Explicit Representation Formula Research Articles

Quantum Exclusion Process driven by Brownian Motions in terms of quantum Bernoulli noises

Quantum Bernoulli noises are the family of annihilation and creation operators on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal time. The paper is devoted to the study of Quantum Exclusion Process driven by Brownian Motions in terms of Quantum Bernoulli Noises. We first study the existence and uniqueness of regular solutions to the classical Quantum Exclusion Process driven by Brownian Motions. We then provide an explicit representation formula by separating the action on off-diagonal and diagonal operators, on which they are reduced to the Quantum Exclusion Process of classical Markov chains. Finally, we prove that the existence of regular invariant measures for Quantum Exclusion Process driven by Brownian Motions in terms of Quantum Bernoulli Noises.

The Born approximation in the three-dimensional Calderón problem Ⅱ: Numerical reconstruction in the radial case

In this work we illustrate a number of properties of the Born approximation in the three-dimensional Calderón inverse conductivity problem by numerical experiments. The results are based on an explicit representation formula for the Born approximation recently introduced by the authors. We focus on the particular case of radial conductivities in the ball $ B_R \subset \mathbb{R}^3 $ of radius $ R $, in which the linearization of the Calderón problem is equivalent to a Hausdorff moment problem. We give numerical evidences that the Born approximation is well defined for $ L^{\infty} $ conductivities, and present a novel numerical algorithm to reconstruct a radial conductivity from the Born approximation under a suitable smallness assumption. We also show that the Born approximation has depth-dependent uniqueness and approximation capabilities depending on the distance (depth) to the boundary $ \partial B_R $. We then investigate how increasing the radius $ R $ affects the quality of the Born approximation, and the existence of a scattering limit as $ R\to \infty $. Similar properties are also illustrated in the inverse boundary problem for the Schrödinger operator $ -\Delta +q $, and strong recovery of singularity results are observed in this case.

Extension of monotone operators and Lipschitz maps invariant for a group of isometries

Abstract We study monotone operators in reflexive Banach spaces that are invariant with respect to a group of suitable isometric isomorphisms, and we show that they always admit a maximal extension which preserves the same invariance. A similar result applies to Lipschitz maps in Hilbert spaces, thus providing an invariant version of Kirszbraun–Valentine extension theorem. We then provide a relevant application to the case of monotone operators in $L^{p}$ -spaces of random variables which are invariant with respect to measure-preserving isomorphisms, proving that they always admit maximal dissipative extensions which are still invariant by measure-preserving isomorphisms. We also show that such operators are law invariant, a much stronger property which is also inherited by their resolvents, the Moreau–Yosida approximations, and the associated semigroup of contractions. These results combine explicit representation formulae for the maximal extension of a monotone operator based on self-dual Lagrangians and a refined study of measure-preserving maps in standard Borel spaces endowed with a nonatomic measure, with applications to the approximation of arbitrary couplings between measures by sequences of maps.

On the Maximum Principle for Optimal Control Problems of Stochastic Volterra Integral Equations with Delay

In this paper, we prove both necessary and sufficient maximum principles for infinite horizon discounted control problems of stochastic Volterra integral equations with finite delay and a convex control domain. The corresponding adjoint equation is a novel class of infinite horizon anticipated backward stochastic Volterra integral equations. Our results can be applied to discounted control problems of stochastic delay differential equations and fractional stochastic delay differential equations. As an example, we consider a stochastic linear-quadratic regulator problem for a delayed fractional system. Based on the maximum principle, we prove the existence and uniqueness of the optimal control for this concrete example and obtain a new type of explicit Gaussian state-feedback representation formula for the optimal control.

Ergodicity of exclusion semigroups constructed from quantum Bernoulli noises

Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy the canonical anti-commutation relation (CAR) in equal time. This paper aimed to discuss the classical reduction and ergodicity of quantum exclusion semigroups constructed by QBN. We first study the classical reduction of the quantum semigroups to an Abelian algebra of diagonal elements and the space of off-diagonal elements. We then provide an explicit representation formula by separating the action on off-diagonal and diagonal operators, on which they are reduced to the semigroups of classical Markov chains. Finally, we prove that the asymptotic behavior of the quantum semigroups is equivalent to one of its associated Markov chains, and that the semigroups restricted to the off diagonal space of operators have a zero limit.

