Representation Formula Research Articles

Volume Integral Equations and Single-Trace Formulations for Acoustic Wave Scattering in an Inhomogeneous Medium

AbstractWe study frequency domain acoustic scattering at a bounded, penetrable, and inhomogeneous obstacleΩ−⊂Rd\Omega^{-}\subset\mathbb{R}^{d},d=2,3d=2,3. By defining constant reference coefficients, a representation formula for the pressure field is derived. It contains a volume integral operator, related to the one in the Lippmann–Schwinger equation. Besides, it features integral operators defined on∂Ω−\partial\Omega^{-}and closely related to boundary integral equations of single-trace formulations (STF) for transmission problems with piecewise constant coefficients. We show well-posedness of the continuous variational formulation and asymptotic convergence of Galerkin discretizations. Numerical experiments in 2D validate our expected convergence rates.

Techniques for the Thermodynamic Consistency of Constitutive Equations

The paper investigates the techniques associated with the exploitation of the second law of thermodynamics as a restriction on the physically admissible processes. Though the exploitation consists of the use of the arbitrariness occurring in the Clausius–Duhem inequality, the approach emphasizes two uncommon features within the thermodynamic analysis: the representation formula, of vectors and tensors, and the entropy production. The representation is shown to be fruitful whenever more terms of the Clausius–Duhem inequality are not independent. Among the examples developed to show this feature, the paper yields the constitutive equation for hypo-elastic solids and for Maxwell–Cattaneo-like equations of heat conduction. The entropy production is assumed to be given by a constitutive function per se and not merely the expression inherited by the other constitutive functions. This feature results in more general expressions of the representation formulae and is crucial for the compact description of hysteretic phenomena.

Weierstrass Representation of Lightlike Surfaces in Lorentz-Minkowski 4-Space

We present a Weierstrass-type representation formula which locally represents every regular two-dimensional lightlike surface in Lorentz-Minkowski 4-Space $\mathbb{M}^4$ by three dual functions $(\rho,f,g)$ and generalizes the representation for regular lightlike surfaces in $\mathbb{M}^3$. We give necessary and sufficient conditions on the functions $\rho$, $f$, $g$ for the surface to be minimal, ruled or $l$-minimal. For ruled lightlike surfaces, we give necessary and sufficient conditions for the representation itself to be ruled. Furthermore, we give a result on totally geodesic half-lightlike surfaces which holds only in $\mathbb{M}^4$.

A representation formula for slice regular functions over slice-cones in several variables

The aim of this paper is to extend the so called slice analysis to a general case in which the codomain is a real vector space of even dimension, i.e. is of the form {mathbb {R}}^{2n}. This is a new setting which contains and encompasses in a nontrivial way other cases already studied in the literature and which requires new tools. To this end, we define a cone {mathcal {W}}_{mathcal {C}}^d in [{text {End}}({mathbb {R}}^{2n})]^d and we extend the slice topology tau _s to this cone. Slice regular functions can be defined on open sets in left( tau _s,{mathcal {W}}_{mathcal {C}}^dright) and a number of results can be proved in this framework, among which a representation formula. This theory can be applied to some real algebras, called left slice complex structure algebras. These algebras include quaternions, octonions, Clifford algebras and real alternative *-algebras but also left-alternative algebras and sedenions, thus providing brand new settings in slice analysis.

A polyanalytic functional calculus of order 2 on the 𝑆-spectrum

The Fueter theorem provides a two step procedure to build an axially monogenic function, i.e. a null-solution of the Cauchy-Riemann operator inR4\mathbb {R}^4, denoted byD\mathcal {D}. In the first step a holomorphic function is extended to a slice hyperholomorphic function, by means of the so-called slice operator. In the second step a monogenic function is built by applying the Laplace operatorΔ\Deltain four real variables to the slice hyperholomorphic function. In this paper we use the factorization of the Laplace operator, i.e.Δ=D¯D\Delta = \mathcal {\overline {D}} \mathcal {D}to split the previous procedure. From this splitting we get a class of functions that lies between the set of slice hyperholomorphic functions and the set of axially monogenic functions: the set of axially polyanalytic functions of order 2, i.e. null-solutions ofD2\mathcal {D}^2. We show an integral representation formula for this kind of functions. The formula obtained is fundamental to define the associated functional calculus on theSS-spectrum.

