Representation Theorem Research Articles

Slepian–Bangs Formula and Cramér–Rao Bound for Circular and Non-Circular Complex Elliptical Symmetric Distributions

This paper is mainly dedicated to an extension of the Slepian-Bangs formula to non-circular complex elliptical symmetric (NC-CES) distributions, which is derived from a new stochastic representation theorem. This formula includes the non-circular complex Gaussian and the circular CES (CCES) distributions. Some general relations between the Cramer Rao bound (CRB) under CES and Gaussian distributions are deduced. It is proved in particular that the Gaussian distribution does not always lead to the largest stochastic CRB (SCRB) as many authors tend to believe it. Finally a particular attention is paid to the noisy mixture where closedform expressions for the SCRBs of the parameters of interest are derived.

Willis theory via graphs

We study the scale and tidy subgroups of an endomorphism of a totally disconnected locally compact group using a geometric framework. This leads to new interpretations of tidy subgroups and the scale function. Foremost, we obtain a geometric tidying procedure which applies to endomorphisms as well as a geometric proof of the fact that tidiness is equivalent to being minimizing for a given endomorphism. Our framework also yields an endomorphism version of the Baumgartner–Willis tree representation theorem. We conclude with a construction of new endomorphisms of totally disconnected locally compact groups from old via HNN-extensions.

The Kempf–Ness Theorem and invariant theory for real reductive representations

This paper does not contain any new result. We give new proofs of the Kempf–Ness Theorem and Hilbert–Mumford criterion for real reductive representations avoiding any algebraic results.

On Janowski functions associated with (n,m)-symmetrical functions

The aim in the present work is to introduce and study new subclasses of analytic functions that are defined by using the generalized classes of Janowski functions combined with the -symmetrical functions, that generalize many others defined by different authors. We gave a representation theorem for these classes, certain inherently properties, while covering and distortion properties are also pointed out.

Adjoint functor theorems for ∞‐categories

Adjoint functor theorems give necessary and sufficient conditions for a functor to admit an adjoint. In this paper we prove general adjoint functor theorems for functors between $\infty$-categories. One of our main results is an $\infty$-categorical generalization of Freyd's classical General Adjoint Functor Theorem. As an application of this result, we recover Lurie's adjoint functor theorems for presentable $\infty$-categories. We also discuss the comparison between adjunctions of $\infty$-categories and homotopy adjunctions, and give a treatment of Brown representability for $\infty$-categories based on Heller's purely categorical formulation of the classical Brown representability theorem.

On the formulation of anisotropic plasticity for polymeric composites‐ rate‐dependent models with non‐linear isotropic/kinematic hardening

AbstractThe use of fiber‐reinforced composites as a primary structural component in automotive and aerospace industries has significantly increased over time, due to their great weight saving potential. Consequently, the predictive modeling of the non‐linear behavior of composites has been a topic of intensive research over the last years. Experimental data shows non‐linear response that is direction dependent due to the fibrous micro‐structure. Since typical fiber materials can be considered essentially as linear elastic, the non‐linearities are attributable to the inelastic deformation in the matrix, occurring at relatively small strains.This study presents a relatively general, homogenized formulation of anisotropic plasticity for application to (mainly) polymeric composites. A constitutive model has been developed to simulate the non‐linearities exhibited by the composite, generating the anisotropic elastic and plastic response functions with the aid of representation theorems. Rate‐independent versions using isotropic hardening have previously been proposed by the authors. The main focus of this work is the extension to include rate dependency and non‐linear kinematic hardening effects. Selected numerical simulations which serve the purpose of illustrating the formulation discussed herein, are presented at the end.

Expected utility theory with probability grids and preference formation

We reformulate expected utility theory, from the viewpoint of bounded rationality, by introducing probability grids and a cognitive bound; we restrict permissible probabilities only to decimal (ell -ary in general) fractions of finite depths up to a given cognitive bound. We distinguish between measurements of utilities from pure alternatives and their extensions to lotteries involving more risks. Our theory is constructive from the viewpoint of the decision maker. When a cognitive bound is small, the preference relation involves many incomparabilities, but these diminish as the cognitive bound is relaxed. Similarly, the EU hypothesis would hold more for a larger bound. The main part of the paper is a study of preferences including incomparabilities in cases with finite cognitive bounds; we give representation theorems in terms of a 2-dimensional vector-valued utility functions. We also exemplify the theory with one experimental result reported by Kahneman and Tversky.

