an efficient strategy for generating all descendant subtree patterns from phylogenetic trees with its implementation abstract phylogenetic trees are an important tool to help in the understanding of relationships between objects that evolve through time in particular molecular sequences in this paper we efficiently solve two subtree comparison problems on a set of phylogenetic trees which have practical applications to analyze the evolution and co evolution genes clustering of genomic sequences let t 1 t 2 t k be a set of k phylogenetic trees such that the leaves of each tree are drawn from 1 2 n and the leaves for two arbitrary trees are not necessary the same where n is the maximum number of the leaves among the k trees we present a linear time algorithm to find all the leaf agreement descendant subtrees by further extending this result we present a linear time algorithm to find all the leaf agreement isomorphic descendant subtrees based on our algorithms a web based system using input tree files from treebase is also implemented

submatrices of non tree realizable distance matrices abstract we deal with distance matrices of real this means not necessarily integer numbers it is known that a distance matrix d of order n is tree realizable if and only if all its principal submatrices of order 4 are tree realizable we discuss bounds for the number denoted qi d of non tree realizable principal submatrices of order i 4 of a non tree realizable distance matrix d of order n i and we construct some distance matrices which meet extremal conditions on qi d our starting point is a proof that a non tree realizable distance matrix of order 5 has at least two non tree realizable principal submatrices of order 4 optimal realizations by graphs with circuits of distance matrices which are not tree realizable are not yet as well known as optimal realizations which are trees using as starting point the optimal realization of the arbitrary distance matrix of order 4 we investigate optimal realizations of non tree realizable distance matrices with the minimum number of non tree realizable principal submatrices of order 4

interval criteria for oscillation of nonlinear second order dynamic equations on time scales interval oscillation criteria are established for a second order nonlinear dynamic equation on time scales by utilizing a generalized riccati technique and an integral averaging technique the theory can be applied to second order dynamic equations regardless of the choice of delta or nabla derivatives

trees tight extensions of metric spaces and the cohomological dimension of certain groups a note on combinatorial properties of metric spaces abstract the concept of tight extensions of a metric space is introduced the existence of an essentially unique maximal tight extension t x the tight span being an abstract analogon of the convex hull is established for any given metric space x and its properties are studied applications with respect to 1 the existence of embeddings of a metric space into trees 2 optimal graphs realizing a metric space and 3 the cohomological dimension of groups with specific length functions are discussed

hyperconvexity and tight span theory for diversities abstract the tight span or injective envelope is an elegant and useful construction that takes a metric space and returns the smallest hyperconvex space into which it can be embedded the concept has stimulated a large body of theory and has applications to metric classification and data visualisation here we introduce a generalisation of metrics called diversities and demonstrate that the rich theory associated to metric tight spans and hyperconvexity extends to a seemingly richer theory of diversity tight spans and hyperconvexity

oscillation criteria for a class of second order emden fowler delay dynamic equations on time scales abstract by means of riccati transformation technique we establish some new oscillation criteria for the second order emden fowler delay dynamic equations x t p t x t 0 on a time scale t here is a quotient of odd positive integers with p t real valued positive rd continuous functions defined on t to the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales our results in this paper not only extend the results given in r p agarwal m bohner s h saker oscillation of second order delay dynamic equations can appl math q 13 1 2005 1 18 but also unify the oscillation of the second order emden fowler delay differential equation and the second order emden fowler delay difference equation

new kamenev type oscillation criteria for second order nonlinear advanced dynamic equations this paper presents a new kamenev type oscillation criterion for a second order advanced dynamic equation on a time scale two examples are included to show that the result obtained complements and improves those reported in the literature

oscillation for second order nonlinear neutral delay difference equations in this paper we obtain sufficient conditions for the oscillation of all solutions of the second order nonlinear delay difference email protected a n d x n p nx g n a f n x s n 0 where dx n x n 1 x n is the forward difference operator p n is a sequence of nonnegative real numbers and a 1 is a quotient of positive odd integers our results extend and improve the known results in the literature an example that dwell up the advantage of our results is included

