Realizability of words in mosaic labyrinths

In this paper we study mosaic labyrinths with the help of words generated by them in the alphabet of labels attached to arcs and vertices of a labyrinth. We consider the problem of the characterization of words generated by a labyrinth. We propose a constructive recognition criterion, it defines whether a word is generated by a labyrinth or not. We establish conditions under which a word can be generated by a unique labyrinth, by a finite number of labyrinths, or by infinitely many labyrinths.     

Well-posedness by perturbations of mixed variational inequalities in Banach spaces

In this paper, we consider an extension of the notion of well-posedness by perturbations, introduced by Zolezzi for a minimization problem, to a mixed variational inequality problem in a Banach space. We establish some metric characterizations of the well-posedness by perturbations. We also show that under suitable conditions, the well-posedness by perturbations of a mixed variational inequality problem is equivalent to the well-posedness by perturbations of a corresponding inclusion problem and a corresponding fixed point problem. Also, we derive some conditions under which the well-posedness by perturbations of a mixed variational inequality is equivalent to the existence and uniqueness of its solution.     

Universal contingent claims in a general market environment and multiplicative measures: Examples and applications

We present and further develop the concept of a universal contingent claim introduced by the author in 1995. This concept provides a unified framework for the analysis of a wide class of financial derivatives.A universal contingent claim describes the time evolution of a contingent payoff. In the simplest case of a European contingent claim, this time evolution is given by a family of nonnegative linear operators, the valuation operators. For more complex contingent claims, the time evolution that is given by the valuation operators can be interrupted by discrete or continuous activation of external influences that are described by, generally speaking, nonlinear operators, the activation operators. For example, Bermudan and American contingent claims represent discretely and continuously activated universal contingent claims with the activation operators being the nonlinear maximum operators.We show that the value of a universal contingent claim is given by a multiplicative measure introduced by the author in 1995. Roughly speaking, a multiplicative measure is an operator-valued (in general, an abstract measure with values in a partial monoid) function on a semiring of sets which is multiplicative on the union of disjoint sets. We also show that the value of a universal contingent claim is determined by a, generally speaking, impulsive semilinear evolution equation.     

Dynamic instability of a non-uniform beam modeled in a two-degree-of-freedom system

The behaviour of a non-uniform beam loaded by a parallel or tangential compressive force respectively is analyzed by exploiting a two-degree-of-freedom dynamic model. The same system has been analyzed previously by Lee and Reissner from a static point of view and by Neer and Baruch by a dynamic approach using a one-degree-of-freedom model. The previous analyses revealed only part of the phenomena by the present approach.Here, for the tangential force a classical flutter instability is obtained and for the parallel force only static instability is possible.     

Proof of the Branner-Hubbard conjecture on Cantor Julia sets

By means of a nested sequence of some critical pieces constructed by Kozlovski, Shen, and van Strien, and by using a covering lemma recently proved by Kahn and Lyubich, we prove that a component of the filled-in Julia set of any polynomial is a point if and only if its forward orbit contains no periodic critical components. It follows immediately that the Julia set of a polynomial is a Cantor set if and only if each critical component of the filled-in Julia set is aperiodic. This result was a conjecture raised by Branner and Hubbard in 1992. This work was supported by the National Natural Science Foundation of China     

Continuous Frames, Function Spaces, and the Discretization Problem

A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous superpositions. Associated to a given continuous frame we construct certain Banach spaces. Many classical function spaces can be identified as such spaces. We provide a general method to derive Banach frames and atomic decompositions for these Banach spaces by sampling the continuous frame. This is done by generalizing the coorbit space theory developed by Feichtinger and Gröchenig. As an important tool the concept of localization of frames is extended to continuous frames. As a byproduct we give a partial answer to the question raised by Ali, Antoine, and Gazeau whether any continuous frame admits a corresponding discrete realization generated by sampling.     

