Nonnegative Function Research Articles

Achromatic solutions of the color constancy problem: the Helmholtz–Kohlrausch effect explained

For given tristimulus values X, Y, Z of the object with reflectance ρ(λ) viewed under an illuminant S(λ) with tristimulus values X0, Y0, Z0, an earlier algorithm constructs the smoothest metameric estimate ρ0(λ) under S(λ) of ρ(λ), independent of the amplitude of S(λ). It satisfies a physical property of ρ(λ), i.e., 0≤ρ0(λ)≤1, on the visual range. The second inequality secures the condition that for no λ the corresponding patch returns more radiation from S(λ) than is incident on it at λ, i.e., ρ0(λ) is a fundamental metameric estimate; ρ0(λ) and ρ(λ) differ by an estimation error causing perceptual variables assigned to ρ0(λ) and ρ(λ) under S(λ) to differ under the universal reference illuminant E(λ)=1 for all λ, tristimulus values X E , Y E , Z E . This color constancy error is suppressed but not nullified by three narrowest nonnegative achromatic response functions A i (λ) defined in this paper, replacing the cone sensitivities and invariant under any nonsingular transformation T of the color matching functions, a demand from theoretical physics. They coincide with three functions numerically constructed by Yule apart from an error corrected here. S(λ) unknown to the visual system as a function of λ is replaced by its nonnegative smoothest metameric estimate S0(λ) with tristimulus values made available in color rendering calculations, by specular reflection, or determined by any educated guess; ρ(λ) under S(λ) is replaced by its corresponding color R0(λ) under S0(λ) like ρ(λ) independent of the amplitude of S0(λ). The visual system attributes to R0(λ)E(λ) one achromatic variable, in the CIE case defined by y(λ)/Y E , replaced by the narrowest middle wave function A2(λ) normalized such that the integral of A2(λ)E(λ) over the visual range equals unity. It defines the achromatic variable ξ2, A(λ), and ξ as described in the paper. The associated definition of present luminance explains the Helmholtz–Kohlrausch effect in the last figure of the paper and rejects CIE 1924 luminance that fails to do so. It can be understood without the mathematical details.

On multi-bump solutions for a class of (N,q)-Laplacian equation with critical exponential growth in [formula omitted]

This article deals with the following (N,q)-Laplacian equation with exponential critical growth in RN of the form:−ΔNu−Δqu+(μV(x)+B(x))(|u|N−2u+|u|q−2u)=h(u)inRN, where 2≤N<q, μ∈(0,∞), h is a continuous function with exponential critical growth and V:RN→R is a continuous function verifying some hypotheses. Assuming that the nonnegative function B has a potential well int(B−1({0})) composed of k disjoint components Ω1,Ω2,⋯,Ωk. With the help of the variational methods and Morse iteration technique, the existence and multiplicity of positive multi-bump solutions are obtained if the parameter μ>0 is large enough.

Well-Posedness of the generalised Dean–Kawasaki Equation with correlated noise on bounded domains

In this paper, we extend the notion of stochastic kinetic solutions introduced in Fehrman and Gess (2024) to establish the well-posedness of stochastic kinetic solutions of generalised Dean–Kawasaki equations with correlated noise on bounded, C2-domains with Dirichlet boundary conditions. The results apply to a wide class of non-negative boundary data, which is based on certain a priori estimates for the solutions, that encompasses all non-negative constant functions including zero and all smooth functions bounded away from zero.

Adaptive Output Formation Tracking for Nonlinear Multiagent Systems With Double Semi-Markovian Switching Topologies and Time-Varying Actuator Faults.

This article investigates adaptive output formation tracking control of nonlinear multiagent systems with time-varying actuator faults and unknown nonidentical control directions under double semi-Markovian switching topologies. Considering the dynamic changes of communication connections in uncertain environments, a double semi-Markov process is first introduced into the leader-follower structure to describe the random switching of communication topologies. Then, a novel adaptive distributed fault-tolerant output formation tracking control framework is established using the backstepping and Nussbaum gain technique to address matched/mismatched uncertainties and disturbances, time-varying actuator faults, and unknown nonidentical control directions. In this control framework, the independent variable of the Nussbaum function is designed as a non-negative function that monotonically increases with respect to time, thereby overcoming the presence of the absolute value of its derivative in the integration process. Based on the distributed structure, an adaptive fault-tolerant controller is further proposed to achieve the asymptotic output formation tracking in mean-square sense. The stability of the closed-loop nonlinear multiagent systems is analysed through the contradiction argument and Lyapunov theorem. The simulation example verifies the effectiveness of the proposed control strategy.

