Uniqueness Properties Research Articles

A generalized inf–sup test for multi-field mixed-variational methods

We present a variational approach towards identifying conditions for the stability of Galerkin methods for multi-field saddle point problems in continuum mechanics and continuum thermodynamics with free energy functions that have positive second variations at critical points. The framework aims to generalize the discrete inf–sup theory and its numerical verification in the context of general problems in an arbitrary number of fields for both linear and nonlinear problems. In utilizing a linearized second derivative test the proposed scheme is purely based on uniqueness properties of a mixed Lagrangian around the solution, thus providing a purely variational requirement for the well-posedness of Galerkin approximations. In maintaining the dependence on parameters and in incorporating the notion of multifold inf–sup conditions the scheme provides a generalization of the classical LBB theory that allows the study of multi-field saddle point problems in the limit of vanishing parameters and their connection to generalized numerical eigenvalue tests. The proposed generalization is trusted to provide a helpful tool for the development of mixed methods that arise in many novel engineering problems due to the coupling of multiple irreversible phenomena or the incorporation of Lagrange multipliers in advanced methods. An emphasis is given with regard to time-dependent problems in irreversible thermodynamics that arise from Biot-type variational formulations if conjugate variables are employed by means of a Legendre–Fenchel transformation. Examples are given for a two-field variational principle in finite deformation poroelasticity in the undrained limit, a three-field variational principle for elasticity in the incompressible limit, and a recent four-field variational principle in gradient-extended plasticity.

RETRACTED ARTICLE: A uniqueness lemma with applications to regularization and incompressible fluid mechanics

In this paper, we extend our previous result from Levy (2016). We prove that transport equations with rough coefficients do possess a uniqueness property, even in the presence of viscosity. Our method relies strongly on duality and bears a strong resemblance to the well-known DiPerna-Lions theory first developed by DiPerna and Lions (1989). This uniqueness result allows us to reprove the celebrated theorem of Serrin (1962) in a novel way. As a byproduct of the techniques, we derive an L1 bound for the vorticity in terms of a critical Lebesgue norm of the velocity field. We also show that the zero solution is unique for the 2D Euler equations on the torus under a mild integrability assumption.

Optimal condition for blow-up of the critical Lq norm for the semilinear heat equation

We shed light on a long-standing open question for the semilinear heat equation ut=Δu+|u|p−1u. Namely, without any restriction on the exponent p>1 nor on the smooth domain Ω, we prove that the critical Lq norm blows up whenever the solution undergoes type I blow-up. A similar property is also obtained for the local critical Lq norm near any blow-up point.In view of recent results of existence of type II blow-up solutions with bounded critical Lq norm, which are counter-examples to the open question, our result seems to be essentially the best possible result in general setting. This close connection between type I blow-up and critical Lq norm blow-up appears to be a completely new observation.Our proof is rather involved and requires the combination of various ingredients. It is based on analysis in similarity variables and suitable rescaling arguments, combined with backward uniqueness and unique continuation properties for parabolic equations.As a by-product, we obtain the nonexistence of self-similar profiles in the critical Lq space. Such properties were up to now only known for p≤pS and in radially symmetric case for p>pS, where pS is the Sobolev exponent.

Steady States in Search-and-Matching Models

Most of the literature that studies frictional search-and-matching models with heterogeneous agents and random search investigates steady-state equilibria. Steady-state equilibrium requires, in particular, that the flows of agents into and out of the population of unmatched agents balance. We investigate the structure of this balance condition, taking agents' matching behavior as given. Building on the fundamental matching lemma for quadratic search technologies in Shimer and Smith (2000), we establish existence, uniqueness, and comparative-static properties of the solution to the balance condition for any search technology satisfying minimal regularity conditions. Implications for the existence and structure of steady-state equilibria are noted.

