Nonnegative Solutions Research Articles

Existence and uniqueness of positive solution for multilinear systems with generalized strong [formula omitted]-tensor

Recently, the existence and uniqueness of positive solution for multilinear systems with H+-tensor were proved by Wang et al. (2019). To expand the range of multilinear systems with positive solutions, a new class of tensors, called generalized strong M-tensor, is proposed while the H+-tensor is a special case of the generalized strong M-tensor. Under certain conditions, the existence and the uniqueness of positive (nonnegative) solutions are proved for multilinear systems with generalized strong M-tensor. Finally, numerical examples are given to illustrate the theoretical results with Venn diagram for the relationship among generalized strong M-tensor with certain conditions, H+-tensor, and strong M-tensor.

A Parabolic Harnack Principle for Balanced Difference Equations in Random Environments

We consider difference equations in balanced, i.i.d. environments which are not necessary elliptic. In this setting we prove a parabolic Harnack inequality (PHI) for non-negative solutions to the discrete heat equation satisfying a (rather mild) growth condition, and we identify the optimal Harnack constant for the PHI. We show by way of an example that a growth condition is necessary and that our growth condition is sharp. Along the way we also prove a parabolic oscillation inequality and a (weak) quantitative homogenization result, which we believe to be of independent interest.

Turing–Hopf Bifurcation and Its Normal Form in a Diffusive Three-Species Food Chain System with Strong Allee Effect

In this paper, we consider a diffusive three-species food chain system with strong Allee effect subject to the Neumann boundary condition. The dynamics of the local system including stability of interior equilibria and Hopf bifurcation are discussed. For the diffusive system, we study the existence of non-negative solutions, stability of a positive homogeneous steady state, diffusion-driven Turing instability and the occurrence of Turing–Hopf bifurcation. Employing the multiple scale analysis, we derive the normal form of Turing–Hopf bifurcation, which helps us classify the dynamical behaviors of the diffusive system near the Turing–Hopf bifurcation point.

Unique non-negative definite solution of the time-varying algebraic Riccati equations with applications to stabilization of LTV systems

In the context of infinite-horizon optimal control problems, the algebraic Riccati equations (ARE) arise when the stability of linear time-varying (LTV) systems is investigated. Using the zeroing neural network (ZNN) approach to solve the time-varying eigendecomposition-based ARE (TVE-ARE) problem, the ZNN model (ZNNTVE-ARE) for solving the TVE-ARE problem is introduced as a result of this research. Since the eigendecomposition approach is employed, the ZNNTVE-ARE model is designed to produce only the unique nonnegative definite solution of the time-varying ARE (TV-ARE) problem. It is worth mentioning that this model follows the principles of the ZNN method, which converges exponentially with time to a theoretical time-varying solution. The ZNNTVE-ARE model can also produce the eigenvector solution of the continuous-time Lyapunov equation (CLE) since the Lyapunov equation is a particular case of ARE. Moreover, this paper introduces a hybrid ZNN model for stabilizing LTV systems in which the ZNNTVE-ARE model is employed to solve the continuous-time ARE (CARE) related to the optimal control law. Experiments show that the ZNNTVE-ARE and HFTZNN-LTVSS models are both effective, and that the HFTZNN-LTVSS model always provides slightly better asymptotic stability than the models from which it is derived.

Factorization in monoids by stratification of atoms and the Elliott problem

In an additive factorial monoid each element can be represented as a linear combination of irreducible elements (atoms) with uniquely determined coefficients running over all natural numbers. In this paper we develop for a wide class of non-factorial monoids a concept of stratification for atoms which allows to represent each element as a linear combination of atoms where the coefficients are uniquely determined when restricted in a particular way. This wide class includes inside factorial monoids and in particular simplicial affine monoids. In the latter case the question of uniqueness is related to a problem studied by E. B. Elliott in a paper from 1903. For the monoid of all nonnegative solutions of a certain linear Diophantine equation in three variables, Elliott considers “simple sets of solutions” (atoms of the monoid) and looks for a method that gives “every set once only”. We show that for simplicial affine monoids in two dimensions a stratification is always possible, which answers to Elliott’s problem also for the cases he left open. The results in this paper on the stratification of atoms for monoids in general may be seen also as an answer to a “generalized Elliott problem”. This manuscript employs very much the concept of extraction monoid which, as many other tools, stems from discrete geometry but it is less well-known.

