On one generalization of a DiPerna and Majda theorem

The weak limits of sequences {f(u^ν)}_ν∈? where u^ν's are vector‐valued µ‐measurable functions defined on a compact set Ω and f is (possibly) discontinuous are investigated. As shown by the author (J. Conv. Anal. (to appear)), they are described in terms of integral formulae involving parametrized measures independent of f, similarly as in the classical theorem by Young and its generalization due to DiPerna and Majda. In the present paper we describe the supports of the involved parametrized measures. Copyright © 2006 John Wiley & Sons, Ltd.     

Gauss‐Green theorem for weakly differentiable vector fields,sets of finite perimeter,and balance laws

We analyze a class of weakly differentiable vector fields F : ?ⁿ → ?ⁿ with the property that F ∈ L^∞ and div F is a (signed) Radon measure. These fields are called bounded divergence‐measure fields. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence‐measure field F over the boundary of an arbitrary set of finite perimeter that ensures the validity of the Gauss‐Green theorem. To achieve this, we first establish a fundamental approximation theorem which states that, given a (signed) Radon measure μ that is absolutely continuous with respect to ??^{N ? 1} on ?^N, any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure‐theoretic interior of the set with respect to the measure ||μ||, the total variation measure. We employ this approximation theorem to derive the normal trace of F on the boundary of any set of finite perimeter E as the limit of the normal traces of F on the boundaries of the approximate sets with smooth boundary so that the Gauss‐Green theorem for F holds on E. With these results, we analyze the Cauchy flux that is bounded by a nonnegative Radon measure over any oriented surface (i.e., an (N ? 1)‐dimensional surface that is a part of the boundary of a set of finite perimeter) and thereby develop a general mathematical formulation of the physical principle of the balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measure‐valued source terms from the formulation of the balance law. This framework also allows the recovery of Cauchy entropy flux through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws. © 2008 Wiley Periodicals, Inc.     

Complex interpolation of Orlicz spaces with respect to a vector measure

We apply the Calderón interpolation methods to Orlicz and weakly Orlicz function spaces with respect to a Banach‐space‐valued measure defined on a σ‐algebra. The results we obtain generalize those in the case of Banach lattices of p‐integrable and weakly p‐integrable functions with respect to such a vector measure.     

Operator-valued positive definite kernels on tubes

In this paper we generalize the concept of an infinite positive measure on a -algebra to a vector valued setting, where we consider measures with values in the compactification of a convex coneC which can be described as the set of monoid homomorphisms of the dual coneC ^* into [0, ]. Applying these concepts to measures on the dual of a vector space leads to generalizations of Bochner's Theorem to operator valued positive definite functions on locally compact abelian groups and likewise to generalizations of Nussbaum's Theorem on positive definite functions on cones. In the latter case we use the Laplace transform to realize the corresponding Hilbert spaces by holomorphic functions on tube domains.     

Archimedean ϕ ‐tolerance graphs

Let ? be a symmetric binary function, positive valued on positive arguments. A graph G = (V,E) is a ?‐tolerance graph if each vertex υ ∈ V can be assigned a closed interval I_υ and a positive tolerance t_υ so that xy ∈ E ? | I_x ∩ I_y|≥ ? (t_x,t_y). An Archimedean function has the property of tending to infinity whenever one of its arguments tends to infinity. Generalizing a known result of [15] for trees, we prove that every graph in a large class (which includes all chordless suns and cacti and the complete bipartite graphs K_2,k) is a ?‐tolerance graph for all Archimedean functions ?. This property does not hold for most graphs. Next, we present the result that every graph G can be represented as a ?_G‐tolerance graph for some Archimedean polynomial ?_G. Finally, we prove that there is a ?universal”? Archimedean function ? ^* such that every graph G is a ?^*‐tolerance graph. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 179–194, 2002     

On realization of the Kreĭn–Langer class Nκ of matrix‐valued functions in Pontryagin spaces