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.

On some properties of functions of the grand Hardy classes

Grand Hardy class H+p), p > 1, is defined and some properties of functions belonging to this class are studied in this work. Namely, the analogs of the Riesz and Smirnov theorems as well as the Cauchy’s formula for representation of function are proved. Necessary and sufficient condition for the validity of Riesz theorem in grand Hardy spaces H+p), p > 1, is found. Subspace GH+p)of the grand Hardy space H+p)generated by this condition is defined.

The analytic extension of solutions to initial-boundary value problems outside their domain of definition

AbstractWe examine the analytic extension of solutions of linear, constant-coefficient initial-boundary value problems outside their spatial domain of definition. We use the Unified Transform Method or Method of Fokas, which gives a representation for solutions to half-line and finite-interval initial-boundary value problems as integrals of kernels with explicit spatial and temporal dependence. These solution representations are defined within the spatial domain of the problem. We obtain the extension of these representation formulae via Taylor series outside these spatial domains and find the extension of the initial condition that gives rise to a whole-line initial-value problem solved by the extended solution. In general, the extended initial condition is not differentiable or continuous unless the boundary and initial conditions satisfy compatibility conditions. We analyse dissipative and dispersive problems, and problems with continuous and discrete spatial variables.

A Gronwall inequality and its applications to the Cauchy-type problem under ψ-Hilfer proportional fractional operators

In this paper, we propose a generalized Gronwall inequality in the context of the ψ-Hilfer proportional fractional derivative. Using Picard’s successive approximation and the definition of Mittag–Leffler functions, we construct the representation formula of the solution for the ψ-Hilfer proportional fractional differential equation with constant coefficient in the form of the Mittag–Leffler kernel. The uniqueness result is proved by using Banach’s fixed-point theorem with some properties of the Mittag–Leffler kernel. Additionally, Ulam–Hyers–Mittag–Leffler stability results are analyzed. Finally, numerical examples are provided to demonstrate the theory’s application.

Half-plane diffraction problems on a triangular lattice

We investigate thin-slit diffraction problems for two-dimensional lattice waves. Namely, Dirichlet problems for the two-dimensional discrete Helmholtz equation on the triangular lattice in a half-plane are studied. Using the notion of the radiating solution, we prove the existence and uniqueness of a solution for the real wave number $$k\in (0,3)\backslash \{2\sqrt{2}\}$$ without passing to the complex wave number. Besides, an exact representation formula for the solution is derived. Here, we develop a numerical calculation method and demonstrate by example the effectiveness of our approach related to the propagation of wavefronts in metamaterials through two small openings.

Rates of convergence in periodic homogenization of nonlocal Hamilton–Jacobi–Bellman equations,

In this paper we provide a rate of convergence for periodic homogenization of Hamilton–Jacobi–Bellman equations with nonlocal diffusion. The result is based on the regularity of the associated effective problem, where convexity plays a crucial role. The necessary regularity estimates are made possible by a representation formula we obtain for the effective Hamiltonian, a result that has an independent interest.

Bilans poprawek do konstytucji Republiki Irlandzkiej dotyczących Seanad Éireann

Seanad Éireann in the Republic of Ireland, being currently one of the few examples of a chamber of parliament implementing the formula of functional representation, is often the object of criticism and proposals for constitutional reforms in terms of its functioning and, in particular, the method of electing its members. The previous amendments related to the Senate – VII of 1979 and XXXII of 2013 – ended in failure and did not implement the accompanying basic ideas regarding the systemic reform. The article introduces the composition of Seanad Éireann, the historical context of the development of both amendments, the context of attempts to implement them, failures in this regard and the consequences of this failure and includes a reflection on the future of the chamber and an indication of further attempts at its reform.

