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Fifty years ago, a small one and a half page paper without a single reference was published in the leading journal Nature. The paper laid out the most general mathematical formulation of natural selection that would work for all kinds of selection processes and under any form of inheritance (not just biological evolution and Mendelian genes), although the paper discussed the issue in a genetical framework. Written by a maverick American expatriate in England, with no prior background of studying evolution or genetics, the paper had initially been turned down by the editor of Nature as too difficult to understand. Largely ignored by the evolutionary biology community till the 1990s, the Price Equation is now widely recognized as an extremely useful conceptualization, permitting the incorporation of non-genetic inheritance into evolutionary models, serving to clarify the relationship between kin-selection and group-selection, unifying varied approaches used in the past to model evolutionary change, and forming the foundation of multi-level selection theory. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of Choquard equations with Hardy–Littlewood–Sobolev lower or upper critical exponent in the whole space R N . We combine an argument of L. Jeanjean and H. Tanaka (see (Proc. Am. Math. Soc. 131:2399–2408, 2003) with a concentration–compactness argument, and then we obtain the existence of ground state solutions, which extends and complements the earlier results. [ABSTRACT FROM AUTHOR]

In this paper, finite time blow-up property of solutions to a p-Kirchhoff-type parabolic equation with general nonlinearity is considered. Some sufficient conditions are given for the weak solutions to blow up in finite time. An upper bound for the blow-up time is also derived. The results partially generalize some recent ones reported by Han and Li (Comput Math Appl. 2018;75:3283–3297). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stability of the solutions of a viscoelastic plate equation with a logarithmic nonlinearity. We assume that the relaxation function g satisfies the minimal condition g ′ (t) ≤ − ξ (t) G (g (t)) , where ξ and G satisfy some properties. With this very general assumption on the behavior of g, we establish explicit and general energy decay results from which we can recover the exponential and polynomial rates when G (s) = s p and p covers the full admissible range [ 1 , 2) . Our new results substantially improve and generalize several earlier related results in the literature such as Gorka (Acta Phys. Pol. 40:59–66, 2009), Hiramatsu et al. (J. Cosmol. Astropart. Phys. 2010(06):008, 2010), Han and Wang (Acta Appl. Math. 110(1):195–207, 2010), Messaoudi and Al-Khulaifi (Appl. Math. Lett. 66:16–22, 2017), Mustafa (Math. Methods Appl. Sci. 41(1):192–204, 2018), and Al-Gharabli et al. (Commun. Pure Appl. Anal. 18(1):159–180, 2019). [ABSTRACT FROM AUTHOR]

In this paper, we study the behavior of solutions of a semilinear elliptic equation in the exterior of a compact set as $|x|\backslash to\; \backslash nomathbreak\; \backslash infty$. Such equations were considered by many authors (for example, Kondrat'ev, Landis, Oleinik, Veron, etc.). In the present paper, we study the case in which in the equation contains lower terms. The coefficients of the lower terms are arbitrary bounded measurable functions. It is shown that the solutions of the equation tend to zero as $|x|\backslash to\; \backslash nomathbreak\; \backslash infty$. [ABSTRACT FROM AUTHOR]

In this paper, we study global regularity for oblique boundary value problems of augmented Hessian equations for a class of general operators. By assuming a natural convexity condition of the domain together with appropriate convexity conditions on the matrix function in the augmented Hessian, we develop a global theory for classical elliptic solutions by establishing global a priori derivative estimates up to second order. Besides the known applications for Monge-Ampère type operators in optimal transportation and geometric optics, the general theory here embraces Neumann problems arising from prescribed mean curvature problems in conformal geometry as well as general oblique boundary value problems for augmented k-Hessian, Hessian quotient equations and certain degenerate equations. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a new system of generalized resolvent equations with corresponding system of variational inclusions in uniformly smooth Banach spaces. We establish an equivalence relation between system of generalized resolvent equations and system of variational inclusions. The iterative algorithms for finding the approximate solutions of system of generalized resolvent equations are proposed. The convergence of approximate solutions of system of generalized resolvent equations obtained by the proposed iterative algorithm is also studied. [ABSTRACT FROM AUTHOR]

