he s variational iteration method for treating nonlinear singular boundary value problems this paper applies he s variational iteration method for solving nonlinear singular boundary value problems the solution process is illustrated and various physically relevant results are obtained comparison of the obtained results with exact solutions shows that the method used is an effective and highly promising method for treating various classes of both linear and nonlinear singular boundary value problems

lehmer s problem for polynomials with odd coefficients we prove that if f x ylkzo akxk is a polynomial with no cyclotomic factors whose coefficients satisfy ak 1 mod 2 for 0 2 we also characterize the polynomials that appear as the noncyclotomic part of a polynomial whose coefficients satisfy ak 1 mod p for each ft for a fixed prime p last we prove that the smallest pisot number whose minimal polynomial has odd coefficients is a limit point from both sides of salem 19 numbers whose minimal polynomials have coef ficients in 1 1

small zeros of quadratic forms over number fields let f be a nontrivial quadratic form in n variables with coefficients in a number field k and let a be a k x n matrix over k we show that if the simultaneous equations f x 0 and ax 0 hold on a subspace x of dimension l and l is maximal then such a subspace x can be found with the height of x relatively small in particular the height of x can be explicitly bounded by an expression depending on the height of f and the height of a we use methods from geometry of numbers over adele spaces and local to global techniques which generalize recent work of h p schlickewei

small solutions to inhomogeneous linear equations over number fields we consider a system of m independent inhomogeneous linear equations in n m variables having coefficients in an algebraic number field k we give a best possible lower bound on the inhomogeneous height of a solution vector in k n and determine when a solution exists in o s n where o s is the ring of s integers in k if such a system has a solution vector in o s n we show that it has a solution in o s n such that the inhomogeneous height of is relatively small we give an explicit upper bound for this height in terms of the heights of the matrices defining the linear system our method uses geometry of numbers over adele spaces and local to global arguments

a behavioral approach to linear exact modeling the behavioral approach to system theory provides a parameter free framework for the study of the general problem of linear exact modeling and recursive modeling the authors present the solution of the continuous time polynomial exponential time series modeling problem both recursive and nonrecursive solutions are provided and classified according to properties like complexity and controllability it is shown in particular that recursive modeling corresponds to updating by means of a cascade inter connection of systems as a special case the solution of several other problems such as rational interpolation realization and modeling of arbitrary discrete time time series is obtained

small zeros of quadratic forms outside a union of varieties let f be a quadratic form in n 2 variables defined on a vector space vk n over a global field k and zk n be a finite union of varieties defined by families of homogeneous polynomials over k we show that if v z contains a nontrivial zero of f then there exists a linearly independent collection of small height zeros of f in v z where the height bound does not depend on the height of z only on the degrees of its defining polynomials as a corollary of this result we show that there exists a small height maximal totally isotropic subspace w of the quadratic space v f such that w is not contained in z our investigation extends previous results on small zeros of quadratic forms including cassels theorem and its various generalizations the paper also contains an appendix with two variations of siegel s lemma all bounds on height are explicit

b spline solution of singular boundary value problems homogeneous and non homogenous singular boundary value problems special case are solved using b splines the original differential equation is modified at singular point then the boundary value problem is treated by using b spline approximation the method is tested on some model problems from the literature and the numerical results are compared with exact solution

a new approach to the rational interpolation problem abstract we shall reformulate the classical newton pade rational interpolation problem nprip to take away several drawbacks of the newton pade approach a new recursive algorithm not reordering the interpolation points will be designed enabling us to give a parametrization of all solutions not only for the linearized rip but also for the proper rip this paper generalizes our earlier work where the pade approximation problem i e when all the interpolation points coincide at the origin was solved by reformulating it as a minimal partial realization problem van barel 1989 van barel and bultheel 1989

b spline method for solving linear system of second order boundary value problems the b spline method is used for the numerical solution of a linear system of second order boundary value problems two examples are considered for the numerical illustration and the method is also compared with the method proposed by j lu j lu variational iteration method for solving a nonlinear system of second order boundary value problems comput math appl 54 2007 1133 1138 the comparison shows that the b spline method is a more efficient and effective tool and yields better results