In this paper the realization problems for the Kre?n–Langer class N_κ of matrix‐valued functions are being considered. We found the criterion when a given matrix‐valued function from the class N_κ can be realized as linear‐fractional transformation of the transfer function of canonical conservative system of the M. Livsic type (Brodskii–Livsic rigged operator colligation) with the main operator acting on a rigged Pontryagin space Π_κ with indefinite metric. We specify three subclasses of the class N_κ (R) of all realizable matrix‐valued functions that correspond to different properties of a realizing system, in particular, when the domains of the main operator of a system and its conjugate coincide, when the domain of the hermitian part of a main operator is dense in Π_κ . Alternatively we show that the class N_κ (R) can be realized as transfer matrix‐functions of some canonical impedance systems with self‐adjoint main operators in rigged spaces Π_κ . The case of scalar functions of the class N_κ (R) is considered in details and some examples are presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Necessary conditions for discontinuities of multidimensional persistent Betti numbers

Topological persistence has proven to be a promising framework for dealing with problems concerning the analysis of data. In this context, it was originally introduced by taking into account 1‐dimensional properties of data, modeled by real‐valued functions. More recently, topological persistence has been generalized to consider multidimensional properties of data, coded by vector‐valued functions. This extension enables the study of multidimensional persistent Betti numbers, which provide a representation of data based on the properties under examination. In this contribution, we establish a new link between multidimensional topological persistence and Pareto optimality, proving that discontinuities of multidimensional persistent Betti numbers are necessarily pseudocritical or special values of the considered functions. Copyright © 2014 John Wiley & Sons, Ltd.     

Intertwining operators for L 2(E)

Let (E,Q) be a finite dimensional quadratic vector space over a finite field. For the natural representation -π of the isometry group G of (E,Q) in the space L ²(E) of all complex valued functions on E, we analyse when the intertwining algebra of π is generated by just one averaging operator.     

Functions of bounded variation and polarization

It is known that, if u is a real valued function on ?^N of bounded variation, then its total variation decreases under polarization. In this paper we identify the difference between the total variation of u and that one of its polar u_Π (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multivariate arrangement increasing functions with applications in probability and statistics

A real valued function of s vector arguments in Rⁿ is said to be arrangement increasing if the function increases in value as the components of the vector arguments become more similarly arranged. Various examples of arrangement increasing functions are given including many joint multivariate densities, measures of concordance between judges and the permanent of a matrix with nonnegative components. Preservation properties of the class of arrangement increasing functions are examined, and applications are given including useful probabilistic inequalities for linear combinations of exchangeable random vectors.     

Sobolev,Besov and Nikolskii fractional spaces: Imbeddings and comparisons for vector valued spaces on an interval

Summary We consider various fractional properties of regularity for vector valued functions defined on an interval I. In other words we study the functions in the Sobolev spaces W^s,p(I;E), in the Nikolskii spaces N^s,p(I;E), or in the Besov spaces B _{s, p} (I; E). Theses spaces are defined by integration and translation, and E is a Banach space. In particular we study the dependence on the parameters s, p and , that is imbeddings for different parameters. Moreover we compare each space to the others, and we give Lipschits continuity, existence of traces and q-integrability properties. These results rely only on integration techniques.     

Self‐Intersection Local Time for 𝒮′(ℝd)‐Ornstein‐Uhlenbeck Processes Arising from Immigration Systems

Fluctuation limits of an immigration branching particle system and an immigration branching measure‐valued process yield different types of 𝒮′(ℝ^d)‐valued Ornstein‐Uhlenbeck processes whose covariances are given in terms of an excessive measure for the underlying motion in Rd, which is taken to be a symmetric α‐stable process. In this paper we prove existence and path continuity results for the self‐intersection local time of these Ornstein‐Uhlenbeck processes. The results depend on relationships between the dimension d and the parameter α.     

