Nonnegative Function Research Articles

Asymptotic stability in a mosquito population suppression model with time delay

In this paper, a delayed mosquito population suppression model, where the number of sexually active sterile mosquitoes released is regarded as a given nonnegative function, and the birth process is density dependent by considering larvae progression and the intra-specific competition within the larvae, is developed and studied. A threshold value [Formula: see text] for the releases of sterile mosquitoes is determined, and it is proved that the origin is globally asymptotically stable if the number of sterile mosquitoes released is above the threshold value [Formula: see text]. Besides, the case when the number of sterile mosquitoes released stays at a constant level [Formula: see text] is also considered. In the special case, it is also proved that the origin is globally asymptotically stable if and only if [Formula: see text] and that the model exhibits other complicated dynamics such as bi-stability and semi-stability when [Formula: see text]. Numerical examples are also provided to illustrate our main theoretical results.

Synthesis of a Control System for a Waste Heat Boiler with Forced Circulation under Restrictions on Control Actions

As is often the case in practice, classical control theory does not fully provide sufficient tools for solving applied problems of a certain specific class. Such problems include automatic control of waste heat boilers where physical restrictions are imposed on controls, namely, they must belong to the class of non-negative functions with restrictions on amplitude and growth rate. In the theoretical part of this paper, within the framework of the block approach, methods for the synthesis of general linear systems with one-sided restrictions on control actions and state variables were developed. Then, the developed algorithms were applied to the waste heat boiler control system under conditions of incomplete measurements of state variables and the action of parametric and external uncontrolled disturbances. The effectiveness of the proposed algorithms was confirmed by numerical simulation.

On an elliptic chemotaxis system with flux limitation and subcritical signal production

In this article we study the existence of solutions of a system of partial differential equations of elliptic type, describing the distribution of a biological species “u” and the density of a chemical stimulus “ψ” in a bounded domain Ω of RN. The equation for u includes a chemotaxis term with nonlinear flux limitation which depends on the exponent p>1.The equation for u is given by −div(M(x)∇u)+u=−χdiv(u|∇ψ|p−2∇ψ)+f(x),where ψ presents a subcritical production term uθ and satisfies the equation −div(M(x)∇ψ)+ψ=uθ.The matrix of coefficients, M, is a known, symmetric and positive defined with coefficients mij∈C1(Ω¯),χ is a given real constant, f is a non-negative function belonging to Lm(Ω), m>max{1,N2}. The production term exponent, θ, is assumed to be positive and fulfills one of the following constrains 1<p<NθNθ−1,1<Nθor max{N,p}<θ+1θ for θ>0.The problem is completed with Dirichlet boundary conditions for u and ψ. The main result of the article includes the existence of positive solutions in H01(Ω)∩L∞(Ω).

Sliding Mode Control of Networked Control Systems: An Auxiliary Matrices-Based Approach

A novel sliding mode control design method is developed to address the robust stabilization problem for uncertain time-varying networked control systems subject to state delay, which utilizes auxiliary matrices to avoid the difficulties of finding quadratic Lyapunov functions and solving the linear matrix inequalities. By embedding the original system into a higher dimensional system, simplified piecewise nonnegative functions can be used to analyze the system. Stability criteria are obtained which can be satisfied and verified easily. An optimization problem is defined based on these criteria, to obtain the sliding mode control parameters to ensure the exponential stability of the closed-loop system.

On a degenerate singular elliptic problem

AbstractIn this article we provide existence, uniqueness and regularity results of a degenerate singular elliptic boundary value problem whose prototype is given by where Ω is a bounded smooth domain in with , w belongs to the Muckenhoupt class for some , f is a nonnegative function belonging to some Lebesgue space and .

Coherent state representation of thermal correlation functions with applications to rate theory.

