Representation Theorem Research Articles

Can a discrete dynamic model ever perfectly simulate a continuum?

This article elaborates on the seemingly impossible notion that a continuous elastic body subjected to dynamic sources on its outer surface could ever be substituted in all of its essential qualities by a discrete model accomplished with finite elements. This topic is taken up herein and discussed in the context of a very simple two-dimensional model involving the propagation of SH shear waves (or acoustic waves) in a homogeneous elastic half-space. It is shown that there exists at least one discrete solid, referred to here as the Guddati Solid, which from its external surface behaves exactly like the continuum and is able to transmit waves of any frequency and any wavelength. This is a rather surprising finding in that it seems to contradict some well-known elastodynamic representation theorems, not to mention falsify the widespread belief that a discrete system can never behave like the continuum it purports to model. The purpose of this article is thus to present one example which disproves this widely believed postulate.

Robust mean–variance hedging via [formula omitted]-expectation

In this paper we study mean–variance hedging under the G-expectation framework. Our analysis is carried out by exploiting the G-martingale representation theorem and the related probabilistic tools, in a continuous financial market with two assets, where the discounted risky one is modeled as a symmetric G-martingale. By tackling progressively larger classes of contingent claims, we are able to explicitly compute the optimal strategy under general assumptions on the form of the contingent claim.

Ordered rate constitutive theories for thermoviscoelastic solids without memory incorporating internal and Cosserat rotations

The work in this paper is based on a non-classical continuum theory in the Lagrangian description for thermoviscoelastic solids without memory in which the conservation and balance laws are derived by incorporating internal rotations ( $${}_i \pmb {\varvec{{\varTheta } }}$$ ) arising from the Jacobian of deformation ( $$ \pmb {\varvec{J }}$$ ), as well as Cosserat rotations ( $${}_e \pmb {\varvec{{\varTheta } }}$$ ) at a material point. Such non-classical solids have additional energy storage due to rotations and additional dissipation due to rotation rates compared to classical continuum theories. Rotations $${}_i \pmb {\varvec{{\varTheta } }}$$ are completely defined by $$ \pmb {\varvec{J }}$$ , whereas displacements $$ \pmb {\varvec{u }}$$ and Cosserat rotations $${}_e \pmb {\varvec{{\varTheta } }}$$ are degrees of freedom at each material point. When $${}_i \pmb {\varvec{{\varTheta } }}$$ and $${}_e \pmb {\varvec{{\varTheta } }}$$ are resisted by the deforming matter, conjugate moments arise, which together with ( $${}_i \pmb {\varvec{{\varTheta } }},{}_e \pmb {\varvec{{\varTheta } }}$$ ) and ( $${}_i\overset{\,\text{. }}{ \pmb {\varvec{{\varTheta } }}},{}_e\overset{\,\text{. }}{ \pmb {\varvec{{\varTheta } }}}$$ ) result in additional work and rate of work. This paper utilizes thermodynamic framework for non-classical solids derived based on internal as well as Cosserat rotations and presents a thermodynamically consistent derivation of the constitutive theories that incorporate the aforementioned deformation physics. The constitutive theories are derived using the conditions resulting from the entropy inequality in conjunction with the representation theorem.

Quantification of Lagrangian travel time statistics under nonstationary random groundwater flow conditions

AbstractThis study presents a Lagrangian description of advection dominated field‐scale transport of inert solutes, namely, the statistics of advective solute travel time and trajectory, in nonstationary random groundwater flow fields. The vertical flow field is generated by the effects of changes in total stress applied in fluid‐saturated heterogeneous deformable porous media. The Lagrangian statistics are developed directly from the velocity statistics based on the application of stochastic methodology such as the representation theorem and Fourier–Stieltjes integral representation. The focus of our numerical evaluation is placed on the influence of the soil compressibility on the Lagrangian statistics. The prediction on the statistics of solute travel time and trajectory may serve as the basic input to the environmental risk assessment.

A note on starlike functions associated with symmetric points

The objective of the present paper is to study results that are defined using the notions of generalization of Janowski classes and k-symmetrical functions. A representation theorem, coefficients inequality, distortion properties and the result on radius of starlikeness are discussed.

A Role for Representation Theorems†

A Role for Representation Theorems†

On stably trivial spin torsors over low-dimensional schemes

The paper discusses stably trivial torsors for spin and orthogonal groups over smooth affine schemes over infinite perfect fields of characteristic unequal to 2. We give a complete description of all the invariants relevant for the classification of such objects over schemes of dimension at most $3$, along with many examples. The results are based on the $\mathbb{A}^1$-representability theorem for torsors and transfer of known computations of $\mathbb{A}^1$-homotopy sheaves along the sporadic isomorphisms to spin groups.

