SCIENTIFIC ABSTRACT GELFAND, I.M. - GELFAND, I.M.

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• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

ton. By taking the
equations is transformed Into the system of ordinary differ-,
ential equations
(2) dv(s, 1)
-
di
where the matrix P(s, 1) has elements which dre polynomials
in S multiplied by continuous funclions of 1, and the initial
condition is v(s, 0) - vo(s) - Vt.
The basic theorems are the following. Let Q(s, 1#. t) be
the matrix of the normal fundamental solution of the system
(2): Q(x, I@, to) -E. 1. If the elements of Q(s, 0, 9) are multi.
pliers in f for all 110, the system (2) has a solution with
arbitrary initial generalized vector-function vo(s;j a 7101(t).
11. If the elements of QU, 1, to) are multipliers in f for all 1,
O:gt;g to, then (2) has a unique solution in the class Pol (0).
For every system, (2), let pe be the greatest oider of the
entire functions of s entering in Q(s, to, 9). 111. Then the
elements of Q(s, it, t) are multipliers in Z,' for all r>po.'
IV, If the vector-function vo(x) satisfies the inequality
1v#(z)j;SCjexp JCjxj-'-$j, o>0,
then the system (1) has a solution in generalized vector
functions belonging to T(z~% where r-po'-5, 6>0. This
solution is unique, A number of other theorems are given.
E. Hewitt (Seattle, Wash.).

M and Graev
Gel'fand, I M. 1. Unituy
of the T;al 'U'nimodular Laroup
incl
e
t nonde
t
M
th pa
g
~F
q"
l
k SSS
i
i
I
Ak
N
d
T
M
a
ematIcal Jbvjqwq ser
zvest
va
a
.
es).
au
R. Ser.
at. I
'
Vol. 15 lio. 3 189-248 (1953). (Russian) 11 . e
March 1954 The authors present a series of continuous unitary repre-'
Algebra 3entations on Hilbert space of the real unimodular group
(=group of space matrices of determinant unity) of arbi.-
trary order, and prove their irreducibility. In the caw of
the 2X2 group. all continuous unitary irreducible repre-
acntafions were obtained by B-aMmalilfAnn. of Math. (2)
46, 568-440 (1947); these Rev. 9, 133] by infinitesimal
methodp. The present series of representations, called the
principal non-degencratc wtes, is obtained by Arlobal'
-
methods parallel to those used by Gelfand and NetKna
rk in
determining all the continuous unitary irredU"C1W-MTM-
WnWiolns of the complex unimodular group [Trudy Mat.
'
Rev. 13, 722]. In particular
Inst. Steklov. 36 (1950); thew
Q
all the present Cepresentations are of multiplier type,
they do not exhaust the representations (of the stated type)
of the rcal,unimodular group,;- r.

1. 14.
UBBR*thewtics Differential Operator I Feb 53
"Certain Simple Identity for Eigenvalues of a Dif-
ferential Operator of Second Order." I.M. Gellfand
and B. M. Levitan
DAN SSSR, Vol 88, No 4, PP 593-596
The authors' purpose is to calculate the sum (Ll_Mj)
+ (Li-Mo) +... where the L'o are the eigenvalues of
Y"- (:i)Z9,y (yl(o)- by(o)~ y, (I) = -Hy(l), 'azxd
the A are the etgenyalues'of y!'-- -Yq (same bound-
arysconditione). This sum is found by simple operi-
tions to equal 16(0)4 q(13) +hH. Presented by Acad
A. X. Kolmogorov 28 Nov 52
24W41

G -e
T. IT.
U=/Hatbewtics Group Tlieory 11 Sep 53
"A General Method for Expanding the Regular Repre-
sentation-of a Lie Group Into Irreducible Represen-
tations,AIN
1. M. Gellfand an~*. I. Grayev
DAil SSSR, Vol 92, No 2, pp 221-224
Note that In the theory of representations of groups
the problem of expanding a regular representation of
a group into irreducible ones is analogous to expand-
Ing a function In a Fourier integral (the analogy
of Plancherel's formula). Find the integrals vi an
arbitrary fixed function x(g) in a Lie group G which
269T76
are given over an arbitrary class of "general posi-
tion" of adjoint cluments e of 0. Also express by
means of this integral the value of x(g) in unit e
of G. Presented by Acad M. V. Kelldysh 3 Jul 53.

0
M,
U
ussR/mathematics - Group Theory, 21 Sep 53
Semisimple
"An Analog of Planch,erell s Formula for, Real Semi-
simple Lie Groups,%-I.M. Gel' fand and4.1. Grayev
DAN SSSR, Vol 92, No 3, pp 461-464
In the preceding issue (DAN 92, No 2 (19513)) the
authors proposed a method vhich enables one to obtain
very simply an analog of Plancherel's formula for
complex semisimple Lie groups. Here they derive an
analog of Plancherel's formula for real Lie group,
for definiteness, only groups of real unimodular
matrices. Note that the analog of Plancherel's
268775
formula for the case a=2 was in essence obtained
first by V. Bargmann (Ann of math.48 (1947)) and
later by Harish-Chandra (Proc Nat Acad Sci.48
(1952)). Presented by Acad M.V. Keldys'"' -1, Jul 53.
268M

I,' A7
TM Cmdttmi m ftelts ftIm (of Vw Cowell of WMatem UM) in the fields of
science md Imyoutims enuounces that the fbLIvAN scienUfte wwksp popUw Was.
tifta boaMp ad tmftooks be" b"w^*"tted fbr oomptitlaim for SWIn PrIzes fbr
the yean 19!M arA 103. (DOV*t!km mmws, jb. 22-4o,, 20 Fob 3 APr 1954)
X~
Gellfard', 1. 11.
Title at Vftk
114ucLurcs on Linear AlCe-
bra" (2d edition, text-
book)
Malmted by
111o:;cow 1'~-thcmlvlcal ~;ocictY
sot W-30604m 7 My 1954

GEWFAND, 1. M.
USSR/kathematics Quantum - Physics
Card 1/1
Authors I Gellfand.. I. H.) Memb. Correa. of Acad. of Se. USSR. and Minlos, R. A.
Title i Solution of equations of quantized fields
Periodical )Dokl. AN SSSR, 97, Ed. 2t 209 - 212, July 1954
Abstract A solution of equations of quantized fields. A somewhat new mathematical
methodt generally outlined in the article, has been applied to the solu-
tion of the quantized field equations. Two references.
Institution :
Submitted : April 28, 1954

SUBJECT USSR/MLTHEMATICS/Topology cAn 112 PG - 141
AUTHOR GILIPAND I.N. and GRAZY V.I.
TITLE ~G analogui-of the Planoherel formula for the classical groupD.
PERIODICAL Trudy Noskovskemat.00. A... 375-404 (1955)
reviewed 7/1956
The generalization of the Plancherel formula to simi-simple Lie groups has
mainly the following diffioultys.The values of the integrals over classes of
conJugate.el.ements of the "general situation" of a sufficiently often
differentlablo-function x(g) (g*-G) which vanishe3 outside of a small
neighborhood of the-unity is knowng then the value of the function in the
unit element shall be obtained. In their investigations on the representation
of classical groups, I.M.Gelffand and M.A.Nejmark have found a complicatea
solution of this.probles for the case of the complex unimo&ular group
(compare 1.14.Gellfand, M.A.Vejaiarks Unitary representations of the classical
grqups, Mo8cow-Leningrad 1950). This process is now replaced by a more general
and claar one and for the classidal group it is carried out in detail. The
result has the following forms X'(e) LI t where I is the integral of
x(g) over the class of.aLl elements being.conjugate to the diagonal matrix
with different eigenvalues, and L is a certain linear homogeneous differentib)
operator. For the proof results of M.Riesz (Act&. Matti. 81L 1-221 (1949))
are used in a somewhat generalized forms Let f(i) be a sufficiently often
differentiable, outside of a compact set vanishing function In an Euclidean

