Regularity of the inverse of a homeomorphism of a Sobolev–Orlicz space

Given a homeomorphism ? ∈ W _M ¹ , we determine the conditions that guarantee the belonging of the inverse of ? in some Sobolev–Orlicz space W _F ¹ . We also obtain necessary and sufficient conditions under which a homeomorphism of domains in a Euclidean space induces the bounded composition operator of Sobolev–Orlicz spaces defined by a special class of N-functions. Using these results, we establish requirements on a mapping under which the inverse homeomorphism also induces the bounded composition operator of another pair of Sobolev–Orlicz spaces which is defined by the first pair.     

On the Covering of the Vertices of a Graph by Cliques

In conversation I was told by Professor R.Brigham the following conjecture [1].Let G(n) be a graph of n vertices.Denote by f(G(n))=t the smallest integer for which the vertices of G(n) can be covered by t cliques. Denote further by h(G(n)) =l the largest integer for which there are l edges of our G(n) no two of Which are in the same clique.Clearly h(G(n)) can be much larger than f(G(n))e.g.if n=2m and G(n) is the complete bipartite graph of m white and m black vertices.Then l(G(n))=m and l(G(n))=m~2. It was conjectured that if G(n).has no isolated vertices then     

Inequality for the Moment of a Function of a Random Variable

ＩｎｅｑｕａｌｉｔｙｆｏｒｔｈｅＭｏｍｅｎｔｏｆａＦｕｎｃｔｉｏｎｏｆａＲａｎｄｏｍＶａｒｉａｂｌｅ￥ＬｉＢａｉｎｉａｎＨｕＳｈｕｈｅ（李柏年，胡舒合）（ＡｎｈｕｉＩｎｓｔｉｔｕｔｅｏｆＦｉｎａｎｃｅａｎｄＴｒａｄｅ）（ＡｎｈｕｉＵｎｉｖｅｒｓｉｔｙ...     

What Is the Object of the Encapsulation of a Process?

Several theories have been proposed to describe the transition from process to object in mathematical thinking. Yet, what is the nature of this “object” produced by the “encapsulation” of a process? Here, we outline the development of some of the theories (including Piaget, Dienes, Davis, Greeno, Dubinsky, Sfard, Gray, and Tall) and consider the nature of the mental objects (apparently) produced through encapsulation and their role in the wider development of mathematical thinking. Does the same developmental route occur in geometry as in arithmetic and algebra? Is the same development used in axiomatic mathematics? What is the role played by imagery?     

Turán Number of the Family Consisting of a Blow-up of a Cycle and a Blow-up of a Star

Let F={H₁,...,H_k}(k> 1) be a family of graphs.The Turán number of the family F is the maximum number of edges in an n-vertex {H₁,...,H_k)-free graph,denoted by ex(n,F) or ex(n,{H₁,H₂,...,H_k}).The blow-up of a graph H is the graph obtained from H by replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different.In this paper we determine the Turán number of the family cons...     

On the differential properties of the support function of the∈-subdifferential of a convex function

We study differential properties of the support function of the-subdifferential of a convex function; applications in algorithmics are also given.     

The σ-algebra of pasts of a random walk over the orbits of a Bernoulli action of the group Z d

In the present paper, we study the σ-algebra of pasts Ξ = {ξ _n } _n of a random walk T over the orbits of a Bernoulli action of the group Z^d. We calculate the proper scaling and the scaling entropy of this sequence of partitions. We show that the proper scaling entropy of the σ-algebra of pasts is equal to . Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 103–112.     

On the Properties of the Solution of a Strongly Degenerate Parabolic Equation

ＯｎｔｈｅＰｒｏｐｅｒｔｉｅｓｏｆｔｈｅＳｏｌｕｔｉｏｎｏｆａＳｔｒｏｎｇｌｙＤｅｇｅｎｅｒａｔｅＰａｒａｂｏｌｉｃＥｑｕａｔｉｏｎＣｈｅｎｇＦｕｘｉａ（ＤｅｐａｒｔｍｅｎｔｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ，ＴｓｉｎｇｈｕａＵｎｉｖｅｒｓｉｔ...     

On solutions of a common generalization of the Go?a?b-Schinzel equation and of the addition formulae

Under some additional assumptions we determine solutions of the equation

f(x+M(f(x))y)=f(x)○f(y),     

On the topology of the set of singularities of a solution to the Hamilton–Jacobi equation

We address the topology of the set of singularities of a solution to a Hamilton–Jacobi equation. For this, we will apply the idea of the first two authors (Cannarsa and Cheng, Generalized characteristics and Lax–Oleinik operators: global result, preprint, arXiv:1605.07581, 2016) to use the positive Lax–Oleinik semi-group to propagate singularities.     