homoclinic orbits and periodic solutions for a class of hamiltonian systems on time scales abstract in this paper we are concerned with a second order non autonomous hamiltonian system on time scales t u t v u t u t f t t t under certain conditions the existence and multiplicity of periodic solutions are obtained for this hamiltonian system on time scales by using the saddle point theory the least action principle as well as the three critical point theorem in addition the existence of homoclinic orbit is obtained as a limit of 2 k t periodic solutions of a given sequence of hamiltonian system on time scales by means of the mountain pass theorem and the standard minimizing argument

a hybrid collocation method for a nonlinear volterra integral equation with weakly singular kernel this work is concerned with the numerical solution of a nonlinear weakly singular volterra integral equation owing to the singular behavior of the solution near the origin the global convergence order of product integration and collocation methods is not optimal in order to recover the optimal orders a hybrid collocation method is used which combines a non polynomial approximation on the first subinterval followed by piecewise polynomial collocation on a graded mesh some numerical examples are presented which illustrate the theoretical results and the performance of the method a comparison is made with the standard graded collocation method

on the oscillation of an n th order nonlinear neutral delay differential equation abstract the authors consider the n th order nonlinear neutral delay differential equation y t p t y g t n q t f y h t 0 where n 1 q t 0 and 1 they give sufficient conditions for solutions to either oscillate or converge to zero

hille and nehari type criteria for third order dynamic equations abstract in this paper we extend the oscillation criteria that have been established by hille e hille non oscillation theorems trans amer math soc 64 1948 234 252 and nehari z nehari oscillation criteria for second order linear differential equations trans amer math soc 85 1957 428 445 for second order differential equations to third order dynamic equations on an arbitrary time scale t which is unbounded above our results are essentially new even for third order differential and difference equations i e when t r and t n we consider several examples to illustrate our results

oscillation criteria for second order nonlinear delay dynamic equations on time scales by employing the generalized riccati transformation we establish some new oscillation criteria for second order delay dynamic equations on time scales the obtained results essentially improve the well know oscillation results for half linear dynamic equations such as kamenev type and philos type oscillation criteria some interesting examples are given to illustrate the versatility of our results

the piecewise polynomial collocation method for nonlinear weakly singular volterra equations second kind volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration using an adaptation of the analysis originally developed for nonlinear weakly singular fredholm integral equations we present a complete discussion of the optimal global and local order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear volterra integral equations with algebraic or logarithmic singularities in their kernels

a new approach to the numerical solution of weakly singular volterra integral equations we consider linear weakly singular volterra integral equations of the second kind with kernels of the form k x v x v k x v 0 1 or k x v log x v k x v k x v being a smooth function the solutions of such equations may exhibit a singular behaviour in the neighbourhood of the initial point of the interval of integration by a transformation of the unknown function we obtain an equation which is still weakly singular but whose solution is as smooth as we like this resulting equation is then solved by standard product integration methods

numerical methods for a volterra integral equation with non smooth solutions we consider the numerical treatment of a singular volterra integral equation with an infinite set of solutions one of which is smooth and all others have infinite gradient at the origin this equation has been the subject of previous works where we have dealt with the approximation of the smooth solution here we present numerical methods which enable us to obtain approximations to any of the infinite class of solutions some numerical examples are given which illustrate the performance of the methods employed

finding maximal leaf agreement isomorphic descendent subtrees from phylogenetic trees with different species phylogenetic trees are an important tool to help in the understanding of relationships between objects that evolve through time in particular molecular sequences in this paper we consider two descendent subtree comparison problems on phylogenetic trees given a set of k phylogenetic trees whose leaves are drawn from 1 2 n and the leaves for two arbitrary trees are not necessary the same we first present a linear time algorithm to final all the maximal leaf agreement descendent subtrees based on this result we also present a linear time algorithm to find all the maximal leaf agreement isomorphic descendent subtrees

on finding common subtrees abstract let t and r be two arbitrary ordered trees t r whose nodes are labelled over an alphabet a we devise a simple solution for detecting all the common subtrees in o t time and space if the size of a is finite and o t log min a t time otherwise we solve the problem of finding in t and r all occurrences if any of any given tree b in either o t b or o b t b time this requires to set up a simple data structure in o t time that allows to find all maximal subtrees of b in o b time and to solve other related problems

