a fast numerical algorithm for the composition of power series with complex coefficients abstract the composition problem for univariate complex power series p q i e the computation of the first n 1 coefficients of the composition q p is numerically solved by interpolation methods using multitape turing machines as a model of computation the composition problem of power series with integer coefficients of modulus n is possible in time o n 2 log n log where m bounds the amount of time for the multiplication of m bit numbers e g m cm log m 1 log log m 2 for multitape turing machines this algorithm is asymptotically faster than an implementation of the brent kung algorithm on a multitape turing machine the improvement is of order n 1 2 up to logarithmic terms

optimal continuous dependence estimates for fractional degenerate parabolic equations we derive continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion the diffusion term involves the fractional laplace operator triangle alpha 2 for alpha in 0 2 our results are quantitative and we exhibit an example for which they are optimal we cover the dependence on the nonlinearities and for the first time the lipschitz dependence on in the bv framework the former estimate dependence on nonlinearity is robust in the sense that it is stable in the limits alpha downarrow 0 and alpha uparrow 2 in the limit alpha uparrow 2 triangle alpha 2 converges to the usual laplacian and we show rigorously that we recover the optimal continuous dependence result of cockburn and gripenberg j differ equ 151 2 231 251 1999 for local degenerate parabolic equations thus providing an alternative proof

julia a fresh approach to numerical computing bridging cultures that have often been distant julia combines expertise from the diverse fields of computer science and computational science to create a new approach to numerical computing julia is designed to be easy and fast and questions notions generally held to be laws of nature by practitioners of numerical computing beginlist item high level dynamic programs have to be slow item one must prototype in one language and then rewrite in another language for speed or deployment item there are parts of a system appropriate for the programmer and other parts that are best left untouched as they have been built by the experts endlist we introduce the julia programming language and its design a dance between specialization and abstraction specialization allows for custom treatment multiple dispatch a technique from computer science picks the right algorithm for the right circumstance abstraction which is what good computation is really about recognizes what remains the same after dif

relaxation based bounds for semi infinite programs finite formulations are presented for the calculation of lower and upper bounds on the optimal solution value of semi infinite programs sips involving smooth potentially nonconvex objective function and constraints the lower bounding problem is obtained by a formulation that combines the first and second order kkt necessary conditions of the lower level problem with a discretization of the parameter set the resulting mathematical program with equilibrium constraints mpec is a relaxation of the original sip and furnishes valid lower bounds if the parameter set is subdivided the optimal solution value of the lower bounding problem converges to the optimal solution value of the sip the upper bounding problem is based on convex and linear relaxations of the lower level problem resulting in a restriction of the sip if the parameter set is subdivided the constructed relaxations converge to the original lower level program the existence of sip slater points ensures convergence of the upper bounding problems to the optimal solution value of the sip several alternatives for the upper bounding problem are presented and discussed numerical results are presented for a number of test problems from the literature

fractal first order partial differential equations the present paper is concerned with semi linear partial differential equations involving a particular pseudo differential operator it investigates both fractal conservation laws and non local hamilton jacobi equations the idea is to combine an integral representation of the operator and duhamel s formula to prove on the one hand the key a priori estimates for the scalar conservation law and the hamilton jacobi equation and on the other hand the smoothing effect of the operator as far as hamilton jacobi equations are concerned a non local vanishing viscosity method is used to construct a viscosity solution when existence of regular solutions fails and a rate of convergence is provided turning to conservation laws global in time existence and uniqueness are established we also show that our formula allows us to obtain entropy inequalities for the non local conservation law and thus to prove the convergence of the solution as the non local term vanishes toward the entropy solution of the pure conservation law

a non local regularization of first order hamilton jacobi equations abstract in this paper we investigate the regularizing effect of a non local operator on first order hamilton jacobi equations we prove that there exists a unique solution that is c 2 in space and c 1 in time in order to do so we combine viscosity solution techniques and green s function techniques viscosity solution theory provides the existence of a w 1 solution as well as uniqueness and stability results a duhamel s integral representation of the equation involving the green s function permits to prove further regularity we also state the existence of c solutions in space and time under suitable assumptions on the hamiltonian we finally give an error estimate in l norm between the viscosity solution of the pure hamilton jacobi equation and the solution of the integro differential equation with a vanishing non local part