Fitted fractional reproducing kernel algorithm for the numerical solutions of ABC – Fractional Volterra integro-differential equations

Abstract This paper focuses on providing a novel high-order algorithm for the numerical solutions of fractional order Volterra integro-differential equations using Atangana–Baleanu approach by employing the reproducing kernel approximation. For this purpose, we investigate couples of Hilbert spaces and kernel functions, as well as, the regularity properties of Atangana–Baleanu derivative, and utilize that the representation theorem of its solution. To remove the singularity in the kernel function, using new Atangana–Baleanu approach the main operator posses smoothing solution with a better regularity properties and the reproducing kernel algorithm is designed for the required equation. The convergence properties of the proposed algorithm are also studied which proves that the new strategy exhibits a high-order of convergence with decreasing error bound. Some numerical examples of single and system formulation illustrate the performance of the approach. Summary and some notes are also provided in the case of conclusion and highlight.

A new way to represent functions as series

Abstract In this paper we will show a new way to represent functions as infinite series, finding some conditions under which a function is expandable with this method, and showing how it allows us to find the values of many interesting series. At the end, we will prove one of the main results of the paper, a Representation Theorem.

Degenerate Laplacian describing topologically constrained diffusion: helicity constraint as an alternative to ellipticity

We study a second-order non-elliptic partial differential operator that generalizes the classical Laplacian with a degenerate coefficient matrix. The theory is motivated by the statistical mechanics of topologically constrained particles. In the context of diffusion models, the coefficient matrix is given by , with a generalized Poisson matrix that dictates particle dynamics. When is the symplectic matrix of canonical Hamiltonian systems, the operator reduces to the classical Laplacian. However, topological constraints make the matrix degenerate. The ellipticity is broken in the direction parallel to the nullity of . We call such a diffusion operator an orthogonal Laplacian and denote it by . When the nullity foliates the space, reduces into a classical Laplacian on the leaf that is orthogonal to the null space. However, when the nullity of is not integrable (which is the case when has helicity), is akin to elliptic operators having some useful properties of convex analysis. We show that defines an inner product of a Sobolev-like Hilbert space, and is a coercive functional in the L2 topology. Constructing a trace theorem, we discuss a boundary value problem (the orthogonal Poisson equation) for the orthogonal Laplacian, and obtain a unique weak solution by application of Riesz’s representation theorem.

Galerkin-type solution for the theory of strain and temperature rate-dependent thermoelasticity

Ultrafast heating has gained serious attention due to technological advancements in recent years. The thermoelastic process accounting for the finite speed of thermal signals is therefore becoming increasingly important. With the help of extended thermodynamics, Yu et al. (Meccanica 53(10):2543–2554, 2018) have established a thermoelastic model by including the strain rate and temperature rate in the constitutive equations. This model is also an attempt to remove the discontinuity in the displacement field under temperature rate-dependent thermoelasticity theory. The present work seeks to derive the representation of a Galerkin-type solution in the context of this recently proposed model in terms of elementary functions. The representation theorem of the Galerkin-type solution of the system of equations for steady oscillation is also proved. In accordance with this theorem, we finally establish a theorem which expresses the general solution of the system of homogeneous equations of steady oscillation in terms of metaharmonic functions.

Generating Wandering Subspaces for Doubly Commuting Covariant Representations

We obtain a Halmos-Richter-type wandering subspace theorem for covariant representations of C*-correspondences. Further the notion of Cauchy dual and a version of Shimorin's Wold-type decomposition for covariant representations of C*-correspondences is explored and as an application a wandering subspace theorem for doubly commuting covariant representations is derived. Using this wandering subspace theorem generating wandering subspaces are characterized for covariant representations of product systems in terms of the doubly commutativity condition.

A Representation Theorem for Change through Composition of Activities

The expanding use of information systems in industrial and commercial settings has increased the need for interoperation between software systems. In particular, many social, industrial, and business information systems require a common basis for a seamless exchange of complex process information. This is, however, inhibited, because different systems may use distinct terminologies or assume different meanings for the same terms. A common solution to this problem is to develop logical theories that act as an intermediate language between different parties. In this article, we characterize a class of activities that can act as intermediate languages between different parties in those cases. We show that for each domain with finite number of elements there exists a class of activities, we called canonical activities, such that all possible changes within the domain can be represented as a sequence of occurrences of those activities. We use an algebraic structure for representing change and characterizing canonical activities, which enables us to abstract away domain-dependent properties of processes and activities, and demonstrate general properties of formalisms required for semantic integration of dynamic information systems.

