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We derive Melnikov type conditions for the persistence of periodic solutions in perturbed slowly varying discontinuous differential equations. In contrast to Battelli and Fečkan (Mathematics 9(19):2449, 2021) we assume that the unperturbed (frozen) equation has a family of periodic solutions depending on some parameters. The result of this paper are motivated by and extend a result in Wiggins and Holmes (SIAM J Math Anal 18:592–611, 1987) where the authors considered a two dimensional hamiltonian family of smooth systems depending on a scalar variable which is the solution of a singularly perturbed equation. [ABSTRACT FROM AUTHOR]

Separation of variables is a common method for producing an analytical based solution to partial differential equations. Despite the wide application of this method, often the physical phenomena described by the differential equations are not adequately involved in the discourse over the appropriate methods to solve a given problem, particularly in mathematics curricula. However, as mathematics is the tool to better understanding of the physical world, the meaning of the differential equation, boundary conditions, and initial conditions cannot be detached from the methods used to solve the differential equations. Failure to recognize the physical conditions being studied can lead to solution methods that are invalid or unphysical. This paper demonstrates how awareness of the physical nature of the system being investigated and its relationship to the mathematics can guide the selection of the relevant solution methods. To illustrate the importance of the comprehension of the physical meaning behind the mathematical equations and representations and the need to avoid rote application a solution technique, the logic behind the selection of the appropriate solution techniques for the one-dimensional transient heat conduction equation is considered under different imposed conditions which lead to different trends in system operation. [ABSTRACT FROM AUTHOR]

In this paper, we study the following two-dimesional system of difference equations ... where the parameters a;b; c;d and the initial values x i; y i, i 2 f1;2;3;4;5;6g, are real numbers. We show that some subclasses of nonlinear two-dimensional system of difference equations are solvable in closed form. We also describe the forbidden set of solutions of the system of differ- ence equations. Some numerical examples are given to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we give a classification of endomorphisms and additive mappings of a prime ring satisfying certain algebraic identities. Moreover, we provide an example proving that the primeness hypothesis imposed in our theorems is not superfluous. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of solutions for a tripled system of fractional differential equations with nonlocal integro multi point boundary conditions by using the p-Laplacian operator and the γ-Caputo derivatives. The presented results are obtained by the two fixed point theorems of Banach and Krasnoselskii. An illustrative example is presented at the end to show the applicability of the obtained results. To the best of our knowledge, this is the first time where such problem is considered. [ABSTRACT FROM AUTHOR]

The analysis of real-world data in classical statistics is commenced by deriving a density function which requires a lengthy function selection and parameter estimation process. The process is supported by difficult integration to obtain the cumulative frequency function. In view of the difficulties in the existing methods, a super function called the Persian Probability Curve is proposed in this paper. This function is based on logical reasoning and mathematical concepts to overcome the shortcomings of the classical method. It is shown that the Persian probability is equivalent to that of the Burr in the governing differential equation. The validity of the work has been verified by comparing the obtained results with those of others. [ABSTRACT FROM AUTHOR]

Mathematics and its applications try to make available a simple and accurate approach to handle nonlinear equations. The present paper investigates for the first time the application of the linearizing method to determine both the approximate frequency and displacement amplitude for the third‐order differential equations. The main merit of the linearized method is a collection of simplicity and accuracy for the solution of high‐order nonlinear conservative or non‐conservative oscillators. The current work sheds light on the approximate proposition to the damped third‐order nonlinear differential equations. The fast estimated solution is simulated to the accurate solution gained through the numerical methods. It is significant to conclude that the non‐perturbative method considered here is simpler and easier to apply than the other perturbation methods. This paper enriches some existing literature. [ABSTRACT FROM AUTHOR]

