.273
3.433
2.871
2.468
2.166
1.931
1.742
1.588
1.460
1.351
1.258
1.178
1.107
1.045
0.989
0.940
0.895
0.854
0.818
0.784
0.753
f2 [cm]
18.778
9.534
6.389
4.805
3.850
3.212
2.756
2.413
2.147
1.933
1.758
1.613
1.489
1.383
1.292
1.212
1.141
1.078
1.022
0.971
0.925
0.884
0.846
0.811
f3 [cm]
19.318
9.759
6.528
4.905
3.928
3.276
2.809
2.459
2.187
1.969
1.790
1.641
1.515
1.408
1.314
1.232
1.160
1.096
1.038
0.986
0.940
0.897
0.858
0.823
f4 [cm]
19.538
9.845
6.580
4.942
3.956
3.299
2.829
2.476
2.201
1.982
1.802
1.652
1.525
1.416
1.322
1.239
1.167
1.102
1.044
0.992
0.945
0.902
0.863
0.827
150
awrynowicz
Fig. 8: Further four cases f5 , ..., f8 of the focal length f dependence on the “lens” thickness d.
Table 2. Focal length f values corresponding to f5 , ..., f8 .
d [cm]
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
f5 [cm]
16.298
8.393
5.648
4.254
3.409
2.842
2.435
2.127
1.887
1.693
1.534
1.400
1.285
1.185
1.096
1.016
0.943
0.872
0.797
f6 [cm]
18.778
9.533
6.388
4.803
3.848
3.209
2.752
2.409
2.142
1.928
1.752
1.606
1.482
1.376
1.284
1.203
1.132
1.068
1.011
0.960
0.913
0.871
0.832
0.796
f7 [cm]
19.318
9.758
6.528
4.904
3.927
3.275
2.808
2.458
2.185
1.967
1.788
1.640
1.513
1.405
1.312
1.230
1.157
1.093
1.035
0.983
0.936
0.893
0.854
0.819
f8 [cm]
19.538
9.845
6.580
4.941
3.956
3.298
2.828
2.475
2.201
1.981
1.801
1.651
1.524
1.415
1.321
1.238
1.166
1.101
1.043
0.991
0.943
0.900
0.861
0.825
151
14. Third conclusion: potentials generating functions dependence vs. immersion electromagnetic “lenses” dependence
Potentials generating functions f and g dependence of the focal length has been
characterized in Sect. 13 by the relation (92) of the form (93) and illustrated by
examples f1 , f2 , . . . , f8 with
1√
(94)
g · cm,
fj (z) = fj (z; aj ) = 1 + aj z 2
s
where
1 for j = 1, 2, 7, 8,
aj =
−1 for j = 3, 4, 5, 6;
(95)
where
(96)
1√
g · cm,
gj (z) = gj (z; bj ) = 1 + bj z 2
s
1 for j = 2, 4, 5, 7,
bj =
−1 for j = 1, 3, 6, 8;
1
g · cm3 = −2, C2 = −4, C3 = 6,
C1 in
s
C4 = 8, C5 = 2, C6 = 4, C7 = −6, C8 = −8.
For 0.05 cm < d < 1.20 cm, let fj− denote the domains consisting of the points
(d, p) situated below the curves fj , j = 1, 2, . . . , 8. Then we can see that
(97)
f1− ⊂ f2− ⊂ f3− ⊂ f4− ,
f5− ⊂ f6− ⊂ f7− ⊂ f8−
and that, at every d ∈ (0.05 cm; 1.20 cm), the differences
(98)
f2− \ f1− , f3− \ f2− , f4− \ f3− ;
f6− \ f5− , f7− \ f6− , f8− \ f7−
are relatively small. On Figs. 9 and 10 we visualize the growth of fj with the help of
arrows joining points:
(99)
(cj , aj ) → (cj+1 , aj+1 ),
◦(cj , aj ) → ◦(cj+1 , aj+1 ),
j = 1, 2, 3;
j = 5, 6, 7,
and
(100)
(cj , bj ) → (cj+1 , bj+1 ),
j = 1, 2, 3;
◦(cj , bj ) → ◦(cj , bj ) → ◦(cj+1 , bj+1 ),
j = 5, 6, 7,
respectively.
The relationship between the potentials generating functions dependence of the
focal length and the immersion electromagnetic “lenses” dependence of that length
is rather a problem in engineering and requires further study. The dependence of
the focal length on various parameters of the immersion electromagnetic “lenses”:
U1 , U2 , ζ(H1 ), ζ(H2 ), h1 , h2 , x1 , x2 , z0 , z, γ1 , γ2 , r1 (z), r2 (z)
were discussed in [IL1], Sects. 4 and 5.
152
awrynowicz
1
1
a 0
b 0
-1
-8
-6
-4
-2
0
C
2
4
6
8
Fig. 9: Growth of the focal length shown
by arrows (99) in the plane (C, a)
according to the formulae (92)-(96).
-1
-8
-6
-4
-2
0
C
2
4
6
8
Fig. 10: Growth of the focal length shown
by arrows (100) in the plane (C, b)
according to the formulae (92)-(96).
Acknowledgments
The authors are deeply indebted to Mrs. Malgorzata Nowak-Kȩpczyk for drawing Figs. 7 and 8, and making the corresponding computer calculations.
This work (J.L) was partially supported by the Ministry of Sciences and Higher
Education grant PB1 P03A 001 26 (Section 9 of the paper) and partially by the
grant of the University of L
ódź no. 505/692 (Sections 10–14).
Errata to [IL1]
On p. 112 in the first formula replace twice C by A.
References
[AIM] P. L. Antonelli, R. S. Ingarden, and M. Matsumoto, The Theory of Sprays and Finsler
Spaces with Applications in Physics and Biology, Kluwer Academic, Dordrecht 1993.
[B]
L. Berwald, Über Finslersche und Cartansche Geometrie. IV. Projektivkrümmung allgemeiner affiner Räume und Finslersche Räume skalarer Krümmung, Ann. of Math.
48 (1947), 755–782.
[C1] É. Cartan, Sur la possibilité de plonger un espace riemannien donné dans un espace
euclidien, Annales de la Société Polonaise de Mathématiques 6 (1921), 1–17.
[C2] —, Observations sur le mémoire précédent {refers to [K]}, Math. Zeitschrift 37
(1933), 619–622.
[C3] —, Les espaces de Finsler, Actualités scientifiques et industrielles, Hermann, Paris
1934; 2nd. ed. 1971.
[G]
W. Glaser, Strenge Berechnung magnetischer Linsen der Feldform H = H0 /[1 +
(z/a)2 ], Zeitschrift für Phys. 117 (1940), 285–315.
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[GLW] B. Gaveau, J. L
awrynowicz, and L. Wojtczak, Statistical mechanics of a tunnelling
electron microscope tip on the surface trajectory. (A complex approximation method),
in: Applied Complex and Quaternionic Approximation. Ed. by Ralitza K. Kovacheva,
J. L
awrynowicz, and S. Marchiafava, Ediz. Nuova Cultura Univ. “La Sapienza”, Roma
2009, pp. 13–39.
[GYN] J. L. Goldstein, H. Yakowitz, D. E. Newburg et all., Practical Scanning Electron Microscopy. Electron and Ion Microprobe Analysis, Plenum Press, New York-London
1975.
[H] R. G. Hutter, Rigorous treatment of the electrostatic immersion lens whose axial potential distribution is given by Φ(z) = Φ0 exp(Karc tan z), J. Appl. Phys. 16, no. 11
(1945), 678–699.
[I]
R. S. Ingarden, On the Geometrically Absolute Optical Representation in the Electron
Microscope (Prace Wroclawskiego Towarzystwa Naukowego [Memoirs of the Wroclaw
Scientific Society] Ser. B 45), WrTN, Wroclaw 1957, 60 pp.
[IL1] — and J. L
awrynowicz, Model of magnetic electron microscope including the scanning
microscope I. Physical model, Bull. Soc. Sci. Lettres L
ódź 59 Sér. Rech. Déform. 59,
no. 2 (2009), 107–117.
[IL2] —, —, Model of magnetic electron microscope including the scanning microscope II.
Mathematical model, ibid. 59 Sér. Rech. Déform. 59, no. 2 (2009), 119–126.
[K] D. Kosambi, Parallelism and path-spaces, Math. Zeitschrift 37 (1933), 608–618.
