Vol. 4 No 7 - Pi Mu Epsilon

P I m i M UEPSILON
-I
'I
I
THE PME MOTTO:
MEANING A N D ETYMOLOGY
Panos D. B a r d i s
P r o f e s s o r o f Sociology and E d i t o r of S o c i a l S c i e n c e
The U n i v e r s i t y o f Toledo, Toledo, Ohio, USA
"Sosicrates says t h a t Pittacus,
a f t e r c u t t i n p o f f a s m a l l fragment,
d e c l a r e d t h a t t h e h a l f was niore t h a n
t h e whole ."i:
Dioeenes L a e r t i u s , P i t t a c u s , 75.
Ten paedeusin c a e t a n a t h e m a t i c a e p i s p e u d e i n :
s h i p and mathematics.
t o iromote scholar-
1. Ten: feminine tender, s i n g u l a r number, a c c u s a t i v e ( o b j e c t i v e )
c a s e o f d e f i n i t e , o r p r e p o s i t i v e , a r t i c l e &, he, to: t h e .
" I n preI n h i s D e f i n i t i o n s , Y I V , 1-6, Hero of Alexandria w r i t e s :
s e n t i n g t o you a s b r i e f l y a s p o s s i b l e , 0 most i l l u s t r i o u s Dionysius, an
o u t l i n e of b a s i c t e c h n i c a l t e r m s i n geometry, I w i l l t a k e a s t h e s t a r t i n g
p o i n t , and w i l l base ,the e n t i r e o r g a n i z a t i o n on,
t e a c h i n g o f Euclid,
t h e a u t h o r o f ~ l e m e n t r d e a l i nw~i t h t h e o r e t i c a l geometry" ( s e e Panos
3. B a r d i s , "~e-~a
V i n c i of Ancient A l e x a n d r i a : His Aeolosphaera
and Other I n v e n t i o n s , " School S c i e n c e and Mathematics, J u n e 1965, pp.
535-5142).
the
2.
Paedeusin: feminine g e n d e r , s i n g u l a r number, a c c u s a t i v e c a s e of
noun p a e d e u s i s : c u l t u r e , e d u c a t i o n , e d u c a t i o n a l system, i n s t r u c t i o n ,
learning, scholarship, training.
P l a t o s t a t e s t h a t "an e x c e l l e n t b r e e d i n c and e d u c a t i o n , i f emphasized
c o n s t a n t l y , g e n e r a t e s good n a t u r e s i n t h e p o l i t y " ( R e p u b l i c , b2lla).
From
paes:
child.
King Nostor, t h e m i l i t a r y f e n i u s and g a r r u l o u s Methuselah of P y l o s ,
s a i d t o King Diomedes of Argos, i h e second g r e a t e s t Greek h e r o - - - a f t e r
Achilles- - - durinp "ie T r o j a n War: " Besides, you a r e s o younp, you could
even be my
my youngest born" (Homer, I l i a d , I X , 57-58].
e,
E n g l i s h words: pedagog, pedagogy ( c h i l d l e a d i n ? ) , p e d i a t r i c i a n ,
~ e d i a t r i c s( c h i l d m e d i c i n e ) . pedobaptism, p e d o c e p h a l i c p e d o g e n e s i s
( i n t r o d u c e d by Von Baer i n 1828). pedology, pedo"orphism, and many
others.
3.
z: c o n j u n c t i o n :
I n E u c l i d ' s Elements.
i n g T h e a e t e t u s (415-369
p h i l o s o p h e r , and one of
t h e 13th, a r e described
and.
Also t r a n s l i t e r a t e d
5,
XIII, Scholium 1. we r e a d t h e f o l l o w i n g r e g a r d 0 . C. ), t h e Athenian a s t r o n o m e r , mathematician,
t h e p u p i l s of S o c r a t e s : " I n t h i s book, namely,
t h e s o - c a l l e d f i v e P l a t o n i c s o l i d s , which,
however, a r e n o t h i s , b u t t h r e e o f t h e aforementioned f i v e s o l i d s a r e
of t h e Pythagoreans, t h a t is, t h e cube & t h e
pyramid & t h e
dodecahedron,
i c o s a h e d r o n being of T h e a e t e t u s . "
t h e octahedron & t h e
t r i a k a i d e k a p h o b i a ( f e a r o f t h e number 1 3 ) .
E n g l i s h word:
4.
Ta: n e u t e r g e n d e r , p l u r a l number, a c c u s a t i v e c a s e o f d e f i n i t e
a r t i c z ho, he, to: t h e .
The famous C a t t l e Problem o f Archimedes ( ? I . which t h e n o t e d i n v e n t o r
and mathematicran of Syracuse s o l v e d i n epigrams and s e n t t o t h e mathemat i c i a n s of A l e x a n d r i a i n a l e t t e r t o E r a t o s t h e n e s , c l o s e s w i t h t h e s e
f o u r l i n e s : "0 s t r a n g e r , i f you f i n d o u t t h e s e t h i n g s and add them up
i n your mind, g i v i n g a l l t h e r e l a t i o n s among t h e s e q u a n t i t i e s , you w i l l
d e p a r t g l o r i o u s l y and v i c t o r i o u s l y , knowing t h a t you have been adjudged
g r e a t i n t h i s kind o f wisdom."
5.
Mathematica: n e u t e r g e n d e r , p l u r a l number, a c c u s a t i v e c a s e of
a d j e c t i v e mathematicus: fond of l e a r n i n g , m a t h e m a t i c a l .
Mathematice e p i s t e m e :
mathematical science.
I n t h e Nicomachean E t h i c s , A r i s t o t l e a s s e r t s t h a t , " As f a r a s cond u c t i s c o n c e r n e d , t h e b a s i c p r i n c i p l e c o i n c i d e s with t h e pursued g o a l ,
which is a n a l o g o u s t o t h e h y p o t h e s e s ( h e r e t h e p h i l o s o p h e r means props i t i o n 9 o f mathematics" ( V I I , v i i i , 17- 18).
From manthanein:
t o comprehend, t o know, t o l e a r n , t o p e r c e i v e .
Mathema: knowledge, something l e a r n e d , l e s s o n , s c i e n c e . Also,
m a t h e m a t i c a l s c i e n c e , e s p e c i a l l y , a r i t h m e t i c , astronomy, geometry
( s e e Panos D. B a r d i s , "Symmentrical Consonance o f P l a y , Rhythm, and
Harmony: An Essay o n P l a t o ' s Mathematics,"School S c i e n c e and Mathematics,
J a n u a r y 1963, pp. 52-67).
E n g l i s h words:
and s o f o r t h .
From + a n d
E&:
P r o f . D a v i d C. K a y , T h e U n i v e r s i t y of O k l a h o m a
( T a l k p r e s e n t e d t o t h e Oklahoma A l p h a c h a p t e r )
I n a moment I am g o i n g t o a c q u a i n t you w i t h a problem which w i l l
l e a d u s q u i t e n a t u r a l l y i n t o two d i f f e r e n t f i e l d s o f mathematics- - - Minkowski geometry, and number t h e o r y . It i s a n e x t r e m e l y e n t e r t a i n i n g
problem, and it may a c t u a l l y have a n a p p l i c a t i o n o u t s i d e mathematics
which r e n d e r s it p a r t i c u l a r l y heart- warming. The problem i s i n t h e
form o f two q u e s t i o n s , t h e second of which makes s e n s e o n l y i f t h e f i r s t
h a s a "No" answer.
C o n s i d e r t h e xy- plane and t h e s e t o f a l l l a t t i c e p o i n t s , p o i n t s
o f t h e form (m,n) where m and n a r e i n t e g e r s . Now imagine a n i n f i n i t e
f o r e s t whose t r e e s a l l have t h e same r a d i u s , s a y r , and c e n t e r e d a t t h e
l a t t i c e p o i n t s (m,n). Now I f e l l t h e t r e e a t (0,O) and s t a n d o n t h e
stump. The q u e s t i o n is, can I s e e t h r o u g h t h e f o r e s t ? G e o m e t r i c a l l y
o f c o u r s e , I am a s k i n g whether I c a n p a s s a l i n e t h r o u g h (0,1) which
d o e s n o t t o u c h any c i r c l e c e n t e r e d a t a l a t t i c e p o i n t and having r a d i u s r
I have p r e p a r e d a l i t t l e s k e t c h o f t h e s i t u a t i o n f o r r = 1 / 1 0 which
I ' l l l e t you s e e . Now i f we examine t h e problem c a r e f u l l y f o r t h i s
c a s e we o b s e r v e s e v e r a l s i m p l i f y i n g p r i n c i p l e s . One is t h a t a s I r o t a t e
a b o u t t h e o r i g i n s e e k i n g a l i n e of v i s i o n o u t o f t h e f o r e s t , it i s n o t
r e a l l y n e c e s s a r y t o c o n s i d e r a l l d i r e c t i o n s . I f I m e r e l y answer t h e
q u e s t i o n f o r t h e r a y s whose a n g l e s w i t h t h e x - a x i s r a n g e from 0 t o
n / 4 , I w i l l be a b l e t o a p p l y t h a t answer t o a l l t h e o t h e r remaining
s e c t o r s , s i n c e t h e s e c t o r o f r a y s from n/4 t o n / 2 c o n t a i n s t h e r e f l e c t e d
image i n t h e l i n e y = x o f t h a t c o n t a i n e d by t h e s e c t o r from 0 t o n/4.
It is t h e n o b v i o u s t h a t , having answered t h e q u e s t i o n f o r t h e f i r s t
m a t h e m a t i c a l , mathematician, mathematics, polymath,
6.
Epispeudein: i n f i n i t i v e o f v e r b epispeudo:
I promote, I u r g e on.
.. a .
ON NOT SEEING THROUGH THE FOREST FOR THE TREES
I h a s t e n onward,
yeudein.
preposition:
by, on, o v e r , upon, and t h e l i k e .
I n d e a l i n g w i t h t h e Pythagorean theorem, E u c l i d r e f e r s t o two of t h e
l i n e s o f t h e c e l e b r a t e d "windmill" f i g u r e w i t h t h e s e words: "two s t r a i g h t
l i n e s , namely, AG, AE, n o t l y i n g on t h e same s i d e , make t h e a d j a c e n t
a n g l e s e q u a l t o two r i g h t o n e s f 9 E l e m e n t s , I , 47).
E n g l i s h words: e p i b l a s t , e p i c a l y x , e p i c a n t h u s , e p i c a r d i u m , e p i c e n e ,
e p i c e n t e r , e p i c r i s i s , epidemic, e p i d e r m i s e p i d i d y m i s , and myriad o t h e r s .
I n mathematics: e p i c y c l o i d , e p i t r o c h o i d , e p i t r o c h o i d a l , and many more.
b. Speudein: i n f i n i t i v e o f v e r b speudo:
promote, I quicken.
I h a s t e n , I p r e s s on, I
Herodotus i n f o r m s u s t h a t Queen Tomyris o f t h e Massagetae s e n t t h e
f o l l o w i n g message t o King Cyrus when h e was marching a g a i n s t h e r kingdom: "0 k i n g o f t h e Medes, c e a s e from promoting what you a r e promoting"
f o r vou cannot know i f t h e s e t a s k s w i l l be f o r your a d v a n t a g e when
f i n i s h e d " ( H i s t o r i e s , I , 206).
NOTE
* A 1 1 quoted p a s s a g e s have been t r a n s l a t e d by t h e p r e s e n t a u t h o r .
q u a d r a n t , I have answered t h e q u e s t i o n f o r a l l q u a d r a n t s . So it s u f f i c e s
t o c o n s i d e r o n l y t h e s e c t o r from 0 t o 17/4 a s i n d i c a t e d by t h e d o t t e d
l i n e s i n t h e diagram I have g i v e n you.
Be making a few shrewd c a l c u l a t i o n s , we c a n a c t u a l l y o b s e r v e a n
upper bound i n t h e c a s e r = 1 / 1 0 , and it is a p p r o x i m a t e l y 9 ~ 8 . A few
o f t h e l o n g e r l i n e s o f v i s i o n a r e shown, l a b e l l e d w i t h t h e i r approximate
l e n g t h s . We may c o n j e c t u r e , t h e r e f o r e , t h a t t h e answer must be "no."!
I c a n n o t s e e t h r o u g h t h e f o r e s t f o r t h e t r e e s . But might we n o t be
j u m x t o c o n c l u s i o n s based o n i n s u f f i c i e n t e v i d e n c e ? What i f r is
e x c e e d i n g l y s m a l l and t h e t r e e s i n t h e f o r e s t a r e m e r e l y v e r y s k i n n y
t o o t h p i c k s ? Is it t h e n c o n c e i v a b l e I might s e e t h r o u g h t h e f o r e s t i n
some c a r e f u l l y chosen d i r e c t i o n ? I t is o b v i o u s t h a t I s h a l l be a b l e
t o s e e a l o t f a r t h e r t h a n I c o u l d when r = 1 / 1 0 , b u t how much f a r t h e r ?
274
The second q u e s t i o n i s , assuming t h a t my v i s i o n is blocked by some
t r e e i n e v e r y d i r e c t i o n , what is t h e f o r t h e s t I can s e e ? I d e a l l y , t h e
s o l u t i o n t o b o t h p r o ~ l e m swould be found by o b t a i n i n g a n e x p r e s s i o n f o r
t h e f u n c t i o n f ( o , r ) which measures t h e d i s t a n c e from t h e o r i g i n t o t h e
f i r s t t r e e which t h e l i n e y = ox t o u c h e s , i f any. I f none touch we
could d e f i n e f ( o , r ) =
f o r t h a t o and t h a t r. But when we t r y t o
w r i t e down what we t'r.ow a l ' r u t f ( o , r ) it becomes c l e a r t h a t t h e r e must
be a n e a s i e r way.
But t h e , a n o t h e r t h o u g h t pops r e a d i l y i n t o mind. Suppose we
r e p h r a s e t h e q u e s t i o n s l i g h t l y . Given a l i n e y = ox, O<ocl, d o e s a
l a t t i c e p o i n t (m,n) come a r b i t r a r i l y c l o s e t o t h i s l i n e , o r c l o s e r t h a n
r , by making some c h o i c e o f t h e p o s i t i v e i n t e g e r s m and n? The answer
i s o b v i o u s l y "yes" i f o i s r a t i o n a l . But suppose o i s i r r a t i o n a l .
L e t u s r,et a n i d e a
how c l o s e t h e l a t t i c e
p o i n t (m,n) i s from
t h e l i n e y = ox. We
can e a s i l y s e e t h a t
t h e d i f f e r e n c e between
t h e o r d i n a t e s om and n, o r
-
i s a measure o f how c l o s e
(m.n)istoy=ox.
If
t h i s is l e s s than r then t h e
l i n e y = ox w i l l d e f i n i t e l y
b e c u t o f f bv t h e t r e e
c e n t e r e d a t (m,n). So o u r
problem becomes t h a t o f choosing p o s i t i v e i n t e g e r s m and n s o a s t o
minimize
om
nl
-
T h i s would be c a l l e d t h e 5 t h sequence o f Farey f r a c t i o n s . A l l
p r e c e d i n g sequences may be o b t a i n e d by d e l e t i n g c e r t a i n f r a c t i o n s .
For example, t h e 4 t h sequence would be
The 3 r d , and 2nd a r e :
Given a p o s i t i v e i n t e g e r n , however, I c a n form t h e n t h sequence independent of t h e others.
F a r e y f r a c t i o n s have a number o f i n t e r e s t i n g p r o p e r t i e s which w i l l
e n a b l e u s t o prove t h e r e s u l t we a r e t r y i n g t o o b t a i n , namely, t h a t
ma
n l c r f o r some c h o i c e o f m and n.
-
The f i r s t o f t h e s e is e a s i l y s p o t t e d from o u r examples* Note t h a t
t h e c r o s s - p r o d u c t s o f numerator t i m e s denominator o f two c o n s e c u t i v e
f r a c t i o n s i n any sequence always d i f f e r by u n i t y . P i c k i n g a few
a t random:
T h i s i s no a c c i d e n t . R e c a l l i n g t h e r u l e o f i n e q u a l i t y f o r f r a c t i o n s
w i t h p o s i t i v e numerator and denominator,
where o is a n arbitrarily g i v e n i r r a t i o n a l , O<o<l.
<;
A s a n example o f t h e t r u e n a t u r e o f t h e problem, I have made a
NOW t h e r a t i o n a l
few c a l c u l a t i o n s f o r o =
6
approximates</5"(which
is 1.41421435...)
t o w i t h i n .0001435,..
5
d
iff
ad < b c ,
we s e e t h a t , s i n c e we have a r r a n g e d t h e f r a c t i o n s i n n u m e r i c a l o r d e r
f o r e a c h sequence, we must minimize t h e d i f f e r e n c e
b
So
given
-b
-
5.
x -ay
Y
b
S i n c e bx - a y i s a p o s i t i v e i n t e g e r , t h e l e a s t it can be is
-
happens t o work b e t t e r f o r t h i s example, f o r t h e n ,
one. Hence bc
ad = 1 i f t h e r e e x i s t s a f r a c t i o n i n o u r list f o r
which t h i s d i f f e r e n c e & u n i t y .
I t may b e t h a t we c a n n o t f i n d s u c h a
f r a c t i o n , and i f we were t o prove t h i s p r o p e r t y o f Farey f r a c t i o n s , we
would have s e t t l e d t h i s i s s u e . However, it may be proved, indeed from
s o b a s i c a p r i n c i p l e a s E n c l i d ' s a l c o r i t h m , t h a t we can f i n d a f r a c t i o n
with t h i s property.
which g i v e s u s a d i f f e r e n c e of .005 a p p r o x i m a t e l y , a s compared w i t h
A second p r o p e r t y , a consequence o f t h e f i r s t , is t h a t i f we t a k e
t h r e e c o n s e c u t i v e Farey f r a c t i o n s a / b , c / d , e / f t h e n c / d = ( a t e ) /
( b t f ) . F o r example, c o n s i d e r ( 3 A , 2 / 3 , 3/41. We have ( 3 t 3 ) /
( 5 t 4 ) = fi/1 = ? / 3 .
We can add t o t h e seeming t r i v i a o f t h e h o u r by
o b s e r v i n g t h a t , s i n c e b c - ad = 1 and de - c f = 1, t h e n
which is n o t v e r y
t h e q u a n t i t y we a r e i n t e r e s t e d i n , e q u a l s .1435....
s m a l l . Thus, c l o s e r and c l o s e r a p p r o x i m a t i o n s t o o by r a t i o n a l numbers
2- d o e s n o t n e c e s s a r i l y make lmo - n l s m a l l . The c h o i c e n = 99, m = 70
.143.