For a quasi-periodically forced differential equation, response solutions are quasi-periodic ones whose frequency vector coincides with that of the forcing function and they are known to play a fundamental role in the harmonic and synchronizing behaviors of quasi-periodically forced oscillators. These solutions are well-understood in quasi-periodically perturbed nonlinear oscillators either in the presence of large damping or in the non-degenerate cases with small or free damping. In this paper, we consider the existence of response solutions in quasi-periodically perturbed, second order differential equations, including nonlinear oscillators, of the form x ¨ + λ x l = ϵ f (ω t , x , x ˙) , x ∈ R , where λ is a constant, 0 < ϵ ≪ 1 is a small parameter, l > 1 is an integer, ω ∈ R d is a frequency vector, and f : T d × R 2 → R 1 is real analytic and non-degenerate in x up to a given order p ≥ 0 , i.e., [ f (· , 0 , 0) ] = [ ∂ f (· , 0 , 0) ∂ x ] = [ ∂ 2 f (· , 0 , 0) ∂ x 2 ] = ⋯ = [ ∂ p - 1 f (· , 0 , 0) ∂ x p - 1 ] = 0 and [ ∂ p f (· , 0 , 0) ∂ x p ] ≠ 0 , where [ ] denotes the average value of a continuous function on T d . In the case that λ = 0 and f is independent of x ˙ , the existence of response solutions was first shown by Gentile (Ergod Theory Dyn Syst 27:427–457, 2007) when p = 1 . This result was later generalized by Corsi and Gentile (Commun Math Phys 316:489–529, 2012; Ergod Theory Dyn Syst 35:1079–1140, 2015; Nonlinear Differ Equ Appl 24(1):article 3, 2017) to the case that p > 1 is odd. In the case λ ≠ 0 , the existence of response solutions is studied by the authors Si and Yi (Nonlinearity 33(11):6072–6099, 2020) when p = 0 . The present paper is devoted to the study of response solutions of the above quasi-periodically perturbed differential equations for the case λ ≠ 0 by allowing p > 0 . Under the conditions that 0 ≤ p < l / 2 and λ [ ∂ p f (· , 0 , 0) ∂ x p ] > 0 when l - p is even, we obtain a general result which particularly implies the following: (1) If either l is odd and λ < 0 or l is even and [ ∂ p f (· , 0 , 0) ∂ x p ] > 0 , then as ϵ sufficiently small response solutions exist for each ω satisfying a Brjuno-like non-resonant condition; (2) If either l is odd and λ > 0 or l is even and [ ∂ p f (· , 0 , 0) ∂ x p ] < 0 , then there exists an ϵ ∗ > 0 sufficiently small and a Cantor set E ∈ (0 , ϵ ∗) with almost full Lebesgue measure such that response solutions exist for each ϵ ∈ E and ω satisfying a Diophantine condition. Similar results are also obtained in the case λ = ± ϵ which particularly concern the existence of large amplitude response solutions. [ABSTRACT FROM AUTHOR]

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes are Hölder continuous at 0 provided that ∫ B 1 | u (x) | 3 d x + ∫ B 1 | f (x) | q d x or ∫ B 1 | ∇ u (x) | 2 d x + ∫ B 1 | ∇ u (x) | 2 d x (∫ B 1 | u (x) | d x) 2 + ∫ B 1 | f (x) | q d x with q > 3 is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points is zero. For the boundary case, we also obtain that 0 is regular provided that ∫ B 1 + | u (x) | 3 d x + ∫ B 1 + | f (x) | 3 d x or ∫ B 1 + | ∇ u (x) | 2 d x + ∫ B 1 + | f (x) | 3 d x is sufficiently small. These results improve previous regularity theorems by Dong-Strain ([8] , Indiana Univ. Math. J., 2012), Dong-Gu ([7] , J. Funct. Anal., 2014), and Liu-Wang ([27] , J. Differential Equations, 2018), where either the smallness of the pressure or the smallness of the scaling invariant quantities on all balls is necessary. [ABSTRACT FROM AUTHOR]

