Search

A scheme for the integration of $ \, {}^{C} \mathit{\boldsymbol{{D}}}^{(1/n)} $-type fractional differential equations (FDEs) is presented in this paper. The approach is based on the expansion of solutions to FDEs via fractional power series. It is proven that $ \, {}^{C} \mathit{\boldsymbol{{D}}}^{(1/n)} $-type FDEs can be transformed into equivalent $ \left(\, {}^{C} \mathit{\boldsymbol{{D}}}^{(1/n)}\right)^n $-type FDEs via operator calculus techniques. The efficacy of the scheme is demonstrated by integrating the fractional Riccati differential equation.

Let $ R $ be a ring with identity. The commuting graph of $ R $ is the graph associated to $ R $ whose vertices are non-central elements in $ R $, and distinct vertices $ A $ and $ B $ are adjacent if and only if $ AB = BA $. In this paper, we completely determine the automorphism group of the commuting graph of $ 2\times 2 $ matrix ring over $ \mathbb{Z}_{p^{s}} $, where $ \mathbb{Z}_{p^{s}} $ is the ring of integers modulo $ p^{s} $, $ p $ is a prime and $ s $ is a positive integer.

Let R be a commutative ring with the identity 1R, and let R* be the multiplicative group of units in R. An element a ∈ R* is called an exceptional unit if there exists a b ∈ R* such that a + b = 1R. We set R** to be the set of all exceptional units in R. In this paper, we consider the residue-class ring Zn. For any positive integers n, s, and c ∈ Zn, let Ns(n, c) :=#{(x1, ..., xs) ∈ (Z** n ) s : x1 + ... + xs(n, c). Later on, Yang and Zhao (Monatsh. Math. 182 (2017)) extended Sander’s theorem to finite terms by using exponential sum theory. In this paper, using matrix theory, we present an explicit formula for N2(n, c). Later on, Yang and Zhao (Monatsh. Math. 182 (2017)) extended Sander’s theorem to finite terms by using exponential sum theory. In this paper, using matrix theory, we present an explicit formula for Ns(n, c). This extends and improves earlier results. [ABSTRACT FROM AUTHOR]

In this paper, we introduced a new contraction principle via altering distance and C -class functions with rational forms which extends and generalizes the existing version provided by Hasan Ranjbar et al. [H. Ranjbar, A. Niknam, A fixed point theorem in Hilbert C ∗ -modules, Korean J. Math. , 30 (2022), 297–304]. Specifically, the rational forms involved in the contraction condition we presented involve the p -th power of the displacements which can exceed the second power mentioned in Hasan Ranjbar et al.'s paper. Moreover, we also proved a fixed point theorem for this type of contraction in the Hilbert C ∗ -module. Some adequate examples were provided to support our results. As an application, we applied our result to prove the existence of a unique solution to an integral equation and a second-order (p , q) -difference equation with integral boundary value conditions. [ABSTRACT FROM AUTHOR]

The irreducible decomposition of a monomial ideal has played an important role in combinatorial commutative algebra, with applications beyond pure mathematics, such as biology. Given a monomial ideal I of a polynomial ring S = k [ x ] over a field k and variables x = { x 1 , ... , x n } , its incidence matrix, is the matrix whose rows are indexed by the variables x and whose columns are indexed by its minimal generators. The main contribution of this paper is the introduction of a novel invariant of a monomial ideal I , termed its signature, which could be thought of as a type of canonical form of its incidence matrix, and the proof that two monomial ideals with the same signature have essentially the same irreducible decomposition. [ABSTRACT FROM AUTHOR]

This work studies a coupled non-linear Schrödinger system with a singular source term. First, we investigate the question of the local existence of solutions. Second, one proves the existence of global solutions which scatter in some Sobolev spaces. Finally, one establishes the existence of non-global solutions. The main difficulty here is to overcome the regularity problem in the non-linearity. Indeed, because of the singularity of the source term, the classical contraction method in the energy space fails in such a regime. So, this paper is to fill such a gap in the literature. The argument follows ideas in T. Cazenave and I. Naumkin (Comm. Contemp. Math. , 19 (2017), 1650038). This consists to remark that the singularity problem is only near the origin. So, one needs to impose that the solution stays away from zero. This is not trivial, since there is no maximum principle for the Schrödinger equation. The existence of global solutions which scatter follows with the pseudo-conformal transformation via the existence of local solutions. Finally, the existence of non-global solutions follows with the classical variance method. [ABSTRACT FROM AUTHOR]