on prouhet s solution to the equal powers problem eugene prouhet is considered to be one of the fathers of combinatorics on words in his pioneering work he constructed a solution to the equal powers problem in number theory later known as the prouhet tarry escott problem based on positions of symbols in words obtained by iteration of a uniform endomorphism of the free monoid over a finite alphabet we show that composition of uniform morphisms of free monoids is a fairly powerful tool to obtain solutions to a more general problem which covers the major generalizations of prouhet s theorem appearing in literature

lower bounds for norms of products of polynomials via bombieri inequality in this paper we give a different interpretation of bombieri s norm this new point of view allows us to work on a problem posed by beauzamy about the behavior of the sequence sn p supqn pqn 2 where p is a fixed m homogeneous polynomial and qn runs over the unit ball of the hilbert space of n homogeneous polynomials we also study the factor problem for homogeneous polynomials defined on c and we obtain sharp inequalities whenever the number of factors is no greater than n in particular we prove that for the product of homogeneous polynomials on infinite dimensional complex hilbert spaces our inequality is sharp finally we use these ideas to prove that any set zk k 1 of unit vectors in a complex hilbert space for which sup z 1 z z1 z zn is minimum must be an orthonormal system

a generalized minimal partial realization problem abstract all solutions of a generalized minimal partial realization problem are parametrized in an easy way using the basis matrix concept this basis matrix can be computed in an efficient way and is also used to solve several other interpolation problems

quintic b spline collocation method for fourth order partial integro differential equations with a weakly singular kernel quintic b spline collocation method for the numerical solution of fourth order partial integro differential equations is considered the scheme is based on the second order backward differential formula in the time direction and the quintic b spline method for the spatial derivative some numerical experiments are included to demonstrate the effectiveness of the discrete technique from the numerical experiments we can find that the present scheme possesses high accuracy and efficiency and the numerical results are in good agreement with the exact solutions

the multiplicity of binary recurrences of the recurrence relation 0 1 it follows from the skolem mahler lech theorem that if u then one at least of and is a root of unity consequently we call um nondegenerate if none of these quantities is a root of unity in the thirties m ward conjectured that a nondegenerate binary recurrence sequence of integers um has multiplicity u 5 5 this conjecture was proved by kubota 4 kubota also proved that any nondegenerate binary recurrence um with elements um in a number eld k of degree d has multiplicity u 5 c d cf 5 beukers and tijdeman 2 established for this case the bound

pure product polynomials and the prouhet tarry escott problem an n factor pure product is a polynomial which can be expressed in the form n i 1 1 x i for some natural numbers 1 n we define the norm of a polynomial to be the sum of the absolute values of the coefficients it is known that every n factor pure product has norm at least 2n we describe three algorithms for determining the least norm an n factor pure product can have we report results of our computations using one of these algorithms which include the result that every n factor pure product has norm strictly greater than 2n if n is 7 9 10 or 11

a reliable method for computing m pade approximants on arbitrary staircases abstract the m pade approximation problem is defined which contains as a special case the hermite pade approximation as well as the newton pade approximation different well known methods for computing m pade approximations are studied they are based on a normal solution table it is shown that the concept of one of them computing bases of sets induced by interpolation conditions can be generalized to the singular case a characterization of special bases leads to a representation of the m pade solution set like in beckermann this journal 1990 a simple efficient and reliable method for computing such bases by so called transfer matrices is derived finally examples of hermite pade and newton pade approximation are given the connection to other well known reliable algorithms for rational interpolation is discussed

numerical solution of singular boundary value problems with logarithmic singularities by pade approximation and collocation methods in this paper we present a method using quintic b spline and pade approximation for numerically solving singular boundary value problems for certain ordinary differential equation having singular coefficient we use the modified quintic b spline to approximate the first coefficients in the taylor expansion on which all the other coefficients depend moreover we introduce the superposed pade approximation the resulting perturbed initial value problem is then efficiently treated

