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This paper is concerned with the qualitative analysis of a diffusion SIR epidemic model with networked delay and nonlinear incidence rate. First, we prove the existence and uniqueness of the model by using the method of upper and lower solutions. Then, we prove that the trivial equilibrium (0, 0) is unstable; the disease‐free equilibrium (N, 0) is locally asymptotically stable if the reproduction number R0≤1 and unstable if R0>1; the endemic equilibrium (S∗, I∗) is locally asymptotically stable if R0≤R1 or if R1τ10∗. Moreover, when R1

A maximum likelihood estimator is guaranteed to exist, provided specific regularity conditions are met to ensure a well-defined likelihood function and achievable maximization. This paper considered the existence and uniqueness of the maximum likelihood estimates (MLE) of the exponentiated BurrXII Poisson (EBXIIP) distribution under some possible conditions. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to study a kind of boundary value problem with an integral boundary condition including Hadamard-type fractional differential equations. To do this, upper and lower solutions are used to guarantee their existence, and Schauder’s fixed point theorem is used to prove the uniqueness of the positive solutions to this problem. An illustrated example is presented to explain the theorems that have been proved. [ABSTRACT FROM AUTHOR]

This paper deals with global well-posedness of the solution to the complex short pulse equation. We first use regularized technology and the approximation argument to prove the local existence and uniqueness of this equation. Then, based on conserved quantities and energy analysis, we show that the solution can be extended globally in time for suitably small initial data. [ABSTRACT FROM AUTHOR]

In this paper, we study an SIRI epidemic model with relapse and free boundaries, which can describe the spreading process of diseases well. At first, the existence and uniqueness of the global solution is proven. And then we obtain some sufficient conditions for the disease spreading and vanishing. We can find an R0 such that the disease will vanish for R0 ≤ 1; otherwise, whether the disease will spread or not is determined by the initial infected region h0 and spreading capacity ho of the spreading front. Moreover, the longtime behavior of the solution is given. Finally, numerical simulations are provided to illustrate the results of the analysis. [ABSTRACT FROM AUTHOR]

In this article, we introduce the notion of cyclic α -admissible mapping with respect to θ with its special cases, which are cyclic α -admissible mapping with respect to θ ∗ and cyclic α ∗ -admissible mapping with respect to θ . We present the notion of orthogonal (α θ - β F) -rational contraction and establish new fixed point results over orthogonal F -metric space. The study includes illustrative examples to support our results. We apply our results to prove the existence and uniqueness of solutions for second-order differential equations. [ABSTRACT FROM AUTHOR]

This paper discusses a new coupled system of Riemann-Liouville fractional differential equations, in which the nonlinear terms include the Riemann-Liouville fractional integrals and the boundary value problems involve three-points. We seek also for the existence and uniqueness of solutions for this new system. We first get some useful properties of the Green's function generated by the system, and then we apply a fixed point theorem of increasing φ - (h, e)-concave operators to this new coupled system. Finally, we gain the existence and uniqueness results of the solution for this problem. In the end, a concrete example is structured to illustrate the main result. [ABSTRACT FROM AUTHOR]

In this paper we present a survey on the uniqueness of limit cycles bifurcating from a center, homoclinic loop or a heterclinic loop in planar systems, introducing five bifurcation theorems, and then apply the theorems to the study of codimension two bifurcations, obtaining a complete analysis on the uniqueness of limit cycles. [ABSTRACT FROM AUTHOR]

In this work, we investigate the existence, uniqueness, and different stability types, focusing on Ulam-Hyers stability, of a solution to a nonlinear fuzzy fractional differential equation with Caputo generalized Hukuhara differentiability of order ℘ ∈ (n - 1, n). We establish explicit criteria to guarantee both the existence and uniqueness of the solution, employing the Banach fixed point theorem. Additionally, we present two examples to enhance the illustration of our findings. [ABSTRACT FROM AUTHOR]