In this paper, the Cauchy problem for a class of coupled system of the four-dimensional cubic focusing nonlinear Schrödinger equations was investigated. By exploiting the double Duhamel method and the long-time Strichartz estimate, the global well-posedness and scattering were proven for the system below the ground state. In our proof, we first established the variational characterization of the ground state, and obtained the threshold of the global well-posedness and scattering. Second, we showed that the non-scattering is equivalent to the existence of an almost periodic solution by following the concentration-compactness/rigidity arguments of Kenig and Merle [17] (Invent. Math., 166 (2006), 645–675). Then, we obtained the global well-posedness and scattering below the threshold by excluding the almost periodic solution. [ABSTRACT FROM AUTHOR]

This paper is devoted to dealing with a kind of new Kirchhoff-type problem in RN that involves a general double-phase variable exponent elliptic operator ϕ. Specifically, the operator ϕ has behaviors like |τ|q(x)−2 τ if |τ| is small and like |τ| p(x)−2 τ if |τ| is large, where 1 < p(x) < q(x) < N. By applying some new analytical tricks, we first establish existence results of solutions for this kind of Kirchhoff-double-phase problem based on variational methods and critical point theory. In particular, we also replace the classical Ambrosetti–Rabinowitz type condition with four different superlinear conditions and weaken some of the assumptions in the previous related works. Our results generalize and improve the ones in [Q. H. Zhang, V. D. Rădulescu, J. Math. Pures Appl., 118 (2018), 159–203.] and other related results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Li-Yau gradient estimate on the positive solution to the following nonlinear parabolic equation ∂/∂t f = Δf + af (ln f)p with Neumann boundary conditions on a compact Riemannian manifold satisfying the integral Ricci curvature assumption, where p ≥ 0 is a real constant. This contrasts Olivé's gradient estimate, which works mainly for the heat equation rather than nonlinear parabolic equations and the result can be regarded as a generalization of the Li-Yau [P. Li, S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153-201] and Olivé [X. R. Olive, Neumann Li-Yau gradient estimate under integral Ricci curvature bounds, Proc. Amer. Math. Soc., 147 (2019), 411-426] gradient estimates. [ABSTRACT FROM AUTHOR]

In this paper, we study variation-inequality initial-boundary value problems with fouth order p(x)-Kirchhoff operators. First, an operator is constructed based on the Leray Schauder principle, and the existence of solutions is obtained. Secondly, the stability and uniqueness of the solution are analyzed after the conditions are appropriately relaxed on the Kirchhoff operators. [ABSTRACT FROM AUTHOR]

This paper investigates the issue of asynchronous H∞ tracking control for nonlinear semi-Markovian jump systems (SMJSs) based on the T-S fuzzy model. Firstly, in order to improve the performance of network control systems (NCSs) and the efficiency of data transmission, this paper adopts a double quantization strategy which quantifies the input and output of the controllers. Secondly, for the purpose of reducing the burden of network communication, an adaptive event-triggered mechanism (AETM) is adopted. Thirdly, due to the influence of network-induce delay, the system mode information can not be transmitted to the controller synchronously, thus, a continuous-time hidden Markov model (HMM) is established to describe the asynchronous phenomenon between the system and the controller. Additionally, with the help of some improved Lyapunov-Krasovski (L-K) functions with fuzzy basis, some sufficient criteria are derived to co-guarantee the state stability and the H∞ performance for the closed-loop tracking control system. Finally, a numerical example and a practical example are given to verify the effectiveness of designed mentality. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish some fixed point results for enriched Kannan-type mappings in convex metric spaces. We first give an affirmative answer to a recent Berinde and Păcurar's question (Remark 2.3) [J. Comput. Appl. Math., 386 (2021), 113217]. Furthermore, we establish the existence and uniqueness of fixed points for Suzuki-enriched Kannan-type mappings in the setting of convex metric spaces. Finally, we present an application to approximate the solution of the Volterra integral equations to support our results. [ABSTRACT FROM AUTHOR]

In this paper, we study normalized solutions of the fractional Schrödinger equation with a critical nonlinearity... we prove the existence of a second normalized solution under some conditions on a, p, s, and N. This is a continuation of our previous work (Z. Angew. Math. Phys., 73 (2022) 149) where only one solution is obtained. [ABSTRACT FROM AUTHOR]

