Representation Theorem Research Articles

Kreĭn space unitary dilations of Hilbert space holomorphic semigroups

The infinitesimal generator A of a strongly continuous semigroup on a Hilbert space is assumed to satisfy that Bβ := A−β is a sectorial operator of angle less than π 2 for some β ≥ 0. If Bβ is dissipative in some equivalent scalar product then the Naimark–Arocena representation theorem is applied to obtain a Krein space unitary dilation of the semigroup.

BSDEs driven by normal martingale

The aim of this paper is to study backward stochastic differential equation driven by a class of normal martingale. We prove the existence and uniqueness results of the solution under the stochastic Lipschitz condition by mean the martingale representation theorem. In addition, we show that the solution of these equations provides a viscosity solution of the associated system with partial differential equations.

Curvature, a mechanical link between the geometrical complexities of a fault: application to bends, kinks and rough faults

SUMMARY Many recent studies have tried to determine the influence of geometry of faults in earthquake mechanics. However, it still remains largely unknown, and it is explored mainly with numerical models. In this paper, we will try to understand how exactly does the geometry come into play in the mechanics of an earthquake from analytical perspective. We suggest a new interpretation of the effect of geometry on the stress on a fault, where the curvatures of the fault that multiply the slip play a major role. Starting from the representation theorem, which links the displacement in a medium to the slip distribution on its boundary, and assuming a 3-D, homogeneous, infinite medium, a regularized boundary-element equation can be obtained. Using this equation, it is possible to separate the influence of geometry, as expressed by the curvatures and torsions of the field lines of slip on the fault surface, which multiply the slip, from the effect of the gradient of slip. This allows us to shed new light on the mechanical effects of geometrical complexities on the fault surface, with the key parameters being the curvatures and torsions of the slip field lines. We have used this new approach to show that, in 2-D static in-plane problems, the shear traction along the fault is mainly controlled by the gradient of slip along the fault, while the normal traction is mainly controlled by the slip that multiplies the curvature along the fault. Finally, we applied this new approach to re-interpret the effect of roughness (why there is a need for a minimum lengthscale in linear elasticity, how to study mechanically the difference of roughness measurements with the direction of slip, scaling of slip distribution versus geometry), bends and kinks (what is the difference between the two, are they sometimes equivalent), as well as to explain further the false paradox between smooth-and-abrupt-bends. This unified framework allows us to improve greatly our understanding of the effect of fault geometry on the mechanics of earthquakes.

On the representation of local indistinguishability operators

This paper studies local indistinguishability operators, i.e., symmetric and transitive fuzzy relations that do not need to be reflexive. This is an important generalization of global indistinguishability relations (fuzzy relations satisfying the reflexivity property in addition) because there are interesting families of fuzzy relations that are non-reflexive. One case are decomposable relations, that are generated by a fuzzy subset and contains the t-norms as an important subfamily. Also the relations associated naturally to fuzzy subgroups are local indistinguishability operators. In this paper these relations will be studied stressing the way they can be generated. A representation theorem will be proved and related to the concepts of extensionality and of fuzzy rough set. Decomposable local indistinguishability operators will also be studied and related with one-dimensional ones in the sense of the previous representation theorem. The presence of these relations in the study of fuzzy subgroups will also be analyzed.

Decomposing Dynamic Risks into Risk Components

The decomposition of dynamic risks a company faces into components associated with various sources of risk, such as financial risks, aggregate economic risks, or industry-specific risk drivers, is of significant relevance in view of risk management and product design, particularly in (life) insurance. Nevertheless, although several decomposition approaches have been proposed, no systematic analysis is available. This paper closes this gap in literature by introducing properties for meaningful risk decompositions and demonstrating that proposed approaches violate at least one of these properties. As an alternative, we propose a novel martingale representation theorem (MRT) decomposition that relies on martingale representation and show that it satisfies all of the properties. We discuss its calculation and present detailed examples illustrating its applicability. This paper was accepted by Baris Ata, stochastic models and simulation.