In this article, for the direct problem, which consists of finding the velocity potential and the displacement of boundary points in the wave equation of memory type with initial and boundary acoustic conditions, the inverse problem of determining the memory kernel by the integral overdetermination condition is studied. By introducing a new function, the problem is reduced to a problem with homogeneous boundary conditions. Using the technique of estimating integral equations and the contraction mappings principle in Sobolev spaces, the existence and uniqueness theorem for the inverse problem is proved. [ABSTRACT FROM AUTHOR]

For the wave equation with inhomogeneity σ(x)umt + q(x)up a forward and an one-dimensional inverse problems are studied. Here m > 1 and p > 1 are real numbers. The forward problem is considered in the domain x > 0; t > 0 with zero initial data and Dirichlet boundary condition at x = 0. An unique solvability theorem of this problem is proved. The inverse problem is devoted to determining the coefficients σ(x) and q(x). As an additional information for recovering this coefficients, two forward problems with different Dirichlet data are considered and traces of the derivative of their solutions with respect to x are given at x = 0 on a finite interval. For the inverse problem a local existence and uniqueness theorem is established. [ABSTRACT FROM AUTHOR]

In this paper, a special system of non-linear Abel integral equations (SNAIEs) is studied, which arises in astrophysics. Here, the well-known collocation method is extended to obtain an approximate solution of the SNAIEs. The existence and uniqueness conditions of the solution are investigated. Finally, some examples are solved to illustrate the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This research paper deals with the numerical method for the solution of high-order Fredholm integro-differential difference equations using Legendre polynomials. We obtain the integral form of the problem, which is transformed into a system of algebraic equations using the collocation method. We then solve the algebraic equation using Newton's method. We establish the uniqueness and convergence of the solution. Numerical problems are considered to test the efficiency of the method, which shows that the method competes favorably with the existing methods and, in some cases, approximates the exact solution. [ABSTRACT FROM AUTHOR]

This paper discusses the large deviations for stochastic predator–prey model driven by multiplicative Lévy noise. Using Galerkin approximation, we initially prove the existence and uniqueness of solution. Due to the equivalence between Laplace principle and large deviation principle under a Polish space, the method of weak convergence has been followed in order to establish our results for this coupled system of equations. [ABSTRACT FROM AUTHOR]

The objective of the current article is to use the Banach fixed point theorem to present proofs for the existence and uniqueness theorems for a nonlinear time-fractional differential equation, namely, the linear and nonlinear, homogeneous and nonhomogeneous time-fractional diffusion equations. In addition, we explore exact solutions to nonlinear time fractional partial differential equations using an effective method called the Adomian natural decomposition method (ANDM), which is a combination of the Adomian decomposition method (ADM) and the natural transform method (NTM). To demonstrate the effectiveness of the suggested scheme, three examples are provided along with graphs and numerical tables to support our work. The outcomes demonstrate how effortless and effective the ANDM is for addressing fractional partial differential equations in multi-dimensional spaces, some of which we examined in this study as special cases.AMS Classifications 34A08, 65P99, 49J15, 35R11, 26A33, 74G10. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of spatially quasi-periodic solutions to the p -generalized KdV equation on the real line with quasi-periodic initial data whose Fourier coefficients are exponentially decaying. In order to solve for the Fourier coefficients of the solution, we first reduce the nonlinear dispersive partial differential equation to a nonlinear infinite system of coupled ordinary differential equations, and then construct the Picard sequence to approximate them. However, we meet, and have to deal with, the difficulty of studying the higher dimensional discrete convolution operation for several functions : c × ⋯ × c ︸ p (total distance) : = ∑ ♣ 1 , ⋯ , ♣ p ∈ Z ν ♣ 1 + ⋯ + ♣ p = total distance ∏ j = 1 p c (♣ j). In order to overcome it, we apply a combinatorial method to reformulate the Picard sequence as a tree. Based on this form, we prove that the Picard sequence is exponentially decaying and fundamental (i.e., a Cauchy sequence). The result has been known for p = 2 [11] , and the combinatorics become harder for larger values of p. For the sake of clarity, we first give a detailed discussion of the proof of the existence and uniqueness result in the simplest case not covered by previous results, p = 3. Next, we prove existence and uniqueness in the general case p ≥ 2 , which then covers the remaining cases p ≥ 4. As a byproduct, we recover the local result from [11]. In the process of proof, we give a combinatorial structure of tensor (multi-linear operator), exhibit the most important combinatorial index σ (it's related to the degree or multiplicity of the power-law nonlinearity), and obtain a relationship with other indices, which is essential to our proofs in the case of general p. [ABSTRACT FROM AUTHOR]

