on the equation y2 x x2 p generators are found for the group of rational points on the title curve for all primes p 5 mod 8 less than 1 000 the rank is always i in accordance with conjectures of selmer and mordell some of the generators are rather large

insufficiency of the brauer manin obstruction applied to etale covers let k be any global field of characteristic not 2 we construct a k variety x such that x k is empty but for which the emptiness cannot be explained by the brauer manin obstruction or even by the brauer manin obstruction applied to finite etale covers

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

linear growth for châtelet surfaces an upper bound of the expected order of magnitude is established for the number of mathbb q rational points of bounded height on ch telet surfaces defined over mathbb q

effectivity of brauer manin obstructions on surfaces abstract we study brauer manin obstructions to the hasse principle and to weak approximation on algebraic surfaces over number fields a technique for constructing azumaya algebra representatives of brauer group elements is given and this is applied to the computation of obstructions

a class of second order difference approximations for solving space fractional diffusion equations a class of second order approximations called the weighted and shifted grunwald difference wsgd operators are proposed for riemann liouville fractional derivatives with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions the stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established several numerical examples are implemented to test the efficiency of the numerical schemes and confirm the convergence order and the numerical results for variable coefficients problem are also presented

global oscillatory waves for second order quasilinear wave equations in this paper we prove the global existence and describe the asymptotic behaviour of a family of oscillatory solutions of cauchy problems for a class of scalar second order quasilinear wave equations when the space dimension is odd and at least equal to 3 if time is bounded corresponding results for quasilinear first order systems were obtained by gues to prove our results we reduce our problems to bounded time problems with the help of a conformal inversion to obtain global results suitable geometric assumptions must be made on the set where the oscillations are concentrated at initial time

compact difference schemes for the modified anomalous fractional sub diffusion equation and the fractional diffusion wave equation in this paper compact finite difference schemes for the modified anomalous fractional sub diffusion equation and fractional diffusion wave equation are studied schemes proposed previously can at most achieve temporal accuracy of order which depends on the order of fractional derivatives in the equations and is usually less than two based on the idea of weighted and shifted grunwald difference operator we establish schemes with temporal and spatial accuracy order equal to two and four respectively

two fully discrete schemes for fractional diffusion and diffusion wave equations with nonsmooth data we consider initial boundary value problems for the subdiffusion and diffusion wave equations involving a caputo fractional derivative in time we develop two fully discrete schemes based on the piecewise linear galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward euler method second order backward difference method and establish error estimates optimal with respect to the regularity of problem data these two schemes are first and second order accurate in time for both smooth and nonsmooth data extensive numerical experiments for two dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity

numerical approximation of distributed order reaction diffusion equations in this paper an implicit scheme for the numerical approximation of the distributed order time fractional reaction diffusion equation with a nonlinear source term is presented the stability and the convergence order of the numerical scheme are analysed and illustrated through some numerical examples

rational points on pencils of conics and quadrics with many degenerate fibres for any pencil of conics or higher dimensional quadrics over q with all degenerate bres dened over q we show that the brauer manin obstruction controls weak approximation the proof is based on the hasse principle and weak approximation for some special intersections of quadrics over q which is a consequence of recent advances in additive combinatorics

a discontinuous petrov galerkin method for time fractional diffusion equations we propose and analyze a time stepping discontinuous petrov galerkin method combined with the continuous conforming finite element method in space for the numerical solution of time fractional subdiffusion problems we prove the existence uniqueness and stability of approximate solutions and derive error estimates to achieve high order convergence rates from the time discretizations the time mesh is graded appropriately near t 0 to compensate for the singular temporal behavior of the exact solution near t 0 caused by the weakly singular kernel but the spatial mesh is quasi uniform in the l infty 0 t l 2 omega norm 0 t is the time domain and omega is the spatial domain for sufficiently graded time meshes a global convergence of order k m alpha 2 h r 1 is shown where 0 alpha 1 is the fractional exponent k is the maximum time step h is the maximum diameter of the elements of the spatial mesh and m and r are the degrees of approximate solutions in time an