An efficient nonstandard computer method to solve a compartmental epidemiological model for COVID-19 with vaccination and population migration

Background and objective: In this manuscript, we consider a compartmental model to describe the dynamics of propagation of an infectious disease in a human population. The population considers the presence of susceptible, exposed, asymptomatic and symptomatic infected, quarantined, recovered and vaccinated individuals. In turn, the mathematical model considers various mechanisms of interaction between the sub-populations in addition to population migration. Methods: The steady-state solutions for the disease-free and endemic scenarios are calculated, and the local stability of the equilibium solutions is determined using linear analysis, Descartes’ rule of signs and the Routh–Hurwitz criterion. We demonstrate rigorously the existence and uniqueness of non-negative solutions for the mathematical model, and we prove that the system has no periodic solutions using Dulac’s criterion. To solve this system, a nonstandard finite-difference method is proposed. Results: As the main results, we show that the computer method presented in this work is uniquely solvable, and that it preserves the non-negativity of initial approximations. Moreover, the steady-state solutions of the continuous model are also constant solutions of the numerical scheme, and the stability properties of those solutions are likewise preserved in the discrete scenario. Furthermore, we establish the consistency of the scheme and, using a discrete form of Gronwall’s inequality, we prove theoretically the stability and the convergence properties of the scheme. For convenience, a Matlab program of our method is provided in the appendix. Conclusions: The computer method presented in this work is a nonstandard scheme with multiple dynamical and numerical properties. Most of those properties are thoroughly confirmed using computer simulations. Its easy implementation make this numerical approach a useful tool in the investigation on the propagation of infectious diseases. From the theoretical point of view, the present work is one of the few papers in which a nonstandard scheme is fully and rigorously analyzed not only for the dynamical properties, but also for consistently, stability and convergence.

A Modeling Framework for Coupling Plasticity with Species Diffusion

This paper presents a modeling framework---mathematical model and computational framework---to study the response of a plastic material due to the presence and transport of a chemical species in the host material. Such a modeling framework is important to a wide variety of problems ranging from Li-ion batteries, moisture diffusion in cementitious materials, hydrogen diffusion in metals, to consolidation of soils under severe loading-unloading regimes. The mathematical model incorporates experimental observations reported in the literature on how (elastic and plastic) material properties change because of the presence and transport of a chemical species. Also, the model accounts for one-way (transport affects the deformation but not vice versa) and two-way couplings between deformation and transport subproblems. The resulting coupled equations are not amenable to analytical solutions; so, we present a robust computational framework for obtaining numerical solutions. Given that popular numerical formulations do not produce nonnegative solutions, the computational framework uses an optimized-based nonnegative formulation that respects physical constraints (e.g., nonnegative concentrations). For completeness, we will also show the effect and propagation of the negative concentrations, often produced by contemporary transport solvers, into the overall predictions of deformation and concentration fields. Notably, anisotropy of the diffusion process exacerbates these unphysical violations. Using representative numerical examples, we will discuss how the concentration field affects plastic deformations of a degrading solid. Based on these numerical examples, we also discuss how plastic zones spread because of material degradation. To illustrate how the proposed computational framework performs, we report various performance metrics such as optimization iterations and time-to-solution.

Blow-up phenomena for a chemotaxis system with flux limitation

In this paper we consider nonnegative solutions of the following parabolic-elliptic cross-diffusion system{ut=Δu−∇(uf(|∇v|2)∇v),0=Δv−μ+u,∫Ωv=0,μ:=1|Ω|∫Ωudx,u(x,0)=u0(x), in Ω×(0,∞), with Ω a ball in RN, N≥3 under homogeneous Neumann boundary conditions and f(ξ)=(1+ξ)−α, 0<α<N−22(N−1), which describes gradient-dependent limitation of cross diffusion fluxes. Under conditions on f and initial data, we prove that a solution which blows up in finite time in L∞-norm, blows up also in Lp-norm for some p>1. Moreover, a lower bound of blow-up time is derived.