sections of the unit ball of ipn let bnp xi rn i li n xi p 1 1 p and let ek be a k dimensional subspace of rn it is proved that if p 2 resp then vol bpn ek vol bpk resp we give some applications to linear forms

a generalization of theorems of faltings and thue siegel roth wirsing in 1929 siegel proved a celebrated theorem on finiteness for integral solutions of certain diophantine equations this theorem applies to systems of polynomial equations which either a describe an irreducible curve whose projective closure has positive genus or b describe an irreducible curve of genus zero with at least three points at infinity for such systems siegel s theorem says that there are only finitely many solutions in the ring of integers of any given number field siegel s proof used the method of diophantine approximations as pioneered by thue in 1909 t to give an example if x y is a large integral solution of the equation x3 2y3 1 then

on the theory and computation of nonperfect pade hermite approximants abstract for a vector of k 1 power series we introduce two new types of rational approximations the weak pade hermite form and the weak pade hermite fraction a recurrence relation is then presented which computes pade hermite forms along with their weak counterparts along a sequence of perfect points in the pade hermite table the recurrence relation results in a fast algorithm for calculating a pade hermite approximant of any given type when the vector of power series is perfect the algorithm is shown to calculate a pade hermite form of type n0 nk in o kn 2 operations where n n0 nk this complexity is the same as that of other fast algorithms the new algorithm also succeeds in the nonperfect case usually with only a moderate increase in cost

a quantitative version of the absolute subspace theorem the celebrated subspace theorem of w m schmidt 12 says the following subspace theorem let l1 ln be linearly independent linear forms in n variables with real or complex algebraic coefficients suppose 0 consider the inequality 1 1 l1 x ln x 0 put 1 5 q c x r li x qi 1 i n then the parametric subspace theorem can be stated as follows parametric subspace theorem let c be a fixed tuple satisfying 1 3 let 0 then there are finitely many proper linear subspaces t1 tb of q n such that for any q which is sufficiently large there exists a subspace ti t1 tb with 1 6 q q c z ti

ideal 9th order multigrades and letac s elliptic curve by showing that the elliptic curve x2 13 y2 13 48 has infinitely many rational points we prove that letac s construction produces infinitely many genuinely different ideal 9th order multigrades we give one not very small new example and by finding the mordell weil group of the curve show how to find all examples obtainable by letac s method

on polynomial exponential equations let m k be the set of absolute values of k normalized such that they extend the standard or a p adic absolute value of q let s c m k be the subset consisting of the archimedean absolute values as well as of the non archimedean absolute values v e m k having i ol 4 1 for some ej thus s is a finite set an s unit is an element e k having i 1 1 for every v s then in particular the o and therefore every a txe with x e z is an s unit therefore 1 2 is related to the s unit equation

on the structure of reduced kernel lattice bases lattice based reformulation techniques have been used successfully both theoretically and computationally one such reformulation is obtained from the kernel lattice associated with an input matrix some of the hard instances in the literature that have been successfully tackled by lattice based techniques have randomly generated input since the considered instances are very hard even in low dimension less experience is available for larger instances recently we have studied larger instances and observed that the lll reduced basis of the kernel lattice has a specific sparse structure in particular this translates into a map in which some of the original variables get a rich translation into a new variable space whereas some variables are only substituted in the new space if an original variable is important in the sense of branching or cutting planes this variable should be translated in a nontrivial way in this paper we partially explain through a probabilistic analysis the obtained struct

a new approach to the rational interpolation problem the vector case we generalize our earlier results on rational interpolation which were given in van barel and bultheel this journal 1990 for the scalar case and in bultheel and van barel 1990 for the vector case when all the interpolation points coincide to the case of vector data given at arbitrary points that may coincide or not this is the vector valued newton pade problem we give a recursive algorithm which has the important advantage over other algorithms that we do not need a reordering of the given interpolation data to overcome a singularity in the interpolation table not even in the nonnormal vector case it also generates all the information needed to give all solutions of the problem