The paper deals with averaging problems for parabolic equations. We prove that exponential dichotomy is preserved without any assumption concerning the almost-periodicity of the coefficients. [ABSTRACT FROM AUTHOR]

The mode-I weight function proposed by Oore and Burns is an efficient and empirical mathematical relationship capable of estimating the Stress Intensity Factor in the presence of a generic planar flaw. In the case of penny shaped cracks or tunnel cracks the Oore-Burns equation provides the classic analytical expressions. In this paper, the authors prove analytically that the Oore-Burns equation is not the weight function for an ellipse in an infinite solid under traction loading. The absolute novelty is given by the analytic evaluation between the Oore-Burns integral and the Irwin solution in terms of an algebraic expression, which is independent from any numerical procedures for an ellipse close to a circle. [ABSTRACT FROM AUTHOR]

In this paper we present some new oscillatory criteria for the n-th order neutral differential equations of the form(x(t){\pm} p(t)x[ \tau(t)])^{(n)}+q(t)x[\sigma(/math>The results obtained extend and improve a number of existing criteria. [ABSTRACT FROM AUTHOR]

The accuracy of analytical obtained solutions of the fractional nonlinear space–time telegraph equation that has been constructed in (Hamed and Khater in J. Math., 2020) is checked through five recent semi-analytical and numerical techniques. Adomian decomposition (AD), El Kalla (EK), cubic B-spline (CBS), extended cubic B-spline (ECBS), and exponential cubic B-spline (ExCBS) schemes are used to explain the matching between analytical and approximate solutions, which shows the accuracy of constructed traveling wave solutions. In 1880, Oliver Heaviside derived the considered model to describe the cutting-edge or voltage of an electrified transmission. The matching between solutions has been explained by plotting them in some different sketches. [ABSTRACT FROM AUTHOR]

Recently, Lal (Appl. Math. Comput. 209:346-350, 2009) has determined the degree of approximation of a function belonging to Lip α and weighted [InlineEquation not available: see fulltext.]-classes using product [InlineEquation not available: see fulltext.] summability with non-increasing weights [InlineEquation not available: see fulltext.]. In this paper, we determine the degree of approximation of function [InlineEquation not available: see fulltext.], conjugate to a 2 π-periodic function f belonging to weighted [InlineEquation not available: see fulltext.]-class by dropping the monotonicity on the generating sequence [InlineEquation not available: see fulltext.] with a new (proper) set of conditions, which in turn generalizes the results of Mishra et al. (Bull. Math. Anal. Appl., 2013) on [InlineEquation not available: see fulltext.]-class and rectifies (removes) the errors of Mishra et al. (Mat. Vesn., 2013). Few examples and applications are also highlighted in this manuscript. MSC: Primary 42B05, 42B08, 40G05, 41A10. [ABSTRACT FROM AUTHOR]

The optimal control of the solutions of the initial-finish problem for Sobolev-type equations is studied. The abstract results obtained in the paper are applied to the linear Hoff model on graphs. [ABSTRACT FROM AUTHOR]

In this paper, we determine the general solution of the functional equation f (kx+ y)+ f (kx-y) = g(x+y)+g(x-y)+h(x)+h(y) for fixed integers k with k ≠= 0,±1 without assuming any regularity condition on the unknown functions f ,g,h,h. The method used for solving these functional equations is elementary but exploits an important result due to Hosszú. The solution of this functional equation can also be determined in certain type of groups using two important results due to Székelyhidi. The results improve and extend some recent results. [ABSTRACT FROM AUTHOR]

