On equivalence of moduli of smoothness of polynomials in ,

It is well known that for functions , 1p∞. For general functions fL_p, it does not hold for 0<p<1, and its inverse is not true for any p in general. It has been shown in the literature, however, that for certain classes of functions the inverse is true, and the terms in the inequalities are all equivalent. Recently, Zhou and Zhou proved the equivalence for polynomials with p=∞. Using a technique by Ditzian, Hristov and Ivanov, we give a simpler proof to their result and extend it to the L_p space for 0<p∞. We then show its analogues for the Ditzian–Totik modulus of smoothness and the weighted Ditzian–Totik modulus of smoothness for polynomials with .     

On the number of primitive representations of integers as sums of squares

Formulas for the number of primitive representations of any integer n as a sum of k squares are given, for 2 ≤ k ≤ 8, and for certain values of n, for 9 ≤ k ≤ 12. The formulas have a similar structure and are striking for their simplicity. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—11E25; Secondary—05A15, 33E05.     

Fast verification of solutions of matrix equations

Summary. In this paper, we are concerned with a matrix equation where A is an real matrix and x and b are n-vectors. Assume that an approximate solution is given together with an approximate LU decomposition. We will present fast algorithms for proving nonsingularity of A and for calculating rigorous error bounds for . The emphasis is on rigour of the bounds. The purpose of this paper is to propose different algorithms, the fastest with flops computational cost for the verification step, the same as for the LU decomposition. The presented algorithms exclusively use library routines for LU decomposition and for all other matrix and vector operations. Received June 16, 1999 / Revised version received January 25, 2001 / Published online June 20, 2001     

Convergence of euler-stokes splitting of the navier-stokes equations

We consider approximation by partial time steps of a smooth solution of the Navier-Stokes equations in a smooth domain in two or three space dimensions with no-slip boundary condition. For small k > 0, we alternate the solution for time k of the inviscid Euler equations, with tangential boundary condition, and the solution of the linear Stokes equations for time k, with the no-slip condition imposed. We show that this approximation remains bounded in H^2,p and is accurate to order k in L^p for p > ∞. The principal difficulty is that the initial state for each Stokes step has tangential velocity at the boundary generated during the Euler step, and thus does not satisfy the boundary condition for the Stokes step. The validity of such a fractional step method or splitting is an underlying principle for some computational methods. © 1994 John Wiley & Sons, Inc.     

Derivation of equations of seismic and acoustic wave propagation and equations of filtration via homogenization of periodic structures

A linear system of differential equations describing a joint motion of elastic porous body and fluid occupying porous space is considered. Although the problem is linear, it is very hard to tackle due to the fact that its main differential equations involve nonsmooth oscillatory coefficients, both big and small, under the differentiation operators. The rigorous justification, under various conditions imposed on physical parameters, is fulfilled for homogenization procedures as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic. As the results for different ratios between physical parameters, we derive Biot’s equations of poroelasticity, a system consisting of nonisotropic Lamé’s equations for the solid component and acoustic equations for the liquid component, nonisotropic Lamé’s equations or equations of viscoelasticity for one-velocity continuum, decoupled system consisting of Darcy’s system of filtration or acoustic equations for the liquid component (first approximation) and nonisotropic Lamé’s equations for the solid component (second approximation), a system consisting of nonisotropic Stokes equations for the liquid component and acoustic equations for the solid component, nonisotropic Stokes equations for one-velocity continuum, or, finally a different type of acoustic equations for one- or two-velocity continuum. The proofs are based on Nguetseng’s two-scale convergence method of homogenization in periodic structures.     

Methods of localization of discontinuities in solution of first kind equation of convolution type

We consider a convolution-type equation of the first kind from L ₁ to L ₁. We construct and investigate regularizing methods for localization (detection of positions) of discontinuities of the first kind of solution to this equation. Under additional conditions of exact solution, we obtain estimates for the accuracy of localization and for the separability threshold, which are another important characteristics of the methods.     

Non-wandering sets of the powers of maps of a tree

Let T be a tree and let Ω ( f ) be the set of non-wandering points of a continuous map f: T→ T. We prove that for a continuous map f: T→ T of a tree T: ( i) if x∈ Ω( f) has an infinite orbit, then x∈ Ω( fⁿ) for each n∈ ℕ; (ii) if the topological entropy of f is zero, then Ω( f) = Ω( fⁿ) for each n∈ ℕ. Furthermore, for each k∈ ℕ we characterize those natural numbers n with the property that Ω(f^k) = Ω(f^kn) for each continuous map f of T.     

Depths of Powers of the Edge Ideal of a Tree

Lower bounds are given for the depths of R/I ^t for t ≥ 1 when I is the edge ideal of a tree or forest. The bounds are given in terms of the diameter of the tree, or in case of a forest, the largest diameter of a connected component and the number of connected components. These lower bounds provide a lower bound on the power for which the depths stabilize.     

On Dehn Functions of Infinite Presentations of Groups

We introduce two new types of Dehn functions of group presentations which seem more suitable (than the standard Dehn function) for infinite group presentations and prove the fundamental equivalence between the solvability of the word problem for a group presentation defined by a decidable set of defining words and the property of being computable for one of the newly introduced functions (this equivalence fails for the standard Dehn function). Elaborating on this equivalence and making use of this function, we obtain a characterization of finitely generated groups for which the word problem can be solved in nondeterministic polynomial time. We also give upper bounds for these functions, as well as for the standard Dehn function, for two well-known periodic groups. In particular, we prove that the (standard) Dehn function of a 2-group Γ of intermediate growth, defined by a system of defining relators due to Lysenok, is bounded from above by C₁x² log₂ x, where C₁ > 1 is a constant. We also show that the (standard) Dehn function of a free m-generator Burnside group B(m, n) of exponent n ≥ 2⁴⁸, where n is either odd or divisible by 2⁹, defined by a minimal system of defining relators, is bounded from above by the subquadratic function x^19/12. Received: September 2007, Revision: March 2008, Accepted: March 2008     

Global dimension of rings of functions

This article deals with the global dimension of rings of functions. An improved lower bound for global dimension is proved for von Neumann regular rings. If Xis a compact, Hausdorff and zero-dimensional space, and its weight and independence character coincide, then the global dimension of (X), its Stone dual, can be calculated. The spaces for which these invariants agree are studied. Finally, it is shown that, except for P-spaces, the global dimension of C(X) is at least 3.     