Affine Volterra processes

We introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classical affine diffusions constitute a special case, but affine Volterra processes are neither semimartingales, nor Markov processes in general. We provide explicit exponential-affine representations of the Fourier–Laplace functional in terms of the solution of an associated system of deterministic integral equations of convolution type, extending well-known formulas for classical affine diffusions. For specific state spaces, we prove existence, uniqueness, and invariance properties of solutions of the corresponding stochastic convolution equations. Our arguments avoid infinite-dimensional stochastic analysis as well as stochastic integration with respect to non-semimartingales, relying instead on tools from the theory of finite-dimensional deterministic convolution equations. Our findings generalize and clarify recent results in the literature on rough volatility models in finance.

Stability and uniqueness properties of Taylor approximations of matrix functions

We consider Taylor approximations of matrix functions and establish bounds controlling the size of a perturbation in terms of the size of the Taylor remainder or, equivalently, the associated spectral shift function(s). This can be viewed as a stability property for the inverse problem of recovering the perturbation from the spectral shift function(s). From our bounds we derive a uniqueness result for spectral sums: if the trace of the Taylor remainder of order n≥2 equals zero for a finite number of monomials, then the perturbation is zero. This uniqueness result is in contrast with nonuniqueness of the first order Taylor remainder. As an application we characterize equality of two graphs in terms of self-returning walks.

A uniqueness property of general Dirichlet series

Let F(s)=∑nan/λns be a general Dirichlet series which is absolutely convergent on ℜ(s)>1. Assume that F(s) has an analytic continuation and satisfies a growth condition, which gives rise to certain invariants namely the degree dF and conductor αF. In this paper, we show that there are at most 2dF general Dirichlet series with a given degree dF, conductor αF and residue ρF at s=1. As a corollary, we get that elements in the extended Selberg class with positive Dirichlet coefficients are determined by their degree, conductor and the residue at s=1.

Backward uniqueness of 3D MHD‐α model driven by linear multiplicative Gaussian noise

In this paper, we devote to proving the backward uniqueness property of the solution to three‐dimensional stochastic magnetohydrodynamic‐α model (3D stochastic MHD‐α model) driven by linear multiplicative Gaussian noise, which involves not only the study of multiplicative noise but also the challenging nonlinear drift terms.

Uniqueness and local properties of positive solutions to a semilinear elliptic equation with singular potentials

In this paper, we study the uniqueness and local behavior at the origin of positive solutions to [Formula: see text] where [Formula: see text], [Formula: see text]) is a bounded smooth domain and [Formula: see text], and [Formula: see text] is a nonnegative continuous function over [Formula: see text]. In the two cases: (i) [Formula: see text], [Formula: see text] and [Formula: see text] on [Formula: see text]; (ii) [Formula: see text], [Formula: see text] and [Formula: see text] on [Formula: see text], we establish the uniqueness of positive solutions and the exact blow-up rate of positive solutions at the origin. This paper seems to be the first one to deal with the latter case.

Heat kernels of non-symmetric Lévy-type operators

We construct the fundamental solution (the heat kernel) pκ to the equation ∂t=Lκ, where under certain assumptions the operator Lκ takes one of the following forms,Lκf(x):=∫Rd(f(x+z)−f(x)−1|z|<1〈z,∇f(x)〉)κ(x,z)J(z)dz,Lκf(x):=∫Rd(f(x+z)−f(x))κ(x,z)J(z)dz,Lκf(x):=12∫Rd(f(x+z)+f(x−z)−2f(x))κ(x,z)J(z)dz. In particular, J:Rd→[0,∞] is a Lévy density, i.e., ∫Rd(1∧|x|2)J(x)dx<∞. The function κ(x,z) is assumed to be Borel measurable on Rd×Rd satisfying 0<κ0⩽κ(x,z)⩽κ1, and |κ(x,z)−κ(y,z)|⩽κ2|x−y|β for some β∈(0,1). We prove the uniqueness, estimates, regularity and other qualitative properties of pκ.