But how c a n we o b t a i n v a l u e s f o r n and m which make t h i s s m a l l e r
than, say 1 0 - 7
C u r i o u s l y enough t h i s problem a p p e a r s t o be s o l v e d i n Niven's
book "An I n t r o d u c t i o n t o t h e Theory o f Numbers" f o l l o w i n g a d i s c u s s i o n
o f Farey f r a c t i o n s . Perhaps you have heard o f t h e s e f r a c t i o n s . For
e a c h p o s i t i v e i n t e g e r n one forms a l l p o s s i b l e f r a c t i o n s p / q between
z e r o and u n i t y , i n c l u s i v e , such t h a t q i n , and r e d u c e t o lowest t e r m s ,
p l a c i n g them i n n u m e r i c a l o r d e r . For example, f o r n = 5 we have
iind p l a c i n g thorn i n n u m e r i c a l o r d e r , we have
Farey f r a c t i o n s have a b i t o f unexpected power, a s we s h a l l now
s e e . Observe t h a t i f we p l o t t h e k t h sequence o f Farey f r a c t i o n s , s i n c e
(k - l)/k, t h e i n t e r v a l s
we have c o n t a i n e d i n t h i s l i s t 1 / k , 2 / k , 3 / k ,
...
between c o n s e c u t i v e f r a c t i o n s is l e s s t h a n l / k .
Consequently, any r e a l
number a between 0 and 1 can be a pp roximated t o w i t h i n 1 / k by a
Farey f r a c t i o n i n tlie k t h sequence. L e t r > o be c i v e n . There is a
p o s i t i v e i n t e g e r k such t h a t 1 < r. L e t u s form t h e k t h sequence of
^T?
Farey f r a c t i o n s .
Choose t h e l a r g e s t
one which is l e s s
0 ;
than o r equal t o
4
.
Ia , s a y n/m. Now
0 1 / k 7/k 3/k
n'
1
I look a t t h e
m B'
n e x t Farey f r a c t i o n
i n t h i s sequence,
c a l l it n'/m'.
I know t h a t mn' - nm' = 1. Hence
n ' - 5 mn' - nm'
m'
m
m'm
=
1
mum
But by o u r above o b s e r v a t i o n t h i s d i f f e r e n c e c a n n o t exceed 1 / k , s o
1
<1
m'm - k .
Hence one o f m,m' must b e a t l e a s t a s l a r g e a s i/K. Let u s s a y t h a t
m' s. ^k. Observing t h e f i g u r e , we have, c e r t a i n l y ,
Hence
A d i f f e r e n t s o r t of i d e a which Minkowski a p p l i e d t o t h e s u c c e s s f u l
s o l u t i o n o f many d i f f i c u l t problems o f number t h e o r y h a s come t o b e
known a s Minkowski's Theorem:
Theorem: Given a convex s e t i n t l e p l a n e symmetric a b o u t one l a t t i c
p o i n t and a r e a g r e a t e r t h a n o r e q u a l t o 4 , t h a t convex s e t must c o n t a i n
a t l e a s t two f u r t h e r l a t t i c e p o i n t s .
To make t h i s p l a u s i b l e , Minkowski reasoned a s f o l l o w s : Let S be
a convex body whose a r e a is K i . 4 and c e n t e r e d a t (0,O). C l e a r l y we
prove t h e theorem i f we prove t h a t S i n c l u d e s a t l e a s t one o t h e r l a t t i c e
p o i n t , f o r it w i l l a u t o m a t i c a l l y c o n t a i n its r e f l e c t i o n i n (0.0).
Using a mapping l i k e x ' = a x , y ' = ay it i s c l e a r we may " enlarge"
S by a f a c t o r a u n t i l it j u s t t o u c h e s one o t h e r l a t t i c e p o i n t , and
i t s r e f l e c t i o n i n ( 0 , 0 ) , b e s i d e s ( 0 , 0 ) . Then s h r i n k t h i s new s e t back
by a f a c t o r 1 / 2 . The r e s u l t i n g s e t S' h a s an a r e a of ( a 2 / 4 ) ~ . L e t S'
b e t r a n s l a t e d t o form a system of c o n c r u e n t symmetric convex s e t s i n
t h e p l a n e c e n t r e d a t each and e v e r y l a t t i c e p o i n t . I t is c l e a r t h a t
t h e s e s e t s look something l i k e t h i s :
They a r e non- overlapping b u t
certain nairs w i l l .
touch
a-.t
1
.
c e r t a i n p o i n t s . By c o n s i d e r i n g
unit squares a l s o centered a t
each l a t t i c e p o i n t , s i n c e t h e
convex s e t s do n o t completely
f i l l up t h e plane- - - there
a r e c e r t a i n gaps left- - - we
must have
.-
--
--
1
or
That i s , we have found, usinp, Farey f r a c t i o n s , a l a t t i c e p o i n t (n.m)
t h a t i s c l o s e r t o t h e l i n e y = ox t h a n r.
I n t e r e s t i n g a s a l l t h i s is I have j u s t shown you t h e wrong way t o
do t h i s problem. For t h e r e i s a way t o answer b o t h q u e s t i o n s we a r e
a s k i n g i n t h e same b r e a t h . A t t h i s p o i n t Minkowski, i f he were h e r e ,
would b e l a u g h i n g u s r i g h t o u t o f t h i s room t h a t we could b e s o i n e p t .
I f I may b e allowed t o p a r a p h r a s e h i s j e e r i n p comments, t h e y would
probably R O something l i k e "You b u n g l i n g i d i o t s ! I can n o t o n l y s o l v e
t h e f i r s t q u e s t i o n a b o u t t h e t r e e s i n t h e f o r e s t , b u t t h e second a s w e l l
and i n c l u d e what you s a i d a b o u t !ma - nl beinp: a r b i t r a r i l y s m a l l , a s a
corollary! "
lie would f e e l e s p e c i a l l y j i l t e d because it s o happens t h a t one o f
h i s famous theorems a p p l i e s t o t h e s i t u a t i o n a t hand. Minkowski made
Â¥lomi n n o v a t i o n s t o number t h e o r y by h i s u s e o f convex b o d i e s and h i s
-.turiy o f a c e r t a i n d i s t a n c e f u n c t i o n .
I
0
I F C i s a convex curve
ymmetric about t h e o r i g i n , d e f i n e
AB
d(A,B) * o,,
whom All I f t h e e u c l i d e a n d i s t a n c e from
A t o R and 0U i s t h e e u c l i d e a n
r a d i u s o f C p a r a l l e l t o l i n e AB.
Tho f u n c t i o n rt(A.0) t u r n s o u t t o be
a m o t r i c f o r t h o p l a n e , and one
may proceed t o s t u d y t h e v a r i o u s p r o p e r t i e s t h i s d i s t a n c e c o n c r p t h a s .
Such a a t u d y b e l o n g s t o t h e a r e a o f mathematics known a s Minkowskian
geometry. I t I s c l o n r t h a t i f C i s any c i r c l e t h e n o u r m e t r i c is
o u c l l d o n n . Rut i t I s n o t t o o h a r d t o r e a s o n t h a t i f C is any e l l i p s e ,
t h e n t h e resulting Minkowskian geometry c o i n c i d e s w i t h E u c l i d e a n geometry.
That i s ( u s i n g k > 4 )
Hence, t h e f a c t o r by which we had t o " s t r e t c h " t h e o r i g i n a l s e t S s o it
j u s t touched two f u r t h e r l a t t i c e p o i n t s was < 1,bearing w i t n e s s t o t h e
f a c t t h a t it a l r e a d y c o n t a i n e d them i n t h e i r i n t e r i o r .
You can e a s i l y make t h i s r i g o r o u s , b u t I ' m s u r e you g e t t h e i d e a
Minkowski had i n mind.
Well, I ' v e k e p t you i n s u s p e n s e l o n g enough. How s h o u l d t h e f o r e s t
problem he s o l v e d ? C o n s i d e r a l i n e 1 i n any d i r e c t i o n p a s s i n g t h r o u r h
Let a r e c t a n g l e be d e s c r i b e d s y m m e t r i c a l l y a b o u t 1 a s a x i s ,
(0,0).
havinp, width 2 r ( t w i c e t h e r a d i u s o f o u r t r e e s ) and l e n f l h 7d where d
is t h e d i s t a n c e we a r e s e e i n g from (0,0) i n t h e d i r e c t i o n o f 1. The
a r e a o f t h i s r e c t a n g l e is 7d
7 r = 4 d r . By o b s e r v i n g a c i r c l e o f
r a d i u s r c e n t e r e d a t one o f t h e
v e r t i c e s , i t i s obvious
t h a t o u r v i s i o n w i l l be blocked by
one o f t h e t r e e s i f
and o n l y i f t h i s r e c t a n g l e
contains a l a t t i c e point.
Hence, i f my v i s i o n is n o t
b l o c k e d i n t h i s d i r e c t i o n , by
Minkowski's theorem, t h e
a r e a of t h e r e c t a n g l e has
t o be l e s s t h a n o r e a u a l
t o four.
Thus
Therefore, the distance I can see i n any direction does not exceed t h e
r e c i p r i c a l of t h e radius of t h e t r e e s . I t ' s Touche' once apain by an
elementary theorem of reometry.
SOME USEFIIL PESIILTS IN APPLICATIONS OF VIE POWER
SERIES TRANSFORM TO THE SOLllTInN OF DIFFERENCE EQUATIONS
UNDERGRADUATE RESEARCH PROJECT
Ray A. Gaskins, Virginia Polytechnic I n s t i t u t e
P r o p o s e d b y K e n n e t h Loewen, U n i v e r s i t y o f Oklahoma
Complex numbers may b e d e s c r i b e d a s a t w o d i m e n s i o n a l
v e c t o r s p a c e o v e r t h e ~ r e a ln u m b e r s w i t h m u l t i p l i c a t i o n g i v e n by
(a
+ b i ) (c + d i )
(a + bi) ( c
+
= (ac - b)
d i ) = ( a c - bd) +(ad
+
be) i:
where a , b , c and d a r e r e a l numbers.
Q u a t e r n i o n s may b e d e f i n e d a s a t w o d i m e n s i o n a l v e c t o r s p a c e
o v e r t h e c o m p l e x n u m b e r s w i t h m u l t i n l i c a t i o n g i v e n by
w i t h m, n , p , q c o m p l e x n u m b e r s a n d i f m = a + h i t h e n ii = a - b i
etc. I t i s c u s t o m a r y t o w r i t e
i j = k; s o t h a t i f n = c + d i , t h e n m + n j = a + b i + c j + dk.
I n t u r n a C a y l e y a l g e b r a may b e d e f i n e d a s a t w o d i m e n s i o n a l
vector space over t h e quaternions with multiolication
(u
+
ve) (x
+
ye) = (ux
-
vv)
+
(yu
+
vx-)e.
H e r e u , v , x , y a r e q u a t e r n i o n s so t h a t i f u = a + b i + c j + d k
ii = a - b i - c j - d k . O r d e r i s i m p o r t a n t i s t h i s d e f i n i t i o n
s i n c e q u a t e r n i o n m u l t i p l i c a t i o n is n o t commutative.
Q u e s t i o n : What s o r t o f s y s t e m s a r i s e i f t h i s p r o c e s s i s c a r r i e d
o u t beqinninq with a s p e c i f i c f i n i t e f i e l d instead of t h e reals?
1. Introduction. Durinp the past several years difference equations
have come i n t o t h e i r own as a techniqae f o r problem solvine in many areas
of science and engineering such as s t o c h a s t i c processes, e l e c t r i c a l
engineering and engineering mechanics. I t i s desirable, therefore, t o
develop a systematic method f o r solving t h e various types of difference
equations which a r i s e . One such method f o r handling a wide range of
difference equations i s the power s e r i e s transform.
The power s e r i e s transform as a tool f o r solvinp difference equations
i s t h e d i s c r e t e counterpart of t h e Laplace Transform which i s used i n
solving d i f f e r e n t i a l equations. The transform method of handling d i f f c r ence equations is superior t o more conventional methods i n t h a t i t i s
able t o handle non-hoirogeneous equations and equations with variable
coefficients. I t i s a l s o possible t o obtain d i r e c t l y a p a r t i c u l a r solution without f i r s t obtaining the general solution.
Equipped with a good t a b l e of transfonrs and a knowledge of p a r t i a l
f r a c t i o n s one i s able t o deal s w i f t l y with complicated difference equations.
P a r t i a l Fractions. In order t o f a c i l i t a t e the use of t h e power s e r i e s
transform i n solving difference equations it is e s s e n t i a l t h a t one be
able t o resolve proper r a t i o n a l f r a c t i o n s i n t o p a r t i a l f r a c t i o n s . To t h i s
end t h e following shortcut i s introduced.
2.
Consider t h e proper r a t i o n a l f r a c t i o n N(x)/D(x), where the polynomial
N(x) i s of lower order than t h e polynomial Dfx):
The e d i t o r s o l i c i t s c o n t r i b u t i o n s f o r t h i s d e o a r t m e n t .
Any
problems which would b e s u i t a b l e u n d e r g r a d u a t e r e s e a r c h a r e
welcome.
1 9 6 8 NATIONAL MEETING
A t w o d a y m e e t i n q o f P i Mu E n s i l o n w i l l b e h e l d i n
c o n j u n c t i o n w i t h t h e r e g u l a r summer m e e t i n g s o f t h e
Mathematical A s s o c i a t i o n o f America and t h e American
Mathematical S o c i e t y a t Madison, Wisconsin sometime t h e
l a s t week i n A u g u s t .
Your c h a p t e r i s e n c o u r a g e d t o n o m i n a t e y o u r b e s t
s p e a k e r who w i l l NOT h a v e a m a s t e r s d e g r e e b y A p r i l ,
Nominations
1 9 6 8 , as a s p e a k e r f o r t h e n a t i o n a l m e e t i n g .
s h o u l d b e m a i l e d t o DR. RICHARD V. ANDREE, P I MU EPSILON,
THE UNIVERSITY OF OKLAHOMA, NORMAN, OKLAHOMA 73069.
where.
Ai =
,m-i
N (XI
(m-i) I (x2+bx+c)
',al (x-a
I
x=a
A and
B a r e found hy p l a c i n g A
1
(i=1,m) and
Hence,
B. ( j = l , n ) i n ( 2 . 2 ) F, s e t t i n g t h e r e s u l t e q u a l
3
t o (2.11.
...
P(k ( k + l )
(k+n-1) y k ) =
(T.4)
(3).
Thus knowing Y ( s ) , t h e c l o s e d form of p{yk), we may
o b t a i n by (T.4) many u s e f u l t r a n s f o r m s .
S e v e r a l examples w i l l be g i v e n i n t h e f o l l o w i n g s e c t i o n s .
EXAMPLE 1. Find P ( k ) .
L e t y k = l and n = l i n (T.4).
3. The Power S e r i e s Transform. L e t ( y k ) r e p r e s e n t t h e se-
quence (yo, y,,...,
yk,
...
1.
We d e f i n e t h e power s e r i e s
-
t r a n s f o r m a t i o n of t h e sequence ( y k ) t o be!
'{yk'
a
Yk
k h 0 ~
k
Now l e t u s c o n s i d e r t h e t r a n s f o r m o f t h e sequence ( r 1
k
(T. 1)
I
From t h e p r o p e r t i e s o f power s e r i e s we have by t h e r a t i o
t e s t t h a t (T.1) converges f o r s > s o=
k+1
which converges and i s a continuous f u n c t i o n of r a s w e l l a s
k
s f o r r>s>so. D i f f e r e n t i a t i n g P ( r In t i m e s w i t h r e s p e c t
-
t o r we o b t a i n !
exists.
A s a s h o r t h a n d n o t a t i o n we s h a l l w r i t e Y ( s ) t o d e n o t e
..
...
k (k-1) , (k-n+l) rkmn
= P(k (k-1)
t h e c l o s e d form of t h e t r a n s f o r m , w h i l e P ( y k ) w i l l s t a n d
(k-n+1) rk-"),
And s i n c e P ' ~ )( r k ) = Y'") (s) we g e t t h e i d e n t i t y :
f o r the s e r i e s representation.
Consider, f o r example, t h e sequence ( 1 ) = (1,1,...,1,...)
and i t s t r a n s f o r m P ( 1 ) =kzol/skwhich h a s t h e c l o s e d form
EXAMPLE 2. Find P(k(k-1) ). L e t n=2 and r5l i n (T.5).
Y ( s ) = s/(s-i) , f o r s > l .
P(k(k-1) 1 = s / ( s - ~ )
The Dower s e r i e s t r a n s f o r m is a l i n e a r o p e r a t o r , i . e .
P(Ayk
+
Bzk) = AP(yk)
+
BP(zk)
(T.2)
which f o l l o w s immediately from t h e s e r i e s d e f i n i t i o n .
-
Consider t h e t r a n s f o r m of t h e sequence (yk+,,)
^k+n
^k n- l y k
p ( ~ k + n ' " k h 0 ~ SEXOF
-kZ0~-n
.
3
With t h e p r e c e e d i n g t h e o r y and examples behind u s , we
a r e i n a p o s i t i o n t o h a n d l e t h e t r a n s f o r m a t i o n o f a l m o s t any
A table
d i f f e r e n c e e q u a t i o n one might encounter.
of t r a n s f o r m s i s g i v e n a t t h e end o f s e c t i o n 5.
4. The I n v e r s e Transform. S i n c e t h e Laurent s e r i e s r e p r e s e n t a t i o n of P ( y k ) i s unique, t h e r e i s a one-to-one c o r respondence between ( y k ) and P ( y k ) , hence t h e i n v e r s e oper-
The Laurent s e r i e s P ( y k ) i s uniformly convergent f o r
ator P
P-^Y(~)I
s>soand i s t h e r e f o r e a c o n t i n u o u s f u n c t i o n o f s f o r t h a t
range.
I t may be d i f f e r e n t i a t e d term by term t o g i v e :
(-1)"k (k+1)
(k+n-1) y,
n
sk-n
a
...
d e f i n e d such t h a t !
i s a l s o unique.
= (yk)
We have by (T.2) t h a t P-I i s a l i n e a r
operator.
EXAMPLE 3. P-I [s2/(s-1) 2 ]
5
P-I [ s / ( s - l )
+
1
P-I[ s / ( s - 1 ) 2 l
by l i n e a r i t y and p a r t i a l f r a c t i o n s . Hence
~ - ~ [ s ~ / ( s - l5) ~( 1] + It).
5. A p p l i c a t i o n s of the Transform t o t h e S o l u t i o n s o f Dif-
f e r e n c e Equations. We c o n s i d e r d i f f e r e n c e e q u a t i o n s o f t h e
tYpc 8
where a i < i = l , n ) and f k may o r may n o t be f u n c t i o n s o f k.
S i n c e t h e d i f f e r e n c e e q u a t i o n i s a c t u a l l y a sequence,
we may a p p l y t h e o p e r a t o r P t o it and o b t a i n an a l g e b r a i c
But l i m P{yk) = yo = limy (s),
ss-
e q u a t i o n i n Y(s) and s.
Hence,
C = yo = 1 and by (T.10)
l
I
S o l v i n g f o r Y ( s ) and u s i n g p a r t i a l
f r a c t i o n s t o f a c t o r t h e r i g h t hand s i d e , we a p p l y t h e operator P
t o b o t h s i d e s Ad o b t a i n a s o l u t i o n f o r yk.
y
(
(91,
04kSn,
Yk = (El.