This paper establishes the global well-posedness issue for the full viscous MHD equations in the axisymmetric setting. Global solutions are obtained in critical Besov spaces uniformly to the viscosity when the resistivity is fixed in the spirit of [Abidi H, Hmidi T, Keraani S. On the global well-posedness for the axisymmetric Euler equations. Math. Ann. 2010;347:15–41.], [Hassainia Z. On the global well-posedness of the 3D axisymmetric resistive MHD equations. Ann. Henri Poincaré. 2022;23:2877-2917], [Hmidi T, Zerguine M. Inviscid limit axisymmetric Navier–Stokes system. Differential and Integral Equations. 2009;22(11–12):1223–1246.]. Furthermore, strong convergence in the resolution spaces with a rate of convergence is also studied. [ABSTRACT FROM AUTHOR]

For a spectrally negative Lévy process, scale functions appear in the solution of twosided exit problems, and in particular in relation with the Laplace transform of the first time it exits a closed interval. In this paper, we consider the Laplace transform of more general functionals, which can depend simultaneously on the values of the process and its supremum up to the exit time. These quantities will be expressed in terms of generalized scale functions, which can be defined using excursion theory. In the case the functional does not depend on the supremum, these scale functions coincide with the ones found on the literature, see e.g. Li and Palmowski (2018) and therefore the results in this work are an extension of them. [ABSTRACT FROM AUTHOR]

This paper is written to provide ideas and guide faculty who want to design a mathematical modeling course for undergraduate mathematics majors and minors. We discuss course goals, assignments, and projects that can be used to help students gain experience relevant for careers and mathematical modeling opportunities. The authors designed this course to build students' mathematical thought processes and toolbox, ability to analyze and evaluate mathematical models, mathematical modeling skills, and teamwork skills. The course described is intended as an upper division undergraduate course with a prerequisite of differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a weak MT-cyclic Reich type contractions and obtain the existence theorems for best proximity point for self-mappings defined on the complete metric spaces. Our results improve and generalize some results in literature. Also, we give some applications of our results to solving some classes of non-linear integral and differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic profiles, uniqueness and orbital stability of McCann's uniformly rotating binary stars (Houston J Math 32(2):603–631, 2006) governed by the Euler–Poisson system. A new uniqueness result will be importantly used in stability analysis. Moreover, we apply our framework to the study of uniformly rotating binary galaxies of the Vlasov–Poisson system through Rein's reduction (Handbook of differential equations: evolutionary equations, vol III, pp 383–476, 2007). [ABSTRACT FROM AUTHOR]

Most undergraduates have limited experience with mathematical modeling. In an effort to respond to various initiatives, such as the recommendations outlined in [S. Garfunkel and M. Montgomery, eds., GAIMME: Guidelines for Assessment & Instruction in Mathematical Modeling Education, SIAM, 2016], this paper describes a course on the mathematical models of cancer growth and treatment. Among its aims is to provide a template for a "bridge" course between the traditional calculus and differential equations sequence and more advanced courses in mathematics and statistics. Prerequisites include a course in ordinary differential equations. Linear algebra is a useful corequisite but no previous programming experience is required. The content includes classical models of tumor growth as well as models for the growth of specific cancer types. Relevant research articles are provided for further study. Material for student projects and effective communication is supplied, as well as suggestions for homework assignments and computer labs. This paper aims to assist instructors in developing their own "Mathematics + Cancer" course. [ABSTRACT FROM AUTHOR]