Quantitative invertibility and approximation for the truncated Hilbert and Riesz transforms

In this article we derive quantitative uniqueness and approximation properties for (perturbations) of Riesz transforms. Seeking to provide robust arguments, we adopt a PDE point of view and realize our operators as harmonic extensions, which makes the problem accessible to PDE tools. In this context we then invoke quantitative propagation of smallness estimates in combination with qualitative Runge approximation results. These results can be viewed as quantifications of the approximation properties which have recently gained prominence in the context of nonlocal operators.

A PUF-based mutual authentication scheme for Cloud-Edges IoT systems

Abstract With the exponential growth of generated and processed Internet of Things (IoT) data, the Cloud-Edges (CE) paradigm, which distributes part of the intelligence and computation burden among edge nodes while adopting cloud services only for heavyweight computation, has emerged as a valuable solution to lower latency and improve efficiency. Despite the clear benefits, the lack of centralized control and the exposure of user sensitive data make CE architectures particularly prone to data tampering and spoofing attacks, thus urging the implementation of effective security mechanisms, including in particular authentication. Unfortunately, due to the relatively small computing resources of involved nodes, implementing traditional cryptography-based security solutions is often not viable. To overcome this issue, in this paper we propose a mutual authentication scheme relying upon the adoption of Physically Unclonable Functions (PUFs), special integrated circuits provided with unclonability, uniqueness and tamper-evident properties. The proposed scheme was specifically designed to meet the computation and storage capabilities of the different devices that are typically involved in a CE system, and was proven to be effective against several known attacks while introducing a relatively low overhead.

Mathematical modeling for the impacts of deforestation on wildlife species using Caputo differential operator

Abstract This research study shows the impacts of deforestation on wildlife species using a newly proposed four dimensional nonlinear mathematical model based upon fractional-order ordinary differential equations. Being a nonlinear model, some theorems using fixed point theory have been proved showing the existence and uniqueness properties for the solution of the fractional-order model. Using an explicit version of Adams–Bashforth–Moulton method devised for the fractional-order ordinary differential equations with convergence order p = min ( 1 + τ , 2 ) , where τ is the order of the differential equations used in the model; some numerical simulations in the form of graphical illustrations have been carried out depicting the better performance of the fractional-order model for being capable enough to capture all the history information of the system under consideration which is a phenomenon not found in the classical (integer-order) differential equations. Varying values for both the fractional-order parameter τ and parameters of the model itself are used during the required numerical simulations.

Decoupling multivariate polynomials: Interconnections between tensorizations

Abstract Decoupling multivariate polynomials is useful for obtaining an insight into the workings of a nonlinear mapping, performing parameter reduction, or approximating nonlinear functions. Several different tensor-based approaches have been proposed independently for this task, involving different tensor representations of the functions, and ultimately leading to a canonical polyadic decomposition. We first show that the involved tensors are related by a linear transformation, and that their CP decompositions and uniqueness properties are closely related. This connection provides a way to better assess which of the methods should be favored in certain problem settings, and may be a starting point to unify the approaches. Second, we show that taking into account the previously ignored intrinsic structure in the tensor decompositions improves the uniqueness properties of the decompositions and thus enlarges the applicability range of the methods.

On algebraic and uniqueness properties of harmonic quaternion fields on 3d manifolds

Let Ω be a smooth compact oriented 3-dimensional Riemannian manifold with boundary. A quaternion field is a pair q = {α, u} of a function α and a vector field u on Ω. A field q is harmonic if α, u are continuous in Ω and ∇α = rot u, div u = 0 holds into Ω. The space 𝒞(Ω) of harmonic fields is a subspace of the Banach algebra 𝒬 (Ω) of continuous quaternion fields with the point-wise multiplication qq′ = {αα′ − u · u ′ , αu′ + α ′u + u ∧ u ′ }. We prove a Stone-Weierstrass type theorem: the subalgebra ∨𝒞(Ω) generated by harmonic fields is dense in 𝒬 (Ω). Some results on 2-jets of harmonic functions and the uniqueness sets of harmonic fields are provided. Comprehensive study of harmonic fields is motivated by possible applications to inverse problems of mathematical physics.