EXAMPLE 4. I n s t o c h a s t i c p r o c e s s e s the u s u a l random walk model w i t h a b s o r b i n g b a r r i e r s a t
TABLE OF POWER SERIES TRANSFORMS
----
x=0 and x=a may be r e p r e s e n t e d by t h e
d i f f e r e n c e e q u a t i o n qk = pqk+l
SEQUENCE
for
l ~ k ~ a - 1where
,
p i s the p r o b a b i l i t y o f a
move one u n i t t o t h e l e f t , q=l-p, and qk
i s t h e p r o b a b i l i t y t h a t x r e a c h e s x=0 bef o r e x=a, g i v e n t h a t i n i t i a l l y x=k.
Sub-
s t i t u t i n g k + l f o r k and a p p l y i n g P:
Using p a r t i a l f r a c t i o n s w i t h 6=0,m=O,n=2:
q/p) + s ( q l
1),
yb) a
J
9-1
s-q/e
a^ -
-
L e t t i n a k=a we f i n d t h a t s i n c e a =0
(T.ll)
(wksin k@)
w sin Ã
(T.12)
k
(W c o s k + )
TRANSFORM
TABLE OF POWER SERIES TRANSFORMS
---SEQUENCE
TRANSFORM
s2
(T.14)
(con't)
-
s
2wbs
-
,
+
,
sL
wbs
2
s2
2wbs + w
-
(wk cosh k + )
b = cosh # > 1
ON HULA HOOPS
w2
Dennis Spellman, Temple U n i v e r s i t y
b = cosh + > 1
REFERENCE:
McFadden, Leonard, Clemens, Paul F. and Campbell, Hugh G.,
The P
o
w
e
and
To
of Difference E a u a t u , Edwards Brothers Inc., 1963.
S e v e r a l months ago L. A i l e e n Hostinsky o f C o n n e c t i c u t C o l l e g e gave
a l e c t u r e a t Temple U n i v e r s i t y e n t i t t l e d "On Groups, Loops, and Hoops"-that's right !
The purpose o f t h i s paper is t o i n v e s t i g a t e t h e prope r t i e s o f hoops - e s p e c i a l l y a c l a s s o f hoops-which we s h a l l c a l l h u l a
hoops.
D e f i n i t i o n 1: Let G be a non-empty s e t . Then a b i n a r y o p e r a t i o n
d e f i n e d on G i s a mapping from t h e C a r t e s i a n p r o d u c t G x G i n t o G. I f
( a , b ) E GxG, i t s image is denoted a 0 b.
o
D e f i n i t i o n 2. Let G b e a non-empty s e t on which a b i n a r y o ~ e r a t i o no
is d e f i n e d . G i s a quasi- group under o i f t h e r e e x i s t unique s o l u t i o n s i n
G t o e a c h o f t h e e q u a t i o n s a 0 x = b and y 0 a = b where a and b a r e f i x e d
e l e m e n t s o f G. A quasi- group G i s a g r o u p i f and o n l y i f it h a s t h e
a s s o c i a t i v e p r o p e r t y . i.e..
i f and o n l y i f (a o b ) o c = a o ( b
c ) for
a l l a , b , c e G. (See ( 1 ) p. 38 Theorem 1 . 4 )
The New Mexico Institute of Mining and Technology, Socorro
New Mexico (enrollment about 500) has found that a mathematical
conference is a good way of stimulating interest in mathematics
among undergraduate students. Papers are presented both by
students and professional mathematicians. Briefing sessions
are held in connection with the conference to supply background
on the material presented i n the conference. Up to 2 hours
may be allowed for each paper.
Students from other schools attend. The first year there
were 12 papers presented and students from 12 institutions in
three states attending. Before beginning this program the
number of students in math was very small but recently as many
as 24 percent of upperclassmen enrolled in the institution
were math majors.
Definition 3:
I f Gl and G
a r e quasi- groups, then t h e d i r e c t
product of G and G is t h e C a r t e s i a n p r o d u c t G x G2 on which a b i n a r y
1
1
o p e r a t i o n i s defined a s follows (al,a2)
a l l (al,a2),
o
i n GI, and C i s t h e unique s o l u t i o n o f a 2
is t h e unique s o l u t i o n o f
equation y
o
(al,a2)
<al,
o
bl,a2
0
b2) f o r
a
2
o
o
x
= b i n 9- t h e n (Cl,C2)
x = ( b , b ) i n GlxG2.
= ( b1 , b 2 h a s a unique s o l u t i o n i n G
Similarly the
x G .
Thus,
t h e d i r e c t p r o d u c t of two quasi- groups is a quasi- group.
of G i n t o G .
Let G1 and G2 be quasi- groups.
a l l a , b e GI.
Then f is a homomorphism i f f ( a
Let f be a mapping
b)
0
= f(a)
0
f(b) for
I f t h e homomorphism f i s a one-one mapping o f Gl o n t o G2,
f is a n isomorphism, and G is ismorphic t o G .
1
The Governing Council of P i Mu Ensilon has approved an
increase in the maximum amount per chapter allowed as a matchinq prize from $70.00 to $25.00.
If your chanter uresents
awards for outstanding mathematical papers and students, you
may apply to the National Office to match the amount spent by
your chapter--i. e., $30.00 of awards, the National Office will
reimburse the chapter for $15.00 etc.--up to a maximum of $25.00.
Chapters are urged to submit their best student papers to the
Editor of the Pi Mu Eosilon Journal for possible publication.
= ( a
( b1'b 2 )
I f C i s t h e u n i q u e s o l u t i o n o f a, o x =
1
b1
(b,b)e G x G .
D e f i n i t i o n 4:
MATCHING PRIZE FUND
2
I n t h i s c a s e we w r i t e
Gl = G
2"
D e f i n i t i o n 5:
L e t Gl and G2 be quasi- groups.
mapping o f Gl o n t o G .
g ( a ) f o r a l l a,b,
E
G .
Let g be a one-one
Then g i s a n anti- ismorphism i f g ( a
0
b) = g(b)
E v i d e n t l y t h e c o n c i p t s o f isomorphism and a n t i -
isomorphism c o i n c i d e i f G and G a r e commutative.
1
2
0
A hoop H s a t i s f y i n g ( A ) is a group s i n c e (H 1 )
Definition 6:
a o a=a.
Let
he a quasi-group.
and (A) a r e s a t i s f i e d simultaneously.
-.H has an i d e n t i t y element.
,\H i s t r i v i a l .
Hence, no n o n - t r i v i a l hoop s a t i s f i e s ( A ) .
Then a e G i s idenpotent i f
Definition 8:
Definition 7: A non-empty s e t tl on which a binary operation a i s defined
i s a hoop ( o r medial systeir) i f t h e following p r o p e r t i e s a r e s a t i s f i e d :
Theorem 1:
11 1)
11 i s a quasi-group under a
I1 2)
a Oa=a f o r a l l a e 11
(every element of H i s idempotent)
tl 3)
( a a h ) a ( c o d) = ( a 0 c)
(the medial property).
0
a 0 (hat) = ( a 0 h ) O (sac) and
?)
(aoh)oc= (aoc)o(boc)
(Kie r i g h t and l e f t s e l f - d i s t r i h u t i v e laws).
fh
0
2
since (al, a ) " '
C)
(i)
can he proven s i m i l a r l y .
over R .
An example o f a hoop is t h e s e t K o f r e a l numhers on which a binary
i f a,b e K, then a o h =
( i i ) We s t i l l have a hoop i f we change t h e hinary operation i n t o a
weighed a r i t h n e t i c mean, i . e . , i f w and w a r e p o s i t i v e r e a l numbers,
1
wla+w h
2 =
"1
then we can d e f i n e aob=
+ (1w1 + w2
w1 + w2
w1 + w2
-
>
t h e r e e x i s t s e E tl such t h a t e o.x=x a e=x f o r a l l x
( a o b ) Â ¡ c = a o ( b ~ cf o r a l l a,h.c e H.
Theorem 2:
Let R
E
r.
Let a he medial i n R and l e t M he a
i s t h e system .(", ? ) where t h e binary operation i s
M"
Let a he medial in R, and l e t M he a module
Let R e T.
Then
V" i s
Proof:
-
a hoop.
Let a, h e
Consider t h e equation a a x = h.
t h i s t r a n s l a t e s i n t o a a + nx = h.
We may v e r i f y by d i r e c t s u b s t i t u t i o n t h a t
=
aa
+
( r l b
a(;
-1
- r1
ma) is
-1
h - a
aa) =
a solution.
aa +
Zh
-1
-aa
aa
- .
We s h a l l prove t h e s e a s s e r t i o n s l a t e r ( i n g r e a t e r generality). Another
example of a hoop i s t h e s i n g l e t o n H=
on which t h e binary operation is
defined hy e oe=e. Such a hoop i s c a x e d t r i v i a l . We note t h a t a t r i v i a l
hoop i s a l s o a group and s a t i s f i e s t h e following p r o p e r t i e s :
E)
A)
,
defined by a o h = a a + ah f o r a l l a , h e V. (b'odules a r e discussed
i n (1) pp. 145-147). We a r e now prepared t o s t a t e our c h i e f r e s u l t .
hy (H 2)
operation is defined a s t h e a r i t h m e t i c mean i.e.,
and R 2 r e s p e c t i v e l y , then (al, a 2) i s medial in Pl 0
-1
= (al
and
=
%-I).
(Dir-
Definition 10:
Theorem 3:
(2)
Moreover, i f a1 and
rings).
module over R.
c) = ( a 0 a) o (b a
i s an clement of I".
R
C
.r,
E
e c t sums a r e discussed in (1) pp. 61-62, 77 ahelian groups, (2) pp. 17-18
Proof of ( I ) :
a
a
a2 a r c medial in R
R
H
1)
.r.
a E R is medial i n R provided:
Let R E
a is i n v e r t i b l e and
2. '5'= 1-a is i n v e r t i b l e .
is medial i n R i f and only i f a is medial i n R. I f Rl, R ,
1.
then t h e i r d i r e c t sum R
( b o d) f o r a l l a,h,c,d e II
E
(R : R is a r i n g with u n i t y 1 # 0 ) .
Definition 9:
He note
I f 1 1 i s a hooq, then f o r a l l a,h,c
r=
E
Suppose x, and x2 a r e any s o l u t i o n s of a
Then a o xl = h and a 0 x2 = h.
0
x = b.
H and
No n o n - t r i v i a l hoop s a t i s f i e s p r o p e r t i e s (E) o r (A).
Proof:
-
Let H be a hoop and a e H.
Let e e H be an i d e n t i t y element.
a 0 a=a
ffl 2)
and a a e=a
by hypothesis
Both a and e s a t i s f y t h e equation a ox=a.
2. a=e
by (H 1)
Since a was a r b i t r a r y , H={e>.
Hence, no n o n - t r i v i a l hoop s a t i s f i e s (E).
.-.XI
= x2
Thus t h e s o l u t i o n of a o x = b i s unique.
Similarly, t h e r e e x i s t s a unique s o l u t i o n of y
I'.M^
s a t i s f i e s (I1 1).
0
a
h.
Let a e
I
a
o a = a a + a a
= a.
s a t i s f i e s (11 2 ) .
:,M^
(a
h)
o
Let a , h, c , d
E
o ( c o d) = a ( a o h) +;(C
d)
o
(See ( 3 ) pp. 158-159 Example 2 ) .
= a ( a a + ah) + a ( a c + ad)
= a2a
+ adh
i
an c
a t b = 1 ;
+ a d
0
2
p)
Then x
0 ( c o d) = (a a + a n c ) + (aah + a d )
= 3 a because
a?i
=a.
( a ), F^
F
:.
d e n o t e t h e hoop o p e r a t i o n s o f F a and F ' ~ )by
R e a r r a n g i n g terms and r e g r o u p i n g , we have
(a
a'l=bandb"
O
y = ax t by and x
commutative s i n c e
ft
= bx
y
t ay.
e SKF). ~ e t ' s
0
and
*
respectively.
F ' ~ )and F ' ~ )a r e non-
0 0 1 = 1 *0 = a S 0 t b .
1 o 0 = 0 * l = a . l t b .
Now c o n s i d e r t h e mapping f :
= a(aa + dc)
+
;(ah
+
ad)
d e f i n i t i o n 11:
!)(I?)
-
(M-)
Let R
F ' ~ ) ¥ F ' ~ )d e f i n e d by f ( x ) = x2.
s a t i s f i e s (H 3) and i s a hoop.
E
h u l a hoop i f 1 1 c IJ Q ( R ) .
Re r
C o r o l l a r y 4:
-
i:
M^
+
M^
Proof:
-
If R
and M ^
E
Let a , h
0
c
+
o
+
+
+
1
b
a
M.
Let M ^
= (M,
o ) and M ^
= (M,
0
.
.'.f(x
).
.
t by)
2
= a2x2
t 2abxy t bZy2
= a2x2
t 0 t bZy2 s i n c e F is o f c h a r a c t e r i s t i c 2,
y ) = a 2x 2 t b 2y 2
= bx2
h) = i ( a a + a h )
t ay
2
= bf(x)
t afty)
= ftx)
* f(y)
Hence, f i s an isomorphism, and t h e non-commutative hoops r a n d
-
= i (b)
.
i(a).
is commutative, t h e n t h e i d e n t i t y mapping i s j u s t t h e
I f M ' ~ )=
i d e n t i t y isomorphism. ( i i i ) F o r example t h e a d d i t i v e group K o f r e a l
numbers i s a module o v e r t h e r i n g K o f r e a l numbers, and 1/2 i s medial
-
i n K. The commutative hoop
b e f o r e i n example ( i ) .
(iv)
We
y ) = f ( a x t by)
= (ax
0 ( R ) , then t h e i d e n t i t y mapping
is an anti- isomorphism.
i(a
f(.x
A hoop I1 i s a
We have a s i m p l e c o r o l l a r y t o theorem 3.
E
o
1
a
b
We n o t e t h a t f i s a one-one mapping o f F ' ~ )o n t o F ( b )
.'I
= M i s an R-module and a is medial i n R l .
0 = a .
c a n look a t t h e m u l t i p l i c a t i o n t a b l e o f F t o w r i t e o u t f e x p l i c i t l y .
f
Finally
1=band
= Kcay = ^\Â¥1'2 i s j u s t t h e one we met
C o n s i d e r t h e f o u r element f i e l d F
F ' ~ )a r e isomorphic.
However, h u l a hoops come i n a n t i - i s o m o r p h i c
p a i r s which i n g e n e r a l need not be isomorphic.
Z
, z")
2 + 4 = 1 ;
emm ma
e "Z)
Z^5
For example
where Z d e n o t e s t h e i n t e g e r s modulo
~-'=3and4'~=4.
5:
(v)
~ ( 4 ?
5.
proof".et
-
h= 2 i 2 )
be a n a r b i t r a r y h m o -
"2
+
We d e n o t e t h e hoop o p e r a t i o n s of
morphism.
Z(2) and 2'
by
5
0
and
*
respectively.
h(O o 1 ) = h ( 2
0 t 4
1)
= h(4)
1. ;.h(O' 0 1 ) = h ( 4 )
But h ( 0 0 1 )
= h(0)
h(l)
= 4h(0) t 2 h ( l )
2.
+;h(O a 1 ) = 4 h ( 0 ) + 2 h ( l )
3.
.%h(4)
= 4h(0) t 2 h ( l )
Now h(O 0 4 )
= h(2 ' 0 t 4
4)
= h(16)
= h(1)
4.
.',h(O 0 4 ) = h ( l )
But
h(O o 4 ) = h ( 0 )
h(4)
= Uh(0) t 2h ( 4 )
5.
,'.h(O o 4) = Uh(0) t 2h(4)
6.
.Â¥.h(l
= 4 h ( 0 ) t 2h(4)
S u b t r a c t i n g ( 3 ) from ( 6 ) we g e t
7. h ( 1 ) - h ( 4 ) = 2h(4)
2h(l)
8.
:.3h(l)
= 3h(4)
*
I f i n theorem (FRl
d e f i n e d by
M u l t i p l y i n g ( 8 ) by 3''
h is n o t one-one.
But h: Z?
"*
(Ml 9
( = 2 ) we have h ( 1 )
= h(4)
M
2
and a l l ( a , a 2 ) c M
Theorem 6:
Let Rl,
M , ) ( ~ ) = (M, 8
is a module o v e r t h e
Theorem 8:
only i f a
~1 ( ~ e1 rfRl),
)
and ~ ~ ( e ~(l(R-1.
2 )
'2).
M2)(("'
a)).
e R and a l l ( a l ,
a ) c
Hence, we have t h e f o l l o w i n g
Let R be a r i n g and M a module o v e r R.
A(M) = h e R: Ax = 0 f o r a l l x e MI.
E MR), t h e n M ( ~is
) commutative i f and
I f R E r a n d M")
- a ( = 2 a -1)
Proof:
-
x ~ ~ ( = ~(Ml2 8) M2)("l'
R, s c a l a r m u l t i p l i c a t i o n being
= Oa1, Aa2) f o r a l l A
A(M) is non-empty s i n c e 0 is always a member; f u r t h e r m o r e . A(M) i s
e a s i l y s e e n t o be a n i d e a l i n R which we s h a l l c a l l t h e M-annihilator i n
R. ( I d e a l s a r e d i s c u s s e d i n ( 2 ) pp. 21-26, and simple r i n g s a r e d i s c u s s e d
i n ( 2 ) pp. 38-39). T h i s d e f i n i t i o n a l l o w s u s t o g i v e a simple c h a r a c t e r i z a t i o n o f commutative h u l a hoops.
0 M .
R2 e l,'
M2
@
I n t h e same s p i r i t a s above, it is e a s i l y s e e n t h a t
D e f i n i t i o n 12:
R where t h e s c a l a r m u l t i p l i c a t i o n is d e f i n e d by
2
r i n g Rl
M^l)
0
t h e n t h e group M1
9
I f H and K a r e hoops, t h e n t h e d i r e c t p r o d u c t H x K is r e a d i l y s e e n t o
be a hoop a l s o . I t is e a s i l y s e e n t h a t i f Ml is a module o v e r Rl and
M2 is a module o v e r R2, t h e n t h e group 'fl
^(al, a2)
,
corollary.
was a n a r b i t r a r y hmomorjihism.
5
M2.
Ml 9
-
Â¥*
=a
is a module o v e r R a s w e l l a s o v e r R 8
*
9.
= R2 = R and al = a
e A(M).
M ( ~ is
) commutative.
Necessity:
a o 0
Then
= 0 o a for a l l a
e M.
Thus, t h e d i r e c t product o f
two h u l a hoops is a h u l a hoop.
Proof:
-
Let ( a l , a 2 ) , (bl, b2 ) be e l e m e n t s o f t h e C a r t e s i a n
product M,x
M l a 1
M2.