In this paper we prove that, after an appropriate rescaling, the sum of moments of an N × N Hermitian matrix H sampled according to the generalized Cauchy (also known as Hua–Pickrell) ensemble with parameter s > 0 is a continuous-Hahn polynomial in the variable k. This completes the picture of the investigation that began in (Cunden et al 2019 Commun. Math. Phys. 369 1091–45) where analogous results were obtained for the other three classical ensembles of random matrices, the Gaussian, the Laguerre and Jacobi. Our strategy of proof is somewhat different from the one in (Cunden et al 2019 Commun. Math. Phys. 369 1091–45) due to the fact that the generalized Cauchy is the only classical ensemble which has a finite number of integer moments. Our arguments also apply, with straightforward modifications, to the other three cases studied in (Cunden et al 2019 Commun. Math. Phys. 369 1091–45) as well. We finally obtain a differential equation for the one-point density function of the eigenvalue distribution of this ensemble and establish the large N asymptotics of the moments. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the long time behavior of the damped forced generalized Ostrovsky equation below the energy space. First, by using Fourier restriction norm method and Tao's [k,Z]-multiplier method, we establish the multi-linear estimates, including the bilinear and trilinear estimates on the Bourgain space Xs,b. Then, combining the multi-linear estimates with the contraction mapping principle as well as L2 energy method, we establish the global well-posedness and existence of the bounded absorbing sets in L2. Finally, we show the existence of global attractor in L2 and its compactness in H5 by means of the high-low frequency decomposition method, cut-off function, tail estimate together with Kuratowski α-measure in order to overcome the non-compactness of the classical Sobolev embedding. This result improves earlier ones in the literatures, such as Goubet and Rosa [J. Differential Equations 185 (2002), no. 1, 25-53], Moise and Rosa [Adv. Differential Equations 2 (1997), no. 2, 251-296], Wang et al. [J. Math. Anal. Appl. 390 (2012), no. 1, 136-150], Wang [Discrete Contin. Dyn. Syst. 35 (2015), no. 8, 3799-3825], and Guo and Huo [J. Math. Anal. App. 329 (2007), no. 1, 392-407]. [ABSTRACT FROM AUTHOR]

The article presents a two-point integral BVP for singularly perturbed systems of linear ordinary differential equations. The integral BVP for singularly perturbed systems of ordinary differential equations previously has not been considered. The paper shows the influence of nonlocal boundary conditions on the asymptotic of the solution of the regarded BVP and the significant effect of integral terms in the definition of the limiting BVP. An explicit constructive formula for the solution of this BVP using initial and boundary functions of the homogeneous perturbed equation is obtained. A theorem on asymptotic estimates of the solution and its derivatives is given. It is established that the solution of the integral BVP at the point t = 0 is infinitely large as µ → 0. From here, it follows that the solution of the considered boundary value problem has an initial jump of zero order. It is found that the solution of the original integral BVP is not close to the solution of the usual limiting unperturbed BVP. A changed limiting BVP is obtained. The presence of integrals in the boundary conditions leads to the fact that the limiting BVP is determined by the changed boundary conditions. This follows from the presence of the jump and its order. A theorem on the close between the solutions of the original perturbed and changed limiting problems is given. [ABSTRACT FROM AUTHOR]

An accurate frequency-amplitude relationship is very needed to elucidate the properties of the oldest device of Fangzhu for collecting water from the air. The Fangzhu oscillator was derived and solved approximately (He et al. in Math Methods Appl Sci, 2020, 10.1002/mma.6384), here we show that the singular Duffing-like oscillator can be more effectively solved by the homotopy perturbation method and a criterion is obtained for the existence of a periodic solution for the singular differential equation. The results obtained in this paper are helpful for the optimal design of the Fangzhu device. [ABSTRACT FROM AUTHOR]

In this paper, we study the genesis and evolution of geometric ideas and techniques in investigations of movable singularities of algebraic ordinary differential equations. This leads us to the work of Mihailo Petrović on algebraic differential equations (ODEs) and in particular the geometric ideas expressed in his polygon method from the final years of the nineteenth century, which have been left completely unnoticed by the experts. This concept, also developed independently and in a somewhat different direction by Henry Fine, generalizes the famous Newton–Puiseux polygonal method and applies to algebraic ODEs rather than algebraic equations. Although remarkable, the Petrović legacy has been practically neglected in the modern literature, although the situation is less severe in the case of results of Fine. Therefore, we study the development of the ideas of Petrović and Fine and their places in contemporary mathematics. [ABSTRACT FROM AUTHOR]

The paper introduces the notions of boundedness and convergence with speed for difference sequences, and characterizes certain matrix classes associating the sets of such classes of sequences involving the operator Δ and two speeds λ = (λk) and µ = (µk) ... The results obtained in this paper should easily extendible to difference sequences of higher orders, and even, in combination with multipliers. [ABSTRACT FROM AUTHOR]

