Sufficient conditions for ε-minimum

The paper contains sufficient conditions for a point x ∈ U, function f X→ R ∪ ∪ {-∞,∞}, and ε≧ 0 to hold the inequality f(x) ≦ inf f=ε on the subset U of a topological vector space X.     

Dirac’s Type Sufficient Conditions for Hamiltonicity and Pancyclicity

Let G = (V, E) be a any simple, undirected graph on n ≥ 3 vertices with the degree sequence . We consider the class of graphs satisfying the condition where , is a positive integer. It is known that is hamiltonian if θ ≤ δ. In this paper,

Benford’s law and distribution functions of sequences in (0, 1)

Applying the theory of distribution functions of sequences x _n ∈ [0, 1], n = 1, 2, ..., we find a functional equation for distribution functions of a sequence x _n and show that the satisfaction of this functional equation for a sequence x _n is equivalent to the fact that the sequence x _n to satisfies the strong Benford law. Examples of distribution functions of sequences satisfying the functional equation are given with an application to the strong Benford law in different bases. Several direct consequences from uniform distribution theory are shown for the strong Benford law.     

Sufficient conditions for functional-limit-theorem versions ofL = λW

The familiar queueing principle expressed by the formulaL=W can be interpreted as a relation among strong laws of large numbers. In a previous paper, we showed that this principle can be extended to include relations among other classical limit theorems such as central limit theorems and laws of the iterated logarithm. Here we provide sufficient conditions for these limit theorems using regenerative structure.Supported by the National Science Foundation under Grant No. ECS-8404809 and by the U.S. Army under Contract No. DAAG29-80-C-0041.     

Some equivalent conditions for Huneke’s two conjectures

Abstract

In this paper we give some equivalent conditions for the C. Huneke’s two conjectures concerning the finiteness properties of the local cohomology module , where R is a regular local ring, I is an ideal of R and i ≥ 0 is an integer.     

Sufficient oscillation conditions for the Sturm–Liouville equation

We use the variational method to establish criteria for the existence of conjugate points and for the oscillation property of the linear differential Sturm–Liouville equation.     

Existence and Weyl’s law for spherical cusp forms

Let G be a split adjoint semisimple group over and a maximal compact subgroup. We shall give a uniform, short and essentially elementary proof of the Weyl law for cusp forms on congruence quotients of . This proves a conjecture of Sarnak for -split groups, previously known only for the case G = PGL(n). The key idea amounts to a new type of simple trace formula. Received: April 2005 Revision: June 2006 Accepted: October 2006     

Rademacher’s reciprocity law for Dedekind sums in function fields

We consider a Dedekind sum \(s(a,c)\) in function fields, defined via the Carlitz module, similar to the classical Dedekind sum \(D(a,c)\) . In this paper, we prove an analog of Pommersheim’s three-term reciprocity law for \(s(a,c)\) .     

Around Schwenninger and Zwart’s zero-two law for cosine families

It is proved that for a cosine family \({\{c(t)\}_{t \in \mathbb{R}}}\) in a normed algebra with a unity e, the following assertions hold: (1) If \({\sup_{t \in \mathbb{R}}\| c(t) - e \| < 2}\), then c(t) =  e for every \({t \in \mathbb{R}}\). (2) If \({\lim sup_{t \to 0}\| c(t) - e \| < 2}\), then \({\lim_{t \to 0} c(t) = e}\). It is also shown that the two respective results, each specific for one of the assertions, are equivalent.     

On Karatsuba’s problem related to Gram’s law

A number of new results related to Gram’s law in the theory of the Riemann zetafunction are proved. In particular, a lower bound is obtained for the number of ordinates of the zeros of the zeta-function that lie in a given interval and satisfy Gram’s law.     

Green’s functions for stationary problems with nonlocal boundary conditions

In this paper, we investigate Green’s functions for various stationary problems with nonlocal boundary conditions. We express the Green’s function per Green’s function for a problem with classical boundary conditions. This property is illustrated by various examples. Properties of Green’s functions with nonlocal boundary conditions are compared with those for classical problems. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-73/09.     

Rothe’s method for an integrodifferential equation with integral conditions

This work deals with an integrodifferential equation with initial, Newman and integral conditions. The existence and uniqueness of the weak solution in an appropriate sense is proved by the method of Rothe.     

Necessary conditions for the convergence of kullback-leibler’s mean information

Summary Two types of necessary conditions are given for the convergence of Kullback-Leibler’s mean information, one of which is connected with an asymptotic equivalence of two sequences of probability measures, and in special cases, with convergence of a sequence of probability distributions. The other is given in terms of the generalized probability density functions.     

Chung’s law of the iterated logarithm for subfractional Brownian motion

Let X~H= {X~H(t), t ∈ R_+} be a subfractional Brownian motion in R~d. We provide a sufficient condition for a self-similar Gaussian process to be strongly locally nondeterministic and show that X~H has the property of strong local nondeterminism. Applying this property and a stochastic integral representation of X~H, we establish Chung's law of the iterated logarithm for X~H.     

Sufficient conditions for uniform convergence on layer-adapted meshes for one-dimensional reaction–diffusion problems

We study convergence properties of a finite element method with lumping for the solution of linear one-dimensional reaction–diffusion problems on arbitrary meshes. We derive conditions that are sufficient for convergence in the L_∞ norm, uniformly in the diffusion parameter, of the method. These conditions are easy to check and enable one to immediately deduce the rate of convergence. The key ingredients of our analysis are sharp estimates for the discrete Green function associated with the discretization. AMS subject classification 65L10, 65L12, 65L15     

Positivity conditions and bounds for Green’s functions for higher order two-point BVP

We consider a linear nonautonomous higher order ordinary differential equation and establish the positivity conditions and two-sided bounds for Green’s function for the two-point boundary value problem. Applications of the obtained results to nonlinear equations are also discussed.     

Necessary and sufficient conditions for the validity of Jensen’s inequality

We consider the d-dimensional Jensen inequality $$ T[\varphi(f_1, \dots, f_d)]\, \ge \, \varphi(T[f_1], \dots, T[f_d])\quad\quad(\ast)$$ T [ φ ( f 1 , … , f d ) ] ≥ φ ( T [ f 1 ] , … , T [ f d ] ) ( * ) as it was established by McShane in 1937r. Here T is a functional, φ is a convex function defined on a closed convex set ${K\subset \mathbb{R}^d}$ K ? R d , and f ₁, . . . , f _d are from some linear space of functions. Our aim is to find necessary and sufficient conditions for the validity of (*). In particular, we show that if we exclude three types of convex sets K, then Jensen’s inequality holds for a sublinear functional T if and only if T is linear, positive, and satisfies T[1] = 1. Furthermore, for each of the excluded types of convex sets, we present nonlinear, sublinear functionals T for which Jensen’s inequality holds. Thus the conditions on K are optimal. Our contributions generalize or complete several known results.     

Global and exponential attractors for mixtures of solids with Fourier’s law

This paper presents a study of the long-time dynamics of the dynamical system generated by a nonlinear system modeling mixture of solids with nonlinear damping and Fourier’s law. By using the recent quasi-stability theory, we prove the existence of a smooth finite dimensional global attractor, which is characterized as an unstable manifold of the set of stationary solutions. The quasi-stability of the system is achieved through an estimated stabilizability. Moreover, the existence of a generalized exponential attractor is shown.     

Chung’s law of the iterated logarithm for anisotropic Gaussian random fields

In this paper we establish Chung’s law of the iterated logarithm for a class of anisotropic Gaussian random fields with stationary increments. This result is applicable to space–time Gaussian random fields and solution to the stochastic fractional heat equation.