Neural-Rendezvous: Provably Robust Guidance and Control to Encounter Interstellar Objects

Interstellar objects (ISOs) are likely representatives of primitive materials invaluable in understanding exoplanetary star systems. Due to their poorly constrained orbits with generally high inclinations and relative velocities, however, exploring ISOs with conventional human-in-the-loop approaches is significantly challenging. This paper presents Neural-Rendezvous—a deep-learning-based guidance and control framework for encountering fast-moving objects, including ISOs, robustly, accurately, and autonomously in real time. It uses pointwise minimum norm tracking control on top of a guidance policy modeled by a spectrally normalized deep neural network, where its hyperparameters are tuned with a loss function directly penalizing the model predictive control state trajectory tracking error. We show that Neural-Rendezvous provides a high probability exponential bound on the expected spacecraft delivery error, the proof of which leverages stochastic incremental stability analysis. In particular, it is used to construct a non-negative function with a supermartingale property, explicitly accounting for the ISO state uncertainty and the local nature of nonlinear state estimation guarantees. In numerical simulations, Neural-Rendezvous is demonstrated to satisfy the expected error bound for 100 ISO candidates. This performance is also empirically validated using our spacecraft simulator and in high-conflict and distributed unmanned aerial vehicle swarm reconfiguration with up to 20 unmanned aerial vehicles.

An Improved Driving Training-Based Optimization Algorithm for the Minimum Gap Graph Connected Partition Problem

In this paper, we consider the minimum gap partition problem in an undirected connected graph with nonnegative vertex weights. This problem involves in dividing the given graph into a specified number of connected subgraphs such that the total difference between the largest and the smallest vertex weights in each subgraph is minimized. Namely, let G = (V, E, W) be a given undirected connected graph with vertex set V , edge set E, and nonnegative vertex weight function W : V → R+, and let k ≥ 2 be a given positive integer. The minimum gap partition problem consists of constructing a partition P = {V1, V2, . . . , Vk} of non-empty and pairwise disjoint subsets of V such that each vertex set Vi, i = 1, 2, . . . , k, induces a connected subgraph G[Vi] of G. The objective of the problem is to find such a partition of V that minimizes the total difference 1≤i≤k maxv∈Viw(v) − minv∈Viw(v). This problem, as many other variants of graph partition problems, is known to be NP-hard problem. We designed an efficient metaheuristic algorithm for finding approximate solution of large-scale instances of this problem. The quality of the proposed algorithm is assessed comparing with the previous algorithms proposed for the problem.

Existence of positive solution for a class of quasilinear Schrödinger equations with potential vanishing at infinity on nonreflexive Orlicz-Sobolev spaces

In this paper we investigate the existence of positive solution for a class of quasilinear problem on an Orlicz-Sobolev space that can be nonreflexive $$ - \Delta_{\Phi} u +V(x)\phi(|u|)u= K(x)f(u)\quad\mbox{in } \mathbb{R}^{N}, $$ where $ N \geq 2 $, $ V, K $ are nonnegative continuous functions and $f$ is a continuous function with a quasicritical growth. Here we extend the Hardy-type inequalities presented in \cite{AlvesandMarco} to nonreflexive Orlicz spaces. Through inequalities together with a variational method for non-differentiable functionals we will obtain a ground state solution. We analyze also the problem with $V=0$.

On submodular prophet inequalities and correlation gap

Prophet inequalities and secretary problems have been extensively studied in recent years due to their elegance, connections to online algorithms, stochastic optimization, and mechanism design problems in game theoretic settings. Rubinstein and Singla [31] developed a notion of combinatorial prophet inequalities in order to generalize the standard prophet inequality setting to combinatorial valuation functions such as submodular and subadditive functions. For non-negative submodular functions they demonstrated a constant factor prophet inequality for matroid constraints. Along the way they showed a variant of the correlation gap for non-negative submodular functions.In this paper we revisit their notion of correlation gap as well as the standard notion of correlation gap and prove much tighter and cleaner bounds. Via these bounds and other insights we obtain substantially improved constant factor combinatorial prophet inequalities for both monotone and non-monotone submodular functions over any constraint that admits an Online Contention Resolution Scheme. In addition to improved bounds we describe efficient polynomial-time algorithms that achieve these bounds.

