Decomposable processes and continuous products of probability spaces

Sequences of independent random variables and products of probability spaces are just two ways of looking at the same thing. The natural generalization of a sequence of independent random variables is a decomposable process. We introduce a corresponding generalization of a product of probability spaces, which will be called a factored probability space, and study the structure and classification of such systems and their relation to decomposable processes.     

Isometry of functional spaces with different number of variables

We construct spaces of real functions of n + k variables that are isometric to spaces of real functions given on an n-dimensional Euclidean space. We present certain properties and examples of delta-like kernels used for the construction of isometric spaces of functions with different number of variables. We prove certain assertions that enable one to construct delta-like kernels with many variables by using delta-like kernels with smaller number of variables. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1027–1045, August, 1998.     

The periodic Cauchy problem of the modified Hunter-Saxton equation

We prove that the periodic initial value problem for the modified Hunter-Saxton equation is locally well-posed for initial data in the space of continuously differentiable functions on the circle and in Sobolev spaces when s > 3/2. We also study the analytic regularity (both in space and time variables) of this problem and prove a Cauchy-Kowalevski type theorem. Our approach is to rewrite the equation and derive the estimates which permit application of o.d.e. techniques in Banach spaces. For the analytic regularity we use a contraction argument on an appropriate scale of Banach spaces to obtain analyticity in both time and space variables.     

A decomposition theorem for solving initial value problems in associated spaces

Initial value problems of Cauchy-Kovalevskaya type can be solved not only in spaces of holomorphic or generalized analytic functions but also in more genereal associated spaces provided the associated space is defined by a partial differential equation whose solutions permit an interior estimate. The present paper is aimed at solving initial value problems with initial functions which can be decomposed in components belonging to a family of associated spaces. The main result is a decomposition theorem obtained in two versions: The first version uses families of scales of Banach spaces (with respect to the spacelike variables), while the second one applies the contraction-mapping principle to a Banach space of functions defined in a conical domain and depending on both the time and the spacelike variables.     

Independence of events and of random variables

The phenomenon of independence of random variables is shown to be singular in that, e.g., there are both finite and infinite sample spaces on which two random variables can be independent iff one is constant. Furthermore on [0,1], with Lebesgue measure for probability, the usual function spaces contain dense subsets each member of which is independent only of constants. Finally, the requirement of independence among a set of orthonormal functions is shown to imply, in all but trivial instances, that the orthogonal complement of the space is infinite-dimensional.     

Malliavin calculus and decoupling inequalities in Banach spaces

We develop a theory of Malliavin calculus for Banach space-valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener-Itô isometry to Banach spaces. In the white noise case we obtain two sided Lp-estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener-Itô chaoses. Our main tools are decoupling inequalities for vector-valued random variables. In the opposite direction we use Meyer's inequalities to give a new proof of a decoupling result for Gaussian chaoses in UMD Banach spaces.     

Vector topologies and the representation of spaces of random variables

Most of the spaces of random variables are non-Hausdorff linear topological spaces, whereas a number of useful results of the theory of vector topologies require the Hausdorff separation axiom for the underlying spaces. Therefore we review and complete the machinery of non-Hausdorff linear topological spaces and the techniques for their representation with suitable Hausdorff linear topological spaces, and then we work out some examples, which emphasize the ideas involved in a consistent application of the topological point of view. The final part of the paper is devoted to the discussion of the pathology, which arises when we try to treat the special case of almost everywhere convergence in the same framework.     

Some Inequalities in Classical Spaces with Mixed Norms

We consider some inequalities in such classical Banach Function Spaces as Lorentz, Marcinkiewicz, and Orlicz spaces. Our aim is to explore connections between the norm of a function of two variables on the product space and the mixed norm of the same function, where mixed norm is calculated in function spaces on coordinate spaces, first in one variable, then in the other. This issue is motivated by various problems of functional analysis and theory of functions. We will currently mention just geometry of spaces of vector-valued functions and embedding theorems for Sobolev and Besov spaces generated by metrics which differ from L ^p. Our main results are actually counterexamples for Lorentz spaces versus the natural intuition that arises from the easier case of Orlicz spaces (Section 2). In the Appendix we give a proof for the Kolmogorov–Nagumo theorem on change of order of mixed norm calculation in its most general form. This result shows that L ^p is the only space where it is possible to change this order.     

Means in metric spaces and the center of mass

In this article k-convex metric spaces are considered where a several variable mapping is provided as a limit point of an iteration scheme based on the midpoint map in the metric space itself. This mapping, considered as a mean of its variables, has some properties which relates it to the center of mass of these variables in the metric space. Sufficient conditions are given here for the two points to be identical, as well as upper bounds on their distances from one another. The asymptotic rate of convergence of the iterative process defining the mean is also determined here. The case of the symmetric space on the convex cone of positive definite matrices related to the geometric mean and the special orthogonal group are also studied here as examples of k-convex metric spaces.     

Measurable linear manifolds in products of linear spaces with measure

We consider some properties of measurable linear manifolds in products of linear spaces with measure which play an important part in the theory of measurable and polylinear functionals in such spaces. An example is adduced of an application of the results obtained in order to investigate the properties of series of independent random variables.Translated from Matematicheskie Zametki, Vol. 5, No. 5, pp. 623–634, May, 1969.In conclusion I should like to thank G. E. Shilov for his attention to my work.     