asymptotic and oscillatory behavior of solutions of first order nonlinear neutral delay differential equations the authors consider the first order nonlinear neutral delay differential equation e y t p t y t q t y t 0 where p q and are continuous q t 0 0 0 and u u 0 if u 0 they give sufficient conditions for all nonoscillatory solutions of e to converge to zero as t two oscillation theorems for equation e are also proved

rigorous convergence analysis of jacobi spectral galerkin methods for volterra integral equations with noncompact kernels abstract this article deals with the implementation of jacobi spectral and pseudospectral galerkin approaches for solving weakly singular volterra integral equations with noncompact kernels the computational domain of the main problem and intervals of the involved integrals are transformed into 1 1 by some appropriate change of variables therefore one can apply the jacobi galerkin method and impose the inner product conditions on the residual of the considered equation also jacobi gauss quadrature rules are implemented to approximate the integral operator and the inner products associated to the jacobi weight function in the pseudospectral galerkin method for both cases of spectral and pseudospectral galerkin schemes rigorous convergence analysis is provided under some mild conditions on the smoothness of kernel and source functions in l and weighted l 2 norms several numerical examples are also given to confirm the theoretical prediction

oscillation of second order damped dynamic equations on time scales abstract the study of dynamic equations on time scales has been created in order to unify the study of differential and difference equations the general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so called time scale which may be an arbitrary closed subset of the reals this way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained in this paper by employing the riccati transformation technique we will establish some oscillation criteria for second order linear and nonlinear dynamic equations with damping terms on a time scale t our results in the special case when t r and t n extend and improve some well known oscillation results for second order linear and nonlinear differential and difference equations and are essentially new on the time scales t h n h 0 t q n for q 1 t n 2 etc some examples are considered to illustrate our main results

oscillation criteria for second order nonlinear neutral variable delay dynamic equations abstract this paper is concerned with oscillation of second order nonlinear neutral dynamic equations on time scales with a variable delay by using the generalized riccati technique and integral averaging technique new oscillation criteria are obtained for all solutions of the equation some results extend known results for difference equations when the time scale is the set z of positive integers and for differential equations when the time scale is r several examples are given to illustrate the results of the paper

oscillation criteria for second order nonlinear neutral delay dynamic equations in this paper we will establish some oscillation criteria for the second order nonlinear neutral delay dynamic equation r t y t p t y t f t y t 0 on a time scale t here 0 is a quotient of odd positive integers with r t and p t real valued positive functions defined on t to the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales so this paper initiates the study

properties of interpolatory product integration rules this paper is concerned with the approximate evaluation of int 1 1 k x f x d x where k is lebesgue integrable and f is at least riemann integrable and preferably smooth the integral is approximated by a rule of the form sum nolimits i 1 n w ni k f x ni where the points x ni are chosen in some prescribed way and the weights w ni k are such that the rule is exact if f is any polynomial of degree n for suitable choices of the points and suitable functions k the rule is shown to converge to the exact integral and the companion rule sum w ni k f x ni to int k x f x dx for all riemann integrable functions f the companion property ensures for example that the weights have the asymptotic positivity property if k is positive error bounds are given that guarantee rapid convergence of the rule if f is smooth finally by establishing a connection with mean convergence of lagrangian interpolation some negative theorems are proved in pa

oscillation criteria for second order matrix dynamic equations on a time scale we obtain oscillation criteria for a second order self adjoint matrix differential equation on a measure chain in terms of the eigenvalues of the coefficient matrices and the graininess function we illustrate our results with some nontrivial examples

high order methods for weakly singular integral equations with nonsmooth input functions to solve one dimensional linear weakly singular integral equations on bounded intervals with input functions which may be smooth or not we propose to introduce first a simple smoothing change of variable and then to apply classical numerical methods such as product integration and collocation based on global polynomial approximations the advantage of this approach is that the order of the methods can be arbitrarily high and that the associated linear systems one has to solve are very well conditioned

product integration for volterra integral equations of the second kind with weakly singular kernels we introduce a new numerical approach for solving volterra integral equations of the second kind when the kernel contains a mild singularity we give a convergence result we also present numerical examples which show the performance and efficiency of our method

