Uniqueness Theorem Research Articles

Limit Load and Collapse Mechanism of Spatial Arched Structures

ABSTRACT A Finite Element Method (FEM) based model is presented that can be applied as a limit-analysis tool for spatial arched structures. A purely geometrical model can find the collapse mechanism of an arbitrary arched structure and the limit load which is the lower bound of the collapse load. The model takes into account the formation of new openings (cracks), which gradually reduce the initial stiffness of the structure, but does not consider sliding failure. The limit load is when the structure becomes a mechanism globally or locally. The effect of the discretisation and the residual torsional stiffness of an opened interface are investigated. The model is verified by planar and spatial examples applying the uniqueness theorem of Heyman. The results of the case study highlight the potential of the applied model: a very good approximation can be given of the collapse mechanism of a real 3D structure. The presented model can be used to evaluate existing or theoretical ribbed vault geometries. It can also be a powerful tool for earthquake assessment of historical arched structures as it gives the magnitude of the ground acceleration that transforms the structure into a mechanism.

A uniqueness theorem for meromorphic functions

In this paper, we prove the uniqueness theorem for a special class of meromorphic functions on the complex plane $\mathbb{C}$. In particular, we study the class of meromorphic functions $f$ in the domain $\mathbb{C}\setminus K'$, where $K'$ is the finite set of limit points of simple poles of the function $f$. In this class, we describe non-trivial subclasses in which every function $f$ can be uniquely determined by the residues of the function $f$ at its poles. The result covered in this paper is a part of a problem in a spectral operator theory.

ON HYPERBOLIC INTEGERS

The purpose of this paper is to investigate an ℓ-ring structure of algebraic integers from an arithmetic point of view. We endow the algebra D of hyperbolic numbers with its standard f-algebra structure [7]. We introduce the ring of hyperbolic integers Zh as a sub f-ring of the ring ZD of integers of D. Next, we prove that Zh is the unique, up to ring isomorphism, Archimedean f-ring of quadratic integers. Our study focuses on arithmetic properties of Zh related to its latticeordered structure. We show that many basic properties of the ring of integers Z such as primes, unique factorization theorem and the notions of floor and ceiling functions can be extended to Zh. A surprising fact is that prime numbers seen as hyperbolic integers are semiprimes. We also obtain some properties of hyperbolic Gaussian integers. As an application, we discuss the Dirichlet divisor problem using hyperbolic intervals.

Rotating metrics and new multipole moments from scattering amplitudes in arbitrary dimensions

We compute the vacuum metric generated by a generic rotating object in arbitrary dimensions up to third post-Minkowskian order by computing the classical contribution of scattering amplitudes describing the graviton emission by massive spin-1 particles up to two loops. The solution depends on the mass, angular momenta, and on up to two parameters related to generic quadrupole moments. In D=4 spacetime dimensions, we recover the vacuum Hartle-Thorne solution describing a generic spinning object to second order in the angular momentum, of which the Kerr metric is a particular case obtained for a specific mass quadrupole moment dictated by the uniqueness theorem. At the level of the effective action, the case of minimal couplings corresponds to the Kerr black hole, while any other mass quadrupole moment requires nonminimal couplings. In D>4, the absence of black-hole uniqueness theorems implies that there are multiple spinning black hole solutions with different topology. Using scattering amplitudes, we find a generic solution depending on the mass, angular momenta, the mass quadrupole moment, and a new stress quadrupole moment which does not exist in D=4. As special cases, we recover the Myers-Perry and the single-angular-momentum black ring solutions, to third and first post-Minkowksian order, respectively. Interestingly, at variance with the four-dimensional case, none of these solutions corresponds to the minimal coupling in the effective action. This shows that, from the point of view of scattering amplitudes, black holes are the “simplest” general relativity vacuum solutions only in D=4. Published by the American Physical Society 2024

AN INVERSE PROBLEM FOR A NONLINEAR HYPERBOLIC EQUATION

For a second-order hyperbolic equation with inhomogeneity |u|m−1u, m > 1, a forward and an one-dimensional inverse problems are studied. The inverse problem is devoted to determining the coefficient under heterogeneity. As an additional information, the trace of the derivative with respect to x of the solution to the forward initial-boundary value problem is given at x = 0 on a finite interval. Conditions for the unique solvability of the forward problem are found. For the inverse problem a local existence and uniqueness theorems are established and a stability estimate of its solutions is found.

