a new numerical method for solving high order fractional eigenvalue problems the paper presents a new numerical method for solving eigenvalue problems for fractional high order differential equations with variable coefficients it combines two techniques the method of external excitation mee and the backward substitution method bsm the first one is a mathematical model of physical measurements when a mechanical electrical or acoustic system is excited by some source and resonant frequencies can be determined by using the growth of the amplitude of oscillations near these frequencies the bsm consists of replacing the original equation by an approximate equation which has an exact analytic solution with a set of free parameters these free parameters are determined by the use of the collocation procedure some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results in particular fractional problems of the fourth sixth and eighth orders are considered the numerical results show that the proposed method is of a high accuracy and is efficient for solving a wide class of eigenvalue problems

modified spline collocation for linear fractional differential equations we propose and analyze a class of high order methods for the numerical solution of initial value problems for linear multi term fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the initial value problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established theoretical results are verified by some numerical examples

quasi bell inequalities from symmetrized products of noncommuting qubit observables noncommuting observables cannot be simultaneously measured however under local hidden variable models they must simultaneously hold premeasurement values implying the existence of a joint probability distribution we study the joint distributions of noncommuting observables on qubits with possible criteria of positivity and the frechet bounds limiting the joint probabilities concluding that the latter may be negative we use symmetrization justified heuristically and then more carefully via the moyal characteristic function to find the quantum operator corresponding to the product of noncommuting observables this is then used to construct quasi bell inequalities bell inequalities containing products of noncommuting observables on two qubits these inequalities place limits on the local hidden variable models that define joint probabilities for noncommuting observables we find that the quasi bell inequalities have a quantum to classical violation as high as 32 on two qubit higher than conventi

lifting bell inequalities a bell inequality defined for a specific experimental configuration can always be extended to a situation involving more observers measurement settings or measurement outcomes in this article such liftings of bell inequalities are studied it is shown that if the original inequality defines a facet of the polytope of local joint outcome probabilities then the lifted one also defines a facet of the more complex polytope

bounds for resultants of univariate and bivariate polynomials the paper considers bounds on the size of the resultant for univariate and bivariate polynomials for univariate polynomials we also extend the traditional representation of the resultant by the zeros of the argument polynomials to formal resultants defined as the determinants of the sylvester matrix for a pair of polynomials whose actual degree may be lower than their formal degree due to vanishing leading coefficients for bivariate polynomials the resultant is a univariate polynomial resulting by the elimination of one of the variables and our main result is a bound on the largest coefficient of this univariate polynomial we bring a simple example that shows that our bound is attainable and that a previous sharper bound is not correct

numerical solution of nonlinear fractional differential equations by spline collocation methods an initial value problem for nonlinear fractional differential equations is considered using an integral equation reformulation of the initial value problem some regularity properties of the exact solution are derived on the basis of these properties the numerical solution of initial value problems by piecewise polynomial collocation methods is discussed in particular the attainable order of convergence of proposed algorithms is studied and a global superconvergence effect for a special choice of collocation points is established theoretical results are verified by means of numerical examples

nonsignaling as the consistency condition for local quasi classical probability modeling of a general multipartite correlation scenario we specify for a general correlation scenario a particular type of local quasi hidden variable lqhv model loubenets 2012 j math phys 53 022201 a deterministic lqhv model where all joint probability distributions of a correlation scenario are simulated via a single measure space with a normalized bounded real valued measure not being necessarily positive and random variables each depending only on a setting of the corresponding measurement at the corresponding site we prove that an arbitrary multipartite correlation scenario admits a deterministic lqhv model if and only if all its joint probability distributions satisfy the consistency condition constituting the general nonsignaling condition formulated in loubenets 2008 j phys a math theor 41 445303 this mathematical result specifies a new probability model that has a measure theoretic structure resembling the structure of the classical probability model but incorporates the latter only as a particular case the local version of this quasi classical probability model covers the probabilistic description of each nonsignaling correlation scenario in particular each correlation scenario on a multipartite quantum state