The concept of contraction mappings plays a significant role in mathematics, particularly in the study of fixed points and the existence of solutions for various equations. In this study, we described two types of enriched contractions: enriched F-contraction and enriched F'-contraction associated with u-fold averaged mapping, which are involved with Kirk's iterative technique with order u. The contractions extracted from this paper generalized and unified many previously common super contractions. Furthermore, u-fold averaged mappings can be seen as a more general form of both averaged mappings and double averaged mappings. Moreover, we demonstrated that the ufold averaged mapping with enriched contractions has a unique fixed point. Our work examined the necessary conditions for the u-fold averaged mapping and weak enriched contractions to have equal sets of fixed points. Additionally, we illustrated that an appropriate Kirk's iterative algorithm can effectively approximate a fixed point of a u-fold averaged mapping as well as the two enriched contractions. Also, we delved into the well-posedness, limit shadowing property, and Ulam-Hyers stability of the u-fold averaged mapping. Furthermore, we established necessary conditions that guaranteed the periodic point property for each of the illustrated strengthened contractions. To underscore the generality of our findings, we presented several examples that aligned with comparable results found in the existing literature. [ABSTRACT FROM AUTHOR]

In this series of papers, we study the crosscap two embedding of a class of multipartite graphs, namely, annihilating-ideal graphs of a lattice. In Part 1 of the series [Class of crosscap two graphs arising from lattices-I, Mathematics, 11 (2023), 1-26], we classified lattices with the number of atoms less than or equal to 4, whose annihilating-ideal graph can be embedded in the Klein bottle. In this paper, which is Part 2 of the series, we classify all finite lattices with at least 5 atoms whose annihilating-ideal graph is embedded in crosscap two surfaces. These characterizations help us to identify classes of multipartite graphs, which are embedded in the Klein bottle. [ABSTRACT FROM AUTHOR]

In this paper, we gave some generalized Young type inequalities due to Zuo and Li [J. Math. Inequal., 16 (2022), 1169-1178], and we also presented a new Young type inequality. As applications, we obtained some operator inequalities and matrix versions inequalities including the Hilbert-Schmidt norm and trace. [ABSTRACT FROM AUTHOR]

The present paper is the result of a contemplative study of a multi-time control problem (MCP) by considering its associated equivalent auxiliary control problem (MCP)ς via the exact l1 penalty method. Further study reveals that the solution set of the considered problem and the auxiliary problem exhibits an equivalence under the KT-pseudoinvexity hypothesis. Moreover, the study is extended towards the saddle point defined for (MCP) to establish the relationship between the solution set of multi-time control problem (MCP) and its associated equivalent auxiliary control problem (MCP)ς. Finally, we present an illustrative application to authenticate the results presented in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following nonlinear Kirchhoff-type problem with sublinear perturbation and steep potential well... ...where a and b are positive constants, λ > 0 is a parameter, 1 < q < 2, the potential V ∈ C(R³;R) and V-1(0) has a nonempty interior. The functions f and g are assumed to obey a certain set of conditions. The existence of two nontrivial solutions are obtained by using variational methods. Furthermore, the concentration behavior of solutions as λ→∞ is also explored. [ABSTRACT FROM AUTHOR]

This paper reports a modified F-iterative process for finding the fixed points of three generalized ff-nonexpansive mappings. We assume certain assumptions to establish the weak and strong convergence of the scheme in the context of a Banach space. We suggest a numerical example of generalized ff-nonexpansive mappings which exceeds, properly, the category of functions furnished with a condition (C). After that, we show that our modified F-iterative scheme of this example converges to a common fixed point of three generalized ff-nonexpansive mappings. As an application of our main findings, we suggest a new projection-type iterative scheme to solve variational inequality problems in the setting of generalized ff-nonexpansive mappings. The main finding of the paper is new and extends many known results of the literature. [ABSTRACT FROM AUTHOR]

In this paper, we derived that a sufficiently large integer N can always be represented as the sum of three generalized D. H. Lehmer numbers. As a consequence, we deduced Lu and Yi's original result (Monatsh. Math., 159 (2010), 45-58). [ABSTRACT FROM AUTHOR]

Our purpose of this paper is to answer several open questions posed by Doan (AIMS Math., 6 (2021), 7895-7908). First, we present two fixed point theorems, which are positive answers to Doan's questions. Second, we establish a new type of Riech's fixed point theorem to improve a result of Doan. Finally, we offer a straightforward example illustrating that a set-valued mapping satisfying the conditions of our fixed point theorem may has more than one fixed point. [ABSTRACT FROM AUTHOR]

In this paper, we showed some generalized refinements and reverses of arithmetic-geometric-harmonic means (AM-GM-HM) inequalities due to Sababheh [J. Math. Inequal. 12 (2018), 901-920]. Among other results, it was shown that if a, b > 0, 0 < p ≤ t < 1 and m ∈ N+, then ...for b ≥ a, and the inequalities are reversed for b ≤ a. As applications, we obtained some inequalities for operators and determinants. [ABSTRACT FROM AUTHOR]