Confluent Appell polynomials

In this paper, we introduce the confluent Appell polynomials and prove a Sheffer type characterization theorem for them by means of the Stieltjes integral of hypergeometric polynomials. We investigate their several properties such as explicit representation, integral representation and finite summation formulas. Moreover, by proving a pure recurrence relation and deriving the lowering and the raising operators, in terms of differential and shift operators, we obtain the equation satisfied the confluent Appell polynomials by using the factorization method. And then, we define the confluent Bernoulli and Hermite polynomials and exhibit their main properties such as explicit representations, recurrence formulas (involving the corresponding usual Bernoulli and Hermite polynomials), finite summation formulas and equations involving differential and shift operators. Finally, we construct approximation operators by using confluent Appell polynomials which helps to approximate to a function defined on the semi infinite interval in a weighted function space. We call these as the confluent Jakimovski–Leviatan operators which includes the confluent version of the well-known Szász–Mirakyan operators. Also, an illustrative example in order to show convergence efficiency of the confluent Szász–Mirakyan operators is given.

On a semilinear wave equation in anti-de Sitter spacetime: The critical case

In the present paper, we prove the blow-up in finite time for local solutions of a semilinear Cauchy problem associated with a wave equation in anti-de Sitter spacetime in the critical case. According to this purpose, we combine a result for ordinary differential inequalities with an iteration argument by using an explicit integral representation formula for the solution to a linear Cauchy problem associated with the wave equation in anti-de Sitter spacetime in one space dimension.

Abstract representation of the SMGTJ equation under rough boundary controls: Optimal interior regularity

We consider the linearized third order SMGTJ equation defined on a sufficiently smooth boundary domain in and subject to either Dirichlet or Neumann rough boundary control. Filling a void in the literature, we present a direct general system approach based on the vector state solution {position, velocity, acceleration}. It yields, in both cases, an explicit representation formula: input solution, based on the s.c. group generator of the boundary homogeneous problem and corresponding elliptic Dirichlet or Neumann map. It is close to, but also distinctly and critically different from, the abstract variation of parameter formula that arises in more traditional boundary control problems for PDEs L‐T.6. Through a duality argument based on this explicit formula, we provide a new proof of the optimal regularity theory: boundary control {position, velocity, acceleration} with low regularity boundary control, square integrable in time and space.

Operator Calculus Approach to Comparison of Elasticity Models for Modelling of Masonry Structures

The solution of any engineering problem starts with a modelling process aimed at formulating a mathematical model, which must describe the problem under consideration with sufficient precision. Because of heterogeneity of modern engineering applications, mathematical modelling scatters nowadays from incredibly precise micro- and even nano-modelling of materials to macro-modelling, which is more appropriate for practical engineering computations. In the field of masonry structures, a macro-model of the material can be constructed based on various elasticity theories, such as classical elasticity, micropolar elasticity and Cosserat elasticity. Evidently, a different macro-behaviour is expected depending on the specific theory used in the background. Although there have been several theoretical studies of different elasticity theories in recent years, there is still a lack of understanding of how modelling assumptions of different elasticity theories influence the modelling results of masonry structures. Therefore, a rigorous approach to comparison of different three-dimensional elasticity models based on quaternionic operator calculus is proposed in this paper. In this way, three elasticity models are described and spatial boundary value problems for these models are discussed. In particular, explicit representation formulae for their solutions are constructed. After that, by using these representation formulae, explicit estimates for the solutions obtained by different elasticity theories are obtained. Finally, several numerical examples are presented, which indicate a practical difference in the solutions.

From low to high-and lower-optimal regularity of the SMGTJ equation with Dirichlet and Neumann boundary control, and with point control, via explicit representation formulae