a quartic b spline for second order singular boundary value problems second order singular boundary value problems are treated using quartic fourth degree b spline approximations the values of the coefficients c i are chosen via optimization error analysis is discussed the efficiency of the method is illustrated with several problems from the literature

a least squares approach to reduce stable discrete linear systems preserving their stability abstract a new stability preserving model reduction algorithm for discrete linear siso systems based on a least squares approach is proposed similar to the pade approximation an equation system for the markov parameters involving a high dimensional hankel matrix is considered it is proved that approximate solutions computed via the moore penrose pseudo inverse give rise to a stability preserving reduction scheme furthermore the proposed algorithm is compared to the balanced truncation method showing comparable performance of the reduced systems

cubic spline for a class of singular two point boundary value problems in this paper we have presented a method based on cubic splines for solving a class of singular two point boundary value problems the original differential equation is modified at the singular point then the boundary value problem is treated by using cubic spline approximation the tridiagonal system resulting from the spline approximation is efficiently solved by thomas algorithm some model problems are solved and the numerical results are compared with exact solution

on the explicit torsion anomalous conjecture the torsion anomalous conjecture states that an irreducible variety v embedded in a semi abelian variety contains only finitely many maximal v torsion anomalous varieties in this paper we consider an irreducible variety embedded in a produc of elliptic curves our main result provides a totally explicit bound for the n eron tate height of all maximal v torsion anomalous points of relative codimension one in the non cm case and an analogous effective result in the cm case as an application we obtain the finiteness of such points in addition we deduce some new explicit results in the context of the effective mordell lang conjecture in particular we bound the n eron tate height of the rational points of an explicit family of curves of increasing genus

mesures d independance lineaire de logarithmes dans un groupe algebrique commutatif this work falls within the theory of linear forms in logarithms over a connected and commutative algebraic group defined over the field of algebraic numbers overline mathbb q let g be such a group let w be a hyperplane of the tangent space at the origin of g defined over overline mathbb q and u be a complex point of this tangent space such that the image of u by the exponential map of the lie group g is an algebraic point then we obtain a lower bound for the distance between u and w which improves the results known before and which is in particular the best possible for the height of the hyperplane w the proof rests on baker s method and hirata s reduction as well as a new arithmetic argument chudnovsky s process of variable change which enables us to give a precise estimate of the ultrametric norms of some algebraic numbers built during the proof

p adic logarithmic forms and a problem of erdős for any natural number m 1 let p m denote the greatest prime divisor of m by the problem of erd s in the title of the present paper we mean the general version of his problem that is his conjecture from 1965 that frac p 2 n 1 n to infty quad rm as n to infty see erd s 10 and its far reaching generalization to lucas and lehmer numbers in the present paper the author delivers three refinements upon yu 40 required by c l stewart for solving completely the problem of erd s see stewart 25 the author gives also some remarks on the solution of this problem aiming to be more streamlined with respect to the p adic theory of logarithmic forms

complexity of makanin s algorithm the exponent of periodicity is an important factor in estimates of complexity of word unification algorithms we prove that the exponent of periodicity of a minimal solution of a word equation is of order 2 1 07d where d is the length of the equation we also give a lower bound 2 0 29d so our upper bound is almost optimal and exponentially better than the original bound 6d 22d4 2 consequently our result implies an exponential improvement of known upper bounds on complexity of word unification algorithms

fifth degree b spline solution for a fourth order parabolic partial differential equations in this paper we find the numerical solution of a fourth order parabolic partial differential equation a family of b spline methods has been considered for the numerical solution of the problems the results show that the present method is an applicable technique and approximates the exact solution very well

the solution of a time dependent problem by the b spline method abstract in this paper we apply the b spline method for solving the boundary value problems involving non homogeneous heat convection diffusion wave and telegraph equations the accuracy of the method for the equation is o t h 2 several examples are given to verify the reliability and efficiency of the method