oscillatory behavior of second order half linear damped dynamic equations by refining the generalized riccati transformation technique we obtain new oscillation criteria for a class of second order half linear delay dynamic equations with damping on a time scale assumptions in our theorems are less restrictive these results complement and improve related contributions to the subject in particular the results obtained amend related results reported in the literature

asymptotic behavior of solutions of third order nonlinear dynamic equations on time scales in this paper we will study asymptotic behavior of solutions to third order nonlinear dynamic equations on time scales of the form 1a 2 t 1a 1 t x d t a 1 d a 2 d q t f x t 0 by using the riccati technique and integral averaging technique two different types of criteria are established one of which extends some existing results and the other is new two examples of dynamic equations on different time scales are given to show the applications of the obtained results

new oscillatory criteria for higher order nonlinear neutral delay differential equation abstract in this paper we consider the higher order nonlinear neutral delay differential equation 1 1 x t p t x t n f t x t 0 where n is an odd number and n 3 0 p c t 0 r t t lim t t f c t 0 r r f t u is nondecreasing in u

the exact solution of a class of volterra integral equation with weakly singular kernel abstract in this paper the weakly singular volterra integral equations with an infinite set of solutions are investigated among the set of solutions only one particular solution is smooth and all others are singular at the origin the numerical solutions of this class of equations have been a difficult topic to analyze and have received much previous investigation the aim of this paper is to present a numerical technique for giving the approximate solution to the only smooth solution based on reproducing kernel theory applying weighted integral we provide a new definition for reproducing kernel space and obtain reproducing kernel function using the good properties of reproducing kernel function the only smooth solution is exactly expressed in the form of series the n term approximate solution is obtained by truncating the series meanwhile we prove that the derivative of approximation converges to the derivative of exact solution uniformly the final numerical examples compared with other methods show that the method is efficient

numerical analysis of a singular integral equation in this paper we give details of the numerical analysis of an integral equation of the form 1 y t f t email protected 0ts m 1t my s email protected 0 the distinctive feature of the equation is the presence of a singularity at t 0 for all values of m 0 and at s 0 for all values of t 0 for 0 m 1 this means that conventional analytical and numerical theory does not apply in fact for 0 m 1 the equation has an infinite family of solutions as background we give details of the analytical results on existence and uniqueness of solutions to 1 and we give new results on the use of numerical schemes that will yield approximations to any specific solution we conclude with some numerical results that show that our methods enable us to find approximations of any chosen order to any of the infinite class of exact solutions to 1

a few logs suffice to build almost all trees part ii abstract inferring evolutionary trees is an interesting and important problem in biology but one that is computationally difficult as most associated optimization problems are np hard although many methods are provably statistically consistent i e the probability of recovering the correct tree converges to 1 as the sequence length increases the actual rate of convergence for different methods has not been well understood in a recent paper we introduced a new method for reconstructing evolutionary trees called the dyadic closure method dcm and we showed that dcm has a very fast convergence rate dcm runs in o n 5 log n time where n is the number of sequences and so although polynomial the computational requirements are potentially too large to be of use in practice in this paper we present another tree reconstruction method the witness antiwitness method wam wam is faster than dcm especially on random trees and converges to the true tree topology at the same rate as dcm we also compare wam to other methods used to reconstruct trees including neighbor joining possibly the most popular method among molecular biologists and new methods introduced in the computer science literature

bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations a new operational matrix for integration of bernoulli polynomials is introduced by using this new operational matrix of integration and the so called collocation method linear volterra and nonlinear volterra fredholm hammerstein integral equations are reduced to systems of algebraic equations with unknown bernoulli coefficients some error estimations are provided and illustrative examples are also included to demonstrate the efficiency and applicability of the technique a new operational matrix for integration of bernoulli polynomials is introduced a collocation method is used to reduce equations to systems of algebraic equations some error bounds and convergence results are provided for the proposed method the efficiency and applicability of the technique are checked by some examples

oscillation criteria for second order quasilinear neutral delay differential equations on time scales in this paper we consider the second order quasilinear neutral dynamic equation r t z d t a 1z d t d q t x d t b 1x d t 0 on a time scale t where z t x t p t x t t a b 0 are constants and obtain oscillation criteria for the equation when b a email protected and b a respectively our results extend some known results when p t 0 in the literature and contain the continuous and discrete cases as special cases