A coherent state phase space representation of operators, based on the Husimi distribution, is used to derive an exact expression for the symmetrized version of thermal correlation functions. In addition to the time and temperature independent phase space representation of the two operators whose correlation function is of interest, the integrand includes a non-negative distribution function where only one imaginary time and one real time propagation are needed to compute it. The methodology is exemplified for the flux side correlation function used in rate theory. The coherent state representation necessitates the use of a smeared Gaussian flux operator whose coherent state phase space representation is identical to the classical flux expression. The resulting coherent state expression for the flux side correlation function has a number of advantages as compared to previous formulations. Since only one time propagation is needed, it is much easier to converge it with a semiclassical initial value representation. There is no need for forward-backward approximations, and in principle, the computation may be implemented on the fly. It also provides a route for analytic semiclassical approximations for the thermal rate, as exemplified by a computation of the transmission factor through symmetric and asymmetric Eckart barriers using a thawed Gaussian approximation for both imaginary and real time propagations. As a by-product, this example shows that one may obtain "good" tunneling rates using only above barrier classical trajectories even in the deep tunneling regime.

Smooth Pycnophylactic Interpolation Produced by Density-Equalising Map Projections

A large amount of quantitative geospatial data is collected and aggregated in discrete enumeration units (e.g. countries or states). Smooth pycnophylactic interpolation aims to find a smooth, nonnegative function such that the area integral over each enumeration unit is equal to the aggregated data. Conventionally, smooth pycnophylactic interpolation is achieved by a cellular automaton algorithm that converts a piecewise constant function into an approximately smooth function defined on a grid of coordinates on an equal-area map. An alternative approach, proposed by Tobler in 1976, is to construct a density-equalising map projection in which areas of enumeration units are proportional to the aggregated data. A pycnophylactic interpolation can be obtained from the Jacobian of this projection. Here, we describe a software implementation of this method. Although solutions are not necessarily optimal in terms of predefined quantitative measures of smoothness, our method is computationally efficient and can potentially be used in tandem with other methods to accelerate convergence towards an optimal solution.

A nonlinear discrete model for approximating a conservative multi-fractional Zakharov system: Analysis and computational simulations

A system of two partial differential equations with fractional diffusion is considered in this study. The system extends the conventional Zakharov system with unknowns being nonlinearly coupled complex- and real-valued functions. The diffusion is understood in the Riesz sense, and suitable initial-boundary conditions are imposed on an open and bounded domain of the real numbers. It is shown that the mass and Higgs’ free energy of the system are conserved. Moreover, the total energy is proven to be dissipated, and that both the free and the total energy are non-negative. As a corollary from the conservation of energy, we find that the solutions of the system are bounded throughout time. Motivated by these properties on the solutions of the system, we propose a numerical model to approximate the fractional Zakharov system via finite-difference approaches. Along with this numerical model for solving the continuous system, discrete analogues for the mass, the Higgs’ free energy and the total energy are we provided. Furthermore, utilizing Browder’s fixed-point theorem, we establish the solubility of the discrete model. It is shown that the discrete total mass and the discrete free energy are conserved, in agreement with the continuous case. The discrete energy functionals (both the discrete free energy and the discrete total energy) are proven to be non-negative functions of the discrete time thoroughly the boundedness of the numerical solutions. Properties of consistency, stability and convergence of the scheme are also studied rigorously. Numerical simulations illustrate some of the anticipated theoretical features of our finite-difference solution procedure.

Metric Growth of the Discrete $q$-Laplacian

The metric growth of Laplacian $\Delta u$ in the theory of potentials on Riemannian manifolds by [13] means the growth of Green potential of $(\Delta u)^2$. This notion was introduced to obtain a Riesz representation for a biharmonic functions. On a hyperbolic network, we investigated a similar problem in [7], by taking the discrete Laplacian $\Delta$ and the (discrete) Green function. Let $q$ be a non-negative function such that $q \not\equiv 0$. Then $- \Delta + q$ is a Schrödinger operator. Let us put $\Delta _q := \Delta - q$ and call it $q$-Laplacian. The $q$-Green function $g_a$ of the network with pole at $a$ always exists. In this paper, we estimate the metric growth of $\Delta _q u$ by $q$-Green potential of $|\Delta _q u|$ and $|\Delta _q u|^2$. As an application, we prove Riesz representation theorem for $q$-biharmonic functions.