new cubic b spline approximation for solving third order emden flower type equations abstract in this article the typical cubic b spline collocation method equipped with new approximations for second and third order derivatives is employed to explore the numerical solution of a class of third order non linear singular boundary value problems the singularity is removed by means of l hospital s rule the taylor s series expansion of the error term reveals that our new scheme is fifth order accurate the proposed technique is tested on several third order emden flower type equations and the numerical results are compared with those found in the current literature it is found that our new approximation technique performs superior to the existing methods due to its simple implementation straight forward interpolation and very less computational cost

on the distribution of points in projective space of bounded height in this paper we consider the uniform distribution of points in compact metric spaces we assume that there exists a probability measure on the borel subsets of the space which is invariant under a suitable group of isometries in this setting we prove the analogue of weyl s criterion and the erdos tur an inequality by using orthogonal polynomials associated with the space and the measure in particular we discuss the special case of pro jective space over completions of number elds in some detail an invariant measure in these projective spaces is introduced and the explicit formulas for the orthogonal polynomials in this case are given finally using the analo gous erdos tur an inequality we prove that the set of all projective points over the number eld with bounded arakelov height is uniformly distributed with respect to the invariant measure as the bound increases

hauteurs de sous espaces sur les corps non commutatifs we study heights of subspaces of d n where d is a finite dimensional rational division algebra and n a positive integer we define them in terms of volumes of euclidean lattices by extending a formula of w schmidt so that we recover the classical height if d is commutative we review basic properties prove a siegel lemma over d a duality theorem and a new formula for the degree of certain abelian varieties we further give matrix versions and compare our notion with the height defined through algebraic groups by j franke y manin and y tschinkel

a new formal approach to the rational interpolation problem an elegant and fast recursive algorithm is developed to solve the rational interpolation problem in a complementary way compared to existing methods we allow confluent interpolation points poles and infinity as one of the interpolation points not only one specific solution is given but a nice parametrization of all solutions we also give a linear algebra interpretation of the problem showing that our algorithm can also be used to handle a specific class of structured matrices

stable row recurrences for the pade table and generically superfast lookahead solvers for non hermitian toeplitz systems abstract we present general recurrences for the pade table that allow us to skip ill conditioned pade approximants while we proceed along a row of the table in conjunction with a certain inversion formula for toeplitz matrices these recurrences form the basis for fast algorithms for solving non hermitian toeplitz systems under the assumption that the lookahead step size i e the number of successive skipped approximants remains bounded we give both o n 2 and o n log 2 n algorithms which are presumably weakly stable with little additional work still in o n 2 operations we can also obtain a decomposition of the toeplitz matrix t according to tr ld where r is upper triangular l is unit lower triangular and d is block diagonal the relation to continued fractions is also discussed

a numerical method for singular two point boundary value problems via chebyshev economizition in this paper we present a numerical method for solving a two point boundary value problem in the interval 0 1 with regular singularity at x 0 by employing the chebyshev economizition on 0 d where d is near the singularity we first replace it by a regular problem on some interval d 1 the stable central difference method is then employed to solve the problem over the reduced interval some numerical results are presented to demonstrate the applicability of the method

an explicit time integration method for structural dynamics using septuple b spline functions in this paper an explicit time integration method with three parameters is proposed for structural dynamics using periodic septuple b spline interpolation polynomial functions in this way by use of septuple bsplines the authors have proceeded to solve the de of motion governing a single dof system and later the presented method has been generalized for a multiple dof system in the proposed method a direct recursive formula for response of the system was formulated on the basis of septuple b spline interpolation approximation in terms of the specific requirements of this proposed method two initialization approaches are given for initial calculation one is called direct initialization and the other is indirect initialization the stability analysis of the proposed method illustrates that by use of adjustable parameters a highfrequency response can be damped out without inducing excessive algorithmic damping in important low frequency modes the computational accuracy and efficiency of the proposed method is demonstrated with three numerical examples and the results from the proposed method are compared with those from some of the existent numerical methods such as the newmark and wilsonmethods the compared results show that the proposed method has high accuracy with low time consumption copyright 2013 john wiley sons ltd

