Necessary and Sufficient Conditions for the Interchange Between Infimum and the Symbol of Integration

We make a study of various notions of decomposability for subsets of measurable functions in relation with the interchange results between infimum and integration. For this we introduce the notions of serial decomposability and of decomposability relatively to an integrand. A characterization of closed serially decomposable subsets of the Lebesgue spaces L _p is given. The second notion of decomposability introduced is characteristic for the interchange property studied. Many examples are presented. The links are made with R. T. Rockafellar’s decomposability, F. Hiai, H. Umegaki’s decomposability, G. Bouchitté and M. Valadier’s stability and normal decomposability introduced by O. Anza Hafsa and J.-P. Mandallena. As applications we obtain exact lower bounds for minimization problems of integral functionals on normally decomposable spaces (spaces of continuous functions for example), and for the minimization of a class of functionals of the Calculus of Variations.     

A proof of the Pfister Factor Conjecture

It is shown that any split product of quaternion algebras with orthogonal involution is adjoint to a Pfister form. This settles the Pfister Factor Conjecture formulated by D.B. Shapiro. A more general problem on decomposability for algebras with involution is posed and solved in the case where the algebra is equivalent to a quaternion algebra.     

Decomposition of finite pseudometric spaces

Here we define decomposable pseudometrics. A pseudometric is decomposable if it can be represented as the sum of two pseudometrics that are obtained in a way other than the multiplication all distances by a positive factor. We consider spaces consisting ofn points. We prove that there exist a finite number of indecomposable pseudometrics (that is, a basis) such that any pseudometric is a linear combination of basic pseudometrics with nonnegative coefficients. Forn ≤ 7, the basic pseudometrics are listed. A decomposability test is derived for finite pseudometric spaces. We also establish some other conditions of decomposability and indecomposability. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 225–234, February, 1998.     

Average homogeneity and dimensions of measures

We introduce the concept of average homogeneity of a measure by comparing the measure to the uniform distribution in a relatively simple way. This leads to a very general notion which may be regarded as an inverse of porosity. In this paper the emphasis is given to relations between homogeneity and dimensions of measures. First we consider the effect of homogeneity on dimensions by proving an upper bound to the Hausdorff dimension as a function of homogeneity and its order. The opposite question of how dimensions effect homogeneity is solved by giving an upper bound to homogeneity in terms of upper packing dimension. We also illustrate by examples that all our results are the best possible ones.Mathematics Subject Classification (2000): 28A75, 28A80MJ acknowledges the support of the Academy of Finland, project #48557.     

Decomposability in queues with background states

A symmetric queue is known to have a nice property, the so-called insensitivity. In this paper, we generalize this for a single node queue with Poisson arrivals and background state, which changes at completion instants of lifetimes as well as at the arrival and departure instants. We study this problem by using the decomposability property of the joint stationary distribution of the queue length and supplementary variables, which implies the insensitivity. We formulate a Markov process representing the state of the queue as an RGSMP (reallocatable generalized semi-Markov process), and give necessary and sufficient conditions for the decomposability. We then establish general criteria to be sufficient for the queue to possess the property. Various symmetric-like queues with background states, including continuous time versions of moving server queues, are shown to have the decomposability.This author is partially supported by NEC C&C Laboratories.     

On decidability of the decomposability problem for finite theories

We consider the decomposability problem for elementary theories, i.e. the problem of deciding whether a theory has a nontrivial representation as a union of two (or several) theories in disjoint signatures. For finite universal Horn theories, we prove that the decomposability problem is $ \sum _1^0 $ \sum _1^0 -complete and, thus, undecidable. We also demonstrate that the decomposability problem is decidable for finite theories in signatures consisting only of monadic predicates and constants.     

Decomposability in the space of HKP‐integrable functions

In this paper we introduce the notion of decomposability in the space of Henstock‐Kurzweil‐Pettis integrable (for short HKP‐integrable) functions. We show representations theorems for decomposable sets of HKP‐integrable or Henstock integrable functions, in terms of the family of selections of suitable multifunctions.     

Classification of designs with nontrivial automorphism groups

In this article, we introduce a new orderly backtrack algorithm with efficient isomorph rejection for classification of t‐designs. As an application, we classify all simple 2‐(13,3,2) designs with nontrivial automorphism groups. The total number of such designs amounts to 1,897,386. The decomposability of the designs is also considered. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 479–489, 2006     

Decomposability of polytopes and polyhedra

In this paper decomposability of polytopes (and polyhedral sets) is studied by investigating the space of affine dependences of the vertices of the dual polytope. This turns out to be a fruitful approach and leads to several new results, as well as to simpler proofs and generalizations of known results. One of the new results is that a 3-polytope with more vertices than facets is decomposable; this leads to a characterization of the decomposability of 3-polytopes.     

