Search

Research on how to pose good problems in mathematical science is rarely touched. Inspired by Kilpatrick's "Where do good problems come from?", the current research investigates the problem of the specific problem posed by mathematicians in mathematical sciences. We select a recent mathematical conjecture of Yang related to periodic functions in the field of functions of one complex variable. These problems are extended to complex differential equations, difference equations, differential-difference equations, etc. Through mathematical analysis, we try to reproduce the effective strategies or techniques used by mathematicians in posing these new problems. The results show that mathematicians often use generalization, constraint manipulation, and specialization when they pose new mathematical problems. Conversely, goal manipulation and targeting a particular solution are rarely used. The results of the study may have a potential impact and promotion on implementing problem-posing teaching in primary and secondary schools. Accordingly, teachers and students can be encouraged to think like mathematicians, posing better problems and learning mathematics better. Then, we give some examples of mathematical teaching at the high school level using problem-posing strategies, which are frequently employed by mathematicians or mathematical researchers, and demonstrate how these strategies work. Therefore, this is a pioneering research that integrates the mathematical problem posing by mathematicians and the mathematical problem posing by elementary and secondary school math teachers and students. In addition, applying constraint manipulation and analogical reasoning, we present four unsolved mathematical problems, including three problems of complex difference-related periodic functions and one problem with complex difference equations. [ABSTRACT FROM AUTHOR]

A mathematical analysis of four-level (three-phase) phase-shifted carrier modulation class-D amplifiers with analog feedback is proposed. The nonlinear difference equations are extracted and solved using the perturbation theorem. In this way, a closed-form function describes the low-frequency part of the output for an arbitrary input signal. The results are generalized for an arbitrary number of levels by analogy, well matched with the previous model of two-, three-, and the proposed four-level ones. The analytical results confirm that, apart from switching frequency, the multilevel class-D amplifiers converge to linear amplifiers as the number of output levels increases. An analytical equation expresses the harmonic distortion of practically interesting five-level amplifiers with bridge-tied load. The model is verified by system-level simulations and circuit-level implementation. In this way, a prototype fully differential five-level circuit is implemented on the printed circuit board to confirm the model. [ABSTRACT FROM AUTHOR]

Classification, boundedness, and existence of solutions of a second order nonlinear difference equation are investigated. First, it is proved that all solutions are eventually monotone. Then, the necessary and sufficient conditions for the boundedness of all solutions are established. Finally, the existence of different types of monotonic solutions are presented. The obtained results have extended and improved some existing ones. [ABSTRACT FROM AUTHOR]

The distributed recursive filtering problem is investigated in this paper for discrete time-delayed nonlinear stochastic systems, where the well-known Takagi–Sugeno (T-S) fuzzy model is used to approximate the nonlinearities. According to obtain the system dynamics, a novel structure of distributed filters is developed, where the difference of estimated states from neighboring sensors is exploited to improve the one-step prediction, and the desired estimation is obtained by fusing the estimation under different rules. Attention is focused on the design of a distributed recursive filter such that, in the presence of time-delays and defuzzifying operations, an upper bound of the filtering error covariance is obtained and then minimized by properly designing filter parameters via elaborate mathematical analysis. With the exception of the desired gains with the online recursive form are dependent on the solutions of two Riccati-type difference equations, and the upper bound is further optimized via the introduced parameters. As a final point, a simulation examples is exploited to show the applicability of the developed filtering scheme. [ABSTRACT FROM AUTHOR]

The present paper constitutes a theoretical review on the relationship between debt and deficit. It is described using mathematical analysis and using a difference equation that relates the current debt stock to the past one and the fiscal deficit to show the conditions regarding the stability and dynamic sustainability of the debt. [ABSTRACT FROM AUTHOR]

