Theory Of Differential Operators Research Articles

Breaking symmetries for equivariant coarse homology theories

We describe a symmetry breaking construction in coarse geometry which allows to obtain information about equivariant coarse homology classes by restriction to smaller groups and spaces. In the case of equivariant coarse K-homology theory we give an analytic interpretation of this construction. As a consequence we obtain applications to the spectral theory of invariant differential operators.

Bernstein collocation technique for a class of Sturm-Liouville problems

Sturm-Liouville problems have yielded the biggest achievement in the spectral theory of ordinary differential operators. Sturm-Liouville boundary value issues appear in many key applications in natural sciences. All the eigenvalues for the standard Sturm-Liouville problem are guaranteed to be real and simple, and the related eigenfunctions form a basis in a suitable Hilbert space. This article uses the weighted residual collocation technique to numerically compute the eigenpairs of both regular and singular Strum Liouville problems. Bernstein polynomials over [0,1] has been used to develop a weighted residual collocation approach to achieve an improved accuracy. The properties of Bernstein polynomials and the differentiation formula based on the Bernstein operational matrix are used to simplify the given singular boundary value problems into a matrix-based linear algebraic system. Keeping this fact in mind such a polynomial with space defined collocation scheme has been studied for Strum Liouville problems. The main reasons to use the collocation technique are its affordability, ease of use, well-conditioned matrices, and flexibility. The weighted residual collocation method is found to be more appealing because Bernstein polynomials vanish at the two interval ends, providing better versatility. A multitude of test problems are offered along with computation errors to demonstrate how the suggested method behaves. The numerical algorithm and its applicability to particular situations are described in detail, along with the convergence behavior and precision of the current technique.

Finite dimensional irreducible representations of Lie superalgebra D (2, 1; α)

This paper focuses on the finite dimensional irreducible representations of Lie superalgebra D(2, 1; α). We explicitly construct the finite dimensional representations of the superalgebra D(2, 1; α) by using the shift operator and differential operator representations. Unlike ordinary Lie algebra representation, there are typical and atypical representations for most superalgebras. Therefore, its typical and atypical representation conditions are also given. Our results are expected to be useful for the construction of primary fields of the corresponding current superalgebra of D(2, 1; α).

Integral Representation of Differential Operators

Integral Representation of Differential Operators

On the uniqueness theorems for transmission problems related to models of elasticity, diffusion and electrocardiography

Abstract We consider a generalization of the inverse problem of the electrocardiography in the framework of the theory of elliptic and parabolic differential operators. More precisely, starting with the standard bidomain mathematical model related to the problem of the reconstruction of the transmembrane potential in the myocardium from known body surface potentials, we formulate a more general transmission problem for elliptic and parabolic equations in the Sobolev type spaces and describe conditions, providing uniqueness theorems for its solutions. Next, the new transmission problem is interpreted in the framework of the elasticity theory applied to composite media. Finally, we prove a uniqueness theorem for an evolutionary transmission problem that can be easily adopted to many models involving the diffusion type equations.

The Renormalization Group

The renormalization group was originally introduced as a multiscale approach to quantum field theory and the theory of critical phenomena, explaining in particular the universality observed e.g. in critical exponents. Over the years it has become a powerful tool in the mathematical analysis of systems with infinitely many interacting degrees of freedom. Its applications include quantum field theories, classical and quantum statistical mechanics, (stochastic) partial differential equations, operator theory, and probability theory. For some important problems, it is the only known tool for mathematical proofs. The last few years have seen further important developments, in particular in the application of the method to probabilistic questions, and to equilibrium and non-equilibrium quantum statistical mechanics. This workshop has given an account of the most important new developments in the last five years, including methodical progress, current applications, relations to other approaches, and identified new challenges that may be tackled in future work with the help of the renormalization group.

Behavior as t → ∞ of Solutions of a Mixed Problem for a Hyperbolic Equation with Periodic Coefficients on the Semi-Axis

In this paper, we consider the asymptotic behavior (as t→∞) of solutions as an initial boundary value problem for a second-order hyperbolic equation with periodic coefficients on the semi-axis (x>0). The main approach to studying the problem under consideration is based on the spectral theory of differential operators, as well as on the properties of the spectrum (σ(H0)) of the one-dimensional Schrödinger operator H0 with periodic coefficients p(x) and q(x).

Properties of the conformal Yangian in scalar and gauge field theories

Properties of the SO(2, n) Yangian acting on scalar and gauge fields are presented. This differential operator representation of the infinite-dimensional extension of the conformal algebra SO(2, n) is proved to satisfy the Serre relation for arbitrary spacetime dimension n for off-shell scalar theory, but only on shell and for n = 4 in the gauge theory. The SO(2, n) Yangian acts simply on the off-shell kinematic invariants (kI + kI+1 + …)2, and it annihilates individual off-shell scalar λϕ3 Feynman tree graphs for n = 6 if the differential operator representation is extended by graph dependent evaluation terms. The SO(2, 4) Yangian level one generators are shown to act in a compact way on pure Yang- Mills gluon tree amplitudes. The action of the Yangian on the scattering polynomials of a CHY formalism is also described.