<p style='text-indent:20px;'>We consider the linear third order (in time) PDE known as the SMGTJ-equation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control <inline-formula><tex-math id="M1">\begin{document}$ g $\end{document}</tex-math></inline-formula>. Optimal interior and boundary regularity results were given in [<xref ref-type="bibr" rid="b1">1</xref>], after [<xref ref-type="bibr" rid="b41">41</xref>], when <inline-formula><tex-math id="M2">\begin{document}$ g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma) $\end{document}</tex-math></inline-formula>, which, moreover, in the canonical case <inline-formula><tex-math id="M3">\begin{document}$ \gamma = 0 $\end{document}</tex-math></inline-formula>, were expressed by the well-known explicit representation formulae of the wave equation in terms of cosine/sine operators [<xref ref-type="bibr" rid="b19">19</xref>], [<xref ref-type="bibr" rid="b17">17</xref>], [<xref ref-type="bibr" rid="b24">24</xref>,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether <inline-formula><tex-math id="M4">\begin{document}$ \gamma = 0 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M5">\begin{document}$ 0 \neq \gamma \in L^{\infty}(\Omega) $\end{document}</tex-math></inline-formula>, since <inline-formula><tex-math id="M6">\begin{document}$ \gamma \neq 0 $\end{document}</tex-math></inline-formula> is responsible only for lower order terms. Here we exploit such cosine operator based-explicit representation formulae to provide optimal interior and boundary regularity results with <inline-formula><tex-math id="M7">\begin{document}$ g $\end{document}</tex-math></inline-formula> "smoother" than <inline-formula><tex-math id="M8">\begin{document}$ L^2(\Sigma) $\end{document}</tex-math></inline-formula>, qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [<xref ref-type="bibr" rid="b17">17</xref>]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [<xref ref-type="bibr" rid="b22">22</xref>], [<xref ref-type="bibr" rid="b23">23</xref>], [<xref ref-type="bibr" rid="b37">37</xref>] for control smoother than <inline-formula><tex-math id="M9">\begin{document}$ L^2(0, T;L^2(\Gamma)) $\end{document}</tex-math></inline-formula>, and [<xref ref-type="bibr" rid="b44">44</xref>] for control less regular in space than <inline-formula><tex-math id="M10">\begin{document}$ L^2(\Gamma) $\end{document}</tex-math></inline-formula>. In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [<xref ref-type="bibr" rid="b42">42</xref>], [<xref ref-type="bibr" rid="b24">24</xref>,Section 9.8.2].</p>

The SMGT equation from the boundary: regularity and stabilization

We consider the third-order (in time) linear equation known as SMGT-equation, as defined on a multidimensional bounded domain. Part A gives optimal interior and boundary regularity results from – Dirichlet or Neumann boundary terms. Explicit representation formulas are given that can be taken to define the notion of solution in the canonical case , while the same regularity results hold for Part B considers the SMGT equation under Neumann dissipative boundary conditions and critical parameter and a.e. in Ω. We provide two results: (i) uniform stabilization under minimal checkable geometric conditions, and (ii) strong stabilization in the absence of geometrical conditions.

A multi-material transport problem with arbitrary marginals

In this paper we study general transportation problems in \({\mathbb {R}}^n\), in which m different goods are moved simultaneously. The initial and final positions of the goods are prescribed by measures \(\mu ^-\), \(\mu ^+\) on \({\mathbb {R}}^n\) with values in \({\mathbb {R}}^m\). When the measures are finite atomic, a discrete transportation network is a measure T on \({\mathbb {R}}^n\) with values in \({\mathbb {R}}^{n\times m}\) represented by an oriented graph \({\mathcal {G}}\) in \({\mathbb {R}}^n\) whose edges carry multiplicities in \({\mathbb {R}}^m\). The constraint is encoded in the relation \({\mathrm{div}}(T)=\mu ^- -\mu ^+\). The cost of the discrete transportation T is obtained integrating on \({\mathcal {G}}\) a general function \({\mathcal {C}}:{\mathbb {R}}^m\rightarrow {\mathbb {R}}\) of the multiplicity. When the initial data \(\left( \mu ^-,\mu ^+\right) \) are arbitrary (possibly diffuse) measures, the cost of a transportation network between them is computed by relaxation of the functional on graphs mentioned above. Our main result establishes the existence of cost-minimizing transportation networks for arbitrary data \(\left( \mu ^-,\mu ^+\right) \). Furthermore, under additional assumptions on the cost integrand \({\mathcal {C}}\), we prove the existence of transportation networks with finite cost and the stability of the minimizers with respect to variations of the given data. Finally, we provide an explicit integral representation formula for the cost of rectifiable transportation networks, and we characterize the costs such that every transportation network with finite cost is rectifiable.

