Random Clarkson inequalities and LP versions of Grothendieck's inequality

In a recent paper KATO [3] used the LITTLEWOOD matrices to generalise CLARKSON'S inequalities. Our first aim is to indicate how KATO'S result can be deduced from a neglected version of the HAUSDORFF -YOUNG inequality which was proved by WELLS and WILLIAMS [12]. We next establish ?random CLARKSON inequalities”?. These show that the expected behaviour of matrices whose coefficients are random ± 1′s is, as one might expect, the same as the behaviour that KATO observed in the LITTLEWOOD matrices. Finally we show how sharp Lp versions of GROTHENDIECK'S inequality can be obtained by combining a KATO -like result with a theorem of BENNETT [1] on SCHUR multipliers.     

Sharp Dirac's theorem for DP‐critical graphs

Correspondence coloring, or DP‐coloring, is a generalization of list coloring introduced recently by Dvořák and Postle [11]. In this article, we establish a version of Dirac's theorem on the minimum number of edges in critical graphs [9] in the framework of DP‐colorings. A corollary of our main result answers a question posed by Kostochka and Stiebitz [15] on classifying list‐critical graphs that satisfy Dirac's bound with equality.     

On the generalizations of Denjoy-Wolff theorem

We consider generalizations of Denjoy-Wolff theorem on strongly pseudo-convex domains. Beardon [3] gave a general Denjoy-Wolff theorem in hyperboli domain in the complex plane. Our main results are generalizations of Beardon's result in a higher dimensional setting.     

S. G Mikhlin (1908 – 1990)

Our concern is to find a representation theorem for operators in B(c(X), c(Y)) where X and Y are Banach spaces with Y containing an isomorphic copy of c₀. Cass and Gao [1] obtained a representation theorem that always applies if Y does not contain an isomorphic copy of c₀. Maddox [3], Melvin - Melvin [4], and Robinson [5] consider operators in B(c(X), c(Y)) that are given by matrices. In this paper we show that Cass's and Gao's result in [1] can be extended, when Y contains an isomorphic copy of c₀, to certain operators that we call represent able. In addition, we show that when Y contains an isomorphic copy of co there are always operators that fall outside the scope of our representation theorem. Light is also cast on a theorem given in Maddox [3, Theorem 4.2] and [5, Theorem IV].     

Swan's theorem for binary tetranomials

Swan's theorem [Pacific J. Math. 12 (1962) 1099–1106] determines the parity of the number of irreducible factors of a binary trinomial. This paper does the same for a binary tetranomial. When phrased in terms of the periodic portion of the factor-parity sequence, the result in several cases is comparable in simplicity to Swan's result for square-free trinomials.     

Automorphisms of Mn,partially ordered by the star order

Let Mn be the space of all n × n matrices with coefficients in [image omitted] or [image omitted], where n ≥ 3. The star order on Mn is defined by [image omitted] iff [image omitted], where A* is the Hermitian adjoint (i.e., the conjugate transpose) of A. We characterize surjective mappings Φ on Mn such that [image omitted] iff [image omitted]. The tools we use are the Fundamental theorem of projective geometry, Wigner's theorem, and the Penrose decomposition, which we need to describe the main result as well.     

Baum–Katz type theorems for martingale arrays

We show convergence rates in the law of large numbers for martingale arrays. The results extend the classical theorems of Baum and Katz (1965) [2] for sums of independent and identically distributed (i.i.d.) random variables. They improve a result of Ghosal and Chandra (1998) [6] for martingale arrays, and generalize a result of Alsmeyer (1990) [1] for a single martingale. As an application, we obtain a new theorem about the convergence rate of Cesàro summation of identically distributed random variables.     

Sudakov's typical marginals, random linear functionals and a conditional central limit theorem

Summary. V.N. Sudakov [Sud78] proved that the one-dimensional marginals of a high-dimensional second order measure are close to each other in most directions. Extending this and a related result in the context of projection pursuit of P. Diaconis and D. Freedman [Dia84], we give for a probability measure and a random (a.s.) linear functional on a Hilbert space simple sufficient conditions under which most of the one-dimensional images of under are close to their canonical mixture which turns out to be almost a mixed normal distribution. Using the concept of approximate conditioning we deduce a conditional central limit theorem (theorem 3) for random averages of triangular arrays of random variables which satisfy only fairly weak asymptotic orthogonality conditions. Received: 25 July 1995 / In revised form: 20 June 1996     

On the stabilization of interconnected stochastic systems

The aim of this work is to prove that if the equilibrium solution of a nonlinear control stochastic system is locally asymptotically stable in probability by means of a continuous state feedback law, then the resulting stochastic system obtained by adding an integrator is also locally asymptotically stable in probability by means of a smooth, except possibly at the equilibrium solution, state feedback law. This result extends to the stabilization of stochastic systems a result proved by Tsinias [9] for deterministic systems. In our proof, we make use of the stochastic version of Artstein's theorem established in [4]     

Limit theorems for random processes with random time substitution

In this paper we prove a theorem on sufficient conditions for the convergence in the Skorokhod space D[0, 1] of a sequence of random processes with random time substitution. We obtain almost sure versions of this theorem.     