F i r s t l e t ' s c o n s i d e r them a s e l e m e n t s of
-
x ~ ( ~ and
2 d) e n o t e t h e hoop o p e r a t i o n by o
Then ( a , a 2 )
0
(bl,
b 2 ) = (al
0
= (aa1
Now l e t ' s c o n s i d e r ( a l , a,)
(MI 0
bl,
t
a2
0
a1b1 'a2 a2
.
Sufficiency: a
b2)
+ ¡,b2)
a2)
(bl,
b2) = (al,
a2)
(al, a 2 ) t
- ;)a
aa
+
.
(al, a2)(bl,b2)
= (a- a l b (=0) f o r a l l a , b
ab
a o b
and ( b , b ) a s e l e m e n t s o f
= ab
=
e M^
+ a;
b o a
f o r a l l a.b c M ^
M(¡ is commutative.
~ ~ ) ( ( "~ 12 ' and d e n o t e t h e hoop o p e r a t i o n by
Then (al,
(a
- a E A(M)
C o r o l l a r y 9:
is
If R e
r,
M^
e B(R), and A(M)
= ( 0 ) t h e n , M^)
commutative i f and o n l y i f 2 is i n v e r t i b l e and a
= 2-l.
Proof:
2a
a
M
-1e
is commutative i f and only i f
A(M) = ( 0 ) .
h ( ( 2 , 1 ) ) = h ( ( 2 . 1 1 - (1,111
= (2, 1 ) x h ((1, 1 ) )
= (2, 1 ) x ( h , h )
-1
Moreover, a = 2
i f and only
= (2h, 2h)
If R e
C o r o l l a r y 10:
r
is simple, 14 # (01, and M")
E
R(R), t h e n
1.
M ( ~is
) commutative i f and only i s 2 i s i n v e r t i b l e and a = 2 " .
Proof:
-
A(M) i s an i d e a l i n R.
E
M , a # 0.
(Recall M # ( 0 ) ) .
'*Â h is n o t one-one.
But h! M + N was a n a r b i t r a r y homorphism.
.Â¥. f N.
hoop K ( " ~ ) appears i n examples ( i ) and ( i i i ) .
( v i i ) Let t h e module N over t h e r i n g K @ K be t h e a b e l i a n group
K on which s c a l a r m u l t i p l i c a t i o n is d e f i n e d by
K
= ( > a , Aa) forall
A(N) = ( ( 0 , u ) : u e K ) .
The r e a d e r may v e r i f y t h a t N l
(1, u ) e K 9
Although we do n o t have a converse t o our r e s u l t , we do have a
p a r t i a l converse a s a consequence of t h e next theorem. We s h a l l need
t h e following lemma.
Kandall ( a , a )
E
2 ) ) is a commutative member of
R(K Ã K) f o r a l l x e K - ( 0 , 1 ) .
( v i i i ) Z9 is a non-simple r i n g s i n c e 3Z9 = ( 0 , 3 , 6 ) i s a n o n - t r i v i a l
prober i d e a l .
Lemma 12:
A(Z)
If S
Proof:
-
E
r
and M"),
+
N")
defined by g ( x )
H + N^
-
d is an
Let yo e N").
Since f is o n t o , t h e r e
= Yo
t d.
Suppose g ( x l ) = g ( x 2 )
If
N ' ~ )e n ( R ) where M and N a r e isomorphic modules over R,
t h e n it is an easy consequence t h a t M '
i s isomorphic t o N u .
The converse o f t h i s statement is not t r u e .
= x2 s i n c e f is one
( l / 2 , 1 / 2 ) i s medial i n
.*g is one-one.
1. g ( a o b)
= f ( a o b)
2. g ( a ) o g ( b ) = ( f ( a )
d)
-
( x i ) Let M be t h e module K 0
(a,,
-d
o ( f ( b ) -d)
a 2 ) , ( x i i ) Let
N be t h e module of example ( v i i ) , and l e t s c a l a r m u l t i p l i c a t i o n be
( ( 1 / 2 , 1 / 2 ) ) ^ .,((1/2, 1 / 2 ) )
denoted by (A, u ) x (al, a ) . C l e a r l y M
M # N.
Proof:
and l e t
- one.
K i n which s c a l a r m u l t i p l i c a t i o n c o i n c i d e s
with r i n g m u l t i p l i c a t i o n and is denoted (A ,v )
Lemma 11:
= f(x)
,: g i s o n t o .
(ix) F ' ~ )of example ( i v ) is non-commutative s i n c e 2 ( = 0 ) is not i n v e r t i b l e .
R
H
e x i s t s xe e H such t h a t f ( x o )
= (0).
= 3 # 2.
F', N ( ~ e) Q(S), H is a hoop, and f :
isomorphism f o r a l l d e N.
The r e a d e r may v e r i f y t h a t z ~ ( is
~ )a commutative member of B(Z9).
of example ( v ) is non-commutative s i n c e 2-I
E
i s an isomorphism, t h e n g :
The group Z9 i s a module over t h e r i n g Z9, s c a l a r m u l t i -
p l i c a t i o n c o i n c i d i n g with r i n g m u l t i p l i c a t i o n i n Z9.
(x) 2")
(1, 1 ) )
Pick
', A(M) = ( 0 ) and i n view of c o r o l l a r y 9 t h e proof is completed.
( v i ) I n t h e r i n g K of r e a l numbers 2 h a s i n v e r s e 1/2. The commutative
(A,p) ( a , a )
= h((2, 2)
Since R i s simple we
must have e i t h e r A(M) = ( 0 ) o r A(M) = R .
a
1)) = (2h. 2 h )
:.h((2,
Now h ( ( 2 , 2 ) )
= f ( a o b ) -d s i n c e f i s a homomorphism.
,' g ( a
Let h: M * N be a n a r b i t r a r y homomorphism,
h((1,l)) = ( h , h ) .
0
Theorem 13:
b)
= g ( a ) o g ( b ) and g is indeed an ismorphism.
I f R,S e
r,
M(') e R(R), N(') e R ( s ) , and M ( a ) =, ( 6 ,)
t h e n M = N where M and N
M and N r e s p e c t i v e l y .
a r e t h e a d d i t i v e groups of t h e modules
Let Q : M ^
Proof:
N(')
+
Then by lemma 12, f : M^
Y(x)
-
= Q (x)
-
Y(0) = Q(0)
be an isomorphism.
-+
N")
defined by
PROBLEM DEPARTMENT
Q ( 0 ) is a l s o an isomorphism, and
Q(0) = 0.
Edited by
Moreover, s i n c e Y is a hoop
U.
ismorphism, it is a one-one mapping of t h e s e t
M onto t h e s e t N.
1. Y(a^x
= ~(aa'lx
0)
o
2.
'i'(a-lx
o
0)
This department welcomes problems believed to be new and, as a rule,
demanding no greater abiISty in problem solving than that of the average
member of the Fraternity, but occasionally we shall publish problems that
should challenge the advanced undergraduate and/or candidate for the
Master's Degree. Solutions of these problems should be submitted on
separate signed sheets within four months after publication.
t Fo)
.
= Y(x)
= ydi'lx)
o
Y(O)
= ~(a^x)
o
o
An asterisk ( * ) placed beside a problem number indicates that the
problem was submitted without a solution.
= ~ ~ ( a - ~+ B
x )0 o
Address all communications concerning problems to M. S. Klamkin,
Ford Scientific Laboratory, P. 0. Box 2 0 5 3 , Dearborn, Michigan 48121.
= Bv(a--'-x)
= Y(X)
3.
BY (a'^x)
4.
~(a'lx)
Similarly
S. Klamkin, Ford Scientific Laboratory
PROBLEMS FOR SOLUTION
= 6'S'(x)
=
Y ( F '^XI
f o r a l l x e M.
6 "YCX) f o r
192.*
a l l x e M.
Proposed by Oystein Ore, Yale University. Albrecht Durer's famous etching "Melancholia" includes the magic square
Since Y is a one-one mapping o f M o n t o N, it s u f f i c e s
t o show t h a t Y is a group homorphism.
Let a.b e Mt.
a(!'
t b)
= ~ ( a a ' l a + EF '^b)
= Y
a -'b)
o
= ~(a^a)
o
Y(F
= BI 'Ma)
o
B
= ~ 6 - l Y(a)
=+
'('a)
t
b)
~ ( b )
IB
~ ( b )
t Y(b)
The boxed-in numbers 15-14 indicate the year in which the picture
was drawn. How many other 4 x 4 magic squares are there which he
could have used in the same way?
I
w
193.
Y : M + Nt is an ismorphism.
Two ships are steaming along at constant velocities (course and
speed). If the motion of one ship is known completely, and if
only the speed of the second ship is known, what is the minimum
number of bearings necessary to be taken by the first ship in order
to determine the course (constant) and range (time-dependent) of
the second ship? Given this requisite number of bearings, show
how to determine the second ship's course and range.
References:
(1.)
Hu, S.T., Elements o f Modern Algebra, ~ o l d i n - ~I nac~. , San
~ r a n c i s c o . 1 9 6-
(2,)
McCoy, N. H.,
York, 1964.
(3. )
McCoy, N. H., I n t r o d u c t i o n t o Modern Algegra, Allyn and Bacon,
Inc. Boston, 1960.
-
The Theory o f Rings, The Macmillan CompanyI New
I thank D r . Hwa Tsang f o r h e r a i d .
Proposed by William H. Pierce, General Dynamics, Electric Boat
Division.
Editorial
Note:
This problem was suggested by problem 186 which was given erroneously. See the comment on 196 (this issue).
194.
Proposed by J. M. Gandhi, University of Alberta.
Show that the equation
h a s no s o l u t i o n s i n i n t e g e r s e x c e p t t h e s o l u t i o n s ( i ) x = '1, y = * l ,
( i i ) x = 3 , y = 9.
195.
196.
Proposed by Leon Bankoff, Los Angeles, C a l i f o r n i a .
Math. Mag. ( J a n . 19631, p. 6 0 , c o n t a i n s
a s h o r t p a p e r by Dov Avishdom, who a s s e r t s
w i t h o u t proof t h a t i n t h e a d j o i n i n g
diagram, AN = NC t CB. Give a p r o o f .
Proposed by R. C. G e b h a r t , P a r s i p p a n y , N. J .
S i n c e 1 = I1 = I2 = 1 ( i n a r e a ) , e a c h o f t h e t r i a n g l e s ABR, BCP, and CAQ would
have g r e a t e r a r e a t h a n AABC c o n t r a d i c t i n g t h e lemma. The c a s e of two p a r a l l e l
s u p p o r t l i n e s is a l s o r u l e d o u t e a s i l y by a r e a c o n s i d e r a t i o n s .
182.
Although E u l e r e s t a b l i s h e d t h e more g e n e r a l r e s u l t t h a t every
E d i t o r i a l *:
odd p e r f e c t number must be of t h e form p'tatl k where p i s a prime o f t h e
fnm 4 n + l , t h e above problem was g i v e n s i n c e it h a s a n e l e g a n t s o l u t i o n i n
a d d i t i o n t o t h e f a c t t h a t t h e r e was a s h o r t a g e of p r o p o s a l s .
/
What is t h e remainder i f xlOO i s d i v i d e d
by x2
197.
-
A--
3x t 2?
S o l u t i o n by H. Kaye, Brooklyn, N . Y.
I f N i s an odd p e r f e c t s q u a r e t h e n
1. The sum o f i t s d i v i s o r s = 2N,
2. The number of i t s d i v i s o r s , e a c h b e i n g odd, must a l s o be odd s i n c e e x c e p t
f o r /N, t h e d i v i s o r s o c c u r i n p a i r s d and N/d.
Proposed by J o s e p h Arkin, Nanuet, N. Y.
A box c o n t a i n s (1600 u t 3200)/3 s o l i d
s p h e r i c a l m e t a l b e a r i n g s . Each b e a r i n g
i n t h e box h a s a c y l i n d r i c a l h o l e o f
l e n g t h .25 c e n t i m e t e r s d r i l l e d s t r a i g h t
through i t s c e n t e r . The b e a r i n g s a r e t h e n melted t o g e t h e r w i t h a l o s s o f 4%
d u r i n g t h e m e l t i n g p r o c e s s and formed i n t o a s p h e r e whose r a d i u s i s an i n t e g r a l
number o f c e n t i m e t e r s . How many b e a r i n g s were t h e r e o r i g i n a l l y i n t h e box?
198.
199.
Proposed by G a b r i e l Rosenberg, Temple U n i v e r s i t y .
Prove t h a t a n odd p e r f e c t number cannot be a p e r f e c t s q u a r e .
Proposed by S t a n l e y Rabinowitz, P o l y t e c h n i c I n s t i t u t e o f Brooklyn. A semir e g u l a r s o l i d is o b t a i n e d by s l i c i n g o f f s e c t i o n s from t h e c o r n e r s of a cube.
It is a s o l i d w i t h 36 congruent e d g e s , 24 v e r t i c e s and 1 4 f a c e s , 6 o f which a r e
r e g u l a r o c t a g o n s and 8 a r e e q u i l a t e r a l t r i a n g l e s . I f t h e l e n g t h o f an edge o f
t h i s p o l y t o p e i s e , what is i t s volume.
T h i s g i v e s a c o n t r a d i c t i o n s i n c e t h e sum o f a n odd number o f odd numbers is
OL...
Also s o l v e d by P a u l J. Campbell ( U n i v e r s i t y o f Dayton), P a u l Myers ( P h i l a d e l p h i a ,
P a . ) , Bob P r i e l i p p ( U n i v e r s i t y o f Wisconsin), S t a n l e y Rabinowitz ( P o l y t e c h n i c
I n s t i t u t e of Brooklyn), Dennis Spellman (Temple U n i v e r s i t y ) , M . Wagner (New York,
N. Y.), F. Z e t t o (Chicago, I l l i n o i s ) and t h e p r o p o s e r .
183.
Proposed by R. Penney, Ford S c i e n t i f i c Laboratory.
If
r ( r r t r r t r r ) = O ,
0 1 2
2 3
3 1
r ( r r + t o r 3 - r r ) = O ,
1 0 2
2 3
Proposed by Larry Forman, Brown U n i v e r s i t y and M. S. Klamkin, Ford S c i e n t i f i c
Laboratory.
m
t
= z ,
Find a l l i n t e g r a l s o l u t i o n s of t h e e q u a t i o n '
3m
r ( r r t r r - r r ) = O ,
2 0 3
0 1
3 1
r ( r r t r 0 r - r r ) = O ,
3 0 1
1 2
SOLUTIONS
181.
Proposed by D. W. Crowe, U n i v e r s i t y o f Wisconsin and M. S. Klamkin, Ford
S c i e n t i f i c Laboratory.
Determine a convex curve c i r c u m s c r i b i n g a g i v e n t r i a n g l e s u c h t h a t
1. The a r e a s o f t h e f o u r r e g i o n s ( 3 segments and a t r i a n g l e ) a r e e q u a l and
2. The c u r v e h a s minimum p e r i m e t e r .
S o l u t i o n by t h e p r o p o s e r s . C o n d i t i o n ( 2 ) is i r r e l e v a n t . The o n l y convex
f i g u r e s a t i s f y i n g ( 1 ) is a c i r c u m s c r i b e d t r i a n g l e whose s i d e s a r e r e s p e c t i v e l y
p a r a l l e l t o t h e s i d e s of t h e g i v e n t r i a n g l e . Our proof depends on t h e known
Lemma* The a r e a o f any i n s c r i b e d t r i a n g l e i n a g i v e n t r i a n g l e cannot be s m a l l e r
t h a n e a c h o f t h e o t h e r t h r e e t r i a n g l e s formed. Consider a convex c u r v e t h a t is
n o t a t r i a n g l e and draw t h e t h r e e l i n e s o f s u p p o r t a t t h e v e r t i c e s .
Show t h a t a t l e a s t one o f t h e q u a n t i t i e s r 0 , r , r 2 , r 3v a n i s h e s .
S o l u t i o n by S t a n l e y Rabinowitz, P o l y t e c h n i c I n s t i t u t e o f Brooklyn.
I f none o f t h e r ' s v a n i s h , t h e n t h e e x p r e s s i o n s i n t h e p a r e n t h e s i s must v a n i s h .
These c o n d i t i o n s may t h e n be w r i t t e n a s
(1)
1/r1 t 1 / r t 1 / r = 0
(2)
1/r
(3)
1 / r t l/rl = l / r o ,
(4)
1/r1 t 1 / r = l / r o
+
,
1 / r = l/ro ,
.
Adding up ( 2 ) . ( 3 1 , and ( 4 ) and u s i n g ( 1 ) g i v e s 3 / r 0 = 0 . w h i c h is i m p o s s i b l e .
Hence, a t l e a s t one o f t h e r ' s must v a n i s h .
E d i t o r i a l E:T h i s problem h a s a r i s e n i n t h e p r o p o s e r ' s p a p e r , "Geometrization
o f a Complex S c a l a r F i e l d . I. Algebra," J o u r . o f Math. P h y s i c s , March 1966,
It a l s o f o l l o w s t h a t a t l e a s t 2 r's must v a n i s h . The problem can
pp. 479-481.
be extended t o any number o f v a r i a b l e s , e . g . , f o r 5 v a r i a b l e s we would have:
For a p r o j e c t i v e p r o o f o f t h i s lemma and o t h e r r e f e r e n c e s , s e e M. S. Klamkin
and D. J. Newman, t h e Philosophy and A p p l i c a t i o n s o f Transform Theory, SIAM
Review, Jan. 1 9 6 1 , p. 14.
If
I t follows i n a s i m i l a r fashion a s before t h a t a t l e a s t two of t h e r's must vanish.
Also solved by Paul J . Campbell (University of Dayton), Mario E. E t t r i c k
(Brooklyn, N. Y.), R. C. Gebhart (Parsippany, N. J . ) , Theodore Jungreis (New
York University), Bruce W. King (Burnt H i l l s , N. Y.), Edward Lacher (Rutgers
u n i v e r s i t y ) , Paul Myers ( P h i l a d e l p h i a , Pa. 1, Charles W. Trigg (San Diego, C a l i f . 1,
F. Zetto (Chicago, 111.) and t h e proposer.
184.
then
and
Proposed by Alan Lambert, University of Miami.
Find a parametric r e p r e s e n t a t i o n f o r transcendental curve given by t h e equation
x" = yx
.
Similarly,
Solution by Bruce W. King (Burnt H i l l s , N . Y.) and P e t e r Lindstrom (Union College).
Let y = x t . Then s i n c e
ylogx = xlogy
we have,
tlogx = logxt
= tl/(t-l)
or
= tt/(t-l)
and
E d i t o r i a l Note: This parametric r e p r e s e n t a t i o n was already known t o Euler.
can a l s o be used t o determine a l l r a t i o n a l s o l u t i o n s of t h e above equation.
185.
It
9 ,,
Also solved by R. C. Gebhart (Parsippany, N. J . ) H. Kaye (Brooklyn, N. Y.),
Stanley Rabinowitz (Polytechnic I n s t i t u t e of Brooklyn, F. Zetto (Chicago, 111.)
and t h e proposer.