Variational approach to scattering of plane elastic waves by diffraction gratings

The scattering of a time‐harmonic plane elastic wave by a two‐dimensional periodic structure is studied. The grating profile is given by a Lipschitz curve on which the displacement vanishes. Using a variational formulation in a bounded periodic cell involving a nonlocal boundary operator, existence of solutions in quasiperiodic Sobolev spaces is investigated by establishing the Fredholmness of the operator generated by the corresponding sesquilinear form. Moreover, by a Rellich identity, uniqueness is proved under the assumption that the grating profile is given by a Lipschitz graph. The direct scattering problem for transmission gratings is also investigated. In this case, uniqueness is proved except for a discrete set of frequencies. Copyright © 2010 John Wiley & Sons, Ltd.     

A recognition algorithm for the intersection graphs of directed paths in directed trees

Consider a finite family of non-empty sets. The intersection graph of this family is obtained by representing each set by a vertex, two vertices being connected by an edge if and only if the corresponding sets intersect. The intersection graph of a family of directed paths in a directed tree is called a directed path graph. In this paper we present an efficient algorithm which constructs to a given graph a representation by a family of directed paths on a directed tree, if one exists. Also, we prove that a graph is a proper directed path graph if and only if it is a directed path graph.     

Randomized system trajectories

The notion of system trajectory of a time-varying input-output, dynamical system is reviewed. By introducing a probability measure on a class of such systems a stochastic system, the randomized system, is defined. The randomized system has a trajectory induced by the trajectories of the original systems. A theorem is proved giving fairly general conditions under which the randomized system trajectory is generated by a strongly continuous semigroup of bounded linear operators in a Banach space. An example is presented for a system represented by a quadratic integral operator.Research supported in part by National Science Foundation under Grant No. ECS-8005960.     

Enhanced and reduced natural convection using magnetic fluid in a square cavity

Natural convection using a magnetic fluid was studied in a square cavity under the influence of a permanent magnet. The aim was to explore the degree by which heat transfer may be controlled, enhanced or reduced, by investigating a set of different distances of a permanent magnet to the cavity. These distances of the magnet were set such that the cavity was in some cases fully dominated by buoyancy or by the magnetic body force and in other cases partly dominated by either of both body forces in different parts of the fluid. The effect on heat transfer was characterised by an averaged Nusselt number, Rayleigh and magnetic Rayleigh number. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Unimodularity of zeros of self-inversive polynomials

We generalise a necessary and sufficient condition given by Cohn for all the zeros of a self-inversive polynomial to be on the unit circle. Our theorem implies some sufficient conditions found by Lakatos, Losonczi and Schinzel. We apply our result to the study of a polynomial family closely related to Ramanujan polynomials, recently introduced by Gun, Murty and Rath, and studied by Murty, Smyth and Wang as well as by Lalín and Rogers. We prove that all polynomials in this family have their zeros on the unit circle, a result conjectured by Lalín and Rogers on computational evidence.     

Singular Double Porosity Model

We consider the linear parabolic equation describing the transport of a contaminant in a porous media crossed by a net of infinitely thin fractures. The permeability is very high in the fractures but very low in the porous blocks. We derive the homogenized model corresponding to a net of infinitely thin fractures, by means of the singular measures technique. We assume that these singular measures are supported by hyperplanes of codimension one. We prove in a second step that this homogenized model could be obtained indistinctly either by letting the fracture thickness, in the standard double porosity model, tend to zero, or by homogenizing a model with infinitely thin fractures.     

Classification of Mixtures of Spatial Point Processes via Partial Bayes Factors

Motivated by the problem of minefield detection, we investigate the problem of classifying mixtures of spatial point processes. In particular we are interested in testing the hypothesis that a given dataset was generated by a Poisson process versus a mixture of a Poisson process and a hard-core Strauss process. We propose testing this hypothesis by comparing the evidence for each model by using partial Bayes factors. We use the term partial Bayes factor to describe a Bayes factor, a ratio of integrated likelihoods, based on only part of the available information, namely that information contained in a small number of functionals of the data. We applied our method to both real and simulated data, and considering the difficulty of classifying these point patterns by eye, our approach overall produced good results.     