A blow-up result for a semilinear wave equation with scale-invariant damping and mass and nonlinearity of derivative type

In this note, we prove blow-up results for semilinear wave models with damping and mass in the scale-invariant case and with nonlinear terms of derivative type. We consider the single equation and the weakly coupled system. In the first case we get a blow-up result for exponents below a certain shift of the Glassey exponent. For the weakly coupled system we find as critical curve a shift of the corresponding curve for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. Our approach follows the one for the respective classical wave equation by Zhou Yi. In particular, an explicit integral representation formula for a solution of the corresponding linear scale-invariant wave equation, which is derived by using Yagdjian's integral transform approach, is employed in the blow-up argument. While in the case of the single equation we may use a comparison argument, for the weakly coupled system an iteration argument is applied.

CVA AND VULNERABLE OPTIONS IN STOCHASTIC VOLATILITY MODELS

This work aims to provide an efficient method to evaluate the Credit Value Adjustment (CVA) for a vulnerable European option, which is an option subject to some default event concerning the issuer solvability. Financial options traded in OTC markets are of this type. In particular, we compute the CVA in some popular stochastic volatility models such as SABR, Hull et al., which have proven to fit quite well market derivatives prices, admitting correlation with the default event. This choice covers the relevant case of Wrong Way Risk (WWR) when a credit deterioration determines an increase in the claim value. Contrary to the structural modeling adopted in [G. Wang, X. Wang & K. Zhu (2017) Pricing vulnerable options with stochastic volatility, Physica A 485, 91–103; C. Ma, S. Yue & Y. Ma (2020) Pricing vulnerable options with Stochastic volatility and Stochastic interest rate, Computational Economics 56, 391–429], we use the reduced-form intensity-based approach to provide an explicit representation formula for the vulnerable option price and related CVA. Later, we specialize the evaluation formula and construct its approximation for the three models mentioned above. Assuming a CIR model for the default intensity process, we run a numerical study to test our approximation, comparing it with Monte Carlo simulations. The results show that for moderate values of the correlation and maturities not exceeding one year, the approximation is very satisfactory as of accuracy and computational time.

Normal approximation and confidence region of singular subspaces

This paper is on the normal approximation of singular subspaces when the noise matrix has i.i.d. entries. Our contributions are three-fold. First, we derive an explicit representation formula of the empirical spectral projectors. The formula is neat and holds for deterministic matrix perturbations. Second, we calculate the expected projection distance between the empirical singular subspaces and true singular subspaces. Our method allows obtaining arbitrary k-th order approximation of the expected projection distance. Third, we prove the non-asymptotical normal approximation of the projection distance with different levels of bias corrections. By the ⌈log(d1+d2)⌉-th order bias corrections, the asymptotical normality holds under optimal signal-to-noise ratio (SNR) condition where d1 and d2 denote the matrix sizes. In addition, it shows that higher order approximations are unnecessary when |d1−d2|=O((d1+d2)1∕2). Finally, we provide comprehensive simulation results to merit our theoretic discoveries. Unlike the existing results, our approach is non-asymptotical and the convergence rates are established. Our method allows the rank r to diverge as fast as o((d1+d2)1∕3). Moreover, our method requires no eigen-gap condition (except the SNR) and no constraints between d1 and d2.

Representation of limit values for nonexpansive stochastic differential games

A classical problem in ergodic control theory consists in the study of the limit behaviour of λVλ(⋅) as λ↘0, when Vλ is the value function of a deterministic or stochastic control problem with discounted cost functional with infinite time horizon and discount factor λ. We study this problem for the lower value function Vλ of a stochastic differential game with recursive cost, i.e., the cost functional is defined through a backward stochastic differential equation with infinite time horizon. But unlike the ergodic control approach, we are interested in the case where the limit can be a function depending on the initial condition. For this we extend the so-called non-expansivity assumption from the case of control problems to that of stochastic differential games and we derive that λVλ(⋅) is bounded and Lipschitz uniformly with respect to λ>0. Using PDE methods and assuming radial monotonicity of the Hamiltonian of the associated Hamilton-Jacobi-Bellman-Isaacs equation we obtain the monotone convergence of λVλ(.) and we characterize its limit W0 as maximal viscosity subsolution of a limit PDE. Using BSDE methods we prove that W0 satisfies a uniform dynamic programming principle involving the supremum and the infimum with respect to the time, and this is the key for an explicit representation formula for W0.