a canonical decomposition theory for metrics on a finite set abstract we consider specific additive decompositions d d 1 d n of metrics defined on a finite set x where a metric may give distance zero to pairs of distinct points the simplest building stones are the slit metrics associated to splits i e bipartitions of the given set x while an additive decomposition of a hamming metric into split metrics is in no way unique we achieve uniqueness by restricting ourselves to coherent decompositions that is decompositions d d 1 d n such that for every map f x r with f x f y d x y for all x y x there exist maps f 1 f n x r with f f 1 f n and f i x f i y d i x y for all i 1 n and all x y x these coherent decompositions are closely related to a geometric decomposition of the injective hull of the given metric a metric with a coherent decomposition into a weighted sum of split metrics will be called totally split decomposable tree metrics and more generally the sum of two tree metrics are particular instances of totally split decomposable metrics our main result confirms that every metric admits a coherent decomposition into a totally split decomposable metric and a split prime residue where all the split summands and hence the decomposition can be determined in polynomial time and that a family of splits can occur this way if and only if it does not induce on any four point subset all three splits with block size two

oscillation by impulses for a second order delay differential equation we consider a certain second order nonlinear delay differential equation and prove that the all solutions oscillate when proper impulse controls are imposed an example is given

oscillation criteria of second order half linear dynamic equations on time scales in this paper by using the riccati transformation technique chain rule and inequalitya l lab l 1 l 1 b l 0 l 1 where a and b are positive constants we will establish some oscillation criteria for the second order half linear dynamic equation p t x d t c d q t x c t 0 t a b on time scales where c 1 is an odd positive integer our results not only unify the oscillation of half linear differential and half linear difference equations but can be applied on different types of time scales and improve some well known results in the difference equation case some examples are considered here to illustrate our main results

meshless methods for one dimensional oscillatory fredholm integral equations abstract in this paper efficient and simple algorithms based on levin s quadrature theory and our earlier work involving local radial basis function rbf and chebyshev differentiation matrices are adopted for numerical solution of one dimensional highly oscillatory fredholm integral equations this work is focused on the comparative performance of local rbf meshless and pseudospectral procedures we have tested the proposed methods on phase functions with and without stationary phase point s both on uniform and chebyshev grid points the proposed procedures are shown accurate and efficient and therefore provide a reliable platform for the numerical solution of integral equations from the numerical results we draw some conclusions about accuracy efficiency and robustness of the proposed approaches

new oscillation criteria for second order nonlinear neutral delay difference equations in this paper we will establish some new oscillation criteria for the second order nonlinear neutral delay difference equation d a n dx n p nx n t c f n x n s 0 n 0 1 2 where c 0 is a quotient of odd positive integers

convergence analysis of spectral collocation methods for a class of weakly singular volterra integral equations abstract in this paper we discuss the application of spectral jacobi collocation methods to a certain class of weakly singular volterra integral equations first we use some function transformations and variable changes to transform the equation into a volterra integral equation defined on the standard interval 1 1 then the jacobi gauss quadrature formula is used to approximate the integral operator for the spectral jacobi collocation method a rigorous error analysis in both the l and weighted l 2 norms is given under the assumption that both the kernel function and the source function are sufficiently smooth finally some numerical examples are provided to illustrate the theoretical results

some pachpatte type inequalities on time scales in this paper using the comparison theorem we investigate some pachpatte type integral inequalities on time scales which provide explicit bounds on unknown functions our results extend some known dynamic inequalities on time scales unify and extend some continuous inequalities and their corresponding discrete analogues some applications of the main results are given in the end of this paper

an explicit computation of the injective hull of certain finite metric spaces in terms of their associated buneman complex an explicit computation of the injective hull of certain finite metric spaces in terms of their associated buneman complex

meshless methods for two dimensional oscillatory fredholm integral equations abstract in this paper a meshless solution procedure incorporating delaminating quadrature method for two dimensional highly oscillatory fredholm integral equation is put forward the proposed method is an extension of our earlier findings of meshless methods for two dimensional oscillatory fredholm integral equation having kernel function free of stationary phase point s siraj ul islam et al 2015 the current method deals not only with the kernels having no stationary phase point s but also with the kernels having stationary phase point s in the context of highly oscillatory integral equations the new method is numerically stable and computationally fast numerical experiments are presented to testify the robustness and efficiency of the proposed method