Nash equilibrium structure of Cox process Hotelling games

AbstractWe study an N-player game where a pure action of each player is to select a nonnegative function on a Polish space supporting a finite diffuse measure, subject to a finite constraint on the integral of the function. This function is used to define the intensity of a Poisson point process on the Polish space. The processes are independent over the players, and the value to a player is the measure of the union of her open Voronoi cells in the superposition point process. Under randomized strategies, the process of points of a player is thus a Cox process, and the nature of competition between the players is akin to that in Hotelling competition games. We characterize when such a game admits Nash equilibria and prove that when a Nash equilibrium exists, it is unique and consists of pure strategies that are proportional in the same proportions as the total intensities. We give examples of such games where Nash equilibria do not exist. A better understanding of the criterion for the existence of Nash equilibria remains an intriguing open problem.

The Gelfand problem on annular domains of double revolution with monotonicity

We consider the following Gelfand problem ( P ) λ { − Δ u = λ a ( x ) f ( u ) a m p ; in Ω , u &gt; 0 a m p ; in Ω , u = 0 a m p ; on ∂ Ω , \begin{equation*} (P)_\lambda \qquad \left \{\begin {array}{ll} -\Delta u = \lambda a(x) f(u) &amp; \text { in } \Omega , \\ u&gt;0 &amp; \text { in } \Omega , \\ u= 0 &amp; \text { on } \partial \Omega , \end{array}\right . \end{equation*} where λ &gt; 0 \lambda &gt;0 is a parameter and f ( u ) = e u f(u)=e^u or f ( u ) = ( u + 1 ) p f(u)=(u+1)^p where p &gt; 1 p&gt;1 and a ( x ) a(x) is a nonnegative function with certain monotonicity (we allow a ( x ) = 1 a(x)=1 ). Here Ω \Omega is an annular domain which is also a double domain of revolution. Our interest will be in the question of the regularity of the extremal solution. We obtain improved compactness because of the annular nature of the domain and we obtain further compactness under some monotonicity assumptions on the domain.

The Mathematical Simulation for the Photocatalytic Fatigue of Polymer Nanocomposites Using the Monte Carlo Methods

We consider an approach to mathematical modeling of photodegradation of polymer nanocomposites with photoactive additives using the Monte Carlo methods. We principally pay attention to the strength decrease of these materials under solar light action. We propose a new term, “photocatalytic fatigue”, which we apply to the particular case when the mechanical strength decreases only owing to the presence of photocatalytically active components in polymeric nanocomposite material. The propriety of the term is based on a relative similarity of photostimulated mechanical destructive processes in nanocomposites with photoactive additives and mechanical destructive processes typical for metal high-cycle fatigue. Formation of the stress concentrations is one of the major causes of fatigue cracks generation in metals. Photocatalytic active nanoparticles of semiconductors initiate a generation of the stress concentrations under sunlight irradiation. The proposed mathematical model is a Wöhler curve analog for the metal high-cycle fatigue. We assume that equations for high-cycle fatigue curves of samples with stress concentrations could be used in mathematical modeling of polymer nanocomposites photodegradation. In this way, we replace the number of loading cycles with the exposition time in the equations. In the case of polypropylene and polyester samples with photoactive titanium dioxide, the experimental parameters of phenomenological equations for “photocatalytic fatigue” are calculated using one of the Monte Carlo methods based on the random search algorithm. The calculating scheme includes a solution of the extreme task of finding of the minimum of nonnegative transcendent multivariable function, which is a relative average quadratic deviation of calculated values of polymeric nanocomposite stress in comparison with corresponding experimental values. The applicability of the “photocatalytic fatigue” model for polymer nanocomposites with photoactive nanoparticles is confirmed by the example of polypropylene and polyester samples. The approximation error of the experimental strength values for them did not exceed 2%.

