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Background: Adult-led organized settings for children (eg, classrooms) provide opportunities for physical activity (PA). The structure of setting time may influence inequalities (ie, unequalness) in the distribution of PA. This study examined differences in PA inequality by setting and time-segment purpose in time-segmented organized group settings for children. Methods: PA and setting meetings were assessed using accelerometer and video observation data from school, before-/after-school, and youth club groups (n = 30) for third- through sixth-grade children (n = 699) in 2 rural US communities. Meetings (n = 130) were time-segmented into smaller units (sessions; n = 835). Each session was assigned a purpose code (eg, PA). Accelerometer data were paired with the meetings and sessions, and the Gini coefficient quantified inequality in activity counts and moderate to vigorous PA minutes for each segment. Beta generalized estimating equations examined differences in PA inequality by setting and session purpose. Results: Activity count inequality was lowest (P <.05) during youth club meetings (Gini = 0.17, 95% CI, 0.14–0.20), and inequality in moderate to vigorous PA minutes was greatest (P <.01) during school (Gini = 0.34, 95% CI, 0.30–0.38). Organized PA sessions (Gini = 0.20, 95% CI, 0.17–0.23) had lower activity count inequality (P <.0001) than academic (Gini = 0.30, 95% CI, 0.27–0.34), enrichment (Gini = 0.31, 95% CI, 0.27–0.36), and nonactive recreation (Gini = 0.30, 95% CI, 0.25–0.34) sessions. Inequality in moderate to vigorous PA minutes was lower (P <.05) in organized PA (Gini = 0.26, 95% CI, 0.20–0.32) and free play (Gini = 0.28, 95% CI, 0.19–0.39) than other sessions. Conclusions: PA inequality differed by setting time structure, with lower inequality during organized PA sessions. The Gini coefficient can illuminate PA inequalities in organized settings and may inform population PA improvement efforts in rural communities. [ABSTRACT FROM AUTHOR]

In this contribution we deal with sequences of polynomials orthogonal with respect to a Sobolev type inner product. A banded symmetric operator is associated with such a sequence of polynomials according to the higher order difference equation they satisfy. Taking into account the Darboux transformation of the corresponding matrix we deduce the connection with a sequence of orthogonal polynomials associated with a Christoffel perturbation of the measure involved in the standard part of the Sobolev inner product. A connection with matrix orthogonal polynomials is stated. The Laguerre-Sobolev type case is studied as an illustrative example. Finally, the bispectrality of such matrix orthogonal polynomials is pointed out. [ABSTRACT FROM AUTHOR]

ABSTRACT This article proposes a novel method to accelerate the boundary feedback control design of cascaded parabolic difference equations (PDEs) through DeepONet. The backstepping method has been widely used in boundary control problems of PDE systems, but solving the backstepping kernel function can be time‐consuming. To address this, a neural operator (NO) learning scheme is leveraged for accelerating the control design of cascaded parabolic PDEs. DeepONet, a class of deep neural networks designed for approximating nonlinear operators, has shown potential for approximating PDE backstepping designs in recent studies. Specifically, we focus on approximating gain kernel PDEs for two cascaded parabolic PDEs. We utilize neural operators to map only two kernel functions, while the other two are computed using the analytical solution, thus simplifying the training process. We establish the continuity and boundedness of the kernels, and demonstrate the existence of arbitrarily close DeepONet approximations to the kernel PDEs. Furthermore, we demonstrate that the DeepONet approximation gain kernels ensure stability when replacing the exact backstepping gain kernels. Notably, DeepONet operator exhibits computation speeds two orders of magnitude faster than PDE solvers for such gain functions, and their theoretically proven stabilizing capability is validated through simulations. [ABSTRACT FROM AUTHOR]

In this study, based on measuring green innovation efficiency (GIE), the regional differences, distribution dynamics and convergences of GIE among 284 Chinese cities from 2005 to 2020 were investigated. It was found that: (1) the GIE showed a decreasing regional difference within and between the eight economic zones and among the whole sample population, with the intensity of transvariation making the major contribution to the differences; (2) the GIE across the country and the eight economic zones appeared to rise slowly, showing a certain level of polarization within the zones, but the gap and polarization tended to narrow gradually on the whole, except for the Northwest Economic Zone; (3) the GIE across the country and among the eight economic zones presented prominent convergences. It was conducive to enrich the research contents and methods of green innovation and provide references for green innovation policies. [ABSTRACT FROM AUTHOR]

