关于非齐次马氏链的一个定理

给出了Csiszar和Krner关于独立随机变量序列的一个定理的一个推广,该定理的推论是关于相对熵的,在统计假设检验及编码理论中起着重要的作用.利用非齐次马氏链的一个强大数定律将这个定理推广到非齐次马氏链上.     

A limit theorem for particle numbers in bounded domains of a branching diffusion process

We shall study the asymptotic behavior of the particle numbers in bounded domains of a binary splitting one-dimensional branching diffusion process. We shall give a Yaglom type limit theorem in the so-called locally subcritical case, and almost sure convergence of the normalized particle number in the locally supercritical case.     

On the Trotter-Kato theorem in a variable space

In this paper, the proof of a Trotter-Kato type theorem in a variable Banach space is given and some special cases and examples are considered.     

二维Vilenkin型系统的Dirichlet核的加权平均

对二维Vilenkin型系统,我们定义加权平均极大算子T(i.e.Tf:=supn=(n1,n2)∈P2,β-1n1/n2β|Hnf|),并证明此算子是弱(1,1)型、强(p,p)型(1p∞)以及(H,L)型,其中Hnf表示部分和的加权平均,H表示Hardy空间.借用此结果得到序列Hnf是几乎处处收敛于可积函数f.     

An infinite-dimensional generalization of the Jung theorem

A complete characterization of the extremal subsets of Hilbert spaces, which is an infinite-dimensional generalization of the classical Jung theorem, is given. The behavior of the set of points near the Chebyshev sphere of such a subset with respect to the Kuratowski and Hausdorff measures of noncompactness is investigated.     

关于任意随机变量序列泛函的一类强偏差定理

通过概率空间上的任意随机变量的分布与独立分布的比较.研究任意随机变量序列泛函的强偏差定理,即小偏差定理.将已有的某些连续型及离散随机变量序列的强偏差定理加以推广.     

Uniqueness in Harper’s vertex-isoperimetric theorem

For a set $A\subseteq Q_n={\left\{0,1\right\}}^n$ the $t$-neighbourhood of $A$ is $N^t\left(A\right)=\left\{x:d\left(x,A\right)\le t\right\}$, where $d$ denotes the usual graph distance on $Q_n$. Harper’s vertex-isoperimetric theorem states that among the subsets $A\subseteq Q_n$ of given size, the size of the $t$-neighbourhood is minimised when $A$ is taken to be an initial segment of the simplicial order. Aubrun and Szarek asked the following question: if $A\subseteq Q_n$ is a subset of given size for which the sizes of both $N^t\left(A\right)$ and $N^t\left(A^c\right)$ are minimal for all $t>0$, does it follow that $A$ is isomorphic to an initial segment of the simplicial order?Our aim is to give a counterexample. Surprisingly it turns out that there is no counterexample that is a Hamming ball, meaning a set that lies between two consecutive exact Hamming balls, i.e. a set $A$ with $B\left(x,r\right)\subseteq A\subseteq B\left(x,r+1\right)$ for some $x\in Q_n$. We go further to classify all the sets $A\subseteq Q_n$ for which the sizes of both $N^t\left(A\right)$ and $N^t\left(A^c\right)$ are minimal for all $t>0$ among the subsets of $Q_n$ of given size. We also prove that, perhaps surprisingly, if $A\subseteq Q_n$ for which the sizes of $N\left(A\right)$ and $N\left(A^c\right)$ are minimal among the subsets of $Q_n$ of given size, then the sizes of both $N^t\left(A\right)$ and $N^t\left(A^c\right)$ are also minimal for all $t>0$ among the subsets of $Q_n$ of given size. Hence the same classification also holds when we only require $N\left(A\right)$ and $N\left(A^c\right)$ to have minimal size among the subsets $A\subseteq Q_n$ of given size.     

Abstract convergence theorem for quasi-convex optimization problems with applications

Abstract

Quasi-convex optimization is fundamental to the modelling of many practical problems in various fields such as economics, finance and industrial organization. Subgradient methods are practical iterative algorithms for solving large-scale quasi-convex optimization problems. In the present paper, focusing on quasi-convex optimization, we develop an abstract convergence theorem for a class of sequences, which satisfy a general basic inequality, under some suitable assumptions on parameters. The convergence properties in both function values and distances of iterates from the optimal solution set are discussed. The abstract convergence theorem covers relevant results of many types of subgradient methods studied in the literature, for either convex or quasi-convex optimization. Furthermore, we propose a new subgradient method, in which a perturbation of the successive direction is employed at each iteration. As an application of the abstract convergence theorem, we obtain the convergence results of the proposed subgradient method under the assumption of the Hölder condition of order p and by using the constant, diminishing or dynamic stepsize rules, respectively. A preliminary numerical study shows that the proposed method outperforms the standard, stochastic and primal-dual subgradient methods in solving the Cobb–Douglas production efficiency problem.     

