(pp. 249-262)
A B S T R A C T S
A SURVEY OF COUNTEREXAMPLES TO HILBERT'S FOURTEENTH PROBLEM
Gene Freudenburg
freudenb@usi.edu
2000 Mathematics Subject Classification: 13A50, 14R20.
Key words:
Hilbert's fourteenth problem, invariant theory,
locally nilpotent derivations.
We survey counterexamples to Hilbert's Fourteenth Problem, beginning
with those of Nagata in the late 1950s, and including recent counterexamples
in low dimension constructed with locally nilpotent derivations. Historical
framework and pertinent references are provided. We also include 8 important
open questions.
ANALOG OF FAVARD'S THEOREM FOR POLYNOMIALS CONNECTED
WITH DIFFERENCE EQUATION OF 4-TH ORDER
S. M. Zagorodniuk
zol@evans.univer.kharkov.ua
2000 Mathematics Subject Classification:42C05, 39A05.
Key words:
Orthogonal polynomials, difference equation.
Orthonormal polynomials on the real line { pn(l)}n = 0¥
satisfy the recurrent relation of the form:
|
ln-1pn-1(l)+an pn(l)+ln pn+1(l) = lpn(l), n = 0,1,2,..., |
|
where ln > 0,an Î R,
n = 0,1,... ;l-1 = p-1 = 0,l Î C.
In this paper we study systems of polynomials
{ pn(l)}n = 0¥ which
satisfy the equation:
|
an-2 pn-2(l)+ |
bn-1
|
pn-1(l)+gn pn(l)+bn pn+1(l)+an pn+2(l) = l2 pn(l), n = 0,1,2,... , |
|
where an > 0, bn Î C, gn Î R,
n = 0,1,2,..., a-1 = a-2 = b-1 = 0,
p-1 = p-2 = 0, p0(l) = 1, p1(l) = cl+b,
c > 0, b Î C, l Î C.
It is shown that they are orthonormal on the real and the imaginary
axes in the complex plane:
|
|
ó
õ
|
RÈT
|
( pn(l), pn(-l))ds(l) |
|
= dn,m, |
|
n,m = [(0,¥)];T = (-¥,¥)
with respect to some matrix measure
.
Also the Green formula for difference equation of 4-th order is built.
COMPLETE SYSTEMS OF HERMITE ASSOCIATED FUNCTIONS
FIRST ORDER CHARACTERIZATIONS OF PSEUDOCONVEX FUNCTIONS
Vsevolod Ivanov Ivanov
vsevolodivanov@yahoo.com
2000 Mathematics Subject Classification:26B25, 90C26, 26E15.
Key words:
Generalized convexity, nonsmooth function,
generalized directional derivative,
pseudoconvex function, quasiconvex function, invex function,
nonsmooth optimization, solution sets,
pseudomonotone generalized directional derivative.
First order characterizations of pseudoconvex functions are
investigated in terms of generalized directional derivatives.
A connection with the invexity is analysed. Well-known first order
characterizations of the solution sets of pseudolinear programs are
generalized to the case of pseudoconvex programs. The concepts of
pseudoconvexity and invexity do not depend on a single definition
of the generalized directional derivative.
ON A CLASS OF GENERALIZED ELLIPTIC-TYPE INTEGRALS
Mridula Garg, Vimal Katta,
S. L. Kalla
Kalla@mcs.sci.kuniv.edu.kw
2000 Mathematics Subject Classification:
33C75, 33E05, 33C70.
Key words:
Elliptic integrals,
hypergeometric functions, integral formulae.
The aim of this paper is to study a generalized form
of elliptic-type integrals which unify and extend various families
of elliptic-type integrals studied recently by several authors. In
a recent communication [1] we have obtained recurrence relations and
asymptotic formula for this generalized elliptic-type integral. Here
we shall obtain some more results which are single and multiple integral
formulae, differentiation formula, fractional integral and approximations
for this class of generalized elliptic-type integrals.
WEAK POLYNOMIAL IDENTITIES FOR M1,1(E)
Onofrio Mario Di Vincenzo
divince@dm.uniba.it,
Roberto La Scala
lascala@dm.uniba.it
2000 Mathematics Subject Classification:
16R10, 16S50.
Key words:
Weak polynomial identities, superalgebras.
We compute the cocharacter sequence and generators of the ideal
of the weak polynomial identities of the superalgebra M1,1(E).
CONTINUITY OF PSEUDO-DIFFERENTIAL OPERATORS ON
BESSEL AND BESOV SPACES
Madani Moussai
mmoussai@yahoo.fr
2000 Mathematics Subject Classification:
47B38, 47G30.
Key words:
Pseudo-differential operators, Bessel and Besov spaces.
We study the continuity of pseudo-differential operators on Bessel potential
spaces Hps( Rn) , and on the corresponding
Besov spaces Bps,q( Rn) . The modulus of
continuity w we use is assumed to satisfy
|
|
å
j ³ 0
|
[ w(2-j) W(2j)]2 < ¥ |
|
where W is a suitable positive function.
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