MIMO multi-relay systems with tensor space-time coding based on coupled nested Tucker decomposition

In this paper, we propose a two-hop MIMO multi-relay system with tensor space-time coding (TSTC) at the source and the relays. The multiple relays use orthogonal channels (parallel relaying) to increase the diversity order, assuming that all the relays can communicate directly with the destination. The signals received at destination form a fifth-order tensor that satisfies a new tensor model, called coupled nested Tucker decomposition (CNTD). This model is a generalization of the nested Tucker decomposition (NTD) and extends the coupling concept, introduced for PARAFAC models, to Tucker-based decompositions. The CNTD consists of a coupling of NTD models that share a common factor, resulting in the concatenation of a generalized Tucker model with a Tucker one. This tensor modeling is exploited to derive a semi-blind receiver for jointly estimating the information symbols and the channels. The uniqueness property of the proposed tensor model and the identifiability conditions of the receiver are established. Monte Carlo simulation results are provided to illustrate the effectiveness of the proposed system and the advantage of exploiting cooperative diversity based on tensor modeling.

Uniqueness results for Zakharov-Kuznetsov equation

In this paper we study uniqueness properties of solutions to the Zakharov-Kuznetsov equation of plasma physic. Given two sufficiently regular solutions u1, u2, we prove that, if decays fast enough at two distinct times, then

Solvability of the clamped Euler–Bernoulli beam equation

In this study, solvability of the initial boundary value problem for general form Euler–Bernoulli beam equation which includes also moving point-loads is investigated. The complete proof of an existence and uniqueness properties of the weak solution of the considered equation with Dirichlet type boundary conditions is derived. The method used here is based on Galerkin approximation which is the main tool for the weak solution theory of linear evolution equations as well as in derivation of a priori estimate for the approximate solutions. All steps of the proposed technique are explained in detail.

Secret Key Generation Over Biased Physical Unclonable Functions With Polar Codes

Internet-of-Things (IoT) devices are usually small, low cost, and have limited resources, which makes them vulnerable to physical and cloning attacks. To secure IoT devices, physical unclonable functions (PUFs) are relatively new security primitives used for device authentication and device-specific secret-key generation. In this paper, we focus on designing a robust construction to derive secret keys from static randomaccess memory (SRAM)-PUFs, which enjoy the uniqueness and randomness properties stemming from the manufacturing variations of SRAM memory cells. We make use of a polar code construction. Based on the fact that SRAM memory can often be found in today's IoT devices, and since polar codes have been selected as error-correction technique in the fifth generation standard, this makes the proposed scheme a promising candidate for reducing the extra cost and securing resource-constrained IoT devices. In this paper, we propose a novel construction method to eliminate the effect of noise and bias in SRAM-PUFs. We shall prove that the secrecy leakage of the helper data about the secretkey can be made negligible due to polarization and proper code construction design. Results show that the proposed scheme provides a significant improvement of the reliability (achieve a failure probability below 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-6</sup> ) and of the realizable secret-key rate, which is also evaluated by the theoretical analysis. In addition, the proposed scheme provides the possibility to tradeoff complexity, secrecy, and reliability with the same code construction for different IoT applications.

Existence, Uniqueness and Qualitative Properties of Global Solutions of Abstract Differential Equations with State-Dependent Delay

AbstractWe study the existence, uniqueness and qualitative properties of global solutions of abstract differential equations with state-dependent delay. Results on the existence of almost periodic-type solutions (including, periodic, almost periodic, asymptotically almost periodic and almost automorphic solutions) are proved. Some examples of partial differential equations with state-dependent delay arising in population dynamics are presented.