Maximal Solution Research Articles

Quasi-Convex Hamilton--Jacobi Equations via Finsler $p$-Laplace--Type Operators

In this paper we show that the maximal viscosity solution of a class of quasi-convex Hamilton--Jacobi equations, coupled with inequality constraints on the boundary, can be recovered by taking the limit as $p\to\infty$ in a family of Finsler $p$-Laplace problems. The approach also enables us to provide an optimal solution to a Beckmann-type problem in the general Finslerian setting and allows recovering a bench of known results based on the Evans--Gangbo technique.

Minimising hulls, p-capacity and isoperimetric inequality on complete Riemannian manifolds

The notion of strictly outward minimising hull is investigated for open sets of finite perimeter sitting inside a complete noncompact Riemannian manifold. Under natural geometric assumptions on the ambient manifold, the strictly outward minimising hull Ω⁎ of a set Ω is characterised as a maximal volume solution of the least area problem with obstacle, where the obstacle is the set itself. In the case where Ω has C1,α-boundary, the area of ∂Ω⁎ is recovered as the limit of the p-capacities of Ω, as p→1+. Finally, building on the existence of strictly outward minimising exhaustions, a sharp isoperimetric inequality is deduced on complete noncompact manifolds with nonnegative Ricci curvature, provided 3≤n≤7.

Monotone iterative technique for solving finite difference systems of time fractional parabolic equations with initial/periodic conditions

This article is devoted to a numerical analysis of two types of fractional nonlinear reaction-diffusion problems with periodic conditions or with initial conditions. We also consider two types of derivative in time, the first one is that of Caputo and the second is the conformable derivative; we discretize the problem via a finite difference method. We construct two iterative schemes by the upper and lower solutions method which converges monotonically towards a maximal solution or a minimal solution of the considered problem when the mesh decreases to zero, depending on whether the initial iteration is an upper solution or a lower solution. A comparison lemma for the different monotone sequences is also proved. The presented iterative scheme is used to show that the finite difference system converges to continuous solutions of the fractional reaction-diffusion problem. Finally to validate the theoretical results some examples with numerical simulations of reaction-diffusion problem are also presented and discussed in detail.

Enumerating teams in first-order team logics

We start the study of the enumeration complexity of different satisfiability problems in first-order team logics. Since many of our problems go beyond DelP, we use a framework for hard enumeration analogous to the polynomial hierarchy, which was recently introduced by Creignou et al. (Discret. Appl. Math. 2019). We show that the problem to enumerate all satisfying teams of a fixed formula in a given first-order structure is DelNP-complete for certain formulas of dependence logic and independence logic. For inclusion logic formulas, this problem is even in DelP. Furthermore, we study the variants of this problem where only maximal or minimal solutions, with respect to cardinality or inclusion, are considered. For the most part these share the same complexity as the original problem. One exception is the cardinality minimum-variant for inclusion logic, which is DelNP-complete, the other is the inclusion maximal-variant for dependence and independence logic, which is in Del+NP and DelNP-hard.

On the analytical aspects of inertial particle motion

In their seminal 1983 paper, M. Maxey and J. Riley introduced an equation for the motion of a sphere through a fluid. Since this equation features the Basset history integral, the popularity of this equation has broadened the use of a certain form of fractional differential equation to study inertial particle motion. In this paper, we give a comprehensive theoretical analysis of the Maxey-Riley equation. In particular, we build on previous local in time existence and uniqueness results to prove that solutions of the Maxey-Riley equation are global in time. In doing so, we also prove that the notion of a maximal solution extends to this equation. We furthermore prove conditions under which solutions are differentiable at the initial time. By considering the derivative of the solution with respect to the initial conditions, we perform a sensitivity analysis and demonstrate that two inertial trajectories can not meet, as well as provide a control on the growth of the distance between a pair of inertial particles. The properties we prove here for the Maxey-Riley equations are also possessed, mutatis mutandis, by a broader class of fractional differential equations of a similar form.

