on the modular curves y e 7 let e denote an elliptic curve over q and y e 7 the modular curve classifying the elliptic curves e over q such that the representations of gal q q in the 7 torsion points of e and of e are symplectically isomorphic in case e is given by a weierstrass equation such that the c 4 invariant is a square we exhibit here nontrivial points of y e 7 q from this we deduce an infinite family of curves e for which y e 7 q has at least four nontrivial points

p torsion monodromy representations of elliptic curves over geometric function fields given a complex quasiprojective curve b and a nonisotrivial familye of elliptic curves over b the p torsione p yields a monodromy representation e p 1 b gl2 fp we prove that if e p e0 p thene ande 0 are isogenous provided p is larger than a constant depending only on the gonality of b this can be viewed as a function eld analog of the frey mazur conjecture which states that an elliptic curve over q is determined up to isogeny by its p torsion galois representation for p 17 the proof relies on hyperbolic geometry and is therefore only applicable in characteristic 0

numerical solution of a class of fractional order integro differential algebraic equations using muntz jacobi tau method abstract this paper presents a spectral tau method based on muntz jacobi basis functions for approximating the solutions of fractional order volterra integro differential algebraic equations we examine the solvability of the considered equation and we show that the exact solutions suffer from a singularity at the origin numerical solvability and convergence analysis of the proposed approximate approach are also studied and the error bounds are obtained in a weighted l 2 norm some illustrative examples are presented to validate the theoretical results

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

selmer companion curves we say that two elliptic curves e1 e2 over a number field k are n selmer companions for a positive integer n if for every quadratic character of k there is an isomorphism seln e 1 k seln e 2 k between the nselmer groups of the quadratic twists e 1 e 2 we give sufficient conditions for two elliptic curves to be n selmer companions and give a number of examples of non isogenous pairs of companions

spectral collocation methods for volterra integro differential equations with noncompact kernels the aim of this paper is to analyze spectral collocation methods for a class of volterra integro differential equations with noncompact kernels the spectral collocation method is proposed to solve the equations after the results on existence uniqueness and regularity of the solutions are established the convergence rates of the methods are proven and a numerical example is provided to verify the theoretical findings

modular diagonal quotient surfaces the main objective of this paper is to analyze the geometry of the modular diagonal quotient surface zn e e x n x n which classifies pairs of elliptic curves e1 and e2 together with an isomorphism of determinant e between their associated modular representations mod n in particular we calculate some of the numerical invariants of its minimal desingularization zn e such as its betti and chern numbers and determine its place in the enriques kodaira classification table

a cutting plane algorithm for the general routing problem the general routing problem grp is the problem of finding a minimum cost route for a single vehicle subject to the condition that the vehicle visits certain vertices and edges of a network it contains the rural postman problem chinese postman problem and graphical travelling salesman problem as special cases we describe a cutting plane algorithm for the grp based on facet inducing inequalities and show that it is capable of providing very strong lower bounds and in most cases optimal solutions

on the undirected rural postman problem tight bounds based on a new formulation the rural postman problem rpp is a classic edge routing problem a mathematical programming formulation of the rpp that differs fundamentally from those in the literature was introduced but not tested computationally by garfinkel and webb 1999 a rudimentary algorithm that yields lower bounds via cutting planes and upper bounds via heuristics is developed and tested for a variation of that formulation computational results are encouraging especially in terms of the relatively small number of added inequalities needed to get good lower bounds and the fact that the vast majority of these have efficient exact separation procedures note that a first algorithm based on this new formulation is computationally competitive allowing the possibility of far more efficient and complex future realizations

dynamics of sl2 ℝ over moduli space in genus two this paper classifies orbit closures and invariant measures for the natural action of sl2 r on um2 the bundle of holomorphic 1 forms over the moduli space of riemann surfaces of genus two

application of the collocation method for solving nonlinear fractional integro differential equations in this paper using the collocation method we solve the nonlinear fractional integro differential equations nfide of the form f t y t a cd t a 0y t a cd t a ry t email protected t y t a tk t s f s y s ds y k a d k k 0 1 m 0 1 we study the convergence and the stability analysis of this method for f t y t a cd t a 0y t a cd t a ry t y t email protected j 0 rb j a cd t a jy t g t some numerical examples are given to show the efficiency of the presented method

