Representation Theorem Research Articles

Revisiting the representation theorem of finite distributive lattices with principal congruences. A \emph{Proof-by-Picture} approach

Revisiting the representation theorem of finite distributive lattices with principal congruences. A \emph{Proof-by-Picture} approach

Sklar’s theorem, copula products, and ordering results in factor models

Abstract We consider a completely specified factor model for a risk vector X = (X 1, . . ., Xd ), where the joint distributions of the components of X with a risk factor Z and the conditional distributions of X given Z are specified. We extend the notion of *-product of d-copulas as introduced for d = 2 and continuous factor distribution in Darsow et al. [6] and Durante et al. [8] to the multivariate and discontinuous case. We give a Sklar-type representation theorem for factor models showing that these *-products determine the copula of a completely specified factor model. We investigate in detail approximation, transformation, and ordering properties of *-products and, based on them, derive general orthant ordering results for completely specified factor models in dependence on their specifications. The paper generalizes previously known ordering results for the worst case partially specified risk factor models to some general classes of positive or negative dependent risk factor models. In particular, it develops some tools to derive sharp worst case dependence bounds in subclasses of completely specified factor models.

Nonconvex noncash risk measures

In 2018, Cong and Zhao introduced noncash risk measures with 1-norm for weak cone-type acceptable sets. In the present paper, we take a further look at nonconvex, noncash risk measures with p-norm (1 ≤ p ≤ ∞) for nonweak cone-type acceptable sets. A convex extension of the nonconvex, noncash risk measures is given and a representation theorem of a convex extension of the nonconvex, noncash risk measures is achieved.

Characterization of the w-Tempered ultradistributions

We use apreviously obtained characterization of test functions of w-Tempered Ultradistributions to charcterize the space w-Tempered Ultradistributions using Riesz representation Theorem.

Random Evolving Lotteries and Intrinsic Preference for Information

We introduce random evolving lotteries to study preference for non‐instrumental information. Each period, the agent enjoys a flow payoff from holding a lottery that will resolve at the terminal date. We provide a representation theorem for non‐separable risk consumption preferences and use it to characterize agents' attitude to non‐instrumental information. To address applications, we characterize peak‐trough utilities that aggregate trajectories of flow utilities linearly but, in addition, put weight on the best (peak) and worst (trough) lotteries along each path. We show that the model is consistent with recent experimental evidence on attitudes to information, including a preference for gradual arrival of good news and the ostrich effect, that is, decision makers' tendency to prefer information after good news to information after bad news.

The Fermionic-bosonic Representations of <I>A</I>(<I>M</I>-1,<I>N</I>-1)-graded Lie Superalgebras

Lie groups, Lie algebras and their representation theories are important parts of mathematical physics. They play a crucial character in symmetries. As a generalization of Lie algebra, Lie superalgebras are from the comprehending and description for supersymmetry of mathematical physics. Unlike the semisimple Lie algebras, understanding the representation theory of Lie superalgebras is a difficult problem. Lie superalgebras graded by root supersystems are Lie superalgebras of great significance. In recent years, the representations of types <I>B</I>(<i>m,n</i>)<i>, C</i>(<i>n</i>)<i>, D</i>(<i>m,n</i>)<i>, P</i>(<i>n</i>) and <I>Q</I>(<i>n</i>)-graded Lie superalgebras coordinatized by quantum tori have been studied. In this paper, we construct fermionic-bosonic representations for a class of <I>A</I>(<I>M</I>-1,<I>N</I>-1)-graded Lie superalgebras coordinatized by quantum tori with nontrivial central extensions. At first, we introduce the background of the research on the graded Lie superalgebras and present some basics on it. Then, a set of bases for <I>A</I>(<I>M</I>-1,<I>N</I>-1)-graded Lie superalgebras and the multiplication operations among them are given specifically to present the construction of the vector space. By using the tensor product of fermionic and bosonic module, the operators and their operation relations are derived. Finally, we obtain a brief and pretty representation theorem of <I>A</I>(<I>M</I>-1,<I>N</I>-1)-graded Lie superalgebras with nontrivial central extensions.

A duality for two-sorted lattices

A series of representation theorems (some of which discovered very recently) present an alternative view of many classes of algebras related to non-classical logics (e.g. bilattices, semi-De Morgan, Nelson and quasi-Nelson algebras) as two-sorted algebras in the sense of many-sorted universal algebra. In all the above-mentioned examples, we are in fact dealing with a pair of lattices related by two meet-preserving maps. We use this insight to develop a Priestley-style duality for such structures, mainly building on the duality for meet-semilattices of G. Bezhanishvili and R. Jansana. Our approach simplifies all the existing dualities for these algebras and is applicable more generally; in particular, we show how it specialises to the class of quasi-Nelson algebras, which has not yet been studied from a duality point of view.