Wavelets based computational algorithms for multidimensional distributed order fractional differential equations with nonlinear source term

In this manuscript we provide effective computational algorithms based on Legendre wavelet (LW), Bernstein wavelet (BW), and standard tau approach to approximate the solution of viscoelasticity damping motion nonlinear distributed order fractional differential equation (VDMNDOFDE) and multi-dimensional distributed order time-fractional nonlinear partial differential equation (DOT-FNPDE). To the best of our understanding, the proposed computational algorithm is new and has not been previously applied for solving VDMNDOFDE and DOT-FNPDE. The matrix representation of distributed order fractional derivatives, integer order derivatives and product term associated with the integral based on LW and BW are established to find the numerical solutions of the proposed VDMNDOFDE and DOT-FNPDE. Moreover, the association of standard tau rule and Legendre-Gauss quadrature (LGQ) techniques along with constructed matrix representation of differential and integral operators diminish VDMNDOFDE and DOT-FNPDE into system of nonlinear algebraic equations. Here, these systems of nonlinear algebraic equations are solved by the Newton's method to get the value of unknown vectors. Error bounds, convergence analysis, numerical algorithms of the VDMNDOFDE and DOT-FNPDE are rigorously investigated. For the reliability of the proposed computational algorithm, numerous test examples have been incorporated in the manuscript to ensure the robustness and theoretical results of proposed technique. Also, the proposed technique found to be more accurate with the existing scheme.

Degenerate kernels of polyharmonic and poly-Helmholtz operators in polar and spherical coordinates

This paper presents the degenerate kernels for the polyharmonic and poly-Helmholtz partial differential operators in the polar and spherical coordinates. These degenerate kernels are essential if the analytical/semi-analytical solutions and the mathematical degeneracies involving these operators are conducted by the boundary integral equations. In addition, a two-dimensional creeping flow problem is considered to illustrate the use of the degenerate kernels for problems governed by coupled partial differential equations. The coupled governing equations of the creeping flow problems are converted into a biharmonic equation by using the Hörmander linear partial differential operator theory and the Cartesian partial derivatives on the harmonic and biharmonic degenerate kernels are used to obtain the degenerate kernels of two-dimensional creeping flow problems for BIE analyses. Finally, numerical experiments are conducted to study the convergence of the polyharmonic and poly-Helmholtz degenerate kernels.

Time-Periodic Thermal Boundary Effects on Porous Media Saturated with Nanofluids: CGLE Model for Oscillatory Mode

Abstract The stability of nonlinear nanofluid convection is examined using the complex matrix differential operator theory. With the help of finite amplitude analysis, nonlinear convection in a porous medium is investigated that has been saturated with nanofluid and subjected to thermal modulation. The complex Ginzburg-Landau equation (CGLE) is used to determine the finite amplitude convection in order to evaluate heat and mass transfer. The small amplitude of convection is considered to determine heat and mass transfer through the porous medium. Thermal modulation of the system is predicted to change sinusoidally over time, as shown at the boundary. Three distinct modulations IPM, OPM, and LBMOhave been investigated and found that OPM and LBMO cases are used to regulate heat and mass transfer. Further, it is found that modulation frequency (ω f varying from 2 to 70) reduces heat and mass transfer while modulation amplitude (δ 1 varying from 0.1 to 0.5 ) enhances both.

Asymptotic Behavior of Solutions of the Cauchy Problem for a Hyperbolic Equation with Periodic Coefficients II

The paper is devoted to studying the behavior of solutions of the Cauchy problem for large values of time—more precisely, obtaining an asymptotic expansion characterizing the behavior of the solution of the Cauchy problem for a one-dimensional second-order hyperbolic equation with periodic coefficients for large values of the time parameter t. To obtain this asymptotic expansion as t→∞, methods of the spectral theory of differential operators are used, as well as the properties of the spectrum of a non-positive Hill operator with periodic coefficients.