[L]
J. L
awrynowicz, Randers and Ingarden spaces with thermodynamical applications,
Bull. Soc. Sci. Lettres L
ódź 59 Sér. Rech. Déform. 58 (2009), 117–134.
[Lu] R. K. Luneburg, Mathematical Theory of Optics, 1st ed.: mimeographed lecture notes,
Braun Univ., Providence, R.I., 1944; 2nd ed.: Univ. of California Press, Berkeley, CA,
1964.
[M1] R. Miron, The geometry of Ingarden spaces, Rep. Math. Phys. 54 (2004), 131–147.
[M2] —, General Randers spaces and general Ingarden spaces, Bull. Soc. Sci. Lettres L
ódź
57 Sér. Rech. Déform. 54 (2007), 81–91.
[M3] —, The geometry of Ingarden spaces, ibid. 57 Sér. Rech. Déform. 54 (2007), 93–100.
[O] C. V. Oatley, The Scanning Electron Microscope I. The Instrument, Cambridge Univ.
Press, Cambridge 1972.
[P]
B. Paszkowski, Optyka elektronowa [Electron Optics], Państwowe Wydawnictwa
Techniczne, Warszawa 1960.
[R] G. Randers, On an asymmetric metric in the four-space of general relativity, Phys.
Rev. 59 (1941), 195–199.
[RK] H. Rothe und W. Kleen, Hochvakuum Elektronenröhren, Bd. I. Physikalische Grundlagen, Akadem. Verlagsgesellschaft, Frankfurt a. Main 1955.
[S]
K. Spangenberg, Vacuum Tubes, McGraw Hill, New York 1948.
[SK] G. Stephenson and C. W. Kilmister, A unified field theory of gravitation and electromanetism; G. Stephenson, Affine field structure of gravitation and electromagnetism,
Il Nuovo Cimento 10 (1953), 230–250 and 354–355.
[SSI] H. Shimada, S. V. Sabau, and R. S. Ingarden, The Randers metric its role in electrodynamics, Bull. Soc. Sci. Lettres L
ódź 58 Sér. Rech. Déform. 57 (2008), 39–49.
[Sch] O. Scherzer, Zur Theorie der elektronenoptischer Linsenfehler, Z. Physik 80 (1933),
193–202.
[V] O. Varga, Über affinzusammenhängende Mannigfaltigkeiten von Linienelementen,
insbesondere deren Äquivalenz, Publ. Math. Debrecen 1 (1949), 7–20.
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[W1] V. V. Wagner, Sur la géometrie différentielle des multiplicités anholonomes, Trudy
Seminara po Vektornomu i Tensornomu Analizu 2–3 (1935), 269–280.
[W2] —, On the embedding of a field of local surfaces in Xn in a constant field of surfaces
in affine space [in Russian], Dokl. Akad. Nauk SSSR N.S. 66 (1949), 785–788.
Institute of Physics
Nicolaus Copernicus University
Grudzia̧dzka 5, PL-87-100 Toruń
Poland
Univeristy of L
ódź
Pomorska 149/153, PL-90-236 L
ódź
Polish Academy of Sciences
L
ódź
Poland
ódź Society of Sciences and Arts on June 25, 2009
MODEL MAGNETYCZNEGO MIKROSKOPU ELEKTRONOWEGO
UWZGLȨDNIAJA̧CY MIKROSKOP SKANINGOWY III
PROCEDURA WARIACYJNA I WYZNACZENIE DLUGOŚCI OGNISKOWEJ
Streszczenie
W pierwszej czȩści pracy przypomnieliśmy ogólne wlasności “soczewek” elektromagnetycznych oraz pewne wzory fenomenologiczne i przybliżone na dlugość ogniskowej jako przygotowanie do konstrukcji przestrzeni elektromagnetycznej. W drugiej czȩści skonstruowaliśmy przestrzeń elektromagnetyczna̧ typu Randersa. W obecnej, końcowej, trzeciej czȩści
przedstawiamy podejście wariacyjne i wyznaczenie dlugości ogniskowej w skonstruowanej tu
geometrii typu Randersa. W podsumowaniu porównujemy zależność wyznaczonej dlugości
ogniskowej od funkcji generuja̧cych potencjaly z zależnościa̧ tejże dlugości od parametrów
użytych imersyjnych “soczewek” elektromagnetycznych.
PL ISSN 0459-6854
BULLETIN
DE
DES
LETTRES DE L
ÓDŹ
2010
Vol. LX
no. 1
pp. 155–174
Roman Stanislaw Ingarden and Julian L
awrynowicz
FINSLER-GEOMETRICAL MODEL OF QUANTUM
ELECTRODYNAMICS I
EXTERNAL FIELD vs. FINSLER GEOMETRY
Summary
After summarizing the physical demands caused by the necessity of including open
systems, we study quantum Dirac-Maxwell equations using a complex-analytical approach
and convolution equations. Then we pass to a more general case of Yang-Mills equations
in the presence of an external field, including the cases of an arbitrary symmetry within
SO(m) or SU(m), the global case, a non-abelian generalization, and a generalization of the
Lagrangian and its embedding in the electroweak model. In the second part of the paper we
shall deal with ferroelectric crystals in a Finsler geometry and the physical interpretation
of solenoidal and nonsolenoidal connections on the canonical principal fibre bundles.
1. Basic physical demands
In analogy to a particular case of an electromagnetic microscope [IL3], and classical
methods of electrodynamics, in the variational procedure involved, we deform in the
first step the electron trajectories, and in the second step the potentials. If we include
thermodynamical effects [L3] or electroweak effects [KBG], we have an additional
motivation of involving a five-dimensional, Kaluża-Klein-like geometry.
Independently [L5], in connection with Kaluża-Klein theories (e.g. [L3]), Beil
[Be1, 2] has shown that the U (1)-symmetry of the electromagnetic field yields a
gauge transformation of the form
1
Yνμ = δνμ − B −2 1 − (1 + kB 2 ) 2 Bμ Bν ,
Yν∗μ = δνμ − B −2
1
1 − (1 + kB 2 )− 2
Bμ Bν ,
156
awrynowicz
where
B2 =
ηαβ B α B β ,
α,β
ηαβ is the initial base space metric in the Lorentz form, an k is, in general, velocity
dependent. The resulting metric
gμν = ημν + kBμ Bν
is, in general, Finslerian, even in the case where k is a universal constant related to
the gravitational constant.
In this direction Kerner [K] had proposed a nonlinear generalization of electrodynamics derived from Kaluża-Klein theory in five dimensions. It is based on
investigation of a Gauss-Bonnet-type addend
Rαβγδ Rαβγδ − 4Rαβ Rαβ + R2 ,
which is no more a topological invariant in five dimensions. When added to the
Einstein-Hilbert Lagrangian, it leads to non-trivial equations of motion of the second
order.
As we already know [L6], the first of us had proposed to consider a Randers
space M with metric F and a Lorentz nonlinear connection N or a Cartan nonlinear
connection N C :
(1)
(M, F, N )
or (M, F, N C ).
Since we are interested in including thermodynamical and electroweak effects, here
we allow (M, F ) to be Finslerian. Now the concept consists in considering the twoelement sequence of mappings
(c, sc, sc)
(c, c, s)
(s, sc, c)
m
c
=⇒
=⇒
= (F0 , N or N c , V ) A = (F# , N or N c , V0 ) B = (F# , N# , V# )
(2)
with
s = F0 – a (relatively simple) Finsler metric,
c = F# – a (relatively complicated) Finsler metric,
sc = N – a Lorentz nonlinear connection or
N C – a Cartan nonlinear connection,
c = N# – a (relatively complicated) nonlinear connection,
s = V# – a (relatively simple) potential corresponding to an external field,
sc = V0 – a (slightly complicated) potential corresponding to an external field,
c = V – a (relatively complicated) potential corresponding to an external field.
Moreover, as in the case of thermodynamics [L6], we are interested in open (nonisolated) systems. If we come over to quantum electrodynamics of the so-called
second quantization, the wave function (spinor) ψ (say) becomes an operator (observable) in the Heisenberg picture (depending on time), while the wave function
(state) ϕ (say) of the whole field is independent of time. Such a treatment enables
one to calculate (after a renormalization for avoiding infinities) only mean values of
Finsler-geometrical model of quantum electrodynamics I
157
these observables for a given time (also correlations for different times) of the whole
field [I]. As it is well known (e.g. [LL], Sections 62–63), the number of particles N
(say) becomes also an observable with integer nonnegative eigenvalues, as it is usual
when following the second quantization (in the non-relativistic case).