A range of new fractional-order love dynamics is modelled in this paper using the Caputo (power-law), Caputo–Fabrizio (exponential decay-law) and Atangana–Baleanu (Mittag-Leffler-law) fractional operators. We utilize the new fractional Adams–Bashforth schemes for the approximation of these derivatives. This method was developed with the standard differentiation technique by applying the fundamental theorem of calculus and taking the difference of two times levels at t = (t n , t n + 1). The stability analysis applies the theory of classical order differential equations and dynamical system. The Existence and uniqueness of solution of fractional dynamics is proved by adopting the fixed-point theorem. Numerical results presented for various α − values justify the theoretical findings. It was observed that modelling of interpersonal and romantic love affairs with fractional derivative could exhibit some strange emotional attractors. • Fractional two-step Adams–Bashforth methods. • Numerical approximation and simulation. • Existence and uniqueness of solutions via fixed-point theorem. • The Caputo, Caputo–Fabrizio and Atangana–Baleanu fractional derivatives. [ABSTRACT FROM AUTHOR]

In this paper we introduce a generalized form of the well known ToDD's difference equation and give the closed form expressions for this generalized form . In other words , we have the following nonlinear rational partial difference equation T ‹X1,X2,X3, …,Xn› = 1 + T ‹X1 - 1,X2 - 1, …,Xn - 1› + T ‹X1 - 2,X2 - 2, …,Xn - 2› T ‹X1 - 3,X2 - 3,X3 - 3, …,Xn - 3› where X1,X2, …:,Xn ∊ ℕ,and the initial values T ‹p1, p2, …:, pn› ,T ‹p2, p1, p3, p4, …, pn› ,T ‹p2, p3, p1, p4, …, pn›,… …,T ‹p2, p3, p4, …p1, pn›,T ‹p2 - 3, p3 - 3, p4 - 3, …pn - 3, p1› are real numbers with p1 ∊ {0,-1,-2} and p2, p3, …, pn ∊ ℕ such that T ‹p1, p2, …:, pn› ≠ 0 ,T ‹p2, p1, p3, p4, …, pn› ≠ 0 , T ‹p2, p3, p1, p4, …, pn› ≠ 0,…,T ‹p2 - 3, p3 - 3, p4 - 3, …pn - 3, p1› ≠ 0. We will use a novel technique to prove the results by using what we call 'piecewise n-dimensional mathematical induction' which we intro- duce here for the first time . We will obvious that this new concept represents generalized form for many types of mathematical induction . As a direct consequences , we investigate and drive the explicit solutions for the well known ordinary ToDD's difference Equation . [ABSTRACT FROM AUTHOR]

Abstract We consider the nonlinear problem of inhomogeneous Allen–Cahn equation ϵ 2 Δ u + V (y) u (1 − u 2) = 0 in Ω , ∂ u ∂ ν = 0 on ∂ Ω , where Ω is a bounded domain in R 2 with smooth boundary, ϵ is a small positive parameter, ν denotes the unit outward normal of ∂Ω, V is a positive smooth function on Ω ¯. Let Γ be a curve intersecting orthogonally with ∂Ω at exactly two points and dividing Ω into two parts. Moreover, Γ satisfies stationary and non-degenerate conditions with respect to the functional ∫ Γ V 1 / 2. We can prove that there exists a solution u ϵ such that: as ϵ → 0 , u ϵ approaches +1 in one part of Ω, while tends to −1 in the other part, except a small neighborhood of Γ. [ABSTRACT FROM AUTHOR]

In this paper, we give some necessary conditions on the existence of meromorphic solutions on Fermat type difference equations. We also consider the properties of transcendental entire solutions on the systems of Fermat type differential-difference equations. [ABSTRACT FROM AUTHOR]

In this article, we investigate the dynamics of the solutions of the following non-linear di¤erence equation xn+1 = xn-2xn-3 - 1; n ∞ N0 with arbitrary initial conditions x-2; x-1; x0. Besides, we have studied periodic behaviours of related di¤erence equation especially asymptotic periodicity and eventually periodicity. Then, we have researched unbounded solutions of di¤erence equation. [ABSTRACT FROM AUTHOR]

Closed form formulas for well-defined complex-valued solutions to a product-type system of difference equations of interest with six parameters are presented. The form of the solutions is described in detail in terms of the parameters and initial values. [ABSTRACT FROM AUTHOR]

In this note, we study a type of nonlinear Riemann‐Liouville fractional q–difference equations. By using the Krasnoselskii fixed‐point theorem, we investigate the existence of solutions of the addressed equation. Prior to the main results and unlike previous theorems, we provide the variation of parameters formula in case the solution starts at some a>0, which belongs to the quantum time scale Tq where the solution is only defined. The case when the solution starts at a=0 is also considered. To apply our result, we provide an existence theorem for the Riemann‐Liouville fractional q–logistic type model. [ABSTRACT FROM AUTHOR]

