Best approximation by downward sets with applications

We develop a theory of downward sets for a class of normed ordered spaces. We study best approximation in a normed ordered space X by elements of downward sets, and give necessary and sufficient conditions for any element of best approximation by a closed downward subset of X. We also characterize strictly downward subsets of X, and prove that a downward subset of X is strictly downward if and only if each its boundary point is Chebyshev. The results obtained are used for examination of some Chebyshev pairs （W,x）, where ∈ X and W is a closed downward subset of X     

The use of phase portraits to visualize and investigate isolated singular points of complex functions

ABSTRACT

Undergraduate students usually study Laurent series in a standard course of Complex Analysis. One of the major applications of Laurent series is the classification of isolated singular points of complex functions. Although students are able to find series representations of functions, they may struggle to understand the meaning of the behaviour of the function near isolated singularities. In this paper, I briefly describe the method of domain colouring to create enhanced phase portraits to visualize and study isolated singularities of complex functions. Ultimately this method for plotting complex functions might help to enhance students' insight, in the spirit of learning by experimentation. By analysing the representations of singularities and the behaviour of the functions near their singularities, students can make conjectures and test them mathematically, which can help to create significant connections between visual representations, algebraic calculations and abstract mathematical concepts.     

Comparison principle and analogues of the phragmen—lindelöf theorem for solutions of parabolic inequalities

It is well known that a Monotonicity Condition and a Coerciveness Condition principally lie in the basis of most results of the Theory of PDE's. The necessity of these important assumptions for the validity of a comparison principle and analogues of the Phragmen-Lindelöf theorem for solutions of quasilinear parabolic inequalities is discussed in the paper. In the first part of the work we introduce a new concept of monotonicity for nonlinear differential operators-nonlinear monotonicity concept-and on its basis we obtain new phenomena for solutions, subsolutions and supersolutions of the well-known quasilinear differential equations. In the second part we omit the current coerciveness condition and change it by a weaker one. In spite of this we obtain a series of new qualitative properties of solutions for wide classes of quasilinear parabolic inequalities. Most of these properties are also new for solutions of the well-known equations, which we consider in the paper.     

Nonexistence in power and gamma density regression, sum of nonnegative terms

When we use the power function α(c x)^b and gamma density αx^be^-cx to fit the data by the least squares method, we have to address the question of existence. The closure of the set of each type of these functions defined on a finite domain is determined. We derive a way to determine the closure of a sum of nonnegative functions if the closures of the summands are available.     

Boundary-Value Problems of the Theory of Thin and Nonthin Orthotropic Shells with Account of Nonlinearly Elastic Properties and Low Shear Rigidity of Composite Materials

The basic geometric and physical relations and resolving equations of the theory of thin and nonthin orthotropic composite shells with account of nonlinear properties and low shear rigidity of their materials are presented. They are derived based on two theories, namely the theory of anisotropic shells employing the Timoshenko or Kirchhoff-Love hypothesis and the nonlinear theory of elasticity and plasticity of anisotropic media in combination with the Lagrange variational principle. The procedure and algorithm for the numerical solution of nonlinear (linear) problems are based on the method of successive approximations, the difference-variational method, and the Lagrange multiplier method. Calculations of the stress-strain state for a spherical shell with a circular opening loaded with internal pressure are presented. The effect of transverse shear strains and physical nonlinearity of the material on the distribution of maximum deflections and circumferential stresses in the shell, obtained according to two variants of the shell theories, is studied. A comparison of the results of the problem solution in linear and nonlinear statements with and without account of the shell shear strains is given. The numerical data obtained for thin and nonthin (medium thick) composite shells are analyzed.     

Sur l'existence, l'unicité, la stabilité et les propriétés de grandes déviations des solutions d'équations différentielles stochastiques rétrogrades à horizon aléatoire. Application à des problèmes de perturbations singulières

We study the existence, uniqueness and stability of solutions of backward stochastic differential equations with random terminal time under new assumptions; then we establish a large deviation principle for the solutions of such equations, related to a family of Markov processes, the diffusion coefficient of which tends to zero. Finally we apply these results to the analysis of some singular perturbation problems for a class of nonlinear partial differential equations.     

Representation of finite groups and the first Betti number of branched coverings of a universal Borromean orbifold

The paper studies the first homology of finite regular branched coverings of a universal Borromean orbifold called B _4,4,4ℍ³. We investigate the irreducible components of the first homology as a representation space of the finite covering transformation group G. This gives information on the first betti number of finite coverings of general 3-manifolds by the universality of B _4,4,4. The main result of the paper is a criterion in terms of the irreducible character whether a given irreducible representation of G is an irreducible component of the first homology when G admits certain symmetries. As a special case of the motivating argument the criterion is applied to principal congruence subgroups of B _4,4,4. The group theoretic computation shows that most of the, possibly nonprincipal, congruence subgroups are of positive first Betti number. This work is partially supported by the Sonderforschungsbereich 288.     