Bun,
Proposed by A. E. Livingston and L. Moser, University of Alberta.
I n oilier t o have convergence,
f (n) = f(n+3)
Given:
,
n = 1,2,
...
5
,
=
limit
{f(l)G(l) + f ( 2 ) H ( l ) + f(3)1(l)}
f(l)+f(2)+f(3)=
,
0
and t h e n
Find
f(2).
S o l u t i o n by
R.
Since f ( 1 ) = 1, we have two l i n e a r equations i n f ( 2 ) and f ( 3 ) , leading t o t h e
previous r e s u l t s . Another way of solving t h e problem is v i a t h e function
C. Gebhart ( ~ a r s i p p a n ~N.
, J. ).
S t a r t i n g from t h e known summations,
2
1
cos n x
n
= -1n2sin
2
, 1E
-=7 ,
sinnx
IT-x
Here, we consider t h e more g e n e r a l problem o f summing
we can o b t a i n t h e p a r t i c u l a r sums
where
r,s
a r e r a t i o n a l and
r > 0, 1 < s
5
r.
For convergence, it i s necessary and s u f f i c i e n t t h a t
Nor m u l t i p l y (1)by -a
JL+%+.-.
and (2) by -b.
+As
=
0
.
T h i s w i l l give t h e i n i t i a l s e r i e s i f
a=f(2),
b=f(3)
,
and
we o b t a i n
= -j =il A3$ ( $ + l >
Whence,
Since
E d i t o r i a l Note: A more d i r e c t way o f summing t h e s e type of s e r i e s would be t o
use g e n e r a t i n g functions which l e a d t o t h e evaluation o f d e f i n i t e i n t e g r a l s .
$(x)
.
is known e x p l i c i t y f o r r a t i o n a l x, t h e s e r i e s can
a l s o b e e x p l i c i t l y evaluated.
$(I)
= -y
- ;.(og3
-2
J
$ ( l ) = -Y
2 /3
.
Using these r e s u l t s on our o r i g i n a l problem, we f i n d t h a t
BOOK REVIEWS
Roy B.
E d i t e d by
D e a l , Oklahoma U n i v e r s i t y M e d i c a l C e n t e r
Also solved by t h e proposers.
1.
186.
Proposed by L t . Joned-Bateman, U.S.C.G. and M. S. Klamkin, Ford S c i e n t i f i c
Laboratory.
Two ships a r e steaming along with constant v e l o c i t i e s . What i s the minimum
number of bearings necessary t o be taken by one ship in order t o determine the
velocity of the other ship? Given t h i s r e q u i s i t e number of bearings, show hob
t o determine the other s h i p ' s velocity. ( I t i s assumed that range-finding
equipment is e i t h e r non-existent o r non-operative).
Solution by William H. Pierce, General Dynamics, E l e c t r i c Boat Division.
Under t h e assumption t h a t both ships a r e proceeding a t constant v e l o c i t i e s
(courses and speeds), then no number of bearings taken by one ship can determine
the velocity of the other ship. To fee t h i s , l e t the parametric coordinates of
the two ships be denoted by
EXements of Real Analysis.
California,
By Sze-Ben Ikl, Holden-ky, Inc.,
San b a n c i s c o ,
1967.
A comprehensive, but readable, beginning graduate l e v e l introduction t o
Real Analysis, following the C U M recommendations with some optional
material. It t r e a t s t h e fundamentals of topological spaces, metric
spaces, topological l i n e a r spaces, including Hilbert spaces "studied
l e i s u r e l y in a s i n g l e section," measures and i n t e g r a l s , including
Daniel1 i n t e g r a l s and. Haar measure, and b r i e f , but modern, introduct i o n s t o d i f f e r e n t i a t i o n and d i s t r i b u t i o n s .
2.
Exercises in Mathematics. Ey Jean Bass, Academic Press, Inc., New York,
w
i
+
k57 pp., $14.75.
A valuable c o l l e c t i o n of c a r e f u l l y worked out problems i n r e a l and complex
analysis, ordinary and p a r t i a l d i f f e r e n t i a l equations, and various aspects
of applied mathematics.
3.
=
+ vgtasgg
,
y' = b'
2
2
+
v't sine'
2
2
where
4.
Editorial
problem.
Note:
5.
P l e a s e b e s u r e t o l e t t h e P i Mu E p s i l o n J o u r n a l know!
Send
y o u r name, o l d a d d r e s s w i t h z i p c o d e a n d new a d d r e s s w i t h
z i p code t o :
P i Mu E p s i l o n J o u r n a l
Department o f Mathematics
The U n i v e r s i t y o f Oklahoma
Norman Oklahoma 73069
LC.,
D i f f e r e n t i a l Geometry.
By Louis Auslander, Harper and Row, New York, 1967.
A f i n e modern treatment of c l a s s i c a l d i f f e r e n t i a l and global geometry
f o r t h e advanced undergraduate or f i r s t year graduate with two years
of calculus and one year of modern algebra. Prerequisite of algebraic
topology i s s k i l l f u l l y avoided. An i n t e r e s t i n g f e a t u r e i s t h e t r e a t ment of matrix Lie groups instead of a b s t r a c t Lie groups.
See problem 193, t h i s issue f o r a corrected version of t h i s
MOVING??,
Ey Paul R. Hahas, Van Nostrand Co.,
A unique c o l l e c t i o n of problems and solutions of varying l e v e l s of
d i f f i c u l t y designed t o i l l u s t r a t e t h e " central techniques and ideas of
The user of t h i s book should have some knowledge of general
t h e subject.
topology, r e a l and complex analysis, l i n e a r topological vector spaces,
including Hilbert spaces, and r e a l and. complex analysis. The f i r s t one
hundred and f o r t y pages l i s t t h e problems with b r i e f h i s t o r i c a l and
mathematical comments and a few d e f i n i t i o n s . The next 22 pages give h i n t s
t o t h e solutions and the l a s t 185 p q e s give f a i r l y complete solutions.
Then i t follows e a s i l y t h a t the following family of t a r g e t ships w i l l generate
the same bearings as the t a r g e t ship above with respect t o the f i r s t ship:
Xg
A Hilbert Space Problem b o k .
Princeton, New Jersey, 1 9 6 7 .
Mundary Value Problem6 of Mathematical Physics.
MacMillan Co., Riverside, New Jersey, 1 9 6 7 .
By I v a r Stakgold,
For t h e graduate students i n applied mathematics, engineering, and physical
sciences, with advanced calculus and elementary complex variables, t h i s
f i r s t of two volumes on l i n e a r boundary value problems i s an excellent introduction t o t h e study of Green's Functions, including t h e theory Of
d i s t r i b u t i o n , and eigenfunction expansions i n l i n e a r i n t e g r a l equations
and. t h e s p e c t r a l theory of second-order d i f f e r e n t i a l operators.
.
A Second
Course I n C
-Inc , New York, ~6f???
Addition Theorems.
x i + U3 pp.7$8.75.
By Henry B. Mann, John Wiley, Inc.,
New York, 1965,
I h i s l i t t l e book on additive theory of s e t s i n number theory and more
general Abelian groups or semi groups affords one of the best opport u n i t i e s f o r t h e interested reader with a l h i t e d bac&round, such as a
course i n modern algebra and a minimal introduction t o number theory,
t o follow some recent, important research r e s u l t s i n mathematics.
Information A S c i e n t i f i c American Book. By W. H. Freeman and Co., San
i l l u s t r a t i o n s.
, 15 -d a t e s , ClothFrancisco, California, 1966,218- - .
bound $5.00, ~aperbou&$2.50.
A collection of the faclnating s e r i e s of a r t i c l e s appearing i n the
September, 1966 S c i e n t i f i c American on the s t a t e of the computer art and
i t s ' impact on society.
By William A Veech, W. A. Benjamin,
=$8.75.
A very good book f o r those with an elementary course i n complex varables
and general topol-y who would l i k e fundamental grounding i n some important aspects of t h e subject including t h e prime numbers theorem.
By Guggenheimer,
Plane Geometr and Its Grou s
Francisco, c a k i ~ &
n ~
Holden-Day, Inc., San
Although t h e generation of motions by reflections i s classic, t h i s
treatment and selection of topics has such a modern flavor t h a t many
students of today w i l l f i n d it interesting and profitable t o pursue.
Foundations g Mechanics. By Ralph Abraham, W. A. Benjamin, Inc.,
York, 1967, ix + 290 pp., $14.75.
New
I h i s i s t r u l y a foundations text. It i s based on a s e r i e s of lectures
directed primarily a t physicists a t the graduate level, but t h e mathematical background i s so well presented t h a t it might also serve a s an
excellent introduction f o r mathematics majors t o a modern treatment of
d i f f e r e n t i a l manifolds vector and tensor bundles, and integration on
manifolds. The study of c e l e c t i a l mechanics contains very recent basic
results.
J
--
Nunbere and Ideals.
California,
By Abraham Robinson, Holden-Cay, Inc.,
San Francisco,
A very elementary introduction t o algebraic number theory with i t s own
introduction t o mdern algebra i n a small book t h a t could be stuiiied,
f o r example, by freshman honor students i n mathematics.
Applied Combinatorial Mathematics. DUted by Etlwir~F. Beckenback, John
Wiley, Inc., New York, 1964, %xi + 591 pp., $13.50.
A collection of eighteen chapters by outstanding mathematicians,
covering a wide v a r i e t y of aspects of t h i s much recently emphasized
aspect of mathematics. Varying levels, generally from advanced calculus
on a r e required t o seriously study t h e a r t i c l e s .
Mathematical L a i c .
1967.
By Stephen Cole KLeene, John Wiley, Inc.,
New York,
A junior ( a l s o written f o r juniors) addition of t h e author's famous
"Introduction t o Metamathematics" which should, prone t o be one of the most
popular books on the subject f o r Mathematics majors.
Evolution-of Mathematical
San Pranciso, California,
NOTE: All correspondence concerning reviews and all books for
review should be sent to PROFESSOR ROY B. DEAL, UNIVERSITY OF
OKLAHOMA MEDICAL CENTER, 800 NE 13th STREET, OKLAHOMA CITY,
OKLAHOMA 73104.
Topological Vector Spaces and Distributions By John Horvath, AddisonWesley Publishing Co., Inc., Reading, Mass., Vol. I., 1966.
Ccmmter Pr ranmi
Wiley, IncN
:ew
.
Y
.,
and Related Mathematics
Z k ~ g pp
m $6-50.
By R. V. Andree, John
Abstract Theory of Groups By 0. U. Schmidt, Translated from t h e Russian
by Fred H o l l h g and J. B. Roberts, m i t e a by J. B. Roberts, W. E. Freeman
and Co., San Francisco, California, 1966, 174 pp., $5.00.
Value Distribution
and x
Klyoshi
N. J. Van
Nostrand
Co., Inc., Theory
Princeton, Leo
N. Sario
J., 1967,
i + 235Noshiro,
pp., $11.50.
Linear
Geomet
K.
Gruenberg
b c . , Princet:,
I?.
J., W.1967,
viii +and
I BA.pp.,
~ J. Weir,
$6.00. Van Nostrand and Co.,
By
I h6 y h t . By &schkowski, Holden-Day, Inc.,
19 5 155 pp., $5.95.
By
This book, translated from the German, i s an interesting discussion
"about" t h e foundations of mathematics f o r a wide range of readers i n
both i n t e r e s t and l e v e l of background.
Geometric Pr
ramm
~ e &i &w
%!78
By Duffin, Peterson and Zener, John Wiley, Inc.,
pp., $12.50.
Although t h e material was collected f o r use in solving engineering design
problems, and many examples a r e given, many readers, interested i n
optimization problems such as generalization of l i n e a r programming
problems, would, f i n d t h i s a profitable book t o study.
Ihe Theoq of Gambli and S t a t i s t i c a l Lcgic. By Richard A. Epstein,
Academic P r ~ s d N e w
York, 1967, x i i i + 485 pp., $10.00.
A scholarly t r e a t i s e on gambling with many thought-provoking ideas on a
wide v a r i e t y of subjects a s well a s some interesting mathematical r e s u l t s
i n probability theory and game theory f o r readers a t about the advanced
calculus l e v e l of mathematical maturity.
Set Theory and the Number System6 By May Risch Klnsolving, International
Textbook Co., Scranton, Penn.
Vector
Belmont,Analytic
C a l i f o Ge-t
rnia,5.
By Paul A. White, Mckenson Fublishing Co.,
Inc.,
.,
Introduction t o Modem Abstract Algebra By David M. Burton, AddisonWesley ~ublishingCo., m a d i n g , ass 1967.
Abstract Algebra By Chih-Hah Sah, Academic Press, Inc.,
x i i i + 335 PP., $9.75.
New York, 1966,
Theory of Functions of a Complex Variable Vol. 11 By A. I. Marhushevlch,
Translated from t h e & h n by Richard A. Silve--,
P r e n t i c e - W , Inc.,
Englewood CU-ffs, N. J., x i i + 329 pp., $12.00.
INITIATES
T h e o r y of F u n c t i o n s of a C m p l e x Variable Vol. I11 By A. I. Markushevich,
T r a n s l a t e d f r o m t h e R u s s i a n by Richard A. Silvern, Printice-Hall, Inc.,
Englewood Cliffs, N. J., 1967, x i
355 pp., $12.95.
+
I n t r o d u c t i o n t o R e a l Analysis
By
Goffman,
Casper
Harper a n d Row,
New
York, 1966.
Vectors, Tensorsand G r o u p s By 'Hior A.
Benjamin, Inc.,
New York, 1967.
of One and Several
W. A. Benjamin,Inc.,Mewrk,
Functions
Bale
a n d J o n a s Lichtenburg,
W. A.
By Thor A. Eak and. Jonas Lichtenburg,
Variables
1967.
York, 1967.
Series, D i f f e r e n t i a l Eouations a n d C o m p l e x F u n c t i o n s
New
W.
A.
ALABAMA ALPHA, University of Alabama
Elaine E. Alexander
Ronald A. ~ t k i n s o n
Charles A. Barnes
Bettve D. Beroen
~ o h n ~ . - ~ l a n dJr.
;
Robert C. Bradley
Cecil W. Bridges
Marion A. Cabaniss
Patrick A. Callahan
Donald A. Campbell
Wavne Heath Causev
~ d w i r d ~ . c h i l d i j~ r .
Ronald James Cornutt
Robert
.- - - - W.. Cowden
-.- Helen D. Darcey
Carolyn Darden
Sheila Anne Denholm
Ralph 0. Doughty
Eleanor Ann Fitts
Joyce Gwin Fox
Mary E. Garrett
Billy Gene Gibbs
Eugene F. Glass
Charles H. Gunselman
James P. Hayes, Jr.
Nevin Julian Henson
Seth J. Hersh
John E. Holleman
James 0. Howell
John W. Howerton
Michael J. Huggins
Margret E. Ingrain
Claud James Irby
James R. Jackson
Daniel J. Jobson
George Thomas Jones
Haim Kaspi
Emogene Knight
Saksiri E. Kridakorn
Jerry Kenneth Lewis
Jimmie R. Lindsay
Larry B. McDonald
Robert Lee Manasco
Linda K. Manduley
Janet Louise Mate
Lianna D. Mulliqan
Michael Terry Newton
Charles M. North, Jr.
Robert R. Parker
Jonathon Raffield
Fred A. Roberson
Ginger Sue Roberts
Robert L. Robinson
Emily Jean Sabo
Mary V. Schorn
Linda Carole Smith
William B. Smith
Billy C. Tankersley
Herman E. Thomason
Donald G. Turner
Claudia E. Vookles
James L. Whitson
Linda Howell Wise
Ronald Lee W w t e n
James Addison Wright, JI
Laura Johnston
Glenda M. Jones
Steven G. Joseph
Judy A. Kennedy
Lynne Mielke
Rex K. Rainer, Jr.
Frank P. Ramsey, Jr.
Beverly J. Upchurch
Patricia F. Villafann
Phillip A. Wallace
Stephanie J. Wallace
Stanley W. Milwee
Karen Anne Monroe
Teresa Ann Morgan
Sarah C. Morris
Helen B. Murphree
Abdul A. Nafoosi
William D. Powell
Jarrett Richardson
Wayne Shaddix
Amy Sue Sims
Shera Lynne Thackery
Barbara E. Thompson
Martha C. Thorpe
Rex
G.
. Walker
-. .
-- -.
Marjorie Watson
Judy L. Wells
Ed Wheeler
John F. Whirley
Aubrev H. White
~rend: P i yarnell
Letitia G. Yeager
Catherine E. Parry
J. Byron McCormick
Michael J. Merchant
Richard E. Rothschild
Dale B. Nixon
Van 0. Parker
Michael A. Plunkett
Mauer D. Schwartz
Gay R. Reid
Richard R. Schurtz
Mavis A. Upton
Chang-Shing Chen
Richard J. Clasen
Carol Grace Cook
Arthur A. Duke
Maxine Eltinqe
John March Gardner
David B. Gauld
Bernard A. Glassman
Jerrold M. Gold
N. Grossman
Benjamin, Inc.,
ALABAMA BETA, Auburn University
Introduction t o Probabilit a n d S t a t i s t i c s
w o r t h P u b ~ s h T n gCo., B e l m t n r C a l i f o r n i a ,
text, $11.35 trade.
By W i l l i a m Menderihall, Wads1967, x i i i + 389 pp., $8.50
I
Guidelines For T e a c h i n g Mathematics By D o n o v a n A. Johnson a n d Gerald R.
Hising, W a d s w o r t h Publishing Co., Inc., Belmont, California, 1967, v i i i
+ 439 PP., $6.95.
Leadbe=,
Informal A r o a c h
8.*-
In
B r o o k s / ~ o l e Publishing Co.,
50.
and R e l a t e d Stochastic
John Wiley, Inc., New
C o m b i n a t o r i a l Methods
Tatoc, John w i n e ;
Processes
7
,
By H a r o l d C r a m e r and
xii
+ 345
pp.,
of Stochastic Processes
York, 1957, x i + 255 -75.
t h e Theory
F i r s t Order Mathematical Loqic
Publishing Co., Waltham, ' l a s s . ,
M.
R.
$12.50.
By L a j o s
B y Angelo 'fargaris, B l a i s d e l l
1967.
NOTE: A l l correspondence concerninq reviews and a l l books f o r
review should be s e n t t o PROFESSOR ROY B DEAL, UNIVERSITY OF
OKLAHOMA MEDICAL CENTER, 800 M
E 13th STREET, OKLAHOMA CITY,
.-by,-
Ruby Dean
Susan A. Donne11
Harold E. Dowler
Janet S. Farris
Martha L. Graves
Lawrence M. Gorham
ALABAMA GAMMA, Samford University
I n t r o d u c t o r G e o m e t r : An
Behont, California, 'ljff~,
Statio
Carolyn H. Adkins
Robert 5. Barnes
Phyllis S. Boozer
Dianne Burgess
Lee R. Christian
..,-..- --...