continuous dependence estimates for viscosity solutions of integro pdes we present a general framework for deriving continuous depen dence estimates for possibly polynomially growing viscosity solutions of fully nonlinear degenerate parabolic integro pdes we use this framework to pro vide explicit estimates for the continuous dependence on the coecients and the levy measure in the bellman isaacs integro pdes arising in stochas tic control dierenti al games moreover these explicit estimates are used to prove regularity results and rates of convergence for some singular perturbation problems finally we illustrate our results on some integro pdes arising when attempting to price european american options in an incomplete stock mar ket driven by a geometric levy process many of the results obtained herein are new even in the convex case where stochastic control theory provides an alternative to our pure pde methods

nonlinear robust optimization via sequential convex bilevel programming in this paper we present a novel sequential convex bilevel programming algorithm for the numerical solution of structured nonlinear min max problems which arise in the context of semi infinite programming here our main motivation are nonlinear inequality constrained robust optimization problems in the first part of the paper we propose a conservative approximation strategy for such nonlinear and non convex robust optimization problems under the assumption that an upper bound for the curvature of the inequality constraints with respect to the uncertainty is given we show how to formulate a lower level concave min max problem which approximates the robust counterpart in a conservative way this approximation turns out to be exact in some relevant special cases and can be proven to be less conservative than existing approximation techniques that are based on linearization with respect to the uncertainties in the second part of the paper we review existing theory on optimality conditions for nonlinear lower level concave min max problems which arise in the context of semi infinite programming regarding the optimality conditions for the concave lower level maximization problems as a constraint of the upper level minimization problem we end up with a structured mathematical program with complementarity constraints mpcc the special hierarchical structure of this mpcc can be exploited in a novel sequential convex bilevel programming algorithm we discuss the surprisingly strong global and locally quadratic convergence properties of this method which can in this form neither be obtained with existing sqp methods nor with interior point relaxation techniques for general mpccs finally we discuss the application fields and implementation details of the new method and demonstrate the performance with a numerical example

semi infinite programming theory methods and applications starting from a number of motivating and abundant applications in 2 including control of robots eigenvalue computations mechanical stress of materials and statistical design the authors describe a class of optimization problems which are referred to as semi infinite because their constraints bound functions of a finite number of variables on a whole region in 3 5 first and second order optimality conditions are derived for general nonlinear problems as well as a procedure for reducing the problem locally to one with only finitely many constraints another main effort for achieving simplification is through duality in 6 there algebraic properties of finite linear programming are brought to bear on duality theory in semi infinite programming section 7 treats numerical methods based on either discretization or local reduction with the emphasis on the design of superlinearly convergent sqp type methods taking this differentiable point of view this paper can be considered to be complementar

arb efficient arbitrary precision midpoint radius interval arithmetic arb is a c library for arbitrary precision interval arithmetic using the midpoint radius representation also known as ball arithmetic it supports real and complex numbers polynomials power series matrices and evaluation of many special functions the core number types are designed for versatility and speed in a range of scenarios allowing performance that is competitive with non interval arbitrary precision types such as mpfr and mpc floating point numbers we discuss the low level number representation strategies for precision and error bounds and the implementation of efficient polynomial arithmetic with interval coefficients

homogenization of first order equations with u varepsilon periodic hamiltonians part i local equations in this paper we present a result of homogenization of first order hamilton jacobi equations with u varepsilon periodic hamiltonians on the one hand under a coercivity assumption on the hamiltonian and some natural regularity assumptions we prove an ergodicity property of this equation and the existence of non periodic approximate correctors on the other hand the proof of the convergence of the solution usually based on the introduction of a perturbed test function in the spirit of evans work uses here a twisted perturbed test function for a higher dimensional problem

