Representation Theorem Research Articles

The density theorem for discrete series representations restricted to lattices

This article considers the relation between the spanning properties of lattice orbits of discrete series representations and the associated lattice co-volume. The focus is on the density theorem, which provides a trichotomy characterizing the existence of cyclic vectors and separating vectors, and frames and Riesz sequences. We provide an elementary exposition of the density theorem, that is based solely on basic tools from harmonic analysis, representation theory, and frame theory, and put the results into context by means of examples.

Quasi-linear functionals on locally compact spaces

This paper combines new and known results in a single convenient source for anyone interested in learning about quasi-linear functionals on locally compact spaces. We define singly generated subalgebras in different settings and study signed and positive quasi-linear functionals. Quasi-linear functionals are, in general, nonlinear, but linear on singly generated subalgebras. The paper gives representation theorems for quasi-linear functionals on C c (X), for bounded quasi-linear functionals on C 0 (X) on a locally compact space, and for quasi-linear functionals on C(X) on a compact space. There is an order-preserving bijection between quasi-linear functionals and compact-finite topological measures, which is also “isometric” when topological measures are finite. We present many properties of quasi-linear functionals and give an explicit example of a quasi-linear functional on ℝ 2 . Results of the paper will be helpful for further study and application of quasi-linear functionals in different areas of mathematics, including symplectic geometry.

Inner Product Groups and Riesz Representation Theorem

In this paper, we introduce an inner product on abelian groups and, after investigating the basic properties of the inner product, we first show that each inner product group is a torsion-free abelian normed group. We give examples of such groups and describe the norms induced by such inner products. Among other results, Hilbert groups, midconvex and orthogonal subgroups are presented, and a Riesz representation theorem on divisible Hilbert groups is proved.

Generalized Benders Decomposition Method to Solve Big Mixed-Integer Nonlinear Optimization Problems with Convex Objective and Constraints Functions

The paper presents the Generalized Benders Decomposition (GBD) method, which is now one of the basic approaches to solve big mixed-integer nonlinear optimization problems. It concentrates on the basic formulation with convex objectives and constraints functions. Apart from the classical projection and representation theorems, a unified formulation of the master problem with nonlinear and linear cuts will be given. For the latter case the most effective and, at the same time, easy to implement computational algorithms will be pointed out.

Derived Hom Spaces in Rigid Analytic Geometry

We construct a derived enhancement of Hom spaces between rigid analytic spaces. It encodes the hidden deformation-theoretic information of the underlying classical moduli space. The main tool in our construction is the representability theorem in derived analytic geometry, which has been established in our previous work. The representability theorem provides us with sufficient and necessary conditions for an analytic moduli functor to possess the structure of a derived analytic stack. In order to verify the conditions of the representability theorem, we prove several general results in the context of derived non-archimedean analytic geometry: derived Tate acyclicity, projection formula, and proper base change. These results also merit independent interest. Our main motivation comes from non-archimedean enumerative geometry. In our subsequent works, we will apply the derived mapping stacks to obtain non-archimedean analytic Gromov–Witten invariants.

Ideals of étale groupoid algebras and Exel’s Effros–Hahn conjecture

We extend to arbitrary commutative base rings a recent result of Demeneghi that every ideal of an ample groupoid algebra over a field is an intersection of kernels of induced representations from isotropy groups, with a much shorter proof, by using the author’s Disintegration Theorem for groupoid representations. We also prove that every primitive ideal is the kernel of an induced representation from an isotropy group; however, we are unable to show, in general, that it is the kernel of an irreducible induced representation. If each isotropy group is finite (e.g., if the groupoid is principal) and if the base ring is Artinian (e.g., a field), then we can show that every primitive ideal is the kernel of an irreducible representation induced from isotropy.

Spectral theory of (n, m)-normal operators on Hilbert space

In this work, we define the set which consists of (n, m)-normal operators. Also, the sets and and defined from . Firstly, the set is proved Banach algebra and establish consists of (n, m)-normal operators. Secondly, we give theorem of the Gelfand representation of the Banach algebra generated by a (n, m)-normal operator, also some theories are proven to reach a circular for the spectral theory. Finally, this study is established and proved generalization the spectral theory of (n, m)-normal operators.