On oriented diameter of [formula omitted]-star graphs

Assignment of one of two possible directions to every edge of an undirected graph G=(V,E) is called an orientation of G. The resulting directed graph is denoted by G⃗. A strong orientation is one in which every vertex is reachable from every other vertex via a directed path in G⃗. The diameter of G⃗, i.e., the maximum distance from any vertex to any other vertex, depends on the particular orientation. The minimum diameter among all possible orientations of G is called the oriented diameterdiam⃗(G) of G. Let n,k be two integers such that 1≤k<n. In the realm of interconnection networks of processing elements, an (n,k)-star graph Sn,k offers a topology that permits to circumvent the lack of scalability of n-star graphs Sn. The oriented diameter quantifies an upper limit on the delay in communication over interconnection networks. In this paper, we present a strong orientation scheme for Sn,k that combines approaches suggested by Cheng and Lipman (2002) for Sn,k with the one proposed by Fujita (2013) for Sn, reaping benefits from both worlds. Next, we propose a distributed routing algorithm for Sn,k⃗ inspired by an algorithm proposed in Kumar et al. (2021) for Sn⃗. With the aid of both the orientation scheme and the routing algorithm, we show that diam⃗(Sn,k)≤⌊n+k2⌋+2k+6−δ(n,k) where δ(n,k) is a non-negative function. The function δ(n,k) takes on values 2k−n, 0, and n−3k2 respectively for three disjoint intervals k>n2, n3<k≤n2 and k≤n3. For every value of n, k, our upper bound performs better than all known bounds in literature.

Optimal singularities of initial functions for solvability of a semilinear parabolic system

Let $(u, v)$ be a nonnegative solution to the semilinear parabolic system $$ \mbox{(P)} \qquad \left\{ \begin{array}{ll} \partial_{t} u = D_{1} \Delta u + v^{p}, \quad x \in \mathbf{R}^{N}, \ t > 0,\\ \partial_{t} v = D_{2} \Delta v + u^{q}, \quad x \in \mathbf{R}^{N}, \ t > 0,\\ (u(\cdot,0), v(\cdot,0)) = (\mu, \nu), \quad x \in \mathbf{R}^{N}, \end{array} \right. $$ where $D_{1}$, $D_{2} > 0$, $0 < p \leq q$ with $pq > 1$ and $(\mu, \nu)$ is a pair of nonnegative Radon measures or nonnegative measurable functions in $\mathbf{R}^{N}$. In this paper we study sufficient conditions on the initial data for the solvability of problem (P) and clarify optimal singularities of the initial functions for the solvability.

Instability of solutions to double‐phase problems with exponential nonlinearity

Let be a weak solution of the double‐phase problem where and are two nonnegative functions such that and for all , where and . We prove that if , or if is bounded from below and , or if has a suitable decay at infinity, then must be unstable.

Monotone k-submodular secretary problems: Cardinality and knapsack constraints

In this paper, we consider the k-submodular secretary problem, in which the items or secretaries arrive one by one in a uniformly random order, and the goal is to select k disjoint sets of items, so as to maximize the expectation of a monotone non-negative k-submodular function. A decision for each item must be made immediately and irrevocably after its arrival: accept and assign it to one of the k dimensions, or reject it. We first show that, in the unconstrained setting, there is an offline algorithm that can be transformed to a k2k−1-competitive online algorithm, which is asymptotically the best possible. For the problem with cardinality constraints, we present online algorithms with provable performance guarantees for the total size constraint and individual size constraint settings that depend on the corresponding budget parameters. For the problem under a knapsack constraint, we provide a constant-competitive online algorithm.