The Completeness of Classical Electromagnetic Theory

The core of classical electromagnetic field theory is Maxwell electromagnetic field equations. Each of its formulas is supported by experimental facts, and its mathematical logic is rigorous and self-consistent, which has been tested for a long time in application.Therefore, its correctness is beyond doubt. In this paper, on the premise of affirming the great significance and important contribution of Maxwell equations(ME), the possible problems and solutions are discussed. Since ME and its derived wave equation (MWE) are vectorized partial differential equations, it is difficult to solve them strictly. In this paper, the solutions of vectorized ME and vectorized MWE are discussed in depth, and the necessity of Hansen function analysis is illustrated before the concrete calculation of engineering structures. Another method is to use vector partial differential operator theory to derive the standard basic equations expressed by state functions, and change the traditional calculation in Euclid space to the calculation in wave function space. Singularities and singularity are also avoided in the process. This paper argues that the completeness and modernity of traditional electromagnetic theory can only be improved by introducing the concept of particle, because the wave field processed by ME is actually a quantum field of light. The motion equation of electromagnetic field in classical theory is ME. In quantum theory, the field vector is an operator, governed by ME. The photon theory successfully explained the photoelectric effect in a way that ME could not. But in the mainstream theory, the photon rest mass is equal to zero, which blocks the way to add particle concept to ME. In 1936, Proca equations assumed ≠0, and with the support of quantum electrodynamics(QED), ME was improved. In this paper, the Proca equations are approved and the Proca wave equation (PWE) is further derived. In short, it reflects the adherence to the concept of wave-particle duality. The extended Maxwell equations proposed by Chinese scientist recently should be emphasized and discussed in this paper.

Asymptotic Behavior of Solutions of the Cauchy Problem for a Hyperbolic Equation with Periodic Coefficients (Case: H0 > 0)

The main goal of this article is to study the behavior of solutions of non-stationary problems at large timescales, namely, to obtain an asymptotic expansion characterizing the behavior of the solution of the Cauchy problem for a one-dimensional second-order hyperbolic equation with periodic coefficients at large values of the time parameter t. To obtain an asymptotic expansion as t→∞, the basic methods of the spectral theory of differential operators are used, as well as the properties of the spectrum of the Hill operator with periodic coefficients in the case when the operator is positive: H0>0.

Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations

We study linear transformations \(T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]\) of the form \(T[x^n]=P_n(x)\) where \(\{P_n(x)\}\) is a real orthogonal polynomial system. With \(T=\sum \tfrac{Q_k(x)}{k!}D^k\), we seek to understand the behavior of the transformation T by studying the roots of the \(Q_k(x)\). We prove four main things. First, we show that the only case where the \(Q_k(x)\) are constant and \(\{P_n(x)\}\) is an orthogonal system is when the \(P_n(x)\) form a shifted set of generalized probabilist Hermite polynomials. Second, we show that the coefficient polynomials \(Q_k(x)\) have real roots when the \(P_n(x)\) are the physicist Hermite polynomials or the Laguerre polynomials. Next, we show that in these cases, the roots of successive polynomials strictly interlace, a property that has not yet been studied for coefficient polynomials. We conclude by discussing the Chebyshev and Legendre polynomials, proving a conjecture of Chasse, and presenting several open problems.

DETERMINATION OF THE TRANSMISSION LINE RESISTANCE MATRIX WITH DEVIATIONS OF DESIGN PARAMETERS FROM NOMINAL

Context. UHF transmission systems make extensive use of transmission line segments, the characteristics of which have a significant impact on the performance of various information technologies. One of the problems of production of transmission lines is to obtain a given wave impedance, which significantly affects the electrical and information characteristics of the entire set of equipment. Currently, there is a burning issue of estimating the influence of disturbing factors on various electrical characteristics of long line segments. To date, the most fully developed methods for assessing the effect of disturbing factors on the wave impedance of a homogeneous line (the wave impedance is constant) under regular perturbations. In this case, the influence of perturbations on the reflection coefficient of matched lines was mainly considered. The effect of perturbations on the other characteristics of homogeneous and, especially, inhomogeneous lines has not been sufficiently studie
 Objective. The purpose of this paper is to determine the effect of wave impedance perturbations on the transmission line impedance matrix. Knowing the perturbed impedance matrix, it is possible to determine the distortion of the characteristics of any device built on transmission line segments. 
 Method. The paper uses the method of perturbation theory of linear differential operators applied to equations describing processes in inhomogeneous long lines.
 Results. The obtained results make it possible to estimate the influence of regular and irregular perturbations of the wave resistance (wave conductance) on the transmission line matrix considered as a quadrupole. Such matrix can be any quadrupole matrix: resistance matrix, conductance matrix, circuit matrix. This makes it possible, according to the desired function of the circuit (gain, input impedance, reflection coefficient), to determine the allowable deviation of the wave impedance from the nominal value in order to select a tolerance for reproducing the wave impedance. 
 Conclusions. The proposed criterion for estimating line parameter deviations using the norm of the four-pole matrix is inherently an integral criterion and can be used to preliminarily estimate the frequency domain of the strongest distortions, regardless of the functional purpose of the transmission line segment. The developed approach is applicable to both homogeneous and heterogeneous transmission lines and covers both regular and irregular wave impedance perturbations.