an alternating direction implicit galerkin finite element method for the distributed order time fractional mobile immobile equation in two dimensions abstract in this paper we shall present the alternating direction implicit adi galerkin finite element method fem for solving the distributed order time fractional mobile immobile equation in two dimensions in the time direction the backward euler method is used to deal with the temporal first order derivative and the weighted and shifted grunwald formula is employed to discretize the distributed order time fractional derivative galerkin fem is used for discretization of the spatial direction and then an adi galerkin finite element scheme is constructed the stability and l 2 norm convergence are proved several numerical experiments are provided to verify our theoretical analysis

error estimates for a semidiscrete finite element method for fractional order parabolic equations we consider the initial boundary value problem for a homogeneous time fractional diffusion equation with an initial condition v x and a homogeneous dirichlet boundary condition in a bounded convex polygonal domain omega we study two semidiscrete approximation schemes i e the galerkin finite element method fem and lumped mass galerkin fem using piecewise linear functions we establish almost optimal with respect to the data regularity error estimates including the cases of smooth and nonsmooth initial data i e v in h 2 omega cap h 1 0 omega and v in l 2 omega for the lumped mass method the optimal l 2 norm error estimate is valid only under an additional assumption on the mesh which in two dimensions is known to be satisfied for symmetric meshes finally we present some numerical results that give insight into the reliability of the theoretical study

boundary value problems for the diffusion equation of the variable order in differential and difference settings abstract solutions of boundary value problems for a diffusion equation of fractional and variable order in differential and difference settings are studied it is shown that the method of the energy inequalities is applicable to obtaining a priori estimates for these problems exactly as in the classical case the credibility of the obtained results is verified by performing numerical calculations for a test problem

time stepping error bounds for fractional diffusion problems with non smooth initial data we apply the piecewise constant discontinuous galerkin method to discretize a fractional diffusion equation with respect to time using laplace transform techniques we show that the method is first order accurate at the nth time level t n but the error bound includes a factor t n 1 if we assume no smoothness of the initial data we also show that for smoother initial data the growth in the error bound as t n decreases is milder and in some cases absent altogether our error bounds generalize known results for the classical heat equation and are illustrated for a model problem

distributed order fractional diffusions on bounded domains fractional derivatives can be used to model time delays in a diffusion process when the order of the fractional derivative is distributed over the unit interval it is useful for modeling a mixture of delay sources this paper provides explicit strong solutions and stochastic analogues for distributed order time fractional diffusion equations on bounded domains with dirichlet boundary conditions

finite difference spectral approximations for the time fractional diffusion equation in this paper we consider the numerical resolution of a time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order a with 0 a 1 the main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time fractional diffusion equation the proposed method is based on a finite difference scheme in time and legendre spectral methods in space stability and convergence of the method are rigourously established we prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order o dt 2 a n m where dt n and m are the time step size polynomial degree and regularity of the exact solution respectively numerical experiments are carried out to support the theoretical claims

finite difference spectral methods for the two dimensional distributed order time fractional cable equation abstract the cable equation plays a significant role in many areas of electrophysiology and in modeling neuronal dynamics in recent years considerable attention has been devoted to distributed order differential equations because they appear to be more effective for modeling complex processes in this work a finite difference legendre spectral method is presented for the numerical simulation of the two dimensional 2d distributed order time fractional cable equation where the finite difference method is employed in the temporal discretization and legendre spectral method is adopted in the spatial discretization the midpoint quadrature rule is used to approximate the distributed order such that the considered equation could be transformed into a multi term fractional equation the stability and convergence analysis of the proposed scheme is established which illustrates that the numerical solution converges to the exact solution with order o 2 2 n s where n are the time step size the step length in the approximation of the distributed order and the polynomial degree respectively furthermore to demonstrate the versatility and applicability of our method we provide numerical results that show good agreement with the theoretical analysis