This research proposes a modified Crank–Nicolson finite difference scheme with local fractional derivatives to approximate the solutions of local fractional LWR traffic flow model. The stability and consistency of the scheme are examined. Further, convergence of the scheme is assured by using Lax’s equivalence theorem. Some exemplary instances are discussed along with their simulations to validate the proposed method. The obtained numerical solutions show the dynamical evolution of traffic density with respect to time and space. The results derived using the proposed numerical scheme establish that they are quite effective in obtaining the numerical solution to the fractal vehicular traffic flow problem. [ABSTRACT FROM AUTHOR]

We show that the system of difference equations x n + k = x n + l y n − e f x n + l + y n − e − f , y n + k = y n + l x n − e f y n + l + x n − e − f , n ∈ N 0 , where k ∈ N , l ∈ N 0 , l < k , e , f ∈ C , and x j , y j ∈ C , j = 0 , k − 1 ‾ , is theoretically solvable and present some cases of the system when the general solutions can be found in a closed form. [ABSTRACT FROM AUTHOR]

Picard's theorem is among the most striking results in complex analysis and plays a decisive role in the development of the theory of entire and meromorphic functions. In this paper, we give an equivalence between Picard's theorem and entire solutions of (i) difference equation Δ η f + a (z) P (f) = 0 in C and (ii) differential equation D k f + h (z) g (f (z)) = 0 in terms of total derivatives in C n . We prove two Picard-type theorems that generalize the results of Lu (Kodai Math J 26:221–229, 2003). In addition, examples have been exhibited to show that certain assumptions in the main result cannot be improved. [ABSTRACT FROM AUTHOR]

In the paper, we consider a one-dimensional difference Schrödinger equation ψ(z+h)+ψ(z−h)+ λυ(z)ψ(z) = Eψ(z) with a periodic potential υ. In the case where the potential in a neighborhood of R is real analytic, as well as in the case where, the potential has a finite number of simple poles per period, for small values of the coupling constant λ, we describe the asymptotics of a monodromy matrix. [ABSTRACT FROM AUTHOR]

The isotropy is a fundamental requirement for solving partial differential equations by the finite difference scheme. In this work, we first introduce the concept of the virtual nodes to construct the finite difference scheme, and propose several stencils of finite difference discretization to optimize the isotropy of the gradient and Laplacian operators. The isotropic error of the optimized gradient stencil is reduced to 4.1% and 7.5% of the conventional scheme and the typical 4th-order isotropic stencil that is widely used in the multiphase lattice Boltzmann method, while the optimized Laplacian stencil displays the best performance in the Fourier analysis. Then, the optimized stencils are applied to suppress the spurious currents in the multiphase lattice Boltzmann method. The spurious current is reduced to only 7.2% of that calculated by the typical 4th-order isotropic stencil at the reduced temperature 0.6, at which the liquid/gas density ratio is near to 1000. Furthermore, they are adopted to rectify the dendrite directions in the alloy solidification simulations by the phase field method. The angle deviation of dendrite growth is reduced to 15.7% of the conventional scheme. [ABSTRACT FROM AUTHOR]

This paper explores the intersection of gender and sports participation in adolescence, focusing on traditionally male-dominated sports like football. We aim to develop a structural model based on the Theory of Planned Behavior (TPB) to examine gender-differentiated patterns and factors influencing adolescent participation in football. We analyzes data from 1,147 adolescents using Structural Equation Modeling (SEM). The results indicate that a blend of attitudes, subjective norms, perceived behavioral control, and past behavior effectively predicts adolescents' willingness to engage in football. Moreover, the study investigates the roles of perceived behavioral control, past behavior, and behavioral intentions in actual football participation, uncovering significant gender disparities in the progression from past behavior to intentions and subsequent behavior. This research highlights the complex role of gender in football participation and provides strategic insights for increasing girls' involvement in the sport. Our study sets the stage for future research on enhancing girls' participation in football. [ABSTRACT FROM AUTHOR]