An Influence Maximization Algorithm for Dynamic Social Networks Based on Effective Links

The connection between users in social networks can be maintained for a certain period of time, and the static network structure formed provides the basic conditions for various kinds of research, especially for discovering customer groups that can generate great influence, which is important for product promotion, epidemic prevention and control, and public opinion supervision, etc. However, the computational process of influence maximization ignores the timeliness of interaction behaviors among users, the screened target users cannot diffuse information well, and the time complexity of relying on greedy rules to handle the influence maximization problem is high. Therefore, this paper analyzes the influence of the interaction between nodes in dynamic social networks on information dissemination, extends the classical independent cascade model to a dynamic social network dissemination model based on effective links, and proposes a two-stage influence maximization solution algorithm (Outdegree Effective Link—OEL) based on node degree and effective links to enhance the efficiency of problem solving. In order to verify the effectiveness of the algorithm, five typical influence maximization methods are compared and analyzed on four real data sets. The results show that the OEL algorithm has good performance in propagation range and running time.

Discounted optimal stopping problems in first-passage time models with random thresholds

AbstractWe derive closed-form solutions to some discounted optimal stopping problems related to the perpetual American cancellable dividend-paying put and call option pricing problems in an extension of the Black–Merton–Scholes model. The cancellation times are assumed to occur when the underlying risky asset price process hits some unobservable random thresholds. The optimal stopping times are shown to be the first times at which the asset price reaches stochastic boundaries depending on the current values of its running maximum and minimum processes. The proof is based on the reduction of the original optimal stopping problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit and modified normal-reflection conditions. We show that the optimal stopping boundaries are characterised as the maximal and minimal solutions of certain first-order nonlinear ordinary differential equations.

Metal-organic frameworks decorated pomelo peel cellulose nanofibers membranes for high performance dye rejection

In this work, the cellulose nanofibers based membranes with both high dye solution flux and rejection rates were obtained. The cellulose nanofibers (PPCNFs) were prepared from pomelo peels by simple H2O2 oxidation process, decorated with ZIF-8 crystals, and vacuum filtrated to form composite membranes (PPCNFs@ZIFx). Ascribing to the highly porous structures of ZIF-8, as well as their uniform and stable deposition on the surfaces of PPCNFs, the resultant membranes were mechanically strong (tensile strength up to ~72.25 MPa), surface area rich (~ 199.46 m2/g), and with a maximal dye solution flux of ~32.3 L m−2 h−1 bar−1 and dye rejection rates larger than 85 % (89.95 % for methyl blue, 98.83 % for methyl violet, 95.91 % for malachite green, 95.36 % for rhodamine 6 G, and a high rejection rate of ~98.6 % toward methyl violet after 30-day cycles). Thus, this work paved a simple and green way to fabricate cellulose nanofibers based membranes with high performance in the remediation of dye contaminated water.

Global solutions of nonlinear fractional diffusion equations with time-singular sources and perturbed orders

In a Hilbert space, we consider a class of nonlinear fractional equations having the Caputo fractional derivative of the time variable t and the space fractional function of the self-adjoint positive unbounded operator. We consider various cases of global Lipschitz and local Lipschitz source with time-singular coefficient. These sources are generalized of the well–known fractional equations such as the fractional Cahn–Allen equation, the fractional Burger equation, the fractional Cahn–Hilliard equation, the fractional Kuramoto–Sivashinsky equation, etc. Under suitable assumptions, we investigate the existence, uniqueness of maximal solution, and stability of solution of the problems with respect to perturbed fractional orders. We also establish some global existence and prove that the global solution can be approximated by known asymptotic functions as \(t\rightarrow \infty \).

On an inversion-free algorithm for the nonlinear matrix problem

We perform extensive theoretical analysis on an inversion-free algorithm to obtain a maximal positive definite solution (PDS) of a general nonlinear matrix equation (NME) with suitable square matrices. We illustrate some estimations for obtaining the solution, rate of convergence and convergence analysis of the presented algorithm. To ensure its validity and usefulness, several different types of test problems, including randomly generated ones, are given. Comparisons with the existing iterative methods are done through error and computational time.

Perpetual American Double Lookback Options on Drawdowns and Drawups with Floating Strikes

We present closed-form solutions to the problems of pricing of the perpetual American double lookback put and call options on the maximum drawdown and the maximum drawup with floating strikes in the Black-Merton-Scholes model. It is shown that the optimal exercise times are the first times at which the underlying risky asset price process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum as well as the maximum drawdown or maximum drawup. The proof is based on the reduction of the original double optimal stopping problems to the appropriate sequences of single optimal stopping problems for the three-dimensional continuous Markov processes. The latter problems are solved as the equivalent free-boundary problems by means of the smooth-fit and normal-reflection conditions for the value functions at the optimal stopping boundaries and the edges of the three-dimensional state spaces. We show that the optimal exercise boundaries are determined as either the unique solutions of the associated systems of arithmetic equations or the minimal and maximal solutions of the appropriate first-order nonlinear ordinary differential equations.