local quasi hidden variable modelling and violations of bell type inequalities by a multipartite quantum state we introduce for a general correlation scenario a new simulation model a local quasi hidden variable lqhv model where locality and the measure theoretic construction inherent to a local hidden variable lhv model are preserved but positivity of a simulation measure is dropped we specify a necessary and sufficient condition for lqhv modelling and based on this prove that every quantum correlation scenario admits a lqhv simulation via the lqhv approach we construct analogs of bell type inequalities for an n partite quantum state and find a new analytical upper bound on the maximal violation by an n partite quantum state of s1 sn setting bell type inequalities either on correlation functions or on joint probabilities and for outcomes of an arbitrary spectral type discrete or continuous this general analytical upper bound is expressed in terms of the new state dilation characteristics introduced in the present paper and not only traces quantum states admitting an s1 sn setting lhv de

quantifying non classical and beyond quantum correlations in the unified operator formalism acin et al 2010 phys rev lett 104 140404 introduced a unified framework for the study of no signalling correlations such a framework is based on the notion of local quantum measurements but in order to account for beyond quantum correlations global pseudo states that are not positive semidefinite are allowed after a short review of the formalism we consider its use in the quantification of both general non local and beyond quantum correlations we argue that the unified framework for correlations provides a simple approach to such a quantification in particular when the quantification is meant to be operational and meaningful in a resource theory scenario i e when considering the processing of resources by means of non resources we relate different notions of robustness of correlations both at the level of pseudo states and abstract probability distributions with particular focus on the beyond quantum robustness of correlations and pseudo states we revisit known results and argue that within the unified framework the relation between the two levels that of operators and that of probability distributions is very strict we point out how the consideration of robustness at the two levels leads to a natural framework for the quantification of entanglement in a device independent way finally we show that the beyond quantum robustness of the non positive operators needed to achieve beyond quantum correlations coincides with their negativity and their distance from the set of quantum states as an example we calculate the beyond quantum robustness for the case of a noisy popescu rohrlich box

piecewise polynomial collocation for linear boundary value problems of fractional differential equations we consider a class of boundary value problems for linear multi term fractional differential equations which involve caputo type fractional derivatives using an integral equation reformulation of the boundary value problem some regularity properties of the exact solution are derived based on these properties the numerical solution of boundary value problems by piecewise polynomial collocation methods is discussed in particular we study the attainable order of convergence of proposed algorithms and show how the convergence rate depends on the choice of the grid and collocation points theoretical results are verified by two numerical examples

algorithms for the fractional calculus a selection of numerical methods many recently developed models in areas like viscoelasticity electrochemistry diffusion processes etc are formulated in terms of derivatives and integrals of fractional non integer order in this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context we believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way

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

a nonpolynomial collocation method for fractional terminal value problems in this paper we propose a nonpolynomial collocation method for solving a class of terminal or boundary value problems for differential equations of fractional order 0 1 the approach used is based on the equivalence between a problem of this type and a fredholm integral equation of a particular form taking into account the asymptotic behaviour of the solution of this problem we propose a nonpolynomial collocation method on a uniform mesh we study the order of convergence of the proposed algorithm and a result on optimal order of convergence is obtained in order to illustrate the theoretical results and the performance of the method we present several numerical examples

simulating all nonsignaling correlations via classical or quantum theory with negative probabilities many party correlations between measurement outcomes in general probabilistic theories are given by conditional probability distributions obeying the nonsignaling condition we show that any such distribution can be obtained from classical or quantum theory by relaxing positivity constraints on either the mixed state shared by the parties or the local functions that generate measurement outcomes our results apply to generic nonsignaling correlations but in particular they yield two distinct quasiclassical models for quantum correlations

an augmented rbf method for solving fractional sturm liouville eigenvalue problems in this work we construct numerical schemes using radial basis functions rbfs to solve sturm liouville eigenvalue problems in which the so called fractional derivatives are involved we take into account the fact that the solutions to such problems might be nonsmooth and consequently we augment the space generated by rbf base functions with some fractional polynomials to overcome this problem we show through some examples the effectiveness of our method