DETERMINATION OF A KERNEL IN A NONLOCAL PROBLEM FOR THE TIME-FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION

In this work, we consider an inverse problem of determining the kernel of a fractional diffusion equation. The backward problem is the initial-boundary/value problem for this equation with non-local initial and homogeneous Dirichlet conditions. To determine the kernel, an overdetermination condition of the integral form is specified for the solution of the backward problem. Using the Fourier method and an ordinary fractional differential equation with a non-local boundary condition, the inverse problem is reduced to an equivalent problem. Further by using the fixed point argument in suitable Sobolev spaces, the global theorems of existence and uniqueness for the solution of the inverse problem are obtained.

Backward doubly stochastic differential equations and SPDEs with quadratic growth

This paper shows the nonlinear stochastic Feynman–Kac formula holds under quadratic growth. For this, we initiate the study of backward doubly stochastic differential equations (BDSDEs, for short) with quadratic growth. The existence, uniqueness, and comparison theorem for one-dimensional BDSDEs are proved when the generator f(t,Y,Z) grows in Z quadratically and the terminal value is bounded, by introducing innovative approaches. Furthermore, in this framework, we utilize BDSDEs to provide a probabilistic representation of solutions to semilinear stochastic partial differential equations (SPDEs, for short) in Sobolev spaces, and use it to prove the existence and uniqueness of such SPDEs, thereby extending the nonlinear stochastic Feynman–Kac formula for linear growth introduced by Pardoux and Peng (1994).

A universal formula for derivation operators and applications

We give a universal formula describing derivation operators on a Hilbert space for a large class of interpolation methods. It is based on a simple new technique on “critical points” where all the derivations attain the maximum. We deduce from this a version of Kalton uniqueness theorem for such methods, in particular, for the real method. As an application of our ideas is the construction of a weak Hilbert space induced by the real J-method. Previously, such space was only known arising from the complex method. To complete the picture, we show, using a breakthrough of Johnson and Szankowski, nontrivial derivations whose values on the critical points grow to infinity as slowly as we wish.

Uniqueness Theorem for Entire Functions of Exponential Type

Uniqueness Theorem for Entire Functions of Exponential Type

On uniqueness and Somigliana theorem for a thermoelastic theory

The main concern of the present article is based on unified generalized thermoelasticity in an isotropic and homogeneous medium. The uniqueness theorem and Betti type reciprocal principle are derived. Based on the reciprocal theorem, the generalization of the Somigliana and Green theorem is also established in the present context.

Some Boundary Problems, in Sobolev Spaces with Constant Exponents, Governed by the Nonlinear Operator of Plate Vibrations

In this paper, we aim to investigate certain nonlinear boundary problems within Sobolev spaces, where the exponents remain constant. We focus on the dynamically modified operator, incorporating a viscosity term into the nonlinear vibrations of plates. Vibrating plates have a broad range of applications. To address user requirements comprehensively, we've taken into account factors such as the geometric configuration, material density, plate thickness, and Poisson's ratio. After formulating the problems, our method involves converting them into hyperbolic-type nonlinear problems. In this study, we examine six boundary value problems, establishing existence and uniqueness theorems for each. Lastly, we establish the existence of a solution for the stationary problem by employing a variation of Brouwer's fixed point theorem.

On sofic approximations of non amenable groups

In this paper we exhibit for every non amenable group that is initially sub-amenable (sometimes also referred to as LEA), two sofic approximations that are not conjugate by any automorphism of the universal sofic group. This addresses a question of Pǎunescu and generalizes the Elek–Szabo uniqueness theorem for sofic approximations.