A multilevel self-similar domination-monotonicity structure is proposed, and a kind of coupled forward-backward stochastic differential equations (FBSDEs) with such structure is proved to be uniquely solvable. Then, this kind of FBSDEs is used to characterize the unique equilibrium of a linear-quadratic generalized Stackelberg game with multilevel hierarchy in a closed form. [ABSTRACT FROM AUTHOR]

In this paper, some new results are obtained for the even order neutral delay difference equation ΔanΔm-1xn+pnxn-kα+qnxn-lβ=0, where m≥2 is an even integer, which ensure that all solutions of the studied equation are oscillatory. Our results extend, include, and correct some of the existing results. Examples are provided to illustrate the importance of the main results. [ABSTRACT FROM AUTHOR]

In this study, we consider two different inequivalent formulations of the logistic difference equation xn+i = βxn (1 - xn), n = 0,1,..., where xn is a sequence of fuzzy numbers and β is a positive fuzzy number. The major contribution of this paper is to study theIn existence, uniqueness and global behavior of the solutions for two corresponding equations, using the concept of Hukuhara difference for fuzzy numbers. Finally, some examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

Abstract A criterion for the existence of a birational embedding of an algebraic curve into a projective plane with two Galois points is presented. Several novel examples of plane curves with two inner Galois points as an application are described. [ABSTRACT FROM AUTHOR]

In this work, sufficient conditions for oscillation of solutions of three dimensional difference systems of the form X(n + 1) = A(n)X(n) are established. Here X(n) = [x1(n), x2(n), x3(n)]T , A(n) = [aij (n)] is a given 3 × 3 matrix, xi : N → R, aij : N → R for i, j ∈ {1, 2, 3}. [ABSTRACT FROM AUTHOR]

We make a case for including difference equations, in particular the discrete logistic equation, in basic differential equations courses. Contrasting the behavior of discrete and continuous models enriches students’ understanding of both modeling and differential equations. To facilitate sharing discrete population models with students, some materials including class handouts and an “exploratory” computer assignment are provided. [ABSTRACT FROM AUTHOR]

We construct solutions to the difference-differential equation that are associated with onepoint commuting difference operators of rank 1 in the case of spectral curves of genus 1. [ABSTRACT FROM AUTHOR]

We consider the chemotaxis-fluid system given by nt + u ·∇n = Δnm - ∇· (n∇c), ct + u ·∇c = Δc - c + n, ut + (u · ∇)u = Δu + ∇P + n∇φ, and ∇ · u = 0, for x ∈ Ω and t > 0, where Ω ⊂ R3 is a bounded domain with smooth boundary and m > 1. Assuming m > 4/3 and sufficiently regular nonnegative initial data, we ensure the existence of global solutions to the no-flux-Dirichlet boundary value problem for this system under a suitable notion of very weak solvability, which in different variations has been utilized in the literature before. Comparing this with known results for the fluid-free setting of the system above the condition appears to be optimal with respect to global existence. In the case of the stronger assumption m > 5/3 we moreover establish the existence of at least one global weak solution in the standard sense. In our analysis we investigate a functional of the form ∫Ωnm-1 + ∫Ωc2 to obtain a spatio-temporal L2 estimate on ∇nm-1, which will be the starting point in deriving a series of compactness properties for a suitably regularized version of the system above. As the regularity information obtainable from these compactness results vary depending on the size of m, we will find that taking m > 5/3 will yield sufficient regularity to pass to the limit in the integrals appearing in the weak formulation, while for m > 4/3 we have to rely on milder regularity requirements making only very weak solutions attainable. [ABSTRACT FROM AUTHOR]

We study the structure of the set of solutions of a certain class of nonlinear functional-difference equations. [ABSTRACT FROM AUTHOR]