The nodes and their connection relationships are the two main bodies for dynamic complex networks. In existing theoretical researches, the phenomena of stabilization and synchronization for complex dynamical networks are generally regarded as the dynamic characteristic behaviors of the nodes, which are mainly caused by coupling effect of connection relationships between nodes. However, the connection relationships between nodes are also one main body of a time-varying dynamic complex network, and thus they may evolve with time and maybe show certain characteristic phenomena. For example, the structural balance in the social networks and the synaptic facilitation in the biological neural networks. Therefore, it is important to investigate theoretically the reasons in dynamics for the occurrence. Especially, from the angle of large-scale systems, how the dynamic behaviors of nodes (such as the individuals, neurons) contribute to the connection relationships is one of worthy research directions. In this paper, according to the structural balance theory of triad proposed by F. Heider, we mainly focus on the connection relationships body, which is regarded as one of the two subsystems (another is the nodes body), and try to find the dynamic mechanism of the structural balance with the internal state behaviors of the nodes. By using the Riccati linear matrix differential equation as the dynamic model of connection relationships subsystem, it is proved under some mathematic conditions that the connection relationships subsystem is asymptotical structural balance via the effects of the coupling roles with the internal state of nodes. Finally, the simulation example is given to show the validity of the method in this paper. [ABSTRACT FROM AUTHOR]

In this paper we consider stochastic evolution equations on UMD-Banach spaces. In a recent paper we proved existence of optimal controls. Here in this paper we develop necessary conditions of optimality whereby one can construct the optimal controls. For illustration we use these results to treat the LQR problem in sufficient details under two sets of alternative and distinct assumptions. [ABSTRACT FROM AUTHOR]

In this paper we prove a sharp trilinear inequality which is motivated by a program to obtain the sharp form of the L²-L6 Tomas- Stein adjoint restriction inequality on the circle. Our method uses intricate estimates for integrals of sixfold products of Bessel functions developed in a companion paper. We also establish that constants are local extremizers of the Tomas-Stein adjoint restriction inequality as well as of another inequality appearing in the program. [ABSTRACT FROM AUTHOR]

In this paper, using the theory of complex differential equations, we study the solution of some non-linear complex differential equations in connection to Brück conjecture which generalized some earlier results due to Pramanik, D. C. and Biswas, M., On solutions of some non-linear differential equations in connection to Bruck conjecture, Tamkang J. Math., 48 (2017), No. 4, 365-375; and Wang, H., Yang, L-Z. and Xu, H-Y., On some complex differential and difference equations concerning sharing function, Adv. Diff. Equ., 2014, 2014:274. [ABSTRACT FROM AUTHOR]

The approach given in this paper leads to numerical methods to find the approximate solution of volterra integro -diff. equ.1st kind. First, we reduce it from integro VIDEs to integral VIEs of the 2nd kind by using the reducing theory, then we use two types of Non-polynomial spline function (linear, and quadratic). Finally, programs for each method are written in MATLAB language and a comparison between these two types of Non-polynomial spline function is made depending on the least square errors and running time. Some test examples and the exact solution are also given. [ABSTRACT FROM AUTHOR]

In this paper, the series solution is applied to solve third order fuzzy differential equations with a fuzzy initial value. The proposed method applies Taylor expansion in solving the system and the approximate solution of the problem which is calculated in the form of a rapid convergent series; some definitions and theorems are reviewed as a basis in solving fuzzy differential equations. An example is applied to illustrate the proposed technical accuracy. Also, a comparison between the obtained results is made, in addition to the application of the crisp solution, when the a-level equals one. [ABSTRACT FROM AUTHOR]