Generalizations of the Riesz convergence theorem for Lorentz spaces

Summary We obtain preservation inequalities for Lipschitz constants of higher order in simultaneous approximation processes by Bernstein type operators. From such inequalities we derive the preservation of the corresponding Lipschitz spaces.     

Converse theorem for the Minkowski inequality

Let (Ω,Σ,μ) a measure space such that 0<μ(A)<1<μ(B)<∞ for some A,B∈Σ. Under some natural conditions on the bijective functions φ,φ₁,φ₂,ψ,ψ₁,ψ₂:(0,∞)→(0,∞) we prove that if

     

Maximal Inequalities and Lebesgue's Differentiation Theorem for Best Approximant by Constant over Balls

For fL_p( ⁿ), with 1p<∞, >0 and x ⁿ we denote by T(f)(x) the set of every best constant approximant to f in the ball B(x, ). In this paper we extend the operators T_p to the space L_p−1( ⁿ)+L_∞( ⁿ), where L₀ is the set of every measurable functions finite almost everywhere. Moreover we consider the maximal operators associated to the operators T_p and we prove maximal inequalities for them. As a consequence of these inequalities we obtain a generalization of Lebesgue's Differentiation Theorem.     

Isotonic Approximation in L1

Let (Ω, ,P) be a measurable space and a sub-σ-lattice of the σ-algebra . For XL₁(Ω, ,P) we denote by P X the set of conditional 1-mean (or best approximants) of X given L₁( ) (the set of all -measurable and integrable functions). In this paper, we obtain characterizations of the elements in P X, similar to those obtained by Landers and Rogge for conditional s-means with 1<s<∞. Moreover, using these characterizations we can extend the operator P to a bigger space L₀(Ω, ,P). When, in certain sense, _n goes to _∞, we will be able to prove theorems about convergence and we will obtain bounds for the maximal function sup_nP _nX. A sharper characterization of conditional 1-means for certain particular σ-lattice was proved in previous papers. In the last section of this paper we generalize those results to all totally ordered σ-lattices.     

An implicit function theorem

Suppose thatF:DR ⁿ×R^mRⁿ, withF(x ⁰,y ⁰)=0. The classical implicit function theorem requires thatF is differentiable with respect tox and moreover that ₁ F(x ⁰,y ⁰) is nonsingular. We strengthen this theorem by removing the nonsingularity and differentiability requirements and by replacing them with a one-to-one condition onF as a function ofx.     

An approximate universal coefficient theorem

An approximate Universal Coefficient Theorem (AUCT) for certain -algebras is established. We present a proof that Kirchberg-Phillips's classification theorem for separable nuclear purely infinite simple -algebras is valid for -algebras satisfying the AUCT instead of the UCT. It is proved that two versions of AUCT are in fact the same. We also show that -algebras that are locally approximated by -algebras satisfying the AUCT satisfy the AUCT. As an application, we prove that certain simple -algebras which are locally type I are in fact isomorphic to simple AH-algebras. As another application, we show that a sequence of residually finite-dimensional -algebras which are asymptotically nuclear and which asymptotically satisfies the AUCT can be embedded into the same simple AF-algebra.

     

A probabilistic proof of the fundamental theorem of algebra

We use Lévy's theorem on invariance of planar Brownian motion under conformal maps and the support theorem for Brownian motion to show that the range of a non-constant polynomial of a complex variable consists of the whole complex plane. In particular, we obtain a probabilistic proof of the fundamental theorem of algebra.

     

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.     

Density versions of Schur's theorem for ideals generated by submeasures

We characterize ideals of subsets of natural numbers for which some versions of Schur's theorem hold. These are similar to generalizations shown by Bergelson (1986) in [1] and Frankl, Graham and Rödl (1990) in [7]. Additionally, we prove a generalization of an iterated version of Ramsey's theorem.     

On Bohr's theorem for multiple Fourier series

The problem of Pringsheim uniform convergence of multiple Fourier series in the trigonometric system is considered. A multidimensional analog of Bohr's theorem on the uniform convergence of the Fourier series of a continuous function after a homeomorphic chance of variable is proved.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 913–924, December, 1998.     

A softer,stronger Lidskii theorem

We provide a new approach to Lidskii’s theorem relating the eigenvalues of the differenceA—B of two self-adjoint matrices to the eigenvalues ofA andB respectively. This approach combines our earlier work on the spectral matching of matrices joined by a normal path with some familiar techniques of functional analysis. It is based, therefore, on general principles and has the additional advantage of extending Lidskii’s result to certain pairs of normal matrices. We are also able to treat some related results on spectral variation stemming from the work of Sunder, Halmos and Bouldin.