k-regular factors and semi-k-regular factors in bipartite graphs

LetG be a graph, andk1 an integer. LetU be a subset ofV(G), and letF be a spanning subgraph ofG such that deg _F (x)=k for allx V(G)–U. If deg _F (x)k for allxU, thenF is called an upper semi-k-regular factor with defect setU, and if deg _F (x)k for allxU, thenF is called a lower semi-k-regular factor with defect setU. Now letG=(X, Y;E(G)) be a bipartite graph with bipartition (X,Y) such that X=Yk+2. We prove the following two results.(1) Suppose that for each subsetU ₁X such that U ₁=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.(2) Suppose that for each subsetU ₁X such that U ₁=X–1/k+1,G has a lower semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=X–1/k+1,G has a lower semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.     

Mean-field critical behaviour for correlation length for percolation in high dimensions

Summary Extending the method of [27], we prove that the corrlation length of independent bond percolation models exhibits mean-field type critical behaviour (i.e. (p(p _c –p)^–1/2 aspp _c ) in two situations: i) for nearest-neighbour independent bond percolation models on ad-dimensional hypercubic lattice ^d , withd sufficiently large, and ii) for a class of spread-out independent bond percolation models, which are believed to belong to the same universality class as the nearest-neighbour model, in more than six dimensions. The proof is based on, and extends, a method developed in [27], where it was used to prove the triangle condition and hence mean-field behaviour of the critical exponents , , , and ₂ for the above two cases.     

The conformal structure of Riemann surfaces with boundary parametrizing minimal surfaces

Let F denote a surface with boundary F, being contained in a Riemann surface R, such that R\F is somedisk. If we vary the boiundary curve _o parametrizing F, we will get a manifold of real dimension 6g–3, such that any bounds some F and any local deformation of F is conformally equivalent to just one F for .This result also implies that none of the conformal invariants of R will be an invariant of this F, since its neighbors {F|} cover all possible deformations of F at all.     

Fixed-width confidence intervals for contrasts in the means

The sequential procedure developed by Bhargava and Srivastava (1973, J. Roy. Statist. Soc. Ser. B, 35, 147–152) to construct fixed-width confidence intervals for contrasts in the means is further analyzed. Second-order approximations for the first two moments of the stopping time and the coverage probability associated with the sequential procedure, are obtained. A lower bound for the number of additional observations after stopping is derived, which ensures the mxact probability of coverage. Moreover, two-stage, three-stage and modified sequential procedures are proposed for the same estimation problem. Relative advantages and disadvantages of these sampling schemes are discussed and their properties are studied.     

An estimate of the norm for self-adjoint operators A and B

One proves that, where is a segment containing the spectra of the self-adjoint operators A and B, is the Lipschitz constant of the function, and · is the operator norm. It is shown on examples that an estimate of the type cannot be true for all with even if the spectra of the operators A, B are contained in the segment [0, 1] the norm A–B is arbitrarily small.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 56, pp. 143–162, 1976.     

Saturation theorems for discretized linear operators

, , , - .

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Part of this research was completed at a conference of Hungarian and American mathematicians held in Madison, Wisconsin, August 1974, and sponsored by the National Science Foundation (USA) and the Institute for Cultural Relations (Hungary).

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The first author gratefully acknowledges NSF support in Grant GP 19620.     

Functorial semantics and HSP type theorems

We show that if ] is the category of models for a theory in the sense of Linton over an arbitrary base category, then a full subcategory of ] is closed under Homomorphic images that split in the underlying category, under -Subobjects for a class of monomorphisms and under Products if and only if it is an intersection of a nest of subcategories, each determined from the preceding by a class of Horns, in which the crucial arrow lies in the class of epimorphisms orthogonal to .Dedicated to the memory of Evelyn Nelson, Alan Day and Alan MeklerPresented by F. E. J. Linton.This research has been supported by grants from the NSERC of Canada and the FCAR du Québec.     

Some complements and corrections to my papers on the theory of attractors for abstract semigroups

In Sec. 1 a correction is given of the estimate of the Hausdorff dimension and an estimate of the fractal dimension of a bounded subset of a Hilbert space, semiinvariant with respect to a flattening transformation. In Sec. 2 the results, proved by the author for semigroups with a continuous group parameter tR⁺[0, ), are carried over to the case when t runs through the semigroup ⁺{tt0} of some additive group R=(–, ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 182, pp. 102–112, 1990.     

Converse results on equiconvergence of interpolating polynomials

, n- n- 1 , . , , ( « ») f .     

