On the Zudilin-Rivoal theorem

We propose a new method for proving the Zudilin-Rivoal theorem stating, in particular, that the sequence of values of the Dirichlet beta function at even natural points contains infinitely many irrational values. For polylogarithms, we use Hermite—Padé approximations of the first type, invariant with respect to the Klein group. Quantitative additions to this theorem are obtained.     

Metrics and tangent cones of uniformly regular Carnot—Carathéodory spaces

Given a uniformly regular Carnot—Carathéodory space, we prove equivalence of the quasimetrics generated by various bases of vector fields which agree with filtration of the space. We prove a theorem on a nilpotent tangent cone for a uniformly regular Carnot—Carathéodory space furnished with quasimetrics. As a consequence, we obtain a theorem on isomorphism of nilpotent tangent cones defined at a common distinguished point.     

Improvement of Hartman''''s linearization theorem

Hartman's linearization theorem tells us that if matrix A has no zero real part and f(x) is bounded and satisfies Lipchitz condition with small Lipchitzian constant, then there exists a homeomorphism of Rn sending the solutions of nonlinear system x' = Ax + f(x) onto the solutions of linear system x = Ax. In this paper, some components of the nonlinear item f(x) are permitted to be unbounded and we prove the result of global topological linearization without any special limitation and adding any condition. Thus, Hartman's linearization theorem is improved essentially.     

A generalization of the classical sphere theorem

In this paper, we prove a sphere theorem for Riemannian manifolds with partially positive curvature which generalizes the classical sphere theorem.

     

A perturbed ergodic theorem

Using a version of an ergodic lemma due to Cuculescu and Foias, we prove a pointwise ergodic theorem for -contractions which can be viewed as a perturbed version of the celebrated ergodic theorem of Chacon and Ornstein. Surprisingly, to some extent, the complex part of the iterates involved have no effect on the ergodic convergence.

     

On the equivalence of the Mizoguchi–Takahashi fixed point theorem to Nadler’s theorem

In this work, we prove that on a metrically convex complete metric space, the Mizoguchi–Takahashi theorem is equivalent to Nadler’s theorem. Also, we obtain its equivalence on a compact metric space.     

A remark about the Orlicz-Pettis theorem and the statistical convergence

In this paper we obtain a new version of the Orlicz-Pettis theorem by using statistical convergence. To obtain this result we prove a theorem of uniform convergence on matrices related to the statistical convergence.     

A pseudospectral mapping theorem

The pseudospectrum has become an important quantity for analyzing stability of nonnormal systems. In this paper, we prove a mapping theorem for pseudospectra, extending an earlier result of Trefethen. Our result consists of two relations that are sharp and contains the spectral mapping theorem as a special case. Necessary and sufficient conditions for these relations to collapse to an equality are demonstrated. The theory is valid for bounded linear operators on Banach spaces. For normal matrices, a special version of the pseudospectral mapping theorem is also shown to be sharp. Some numerical examples illustrate the theory.

     

Vietoris-Begle theorem for spectral pro-homology

Dydak and Kozlowski (1991) obtained a generalization of the Vietoris-Begle theorem for the cohomology theories induced by CW spectra. In this note we prove a dual of their theorem involving the pro-homology theories induced by CW spectra.     

A refined Gallai-Edmonds structure theorem for weighted matching polynomials

In this work, we prove a refinement of the Gallai-Edmonds structure theorem for weighted matching polynomials by Ku and Wong. Our proof uses a connection between matching polynomials and branched continued fractions. We also show how this is related to a modification by Sylvester of the classical Sturm's theorem on the number of zeros of a real polynomial in an interval. In addition, we obtain some other results about zeros of matching polynomials.     

The equivariant family index theorem in odd dimensions

In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed's odd dimensional index formula. Then we extend this theorem to the noncommutative geometry framework. As a corollary, we get the odd family Lichnerowicz vanishing theorem and the odd family Atiyah-Hirzebruch vanishing theorem.     

L0-线性函数的Hahn-Banach扩张定理的几何形式与随机赋范模中的Goldstine-Weston定理

本文给出了关于L⁰- 线性函数的Hahn-Banach 扩张定理的几何形式并证明这个几何形式等价于它的代数形式. 进一步, 我们利用这个几何形式给出了随机局部凸模中熟知的基本分离定理的一个新的且简单的证明. 最后, 利用这个分离定理, 我们同时在两种拓扑 —(ε, λ)- 拓扑和局部L⁰- 凸拓扑下证明了随机赋范模中的Goldstine-Weston 稠密性定理, 并举出一个反例说明在局部L⁰- 凸拓扑下如果随机赋范模不具有可数连接性质, 则Goldstine-Weston 稠密性定理不一定成立.     

Weyl’s theorem and thin spectra

Using the notion of thin sets we prove a theorem of Weyl type for the Wolf essential spectrum ofT ∈β (H). *Further we show that Weyl’s theorem holds for a restriction convexoid operator and consequently modify some results of Berberian. Finally we show that Weyl’s theorem holds for a paranormal operator and that a polynomially compact paranormal operator is a compact perturbation of a diagnoal normal operator. A structure theorem for polynomially compact paranormal operators is also given.     

Mechanical theorem proving in the surfaces using the characteristic set method and Wronskian determinant

In this paper, we generalize the method of mechanical theorem proving in curves to prove theorems about surfaces in differential geometry with a mechanical procedure. We improve the classical result on Wronskian determinant, which can be used to decide whether the elements in a partial differential field are linearly dependent over its constant field. Based on Wronskian determinant, we can describe the geometry statements in the surfaces by an algebraic language and then prove them by the characteristic set method.     

Multivariate irregular sampling theorem

In this paper,we prove a Marcinkiewicz-Zygmund type inequality for multivariate entire functions of exponential type with non-equidistant spaced sampling points. And from this result,we establish a multivariate irregular Whittaker-Kotelnikov-Shannon type sampling theorem.     

The Chang-Łoś-Suszko theorem in a topological setting

The Chang-Łoś-Suszko theorem of first-order model theory characterizes universal-existential classes of models as just those elementary classes that are closed under unions of chains. This theorem can then be used to equate two model-theoretic closure conditions for elementary classes; namely unions of chains and existential substructures. In the present paper we prove a topological analogue and indicate some applications.     

An analogue of Beurling's theorem for the Laguerre hypergroup

In this paper, we prove an analogue of Beurling's theorem for the Laguerre hypergroup, then we derive some other versions of uncertainty principle.     

A local convergence theorem for partial sums of stochastic adapted sequences

In this paper we establish a new local convergence theorem for partial sums of arbitrary stochastic adapted sequences. As corollaries, we generalize some recently obtained results and prove a limit theorem for the entropy density of an arbitrary information source, which is an extension of case of nonhomogeneous Markov chains.     

On a tauberian theorem of Hardy and Littlewood

In this paper, we give a simple alternative proof of a Tauberian theorem of Hardy and Littlewood (Theorem E stated below, [3]).     

The heat kernel and Hardy’s theorem on symmetric spaces of noncompact type

For symmetric spaces of noncompact type we prove an analogue of Hardy’s theorem which characterizes the heat kernel in terms of its order of magnitude and that of its Fourier transform.