oscillation of nonlinear dynamic equations on time scales by means of riccati transformation techniques we establish some new oscillation criteria for the second order nonlinear dynamic equation p t x d t d q t f x s t 0 email protected a b on time scales in terms of the coefficients and the graininess function we illustrate our results with a nontrivial example

analysis of general quadrature methods for integral equations of the second kind this paper is concerned with a class of approximation methods for integral equations of the form y t f t int limits a b k t s y s ds wherea andb are finite f andy are continuous and the kernelk may be weakly singular the methods are characterized by approximate equations of the form y n t f t sum limits i 1 n w ni t y n s ni such methods include the nystrom method and a variety of product integration methods a general convergence theory is developed for methods of this type in suitable cases it has the feature that its application to a specific method depends only on a knowledge of convergence properties of the underlying quadrature rule the theory is used to deduce convergence results some of them new for a number of specific methods

oscillation of second order nonlinear neutral dynamic equations on time scales abstract this paper is concerned with the oscillation of second order nonlinear neutral dynamic equations of the form r t y t p t y t f t y t 0 on a time scale t by using a generalized riccati technique and integral averaging techniques we establish new oscillation results which handle some cases not covered by known criteria

numerical solutions of weakly singular volterra integral equations using the optimal homotopy asymptotic method the purpose of this paper is to apply a numerical technique namely the optimal homotopy asymptotic method oham for finding the approximate solutions of a class of volterra integral equations with weakly singular kernels this method uses simple computations with quite acceptable approximate solutions which has close agreement with exact solutions illustrative examples are included to demonstrate the validity and applicability of the present method and a comparison has been made with existing results

dynamic equations on time scales a survey the study of dynamic equations on time scales which goes back to its founder stefan hilger 1988 is an area of mathematics that has recently received a lot of attention it has been created in order to unify the study of differential and difference equations in this paper we give an introduction to the time scales calculus we also present various properties of the exponential function on an arbitrary time scale and use it to solve linear dynamic equations of first order several examples and applications among them an insect population model are considered we then use the exponential function to define hyperbolic and trigonometric functions and use those to solve linear dynamic equations of second order with constant coefficients finally we consider self adjoint equations and more generally so called symplectic systems and present several results on the positivity of quadratic functionals

a necessary and sufficient condition for oscillation of the sturm liouville dynamic equation on time scales in this paper we investigate oscillatory properties of the second order sturm liouville dynamic equation on a time scale r t x c t x 0 using the so called trigonometric transformation we establish a necessary and sufficient condition for oscillation of this condition is then used to derive an explicit oscillation criterion for this equation

numerical solutions of volterra integral equation with weakly singular kernel using scw method numerical methods for weakly singular volterra integral equations are rarely considered in the literature the solutions of such equations may exhibit a singular behaviour in the neighbourhood of the initial point of the interval of integration and bring some difficulties in numerical computation in this paper we present a numerical solution of weakly singular volterra integral equations including the abels equations by the second chebyshev wavelet scw we give the scw operational matrix of fractional integration and combine with the block pulse functions bpfs to derive the procedure of solving this kind integral equations the proposed method is illustrated with numerical examples the results reveal that the method is accurate and easy to implement

a oscillation theorem for a class of even order neutral differential equations abstract the even order neutral differential equation 1 d n dt n x t h t x t f t x g t 0 is considered under the following conditions n 2 is even 0 h c r g c t 0 lim t g t f c t 0 r uf t u 0 for t u t 0 r and f t u is nondecreasing in u r for each fixed t t 0 it is shown that 1 is oscillatory if and only if the certain non neutral differential equation is oscillatory for the case where 0 h t h t