In this more ambitious theory, the concept of a particle as an independent and
constant physical object actually disappears in contradiction to the first quantization
theory: particles are created and annihilated as excitations of the whole field and
the number of particles is, in general, not preserved. However, the total electric
charge of the field is preserved as the difference of the positive and negative charges
of positrons and electrons. Moreover, special quantum particles and their quantum
agglomerations, as atoms, molecules, gases, fluids, and solid states, can sometimes
exist, in special conditions, for billions of years. Atoms and molecules are examples
of open systems, more complicated are maser and laser systems.
2. A generalized Dirac-Maxwell system
Let (M, g) be a curved space-time: a C ∞ -differentiable paracompact connected fourdimensional manifold endowed with a pseudoriemannian metric g: a symmetric C ∞
tensor field of type (0, 2) which is nondegenerate and has at each point the index
1. (It is not difficult to generalize the staff of Sections 2–4 to the case of locally
Minkowskian Ingarden spaces; cf. [L6], pp. 121–123 and papers quoted therein. It
seems to us better to generalize first the staff to the Yang-Mills system.) Let ψ :
M → C4 be a spinor, i.e. a C ∞ -differentiable function which is supposed to satisfy
the generalized Dirac equation in the sense of [Sc] and [L1, 3], in presence of its
C ∞ -differentiable self-electromagnetic field A : M → R4 :
(3)
A = j,
Dψ = 0,
1
2
3
Div A = 0.
0
In a local co-ordinate system (x , x , x , x ) we can express the generalized Dirac
equation as
(4)
D = γ̂ k (∂/∂xk ) + ieAk + Γk + m,
where m = m0 c/ and e = e0 /c, m0 and e0 denoting the rest mass and electric
charge of the particle in question, respectively. The symbols γ̂ k denote the Dirac
matrices obtained by the commutation rules
γ̂ j γ̂ k + γ̂ k γ̂ j = 2g jk I4 ,
and Γk are the spinor connections (the generalized Christoffel symbols), determined
by
1 1
Γk = ψ j (∂/∂xk )γ̂j − γ̂ Γjk − Tr γ̂γ̂ j (∂/∂xk )γ̂j γ̂
4
32
with
1
εjkrs γ̂ j γ̂ k γ̂ r γ̂ s ,
γ̂k = gjk γ̂ k and γ̂ =
24
158
awrynowicz
where [εjkrs ] denotes the totally antisymmetric Levi-Civita tensor and Γjk denote
the usual Christoffel symbols
∂
1
∂
∂
g
+
g
−
g
(5)
grs Γrjk =
sk
js
jk .
2 ∂xj
∂xk
∂xs
For a study of the operator D we refer to [LW2, S]; in case of the Minkowski
space-time we may take the representation (cf. [L3]):
γ̂ μ = −iαμ , μ = 1, 2, 3, γ̂ 0 = −iγ0 ;
0 σμ
I2
0
αμ =
,
,
γ0 =
σμ 0
0 −I2
σμ denoting the familiar Pauli matrices.
The symbol j in (3) denotes the current generated by ψ:
1 2 3 0
, γ̂jk , γ̂jk , γ̂jk ,
j = eψ̄ j γ̂jk ψ k , γ̂jk = γ̂jk
where ψ̄ j stands for the complex conjugate of ψ j . The divergence Div and the
Laplace-Beltrami operator can be expressed in a local co-ordinate system as
Div A = (∂/∂xj )uj + uk Γjjk
and Ak = Div(Grad Ak ),
where
Ajk = g j (∂/∂x )Ak .
The system (3) satisfies a compatibility condition being the continuity equation
Div j = 0 which can be checked directly. The choice (4) for the generalized Dirac
operator is not the only reasonable possibility. For instance one may take [LW1]:
(6)
D = gjk γ̂ j {g k [(∂/∂x ) + ieAk + Γk ] + δ k0 m},
where δk denotes the Kronecker symbol.
3. A complex-analytical approach
Let pk = i[(∂/∂xk ) + ieAk ]. In case of the usual Minkowski space-time, following
[GLW1]. we write the Dirac equation in (3) as the system
(p1 − ip2 )ψ 0 + p3 ψ 3 + (p0 − m)ψ 1 = 0,
(p1 + ip2 )ψ 3 − p3 ψ 0 + (p0 − m)ψ 2 = 0,
(p1 − ip2 )ψ 2 + p3 ψ 1 − (p0 + m)ψ 3 = 0,
(p1 + ip2 )ψ 1 − p3 ψ 2 − (p0 + m)ψ 0 = 0,
which is equivalent to
(p1 + ip2 ) (ψ 1 + ψ 3 ) = (p3 + m) (ψ 2 + ψ 0 ) − p0 (ψ 2 − ψ 0 ),
(p1 − ip2 ) (ψ 2 + ψ 0 ) = −(p3 − m) (ψ 1 + ψ 3 ) − p0 (ψ 1 − ψ 3 ),
(p1 + ip2 ) (ψ 1 − ψ 3 ) = (p3 − m) (ψ 2 − ψ 0 ) + p0 (ψ 2 + ψ 0 ),
(p1 − ip2 ) (ψ 2 − ψ 0 ) = −(p3 + m) (ψ 1 − ψ 3 ) + p0 (ψ 1 + ψ 3 ).
159
In the case of an arbitrary curved space-time with metric g we replace the above
system by a system of the form
1
a1 (∂1 + ieA01 ) + ia12 ∂2 + ieA02 ϕ1 = a13 ∂3 + a10 ∂0 + c1 (e, m) ϕ2
+b1 (∂ 0 + ieA00 )ϕ0 ,
(7)
+b1 (∂ 0 + ieA00 )ϕ3 ,
+b2 (∂ 0 + ieA00 )ϕ2 ,
(8)
+b2 (∂ 0 + ieA00 )ϕ1 ,
where the real-valued functionals ajk , bj , and cj (e, m) depend, in general, on g, but
do not depend on the operators ∂ k = ∂/∂y k and, moreover, ajk and bj do not depend
on the normalized electric charge e and the normalized mass m.
We require that
(9)
ajk [g] = 1
and bj [g] = 0
for k = 1, 2,
for a suitable representation of the Dirac matrices γ̂ k , together with
(10)
ΔA0 = j0 ,
div A0 = 0,
div j0 = 0,
where div and Δ are the two-dimensional analogues of Div and ; j0 standing for
the two-dimensional current generated by the spinor (ϕ1 , ϕ2 )T (T – transposed)
satisfying the generalized Dirac equation (7) with restrictions (9), (10), and
(11)
ajk [g] = 0 for
k = 3, 0 and cj (e, m) = 0.
Therefore we suppose that in each co-ordinate neighbourhood on M the fourteen
real-valued functions
gjk ,
0 ≤ j ≤ k ≤ 3;
A01 , A02 , f10 , f20
satisfy the fourteen real differential equations (9) and (10). We call (9) and (10) the
saparability conditions for the Dirac-Maxwell system (3), equivalent to the system
of equations (7), (8), and
(12)
A = j,
Div A = 0.
The further procedure with equations (7), (8), and (12) may start with considering the Fourier transform with respect to the variable e. Denote by ε the conjugate
variable and set
Z = ∂ 1 − i∂ 2 + 2 A01 − iA02 (∂/∂ε) , Z̄ = ∂ 1 + i∂ 2 + 2 A01 + iA02 (∂/∂ε).
160
awrynowicz
Then the system of (7) and (8), subjected to conditions (9) and (11), implies
(13)
Z̄ ϕ̂1 = 0,
Z ϕ̂2 = 0,
Z̄ ϕ̂3 = 0,
Z ϕ̂0 = 0,
whereˆstands for the partial Fourier transform with respect to ε.
Now, for every co-ordinate neighbourhood on M , let us introduce the space of
the two complex variables (y 1 + iy 2 , ε + iϑ), where ϑ is the complex conjugate of ε.
Consider the hypersurface
Z = y 1 + iy 2 , ε + ϑ ∈ C2 : ϑ = F (y 1 , y 2 ), F independent of ε .