In the study done here regarding the theory of summability, we introduce some new concepts in fuzzy normed spaces. First, at the beginning of the original part of our study, we define the lacunary invariant statistical convergence. Then, we examine some characteristic features like uniqueness, linearity of this new notion and give its important relation with pre-given concepts. [ABSTRACT FROM AUTHOR]

The Yang Adomian decomposition technique (YADM) is an excellent analytical tool employed in this study to solve the partial differential equations (PDEs). The result of the suggested approach is stated as a series of Adomian components that converges to the precise solution of the problem. To show the applicability of the recommended technique, examples are presented. [ABSTRACT FROM AUTHOR]

The main purpose of the present paper is to prove the existence of periodic solutions of the three-body problem in the 3D Kepler formulation. We have solved the same problem in the case when the three particles are considered in an external inertial system. We start with the three-body equations of motion, which are a subset of the equations of motion (previously derived by us) for any number of bodies. In the Minkowski space, there are 12 equations of motion. It is proved that three of them are consequences of the other nine, so their number becomes nine, as much as the unknown trajectories are. The Kepler formulation assumes that one particle (the nucleus) is placed at the coordinate origin. The motion of the other two particles is described by a neutral system with respect to the unknown velocities. The state-dependent delays arise as a consequence of the finite vacuum speed of light. We obtain the equations of motion in spherical coordinates and split them into two groups. In the first group all arguments of the unknown functions are delays. We take their solutions as initial functions. Then, the equations of motion for the remaining two particles must be solved to the right of the initial point. To prove the existence–uniqueness of a periodic solution, we choose a space consisting of periodic infinitely smooth functions satisfying some supplementary conditions. Then, we use a suitable operator which acts on these spaces and whose fixed points are periodic solutions. We apply the fixed point theorem for the operators acting on the spaces of periodic functions. In this manner, we show the stability of the He atom in the frame of classical electrodynamics. In a previous paper of ours, we proved the existence of spin functions for plane motion. Thus, we confirm the Bohr and Sommerfeld's hypothesis for the He atom. [ABSTRACT FROM AUTHOR]

The existence and uniqueness of the global positive solution are proved for the system of stochastic differential equations describing a two-species Lotka–Volterra mutualism model disturbed by white noise, centered and noncentered Poisson noises. For the considered system, sufficient conditions of stochastic ultimate boundedness, stochastic permanence, nonpersistence and strong persistence in the mean are obtained. [ABSTRACT FROM AUTHOR]

We investigate the conditions for the existence and uniqueness of solutions in a nonlinear system of sequential fractional differential equations using the Liouville–Caputo type with varying orders. This system is enriched by nonlocal coupled integral boundary conditions. The desired outcomes are attained by employing traditional fixed-point theorems. It is essential to emphasize that the fixed-point approach proves to be an effective method for establishing the existence of solutions in boundary value problems. Furthermore, we provide constructed examples to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