A Practical Guide to Updating Beliefs From Contradictory Evidence

We often make high stakes choices based on complex information that we have no way to verify. Careful Bayesian reasoning—assessing every reason why a claim could be false or misleading—is not feasible, so we necessarily act on faith: we trust certain sources and treat claims as if they were direct observations of payoff relevant events. This creates a challenge when trusted sources conflict: Practically speaking, is there a principled way to update beliefs in response to contradictory claims? I propose a model of belief formation along with several updating axioms. An impossibility theorem shows there is no obvious best answer, while a representation theorem delineates the boundary of what is possible.

On a certain compactification of an arbitrary subset of ℝm and its applications to DiPerna-Majda measures theory

We present a constructive proof of the fact, that for any subset 𝒜 ⊆ ℝm and a countable family ℱ of bounded functions f : 𝒜 → ℝ there exists a compactification 𝒜′ ⊂ ℓ2 of 𝒜 such that every function f ∈ ℱ possesses a continuous extension to a function f̅ : 𝒜′→ℝ. However related to some classical theorems, our result is direct and hence applicable in Calculus of Variations. Our construction is then used to represent limits of weakly convergent sequences {f(uν)} via methods related to DiPerna-Majda measures. In particular, as our main application, we generalise the Representation Theorem from the Calculus of Variations due to Kałamajska.

Three-Dimensional Generalized Discrete Fuzzy Number and Applications in Color Mathematical Morphology

In this paper, the definition of three-dimensional generalized discrete fuzzy number (3-GDFN) is introduced based on the representation theorem of one-dimensional discrete fuzzy number and the similarity measure definition of two 3-GDFNs is given. Then the concept above mentioned is applied to color image representation and color mathematical morphology (CMM) in RGB space. The basic morphology operators, erosion and dilation, are extended to the CMM by defining the total preorder relation based on similarity measure between two 3-GDFNs instead of general vector sorting methods. The corresponding structuring elements in CMM are variable. Finally, the effectiveness and potential of the theoretical results are verified by comparative experiments. The proposed CMM operators are efficiently used in color image processing.

Ornstein-Uhlenbeck Semigroup on the dual space of Gelfand-Shilov Spaces of Beurling type

We use a previously obtained topological characterization of Gelfand-Shilov spaces of Beurling type to characterize its dual using Riesz representation theorem. Using the characterization of the dual space equipped with the weak topology, we study the action of Ornstein-Uhlenbeck Semigroup on the dual space.

Three classes of decomposable distributions

Abstract In this work, we refine the results of Sendov and Shan [New representation theorems for completely monotone and Bernstein functions with convexity properties on their measures, J. Theor. Probab. 28 (2015), 1689–1725] on subordinators obtained by the class of Bernstein functions stable by the Mellin-Euler differential operator I − x d d x I-x\tfrac{\text{d}}{\text{d}x} by giving a stochastic interpretation, proving monotonicity properties of the related distributions and providing additional extensions.

SOME NOTES ABOUT THE MARTINGALE REPRESENTATION THEOREM AND THEIR APPLICATIONS

An important theorem in stochastic finance field is the martingale representation theorem. It is useful in the stage of making hedging strategies (such as cross hedging and replicating hedge) in the presence of different assets with different stochastic dynamics models. In the current paper, some new theoretical results about this theorem including derivation of serial correlation function of a martingale process and its conditional expectations approximation are proposed. Applications in optimal hedge ratio and financial derivative pricing are presented and sensitivity analyses are studied. Throughout theoretical results, simulation-based results are also proposed. Two real data sets are analyzed and concluding remarks are given. Finally, a conclusion section is given.

The Class Equation and the Commutativity Degree for Complete Hypergroups

The aim of this paper is to extend, from group theory to hypergroup theory, the class equation and the concept of commutativity degree. Both of them are studied in depth for complete hypergroups because we want to stress the similarities and the differences with respect to group theory, and the representation theorem of complete hypergroups helps us in this direction. We also find conditions under which the commutativity degree can be expressed by using the class equation.

A remark on a trace Paley–Wiener theorem

In this paper we prove a version of a trace Paley--Wiener theorem for tempered representations of a reductive $p$--adic group. This is applied to complete certain investigation of Shahidi on the proof that a Plancherel measure is invariant of a $L$--packet of discrete series.