By (10), div A0 = 0, so we can choose a real-valued function F such that
A01 = ∂/∂y 2 F, A02 = −(∂/∂y 1)F
(14)
and hence
1 0
A + iA02 = −i(∂/∂ z̄)F, where z = y 1 + iy 2 .
2 1
¯
In consequence, Z̄ and Z can be indentified with the ∂-tangential
and ∂-tangential
on S, respectively.
Choose now ϕ̂k so that
(15)
ϕ̂k = Φk |S,
k = 1, 2, 3, 0,
Φk : C → C being holomorphic for k odd and antiholomorphic for k even. Then we
get
Lemma 1. With the previous notation, the mapping (15) provides a solution to the
system (13) for any choice of the potentials A01 and A02 .
4. Convolution equations
Before investigating the general system of equations (7), (8), and (12) subjected to
conditions (9) and (10), we concentrate on the equations (10). Let G denote the
Green operator in two dimensions and the asterisk * the convolution. Then we can
express the potentials A01 and A02 as
(16)
A01 = G ∗ f10 ,
A02 ∗ f20 .
The procedure, originally due to [GLW1] and generalizing that of [GL], follows
the same main idea as the method of generating functions used in [IL3] for a model
of magnetic electron microscope. On the other hand, by (10), we have div j0 = 0.
Consequently, relations (14) and (16) yield
dF = −A02 dy 1 + A01 dy 2 = G ∗ −f20 dy 1 + f10 dy 2 .
In turn we consider four functions Ψk : C2 → C, holomorphic for k odd:
(17)
Ψk = Ψk y 1 + iy 2 , ε + iϑ , k = 1, 3,
and antiholomorphic for k even:
Ψk = Ψk y 1 − iy 2 , ε − ϑ ,
(18)
k = 2, 0,
161
and calculate their Fourier transforms in the variable ε:
Φk = Φk y 1 + iy 2 , e, ϑ , k = 1, 3,
(19)
Φk = Φk (y 1 − iy 2 , ε, ϑ),
k = 2, 0,
respectively. This yields the differential form
1
2
2
1
dJ(y 1 , y 2 , e, ϑ) = Φ Φ2 + Φ Φ1 dy 2 − i Φ Φ1 − Φ Φ2 dy 1
(20)
and we are led to solve the equation
dF (y 1 , y 2 ) = (G ∗ dJ) (y 1 , y 2 , F (y 1 , y 2 )).
(21)
Equation (21) implies
(22)
F (y01 , y02 ) =
2
1 2
2
1
K y 1 , y 2 , y01 , y02
Φ Φ + Φ Φ1 − i Φ Φ1 − Φ Φ2 dy 1 dy 2
C
with [GL]:
1
K(y , y
2
, y01 , y02 ) =
1
0
(23)
−
z0 − z z0 z̄ + z z̄0 z0 − z Ln|tz0 − z|dt = −1 + ln ln −
z 4|z0 |2
z 4|z0 |2 |z|2 + (z0 z̄ + z z̄0 )
|z0 |2 [4|z0 |2 |z|2 − (z0 z̄ + z z̄0 )]1/2
|z0 |2
×Arctan
.
2
2
2
2|z0 | |z| − |z0 | (z0 z̄ + z z̄0 ) − (z0 z̄ + z z̄0 )2
Therefore the solution to the system of equations (7), (8), and (10), subjected to
conditions (9) and (11), is given by the formulae (14) and
for k = 1, 3,
ϕk0 = Φk y 1 + iy 2 , e, F (y 1 , y 2 )
(24)
ϕk0 = Φk y 1 − iy 2 , e, F (y 1 , y 2 )
for k = 2, 0,
where F and K are calculated in (22) and (23), respectively.
Finally we come back to the general system of equations (7), (8), and (12) with
conditions (9) and (10). We express any solution of this system in the form
(25)
ϕ1 − ϕ10 = Y1 ϕ2 − ϕ20 , ϕ2 − ϕ20 = Y1−1 ϕ1 − ϕ10 ,
(26)
where
(27)
ϕ3 − ϕ30 = Y2 ϕ0 − ϕ00 ,
ϕ0 − ϕ00 = Y2−1 ϕ2 − ϕ20 ,
Yj [Ψ] = Yj y 1 , y 2 , y 3 , y 0 , e, m, ψ(y 1 , y 2 , y 3 , y 0 , e, m) ,
j = 1, 2.
162
awrynowicz
The transformation Y1 is determined by inserting the expressions (25) into equations
(7) and, similarly, Y2 is determined by inserting (26) into (8). Hence we have proved
Theorem 1. Let (M, g) be a C ∞ -differentiable four-dimensional pseudoriemannian
manifold with metric g of index 1. Let further ψ : M → C1 be a C ∞ -differentiable
solution to the system of equations (3) with the operator D given in a local co-ordinate
system (x1 , x2 , x3 , x0 ), by (4), where the vector field (potential) A is supposed to be
C ∞ -differentiable.
(i) Then this system is equivalent to the system of the form (7), (8), and (12)
under the conditions (9) and (10), where the real-valued functionals ajk , bj , cj (e, m)
depend, in general, on g, but do not depend on the operators ∂ k = ∂/∂y k and, in
addition, ajk and bj do not depend on the parameters e and m.
(ii) Moreover, any solution of the system can be expressed in the form (25) and
(26) with Yj as in (27), where ϕk0 , F , and K are given by (24), (22), and (23),
respectively, the transform (19) are four partial Fourier transforms with respect to
the variable ε of arbitrary four corresponding functions (17) and (18), holomorphic
for k odd (k = 1, 3) and antiholomorphic for k even (k = 1, 0). The transformation Y1
is determined by inserting the expressions (23) into the equations (7) and, similarly,
Y2 is determined by inserting (24) into (8).
The above method has an advantage of being applicable for certain curved spacetimes because its crucial point of applying the global Fourier transform together
with suitable convolution equation concerns the electric charge e treated as a real
variable whose conjugate variable ε is considered as the real part of the complex
variable ε + iϑ. In fact, this means that the space-time in question is treated as the
typical fibre in the fibre bundle [St] generated by the spectrum of all possible electric
charges. For further generalization we refer to
5. A generalization: Yang-Mills system in the presence
of an external field
We are going to derive rigorously the system of local Yang-Mills equations in the case
of an arbitrary (pseudo)riemannian metric or, differently speaking, the Yang-Mills
equations in the presence of external fields.
Let su(2) be the Lie algebra corresponding to SU(2). By a Yang-Mills field we
mean any vector foeld A = (Ak ), k = 1, 2, . . . , n, in an open set U in Rn , with values
in su(2); [AL], pp. 12–13. Let further T be a representation of SU(2) in a vector space
M and let Ψ : U → M be a C ∞ -mapping. Then the covariant derivative of the vector
field Ψ is given by ∇k Ψ = (∂/∂xk )Ψ + t(Ak )Ψ, t being the representation of su(2)
corresponding to T and x = (xk ), k = 1, 2, . . . , n, denoting a co-ordinate system in
U . It satisfies the identity (Vj ◦ Vk − Vk ◦ Vj )Ψ = t(Fjk )Ψ, where
(28)
Fjk = ∂/∂xj Ak − ∂/∂xk Aj + [Aj , Ak ].
163
If SU(2) is replaced by SU(1), the [Aj , Ak ] = 0 and [Fjk ] is the electromagnetic
tensor.
Next, let us consider a C ∞ -function ρ : U → SU(2) and the following gauge
transformation of the pair (Ψ(x), Ak (x)), x ∈ U :
(29)
(30)
Ψ(x) → Ψ (x) = T (ρ(x))Ψ(x),
Ak (x) → Ak (x) = ρ(x)Ak (x)ρ−1 (x) − (∂/∂xk ρ(x) ρ−1 (x).
With each functional of the form (29)–(30) we may associate the functional [GH]:
L∼ (Ψ, A) = L [Ψ(x), (∇j Ψ(x))] dx + L[A],
(31)
where L[A] is a volume integral involving the Killing form acting on [Fjk ]:
1
L[A] = −
(32)
g js g rk Tr (Fjr Fsk ) dV ;
4
here g = [gjk ] is a pseudoriemannian (in particular, Riemannian) tensor on U , and
dV is the volume element in the manifold (U, g). We observe that (31) is invariant
with respect to (29)–(30). Therefore L∼ expresses the action of the vector field Ψ on
the Yang-Mills field A, which may be interpreted as a compensation field, whereas
L expresses the action of a free compensation field.