New results on slowly varying functions in the Zygmund sense

Very recently Seneta [15] has provided a characterization of slowly varying functions $L$ in the Zygmund sense by using the condition, for each $y>0$, \begin{equation} x\left(\frac{L(x+y)}{L(x)}-1\right)\to0 \text{ as } x\to∞. \tag{1} \end{equation} We extend this result by considering a wider class of functions and a more general condition than (1). Further, a representation theorem for this wider class is provided.

Local and global pathwise solutions for a stochastically perturbed nonlinear dispersive PDE

In this paper, we consider the periodic Cauchy problem for a stochastically perturbed nonlinear dispersive partial differential equation with cubic nonlinearity, which involves the integrable Novikov equation arising from the shallow water wave theory as a special case. We first establish the existence and uniqueness of local pathwise solutions in Sobolev spaces Hs(T)(s>32) with nonlinear multiplicative noise, where the key ingredients are the stochastic compactness method, the Skorokhod representation theorem and the Gyöngy–Krylov characterization of convergence in probability. In the case of linear multiplicative noise, we investigate the conditions which lead to the blow-up phenomena and global existence of pathwise solution. Finally, we show that the linear multiplicative noise has a dissipative effect on the periodic peakon solutions to the associated deterministic Novikov equation.

Locally defined operators in the space of C k,ω -functions

Abstract For a closed set A ⊂ ℝ n a representation theorem for locally defined operators maping the space C k,ω (A) consisting of all k-times continuously differentiable functions on A whose k-th derivatives have modulus of continuity ω into C 0,ω (A) is presented.

Some new results on Dieudonné-type theorems for k-triangular lattice group-valued set functions

Using the Maeda–Ogasawara–Vulikh representation theorem and sliding hump-type techniques, we prove some Dieudonne-type theorems for k-triangular set functions, taking values in lattice groups.

Efficient spatio-temporal Gaussian regression via Kalman filtering

We study the non-parametric reconstruction of spatio-temporal dynamical processes via Gaussian Processes (GPs) regression from sparse and noisy data. GPs have been mainly applied to spatial regression where they represent one of the most powerful estimation approaches also thanks to their universal representing properties. Their extension to dynamical processes has been instead elusive so far since classical implementations lead to unscalable algorithms or require some sort of approximation. We propose a novel procedure to address this problem by coupling GPs regression and Kalman filtering. In particular, assuming space/time separability of the covariance (kernel) of the process and rational time spectrum, we build a finite-dimensional discrete-time state-space process representation amenable to Kalman filtering. With sampling over a finite set of fixed spatial locations, our major finding is that the current Kalman filter state represents a sufficient statistic to compute the minimum variance estimate of the process at any future time over the entire spatial domain. In machine learning, a representer theorem states that an important class of infinite-dimensional variational problems admits a computable and finite-dimensional exact solution. In view of this, our result can be interpreted as a novel Dynamic Representer Theorem for GPs. We then extend the study to situations where the spatial input locations set varies over time. The proposed algorithms are tested on both synthetic and real field data, providing comparisons with standard GP and truncated GP regression techniques.

A Representation Theorem for Generic Line Arrangements in the Plane

In this article, we prove a representation theorem that any generic line arrangement in the plane over an ordered field can be represented isomorphically by a very generic line arrangement in the sense of C. A. Athanasiadis [2] with a given set of distinct slopes of the same cardinality.

Categorical consideration and perception complementarity

We introduce the notion of perception complementarity in a representation for random choice rules, which we call Complemented Perception Categorical Consideration Representation. The model is similar in nature to the perception-adjusted Luce model of Echenique et al. (Math Soc Sci 93:67–76, 2018), but allows perception to be menu-dependent. We provide a representation theorem and an identification result. Furthermore, we consider special cases of the model, including the case when it reduces down to the Random Consideration Set Rule of Manzini and Mariotti (Econometrica 82(3):1153–1176, 2014).