In this paper, we construct and analyze a mathematically reasonable and simplest population dynamics model based on Mark Granovetter’s idea for the spread of a matter (rumor, innovation, psychological state, etc.) in a population. The model is described by a one-dimensional difference equation. Individual threshold values with respect to the decision-making on the acceptance of a spreading matter are distributed throughout the population ranging from low (easily accepts it) to high (hardly accepts). Mathematical analysis on our model with some general threshold distributions (uniform; monotonically decreasing/increasing; unimodal) shows that a critical value necessarily exists for the initial frequency of acceptors. Only when the initial frequency of acceptors is beyond the critical, the matter eventually spreads over the population. Further, we give the mathematical results on how the equilibrium acceptor frequency depends on the nature of threshold distribution. [ABSTRACT FROM AUTHOR]

Second order accurate Cartesian grid methods have been well developed for interface problems in the literature. However, it is challenging to develop third or higher order accurate methods for problems with curved interfaces and internal boundaries. In this paper, alternative approaches are developed for some interface and internal layer problems based on non-matching grids, in which two mesh sizes are used to obtain comparable high order accuracy near and away from an interface or an internal layer. For one-dimensional, or two-dimensional problems with straight interfaces or boundary layers that are parallel to one of the axes, the discussion is relatively easy. One of the challenges is how to construct a fourth order compact finite difference scheme at border grid points that connect two meshes. The idea is to employ a second order discretization near the interface in the fine mesh and a fourth order discretization away from the interface in the coarse and border grid points. For two-dimensional problems with a curved interface or an internal layer, a level set representation is utilized for which we can build a fine mesh within a tube | φ (x) | ≤ δ h of the interface. A new super-third seven-point discretization that can guarantee the discrete maximum principle has been developed at hanging nodes. The coefficient matrices of the finite difference equations developed in this paper are M-matrices, which leads to the convergence of the finite difference schemes. Non-trivial numerical examples have confirmed the desired accuracy and convergence of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, the author investigates particular disconjugate even‐order linear difference equations with two terms and classify them based on the properties of their recessive solutions at plus and minus infinity. The main theorem described states that the studied equation is (k−p+1)$(k-p+1)$‐critical whenever a specific second‐order linear difference equation is p$p$‐critical. In the proof, the author derived closed‐form solutions for the studied equation wherein the solutions of the said second‐order equation appear. Furthermore, the solutions were organized, in order to determine recessive solutions, into a linear chain by sequence ordering that compares the solutions at ±∞$\pm \infty$. [ABSTRACT FROM AUTHOR]

In this paper we show that the following four-dimensional system of difference equations x n + 1 = y n α z n − 1 β , y n + 1 = z n γ t n − 1 δ , z n + 1 = t n ϵ x n − 1 μ , t n + 1 = x n ξ y n − 1 ρ , n ∈ N 0 , where the parameters α, β, γ, δ, ϵ, μ, ξ, ρ ∈ ℤ and the initial values x–i, y–i, z–i, t–i, i ∈ {0, 1}, are real numbers, can be solved in closed forms, extending further some results in literature. [ABSTRACT FROM AUTHOR]

The results in this paper are related to the existence of solutions to hybrid and non-hybrid discrete fractional three-point boundary value inclusion problems for the elastic beam equation. The development of our results is attributed to the use of the Caputo and difference operators. The existence results for the non-hybrid discrete fractional inclusion problem are established by using fixed point theory for multi-valued upper semi-continuous maps, and the case of the hybrid discrete fractional inclusion problem is treated by Dhage's fixed point theory. Additionally, we present two examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this present study, series-parallel system composed of five subsystems with the following specifications were analyzed: subsystem 1 consists of two dissimilar clients that are connected to a single unit load balancer I which made up subsystem 2, whereas subsystem 3 consist of two active fog node working in parallel, subsystem 4 comprises of a load balancer II and subsystem 5 is made up of two similar units/components of cloud server. Cloud server, load balancer, fog node and clients failure and repair rate are assumed to be exponentially distributed. The system is under four different scenarios as follows: Scenario 1 system with replacement at complete, scenario 2 system with replacement at partial failure and complete, scenario 3 system without failure detection and replacement repair at complete and lastly, scenario 4 system with undetected failure and replacement at complete. This system is susceptible under first order differential difference equation to formulate the expression of availability and MTTF. The steady state availability, MTTF, sensitivity and expected profit based on general were compared and presented. This study is important to system engineers, designers, plant management, developers and maintenance personnel in the suitable designing and analysis of maintenance policy or processes and also in the assessment of performance and safety of the systems in general during and after the burn-in period. [ABSTRACT FROM AUTHOR]