Extremal feedback perturbations for families of closed operators

Let T- S, be a family of not necessarily bounded semi-Fredholm operators, where T and S are operators acting between Banach spaces X and Y, and where S is bounded with D(S) D(T). For compact sets , as well as for certain open sets , we investigate existence and minimal rank of bounded feedback perturbations of the form F=BE such that min.ind (T-S+F)=0 for all . Here B is a given operator from a linear space Z to Y and E is some operator from X to Z.We give a simple characterization of that situation, when such regularizing feedback perturbations exist and show that for compact sets the minimal rank never exceeds max { min.ind (T-S) }+1. Moreover, an example shows that the minimal rank, in fact, may increase from max {...} to max {...}+1, if the given B enforces a certain structure of the feedbachk perturbation F.However, the minimal rank is equal to max { min.ind (T-S) }, if is an open set such that min.ind (T-S) already vanishes for all but finitely many points . We illustrate this result by applying it to the stabilization of certain infinite-dimensional dynamical systems in Hilbert space.     

Замечания о Сходкмости лагранжева интерполрования

( qP. )     

Primitive Inner Illuminating Systems for Convex Bodies

A set X of boundary points of a (possibly unbounded) convex body KE ^d illuminating K from within is called primitive if no proper subset of X still illuminates K from within. We prove that for such a primitive set X of an unbounded, convex set KE ^d (distinct from a cone) one has X=2 if d=2, X6 if d=3, and that there is no upper bound for X if d4.     

A finite element method for a singularly perturbed boundary value problem

Summary We examine the problem:u+a(x)u–b(x)u=f(x) for 0<x<1,a(x)>0,b(x)>, ² = 4>0,a, b andf inC ² [0, 1], in (0, 1],u(0) andu(1) given. Using finite elements and a discretized Green's function, we show that the El-Mistikawy and Werle difference scheme on an equidistant mesh of widthh is uniformly second order accurate for this problem (i.e., the nodal errors are bounded byCh ², whereC is independent ofh and ). With a natural choice of trial functions, uniform first order accuracy is obtained in theL (0, 1) norm. On choosing piecewise linear trial functions (hat functions), uniform first order accuracy is obtained in theL ¹ (0, 1) norm.     

On Riesz means with respect to a cylindric distance function

(1–) ₊ , — R ⁿ =R ^j ×R ^k , ()=max{¦ ₁¦, ¦ ₁¦},=( ₁, ₂), ₁R ^J , ₂R ^k ,j,k1,n=j+k. n=3 , (1–) ₊ [L ¹(R ⁿ )]¹, >1/2; j=4, (1–) ₊ R L ^p (R ⁿ ). .

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The author would like to thank Professor W. Trebels for encouragement and valuable advice.     

On discrete and continuous norms in Paley-Wiener spaces and consequences for exponential frames

It is well known that for certain sequences {t_n}_n the usual L^p norm ·_p in the Paley-Wiener space PW ^p is equivalent to the discrete norm f_p,{tn}:=( _n=– |f(t_n)|^p)^1/p for 1 p = < and f_,{tn}:=sup _n|f(t_n| for p=). We estimate f_p from above by Cf_p, _n and give an explicit value for C depending only on p, , and characteristic parameters of the sequence {t_n}_n. This includes an explicit lower frame bound in a famous theorem of Duffin and Schaeffer.     

Transferring estimates for zonal convolution operators

_p - , . Ad- .     

A representation of symmetric functions in Carleman-Gevrey spaces

For a symmetric function t(x)(x^d) one investigates the representation, where _j(x) is the elementary symmetric polynomial of degree j. Let be the closure of the domain in ^d, let be a numerical sequence such that (n) does not decrease, let be the Carleman-Gevrey space, i.e. the collection of functions (n+1)/(n) such that for any bounded subdomain there exists a constant tC() with which one has the inequality _x t(x)H⁺¹!() (x*#x03A9;'). Let S be the image of ^d under the mapping x(₁(x), ..., _d(x)). One proves the following theorem: For any tk(^d) there exists such that, if and only if (n)(nd)ⁿ⁺¹, where is some positive number, independent of n.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 116–126, 1986.     

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.     

The problem of controlling a Wiener process when the number of switchings is bounded

We are considering the problem of controlling a one-dimensional Wiener process (t) (0)=0,E=0,D= ²t.Translated fromProblemy Ustoichivosti Stokhasticheskikh Modelei. Trudy Seminara, 1988, pp. 53–55.     

On the optimal mapping of distributions

We consider the problem of mappingXY, whereX andY have given distributions, so as to minimize the expected value of X–Y². This is equivalent to finding the joint distribution of the random variable (X, Y), with specified marginal distributions forX andY, such that the expected value of X–Y² is minimized. We give a sufficient condition for the minimizing joint distribution and supply numerical results for two special cases.