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Text Let K be a complete, algebraically closed, non-Archimedean valued field, and let ϕ ∈ K ( z ) with deg ⁡ ( ϕ ) ≥ 2 . In this paper we consider the family of functions ord Res ϕ n ( x ) , which measure the resultant of ϕ n at points x in P K 1 , the Berkovich projective line, and show that they converge locally uniformly to the diagonal values of the Arakelov–Green's function g μ ϕ ( x , x ) attached to the canonical measure of ϕ . Following this, we are able to prove an equidistribution result for Rumely's crucial measures ν ϕ n , each of which is a probability measure supported at finitely many points whose weights are determined by dynamical properties of ϕ . Video For a video summary of this paper, please visit https://youtu.be/YCCZD1iwe00 . [ABSTRACT FROM AUTHOR]

Abstract: One of the particularities of information encoded as DNA strands is that a string u contains basically the same information as its Watson–Crick complement, denoted here as . Thus, any expression consisting of repetitions of u and can be considered in some sense periodic. In this paper, we give a generalization of Lyndon and Schützenberger’s classical result about equations of the form , to cases where both sides involve repetitions of words as well as their complements. Our main results show that, for such extended equations, if , then all three words involved can be expressed in terms of a common word t and its complement . Moreover, if , then is an optimal bound. These results are established based on a complete characterization of all possible overlaps between two expressions that involve only some word u and its complement , which is also obtained in this paper. [Copyright &y& Elsevier]

Abstract: This paper distinguishes two categories of questions that the Price equation can help us answer. The two different types of questions require two different disciplines that are related, but nonetheless move in opposite directions. These disciplines are probability theory on the one hand and statistical inference on the other. In the literature on the Price equation this distinction is not made. As a result of this, questions that require a probability model are regularly approached with statistical tools. In this paper, we examine the possibilities of the Price equation for answering questions of either type. By spending extra attention on mathematical formalities, we avoid the two disciplines to get mixed up. After that, we look at some examples, both from kin selection and from group selection, that show how the inappropriate use of statistical terminology can put us on the wrong track. Statements that are ‘derived’ with the help of the Price equation are, therefore, in many cases not the answers they seem to be. Going through the derivations in reverse can, however, be helpful as a guide how to build proper (probabilistic) models that do give answers. [Copyright &y& Elsevier]

Let F q denote the finite field of order q . In this paper, some new classes of permutation polynomials of the form ( x p m − x + δ ) s + x over F p 2 m are obtained by determining the number of solutions of certain equations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we use basic properties of superquadratic functions to obtain new inequalities including Fejer''s type and Hermite–Hadamard type inequalities. For superquadratic functions which are also convex, we get refinements of known results. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to provide a new approach concerning the characterization of exponential dichotomy of difference equations by means of admissible pair of sequence spaces. We classify the classes of input and output spaces, respectively, and deduce necessary and sufficient conditions for exponential dichotomy applicable for a large variety of systems. By an example we show that the obtained results are the most general in this topic. As an application we deduce a general lower bound for the dichotomy radius of difference equations in terms of input–output operators acting on sequence spaces which are invariant under translations. [Copyright &y& Elsevier]

Abstract: In this paper, we study the free boundary problem for 1D compressible Navier–Stokes equations with density-dependent viscosity. We focus on the case where the viscosity coefficient vanishes on vacuum. We prove the global existence and uniqueness for discontinuous solutions to the Navier–Stokes equations when the initial density is a bounded variation function, and give a decay result for the density as . [Copyright &y& Elsevier]

Abstract: In this paper, we consider dynamical behavior of non-autonomous plate-type evolutionary equations with critical nonlinearity. We prove the existence of a uniform attractor in the space . [Copyright &y& Elsevier]

Abstract: In this paper we study the global asymptotic stability for a class of delay logistic equations. A sufficient and necessary condition is derived. As a result, we obtain a full affirmative answer to the Gopalsamy and Liu'' conjecture. [Copyright &y& Elsevier]

Abstract: This paper investigates the regularity of solutions of convolution equations in the frame of classes of ultradifferentiable functions and ultradistributions. We improve previous work by Bonet, Chou, Fernández, Galbis, Meise and others. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a new generalized discontinuous Galerkin (GDG) method for Schrödinger equations with nonsmooth solutions. The numerical method is based on a reformulation of Schrödinger equations, using split distributional variables and their related integration by parts formulae to account for solution jumps across material interfaces. The proposed GDG method can handle time dependent and nonlinear jump conditions . Numerical results for 1D and 2D time dependent Schrödinger equations validate the high order accuracy and the flexibility of the method for various types of interface conditions. [Copyright &y& Elsevier]