new special cases of the quadratic assignment problem with diagonally structured coefficient matrices abstract we consider new polynomially solvable cases of the well known quadratic assignment problem involving coefficient matrices with a special diagonal structure by combining the new special cases with polynomially solvable special cases known in the literature we obtain a new and larger class of polynomially solvable special cases of the qap where one of the two coefficient matrices involved is a robinson matrix with an additional structural property this matrix can be represented as a conic combination of cut matrices in a certain normal form the other matrix is a conic combination of a monotone anti monge matrix and a down benevolent toeplitz matrix we consider the recognition problem for the special class of robinson matrices mentioned above and show that it can be solved in polynomial time

on a class of metrics related to graph layout problems we examine the metrics that arise when a finite set of points is embedded in the real line in such a way that the distance between each pair of points is at least 1 these metrics are closely related to some other known metrics in the literature and also to a class of combinatorial optimization problems known as graph layout problems we prove several results about the structure of these metrics in particular it is shown that their convex hull is not closed in general we then show that certain linear inequalities define facets of the closure of the convex hull finally we characterize the unbounded edges of the convex hull and of its closure

inferring evolutionary trees with strong combinatorial evidence we consider the problem of inferring the evolutionary tree of a set of n species we propose a quartet reconstruction method which specifically produces trees whose edges have strong combinatorial evidence let q be a set of resolved quartets defined on the studied species the method computes the unique maximum subset q of q which is equivalent to a tree and outputs the corresponding tree as an estimate of the species phylogeny we use a characterization of the subset q due to bandelt and dress adv appl math 7 1986 309 343 to provide an o n4 incremental algorithm for this variant of the np hard quartet consistency problem moreover when chosing the resolution of the quartets by the four point method fpm and considering the cavender farris model of evolution we show that the convergence rate of the q method is at worst polynomial when the maximum evolutive distance between two species is bounded we complete these theoretical results by an experimental study on real and simulated data sets the results show that i as expected the strong combinatorial constraints it imposes on each edge leads the q method to propose very few incorrect edges ii more surprisingly the method infers trees with a relatively high degree of resolution

the asymptotic and oscillatory behavior of the solutions of higher order neutral equations abstract consider the higher order neutral differential equation y t i 1 n1 p i y t i j 1 n2 r j y t j n s 1 n3 q s y t 0 we study the asymptotic behavior of solutions of eq 1 and we obtain several sufficient conditions for the oscillation of solutions of eq 1 this paper generalizes all results of ladas and sficas 1

asymptotic and oscillatory behavior of solutions of nonlinear neutral delay equations of arbitrary order abstract the authors obtain results on the oscillatory and asymptotic behavior of solutions of the nonlinear neutral delay differential equation y t p t y g t n f t y h t 0 where n 1 1 and uf t u 0 for u 0 several known results of this type are included as special cases examples illustrating their results are included

linearized oscillations for even order neutral differential equations abstract we are concerned with the oscillation of all bounded solutions of some even order linear neutral delay differential equations we establish a comparison theorem and a linearized oscillation result and we prove the existence of a bounded positive solution

infinitely many nonoscillatory solutions for second order nonlinear neutral delay differential equations abstract in this paper we consider the second order nonlinear neutral delay differential equation a t x t b t x t h t x h 1 t x h 2 t x h k t f t x f 1 t x f 2 t x f k t g t t t 0 where 0 a b g c t 0 r with a t 0 for t t 0 h c 1 t 0 r k r f c t 0 r k r h l c 1 t 0 r and f l c t 0 r with lim t h l t lim t f l t l 1 k under suitable conditions by making use of the banach fixed point theorem we show the existence of infinitely many nonoscillatory solutions which are uncountable for the above equation suggest several mann type iterative approximation sequences with errors for these nonoscillatory solutions and establish some error estimates between the approximate solutions and the nonoscillatory solutions five nontrivial examples are given to illustrate the advantages of our results

convergence properties of a class of product formulas for weakly singular integral equations we examine the convergence of product quadrature formulas of interpolatory type based on the zeros of certain generalized jacobi polynomials for the discretization of integrals of the type f k x y f x dx l 1 and f x has algebraic singularities of the form 1 x a a 1 we prove that the uniform rate of convergence of the rules is o m log2 m in the case of the first kernel and o m 2 a2v logim if v 0 for the second where m is the number of points in the quadrature rule

