Uniqueness Theorem Research Articles

Recovery of a coefficient in a diffusion equation from large time data

Abstract This paper considers the determination of a spatially varying coefficient in a parabolic equation from time trace data. There are many uniqueness theorems known for such problems but the reconstruction step is severally ill-posed: essentially the problem comes down to trying to reconstruct an analytic function from values on a strip. However, we look at an even more restricted data where the measurements are not made on the whole time axis but only for large values adding further to the ill-conditioning situation. In addition, we do not assume the initial state is known. Uniqueness is restored by making changes to the boundary condition, in particular, to the impedance parameter, for each of a series of measurements. We show that an implementation of the above paradigm leads to both uniqueness and an effective reconstruction algorithm. Extension is also made to the case of fractional model and to replacing the parabolic equation with a damped wave equation.

On the Laplace equation on bounded subanalytic manifolds

We prove a trace formula for integration by parts on subanalytic bounded submanifolds of Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document}, possibly non closed. We also establish density results for W∇1,p(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{W}}^{1,p}_\ abla (M)$$\\end{document}, M bounded subanalytic manifold, which is the space of the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} tangent vector fields v on M for which ∇v\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ abla v$$\\end{document} is Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}, where ∇\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ abla $$\\end{document} is the divergence operator. We derive from these results some theorems of existence and uniqueness of solutions of the Laplace equation with Dirichlet and Neumann boundary type conditions. We then study the p-Laplace equation, for p∈[1,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in [1,\\infty )$$\\end{document} large.

Hopf bifurcation control of memristor-based fractional delayed tri-diagonal bidirectional associative memory neural networks under various controllers

In this paper, authors present a memristor-based fractional order system of tri-diagonal bidirectional associative memory neural networks (TdBAMNNs) incorporating leakage and communication delays. The existence and uniqueness theorem is established for the memristor-based fractional-order TdBAMNNs system. Analysis of Hopf bifurcation anti-control is conducted using various feedback controllers by exploring single-parameter and double-parameter approaches. A study of the stability switching curves is undertaken to determine the direction of stability transitions in the system. The theoretical results and graphical illustrations are validated through numerical simulations to exhibit their feasibility. Comparative analysis of the system’s dynamical behavior under different controllers is performed. This paper comprehensively explores the proposed memristor-based system, proving its theoretical underpinnings, and analyzing stability through bifurcation control strategies and comparative assessments under different control scenarios.

CHAOTIC REGIMES GENERATED BY SPECIAL POINTS AND SINGULAR SOLUTIONS

The role of singular solutions and some singular points of ordinary differential equations with respect to the occurrence of chaotic regimes in physical systems is considered. It is shown that in those physical systems in which singular solutions exist, they can cause the occurrence of chaotic regimes. Examples of such systems are given. Of particular interest are the results showing that chaotic regimes can occur even in systems with one degree of freedom and even in completely integrable systems. These results, at first glance, violate known mathematical theorems (Poincare-Bendixson), and also destroy the paradigms of dynamic chaos. It is shown that this contradiction is an apparent contradiction. The reason for the appearance of chaotic dynamics in these cases are special solutions. At the points of this solution, the conditions of the uniqueness theorem are violated. In numerical calculations, this is manifested in the fact that a random component (numerical noise) appears in the calculations. As a result, the system under study becomes a system with 1.5 degrees of freedom. It is shown that in almost all cases, increasing the accuracy of the calculations allows suppressing chaotic dynamics in systems with one degree of freedom.

Lagrange expansion formula and its application in combinatorial mathematics

There is a uniqueness theorem for an implicit function. When a function satisfies certain conditions, the equation can determine a unique implicit function. However, we cannot determine the display expression of the implicit function according to this theorem. The Lagrange expansion formula just compensates for this disadvantage, which explicitly gives the display expression of the implicit function. This paper will give a simpler proof of the Lagrange expansion formula and provide examples to illustrate its application in combinatorial mathematics.