Representation theorem of set valued regular martingale: Application to the convergence of set valued martingale

In this paper, we prove a new theorem concerning a representation of set valued regular martingales, the proof is based on martingale selectors approach. As applications, various convergence results of set valued martingales are provided.

Constitutive modeling of anisotropic plasticity with application to fiber-reinforced composites

This article proposes two constitutive models to describe the nonlinear, elastic-plastic behavior of unidirectional fiber-reinforced composites at the meso-level. Key ingredients are the formulation of two anisotropic yield functions with the aid of representation theorems (invariant formulation), and the set-up of governing equations for plastic variables with respect to associative and non-associative flow response. The governing equations are solved using an elastic predictor-plastic corrector algorithm which imposes the constraint posed by the yield condition. Both the models are evaluated qualitatively and quantitatively by comparison to micromechanics simulations as well as to experimental data.

Optimal Incentive Contract for Sales Team with Loss Aversion Preference

When manufacturing enterprises employ sales team (or multiple salesmen) to sell products, there is asymmetric information such as the ability and efforts salesmen. Enterprises can use contracts to incentivize salesmen to work hard to maximize their profits. Assuming that market demand is sensitive to effort, and the salesman can exploit the market by increasing effort, a multi-agent model is established for the case of symmetrical information and asymmetrical information, in which the sales team has a loss aversion preference. In this multi-agent model, the agents’ utility function is non-concave and cannot be solved by traditional methods. We use a backward stochastic differential equation (BSDE) to represent agents’ contract through the martingale representation theorem and use the stochastic optimal control and matrix method to obtain the explicit solution of the optimal contract. Based on the conclusions of the research, an empirical analysis is made on the sales team of an enterprise.

Representation theorems of ℝ-trees and Brownian motions indexed by ℝ-trees

We provide a new representation of an [Formula: see text]-tree by using a special set of metric rays. We have captured the four-point condition from these metric rays and shown an equivalence between the [Formula: see text]-trees with radial and river metrics, and these sets of metric rays. In stochastic analysis, these graphical representation theorems are of particular interest in identifying Brownian motions indexed by [Formula: see text]-trees.

Local representation and construction of Beltrami fields II.solenoidal Beltrami fields and ideal MHD equilibria

Object of the present paper is the local theory of solution for steady ideal Euler flows and ideal MHD equilibria. The present analysis relies on the Lie-Darboux theorem of differential geometry and the local theory of representation and construction of Beltrami fields developed in [1]. A theorem for the construction of harmonic orthogonal coordinates is proved. Using such coordinates families of solenoidal Beltrami fields with different topologies are obtained in analytic form. Existence of global solenoidal Beltrami fields satisfying prescribed boundary conditions while preserving the local representation is considered. It is shown that only singular solutions are admissible, an explicit example is given in a spherical domain, and a theorem on existence of singular solutions is proven. Local conditions for existence of solutions, and local representation theorems are derived for generalized Beltrami fields, ideal MHD equilibria, and general steady ideal Euler flows. The theory is applied to construct analytic examples.

Matrix operators on function-valued function spaces

We study spaces of continuous-function-valued functions that have the property that composition with evaluation functionals induce weak$^{^*}$ to norm continuous maps to $l^p $ space $ (p\in (1, \infty))$. Versions of H\older's inequality and Riesz representation theorem are proved to hold on these spaces. We prove a version of Dixmier's theorem for spaces of function-valued matrix operators on these spaces, and an analogue of the trace formula for operators on Hilbert spaces. When the function space is taken to be the complex field, the spaces are just the $l^p$ spaces and the well-known classical theorems follow from our results.

Sesquilinear forms associated to sequences on Hilbert spaces

The possibility of defining sesquilinear forms starting from one or two sequences of elements of a Hilbert space is investigated. One can associate operators to these forms and in particular look for conditions to apply representation theorems of sesquilinear forms, such as Kato’s theorems. The associated operators correspond to classical frame operators or weakly-defined multipliers in the bounded context. In general some properties of them, such as the invertibility and the resolvent set, are related to properties of the sesquilinear forms. As an upshot of this approach new features of sequences (or pairs of sequences) which are semi-frames (or reproducing pairs) are obtained.