Spectral properties of some differential operators of Sturm-Liouville type with homogeneous delay

In this paper we observe the operator $ D^{2}=D^{2}(h,H,q,\alpha)$, $h, H\in \overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,\infty\}$, $q(x)\in L_{2}\left[0,\pi \right]$, $\alpha\in (0,1) $ and construct and partially transform its characteristic function. Those transformations enable more complete asymptotic decomposition of the zeroes and eigenvalues of the operator. The goal of this paper is to contribute to the development of the spectral theory of differential operators with homogeneous delay.

The killed Brox diffusion

We are proposing to study a diffusion in random environment confined in bounded interval. As for many standard diffusions, we build in natural way a stochastic process with bounded domain, but with addition of considering a random environment. We carry out this construction using the Brox diffusion and applying well-known diffusion theory in a quenched fashion, which is a natural way to deal with random environment. The outcome of this procedure is an object that we may call the killed Brox diffusion. Since the generator of this process is initially an ill-posed expression we develop a Sturm–Liouville theory for one-dimensional second-order differential operators with white-noise coefficients. Our first main result is to give a close form of the Green operator associated to the generator, i.e., the inverse of the generator. We do so by setting the Lagrange identity in this context. Then, we give explicit expressions in quenched form of the probability density function of the process; such an object is given theoretically in terms of the spectral decomposition using the eigenvalues and eigenfuntions of the infinitesimal generator of the diffusion. Moreover, we characterize the eigenvalues and eigenfunctions using some integro-differential equations.

A study on the effect of temperature modulation on Darcy–Bénard convection using a local thermal non-equilibrium model

The criterion for the onset of Darcy–Bénard convection is analyzed when the fluid and porous medium are out of thermal equilibrium and the temperatures of the boundaries vary sinusoidally with time in either a synchronous or an asynchronous manner. A stability analysis of the linearized governing equations is performed by using the matrix differential operator theory. The shift in the critical Darcy–Rayleigh number is evaluated in terms of system parameters, and the effect of those on the shift is depicted graphically to realize the significant effect of temperature modulation on the onset, especially when the thermal non-equilibrium effects are prominent.

The Witten index and the spectral shift function

In [Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976) 71–99] Atiyah, Patodi and Singer introduced spectral flow for elliptic operators on odd-dimensional compact manifolds. They argued that it could be computed from the Fredholm index of an elliptic operator on a manifold of one higher dimension. A general proof of this fact was produced by Robbin–Salamon [The spectral flow and the Maslov index, Bull. London Math. Soc. 27(1) (1995) 1–33, MR 1331677]. In [F. Gesztesy, Y. Latushkin, K. Makarov, F. Sukochev and Y. Tomilov, The index formula and the spectral shift function for relatively trace class perturbations, Adv. Math. 227(1) (2011) 319–420, MR 2782197], a start was made on extending these ideas to operators with some essential spectrum as occurs on non-compact manifolds. The new ingredient introduced there was to exploit scattering theory following the fundamental paper [A. Pushnitski, The spectral flow, the Fredholm index, and the spectral shift function, in Spectral Theory of Differential Operators, American Mathematical Society Translations: Series 2, Vol. 225 (American Mathematical Society, Providence, RI, 2008), pp. 141–155, MR 2509781]. These results do not apply to differential operators directly, only to pseudo-differential operators on manifolds, due to the restrictive assumption that spectral flow is considered between an operator and its perturbation by a relatively trace-class operator. In this paper, we extend the main results of these earlier papers to spectral flow between an operator and a perturbation satisfying a higher pth Schatten class condition for [Formula: see text], thus allowing differential operators on manifolds of any dimension [Formula: see text]. In fact our main result does not assume any ellipticity or Fredholm properties at all and proves an operator theoretic trace formula motivated by [M.-T. Benameur, A. Carey, J. Phillips, A. Rennie, F. Sukochev and K. Wojciechowski, An analytic approach to spectral flow in von Neumann algebras, in Analysis, Geometry and Topology of Elliptic Operators (World Scientific Publisher, Hackensack, NJ, 2006), pp. 297–352, MR 2246773; A. Carey, H. Grosse and J. Kaad, On a spectral flow formula for the homological index, Adv. Math. 289 (2016) 1106–1156, MR 3439708]. We illustrate our results using Dirac type operators on [Formula: see text] for arbitrary [Formula: see text] (see Sec. 8). In this setting Theorem 6.4 substantially extends Theorem 3.5 of [A. Carey, F. Gesztesy, H. Grosse, G. Levitina, D. Potapov, F. Sukochev and D. Zanin, Trace formulas for a class of non-Fredholm operators: a review, Rev. Math. Phys. 28(10) (2016) 1630002, MR 3572626], where the case d = 1 was treated.