A note on quasi-weakly almost periodic point

Recently, He et al. [On quasi-weakly almost periodic points. Sci. China Math., 56, 597–606(2013)] constructed two binary sub-shifts to solve an open problem posed by Zhou and Feng in[Twelve open problems on the exact value of the Hausdorff measure and on topological entropy: A brief survey of recent results. Nonlinearity, 17, 493–502(2004)]. In this paper, we study more dynamical properties of those two binary sub-shifts. We show that the first one has zero topological entropy and is transitive but not weakly mixing, while the second one has positive topological entropy and is strongly mixing.     

A note on causality in reflexive Banach spaces.

In this short note we discuss the causality of linear closable operators in reflexive Banach spaces. We show that in general causality is not preserved under closure procedures. We shall give an alternative definition of causality for closable operators by means of a certain continuity poperty that carries over to the closure. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on rigidity of the almost Ricci soliton

The aim of this paper is to prove that a gradient almost Ricci soliton ${(M^{n}, g, \nabla f, \lambda )}$ whose Ricci tensor is Codazzi has constant sectional curvature. In particular, in the compact case, we deduce that (M ⁿ , g) is isometric to a Euclidean sphere and f is a height function. Moreover, we also classify gradient almost Ricci solitons with constant scalar curvature R provided a suitable function achieves a maximum in M ⁿ .     

A note on almost difference sets in nonabelian groups

We present a new construction of almost difference sets. The construction occurs in nonabelian groups of order 4N with a subgroup H of order N so that H has an (N,\(\frac{N-1}{2}\),\(\frac{N-3}{4}\)) difference set (and hence N must be an integer that is 3 (mod 4)).     

A note on the almost sure limit theorem for U-statistic

Summary In this note we prove an almost sure limit theorem for the products of U-statistics.     

An equivalent definition of the strong causality of space-time

Omsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 33, No. 2, pp. 200–201, March–April, 1992.     

A note on a conjecture on maximum matching in almost regular graphs

Mkrtchyan, Petrosyan, and Vardanyan made the following conjecture: Every graph G with Δ(G)−δ(G)≤1 has a maximum matching whose unsaturated vertices do not have a common neighbor. We disprove this conjecture.     

A note on the almost sure approximation of weakly dependent random variables

A strong approximation theorem for Hilbert space valued martingales, due to Morrow and Philipp, is improved. This result is then used to strengthen and, at the same time, to extend a recent strong approximation theorem for weakly dependent ^d -valued random variables, due to Eberlein, to Hilbert space valued random variables.Dedicated to Professor E. Hlawka on the occasion of his seventieth birthday     

A note on the fundamental groups of manifolds with almost nonnegative curvature

We show that given and , there exists a positive number such that if a closed -manifold satisfies and , then is almost abelian.

     

A note on Arzela–Ascoli's lemma in almost periodic problems

This paper is concerned with a generalized Arzela–Ascoli's lemma, which has been extensively applied in almost periodic problems by the continuation theorem of degree theory. We give a counter example to show that this lemma is incorrect, and there is a gap in the proof of some existing literature, where the addressed generalized Arzela–Ascoli's lemma was used. Moreover, we make some final comments and introduce an open problem. Copyright © 2016 John Wiley & Sons, Ltd.     

A note on almost sure uniform and complete convergences of a sequence of random variables

The aim of this note is to introduce another way of defining the almost sure uniform convergence, which is necessary when studying some mathematical results on the existence of price bubbles in certain scenarios of trading securities. This mode of convergence of random variables' sequences is intermediate between the uniform and the almost sure ones, and, more specifically, between the uniform and the complete convergences. In this way, this paper presents some mathematical characterizations of both almost sure uniform and complete convergences, and shows that the almost sure uniform convergence is a particular case of complete convergence, when the number of summands in the series defining this mode of convergence is finite. Finally, this paper presents the relation of almost surely uniform convergence with convergence in mean when the random variable limit is integrable. Moreover, almost surely convergence and local boundedness of the sequence of random variables minus its limit are sufficient to derive convergence in mean.     

A note on the almost sure limit theorem for the product of partial sums

We present an almost sure limit theorem for the product of the partial sums of i.i.d. positive random variables. We also prove a corresponding almost sure limit theorem for a triangular array.     

An almost sure invariance principle for random walks in a space-time random environment

We consider a discrete time random walk in a space-time i.i.d. random environment. We use a martingale approach to show that the walk is diffusive in almost every fixed environment. We improve on existing results by proving an invariance principle and considering environments with an L² averaged drift. We also state an a.s. invariance principle for random walks in general random environments whose hypothesis requires a subdiffusive bound on the variance of the quenched mean, under an ergodic invariant measure for the environment chain. T. Sepp?l?inen was partially supported by National Science Foundation grant DMS-0402231.     

A remark on totally vicious space-time

If a space-time is vicious at a some point, then it is totally vicious.     

A remark on almost complete intersections

Let k be a perfect field and S the quotient ring of a polynomial ring k[X₁,...,X_t] with respect to a prime ideal. Let I be a prime ideal of S such that R=S/I is an almost complete intersection. Then, in his paper [2], Matsuoka proves that the homological dimension of the differential module _R/Kis infinite under the assumption that R is Cohen-Macaulay and I² is a primary idea]. In this paper we prove that the result is valid without the above assumption.     

A remark on almost periodic transition operators

Results of Rosenblatt on almost periodic transition operators are extended to the reducible case.