Investigations of isotropy and homogeneity of spacetime in first-order logic

We investigate the logical connection between (spatial) isotropy, homogeneity of space, and homogeneity of time within a general axiomatic framework. We show that isotropy not only entails homogeneity of space, but also, in certain cases, homogeneity of time. In turn, homogeneity of time implies homogeneity of space in general, and the converse also holds true in certain cases.An important innovation in our approach is that formulations of physical properties are simultaneously empirical and axiomatic (in the sense of first-order mathematical logic). In this case, for example, rather than presuppose the existence of spacetime metrics – together with all the continuity and smoothness apparatus that would entail – the basic logical formulas underpinning our work refer instead to the sets of (idealised) experiments that support the properties in question, e.g., isotropy is axiomatised by considering a set of experiments whose outcomes remain unchanged under spatial rotation. Higher-order constructs are not needed.     

Existence and sum decomposition of vertex polyhedral convex envelopes

Convex envelopes are a very useful tool in global optimization. However finding the exact convex envelope of a function is a difficult task in general. This task becomes considerably simpler in the case where the domain is a polyhedron and the convex envelope is vertex polyhedral, i.e., has a polyhedral epigraph whose vertices correspond to the vertices of the domain. A further simplification is possible when the convex envelope is sum decomposable, i.e., the convex envelope of a sum of functions coincides with the sum of the convex envelopes of the summands. In this paper we provide characterizations and sufficient conditions for the existence of a vertex polyhedral convex envelope. Our results extend and unify several results previously obtained for special cases of this problem. We then characterize sum decomposability of vertex polyhedral convex envelopes, and we show, among else, that the vertex polyhedral convex envelope of a sum of functions coincides with the sum of the vertex polyhedral convex envelopes of the summands if and only if the latter sum is vertex polyhedral.     

Local Spectral Theory of Linear Operators RS and SR

In this paper, we study the relation between local spectral properties of the linear operators RS and SR. We show that RS and SR share the same local spectral properties SVEP, (β), (δ) and decomposability. We also show that RS is subscalar if and only if SR is subscalar. We recapture some known results on spectral properties of Aluthge transforms.     

Differential topology of quotients of complex surfaces by complex conjugation

The paper contains a brief survey of the author’s results on the diffeomorphism type of quotients of complex surfaces by anti-holomorphic involutions. The conjecture of complete decomposability is discussed, which says that if such a quotient is simply connected, then it is completely decomposable, i.e., is diffeomorphic to the connected sum of several copies of the projective plane (possibly, with reversed orientation) and the quadric. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 215–221. Translated by O. A. Ivanov.     

Decomposability in fixed point computation with applications and acceleration techniques

In the past decade, several complementary pivot algorithms have been developed to search for fixed points of certain functions and point to set maps on unbounded regions. This paper develops a structure (called decomposability), which, when present, enables one to work in a lower dimensional space when solving these problems. Several examples of where this structure arises in applications are presented. It is shown that under suitable circumstances, the general constrained optimization problem (that of optimizing an objective function subject to both equality and inequality constraints) may be formulated as a decomposèble fixed point problem. At the same time, an approximation technique is developed to potentially improve the efficiency of the complementary pivot algorithms.     

How homogeneous can the last appearing pattern be?

Given a sequence of indepent, uniformly distributed random letters coming from a finite alphabet, we are interested in the word (pattern) which is the last to be observed for the first time among all words of length n. We derive anas. result on the homogeneity of this last appearing pattern as a corollary of a more general theorem. © 1993 John Wiley & Sons, Inc.     

Decomposability criterion for linear sheaves

We establish a decomposability criterion for linear sheaves on ℙ ⁿ . Applying it to instanton bundles, we show, in particular, that every rank 2n instanton bundle of charge 1 on ℙ ⁿ is decomposable. Moreover, we provide an example of an indecomposable instanton bundle of rank 2n − 1 and charge 1, thus showing that our criterion is sharp.     

On isotropy conditions in second gradient materials

In gradient elasticity, isotropy and frame-indifference requirements are sensitive to the homogeneity of the applied rotation field Q ∈ SO(3). This is in contrast to standard elasticity, where only first gradients of the deformation are under consideration. We use a diffeomorphism to show the effect of inhomogeneous coordinate transformation to the form-invariance requirement of elastic energy. From a classical geometric rigidity result follows that the appearance of a right-local SO(3)-invariance condition is not the general condition for isotropy. The correct statement for isotropy in second gradient elasticity should be a right-global SO(3)-invariance condition. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A class of asymptotically normal degenerate quasi <Emphasis Type="Italic">U</Emphasis>-statistics

Some quasi U-statistics, unlike other variants of U-statistics, arising in distance based tests for homogeneity of groups, have first-order stationary kernels of degree 2, and yet they enjoy asymptotic normality under suitable hypotheses of invariance. Central limit theorems for a more general class of quasi U-statistics with possibly higher order stationarity (and degree) are formulated with the aid of appropriate martingale (array) characterizations as well as permutational invariance structures.     

Diversity in inside factorial monoids

In a recent paper (Diversity in Monoids, Czech. Math. J. 62 (2012), 795–809), the last two authors introduced and developed the monoid invariant “diversity” and related properties “homogeneity” and “strong homogeneity”. We investigate these properties within the context of inside factorial monoids, in which the diversity of an element counts the number of its different almost primary components. Inside factorial monoids are characterized via diversity and strong homogeneity. A new invariant complementary to diversity, height, is introduced. These two invariants are connected with the well-known invariant of elasticity.