The operation space of the vertical lift shaft is small, the components are complex, the occluding and different behavior space characteristics are similar, and the unsafe behavior is not easy to detect, which makes the operation safety of maintenance personnel in the elevator greatly threatened. This paper proposes an elevator maintenance personnel behavior detection algorithm based on the first-order deep network architecture (FOA-BDNet). First, a lightweight backbone feature extraction network is designed to meet the online real-time requirements of elevator maintenance environment monitoring video stream detection. Then, the feature fusion network structure of "far intersection and close connection" is proposed to fuse the fine-grained information with the coarse-grained information and to enhance the expression ability of deep semantic features. Finally, a first-order deep target detection algorithm adapted to the elevator scene is designed to identify and locate the behavior of maintenance personnel and to correctly detect unsafe behaviors. Experiments show that the detection accuracy rate on the self-built data set in this paper is 98.68%, which is 4.41% higher than that of the latest target detection model YOLOv8-s, and the reasoning speed reaches 69.51fps/s, which can be easily deployed in common edge devices and meet the real-time detection requirements for the unsafe behaviors of elevator scene maintenance personnel.

Recently, in the field of periodic nonuniform sampling, researchers (Wang et al., 2019; Asharabi, 2023) have investigated the incorporation of a Gaussian multiplier in the one-dimensional series to improve its convergence rate. Building on these developments, this paper aimed to accelerate the convergence of the two-dimensional periodic nonuniform sampling series by incorporating a bivariate Gaussian multiplier. This approach utilized a complex-analytic technique and is applicable to a wide range of functions. Specifically, it applies to the class of bivariate entire functions of exponential type that satisfy a decay condition, as well as to the class of bivariate analytic functions defined on a bivariate horizontal strip. The original convergence rate of the two-dimensional periodic nonuniform sampling is given by $ O(N^{-p}) $, where $ p \geq 1 $. However, through the implementation of this acceleration technique, the convergence rate improved drastically and followed an exponential order, specifically $ \mathrm{e}^{-\alpha N} $, where $ \alpha > 0 $. To validate the theoretical analysis presented, the paper conducted rigorous numerical experiments.

In this current paper, we developed a new predator-prey model accompanying delay based on the earlier works. By applying inequality strategies, fixed point theorem, and a suitable function, we got new necessary conditions for the existence, uniqueness, nonnegativeness, and boundedness of the solution to the developed delayed predator-prey model. The bifurcation behavior and stability nature of the defined delayed predator-prey model were investigated by using stability and bifurcation theory of delayed differential equations. We have modified the Hopf bifurcation's appearance time and stability domain by building two distinct hybrid delayed feedback controllers for the delayed predator-prey model. The time of Hopf bifurcation appearance and stability domain of the model were explored. Matlab experiment diagrams were given to support the learned important results. The derived outcomes in this paper were original and have significant theoretical implications for maintaining equilibrium between the densities of the three species.

In this paper, we first introduced systems of generalized vector quasi-variational equilibrium problems as well as systems of vector quasi-variational equilibrium problems as their special cases in abstract convex spaces. Next, we established some existence theorems of solutions for systems of generalized vector quasi-variational equilibrium problems and systems of vector quasi-variational equilibrium problems in non-compact abstract convex spaces. Furthermore, we applied the obtained existence theorem of solutions for systems of vector quasi-variational equilibrium problems to solve the existence problem of Nash equilibria for noncooperative games. Then, as applications of the existence result of Nash equilibria for noncooperative games, we studied the existence of weighted Nash equilibria and Pareto Nash equilibria for multi-objective games. The results derived in this paper extended and unified the primary findings presented by some authors in the literature.

This paper introduces a subclass of Markov renewal processes (MRPs) and presents a solution to the optimal filtering problem in a stochastic observation system, where the state is modeled by an MRP and observed indirectly through noisy measurements. The MRPs considered here can be interpreted as continuous-time Markov chains (CTMCs) with a finite set of abstract states representing distributions of random vectors. The paper outlines the probabilistic properties of MRPs, emphasizing the ability to express any arbitrary function of the MRP as the solution to a linear stochastic differential system (SDS) with a martingale on the right-hand side (RHS). Using these properties, an optimal filtering problem is formulated in stochastic observation systems, where the hidden state belongs to the class of MRPs, and the observations consist of both diffusion and counting components. The drift terms in all observations depend on the system state. An optimal filtering estimate for a scalar function of the MRP is provided through the solution of an SDS with innovation processes on the RHS. Additionally, the paper presents a version of the Kushner-Stratonovich equation, describing the evolution of the conditional probability density function (PDF). To demonstrate the practical application of the estimation method, the paper presents a communications-related example, focusing on monitoring the qualitative state and numerical characteristics of a network channel using noisy observations of round-trip time (RTT) and packet loss flow. The paper also highlights the robustness of the filtering algorithm in scenarios where the MRP distribution is uncertain.