Belief consistency and invariant risk preferences

Karni and Vierø (2013) introduce a novel theory characterizing the evolution of decision makers’ beliefs in response to growing awareness of new conceivable contingencies. As awareness grows, the original subjective expected utility (SEU) preferences need to be reformulated. Under two key axioms linking the original and new preferences, Invariant Risk Preferences and Consistency, the authors derive representation theorems and updating rules for beliefs referred to as reverse Bayesianism. We prove that their Consistency axiom together with the SEU axioms implies Invariant Risk Preferences. In other words, we show that the theory of reverse Bayesianism can be behaviorally underpinned without directly imposing Invariant Risk Preferences.

Asymmetric norms given by symmetrisation and specialisation order

In this paper we continue the investigations of the relationship between T0-quasi-metric spaces and partially ordered metric spaces. Among other things, we establish an equivalence of categories between so-called maximal T0-quasi-metric spaces and partially ordered metric spaces produced by T0-quasi-metrics. In the linear context we give geometric interpretations of the obtained results. In particular, we also derive a representation theorem for injective asymmetrically normed spaces.

On some subfamilies of holomorphic mappings associated with symmetrical points

In this paper, we define two families of analytic functions using the concept of the starlikeness with respect to symmetrical points, Janowski functions and functions of bounded boundary rotation. We study here integral representation theorem, coefficient bounds, coefficient difference and arc-length problem of the newly defined class.

Relation algebras and groups

Generalizing results of Jonsson and Tarski, Maddux introduced the notion of a pair-dense relation algebra and proved that every pair-dense relation algebra is representable. The notion of a pair below the identity element is readily definable within the equational framework of relation algebras. The notion of a triple, a quadruple, or more generally, an element of size (or measure) $$n>2$$ is not definable within this framework, and therefore it seems at first glance that Maddux’s theorem cannot be generalized. It turns out, however, that a very far-reaching generalization of Maddux’s result is possible if one is willing to go outside of the equational framework of relation algebras, and work instead within the framework of the first-order theory. Moreover, this generalization sheds a great deal of light not only on Maddux’s theorem, but on the earlier results of Jonsson and Tarski. In the present paper, we define the notion of an atom below the identity element in a relation algebra having measure n for an arbitrary cardinal number $$n>0$$ , and we define a relation algebra to be measurable if it’s identity element is the sum of atoms each of which has some (finite or infinite) measure. The main purpose of the present paper is to construct a large class of new examples of group relation algebras using systems of groups and corresponding systems of quotient isomorphisms (instead of the classic example of using a single group and forming its complex algebra), and to prove that each of these algebras is an example of a measurable set relation algebra. In a subsequent paper, the class of examples will be greatly expanded by adding a third ingredient to the mix, namely systems of “shifting” cosets. The expanded class of examples—called coset relation algebras—will be large enough to prove a representation theorem saying that every atomic, measurable relation algebra is essentially isomorphic to a coset relation algebra.

On generalized fractional cosine and sine transforms

Abstract In this paper, we define and investigate fractional cosine and sine transforms for functions as well as distributions. For these transforms, several properties are provided, the representation theorems are proved and the analyticity is established.

A new representation theorem of L-subsets

In the paper, the implication and tensor operations of strong L-nested systems are found, and further we show that the family of all strong L-nested systems on a universal set forms a complete Girard residuated lattice provided the underlying complete residuated lattice is Girard. Finally, an isomorphism of complete Girard residuated lattices between the family of all strong L-nested systems and the L-power set for any fixed universal set is established, and thus a new representation theorem of L-subsets is obtained.

Weak rationalizability and Arrovian impossibility theorems for responsive social choice

This paper provides representation theorems for choice functions satisfying weak rationality conditions: a choice function satisfies $$\alpha$$ if and only if it can be expressed as the union of intersections of maximal sets of a fixed collection of acyclic relations, and a choice function satisfies $$\gamma$$ if and only if it consists of the maximal elements of a relation that can depend on the feasible set in a particular, well-behaved way. Other rationality conditions are investigated, and these results are applied to deduce impossibility theorems for social choice functions satisfying weak rationality conditions along with positive responsiveness conditions. For example, under standard assumptions on the set of alternatives and domain of preferences, if a social choice function satisfies Pareto optimality, independence of irrelevant alternatives, a positive responsiveness condition for revealed social preferences, and a new rationality condition $$\delta ^{*}$$ (a strengthening of $$\gamma$$ ), then some individual must have near dictatorial power.

Complete Axiomatization for the Total Variation Distance of Markov Chains

We propose a complete axiomatization for the total variation distance of finite labelled Markov chains. Our axiomatization is given in the form of a quantitative deduction system, a framework recently proposed by Mardare, Panangaden, and Plotkin (LICS 2016) to extend classical equational deduction systems by means of inferences of equality relations t≡εs indexed by rationals, expressing that “t is approximately equal to s up to an error ε”. Notably, the quantitative equational system is obtained by extending our previous axiomatization (CONCUR 2016) for the probabilistic bisimilarity distance with a distributivity axiom for the prefix operator over the probabilistic choice inspired by Rabinovich's (MFPS 1983). Finally, we propose a metric extension to the Kleene-style representation theorem for finite labelled Markov chains w.r.t. trace equivalence due to Silva and Sokolova (MFPS 2011).

