An Extension of a Class of Iterative Procedures for Nonlinear Quasivariational Inequalities

Consider the convergence of the projection methods based on an extension of a special class of algorithms for the approximation--solvability of the following class of nonlinear quasivariational inequality (NQVI) problems: find an element such that and

Solvability Theory and Projection Methods for a Class of Singular Variational Inequalities: Elastostatic Unilateral Contact Applications

The mathematical modeling of engineering structures containing members capable of transmitting only certain type of stresses or subjected to noninterpenetration conditions along their boundaries leads generally to variational inequalities of the form , where C is a closed convex set of (kinematically admissible set), (loading strain vector), and (stiffness matrix). If rigid body displacements and rotations cannot be excluded from these applications, then the resulting matrix M is singular and serious mathematical difficulties occur. The aim of this paper is to discuss the existence and the numerical computation of the solutions of problem (P) for the class of cocoercive matrices. Our theoretical results are applied to two concrete engineering problems: the unilateral cantilever problem and the elastic stamp problem.     

Existence and Uniqueness of a Weak Solution to the Initial Mixed Boundary-Value Problem for Quasilinear Elliptic-Parabolic Equations

We prove the existence and the uniqueness of a weak solution to the mixed boundary problem for the elliptic-parabolic equation

Spectral Estimations for the Laplace Operator on the Discrete Heisenberg Group

We estimate the spectral measure of the Laplace operator for the discrete Heisenberg group with generators x and y in the vicinity of the unity. Bibliography: 7 titles.     

The pressure equation for fluid flow in a stochastic medium

An equation modelling the pressurep(x) =p(x, w) atx ∈D ⊂R ^d of an incompressible fluid in a heterogeneous, isotropic medium with a stochastic permeabilityk(x, w) ≥ 0 is the stochastic partial differential equation

On Some Proximinal Subspaces of Modular Spaces

Let (, , ) be a complete measure space, L⁰ the vector lattice of -measurable real functions on , : L⁰ [0, )] a lattice semimodular, the corresponding modular space, S₀ the ideal generated by and 0,{\text{ }}\exists {\text{ }}s \in {\text{ }}S_{\text{0}} {\text{ such that }}\rho \left( {\frac{{x - s}}{\user1{\lambda }}} \right) < \infty } \right\}$$ " align="middle" border="0"> . In X consider the distance 0:\rho \left( {\frac{{x - y}}{\user1{\lambda }}} \right) \leqq \user1{\lambda }} \right\}$$ " align="middle" border="0"> and, if is convex, the distances d_L, d_o subordinated to the Luxemburg and Amemiya-Orlicz norms, respectively. We give necessary and sufficient conditions for H(S_o) in order to be proximinal in X with the distances d, d_L and d_o.     

On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives

For 2-periodic functions and arbitrary q [1, ] and p (0, ], we obtain the new exact Kolmogorov-type inequality which takes into account the number of changes in the sign of the derivatives (x ^(k)) over the period. Here, = (r – k + 1/q)/(r + 1/p), _r is the Euler perfect spline of degree r, and . The inequality indicated turns into the equality for functions of the form x(t) = a _r (nt + b), a, b R, n N. We also obtain an analog of this inequality in the case where k = 0 and q = and prove new exact Bernstein-type inequalities for trigonometric polynomials and splines.     

Effective Reducibility of a Class of Linear Differential Equations with Quasiperiodic Coefficients

In this paper we consider the effective reducibility of the following linear differentialequation: x = (A ∈Q(t,∈))x, |∈| ≤ ∈0, where A is a constant matrix, Q(t,e) is quasiperiodic in t, and e is a small perturbation parameter. We prove that if the eigenvalues of A and the basic frequencies of Q satisfy some non-resonant conditions, the linear differential equation can be reduced to y = (A^*(∈) R^*(t, ∈))y, |∈| ≤ ∈o, where R^* is exponentially small in ∈.     

A Problem with Nonlocal Boundary Conditions for a Quasilinear Parabolic Equation

The solvability of the nonlocal boundary value problem

Polynomial solutions of quasi-homogeneous partial differential equations

By means of a method of analytic number theory the following theorem is proved. Letp be a quasi-homogeneous linear partial differential operator with degreem,m > 0, w.r.t a dilation given by ( a₁, …, a_n). Assume that either a₁, …, a_n are positive rational numbers or for some Then the dimension of the space of polynomial solutions of the equationp[u] = 0 on ℝⁿ must be infinite     

Correction to: ‘The UCT, the Milnor Sequence, and a Canonical Decomposition of the Kasparov Groups’