Abstract: The present paper is concerned with asymptotic behaviours of the solutions to the micropolar fluid motion equations in . Upper and lower bounds are derived for the decay rates of higher order derivatives of solutions to the micropolar fluid flows. The findings are mainly based on the basic estimates of the linearized micropolar fluid motion equations and generalized Gronwall type argument. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a food-limited population model with impulsive effect. Several explicit sufficient conditions are established for oscillation and nonoscillation of solutions of the equations. [Copyright &y& Elsevier]

Abstract: We study the optimal approximation of the solution of an operator equation by certain n-term approximations with respect to specific classes of frames. We consider worst case errors, where f is an element of the unit ball of a Sobolev or Besov space and is a bounded Lipschitz domain; the error is always measured in the -norm. We study the order of convergence of the corresponding nonlinear frame widths and compare it with several other approximation schemes. Our main result is that the approximation order is the same as for the nonlinear widths associated with Riesz bases, the Gelfand widths, and the manifold widths. This order is better than the order of the linear widths iff . The main advantage of frames compared to Riesz bases, which were studied in our earlier papers, is the fact that we can now handle arbitrary bounded Lipschitz domains—also for the upper bounds. [Copyright &y& Elsevier]

Abstract: We are interested in spatially extended pattern forming systems close to the threshold of the first instability in case when the so-called degenerated Ginzburg–Landau equation takes the role of the classical Ginzburg–Landau equation as the amplitude equation of the system. This is the case when the relevant nonlinear terms vanish at the bifurcation point. Here we prove that in this situation every small solution of the pattern forming system develops in such a way that after a certain time it can be approximated by the solutions of the degenerated Ginzburg–Landau equation. In this paper we restrict ourselves to a Swift–Hohenberg–Kuramoto–Shivashinsky equation as a model for such a pattern forming system. [Copyright &y& Elsevier]

Abstract: This paper presents a systematic investigation of the integrability conditions for nonautonomous quad-graph maps, using the Lax pair approach, the ultra-local singularity confinement criterion and direct construction of conservation laws. We show that the integrability conditions derived from each of the methods are the one and the same, suggesting that there exists a deep connection between these techniques for partial difference equations. [Copyright &y& Elsevier]

Abstract: The general case of the Nagell–Ljunggren equation is and odd primes. In this paper we derive a new bound for which there are no solutions. For small q the problem is harder and we achieve a conditional result: for and some additional condition on must hold. Both functions are quadratic and they leave a small gap for , on which we have no general result. The picture is completed by some explicit, unconditional upper bounds for the absolute values , . [Copyright &y& Elsevier]

Abstract: This paper deals with the global attraction and global weak attraction of the origin for the generalized Liénard equation. The problem proposed by Jiang is solved by a negative answer and some new criteria for the globally asymptotically stability are obtained by the ways different from the usual Filippov''s conditions. [Copyright &y& Elsevier]

Abstract: In this paper we establish the exact blow-up rate of the large solutions of a porous media logistic equation. We consider the carrying capacity function with a general decay rate at the boundary instead of the usual cases when it can be approximated by a distant function. Obtaining the accurate blow-up rate allows us to establish the uniqueness result. Our result covers all previous results on the ball domain and can be further adapted in a more general domain. [Copyright &y& Elsevier]

Abstract: In this paper we study uniqueness properties of solutions of the so-called k-generalized Korteweg–de Vries equations. Our goal is to obtain sufficient conditions on the behavior of the difference of two solutions of (1.1) at two different times and which guarantee that . [Copyright &y& Elsevier]

Abstract: In this paper we consider the existence of nontrivial solutions for an elliptic system, where the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing two cones and computing the fixed point index in , and , we obtain that the elliptic system has three nontrivial solutions , and . It is remarkable that the third nontrivial solution is established on the Cartesian product of two cones, in which the feature of two equations can be exploited better. [Copyright &y& Elsevier]

Abstract: In this paper, we establish sufficient conditions for the oscillation of the linear non-autonomous systems of difference equations with continuous arguments where , , , , ; τ, γ and , , are non-negative constants. Furthermore, assume , where , , , are continuous, bounded and do not change signs on . [Copyright &y& Elsevier]

