Converse results on equiconvergence of interpolating polynomials

, n- n- 1 , . , , ( « ») f .     

On the convergence in a restricted sense of multiple series

Z ^d — k=(k ₁, ...,k _d) k _j,d1.d- (8), . . a _k s _m= a _k s, >0 N, min (m ₁,...,m _d)N, ¦s _m–s¦. , , >0 N, min (m ₁,...,m _d)N min (n ₁,...,n _d)N, ¦s _m–s _n. . , (8) , >0 N, max (b ₁,...,b _d) N, m — Z ^d , m1, ¦s(b, m)¦ where      

On a retract of an integral domain

Let A R be commutative integral domains with identity. We say A is a retract of R when there exists a ring homomorphism : RA such that |_A=id_A. We Prove two theorems: Theorem 3.1 says when A is a retract of R via , with Kernel ()= =t·R (O), then t is transcendental over A and if A is sub-inert in R, then RK(t)=A[t], where K is the quotient field of A. This theorem with the hypotheses strengthened to A being a 2-valuation algebra of R and the transcendence degree of R over A equal to one, gives Theorem 4.1, which now says R=A[t]. Given the other hypotheses of Theorem 4.1, A being a 2-valuation algebra of R is equivalent to the existence of a valuation ring V of L, the quotient field of R, such that VR=A. Some counterexamples are given and some variations on the hypotheses of Theorem 4.1 are discussed.     

Interpolation of weighted Orlicz-valued function spaces

- L. , .

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This paper is to be part of the author's doctoral dissertation written at the University of Campinas under the supervision of Prof. D. L. Fernandez.     

Folgenräume als universelle Generatoren für Klassen lokalkonvexer Räume

Motivated by the known characterizations of equicontinuity in the dual of a Schwartz space, a nuclear space, or a strongly nuclear space,we introduce the concepts of a -sequence and of a ()-sequence in the dual of an arbitrary lcs [E,], and we investigate the corresponding topologies and ₍₎ on E of uniform convergence on these sequences. Here is a normal sequence space such that . Under favorable enough conditions on , including the nuclearity of its normal topology , [,] acts as a universal generator for those lcs [E,] which satisfy =. Under somewhat weaker assumptions on , [,₍₎] is a universal generator for the lcs [E,] with =₍₎. These results cover e.g. the cases of -nuclear spaces and of nuclear spaces known from the recent literature. As an application we show that every non-trivial ultrabornological lcs is representable as an inductive limit of isomorphic copies of [, ( , )], where is any nuclear power series space of infinite type with stable exponent sequence.     

Global bifurcation of fixed points and the Poincaré translation operator on manifolds

Let M be a differentiable manifold and [0, )×MM be a C¹ map satisfying the condition (0, p)=p for all pM. Among other results, we prove that when the degree (also called Hopf index or Euler characteristic) of the tangent vector field wMTM, given by w(p)=(/)(0, p), is well defined and nonzero, then the set (of nontrivial pairs) admits a connected subset whose closure is not compact and meets the slice {0}×M of [0, )×M. This extends known results regarding the existence of harmonic solutions of periodic ordinary differential equations on manifolds.     

Exchange Rings with Artinian Primitive Factors

We prove that every exchange ring with primitive factors Artinian is clean. Also, it is shown that for exchange rings with Artinian primitive factors, the following are equivalent: (1) Every element in R is a sum of two units. (2) There exist , U(R) such that + = 1. (3) R does not have Z / 2 Z as a homomorphic image. Finally, we prove that exchange ring R is strongly -regular if the Jacobson radical of any homomorphic image of R is T-nilpotent or locally nilpotent. These are generalizations of the corresponding results of A. Badawi, W. D. Burgess and P. Menal, Fisher and Snider, and J. Stock.     

A local comparison theorem with applications to approximation in certain group and multiplier algebras

X- E — -. , X . . Rⁿ Rⁿ/(2Z)ⁿ L ^p(Rⁿ) . -R _t f , (-1)/2.

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The contribution of the first author was supported in part by Grant No. II B 7 — FA 5334 awarded by the Minister für Wissenschaft und Forschung des Landes Nordrhein-Westfalen.     

О Распределении Значений Целой Функции Ро Аргументам

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K1 of Twisted Rings of Polynomials

We prove that for an arbitrary endomorphism of a ring R the group K1(R[t]) splits into the direct sum of K1(R) and Ñil (r;). Moreover, for any such R and Ñil (R; ) is isomorphic to Ñil (R ; ) for some ring R with : R R – an isomorphism.     