A Thermodynamically Consistent Formulation for Dynamic Response of Thermoviscoelastic Plate/Shell Based on Classical Continuum Mechanics (CCM)

This paper presents a thermodynamically consistent and kinematic assumption free formulation for dynamics of thermoviscoelastic plates/shells based on the conservation and balance laws of classical continuum mechanics (CCM) in which dissipation mechanism has been incorporated through ordered rate constitutive theory for deviatoric stress tensor. In this paper, we consider small deformation, small strain. The conservation and balance laws of CCM in [Formula: see text] in Lagrangian description using Cauchy stress tensor ([Formula: see text]) and linearized Green strain tensor ([Formula: see text]) constitute the mathematical model. The constitutive theory for the deviatoric Cauchy stress ([Formula: see text]) is derived using conjugate pairs in entropy inequality in conjunction with representation theorem. The argument tensors of [Formula: see text] are [Formula: see text] and rates of [Formula: see text] up to order [Formula: see text]. This yields a constitutive theory with dissipation mechanism based on rates of strain up to order [Formula: see text]. Constitutive theory for heat vector is also derived using the conjugate pairs in the entropy inequality and representation theorem. Finite element method is used to obtain solutions of the initial value problems descried by the balance of linear momenta (BLM), energy equation and the constitutive theories. The shell element geometry is described by the middle surface and the nodal vectors at the middle surface defining bottom and top surfaces of the element. The local approximation for the displacement field is [Formula: see text] - version hierarchical in the plane of the element as well as in the transverse direction. A space-time decoupled finite element formulation using Galerkin Method with Weak Form (GM/WF) in space is constructed for BLM as well as energy equation, both resulting in ordinary differential equations (ODEs) in time. The ordinary differential equations (ODEs) in time resulting from the finite element formulation of BLM are used to study: (i) natural undamped modes of vibration (ii) the transient dynamic response using the ODEs in time recast in modal basis: (a) using Rayleigh damping (b) using the ordered rate damping proposed in this paper. Time response is calculated using modal damping based on Rayleigh damping as well as using the proposed ordered rate damping mechanism. Model problem studies are presented to demonstrate: (1) accuracy of the natural frequencies obtained from the present formulation for thin and thick plates/shells (in which shear deformation is significant) and the results are compared with the currently used plate formulations (2) accuracy of damped transient response using proposed damping mechanism is compared with time response using Rayleigh damping (3) it is shown that Rayleigh damping has no physical basis and leads to spurious stationary states. The proposed damping yields accurate stationary states that are in exact agreement with the solution of corresponding BVP. A single formulation presented in this paper remains valid and accurate for very thin as well as very thick plates/shells and correctly simulates 3D state of deformation regardless of plate/shell thickness and is free of shear locking problems as well as need for shear corrections. When obtaining the time response, solution for an increment of time alternates between the solution of BLM followed by the solution of the energy equation. Details are presented in the paper.

Maxmin with multiple tastes and beliefs

This paper presents the main results of my doctoral thesis written in 2016 as part of the requirements to obtain the Doctor degree in the Doctoral Program in Economics at the University of Brasilia (UnB). It regards the axiomatization of a model of complete preferences under uncertainty that allows for a multiplicity of tastes and beliefs. As in Gilboa and Schmeidler (1989), the analysis is made in an Anscombe-Aumann setup and each act is evaluated by the worst-case scenario. The difference is that while in Gilboa and Schmeidler (1989) the multiple scenarios are composed of a set of priors, here they are composed of a set of probability-utility pairs. Thus, after imposing some postulates on preferences, we come to our main result, described in the form of a representation theorem (Theorem 1). A special case in which the multiple scenarios are a consequence of the multiplicity of tastes only is also axiomatized (Theorem 2).

Monadic classes of quantum B-algebras

The aim of this paper is to define the monadic quantum B-algebras and to investigate their properties. If the monadic operators are isotone, we show that they form a residuated pair. Special properties are studied for the particular case of monadic quantum B-algebras with pseudo-product, and a representation theorem for monadic quantum B-algebras with pseudo-product is proved. The monadic filters of monadic quantum B-algebras are defined, and their properties are studied. We prove that there is an isomorphism between the lattice of all filters of a monadic quantum B-algebra and the lattice of all filters of its subalgebra of fixed elements. Monadic operators on unital quantales are introduced, and the functional monadic quantale is constructed.

Representation of non-semibounded quadratic forms and orthogonal additivity

A representation theorem for non-semibounded Hermitian quadratic forms in terms of a (non-semibounded) self-adjoint operator is proven. The main assumptions are closability of the Hermitian quadratic form, the direct integral structure of the underlying Hilbert space and orthogonal additivity. We apply this result to several examples, including the position operator in quantum mechanics and quadratic forms invariant under a unitary representation of a separable locally compact group. The case of invariance under a compact group is also discussed in detail.

Some Results for the Two-Sided Quaternionic Gabor Fourier Transform and Quaternionic Gabor Frame Operator

In this paper, we first present some properties of the two-sided quaternionic Gabor Fourier transform (QGFT) on quaternion valued function space $$L^2({\mathbb {R}}^2,{\mathbb {H}})$$ , such as Parseval’s formula and the characterization of the range of the two-sided QGFT. Then, we give the definitions of quaternionic Wiener space and quaternionic Gabor frame operator (QGFO), which are the generalizations in the quaternionic settings. Finally, we prove Walnut’s and Janssen’s representation theorems and other boundedness results of the QGFO.