Regularity of solutions of elliptic equations in divergence form in modified local generalized Morrey spaces

Aim of this paper is to prove regularity results, in some Modified Local Generalized Morrey Spaces, for the first derivatives of the solutions of a divergence elliptic second order equation of the form Lu:=∑i,j=1naij(x)uxixj=∇·f,for almost allx∈Ω\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\mathscr {L}u{:}{=}\\sum _{i,j=1}^{n}\\left( a_{ij}(x)u_{x_{i}}\\right) _{x_{j}}=\\nabla \\cdot f,\\qquad \\hbox {for almost all }x\\in \\Omega \\end{aligned}$$\\end{document}where the coefficients a_{ij} belong to the Central (that is, Local) Sarason class CVMO and f is assumed to be in some Modified Local Generalized Morrey Spaces widetilde{LM}_{{x_{0}}}^{p,varphi }. Heart of the paper is to use an explicit representation formula for the first derivatives of the solutions of the elliptic equation in divergence form, in terms of singular integral operators and commutators with Calderón–Zygmund kernels. Combining the representation formula with some Morrey-type estimates for each operator that appears in it, we derive several regularity results.

Spectral Numerical Algorithm for Solving Optimal Control Using Boubaker-Turki Operational Matrices

The aim of the present research is to propose a spectral method for solving optimal control problem indirectly using Boubaker - Turki polynomial functions as basis functions. To achieve this goal, explicit representation formulas for some interesting operational matrices for Boubaker - Turki polynomials functions are first derived which play an important role in dealing with the problem of optimal control. They are operational matrix of derivative, operational matrix of product. By applying the obtained operational matrices and spectral scheme, the main problem is transformed to a set of linear algebraic equations that greatly simplifies the problem. The presented method in details by solving numerical example has been investigated. A new recursive relation of the Boubaker - Turki and Chebyshev polynomials of the second kind as well as a general formula for power function as a linear combination of the Boubaker - Turki polynomial are also included in this work.

ON ASYMPTOTIC PROPERTIES OF THE CAUCHY FUNCTION FOR AUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATION OF NEUTRAL TYPE

We investigate stability of a linear autonomous functional differential equation of neutral type. The basis of the study is the well-known explicit solution representation formula including an integral operator, the kernel of which is called the Cauchy function of the equation under study. It is shown that the definitions of Lyapunov, asymptotic and exponential stabilities can be formulated without loss of generality in terms of the corresponding properties of the Cauchy function. The conclusion is drawn that stability with respect to initial data depends on the functional space which the initial function belongs to, and, as a consequence, that there is the need to indicate this space in the definition of stability. It is shown that, along with the concept of asymptotic stability, a certain stronger property should be introduced, which we call strong asymptotic stability. The main study is devoted to stability with respect to initial function from spaces of integrable functions. Special attention is paid to the study of asymptotic and exponential stability. We use the following known properties of the Cauchy function of an equation of neutral type: this function is piecewise continuous, and its jumps are determined by a Cauchy problem for a linear difference equation. We obtain that the strong asymptotic stability of the equation under consideration for initial data from the space L1 is equivalent to an exponential estimate of the Cauchy function and; moreover, we show that these properties are equivalent to the exponential stability with respect to initial data from the spaces Lp for all p from 1 to infinity inclusive. However, we show that strong asymptotic stability with respect to the initial data from the space Lp for p greater than one may not coincide with exponential stability.

Logistic Neumann problems with discontinuous coefficients

The purpose of this paper is for the first time to provide a careful and accessible exposition of the study of the existence of positive solutions of semilinear Neumann problems for diffusive logistic equations with discontinuous coefficients, which models population dynamics in environments with spatial heterogeneity. A biological interpretation of our main result is that when the environment has an impassable boundary and is on the average unfavorable, then high diffusion rates have the same effect (that is, the ultimate extinction of the population) as they always have when the boundary is deadly; but if the boundary is impassable and the environment is on the average neutral or favorable, then the population can persist, no matter what its rate of diffusion. The approach here is based on explicit representation formulas for the solutions of the Neumann problem and also on the $$L^{p}$$ boundedness of Calderon–Zygmund singular integral operators appearing in those representation formulas. That is why we consider the case where the space dimension is greater than 2. Moreover, we make use of an $$L^{p}$$ variant of an estimate for the Green operator of the Neumann problem introduced in the study of Feller semigroups.