Abstract: The classical criterion of asymptotic stability of the zero solution of equations is that there exists a function , for some , such that for some . In this paper we prove that if is bounded, is uniformly continuous and bounded, then the condition that can be weakened and replaced by and contains no complete trajectory of , , where , uniformly for . [Copyright &y& Elsevier]

Abstract: This paper discusses a class of second-order derivative nonlinear Schrödinger equations which are used to describe the upper-hybrid oscillation propagation. By establishing a variational problem, applying the potential well argument and the concavity method, we prove that there exists a sharp condition for global existence and blow-up of the solutions to the nonlinear Schrödinger equation. In addition, we also answer the question: how small are the initial data, the global solutions exist? [Copyright &y& Elsevier]

Abstract: In the paper we examine Pexiderized ϕ-homogeneity equation almost everywhere. Assume that G and H are groups with zero, and are a G- and an H-space, respectively. We prove, under some assumption on , that if functions and satisfy Pexiderized ϕ-homogeneity equation almost everywhere in then either almost everywhere in G or almost everywhere in X or there exists a homomorphism such that almost everywhere in G and there exists a function such that and where is a constant. From this result we derive solution of the classical Pexider equation almost everywhere. [Copyright &y& Elsevier]

Abstract: In this paper we study several qualitative properties of the Degasperis–Procesi equation. We first established the precise blow-up rate and then determine the blow-up set of blow-up strong solutions to this equation for a large class of initial data. We finally prove the existence and uniqueness of global weak solutions to the equation provided the initial data satisfies appropriate conditions. [Copyright &y& Elsevier]

Abstract: In this paper analytical approximations for the period of a generalized nonlinear van der Pol equation will be obtained by using various asymptotic methods. [Copyright &y& Elsevier]

Abstract: In this paper we consider the blow up phenomenon of critical nonlinear Schrödinger equations in dimension 1D and 2D. We define the minimal mass as the norm necessary to ignite a wave collapse and we stress its role in the blow up mechanism. Asymptotic compactness properties and -concentration are proved. The proof relies on linear and nonlinear profile decompositions. [Copyright &y& Elsevier]

Abstract: Let be a field and a finite directed multi-graph. The focus of this paper is to offer a complete description of all path algebras and admissible orders with the property that all of their finitely generated ideals have finite Gröbner bases and of those which contain a finitely generated ideal whose Gröbner bases are all infinite. [Copyright &y& Elsevier]

Abstract: In this paper, we use a method different from the known literature to investigate the qualitative properties of the following fourth-order rational difference equation: where and the initial values . The successive lengths of positive and negative semicycles of nontrivial solutions of the above equation is found to periodically occur, that is, , or, . By using the rule, the positive equilibrium point of the equation is verified to be globally asymptotically stable. [Copyright &y& Elsevier]

Abstract: In this paper, we describe in detail the computational algorithm used by our parallel multigrid elliptic equation solver with adaptive mesh refinement. Our code uses truncation error estimates to adaptively refine the grid as part of the solution process. The presentation includes a discussion of the orders of accuracy that we use for prolongation and restriction operators to ensure second order accurate results and to minimize computational work. Code tests are presented that confirm the overall second order accuracy and demonstrate the savings in computational resources provided by adaptive mesh refinement. [Copyright &y& Elsevier]

Abstract: In this paper, we establish certain new Ramanujan''s Schläfli-type mixed modular equations. [Copyright &y& Elsevier]

In this paper the approach via time-map is used to investigate the existence of periodic solutions for second order equation with time-dependent potential. Our theorem generalized the results obtained by Ding, Habets, Omari and Zanolin which are for time-independent potential and relate the behavior of the primitive of the nonlinearity with respect to the Fučik spectrum of the periodic problem. [Copyright &y& Elsevier]

In the paper we shall present a survey of recent results on substochastic semigroups and provide new methods for determining their honesty. These methods are applied to the fragmentation equation with mass loss, yielding sufficient conditions for the existence of conservative and shattering solutions. Our results provide a mathematical framework that clarifies the discussion of [Phys. Rev. A 43 (1991) 656, Phys. Rev. A 41 (1990) 5755, J. Phys. A 24 (1991) 3967] on shattering fragmentation in such models showing, among others, that there occurs an unexpected mass loss associated with shattering which is not accounted for by the discrete and continuous mass loss, contrary to the conjecture of [Phys. Rev. A 43 (1991) 656, Phys. Rev A 41 (1990) 5755]. [Copyright &y& Elsevier]