Necessary and sufficient condition of the existence of global Hopf bifurcation for Hamiltonian systems

ForH C ² (,R) where 0 R ²ⁿ ,H (0)=0 and detH(0)0, the paper proves that there is a global Hopf bifurcation fromx=0 for Hamiltonian systemx=JH(x) iffJH(0)possesses purely imaginary eigenvalues. The work improves the corresponding result of J.C.Alexander and J. Yorke (Amer. J. Math., 100 (1978), 263–292).     

On the Irrationality of Certain Series

In this note we consider the irrationality of certain series of the form . A conjecture of P. Erds in [Sur lirrationalité dune certaine série, C. R. Acad. Sci., Paris, Sér I 292(1981), 765-768] is confirmed, and the theorem of the same paper is generalized.     

The product of finitely based varieties of lattice-ordered groups

The question as to whether a product of two finitely based varieties of lattice-ordered groups is finitely based is considered. It is proved that varieties and are finitely based; here is a variety of lattice-ordered groups defined by identities [x ⁿ,y ⁿ] =e and [[x,y] z, [x ₁,y ₁] z ₁] =e; is a variety of lattice-ordered nilpotent groups of class s, defined by an identity [x ₁,x ₂,...,x _(s+1)] =e; V is an arbitrary finitely based variety of lattice-ordered groups. Translated fromAlgebra i Logika, Vol. 33, No. 3, pp. 255–263, May–June, 1994.Supported by the Russian Foundation for Fundamental Research, grant No. 93-011-1524.     

Identities of an algebra of triangular matrices

This paper deals with the ideals of identities of certain associative algebras over a field F of characteristic zero. An algebra W of matrices of the form ,,,M, where and , are F-algebras with unity and M is a (,)-bimodule, is considered. Under certain natural restrictions on M one obtains the equality of ideals of identities T(W)=T()T(), if [[x₁,x₂], x₃[x₄,x₅]]T().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 7–27, 1982.     

Covering the Hecke Triangle Surfaces

In the mid-1980s an equivalence was established between the simple closed geodesics on the Riemann surfaces obtained as quotients of the upper half plane H by any of the following subgroups of the modular group (1) : , (3), and ³. An axis of a hyperbolic element of (1) projects to a simple closed geodesic on one of these surfaces if and only if it does so on the other two.This equivalence was used to obtain a variety of Diophantine and geometric results. In subsequent related investigations, the role of (1) was assumed by the Hecke triangle group G_q for q 3. (For q = 3, we have (1) = G₃.) These works employed the analog of ³, denoted _q.In the context of the G_q, the present paper gives the analog of , which we denote _q. As in the case q = 3, we have [_q:_q] = 2. A rather full discussion of geometry of _q\ H is given. In particular, we demonstrate that the equivalence of simple closed geodesics on _q\ H and _q\ H does not hold for q 7.As of this writing, we have not been able to obtain an appropriate analog of (3).     

On certain imbedding and extension theorems for a weighted class of functions

n- , S n–1 m (0m<n), () S . () , ().     

The Parametric Buffer Phenomenon for a Singularly Perturbed Telegraph Equation with a Pendulum Nonlinearity

We consider the boundary-value problem u _tt + u _t + (1 + cos2)sin u =² u _xx, u _x|_x=0=u_x|_x==0, where 0<1, =(1+)t, ,> 0, and the sign of is arbitrary. It is proved that for an appropriate choice of the external parameters and and for sufficiently small the number of exponentially stable solutions 2-periodic in can be made equal to an arbitrary predefined number.     

Bivariate distributions with Weibull conditionals

, . , - . , , , .     

‘Outside’ as a primitive notion in constructive projective geometry

In intuitionistic (or constructive) geometry there are positive counterparts, apart and outside, of the relations = and incident. In this paper it is shown that the relation outside suffices to define incident, apart and equality. The equivalence of the new system with Heyting's system is shown and as a simple corollary one obtains duality for intuitionistic projective geometry.     

Extension of functions,which are traces of functions belonging to 249-01249-01249-01on an arbitrary subset of the line

( , k) , ER f(x) f(x)H _k (R) (k2— ,=(t) — k- ). f(x) E (t).