Representation Formula Research Articles

Slice Dirac operator over octonions

The slice Dirac operator over octonions is a slice counterpart of the Dirac operator over quaternions. It involves a new theory of stem functions, which is the extension from the commutative O(1) case to the non-commutative O(3) case. For functions in the kernel of the slice Dirac operator over octonions, we establish the representation formula, the Cauchy integral formula (and, more in general, the Cauchy-Pompeiu formula), and the Taylor as well as the Laurent series expansion formulas.

Harmonic Green and Neumann functions for domains bounded by two intersecting circular arcs

ABSTRACT The parqueting-reflection principle is shown to work for constructing the harmonic Green functions and harmonic Neumann functions for a class of domains, which are bounded by two intersecting circular arcs in with a particular angle . With the explicit expressions of the harmonic Green and Neumann functions, we solve the Dirichlet and Neumann boundary problems to the Poisson equation in these domains by applying the Green and Neumann representation formulas.

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.

A representation formula for the distributional normal derivative

We prove an integral representation formula for the distributional normal derivative of solutions of $$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} - \Delta u + V u &{}= \mu &{}&{} \text {in }\Omega , u &{}= 0 &{}&{} \text {on }\partial \Omega , \end{aligned} \end{array}\right. } \end{aligned}$$ where $$V \in L_\mathrm {loc}^1(\Omega )$$ is a nonnegative function and $$\mu $$ is a finite Borel measure on $$\Omega $$ . As an application, we show that the Hopf lemma holds almost everywhere on $$\partial \Omega $$ when $$V$$ is a nonnegative Hopf potential.

On Holomorphic Functions in the Upper Half-Plane Representable by Carleman Formula

Let $$\Pi =\{z=x+iy\in \mathbb {C}:\;y>0\} $$ be the upper half-plane and the interval [a, b] be a subset of $$ \partial \Pi =\mathbb {R}$$ . We derive a Carleman integral representation formula for all holomorphic functions $$f\in {\mathcal H}(\Pi )$$ that have angular boundary values on [a, b] and which belong to the class $$\mathcal { N H}^1_{[a,b]}(\Pi )$$ . The class $$\mathcal {NH}^1_{[a,b]}(\Pi )$$ is the class of holomorphic functions in $$\Pi $$ which belong to the Hardy class $${\mathcal H}^1$$ near the interval [a, b] (“The Class of Functions Representable by Carleman Integral Representation Formula” section). As an application of the above characterization, our main result is an extension theorem for a function $$f\in L^1([a,b])$$ to a function $$f\in \mathcal {NH}^1_{[c,d]}(\Pi )$$ , for almost all intervals $$[c,d]\subset (a,b)$$ . Similar results can be proved for a function f holomorphic in a slanted disc, with integrable boundary values on the horizontal part of the boundary.

Stability of Ulam–Hyers and Existence of Solutions for Impulsive Time-Delay Semi-Linear Systems with Non-Permutable Matrices

In this paper, the stability of Ulam–Hyers and existence of solutions for semi-linear time-delay systems with linear impulsive conditions are studied. The linear parts of the impulsive systems are defined by non-permutable matrices. To obtain solution for linear impulsive delay systems with non-permutable matrices in explicit form, a new concept of impulsive delayed matrix exponential is introduced. Using the representation formula and norm estimation of the impulsive delayed matrix exponential, sufficient conditions for stability of Ulam–Hyers and existence of solutions are obtained.

Convergence Rates of Solutions for Elliptic Reiterated Homogenization Problems

In this paper, we study reiterated homogenization problems for equations −div(A(x/e, x/e2)∇ue) = f (x). We introduce auxiliary functions and obtain the representation formula satisfied by ue and homogenized solution u0. Then we utilize this formula in combination with the asymptotic estimates of Neumann functions for operators and uniform regularity estimates of solutions to obtain convergence rates in Lp for solutions as well as gradient error estimates for Neumann problems.

Multiple Convolution Formulae of Bernoulli and Euler Numbers

By examining higher derivatives of hyperbolic functions, we derive monomial and binomial representation formulae, that are utilized to establish several multiple convolution identities for the Bernoulli numbers Bn, Euler numbers En and two variants of Bn

Rayleigh-Quotient Representation of the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of General Matrices

In the present paper, formulas for the Rayleigh-quotient representation of the real parts, imaginary parts, and moduli of the eigenvalues of general matrices are obtained that resemble corresponding formulas for the eigenvalues of self-adjoint matrices. These formulas are of interest in Linear Algebra and in the theory of linear dynamical systems. The key point is that a weighted scalar product is used that is defined by means of a special positive definite matrix. As applications, one obtains convexity properties of newly-defined numerical ranges of a matrix. A numerical example underpins the theoretical findings.

Symplectic quandles and parabolic representations of 2-bridge knots and links

In this paper, we study the parabolic representations of 2-bridge links by finiding arc coloring vectors on the Conway diagram. The method we use is to convert the system of conjugation quandle equations to that of symplectic quandle equations. In this approach, we have an integer coefficient monic polynomial [Formula: see text] for each 2-bridge link [Formula: see text], and each zero of this polynomial gives a set of arc coloring vectors on the diagram of [Formula: see text] satisfying the system of symplectic quandle equations, which gives an explicit formula for a parabolic representation of [Formula: see text]. We then explain how these arc coloring vectors give us the closed form formulas of the complex volume and the cusp shape of the representation. As other applications of this method, we show some interesting arithmetic properties of the Riley polynomial and of the trace field, and also describe a necessary and sufficient condition for the existence of epimorphisms between 2-bridge link groups in terms of divisibility of the corresponding Riley polynomials.