Operator ∗‐correspondences in analysis and geometry

An operator *-algebra is a non-selfadjoint operator algebra with completely isometric involution. We show that any operator *-algebra admits a faithful representation on a Hilbert space in such a way that the involution coincides with the operator adjoint up to conjugation by a symmetry. We introduce operator *-correspondences as a general class of inner product modules over operator *-algebras and prove a similar representation theorem for them. From this we derive the existence of linking operator *-algebras for operator *-correspondences. We illustrate the relevance of this class of inner product modules by providing numerous examples arising from noncommutative geometry.

Fatou closedness under model uncertainty

We provide a characterization in terms of Fatou closedness for weakly closed monotone convex sets in the space of $${\mathcal P}$$ -quasisure bounded random variables, where $${\mathcal P}$$ is a (possibly non-dominated) class of probability measures. Applications of our results lie within robust versions the Fundamental Theorem of Asset Pricing or dual representation of convex risk measures.

Dynamic consistency of expected utility under non-classical (quantum) uncertainty

Quantum cognition in decision making is a recent and rapidly growing field. In this paper, we develop an expected utility theory in a context of non-classical (quantum) uncertainty. We replace the classical state space with a Hilbert space which allows introducing the concept of quantum lottery. Within that framework, we formulate axioms on preferences over quantum lotteries to establish a representation theorem. We show that demanding the consistency of choice behavior conditional on new information is equivalent to the von Neumann–Luders postulate applied to beliefs. A dynamically consistent quantum-like agent may violate dynamic recursive consistency, however. This feature suggests interesting applications in behavioral economics as we illustrate in an example of persuasion.

An average queue‐length‐difference–based congestion detection algorithm in TCP/AQM network

SummaryCongestion detection in transmission control protocol/active queue management networks remains a challenging problem in which the choosing of congestion signal is one of the most important factors. Exponentially weighted moving average of queue length, the most widely used congestion signal, is facing difficulties in detecting incipient congestion and quantifying the optimal forgetting factor. Aiming at these 2 disadvantages, we propose an average queue‐length‐difference–based congestion detection algorithm where exponentially weighted moving average of queue‐length difference is chosen as the congestion signal with the theoretical optimal forgetting factor deduced. First, by defining the queue‐length difference as the state, the corresponding state equation is derived from the fluid model. Second, we prove that the inflow traffic in state equation is a discrete‐time martingale, which can be transformed to a Wiener process according to the martingale representation theorem. Noticing that the observation of state will be coupled with noise because of the unstable transmission, the state estimation is then derived with the application of recursive least squares filter. The filter gain of state estimation, which is a function of the noise‐signal ratio, corresponds to the optimal forgetting factor in average queue‐length‐difference–based congestion detection algorithm. Simulation results in NS‐3 and MATLAB illustrate the effectiveness of the proposed algorithm.

Ordered Rate Constitutive Theories for Non-Classical Thermoviscoelastic Fluids Incorporating Internal and Cosserat Rotation Rates

This paper considers conservation and balance laws for non-classical fluent continua in the presence of internal rotation rates due to the velocity gradient tensor and the rotation rates due to Cosserat rotations. In these balance laws, the internal rotation rates are completely defined as functions of the velocity gradient tensor, but the Cosserat rotation rates are additional three degrees of freedom at each material point. When these rotation rates are resisted by the deforming continua, conjugate moments are created. For thermoviscoelastic fluent continua, these result in additional dissipation mechanism as well as rheology. This paper presents a thermodynamically consistent derivation of constitutive theories for such fluids based on the entropy inequality in conjunction with the representation theorem using integrity, i.e., complete basis. Material coefficients are derived and discussed. The constitutive theories are presented in the absence as well as presence of the balance of moments of moments as a balance law and are compared with the classical continuum theories. Retardation moduli corresponding to the Cauchy stress tensor, both symmetric and antisymmetric, as well as the symmetric and antisymmetric Cauchy moment tensors are derived. The constitutive theories presented in this paper are ordered rate theories. These incorporate convected time derivatives of the strain tensor up to order [Formula: see text] and the convected time derivatives of the stress and moment tensors up to certain orders. Simplified forms of the constitutive theories are also derived. It is shown and concluded that the constitutive theories for thermoviscoelastic fluids based on classical continuum theory such as Maxwell model, Oldroyd-B model, Giesekus model, etc., are all a subset of the constitutive theories presented in this paper for non-classical physics. Both compressible and incompressible thermoviscoelastic fluids are considered.