This article mainly focuses on presenting a new accurate technique (Fourier Transform Method) for solving linear of mth order Difference Equations with constant coefficients. Also, a new lower triangular matrix was introduced to overcome problems related to finding the Fourier Transform of polynomials by rewriting standard-based polynomials through the fallen power polynomial base. Besides, five examples have been presented to illustrate the validity and accuracy of this method. The results reveal that the Fourier transform method is very effective and attractive in solving the difference equations. [ABSTRACT FROM AUTHOR]

This study aimed to increase textile manufacturing system dependability, reliability, maintainability, availability, and metrics like MTBF and MTTF by boosting RAMD. The textile system under investigation is a serial system consisting of five subsystems, which are; subsystem A is weaving section, subsystem B is the dry clean section, subsystem C is the cross cut section, subsystem D is the side seam section and subsystem E is the cleaning section. Each of the subsystem consist of main unit, warm standby unit and cold standby unit. For design and prediction, the Markovian birth-death method is employed to assemble the system governing the differential difference equation from the state-to-state transition diagram. The rates of repair and failure of each subsystem are exponentially distributed and statistically independent. For several subsystems of the system, the findings for RAMD, all of which are crucial to system performance, have been acquired and shown in figures and tables. Furthermore, the results of this study reveal that the highest system performance and dependability may be achieved when the overall system failure rate is low. The findings of this research are thought to be valuable for analyzing performance and determining the best system design and feasible maintenance strategies that may be used in the future to improve system performance, strength, effectiveness, production output as well as revenue mobilization. [ABSTRACT FROM AUTHOR]

This paper presents a new third-order symmetric difference equation transformed into a 3D discrete symmetric map. The nonlinear dynamics and symmetry of the proposed map are analyzed with two initial conditions for exploring the sensitivity of the map and highlighting the influence of the map parameters on its behaviors, thus comparing the findings. Moreover, the stability of the zero fixed point and symmetry are examined by theoretical analysis, and it is proved that the map generates diverse nonlinear traits comprising multistability, chaos, and hyperchaos, which is confirmed by phase attractors in 2D and 3D space, Lyapunov exponents ( L E s ) analysis and bifurcation diagrams; also, 0-1 test and sample entropy (SampEn) are used to confirm the existence and measure the complexity of chaos. In addition, a nonlinear controller is introduced to stabilize the symmetry map and synchronize a duo of unified symmetry maps. Finally, numerical results are provided to illustrate the findings. [ABSTRACT FROM AUTHOR]

This paper presents an analytical study on the in-plane vibrations of a rectangular elastic lattice plate. The plane lattice is modelled considering central and angular interactions. The lattice difference equations are shown to coincide with a spatial finite difference scheme of the corresponding continuous plate. The considered lattice converges to a 2D linear isotropic elastic continuum at the asymptotic limit for a sufficiently small lattice spacing. This continuum has a free Poisson's ratio, which must be lower than that foreseen by the rare-constant theory, to preserve the definite positiveness of the associated discrete energy. Exact solutions for the in-plane eigenfrequencies and modes are analytically derived for the discrete plate. The stiffness characterising the lattice interactions at the boundary is corrected to preserve the symmetry properties of the discrete displacement field. Two classes of constraints are considered, i.e., sliding supports at the nodes, one normal and the other parallel to the boundary. For both boundary conditions, a single equation for the eigenfrequency spectrum is derived, with two families of eigenmodes. Such behaviour of the lattice plate is like that of the continuous plate, the eigenfrequency spectrum of which has been given by Rayleigh. The convergence of the spectrum of the lattice plate towards the spectrum of the continuous plate from below is confirmed. Two continuous size-dependent plate models, considering the strain gradient elasticity and non-local elasticity, respectively, are built from the lattice difference equations and are shown to approximate the plane lattice accurately. [ABSTRACT FROM AUTHOR]