Let us assume that f is a continuous function defined on the unit ball of ℝ, of the form f( x)= g( Ax), where A is a k× d matrix and g is a function of k variables for k≪ d. We are given a budget m∈ℕ of possible point evaluations f( x), i=1,..., m, of f, which we are allowed to query in order to construct a uniform approximating function. Under certain smoothness and variation assumptions on the function g, and an arbitrary choice of the matrix A, we present in this paper Due to the arbitrariness of A, the sampling points will be chosen according to suitable random distributions, and our results hold with overwhelming probability. Our approach uses tools taken from the compressed sensing framework, recent Chernoff bounds for sums of positive semidefinite matrices, and classical stability bounds for invariant subspaces of singular value decompositions. [ABSTRACT FROM AUTHOR]

This paper gives a brief history of the chord function - a precursor of the sine function - from the mathematics/astronomy of Hipparchus and Ptolemy to the early twentieth century. [ABSTRACT FROM AUTHOR]

Abstract  In this paper, we consider the action of (2, q) on the finite projective line $${\mathbb{F}}_q\cup\{\infty\}$$ for q ≡ 1 (mod 4) and construct several infinite families of simple 3-designs which admit PSL(2, q) as an automorphism group. Some of the designs are also minimal. We also indicate a general outline to obtain some other algebraic constructions of simple 3-designs. [ABSTRACT FROM AUTHOR]

The progress of Mathematics during the nineteenth century was characterised both by an enormous acquisition of new knowledge and by the attempts to introduce rigour in reasoning patterns and mathematical writing. Cauchy’s presentation of Mathematical Analysis was not immediately accepted, and many writers, though aware of that new style, did not use it in their own mathematical production. This paper is devoted to an episode of this sort that took place in Spain during the first half of the century: It deals with the presentation of a method for numerically solving algebraic equations by José Mariano Vallejo, a late Spanish follower of the Enlightenment ideas, politician, writer, and mathematician who published it in the fourth (1840) edition of his book Compendio de Matemáticas Puras y Mistas, claiming to have discovered it on his own. Vallejo’s main achievement was to write down the whole procedure in a very careful way taking into account the different types of roots, although he paid little attention to questions such as convergence checks and the fulfilment of the hypotheses of Rolle’s Theorem. For sure this lack of mathematical care prevented Vallejo to occupy a place among the forerunners of Computational Algebra. [ABSTRACT FROM AUTHOR]

The paper is concerned with the problem of inverting block matrices to which the well-known Frobenius— Schur formulas are not applicable. These can be square matrices with four noninvertible square or rectangular blocks as well as square or rectangular matrices with two blocks. With regard to rectangular matrices, the results obtained are a further step in the development of the canonization method, which is used for solving arbitrary matrix equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the Darboux transformation of the Darboux-Treibich-Verdier equation. On the basis of this transformation, we construct a generalization of the Darboux transformation to the case of the Heun equation and to other linear ordinary differential equations of second order. Examples are given. [ABSTRACT FROM AUTHOR]

This paper is devoted to the proof of the existence of a solution of the Cauchy problem for a system of equations of ultraparabolic type. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with a variational formulation for the sensitivity problem of beam systems in the context of deformable solids with cracks. Natural frequencies are defined as state variables involved in the energy functional of the system, while the cracks length and position are treated as system parameters. The hierarchical equation system is formed and solved for the first and second derivatives of the natural frequency functions with respect to the cracks length and position. An analytical procedure for calculations of the second-order sensitivities of natural frequencies is proposed for the non-symmetrical equation system operator. Numerical algorithms are worked out and implemented computationally. Analytical and numerical aspects of the problem are discussed in detail through a number of illustrative results. [ABSTRACT FROM AUTHOR]

For the linear stationary dynamic systems, the necessary and sufficient conditions for output invariance to arbitrary external perturbations were substantiated by solving the matrix equations with the help of the canonization method. The conditions obtained were used to design controllers maintaining this invariance. The results were extended to the problem of parametric invariance. The paper was confined to the linear stationary systems without delays and constraints on control. An illustrating example was presented. [ABSTRACT FROM AUTHOR]