The level of difficulty of mathematical test items is a critical aspect for evaluating test quality and educational outcomes. Accurately predicting item difficulty during test creation is thus significantly important for producing effective test papers. This study used more than ten years of content and score data from China's Henan Provincial College Entrance Examination in Mathematics as an evaluation criterion for test difficulty, and all data were obtained from the Henan Provincial Department of Education. Based on the framework established by the National Center for Education Statistics (NCES) for test item assessment methodology, this paper proposes a new framework containing eight features considering the uniqueness of mathematics. Next, this paper proposes an XGBoost-based SHAP model for analyzing the difficulty of mathematics tests. By coupling the XGBoost method with the SHAP method, the model not only evaluates the difficulty of mathematics tests but also analyzes the contribution of specific features to item difficulty, thereby increasing transparency and mitigating the "black box" nature of machine learning models. The model has a high prediction accuracy of 0.99 for the training set and 0.806 for the test set. With the model, we found that parameter-level features and reasoning-level features are significant factors influencing the difficulty of subjective items in the exam. In addition, we divided senior secondary mathematics knowledge into nine units based on Chinese curriculum standards and found significant differences in the distribution of the eight features across these different knowledge units, which can help teachers place different emphasis on different units during the teaching process. In summary, our proposed approach significantly improves the accuracy of item difficulty prediction, which is crucial for intelligent educational applications such as knowledge tracking, automatic test item generation, and intelligent paper generation. These results provide tools that are better aligned with and responsive to students' learning needs, thus effectively informing educational practice. [ABSTRACT FROM AUTHOR]

Random environment and human activities have important effects on the survival of listeria. In this paper, treating infected people and removing bacteria from the environment as control strategies, we developed a listeriosis model that considers random noise and spatial diffusion. By constructing a Lyapunov function, we demonstrated the existence and uniqueness of the global positive solution of the model. However, it was a challenging task to realize the optimal control of the model by solving the Pontryagin random maximum principle with the lowest control cost. Therefore, our study on near-optimal controls is of great significance for controlling the spread of listeriosis. Initially, we gave some adjoint equations and a priori estimates. Subsequently, the Pontryagin random maximum principle was utilized to establish the sufficient and necessary conditions for achieving near-optimal controls. Ultimately, the theoretical findings are corroborated through numerical analysis. [ABSTRACT FROM AUTHOR]

For this study, we investigate the existence and uniqueness of local solutions and derive a blow-up solution for a quasi-linear bi-hyperbolic equation under dynamic boundary conditions. We utilize the contraction mapping concept to demonstrate the solution's local well-posedness and employ a concavity approach to establish the blow-up result. [ABSTRACT FROM AUTHOR]

In this paper, we study an initial value problem with a weakly singular nonlinear fractional differential equation of higher order. First, we establish the existence of global solutions to the problem within the appropriate function space. We then introduce a generalized Riemann-Liouville mean value theorem. Using this theorem, we prove the Nagumo-type uniqueness theorem for the stated problem. Moreover, we give two examples to illustrate the applicability of the existence and uniqueness theorems. [ABSTRACT FROM AUTHOR]

This article considered two generalized Nicholson's blowflies equations with iteration term and time delay, as well as with immigration, and Nicholson's blowflies equation with iteration term and time delay, as well as harvesting term, respectively. Under appropriate conditions, the existence and uniqueness of almost periodic positive solutions for these two equations were established, respectively, by employing Banach's fixed point theorem. These results were brand new. [ABSTRACT FROM AUTHOR]

In this manuscript, we introduce the idea of complex-valued Neutrosophic b-metric spaces along with numerous significant illustrations. We provide fixed-point results for contraction maps. To support the main result, we establish the existence and uniqueness of solutions for nonlinear integral equations after the work. [ABSTRACT FROM AUTHOR]

In this work, we study the existence and uniqueness of a common best proximity point for a pair of nonself functions that are not necessarily continuous using the simulation function. In the following, we state important common best proximity point theorems as results of the main theorems of this article. This achievement allows us to have an example that covers our main theorem but does not apply to the Banach contraction principle. Finally, an application of a nonlinear fractional differential equation to support the obtained conclusions. [ABSTRACT FROM AUTHOR]