In this note we correct a mistake in K-Theory 10 (1996), 49–72. In that paper we asserted that under bootstrap hypotheses the short exact sequence

Existence results for nonlinear parabolic equations via strong convergence of truncations

Summary  We prove existence results for the initial-boundary value problem for parabolic equations of the type

Two families of approximations for the gamma function

In this paper, we establish two families of approximations for the gamma function: $$ \begin{array}{lll} {\varGamma}(x+1)&=\sqrt{2\pi x}{\left({\frac{x+a}{{\mathrm{e}}}}\right)}^x {\left({\frac{x+a}{x-a}}\right)}^{-\frac{x}{2}+\frac{1}{4}} {\left({\frac{x+b}{x-b}}\right)}^{\sum\limits_{k=0}^m\frac{{\beta}_k}{x^{2k}}+O{{\left(\frac{1}{x^{2m+2}}\right)}}},\\ {\varGamma}(x+1)&=\sqrt{2\pi x}\cdot(x+a)^{\frac{x}{2}+\frac{1}{4}}(x-a)^{\frac{x}{2}-\frac{1}{4}} {\left({\frac{x-1}{x+1}}\right)}^{\frac{x^2}{2}}\\ &\quad\times {\left({\frac{x-c}{x+c}}\right)}^{\sum\limits_{k=0}^m\frac{{\gamma}_k}{x^{2k}}+O{\left({\frac{1}{x^{2m+2}}}\right)}}, \end{array}$$ where the constants ${\beta }_k$ and ${\gamma }_k$ can be determined by recurrences, and $a$ , $b$ , $c$ are parameters. Numerical comparison shows that our results are more accurate than Stieltjes, Luschny and Nemes’ formulae, which, to our knowledge, are better than other approximations in the literature.     

Projection in a Space of Solenoidal Vector Fields

Let R ³ be a bounded domain, 0$$ " align="middle" border="0"> , a family of extending subdomains, and =(x) a positive function in be a space of -solenoidal vector fields, 0$$ " align="middle" border="0"> , a family of subspaces, G orthogonal projectors in onto . A unitary transformation that diagonalizes the family of projectors {G} is constructed: it takes to the operator of multiplication by the independent variable. The isometry of this transformation is proved with the help of the operator Riccati equation for the NeumanntoDirichlet mapping. Bibliography: 8 titles.     

Whitney Interpolation Constants Bounded by 2 for k = 5, 6, 7

Let f C[0, 1], k = 5, 6, 7. We prove that if f(i/(k – 1)) = 0, i = 0, 1,..., k – 1, then      

Hoffman’s Least Error Bounds for Systems of Linear Inequalities

Let E be a normed space, and . Let . We give some exact formulas for 7#x03C4;.     

Strengthening of the Kolmogorov Comparison Theorem and Kolmogorov Inequality and Their Applications

We obtain a strengthened version of the Kolmogorov comparison theorem. In particular, this enables us to obtain a strengthened Kolmogorov inequality for functions x L ^x (r), namely,

On the dimension of the cartesian product of relations and orders

It is known that for incidence structures and , max , wheref dim stands for Ferrers relation. We shall show that under additional assumptions on and , both bounds can be improved. Especially it will be shown that the square of a three-dimensional ordered set is at least four-dimensional.     

Three primes theorem in a short interval (V)

In this paper, we shall prove that for a sufficiently large odd numberN, the equation has solutions. The Project Supported by National Natural Science Foundation of China     

<Emphasis Type="Italic">q</Emphasis>-Besselian Frames in Banach Spaces

In this paper, we introduce the concepts of q-Besselian frame and （p, σ）-near Riesz basis in a Banach space, where a is a finite subset of positive integers and 1/p＋1/q = 1 with p 〉 1, q 〉 1, and determine the relations among q-frame, p-Riesz basis, q-Besselian frame and （p, σ）-near Riesz basis in a Banach space. We also give some sufficient and necessary conditions on a q-Besselian frame for a Banach space. In particular, we prove reconstruction formulas for Banach spaces X and X^＊ that if {xn}n=1^∞ C X is a q-Besselian frame for X, then there exists a p-Besselian frame {y＆＊}n=1^∞ belong to X^＊ for X^＊ such that x = ∑n=1^∞ yn^＊（x）xn for all x ∈ X, and x^＊ =∑n=1^∞ x^＊（xn）yn^＊ for all x^＊ ∈ X^＊. Lastly, we consider the stability of a q-Besselian frame for the Banach space X under perturbation. Some results of J. R. Holub, P. G. Casazza, O. Christensen and others in Hilbert spaces are extended to Banach spaces.