Integral representation of unbounded variational functionals on Sobolev spaces

In this paper we establish an unbounded version of the integral representation theorem by Buttazzo and Dal Maso [see Buttazzo Dal Maso (Nonlinear Anal 9(6):515–532, 1985) and also Bouchitté et al. (Arch Ration Mech Anal 165(3), 187–242, 2002)]. More precisely, we prove an integral representation theorem (with a formula for the integrand) for functionals defined on $$W^{1,p}$$ with $$p>N$$ (N being the dimension) that do not satisfy a standard p-growth condition from above and can take infinite values. Applications to $$\Gamma $$ -convergence, relaxation and homogenization are also developed.

Exchangeable coalescents, ultrametric spaces, nested interval-partitions: A unifying approach

Kingman’s (1978) representation theorem (J. Lond. Math. Soc. (2) 18 (1978) 374–380) states that any exchangeable partition of ℕ can be represented as a paintbox based on a random mass-partition. Similarly, any exchangeable composition (i.e., ordered partition of ℕ) can be represented as a paintbox based on an interval-partition (Gnedin (1997) Ann. Probab. 25 (1997) 1437–1450). Our first main result is that any exchangeable coalescent process (not necessarily Markovian) can be represented as a paintbox based on a random nondecreasing process valued in interval-partitions, called nested interval-partition, generalizing the notion of comb metric space introduced in Lambert and Uribe Bravo (2017) (p-Adic Numbers Ultrametric Anal. Appl. 9 (2017) 22–38) to represent compact ultrametric spaces. As a special case, we show that any Λ-coalescent can be obtained from a paintbox based on a unique random nested interval partition called Λ-comb, which is Markovian with explicit transitions. This nested interval-partition directly relates to the flow of bridges of Bertoin and Le Gall (2003) (Probab. Theory Related Fields 126 (2003) 261–288). We also display a particularly simple description of the so-called evolving coalescent (Pfaffelhuber and Wakolbinger (2006) Stochastic Process. Appl. 116 (2006) 1836–1859) by a comb-valued Markov process. Next, we prove that any ultrametric measure space U, under mild measure-theoretic assumptions on U, is the leaf set of a tree composed of a separable subtree called the backbone, on which are grafted additional subtrees, which act as star-trees from the standpoint of sampling. Displaying this so-called weak isometry requires us to extend the Gromov-weak topology of Greven, Pfaffelhuber and Winter (2009) (Probab. Theory Related Fields 145 (2009) 285–322), that was initially designed for separable metric spaces, to nonseparable ultrametric spaces. It allows us to show that for any such ultrametric space U, there is a nested interval-partition which is (1) indistinguishable from U in the Gromov-weak topology; (2) weakly isometric to U if U has a complete backbone; (3) isometric to U if U is complete and separable.

XII—A New Method for Value Aggregation

Abstract Many axiological theories ground the goodness or badness of options in the aggregate of the goodness or the badness of these options for individuals. Most commonly, this works by summing (or averaging), and taking the expectation of this result if there is uncertainty. Such theories face problems dealing with infinite populations, for which sums or averages are infinite or undefined. They fetishize certain mathematical operations, in a subject that is not inherently mathematical. The fact that the sum is the target of maximization is said to mean that the theory ignores the separateness of persons. I propose an extension of my 2014 paper, ‘Decision Theory without Representation Theorems’. I illustrate how the resulting theory can account for a case involving an infinite population, dealing with the objections. I connect this to the theory of measurement, and explain why the mathematical operations are co-extensive with the result without grounding it. Because there is no sum in these infinite populations, it avoids the worry that aggregative theories treat individuals as secondary to some aggregate.

Well-posedness and large deviations for 2D stochastic constrained Navier-Stokes equations driven by Lévy noise in the Marcus canonical form

We consider stochastic two-dimensional constrained Navier-Stokes equations driven by Lévy noise in the Marcus canonical form. The aim of this work is two-fold. At first, we prove the existence of a martingale solution based on the construction relying on classical Faedo-Galerkin approximations, compactness method and the Jakubowski's version of Skorokhod representation theorem for non-metric spaces. We further prove that the martingale solution is pathwise unique and deduces the existence of a strong solution. In the second part of the paper, we establish a Wentzell-Freidlin type large deviations principle for the small noise asymptotic of solutions using weak convergence method.