An accelerated deterministic algorithm for maximizing monotone submodular minus modular function with cardinality constraint

Submodular optimization not only covers some classical combinatorial optimization problems, but also has a wide range of applications in fields such as machine learning and artificial intelligence. For submodular maximization problems with constraints, some work has been done including the design of approximation algorithms, the measurement of approximation algorithms in terms of quality and efficiency, etc. In this paper, we consider the problem of maximizing a non-negative monotone submodular function minus a non-negative modular function with the cardinality constraint. This model has been applied to many scenarios, such as team formation problem, influence maximization problem, recommender systems problem, etc. We propose a threshold algorithm that achieve a (1/2−O(ε),2)-bicriteria approximation ratio and query complexity O(nlog⁡n). Our algorithm makes a small sacrifice in the approximation ratio but improves the best query complexity result of existing deterministic algorithms from O(n2) to O(nlog⁡n) in the worst case.

Liouville-type results for semilinear inequalities involving the Dunkl Laplacian operator

Let Δ k be the Dunkl generalized Laplacian operator associated to a root system R of R N and a nonnegative multiplicity function k defined on R and invariant by the finite reflection group W. In this paper, we establish Liouville-type theorems for the semilinear inequality − Δ k u ≥ | u | p in R N and the system of inequalities − Δ k u ≥ | v | p , − Δ k v ≥ | u | q in R N , where N ≥ 1 and p,q>1. To the best of our knowledge, this contribution is the first work dealing with Liouville-type results for nonlinear problems involving the Dunkl Laplacian.

Analysis of a stochastic SEIuIrR epidemic model incorporating the Ornstein-Uhlenbeck process

This article aims to analyze a stochastic epidemic model SEIuIrR (Susceptible-exposed-undetected infected-detected infected (reported -recovered) assuming that the transmission rate at which people undetected become detected is perturbed by the Ornstein–Uhlenbeck process. Our first objective is to prove that the stochastic model has a unique positive global solution by constructing a nonnegative Lyapunov function. Afterward, we provide a sufficient criterion to prove the existence of an ergodic stationary distribution of the mode by constructing a suitable series of Lyapunov functions. Subsequently, we establish sufficient conditions for the extinction of the disease. Finally, a series of numerical simulations are carried out to illustrate the theoretical results.

On Some Positive Linear Operators Preserving the B^{-1}-Convexity of Functions

In the study, we first give an inequality that non-negative differentiable functions must satisfy to be B^{-1}-convex. Then, using the inequality, we show that the B^{-1}-convexity property of functions is preserved by Bernstein-Stancu operators, Sz'asz-Mirakjan operators and Baskakov operators. In addition, we compare the concepts B^{-1}-convexity and B-concavity of functions.

Integral means of solutions of the one-dimensional Poisson equation with Robin boundary conditions

Consider the one-dimensional Poisson equation −u″=f on the interval [−π,π], where f is an non-negative integrable function, with Robin boundary conditions −u′(−π)+αu(−π)=u′(π)+αu(π)=0, where α>0 is a constant. In this paper we prove inequalities for the convex integral means of solutions of this problem, using the polarization of functions and its properties. We find solutions with maximal convex integral means and prove their uniqueness.

Caloric functions and boundary regularity for the fractional Laplacian in Lipschitz open sets

AbstractWe give Martin representation of nonnegative functions caloric with respect to the fractional Laplacian in Lipschitz open sets. The caloric functions are defined in terms of the mean value property for the space-time isotropic $$\alpha $$ α -stable Lévy process. To derive the representation, we first establish the existence of the parabolic Martin kernel. This involves proving new boundary regularity results for both the fractional heat equation and the fractional Poisson equation with Dirichlet exterior conditions. Specifically, we demonstrate that the ratio of the solution and the Green function is Hölder continuous up to the boundary.

Existence and regularity results of parabolic problems with convection term and singular nonlinearity

In this work, we investigate the influence of the convection term and the singular lower order term on the existence and regularity of solutions to the following parabolic problem: \begin{cases}\frac{\partial u}{\partial t}-\operatorname{div}(M(x,t)\nabla u) =-\operatorname{div}(uE(x,t))+\frac{f}{u^{\theta}}&amp;\text{in }\Omega\times(0,T),\\ u(x,t)=0&amp;\text{on }\partial \Omega\times (0,T),\\ u(x,0)=u_{0}(x)&amp;\text{in }\Omega,\end{cases} where \theta&gt;0 , \Omega\subset \mathbb{R}^{N}\ (N&gt;2) is a bounded smooth domain with 0\in \Omega , and f\in L^{m}(\Omega\times (0,T)) with m\geq 1 is a non-negative function. The function u_{0} is a non-negative function that belongs to the space L^{\infty}(\Omega) such that \forall \omega\subset\subset \Omega,\ \exists c_{\omega}&gt;0,\quad u_{0}\geq c_{\omega}\text{ in }\omega. The main idea of this research explains the combined impact of the convection term and the singular lower order term on the existence and regularity of a solution to the above problem.