The hypothesis of chaotic population dynamics was proposed in ecology by Robert May in the mid-1970s. At that time the idea was controversial, and it remains a fascinating and unsettled issue today. We report the results of a 20-year laboratory research programme that continued in the tradition of the pioneering ecologist Thomas Park using the Tribolium flour beetle model. We present biological evidence of complex population dynamics – including bifurcations, chaos, saddle nodes, phase switching, resonance effects, and multiple attractors – by using a low-dimensional difference equation model for Tribolium together with carefully designed, conducted, and statistically analysed experiments. The model, parameterized with data, also explains the results of historical Tribolium experiments, such as the classical competition studies of Thomas Park and his colleagues. Our research programme has inspired other studies using the Tribolium mathematical and laboratory model. This work was conducted by a multidisciplinary team, which included Jim Cushing. [ABSTRACT FROM AUTHOR]

If a structured discrete-time population model $ x_n = F(x_{n-1}) $ x n = F (x n − 1) , $ n \in {\mathbb {N}} $ n ∈ N , takes account of two-sexes, the year-to-year turnover operator F often has a first-order approximation B that is homogeneous and order-preserving rather than linear and positive. Still, one can define the spectral radius $ {\mathcal {T}} = \mathbf{r}(B) $ T = r (B) , the basic turnover number, which plays the role of a threshold parameter between extinction and persistence. If $ {\mathcal {T}} \lt 1 $ T < 1 , the extinction state is locally stable, and if $ {\mathcal {T}} \gt 1 $ T > 1 the population shows various degrees of persistence that depend on the irreducibility properties of the dispersal kernels. This threshold property raises interest in the continuous dependence of the basic turnover number on the parameters in the population model. In this paper, we concentrate on the continuous dependence of $ {\mathcal {T}}=\mathbf{r}(B) $ T = r (B) on mating functions. [ABSTRACT FROM AUTHOR]

A resource administrator (RM) can distribute tasks processing time between heterogeneous processors is presented. Its design is based on analyzing of the possible forms of attention to pending tasks and in the previous knowledge about them. A model is proposed based on difference equations that quantify the resource allocation of each task and choose the best processor where the task can be executed. The scheme adapts to dynamic changes in the requirements or the number of tasks. [ABSTRACT FROM AUTHOR]

In the present study, we show that the second-order nonlinear difference equation x n = x n - 1 x n - 2 - a x n - 1 - x n - 2 , n ∈ N 0 , where a ∈ 0 , ∞ and the initial values x - 2 , x - 1 are real numbers, is solvable in closed form and that solutions can be analyzed in detail by the Fibonacci numbers. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a discrete-time prey-predator model. The model is formulated by using the piecewise constant argument method for differential equations and taking into account Holling type III. The existence and local behavior of equilibria are studied. We established that the system experienced both Neimark-Sacker and perioddoubling bifurcations analytically by using bifurcation theory and the center manifold theorem. In order to control chaos and bifurcations, the state feedback method is implemented. Numerical simulations are also provided for the theoretical discussion. [ABSTRACT FROM AUTHOR]

In the present paper, by the variational method coupled with mountain pass type theorems, we study nontrivial homoclinic solutions of nonlinear $ (p, q) $-Laplacian partial difference equations with a parameter $ \lambda>0 $:$ \begin{equation*} \begin{split} &\Delta^2_1(\phi_{p_2}(\Delta^2_1u(k-2, l)))+\Delta^2_2(\phi_{p_2}(\Delta^2_2u(k, l-2)))-a[\Delta_1(\phi_{p_1}(\Delta_1u(k-1, l)))\\ &+\Delta_2(\phi_{p_1}(\Delta_2u(k, l-1)))]+V(k, l)\phi_q(u(k, l))\\& = \lambda f((k, l), u(k, l)). \end{split} \end{equation*} $We prove that the above equation admits two nontrivial homoclinic solutions if $ \lambda $ is sufficiently large and one nontrivial homoclinic solutions if $ \lambda>0 $. Our obtained results generalize and improve some existing ones. [ABSTRACT FROM AUTHOR]