In this paper we show that all totally real minimal tori in correspond to doubly periodic finite gap solutions of the Tzitzéica equation Using the results on the Tzitzéica equation in integrable systems theory, we describe explicitly these tori by Prym-theta functions. [ABSTRACT FROM AUTHOR]

The paper derives a normative model for partial fiscal equalisation based on a number of axioms and makes special allowance for the existence of a specific fiscal need in the jurisdictions. A simple version of this idealised equalisation scheme relates net contributions to the equalisation funds to deviations of a jurisdiction’s gross income from average gross income and a jurisdiction’s specific needs from average specific needs. The theoretical model is then empirically tested for the case of the European Union using data from 1986-97. It is found that most restrictions of the model appear to hold, in particular, relatively richer countries contribute more and those with greater fiscal needs, approximated by the importance of the agricultural sector, pay less. However, in the EU, an adjustment of net payments to changes in the actual importance of the specific fiscal need for a country is lacking. [ABSTRACT FROM AUTHOR]

In economic systems, reactions to external shocks often come with a delay. On the other hand, agents try to anticipate future developments. Both can lead to difference-differential equations with an advancing argument. These are more difficult to handle than either difference or differential equations, but they have the merit of added realism and increased credibility. This paper generalizes a model from monetary economics by von Kalckreuth and Schröder. Working out its stability properties, we present a general method for determining the stability of any solution to a homogeneous linear difference-differential equation with constant coefficients and advancing arguments. [ABSTRACT FROM AUTHOR]

In this paper, we study the inverse problem of the reconstruction of the right-hand side of special form for a parabolic equation in $u$ in which the coefficients of $u\_t$ and $u$ depend on $(x,t)$, with overdetermination given by the integral of the solution over time. The Fredholm property for this problem and the existence and uniqueness theorems in Sobolev spaces are established. [ABSTRACT FROM AUTHOR]

In this paper, we derive a discontinuous Galerkin finite element formulation for the Stokes equations and a group of stable elements associated with the formulation. We prove that these elements satisfy the new inf–sup condition and can be used to solve incompressible flow problems. Associated with these stable elements, optimal error estimates for the approximation of both velocity and pressure in L2 norm are obtained for the Stokes problems, as well as an optimal error estimate for the approximation of velocity in a mesh dependent norm. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the connection between the relative Dixmier property and the complete boundedness of bounded modular maps. For an inclusion of II 1-factors with the relative Dixmier property, we will show that is properly infinite where p is the GNS representation associated with a singular state on . We show that a bounded -multimodular bilinear map is completely bounded and that a bounded -bimodule map is completely bounded. Finally, we prove the row boundedness of a normal -bimodule map where is hyperfinite without the relative Dixmier property. [ABSTRACT FROM AUTHOR]

This paper gives a solution of the Laplace problem, which consists of finding all invariants of the hyperbolic equations and constructing a basis of the invariants. Three new invariants of the first and second orders are found, and invariant‐differentiation operators are constructed. It is shown that the new invariants, together with the two invariants detected by Ovsyannikov, form a basis such that any invariant of any order is a function of the basis invariants and their invariant derivatives. [ABSTRACT FROM AUTHOR]

The paper presents an approach based on three-dimensional elastic equations to solve boundary-value stress problems for hollow cylinders with corrugated elliptical cross section. Discrete Fourier series are used to make the problem one-dimensional and then to solve it by the stable discrete orthogonalization method. Solutions for cylinders of different thicknesses are presented [ABSTRACT FROM AUTHOR]

The paper deals with the three-dimensional dynamic problem for an elliptic crack interacting with a normally incident harmonic compression–expansion wave, considering the contact interaction of the crack faces. An asymmetric solution is obtained using an iteration algorithm developed earlier. Numerical results are presented [ABSTRACT FROM AUTHOR]