the cassels tate pairing on polarized abelian varieties let a e be a principally polarized abelian variety defined over a global field k and let a be its shafarevich tate group let a nd denote the quotient of a by its maximal divisible subgroup cassels and tate constructed a nondegenerate pairing a nd a nd q z if a is an elliptic curve then by a result of cassels the pairing is alternating but in general it is only antisymmetric using some new but equivalent definitions of the pairing we derive general criteria deciding whether it is alternating and whether there exists some alternating nondegenerate pairing on a nd these criteria are expressed in terms of an element c a nd that is canonically associated to the polarization e in the case that a is the jacobian of some curve a down to earth version of the result allows us to determine effectively whether a if finite is a square or twice a square we then apply this to prove that a positive proportion in some precise sense of all hyperelliptic curves of even genus g 2 over q have a jacobian with nonsquare if finite for example it appears that this density is about 13 for curves of genus 2 the proof makes use of a general result relating global and local densities this result can be applied in other situations

on the differentiability of the solution to the hamilton jacobi equation with critical fractional diffusion abstract we prove that the hamilton jacobi equation for an arbitrary hamiltonian h locally lipschitz but not necessarily convex and fractional diffusion of order one critical has classical c 1 solutions the proof is achieved using a new holder estimate for solutions of advection diffusion equations of order one with bounded vector fields that are not necessarily divergence free

the sphere packing problem in dimension 8the sphere packing problem in dimension 8 in this paper we prove that no packing of unit balls in euclidean space r 8 has density greater than that of the e8 lattice packing

discretization in semi infinite programming the rate of convergence the discretization approach for solving semi infinite optimization problems is considered we are interested in the convergence rate of the error between the solution of the semi infinite problem and the solution of the discretized program depending on the discretization mesh size it will be shown how this rate depends on whether the minimizer is strict of order one or two and on whether the discretization includes boundary points of the index set in a specific way this is done for ordinary and for generalized semi infinite problems

the sphere packing problem in dimension 24 building on viazovska s recent solution of the sphere packing problem in eight dimensions we prove that the leech lattice is the densest packing of congruent spheres in twenty four dimensions and that it is the unique optimal periodic packing in particular we find an optimal auxiliary function for the linear programming bounds which is an analogue of viazovska s function for the eight dimensional case

the complexity of class polynomial computation via floating point approximations we analyse the complexity of computing class polynomials that are an important ingredient for cm constructions of elliptic curves via complex floating point approximations of their roots the heart of the algorithm is the evaluation of modular functions in several arguments the fastest one of the presented approaches uses a technique devised by dupont to evaluate modular functions by newton iterations on an expression involving the arithmetic geometric mean it runs in time o d log 5 d log log d o d 1 epsilon o h 2 epsilon for any epsilon 0 where d is the cm discriminant and h is the degree of the class polynomial another fast algorithm uses multipoint evaluation techniques known from symbolic computation its asymptotic complexity is worse by a factor of log d up to logarithmic factors this running time matches the size of the constructed polynomials the estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary quadratic order and on a rigorously proven upper bound for the height of class polynomials

a fast algorithm for reversion of power series we give an algorithm for reversion of formal power series based on an efficient way to evaluate the lagrange inversion formula our algorithm requires o n 1 2 m n mm n 1 2 operations where m n and mm n are the costs of polynomial and matrix multiplication respectively this matches an algorithm of brent and kung but we achieve a constant factor speedup whose magnitude depends on the polynomial and matrix multiplication algo rithms used benchmarks confirm that the algorithm performs well in practice

on the pair correlation of zeros of the riemann zeta function in 1972 montgomery 20 21 introduced a new method for studying the zeros of the riemann zeta function one of his main accomplishments was to determine partially the pair correlation of zeros and to apply his results to obtain new information on multiplicity of zeros and gaps between zeros perhaps more importantly he conjectured on number theoretic grounds an asymptotic formula for the pair correlation of zeros and found that the form of this correlation exactly agreed with the gaussian unitary ensemble gue model for random hermitian matrices which had been studied earlier by physicists he was therefore able to formulate a general n correlation conjecture for zeros during the 1980s odlyzko 23 24 performed extensive numerical calculations of the correlations for zeros in ranges up to the 10th zero and found excellent agreement with the gue model more recently hejhal 17 was able to prove the same partial result for triple correlation as montgomery proved for pair correlation and rudnick and sarnak 25 have done the same for n correlation rudnick and sarnak extended their results to a large class of l functions and also showed that the riemann hypothesis rh is not needed for smoother forms of the asymptotic result very recently bogomolny and keating 1 2 have used a prime twin type conjecture to derive heuristically the n correlation conjecture beyond the range where the results of rudnick and sarnak apply the conclusion of all this work is to rmly establish but not prove the gue distribution for zeros of many zeta functions there is a dual relationship between zeros of the riemann zeta function and prime numbers following montgomery s work it was realized that information on pair correlation of zeros could be used to obtain information on primes this connection was developed by gallagher and mueller 10 and heath brown 15 later goldston and montgomery 14 found an equivalence under the riemann hypothesis between the pair correlation of zeros and the variance for the number of primes in short intervals this equivalence arises out of the explicit formula via a parseval relation together with a tauberian theorem from this work one sees that to obtain new information on primes from zeros will require some new insight on the zeros and while the connections to statistical physics mentioned above are a possible source of this insight so far no progress has been made on this fundamental problem the motivation for this paper is the observation that there are additional tools