Diff(Rn) as a Milnor–Lie group

We describe a construction of the Lie group structure on the diffeomorphism group Diff( R ⁿ), modelled on the space D( R ⁿ, R ⁿ) of R ⁿ‐valued test functions on R ⁿ, in John Milnor's setting of infinite‐dimensional Lie groups. New tools are introduced to simplify this task. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A stability of the steady flow of compressible viscous fluid with respect to initial disturbance (v∞ ≠ 0)

We consider a compressible viscous fluid with the velocity at infinity equal to a strictly non‐zero constant vector in ?³. Under the assumptions on the smallness of the external force and velocity at infinity, Novotny–Padula (Math. Ann. 1997; 308 :439– 489) proved the existence and uniqueness of steady flow in the class of functions possessing some pointwise decay. In this paper, we study stability of the steady flow with respect to the initial disturbance. We proved that if H³‐norm of the initial disturbance is small enough, then the solution to the non‐stationary problem exists uniquely and globally in time, which satisfies a uniform estimate on prescribed velocity at infinity and converges to the steady flow in L_q‐norm for any number q? 2. Copyright © 2006 John Wiley & Sons, Ltd.     

On the Laplacian vector fields theory in domains with rectifiable boundary

Given a domain Ω in ?³ with rectifiable boundary, we consider main integral, and some other, theorems for the theory of Laplacian (sometimes called solenoidal and irrotational, or harmonic) vector fields paying a special attention to the problem of decomposing a continuous vector field, with an additional condition, u on the boundary Γof Ω ? ?³ into a sum u = u⁺+u^? were u^± are boundary values of vector fields which are Laplacian in Ω and its complement respectively. Our proofs are based on the intimate relations between Laplacian vector fields theory and quaternionic analysis for the Moisil–Theodorescu operator. Copyright © 2006 John Wiley & Sons, Ltd.     

The Szeg? Metric Associated to Hardy Spaces of Clifford Algebra Valued Functions and Some Geometric Properties

In analogy to complex function theory we introduce a Szeg? metric in the context of hypercomplex function theory dealing with functions that take values in a Clifford algebra. In particular, we are dealing with Clifford algebra valued functions that are annihilated by the Euclidean Dirac operator in \mathbbR^m+1{\mathbb{R}^{m+1}} . These are often called monogenic functions. As a consequence of the isometry between two Hardy spaces of monogenic functions on domains that are related to each other by a conformal map, the generalized Szeg? metric turns out to have a pseudo-invariance under M?bius transformations. This property is crucially applied to show that the curvature of this metric is always negative on bounded domains. Furthermore, it allows us to establish that this metric is complete on bounded domains.     

On the stability of the completeness of L α spaces

The object of this paper is to study the stability of the quasi-completeness of theL ^α spaces for locally convex vector valued functions, by the inductive limits and the countable products.     

Universal harmonic functions

In this paper, we prove that there exists an infinite dimensional closed vector space M of harmonic functions in R ⁿ such that each v ? M \{0} is a universal harmonic function.     

General Littlewood–Paley functions and singular integral operators on product spaces

This paper is devoted to the study on the L^p ‐mapping properties for certain singular integral operators with rough kernels and related Littlewood–Paley functions along “polynomial curves” on product spaces ?^m × ?ⁿ (m ≥ 2, n ≥ 2). By means of the method of block decomposition for kernel functions and some delicate estimates on Fourier transforms, the author proves that the singular integral operators and Littlewood–Paley functions are bounded on L^p (?^m × ?ⁿ ), p ∈ (1, ∞), and the bounds are independent of the coefficients of the polynomials. These results essentially improve or extend some well‐known results. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Zeros of Fredholm Operator Valued Hp–Functions

This paper is concerned with Fredholm operator valued H^p – functions on the unit disc, where the Fredholm operators action a Banach space. Sufficient conditions are presented which guarantee that Fatou's theorem is valid. Using the theory of traces and determinants on quasi – Banach operator ideals, we develop conditions that guarantee that the zeros of Fredholm operator valued H^p – functions satisfy the Blaschke condition.