In the present research, we investigate the existence and uniqueness of solutions for a system of (k; Χ)-Hilfer fractional differential equations, subject to coupled nonlocal boundary conditions, which contain various fractional derivatives and Riemann-Stieltjes integrals. The uniqueness result relies on the Banach contraction mapping principle, while the existence results depend on the Leray-Schauder alternative and Krasnosel'skiı fixed point theorem. Examples are also constructed to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In [6], the authors studied an SIR epidemic model with free boundary. They first proved the global existence and uniqueness of solution, and then gave the criteria for spreading and vanishing. They also obtained the long-time behavior for the case of vanishing. However, the longtime behavior when spreading happens remains open. In this short paper, we aim to solve this open problem. [ABSTRACT FROM AUTHOR]

The study of fractional Langevin equation has obtained abundant results in recent years. However, there are few studies on resonant fractional Langevin equation. In this paper, we investigate boundary value problems for fractional Langevin equation at resonance. By virtue of Banach contraction mapping principle and Leray-Schauder fixed point theorem, we obtain the uniqueness and existence of solutions. In addition, we get different stability results, including Ulam-Hyres stability and generalized Ulam-Hyres stability. Finally, give relevant examples to demonstrate the main results. [ABSTRACT FROM AUTHOR]

Using the method of characteristics and the Banach fixed point theorem (for the Bielecki metric), in the paper it is established the existence and uniqueness of a global (continuous) solution of the mixed problem in the rectangle Π = {(x, t): 0 < x < l < ∞, 0 < t < T < ∞} for the first order hyperbolic system of semi-linear equations of the form... for a singular with orthogonal (degenerate) and non-orthogonal to the coordinate axes characteristics and with nonlinear boundary conditions, where Λ(x, t) = diag(λ1(x, t),. . ., λk(x, t)), u = (u1, . . ., uk), v = (v1, . . ., vm), w = (w1, . . ., wn), f = (f1, . . ., fk), g = (g1, . . ., gm), h = (h1, . . ., hn) and besides sign λi(0, t) = const ̸= 0, sign λi(l, t) = const ̸= 0 for all t ∈ [0, T] and for all i ∈ {1, . . ., k}. The presence of non-orthogonal and degenerate characteristics of the hyperbolic system for physical reasons indicates that part of the oscillatory disturbances in the medium propagates with a finite speed, and part with an unlimited one. Such a singularity (degeneracy of characteristics) of the hyperbolic system allows mathematical interpretation of many physical processes, or act as auxiliary equations in the analysis of multidimensional problems. [ABSTRACT FROM AUTHOR]

In this paper, we prove the uniqueness theorem for a special class of meromorphic functions on the complex plane C. In particular, we study the class of meromorphic functions f in the domain C\K′, where K′ is the finite set of limit points of simple poles of the function f. In this class, we describe non-trivial subclasses in which every function f can be uniquely determined by the residues of the function f at its poles. The result covered in this paper is a part of a problem in a spectral operator theory. [ABSTRACT FROM AUTHOR]

This paper addresses the uniqueness problem concerning the j-th derivative of a meromorphic function f(z) and the k-th derivative of its shift, f(z + c), where j, k are integers with 0 ≤ j < k. In this regard, our work surpasses the achievements of [2], as we have improved upon the existing results and provided a more refined understanding of this specific aspect. We give some illustrative examples to enhance the realism of the obtained outcomes. Denote by E(a, f) the set of all zeros of f-a, where each zero with multiplicity m is counted m times. In the paper proved, in particular, the following statement: Let f(z) be a non-constant meromorphic function of finite order, c be a non-zero finite complex number and j, k be integers such that 0 ≤ j < k. If f(j)(z) and f(k)(z + c) have the same a-points for a finite value a(̸= 0) and satisfy conditions... [ABSTRACT FROM AUTHOR]

In this work, we consider an inverse problem of determining the kernel of a fractional diffusion equation. The backward problem is the initial-boundary/value problem for this equation with non-local initial and homogeneous Dirichlet conditions. To determine the kernel, an overdetermination condition of the integral form is specified for the solution of the backward problem. Using the Fourier method and an ordinary fractional differential equation with a non-local boundary condition, the inverse problem is reduced to an equivalent problem. Further by using the fixed point argument in suitable Sobolev spaces, the global theorems of existence and uniqueness for the solution of the inverse problem are obtained. [ABSTRACT FROM AUTHOR]