In this paper, we focus on two relative aging classes, namely increasing (decreasing) relative failure rate and increasing (decreasing) failure rate relative to average failure rate. We studied some reliability properties and connections with other classes of lifetime distributions. The main objective of this paper was to investigate the preservation properties of decreasing relative failure rate class and decreasing failure rate relative to average failure rate class under the structure of ($ n-k+1 $)-out-of-$ n $ system. We give some examples of parametric distributions to evaluate the correctness of the results.

In this paper, polynomial equations with real coefficients and in one variable were considered which contained a small, positive but specified and fixed parameter $ \varepsilon_0 \neq 0 $. By using the classical asymptotic method, roots of the polynomial equations have been constructed in the literature, which were proved to be valid for sufficiently small $ \varepsilon $-values (or equivalently for $ \varepsilon \to 0 $). In this paper, it was assumed that for some or all roots of a polynomial equation, the first few terms in a Taylor or Laurent series in a small parameter depending on $ \varepsilon $ exist and can be constructed. We also assumed that at least two approximations $ x_1(\varepsilon) $ and $ x_2(\varepsilon) $ for the real roots exist and can be constructed. For a complex root, we assumed that at least two real approximations $ a_1(\varepsilon) $ and $ a_2(\varepsilon) $ for the real part of this root, and that at least two real approximations $ b_1(\varepsilon) $ and $ b_2(\varepsilon) $ for the imaginary part of this root, exist and can be constructed. Usually it was not clear whether for $ \varepsilon = \varepsilon_0 $ the approximations were valid or not. It was shown in this paper how the classical asymptotic method in combination with the bisection method could be used to prove how accurate the constructed approximations of the roots were for a given interval in $ \varepsilon $ (usually including the specified and fixed value $ \varepsilon_0 \neq 0 $). The method was illustrated by studying a polynomial equation of degree five with a small but fixed parameter $ \varepsilon_0 = 0.1 $. It was shown how (absolute and relative) error estimates for the real and imaginary parts of the roots could be obtained for all values of the small parameter in the interval $ (0, \varepsilon_0] $.

In this paper, we introduced a new contraction principle via altering distance and $ C $-class functions with rational forms which extends and generalizes the existing version provided by Hasan Ranjbar et al. [H. Ranjbar, A. Niknam, A fixed point theorem in Hilbert $ C^\ast $-modules, Korean J. Math., 30 (2022), 297–304]. Specifically, the rational forms involved in the contraction condition we presented involve the $ p $-th power of the displacements which can exceed the second power mentioned in Hasan Ranjbar et al.'s paper. Moreover, we also proved a fixed point theorem for this type of contraction in the Hilbert $ C^\ast $-module. Some adequate examples were provided to support our results. As an application, we applied our result to prove the existence of a unique solution to an integral equation and a second-order $ (p, q) $-difference equation with integral boundary value conditions.

The discrete Hopfield neural network 3-satisfiability (DHNN-3SAT) model represents an innovative application of deep learning techniques to the Boolean SAT problem. Existing research indicated that the DHNN-3SAT model demonstrated significant advantages in handling 3SAT problem instances of varying scales and complexities. Compared to traditional heuristic algorithms, this model converged to local minima more rapidly and exhibited enhanced exploration capabilities within the global search space. However, the model faced several challenges and limitations. As constraints in SAT problems dynamically increased, decreased, or changed, and as problem scales expanded, the model's computational complexity and storage requirements may increase dramatically, leading to reduced performance in handling large-scale SAT problems. To address these challenges, this paper first introduced a method for designing network synaptic weights based on fundamental logical clauses. This method effectively utilized the synaptic weight information from the original SAT problem within the DHNN network, thereby significantly reducing redundant computations. Concrete examples illustrated the design process of network synaptic weights when constraints were added, removed, or updated, offering new approaches for managing the evolving constraints in SAT problems. Subsequently, the paper presented a DHNN-3SAT model optimized by genetic algorithms combined with K-modes clustering. This model employed genetic algorithm-optimized K-modes clustering to effectively cluster the initial space, significantly reducing the search space. This approach minimized the likelihood of redundant searches and reduced the risk of getting trapped in local minima, thus improving search efficiency. Experimental tests on benchmark datasets showed that the proposed model outperformed traditional DHNN-3SAT models, DHNN-3SAT models combined with genetic algorithms, and DHNN-3SAT models combined with imperialist competitive algorithms across four evaluation metrics. This study not only broadened the application of DHNN in solving 3SAT problems but also provided valuable insights and guidance for future research.