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In this paper, we examine the existence and uniqueness of solutions for a system of the first-order q-difference equations with multi-point and q-integral boundary conditions using various fixed point (fp) theorems. Also, we give two examples to support our results. [ABSTRACT FROM AUTHOR]

The cloud storage service, known for its flexible and expandable nature, often has difficulties managing operating costs while ensuring dependable service and quick response times. This investigation presents a novel approach to optimizing cost efficiency in cloud storage systems by applying particle swarm optimization of the Markovian retrial queueing model in a generic setup by incorporating the working vacation and users' discouragement behavior. Some users may opt not to enter the system or join the retry pool to wait for their turn if the server is occupied. After returning from working vacation, if there is one user available for service, the server can interrupt the vacation period. The server is subject to breakdown and can be recovered after getting the repair. In the proposed model, the server is prone to catastrophes and can fail at any time, leading to the entire system breaking down, and no users being able to access it during this period. Chapman–Kolmogorov (CK) steady-state equations associated with the quasi-birth-death (QBD) process are constructed to make a mathematical design. The governing equations framed to derive the queue length distributions and various performance indices are solved using the recursive method and difference equation theory. The fuzzified parameters are used to develop the FM/FM/1/WV model, which is analyzed using a parametric nonlinear programming approach. To determine the optimal design parameters, the cost minimization problem has been done using the quasi-Newton method and particle swarm optimization. This model incorporates features such as server failures, retrials, and catastrophes, thereby reflecting the complex nature of cloud storage operations. A suitable illustration of cloud storage is taken for both classical and fuzzified models to facilitate the numerical results of performance indices and optimal decision descriptors. [ABSTRACT FROM AUTHOR]

This article addresses the boundary control problem of the transport equation. Namely, we propose a control method, which is merely a delayed output feedback relying on a partial pole placement idea, that consists in assigning an appropriate exponential decay rate to the closed‐loop system's solution. The proposed control structure appearing in the transport boundary, which has proven its effectiveness in controlling finite dimensional systems, consists of an autoregressive relation linking the transport equation's input and output. The obtained result provides an analytical lower bound for the solution's exponential decay. The contribution is concluded with a numeric discussion on the concrete implementation of the method with a special emphasis on the delay seen as a control parameter. [ABSTRACT FROM AUTHOR]

We study difference approximations of an optimal control problem with a boundary observation of the conormal derivative of the state described by the Dirichlet problem for semilinear elliptic equations with controls involved in coefficients of the convective transport operator and in the nonlinear term of the equation. The well-posedness of the optimal control problem is examined. Difference approximations for the optimal control problem are constructed. The convergence of the approximations with respect to the functional and control is analyzed. A regularization of the approximations is constructed. [ABSTRACT FROM AUTHOR]

This paper deals with the host–parasitoid model, where the logistic equation governs the host population growth, and a proportion of the host population can find refuge. The equilibrium points' existence, number, and local character are discussed. Taking the parameter regulating the parasitoid's growth as a bifurcation parameter, we prove that Neimark–Sacker and period-doubling bifurcations occur. Despite the complex behavior, it can be proved that the system is permanent, ensuring the long-term survival of both populations. Furthermore, it was observed that the presence of the proportional refuge does not significantly influence the system's behavior compared to the system without a proportional refuge. [ABSTRACT FROM AUTHOR]

Pressure and flow rate are the most important hydraulic parameters in natural gas pipeline flow, and leak rate is the most crucial parameter after a leak accident occurs. The study of these parameters is vital to the safey of natural gas pipelines and risk assessment after accidents. In this research, based on the conservation laws universally applicable to the motion of objects, we establish the fundamental control equation group for natural gas flow. Then, for both normal and leak conditions of pipelines, we use the characteristic line method to derive the corresponding difference equations for the fundamental control equations, thereby providing calculation methods for pressure and flow rate. Finally, we investigate the calculation method of natural gas pipeline leak rate under different leak aperture sizes, and validate the accuracy of this method through simulation examples. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a discretization scheme for the Yang–Mills equations in the two-dimensional case using a framework based on discrete exterior calculus. Within this framework, we define discrete versions of the exterior covariant derivative operator and its adjoint, which capture essential geometric features similar to their continuous counterparts. Our focus is on discrete models defined on a combinatorial torus, where the discrete Yang–Mills equations are presented in the form of both a system of difference equations and a matrix form. [ABSTRACT FROM AUTHOR]