In the case of Minkowski space-time we have
g jj = −1 for
j = 1, 2, 3,
g 00 = 1, g jk = 0
for j = k;
j, k = 1, 2, 3, 0,
and the extremals for L satisfy the Yang-Mills equations [1], pp. 12–13:
(33)
∇k Fjk = 0,
with
(34)
∇k = g k ∇ ,
j = 1, 2, 3, 0,
∇ Fjk = ∂/∂x Fjk + [A , Fjk ].
We have [GKL]:
Lemma 2. Suppose that M# is an n-dimensional compact orientable (pseudo)riemannian manifold with metric g of index 0 or 1. Let su(2) be the Lie algebra corresponding to SU(2) and A = (Ak ) a C ∞ Yang-Mills field in a co-ordinate neighbourhood U of M# . Further, in a co-ordinate system (xk ) in U , let Fjk be given by (28).
Consider the functional (32), where dV is the volume element in U . If the functional
attains its stationary value for some Yang-Mills field A, then this field satisfies the
generalized Yang-Mills equations
(35)
(DivF )j + Ak , F jk = 0, j = 1, 2, . . . , n,
where
(36)
(37)
(DivF )j = g sk (∂/∂xs )Fkj − Γrsk Frj ,
F jk = g jr g sk Frs ,
and Γrsk are the usual Christoffel symbols.
Fkj = g j Fk ,
164
awrynowicz
Proof. Denote by Fjr , Fsk the Killing form −Tr(Fjr Fsk ) in (32) and calculate
the first local Gâteaux variation δL[A]; this means that the supports of δAj , j =
1, 2, . . . , n, are compact:
1
1
g js g rk (δFjr , Fsk + Fjr , δFsk ) dV = −
g js g rk δFjr , Fsk dV.
δL[A] = −
4
2
Yet,
∂
∂
δFjr =
δAr − r δAj + [δAj , Ar ] + [Aj , δAr ] .
j
∂x
∂x
Thus, integrating the integrand by parts under the assumption that the supports of
δAj are compact, we obtain
(38)
δL[A] =
10
δk ,
k=1
where, with the notation |j = ∂/∂xj and dVeucl = (±detg)−1/2 dV ,
1
δ1 =
g js g rk δAr , (∂/∂xj )Fsk (±detg)1/2 dVeucl ,
2
1
js rk
δ2 =
g δAr , Fsk (±detg)1/2 dVeucl ,
g|j
2
1
rk
δ3 =
δAr , Fsk (±detg)1/2 dVeucl ,
g js g|j
2
1
g js g rk δAr , Fsk (∂/∂xj )(±detg)1/2 dVeucl ,
δ4 =
2
1
δ5 = −
g js g rk δAj , (∂/∂xr )Fsk (±detg)1/2 dVeucl ,
2
1
δ6 = −
g js g rk δAj , Fsk (±detg)1/2 dVeucl ,
2
1
δ7 = −
g js g rk δAj , Fsk (±detg)1/2 dVeucl ,
2
1
δ8 = −
g js g rk δAj , Fsk (∂/∂xr )(±detg)1/2 dVeucl ,
2
1
δ9 = −
g js g rk [Aj , δAr ]Fsk (±detg)1/2 dVeucl ,
2
1
δ10 = −
g js g rk [δAj , Ar ]Fsk (±detg)1/2 dVeucl .
2
By the properties of the Killing form, we have
1
δ10 =
g js g rk δAj , [Ar , Fsk ](±detg)1/2 dVeucl
2
1
g js g rk δAr , [Fsk , Aj ](±detg)1/2 dVeucl
=−
2
1
=
g js g rk [δAr , Aj ], Fsk (±detg)1/2 dVeucl = δ9 .
2
165
Besides,
δ5 = δ1 ,
δ7 = δ2 ,
δ6 = δ3 ,
δ8 = δ4 .
Yet, with the notation Δjs = g js det g, we get
∂
∂
∂
1
1
(±detg)1/2 = (±detg)1/2 g js r gjm = − (±detg)1/2 gjs r g js ,
∂xr
2
∂x
2
∂x
where the latter equality follows from the identity g js gjs = 4. Next we express the
derivatives of g js in terms of g and the related Christoffel symbols:
js
= −Γjqr g qs − Γsqr g jq .
g|r
(39)
Hence
(40)
1/2
(∂/∂xr ) (±detg)
1/2
= (±detg)
Γqqr ,
so the corresponding addends δk become:
1
g js Γrqr g qk − Γkqr g rq δAj , Fsk (±detg)1/2 dVeucl
δ7 = δ2 =
2
1 js qk r
=
g g Γqr + g js g rq Γkqr δAj , Fsk (±detg)1/2 dVeucl ,
2
1 rk qs j
δ6 = δ3 =
g g Γqr + g rk g jq Γsqr δAj , Fsk (±detg)1/2 dVeucl ,
2
1
δ8 = δ4 = −
g js g rk Γqqr δAj , Fsk (±detg)1/2 dVeucl .
2
Consequently, we obtain
δ1 + δ5 + δ9 + δ10 = − g js g rk δAj , Δr Fsk (±detg)1/2 dVeucl ,
δ2 + δ3 + δ4 + δ6 + δ7 + δ8 = 2(δ2 + δ3 + δ4 ) = Rjsk δAj , Fsk (±detg)1/2 dVeucl
with ∇Fsk given by (34) and
Rjsk = g js g rq Γkqr + g rk g qs Γjqr + g rk g jq Γsqr .
Since δAj , j = 1, 2, . . . , n, are arbitrary and our Killing form is nondegenerate, by
(36) and (37) the relation δL[A] = 0 implies
(41)
g js g rk ∇r Fsk = Rjsk ,
j = 1, 2, . . . , n.
We need to prove that the systems (41) and (35) are equivalent. For this we
introduce the notation (38) and apply again formulae (39). In this direction from
(41) we deduce that
g rk (∂/∂xr )Fkj + Ar , Fkj
= g rq Γkqr Fkj + g rk Γjqr Fkq − g jq Γsqr Fsr + g rk [(∂/∂xr )g rs ]Fsk
= g rq Γkqr Fkj + g rk Γjqr Fkq − g jq Γsqr Fsr + g qs Γjqr Fsr − g jq Γsqr Fsr
166
awrynowicz
and distinguish in a natural way five addends a1 , . . . , a5 in the expression obtained.
Evidently, a5 = −a3 . Since F rq = g sq Fsr equals −F qr = g rk Fkq , we also get a4 = −a2 .
Hence
g rk (∂/∂xr )Fkj + Ar , Fkj = g lq Γkqr Fkj , j = 1, 2, . . . , n.
and, consequently,
g rk (∂/∂xr )Fkj − Γqrk Fqj + g rk Ar , Fkj
(42)
= 0,
j = 1, 2, . . . , n.
Therefore we arrive indeed at (35) with the notation (36) and (37). Of course we
can also proceed in the opposite way, and this completes the proof.
6. The case of an arbitrary symmetry within SO(m) or SU(m)
We proceed to generalize our staff to any symmetry within SO(m) or SU(m), m =
2, 3, . . . . Take an arbitrary compact subgroup G of SO(m) or SU(m) and consider
a G-vector bundle E = (E, π, M# ) over the base space M# – an n-dimensional
compact orientable (pseudo)riemannian manifold with metric g, E standing for the
bundle space and π : E → M# for the projection. We include the case of complex,
in particular, holomorphic vector bundles. It is well known that it exists a covering
π = {Uj , j ∈ I} of M# with local frames over Uj such that the corresponding
transition matrices have their values in SU(2). Then a connection N# on E is called
a G-connection if for every local frame in question the connection matrix ω has its
values in the Lie algebra G corresponding to G. It is well known (e.g. [W]) that,
with the help of a partition of unity, for a given G-vector bundle we can show the
existence of a G-connection.
For applications in quantum electrodynamics it is worth-while to distinguish the
notion of a G-vector field defined as any vector field A on M# with values in G.