Asymptotic expansions for Markov processes with lévy generators

This paper considers a deterministic flow inn-dimensional space, perturbed by a Markov jump process with small variance. Asymptotic expansions are obtained for certain functionals of Feynman—Kac type, in powers of a small parameter representing a noise intensity. The methods are analytical rather than probabilistic.The research of the first author was partly supported by AFOSR under Contract No. 91-0116-0, by ONR under Contract No. N0014-83-K-0542, and by the Institute for Mathematics and Its Applications with funds provided by the NSF and ONR. The second author's research was partly supported by NSF under Contract No. DMS-8702537, and by the Institute for Mathematics and Its Applications with funds provided by the NSF and ONR.     

Automation and group structures in certain economic adjustment mechanisms

It is shown that the economic adjustment mechanism developed by Hurwicz and his associates has the structure of automata. It is then shown that certain price adjustment mechanisms, having an acceptability condition, impose a group structure upon the automaton. This condition is a bilinear invariance implied by a budget constraint. Then the automaton is defined by a subgroup, depending on agents' tastes, technologies and strategies, and by the representations of the subgroup imposed by the automaton.     

On the covering relation in the interpretability lattice of equational theories

We prove that the chapter in the interpretability lattice represented by the equational theory of Boolean algebras has a unique cover in the lattice. We conjecture that, among chapters represented by equational theories of two-element algebras, this is the only one to have a cover. Also, we prove that the chapter represented by the equational theory of Abelian groups has no cover.Presented by W. Taylor.Research supported by a Ulam Research Professorship at the University of Colorado and by NSF Grant DMS 89 04014.     

Normal ergodic actions

The von Neumann-Halmos theory of ergodic transformations with discrete spectrum makes use of the duality theory of locally compact abelian groups to characterize those transformations preserving a probability measure, which are defined by a rotation on a compact abelian group. We use the recently developed duality between general locally compact groups and Hopf-von Neumann algebras to characterize those actions of a locally compact group, preserving a σ-finite measure, which are defined by a dense embedding in another group. They are characterized by the property of normality, previously introduced by the author, and motivated by Mackey's theory of virtual groups. The discrete spectrum theory is readily seen to come out as the special case in which the invariant measure is finite.     

Spaces of domino tilings

We consider the set of all tilings by dominoes (2×1 rectangles) of a surface, possibly with boundary, consisting of unit squares. Convert this set into a graph by joining two tilings by an edge if they differ by aflip, i.e., a 90° rotation of a pair of side-by-side dominoes. We give a criterion to decide if two tilings are in the same connected component, a simple formula for distances, and a method to construct geodesics in this graph. For simply connected surfaces, the graph is connected. By naturally adjoining to this graph higher-dimensional cells, we obtain a CW-complex whose connected components are homotopically equivalent to points or circles. As a consequence, for any region different from a torus or Klein bottle, all geodesics with common endpoints are equivalent in the following sense. Build a graph whose vertices are these geodesics, adjacent if they differ only by the order of two flips on disjoint squares: this graph is connected. The first two authors received support from SCT and CNPq, Brazil. The other two were supported by a grant for undergraduates by CNPq.     

Relatively optimal filtering on a Hilbert space for measure driven stochastic systems

In this paper we consider linear filtering for discontinuous processes determined by stochastic differential equations on a Hilbert space driven by signed measures in addition to Brownian motion. The dynamics of the observed data is governed by a differential equation driven by a square integrable martingale (not necessarily continuous) while perturbed by a signed measure. We formulate the filtering problem as an optimization problem on the space of bounded linear operator valued functions and present necessary and sufficient conditions for optimality. Further, we prove, under the assumption of finite dimensionality of the output space, that a Kalman-like filter exists and it is explicitly determined by a Riccati type evolution equation.