Sharon Blice
Larry T. Bolton
E. Delilah Carter
Barbara P. Chandler
Floyd L. Christian
Richard H. Collier
Janice E. Cox
Wayne B. Faulkner
Ernest G. Garrick
Charles A. Greathouse
Robert B. Hall
Carey R. Herring
Charlotte Jarrett
M. Carolyn Keyees
Dorothy C. King
Phyllis I. Laatsch
Chester R. Killy
Dianne Crews McDaniel
Wayne K. Meshejian
Helen B. Miller
ARIZONA ALPHA, University of Arizona
H. Bradford Barber
Arts E. Dameani
Robert E. Darwin
Jack M. Kemler
James I. Leong
ARKANSAS AuPHA, University of Arkansas
John R. Bass
James P i Douglas
Russell H. Ewing
Jerry A. Jenkins
James P. Luette
William H. Ma
Jerrel D. Martin
Susan N. McManus
CALIFORNIA ALPHA, University of California
NEED MONEY?
The Governing Council o f Pi Mu Epsilon announces a c o n t e s t
f o r the b e s t e x p o s i t o r y paper by a student (who has n o t y e t
r e c e i v e d a masters degree) s u i t a b l e f o r p u b l i c a t i o n i n t h e
P i Mu Epsilon Journal.
There w i l l be t h e f o l l o w i n g p r i z e s
$200 f i r s t p r i z e
$100 second p r i z e
$50 t h i r d p r i z e
providing a t l e a s t t e n papers are r e c e i v e d f o r t h e c o n t e s t .
In a d d i t i o n there w i l l be a $20 p r i z e f o r t h e b e s t paper
from any one chapter, provided t h a t chapter submits a t l e a s t
f i v e papers.
Nguyen Huu Anh
David J. Bailey
M. Kenneth Bates
John D. Billings
Sonia Munn Bloom
Charles R. Bosworth
Jeanne I. Brown
Patrick M. Brown
Edgar Coleman
Lois Ann Chaffee
A. Hales
Thomas L. Henderson
Barry Connick Howard
Keith Joseph
Jeffrey Asch Karas
Martha R. Katzin
Melissa Knapp
Don Krakowski
,
Young Koan-Kwon
Michael King-Xing Lai
Richard Lewis
Wickham Low
Victor Marfoe
Yashaswini Deval Mittal
Hadi J. Mustafa
Samuel D. Oman
Nicholas Perceval-Price Arthur Tashima
James Thompson
Gabriel 5. Salzman
Edward Turner
U. Kurt Scholz
Leonard 0. Weisberg
Donald J. Secor
Norman Weiss
Charles B. Sheckells
Ellen L. Wong
Hiroshi Shimada
Herbert Siege1
Kenneth J. Wong
Daniel A. Stock
Virginia Zwnkovich
Leon M. Traister
CALIFORNIA ZETA, University of California at Riverside
Brian W. Bauske
Donald P. Brithinee
Wallace P. Brithinee
~ o b e r tEarle Carr
Alfred James Delay
Lynnette Rosalie Daroca
James Dunbrack
Abraham Dykstra
William A. Fanner
Sheryl I. Hampton
John Dale Harris
David George Hazard
Jfadith Marie James
Kent D. Mason
Kenneth Lee Holnar
Robert Norman Reber
Greg W. Scragg
Pamela Kay Sierer
Suzanne Smith
Michael J. Stimson
Julianne Tracy
Robert James Vrtis
FLORIDA ALPHA, University of Miami
Marilyn Bennan
Harvey L. Bernstein
Jeffry Ben Fuqua
Habib H. Jabali
William F. Kern
Jeanne Eve Melvin
William Frank Page
Geoffrey L. Robb
Vaughn D. Rockney, Jr.
Juan Luis Sanchez
Kenneth M. Simon
John A. Moore
Daniel J. Mundrick
Sam S. Narkinsky
Brian R. O'Callaghan
Susan E. Richardson
Kennit Rose
Richard W. Ross
Yaichi Shinohara
Ilk0 P. Shulhan
James Richard Smotryeki
Bennett M. Stern
L. Lee Williams
Thomas G. Parish
Harriet C. Seligsohn
FLORIDA BETA, Florida State University
Gary L. Alderman
William Walton Alfred
Wayne Curtis Barksdale
John M. Bowling
Charles M. Bundrick
Tanya A. Connor
William R. Crowson
Leila E. Donahue
Martin J. Fischer
Mary Ellen Gaston
Charles W. Jernigan
Douglas Julian Jones
Robert E. Lorraine
Mary Louise Miner
FLORIDA EPSILON, University of South Florida
LaRay E. Geist
Peter T. Laseau
CALIFORNIA ETA, The University of Santa Clara
Gerald Alexanderson
Karel De Bouvere
Sherrill Ford
Leonard Klosinski
Thomas Kropp
Bruce McLemore
Russell Meredith
Richard Schweickert
Dennis Smolarski
Irving Sussman
Charles Swart
Kathleen Weland
Ilgee Moon
3EORGIA ALPHA, University of Georgia
Susan Gerald
John P. Holmes
COLORADD GAMMA, United States Air Force Academy
Robert J. Barnum
Charles L. Clements
Michael N. Gilea
Edward A. Greene, I1
Joseph Hasek
Ronald L. Kerchner
Peter L. Knepell
Edward C. Land, I1
Frederick K. Olafson
Howard L. Parris, Jr.
Arthur R. Miller
Eugene A. Rose, 111
Jeffrey E. Schofield
Robert Rue
Craig C. Squier
John J. Watkins, Jr.
Gary N. Wiedenmann
Roger L. Wiles
Michael C. Wirth
Donald R. Windham
CONNECTICUT ALPHA, University of Connecticut
Joseph Adamski
Elizabeth Baer
Bruce Barrett
Donald Bleich
Jane Chapitis
Sandra Duchnok
John Kuzma
Linda Musco
Gail Perry
Nola Reinhardt
Judith Rosenberq
Sharon Sluboski
DISTRICT OF COLUMBIA Alpha, Howard University
Hayford A. Alile
Cleo L. Bentley
Mohammed A. Bilal
Millicent S. Carter
Marie A. Cloyd
Laura E. Parley
John A. Harris
William A. Hawkins
James A. Jackson
Beverly A. Jacques
Cornel J. Rigby
William H. Jones
Timothy 0. Olateju
Fred D. Price
Linda M. Raiford
Pericles D. Stabekis
Sudershan K. Vadehra
Quinton E. Worrell
James T. Yeh
Arthur H. Gardner
Jerry W. Gaskill
Stephen Z. Kahn
R. E. Einziqer
J. T. Gill
D. F. Jackson
w. G. Kelley
J. 0. Scardina
J. H i S c h m e r s
Y. S. Yen
Ruth Brady
Ronald J. Dale
Roberta S. Haqer
Janice E. Hunter
Edwin H. Kaufman, Jr.
Marie S. Ludmar
William P. Mech
Edward B. Noffsinger
Bilram 5. Rajput
Stephen M i Taylor
Constance L. Thompson
William L. Young
Rodney R. Oldehoeft
Jane M. Pinkstaff
Lawrence E. Prichard
George Richards
Richard R. Setzekorn
Michel B. Samokhine
Gloria E. Thurston
John Robert Rhoadee
Judith Ann Sharps
Gerald Slack
Artemese E. Smith
Barbara Spangle
Minnie M. Tashima
Larry Tredway
Jim Tremper
Tommy Tomlinson
Enmitt Tyler
Lee Vercoe
Timothy D. Voorhies
ILLINOIS DELTA, Southern Illinois University
Johnny M. Brown
John R. Crede
~ a l t0. Diltinstraat
John R. Graef
Melvin D. Hall
Robert
Kohn
- - -- - .
....
.
James L. Mazander
Abraham Mazliach
~
Dale
- -- - G
-.. .Roedl
.- --
ILLINOIS GAMMA, DePaul University
INDIANNA BETA, Indiana University
Thorns P i Haggerty
James Monaqan
Denise Huet
Larry Panasci
Eleanor Christine Iorio Marilyn J. Power
James Jenkins
Donald J. Propst
Svetoza Kurepa
Kennard W. Reed
John B. Manning
Stephen M. Soldz
Mary Helen Sullivan
Michael J. Sullivan
Lewis Suskiewicz
John Tallman
Edmond Villani
DISTRICT OF COLUMBIA GAMMA, George Washington University
Sarah D. Barber
Timothy Boehn
Richard Epstein
E. H. Brownlee
D. N. Davis
Gerhard Perschke
DISTRICT OF COLUMBIA BETA, Georgetown University
Robert Baldwin
Joseph A. Ball
Charles Byrne
Maureen Castle
Elroy W. Eckhardt
Andrew Grekas
GEORGIA BETA, Georgia Institute of Technology
Ruth D. Koidan
Linda K. Larsen
Thierry Livennan
Kent Minichiello
Randy Boss
David A. Schedler
Anne Marie Hocker
INDIANA DELTA, Indiana State University
Phillip W. Boqan
Bill Fisher
Mrs. Charles Flory
Julia Hamilton
Norbert Johnson
David Keusch
James R. Mayrose
Ronald L. Melbert
Gloria Kay Norton
Gerald Pearson
INDIANA GAMMA, Rose Pol$technic Institute
David H.
Herbert
Karl W.
John D.
William
Badtke
R. Bailey
Beeson
Borst
R. Brown
Earl E. Creekmore
Charles H. Divine
Randall E. Drew
Michael J. Hanley
G. Felda Hardymon
Alan F. Hoskin
Robert E. Janes
Gary Roy Johnson
Terrence M. Joyce
William D. Schindel
Thomas George Schultz
Gholam Ali Semsarzade'
George R. Scherfick
Robert W. Walden
Russell A. Zimennan
David P. Meyer
Alan D. Miller
Loren K. Miller
Michael G. Milligan
Eugene J. Mock
Judith K. Noordsy
Mary S. Pickett
David W. Porter
James A. Schuttinga
Albert D. Sherick
Elaine B. Vance
Harold R. Warner, Jr.
Dennis A. Watts
James L. Woo
Martin R. Holmer
Lee Martin Hubbell
Joseph R. Jacobs
David A. Kikel
Kurt L. Krey
Donald E. Kruse
Maria Y. C. Liu
Judith Magnuson
Dave Mcclain
Sue Meredith
Jeff S. Nichols
Kathie Phillips
Richard E. Sweet
Michael C. Rasmussen
Louis A. Talman
Emil G . Trott, Jr.
Roger K. Alexander
IOWA ALPHA, Iowa State University
Sudhir Aqgarwal
Robert F. Baur
Robert E. Bonnewell
Michael M. Burns
James F. Frahm
David M. Gustafson
John S. Hartman
Dean E. Harvey
Gregory F. Hutchinson
Dean 0. Keppy
Thomas B. Leffler
Steven P. LeMett
Jerry Malone
Roy E. MeLuen
LOUISIANA GAMMA, Tulane University
Jack Aiken
Evelyn Alband
Jessica Aucoin
Claudia G. Avner
Don Barrvs
Bruce Baguley
Wayne Bru
Prank R. Buchanan
Betty Carriere
Mrs. Joan copping
Elizabeth Ford
Kenneth Goldberg
Richard J. Gonzalez
Charles E. Gow
Patricia Greene
Frances Hayes
Thomas W. Kimbrell
Karen Klingman
Carolyn Kutcher
James M. Laborde
Mary Ann Lemley
Jonathan Levin
Paul E. Livaudais
william McCue, Jr.
Colyin G. Norwood, Jr.
Richard C. Penney
stots B. Reels
Nicola Ricciuti
Roslyn B. Robert
Brenda M i Robinson
Christine Robinson
Michael P. Saizan
Ross Serold
Gayle Singer
George A. Swan
Raymond L. Walker
Jocelyn Weinberg
John C. Woodward
Glenn J. Rrejean
Ronald Shirley
LOUISIANA EPSILON, McMeese State College
Terry Franks
Peter Leathard
Theodore Laftin
KANSAS ALPHA, University of Kansas
Allen Leon Ambler
Charles J. Banqert
Martin R. Bebb
David L. Bitters
Sara Ann Bly
Paul W. Bock
John W. Boushka
Ronald R. Brown
Michael Sterling Cann
Jose J. Chaparro
Mary Ruth Church
Barbara Ann Crow
Jerald P. Oauer
James W, Frane
Ned Gibb ns
Edward C? Gordon
Danille L. Harder
John 0. Harris
R. Phillip DuBois
Joseph Arimja Elukpo
Gary D. Gabrielson
John David Hartman
Y. 0. Koh
Louis D. Kottmann
David D. Kreuger
Raja Nassar
Marvin C. Papenfuss
Thomas K. Plant
Ranjit S. Sabharwal
Roae K. Shaw
Darrol H. Timmona
Jerry D. Wheeler
Micha Yadin
Shelemyahu Zacks
J i m i e J. McKim
William 11. McMillian
Stuart E. Mills
Ronald H. Getting
Paul I. Pedroso
James E. Phillips
Carlton W. Pruden
Owen D. Roberts
Keith N. Roussel
Gail I. Rusoff
Florence M. Sistrunk
Ray C. Shipley
Edwin A. Smith
Carol Seasums
Harold 7 . Toups
Ernestine Watson
Ronnie E. Zammit
Judy M. Dyson
Morgan C. Eiland
Daniel J. Fourrier
Patricia A. Fults
Charles R. Guin
Andrew J. Hargooa
Earl W. H o m e
loy ye 0. Howard
Thomas V. McCauley
LOUISIANA BETA, Southern University
James Altemus
James Handy
Isiah Moore
Kathleen Morris
J. Ray Comeau, Jr.
Remel R. Cooper, Jr.
Robert W. H a m
Ossie J. Huval
Rose M. Lillie
Linda A. Looq
Anne 8. McGampbell
Cecil H. McGregor
Robert E. Mooney
John C. Peck
Matthew D. Peffarkorn
James J. Phillips
Gospard T. RiZZut0
Constance D. Watler
Karen Kristine Anderson
Barry Michael Beganny
Mary Ann Carson
Dorothy Rose Dobbins
Clayton W. Dodge
Meriel F. Duckett
Sally Ann Emery
Elizabeth D. Giusani
Stephen Eric Godsoe
Gary Leroy Goss
Jane Rachelle Huard
Thomas Peter Hussey
Thomas Childs Jane
Carroll F. Merrill
Van Mourdian
Martha Ann Parry
Joanne S. Perry
Stephen M. Perry
Carl Henry Rasmussen
Sandra Joyce Rogers
Jeffrey Cole Runnels
Stanley J. Sawyer
Madeline C. S O U C ~ ~
william L. soule, Jr.
WilliamFranklin S t e a m
Wilfred Paul Turgeon
Helen P. Whitten
Janet Marie Viger
Clover Jane Willett
Gail Laura L m g e
Roselyn Lowenbach
Alan M. Mathau
John Wayne Molino
Bruce Whitmore Parkinson
Ralph P. Pass I11
Terri Sue Pierce
Rhoda Shaller
Sue Ellen Stiefel
Nancy Wolf
Andrew Anthony Lesick
Edward A. Olszewski
Michael R. Paige
Richard Arthur SnaY
Leo T. Sprecher
John P. van Alstyne
MARYLAND ALPHA, University of Maryland
LOUISIANA ALPHA, Louisiana State University
William F. Beyer
Thomas C. Bleach
David M. Bock
Joan J. Capana
Robert P. Cifreo
Matthew B. Collins
Howard B. Davis
Stanley J. DeMoss
Charles N. Dyer
McKenzie W. Allen
Claudia A. Barras
~ e w i sE. Batson
John C. Bell
Grace A. Blanchard
MAINE ALPHA, University of Maine
KANSAS BETA, Kansas State University
John H i Brand I1
Anna Bruce
Patricia A. Buchan
John (Chang-Ho) Chu
Terry J. Dahlquist
Arthur Dale Dayton
LOUISIANA ZETA, University of Southwestern Louisiana
Dan V. Snedecor
Arthur Turnmr
Butrus G. Basryaji
John Gerald Belt
Anne Bernstein
William Edward Caswell
Gary Robert Catzva
Bryant Stewart Centofanti
Paul Gilbert Clemens
Robert L. Epstein
Ellen J. Feinglass
Marianne T. Hill
Paul Don Yon9 Hong
Fred Jacobson
MASSACHUSETTS ALPHA, Wocester Polytechnic Institute
George M. Banks
George R. Bazinet
Warren Bentley
John F. Cyranski
Wayne N. Fabricius
Berton Hal Gunter
Mark Hubelbank
Ronald E. Jodoin
MASSACHUSETTS BETA, College of the Holy Cross
LOUISIANA DELTA, Southeastern Louisiana College
Judith Ann Alston
Thomas Lynn Dieterich
James Russell Dupuy
Pearl Jo Gebauer
Sharon Claire Hein
Karen Raye Kistrup
Daniel G. Dewey
Sandra Kay Morere
David Edward Thomas
Naomi Barbara Richardson Harvey L. Yarborough
Donna L. Schilling
Vincent 0. McBrian
Patrick Shanahan
MINNESOTA ALPHA, Carleton College
Bruce L. Anderson
David R. Barstow
Ralph W. Carr
Janice Clarke
John W. Gage
Jeffrey H. Hoe1
Eric S. Janus
Frank M. Markel
J. Paul Marks
Thomas L. Moore
Stephen R. s a v i t z k y
Edward F. Schlenk
Edwin D. ~ c o t t
Timothy I . Wegner
Mary G. Whitaker
Barbara L. Whitten
James Louis Morley I1
Terry A . McKee
Dennis A. Novacek
Donna Novotny
Dale Osborn
Vernon
Marvin
Donald
Robert
Lee Pankonin
Rowher
Joe Saal
John Schmucker
Elden H. S t e e v e s
James Robert S t o r k
J a n i c e M. S t r a s b u r g
J o e l Richard Swanson
Paul C. Tangeman
Joseph E. Tyer
Keith W i l l i s
Reginald Lee Wyatt
MINNESOTA BETA, College of S t . C a t h e r i n e
John R. Chuchel
S r . C e c i l y Debelak
HEW HAMPSHIRE ALPHA, U n i v e r s i t y of New Hampshire
C a r o l Opatrny
MISSOURI ALPHA, Universit y of Missouri
Rose Marie Ahee
T r u d i e Akers
Bonita J . Allgeyer
Manouchehr Assadi
Paul W. Barker
Stephen B a u d e n d i s t i l
Jane Brightwell
Chi Keng Chen
Javad S i r u s C h i t s a z
S a l l y Clayton
John A. C r i s c u o l o
P h i l l i p Embree
Max Engelhardt
Ronald 0. E u l i n q e r
Ed W. Gray
Robert B. G r i f f i t h s
Leland R. Hensley
Robert A. H a r r i s
James R. HOUX. J r .
Henry B. Ivy, J r .