Hence a Yang-Mills field is an SU(2)-vector field. A G-connection N# is called the
connection corresponding to a G-vector field if in any local frame belonging to the
G-structure the curvature form corresponding to A satisfies the differential equation
(43)
δF + 2Trg (A ⊗G F) = 0,
where ⊗G denotes the G-dependent tensor product operator and, locally,
(44)
F = Fjk dxj ∧ dxk ,
A = A dx ,
with Fjk and A given by (28) and A = (Aj ), respectively, and ∧ denoting the wedge
product operator. These definitions are motivated by
Lemma 3. Suppose M# , g, G, G and E are as before. Then the system of differential
equations (35) with the notation (36) and (37) is well-posed and equivalent to (43).
Proof. By the relations (42) and
(45)
(gjs )|r = Γqrs gjq + Γrj gqs ,
167
the latter being analogous to (39), the equations (35) with the notation (36) and
(37) become
g rk [(∂/∂xr )Fsk − Γqrk Fsq ] + g rk [Ar , Fsk ] = 0,
s = 1, 2, . . . , n.
Therefore
g rk (∂/∂xr )Fsk − Γqrk Fsq − Γqrs gjq Fkj − Γqrj gqs Fkj + g rk [Ar , Fsk ] = 0.
Thus
g rk (∂/∂xr )Fsk − Γqrk Fsq − Γqrs Fqk − Γqrj gqs Fkj + g rk [Ar , Fsk ] = 0.
Yet, by the anitisymmetry of F rj ,
g rk Γqrj gqs Fkj = Γqrj gqs F rj = Γqrj F rj gqs = 0,
and hence
−g rk (∂/∂xr )Fsk − Γqrk Fsq − Γqrs Fqk + g rk [Ar , Fks ] = 0,
s = 1, 2, . . . , n.
By the definitions of the first Gâteaux variation δ and the G-structure-dependent
tensor product ⊗G , the above system is identical with (43), as desired.
7. The global Yang-Mills system
We are now prepared to derive the global Yang-Mills system which depends on an
arbitrary (pseudo)riemannian metric g and an arbitrary non-abelian compact Lie
group G [GKL]:
Theorem 2. Take an arbitrary compact subgroup of SO(m) or SU(m) and consider
a real or complex G-vector bundle E = (E, π, M# ) over the base space M# – an
n-dimensional compact orientable (pseudo)riemannian manifold with metric g of
index 0 or 1, E standing for the bundle space and π : E → M# for the projection.
Suppose further that, in a local frame,
(46)
F = dA + A ∧ A,
A = Ak dxk ,
where (X k ) is a co-ordinate system in a co-ordinate neighbourhood of M# and ∧
denotes the wedge product operator. Consider the functional
1
L[A] = −
(47)
Tr(F ∧ ∗g F),
4
M#
where A is the G-vector field of class C ∞ on M# such that, locally, A = (Ak ) with
A = Ak dxk , and ∗g stands for the g-dependent Hodge ∗-operator. Finally, suppose
that the functional (47) attains its stationary value for some G-vector field A. Let
P (M# , G) be the bundle of orthonormal frames of E, endowed with the connection
N0 induced by a given G-connection N# of E, corresponding to A. Then the field A
satisfies on M# the system of generalized Yang-Mills equation
(48)
D(G Ω2 ) = 0,
168
awrynowicz
where D is the covariant derivative operator related to N0 ,
G : Ap (P ) → An−p (P ),
(49)
Aν (P ) being the modulus of horizontal ν-forms on P , of the type adG, and where
Ω2 stands for the curvature form corresponding to A.
Proof. By Lemmas 2 and 3, the G-vector field A satisfies the differential equation (42).
Denote by ω the connection matrix corresponding to A. Then for any cross-section
σ of P (M# , G) we have
σ ∗ Ω2 = F,
(50)
σ ∗ ω = A,
where F and A are locally given by (46), and also
(51)
D(G Ω2 ) = d(G Ω2 ) + ω ∧ G Ω2 .
Therefore
σ ∗ D(G Ω2 ) = σ ∗ d(G Ω2 ) + (σ ∗ ω) ∧ (σ ∗ G Ω2 ) = dσ ∗ (G Ω2 ) + (σ ∗ ω) ∧ (σ ∗ G Ω2 ).
It can be easily seen that σ ∗ G = ∗g σ ∗ , so relation (51) gives
(52)
∗g σ ∗ D(G Ω2 ) = (∗g d∗g )σ ∗ Ω2 + ∗g (δ ∗ ω) ∧ ∗g (δ ∗ Ω2 ) .
Consequently, by the definition of the codifferential operator: δg = ∗g d∗g and relations (50), the condition (52) takes the form
(53)
∗g σ ∗ D G Ω2 = δg F + ∗g (A ∧ ∗g F).
In order to prove that the global differential equation (43) is equivalent to
∗g σ ∗ D G Ω2 = 0,
(54)
it suffices to verify the relation
∗g (A ∧ ∗g F) = 2Trg (A ⊗G F).
(55)
For this, let us consider an orthonormal system (eq ), q = 1, 2, . . . , m, of vector fields
in any co-ordinate neighborhood u of M# , mentioned in Lemma 2. Consider also
the system (e∗j ), j = 1, 2, . . . , n, of one-forms on U such that e∗j [ek ] = δjk for each
(j, k). Then from (44) we infer
1
(56)
F = f jk e∗j , A = a e∗
2
and, furthermore,
1 jk
1
∗
∗
f ∗g (e∗j ∧ e∗k ) = f jk εrs
jk er ∧ es ,
2
4
where εrs
jm stands for the totally antisymmetric Levi-Civita tensor, and
1 q jk rs ∗
1 q jk rs q ∗
a ,f
a ,f
εjk er ∧ e∗s ∧ e∗q , ∗g (∗g F ∧ A) =
εjk εrs e ,
A ∧ ∗g F =
4
4
where
q q
q
εrs
jk εrs = 2 δj δk − δj δk
∗g F =
169
and δjk denotes the Kronecker symbol. Therefore
1
· 2 · 2 ar , f q δrq e∗ = 2Trg (A ⊗G F),
4
δjk being again the Kronecker symbol. Consequently, the global differential equation
(43) is indeed equivalent to (54).
The system of differential equations (54) holds, in particular, for any local crosssection σ on P (M# , G). Yet, the Hodge operator ∗g is an isomorphism, so the Gvector field A satisfies on M# the system of differential equations (48), as desired.
∗g (A ∧ ∗g F) =
8. An SU(2)-based non-abelian generalization
In Sect. 7 our non-abelian generalization was related with consideration of an arbitrary non-abelian compact Lie group G. Now let us concentrate, following [BI1, 2],
on finding non-abelian solutions of finite mass to the Yang-Mills equations. Let us
introduce the following generalization of the U(1)-action:
1 α μν
1 α μν 2
1 2
Fβγ Fα
β
(57) S =
Fα −
,
(1 − )d4 x with = 1 + 2 Fβγ
4π
2β
16β 4
where the constant β appears for dimensional reasons, meaning essentially the limiting value of the electric field in Mie’s nonlinear electrodynamics [Mi]. The nonlinearity breaks the conformal symmetry and the stress-energy tensor [Tνμ ] has the
non-zero trace:
α
Fαμν = 0.
Tμμ = −1 4β 2 (1 − ) − Fμν
The trace vanishes as β → 0, and the approach reduces to the standard one.
We assume that
Aα
0 = 0,
k
Aα
j = εαjk (n /r)[1 − w(r)],
where
nk = xk /r,
r=
x2 + y 2 + z 2 ,
and the function w is real-valued. Integration in (57) over the sphere gives a twodimensional action from which we eliminate the constant β by the co-ordinate rescal√
√
ing βt → t, βr → r. Then we obtain
w 2
(1 − w2 )2
S = r2 (1 − )dr with = 1 + 2 2 +
,
r
r4
where stands for the differentiation with respect to r. The nonlinearity appears
α
] on the potentials Ajμ .
here because of the nonlinear dependence of the tensor [Fμν
The corresponding equation of motion reads [KBG]:
(58)
(w /) = (1/r2 )w(w2 − 1).
170
awrynowicz
For time-dependent configurations the energy density is equal to minus the Lagrangian, so the total energy (mass) is given by
∞
M [w] = (2 − 1)r2 dr.