Unpiloted technology requires attitude measurement from global positioning system to adjust the moving direction of car. However, the measurement result from the satellites suffer from fluctuation, which reduces the control ability of system. Thus, we propose our novel algorithm. First, it builds double difference observation equation based the measurements from two satellites. Second, the least squares algorithm is utilized to calculate double difference integer ambiguity. Third, it constructs the length of baseline to calculate baseline vector, which improves the solution success rate of integer ambiguity and reduces frequency of mutation occurs in attitude angle solving process. Finally, baseline vector is employed to solve the attitude angle of unpiloted car. The extensive experimental results demonstrate that the success rate of the attitude reach 95% within 2° of error under the constraint of the baseline, and the error of the attitude is less than 1° under the dynamic condition. [ABSTRACT FROM AUTHOR]

A new interpretation and applications of the 'Diophantine' and factorization properties of finite orthogonal polynomials in the Askey scheme are explored. The corresponding twelve polynomials are the (q-)Racah, (dual, q-)Hahn, Krawtchouk and five types of q-Krawtchouk. These (q-)hypergeometric polynomials, defined only for the degrees of $ 0,1,\ldots,N $ 0 , 1 , ... , N , constitute the main part of the eigenvectors of N + 1-dimensional tri-diagonal real symmetric matrices, which correspond to the difference equations governing the polynomials. The monic versions of these polynomials all exhibit the 'Diophantine' and factorization properties at higher degrees than N. This simply means that these higher degree polynomials are zero-norm 'eigenvectors' of the N + 1-dimensional tri-diagonal real symmetric matrices. A new type of multi-indexed orthogonal polynomials belonging to these twelve polynomials could be introduced by using the higher degree polynomials as the seed solutions of the multiple Darboux transformations for the corresponding matrix eigenvalue problems. The shape-invariance properties of the simplest type of the multi-indexed polynomials are demonstrated. The explicit transformation formulas are presented. [ABSTRACT FROM AUTHOR]

Let $ F(n,k) $ F (n , k) be a hypergeometric function that may be expressed so that n appears within initial arguments of inverted Pochhammer symbols, as in factors of the form $ \frac {1}{(n)_{k}} $ 1 (n) k . Only in exceptional cases is $ F(n, k) $ F (n , k) such that Zeilberger's algorithm produces a two-term recursion for $ \sum _{k = 0}^{\infty } F(n, k) $ ∑ k = 0 ∞ F (n , k) obtained via the telescoping of the right-hand side of a difference equation of the form $ p_{1}(n) F(n + r, k) + p_{2}(n) F(n, k) = G(n, k+1) - G(n, k) $ p 1 (n) F (n + r , k) + p 2 (n) F (n , k) = G (n , k + 1) − G (n , k) for fixed $ r \in \mathbb {N} $ r ∈ N and polynomials $ p_{1} $ p 1 and $ p_{2} $ p 2 . Building on the work of Wilf, we apply a series acceleration technique based on two-term hypergeometric recursions derived via Zeilberger's algorithm. Fast converging series previously given by Ramanujan, Guillera, Chu and Zhang, Chu, Lupaş, and Amdeberhan are special cases of hypergeometric transforms introduced in our article. [ABSTRACT FROM AUTHOR]

This paper presents a modified version of Leibman's group-theoretic generalizations of the difference calculus for polynomial maps from nonempty commutative semigroups to groups, and proves that it has many desirable formal properties when the target group is locally nilpotent and also when the semigroup is the set of nonnegative integers. We will apply it to solve Waring's problem for general discrete Heisenberg groups in a sequel to this paper. [ABSTRACT FROM AUTHOR]