Trudy Moskovsk.mat.0b9t _IL 375-404 (1955) CARD 2/2 PG - 141
space of odd number of dimension and Ca(x) a non-degenerated and non-definits
quadratic form with an odd number of positive squares. Then the function
R(W ) being defined by the integral
IN
f (1) 10 (X) dx,
Re A > 0 and by the derivatIve of uhich, respectively, has sImple poles at
the places 'A. -m;-2k and we have Res R(N) . 0 k( Akf)(0) (k-1,2,...).
A--m-2k ralL
Here the ck are c93~tain constants.and A w Z a pq '; X a I , where I Cpq
piq p q
is the inverse of the matrix of the quadratic form ta(x). An other derivative
of the Plancherel formula for complex semi-simple groups was given by Harish-
Chandra (Proc.Nat.Acad.Sci. USA 813-818 (1951)1 Trana.Amer.Math.Soc. 16,jL
485-528 (1954))-

-Aj'
kerfial 14 Pf and I MAJ) V it On Ov, stwdm~ uf
disffrtel.-MAI C~tali4
it r.i
4t,
t
matii:L i I if f-i eno it,,l cpiatic-ii.
i.-, real, cyumetric. confilluejus aml of
jn t and I ha -E
s th .e torm
bt.infl the fth rml~r w1ii w4tlis. The. Arits-'a ii to f
3
"arongly stibk" in th(- Rw~i! that ~dl ol
are boalidtA fo; for any h(f) of t~ln
tto for ~,mt: r>O.
OrTn S1 c
ploblvm ii to enurnevac the cunn~ctcd lei-jons fcwlt~-l I)%.
tilt- Ise'. W 11101 //((1, W:thill orvt kA wWrIl any '-w It ,,fjfi
call b..! Colitilt-aumay d~(UrIlIQL1, into -ally:oii.rt- lti,,!,,,~~
YY) by , 1"(0) -- b,"', 'i V it k J.-
rurvi, from L,-,, if) Y"C.) %N"Ovil 01.,
giuul)' of in.-mic."s, and I.Llt mly s:kkll
if lids t~l :m /fit)
'MON1, the Mrcmm, sta0ii(y J-p-~ k "n!y wj

a to
66 i~ ann.
A
011 L~g! l1pri-,
loll:i[ ~ ~ , ' - - ( I I
(,;,Ii' :_WO Ii
phi.- :-.0"
111j~ A fllil! ;5
till-
'Xv. I
'or it; i'~ )t
t c n nn h.:

ilmla' V, D. fOT CAII~,".'
of
thf~~ 21 11.14~T4V,:-
0.
vvh:uf~ 11(f) :_.i ftA .-ad symoneuir, ti.l. m;:1 .'fl!
clifferinitia) Itkirl d's"ll 1110 '4" Yii;~
J-4 'kith Ow LJ*11~1~lf',Cl: '7
of Ille
co I', I pari son r~Tll P) d Y I id, It (.`1
ration '6v(l hiAlvl- all!, p
y',lt) W~7

Yak u L,' V lit, 15111tv of SAutions
iff-ei of of
inrul "~W:L rej;A": All. S')~
'I hr~ fodhur ~"ivv;
ZOIV~-~t;lv,d In" hk)'. id -n-'1 INailk
;A (19111), 1 NI '13,
I of the
r"o
"E; lw~w%v' for C:x'a t
*11wili (fiffuclit owillud:" I:,
uqii tcd of
Oi ."i
A t~-
1"w 'W-'dA'?f condolOW
oil I-I tf~o to Z'
particular r~-gimi clr.t-na arc
tlilditud rcip'ming Thc stability rd
ri soil
i1worom", !;itlt"Jar co thatof 1.-
and ib; Fol v.'.i t~~.Alix ftimt,

U. Af b- t;:,
batty ~-f itervi for
illieffiled for m),4 "atien,
with a ctl the ".. m.01.11j,zi
*t;ol-yalu(:f~
I Ed jy)4~ivc, rllctf~ltrc.

01
M"- and gapfro, Z. Va. 1hwu.g*eo,-,n-11--
muni-finnt Ana lbek extenVong. LI 'hi Nfaj:. Njuk
I9.S.) 10 (19551t_no. 3(65), 3-70. ~assian)
.et article utilise [a thdorie des distributiors. tic L.
~
Schwartz (t. I et 11, Hermann, Paris, 1950, 1951 : MR, 12,
31, 833]. Les n1sultats essentiels sont relatifit h des 1), rti
finics WiWgralcs divergentes (cf. fla,6mard, U.-- prvbl~-
me de Cauchy, Hermann, Paris, 19-'-U; Bureau notam-
ment Conim. Pure A1)p1. Math, 8 (1955), )43-;)02; MR
46, 826]; les r&jltats sont obtenus par ],It m&:hode de
W Riesz de polongerneat analytiquc (AcU,. Math,
81 (1249~ 1-223;,MR 10, 71~1._

77777-71-~7~
1. solt Ii -xi
a ofictio
> -- I dans Yespace D' des diritributiowi ~ui R, 4,!St holo.
morpile ct !tc prolonge analytiquenwnt wi plan cnfi,r , ii
une fonction w1romorplic, a%,(:c des p,)I(,s simpIrs aux
points - 1, - 7, Le Osidu au point -k est
(d(1'=d6riv& d'ordra A de la muse dc Dirac 4). I?6sijIta-:s
analogues pour jxj', etc.
f f 2. Soit dans R" une fanction coutinue F. hoinoghic
do dehrrd 1, >0 m delmrs dc. I'origine (exerPlAe ~ jxl -I);
pour Re --n, z-j-,F~x) definit unu (list I ibotion T'j;
A-,,I-'j e:,t bolarmIrphe'dans Rel>--i, A valepts (Lini,
V; ulle se prolonirr analytiqu(mcnt ra une fork-tion ni~-
forn III I i1jr, a -~ vc I jolv-l 61-tplri alu I"Ilt t --- ", --0 - 1, - - ..
h 1-0
o"Ic-&nPq tie d 'jA A (1~1-
O. L. Lions (Nancy).