manin s conjecture for quartic del pezzo surfaces with a conic fibration an asymptotic formula is established for the number of q rational points of bounded height on a non singular quartic del pezzo surface with a conic bundle structure

hasse principle for pencils of curves of genus one whose jacobians have rational 2 division points une serie d articles exploite une nouvelle technique qui mene a des conditions suffisantes d existence et de densite des points rationnels sur certaines surfaces fibrees en courbes de genre un au dessus de la droite projective dans les premiers articles de cette serie plusieurs articles de swinnerton dyer un article en collaboration de skorobogatov swinnerton dyer et l auteur la jacobienne de la fibre generique des surfaces considerees a tous ses points d ordre 2 rationnels un article recent de bender et swinnerton dyer traite de cas ou cette jacobienne possede seulement un point d ordre 2 non trivial pour que la methode fonctionne il semble necessaire que la jacobienne possede un point de torsion rationnel non trivial le present article est une reecriture de celui de bender et swinnerton dyer la principale contribution est une reformulation plus abstraite des hypotheses principales des theoremes la premiere hypothese est formulee de facon entierement algebrique certains groupes de selmer algebrico geometriques sont supposes petits et la seconde hypothese est simplement il n y a pas d obstruction de brauer manin verticale comme dans la plupart des articles de cette serie les resultats dependent de deux conjectures difficiles l hypothese de schinzel et la finitude des groupes de tate shafarevich

exponentially fitted methods for solving time fractional nonlinear reaction diffusion equation abstract in this article we will develop a new numerical scheme with the second order in time and a class of fourth order or sixth order in space based on the exponential fitting techniques to approximate the nonlinear time fractional reaction diffusion equation with fixed order and distributed order derivatives these techniques depend on a parameter which will be used to annihilate the error and increase the order of accuracy the proposed methods are proved to be unconditionally stable and convergent by fourier analysis also the theoretical results and the effectiveness of the numerical scheme are confirmed by numerical test problems and a comparison with other methods is presented

numerical algorithms for time fractional subdiffusion equation with second order accuracy this article aims to fill in the gap of the second order accurate schemes for the time fractional subdiffusion equation with unconditional stability two fully discrete schemes are first proposed f

families of rationally connected varieties recall that a proper variety x is said to be rationally connected if two general points p q x are contained in the image of a map g p x this is clearly a birationally invariant property when x is smooth this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the a priori stronger condition that for any finite subset x there is a map g p x whose image contains and such that gtx is an ample bundle

on the fibration method for zero cycles and rational points conjectures on the existence of zero cycles on arbitrary smooth projective varieties over number elds were proposed by colliot th el ene sansuc kato and saito in the 1980 s we prove that these conjectures are compatible with brations for brations into rationally connected varieties over a curve in particular they hold for the total space of families of homogeneous spaces of linear groups with connected geometric stabilisers we prove the analogous result for rational points conditionally on a conjecture on locally split values of polynomials which a recent work of matthiesen establishes in the case of linear polynomials over the rationals

simultaneous integer values of pairs of quadratic forms we prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent almost all pairs of integers that satisfy the necessary local conditions provided that the forms satisfy a suitable nonsingularity condition in particular such forms simultaneously attain prime values if the obvious local conditions hold the proof uses the circle method and in particular pioneers a two dimensional version of a kloosterman refinement

on the hilbert property and the fundamental group of algebraic varieties in this paper we link the so called hilbert property hp for an algebraic variety over a number field with its fundamental group in a perspective which appears new the notion of hp comes from hilbert s irreducibility theorem and has important implications for instance towards the inverse galois problem we shall observe that the hp is in a sense opposite to the chevalley weil theorem this shall immediately entail the result that the hp can possibly hold only for simply connected varieties in the appropriate sense in turn this leads to new counterexamples to the hp involving enriques surfaces in this view we shall also formulate an alternative related property possibly holding in full generality and some conjectures which unify part of what is known in the topic these predict that for a variety with a zariski dense set of rational points the validity of the hp which is defined arithmetically is indeed of purely topological nature also a consequence of these conjectures would be a positive solution of the inverse galois problem in the paper we shall also prove the hp for a k3 surface related to the above enriques surface providing what appears to be the first example of a variety non rational over mathbb c for which the hp can be proved in an appendix we shall further discuss among other things the hp also for kummer surfaces all of this basically exhausts the study of the hp for surfaces with a zariski dense set of rational points