Temporal-aware influence maximization solution in artificial intelligent edge application

Temporal-aware influence maximization solution in artificial intelligent edge application

An Inertial Algorithm of Generalized f -Projection for Maximal Monotone Operators and Generalized Mixed Equilibrium Problems in Banach Spaces

In this paper, we study a modified hybrid inertial algorithm of generalized f-projection for approximating maximal monotone operators and solutions of generalized mixed equilibrium problems in Banach spaces. Our results generalize and improve many recent announced results in the literature.

Lp -Solutions of backward doubly SDEs with continuous and stochastic linear growth coefficients and p ∈ (1,2)

In this paper, we are interested in solving backward doubly stochastic differential equations in Lp , for any p ∈ ( 1 , 2 ) , when the coefficients are continuous with stochastic linear growth. Via approximation and comparison theorem, the existence of L p − minimal and L p − maximal solutions are obtained.

Outage Energy Efficiency Maximization for UAV-Assisted Energy Harvesting Cognitive Radio Networks

The UAV-assisted energy harvesting cognitive wireless network has the feature of fast layout, remarkable spectrum efficiency (SE) and energy efficiency (EE), thereby being envisioned as one significant cutting-edge communication technology in the 5G/6G era. In this study, we consider an UAV-assisted energy harvesting cognitive radio network (UAV-EH-CRN), where an UAV is employed as one cognitive user (CU), hovering in the air to perform spectrum sensing and communicating with the dedicated receiver (DR) on the ground. Based on the sensing results of the primary user (PU), the UAV can adaptively adjust power to transmit with the DR. Moreover, by harvesting renewable energy, the UAV can replenish energy actively for transmission. By introducing outage rate (OR) for guaranteeing the UAVs SE requirements, our goal is to maximize the outage energy efficiency (OEE) for the UAV-EH-CRN, subject to constraints of energy, transmission power, and interference power. To make the non-convex OEE maximization problem solvable, one resource allocation policy portrayed as outage related convex approximation strategy (OR-CAS) is presented. Specifically, the OEE problem is firstly converted into a tractable counterpart by the proposed OR-CAS. Then, making use of the Lagrange duality method and one-dimensional linear programming, the optimal resource allocation solution for the OEE maximization is obtained. Simulation results show that our scheme enable the UAV-EH-CRN to adapt to various on-demand outage SE scenarios by setting different OR thresholds. Moreover, our algorithm can obtain significant EE gains and less time expenditure in comparison with the traditional exhaustive search approach.

Generalized Monotone Method for Caputo Fractional Reaction-Diffusion Equation

In this paper, our aim is to obtain the integral representation for the solution of non-linear Caputo reaction-diffusion equation of order q, where 0 < q < 1, in term of Green’s function. We have developed a generalized monotone method for non-linear weakly coupled Caputo reaction-diffusion equation. The generalized monotone method yields monotone sequences which converges uniformly and monotonically to coupled minimal and maximal solutions. The existence of a unique solution for the non-linear Caputo reaction-diffusion equation is obtained.

Simultaneous supplementation of magnetite and polyurethane foam carrier can reach a Pareto-optimal point to alleviate ammonia inhibition during anaerobic digestion

Material addition is a potential method to alleviate ammonia inhibition in the anaerobic digestion process. In this study, seven materials were selected to examine their impact on methane production (MP). In the batch experiment, the reactors with “Magnetite + foam carrier” exhibited nearly a 56% increase in maximum MP rate. Different strategies of magnetite and foam carrier addition were further explored in semi-continuous experiment. Adding two materials together (simultaneous supplementation) resulted in almost 60% increase in MP, 28–41% higher than when supplemented separately. Simultaneous supplementation of the two materials was considered as the Pareto-optimal solution for MP maximization. The acetogenic and methanogenic activities analyses revealed that foam carrier addition accelerated propionate transformation (from propionate to acetate or valerate) while magnetite addition intensified the methanogenesis step. Direct interspecies electron transfer (DIET) was strengthened with over 50% enrichment in DIET-related bacteria. The Anaerobic Digestion Model No.1 was modified with the development of liquid metabolite and biofilm, result in a goodness-of-fit of R2 > 0.96.16sRNA gene sequencing demonstrated that the microbes in the biofilm had kinetic advantages over those in the liquid phase, coupled with higher km and lower Ks values, which contributed to the enhanced MP.