In a real Hilbert space H, let us consider the boundary-value problem -εu"(t) + βu'(t) + Au(t) + Bu(t) ∋ f(t), t ∈ [0; T]; u(0) = u0, u'(T) = 0, where T > 0 is a given time instant, ε,μ are positive parameters, A: D(A) ⊂ H → H is a (possibly set-valued) maximal monotone operator, and B: H → H is a Lipschitz operator. In this paper, we investigate the behavior of the solutions to this problem in two cases: (i) μ > 0 fixed, 0 < ε → 0, and (ii) ε > 0 fixed and 0 < μ → 0. Notice that if μ = 1 and ε is a positive small parameter, the above problem is a Lions-type regularization of the Cauchy problem u0(t) + Au(t) + Bu(t) ∋ f(t); t 2 [0; T]; u(0) = u0, which was recently studied by L. Barbu and G. Moroşanu [Commun. Contemp. Math. 19 (2017)]. Our abstract results are illustrated with examples related to the heat equation and the telegraph differential system. [ABSTRACT FROM AUTHOR]

The inverse data envelopment analysis is an inverse optimization problem, which can be used as an appropriate planning tool for management decisions. The typical DEA mainly focuses on post-operative evaluation of an organizational performance. Sometimes economic conditions such as economic prohibitions on exports or imports are imposed on a system. These prohibitions prevent decision-making units from best performance (efficiency one). In this case, if the system has the best performance (with a less than one efficiency score) then it will be considered as an efficient system. So, the efficiency frontier changes problem must be studied. So by making change in definition of the best efficiency amount of a system, it still has the best performance. In these situations, the inefficient units can select a real pattern instead of reaching an unrealistic pattern that is presented in ideal terms to achieve the best conditions (the best efficiency value is one). So a long-term management plan can be developed. The efficiency frontier change will be expressed inputs and outputs as a coefficient of efficiency. The frontier change looks at the changes in inputs and outputs to reach the new frontier. One of the purposes of the data envelopment analysis is the investigation of inputs and outputs amounts by changing the amount of efficiency. So far, many models must be solved to calculating these changes. Efficiency frontier problem can replace a simple mathematical model with these models. All of these advantages can improve calculating input and outputs changes and RTS will be unchanged and decision maker can estimate units RTS without solving any model. So a unit will be stayed MPSS by reducing inputs. In other frontier change methods some hyperplanes and extreme units had been deleting but our method transforms them on new frontier. So all extreme units and RTS can estimate easily. The efficiency frontier changes can delete some inefficient units so systems cost will be reduced. For this purpose, in this paper, the change in the efficiency frontier, its properties and its effect on the inverse data envelopment analysis is examined. [ABSTRACT FROM AUTHOR]

In this paper we investigate the stability and asymptotic stability of the zero solution for the first order delay differential equation þ(t)=-ΣNj=1(t)y(t-τj(t))+f(t,y(t-964;(t)) where the delay is variable and by using Banach fixed point theorem. We give new conditions to ensure the stability and asymptotic stability of the zero solution of this equation. [ABSTRACT FROM AUTHOR]

To balance the supply and demand in China's beef market, beef consumption must be scientifically and effectively forecasted. Beef consumption is affected by many factors and is characterized by gray uncertainty. Therefore, gray theory can be used to forecast the beef consumption, In this paper, the structural defects and unreasonable parameter design of the traditional gray model are analyzed. Then, a new gray model termed, EGM(1,1,r), is built, and the modeling conditions and error checking methods of EGM(1,1,r) are studied. Then, EGM(1,1,r) is used to simulate and forecast China’s beef consumption. The results show that both the simulation and prediction precisions of the new model are better than those of other gray models. Finally, the new model is used to forecast China’s beef consumption for the period from 2019–2025. The findings will serve as an important reference for the Chinese government in formulating policies to ensure the balance between the supply and demand for Chinese beef. [ABSTRACT FROM AUTHOR]