pathologies of the brauer manin obstruction it had long been suspected that varieties x over a number field k without k points but with a non empty brauer manin set x ak are fairly common the first examples were found in 27 and then in 2 an earlier conditional example is given in 21 one should also expect that there are many varieties x without k points for which the etale brauer manin set x ak x ak is non empty we refer to 19 or 29 for the definition of these subsets of the space x ak of adeles of x different methods to construct such varieties have been found recently in 19 poonen constructs a threefold x with a surjective morphism to a curve c that has exactly one k point p and the fibre xp has points everywhere locally but not globally in poonen s example xp is a smooth ch telet surface the trick with a curve with just one rational point was also used in 11 where the fibres of x c are curves of high genus and xp is a singular curve which geometrically is a union of projective lines in retrospect one could note that the examples in 27 and 2 are families of genus 1 curves parameterised by elliptic curves of mordell weil rank 0 in this paper we propose more flexible methods to construct such examples we show that the varieties x such that x k and x ak include the following

new solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation a physical mathematical approach to anomalous diffusion is based on a generalized diffusion equation containing derivatives of fractional order in this paper an anomalous subdiffusion equation asub de is considered a new implicit numerical method inm and two solution techniques for improving the order of convergence of the inm for solving the asub de are proposed the stability and convergence of the inm are investigated by the energy method some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis these methods and supporting theoretical results can also be applied to other fractional integro differential equations and higher dimensional problems

a new difference scheme for the time fractional diffusion equation abstract in this paper we construct a new difference analog of the caputo fractional derivative called the l 2 1 formula the basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second and fourth order in space and the second order in time for the time fractional diffusion equation with variable coefficients are considered stability of the suggested schemes and also their convergence in the grid l 2 norm with the rate equal to the order of the approximation error are proved the obtained results are supported by the numerical calculations carried out for some test problems

arithmetic of del pezzo surfaces of degree 4 and vertical brauer groups we show that brauer classes of a locally solvable degree 4 del pezzo surface x are vertical for some projection away from a plane g x p 1 i e that every brauer class is obtained by pullback from an element of br k p 1 as a consequence we prove that a brauer class obstructs the existence of a k rational point if and only if all k fibers of g fail to be locally solvable or in other words if and only if x is covered by curves that each have no adelic points using work of wittenberg we deduce that for certain quartic del pezzo surfaces with nontrivial brauer group the algebraic brauer manin obstruction is sufficient to explain all failures of the hasse principle conditional on schinzel s hypothesis and the finiteness of tate shafarevich groups the proof of the main theorem is constructive and gives a simple and practical algorithm distinct from that in 5 for computing all classes in the brauer group of x modulo constant algebras c 2014 elsevier inc all rights reserved

universal torsors and values of quadratic polynomials represented by norms let k k be an extension of number fields and let p t be a quadratic polynomial over k let x be the affine variety defined by p t n k k mathbf z we study the hasse principle and weak approximation for x in three cases for k k 4 and p t irreducible over k and split in k we prove the hasse principle and weak approximation for k mathbb q with arbitrary k we show that the brauer manin obstruction to the hasse principle and weak approximation is the only one for k k 4 and p t irreducible over k we determine the brauer group of smooth proper models of x in a case where it is non trivial we exhibit a counterexample to weak approximation

cubic hypersurfaces and a version of the circle method for number fields a version of the hardy littlewood circle method is developed for number fields k q and is used to show that non singular projective cubic hypersurfaces over k always have a k rational point when they have dimension at least 8