We obtain in terms of additive and multi-additive functions the solutions f, h: S → H of the functional equation ..., x,y ∊ S, where (S, +) is an abelian monoid, Φ is a finite group of automorphisms of S, N = |Φ| designates the number of its elements, {aλ,λ ∊ Φ} are arbitrary elements of S and (H, +) is an abelian group. In addition, some applications are given. This equation provides a joint generalization of many functional equations such as Cauchy's, Jensen's, Lukasik's, quadratic or Φ-quadratic equations. [ABSTRACT FROM AUTHOR]

In this paper, we address the mathematical analysis of a partial differential equation model for pricing fixed-rate mortgages with prepayment and default options, where the underlying stochastic factors are the house price and the interest rate. The mathematical model is posed in terms of a sequence of linked complementarity problems, one for each month of the loan life, associated with a uniformly parabolic operator. We study the existence of a strong solution to each one of the obstacle problems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new modelling approach for the dynamic granular matter formation process in the form of a system of difference equations, directly tailored to the physical nature of the process at hand. Respectively, the dynamic 1D and 2D discrete models, proposed in this paper, are not constructed as numerical schemes approximating some partial differential equations (PDEs).We propose here to look for the functions describing the standing and the rolling layers of the granular matter as the limits of discrete solutions to the aforementioned model equations as the size of the mesh tends to zero. In particular, this approach allows us to differentiate between the influx of the rolling layer coming down from different directions to the corner points of the standing layer. Such points are difficult to adequately describe by means of PDEs and their straightforward numerical approximations, typically 'ignoring' the system's behaviour on the sets of zero measure. However, these points are critical for understanding the dynamics of formation process when the standing layer is created by the moving front of the rolling matter or when the latter is filling a cavity and/or stops rolling. The existence of distributed (infinite-dimensional) limit solutions to our discrete models as the size of the mesh tends to zero is also discussed. We illustrate our findings by numerical examples which use our models as the direct algorithm. [ABSTRACT FROM AUTHOR]

We investigate global dynamics of the following second order rational difference equation xn+1... where the parameters α, β, a, b are positive real numbers and initial conditions x-1 and x0 are arbitrary positive real numbers. The map associated to the right-hand side of this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the corresponding parametric space. In most cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability. [ABSTRACT FROM AUTHOR]

In this paper we analyze the existence of bounded solutions for a nonlinear second-order neutral difference equation, which is more general than other equations of this type studied recently. Moreover, our analysis is based under more general conditions than the required in other works, as we only assume the continuity of the involved functions. The main tool here are the so called measures of noncompactness, and more specifically, the celebrated Darbo fixed point theorem. Also we will state, in the specified sense, the stability of the solutions. [ABSTRACT FROM AUTHOR]

In this paper, we establish the Freidlin–Wentzell's large deviations for 3D stochastic primitive equations with small noise perturbation. The weak convergence approach plays an important role. [ABSTRACT FROM AUTHOR]

Two (p, q)-Laplace transforms are introduced and their relative properties are stated and proved. Applications are made to solve some (p, q)-linear difference equations. [ABSTRACT FROM AUTHOR]

In this study,we consider a stochastic optimal control problem whose state variables are described by the system of forward and backward stochastic differential equations with a general cost functional which relies on the global terminal condition. In the framework of Fréchet derivatives, we derive the corresponding maximum principle via constructing a series of adjoint equations which need to be solved step by step. [ABSTRACT FROM AUTHOR]

The solution form of the system of nonlinear difference equations [ABSTRACT FROM AUTHOR]

This paper discusses the asymptotic stability of Markov switched stochastic differential equations. By using the method of multiple Lyapunov functions, we provide sufficient conditions for stochastic asymptotic stability of Markov switched stochastic differential equations with both stable and unstable subsystems via the inequality based on the multiple Lyapunov functions and the stationary distribution of Markovian switching process. Particularly, our results include some existing results as special cases and improve some results in the literature. Two examples are given to illustrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

In this article, we study the oscillatory behavior of the third-order neutral type difference equation ∆(an(∆² (xn + pnxn-k))α) + qnƒ(xn-l) = 0 where α > 0, an > 0, qn ≥ 0 and 0 ≤ pn ≤ p < ∞. By using generalized Ricatti type transformation we present some new criteria which ensure that every solution is oscillatory. Also we provide examples that illustrate the importance of our results. [ABSTRACT FROM AUTHOR]