A/
USSR/Theoretical A48ica - General Problems
Abst Journal t Referat Zhur - rl;gika, No 12, 1956, 33707
AuthorI Gellfand, 1. M., Taglow, A. M.
Institution none
Title Methods of the Theory of Random Processes in Quantum PhysiC5
Original
Periodical Vestn. Leningr. Un-ta, 1956, No 1, 33-34
Abstracti Brief Discussion of the possible utilization of methods of the
theory of integration in functional spaces in problems of quantum
mechanics.
B-1
Card 1/1

~wte 1-F\W
Ber"in, F. Ali; aticGOM4.~UA -5onle refflarks o
theory 6fiPhtrlW fundions on synimelric Riemannian
351. Rumian)
Let G T M, . 'mple Lie -,x, be an
a se -sti group, and let x
involutary miti-atitomorphism of G such that the sub-
group Go (if all x with is compact and Such that
.~very.lr in G may Im written in the form uh, where is E Ga
and h*=h. Then every (;o-,Go double coset is invirimt
under *, Ind it follows that tho5e members of the group
alge'ara LI(G) wNch tire constant mj till! Go:Go double
cosets form a commutative subalgebra R of LI(G) Now
let L be an arbitrary irTeducible unitary representation
of G whost: restriction to Go contains the identity as a
direct summand.it follows from the comintitativit'
y of R
that tii3 restriction contains the identity just once. Thils
if 0 is a unit vector in the space of L Ruch that L(,'~)
for ill 4 4 Go, then the filliethill x - 0) is tilliquely
dett,rinined by L. It is a continuous fanction, roaitant
on the GO:Go double cosels, which we. sim)) th-nott, by
OL. 'ne functions of the form Ojare called zonal sj)'Aerical
functions. If ~,& is a Zonal Spherical ftilictioll, thell t1le 111ap
numbers, and every such hoinomorphism may be Y) ob-
tained.

rt_4 e/ 1"'417-A
'These known facts and notions and'others it,11,1toi to I
them are recalled In the introductory fw-ction of the 11);iper'l-F~W
In the second section the law of multiplication in the ring
R is studied, ConsideriP& first the case in which G is the
dircvt product with itself of the group of all ?I x n unitary
unim(xlular matrices and Go is tile diagonal subgroul), the
members of R are considered as functionq oti the set 1) of
Go*Gn double cosets and the multiplication law thrown
into the form V11a)(1) -ff 12, 1)j1ljf2, where
1, 11, and Is are in D and it is an explicitly dc,4xribed 'iflinc-
tion on DxDxD, There are relations hetwe(m the unc-
tion a, tile zonal spherical functions and certain differ-
ential operators, and thc:;e are explored in detail. In this
ame (and In generul when G is thr direct product with
itself of a compact group and Go is the diagonal snbgroup)
the zonal spherical fuixtions are in a natural onmD-011C
correspondence with the charactets of the original
ct group. Mcxt if is briefly indicated how these
cre(TU11tas may be generalized to the case in which the unitary
unlinodular roup is re.?Iaced by ait arbitrary compact
serni-slinple rip. group. Futther indications then tre.1t tile
tiosely Vatallel facts which hold in the case in which G is,
a complex serni-Ample Lie group and Go is a nv.ximal

1`1 If
;A. t, 1
C0111 Pict s-ubgtou- of G fly way of application a thil,jrcm,
on tille proper varm's (if' sums and produrt,4 (if mfdfivil,; ii
cd. (For a statement of this theorem the vvie-w 07
rov
P
of a paper of Li&kil [Dokl~ Ak-act. Nlauk SS5[t CN.S.) 75 -
(1950), 769-772; MR 12, 5,911 ill which anotlivr' pi(oif ii
In the third sectirm it is shown that t~e zon-a sph,~ricalI
functions mtisfy tho functional ecju:ttion
fG, and tht tho light-litt'.11 Spb, 1110V 1W
Interpreted as the avorage valuo of 01, over a of
"radiiii" GOxlGo with center it "distmro" from t1w
origin Go. The GC:60 d(mble cos,!ts are ill 111itulal (Inc.' to
one corresponderce with the c(jtd%-aI,:nc;! vla~s,,:i of pairs
of points ill the hon-logeneolls space 6,!G,) unoer tht, iction
of G and in tbi; sivnse can be regardcd ai go:vtalit"d
distances. This fandional equation is examincd in detail
in the special case in which GjG,, ii the n-diuwrAonal
spliere.
rhe results of section 4 are also preqcnted in terms of
generalized distances. Let Vj, devote the operator which
tahes a functim / on G/Go into its mean value on a sphere
115-
with nter at x and "tadiw," t. Let Al, A2, A
ator, , 1. 1-0
for GIIt, i t

'-Wrt~~ I r4,,r- iari
[GelTand, ibid. 70 (1950) - 5--8:
duced in carlier art W
-
'
-e2g
1111111, 498; Gel'tand anCi Cellin, ibi(L 71 (1950), 825
;
IM11. 12, 91, Then if V(1, x)--=Vi,,l for somit fittiction f, wo
have Axkjp(t, x) x), NvItcre oil 61C left Sid(., I k behl
-
*
- t
xed and u
h
e
n right sWe x ii h(qd fixcd aM--ip iis a-
function of I is regardo'd its a (mictiun of a point in GlGo a
distance 9 froin Go 1j) ;o1flitior it is ~4,owjj that V,,, may
bc exPTC5Wd as a fond i0l) of the oplimtor5 A', A", An. M-
the fanctinn being cmintructed fl-11111 tll(! ~1.011:11 t)pllcrical
1.(A G b", a compact sumi-illipti. l'i" 1"Fillip '111(l ~i
dunote thr Agy-Imi (if all

L.~, Ly

LI 1, 1 i4 Ow llimflj~ I if iiii-ci thA L k run-
III 01E., ri-liu(ti-:11 of Ow krml(Tla-i pi,d!mt t)f Ll
-Mid srcHml five ;. t:ltivflt~ ot-%-;r~t -1 to thi. q!1 11(jut'- Imil
p1q)r(ti; ~ r'! thl- '111'rifta.((It 161kil In it IN(Wrt-fic twilt

ljvv~k":cll it 1111d th': 1141 !,!,1 N A -it It"', tva' llill~

the function G playi the role of the function a in secflon 141W
two and ha, rhany :tnalogom propertiei. Again a spe(ial
case iq treated in di-tail and generalizations 1110[e briefly
indicated. At the ead of the ,section some of the resvIts
of section 2 on complux semi-siniple groups are jtcner-
alized to the real case and to certain non-semi-simple
groups. G. I,V. Mackcy (Carnbridgi-, M,Iss,).

SUBJZCT USSR/NATHMTICS/Topology CARD 1A PG - 991
AUTHOR BWSIN F,Aot ONLYAND I.M.
TITLZ Som* remarks t-5-UV-%W66Fy-of spherical functiona on
symmetric Riemannian manifolds.
PERIODICAL Unpechi mat.Nank 11* 3f 211-218 (1956)
reviewed 7 1957
To a symmetric Riemannian manifold Qj/ Ck with the group 9 the authors
consider spherical funotionsl if g __joft is a unitary representation such
that for a certain 10 ~ 0 holds (Tg) 10 - So for all ge%o , then the
functions ( ~,(Tg)n) are called spherical functions; they are called zonal
If they are constant on the "spheres" with the "center" x 0. Sepocially for
the manifolds of the semi-simple groups ( 1consists of the ae---Ya- Ivt b) and
for the cases 14 - complex s*mi-simplA gro p, 40 - maximal compact group"
the law of multiplication of the zonal functions is given. Further the
Laplace-operators (i.e. exchangeable with the f(x)--jpf(gx:), ge %) are brought
in relation with the establishment of mean values, and finally a duality
between the ring of the class functions of the compact group a and the
algebra of the representations of 4 is explained. No proofs Ire given.