Clustering Methods in Microarrays

Summary DCT Given a finite set of points in an Euclidean space the \emph{spanning tree} is a tree of minimal length having the given points as vertices. The length of the tree is the sum of the distances of all connected point pairs of the tree. The clustering tree with a given length of a given finite set of points is the spanning tree of an appropriately chosen other set of points approximating the given set of points with minimal sum of square distances among all spanning trees with the given length. DCM A matrix of real numbers is said to be column monotone orderable if there exists an ordering of columns of the matrix such that all rows of the matrix become monotone after ordering. The {\emph{monotone sum of squares of a matrix}} is the minimum of sum of squares of differences of the elements of the matrix and a column monotone orderable matrix where the minimum is taken on the set of all column monotone orderable matrices. Decomposition clusters of monotone orderings of a matrix is a clustering ofthe rows of the matrix into given number of clusters such that thesum of monotone sum of squares of the matrices formed by the rowsof the same cluster is minimal.DCP A matrix of real numbers is said to be column partitionable if there exists a partition of the columns such that the elements belonging to the same subset of the partition are equal in each row. Given a partition of the columns of a matrix the partition sum of squares of the matrix is the minimum of the sum of square of differences of the elements of the matrix and a column partitionable matrix where the minimum is taken on the set of all column partitionable matrices. Decomposition of the rows of a matrix into clusters of partitions is the minimization of the corresponding partition sum of squares given the number of clusters and the sizes of the subsets of the partitions.     

Absolute Continuity of Measures in the Class of Markov and Semi-Markov Processes of Diffusion Type

The property of absolute continuity of measures in the class of one-dimensional semi-Markov processes of diffusion type is investigated. The measure of such a process can be composed of two measures. The first one is a distribution of a random track, and the second one is a conditional distribution of a time run along the track. The desired density is represented in the form of product of two corresponding densities.     

Hausdorff measures of a class of Sierpinski carpets

thenandIn this paper, a lemma as a new method to calculate the Hausdorff measure of fractal is given. And the exact values of Hausdorff measure of a class of Sierpinski sets which satisfy balance distribution ang dimension ≤1 are obtained     

Effects of permeability and viscosity in linear polymeric gels

We propose and analyze a mathematical model of the mechanics of gels, consisting of the laws of balance of mass and linear momentum of the polymer and liquid components of the gel. We consider a gel to be an immiscible and incompressible mixture of a nonlinearly elastic polymer and a fluid. The problems that we study are motivated by predictions of the life cycle of body‐implantable medical devices. Scaling arguments suggest neglecting inertia terms, and therefore, we consider the quasi‐static approximation to the dynamics. We focus on the linearized system about stress‐free states, uniform expansions, and compressions and derive sufficient conditions for the solvability of the time‐dependent problems. These turn out to be conditions that guarantee local stability of the equilibrium solutions. We also consider non‐stress free equilibria and states with residual stress and derive an energy law for the corresponding time‐dependent system. The conditions that guarantee stability of solutions provide a selection criteria of the material parameters of devices. The boundary conditions that we consider are of two types, displacement‐traction and permeability of the gel surface to the fluid. We address the cases of viscous and inviscid solvent, assume Newtonian dissipation for the polymer component, and establish existence of weak solutions for the different boundary permeability conditions and viscosity assumptions. We present two‐dimensional, finite element numerical simulations to study stress concentration on edges, this being the precursor to debonding of the gel from its substrate. Copyright © 2015 John Wiley & Sons, Ltd.     

Lie algebraic characterization of manifolds

Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operators, i.e. isomorphisms of such Lie algebras are induced by the appropriate class of diffeomorphisms of the underlying manifolds. The research of Janusz Grabowski supported by the Polish Ministry of Scientific Research and Information Technology under the grant No. 2 P03A 020 24, that of Norbert Poncin by grant C.U.L./02/010.     