Black hole spectroscopy with ground-based atom interferometer and space-based laser interferometer gravitational wave detectors

Gravitational wave (GW) detection allows us to test general relativity in entirely new regimes. A prominent role takes the detection of quasi-normal modes (QNMs), which are emitted after the merger of a binary black hole (BBH) when the highly distorted remnant emits GWs to become a regular Kerr black hole (BH). The BH uniqueness theorems of Kerr black hole solutions in general relativity imply that the frequencies and damping times of QNMs are determined solely by the mass and spin of the remnant BH. Therefore, detecting QNMs offers a unique way to probe the nature of the remnant BH and to test general relativity. We study the detection of a merging BBH in the intermediate-mass range, where the inspiral–merger phase is detected by space-based laser interferometer detectors TianQin and LISA, while the ringdown is detected by the ground-based atom interferometer (AI) observatory AION. The analysis of the ringdown is done using the regular broadband mode of AI detectors as well as the resonant mode optimizing it to the frequencies of the QNMs predicted from the inspiral–merger phase. We find that the regular broadband mode allows constraining the parameters of the BBH with relative errors of the order 10−1 and below from the ringdown. Moreover, for a variety of systems considered, the frequencies and the damping times of the QNMs can be determined with relative errors below 0.1 and 0.2, respectively. We further find that using the resonant mode can improve the parameter estimation for the BBH from the ringdown by a factor of up to three. Utilizing the resonant mode significantly limits the detection of the frequency of the QNMs but improves the detection error of the damping times by around two orders of magnitude.

A uniqueness theorem for orthonormal spline series

A uniqueness theorem for orthonormal spline series

On the uniqueness and computation of commuting extensions

A tuple (Z1,…,Zp) of matrices of size r×r is said to be a commuting extension of a tuple (A1,…,Ap) of matrices of size n×n if the Zi pairwise commute and each Ai sits in the upper left corner of a block decomposition of Zi (here, r and n are two arbitrary integers with n<r). This notion was discovered and rediscovered in several contexts including algebraic complexity theory (in Strassen's work on tensor rank), in numerical analysis for the construction of cubature formulas and in quantum mechanics for the study of computational methods and the study of the so-called “quantum Zeno dynamics.” Commuting extensions have also attracted the attention of the linear algebra community. In this paper we present 3 types of results:(i)Theorems on the uniqueness of commuting extensions for three matrices or more.(ii)Algorithms for the computation of commuting extensions of minimal size. These algorithms work under the same assumptions as our uniqueness theorems. They are applicable up to r=4n/3, and are apparently the first provably efficient algorithms for this problem applicable beyond r=n+1.(iii)A genericity theorem showing that our algorithms and uniqueness theorems can be applied to a wide range of input matrices.

Local and Global Uniqueness Theorems of the N-th Order Partial Differential Equations

In this paper, we consider inequalities in which the function is an element of n-th partially order space. Local and Global uniqueness theorem of solutions of the n-the order Partial differential equation Obtained which are applications of Gronwall's inequalities.

Existence and uniqueness theorems for nonlinear coupled boundary value problem of the A B C fractional differential equation

Existence and uniqueness theorems for nonlinear coupled boundary value problem of the A B C fractional differential equation

Application of the Shape Uniqueness Theorem to the H-Bézier Curve

Application of the Shape Uniqueness Theorem to the H-Bézier Curve

Numerical simulation of an effective transform mechanism with convergence analysis of the fractional diffusion-wave equations

In the current study, we solve two very important mathematical models, such as the time fractional-order space-fractional telegraph and diffusion-wave equations using a reliable technique called the Adomian decomposition natural method (ADNM), which combines Adomian decomposition and natural transform. The diffusion wave equation describes the flood wave propagation, which is used in solving overland and open channel flow problems. For this reason, it is critical to fully understand and effectively solve the diffusion wave equations. Because telegraph equations are crucial for modeling and developing voltage or frequency transmission, they are widely used in physics and engineering. Furthermore, the designing process is greatly impacted by the uncertainty in the system parameters. For nonlinear ordinary differential equations based on the theorem of Banach fixed point, we provide existence and uniqueness theorem proofs. The present approach has been successfully used to explore exact solutions for time fractional-order and space fractional-order applications. The results show how effective and valuable the ADNM. This paper presents a methodology that will be used in future work to address similar nonlinear problems related to fractional calculus.

