Galois theory for the family of partial differential equationsΔ α ψ=0

Summary The partial differential fields most suited for the purpose of construction of Galois theory for the family (1) are endowed with the symmetric bilinear form (2iv) and are called -differential fields. In Section 1 are defined certain algebraic notions related to the symmetric bilinear form (2iv) and which are necessary for the construction of any Galois theory. Necessary and sufficient condition for the extension of the domain of the operator (this operator is not a derivation although it commutes with the partial derivations of the -differential field) from an -differential fieldK to a finitely generated -differential extension field is given in Theorem 1.Section 2 defines the notion of -differential mapping as linear mappings which preserve the symmetric bilinear form and commute with the partial derivations. The group properties of the set of -differential mappings are discussed and the Galois correspondence theorems set up for -differential fields.Section 3 sets up the notion of -Liouvillian extensions of -differential fields and briefly discusses the Galois groups associated with these -Liouvillian extension fields.Section 4 points to the procedure for the algebraic characterization of -simple--differential field extensions by elementary solutions of the partial differential equation _m =0.     

Orthogonal systems invariant under an automorphism

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On extremal subspaces and approximation of periodic functions by splines of minimal defect

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Notes on best constants in some Sobolev inequalities

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On convergence of multiparameter strong submartingales in Banach lattices

- , ., B - — , B- {M _n ,nN ^d } n. .     

The Laplacian with Robin Boundary Conditions on Arbitrary Domains

Using a capacity approach, we prove in this article that it is always possible to define a realization of the Laplacian on L ²() with generalized Robin boundary conditions where is an arbitrary open subset of R ⁿ and is a Borel measure on the boundary of . This operator generates a sub-Markovian C ₀-semigroup on L ²(). If d=d where is a strictly positive bounded Borel measurable function defined on the boundary and the (n–1)-dimensional Hausdorff measure on , we show that the semigroup generated by the Laplacian with Robin boundary conditions has always Gaussian estimates with modified exponents. We also obtain that the spectrum of the Laplacian with Robin boundary conditions in L ^p () is independent of p[1,). Our approach constitutes an alternative way to Daners who considers the (n–1)-dimensional Hausdorff measure on the boundary. In particular, it allows us to construct a conterexample disproving Daners' closability conjecture.     

An approach to infinitary temporal proof theory

Aim of this work is to investigate from a proof-theoretic viewpoint a propositional and a predicate sequent calculus with an –type schema of inference that naturally interpret the propositional and the predicate until–free fragments of Linear Time Logic LTL respectively. The two calculi are based on a natural extension of ordinary sequents and of standard modal rules. We examine the pure propositional case (no extralogical axioms), the propositional and the first order predicate cases (both with a possibly infinite set of extralogical axioms). For each system we provide a syntactic proof of cut elimination and a proof of completeness.Supported by MIUR COFIN 02 Teoria dei Modelli e Teoria degli Insiemi, loro interazioni ed applicazioni.Supported by MIUR COFIN 02 PROTOCOLLO.Mathematics Subject Classification (2000):03B22, 03B45, 03F05     

Groups acting simply transitively on the vertices of a building of typeà 2, I

If a group acts simply transitively on the vertices of an affine building with connected diagram, then must be of typeà _n–1 for somen2, and must have a presentation of a simple type. The casen=2, when is a tree, has been studied in detail. We consider the casen=3, motivated particularly by the case when is the building ofG=PGL(3,K),K a local field, and when G. We exhibit such a group whenK=F _q ((X)),q any prime power. Our study leads to combinatorial objects which we calltriangle presentations. These triangle presentations give rise to some new buildings of typeà ₂.     

On strong summability of polynomial expansions

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Dedicated to Professor L. Leindler on his 50^th birthday     

Some expansion formulae for Fox'sH-function of two variables involving associated Legendre functions

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Approximation of functions of several variables by trigonometric polynomials with given number of harmonics,and estimates of ε-entropy

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WT measure vector spaces

n- M WT- , M n–1 . . WT- . .     

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 exactness of a theorem of G.A. Fomin

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О лагранжевом интерполировании с узлами в корнях многочленов сонина-маркова

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Асимптоический Анализ Распределения Статистики Диксона

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Moduli of smoothness and best approximation in the spacesL p , 0<p<1

L ^p , 0<<1, .     

Interpolation and best polynomial approximation in the domain ofp-adic integers

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The authors are indebted to Professor R. Bojanic for his valuable remarks and suggestions, especially for the simplification of the proof of Theorem 4.     

Determinants in net subgroups

Suppose R is a commutative ring with 1, =( _ij ) is a fixedD-net of ideals of R of ordern, and G is the corresponding net subgroup of the general linear group GL (n, R). There is constructed for a homomorphismdet of the subgroup G() into a certain Abelian group (). Let I be the index set {1...,n}. For each subset I let ()= _{ij ji} , wherei, ranges over all indices in and j independently over the indices in the complement I ((I) is the zero ideal). Letdet (a) denote the principal minor of order ||n of the matrixa G () corresponding to the indices in , and let' () be the Cartesian product of the multiplicative groups of the quotient rings R/() over all subsets I. The homomorphismdet is defined as follows: It is proved that if R is a semilocal commutative Bezout ring, then the kernelKer det coincides with the subgroup E() generated by all transvections in G(). For these R is also definedTm det .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 37–49, 1982.     

On the admissibility of function spaces of typeB p,q s andF p,q s,and boundary value problems for non-linear partial differential equations

u=f(x)+S(u), S — , u-G(u), G — . B _p,q ^s () -F _p,q ^s (). R _n — . — . _p,q ^s F _p,q ^s .