On Convergence Rates of Some Limits

In 2019 Seneta has provided a characterization of slowly varying functions L in the Zygmund sense by using the condition, for each y > 0 , x L ( x + y ) L ( x ) − 1 → 0 as x → ∞ . Very recently, we have extended this result by considering a wider class of functions U related to the following more general condition. For each y > 0 , r ( x ) U ( x + y g ( x ) ) U ( x ) − 1 → 0 as x → ∞ , for some functions r and g. In this paper, we examine this last result by considering a much more general convergence condition. A wider class related to this new condition is presented. Further, a representation theorem for this wider class is provided.

Constrained consistency enforcement in AHP

Decision-making in the presence of intangible elements must be based on a robust, but subtle, balance between expert know-how and judgment consistency when eliciting that know-how. This balance is frequently achieved as a trade-off reached after a feedback process softens the tension frequently found between one force steadily pulling towards (full) consistency, and another force driven by expert feeling and opinion. The linearization method, developed by the authors in the framework of the analytic hierarchy process, is a pull-towards-consistency mechanism that shows the path from an inconsistent body of judgment elicited from an expert towards consistency, by suggesting optimal changes to the expert opinions. However, experts may be reluctant to alter some of their issued opinions, and may wish to impose constraints on the adjustments suggested by the consistency-enforcement mechanism. In this paper, using the classical Riesz representation theorem, the linearization method is accommodated to consider various types of constraints imposed by experts during the abovementioned feedback process.

A thermodynamically consistent formulation for bending of thermoviscoelastic beams for small deformation, small strain based on classical continuum mechanics

The work presented in this paper extends the kinematic assumption free and thermodynamically consistent formulation for bending of thermoelastic beams presented by Surana et al. for bending of thermoviscoelastic beams with dissipation mechanism without memory. We consider small strain, small deformation physics in Lagrangian description. Conservation and balance laws of classical continuum mechanics (CCM) constitute the mathematical model for the physics considered in this paper. Constitutive variables and their argument tensors are established using conjugate pairs in the entropy inequality, additional desired physics and the principle of equipresence. Cauchy stress tensor is decomposed into equilibrium stress tensor () and deviatoric stress tensor (). Constitutive theory for is derived using Helmholtz free energy density in conjunction with incompressibility condition. The constitutive theory for is derived in by first establishing its argument tensors using conjugate pairs in the entropy inequality and other desired physics and then using the representation theorem with complete basis (integrity). The constitutive theory for is a nonlinear constitutive theory in terms of strain tensor containing up to fifth degree terms in the components of the strain tensor and an ordered rate theory in the strain rate tensors up to order n. Simplified linear ordered rate theory for is also presented. The formulation presented here for thermoviscoelastic beams is based on the conservation and balance laws of classical continuum mechanics and construction of beam finite element formulation using hpk framework with variationally consistent integral form. In this approach the mathematical model consist of true conservation and balance laws and the choice of local approximations for the beam finite elements facilitates incorporation of required kinematic description based on the application. This approach is free of the a priori assumptions of kinematic relations, computations are unconditionally stable and the local approximations can be of higher degree (p) and of higher order (k). This approach addresses slender as well as deep beam physics and can be used to measure error in the computed solution through residual functional. The rate constitutive theory is based on the second law thermodynamics consistent with physics of deformation. Mathematical details of new formulation and the model problem studies and comparisons with currently used beam models are presented for slender as well as deep beams. This approach permits consideration of reversible (thermoelastic) as well as irreversible (thermoviscoelastic) processes. It is shown that Rayleigh damping used currently to derive modal damping has no physical basis and can lead to spurious solution. The dissipation mechanism presented here has physical basis and yields non-spurious and valid solutions Model problem studies are presented for undamped natural modes of vibration, damped and undamped transient dynamic response. The results obtained from the formulation presented here are compared with published works.

Ordered Rate Constitutive Theories for Non-Classical Thermofluids Based on Convected Time Derivatives of the Strain and Higher Order Rotation Rate Tensors Using Entropy Inequality.