minima of quasisuperminimizers abstract we show that unlike minima of superharmonic functions which are again superharmonic the same property fails for q quasisuperminimizers more precisely if u i is a q i quasisuperminimizer i 1 2 where 1 q 1 q 2 then u min u 1 u 2 is a q quasisuperminimizer but there is an increase in the optimal quasisuperminimizing constant q we provide the first examples of this phenomenon i e that q q 2 in addition to lower bounds for the optimal quasisuperminimizing constant of u we also improve upon the earlier upper bounds due to kinnunen and martio moreover our lower and upper bounds turn out to be quite close we also study a similar phenomenon in pasting lemmas for quasisuperminimizers where q q 1 q 2 turns out to be optimal and provide results on exact quasiminimizing constants of piecewise linear functions on the real line which can serve as approximations of more general quasiminimizers

a p adic algorithm to compute the hilbert class polynomial classically the hilbert class polynomial p z x of an imaginary quadratic discriminant a is computed using complex analytic techniques in 2002 couveignes and henocq suggested a p adic algorithm to compute p unlike the complex analytic method it does not suffer from problems caused by rounding errors in this paper we give a detailed description of the algorithm in the paper by couveignes and henocq and our careful study of the complexity shows that if the generalized riemann hypothesis holds true the expected runtime of the p adic algorithm is o log 8 e instead of o 1 e we illustrate the algorithm by computing the polynomial p 639 using a 643 adic algorithm

computing igusa class polynomials we give an algorithm that computes the genus two class polynomials of a primitive quartic cm eld k and we give a running time bound and a proof of correctness of this algorithm this is the rst proof of correctness and the rst running time bound of any algorithm that computes these polynomials our algorithm is based on the complex analytic method of spallek and van wamelen and runs in time e o 7 2 where is the discriminant of k

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

error estimates for approximate solutions to bellman equations associated with controlled jump diffusions we derive error estimates for approximate viscosity solutions of bellman equations associated to controlled jump diffusion processes which are fully nonlinear integro partial differential equations two main results are obtained i error bounds for a class of monotone approximation schemes which under some assumptions includes finite difference schemes and ii bounds on the error induced when the original levy measure is replaced by a finite measure with compact support an approximation process that is commonly used when designing numerical schemes for integro partial differential equations our proofs use and extend techniques introduced by krylov and barles jakobsen

explicit arithmetic intersection theory and computation of neron tate heights we describe a general algorithm for computing intersection pairings on arithmetic surfaces we have implemented our algorithm for curves over q and we show how to use it to compute regulators for a number of jacobians of smooth plane quartics and to numerically verify the conjecture of birch and swinnerton dyer for the jacobian of the split cartan curve of level 13 up to squares

bmo and morrey campanato estimates for stochastic convolutions and schauder estimates for stochastic parabolic equations abstract in this paper we are aiming to prove several regularity results for the following stochastic fractional heat equations with additive noises d u t x 2 u t x d t g t x d t u 0 0 t 0 t x g for a random field u t x 0 t g u t x u t x r where 2 2 0 2 is the fractional laplacian t 0 is arbitrarily fixed g r d is a bounded domain g 0 t g r is a joint measurable coefficient and t t 0 is either a brownian motion or a levy process on a given filtered probability space f p f t t 0 t to this end we derive the bmo estimates and morrey campanato estimates respectively for stochastic singular integral operators arising from the equations concerned then by utilizing the embedding theory between the campanato space and the holder space we establish the controllability of the norm of the space c 2 d where 0 d 0 t g with all these in hand we are able to show that the q th order bmo quasi norm of the q 0 order derivative of the solution u is controlled by the norm of g under the condition that t is a levy process finally we derive the schauder estimate for the p moments of the solution of the above stochastic fractional heat equations driven by levy noise