n some probleals of (III)CI
ki,fitudil-M. 0
3_ 1274
-V7sj-)vIfi Mut. Naxik (N,S.) 11 (1956), rm. ~(7 ~2), t-F\W
(Rlmian) 001
'file author PAWS out tilt! Widellin- dolmull of f LITIC-.
Distri 4&tfXF1 e.,
tional anall-dis and ("ints to the applications of fimc:ianal
analysis to numerical analysis, theci-y of probability,
theory of differential eyndiom;, and informitirm theory.
lie tidievvs that hydrodynan-irc-s (th,! flow (if ,-mucous
and
fluids, collipt"'Able gaws. theory (-if 611711,77.,
theoretical J)hY.iic:s (qualitum throry Awl thtoly of uk-
mentary partict") will have a mrcinij influrn, on the
filture collvv.~ of development of fillictional allaly~,is.
Problems in t1w fullu%ving huldi art: ukdicalcil ~1) luicar
topological sjyac(~S, (2) ~jnear and titivii-fine.ki partial
differential equations, (3) measure theoty, 1,4 (if
pe 111, and (5) hyptrgeornetric fuitefions. No spccific
r;.-ferellus aw givell. J. P, LaSalle (Noti-c Damc, Ind..),

BBRIZIN, F.A.; GILIFAND I K, GRATIT, Mel.; xjrNAm, N.A.
1111,449hAAW,
Representation of groups. Usp.mat.Aauk 11 Ao.6:13-W X-D 056.
(MIRA 10:3)
(Groups. Theory of)

.7-
1, VNI11F, NINVACAL CALCULATION OF
31, No. 41112).'1161.7 (0501. M
Curdirwois Imlefrxis tro expreived &a Stlellj&a ItAcgrale of
tufflacrAly Mah r.Wtipliatty. Ah examrAt Is givom Its %hien a cun.
Irlogral it) reploud t;T IW- o"14 230-VAO Mccrall. swrnn..-l
10,4m) V/ &.1,durta CLOP wft~~J.

F- r I
SUBJECT USSR/XLTMgKATICS/Theory of probability CARD 1/2 PG - 995
LUTHOR GZLPAND I.N., KOLMOGOROV A.Y., TAGLON A.M.
TITLE On a general definition of an amount of information.
PERIODICAL Doklady Akad.Nauk 111, 745-748 (1956)
reviewed 7/1957
Lot T be a Boolean algebra and lot F denote a probability on If Otsnd
are two finite subalgebras of T , thon the expression
1(010 ?(AiS log P(A 2
is by definition "the amount of information contained in the results of the
experiment Ot relative to the results of the experiment Mot, C -
- I( t, Ot)). generally,
(1) 10111e) - sup I( 0110&l)?
616%, Wis-Z
where OL1 and 4 1 are finite subalgebrasl symbolically (1) can be written

D6klady Akad.Nauk 111 745-748 (1956) CAU 2/2 PC - 995
I(00t) P(dOtdt) log P(dgtdf-)
ff P(dOQP(d J)
at f,
SU so now that I Is a Boolean 6-algebra, P a 6-additive probability
onTand (X,Sx) (SX a V -algebra) a measurable space. A random elemont
of the space X is a homomorphism -~*(A) - B of 3 X into 'r. The expression
is taken as definition of the amount of information, r(SX).
A condition under which 1( 1,17) is finite is given. finally some properties
of this expression are discussed.

L):fz tr r Wd
0,4 QU,*Jrn=g WIT'If ANOMALMS PARITY AND 014 A
P099BLE "PLALMAnom oF PAarry vioutcaAcy
IdESON, j ", 04WUNI I rA W 1. Tortilm. sovta
40,
L,

C71 L
SUBJECT USSR/MATHWTICS/Functional
AUTHOR GELIPAND,7.U., JAGLOM A.M.
TITLE Th-e-UnTe-g-r-&Nron in function
quantum physics.
analysis CARD 1/1 PG -679
spaces and its applications in
PERIODICAL Uspechi mut.Nauk 11, It 77-114 (1956)
reviewed 4 1957
In the present paper the authors give a detailed establishment of Feynsan's
method of the path integrals (Rev.modern Phya. 20L 367-387 (1948)) by aid
of the Wiener integrals (Acta, math. 'Uppsala 2~x 111-25B (1930)). After an
introducing section in which the Wiener integrals are introduced, in the
second section the application to quantum mechanics is treated in datail.
The application of similar methods in the quantum field theory is treated
very short. The third section brings the treatment of some fundamental
properties and some useful computing rules for the Wiener integrals. The
last section contains some special problems the treatment of which is very
simple with the methods developed here. A translation of this interesting
paper will be published soon in the "Fortschritte der Physik".

SUBjECT USSR/MATHELUTICS/Theory of probability CARD '/4 PG - 731
AUTHOR GELVAND I.M., JAGLON A~M.
TITLE TKe `Binip-dTnion of the eet of communication about a random
function contained in an other random function-
PERIODICAL Uspechi mat.Nauk IZ.. 1, 3-52 (1957)
reviewed 5/195T
The first chapter in essential corresponds to the appendix 7 of Shannon and
Weaver 'A mathematical theory of communioation" but it oontains some now
results. Lot ~ and -1 be discrete random terms which can attain the values x
(1.1,2,. n) and Yk with the probabilities P,(i) and P 11 (k).
Let PVq (i,k) be the probability that at the same time ~ attains the value x i
and i the value Yk ~ The set of communication about Yi wontained in -~ then reads
n M Pti (i,k)
;_~ 2: P (i,k) log
i-1 k-1 P (i) P (kT
S It
For arbitrary (not discrete) and I the set of communication is defined as
1( ~, 1) - sup [9( 4, , A2' ~ - - 'Ad ~ -q( A 1' 62' 1 . - ~ 6 , $

tspechi mat,Nnuk 12, 1, 3-52 (1957) CLRD 214 PG - 751,
where the sup has to be taken over all poasible subdivisions of the ranges
of values of 9 and q into finite numbers of free of common points intervals
.&I and Aj, respectively.
It is stated that in general the dependence of the set of communication
Vt a) on the probability distribution of the pair of vectors is
disco inuous, but that always
I( lim 1( Sn' Id
if the sequence converges to with respect to the probability
distribution.
A further'new resUlt 19 contained in the theorazi In order that I(S is
finite, it is necessary and sufficient that the probability distribution
PE I is absolutely continuous with respect to the distribution P. -P7 . Then
f o( (x, y) log A (z, y) 0 1 W dP YL (y), where
dP 12 (XVY)
dpf(x)dP,L(y)
by -pu t fl -n-g uo" u '"mula which f0l'A.Ows from
C, A 0 A-1 0
0 B C 0 D-0.
Then

U.apP-.:.hI mat.. Nauk ' 21 9":7 CkHD 4/4 FG - 73~
det I dez B r ..1 IE DB'1
de C do (cc I ' 'I
D I,& N
and dot (S DB"'D IA-
2 -
On the basis of these preparations several oxamples for the ccoputation of
the set of communication are computed,

17
CARD 1/2 PG - 815
SQBJZCO IISSR/MATHEMATICS/Algebra
AUTHOR GEL I ?AnLJX~
TITLE -Un-subrings of the ring of continuous functions.
PERIODIM Uspechi mat.Nauk 11, (1957) 249-251
reviewed 6 (1957)7-
The present communication is a completion to the preceding note of Wilov.
The author puts seven questionst
1. Does there exist an antisymmetric ring with an one-dimensional carrier ?
2. Is it possible in the case of an arbitrary antisymmetric ring with a
two-dimensional carrier to transform the carrier homeomophically such
that the ring coincides with a ring of the type A(a) ?
(A(G) is the totality of the functions which are analytic in the plane
domain G and continuous in 6).
3- Is the ring A(G) the maximal antisymmetric ring with the carrier G ?
4. Does there exist antisymmetrio rings the carriers of which are atwo-
diwnsional sphere or a torus ?
5. Does there exist an antisymmetric ring with a carrier being honeonorphic
to the three-dimensional cube ?
6. In four dimensions antisymmetric rings are well-known, which consist of
luncrions Or--tW6--Waiiables in the _-c 1--o-se-id- domain G of the two-
dimensional complex space. Does there exist further antisymmetric rings
with four-dimensional carriers ? Are the mentioned rings the maximal
Rntisymmetric rings with the carrier -6 ?
7. Are the notions antisymmetric and analytic identical for a ring with
uniform convergence ?