Pro-Inflammatory Cytokines are Dysregulated in Depression and Correlated With Measures of Psychopathology

This paper presents a kinematic assumption free and thermodynamically consistent non-linear formulation incorporating finite strain and finite deformation for thermoviscoelastic plates/shells based on the conservation and balance laws of the classical continuum mechanics (CCM) in R3 (see Surana and Mathi, (2020) for linear theory). The conservation and balance laws in Lagrangian description with finite strain measure and the conjugate stress measure in R3 are considered. The conjugate pairs in the second law of thermodynamics (SLT), the consideration of additional physics and the principle of equipresence are utilized to determine the constitutive variables and their argument tensors. The constitutive theory for the contravariant second Piola–Kirchhoff stress tensor is derived using Green’s strain tensor and its convected time derivatives of up to order n as its argument tensors using representation theorem with complete basis (i.e. using integrity). The convected time derivatives of the Green’s strain tensor up to order n provide ordered rate constitutive theory for the dissipation mechanism that is naturally non-linear due to Green’s strain measure. The constitutive theory for heat vector derived using representation theorem and integrity is also a non-linear constitutive theory. Simplified linear forms of these constitutive theories are also presented. The solution methods for the mathematical model for the BVPs as well as the IVPs using p-version hierarchical higher degree and higher order finite element method are presented. Due to dissipation, the energy equation is integral part of the mathematical model that accounts for rate of mechanical work resulting in rate of entropy production, hence heat.The plate/shell geometry (flat or curved) is described by its middle surface containing the nodal vectors (at each of the nine nodes), the ends of which define bottom and top surfaces of the plate/shell (Surana and Mathi, 2020). The geometry is mapped into ξηζ natural coordinate space in a two unit cube. The p-version hierarchical local approximations in ξηζ are constructed to describe the deformation of the plate/shell middle surface as well as its faces that is controlled by p-levels in ξ,η and ζ directions (element local approximation). The formulation presented here is accurate for very thin as well as very thick plate/shells and for small as well as finite deformation and finite strains. Non-linear constitutive theories based on integrity allow more complex constitutive behavior in term of strains as well as strain rates, hence dissipation mechanism. Dissipation mechanism is described by an ordered rate theory based on SLT and additional physics. The formulation presented here always ensures thermodynamic equilibrium during the evolution as it is derived using the conservation and balance laws of CCM in R3. The formulation always preserves three dimensional nature of the deformation physics regardless of the plate/shell thickness and is free of locking problems and shear correction factors that plague most of the currently used plate/shell formulations.

A representation theorem for finite best–worst random utility models

This paper investigates the representation of best–worst choice probabilities (picking the best and the worst alternative from an offered set). It is shown that non-negativity of best–worst Block–Marschak polynomials is necessary and sufficient for the existence of a random utility representation. The representation theorem is obtained by extending proof techniques for a corresponding result on best choices (picking the best alternative from an offered set) developed by Falmagne (1978).

Relativistic Extended Thermodynamics of Polyatomic Gases with Rotational and Vibrational Modes

In a recent article an infinite set of balance equations has been proposed to modelize polyatomic gases with rotational and vibrational modes in the non-relativistic context. To obtain particular cases, it has been truncated to obtain a model with 7 or 15 moments. Here the following objectives are pursued: 1) to obtain the relativistic counterpart of this model which, at the non-relativistic limit, gives the same balance equations as in the known classical case; 2) to obtain the previous result for the model with an arbitrary but fixed number of moments, 3) to obtain the closure of the resulting relativistic model so that all the functions appearing in the balance equations are expressed in terms of the independent variables. To achieve these goals, the following methods are used: 1) The Entropy Principle is imposed. As a result is obtained that the closure is determined up to a single 4-vectorial function usually called 4-potential. 2) To determine this last function, a more restrictive principle is imposed, namely the Maximum Entropy Principle (MEP). 3) Since all the functions involved must be expressed in the covariant form, so as not to depend on the observer, the Representation Theorems are used. Findings of this article are exactly the goals outlined earlier. They are clearly novelty because they had never been achieved before. They can be considered also improvements because, if the aforementioned arbitrary number of moments is restricted to 16, the present work coincide with that already known in literature. Doi: 10.28991/HIJ-2021-02-03-04 Full Text: PDF

