Observations on the tanh-coth expansion method for finding solutions to nonlinear evolution equations

The ‘tanh-coth expansion method’ for finding solitary travelling-wave solutions to nonlinear evolution equations has been used extensively in the literature. It is a natural extension to the basic tanh-function expansion method which was developed in the 1990s. It usually delivers three types of solution, namely a tanh-function expansion, a coth-function expansion, and a tanh-coth expansion. It is known that, for every tanh-function expansion solution, there is a corresponding coth-function expansion solution. It is shown that there is a tanh-coth expansion solution that is merely a disguised version of the coth solution. In many papers, such tanh-coth solutions are erroneously claimed to be ‘new’. However, other tanh-coth solutions may be delivered that are genuinely new in the sense that they would not be delivered via the basic tanh-function method. Similar remarks apply to tan, cot and tan-cot expansion solutions.     

Existence and uniqueness of v-asymptotic expansions and Colombeau's generalized numbers

We define a type of generalized asymptotic series called v-asymptotic. We show that every function with moderate growth at infinity has a v-asymptotic expansion. We also describe the set of v-asymptotic series, where a given function with moderate growth has a unique v-asymptotic expansion. As an application to random matrix theory we calculate the coefficients and establish the uniqueness of the v-asymptotic expansion of an integral with a large parameter. As another application (with significance in the non-linear theory of generalized functions) we show that every Colombeau's generalized number has a v-asymptotic expansion. A similar result follows for Colombeau's generalized functions, in particular, for all Schwartz distributions.     

Multiple solutions for the laminar flow in a porous pipe with suction at slowly expanding or contracting wall

In this paper, the asymptotic solution for the similarity equation of the laminar flow in a porous pipe with suction at expanding and contracting wall has been obtained using the singular perturbation method. However, this solution neglects exponentially small terms in the matching process. To take into account these exponentially small terms, a method involving the inclusion of exponentially small terms in a perturbation series was used to find the two solutions analytically. The series involving the exponentially small terms and expansion ratio predicts dual solutions. Furthermore, the result indicates that the expansion ratio has much important influence on the solutions. When the expansion ratio is zero, it is a special case that Terrill has discussed.     

Note on the coefficients of Ramanujan’s expansion for the logarithm of the factorial

Similar to Ramanujan’s expansion for the nth harmonic number, Villarino suggested that there might exist a series expansion for the logarithm of the factorial in terms of the reciprocal of a triangular number. This has been proved in 2010 by Nemes, who gave a complete asymptotic expansion with explicit coefficients and error terms. In this short note, we provide a recursive formula for successively determining the coefficients of the asymptotic expansion by using combinatorial technique.     

An asymptotic expansion for product integration applied to Cauchy principal value integrals

Product integration methods for Cauchy principal value integrals based on piecewise Lagrangian interpolation are studied. It is shown that for this class of quadrature methods the truncation error has an asymptotic expansion in integer powers of the step-size, and that a method with an asymptotic expansion in even powers of the step-size does not exist. The relative merits of a quadrature method which employs values of both the integrand and its first derivative and for which the truncation error has an asymptotic expansion in even powers of the step-size are discussed.     

Expanding topological space,study and applications

In this paper, we introduce the notion of expanding topological space. We define the topological expansion of a topological space via local multi-homeomorphism over coproduct topology, and we prove that the coproduct family associated to any fractal family of topological spaces is expanding. In particular, we prove that the more a topological space expands, the finer the topology of its indexed states is. Using multi-homeomorphisms over associated coproduct topological spaces, we define a locally expandable topological space and we prove that a locally expandable topological space has a topological expansion. Specifically, we prove that the fractal manifold is locally expandable and has a topological expansion.     

The Behaviors of Expansion Functor on Monomial Ideals and Toric Rings

In this article, we study some algebraic and combinatorial behaviors of expansion functor. We show that on monomial ideals some properties like polymatroidalness, weakly polymatroidalness, and having linear quotients are preserved under taking the expansion functor.

The main part of the article is devoted to study of toric ideals associated to the expansion of subsets of monomials which are minimal with respect to divisibility. It is shown that, for a given discrete polymatroid P, if toric ideal of P is generated by double swaps, then toric ideal of any expansion of P has such a property. This result, in a special case, says that White's conjecture is preserved under taking the expansion functor. Finally, the construction of Gröbner bases and some homological properties of toric ideals associated to expansions of subsets of monomials is investigated.     

Finite sections of toeplitz-like matrices

Determinants of large finite sections of Toeplitz and Toeplitz-like matrices are evaluated by an expansion in which the deviation from the identity is parametrically increased. Classical results are reproduced and the inverse matrix expanded as well. The expansion is asymptotically valid only for low-order terms, and so a reordered expansion is introduced. It has the desired asymptotic character and suggests that sharp bounds are also available.     