~- k~ I ' V IZ-t V, L , -L I-(
A UT HOR i KULAKOV,ApM., ~~F Hklj.U Nzigineera, FA - 2385
Metallurgical Combine of Magnitogorsk.
TITLEs Performance Practice of Automation of the Rolling Mille Heating
Installations. (Opyt ekepluatatoll avtomattki nagrevatellnykh
ustroystv prokatnykh stanoy, Russian).
PERIODICALs Stall, 1957, Vol 17, fir 1, pp 80 - 83 (U.S.S.R.)
Received, 5 / 1957 Reviewe~t 5 / 1957
ABSTRACTs About 140 vertical Ingot heating furnaces are available foi
the rolling mill train of Magnitogorsk. Saving of I 9C fuel
amounts to 800.000 roubles per annum. First the attempt at thermal
control and automation in connection with the regeneration ingot
heating furnaces of the blooming mill train Nr 3 is described.
Waste caused by heating these furnaces consists to 85 - 90 ~ of
the ingots of quiet steel. The best means of reducing waste caused
by heating is the automation of the heating process of the ingots.
It consists in regulating the temperature in the ingot heating
furnaces, control of the consumption ratio of air and gas, pressure
regulation in the working chamber of the furnace, automatic swith-
ing of valves and automatic switching off of gas and air. For an
optimum control and regulation thermoelements and radiation pyro-
meters are used simultaneously. In the Pecond part of this paper
the thermal control and automation of the three-zonal methodical
Card 1/2 furnaces LPTs-1 of the xheet~olling mill Nr 1 are described. It

Prof
G:-,V17A11Do
3cot ix'.~'d
tlit: In llburE""
t i
Cur"6v"'Sa (If
"ape'..
r
14-21 Aug 5E",

BRAGINSKIY, S. I-, GF-L'FATIDs L M- an(' F1';DORE1;Y-OI "' P-
"The Theory of the Compression and Pulsation of a Plasma Column in a Strong
Pulse Discharge," (Work carried out 1957-1958); pp. 201-221.
"The Phynics of Plasmas; Problems of Controlled Thermonuclear Reactions." VOI. IV.
1958, published by Inst. Atomic Energy, Acad. Sci. USSR.
resp. ed. M. A. Locontovich, editorial work V, I. Koean.
Available in Library.

PHASE I BOOK EXPLOITATION SO'1/1325
Gellfand, Izraill Moieeyevich and Geor-gly Yevgenlyevich Shilov
-Nekotoryye voprosy teorli differentsiallnykh uravneniy (Some
problems of the Theory of Differential Equations) Moscow,
Fizmat iz 1958. 274 p. (Seriesi Obobshchennyye funktaii,
vyp- 31 '6j,000 copies printed.
Eds.: Agranovich, M.S. and Stebakova, L.A.; Tech. Ed.: Kryuchko,,ia,
V.N.
PURPOSE:. This book Is the third of a series of five monographs on
functional analysis and is intended for mathematicians and for
specialists in allied sciences* To read the book it is necessary
to have a good background in mathematics and a knowledge of the
results presented in the second book of the series.
COVERAGE: The book deals with the application of the theory of
'generalized functions to two classical problems of mathematical
Card 1/9

3ome'Froblems of the Theory of Differential Equations SOV/1325
analysis: the problem of expansion of differential operators in
eigenfunctions and the Cauchy problem for partial differential
equations with constant coefficients or with caeffiaients
dependent only on time. The theory of fundamental spaces of type
W which in heeded to study the Cauchy problem,is also presented.
The authors thank those who participated in the Moscow State
University seminar on generalized functions and differential
equations, where many sections of this book were discussed.
Special gratitude is expressed to VA. Borok, Ya. I. Zhitomirskly,
G.N. Zolotarev, and A.G# Kostyuchenko. There are 64 referencess
of which 39 are Soviet, 13 English, 6 French, and 6 German.
TABLE OF CONTENTS:
Preface
Ch. I. Spaces of Type W
6
1. Definitions 7
1. W" spaces. 2. e. spaces. 3. WPI spaces. 4. Problem
of nontrivial t of spaces. 5. On the abound of
functions in spaces.
Card 2/9

416(1) PHASE I BOOK EXPLOITATION SOV/1218
'~~a.Td Izrail' Moiseyevich and Shilov, Georgly Yevgenlyevich
1~
-Prostranstva oanovnykh I obobshchennykh funktsiy (spaces of
Fundamental and Generalized Functions) Moscow, Fizmatgiz, 1958.
307 p. (Series: Obobshchennyye funktsii, vyp. 2) 7,000 copies
printed.
Eds.: Agranovich, M.S. and Stebakova, L.A.; Tech. Ed.; Oavrilov,
S.S.
PURPOSEi This book is the second of a seftes of five monographs on
functional analysis. primarily Intended for mathematicians,
although It may be useful to others having a good mathematical
background and a knowledge of the fundamentals of functional
analysis.
COVERAGE: This book is devoted to the further development of the
theory of generalized functions presented in the first book
Card-174"-

Sphces of Fundamental and Generalized Functions 30v/*-!218
of the series. In particular the book deals with the transfer
of the technique of operation with generalized functions studied
in the first book to more extensive classes of spaces. The
basis of the theory of generalized functions is the theory of
countable normal spaces, the presentation of which makes U'P the
greater part of the book* The class of all countable normed
spaces in many problems is too extensive for the theory of
generalized functions. For this reason certain special types
of countable normed spaces are introduced and studied. The
spaces studied in this book are to be used in the third book of
the series., which is to be devoted to certain applications of
the theory of generalized functions to differential equations.
The authors thank D.A. Raykov, B. Ya. Levin, G.N. Zolotarev,
N. Ya. Vilenkina, and M.S. Agranovich for assistance in preparing
the book. There are 39 references, of which 10 are Soviet. 7
English, 17 French, and 5 German.
Card--2~L --
10

16(1) MASE I BOOK MUNTATION 7(,-l
Gellfand lzraill Moiseyevich, Robert Adollfovich Minlos, and Zorya Yakovlezna
Shapiro
Predstavlen1ya gruppy vrashcheniy I gruppy Lorentsa, 1kh primeniya (Rotation
Group and Lorentz Group Representations and Their AppIleations) Moscow,
Fizmatgizo 1956. 368 P. 7,000 copies printed.
M,; F. A, Berezin and L. A, Stebakova; Tech. Ed.: S. S. rayrilov.
PURPOSE: This book is intended for mathematicians and physicists and for
students of mathematics and physics.
COVERAGE: This book is devoted to a detailed study of the representations
of rotation groups In 3-dimensional space and to the Lorentz group. For
the benefit of physicists and physics students the nixthors have Included
in the book all basic material on representation theory uhich Is applicable
to quantum mechanics. Matbamaticians and mathematics students who are
studying representation of Me groupt. may use the book as an introduction
to the general theory of represental"ons. In addition the material included