the method of external excitation for solving generalized sturm liouville problems a new numerical technique for solving the generalized sturm liouville problem d 2wdx 2 q x l w 0 b l w 0 l b r w 1 l 0 is presented in particular we consider the problems when the coefficient q x l or the boundary conditions depend on the spectral parameter l in an arbitrary nonlinear manner the method presented is based on mathematically modelling the physical response of a system to excitation over a range of frequencies the response amplitudes are then used to determine the eigenvalues the results of the numerical experiments justifying the method are presented

an efficient algorithm for solving higher order fractional sturm liouville eigenvalue problems abstract in this paper we present a simple and efficient computational algorithm for solving eigenvalue problems of high fractional order differential equations with variable coefficients the method of solution is based on utilizing the series solution to convert the governing fractional differential equation into a linear system of algebraic equations then the eigenvalues can be calculated by finding the roots of the corresponding characteristic polynomial notice that this class of eigenvalue problems is very promising to the solution of linear fractional partial differential equations fpde the numerical results demonstrate reliability and efficiency of the proposed algorithm based on our simulations some theoretical conjectures are reported

number of common roots and resultant of two tropical univariate polynomials abstract it is well known that for two univariate polynomials over the complex number field the number of their common roots is equal to the order of their resultant in this paper we show that this fundamental relationship still holds for the tropical polynomials under suitable adaptation of the notion of order if the roots are simple and non zero

spline collocation for nonlinear fractional boundary value problems abstract we consider a class of boundary value problems for nonlinear fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the boundary value problem some regularity properties of the exact solution are derived based on these properties and spline collocation techniques the numerical solution of boundary value problems by suitable non polynomial approximations is discussed optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given theoretical results are tested by two numerical examples

effective polynomial computation 1 euclid s algorithm 2 continued fractions 3 diophantine equations 4 lattice techniques 5 arithmetic functions 6 residue rings 7 polynomial arithmetic 8 polynomial gcd s classical algorithms 9 polynomial elimination 10 formal power series 11 bounds on polynomials 12 zero equivalence testing 13 univariate interpolation 14 multivariate interpolation 15 polynomial gcd s interpolation algorithms 16 hensel algorithms 17 sparse hensel algorithms 18 factoring over finite fields 19 irreducibility of polynomials 20 univariate factorization 21 multivariate factorization list of symbols bibliography index

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

zero testing of p adic and modular polynomials we obtain new algorithms for testing whether a given by a black box multivariate polynomial over p adic fields given by a black box is identical to zero we also remark on the zero testing of polynomials in residue rings our results complement a known results on the zero testing of polynomials over the integers the rationals and over finite fields

a novel method for solving second order fractional eigenvalue problems the paper presents a new numerical method for solving eigenvalue problems for fractional differential equations it combines two techniques the method of external excitation mee and the backward substitution method bsm the first one is a mathematical model of physical measurements when a mechanical electrical or acoustic system is excited by some source and resonant frequencies can be determined by using the growth of the amplitude of oscillations near these frequencies the bsm consists of replacing the original equation by an approximate equation which has an exact analytic solution with a set of free parameters these free parameters are determined by the use of the collocation procedure some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results the numerical results show that the proposed method is of a high accuracy and is efficient for solving of a wide class of eigenvalue problems

a method of particular solutions for multi point boundary value problems the paper presents a new semi analytic numerical method for solving multi point boundary value problems with linear and nonlinear equations of the second and higher orders the method is based on the use of the particular solutions of the linearized equation presented in the paper examples include the solution of equations with differential operators of the second third and fourth orders

on shifted jacobi spectral approximations for solving fractional differential equations in this paper a new formula of caputo fractional order derivatives of shifted jacobi polynomials of any degree in terms of shifted jacobi polynomials themselves is proved we discuss a direct solution technique for linear multi order fractional differential equations fdes subject to nonhomogeneous initial conditions using a shifted jacobi tau approximation a quadrature shifted jacobi tau q sjt approximation is introduced for the solution of linear multi order fdes with variable coefficients we also propose a shifted jacobi collocation technique for solving nonlinear multi order fractional initial value problems the advantages of using the proposed techniques are discussed and we compare them with other existing methods we investigate some illustrative examples of fdes including linear and nonlinear terms we demonstrate the high accuracy and the efficiency of the proposed techniques