on a disconjugacy criterion for second order dynamic equations on time scales in this paper a sufficient condition for disconjugacy of second order differential equations on time scales is given for this purpose a mean value result on time scales is employed

some oscillation results for second order nonlinear dynamic equations of neutral type abstract in this paper some oscillation results for the second order nonlinear neutral delay dynamic equation r t y t p t y t q t y t sgn y t 0 are obtained on a time scale t under the assumption t 0 1 r t 1 t where 0 is a quotient of odd positive integers

oscillation of second order nonlinear neutral dynamic equations on time scales with distributed deviating arguments this article is concerned with oscillation of second order neutral dynamic equations with distributed deviating arguments of the form r t y t p t y t t d c email protected c df t y q t x email protected 0 where c 0 is a ratio of odd positive integers with r t and p t real valued rd continuous positive functions defined on t we establish some new oscillation criteria and give sufficient conditions to insure that all solutions of nonlinear neutral dynamic equation are oscillatory on a time scale t

numerical solution of a nonuniquely solvable volterra integral equation using extrapolation methods in this work the numerical solution of a volterra integral equation with a certain weakly singular kernel depending on a real parameter is considered although for certain values of this equation possesses an infinite set of solutions we have been able to prove that euler s method converges to a particular solution it is also shown that the error allows an asymptotic expansion in fractional powers of the stepsize so that general extrapolation algorithms like the e algorithm can be applied to improve the numerical results this is illustrated by means of some examples

interval oscillation criteria for second order forced delay dynamic equations with mixed nonlinearities interval oscillation criteria are established for second order forced delay dynamic equations on time scales containing mixed nonlinearities of the form r t f a x d t d p 0 t f a x t 0 t i 1np i t f b i x t i t e t t t 0 t where t is a time scale t 0 t a fixed number t 0 t is a time scale interval f u u 1u the functions r p i e t 0 t r are right dense continuous with r 0 nondecreasing t k t t are nondecreasing right dense continuous with t k t t lim t t k t and the exponents satisfy b 1 b m a b m 1 b n 0 all results are new even for t r and t z analogous results for related advance type equations are also given as well as extended delay and advance equations the theory can be applied to second order dynamic equations regardless of the choice of delta or nabla derivatives two examples are provided to illustrate one of the theorems

oscillation of second order nonlinear neutral delay dynamic equations on time scales in this paper some sufficient conditions for oscillation of the second order nonlinear neutral delay dynamic equation r t y t p t y t f t y t 0 on a time scale t are established here 1 is an odd positive integer with r t and p t are rd continuous functions defined on t our results as a special case when t r and t n involve and improve some well known oscillation results for second order neutral delay differential and difference equations when t hn and t qn t t qk k n q 1 i e for generalized neutral delay difference and q neutral delay difference equations our results are essentially new and also can be applied on different types of time scales e g t n2 t2 t n and t tn tn n n0 where tn is the set of harmonic numbers some examples illustrating our main results are given

a nystrom interpolant for some weakly singular nonlinear volterra integral equations we consider a second kind weakly singular nonlinear volterra hammerstein integral equation defined by a compact operator and derive a nystrom type interpolant of the solution based on gauss radau nodes we prove the convergence of the interpolant and derive convergence estimates for equations with nonlinearity of algebraic kind we improve the rate of convergence by using a smoothing transformation some numerical examples are given

oscillation criteria for second order nonlinear delay dynamic equations abstract in this paper we consider the second order nonlinear delay dynamic equation r t x t p t f x t 0 on a time scale t by employing the generalized riccati technique we will establish some new sufficient conditions which ensure that every solution oscillates or converges to zero the obtained results improve the well known oscillation results for dynamic equations and include as special cases the oscillation results for differential equations some applications and examples are considered to illustrate the main results

some new dynamic inequalities on time scales in this paper based on some known dynamic inequalities we investigate certain new dynamic inequalities on time scales which provide explicit bounds on unknown functions our results unify and extend some continuous inequalities and their corresponding discrete analogues

fractional linear multistep methods for abel volterra integral equations of the second kind fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular volterra integral equations the proposed methods are convergent of the order of the underlying multistep method also in the generic case of solutions which are not smooth at the origin the stability properties stability region a stability a a stability are closely related to those of the underlying linear multistep method