On the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations

An analytical formula expressing the ultraspherical coefficients of an expansion for an infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is stated in a more compact form and proved in a simpler way than the formula suggested by Phillips and Karageorghis (27 (1990) 823). A new formula expressing explicitly the integrals of ultraspherical polynomials of any degree that has been integrated an arbitrary number of times of ultraspherical polynomials is given. The tensor product of ultraspherical polynomials is used to approximate a function of more than one variable. Formulae expressing the coefficients of differentiated expansions of double and triple ultraspherical polynomials in terms of the original expansion are stated and proved. Some applications of how to use ultraspherical polynomials for solving ordinary and partial differential equations are described.     

Magnus’ expansion for time-periodic systems: Parameter-dependent approximations

Magnus’ expansion solves the nonlinear Hausdorff equation associated with a linear time-varying system of ordinary differential equations by forming the matrix exponential of a series of integrated commutators of the matrix-valued coefficient. Instead of expanding the fundamental solution itself, that is, the logarithm is expanded. Within some finite interval in the time variable, such an expansion converges faster than direct methods like Picard iteration and it preserves symmetries of the ODE system, if present. For time-periodic systems, Magnus expansion, in some cases, allows one to symbolically approximate the logarithm of the Floquet transition matrix (monodromy matrix) in terms of parameters. Although it has been successfully used as a numerical tool, this use of the Magnus expansion is new. Here we use a version of Magnus’ expansion due to Iserles [Iserles A. Expansions that grow on trees. Not Am Math Soc 2002;49:430–40], who reordered the terms of Magnus’ expansion for more efficient computation. Though much about the convergence of the Magnus expansion is not known, we explore the convergence of the expansion and apply known convergence estimates. We discuss the possible benefits to using it for time-periodic systems, and we demonstrate the expansion on several examples of periodic systems through the use of a computer algebra system, showing how the convergence depends on parameters.     

Asymptotic expansion for log <Emphasis Type="Italic">n</Emphasis>! in terms of the reciprocal of a triangular number

Ramanujan suggested an expansion for the nth partial sum of the harmonic series which employs the reciprocal of the nth triangular number. This has been proved in 2006 by Villarino, who speculated that there might also exist a similar expansion for the logarithm of the factorial. This study shows that such an asymptotic expansion indeed exists and provides formulas for its generic coefficient and for the bounds on its errors.     

On a singularly perturbed initial value problem in the case of a double root of the degenerate equation

We study the initial value problem of a singularly perturbed first order ordinary differential equation in case that the degenerate equation has a double root. We construct the formal asymptotic expansion of the solution such that the boundary layer functions decay exponentially. This requires a modification of the standard procedure. The asymptotic solution will be used to construct lower and upper solutions guaranteeing the existence of a unique solution and justifying its asymptotic expansion.     

Laminar flow separation in an axi-symmetric sudden smooth expanded circular tube

This paper concerns with the investigation of laminar flow separation and its consequences in a tube over a smooth expansion under the axi-symmetric approximations. A co-ordinate stretching has been made to map the expanded tube into a straight tube. The two-dimensional unsteady Navier-Stokes equations are solved approximately by using primitive variables in staggered grid. A thorough quantitative analysis is performed through numerical simulations of the desired quantities such as wall shear stress, axial velocity, pressure distribution etc. These quantities are presented graphically and their consequences in the flow field are analysed in details. The dependence of the flow field on the physical parameter like expansion height d and on the Reynolds number has been investigated in details. It is interesting to note that the peak value of wall shear stress decreases with increasing height of expansion and also with the increasing Reynolds number.     

On ANOVA expansions and strategies for choosing the anchor point

The classic Lebesgue ANOVA expansion offers an elegant way to represent functions that depend on a high-dimensional set of parameters and it often enables a substantial reduction in the evaluation cost of such functions once the ANOVA representation is constructed. Unfortunately, the construction of the expansion itself is expensive due to the need to evaluate high-dimensional integrals. A way around this is to consider an alternative formulation, known as the anchored ANOVA expansion. This formulation requires no integrals but has an accuracy that depends sensitively on the choice of a special parameter, known as the anchor point.We present a comparative study of several strategies for the choice of this anchor point and argue that the optimal choice of this anchor point is the center point of a sparse grid quadrature. This choice induces no additional cost and, as we shall show, results in a natural truncation of the ANOVA expansion. The efficiency and accuracy is illustrated through several standard benchmarks and this choice is shown to outperform the alternatives over a range of applications.     