mechanical quadrature methods and their extrapolation for solving first kind abel integral equations this paper presents high accuracy mechanical quadrature methods for solving first kind abel integral equations to avoid the ill posedness of problem the first kind abel integral equation is transformed to the second kind volterra integral equation with a continuous kernel and a smooth right hand side term expressed by weakly singular integrals by using periodization method and modified trapezoidal integration rule not only high accuracy approximation of the kernel and the right hand side term can be easily computed but also two quadrature algorithms for solving first kind abel integral equations are proposed which have the high accuracy o h 2 and asymptotic expansion of the errors then by means of richardson extrapolation an approximation with higher accuracy order o h 3 is obtained moreover an a posteriori error estimate for the algorithms is derived some numerical results show the efficiency of our methods

solving the prize collecting rural postman problem abstract in this work we present an algorithm for solving the prize collecting rural postman problem this problem was recently defined and is a generalization of other arc routing problems like for instance the rural postman problem the main difference is that there are no required edges instead there is a profit function on the edges that must be taken into account only the first time that an edge is traversed the problem is modeled as a linear integer program where the system has an exponential number of inequalities we propose a solution algorithm where iteratively we solve relaxed models with a small number of inequalities that provide upper bounds and we propose exact separation procedures for generating violated cuts when possible we also propose a simple heuristic to generate feasible solutions that provide lower bounds at each iteration we use a two phase method with different solvers at each phase despite the difficulty of the problem the numerical results from a series of computational experiments with various types of instances assess the good behavior of the algorithm in particular 75 of the instances were optimally solved with the lp relaxation of the model the remaining instances were optimally solved on a second phase most of them in small computation times

high accuracy combination algorithm and a posteriori error estimation for solving the first kind abel integral equations abstract this paper presents high accuracy combination algorithm for solving the first kind abel integral equations to avoid solving ill posed problems we transform the first kind abel integral equation to the second kind volterra integral equation with a continuous kernel expressed by a weakly singular integral by using integration rules the approximation of this kernel can be easily computed then two quadrature algorithms for solving abel integral equations are proposed which possess accuracy order o h 1 0 h 2 moreover a posteriori error estimate for the algorithms is derived both theory and numerical examples show that it is effective and saves storage capacity and computational work

solving the hierarchical chinese postman problem as a rural postman problem abstract in the undirected hierarchical chinese postman problem hcpp the edges of a graph are partitioned into clusters and must be serviced while respecting a hierarchy or precedence relation between clusters two objectives are considered a hierarchical objective and a makespan objective in this article a transformation of the hcpp into an equivalent rural postman problem rpp is presented the hcpp is solved optimally for both objectives by applying an exact branch and cut rpp algorithm to the transformed problem two heuristics based on the rpp algorithm are also described and assessed computationally

the hierarchical mixed rural postman problem polyhedral analysis and a branch and cut algorithm abstract the hierarchical mixed rural postman problem is defined on a mixed graph where arcs and edges that require a service are divided into clusters that have to be serviced in a hierarchical order the problem generalizes the mixed rural postman problem and thus is np hard in this paper we provide a polyhedral analysis of the problem and propose a branch and cut algorithm for its solution based on the introduced classes of valid inequalities extensive computational experiments are reported on benchmark instances the exact approach allows to find the optimal solutions in less than 1 hour for instances with up to 999 vertices 2678 links and five clusters

reconstruction of exponentially rate of convergence to legendre collocation solution of a class of fractional integro differential equations in this paper legendre collocation method an easy to use variant of the spectral methods for the numerical solution of a class of fractional integro differential equations fide s is researched in order to obtain high order accuracy for the approximation the integral term in the resulting equation is approximated by using legendre gauss quadrature formula an efficient convergence analysis of the proposed method is given and rate of convergence is established in the l 2 norm due to the fact that the solutions of fide s usually have a weak singularity at origin we use a variable transformation to change the original equation into a new equation with a smooth solution we prove that after this regularization technique numerical solution of the new equation by adopting the legendre collocation method has exponentially rate of convergence numerical results are presented which clarify the high accuracy of the proposed method