We derive a deterministic mathematical model that scrutinizes the dynamics of cholera pathogen carriers and the hygiene consciousness of individuals, before the illness, during its prevalence, and after the disease's outbreaks. The dynamics can effectively help in curtailing the disease, but its effects had less coverage in the literature. Boundedness of the solution of the model, its existence, and uniqueness are ascertained. Effects of cholera pathogen carriers and hygiene consciousness of individuals in controlling the disease or allowing its further spread are analyzed. The differential transformation method is used to obtain series solutions of the differential equations that make the system of the model. Simulations of the series solutions of the model are carried out and displayed in graphs. The dynamics of the concerned state variables and parameters in the model are interpreted via the obtained graphs. It is observed that higher hygiene consciousness of individuals can drastically reduce catching cholera disease at onset and further spread of its infections in the population, this in turn, shortens the period of cholera epidemic. [ABSTRACT FROM AUTHOR]

In this article, we investigate the problem of sharing values between entire functions f(z) and f1(z) = b-1 + ... bi f(ki)(z+iη) share two distinct values a with counted and b with ignoring multiplicities, where b-1 and bi(i = 0,1· · ·, n) are small meromorphic functions of f(z), ki ≥ 0 (i = 0,1, · · ·, n) are integers. In relation to previous research, we obtain results that improve and generalise the findings conducted by Yang and Qi [CMFT, 20.1 (2020): 159-178]. [ABSTRACT FROM AUTHOR]

Let X be an Abelian group, Y be a commutative monoid, K ⊂Y be a submonoid and F : X → 2Y \ {∅} be a set-valued map. Under some additional assumptions on ideals ℐ1 in X and ℐ2 in X2, we prove that if F is ℐ2-almost everywhere K-additive, then there exists a unique up to K K-additive set-valued map G : X → 2Y \{∅} such that F = G ℐ1-almost everywhere in X. Our considerations refers to the well known de Bruijn's result [1]. [ABSTRACT FROM AUTHOR]

In the present paper, a nonlinear system of sine-Gordon equations that describes the DNA dynamics is considered. A novel unconditionally stable second-order accuracy difference scheme corresponding to the system of sine-Gordon equations is presented. In this work, for the first time in the literature, weak solution of this difference scheme is studied. The existence and uniqueness of the weak solution for the difference scheme are proved in the space of distributions, and the methods of variational calculus are applied. The finite-difference method and the fixed point theory are used in combination to perform numerical experiments that verify the theoretical statements. [ABSTRACT FROM AUTHOR]

In this paper, we consider the first boundary value problem for a class of steady non-Newtonian micropolar fluid equations with heat convection in the three-dimensional smooth and bounded domain Ω. By using the fixed-point theorem and introducing a family of penalized problems, under the condition that the external force term and the vortex viscosity coefficient are appropriately small, the existence and uniqueness of strong solutions of the problem are obtained. [ABSTRACT FROM AUTHOR]

In this article, we obtain new results on the unconditional well-posedness for a pair of periodic nonlinear dispersive equations using an abstract framework introduced by Kishimoto. This framework is based on a normal form reductions argument coupled with a number of crucial multilinear estimates. [ABSTRACT FROM AUTHOR]

The parametrized approach is extended in this study to find solutions to differential equations with fractal, fractional, fractal-fractional, and piecewise derivatives with the inclusion of a stochastic component. The existence and uniqueness of the solution to the stochastic Atangana-Baleanu fractional differential equation are established using Caratheodory's existence theorem. For the solution of differential equations using piecewise differential operators, which take into account combining deterministic and stochastic processes utilizing certain significant mathematical tools such as fractal and fractal-fractional derivatives, the applicability of the parametrized technique is being examined. We discuss the crossover behaviors of the model obtained by including these operators and we present some illustrative examples for some problems with piecewise differential operators. [ABSTRACT FROM AUTHOR]