On Two Classes of Exponential Diophantine Equations

In this paper, we study on the exponential Diophantine equations: \(n^{x}+24^{y}=z^{2}\), for \(n \equiv 5\) or 7 (mod 8). We show that \(5^{x}+24^{y}=z^{2}\) has a unique positive integral solution \((2,1,7)\). Further, we show that for \(k \in \mathbb{N}\), \((8k+5)^{x}+24^{y}=z^{2}\) has a unique solution \((0,1,5)\) in non-negative integers. We also show that for a perfect square \(8m\), the exponential Diophantine equation \((8m-1)^{x}+24^{y}=z^{2}\), \(m \in \mathbb{N}\) has exactly two non-negative integral solutions \((0,1,5)\) and \((1,0,\sqrt{8m})\). Otherwise, it has a unique solution \((0,1,5)\). Finally, we illustrate our results with some examples and non-examples.

Another operator-theoretical proof for the second-order phase transition in the BCS-Bogoliubov model of superconductivity

In the preceding papers, imposing certain complicated and strong conditions, the present author showed that the solution to the BCS-Bogoliubov gap equation in superconductivity is twice differentiable only on the neighborhoods of absolute zero temperature and the transition temperature so as to show that the phase transition is of the second order from the viewpoint of operator theory. Instead, we impose a certain simple and weak condition in this paper, and show that there is a unique nonnegative solution and that the solution is indeed twice differentiable on a closed interval from a certain positive temperature to the transition temperature as well as pointing out several properties of the solution. We then give another operator-theoretical proof for the second-order phase transition in the BCS-Bogoliubov model. Since the thermodynamic potential has the squared solution in its form, we deal with the squared BCS-Bogoliubov gap equation. Here, the potential in the BCS-Bogoliubov gap equation is a function and need not be a constant.

Infinitely many vector solutions of a fractional nonlinear Schrödinger system with strong competition

In this paper, we prove the existence of infinitely many nonnegative vector solutions to a symmetric fractional nonlinear Schrödinger system, where the coupling parameter is sufficiently large (strong competition). Our arguments can be used to refine the multiplicity results of the corresponding local system, in the sensed that more general exponents and space dimensions are allowed.

Enforced block diagonal subspace clustering with closed form solution

Subspace clustering aims to fit each category of data points by learning an underlying subspace and then conduct clustering according to the learned subspace. Ideally, the learned subspace is expected to be block diagonal such that the similarities between clusters are zeros. In this paper, we provide the explicit theoretical connection between spectral clustering and the subspace clustering based on block diagonal representation. We propose Enforced Block Diagonal Subspace Clustering (EBDSC) and show that the spectral clustering with the Radial Basis Function kernel can be regarded as EBDSC. Compared with the exiting subspace clustering methods, an analytical, nonnegative and symmetrical solution can be obtained by EBDSC. An important difference with respect to the existing ones is that our model is a more general case. EBDSC directly uses the obtained solution as the similarity matrix, which can avoid the complex computation of the optimization program. Then the solution obtained by the proposed method can be used for the final clustering. Finally, we provide the experimental analysis to show the efficiency and effectiveness of our method on the synthetic data and several benchmark data sets in terms of different metrics.

The Cauchy problem for the fast p-Laplacian evolution equation. Characterization of the global Harnack principle and fine asymptotic behaviour

We study fine global properties of nonnegative solutions to the Cauchy Problem for the fast p-Laplacian evolution equation ut=Δpu on the whole Euclidean space, in the so-called “good fast diffusion range” 2NN+1<p<2. It is well-known that non-negative solutions behave for large times as B, the Barenblatt (or fundamental) solution, which has an explicit expression.We prove the so-called Global Harnack Principle (GHP), that is, precise global pointwise upper and lower estimates of nonnegative solutions in terms of B. This can be considered the nonlinear counterpart of the celebrated Gaussian estimates for the linear heat equation.We characterize the maximal (hence optimal) class of initial data such that the GHP holds, by means of an integral tail condition, easy to check. The GHP is then used as a tool to analyze the fine asymptotic behaviour for large times. For initial data that satisfy the same integral condition, we prove that the corresponding solutions behave like the Barenblatt with the same mass, uniformly in relative error.When the integral tail condition is not satisfied we show that both the GHP and the uniform convergence in relative error, do not hold anymore, and we provide also explicit counterexamples. We then prove a “generalized GHP”, that is, pointwise upper and lower bounds in terms of explicit profiles with a tail different from B. Finally, we derive sharp global quantitative upper bounds of the modulus of the gradient of the solution, and, when data are radially decreasing, we show uniform convergence in relative error for the gradients.To the best of our knowledge, analogous issues for the linear heat equation p=2, do not possess such clear answers, only partial results are known.