Hoeffding’s Inequality for Sums of Dependent Random Variables

Let \(X_1,\ldots ,X_n\) be, possibly dependent, [0, 1]-valued random variables. What is a sharp upper bound on the probability that their sum is significantly larger than their mean? In the case of independent random variables, a fundamental tool for bounding such probabilities is devised by Wassily Hoeffding. In this paper, we provide a generalisation of Hoeffding’s theorem. We obtain an estimate on the aforementioned probability that is described in terms of the expectation, with respect to convex functions, of a random variable that concentrates mass on the set \(\{0,1,\ldots ,n\}\). Our main result yields concentration inequalities for several sums of dependent random variables such as sums of martingale difference sequences, sums of k-wise independent random variables, as well as for sums of arbitrary [0, 1]-valued random variables.     

A theorem on paths in planar graphs

We prove a theorem on paths with prescribed ends in a planar graph which extends Tutte's theorem on cycles in planar graphs [9] and implies the conjecture of Plummer [5] asserting that every 4-connected planar graph is Hamiltonian-connected.     

一般随机不动点定理及其应用

在本文中我们得到了一个一般的随机不动点定理,推广了Engl[4,7]和Bocsan[8]的主要结果.这一定理的有用性在于目前由许多作者用特殊方法得到的随机不动点定理[1,4,5-13]均能利用我们的一般定理(定理1和系1,2)得到,最后给出了我们的定理对随机积分和微分方程的应用.     

Caratheodory-type selections and random fixed point theorems

We provide some new Caratheodory-type selection theorems, i.e., selections for correspondences of two variables which are continuous with respect to one variable and measurable with respect to the other. These results generalize simultaneously Michael's [21]continuous selection theorem for lower-semicontinuous correspondences as well as a Caratheodory-type selection theorem of Fryszkowski [10]. Random fixed point theorems (which generalize ordinary fixed point theorems, e.g., Browder's [6]) follow as easy corollaries of our results.     

Ingham-Beurling type theorems with weakened gap conditions

Completing a series of works begun by Wiener [34], Paley and Wiener [28] and Ingham [9], a far-reaching generalization of Parseval"s identity was obtained by Beurling [4] for nonharmonic Fourier series whose exponents satisfy a uniform gap condition. Later this gap condition was weakened by Ullrich [33], Castro and Zuazua [5], Jaffard, Tucsnak and Zuazua [11] and then in [2] in some particular cases. In this paper we prove a general theorem which contains all previous results. Furthermore, applying a different method, we prove a variant of this theorem for nonharmonic Fourier series with vector coefficients. This result, partly motivated by control-theoretical applications, extends several earlier results obtained in [15] and [2]. Finally, applying these results we obtain an optimal simultaneous observability theorem concerning a system of vibrating strings. This revised version was published online in June 2006 with corrections to the Cover Date.     

Remarks, du côté de chez Tarski, on symmetric ternary relations

We point out that the results in [12] are the model-theoretic counterpart of results established syntactically in [3] and [10], and that Martin’s theorem for the Euclidean 3- or higher-dimensional case, established in [5], does not depend on the Beckman-Quarles theorem, and can be rephrased as a result about axiomatizability and definability.     

Walks and paths in trees

Recently Csikvári [Combinatorica 30(2) 2010, 125–137] proved a conjecture of Nikiforov concerning the number of closed walks on trees. Our aim is to extend this theorem to all walks. In addition, we give a simpler proof of Csikvári's result and answer one of his questions in the negative. Finally we consider an analogous question for paths rather than walks. © 2011 Wiley Periodicals, Inc. J Graph Theory     

«Assoluta continuità» e «Quasi sub-additività rispetto a una coppia» in spazi di misura

A well-known theorem due to J. C. Burkill and some existence theorems for Cesari's integral obtained in [1] are generalized here in measure spaces. Moreover, the last result, where is used the concept of quasi sub-additivity with respect to two families \((\mathfrak{D},\mathfrak{D}'),\mathfrak{D}' \subset \mathfrak{D}\) , and to the mesh for a set function ψ:I?ψ(I),I ∈ {I}, introduced in [1], is an extension of a theorem obtained by L. Cesari in [5].     

CLT for linear random fields with stationary martingale-difference innovation

In [V. Paulauskas, On Beveridge–Nelson decomposition and limit theorems for linear random fields, J. Multivariate Anal., 101:621–639, 2010], limit theorems for linear random fields generated by independent identically distributed innovations were proved. In this paper, we present the central limit theorem for linear random fields with martingale-differences innovations satisfying the central limit theorem from [J. Dedecker, A central limit theorem for stationary random fields, Probab. Theory Relat. Fields, 110(3):397–426, 1998] and arranged in lexicographical order.