A general theory of self-similarity

A little-known and highly economical characterization of the real interval [0,1], essentially due to Freyd, states that the interval is homeomorphic to two copies of itself glued end to end, and, in a precise sense, is universal as such. Other familiar spaces have similar universal properties; for example, the topological simplices Δn may be defined as the universal family of spaces admitting barycentric subdivision. We develop a general theory of such universal characterizations.This can also be regarded as a categorification of the theory of simultaneous linear equations. We study systems of equations in which the variables represent spaces and each space is equated to a gluing-together of the others. One seeks the universal family of spaces satisfying the equations. We answer all the basic questions about such systems, giving an explicit condition equivalent to the existence of a universal solution, and an explicit construction of it whenever it does exist.     

Quadrics on Riemannian spaces of constant curvature. Separation of variables and connection with the Gaudin magnet

Integrable systems associated with separation of the variables in real Riemannian spaces of constant curvature are considered. An isomorphism between all such systems and the hyperbolic Gaudin magnet is established. This isomorphism is used in a classification of all coordinate systems that admit separation of the variables, the basis of which is the classification of the correspondingL operators of the Gaudin magnet.Leningrad State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 1, pp. 83–111, April, 1992.     

On Almost Sure Behavior of Sums of Independent Banach Space Valued Random Variables

Levy's strong law of large numbers is extended to the Banach space and Chover type laws of the iterated logarithm are proved for random variables which do not necessarilly belong to the domain of normal attraction of a stable law. Also characterizations of Banach spaces in which conditions on the summands imply the above strong laws are given.     

拟牛顿流的一种三变量域模型的有限元方法的数值分析

0．引言目前，涉及高温条件下材料蠕变性质的粘弹性流动问题已引起人们广泛的研究兴趣，不少文章讨论了如何对其进行数值求解（见［1］－－［41），首先，人们研究了较简单的仅以速度，压力两个变量来表述此现象的模型问题（如［1，2］）等．鉴于应力变量在材料性质方面的特殊重要性，最近J．Baxanzer等人在［3］中首次对应力满足幂函数规律的蠕变流研究了包含应力、速度和压力三种变量的模型问题的有限元逼近，当粘性的牛顿部分为零时（详见下述）在假定速度与应力、速度与压力有限元空间之间同时满足两种＊＊B条件以后，证明了有限元解…     

On the geometry in projective limits of Hilbert spaces

In this paper we consider the concept of orthogonality with respect to infinitely many inner products. We describe geometric properties related to this concept of orthogonality in certain Köthe sequence spaces (power series spaces), spaces of holomorphic functions in one and several variables and spaces of infinitely differentiable functions. The methods are required from a mixture of functional analysis (theory of bases), theory of functions of one complex variable, Fourier analysis and interpolation theory.     

Random processes in Sobolev-Orlicz spaces

We establish conditions under which the trajectories of random processes from Orlicz spaces of random variables belong with probability one to Sobolev-Orlicz functional spaces, in particular to the classical Sobolev spaces defined on the entire real axis. This enables us to estimate the rate of convergence of wavelet expansions of random processes from the spaces L _p (Ω) and L ₂ (Ω) in the norm of the space L _q (ℝ). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1340–1356, October, 2006.     

Nonlinear differential equations as invariants under group action on coset bundles: Burgers and Korteweg-de Vries equation families

Given a group, its coset spaces provide all homogeneous spaces for its action. A subgroup chain allows for the construction of a bundle of sections over a coset space of independent variables, where the fiber coordinates are dependent variables and all their partial derivatives up to some order, (i.e., the kth order jet). In this coset bundle, group invariants take the form of differential equations. We present two families of group-subgroup chains, one leading to various tensor Burgers-type differential equations, and the other to Korteweg-de Vries equations with an nth space derivative. Maps of the Hopf-Cole type appear in both families as transformations which intertwine the original group action to a multiplier realization of a normally extended group, yielding a new differential equation with greater symmetry.     

Differential equations in spaces of abstract stochastic distributions

Stochastic Itô equations with additive and multiplicative noise in separable Hilbert spaces are studied by reducing them to differential-operator equations in spaces of generalized Hilbert space-valued random variables. Results on the existence and uniqueness of solutions in these spaces are obtained by using the S-transform technique and methods of the theory of semigroups of linear operators.     

Some consequences of Pisier’s approach to interpolation

A recent result of Pisier onK-functionals is exploited in the context of function spaces defined from singular integrals. The main consequences appear forH ^p spaces on the complex ball (1≤p≤∞) and spaces of differentiable functions in several variables. It turns out that the complex variable arguments used in [P1] may in certain cases be substituted for real variable methods, which of course have a wider range of applicability. We discuss interpolation of vector valued Hardy spaces on the ball and prove thatL ¹(S)/H ¹(B) satisfies Grothendieck’s theorem, similarly as in the disc case.     

Sur une inegalite de H.P. Rosenthal et le theoreme limite central dans les espaces de Banach

This paper studies an inequality of H. P. Rosenthal for vector valued random variables, its relations with some geometric properties of Banach spaces and its applications to the study of the central limit theorem in Banach spaces.