C h r i s t o p h e r W. J o n e s
John E. J o n e s
Harvey L. Johnston, J r .
F r a n k l i n Kahl
Larry Livengood
Larry J. Logan
Sandra Meinershaqen
Larry Mot2
E l i z a b e t h Ann M i l l e r
Edward J. O'Dell
Ronald W. Oxenhandler
Robert W. Ramsey
Dan Rea
J a c q u e l i n e Roberts
D. M. Robinson
Walter J . Rychlewski
Husaini S a i f e e
B i l l Sangster
J e r r y Schatzer
C a r o l Scrivnmr
Larry Smarr
Robert H. Smith
John B. Spalding
Roy G. S t u e r z l
Lynn Walley
Gary Walling
Thomas F. Walton
Samuel R. Weaver
Edward West
E l a i n e Wyett
Steven W. Yates
Greg Z i e r c h e r
David R. Lammlein
Thomas Madigan
Margaret A. Lynn
Mary Ann McClintock
J o e T. McEwen
Margie McNamee
E l i z a b e t h Ann Meyer
Ann K. M i l l e r
Linda K. Montani
E r n e s t R. Moses
Robert Murdock
Katherine V. Murphy
J a n e t Nelson
Lawrence W. Oberhausen
E l l e n O'Hara
David P i s a r k i e w i c z
Nancy C. Poe
Mary Louise Radersdorf
Maureen R a f t e r
Donna R i z z u t i
Michael J. Z l a t i c
James J. Ruder
Joseph E. Snyder
F r a n c i s E. Sohm, J r .
J a r o s l a w Sobieszczanski
Warren F. Speh
J o e l J. Tannenholz
Brian C. Ternoey
Nancy Anne Tlapek
P a t r i c k L. Trammel1
Lourdes T r i a n a
Benjamin H. U l r i c h , J r .
Joseph C. Vago
Gayle Vincent
Robert J. Walsh
Barbara A. Wanchek
John G. Weber
Marie A. Werkmeister
William C. Wester
Steven R. Wood
Mary Joan Woods
James R. Wyrnch
Spencer E. Minear
John B. Mueller
Judson L. Temple
Edward A. Teppo
William J. Weber
Larry R. Wedel
Herman J . G i t s c h i e r
Carolyn Gorski
Joan A. Kearney
Marie LaFrance
Robert G. L e a v i t t
Robert Ludwig
Michael A. Minor
Charles G . MCLure
Michael C. Malcolm
George B. Phalen
John L. Sanborn
Leon R. T e s s i e r
Joane Wakefield
NEVADA ALPHA, U n i v e r s i t y of Nevada
J e r r y Ivan B l a i r
Douglas M. C o l b a r t
Michael Paul Ekstrom
William B. Jacky
Thomas 11. Lambert
Erwin McPherson, J r .
Richard Ross O l i v e r
Larry Anthony Rosa
Richard M. H a r r i s
C l a i r e C. Jacobs
Mary T. Kondash
Leon P. Polek
A l i c e C. P r u t z e r
Frank G. ROSS
Robert D. Turkelson
David T. Lindsey
D a r r e l Lotz
Dennis R. Moon
Elmo LaMont Palmer
Dave Peercy
Tommy V. Ragland
NEW JERSEY GAMMA, Rutqers Collage of South J e r s e y
Orhan H. Alisbah
Jonathan C. Barron
Robert H. Barron
Leonard N. Bidwell
Karen T. Cinkay
J e s s e P. Clay
Joseph R. DeMeo
E l l e n 11. Fenwick
MISSOURI GAMMA, S t . Louis U n i v e r s i t y
Theodore C. Appelbaum
J. Frank Armijo
Lynn M i B a r t e l s
Mary G. Barton
Yvonne Bauer
S i s t e r M. Luke Bauman
David N. Becker
Sandra Bellon
P e t e r C. Bishop
James F. Boyer
A u r e l i a Brennan
Axel F. Brisken
C a t h e r i n e P. Browman
Kathryn A. Cross
Dela J e a n Doerr
Mary Donnelly
Thomas J. Douglas
Robert J. Dufford
Jeanne Edwards
Dennis M. Elking
Francine E n d i c o t t
Eugene A. Enneking
Mrs. Eugene A. Enneking
V a l e r i e Erickson
J u l i a E. Fusiek
Sue E l l e n Goodman
John W. Green
Veronica Grob
M a r j o r i e E. Hicks
E i l e e n M. Kurd
F e l i x 0. I s i b o r
Thomas I v e r s o n
Anthony Karrenbrock, Jr.
William D. Keeley
C a t h e r i n e Kestly
Diane Kennedy
J . Stephen Kennedy
S i s t e r M. Ronald Kozicki
Douglas J. Kuhlmann
P a t r i c i a A. Laffoon
MONTANA BETA, Montana S t a t e U n i v e r s i t y
Fred J. Balkovetz
Kathleen M. Brownell
Roger L. Brewer
Eugene A. Enebo
William J. Haley
John W. Hemmer
David P. Jacobson
Karen McNeal
Thomas Wesley Copenhaver
Daniel E. Crawford
Thomas B. Cunningham
Wendell Dam
Robert N. Dawaon
Danne E. C o i l i n g
Charles P. Oowney
Bernard F. Engebos
Neal Hart
Jim Olin Howard
Michele Kravitz
David W. LeJeune
NEW MEXICO BETA, New Mexico I n s t i t u t e of Mining and Technology
R u s s e l l E. Erbes
Lawrence E. F i e l d s
F r e d e r i c k C. Frank
C h a r l e s D. Glazer
Lawrence Lake
Richard W. Lenninq
Nicholas J. Nagy
Thomas A. Nartker
Paul M. P e r r y
Robert B. P i r o
Ronald D. Reynolds
Linda N . Sharp
Surendra Singh
Henry L. Stuck
James W. Tyree, J r .
Dennis E. Williams
Robert W. L i s s i t z
Meryll Lizan
Joseph E. Paris!
Michael R. Pike
Carolyn A. Quine
C h a r l e s D. Reinsuer
Henry E. Riqer
John A. R o u l i e r
John D. Schoenfeld
Steven Schulman
James S h o r t
P e t e r M. S i l v a g g i o
Richard C. S i n s
Robert L. Smith
J o e l M. Tendler
Richard Tuacione
Emily W i l i e r
David G. West
Linda Schiffman
Carol J a n e S h e v i t z
Leopoldo T o r a l b a l l a
A l i c e Whitternore
William Xanthos
Martin Zuckerman
NEW YORK ALPHA, Syrocusie U n i v e r s i t y
Walter W. Carey
Thomas B. C l a r k e
Anthony Culmone
P e t e r M. Danilchick
Hai Hong Doan
Alexander J . Donadoni
Anne L e s l i e Drazen
Joann F a c o n t i
Rosalind F. Friedman
S h a r i Frankel
Marilyn L. Freedman
Halfon N. Hamaoui
Donna L. H i r s t
Wallace F. J e w e l l , J r .,
F r e d e r i c H. Katz
Michael Kreiqer
Robert A. K r e l l
J u d i t h E l l e n Ladwig
Alexander J . Lindsey
NEW YORK BETA, Hunter College of The Cuny
NEBRASKA ALPHA, U n i v e r s i t y o f Nebraska
Gary W. A h l q u i s t
Claude Bolton, Jr.
John E. Boyer, Jr.
Thomas Louis Burger
Richard L. C h i b u r i s
NEW MEXICO ALPHA, New Mexico S t a t e U n i v e r s i t y
Larry Dean E l d r i d q e
Stephen Ray Gold
Larry E. Groff
Linda E. Hammer
A l l e n Lee Harms
J e a n Marie K h e s
Alan Lee Larson
John S. Lehigh, Jr.
Ronald Eugene Lux
Norman L. M e j s t r i k
Dennis C a l l a s
Rose Ann G a t t o
Dianne M i l s t e i n
Norbert Moelke
Diane P e s h l e r
Shlomo Piontkowaki
,
NEW YORK GAMMA, Brooklyn College
Moshe Augenstein
Robert Chalik
Joseph C h e s i r
Carl C i r i l l o
Barbara R. Comisar
Aaron D r i l l i c k
Hsrmsnn E d e l s t e i n
Martin E d e l s t e i n
Benjamin Fine
Sheldon J. F i n k e l s t e i n
Allen M. Flusberg
Marilyn S. F r a n k e n s t e i n
David Freedman
Marc M. F r i e d l a n d e r
S y l v i a Gelb
Sonja M. Gideon
Lois Goldstein
Seymour N. G o l d s t e i n
Keith Narrow
Brian Hingerty
Mayer M. ~ s c o b o v i t z
M a r g a r i t a Jimenez
A l l e n Lipmsn
Dennis C. Lynch
Kenneth Mandelberg
J u d i t h R. Markowitz
I r a n e L. May
Michael L. ~ u c h l i s
Harry Rutmsn
A l b e r t Schon
Ellen N. schultz
I s r a e e Sendrovic
Janet Spitzer
Susan L. T i r s c h w e l l
George Weinberger
Morris Zyskind
NEW YORK EPSILON, S t . Lawrence U n i v e r s i t y
Joyce P. Gurzynski
Thomas C. L i t t l e
Robert J . Sokol
NEW YORK LAMBDA, Manhattan College
John A l i n i
Diego X. Alvarez
Andrew Arenth
Richard Basener
A l f r e d Buonaguaro
J e r r y Carey
Gerard Chingas
Frank C i o f f i
David Concis
Lawrence Cox
Frank O o t z l e r
Joseph Femia
Alphonse Fennelly
Thomas Finn
James F. Garside
Thomas Heenan
Raymond Huntington
Michael A. I n f r a n c o
Fred I p a v i c h
C h r i s t o p h e r Kuretz
David Lee
Steven Lincks
F r a n c i s J . Martin
John Menick
Kevin M. O'Brien
Daniel O'Donnell
Robert E. Padian
P e t e r Savino
Robert S c i a f f i n o
Vincent S g r o i
John Slowik
V i c t o r C. S t a s i o
Fred Szostek
Mary Shiffman
Karen Sidelmsn
Joseph Tosto
I r a j Yaqhoobia
Steven Levinson
Trueman MacHenry
Hubert Msdor
B o r i s Mitrohin
Vladimir Morosoff
Steven P f l a n z e r
Barry Reed
Martin Rudolph
Judith Russell
Gsspar Schweiger
Benedict S c o t t
Robert S i l v e r s t o n e
John Stevenson
Harry S t r a u s s
J a n e t Tierney
Marian Weinstein
Arthur Whelan
S t u a r t Wiitamak
NEW YORK NU, New York U n i v e r s i t y
Rostom Dagazian
F r e d e r i c k Fein
William Gesick
Nancy Makiesky
Richard McNulty
Melvin Melchner
Robert E l d i
Doris Fleishman
Geoffrey Goldbogen
Michael Gorelick
Whitney H a r r i s
Susan Herman
James Holwell
Alan Hulsaver
Robert Iosue
John Kruckeberg
Dale K r e i s l e r
Robert Schular
David Silberman
Richard S t r a k o
Mark S t e i n e r
Robert V e t t e r
Cameron Wood
Grady M i l l e r
Martha Moore
Gwynne Ormaby
P
a -t .
Paliner
- -- - Lucy Roberts
Jerome Saks
Richard S c o t t
L e s l i e Stanford
Ron staift-sy
~d S u l l i v a n
B i l l Taylor
Mary E. T h r a l l
Robert S. T r e m l e t t
William W a t e r f i e l d
C h a r l e s Whits
Ken Young
NORTH CAROLINA ALPHA, Duke U n i v e r s i t y
Michael D. Holloway
J e f f Kay
Freddy Knape
Cheryl Kohl
Robert Lees
Cathy Losey
Dennis H. Matthias
Linda McKissack
Ware BotsfOrd
Don W. Brown
Marshall Checket
J a c k Davis
Peggy E r l a n g e r
Katharine K. Flory
Harvey S. G o t t s
Thomas A. H a r r i s
Corinna Head
NORTH CAROLINA GAMMA,
North C a r o l i n a S t a t e U n i v e r s i t y
~ o b yM. Anthony
James A. Blalock
Gary M. Brady
Dinnis L. C a r r o l l
P a t r i c i a M. Beauchanip
C u r t i s R. Bring
C h a r l e s L. Comstock
Kenneth L. Daniels
Kenneth M. Ehley
Edward D. Elqsthun
Edward D. E u s t i c e
Wayne W. Fercho
Doris M. F i s h e r
C h a r l e s R. F r i e s e
Betty L. Frigen
Vernon L. Gallagher
Carol J . G e l l n e r
Joseph A. Middleton
Tom L. Murdock
W i l b e r t L. Murphy
F r e d e r i c k Naqle
Douglas M. Tennant
w i l l i a m W. Wilfong, J r
Paul L. J u e l l
Donald J. Lacher
Thomas J. Lien
R u s s e l l E. Long
Kenneth A. Losee
Q u e n t i n D. Lundquist
Bruce C. MacDonald
Cheryl M. McDougal
Dalton D. Meske
Terrance E. Nelson
E s t e l l e R. Neuharth
Donovan M. Olson
John F. Ovrebo
Marcia L. Parker
Westly Parker
Lyle R. Paton
Harold Paulsen
A. rillan Riveland
Carol Rudolph
Franklyn Schoenbeck
J e f f M. Siemers
Madeleine Skogen
R. Vslborg Smith
Larry A. Wurdeman
Trudy A. Westrick
J . Ingram
Jewett
Johnson
Jones
Sherri-Ann Lancton
Edward McCleIlan
I d a E c k l e r Pugh
L e s l i e A. Rodgers
Lew H. Walters
Susan K. Woernsr
Peggy Wurzburger
Robert R. Chandler
Michael B. D i l l e
Linda J. F r i t z
E. James Gunn
Dale E. McCUnn
J e r r y L. P i t t e n g e r
Richard D. S e l c e r
Helen M. Smith
Lawrence A. White
Marilyn H i Zanni
Ted R. Clem
Michael L. Kelly
Richard P. Kitson
James D. McAdams
Frank J . Lambiase
Max 0. Gerling
Jeanne M. Glasoe
Dale J. Haaland
Richard G. Haedt
w i l l i a m R. Nauqen
Amelia R. Hoffman
David L. Hoffman
Martin 0. Holoien
Lynn H. Jennings
Caye M. Johnson
C l a r Rene Johnson
J e r r e l W. Johnson
Keith W. Johnson
OHIOBETA, Ohio Weslayan U n i v e r s i t y
Stephen D. Clements
Barbara S. Combs
Marjorie E. F e r s t e r
william N. Grunow
NEW YORK OMICRON, Clarkson College of Technology
William Richard F r s n i c s
Harold King, J r .
James Markowsky
Norman P e a r l
Steven Pincus
Michael P o t t e r
p h i l i p Rossomando
NORTH DAKOTA ALPHA, North Dakota S t a t e U n i v e r s i t y
NEW YORK XI, Adelphi U n i v e r s i t y
Pasquale Arpaia
Lynne Bensol
Richard Bentz
Dorothy Bergmann
P h i l i p Bookman
Douglas Brsz
Alan Chutsky
Penelope D'Antonio
Joseph Dunn
Bernard Eisenberg
Masako K i t a t a n i
Drew Koschek
John Majewski
Fred Mallen
David Matos
Richard Freedman
P e t e r M. G s b r i e l e
Dennis J. Holovach
Frank James Jordan
John Kiely
Marjorie
James E.
Gregg W.
susan E.
P h i l l i p Washburn
OHIO DELTA, Miami U n i v e r s i t y
NEW YORK RHO, S t . J o h n ' s U n i v e r s i t y
Theresa M. Boyd
Donald R. Coscia
S i s t e r Anne Daniel
Gerald S. Korb
Dennis J. Korchinski
John C. Lumia
David J. Nolan
Diana M. P a t e r n o s t e r
P a t r i c i a L. S a b a t e l l i
Beatrice J . S e t t e
Cristiano S t e i d l e r
Marty T r i o l a
OHIO GAMMA, U n i v e r s i t y of Toledo
NEW YORK SIGMA, P r a t t I n s t i t u t e
Lawrence Adams
Gerald A. A i e l l o
F r a n c i s Atkins
William B e l l
Ray L. Bieber
Gordon Brookman
Michael F. Adolf
Candy L. Anthony
r o s e C. Antunes
James E. Brockhoume
J e f f r e y Bromberq
Frank Corredine
Rudolf Diener
Mary E l l i e n
William Erny
S t a n l e y Finger
William V i c t o r Backensto
Richard C. B e l l
Donna Bellmaier
Roberta Boszor
I r i s E l i z a b e t h Coffman
Alan Roy Cohen
P a u l Richard C o l l i e r
John Gerald Duda
James E a r l G e i s e r t
P e r s i s Ann Giebel
James Anthony Grau
Ray A. Grove
Lawrence I r v i n H i l l
Wendell C. H U l ?
James E. K i t t l e
Richard E. Knauer
Denise Ann Krauss
Martin P e t e r Kummer
. w a s G. La Pointm
James E. Livingston
Msnikant Lodaya
Cathay M. McHenry
Lawrence E. Mlinac
m b e r t Pollex
114
Gerald m a Babbitt
L o u i s e Ann R i d e o u t
1
S t e v e n It. S e l m a n
R i c h a r d N. S n y d a r
-
Thomas T e n n e y
Maryann W o j c i e c h o w s k i
OHIO EPSILON, K e n t S t a t e U n i v e r s i t y
Roger Barnard
Susan Brakua
Michael Cleckner
V i t t o r i a Consentino
J o e l Edelman
Donald Federchek
Karen F i s h e r
Carol Qorrer
Linda F r y e
R o b e r t B. Good, Jr.
J e a n Hannah
S u s a n Hay
Gary Holues
Marilyn Kirschner
Eunice Krinsky
Nancy L y o n s
Ann M a l l e t t
Richard Mathias
Myra P a t t e r s o n
Theresa Recchio
R i c h a r d Shoop
Howard S i l v e r
Wei C h i n q Chang
Mark C h r i s t i e
J i h J i a n g Chyu
E l i z a b e t h Ann C l i f t o n
D a v i d S. Cohen
C l i f f o r d Comisky
L a r r y Cowan
W i n s t o n Dean
J e r r y Eagle
Gordon E v a n s
D a v i d L. F i s h
Harold F i t z p a t r i c k
--.