0
In particular,
2
∞ 3
1 √ 3
2
4
M [0] =
r + 1 − r dr =
π
≈ 1.236
Γ
3
4
0
corresponds to the point-like magnetic monopole with the unit magnetic charge
being an embedded U(1)-solution. In order to assure the convergence of M [w], the
ratio of its integrand to r has to tend to zero as r → ∞. Then the equation (58)
should reduce to the ordinary Yang-Mills equation, equivalent to the autonomous
system
(59)
ẇ = u,
u̇ = u + (w2 − 1)w,
where the dot stands for differentiation with respect to τ = ln r. We have thus constructed a dynamical system with three non-degenerate stationary points (u, w) =
(0, −1), (0, 0), (0, 1), where (0, 0) is a focus and two others are saddle points with
eigenvalues λ = −1, 2. The separatrices along the directions λ = −1 go from ∞ to
the focus with eigenvalues
√
√
1
1
λ = (1 − 3i), (1 + 3i)
2
2
passing through the saddle points.
Following [GK], let us consider finite-energy configurations with nonvanishing
magnetic charge. Such solutions have w = 0 asymptotically which does not correspond to bounded solutions unless w ≡ 0, or w = −1, 0, 1 asymptotically, and this
corresponds to zero magnetic charge. Consequently, the only finite-energy configurations with nonvanishing magnetic charge are the embedded U(1)-monopoles. In
terms of the variable r, (58) implies
w = −1 + cr−1 + O(r−2 )
or w = 1 + cr−1 + O(r−2 ),
where c is a constant. Hence the corresponding integral M [w] converges as r → ∞.
The values w = −1, 1 correspond to two neighbouring topologically distinct YangMills vacua.
It seems important to consider the local solutions of (58) near r = 0. If M [w]
converges, then w tends to a finite limit as r → 0. By (58) the only allowed limiting
values are w = −1, 1. By the symmetry of (59) under reflection w → ±w, without
any loss of generality we may take w(0) = 1, and then from (59) we get
w = 1 − br2 +
b2 (44b2 + 3) 4
r + O(r6 ),
10(4b2 + 1)
171
where b is a constant. At r → 0, → 1 + 12b2 + O(r2 ), so this time we have not a
solution of the initial system (58). It remains to find proper values of b to smooth
finite-energy solutions by gluing together the two asymptotic solutions between 0
and ∞. This is effectively done in [GK, KBG].
9. A generalization of the Lagrangian and its embedding
in the electroweak model
It seems instructive to compare the pure electromagnetic (abelian) Lagrangian of
Born and Infeld, used in Sect. 8, with what can be extracted from its non-abelian
version based on the symmetry group G = SU(2)× U(1) after defining physical fields
as linear combinations of the U(1) and SU(2) gauge fields; see [KBG] for details.
Write the non-abelian generalizations P ans S of Maxwell’s tensor invariants P
and S as
1
P = Fμν F μν , S = Fμν F̃ μν = εμνρσ Fμν Fρσ ,
2
where
a
Fμν = Fμν
Ja ; a = 0 for U(1), a = 1, 2, 3 for SU(2).
We get
1 a aμν
0
LG = 2aFμν
F 0μν + aFμν
F
2
1 a aμν c cρσ
0
0
+β −2 M 2 b 2Fμν
F 0μν Fρσ
F 0ρσ + Fμν
F
Fρσ F
8
0
a
0
0
0
0
F̃ 0μν Fρσ
F̃ 0ρσ
+ Fμν
F 0μν Fρσ
F aρσ + 2Fμν
F aμν Fρσ
F aρσ + c 2Fμν
1 a aμν c cρσ
0
a
0
0
F̃
F̃ 0μν Fρσ
F̃ aρσ + 2Fμν
F̃ aμν Fρσ
F̃ aρσ
+ Fμν
Fρσ F̃
+ Fμν
8
+ ...
2
1
1
1
2
LBI = − Fμν F μν + β −2 (Fμν F μν ) + β −2 Fμν F̃ μν
4
32
32
−
2
1 −4
1 −4
5 −6
3
4
β (Fμν F μν ) −
β Fμν F μν Fμν F̃ μν +
β (Fμν F μν )
128
128
2048
+
2
4
3 −6
1 −6 Fμν F̃ μν + . . . ,
β (Fμν F μν )2 Fμν F̃ μν +
β
1024
2048
where a, b, c, . . . are representation-dependent coefficients coming from the traces.
Next we introduce physical fields with linear combinations of the U(1) and SU(2)
gauge fields and compare the two series term by term, fixing the coefficients. In
particular, for the pure electromagnetic sector LEM in LBI , we arrive at the formula
172
awrynowicz
(related to rotation with the so-called Weinberg angle θ within the linear combinations of the U(1) and SU(2) gauge fields):
1
LG − LEM = a 2 cos2 θ + sin2 θ Fμν F μν
2
1
−2
2
4
4
2
2
+β M b 2 cos θ + sin θ + 3 sin θ cos θ Fμν F μν Fρσ F ρσ
8
(60)
1
+c 2 cos2 θ + sin4 θ + 3 sin2 θ cos2 θ Fμν F̃ μν Fρσ F̃ρσ
8
1
15
15
6
4
2
−4
3
2
4
6
sin θ +
cos θ sin θ +
cos θ sin θ + 2 cos θ + . . . .
+β M g
32
8
2
For other instructive examples we recommend [To, Ma1, 2]. Examples strictly
related with the Finsler geometry will be given in the second part of this paper
[IL5].
Acknowledgments
This work (J. L
) was partially supported by the Ministry of Scinces and Higher
Education grant PB1 P03A 001 26 (Sections 1–2 of the paper) and partially by the
grant of the University of L
ódź no. 505/692 (Sections 3–9).
References
M. F. Atiyah, Geometry of Yang-Mills Fields (Lezioni Fermiane), Accademia
Nazionale dei lincei – Scuola Normale Superiore, Pisa 1979.
[BI1] M. Born and L. Infeld, Electromagnetic mass, Nature 132 (1932), 970–970.
[BI2] —, —, Foundations of the new field theory, Proc. Roy. Soc. London A 144 (1934),
425–451.
[Be1] R. G. Beil, Finsler and Kaluza-Klein gauge theories, Internat. J. Theor. Phys. 32
(1933), 1021–1031.
[Be2] —, Moving frame transport and gauge transformations, Found. Phys. 25 (1995),
717–742.
[GH] S. G. Gindikin and G. M. Henkin, Penrose’s transformation and complex integral geometry [in Russian], in: Itogi nauki i tekhniki. Seriya: Sovremennye problemy matematiki 17, Moskva 1981, pp. 57–111.
[GK] D. V. Gal’tsov and R. Kerner, Classical glueballs in non-abelian Born-Infeld theory,
Phys. Rev. Lett. 84, no. 26 (2000), 5955–5958.
[GKL] B. Gaveau, J. Kalina, J. L
awrynowicz, and L. Wojtczak. The global Yang-Mills equations depending on an arbitrary metric, Ann. Polon. Math. 46 (1985), 105–114.
[GL] — and G. Laville, On the Maxwell-Dirac system in two dimensions, Bull. Soc. Sci.
Lettres L
ódź 32 (1982), 1–8.
[At]
173
[GLW1] —, J. L
awrynowicz, and L. Wojtczak, A complex-analytical procedure for the generalized Dirac-Maxwell system, in: Mathematical Structures – Computational Mathematics – Mathematical Modelling 11. papers dedicated to Professor L. Iliev’s 70th
anniversary. Publ. House of the Bulgarian Acad. of Sciences, Sofia 1984, pp. 170–176.
[I]
R. S. Ingarden, Open systems and consciousness. A physical discussion, Open Systems
and Information Dynamics 9 (2002), 339–369.
[IL1] — and J. L
awrynowicz, Randers geometry and gauge theories II. Randers and KalużaKlein gauge theories, self-duality and homogeneity, Bull. Soc. Sci. Lettres L
ódź 58
Sér. Rech. Déform. 56 (2008), 49–60.
[IL2] —, —, Model of magnetic electron microscope including the scanning microscope II.
Mathematical model, ibid. 59 Sér. Rech. Déform. 59, no. 2 (2009), 119–126.
[IL3] —, —, Model of magnetic electron microscope including the scanning microscope III.
Variational approach and caculation of the focal length in a Randers-type geometry,
ibid. 60 Sér. Rech. Déform. 60, no. 1 (2010), 137–154.