an optimal uncertainty principle in twelve dimensions via modular forms we prove an optimal bound in twelve dimensions for the uncertainty principle of bourgain clozel and kahane suppose f mathbb r 12 rightarrow mathbb r is an integrable function that is not identically zero normalize its fourier transform widehat f by widehat f xi int mathbb r d f x e 2 pi i langle x xi rangle dx and suppose widehat f is real valued and integrable we show that if f 0 le 0 widehat f 0 le 0 f x ge 0 for x ge r 1 and widehat f xi ge 0 for xi ge r 2 then r 1r 2 ge 2 and this bound is sharp the construction of a function attaining the bound is based on viazovska s modular form techniques and its optimality follows from the existence of the eisenstein series e 6 no sharp bound is known or even conjectured in any other dimension we also develop a connection with the linear programming bound of cohn and elkies which lets us generalize the sign pattern of f and widehat f to develop a complementary uncertainty principle this generalization unites the uncertainty principle with the linear programming bound as aspects of a broader theory

existence and uniqueness for parabolic problems with caputo time derivative abstract in this paper we are interested in the well posedness of fully nonlinear cauchy problems in which the time derivative is of caputo type we address this question in the framework of viscosity solutions obtaining the existence via perron s method and comparison for bounded sub and supersolutions by a suitable regularization through inf and sup convolution in time as an application we prove the steady state large time behavior in the case of proper nonlinearities and provide a rate of convergence by using the mittag leffler operator

prime sieves using binary quadratic forms we introduce an algorithm that computes the prime numbers up to n using o n log log n additions and n 1 2 o 1 bits of memory tie algorithm enumerates representations of integers by certain binary quadratic forms we present implementation results for this algorithm and one of the best previous algorithms

efficient cm constructions of elliptic curves over finite fields we present an algorithm that on input of an integer n 1 together with its prime factorization constructs a finite field f and an elliptic curve e over f for which e f has order n although it is unproved that this can be done for all n a heuristic analysis shows that the algorithm has an expected run time that is polynomial in 2 n log n where n is the number of distinct prime factors of n in the cryptographically relevant case where n is prime an expected run time o logn 4 e can be achieved we illustrate the efficiency of the algorithm by constructing elliptic curves with point groups of order n 10 2004 and n nextprime 102004 10 2004 4863

explicit bounds for primality testing and related problems many number theoretic algorithms rely on a result of ankeny which states that if the extended riemann hypothesis erh is true any nontrivial multiplicative subgroup of the integers modulo m omits a number that is o log2 m this has been generalized by lagarias montgomery and odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of number fields this paper gives a different proof of this theorem in which explicit constants are supplied the bounds imply that if the erh holds a composite number m has a witness for its compositeness in the sense of miller or solovay strassen that is at most 2 log2m

elliptic curves and primality proving this report describes the theory and implementation of the elliptic curve primality proving algorithm ecpp algorithm this includes the relationships between representing primes by quadratic forms and the explicit construction of class fields of imaginary quadratic fields the theory of elliptic curves with complex multiplication over the field of complex numbers as well as over finite fields we then use this theory to design a very powerful primality proving algorithm half of the paper is devoted to the description of its implementation in particular we give the currently best algorithms to speed up each part of the program the resulting program is very fast we can prove the primality of 100 digit numbers in less than five minutes on a sun 3 60 workstation and we can treat all numbers with less than 1000 digits in a reasonable amount of time using a distributed implementation

bounding s t and s_1 t on the riemann hypothesis let pi s t denote the argument of the riemann zeta function zeta s at the point s frac 1 2 it assuming the riemann hypothesis we present two proofs of the bound begin aligned s t le left frac 1 4 o 1 right frac log t log log t end aligned for large t this improves a result of goldston and gonek by a factor of 2 the first method consists of bounding the auxiliary function s 1 t int 0 t s u text d u using extremal functions constructed by carneiro littmann and vaaler we then relate the size of s t to the size of the functions s 1 t pm h s 1 t when h asymp 1 log log t the alternative approach bounds s t directly relying on the solution of the beurling selberg extremal problem for the odd function f x arctan left frac 1 x right frac x 1 x 2 this draws upon recent work by carneiro and littmann