Inverse Problem of Determining the Unknown Coefficients in an Elliptic Equation

The inverse problem of determining the coefficients of an elliptic equation under different boundary conditions in a given rectangle is considered. These problems lead to the necessity of approximate solution of inverse problems of mathematical physics, which are incorrect in the classical sense. The existence, uniqueness, and stability theorems for the solution of the set inverse problem are proved and a regularizing algorithm for determining the coefficient is constructed.

Stability estimates for some parabolic inverse problems with the final overdetermination via a new Carleman estimate

This paper is about Hölder and Lipschitz stability estimates and uniqueness theorems for some coefficient inverse problems and associated inverse source problems for a general linear parabolic equation of the second order with variable coefficients. The data for the inverse problem are given at the final moment of time . In addition, both Dirichlet and Neumann boundary conditions are given either on a part or on the entire lateral boundary. Thus, if these boundary conditions are given only at a part of the boundary, then even if the target coefficient is known, still the forward problem is not a classical initial boundary value problem.

Holomorphic curves into projective spaces with some special hypersurfaces

Abstract In this article, we establish some truncated second main theorems for holomorphic curves into projective spaces with some special hypersurfaces and give some applications. In addition, the defect relation, the algebraically degenerate conditions and uniqueness theorem for holomorphic curves with some special divisors may be improved.

New Results for Existence, Uniqueness, and Ulam Stable Theorem to Caputo–Fabrizio Fractional Differential Equations with Periodic Boundary Conditions

New Results for Existence, Uniqueness, and Ulam Stable Theorem to Caputo–Fabrizio Fractional Differential Equations with Periodic Boundary Conditions

New solitary waves in a convecting fluid

It is critical to examine the effects of small perturbations on solitary wave solutions to nonlinear wave equations because of the existence of the unavoidable perturbations in real applications. In this work, we aim to study the solitary wave solutions for the dissipative perturbed mKdV equation which is proposed to model the evolution of shallow wave in a convecting fluid. By using the geometric singular perturbation theorem, the three-dimensional traveling wave system is reduced to a planar dynamical system with regular perturbation. By examining the homoclinic bifurcations via Melnikov method, we show that not only some solitary waves with particularly chosen wave speed persist but also some new type of solitary waves appear under small perturbation. The theoretical analysis results are illustrated by numerical simulations.

All separable supersymmetric AdS5 black holes

We consider the classification of supersymmetric black hole solutions to five-dimensional STU gauged supergravity that admit torus symmetry. This reduces to a problem in toric Kähler geometry on the base space. We introduce the class of separable toric Kähler surfaces that unify product-toric, Calabi-toric and orthotoric Kähler surfaces, together with an associated class of separable 2-forms. We prove that any supersymmetric toric solution that is timelike, with a separable Kähler base space and Maxwell fields, outside a horizon with a compact (locally) spherical cross-section, must be locally isometric to the known black hole or its near-horizon geometry. An essential part of the proof is a near-horizon analysis which shows that the only possible separable Kähler base space is Calabi-toric. In particular, this also implies that our previous black hole uniqueness theorem for minimal gauged supergravity applies to the larger class of separable Kähler base spaces.

Weekly Ill-posed integral geometry problems of Volterra type in three-dimensional space

In this paper considers the problem of recovering a function from families of spheres in space. The uniqueness of the solution of the problem is proved by reducing it to the Volterra integral equation of the first and then the second kind. Fourier transform methods are also used. Uniqueness theorems are proved for some new classes of operator equations of Volterra type in three-dimensional space.

Short-time existence of Ricci flow in standard static spacetimes

We prove the short-time existence of Ricci flow in the class of standard static spacetimes with a closed and compact Riemannian spatial component. The proof constitutes establishing parabolicity of the Ricci-DeTurck flow system which therefore, by existence and uniqueness theorem, admits a short-time solution. Finally pulling this solution back via an appropriate family of diffeomorphisms yields a solution to the original Ricci flow.