rationally connected varieties over local fields let x be a proper variety defined over a field k following ma we say that two points x x x k are r equivalent if they can be connected by a chain of rational curves defined over k cf 4 1 in essence two points are r equivalent if they are obviously rationally equivalent several authors have proved finiteness results over local and global fields cubic hypersurfaces ma sd linear algebraic groups ct sa vo1 gi vo2 rational surfaces ct co ct1 ct sk1 quadric bundles and intersections of two quadrics ct sa sd pa su r equivalence is only interesting if there are plenty of rational curves on x at least over k such varieties have been studied in the series of papers komi mo1 ko mi mo3 see also ko1 there are many a priori different ways of defining what plenty of rational curves should mean fortunately many of these turn out to be equivalent and this leads to the notion of rationally connected varieties see ko mi mo2 ko1 iv 3 ko2 4 1 2

rational solutions of certain equations involving norms let k be an algebraic closure of k in the case when p t has at most one root in k the open subset of the affine variety 1 given by p t y o is a principal homogeneous space under an algebraic k torus in this case it is well known that the brauer manin obstruction is the only obstruction to the hasse principle and weak approximation on any smooth projective model of this variety colliot th l ne and sansuc csanl in this paper we prove the same result when p t has exactly two roots in k and no other roots in k and k is the field of rational numbers q an immediate change of variables then reduces 1 to the equation ta

finite difference spectral approximations for the distributed order time fractional reaction diffusion equation on an unbounded domain the numerical approximation of the distributed order time fractional reaction diffusion equation on a semi infinite spatial domain is discussed in this paper a fully discrete scheme based on finite difference method in time and spectral approximation using laguerre functions in space is proposed the scheme is unconditionally stable and convergent with order o 2 2 n 1 m 2 where n and m are the time step size step size in distributed order variable polynomial degree and regularity in the space variable of the exact solution respectively a pseudospectral scheme is also proposed and analyzed some numerical examples are presented to demonstrate the efficiency of the proposed scheme

two alternating direction implicit difference schemes with the extrapolation method for the two dimensional distributed order differential equations the grunwald formula is used to solve the two dimensional distributed order differential equations two alternating direction implicit adi difference schemes are derived it is proved that the schemes are unconditionally stable and convergent with the convergence orders o h 1 2 h 2 2 2 and o h 1 4 h 2 4 4 in h 1 norm respectively where h 1 h 2 and are the step sizes in time space in x and y direction and distributed order the extrapolation method is applied to improve the approximate accuracy to the orders o 2 ln 2 h 1 2 h 2 2 2 and o 2 ln 2 h 1 4 h 2 4 4 respectively several numerical examples are given to confirm the theoretical results

zero cycles on rational surfaces over number fields let x be a smooth projective geometrically integral surface defined over a perfect field k let kbe an algebraic closure of k and f be the function field of x k x then x is said to be rational if f is a purely transcendental extension of k the tame symbols define a map from ke f to the sum of function fields h k 7 of all irreducible divisors 7 on x let m be the cokernel of this map if k is a number field and kv the completion at a place v of k let kv be an algebraic closure of kv and m the corresponding cokernel for k x x further let i i i l k m be the kernel of the natural map from hl gal k k m to i h1 gal k v kv my in 5 w colliot th616ne and sansuc conjectured that

multi dimensional spectral tau methods for distributed order fractional diffusion equations abstract the distributed order fractional diffusion equation is a generalization of the standard fractional diffusion equation that can model processes lacking power law scaling over the whole time domain an important application of distributed order diffusions is to model ultraslow diffusion where a plume of particles spreads at a logarithmic rate to broaden the range of applicability of distributed order fractional diffusion models efficient numerical methods are needed to solve the model equation in this work we develop spectral tau schemes to discretize the fractional diffusion equation with distributed order fractional derivative in time and dirichlet boundary conditions the model solution is expanded in multi dimensions in terms of legendre polynomials and the discrete equations are obtained with the tau method numerical examples are provided to highlight the convergence rate and the flexibility of this approach the proposed spectral tau methods yield an exponential rate of convergence when the solution is smooth our results confirm that nonlocal numerical methods are best suited to discretize distributed order fractional differential equations as they naturally take the global behavior of the solution into account