2D Toda τ functions, weighted Hurwitz numbers and the Cayley graph: Determinant representation and recursion formula

We generalize the determinant representation of the Kadomtsev–Petviashvili τ functions to the case of the 2D Toda τ functions. The generating functions for the weighted Hurwitz numbers are a parametric family of 2D Toda τ functions, for which we give a determinant representation of weighted Hurwitz numbers. Then, we can get a finite-dimensional equation system for the weighted Hurwitz numbers HGd(σ,ω) with the same dimension |σ| = |ω| = n. Using this equation system, we calculated the value of the weighted Hurwitz numbers with dimension 0, 1, 2, 3 and give a recursion formula for calculating the higher dimensional weighted Hurwitz numbers. Finally, we get a matrix representation for the Hurwitz numbers and obtain a determinant representation of weighted paths in the Cayley graph.

Some identities of degenerate higher-order Daehee polynomials based on $ \lambda $-umbral calculus

<abstract><p>The degenerate versions of special polynomials and numbers, initiated by Carlitz, have regained the attention of some mathematicians by replacing the usual exponential function in the generating function of special polynomials with the degenerate exponential function. To study the relations between degenerate special polynomials, $ \lambda $-umbral calculus, an analogue of umbral calculus, is intensively applied to obtain related formulas for expressing one $ \lambda $-Sheffer polynomial in terms of other $ \lambda $-Sheffer polynomials. In this paper, we study the connection between degenerate higher-order Daehee polynomials and other degenerate type of special polynomials. We present explicit formulas for representations of the polynomials using $ \lambda $-umbral calculus and confirm the presented formulas between the degenerate higher-order Daehee polynomials and the degenerate Bernoulli polynomials, for example. Additionally, we investigate the pattern of the root distribution of these polynomials.</p></abstract>

On the relaxation of functionals with contact terms on non-smooth domains

We provide the integral representation formula for the relaxation in $BV(\Omega; \mathbb{R}^M)$ with respect to strong convergence in $L^1(\Omega; \mathbb{R}^M)$ of a functional with a boundary contact energy term. This characterization is valid for a large class of surface energy densities, and for domains satisfying mild regularity assumptions. Motivated by some classical examples where lower semicontinuity fails, we analyze the extent to which the geometry of the set enters the relaxation procedure.

Sharp Adams type inequalities in Lorentz-Sobole space

<abstract><p>This article addresses several sharp weighted Adams type inequalities in Lorentz-Sobolev spaces by using symmetry, rearrangement and the Riesz representation formula. In particular, the sharpness of these inequalities were also obtained by constructing a proper test sequence.</p></abstract>

CVA in fractional and rough volatility models

In this work we present a general representation formula for the price of a vulnerable European option, and the related CVA in stochastic (either rough or not) volatility models for the underlying’s price, when admitting correlation with the default event. We specialize it for some volatility models and we provide price approximations, based on the representation formula. We study numerically their accuracy, comparing the results with Monte Carlo simulations, and we run a theoretical study of the error. We also introduce a seminal study of roughness influence on the claim’s price.

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.

Three boundary value problems of the Cauchy–Riemann equation in the Eclipse domain

In this article, by the parqueting-reflection technique, we obtain a covering of the complex plane from reflecting the Eclipse domain. Then, by the Cauchy–Pompeiu formula, we derive the generalized Cauchy–Pompeiu representation formula in the Eclipse domain. In addition, we consider the Schwarz boundary value problem (BVP) for the nonhomogeneous Cauchy–Riemann equation and the explicit expressions of solution and the condition of solvability. Besides, by using the Schwarz BVP, we investigate the inhomogeneous Dirichlet and homogeneous Neumann BVPs.