In this paper, we apply the powerful sine-Gordon expansion method (SGEM), along with a computational program, to construct some new traveling wave soliton solutions for two models, including the higher-order nonlinear Boussinesq dynamical wave equation, which is a well-known nonlinear evolution model in mathematical physics, and the (1+1)-dimensional framework of the Van der Waals gas system. This study presents some new complex traveling wave solutions, as well as logarithmic and complex function properties. The 3D and 2D graphical representations of all obtained solutions, unveiling new properties of the considered model are simulated. Additionally, several simulations, including contour surfaces of the results, are performed, and we discuss their physical implications. A comprehensive conclusion is provided at the end of this paper.

In this computational paper, we focused on the efficient numerical implementation of semi-implicit methods for models in materials science. In particular, we were interested in a class of nonlinear higher-order parabolic partial differential equations. The Cahn-Hilliard (CH) equation was chosen as a benchmark problem for our proposed methods. We first considered the Cahn-Hilliard equation with a convexity-splitting (CS) approach coupled with a backward Euler approximation of the time derivative and tested the performance against the bi-harmonic-modified (BHM) approach in terms of accuracy, order of convergence, and computation time. Higher-order time-stepping techniques that allow for the methods to increase their accuracy and order of convergence were then introduced. The proposed schemes in this paper were found to be very efficient for 2D computations. Computed dynamics in 2D and 3D are presented to demonstrate the energy-decreasing property and overall performance of the methods for longer simulation runs with a variety of initial conditions. In addition, we also present a simple yet powerful way to accelerate the computations by using MATLAB built-in commands to perform GPU implementations of the schemes. We show that it is possible to accelerate computations for the CH equation in 3D by a factor of 80, provided the hardware is capable enough.

This paper aims to formulate a class of nonlinear hybrid stochastic time-delay neural networks (STDNNs) with Lévy noise. Specifically, the coefficients of networks grow polynomially instead of linearly, and the time delay of given neural networks is non-differentiable. In many practical situations, nonlinear hybrid STDNNs with Lévy noise are unstable. Hence, this paper uses feedback control based on discrete-time state and mode observations to stabilize the considered nonlinear hybrid STDNNs with Lévy noise. Then, we establish stabilization criteria of $ H_{\infty} $ stability, asymptotic stability, and exponential stability for the controlled nonlinear hybrid STDNNs with Lévy noise. Finally, a numerical example illustrating the usefulness of theoretical results is provided.

In this paper, we study the weighted spaces $ L^p(\omega, \mathbb{R}^d) $ boundedness of certain class of maximal operators when their kernels belong to the space $ L^{q}(\mathbb{S} ^{d-1}) $, $ q > 1 $. Our results in this paper are improvements and extensions of some previously known results.

This paper contributes to the classification of flag-transitive 2-$ (v, k, \lambda) $ designs. Let $ \mathcal{D} $ be a non-trivial and non-symmetric $ 2 $-$ (v, k, \lambda) $ design with $ \lambda $ prime and $ G $ be a flag-transitive point-primitive automorphism group of $ \mathcal{D} $. A recent work by the first author and Chen has proven that the socle of $G$ is either a nonabelian simple group or an elementary abelian $ p $-group for some prime $ p $. In this paper, we focus on the case where the socle of $G$ is an exceptional group of Lie type and give all possible parameters of such 2-designs.

Let $ R $ be a commutative ring with the identity $ 1_{R} $, and let $ R^* $ be the multiplicative group of units in $ R $. An element $ a\in R^* $ is called an exceptional unit if there exists a $ b\in R^* $ such that $ a+b = 1_{R} $. We set $ R^{**} $ to be the set of all exceptional units in $ R $. In this paper, we consider the residue-class ring $ \mathbb{Z}_n $. For any positive integers $ n, s $, and $ c\in\mathbb{Z}_n $, let $ {\mathcal N}_{s}(n, c): = \sharp\big\{(x_1, ..., x_s)\in (\mathbb{Z}_n^{**})^s : x_1+...+x_s\equiv c \pmod n\big\} $. In 2016, Sander (J.Number Theory 159 (2016)) got a formula for $ {\mathcal N}_{2}(n, c) $. Later on, Yang and Zhao (Monatsh. Math. 182 (2017)) extended Sander's theorem to finite terms by using exponential sum theory. In this paper, using matrix theory, we present an explicit formula for $ {\mathcal N}_{s}(n, c) $. This extends and improves earlier results.