This paper considers non-classical continuum theory for thermoviscous fluids without memory incorporating internal rotation rates resulting from the antisymmetric part of the velocity gradient tensor to derive ordered rate constitutive theories for the Cauchy stress and the Cauchy moment tensor based on entropy inequality and representation theorem. Using the generalization of the conjugate pairs in the entropy inequality, the ordered rate constitutive theory for Cauchy stress tensor considers convected time derivatives of the Green’s strain tensor (or Almansi strain tensor) of up to orders as its argument tensors and the ordered rate constitutive theory for the Cauchy moment tensor considers convected time derivatives of the symmetric part of the rotation gradient tensor up to orders . While the convected time derivatives of the strain tensors are well known the convected time derivatives of higher orders of the symmetric part of the rotation gradient tensor need to be derived and are presented in this paper. Complete and general constitutive theories based on integrity using conjugate pairs in the entropy inequality and the generalization of the argument tensors of the constitutive variables and the representation theorem are derived and the material coefficients are established. It is shown that for the type of non-classical thermofluids considered in this paper the dissipation mechanism is an ordered rate mechanism due to convected time derivatives of the strain tensor as well as the convected time derivatives of the symmetric part of the rotation gradient tensor. The derivations of the constitutive theories presented in the paper is basis independent but can be made basis specific depending upon the choice of the specific basis for the constitutive variables and the argument tensors. Simplified linear theories are also presented as subset of the general constitutive theories and are compared with published works.

Non-Archimedean preferences over countable lotteries

We prove a representation theorem for preference relations over countably infinite lotteries that satisfy a generalized form of the Independence axiom, without assuming Continuity. The representing space consists of lexicographically ordered transfinite sequences of bounded real numbers. This result is generalized to preference orders on abstract superconvex spaces.

A martingale representation theorem and valuation of defaultable securities

AbstractWe consider a financial framework with two levels of information: the public information generated by the financial assets, and a larger flow of information that contains additional knowledge about a random time. This random time can represent many economic and financial settings, such as the default time of a firm for credit risk, and the death time of an insured for life insurance. As the random time cannot be seen before its occurrence, the progressive enlargement of filtration seems tailor‐fit to model the larger flow of information that incorporates both the public flow and the information about the random time. In this context, our interest focuses on the following challenges: (a) How to single out the various risks coming from the financial assets, the random time, and their correlations? (b) How these risks interplay and lead to the formation of any risk in the larger flow of information? It is clear that understanding how risks build‐up and interact, when one enlarges the flow of information, is vital for an efficient risk management and derivatives' evaluation in those informational markets. Our answers to these challenges are full and complete no matter what the model for the random time is and no matter how the random time is related to the public flow. In fact, we introduce “pure default” risks, and quantify and classify these risks afterward. Then we elaborate our martingale representation results, which state that any martingale in the large filtration stopped at the random time can be decomposed into orthogonal local martingales (i.e., local martingales whose product remains a local martingale). This constitutes our first principal contribution, while our second contribution consists in evaluating various defaultable securities according to the recovery policy, within our financial setting that encompasses any default model, using a martingale “basis.” Our pricing formulas explain the impact of various recovery policies on securities and determine the types of pure default risk they entail.

Representation of De Morgan and (Semi-)Kleene Lattices

Twist-structure representation theorems are established for De Morgan and Kleene lattices. While the former result relies essentially on the quasivariety of De Morgan lattices being finitely generated, the representation for Kleene lattices does not and can be extended to more general algebras. In particular, one can drop the double negation identity (involutivity). The resulting class of algebras, named semi-Kleene lattices by analogy with Sankappanavar’s semi-De Morgan lattices, is shown to be representable through a twist-structure construction inspired by the Cornish–Fowler duality for Kleene lattices. Quasi-Kleene lattices, a subvariety of semi-Kleene, are also defined and investigated, showing that they are precisely the implication-free subreducts of the recently introduced class of quasi-Nelson lattices.

Minimal modularity lifting for nonregular symplectic representations

In this paper, we prove a minimal modularity lifting theorem for Galois representations (conjecturally) associated to Siegel modular forms of genus two which are holomorphic limits of discrete series at infinity.