A Uniqueness Theorem for Stability Problems of Functional Equations

In this paper, we present a uniqueness theorem obtained by using direct calculation. This theorem is applicable to stability problems of functional equations whose solutions are monomial or generalized polynomial mappings of degree n. The advantage of this uniqueness theorem is that it simplifies the proof by eliminating the need to repeatedly and cumbersomely prove uniqueness in stability studies.

A Note on Injective Factors with Trivial Bicentralizer

We give an alternative proof that an injective factor on a Hilbert space with trivial bicentralizer is an infinite tensor product of factors of finte type I (ITPFI factor). Our proof is given in parallel with each type of factor and it is based on the strategy of Haagerup. As a consequence, the uniqueness theorem of injective factors, except for type III _{0} , follows from Araki–Woods’ result.

On the (k,φ)-Hilfer Langevin fractional coupled system having multi point boundary conditions and fractional integrals

In this manuscript, we investigate the (k,φ)-Hilfer Langevin fractional coupled system having multi point boundary conditions and fractional integrals. This study is crucial as it addresses the complexities of (k,φ)- Hilfer Langevin coupled system with multi point conditions, which have applications in many scientific and engineering fields. Unlike previous research, our work introduces novel techniques for analyzing these system, leading to more accurate and comprehensive results. The major findings include new existence and uniqueness theorems, along with improved stability conditions, which enhance the understanding and potential applications of these systems. We demonstrate the existence, uniqueness and different kinds of Ulam stability for our suggested model. We prove the uniqueness of the solution by utilizing Banach contraction mapping principle. The existence of solution is studied by using Krasnoselskii's fixed point theorem. We also explore the different types of Ulam stability under the specific conditions. We presented an example at the end to support our main results.

Toward singularity theorems with torsion

Toward singularity theorems with torsion

Dynamical mixed boundary-transmission problems of the generalized thermo-electro-magneto-elasticity theory for composed structures

Abstract In the present paper we investigate three-dimensional dynamical mixed boundary-transmission problems for composed body consisting of two adjacent anisotropic elastic components having a common interface surface. The two contacting elastic components are subject to different mathematical models: Green–Lindsay’s model of generalized thermo-electro-magneto-elasticity in one component and Green–Lindsay’s model of generalized thermo-elasticity in the other one. We assume that the composed elastic structure under consideration contains an interfacial crack. We prove the uniqueness and existence theorems in appropriate function spaces.

An one-dimensional inverse problem for the wave equation

For the wave equation with inhomogeneity σ(x)um t + q(x)up a forward and an one-dimensional inverse problems are studied. Here m &gt; 1 and p &gt; 1 are real numbers. The forward problem is considered in the domain x &gt; 0,t &gt; 0 with zero initial data and Dirichlet boundary condition at x = 0. An unique solvability theorem of this problem is proved. The inverse problem is devoted to determining the coefficients σ(x) and q(x). As an additional information for recovering this coefficients, two forward problems with different Dirichlet data are considered and traces of the derivative of their solutions with respect to x are given at x =0 on a finite interval. For the inverse problem a local existence and uniqueness theorem is established.

A uniqueness theorem for holomorphic curves on annulus sharing hypersurfaces

Recently, some authors have given the uniqueness theorems for the holomorphic curves on an annulus in hyperplanes in general position and hypersurfaces in general position for Veronese embedding. In this paper, we will give a uniqueness theorem in hypersurfaces in general position.

Inverse problem for wave equation of memory type with acoustic boundary conditions

In this article, for the direct problem, which consists of finding the velocity potential and the displacement of boundary points in the wave equation of memory type with initial and boundary acoustic conditions, the inverse problem of determining the memory kernel by the integral overdetermination condition is studied. By introducing a new function, the problem is reduced to a problem with homogeneous boundary conditions. Using the technique of estimating integral equations and the contraction mappings principle in Sobolev spaces, the existence and uniqueness theorem for the inverse problem is proved.