The paper studies the dependence of edge effects in a laminated composite of regular structure with a periodic system of symmetric interfacial cracks on the mechanical characteristics of the composite components. Loading the composite induces constant strain in the reinforcement direction. The problem is solved in exact formulation using the linear elastic equations for piecewise-homogeneous media and a criterion for quantitative evaluation of edge effects. An approximate solution of the problem is found by the mesh approach. A difference scheme for the mixed problem under consideration is derived. The edge-effect zones for normal stresses are constructed. The maximum values of the normal stress concentration factors are determined [ABSTRACT FROM AUTHOR]

The paper deals with numerical solution of internal flow problems. It mentions a long tradition of mathematical modeling of internal flow, especially transonic flow at our department. Several models of flow based on potential equation, Euler equations, Navier-Stokes and Reynolds averaged Navier-Stokes equations with proper closure are considered. Some mathematical and numerical properties of the model are mentioned and numerical results achieved by in-house developed methods are presented. [ABSTRACT FROM AUTHOR]

When mathematical models describing various processes are analysed, the fact of existence of a positive solution is often among the basic features. In this paper, a general delayed discrete equation\Delta u(k+n)=f(k,u(k),u(k+1),\ldots,u/math>is considered. Sufficient conditions concerning f are formulated in order to guarantee the existence of a positive solution for k → ∞. An upper estimate for it is given as well. The appearance of the positive solution takes its origin in the nature of the equation considered since the results hold only for delayed equations (i.e. for n > 0) and not for the case of an ordinary equation (with n = 0). [ABSTRACT FROM AUTHOR]

In this paper, we study singular parabolic equations with “quadratic nonlinearity.” We consider questions of unique solvability, of stabilization of solutions at infinity, and of the presence of positive solutions. [ABSTRACT FROM AUTHOR]

In this paper, four methods for the through the thickness response evaluation of laminated composite and sandwich plates are comparatively evaluated in terms of their characteristics and numerical accuracy. The laminated composite and sandwich plates are subjected to thermo-mechanical loading. The four methods include two post-processor-type methods and two re-analysis-type methods. The post-processor-type methods based on the three-dimensional stress equilibrium equations and the three-dimensional thermo-elasticity equations. And the re-analysis methods are based on the incorporation of the nonlinear through the thickness displacements recovered using post-processing technique in the re-analysis phase. The accuracy of the proposed methods for the evaluation of the through the thickness responses of the laminated plates are assessed via numerical examples of various configurations of laminated plates. Also, applicability and computational efficiencies of the methods are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we employ some new techniques to study the existence of positive periodic solution of n-species neutral delay system N_{i}^{\prime}(t)=N_i(t)\left[ a_i(t)-\sum_{j=1}^n \beta_{ij}(t)N_j(t)-\sum_{j=1}^n b_{ij}(t)N_j(t-\tau_{ij}(t))-\sum_{j=1}^n c_{ij}(t)N_{j}^\prime(t-\tau_{ij}(t))\r/math>As a corollary, we answer an open problem proposed by Y. Kuang. [ABSTRACT FROM AUTHOR]

In this paper, we study necessary and sufficient conditions for functions to be approximated uniformly on plane compact sets by polynomial solutions to second-order homogeneous elliptic equations with constant coefficients. Sufficient conditions for approximability are of reductive character, i.e., the possibility of approximating on some (simpler) parts of the compact set implies approximability on the entire compact set. [ABSTRACT FROM AUTHOR]

Quasi-Newton equations play a central role in quasi-Newton methods for optimization and various quasi-Newton equations are available. This paper gives a survey on these quasi Newton equations and studies properties of quasi-Newton methods with updates satisfying different quasi-Newton equations. These include single-step quasi-Newton equations that use only gradient information and that use both gradient and function value information in one step, and multi-step quasi-Newton equations that use the gradient information in last m steps. Main properties of quasi-Newton methods with updates satisfying different quasi-Newton equations are studied. These properties include the finite termination property, invariance, heredity of positive definite updates, consistency of search directions, global convergence and local superlinear convergence properties. [ABSTRACT FROM AUTHOR]