[IL5] —, —, Finsler-geometrical model of quantum electrodynamics II. Physical interpretation of solenoidal and nonsoleidal connections on the canonical principal fibre bundles,
ibid. 60 Sér. Rech. Déform. 60, no. 2 (2010), to appear.
[K] R. Kerner, Non-linear electrodynamics derived from the Kaluża-Klein theory,
Comptes Rendus Acad. Sci. Paris 304, série 2 (1987), 621–624.
[KBG] —, A. L. Barbosa, and D. V. Gal’tsov, Topics in Born-Infeld electrodynamics, in:
New Developments in Fundamental Interaction Theories, ed. by J. Lukierski and
J. Rembieliński (AIP Conference Proceedings 589), American Institute of Physics,
Melville, NY 2001, pp. 377–389.
[LL] L. D. Landau and E. M. Lifschitz, (a) Lehrbuch der theoretischen Physik 8. Band II.
Klassiche Feldtheorie, 6th ed. [transl. from Russian], Akademie-Verlag, Berlin 1973;
b) Course of Theoretical Physics. Vol. II. The Classical Theory of Fields [transl. from
Russian], Pergamon Press, London 1959.
[L1] J. L
awrynowicz, Application à une théorie non linéaire de particules élémentaires,
Appendice pour le travail de B. Gaveau et J. L
awrynowicz: Intégrale de Dirichlet
sur une variété complexe. I, in: Séminaire Pierre Lelong – Henri Skoda (Analyse),
Années 1980/81 (Lecture Notes in Mathematics 919), Springer, Berlin-Heidelberg
1982, pp. 152–166.
[L3] —, Quaternions and fractals vs. Finslerian geometry. A. Quaternions, noncommutativity, and C ∗ -algebras; B. Algebraic possibilities (quaternions, octonions, sedenions);
1. Quaternion-based modifications of the Cayley-Dickson process, Bull. Soc. Sci. Lettres L
ódź 55 Sér. Rech. Déform. 47 (2005), 55–66, 67–82, and 83–98.
[L5] —, Quaternions and fractals vs. Finslerian geometry. 4. Isospectral deformations in
complex Finslerian quantum mechanics; 5. Clifford-analytical description of quantummechanical gauge theories; 6. Jordan algebras and special Jordan algebras, ibid. 56
Sér. Rech. Déform. 49 (2006), 35–43, 45–602, and 61–77.
[L6] —, Randers and Ingarden spaces with thermodynamical applications, ibid 59 Sér.
Rech. Déform. 58 (2009), 117–134.
[LW1] — and L. Wojtczak, A concept of explaining the properties of elementary particles
in terms of manifolds, Z. Naturforsch. 29a (1974), 1407–1417.
[LW2] —, —, On an almost complex manifold approach to elementary particles, ibid. 32a
(1977), 1215–1221.
[LW3] —, —, A complex-analytical method of solving the generalized Dirac-Maxwell system
for complex-Riemannian metric, Rev. Roumaine Math. Pures Appl. 33 (1988), 89–95.
174
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[Ma1] Yu. I. Manin, Quantum Groups and Non-Commutative Geometry, Centre de
Recherches Mathématiques, Univ. de Montréal, Montréal 1988.
[Ma2] —, Gauge Field Theory and Complex Geometry (Grundlehren der mathematischen
Wissenschaften 289), Springer, Berlin-Heidelberg-New York-Paris-Tokyo 1989.
[Mi] G. Mie, Grundlagen einer Theorie der Materie. Erste Mitteilung, Annalen der Physik
37 (1912), 511–534; Ditto. Zweite Mitteilung, ibid. 39 (1912), 1–40; Ditto. Dritte
Mitteilung, ibid. 40 (1913), 1–66.
[S]
N. Salingaros, Electromagnetism and the holomorphic properties of spacetime, J.
Math. Phys. 22 (1981), 1919–1925.
[Sc] E. Schmutzer, Relativistische Physik, 2. Aufl., B. G. Teubner, Leipzig 1968.
[St] N. E. Steenrod, The Topology of Fibre Bundles, Princeton Univ. Press, Princeton,
N. J. 1951.
[To] I. T. Todorov, Conformal covariant instanton solutions of Euclidean Yang-Mills equations (An elementary introduction), in: III School of Elementary Particles and High
Energy Physics, Primorsko, Bulgaria 1977, Bulgarian Acad. of Sci., Sofia 1978,
pp. 131–182.
[W] R. O. Wells, Jr., Differential Analysis on Complex Manifolds, Prentice Hall, Englewood Cliffs, N. J. 1973.
Nicolaus Copernicus University
Grudzia̧dzka 5, PL-87-100 Toruń
Poland
Univeristy of L
ódź
Pomorska 149/153, PL-90-236 L
ódź
Polish Academy of Sciences
L
ódź
Poland
MODEL FINSLEROWSKO-GEOMETRYCZNY
ELEKTRODYNAMIKI KWANTOWEJ I
POLE ZEWNȨTRZNE A GEOMETRIA FINSLERA
Streszczenie
Po podsumowaniu fizycznych oczekiwań spowodowanych koniecznościa̧ uwzglȩdnienia
ukladów otwartych (nie izolowanych), rozważamy kwantowe równania Diraca-Maxwella
przy użyciu podejścia analitycznego zespolonego oraz równań splotowych. Z kolei przechodzimy do ogólniejszego przypadku równań Yanga-Millsa w obecności pola zewnȩtrznego,
z wla̧czeniem przypadków dowolnej symetrii w zakresie grup SO(m) lub SU(m), przypadku
globalnego, uogólnienia nie-abelowego oraz uogólnienia lagranżianu i jego wlożenia w model
elektro-slaby. W drugiej czȩści pracy bȩdziemy zajmowali siȩ krysztalami ferrelektrycznymi
w geometrii Finslera oraz interpretacja̧ fizyczna̧ koneksji solenoidalnych i niesolenoidalnych
w kanonicznych glównych wia̧zkach wlóknistych.
!" #$ %#
& ' (% )%*!$ %#
% (+ ,-.
/*! ($0*!$ ,-.
1% 23$ ,-.
)%% 2!% 10#
3 2% 0*
% 2 2 40 * '0
%+ ,0%$ ,-.
!$5$ ,-.
6 6 %+ $ 0*
$"+%$ ,-.
$ ,-.
70 '8 96
20 ': 0$0$
10 ; 1$
%$!% ,-.
2%%6 $ ,-.
7 ) 0 2
9$ 0%$$ %#
<0 ($0* *#$ ,-.
"! !$ ,-.
(% 0$!$ ,-.
25= $ ;$
9 33
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+ $!%#$ >%!
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>"%% >%!$ ,-.
)%% >!$ ,-.
>""! >%#$ ,-.
9 /0 0
! / 5)
70 /$AB2
' /AB2
CONTENU DU VOLUME LX, no. 2
1. Yu. Zelinskiı̌, Continuous mappings between domains of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ca. 4 pp.
2. A. Touzaline, On the solvability of a quasistatic contact problem for elastic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ca. 20 pp.
3. J. Rutkowski and C. Surry, Melting and related phenomena
in thin lead films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ca. 8 pp.
4. J. Zaja̧c and B. Falda, Influence of Professor Julian L
awrynowicz and his Lublin colleagues during 20 years of PolishMexican collaboration in generalized complex analysis . . . . . . . .
ca. 16 pp.
5. V. S. Shpakivskyi and S. A. Plaksa, Integral theorems and
a Cauchy formula in a commutative three-dimensional harmonic
algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ca. 10 pp.
6. D. Mierzejewski, The dimensions of sections of the sets of the
solutions of some quadratic quaternionic equations . . . . . . . . . . . .
ca. 12 pp.
7. A. K. Kwaśniewski, Some Cobweb posets digraphs elementary
properties and questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ca. 7 pp.
8. M. Nowak-Kȩpczyk, Binary alloy thin films vs. LennardJones and Morse potentials. A note on binary alloys with arbitrary atoms concentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ca. 16 pp.
awrynowicz, Finsler-geometrical model of quantum electrodynamics II. Physical interpretation of
solenoidal and nonsolenoidal connections on the cannonical principal fibre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ca. 16 pp.
awrynowicz, Finsler geometry and
physics. Physical overwiev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ca. 16 pp.

B U L L E T I N

Transcription

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