Rotation Group and Lorentz Group (Cont.)
SOV/22h2
In the book renders sufficiently clear the connection between representation
theory and other branches of mathematics., such As spherical lunctions, tensors,
differential equationsp etc., vhich bad not previously been analyzed in the
general came. 1. M. Gellfand and Z, Ya. Shapiro wrote the first part of
the book on rotation groups, K, A* Minlos wrote the second part on
representations of the Lorentz group and relativistic-invariant equations,
This part vas based mainly on the work of I. M. Gellfand and A. M. Yaglom
"General Relativistic-invariant Equations and Infinite Dimensional Repre-
sentations of a Lorentz Group" (Zh=al ekBperimentallnoy I teoreticheskoy
fiziki, Vol 18, No 8. 1948). The authors thank F. A. Berezin, editor of
the book, for his assistance. There are 25 referenc,e*.-;: 23 Soviet, I German,
and 1 English.
TAKE OF COMM:
Preface 7
PART I, FMIMMATrONs OF A Ro,,"AnON
GROUP OF -DIKEMIONAI, SPACT.
Ch. 1. The Rotation group and Ito Representatlon 9
CarA=e/jo.

Pam I BM MCPLO="I0N 629
2!1!fandj_4r!!K 1se T~Ic~d Shilov,, Georgly Yevpnlyevich
Obobahchennyye funktaii I deystylya nod nind (Generalized Functions and
Operations With Them) Moscow, Go** izd-vo f1zIko-wtemticheskoY lit-
ry.,
1958- 439 pe (Series: Obobahchennyye funktoli.. v". 1) 8,000 copies
printed.
Eds.: Agmavich, M. S. and Poftin, L Z*0 Tech., Ed.: Brudno.,, Ke F.
PMOSE: This book In the first of a series of f1ve mnogrsphm on ftoctional
analysis Intended for scientific workers,, graduate students and senior
un1yersitj students In natheiiatics, physics and allied sciences. It can
also be useful for enegineers.
COVERAGE: The basic concepts and definItIons of generalized functions
(distributions) are introduced, their properties described and operstAous
with then demonstrated. Fourier transformations of generalized functIous
of one and of several vaAables, and Fourier transformtions in connection

Generalized ftwtions (coat.)
629
with-vwrtab differential equations an analyzed. Generalized functions
on surfaces and fundamental solutions of differential equations VIth
constant coefficients am studied. The general theory of homogeneous.
generalized functions Is presented. In the preface, Soviet mathematicians
Be L. Sobolev., Z. Ta.Shapiro, 0. Yee Shiloy and N. T& MeWdo are mentioned
in connection with publications on generalized functions. The authors
thank their co-workers, in partIcular V, A. Borovikov, N. Tae Vuenkln,,
M. L Grayev, M. Be Agranovich and Z. Yee Shapiro for their assistance in
preparing the book. There are 22 references, 6 of vhIch am Soviet,
5 English# 8 French and 3 Germn.
TAMZ (W CONIMM:
Preface T
Ch. 1. DefInItIon and SIMle Properties of Generalized Functions
1e rundmimental and generalized functions 2.1
1. Introductor7 remarks U
2. ftndmental functions 12
3. Generalized functions 13
4. Local properties of generalzed functions 16
5. Operations of addition and multiplication vith a number
and with a function 18

WXY10. I.I.; GILIFAND, I,M.
Several remarks on hyperbolic system. Bauch. dokl. v7s. skoly;
fis.-mat. n&Wd no.1:12-18 '58. tKIRA 12:3)
lAoskovskly gosadaretvennyy universitet in. N.V. Losonomova.
(Differential equations. ftrtial)

AUTHORSt e Pralov, A.S* and Chentsov,N-N. SOY/140-58-5-4/14
TITLEt Calculation of ContJnuous Integrals With the Monte-Carlo Method
(Vychisleniye kontinuallnykh integralov metodom Monte-Karlo)
PERIODICALs Izvestiya vysshikh uchabnykh zavedeniy. Matematikap 1958, Nr 51
PP 32-45 (USSR)
ABSTRACTs This is a survey consisting of 10 paragraphs and a summary.
The application of the Monte-Carlo method for the calculation
of integrals of high ( even of denumerable) number of variables
is discussed in many aspects. The Soviet contributions
(Bakhvalov, Korobov, the authors, Kolmogorov, Soboll) as well
as the western contributions in this now direction are ap-
preciated. The authors present some interesting examples
(diminution of dispersion, determination of the trajectory
for the Brownian motion eta.). In the text 4 Soviet and 7
American papers are mentioned.
ASSOCIATIOUt Matematicheskiy institut imeni V.A.Steklova AN SSSR (Mathe-
matical Institute imeni V.A.Steklov AS USSR)
SUBMITTED% December 6, 1957 (Date of LectureLeningrad)
Card 1/1

GALIAND,.1sraill Holseyevich; SHIWV, Goorgiy Tevgenlyavich: RYVKIN,
--- ", BRUDY0, K.F., t*.hn.red.
0 06
(Gonerallsed. functions and operations on them] Obshchonxro
funktall I deletvita rAd n1al. Moskva, Gos.lxd-vo fisiko-
smitsmattabeekol lit-ry, 1959. 470 P, (ObshchenrWa funktall,
ROM. (MIRA 13t4)
(IFUSO'nonal analysis)

A IV
v lip :1
51 [oI ~3,
F. 2 1 1 8 1 z
33. ,,5

Eli, AJ.
lk I't .4
chal
~K! lilt',
will
Hill
i-IN
2; 0. 'Ag
'11A
g
116
a In 4
At
11,2 .9-
I 1'-A
Ell
a- Aj t4j= j a
1 e I-I i
A 1.
au 1 .11! - I
'J&
PCs
A
A' 10
rot

8 (2)
AUTHOR: Gellfald Engineer
j_ I. Moo SOV11 19-59-4-4/16
TITLE: An Automatic Controller of the Air-fuel Ratio Under Simultaneous
Supply of Three Kinds of Puel for Burning (Regulyator soot-
nosheniya toplivo-vozdukh pri odnovremennoy podache dlya
gorcaiya topliva trekh vidov)
PERIODICAL: Priborostroyeniye, 1959, Nr 4, PP 9 - 10 (USSR)
ABSTRACT: The automatic controller-investigated in this paper belongs
to the double-zone soaking furnace of the Magnitogorskiy
metal lurgicheskiy kombinat (Magnitogorsk Metallurgical Kombinat~
It is supplied with a gas mixture (blast furnace gas + coke
oven gas, coke oven gas and mazut). The air necessary for the
combustion is provided by a medium-pressure centrifugal fan.
The unit used for the automatic control is composed of in-
strumente and controllers coming from series production. In
figure 2 the principal circuit diagram of the air-fuel mixture
control is presented. The principle of operation of the in-
dividual elements is briefly discussed. The actual air-fuel
mixture controller is based upon an electronic zero relay for
Card 1/2 the astatic part of an isodrome relay of the type IR 130 with a

