The Nikodym boundedness theorem for lattice-valued measures

The author would like to thank the referee for several valuable suggestions, onr of which led to the present form of Theorem 4.     

On stochastic boundedness and stationary measures for Markov processes

In this report we relate the property of stochastic boundedness to the existence of stationary measures for arbitrary Markov processes on the positive real line. We further develop a sufficiency criterion for the independence of such measures from initial conditions. The results are then applied to the question of approximating the fixed point vector of an irreducible infinite stochastic matrix by the solutions of finite ones.     

Projective boundedness and convolution of Fréchet measures

Fréchet measures of order ( -measures) are the measure-theoretic analogues of bounded -linear forms on products of spaces. In an LCA setting, convolution of -measures is always defined, while there exist -measures whose convolution cannot be defined. In a three-dimensional setting, we demonstrate the existence of an -measure which cannot be convolved with arbitrary -measures.

     

Razumikhin type boundedness theorems in terms of two measures for impulsive integro-differential systems

Some new direct criteria of boundedness in terms of two measures for impulsive integro-differential systems with fixed moments of impulse effects are established by Lyapunov functions coupled with Razumikhin techniques.     

非倍测度条件下Marcinkiewicz积分及其交换子在Morrey空间中的有界性

讨论了测度μ在满足非倍条件下,Marcinkiewicz积分算子及其与RBMO（μ）函数、Lipschitz函数生成的交换子的有界性,通过Marcinkiewica积分及该交换子在Lebesgue空间中的有界性,得到了该算子及交换子在非齐型空间上的Morrey空间中的有界性．     

Characterization of integrals with respect to arbitrary radon measures by the boundedness indices

The problem of characterization of integrals as linear functionals is considered in the paper. It starts from the familiar results of F. Riesz (1909) and J. Radon (1913) on integral representation of bounded linear functionals by Riemann?CStieltjes integrals on a segment and by Lebesgue integrals on a compact in $ {\mathbb{R}^n} $ , respectively. After works of J. Radon, M. Fréchet, and F. Hausdorff the problem of characterization of integrals as linear functionals took the particular form of the problem of extension of Radon??s theorem from $ {\mathbb{R}^n} $ to more general topological spaces with Radon measures. This problem has turned out difficult and its solution has a long and rich history. Therefore, it may be naturally called the Riesz?CRadon?CFréchet problem of characterization of integrals. The important stages of its solution are connected with such mathematicians as S. Banach, S. Saks, S. Kakutani, P. Halmos, E. Hewitt, R. E. Edwards, N. Bourbaki, V. K. Zakharov, A. V. Mikhalev, et al. In this paper, the Riesz?CRadon?CFr??echet problem is solved for the general case of arbitrary Radon measures on Hausdorff spaces. The solution is given in the form of a general parametric theorem in terms of a new notion of the boundedness index of a functional. The theorem implies as particular cases well-known results of the indicated authors characterizing Radon integrals for various classes of Radon measures and topological spaces.     

Automatic relative boundedness of derivations in C1-algebras

We show that a derivation of a C¹-algebra $\text{A}$ is automatically relative bounded with respect to any closed ¹derivation of $\text{A}$ with smaller domain. We give also some related results on the automatic continuity of derivations in certain Banach algebras.     

Group-valued measures on coarse-grained quantum logics

In [3] it was shown that a (real) signed measure on a cyclic coarse-grained quantum logic can be extended, as a signed measure, over the entire power algebra. Later ([9]) this result was re-proved (and further improved on) and, moreover, the non-negative measures were shown to allow for extensions as non-negative measures. In both cases the proof technique used was the technique of linear algebra. In this paper we further generalize the results cited by extending group-valued measures on cyclic coarse-grained quantum logics (or non-negative group-valued measures for lattice-ordered groups). Obviously, the proof technique is entirely different from that of the preceding papers. In addition, we provide a new combinatorial argument for describing all atoms of cyclic coarse-grained quantum logics.     

