On the Distribution of the Limit of Products of I.I.D. 2×2 Random Stochastic Matrices

This article gives sufficient conditions for the limit distribution of products of i.i.d. 2×2 random stochastic matrices to be continuous singular, when the support of the distribution of the individual random matrices is finite.     

On the D.H. Lehmer Problem and its Generalization

For each x(0相似文献   

On the Continuous Singularity of the Limit Distribution of Products of I.I.D. d×d Stochastic Matrices

This article gives sufficient conditions for the limit distribution of products of i.i.d. d×d random stochastic matrices, d finite and 2, to be continuous singular, when the support of the distribution of the individual random matrices is finite or countably infinite. Proofs are based on applications of the multivariate Central Limit Theorem.     

A Berry–Esseen Bound for U-Statistics in the Non-I.I.D. Case

Let X ₁,..., X _n be independent, not necessarily identically distributed random variables. An optimal Berry–Esseen bound is derived for U-statistics of order 2, that is, statistics of the form T=_1i<jn g _ij(X _i, X _j), where the g _ij are measurable functions such that |g _ij(X _i, X _j)|<. An application is given concerning Wilcoxon's rank-sum test.     

One Hundred and Eighty-nine Homeomorphism Classes of the G. M. 

InG.M.@,asshowedinfigure1,alltheedgescanbedividedintothreetiers:innertierABCDEA,outertierA'B'C'D'E'A'andmiddletierAD'BE'CAIDBIEC'A(radiationedges).Bytwistingwepushthenegativeedgesininnertierand0utertierintothemiddletierasfaraspossible.IfDEisanegativeedge,wecantwistD,tomakeitbecameap0sitiveedgelifCDturnedint0thenegativeedge,wecantwistCt0makeitbecometheP0sitiveedge,ands00n.Atlast,ifABisanegativeedge,twistA,tomakeitbecomeaPOsitiveedge.Inthisway,excepttheedgeAEcouldbePOsitiveorne…     

On the Periodic Zeta‐Function. II

We obtain a more precise asymptotic formula than in [Lith. Math. J., 41(2), 168–177 (2001)] for the mean square of the periodic zetafunction.     

The Convergence of the Steepest Descent Algorithm for D.C.Optimization

Some properties of a class of quasi-differentiable functions(the difference of two finite convex functions) are considered in this paper.And the convergence of the steepest descent algorithm for unconstrained and constrained quasi-differentiable programming is proved.     

A constructive approach to the quantum (cosh ϕ)2 model. I. The method of the Gel'fand-Levitan-Marchenko equations

In a Hubert , with the aid of the postulated Gel'fand-Levitan-Marchenko quantum equations, one introduces the fields ₁(x) and ₂(x), which are the quantum analogues of the classical fields cosh (x) and sinh (x) in the sinh-Gordon model. It is shown that the fields _j(x) satisfy the Wightman axioms, including the invariance relative to reflections of space-time and mutual local commutativity. In addition, one proves the asymptotic completeness of the theory and one computes explicitly the scattering operator. In the developed approach, no cut-offs are used and, therefore, there are no renormalization effects.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 146, pp. 147–190, 1985.     

On a J. C. C. Nitsche’s Type Inequality for the Hyperbolic Space H3

Let H ³ be the hyperbolic space identified with the unit ball B ³ = {x ∈ R ³ : |x| < 1} with the Poincaré metric d _h and let ??(x ₀, p, q) : = {x : p < d _h (x, x ₀) < q} ? H ³ be a hyperbolic annulus with the inner and outer radii 0 < p < q < ∞. We prove that if there exists a hyperbolic harmonic diffeomorphism between annuli ??(x ₀, a, b) and ?? (y ₀, α, β) in the hyperbolic space H ³, then β/α>1+ψ(a,b), where ψ is a positive function. In addition, for given two annuli in the hyperbolic space H ³, satisfying certain conditions, we construct radial harmonic mappings between them.     

A sharp inequality for the tail probabilities of sums of i.i.d. r.v.’s with dominatedly varying tails

Let F be a distribution function supported on (-∞, ∞) with a finite mean μ. In this note weshow that if its tail F = 1 - F is dominatedly varying, then for any γ＞ max{μ, 0}, there exist C(γ) ＞ 0 and D(γ) ＞ 0 such thatC(γ)nF(x) ≤ Fn*(x) ≤ D(γ)nF(x),for all n ≥ 1 and all x ≥γn. This inequality sharpens a classical inequality for the subexponential distributioncase.     

A Geometric Problem and the Hopf Lemma. Ⅱ

Abstract A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in ℝⁿ⁺¹, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane X_n+1 =constant in case M satisfies: for any two points (X′,X_n+1), on M, with , the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional conditions. Some variations of the Hopf Lemma are also presented. Several open problems are described. Part I dealt with corresponding one dimensional problems. (Dedicated to the memory of Shiing-Shen Chern) * Partially supported by NSF grant DMS-0401118.     

Is the Big Bang the Sole Cause of the Universe? A Response to John J. Park

In a recent paper, John J. Park argues (1) that an abstract object can bring a universe into existence, and (2) that, according to the Big Bang Theory, the initial singularity is an abstract object that brought the universe into existence. According to Park, if (1) and (2) are true, then the kalam cosmological argument fails to show that the cause of the universe must be divine. I argue, however, that both (1) and (2) are false. In my argument I analyse the abstract/concrete distinction and conclude that, by its nature, an abstract object is causally inefficacious in the sense that it cannot bring something into existence.     

Scattering problem for the Schrödinger equation in the case of a potential linear in time and coordinate. I. Asymptotics in the shadow zone

The formal asymptotics of the scattering problem for the Schrödinger equation with a linear potential as x+¦t¦+ is studied. In the shadow zone a formal asymptotic expansion is constructed which is matched with the known asymptotics as t– The expansion constructed loses asymptotic character near the curve x=1/6 t³ (in the so-called projector zone). An assumption regarding the analogous behavior of the asymptotic series in the projector zone makes it possible to construct an expansion in the post-projection zone which goes over into the formulas for creeping waves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 6–17, 1984.In conclusion, the authors would like to bring to the reader's attention another approach to asymptotics in the projector zone proposed by M. M. Popov (see the present collection).     

A Comment the BooK“continuous-Time Markov Chains” by W.J.Anderson

The book "Continuous-Time Markov Chains" by W. J. Anderson collects a large part of the development in the past thirty years. It is now a popular reference for the researchers on this subject or related fields. Unfortunately, due to a misunderstanding of the approximating methods, several results in the book are incorrectly stated or proved. Since the results are related to the present author's work, to whom it may be a duty to correct the mistakes in order to avoid further confusion. We emphasize the approximating methods because they are useful in many situations.     

An Answer to the Conjecture of I. Babuška and T. Janik

In this paper, we verify the conjecture in the recent paper of I. Babuška and T. Janik.     

Generalization of B.M. Levitan and M.G. Gasymov’s solvability theorems to the case of indecomposable boundary conditions

Doklady Mathematics -     

A commentary “on the periodic solutions of a forced second-order equation” by S. P. Hastings and J. B. McLeod

Summary We discuss the relationship of the shooting and perturbation methods used by Hastings and McLeod in the paper On the Periodic Solutions of a Forced Second-Order Equation to the geometrical techniques of nonlinear dynamical systems theory.     

Winning “Big” in the St. Petersburg Game

Mathematical Notes - We present a self-contained and easy to follow proof for the value of the probability of large gains of a player in the celebrated St. Petersburg game.