This paper investigates rumor propagation in a multilingual environment, taking into account language usage variations. Firstly, a 2I2S2R model is proposed within a heterogeneous network framework that incorporates both immunologic and cross-transmitted mechanisms. Secondly, the paper calculates the basic reproduction number $ R_0 $ by the next-generation matrix method. Thirdly, the local asymptotic stability and the global asymptotic stability are further explored, which indicate that whether the rumor continuously spreads or becomes extinct is determined by the threshold. Finally, the numerical simulation and sensitivity analysis are given to illustrate the effectiveness of theoretical results and the influence of model parameters on rumor spreading.

Cholera, as an endemic disease around the world, has imposed great harmful effects on human health. In addition, from a microscopic viewpoint, the interference of random factors exists in the process of virus replication. However, there are few theoretical studies of viral infection models with biologically reasonable stochastic effects. This paper studied a stochastic cholera model used to describe transmission dynamics in China. In this paper, we adopted a special method to simulate the effect of environmental perturbations to the system instead of using linear functions of white noise, i.e., the transmission rate of environment to human was satisfied Ornstein–Uhlenbeck processes, which is a more practical and interesting. First, it was theoretically proved that the solution to the stochastic model is unique and global, with an ergodic stationary distribution. Moreover, by solving the corresponding Fokker–Planck equation and using our developed algebraic equation theory, we obtain the exact expression of probability density function around the quasi-equilibrium of the stochastic model. Finally, several numerical simulations are provided to confirm our analytical results.

In this paper, our study is divided into two parts. The first part involves analyzing a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo derivatives. The also system incorporates the Caputo derivatives in the initial conditions, which adds a layer of complexity and realism to the problem. We focus on proving the existence of a unique solution for this system, and highlighting the robustness and applicability of fractional derivatives in modeling complex physical phenomena. In the second part of the paper, we employ conformable fractional derivatives, as defined by Khalil, to examine another system consisting of two coupled evolution equations. By the Tanh method, we derive new progressive waves. The connection between these two parts lies in the use of fractional calculus to extend and enhance classical problems.

The strong geodetic number of a graph and its edge counterpart are recent variations of the pioneering geodetic number problem. Covering every vertex and edge of $ G $, respectively, using a minimum number of vertices and the geodesics connecting them, while ensuring that one geodesic is fixed between each pair of these vertices, is the objective of the strong geodetic number problem and its edge version. This paper investigates the strong geodetic number of the lexicographic product involving graph classes that include complete graph $ K_{m} $, path $ P_{m} $, cycle $ C_{m} $ and star $ K_{1, \, m} $ paired with $ P_{n} $ and with $ C_{n} $. Furthermore, the parameter is studied in the lexicographic product of, arbitrary trees with diameter-2 graphs whose geodetic number is equal to 2, $ K_{n}-e $ with $ K_{2} $ and their converses. Upper and lower bounds for the parameter are established for the lexicographic product of general graphs and in addition, the edge variant of the aforementioned problem is studied in certain lexicographic products. The strong geodetic parameters considered in this paper have pivotal applications in social network problems, thereby making them indispensable in the realm of graph theoretical research. This work contributes to the expansion of the current state of research pertaining to strong geodetic parameters in product graphs.

The study of Large Language Models (LLMs), as an interdisciplinary discipline involving multiple fields such as computer science, artificial intelligence, and linguistics, has diverse collaborations within its field. In this study, papers related to LLMs in the SSCI and SCI sub-collections of the Web of Science core database from January 2020 to April 2024 are selected, and a mixed linear regression model is used to assess the impact of scientific collaborations on the application of LLMs. On this basis, the paper further considers factors such as financial support and dominant countries to deeply explore the heterogeneous impact of scientific collaborations on the application of LLMs. The findings show that (1) excessive involvement of academic institutions limits the research and application of LLMs, and the number of authors does not have a significant effect on the application of LLMs; (2) with or without financial support, the role played by scientific collaborations in the application of LLMs does not significantly change; and (3) differences in the dominant countries of scientific collaborations have a slightly heterogeneous effect on the role of LLMs applications, which are mainly reflected in the number of collaborators.