hilbert spaces and the pair correlation of zeros of the riemann zeta function montgomery s pair correlation conjecture predicts the asymptotic behavior of the function n t defined to be the number of pairs and of ordinates of nontrivial zeros of the riemann zetafunction satisfying 0 0 using montgomery s formula and some extremal functions of exponential type these functions are optimal in the sense that they majorize and minorize the characteristic function of the interval in a way to minimize the l1 r 1 sin x x 2 dx error we give a complete solution for this extremal problem using the framework of reproducing kernel hilbert spaces of entire functions this extends previous work of p x gallagher 18 in 1985 where the case 1 2 n was considered using non extremal majorants and minorants

calculus on arithmetic surfaces in a2 and a3 arakelov introduces an intersection calculus for arithmetic surfaces that is for stable models of curves over a number field in this paper we intend to show that his intersection product has a lot of useful properties more precisely we show that the following properties from the theory of algebraic surfaces have an analogue in our situation

fractional semi linear parabolic equations with unbounded data this paper is devoted to the study of semi linear parabolic equations whose principal term is fractional i e is integral and eventually singular a typical example is the fractional laplace operator this work sheds light on the fact that if the initial datum is not bounded assumptions on the non linearity are closely related to its behavior at infinity the sub linear and super linear cases are first treated by classical techniques we next present a third original case if the associated first order hamilton jacobi equation is such that perturbations propagate at finite speed then the semi linear parabolic equation somehow keeps memory of this property by using such a result locally bounded initial data that are merely integrable at infinity can be handled next regularity of the solution is proved eventually strong convergence of gradients as the fractional term disappears is proved for strictly convex non linearity

computing heights on elliptic curves we describe how to compute the canonical height of points orn elliptic curves tate has given a rapidly converging series for archimedean local heights over r we describe a modified version of tate s series which also converges over c and give an efficient procedure for calculating local heights at non archimedean places in this way we can calculate heights over number fields having complex embeddings we also give explicit estimates for the tail of our series and present several examples let e be an elliptic curve defined over a number field k say given by a weierstrass equation 1 y2 alxy a3y x3 a2x2 a4x a6 the canonical height on e is a quadratic form h e k r for the definition and basic properties of h see 11 viii section 9 or 6 chapter vi the canonical height is an extremely important theoretical tool in the arithmetic theory of elliptic curves being used for such diverse purposes as studying values of l functions 5 numbers of integral points 12 and transcendence theory 9 it is also important as a computational tool such as its use in zagier s algorithm for finding integral points up to large bounds 18 it is thus of interest to have an efficient method for calculating the canonical height of a point the usual definition of h as a limit h p limn 0 4 nh x 2np is not practical for computation instead one uses the fact that the canonical height can be written as a sum of local heights one term for each distinct absolute value on k 2 h p e na p vemk for example if k q then mk can be identified with the set of rational primes together with the usual absolute value on q the multiplicities nv are chosen so that the product formula holds and so that h is independent of the choice of the field k the local height corresponding to a non archimedean absolute value is given by intersection theory in a well known manner see e g 2 4 or 7 chapter 11 section 5 we will describe a quick way to compute non archimedean local heights in section 5 the local height for an archimedean absolute value is given by a transcendental function and so efficient computation is somewhat more difficult j tate 15 1988 american mathematical society 0025 5718 88 1 00 25 per page 339 received august 20 1987 revised october 21 1987 1980 mathematics subject classification 1985 revision primary 11g05 14k07 11d25 this work was partially supported by nsf grant dms 8612393 current address mathematics department brown university providence ri 02912 this content downloaded from 207 46 13 103 on thu 20 oct 2016 04 37 48 utc all use subject to http about jstor org terms 340 joseph h silverman has given an easily computed power series which works for real absolute values precisely for a given curve e and point p x y he gives a sequence of easily computed numbers co cl so that