a branch and cut algorithm for the undirected rural postman problem the well known undirected rural postman problem is considered and a binary linear problem using new dominance relations is presented polyhedral properties are investigated and a branch and cut algorithm is developed extensive computational results indicate that the algorithm is capable of solving much larger instances than previously reported

application of the fractional differential transform method to fractional order integro differential equations with nonlocal boundary conditions in this paper the fractional differential transform method is developed to solve fractional integro differential equations with nonlocal boundary conditions the method is described and illustrated with numerical examples the results reveal that the method is accurate and easy to apply

explicit moduli spaces for congruences of elliptic curves we determine explicit birational models over mathbb q for the modular surfaces parametrising pairs of n congruent elliptic curves in all cases where this surface is an elliptic surface in each case we also determine the rank of the mordell weil lattice and the geometric picard number

a branch and cut algorithm for the maximum benefit chinese postman problem the maximum benefit chinese postman problem mbcpp is an np hard problem that considers several benefits associated with each edge one for each time the edge is traversed with a service the objective is to find a closed walk with maximum benefit we propose an ip formulation for the undirected mbcpp and based on the description of its associated polyhedron we propose a branch and cut algorithm and present computational results on instances with up to 1 000 vertices and 3 000 edges

numerical solution of fractional integro differential equations by collocation method abstract this paper deals with the numerical solution of fractional integro differential equations of the type d q y t p t y t f t 0 t k t s y s d s t i 0 1 by polynomial spline functions we derive a system of equations that characterizing the numerical solution some numerical examples are also provided to illustrated our results

convergence analysis of the jacobi spectral collocation methods for volterra integral equations with a weakly singular kernel in this paper a jacobi collocation spectral method is developed for volterra integral equations of second kind with a weakly singular kernel we use some function transformation and variable transformations to change the equation into a new volterra integral equation deflned on the standard interval i1 1 so that the solution of the new equation possesses better regularity and the jacobi orthogonal polynomial theory can be applied conveniently in order to obtain high order accuracy for the approx imation the integral term in the resulting equation is approximated by using jacobi spectral quadrature rules the convergence analysis of this novel method is based on the lebesgue constants corresponding to the lagrange interpolation polynomials poly nomials approximation theory for orthogonal polynomials and the operator theory the spectral rate of convergence for the proposed method is established in the l 1 norm and weighted l 2 norm numerical results are presented to demonstrate the efiectiveness of the proposed method

exploring variants of 2 opt and 3 opt for the general routing problem the general routing problem grp is the problem of finding a minimum length tour visiting a number of specified vertices and edges in an undirected graph in this paper we describe how the well known 2 opt and 3 opt local search procedures for node routing problems can be adapted to solve arc and general routing problems successfully two forms of the 2 opt and 3 opt approaches are applied to the grp the first version is similar to the conventional approach for the traveling salesman problem the second version includes a dynamic programming procedure and explores a larger neighborhood at the expense of higher running times extensive computational tests including ones on larger instances than previously reported in the arc routing literature are performed with variants of both algorithms in combination with the guided local search metaheuristic and the mechanisms of marking and neighbor lists the procedures systematically detect optimal or high quality solutions within limited computation time

numerical solution of fractional integro differential equations by a hybrid collocation method fractional integro differential equations have been recently solved by many methods such as adomian decomposition method differential transform method collocation method and taylor expansion approach in this paper a hybrid collocation method is used which combines a non polynomial collocation used on the first subinterval and graded piecewise polynomial collocation used on the rest of the interval a theoretical analysis for the convergence order of the method is presented some numerical examples are given which confirm the theoretical results

approximate solution of fractional integro differential equations by taylor expansion method in this paper taylor expansion approach is presented for solving approximately a class of linear fractional integro differential equations including those of fredholm and of volterra types by means of the mth order taylor expansion of the unknown function at an arbitrary point the linear fractional integro differential equation can be converted approximately to a system of equations for the unknown function itself and its m derivatives under initial conditions this method gives a simple and closed form solution for a linear fractional integro differential equation in addition illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method