Roots, Trace, and Extendability of Flat Nonnegative Smooth Functions

Abstract Building on the univariate techniques developed by Ray and Schmidt-Hieber, we study the class $\mathcal{F}^{s}(\mathbb{R}^{n})$ of multivariate nonnegative smooth functions that are sufficiently flat near their zeroes, which guarantees that $F^{r}$ has Hölder differentiability $rs$ whenever $F \in \mathcal{F}^{s}$. We then construct a continuous Whitney extension map that recovers an $\mathcal{F}^{s}$ function from prescribed jets. Finally, we prove a Brudnyi–Shvartsman Finiteness Principle for the class $\mathcal{F}^{s}$, thereby providing a necessary and sufficient condition for a nonnegative function defined on an arbitrary subset of $\mathbb{R}^{n}$ to be $\mathcal{F}^{s}$-extendable to all of $\mathbb{R}^{n}$.

Rearrangement inequalities on the lattice graph

AbstractThe Polya–Szegő inequality in states that, given a nonnegative function , its spherically symmetric decreasing rearrangement is ‘smoother’ in the sense of for all . We study analogues on the lattice grid graph . The spiral rearrangement is known to satisfy the Polya–Szegő inequality for , the Wang‐Wang rearrangement satisfies it for and no rearrangement can satisfy it for . We develop a robust approach to show that both these rearrangements satisfy the Polya–Szegő inequality up to a constant for all . In particular, the Wang‐Wang rearrangement satisfies for all . We also show the existence of (many) rearrangements on such that for all .

Multiple bound states for a class of fractional critical Schrödinger–Poisson systems with critical frequency

In this paper we study the fractional Schrödinger–Poisson system ε2s(−Δ)su+V(x)u=ϕ|u|2s*−3u+|u|2s*−2u,ε2s(−Δ)sϕ=|u|2s*−1,x∈R3, where s ∈ (0, 1), ɛ &amp;gt; 0 is a small parameter, 2s*=63−2s is the critical Sobolev exponent and V∈L32s(R3) is a nonnegative function which may be zero in some regions of R3, e.g., it is of the critical frequency case. By virtue of a new global compactness lemma, and the Lusternik–Schnirelmann category theory, we relate the number of bound state solutions with the topology of the zero set where V attains its minimum for small values of ɛ.

Sufficient Conditions for a Graph k G Admitting all [ 1 , k ] -factors

Let g,f:V(G)→{0,1,2,3,⋯} be two functions satisfying g(x)≤f(x) for every x∈V(G). A (g,f)-factor of G is defined as a spanning subgraph F of G such that g(x)≤dF(x)≤f(x) for every x∈V(G). An (f,f)-factor is simply called an f-factor. Let φ be a nonnegative integer-valued function defined on V(G). Set Deveng,f={φ:g(x)≤φ(x)≤f(x) for every x∈V(G) and ∑x∈V(G)φ(x) is even}. If for each φ∈Deveng,f, G admits a φ-factor, then we say that G admits all (g,f)-factors. All (g,f)-factors are said to be all [1,k]-factors if g(x)≡1 and f(x)≡k for any x∈V(G). In this paper, we verify that for a connected multigraph G satisfying NG(X)=V(G) or |NG(X)|&gt;(1+1k+1)|X|−1 for every X⊂V(G), kG admits all [1,k]-factors, where k≥2 is an integer and kG denotes the graph derived from G by replacing every edge of G with k parallel edges.

Approximation Algorithms for the Submodular Hitting Set Problem

In this paper, we study the submodular hitting set problem (SHSP), which is a variant of the hitting set problem. In the SHSP, we are given a supergraph H = ( V , C ) and a nonnegative submodular function on the set 2 V . The objective is to determine a vertex subset to cover all hyperedges such that the cost of submodular covering is minimized. Our main work is to present a rounding algorithm and a primal-dual algorithm respectively for the SHSP and prove that they both have the approximation ratio k , where k is the maximum number of vertices in all hyperedges.