This article concerns the uniqueness of an inverse problem of simultaneously identifying the space‐dependent coefficient and source in a one‐dimensional time‐fractional diffusion equation with derivative order α∈(0,1)$$ \alpha \in \left(0,1\right) $$ and the zero Neumann boundary value. By additional boundary measurements, we first obtain the uniqueness of the coefficient from the Laplace transform and a transformation formula. Then, we further show the uniqueness of the source through the asymptotic behavior of solutions to the corresponding forward problem. The result shows that the uniqueness of the simultaneous identification can be obtained under the condition that the prior information only on one set of parameters in the model is given other than that of two sets. [ABSTRACT FROM AUTHOR]

In this paper, we study the uniqueness problem of meromorphic functions when two differential polynomials share a set of roots of unity with finite weight. Our results not only generalize a result of Lahiri and Sinha [Commun. Korean Math. Soc., 35 (2020), 773-787], at the same time improve the recent results of Khoai and An [Advanced Studies: Euro-Tbilisi Math. J., 15 (2022), 39-51]. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new existence results for certain coupled systems of nonlinear Ψ-fractional differential equations with four-point fractional integral boundary conditions. The proofs are derived from Banach fixed point theorem. As an application, a nontrivial example is presented to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

Linear complementarity problems are fundamental in various mathematical fields, and it is widely recognized that these problems have a unique solution if and only if the matrix related to them is a P-matrix. However, determining whether a matrix is a P-matrix can be extremely challenging or even impossible, especially for large matrices. In this paper, we present a method for establishing the existence and uniqueness of the solution to linear complementarity problems without the need for P-matrix verification. [ABSTRACT FROM AUTHOR]

In [6, 7], it was shown that mixed problems can be both ill-posed for Petrovsky well-conditioned equations and well-posed for ill-conditioned equations. In the present paper we study the existence and uniqueness of the solution of a mixed problem for a class of equations with complex-valued coefficients that behave as parabolic ones, despite the fact that they can change over "time" from parabolic type to Schrödinger type, or even to antiparabolic type. [ABSTRACT FROM AUTHOR]

We consider the stochastic Navier–Stokes equations in three dimensions and prove that the law of analytically weak solutions is not unique. In particular, we focus on three examples of a stochastic perturbation: an additive, a linear multiplicative and a nonlinear noise of cylindrical type, all driven by a Wiener process. In these settings, we develop a stochastic counterpart of the convex integration method introduced recently by Buckmaster and Vicol. This permits us to construct probabilistically strong and analytically weak solutions defined up to a suitable stopping time. In addition, these solutions fail to satisfy the corresponding energy inequality at a prescribed time with a prescribed probability. Then we introduce a general probabilistic construction used to extend the convex integration solutions beyond the stopping time and in particular to the whole time interval [0,∞). Finally, we show that their law is distinct from the law of solutions obtained by Galerkin approximation. In particular, non uniqueness in law holds on an arbitrary time interval[0,T], T > 0. [ABSTRACT FROM AUTHOR]

This paper aims to solve two enduring challenges in existing trajectory similarity measures: computational inefficiency and the absence of the ‘uniqueness’ property that should be guaranteed in a distance function: dist(X, Y ) = 0 if and only if X = Y, where X and Y are two trajectories. In this work, we present a novel approach utilizing a distributional kernel for trajectory representation and similarity measurement, based on the kernel mean embedding framework. It is the very first time a distributional kernel is used for trajectory representation and similarity measurement. Our method does not rely on point-to-point distances which are used in most existing distances for trajectories. Unlike prevalent learning and deep learning approaches, our method requires no learning. We show the generality of this new approach in anomalous trajectory and sub-trajectory detection. We identify that the distributional kernel has (i) a data-dependent property and the ‘uniqueness’ property which are the key factors that lead to its superior task-specific performance, and (ii) runtime orders of magnitude faster than existing distance measures. [ABSTRACT FROM AUTHOR]