Formulas for sums of powers of integers by functional equations

This paper presents a direct and simple approach to obtaining the formulas forS _k(n)= 1 ^k + 2 ^k + ... +n ^k wheren andk are nonnegative integers. A functional equation is written based on the functional properties ofS _k (n) and several methods of solution are presented. These lead to several recurrence relations for the functions and a simple one-step differential-recurrence relation from which the polynomials can easily be computed successively. Arbitrary constants which arise are (almost) the Bernoulli numbers when evaluated and identities for these modified Bernoulli numbers are obtained. The functional equation for the formulas leads to another functional equation for the generating function for these formulas and this is used to obtain the generating functions for theS _k 's and for the modified Bernoulli numbers. This leads to an explicit representation, not a recurrence relation, for the modified Bernoulli numbers which then yields an explicit formula for eachS _k not depending on the earlier ones. This functional equation approach has been and can be applied to more general types of arithmetic sequences and many other types of combinatorial functions, sequences, and problems.     

On the sum of powers of Laplacian eigenvalues of bipartite graphs

For a bipartite graph G and a non-zero real α, we give bounds for the sum of the αth powers of the Laplacian eigenvalues of G using the sum of the squares of degrees, from which lower and upper bounds for the incidence energy, and lower bounds for the Kirchhoff index and the Laplacian Estrada index are deduced.     

Decomposition of the lattice of pseudovarieties of finite semigroups induced by bands

It has been shown by the authors that the mapping,where B is the pseudovariety of finite bands, is a complete retraction of the lattice L(F) of pseudovarieties of finite semigroups onto the lattice of pseudovarieties of bands. It follows that the classes of the induced congruence on L(F), or on the lattice of subpseudovarieties L(W) for any subpseudovariety W of F, are intervals. In this paper we solve the membership problem for the upper limit of the classes of restricted to L(W) for various , including F itself, and provide bases of pseudoidentities for certain cases. Received April 7, 1999; accepted in final form April 30, 2000.     

K-groups of quadratic extensions of rings

We construct the Levin braid connecting the Tate cohomology ofK-groups for quadratic extensions of rings with antistructures. Explicit formulas for isomorphisms of relative cohomology groups for the induced mapping and for the transfer mapping are obtained; these formulas are necessary in the construction of the Levin braid.Translated fromMatematicheskie Zametki, Vol. 58, No. 2, pp. 272–280, August, 1995.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-011-1402.     

Mixed multiplicities and the minimal number of generator of modules

Let (R,m) be a d-dimensional Noetherian local ring. In this work we prove that the mixed Buchsbaum-Rim multiplicity for a finite family of R-submodules of R^p of finite colength coincides with the Buchsbaum-Rim multiplicity of the module generated by a suitable superficial sequence, that is, we generalize for modules the well-known Risler-Teissier theorem. As a consequence, we give a new proof of a generalization for modules of the fundamental Rees’ mixed multiplicity theorem, which was first proved by Kirby and Rees in (1994, [8]). We use the above result to give an upper bound for the minimal number of generators of a finite colength R-submodule of R^p in terms of mixed multiplicities for modules, which generalize a similar bound obtained by Cruz and Verma in (2000, [5]) for m-primary ideals.     

Controllability of nonlinear systems

Some sufficient conditions are presented for the controllability of general nonlinear systems. First, the controllability problem is transformed into the problem of existence of fixed points for some operator; using Schauder's theorem, it is derived that a sufficient condition for controllability is the existence of a subsetS inC _n+m (T) which is invariant for a derived operator. Secondly, with the aid of the notion of comparison principle, the existence of the subsetS is guaranteed by the existence of solutions for some nonlinear integral inequality or equality equations. For example, one solution for such nonlinear integral equations is obtained under the assumption of the uniform boundedness for a nonlinear term of the differential equation.     

Excess of a class of g-frames

In this paper, we study the relationship between frames for the super Hilbert space H⊕H and g-frames for H with respect to C². We show that a g-frame associated with a frame for H⊕H remains a g-frame whenever any one of its elements is removed. Furthermore, we show that the excess of such a g-frame is at least dimH.     

The recognition of a class of bifurcation problems

In bifurcation theory there are two recognition problems concerning a given normal form, the recognition for the normal form and the recognition for universal unfoldings of bifurcation problems which are equivalent to the normal form. The two recognition problems for the normal forms εx²+δλ^k were only partially solved. In this paper we give a complete solution of the two problems for all k?1 uniformly.     

Toward determination of the eigenvalues of problems on description of the rotation of deep-water drill columns

This paper presents a method for computing the complex eigenvalues for a boundary-value problem describing the rotation of deep-water drill columns for drilling at depths of thousands of meters. A solution is sought for a singularly perturbed boundary-value problem in the form of an asymptotic expansion in a small parameter ε for two parts of the of the drill column: a lower compressed length, and the upper length. The method can be used to investigate the stability of drill columns used for deep-water drilling. Translated fromDinamicheskie Sistemy. Vol. 12. pp. 29–36, 1993.