.. 11
An Automatic Controller of the Air-fuel Ratio Under SOV/119-59-4-4/18
Simultaneous Supply of Three Kinds of Fuel for Burninj;
voltage amplification by means of two cascades. The controller
discussed permits to reduce the number of secondary recording
instruments and the size of the pertainin,- switchboard panel.
The rate of consumption of each of the three fuels can be re-
corded on one ainCle diagram. This controller exhibits a simple
dosiGn and is reliable in operation. There are 3 fi(;urea, and
2 Soviet references.
Card 2/2

'
16(1 ) *
-AUTHOR: -11 _Ge 11 f and, I.M., 30'1142-1-1-2-2119
TTTLE- Some Problems of Theory of Quasilinear Equations
PERIODICALt Uspekhi matematicheskikh nauk,1959,vol 14,11r 2,pp 87-158 (USSR)
ABSTRACT: The present paper originated in lectures given by the author
1957-1958 at the Moscow State University and which were written
down by K.Y.Brushlinskiy and V.P.Dlyachenko. The author does not
aim at giving an exact representation of final results but he
describes single, not finally solved statements of the theory
which shall be developed. The author hopes that his paper will.
incite to solve the given problems. The author thanks K.I.Baberko,
S.K.Godunov, and V.P.Dlyachenko for the discussion of several,
questions, and G.I.Barenblatt and O.A.Oleynik for writing some
parts of the manuscript. Contents; Introduction; �1. Evolution
systems of equational �2. Quasilinear system of equations of
first order. Discontinuous solutionst D. Fundamental relations
of the thermodynamics; �4. Equations of the hydrodynamics ' of
ideal fluidal �5. Motion equations for the presence of tenacityl
�6. System of Lagrange coordinatesl V. Characteristics; �8.
Stability of discontinuous solutions; �9. Decay of an arbitrary
discontinuityl �10. On uniqueness and existence theorems; �11.
Card 1/2 Two-dimensional evolution systems; �12. Linear equations with

Some Problems of Theory of quasilinear Equations SOII/41-14-2-2/19
disoontinuous coefficients; �13. Equations of magnetic hydro-
dynamical �14. Characteristic cone in the equations of magnetic
hydrodynamical �15. Problem of the thermal solf-ignition; �16.
Problem of becoming stationary of chemical processes; �17. Normal
flame propagation. Additions.
The author mentions N.N.Semenov, Ya.B.Zelldovi,,h, and I.G.
Petrovskiy.
There are 20 figires, and 17 references, 16 of which are Soviet,
and I American.
SUBMITTEDt November 17, 1958
Card 2/2

6~
AUTHORSj Gellfand, I..,.14.,and Pyatetakiy-Shapiro,I.I. SOV/42-14-----5/"9
TITLE: Theory of the Representations and Theory of Automorphic Fun~-tio%z
11"ERIODICALt Uspekhi matematicneakikh nauk, 19599VOl 14,Nr 2,pp 171-194 (USSR)
AWJTRACT3 In the paper of the authors and S.V.Fomin fRef 4_7 firntly th-
connection between the theory of infinite-dimonsional
representations of Lie groups and the theory oil automorphic
functions was pointed out. In the presont paper th13 connoction
is investigated furthermore. It in shown that many quoationj of
the theory of automorphic functions can be treated uniformly
with the aid of infinite-dimensional representations. The
calculation of the dimension of the space of automorphiu firml
is reduced to the determination of the multiplicity with which
the corresponding irreducible representation appears in the
decomposition of a certain representation. In a simple manner
the authors introduce nonanalytic automorphic funct4ons. The
trace formula of fRef 4 to the ca
_7 is generalized ue of an
arbitrary non-compact Lie group. The trace formula of Selberg
Card 1/2

Thcory of' the Representations and Theory of Automorphic 3011/42-14-2-5M
, Fun(~ t ions
Z-Ref 2_7 is a apecial case. The duality thqorem of Cartan it;
transferred to the considered case.
The authors mention A.G.Kostyuchenko and MoA.Naymark.
There are 11 references, 5 of whi'ch are Soviet, 4 American,
1 German, and I French.
SUBMITTEED: December 2, 1958
Card 2/2

6(1')
AUTHORs SOV/42-14-3-1/22
TITLEt Some Questions of Analysis and Differential Equationa
iEUODICALt Uspekhi matematicheskikh nauk, 1959,Vol 14,Nr 3,pp 3-20 (U-"3R)
ABSTRAM The author formulates a large number of interesting unsolved
problems from the theory of partial differential equations
and from the analysis ; e.g. s
1. Into which connected components decomposes the set of weak-
ly elliptic systems
Apq ? u (p) f (p9q - 1,2,...,n) ?
ik ~x i ax k q
2. If the system belongs to the same component as z-lui . 0
then it is to be decided whether the Dirichlet problem is
uniquely solvable for this system.
3. All the boundary value problems for elliptic systems are
to be given, the solution of which is unique with the ex-
ception of one finitely dimensional subspace and correctness.
4. The boundary conditions are to be given under which
Card 1/3

liome Questions of Analysis and Differential 2:quations SOV/42-14-3-1/22
R_U 1, ? 9 u admits a correct Cauchy problem with a
?t (PX i X)
unique solution.
5- Is it true that a boundary value problem is correct, if
and only if it is correct in every point ?
6. Is it possible to obtain every theorem of existence for
hyperbolic equations with the aid of energy integrals, if ar-
bitrary correct boundary conditions are given ?
7. The existence of the vacuum measure for the Tirring
equation Cjp+ k 2 'V A *3
is to be proved.
S. What kind of general methods can be developed in quantum
theory, if the existence of the vacuum measure of interacting
fields is assuzed7
The following authors are mentioned: Mityagin, Gorin, V.S.
Livehits, M.V. Keldysh, V.D. Lidskiy, I.G. Petrovskiy, Ye.11.
Landis, Biteadze, I.H. Vekua, Y.I. Vishik.
Card 2/3

2
. Some Questions of Analysis and Differential SOV/42-14-3-1/22
Equations
The author thanks S.V. Pomin for valuable discussions and
assistance in writing the final text.
There are 10 references, 0 of which are 3oviet, 1 American,
and I French.
SUBIaTTEDt February 23, 1959
Card 3/3

16(1)
LUTHORt- Ceilland# T-v- Corresponding Member, SOV/20-124-1-3/69
Academy of Sciences, USSR
TXTLEs On the Structure of the Ring of Rapidly lecreasing Functions
on a Lie Group (+trukture kolltsa bystro ubyvayushchikh
funktaijr n& gruppo Li)
PERIODICALt Doklady Akadenii nauk 85SH919599Vol 124,Nr 1,pp 19-21 (USSR)
ABSTRACT: Lot G be the group of the complex matrices g 11.4 a 11 ,
r G
4,~ - 8 r - I . The irreducible representations are given by
z + I n1-1 - n 2-1
(1) T 9f(z) W f "BZZ*+ J (8z + 9) (8z +9) ni-n2 - n in-
teger. A function is called quickly decreasing if for all n
it holdst jx(g)l - O(jjgjj_n) . The expressions P(D)x(g) are
called derivatives of x(g), where P is a polynomial in the
Lie operator Do Lot I' be the set of all those infinitely
differentiable x(g), the derivatives of which are quickly de-
creasing* If in r the multiplication is defined as con -
Card 1/3

On the Structure of the Ring of FAId4ly Decreasing SOV/20-124-1-3/0
Functions on a Lis group
volutionj and if a natural topology in introducedo then the
fundamental group ring of 0 arises. if x(g),ff r , then the
kernel K(sloz,,In,,n2) of the overator ~x(g)T 9 dg is denoted as
Fourier transfo= of x(g). The ring r tAnsforms into & ring
of kernels with usually defined multiplication. As author
formulates necessary and sufficient conditions which are to
be satisfied by a function K(z I'Z2 nl,n2) in order to b* a
Fourier transform of x(g); e.g. 1.)-K has to be infinitely
differentiable with respect to z Iand z2. 2*) There hold the
relations I
9L(S,,x2-xjn I n2)z IF 2 d z cr-z
SK(S,-ZRZ 2; - nj,-n2)z'nI-1 x -n2.1 dz -, dT
K(z Inq n,)-(-1)1' 1 R K(Zl,z 21-'1"2)
ni 1022 ?z n,
ZI 2
Card 2/3 n2 integer n, 1,2,....