This paper introduces a novel numerical scheme, the conformable finite difference method (CFDM), for solving time-fractional gas dynamics equations. The method was developed by integrating the finite difference method with conformable derivatives, offering a unique approach to tackle the challenges posed by time-fractional gas dynamics models. The study explores the significance of such equations in capturing physical phenomena like explosions, detonation, condensation in a moving flow, and combustion. The numerical stability of the proposed scheme is rigorously investigated, revealing its conditional stability under certain constraints. A comparative analysis is conducted by benchmarking the CFDM against existing methodologies, including the quadratic B-spline Galerkin and the trigonometric B-spline functions methods. The comparisons are performed using $ {L}_{2} $ and $ {L}_{\infty } $ norms to assess the accuracy and efficiency of the proposed method. To demonstrate the effectiveness of the CFDM, several illustrative examples are solved, and the results are presented graphically. Through these examples, the paper showcases the capability of the proposed methodology to accurately capture the behavior of time-fractional gas dynamics equations. The findings underscore the versatility and computational efficiency of the CFDM in addressing complex phenomena. In conclusion, the study affirms that the conformable finite difference method is well-suited for solving differential equations with time-fractional derivatives arising in the physical model.

The present paper dealt with the study of solitons of Lorentzian Kähler space-time manifolds. In this paper, we have discussed different conditions for solitons to be steady, expanding, or shrinking in terms of isotropic pressure, the cosmological constant, energy density, nonlinear equations, and gravitational constant in pseudo-quasi-conformally flat and pseudo-$ W_8 $ flat Lorentzian Kähler space-time manifolds.

The filled function method is a deterministic algorithm for finding a global minimizer of global optimization problems, and its effectiveness is closely related to the form of the constructed filled function. Currently, the filled functions mainly have three drawbacks in form, namely, parameter adjustment and control (if any), inclusion of exponential or logarithmic functions, and properties that are discontinuous and non-differentiable. In order to overcome these limitations, this paper proposed a parameter-free filled function that does not include exponential or logarithmic functions and is continuous and differentiable. Based on the new filled function, a filled function method for solving unconstrained global optimization problems was designed. The algorithm selected points in the feasible domain that were far from the global minimum point as initial points, and improved the setting of the step size in the stage of minimizing the filled function to enhance the algorithm's global optimization capability. In addition, tests were conducted on 14 benchmark functions and compared with existing filled function algorithms. The numerical experimental results showed that the new algorithm proposed in this paper was feasible and effective.

This research paper investigated weighted Milne-type inequalities utilizing Riemann-Liouville fractional integrals across diverse function classes. A key contribution lies in the establishment of a fundamental integral equality, facilitated by the use of a nonnegative weighted function, which is pivotal for deriving the main results. The paper systematically proved weighted Milne-type inequalities for various function classes, including differentiable convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation. The obtained results not only contribute to the understanding of Milne-type inequalities but also offer insights that pave the way for potential future research in the considered topics. Furthermore, it is evident that the results obtained encompass numerous findings that were previously presented in various studies as special cases.

By employing the operator theory, the Lyapunov function on time scales and the famous Gronwall's inequality, this paper addresses some dynamic properties of almost periodic solutions for a class of two species co-existence delayed model on time scales with almost periodic coeffcients and Ricker, as well as the Beverton-Holt type function. First, we establish the existence and uniqueness of the almost periodic solution with a positive infimum by transforming the initial model into an equivalent integral equation. Second, we investigate the global exponential stability and uniformly asymptotic stability of the positive almost periodic solution. Finally, we give two examples to illustrate the main presented results. [ABSTRACT FROM AUTHOR]

It is undeniable that convex and non-convex functions play an important role in optimization. As a result of its behavior, convexity also plays a significant role in discussing inequalities. It is clear that convexity and stochastic processes are intertwined. The stochastic process is a mathematical model that describes how systems or phenomena fluctuate randomly. Probability theory generally says that the convex function applied to the expected value of a random variable is bounded above by the expected value of the random variable's convex function. Furthermore, the deep connection between convex inequalities and stochastic processes offers a whole new perspective on the study of inequality. Although Godunova-Levin functions are well known in convex theory, their properties enable us to determine inequality terms with greater accuracy than those obtained from convex functions. In this paper, we established a more refined form of Hermite-Hadamard and Jensen type inequalities for generalized interval-valued h-Godunova-Levin stochastic processes. In addition, we provide some examples to demonstrate the validity of our main findings. [ABSTRACT FROM AUTHOR]

In this paper, we take advantage of implicit relationships to come up with a new concept called "Aϑ-α-contraction mapping". We utilized our new notion to formulate and prove some common fixed point theorems for two and four self-mappings over complete extended quasi b-metric spaces under a set of conditions. Our main results widen and improve many existing results in the literature. To support our research, we present some examples as applications to our main findings. [ABSTRACT FROM AUTHOR]