Faces and Support Functions for the Values of Maximal Monotone Operators

Fondecyt Postdoc Project 3180080 Basal Program from CONICYT-Chile CMM-AFB 170001 National Foundation for Science & Technology Development (NAFOSTED) 101.01-2017.325

Representation formula for viscosity solution to a PDE problem involving Pucci’s extremal operator

We provide a representation formula for viscosity solutions to an elliptic Dirichlet problem involving Pucci’s extremal operators. This is done through a dynamic programming principle derived from Denis et al. (2010). The formula can be seen as a nonlinear extension of the Feynman–Kac formula.

On Representation Formulas for Solutions of Linear Differential Equations with Caputo Fractional Derivatives

Abstract In the paper, a linear differential equation with variable coefficients and a Caputo fractional derivative is considered. For this equation, a Cauchy problem is studied, when an initial condition is given at an intermediate point that does not necessarily coincide with the initial point of the fractional differential operator. A detailed analysis of basic properties of the fundamental solution matrix is carried out. In particular, the Hölder continuity of this matrix with respect to both variables is proved, and its dual definition is given. Based on this, two representation formulas for the solution of the Cauchy problem are proposed and justified.

Sobolev regular solutions for the incompressible Navier\u2013Stokes equations in higher dimensions: asymptotics and representation formulae

In this paper, we consider the steady incompressible Navier–Stokes equations in a smooth bounded domain Omega subset mathbb R^n with the dimension nge 3. We first establish asymptotic expansion formulae of Sobolev regular finite energy solutions in Omega. In the second part of this paper, explicit representation formulae of Sobolev regular solutions are showed in the regular polyhedron Omega :=[0,T]^n.

Certain Image Formulae and Fractional Kinetic Equations of Generalized $$\mathtt {k}$$-Bessel Functions via the Sumudu Transform

The monotonicity and the representation formulae of generalized $$\mathtt {k}$$ -Bessel functions, $$W_{\nu ,c}^{\mathtt {k}}$$ , studied by SR Mondal. This paper establishes the image formulae and then extract the solutions for fractional kinetic equations, involving $$W_{\nu ,c}^{\mathtt {k}}$$ utilizing their Sumudu transforms. Some significant particular cases are then deduced and analyzed.

Herglotz' variational principle and Lax-Oleinik evolution

We develop an elementary method to give a Lipschitz estimate for the minimizers in the problem of Herglotz' variational principle proposed in the paper (P. Cannarsa, W. Cheng, K. Wang, J. Yan, 2019 [17]) in the time-dependent case. We deduce Erdmann's condition and the Euler-Lagrange equation separately under different sets of assumptions, by using a generalized du Bois-Reymond lemma. As an application, we obtain a representation formula for the viscosity solution of the Cauchy problem for the Hamilton-Jacobi equationDtu(t,x)+H(t,x,Dxu(t,x),u(t,x))=0 and study the related Lax-Oleinik evolution.

Overcoming the curse of dimensionality for some Hamilton–Jacobi partial differential equations via neural network architectures

We propose new and original mathematical connections between Hamilton–Jacobi (HJ) partial differential equations (PDEs) with initial data and neural network architectures. Specifically, we prove that some classes of neural networks correspond to representation formulas of HJ PDE solutions whose Hamiltonians and initial data are obtained from the parameters of the neural networks. These results do not rely on universal approximation properties of neural networks; rather, our results show that some classes of neural network architectures naturally encode the physics contained in some HJ PDEs. Our results naturally yield efficient neural network-based methods for evaluating solutions of some HJ PDEs in high dimension without using grids or numerical approximations. We also present some numerical results for solving some inverse problems involving HJ PDEs using our proposed architectures.

Möbius transformation and a version of Schwarz lemma in octonionic analysis

The octonion is a generalization of complex to noncommutative and nonassociative space which has closed relation with exception geometries, wave equation, Yang‐Mills equations, black hole, string theory, and special relativity. In this paper, the Möbius transformation in this manner is first introduced, and some properties are discussed about the transformation in octonionic analysis. Some technique lemmas will be given to solve the problems caused by the weak form of associativity. These versions of Schwarz lemma and Schwarz‐Pick lemma are first studied in octonionic setting which will invoke integral representation formula for harmonic function and Möbius transformations. This will generalize the corresponding results which appear in the classical function theory to nonassociative space and may give new energy for the development of physics.

Stationary scattering theory for unitary operators with an application to quantum walks

We present a general account on the stationary scattering theory for unitary operators in a two-Hilbert spaces setting. For unitary operators U0,U in Hilbert spaces H0,H and an identification operator J:H0→H, we give the definitions and collect properties of the stationary wave operators, the strong wave operators, the scattering operator and the scattering matrix for the triple (U,U0,J). In particular, we exhibit conditions under which the stationary wave operators and the strong wave operators exist and coincide, and we derive representation formulas for the stationary wave operators and the scattering matrix. As an application, we show that these representation formulas are satisfied for a class of anisotropic quantum walks recently introduced in the literature.

Direct and Inverse Results for Kantorovich Type Exponential Sampling Series

In this article, we analyse the behaviour of the new family of Kantorovich type exponential sampling series. We derive the point-wise approximation theorem and Voronovskaya type theorem for the series $$(I_{w}^{\chi })_{w>0}.$$ Further, we establish a representation formula and an inverse result of approximation for these operators. Finally, we give some examples of kernel functions to which the theory can be applied along with the graphical representation.