The Stablility of a Loop Formed the Separatrix of a Saddle Point and the Condition to Produce a Limit Cycle

In this paper,we discuss a loop by the separatrix of a saddle point of two dimensional autonmoussystem.It contains two sections.In section 1,a short proof of the criterion theorem of stability ofthe loop of saparatrix is given wi~h ~he help o~th~ ~h~orjr 0~ i'otatin~ vector field, and some in~ormatioafor the study of stability of that loop in critical case is supplied.In section 2,we give a sufficientcondition under which a limit cycle is produced from the loop of separatrix.     

Computation of the number representations of the elements of the ringℤ/dℤ as a sum of squares

The number of representations of the elements of the ringℤ/dℤ as a sum of invertible squares is computed, provided that each square occurs in the sum no more tha a fized number of times. For prime d an exhaustive answer is given in term of the class number and the fundamental unit of the real quadratic field . Biblography: 5 titles. Translted fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 5–8.     

Correction to the paper “on the curvature of a generalization of a contact metric manifolds”

We considered in Example 3.1 of the paper [1] an S-structure on R^2n+s . We concluded that when s > 1 this manifold cannot be of constant φ-sectional curvature. Unfortunately this result is wrong. In fact, essentially due to a sign mistake in defining the φ-structure and a consequent transposition of the elements of the φ-basis (3.2), some of the Christoffel’s symbols were incorrect. In the present rectification, using a more slendler tecnique, we prove that our manifold is of constant φ-sectional curvature −3s and then it is η-Einstein.     

Proof of a conjecture of Zahariuta concerning a problem of Kolmogorov on the ε-entropy

We prove a conjecture of Zahariuta which itself solves a problem of Kolmogorov on the -entropy of some classes of analytic functions. For a given holomorphically convex compact subset K in a pseudoconvex domain D in Cⁿ, Zahariutas conjecture consists in approximating the relative extremal function u^*_K,D, uniformly on any compact subset of DK, by pluricomplex Green functions on D with logarithmic poles in the compact subset K.     

On the Monotonicity of the Speed of Random Walks on a Percolation Cluster of Trees

The authors consider the simple random walk on the infinite cluster of the Bernoulli bond percolation of trees, and investigate the relation between the speed of the simple random walk and the retaining probability p by studying three classes of trees. A sufficient condition is established for Galton-Watson trees.     

Boundedness of the gradient of a solution to the Neumann–Laplace problem in a convex domain

It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex n-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem on convex polyhedra are given. To cite this article: V. Maz'ya, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Cardinality of the set of extreme extensions¶of a quasi-measure

Cardinals that arise as the number of extreme quasi-measure extensions of a quasi-measure [resp., measure] μ defined on an algebra [resp., σ-algebra] of sets to a larger algebra [resp., σ-algebra] of sets are characterized in the general case as well as under some natural assumptions on μ. Received: 19 September 2000     

On a transformation of the wiener process in ℝ m by a functional of the local time type on a surface

A transformation of the Wiener process _t in ^m is considered. This transformation is realized by a multiplicative functional _l=u(_l/u(₀), where the functionu is constructed in a certain way by using a functional of the local time type on a surface. It is proved that this transformation is equivalent to the successive application of an absolutely continuous change of a measure and killing on the surface.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 863–866, June, 1993.     

On the Truth of a Conjecture Concerning the q-Derangement Numbers

ＯｎｔｈｅＴｒｕｔｈｏｆａＣｏｎｊｅｃｔｕｒｅＣｏｎｃｅｒｎｉｎｇｔｈｅｑ－ＤｅｒａｎｇｅｍｅｎｔＮｕｍｂｅｒｓＺｈａｎｇＸｉａｎｇｄｅ（Ｄｅｐｔ．ｏｆＡｕｔｏｍａｔｉｃＣｏｎｔｒｏｌ，ＮｏｒｔｈｅａｓｔｅｒｎＵｎｉｖ，Ｓｈｅｎｙａｎｇ１１０００６）...     

Schrödinger equation. The theorem concerning the ansatz representation of a solution concentrated in a neighborhood of a minimum of the potential

The one-dimensional Schrödinger equation is considered on the segment [–l,l] It is assumed that the potential v(x) of this equation has one minimum v(0)=v(0)=0, v(0)>0 v(x)>0 for x0; v(x)h>0 outside some neighborhood of zero. It is proved that there exists a solution of the form where is a parabolic cylinder function, and is a smooth function which is bounded on [–l,l] together with derivatives through third order by a constant not depending on . The function and the real number E admit a known asymptotic expansion as 0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 137–150, 1984.