Preconditioners for fractional diffusion equations based on the spectral symbol

AbstractIt is well known that the discretization of fractional diffusion equations with fractional derivatives , using the so‐called weighted and shifted Grünwald formula, leads to linear systems whose coefficient matrices show a Toeplitz‐like structure. More precisely, in the case of variable coefficients, the related matrix sequences belong to the so‐called generalized locally Toeplitz class. Conversely, when the given FDE has constant coefficients, using a suitable discretization, we encounter a Toeplitz structure associated to a nonnegative function , called the spectral symbol, having a unique zero at zero of real positive order between one and two. For the fast solution of such systems by preconditioned Krylov methods, several preconditioning techniques have been proposed in both the one‐ and two‐dimensional cases. In this article we propose a new preconditioner denoted by which belongs to the ‐algebra and it is based on the spectral symbol . Comparing with some of the previously proposed preconditioners, we show that although the low band structure preserving preconditioners are more effective in the one‐dimensional case, the new preconditioner performs better in the more challenging multi‐dimensional setting.

On the Solvability in the Class of Nonnegative Functions of a Nonlinear Parabolic Equation with Double Degeneration and a Nonlocal Spatial Operator

On the Solvability in the Class of Nonnegative Functions of a Nonlinear Parabolic Equation with Double Degeneration and a Nonlocal Spatial Operator

Rare event asymptotics for exploration processes for random graphs

Large deviations for random graph models has been a topic of significant recent research activity. Much work in this area is focused on the class of dense random graph models (number of edges in the graph scale as n2, where n is the number of vertices) where the theory of graphons has emerged as a principal tool in the study of large deviation properties. These tools do not give a good approach to large deviation problems for random graph models in the sparse regime. The aim of this paper is to study an approach for large deviation problems in this regime by establishing large deviation principles (LDP) on suitable path spaces for certain exploration processes of the associated random graph sequence. Exploration processes are an important tool in the study of sparse random graph models and have been used to understand detailed asymptotics of many functionals of sparse random graphs, such as component sizes, surplus, deviations from trees, etc. In the context of rare event asymptotics of interest here, the point of view of exploration process transforms a large deviation analysis of a static random combinatorial structure to the study of a small noise LDP for certain stochastic dynamical systems with jumps. Our work focuses on one particular class of random graph models, namely the configuration model; however, the general approach of using exploration processes for studying large deviation properties of sparse random graph models has broader applicability. The goal is to study asymptotics of probabilities of nontypical behavior in the large network limit. The first key step for this is to establish a LDP for an exploration process associated with the configuration model. A suitable exploration process here turns out to be an infinite-dimensional Markov process with transition probability rates that diminish to zero in certain parts of the state space. Large deviation properties of such Markovian models is challenging due to poor regularity behavior of the associated local rate functions. Our proof of the LDP relies on a representation of the exploration process in terms of a system of stochastic differential equations driven by Poisson random measures and variational formulas for moments of nonnegative functionals of Poisson random measures. Uniqueness results for certain controlled systems of deterministic equations play a key role in the analysis. Next, using the rate function in the LDP for the exploration process we formulate a calculus of variations problem associated with the asymptotics of component degree distributions. The second key ingredient in our study is a careful analysis of the infinite-dimensional Euler–Lagrange equations associated with this calculus of variations problem. Exact solutions of these systems of nonlinear differential equations are identified which then provide explicit formulas for decay rates of probabilities of nontypical component degree distributions and related quantities.

Generalized relative Nevanlinna order (α,β) and generalized relative Nevanlinna type (α,β) based some growth properties of composite analytic functions in the unit disc

Our aim in this paper is to introduce some idea about generalized relative Nevanlinna order (α,β) and generalized relative Nevanlinna type (α,β) of an analytic function with respect to another analytic function in the unit disc where α and β are continuous non-negative functions on (-∞,+∞). So we discuss about some growth properties relating to the composition of two analytic functions in the unit disc on the basis of generalized relative Nevanlinna order (α,β) and generalized relative Nevanlinna type (α,β) as compared to the growth of their corresponding left and right factors.