An application of interactive fuzzy optimization model for redesigning supply chain for resilience

Supply chain disruptions compel professionals all over the world to consider alternate strategies for addressing these issues and remaining profitable in the future. In this study, we considered a four-stage global supply chain and designed the network with the objectives of maximizing profit and minimizing disruption risk. We quantified and modeled disruption risk as a function of the geographic diversification of facilities called supply density (evaluated based on the interstage distance between nodes) to mitigate the risk caused by disruptions. Furthermore, we developed a bi-criteria mixed-integer linear programming model for designing the supply chain in order to maximize profit and supply density. We propose an interactive fuzzy optimization algorithm that generates efficient frontiers by systematically taking decision-maker inputs and solves the bi-criteria model problem in the context of a realistic example. We also conducted disruption analysis using a discrete set of disruption scenarios to determine the advantages of the network design from the bi-criteria model over the traditional profit maximization model. Our study demonstrates that the network design from the bi-criteria model has a 2% higher expected profit and a 2.2% lower profit variance under disruption than the traditional profit maximization solution. We envisage that this model will help firms evaluate the trade-offs between mitigation benefits and mitigation costs.

Regular matching problems for infinite trees

We study the matching problem of regular tree languages, that is, "$\exists \sigma:\sigma(L)\subseteq R$?" where $L,R$ are regular tree languages over the union of finite ranked alphabets $\Sigma$ and $\mathcal{X}$ where $\mathcal{X}$ is an alphabet of variables and $\sigma$ is a substitution such that $\sigma(x)$ is a set of trees in $T(\Sigma\cup H)\setminus H$ for all $x\in \mathcal{X}$. Here, $H$ denotes a set of "holes" which are used to define a "sorted" concatenation of trees. Conway studied this problem in the special case for languages of finite words in his classical textbook "Regular algebra and finite machines" published in 1971. He showed that if $L$ and $R$ are regular, then the problem "$\exists \sigma \forall x\in \mathcal{X}: \sigma(x)\neq \emptyset\wedge \sigma(L)\subseteq R$?" is decidable. Moreover, there are only finitely many maximal solutions, the maximal solutions are regular substitutions, and they are effectively computable. We extend Conway's results when $L,R$ are regular languages of finite and infinite trees, and language substitution is applied inside-out, in the sense of Engelfriet and Schmidt (1977/78). More precisely, we show that if $L\subseteq T(\Sigma\cup\mathcal{X})$ and $R\subseteq T(\Sigma)$ are regular tree languages over finite or infinite trees, then the problem "$\exists \sigma \forall x\in \mathcal{X}: \sigma(x)\neq \emptyset\wedge \sigma_{\mathrm{io}}(L)\subseteq R$?" is decidable. Here, the subscript "$\mathrm{io}$" in $\sigma_{\mathrm{io}}(L)$ refers to "inside-out". Moreover, there are only finitely many maximal solutions $\sigma$, the maximal solutions are regular substitutions and effectively computable. The corresponding question for the outside-in extension $\sigma_{\mathrm{oi}}$ remains open, even in the restricted setting of finite trees.

Approximation by lexicographically maximal solutions in matching and matroid intersection problems

We study how good a lexicographically maximal solution is in the weighted matching and matroid intersection problems. A solution is lexicographically maximal if it takes as many heaviest elements as possible, and subject to this, it takes as many second heaviest elements as possible, and so on. If the distinct weight values are sufficiently dispersed, e.g., the minimum ratio of two distinct weight values is at least the ground set size, then the lexicographical maximality and the usual weighted optimality are equivalent. We show that the threshold of the ratio for this equivalence to hold is exactly 2. Furthermore, we prove that if the ratio is less than 2, say α, then a lexicographically maximal solution achieves (α/2)-approximation, and this bound is tight.