On the Structure of the Ring of AsSIAly Docreasine SOV120-124-1-3169
Functions on a Lio Oroup
@t0.
The integral@ ~x(g)a(g)dg are denoted as moments of x(g),,
where &(g) is a matrix element of a finite-dimensional re-
presentation of 0 . The author directs to the importance of
the results for the theory of representations.
There are 4 references# 2 of which are Soviet and 2 American.
SUBMITTEDs September 24o 1958
Card 3/3

16(1)
AUTH0113i Gelltg"J..o Corresponding Member of the SOV/20-127-2-00
-rT-V-s-3R# and Grayev,X.I.
TITLE% Resolution of Lorents Group Representations Into Irreducible
Representations In Spaces of Functions Defined on Symmetrical
Spaces
PERIODICALt Doklady Akademii nauk SS3Rg1959tVo1 127FNr 2tpp 250-253 (USSR)
ABSTRAM Lot 0 be a Lorentz group$ i.e. the group of complex matrices of
second order C with the determinant Is Lot X be a
symmetrical ith the motion group G. In the space of
9 let to every g F_G correspond a translation
function f(x on X
operator T 9t T9f(x) - f(xg). The obtained representation of 0
shall be r4*oompo 'ed into Irreducible representations. The
authors 2 j v9 solved this problem if X is a Lobaohevokiy
ha
space. In the present paper the *&is problem in treated In an
other X. As a model of this space there say serve 9.g. the
exterior of the sphere (the "absolute") in the real projective
Card 1/2

4
Resolution of Lorentz Group Representations Into SOV/20-127-2-4/10
Irreducible Representations in Spaces of Functions
Defined on Symmetrical Spaces
space. Since in the present case the subgroup of the revolutions
is not compact ( this was used essentially in rRef 2_7 for the
case of the Lobachevskiy space) the authors propose a different
method which in essential bases on a certain decomposition for
the 6-funotion.
There are 3 references, 2 of which are Soviet, and I German.
SUBUITTED: U&Y 5, 1959
card 2/2

AUTIMS: Cellfanct,-laYsiGorresponding, Ilember, 30-1/20-127--l-2/7"
Academy of Sciences, USBR, Pyatetskiy-
Shapiro, I.I.
TITLE: On a Poincarb Theorem
1','M'C:)XILt D-oklady Akadenii nauk 'ISSR,1959,vol 127,.Nr 3,PP 490-40~3 (U:-,3R)
AI3"V:"tAI'TI The authors generalize a well-known result of roincar6 on the
mean number or revolutiona to ergodic dynamic systems with a
summable directional field.
Let V be a compact differentiable cuinifold; )e1,
the base of the group of integral homologies of I'. Let x be a
point of IM. Lot a dynamic system, i.e. a one-paraz-,eter Croup
X- -. Xtwith the invariant neasure, u. be dofined on 14. A tra-
jectory rising in x is ausumed to return infinitely often into
a neighborhood U of x, and this may happen for the parameter
values t1 < t2 -O fcr i S 'M
it holds uniformly in fS
D (f ( I Is o( Is I _m)
(here I varies in a compact domain of the space with the point
0 slackened; 4.) the integral
00 T( I Is) ak ds
is a homogeneous polynomial in I of the degree k (k a 0,1 ...
Theorem 2 contains the analogous statement for the complex case.
The Radon transformation of a generalized function in the complex
space is defined so that the usual definition is obtainec! for the
fundamental functions. The furmulu
Card V4

87386
S/020/60/135/006/002/037
C 111/ C 333
Integrals Over Hyperplanes of Fundamental and Generalized Functions
(-1) n-1 r V V
(5) fF(z)f(z)dtdf - (2 yr )2n-2 j F( 3 is) f S ;s)
is the starting point. (5) is briefly written as do di
V " (n-1, n-1)
(F,f) a F, f4 ) . As Radon transform,,of a generalized
function F the authors denote the functional F which is defined
V V (n-1, n-1
by the equation (F, f -(F, f) on the set of' the functions
V (n-10 n-1) 5 k
f,5 , where I runs thro gh the Ration tt~..i,4;forms of the
fundamental functions . Thereby ~ in ehiefl~ defiried in the subspace
of tho fundamental functions which satiofi cortain additional re-
lations. ~ can be continued to the whole space of the fundamental
functions in different ways. The authors give 10 examples of Radon
transformations.
There are 5 referencess 3 Soviet and 2 German.
SUBMITTED% September 26, 1960
Card 4/4

,gWJAW)p.~I. and USTLINg M. L.
"Mathematical Nodel of the Work of the Heart"
prenented at the All-Unlon Conference on Computational Mathematics and
Computational Techniques, Moscow, 16-28 November 1961
So: ZMWM k1k2rafi=. Issue 3. 1961. pp, 289-294

GEL I -I-zrai Sergey Vasillyevich; FOLOVIUKIN, S.M.,
red.; IUMARKIIIA, N.A4p tekhn, red,
(Calculus oC variations] Variatsionnoo inchislenie. Moskva, Gos.
izd-vo fiziko-matem.lit-ry, 1961. 228 (MURA 14-12)
(Calculus of variatlanX

GEL'FP11D, Izraill I-01oiseyevicb; YILF14KIN, N.Ya.
(Some applications of harmonic analysis- Adapted Hilbert
spaces] Nekotoryo primeneniia garmonicheakogo analiza. Os-
nashcheruWe gillbortovy prostranstva. Moskva, Fimatgizt
1961. 472 p. (Obobnhchenie funktf;ii,, no 4)
?MIRA 34:12)
(Hilbert space) (Harmonic analysis)

GGW4=,, I.M, I MWAXWv L.F.
Automatic contriol of heat-treating furnaces on a progr=wd
OF07"k , * Jbiallurg 6 no.8:29-31 Ag t61. (11IRA 14:8)
1. Muchno-iosledovatellakiy inotitut mitiznoy proinyshlonnosti.
(Furnaces, Iloat-treating)
(Automatic control)

KEDOVIZITP I.N. inzh.; GELIFAND. IJK.0 inzh.; ALITM, V.F.) lnzh.
P
Using an electric model for temperature deternInation, in the
center of defsWation during drawing. Stall 21 no.6:567-570
h 161, (MIRA 14: 5)
1. Nauchno-iosledovatellskiy institut metiznoy promyshlennosti.
(Drawing (Metalwork)-Electromechanical analogies)