We consider the problem of determining an optimal driving strategy in a train control problem with a generalised equation of motion. We assume that the journey must be completed within a given time and seek a strategy that minimises fuel consumption. On the one hand we consider the case where continuous control can be used and on the other hand we consider the case where only discrete control is available. We pay particular attention to a unified development of the two cases. For the continuous control problem we use the Pontryagín principle to find necessary conditions on an optimal strategy and show that these conditions yield key equations that determine the optimal switching points. In the discrete control problem, which is the typical situation with diesel-electric locomotives, we show that for each fixed control sequence the cost of fuel can be minimised by finding the optimal switching times. The corresponding strategies are called strategies of optimal type and in this case we use the Kuhn-Tucker equations to find key equations that determine the optimal switching times. We note that the strategies of optimal type can be used to approximate as closely as we please the optimal strategy obtained using continuous control and we present two new derivations of the key equations. We illustrate our general remarks by reference to a typical train control problem. [ABSTRACT FROM AUTHOR]

Nontransitive additive conjoint measurement for a binary relation » on a set X1 ⊗ X2 ⊗ ··· ⊗ Xn of n-tuples (x1, …, xn), (y1, …, yn), … is concerned with the representation (x1, …, xn) » (y1, …, yn) ⇔ [This symbol cannot be presented in UNICODE format] φi(xi, yi) »0, where φi is a skew symmetric real valued function on Xi ⊗ Xi. The representation is said to be unique if, whenever (φ1, …, φn) and (φ′1, …, φ′n) satisfy it, there is a positive constant λ such that φ′i = λφi for all i. This paper investigates unique representation for nontransitive additive conjoint measurement when every Xi is finite. It begins with background on measurement theory, followed by conditions for uniqueness that are based on equations ∑φi(xi, yi) = 0 that correspond to the symmetric complement of ». We then examine aspects of sets of unique solutions for n = 2, for arbitrary n with ¦Xi¦ = 2 for all i, and for n = 3. Various combinatorial and number-theoretic arguments are used to derive results that count and characterize sets of unique solutions. [ABSTRACT FROM AUTHOR]

Low-probability high-consequence events play an important role is assessing the risk of catastrophic loss. Their risk profiles, however, can be difficult to obtain. This paper obtains the risk profiles of low-probability high-consequence events where the final consequence results from a number of intermediate events. Called composite events, these events occur, for example, in accidents releasing hazardous material. The structure of composite events is described and risk profile equations developed, both on a per-event and per-annum basis. The many extremely low valued terms, especially in the tail of the risk profile, make calculation nontrivial. Accordingly, algorithm are developed to compute these equations. In addition, formulae for means and variances are obtained, and an illustrative example is provided. [ABSTRACT FROM AUTHOR]

We develop a dynamic generalized conditional gradient method (DGCG) for dynamic inverse problems with optimal transport regularization. We consider the framework introduced in Bredies and Fanzon (ESAIM: M2AN 54:2351–2382, 2020), where the objective functional is comprised of a fidelity term, penalizing the pointwise in time discrepancy between the observation and the unknown in time-varying Hilbert spaces, and a regularizer keeping track of the dynamics, given by the Benamou–Brenier energy constrained via the homogeneous continuity equation. Employing the characterization of the extremal points of the Benamou–Brenier energy (Bredies et al. in Bull Lond Math Soc 53(5):1436–1452, 2021), we define the atoms of the problem as measures concentrated on absolutely continuous curves in the domain. We propose a dynamic generalization of a conditional gradient method that consists of iteratively adding suitably chosen atoms to the current sparse iterate, and subsequently optimizing the coefficients in the resulting linear combination. We prove that the method converges with a sublinear rate to a minimizer of the objective functional. Additionally, we propose heuristic strategies and acceleration steps that allow to implement the algorithm efficiently. Finally, we provide numerical examples that demonstrate the effectiveness of our algorithm and model in reconstructing heavily undersampled dynamic data, together with the presence of noise. [ABSTRACT FROM AUTHOR]