Toward eudaimonology: notes on a quantitative framework for the study of happiness

After noting factors (concern for others, ignorance, irrationality) accounting for the divergences between preference and happiness, the question of representing the preference of an individual by a utility function is discussed, taking account of lexicographic ordering, imperfect discrimination and the corresponding concepts of semiorder and sub-semiorder. Methods to improve upon the interpersonal comparability of measures of happiness such as pinning down the dividing line of zero happiness and the use of a just perceivable increment of happiness are discussed. The relation of social welfare to individual welfare (i.e. happiness) is then considered. Some reasonable set of axioms ensuring that social welfare is a separable function of and indeed an unweighted sum of individual welfares are reviewed. Finally, happiness is regarded as a function of objective, institutional and subjective factors; an interdisciplinary approach is needed even for an incomplete analysis.     

Bounds for the zeros of polynomials

Let P(z)=∑↓j=0↑n ajx^j be a polynomial of degree n. In this paper we prove a more general result which interalia improves upon the bounds of a class of polynomials. We also prove a result which includes some extensions and generalizations of Enestrǒm-Kakeya theorem.     

Noncommutative geometry and motives: The thermodynamics of endomotives

We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of L-functions. The analogue in characteristic zero of the action of the Frobenius on ?-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative generalization of Artin motives). Semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives, with an action of the absolute Galois group. The semigroup of endomorphisms of the multiplicative group yields the Bost-Connes system, from which one obtains, through the above procedure, the desired cohomological interpretation of the zeros of the Riemann zeta function. In the last section we also give a Lefschetz formula for the archimedean local L-factors of arithmetic varieties.     

Rigid isotopy classification of real algebraic curves of bidegree (3,3) on quadrics

A rigid isotopy of nonsingular real algebraic curves on a quadric is a path in the space of such curves of a given bidegree. We obtain the rigid isotopy classification of nonsingular real algebraic curves of bidegree (3, 3) on a hyperboloid and on an ellipsoid. We also study of the space of real algebraic curves of bidegree (3, 3) with a single node or cusp. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 810–815, December, 1999.     

Cocyclic Codes of Length 40

There are 5 groups of order 20. This paper reports on the search for binary self-dual codes of length 40, cocyclic over any one of the first four groups, using cocyclic Hadamard matrices and the [I, A] construction. The fifth group is not investigated here. A total of 28 classes of extremal cocyclic self-dual codes were found—27 of these are doubly-even and one singly-even. The majority of these classes arise from the dihedral-cocyclic Hadamard matrices. There is also a class of dihedral-cocyclic Hadamard matrices which gives a large collection of [40, 20] codes with only one codeword of length 4.     

An Ansatz for the asymptotics of hypergeometric multisums

Sequences that are defined by multisums of hypergeometric terms with compact support occur frequently in enumeration problems of combinatorics, algebraic geometry and perturbative quantum field theory. The standard recipe to study the asymptotic expansion of such sequences is to find a recurrence satisfied by them, convert it into a differential equation satisfied by their generating series, and analyze the singularities in the complex plane. We propose a shortcut by constructing directly from the structure of the hypergeometric term a finite set, for which we conjecture (and in some cases prove) that it contains all the singularities of the generating series. Our construction of this finite set is given by the solution set of a balanced system of polynomial equations of a rather special form, reminiscent of the Bethe ansatz. The finite set can also be identified with the set of critical values of a potential function, as well as with the evaluation of elements of an additive K-theory group by a regulator function. We give a proof of our conjecture in some special cases, and we illustrate our results with numerous examples.     

Structure and Properties of Nanocrystalline Oxide Powders

The crystalline and porous structure of superfine powders and fibers made of alumina and zirconia doped with yttria is investigated. The particle sizes were determined by the methods of coherent dispersion and scanning transmission electron microscopy. Based on data on the sizes of crystallites of the metal oxides, which were obtained by different methods, it is found that crystalline transformations of the oxides lead to dispersion of the materials. The structural transformations of the superfine oxides are accompanied by changes in crystallite sizes and in the character and size of pores. The laws established allow one to purposefully control the process of producing nanostructured oxide powders and fibers, which can be utilized as active fillers for composite materials based on various matrices. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 6, pp. 831–840, November–December, 2005.     

Vagueness, ignorance, and margins for error

We argue that the epistemic theory of vagueness cannot adequately justify its key tenet-that vague predicates have precisely bounded extensions, of which we are necessarily ignorant. Nor can the theory adequately account for our ignorance of the truth values of borderline cases. Furthermore, we argue that Williamson’s promising attempt to explicate our understanding of vague language on the model of a certain sort of “inexact knowledge” is at best incomplete, since certain forms of vagueness do not fit Williamson’s model, and in fact fit an alternative model. Finally, we point out that a certain kind of irremediable inexactitude postulated by physics need not be-and is not commonly-interpreted as epistemic. Thus, there are aspects of contemporary science that do not accord well with the epistemicist outlook.