This paper studies how the temperature in the bathroom keeps changing when people bathe in the bathtub. The shape, capacity, behavior and body posture of people in the bathroom are considered. In fluid mechanics point of view, we consider the impact of the flow of heat transfer, by the energy differential equations and boundary-layer momentum to establish a set of PDE, and use Laplace operator rewrite it. We use finite difference method, with Taylor series expansion. We use the function value of grid nodes of difference quotient instead of control equation of derivative, and discretize it, and solve the heat conduction equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the optimal control problem for a class of systems governed by nonlinear time-varying partially observed interval differential equations. The control process is assumed to be governed by linear time varying interval differential equation driven by the observed process. Using the fact that the state, observation, and control processes possess lower and upper bounds, we have developed sets of (ordinary) differential equations that describe the behavior of the bounds of these processes. Using these differential equations, the interval control problem can be transformed into an equivalent ordinary control problem in which interval mathematics and extension principle of Moore are not required. Using variational arguments, we have developed the necessary conditions of optimality for the equivalent (ordinary) control problem. Finally, we present some numerical simulations to illustrate the effectiveness of the proposed control scheme. [ABSTRACT FROM AUTHOR]

This paper presents a new method to prove the nonuniform dichotomy spectrum theorem. Chu et al. [Bull. Sci. Math. 139 (2015), pp. 538-557] and Zhang [J. Funct. Anal. 267 (2014), pp. 1889-1916] generalized the dichotomy spectrum in Siegmund [J. Dynam. Differential Equations 14 (2002), pp. 243-258] to the nonuniform dichotomy spectrum and the authors in these works employed linear integral manifolds (stable and unstable) to establish the spectral theorem. They then used the spectrum theorem to study reducibility. We prove the nonuniform dichotomy spectrum by way of contradiction. In particular, we employ the nonuniform kinematically similarity (nonuniform reducibility) to reduce the shift system into two blocks and then we get a contradiction based on a technique in mathematical analysis. The method in the proof is completely different from previous works. [ABSTRACT FROM AUTHOR]

To eliminate and eradicate gambiense human African trypanosomiasis (HAT), maximizing the effectiveness of active case finding is of key importance. The progression of the epidemic is largely influenced by the planning of these operations. This paper introduces and analyzes five models for predicting HAT prevalence in a given village based on past observed prevalence levels and past screening activities in that village. Based on the quality of prevalence level predictions in 143 villages in Kwamouth (DRC), and based on the theoretical foundation underlying the models, we consider variants of the Logistic Model—a model inspired by the SIS epidemic model—to be most suitable for predicting HAT prevalence levels. Furthermore, we demonstrate the applicability of this model to predict the effects of planning policies for screening operations. Our analysis yields an analytical expression for the screening frequency required to reach eradication (zero prevalence) and a simple approach for determining the frequency required to reach elimination within a given time frame (one case per 10000). Furthermore, the model predictions suggest that annual screening is only expected to lead to eradication if at least half of the cases are detected during the screening rounds. This paper extends knowledge on control strategies for HAT and serves as a basis for further modeling and optimization studies. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the form of the solutions and the periodicity character of the following systems of nonlinear difference equations of order two zn+1=zn+tn-l/±tn±tn-1,tn+1=tnzn-l/±zn±zn-l where the initial conditions z-1, z0, t-1 and t0 are nonzero real numbers. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of analytic functions associated with Mittag-Leffler fuction in the open unit disk. Several properties of functions belonging to this class are derived. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to study the existence of solution of a system of differential equations using fixed point technique. In this regard, in the first part of this article, along with some properties of partial b-metric topology, we prove a common fixed point theorem for generalized Geraghty type contraction mappings in a complete partial b-metric spaces. Then in second part we apply this result to show the existence of the solution of a system of ordinary differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we study differential equations arising from the generating functions of generalized degenerate tangent polynomials. We give explicit identities for the generalized degenerate tangent polynomials arising from differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of positive solutions of the fully third-order boundary value problem... is continuous. Some inequality conditions on f to guarantee the existence of positive solution are presented. These inequality conditions allow that ƒ(t; x; y; z) may be superlinear or sublinear growth on x, y and z as |(x; y; z)| → 0 and |(x; y; z)| → ∞. [ABSTRACT FROM AUTHOR]