uniform decay estimates and the lorentz invariance of the classical wave equation on etudie le comportement asymptotique des solutions de u 0 ou t 2 1 2 n 2 pour des conditions initiales u 0 u t g x en t 0 avec g reguliere a support compact dans r n

global solutions of nonlinear hyperbolic equations for small initial data on considere des systemes quasilineaires d equations hyperboliques d ordre 2 qui sont des deformations non lineaires de l equation d onde on etudie l existence globale et le comportement asymptotique des solutions du probleme de cauchy quand la donnee initiale est suffisamment petite

superconvergence of a discontinuous galerkin method for fractional diffusion and wave equations we consider an initial boundary value problem for partial tu partial t alpha nabla 2u f t that is for a fractional diffusion 1 alpha 0 or wave 0 alpha 1 equation a numerical solution is found by applying a piecewise linear discontinuous galerkin dg method in time combined with a piecewise linear conforming finite element method in space the time mesh is graded appropriately near t 0 but the spatial mesh is quasi uniform previously we proved that the error measured in the spatial l 2 norm is of order k 2 alpha h 2 ell k uniformly in t where k is the maximum time step h is the maximum diameter of the spatial finite elements alpha min alpha 0 le0 and ell k max 1 log k here we prove convergence of order k 3 2 alpha ell k h 2 at each time level t n for 1 alpha 1 thus if 1 2 alpha 1 then the dg solution is superconvergent which generalizes a known result for the classical heat equation i e the case alpha 0 a

towards the brauer manin obstruction on varieties fibred over the projective line abstract recently dasheng wei proved that the brauer manin obstruction is the only obstruction to the hasse principle for 0 cycles of degree 1 on some fibrations over the projective line defined by bi cyclic normic equations in the present paper we prove the exactness of the global to local sequence for chow groups of 0 cycles of such varieties this signifies that the brauer manin obstruction is also the only obstruction to weak approximation for 0 cycles of arbitrary degree our main theorem also generalizes several existing results

on manin s conjecture for a family of châtelet surfaces the manin conjecture is established for ch telet surfaces over q arising as minimal proper smooth models of the surface y 2 z 2 f x where f is a totally reducible polynomial of degree 3 without repeated roots these surfaces do not satisfy weak approximation

descent on fibrations over the projective line we formulate a general set up for the descent method of j l colliot thelene and j j sansuc applied to varieties fibred over the projective line over a number field this makes it possible to prove that the manin obstruction to the hasse principle and weak approximation is the only one provided there are only few degenerate fibres usually two three in some cases and that sufficiently many smooth fc fibres satisfy the hasse principle and weak approximation we introduce a new concept of what should be called a degenerate fibre the so called split fibres the property which depends only on the generic fibre and not on the choice of a particular model this yields a simpler and more general approach to the previous results of that kind 0 introduction motivated by hasse s theorem on quadratic forms one says that the hasse principle holds for a family of geometrically integral smooth and proper varieties over a number field k9 if for every member of this family the existence of rational points over all the completions of k implies the existence of a rational point the fact that the hasse principle holds for some family signif icantly simplifies the study of its arithmetic because the problem of determining whether or not there is a rational point is then reduced to a finite number of

rationally connected varieties over finite fields let x be a geometrically rational or more generally separably rationally connected variety over a finite field k we prove that if k is large enough then x contains many rational curves defined over k as a consequence we prove that r equivalence is trivial on x if k is large enough these results imply the following conjecture of j l colliot thelene let y be a rationally connected variety over a number field f for a prime p let y p denote the corresponding variety over the local field f p then for almost all primes p the chow group of 0 cycles on y p is trivial and r equivalence is trivial on y p