Jndv
P
uller
- -----
R o b e r t S. G a r n e r 0
Ronald G r i s e l l
D e n n i s Ileckman
D o n a l d G. I l o f f a r d
Dean H a y a s h i d a
Kuo-Cheng H s i e h
.Mei-Lee
.- - -. H-s-i e- h
W a l t e r E. I s l e r
Ben J. J o n e s
P a u l C. Kang
JWayne
u C. Teng
Trucks
J o h n P. K e a t i n g
Gary K o l s t o e
Bruce Link
Eugene C. McKay
S i s t e r Andrea Nenzel
william Peterson
James P h i l l i p s
Gerald Saunders
Jonathan Stafford
Evan Simmons
R o b e r t 11. T a r s
G r e g o r y Tye
J a m e s R. T y e
R o n a l d Wagoner
M i l t o n Wannier
J u d i t h E. W e i n b e r q
Nick Wengreniuk
L e s l i e Wilson
J a m e s Yu
Barbara Z e l l e r
M i c h a e l HcDonald
S u e Ann McGraw
B y r a n F. H a c P h e r s o n
C a r o l G. M a u r a t t
Danny M. N e s s e t t
M a r s h a l l E. N o e l
S t e v e n Q. R a f o t h
A l a n T. R i c h a r d s
L a r r y W. S a s s e s
H a r v e y W. S c h u b o t h e
C a r o l e Anne S e t t l e s
M a r b a r e t E. S h a y
M a r i o n M. S t a l n a k e r
Mary J. T a y l o r
N e i l 11. T o k e r u d
Susan M i Woerndle
L o r e n P. H o t c h k i a s
L a r r y G. Meyer
Joyce Schlieter
J a m e s R. McSkimin
L o u i s J. M o r a y t i s
B r i a n R. Mund
Ned W. P o l a n
J e f f r e y B. P r e s s
Allan
Reichenbach
S t e p h e n W. R i d d e l l
D a v i d A. R o b b i n s
R o b e r t G. Simons
Christopher Schell
Robert Taylor
Mark T o r r e n c e
OREGON BETA, O r e g o n S t a t e U n i v e r s i t y
OHIO ZETA, U n i v e r s i t y o f D a y t o n
N e i l A. B i t z e n h o f e r
D e n n i s M. C o n t i
C h a r l e s J. D e B r o s s e
D a n i e l Q. D u l a
R o b e r t C. F o r n e y
Bernard J. G a l i l e y
K e n n e t h M. Herman
Mary H. K a l a f u s
J o s e p h W. Keim
Nancy C. ~ o c h
J e f f r e y A. L i n c k
C h a r l e s A. Merk
Maureen M o r r i s o n
R o b e r t A. N e r o
Edward N e u s c h l e r
P a u l E. P e t e r s
Thomas S a n t n e r
J a m e s A. S c h o e n
Roger S t e i n l a g e
I n e z Young
B a r b a r a Zimmerman
D a r r e l A l l e n Nixon
Thomas M. P e c h n i k
V i n c e n t T. P h i l l i p s
G e r a l d H. Rosen
Roger ~ o f o
L a w r e n c e M. T a u s c h
K e i t h N. Ward
K e n n e t h H. A n d e r s o n
Thomas
Danny R.M. ABv ea ri yl e y
S c o t t A.
D a v i d G.
D a v i d A.
R o b e r t D.
J a m e s N.
Beuvais
Bennett
Butler
Erwin
llaek
J a m e s R. Hagan
A r t h u r D. Hudson
S h i r l e y D. Hudson
F r a n c i s C. Hung
J o h n R. J a c o b s o n
M i c h a e l L. J o h n s o n
John Leonard
L e w i s Lum
OHIO ETA, C l e v e l a n d S t a t a ~ n i v e k s i t ~
OREGON GAMMA, P o r t l a n d S t a t e C o l l e g e
D o u g l a s S. Bouck
David C h e y f i t z
Ralph Gigante
Joanne Hart
~ i c h a r dl l u t b e r
J a m e s Reed K e s u l l
P h i l l i p Kramer
D a v i d M. L Ã § w i
H.
Richard David B a l l
J a m e s M. B r a u n
K e n t H a l e Byron
T r o y J. Downey
PENNSYLVANIA BETA, B u c k n e l l U n i v e r s i t y
OHIO IOTA, D e n i s o n U n i v e r s i t y
J a n e t Atkinson
Richard Bedient
M a r g a r e t Enke
S u s a n Hardy
Marilyn
Stephen
Richard
Richard
Knox
H. K r i e b e l
C. McBurney
H a r p e r Meeks
P a u l David Neidhardt
Margaret Prica
J a n i c e ShilcoOk
B r u c e M. S h m a n
A r t h u r Staddon
C l a i r Wvichet
D a v i d L. Ackermann
F r a n k A r e n t o w i c z , Jr.
J e f f r e y C. B a r n e t t
w i l l i a m B. B e r t h o l f
G e o r g e 8. B r i n s e r
S a m u e l E. C l o p p e r , Jr.
B a r b a r a D. C l o y d
P h i l l i p R. D a n t i n i
S y l v i a A. G a r d n e r
Thomas L. H o t s l e n
J u d i t h A. J a c o b s o n
P e t e r J. J o h a n s s o n
B a r b a r a A. M a t t i c k
V i v i a n K. McDonough
J.
OKLAHOMA ALPHA, U n i v e r s i t y o f Oklahoma
S t e v e Abney
C l a r e n c e H. B e s t
D a l e T. Bradshaw
J o h n W. B y m e
Thadeus C a l i n s o n
L e l i a Chance
J a m e s Chapman
J i m m i e W. C h o a t e
Mary E. C o a t e s
S t e v e n Doughty
Rodney G r i s h a m
Gary H e n s l e r
D a v i d Hughes
Jack Johnson
D a v i d Kay
S i d n e y W. K i t c h e l
R u t h A. Loyd
C e c i l M. McLaury
J.
George M i t c h e l l
G e o r g e Morrow
J a m e s S. Muldownmy
David 0. N i e l s s o n
John Norris
Joanne Patterson
P a u l 11. P o w e l l
M o r r i s E. R i l l
Thomas J. S a n d e r s
H. P. S a n k a p p a n a v a r
G e r a l d D. S m e t z e r
Tom T r e v a t h a n
M e l i s s a Uhlenhop
L o n n i e Vance
James Wantland
H a r o l d Wiebe
E l l i n o r Wiest
L a r r y J. Y a r t z
Benny E v a n s
S i s t e r Ann Mark
D o n a l d R. P o r t e r
Rodney S c h u l t z
William Joseph Davis
T e r r y Eastham
OREGON ALPHA, U n i v e r s i t y o f O r e g o n
Michael Arcidiacono
Norman Beck
Carlos Berkovics
R o b e r t Bohac
R i c h a r d L. A t k i n s o n
Clifford Bastuscheck
C a r l Beiswenger
J o h n Bobick
Lawrence Dunst
George P r i c k
H a r o l d W. G a l b r a i t h
N e l s o n L. Seaman
F r a n k l i n P. S h u l o c k
Ted S i m s
A l l a n K. S w e t t
Dennis T o r r e t t i
S a n d r a M. Z e i l e r
Barbara Gross
James Harvey
Lee H i v e l y
T e r r y Ann L a n d e r s
K e n n e t h S. Law, Jr.
J o h n R u s s e l l Mashey
A l b e r t Lehman, Jr.
PENNSYLVANIA THETA, D r e x e l I n s t i t e o f T e c h n o l o g y
OKLAHOMA BETA, Oklahoma S t a t e U n i v e r s i t y
J a m e s L. Alexander!
C h a r l e s M. B a k e r
P e t e r Chevng
PENNSYLVANIA DELTA, P e n n s y l v a n i a S t a t e U n i v e r s i t y
David S. Browder
Theodore Burrowes
Robert Codergreen
Chung H a i Chang
Norman E. A u s p i t z
F r a n k Gamblin
Alan G o l d i n
R i c h a r d E. Myers
D a v i d P. P e t e r s
R i c h a r d E. R u s s e l
J u d i t h E. S i s s o n
c
TEXAS ALPHA, Texas C h r i s t i a n U n i v e r s i t y
PENNSYLVANIE IOTA, Villanova U n i v e r s i t y
Murray Adelman
mi1 Ame10tti
Nicholas P. A u g e l l i
Robert E. Beck
Joseph E. Brennan
James 0. Brooks
Leo P. Comerford
John F. D'Arcy
Richard A. D e r r i g
Robert DeVos
William H. DeLone
Thomas Dougherty
h a n d Innocenzi
Richard L. I r v i n e
S h e l l e y M. Kam
Michael Makynen
Walter 11. Marter
Robert J. McDevitt
Joseph R. McEnerney
F r a n c i s c o Migliore
Michael J . Hruz
Lawrence Muth
Joseph P e r f e t t e
Leon C. Robbins, Jr.
Lucien R. Roy
August A. S a r d i n a s
Robert L. Smith
David Sprows
Joseph A. S t r a d a
James Vincenzo
John E. Zurad
Constance A. Magner
Susan Nadeau
Gregory P. Niedzwecki
Richard W. Palczynski
A l i c i a Parente
P h y l l i s A. P e l l a
John F r a n c i s Ryan
Lynne Helen S t e w a r t
S a l l y Taylor
M. L. Verrecchia
Robert A. Wilkinson
Doris E. Wise
Benjamin M. Adams
Hull Belmore
L i l l a s K. Berzsenyi
Ray J o e l Chandler
Roy J. Chandler
C h a r l o t t e E. Dickerson
Jeanne M. Ericson
R u s s e l l Floyd
James Fugate
Steven D. H i t t
Nazem M. Marrach
Charles W. Mastin
Donald F. Reynolds
H. Edward Stone
Wanda L. Savage
Johnny D. Walker
Michael Dennis J u l i a n
William C. King
John P. Lamb
Roger G. Larson
Marilyn Kay Latham
Barbara J e a n Laucher
N. Wayne L a u r i t z e n
David Payne
Alexander Peyerimhoff
Walter Reid
Robert J. Rich
Don K. Richards
George W. S h e l l
David A. Simpson
Glen Kerr Tolman
Allen L. T u f t
Sheldon F. Whitaker
William A. Williams
Frank Wanq
Kai Tak Wong
Marion R. Reynolds, Jr.
John Rogers
Robert S n i d e r
George T. Vance
C a r l t o n R. Wine
John A. Wixson
Thomas Woteki
Mary Sue Younger
Gary R. Houlahan
Paul Joppa
J a n e t Lamberg
Susan McDougall
Heidi J . Madden
Gary F. Martin
George J. N e t t
Arthur Pearson
Charles J . Schleqel
Michael D. Smeltzer
Richard Waggoner
Mary Ann Hindery
Michael G. Severance
C e l e s t a J. Gardner
Margaret H. Garlow
Henry W. Gould
William K. Hale
Kathryn H e l l e r
James B. Hickman
Franz
Hiergeist
John W. Hogan
Florence Hunter
James M. Handlan
Margaret K i r t l e y
John Klemm
Tze-fen Li
Stephen Lipscomb
Brenda McFrederick
Betty L. M i l l e r
TEXAS BETA, Lamar S t a t e College of Technology
John S t Bagby
Georgia A. F a r r
Muriel D. Huckaby
Mary Ann F. Mathews
RHODE ISLAND ALPHA, U n i v e r s i t y o f Rhode I s l a n d
UTAH ALPHA, U n i v e r s i t y of Utah
Herbert P. Adams
Harry Aharonian, Jr.
Grace Ann B e i n e r t
Sandra Ann Campo
J u d i t h Anne C a r r o l l
Carole C a r l s t e n
Joyce Davidson
Ubiratan D'Ambrosio
Leo A. Dextradeur
Norman J. F i n i z i o
Cheryl Anne Gerry
Dennis M. Hagqerty
MODE ISLAND BETA, Rhode I s l a n d College
Clancy P. A s s e l i n
Bryan L. Barnes
J u d i t h A. Bruno
Katherine J. Camara
Joyce C o l l i n s
Frank B. C o r r e i a
Michael D i B a t t i s t a
J u l i e A. Fidrych
Edward T. Ford
F r a n c i s P. Ford
Edmund B. Gamea
Valmord Guernon
Henry P. G u i l l o t t e
Ronald Hamill
Nancy L. Humes
William E. I d e
David L a n c a s t e r
Marie L. LaPrade
George A. Lozy
Raynor Marsland, J r .
Ann Maynard
Kathleen II. McLee
Katherine M. Morgan
John Nazarian
P a t r i c k J. O'Regan
Constance Pasch
Linda L. Read
Mariano Rodrigues
Arthur F. Smith
E l a i n e Sobodacha
Robert Steward
Thomas H. S u l l i v a n
Gertrude M. T e s s i e r
P h i l l i p M. Whitman
SOUTH CAROLINA ALPHA, U n i v e r s i t y of South C a r o l i n a
C e l i a Lane Adair
Ronald E. Bowers
Willys Brotherton
Alpheus Burner
William Burns
Linda M. Chatham
Dan Fabrick
Dugenia F e l k e l
David J . F r o n t z
Robart L. H a r t
P e t e r D. Hoffmann
D e l l a Hicks
Michael D. Hyman
William S. Jacobs
E i l e e n B. K i l p a t r i c k
Ada D. Kirkman
Gary D. LaBruyere
Alonzo Layton
0. William Lever
Cynthia S. L y t l e
L o r r i n F. Martin
Gerald H. Morton
Paul H. Pinson
V i r g i n i a Ann Reeves
B e t t y Ann Shannon
John H. S i g l e r
E d i t h Smoak
John J. Smoak, J r .
P h i l i p Stowell
S a l l y S a i Cam Lau
Mary Agnes Lundburq
Kathleen E. McMinemee
Roy F. Myers
Jung J a Park
Mona Redmond
Elaine Rietz
S h i r l e y J e a n Schempp
Gary C l i n t o n Stamann
Alan Thorson
Thomas Tucker
David Vlasak
Gerald Winckler
V i r g i n i a Mae W i t t
SOUTH DAKOTA ALPHA, U n i v e r s i t y o f South Dakota
Alyn N . Dull
Faryl Foster
James F. FOX
John H a l l
Cheryl Marie Hone
Cynthia Ann Jensen
Stephen E. Jones
William J . Kabeisman
TENNESSEE ALPHA, Memphis S t a t e U n i v e r s i t y
G a i l D. Clement
Carolyn A. Cothran
Ada J. Culbreath
Betty L. Downs
Ronald L. F o s t e r
Dorothy A. Heidbrink
James D. Holder
J e r r y A. Matthews
Ronald L. McCalla
Henry H . Nabors
John L. P e l 1
Ben F. P r e w i t t
C h a r l e s E. S t a v e l y
Nancy C. T i c e
Robert Lee Askew
George Cleo Barton
S h i r l M. B r e i t l i n g
Lorin W. Brown
A. E a r l Catmull
Gary James Clark
Eva
A.
-. - .
.. Cranale
Gerald L. Despain
Sharon Lee Doran
.--
Thompson E. Fehr
Robert M. F l e g a l
Richard W. F u l l e r
Robert Gordon
N e i l s H. Ilansen
Joseph H. H a r r i s
Paul W. Heaton
Kenneth E. IIooton
Wayne R. Jones
UTAH BETA, Utah S t a t e U n i v e r s i t y
Wen Ming Chen
Robert Michael H a r r i s
Dale M.
Rasmuson
VIRGINIA BETA, V i r g i n i a P o l y t e c h n i c I n s t i t u t e
Margie C. Clevinger
John A. C o r n e l l
Daniel Denby
Ralph C u r t i s Epperly
Samuel V. Givens
S t e v e J . Lahoda
Gene Lowrimore
William T. McClelland
J e r r y Moore
Glenda Gayle Pace
WASHINGTON BETA, U n i v e r s i t y o f Washington
Alan Aronson
Sandra Burroughs
P e t e r W. Cole
Gael A. Curry
Bradley C. Fowler
P a t r i c i a J . Gale
Glenn Goodrich
C l i f f o r d G. Hale
Ronald L. Hebron
M e r r i l e e K. Helmers
Sidney L. Hendrickson
Ken R. Hinkleman
WASHINGTON GAMMA, S e a t t l e University
William Alan Ayres
G a i l Marie H a r r i s
WEST VIRGINIA ALPHA, West V i r g i n i a U n i v e r s i t y
Robert A l l e n
John Atkins
S a l l y Beile
Donald Beken
Larry Brumbaugh
~ o s e p h~ a r t i s i n o
Richard C e r u l l o
Charles Cochran
Barbara Conaway
James C o t t r i l l
Alyson Crogan
A. B. Cunningham
S a r a Debolt
Paul Deskevich
Joy B. Easton
Roger F l o r a
'
David Moran
William C. Nordai
I. D. Peters
John Perrine
Deborah Seltz
'
Dawn R. Shelkey
John S. Skocik
Joseph K. Stewart
William Stewart
Mary Squire
Samuel Swallow
Ronald Tomaschko
Ann Trump
Ronald Vento
M. L. Vest
Ronald Virden
Dorothy Watt
Stanley Wearden
Lynn Reppa
Anthea Smith
Peter Stockllausen
WISCONSIN ALPHA, Marquette University
Kathleen McElligott
Sharon Parrot
Jon Davis
Ronald Jodat
NEW CHAPTERS OF PI MU EPSILON
ALABAMA GAMMA
125-1967
Prof. William Peeples, Jr., Dept of Math, Samford
University, Birmingham 35209
CALIFORNIA ETA
129-1967
Prof. L. F. Klosinki, Dept of Math., The University
of Santa Clara, Santa Clara 95053
MASSACHUSETTS BETA
122-1967
Prof. Daniel R. Dewey, Dent of Math, Collage of the
Holy Cross. Worcester 01610
NEW JERSEY GAMMA
131-1967
Prof. Claire C. Jacobs, Dept of Math, Rutqers Colleqe
of South Jersey, Camden 08102
NEW MEXICO BETA
124-1967
Prof. Merry M. Lomanitz, Dept of Math, New Mexico
Inst. of Mining 6 Technology, Socorro 97801
NEW YORK SIGMA
132-1967
Prof. A. B. Finkelstein, Dept of Math, Pratt Institute
of Brooklyn, Brooklyn 11205
NORTH DAKOTA ALPHA
127-1967
Prof. Martin 0. Holoien, Dept of Math, North Dakota
State University, Fargo 58102
PENNSYLVANIA IOTA
130-1967
Prof. James 0. Brooks, Dept of Math, Villanova
University, Villanova 19085
RHODE ISLAND BETA
128-1967
Dr. Frank B. Correia, Dept of Math, Rhode Island
State College, Providence 02908
TENNESSEE ALPHA
126-1967
Prof. Henry L. Reeves, Dept of Math, Memphis State
University, Memphis 38111
WEST VIRGINIA ALPHA
123-1967
Dr. Franz X. Hiergelst, Dept of Math, Weç Virginia
University, Morqantown 26506
BACK ISSUES AVAILABLE
Because of special reprinting, ALL back issues of the Pi
Mu Epsilon Journal are now available. This is your chance
to complete your set.
VOLUME
VOLUME
VOLUME
VOLUME
PRICE:
I, NUMBERS 1
2, NUMBERS 1
3, NUMBERS 1
4, Numbers 1
- 10 (complete)
-- 10
(complete)
10 (complete)
- 7 (complete)
+ Index
+ Index
+
Index
fifty cents (50t) each issue; indexes are free.
SEND REQUESTS PEE-PAID TO:
Pi Mu Epsilon Journal
Department of Mathematics
The University of Oklahoma
Norman Oklahoma 73069