We explore extensions of tropical methods to arithmetic enumerative problems such as A 1 -enumeration with values in the Grothendieck–Witt ring and rationality over Henselian valued fields, using bitangents to plane quartics as a test case. We consider quartic curves over valued fields whose tropicalizations are smooth and satisfy a mild genericity condition. We then express obstructions to rationality of bitangents and their points of tangency in terms of twisting of edges of the tropicalization; the latter depends only on the tropicalization and the initial coefficients of the defining equation modulo squares. We also show that the GW -multiplicity of a tropical bitangent, i.e. the multiplicity with which its lifts contribute to the A 1 -enumeration of bitangents as defined by Larson and Vogt (Res Math Sci 8(26):1–21, 2021), can be computed from the tropicalization of the quartic together with the initial coefficients of the defining equation. As an application, we show that the four lifts of most tropical bitangent classes contribute 2 H , twice the class of the hyperbolic plane, to the A 1 -enumeration. These results rely on a degeneration theorem relating the Grothendieck–Witt ring of a Henselian valued field to the Grothendieck–Witt ring of its residue field, in residue characteristic not equal to two. [ABSTRACT FROM AUTHOR]

This work illustrates how one could apply Lie point symmetries for finding the analytical solution to first-order mechanical systems. Although the classical Lie method constitutes a powerful tool for solving differential equations, an obstacle appears in the case of systems of first-order equations because they admit an infinite number of symmetries, and it is not possible to compute them by following a systematic procedure. To overcome this difficulty, we follow the idea exposed in Nucci (J. Math. Phys. 37: 1772–1775, 1996), Nucci (Electr. J. Diff. Eqn. 12: 87–101, 2005), Nucci (J. Math. Phys. 42: 746, 2001) consisting of transforming the original system to an equivalent system in which one of the equations is of second–order. The presented approach is applied to Hathaway's circular pursuit problem, leading to the analytical solution of the system expressed in terms of the general solution to an Abel equation. [ABSTRACT FROM AUTHOR]

A symmetry classification of integrable vector evolution equations of third order admitting Miura-type transformations is presented. We obtain the Bäcklund autotransformation for the new equation as well as differential substitutions relating the solutions of some integrable isotropic equations. [ABSTRACT FROM AUTHOR]

The present work is concerned with the oscillation and asymptotic properties of the third-order mixed neutral differential equation a(a(t) (x(t)+ p1(t)x(t -τ1)+ p2(t)x(t+τ2))″)′ +q1(t)x(t-τ3)+q2(t)x(t+τ4)=0; t ≥t0. We establish two theorems which guarantee that every solution x of the above equation oscillates or limt→∞x(t) = 0. These results complement some known results obtained in the literature. Some examples are considered to illustrate the main results. [ABSTRACT FROM AUTHOR]

The article features the chaotic behavior of cubic p-Adic Dynamical Systems in Portugal. Such scheme was developed since 1993. A study was conducted by researchers on non-Archimedean dynamics in the neighborhood of a periodic and an almost periodic point over global fields using local fields, thus the Siegel discs of pertubated monomial dynamical systems were formulated. Presented are the details of the issue.

The article investigates some delay integral inequalities on time scales which provide explicit bounds on unknown functions. Researchers have used Gronwall's inequality and comparison theorem. Results of the study unify and extend some delay integral inequalities and their corresponding discrete analogues. It was found that the numerical solution agrees with the analytical solution for some discrete inequalities. The inequalities given can be used as handy tools in the qualitative theory of certain classes of delay dynamic equations on time scales.