Stability and boundedness of nonlinear impulsive systems in terms of two measures via perturbing Lyapunov functions

This paper develops the concepts of stability, practical stability and boundedness in terms of two measures for nonlinear impulsive differential systems using the method of perturbing Lyapunov functions. The notion of perturbing Lyapunov functions enables us to discuss stability properties of solutions of nonlinear impulsive differential systems in terms of two measures under much weaker assumptions. The novel results offer a way to unify a variety of stability results found in the relative literature.     

Evolution of probability measures associated with quantum systems

We derive the evolution equation for probability distributions and characteristic functions of the quantum tomograms associated with the linear and nonlinear evolutions of quantum states.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 365–370, February, 2005.     

Crystalline boundedness principle

We prove that an F-crystal (M,φ) over an algebraically closed field k of characteristic p>0 is determined by , where n?1 depends only on the rank of M and on the greatest Hodge slope of (M,φ). We also extend this result to triples (M,φ,G), where G is a flat, closed subgroup scheme of GLM whose generic fibre is connected and has a Lie algebra normalized by φ. We get two purity results. If C is an F-crystal over a reduced Fp-scheme S, then each stratum of the Newton polygon stratification of S defined by C, is an affine S-scheme (a weaker result was known before for S noetherian). The locally closed subscheme of the Mumford scheme Ad,1,Nk defined by the isomorphism class of a principally quasi-polarized p-divisible group over k of height 2d, is an affine Ad,1,Nk-scheme.     

Hilbert space representations of decoherence functionals and quantum measures

We show that any decoherence functional D can be represented by a spanning vector-valued measure on a complex Hilbert space. Moreover, this representation is unique up to an isomorphism when the system is finite. We consider the natural map U from the history Hilbert space K to the standard Hilbert space H of the usual quantum formulation. We show that U is an isomorphism from K onto a closed subspace of H and that U is an isomorphism from K onto H if and only if the representation is spanning. We then apply this work to show that a quantum measure has a Hilbert space representation if and only if it is strongly positive. We also discuss classical decoherence functionals, operator-valued measures and quantum operator measures.     

A topological notion of boundedness

A topological boundedness notion is studied, which is proved to be productive. Classical theorems on compactness of Tychonoff, Alexander and Obreanu are generalized. A boundedness operator is defined and studied. Finally, a classification of all topological spaces is obtained according to boundedness criteria.The author is grateful to prof. N. Oeconomidis, who suggested the topic, for his continuous interest.     

Total boundedness and bornologies

A set A in a metric space is called totally bounded if for each ε>0 the set can be ε-approximated by a finite set. If this can be done, the finite set can always be chosen inside A. If the finite sets are replaced by an arbitrary approximating family of sets, this coincidence may disappear. We present necessary and sufficient conditions for the coincidence assuming only that the family is closed under finite unions. A complete analysis of the structure of totally bounded sets is presented in the case that the approximating family is a bornology, where approximation in either sense amounts to approximation in Hausdorff distance by members of the bornology.     

Liapunov functions and boundedness

In this paper we give conditions under which one can conclude that all solutions of a differential equation are bounded when there is a Liapunov function which is not radially unbounded. The problem of Lurie and the Liénard equation are given as examples.     

Local boundedness of monotone bifunctions

We consider bifunctions ${F : C\times C\rightarrow \mathbb{R}}$ where C is an arbitrary subset of a Banach space. We show that under weak assumptions, monotone bifunctions are locally bounded in the interior of their domain. As an immediate corollary, we obtain the corresponding property for monotone operators. Also, we show that in contrast to maximal monotone operators, monotone bifunctions (maximal or not maximal) can also be locally bounded at the boundary of their domain; in fact, this is always the case whenever C is a locally polyhedral subset of ${\mathbb{R}^{n}}$ and F(x, ·) is quasiconvex and lower semicontinuous.