Octonionic Kerzman\u2013Stein Operators

In this paper we consider generalized Hardy spaces in the octonionic setting associated to arbitrary Lipschitz domains where the unit normal field exists almost everywhere. First we discuss some basic properties and explain structural differences to the associative Clifford analysis setting. The non-associativity requires special attention in the definition of an appropriate inner product and hence in the definition of a generalized Szegö projection. Whenever we want to apply classical theorems from reproducing kernel Hilbert spaces we first need to switch to the consideration of real-valued inner products where the Riesz representation theorem holds. Then we introduce a generalization of the dual Cauchy transform for octonionic monogenic functions which represents the adjoint transform with respect to the real-valued inner product langle cdot , cdot rangle _0 together with an associated octonionic Kerzman–Stein operator and related kernel functions. Also in the octonionic setting, the Kerzman–Stein operator that we introduce turns out to be a compact operator. A motivation behind this approach is to find an approximative method to compute the Szegö projection of octonionic monogenic functions offering a possibility to tackle BVP in the octonions without the explicit knowledge of the octonionic Szegö kernel which is extremely difficult to determine in general. We also discuss the particular cases of the octonionic unit ball and the half-space. Finally, we relate our octonionic Kerzman–Stein operator to the Hilbert transform and particularly to the Hilbert–Riesz transform in the half-space case.

Solving and learning nonlinear PDEs with Gaussian processes

We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), and for solving inverse problems (IPs) involving the identification of parameters in PDEs, using the framework of Gaussian processes. The proposed approach: (1) provides a natural generalization of collocation kernel methods to nonlinear PDEs and IPs; (2) has guaranteed convergence for a very general class of PDEs, and comes equipped with a path to compute error bounds for specific PDE approximations; (3) inherits the state-of-the-art computational complexity of linear solvers for dense kernel matrices. The main idea of our method is to approximate the solution of a given PDE as the maximum a posteriori (MAP) estimator of a Gaussian process conditioned on solving the PDE at a finite number of collocation points. Although this optimization problem is infinite-dimensional, it can be reduced to a finite-dimensional one by introducing additional variables corresponding to the values of the derivatives of the solution at collocation points; this generalizes the representer theorem arising in Gaussian process regression. The reduced optimization problem has the form of a quadratic objective function subject to nonlinear constraints; it is solved with a variant of the Gauss–Newton method. The resulting algorithm (a) can be interpreted as solving successive linearizations of the nonlinear PDE, and (b) in practice is found to converge in a small number of iterations (2 to 10), for a wide range of PDEs. Most traditional approaches to IPs interleave parameter updates with numerical solution of the PDE; our algorithm solves for both parameter and PDE solution simultaneously. Experiments on nonlinear elliptic PDEs, Burgers' equation, a regularized Eikonal equation, and an IP for permeability identification in Darcy flow illustrate the efficacy and scope of our framework.

Global well-posedness and long-time behavior of the fractional NLS

In this paper, our discussion mainly focuses on equations with energy supercritical nonlinearities. We establish probabilistic global well-posedness (GWP) results for the cubic Schrödinger equation with any fractional power of the Laplacian in all dimensions. We consider both low and high regularities in the radial setting, in dimension ge 2. In the high regularity result, an Inviscid - Infinite dimensional (IID) limit is employed while in the low regularity global well-posedness result, we make use of the Skorokhod representation theorem. The IID limit is presented in details as an independent approach that applies to a wide range of Hamiltonian PDEs. Moreover we discuss the adaptation to the periodic settings, in any dimension, for smooth regularities.

Certain Class of Analytic Functions with respect to Symmetric Points Defined by Q-Calculus

In this study, we familiarise a novel class of Janowski-type star-like functions of complex order with regard to j , k -symmetric points based on quantum calculus by subordinating with pedal-shaped regions. We found integral representation theorem and conditions for starlikeness. Furthermore, with regard to j , k -symmetric points, we successfully obtained the coefficient bounds for functions in the newly specified class. We also quantified few applications as special cases which are new (or known).

A complete representation theorem for nullnorms on bounded lattices with ample illustrations

We present a complete representation theorem for nullnorms on bounded lattices. Explicitly, any nullnorm on a bounded lattice can be represented in terms of two order-preserving maps, a triangular conorm, a triangular norm and a conditionally associative function. As a particular case, we retrieve the known representation of nullnorms only taking values that are comparable with the annihilator (in terms of two order-preserving maps, a triangular conorm and a triangular norm). As an illustration of the representation theorem, we generate construction methods for nullnorms, especially those taking at least one value that is incomparable with the annihilator.

Ternary operator categories

T*-categories are introduced as a ternary generalization of C*-categories. Their linking C*-categories are constructed and the Gelfand-Naimark representation theorems of Zettl for C*-ternary rings and for W*-ternary rings, are generalized to T*-categories. Biduals of C*-categories and of T*-categories are considered.