on recursiveness and related topics in linear systems a decomposition of matrix formal power series linear systems is developed the interconnections between formal power series realization pade approximation recursiveness berlekamp massey algorithm feedback interconnections matrix linear fractional representations matrix continued fractions and the matrix euclidean algorithm are thereby revealed

bounded height in pencils of finitely generated subgroups in this paper we prove a general bounded height result for specializations in finitely generated subgroups varying in families which complements and sharpens the toric mordell lang theorem by replacing finiteness by emptyness for the intersection of varieties and subgroups all moving in a pencil except for bounded height values of the parameters and excluding identical relations more precisely an instance of the results is as follows consider the torus scheme g r m c over a curve c defined over q and let be a subgroup scheme generated by finitely many sections satisfying some necessary conditions further let v be any subscheme then there is a bound for the height of the points p c q such that for some which does not generically lie in v p lies in the fiber vp we further offer some direct diophantine applications to illustrate once again that the results implicitly contain information absent from the previous bounds in this context

small zeros of quadratic forms we give upper and lower bounds for zeros of quadratic forms in the rational real and p adic fields for example given r 0 s 0 there are infinitely many forms a with integer coefficients in r s variables of the type r s i e equivalent over r to x2 xr2 3 x2 j such that every nontrivial integer zero x has lxl fr 28 where f is the maximum modulus of the coefficients of a

a lookahead algorithm for the solution of block toeplitz systems abstract we give a weakly stable algorithm to solve a block toeplitz system of linear equations if the lookahead steps taken to compute the parameters of the inversion formula for the block toeplitz matrix are small compared to the order n of the matrix the algorithm requires o n2 floating point operations the parameters of the inversion formula are interpreted and computed in a recursive way as solutions of certain interpolation problems given the formal laurent series based on the data of the block toeplitz matrix

the number of solutions of norm form equations a norm form is a form f xi xn with rational coefficients which factors into linear forms over c but is irreducible or a power of an irreducible form over q it is known that a nondegenerate norm form equation f xl x m has only finitely many solutions xl x e zn we derive explicit bounds for the number of solutions when f has coefficients in z these bounds depend only on n m and the degree of f but are independent of the size of the coefficients of f

a further improvement of the quantitative subspace theorem in 2002 evertse and schlickewei 11 obtained a quantitative version of the so called absolute parametric subspace theorem this result deals with a parametrized class of twisted heights one of the consequences of this result is a quantitative version of the absolute subspace theorem giving an explicit upper bound for the number of subspaces containing the solutions of the diophantine inequality under consideration in the present paper we further improve evertse s and schlickewei s quantitative version of the absolute parametric subspace theorem and deduce an improved quantitative version of the absolute subspace theorem we combine ideas from the proof of evertse and schlickewei which is basically a substantial refinement of schmidt s proof of his subspace theorem from 1972 22 with ideas from faltings and wustholz proof of the subspace theorem 14 a new feature is an interval result which gives more precise information on the distribution of the heights of the solutions of the system of inequalities considered in the subspace theorem

solution of fifth order boundary value problems by using local polynomial regression abstract in this paper we present a novel method based on the local polynomial regression for solving of fifth order boundary value problems the method is tested on numerical example to demonstrate its usefulness the method presented in this paper is also compared with those developed by siddiqi and akram solution of fifth order boundary value problems using nonpolynomial spline technique appl math comput 175 2006 1575 1581 as well and is observed to be better

b spline interpolation compared with finite difference finite element and finite volume methods which applied to two point boundary value problems abstract this paper considers the b spline interpolation and compares this method with finite difference finite element and finite volume methods which applied to the two point boundary value problem d d x p x d u d x f x a x b u a u b 0