The unified Taylor-Ito expansion

We consider the problem of the Taylor-Ito expansion for Ito processes in a neighborhood of a fixed time moment. The Taylor-Ito expansion known in literature is unified by a canonical system of repeated stochastic Ito integrals with polynomial weight functions. The unified expansion has some computational advantages, such as recurrent relations between the expansion coefficients, ordering of the expansion with respect to smallness of its terms, and a smaller number of applied repeated stochastic integrals of different types. The unified expansion is more convenient in constructing algorithms of numerical solution for stochastic Ito differential equations. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 186–204. Translated by S. Yu. Pilyugin.     

Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions

 This paper describes the cutting sequences of geodesic flow on the modular surface with respect to the standard fundamental domain of . The cutting sequence for a vertical geodesic is related to a one-dimensional continued fraction expansion for θ, called the one-dimensional Minkowski geodesic continued fraction (MGCF) expansion, which is associated to a parametrized family of reduced bases of a family of 2-dimensional lattices. The set of cutting sequences for all geodesics forms a two-sided shift in a symbol space which has the same set of forbidden blocks as for vertical geodesics. We show that this shift is not a sofic shift, and that it characterizes the fundamental domain ℱ up to an isometry of the hyperbolic plane . We give conversion methods between the cutting sequence for the vertical geodesic , the MGCF expansion of θ and the additive ordinary continued fraction (ACF) expansion of θ. We show that the cutting sequence and MGCF expansions can each be computed from the other by a finite automaton, and the ACF expansion of θ can be computed from the cutting sequence for the vertical geodesic θ + it by a finite automaton. However, the cutting sequence for a vertical geodesic cannot be computed from the ACF expansion by any finite automaton, but there is an algorithm to compute its first symbols when given as input the first symbols of the ACF expansion, which takes time and space . (Received 11 August 2000; in revised form 18 April 2001)     

Associativity in multiary quasigroups: the way of biased expansions

A multiary (polyadic, n-ary) quasigroup is an n-ary operation which is invertible with respect to each of its variables. A biased expansion of a graph is a kind of branched covering graph with an additional structure similar to the combinatorial homotopy of circles. A biased expansion of a circle with chords encodes a multiary quasigroup, the chords corresponding to factorizations, i.e., associative structure. Some but not all biased expansions are constructed from groups (group expansions); these include all biased expansions of complete graphs (with at least four nodes), which correspond to Dowling’s lattices of a group and encode an iterated group operation. We show that any biased expansion of a 3-connected graph (with at least four nodes) is a group expansion, and that all 2-connected biased expansions are constructed by the identification of edges from group expansions and irreducible multiary quasigroups. If a 2-connected biased expansion covers every base edge at most three times, or if every four-node minor that contains a fixed edge is a group expansion, then the whole biased expansion is a group expansion. We deduce that if a multiary quasigroup has a factorization graph that is 3-connected, or if every ternary principal retract is an iterated group isotope, it is isotopic to an iterated group. We mention applications of generalizing Dowling geometries and of transversal designs of high strength.     

Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions

 This paper describes the cutting sequences of geodesic flow on the modular surface with respect to the standard fundamental domain of . The cutting sequence for a vertical geodesic is related to a one-dimensional continued fraction expansion for θ, called the one-dimensional Minkowski geodesic continued fraction (MGCF) expansion, which is associated to a parametrized family of reduced bases of a family of 2-dimensional lattices. The set of cutting sequences for all geodesics forms a two-sided shift in a symbol space which has the same set of forbidden blocks as for vertical geodesics. We show that this shift is not a sofic shift, and that it characterizes the fundamental domain ℱ up to an isometry of the hyperbolic plane . We give conversion methods between the cutting sequence for the vertical geodesic , the MGCF expansion of θ and the additive ordinary continued fraction (ACF) expansion of θ. We show that the cutting sequence and MGCF expansions can each be computed from the other by a finite automaton, and the ACF expansion of θ can be computed from the cutting sequence for the vertical geodesic θ + it by a finite automaton. However, the cutting sequence for a vertical geodesic cannot be computed from the ACF expansion by any finite automaton, but there is an algorithm to compute its first symbols when given as input the first symbols of the ACF expansion, which takes time and space .     

Asymptotic expansions of solutions to singularly perturbed systems

Under the condition that a degenerate system has an exponentially stable integral manifold, an asymptotic expansion of the Cauchy problem that generalizes the well known Vasil'eva expansion is constructed for a perturbed system.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 4, pp. 552–560, April, 1993.     

Quasirandom Rumor Spreading on Expanders

Randomized rumor spreading is an efficient way to distribute information in networks. Recently, a quasirandom version of this protocol has been proposed. It was proven that it works equally well or even better in many settings.In this work, we exhibit a natural expansion property for networks, which ensures that quasirandom rumor spreading informs all nodes of the network in logarithmic time with high probability. This expansion property is satisfied, among others, by many expander graphs, random regular graphs, and Erdős-Rényi random graphs.