In this paper, some qualitative properties are discussed such as the boundedness, the periodicity and the global stability of the positive solutions of the nonlinear difference equation... where the coefficients A; αi; βi 2 (0,1), i = 1,...., 5; while the initial conditions y-5,y-4,y-3,y-2; y-1; y0 are arbitrary positive real numbers. Some numerical examples will be given to illustrate our results. [ABSTRACT FROM AUTHOR]

The self-synchronization of rotors mounted on different vibrating bodies can be easily controlled by adjusting the coupling parameters. To reveal the synchronization characteristics of a weakly damped system with two rotors mounted on different vibrating bodies, we propose a simplified physical model. The topics described in this paper are related to coupling dynamic problems between two vibrating systems. Both synchronization and vibratory synchronization transmission of the system are studied. The coupling mechanism between the two rotors is analyzed to derive the synchronization condition and the stability criterion of the system. The vibration of the system is described by an averaging method that can separate fast motion (high frequency) from slow motion (low frequency). Theoretical research shows that vibration torque is the key factor in balancing the energy distribution between the rotors. Taking the maximum vibration torque (MVT) as a critical parameter, we investigate the synchronization characteristics of the vibrating system in different cases. The curve of the maximum vibration torque (MVT) versus coupling frequency is divided into several parts by the coupling characteristic frequency and the input torque difference between the rotors. Simulations of the system with coupling frequencies from different parts are carried out. For the system with rotational frequencies larger than the natural frequencies, the coupling characteristic frequency or characteristic frequency curve should be considered. When the coupling frequency is close to the characteristic frequency or the vibration state is close to the characteristic frequency curve, self-synchronization of the two rotors can be obtained easily. Under certain conditions when the coupling effect between the rotors is strong enough, the rotors can maintain synchronous rotation even when one of the two motors is shut off after synchronization is achieved, which is called vibratory synchronization transmission. Vibratory synchronization transmission of the system occurs in a new synchronous condition, and the phase difference between the rotors takes on a new value, that is, the system approaches a new synchronization state. [ABSTRACT FROM AUTHOR]

In this paper, we consider three types of functions related to higher-order Genocchi functions and derive their Fourier series expansions. In addition, we express each of them in terms of Bernoulli functions. [ABSTRACT FROM AUTHOR]

For PM functions of height 1, the existence of continuous iterative roots of any order was obtained under the characteristic endpoints condition. A natural open question about iterative roots without that condition was raised. This question was answered partially in the case that the function is increasing on its characteristic interval. In this paper, to the opposite, we consider the decreasing case and give the existence and nonexistence results for their iterative roots. [ABSTRACT FROM AUTHOR]

In this paper we derive the famous Fekete-Szegö inequality for the class of p-valent non-bazilevič function using differential operator. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the nonlocal symmetries and exact interaction solutions of the variable coefficient Korteweg–de Vries (KdV) equation are studied. With the help of pseudo-potential, we construct the high order nonlocal symmetries of the time-dependent coefficient KdV equation for the first time. In order to construct the new exact interaction solutions, two auxiliary variables are introduced, which can transform nonlocal symmetries into Lie point symmetries. Furthermore, using the Lie point symmetries of the closed system, some exact interaction solutions are obtained. For some interesting solutions, such as the soliton–cnoidal wave solutions are discussed in detail, and the corresponding 2D and 3D figures are given to illustrate their dynamic behavior. [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is to complement existing oscillation results for third-order neutral delay differential equations by establishing sufficient conditions for nonexistence of so-called Kneser solutions. Combining newly obtained results with existing ones, we attain oscillation of all solutions of the studied equations. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to study the existence of solution of a system of differential equations using fixed point technique. In this regard, in the first part of this article, along with some properties of partial b-metric topology, we prove a common fixed point theorem for generalized Geraghty type contraction mappings in a complete partial b-metric spaces. Then in second part we apply this result to show the existence of the solution of a system of ordinary differential equations. [ABSTRACT FROM AUTHOR]