A Hubert space characterization among function spaces

— [0,1] ,E — - e=1 [0,1]. I — _E =1, E=L ₂ x _e =xL ₂ x E.

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This work was prepared when the second author was a visiting professor of the CNR at the University of Firenze. He was supported by the Soros International Fund.     

On the representation type of one point extensions of tame concealed algebras

Letk be an algebraically closed field and a finite dimensionalk-algebra. Letq be the quadratic Tits form associated with . If is tame we show thatq is weakly semipositive. Let be a one-point extension of a tame concealed algebra, then is tame iffq is weakly semipositive.     

Additive functions

1<q<2 L:= _n=1 1/q ⁿ=1/q–1. [0,1] _n()=1, A _n:= _i=1 ^n–1 _i(x)/q^i+1/n x _n(x)=0, _n>. , = _{n=1 n}(x)/qⁿ. F: [0,L]R , F(x)= _{n=1 n}(x)a_n, _n=1 ¦a _n¦<. [0,L]. q(1,2), . , q(1, 2), . .     

On the growth order of partial sums of non-orthogonal series

a _k f _k , f _k L ₂, w-, (2), w(n) — . a _k f _k N {a _k }l ₂, {a _k }l ₂ ( 1, 2, 1a, 2a). ( 2) [8]. , {a _k } w-.     

Limiting behavior of certain mixtures of distributions

U — [0, 1] Y — . X=[1–U ^1/v /Y], U Y.     

On the parameter estimation of diffusional type processes with constant coefficients (Elementary Gaussian Processes)

, (1). 3, , ()=, (8) (16). [1], . (28) (31) ( 5), - (. [3]).

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The author thanks Professor M.Arató for having pointed out this problem, and for his valuable suggestions.     

Division Algebras that Ramify only on a Plane Quartic Curve with Simply Connected Components

A central division algebra over the field of rational functions in two variables with coefficients over an algebraically closed field ramifies along a divisor on P ². If the ramification divisor of is a quartic curve which is the union of simply connected curves, we show that is a symbol algebra and satisfies the index equals exponent equation.     

On convergence and summability of Fourier series

. , , –1<<0. .

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The present work was written on the basis of two earlier works received byAnalysis Mathematica on January 16, 1979, and July 20, 1979.     

On rearrangements of the Haar system in the space BMO

, .     

Onp-Helson sets in R n

p- . E R ⁿ -, f () _p(R ⁿ)., ER ⁿ 2nq ₀, E— - q ₀(q ₀-1). : q₀>2 n1 E R ⁿ 2nq ₀, p- p<₀. , f-[-, ]ⁿ, f A _p(R ⁿ) , _p([-, ]ⁿ) (1 << ).     

The convolution equation of Choquet and Deny on [IN]-groups

Let be a probability measure on a locally compact groupG. A real Borel functionf onG is called -harmonic if it satisfies the convolution equation *f=f. Given that isnonsingular with its translates, we show that the bounded -harmonic functions are constant on a class of groups including the almost connected [IN]-groups. If is nondegenerate and absolutely continuous, we solve the more general equation *= for positive measure on those groups which are metrizable and separable.Supported by Hong Kong RGC Earmarked Grant and CUHK Direct Grant     

An evolution problem for plastic antiplanar shear

We study the parabolic problemu/t=div_{x p}() in the function space L²(0, T; BV()). Here is the linear growth functional arising from the study of plastic antiplanar shear deformations.     

Generalization of Gδ*-Diagonals and ωΔ-Spaces

In this paper we introduce the concepts of a quasi-G*-diagonal and quasi--space as generalizations of the concepts of G*-diagonal and -space respectively. It is shown that a quasi-Moore space may be characterized in terms of these concepts. As a consequence we obtain the following metrization theorems: every paracompact -space with quasi-G-diagonal is metrizable and every collectionwise normal quasi--space is metrizable.     

On ideals of the group algebra of a free group of degree of freedom two over the field of complex numbers

In this paper, we prove the existence of an element of the group algebra A=F of a free groupF with two generatorsx andy over the field of complex numbersC such that, for any complexa andb for which ¦a¦=¦b¦=1, we haveA _a,b ()A=0, where _a,b ( is an automorphism ofA that mapsx,y intoax, by, respectively. Thus, we give a negative answer to question 12.46 of P. A. Linnel from Kourovka Notebook.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No.4, pp. 571–572, April, 1995.     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

The order of magnitude of the arithmetic means of double orthogonal series

, - , (C, 1,1), (C, 1,0) (C, 0,1)- . , , . - .

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Dedicated to Academician S. M. Nikol'skii on his 80th birthday     

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

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

On summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).     

Martingale Hardy spaces,BMO and VMO spaces with nonlinearly ordered stochastic basis

VMO. , ⁺, «» VMO . « » .     

Free Algebras of Automorphic Forms on the Upper Half-Plane

We find all pairs (,a) consisting of a cocompact Fuchsian group of genus zero and an automorphy factor a of for which the graded algebra of a--automorphic forms is free.