Summary: This study focuses on the analysis of a linear quadratic zero‐sum game (LQZSG) for time‐delayed uncertain stochastic systems. To begin, we introduce a general time‐delayed zero‐sum game. Employing an algebraic transformation method, we transform the time‐delayed zero‐sum game into an equivalent uncertain random zero‐sum game without time delay. Subsequently, we present equilibrium equations that streamline the transformation of the uncertain random zero‐sum game into problems solvable as deterministic difference equations. We then investigate the LQZSG involving a linear time‐delayed uncertain stochastic system with a quadratic objective function. Within this framework, we provide a unified framework for solving this type of game and obtaining the analytic expression for the saddle‐point equilibrium of the LQZSG. Additionally, we present a numerical example and a counter‐terrorism economic game to illustrate the applicability of our findings. [ABSTRACT FROM AUTHOR]

We study in this paper the global attractivity for a higher order rational difference equation. As application, our results not only include and generalize many known ones, but also formulate some new results for several conjectures presented by Camouzis and Ladas, et al. [ABSTRACT FROM AUTHOR]

We deal with a population model with renewal growth of individuals under mild (geometric) catastrophes. We study the model in a discrete-time setup and consider both an early arrival system and late arrival system with delayed access. Our main objective is to derive the time-dependent solution of the population size distribution at an arbitrary point of time using supplementary variables and some properties of difference equations. Further, we obtain the time-dependent factorial moments and generate certain numerical results to study the system characteristics. [ABSTRACT FROM AUTHOR]

This paper makes a game-theoretic analysis of an M/M/1 queue with variable vacation and vacation interruptions. The Nash equilibrium mixed strategy and social utility maximization are derived based on a non-cooperative game theory and an optimization theory under different information precision levels, namely almost unobservable and fully unobservable cases. The explicit solutions of the entrance probabilities and the steady-state probability distribution of the system are investigated using the probability generating function and nonhomogeneous linear difference equations. Furthermore, the effects of the information levels and diverse system parameters on the equilibrium strategies, the arrival rates, and the expected benefits are explicitly illustrated by numerical comparisons. The research results can provide a theoretical basis and performance analysis tool for the optimal design in the wireless transmission and network communication system. [ABSTRACT FROM AUTHOR]

Differential equations are used to represent different scientific problems are handled efficiently by integral transformations, where integral transforms represent an easy and effective tool for solving many problems in the mentioned fields. This work utilizes the integral transform of the Complex SEE integral transformation to provide an efficient solution method for the difference and differential-difference equations by benefiting from the properties of this complex transform to solve some problems related to difference and differential-difference equations. The 3D, contour and 2D surfaces, as well as the related density plot surfaces of some acquired data, are used to draw the physical aspect of the obtained findings. The proposed approach offers an efficient and rapid solution for addressing the inherent complexity of differential-difference problems with initial conditions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a GI X /M/1 queue with the two-stage vacation policy. The system will first enter a working vacation once the system becomes empty in the normal busy period, during which the service rate will be switched to a lower rate. After this working vacation, if the system is nonempty, the service rate will be switched to the normal rate immediately; otherwise, the system will enter multiple vacations during which the service rate reduces to zero until the system is nonempty after one vacation and a new normal busy period starts. The distinguishing feature of this paper is the use of a general algorithm for solving the difference equation based on the right shift operator method to obtain the queue length distribution at pre-arrival time and arbitrary time. Moreover, the sojourn time of arbitrary customer is analyzed. Finally, several typical numerical examples are provided to validate our computational algorithm. [ABSTRACT FROM AUTHOR]

In this article, we establish a new closed formula for the solution of homogeneous second-order linear difference equations with constant coefficients by using matrix theory. This, in turn, gives new closed formulas concerning all sequences of this type such as the Fibonacci and Lucas sequences. Next, we show the main advantage of our formula which is based on the fact that future calculations rely on the previous ones and this is true from any desired starting point. As applications, we show that Binet’s formula, in this case, is valid for negative integers as well. Finally, new summation formulas relating elements of such sequences. As a conclusion, we present a formula for the sum of squares of Chebychev polynomials of the first and second kind.