We obtain sufficient conditions for the existence of a nontrivial homoclinic solution to a second-order nonlinear difference equation with Jacobi operator. To do this, we use variational methods and critical point theory. An example is provided to illustrate our main result. [ABSTRACT FROM AUTHOR]

We consider continuous-state branching processes that are perturbed by a Brownian motion. These processes are constructed as the unique strong solution of a stochastic differential equation. The long-term behaviours are studied. In the stable case, the extinction and explosion probabilities are given explicitly. We find three regimes for the asymptotic behaviour of the explosion probability and five regimes for the asymptotic behaviour of the extinction probability. In the supercritical regime, the process conditioned on eventual extinction has three regimes for the asymptotic behaviour of the extinction probability. Finally, the process conditioned on non-extinction and the process with immigration are given. [ABSTRACT FROM AUTHOR]

We study isomonodromic deformation of Fuchsian linear q-difference systems. Furthermore, we are looking at the behaviour of the Birkhoff connection matrix when q goes to 1. We use our results to study the convergence of the Birkhoff connection matrix that appears in the definition of the q-analogue of the sixth Painlevé equation. [ABSTRACT FROM AUTHOR]

In this paper, a novel nonlinear differential equation in the field of heat transfer has been investigated and solved completely by a new method that we called it Akbari-Ganjis Method (AGM). Regarding to the previously published papers, investigating this kind of equations is a very hard project to do and the obtained solution is not accurate and reliable. This issue will be appeared after comparing the obtained solution by Numerical Method or the Exact Solution. Based on the comparison which has been made between the achieved solutions by AGM and Numerical Method (Runge-Kutte 4th), it is possible to indicate that AGM can be successfully applied to various differential equations particularly for difficult ones. Furthermore, It is necessary to mention that a summary of the excellence of this method in comparison with other approaches can be considered as follows: Boundary conditions are required in accordance with order of the differential equation, this approach can create additional new boundary conditions in regard to the own differential equation and its derivatives. Therefore, it is logical to mention which AGM is operational for miscellaneous nonlinear differential equations in comparison with the other methods. [ABSTRACT FROM AUTHOR]

We present a new class of solvable systems of difference equations of interest by describing a method for finding its general solution. [ABSTRACT FROM AUTHOR]

In this paper, we use a method based on orthogonal polynomials to give closed-form evaluations of the Hankel transform of sequences based on the Motzkin numbers. It includes linear combinations of consecutive two, three, and four Motzkin numbers. In some cases, we were able to derive the closed-form evaluation of the Hankel transform, while in the others we showed that the Hankel transform satisfies a particular difference equation. As a corollary, we reobtain known results and show some new results regarding the Hankel transform of Motzkin and shifted Motzkin numbers. Those evaluations also give an idea on how to apply the method based on orthogonal polynomials on the sequences having zero entries in their Hankel transform. [ABSTRACT FROM AUTHOR]

The paper is devoted to properties of set-valued stochastic differential equations. The main result of the paper deals with existence and uniqueness of solutions. Furthermore, a connection between solutions of stochastic differential inclusions and solutions of set-valued stochastic differential equations are given. The result of the paper extends a lot of particular results dealing with such type equations. [ABSTRACT FROM PUBLISHER]

We consider the class of entire functions of exponential type in relation to the scattering theory for the Schrödinger equation with a finite potential that is a finite Borel measure. These functions have a special self-similarity and satisfy q-difference functional equations. We study their asymptotic behavior and the distribution of zeros. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability of the zero equilibria of the following systems of difference equations: [ABSTRACT FROM AUTHOR]

We obtain in this paper the expression of the solutions of the following recursive sequences: [ABSTRACT FROM AUTHOR]

In this paper, a nonperiodic discrete nonlinear equation with Jacobi operators is considered. By using the critical point theory, we establish some new sufficient conditions on the existence and multiplicity of homoclinic solutions. Recent results are generalized and significantly improved. Furthermore, our results greatly improve some existing ones even for some special cases. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A solvable class of product-type systems of difference equations with two dependent variables on the complex domain is presented. The main results complement some recent ones in the literature, while their proofs contain some refined methodological details. We provide closed form formulas for general solutions to the system or give procedures for how to get them. [ABSTRACT FROM AUTHOR]