Dynamical behavior in a reaction-diffusion system with prey-taxis

In this article, we study a diffusive predator-prey system with prey-taxis under homogeneous Neumann boundary conditions. We establish the existence and boundedness of nonnegative global solutions. Through comparison with the system without prey-taxis, we find that the positive constant equilibrium remains stable for positive prey-taxis, while negative prey-taxis makes it unstable.

Fisher–KPP equation with Robin boundary conditions on the real half line

We consider the Fisher–KPP equation with Robin boundary conditions on the half line. We show that this problem has even number of nonnegative stationary solutions, which are ordered, a stable one is sandwiched by two unstable ones. Then we construct several types of entire solutions between them, each entire solution connects an unstable stationary solution with a stable one. In particular, we construct two types of entire solutions Uc(x,t) and U(x,t) connecting 0 and the smallest positive stationary solution. In addition, Uc tend to the traveling wave solution with speed c as t→−∞ in a moving frame, and U(x,t) enjoys convexity. This paper extends the recent results of Lou, Lu and Morita in Lou et al. (2020) from Dirichlet boundary condition to Robin boundary condition.

An elliptic Harnack inequality for difference equations with random balanced coefficients

We prove an elliptic Harnack inequality at large scale on the lattice Zd for nonnegative solutions of a difference equation with balanced i.i.d. coefficients which are not necessarily elliptic. We also identify the optimal constant in the Harnack inequality. Our proof relies on a quantitative homogenization result of the corresponding invariance principle to Brownian motion and on percolation estimates. As a corollary of our main theorem, we derive an almost optimal Hölder estimate.

The policy iteration algorithm for a compound Poisson process applied to optimal dividend strategies under a Cramér–Lundberg risk model

In this paper, we focus on the policy iteration algorithm (PIA) for the optimal dividend problem under the Cramér–Lundberg risk model. We conclude that the optimal value function is the minimum nonnegative solution of an optimization equation. Under any conditions, it can be approximated by iteration starting with the initial zero-valued policy, i.e., the policy of no dividend payment at all. An auxiliary optimization problem of maximizing dividend payment with terminal reward up to the first claim arrival time is introduced. This auxiliary optimization problem is solved completely in the sense that, no matter what the claim-size distribution is, the explicit formulae for both optimal strategy and value function are presented in general situation. These ensure that, at each iteration, we can get the explicit formulae of the improved strategy and its evaluation function. Finally, the PIA is illustrated with some numerical examples.

Onpz–Laplacian System Involving Critical Nonlinearities

In this paper, we deal with the existence of at least two nonnegative nontrivial solutions to apz–Laplacian system involving critical nonlinearity in the context of Sobolev spaces with variable exponents on complete manifolds. We have established our main results by exploring both Nehari’s method and doing a refined analysis on the associated fiber map and some variational techniques.

The Existence and Multiplicity of Homoclinic Solutions for a Fractional Discrete p−Laplacian Equation

In this study, we investigate the existence and multiplicity of solutions for a fractional discrete p−Laplacian equation on Z. Under suitable hypotheses on the potential function V and the nonlinearity f, with the aid of Ekeland’s variational principle, via mountain pass lemma, we obtain that this equation exists at least two nonnegative and nontrivial homoclinic solutions when the real parameter λ&gt;0 is large enough.

On the extinction-extinguishing dichotomy for a stochastic Lotka–Volterra type population dynamical system

Applying the Foster–Lyapunov type criteria and a martingale method, we study a two-dimensional process (X,Y) arising as the unique nonnegative solution to a pair of stochastic differential equations driven by independent Brownian motions and compensated spectrally positive Lévy random measures. Both processes X and Y can be identified as continuous-state nonlinear branching processes where the evolution of Y is negatively affected by X. Assuming that process X extinguishes, i.e. it converges to 0 but never reaches 0 in finite time, and process Y converges to 0, we identify rather sharp conditions under which the process Y exhibits, respectively, one of the following behaviors: extinction with probability one, extinguishing with probability one or both extinction and extinguishing occurring with strictly positive probabilities.