pair correlation estimates for the zeros of the zeta function via semidefinite programming abstract in this paper we study the distribution of the non trivial zeros of the riemann zeta function s and other l functions using montgomery s pair correlation approach we use semidefinite programming to improve upon numerous asymptotic bounds in the theory of s including the proportion of distinct zeros counts of small gaps between zeros and sums involving multiplicities of zeros

modular polynomials via isogeny volcanoes we present a new algorithm to compute the classical modular polynomial l in the rings z x y and z mz x y for a prime l and any positive integer m our approach uses the graph of l isogenies to eciently compute l mod p for many primes p of a suitable form and then applies the chinese remainder theorem crt under the generalized riemann hypoth esis grh we achieve an expected running time of o l 3 logl 3 log logl and compute l mod m using o l 2 logl 2 l 2 logm space we have used the new algorithm to compute l with l over 5000 and l mod m with l over 20000 we also consider several modular functions g for which g is smaller than l allowing us to handle l over 60000

existence of viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games recently m g crandall and p l lions trans amer math soc 277 1983 1 42 introduced the class of viscosity solutions of these equations and proved uniqueness within this class this paper discusses the existence of these solutions under assumptions closely related to the ones which guarantee the uniqueness

the adaptive convexification algorithm a feasible point method for semi infinite programming we present a new numerical solution method for semi infinite optimization problems its main idea is to adaptively construct convex relaxations of the lower level problem replace the relaxed lower level problems equivalently by their karush kuhn tucker conditions and solve the resulting mathematical programs with complementarity constraints this approximation produces feasible iterates for the original problem the convex relaxations are constructed with ideas from the alpha bb method of global optimization the necessary upper bounds for second derivatives of functions on box domains can be determined using the techniques of interval arithmetic where our algorithm already works if only one such bound is available for the problem we show convergence of stationary points of the approximating problems to a stationary point of the original semi infinite problem within arbitrarily given tolerances numerical examples from chebyshev approximation and design centering illustrate the performance of the method

regularization by 12 laplacian and vanishing viscosity approximation of hjb equations abstract we investigate the regularizing effect of adding small fractional laplacian with critical fractional exponent 1 2 to a general first order hjb equation our results include some regularity estimates for the viscosity solutions of such perturbations making the solutions classically well defined most importantly we use these regularity estimates to study the vanishing viscosity approximation to first order hjb equations by 1 2 laplacian and derive an explicit rate convergence for the vanishing viscosity limit

extremal functions in de branges and euclidean spaces abstract in this work we obtain optimal majorants and minorants of exponential type for a wide class of radial functions on r n these extremal functions minimize the l 1 r n x 2 2 n d x distance to the original function where 1 is a free parameter to achieve this result we develop new interpolation tools to solve an associated extremal problem for the exponential function f x e x where 0 in the general framework of de branges spaces of entire functions we then specialize the construction to a particular family of homogeneous de branges spaces to approach the multidimensional euclidean case finally we extend the result from the exponential function to a class of subordinated radial functions via integration on the parameter 0 against suitable measures applications of the results presented here include multidimensional versions of hilbert type inequalities extremal one sided approximations by trigonometric polynomials for a class of even periodic functions and extremal one sided approximations by polynomials for a class of functions on the sphere s n 1 with an axis of symmetry

empirical evidence for the birch and swinnerton dyer conjectures for modular jacobians of genus 2 curves this paper provides empirical evidence for the birch and swinnerton dyer conjectures for modular jacobians of genus 2 curves the second of these conjectures relates six quantities associated to a jacobian over the rational numbers one of these six quantities is the size of the shafarevich tate group unable to compute that we computed the five other quantities and solved for the last one in all 32 cases the result is very close to an integer that is a power of 2 in addition this power of 2 agrees with the size of the 2 torsion of the shafarevich tate group which we could compute

a widder s type theorem for the heat equation with nonlocal diffusion the main goal of this work is to prove that every non negative strong solutionu x t to the problem u t delta alpha 2 u 0 rm for x t in mathbb r n times 0 t 0 alpha 2 can be written as u x t int mathbb r n p t x y u y 0 dy where p t x frac 1 t n alpha p left frac x t 1 alpha right and p x int mathbb r n e i x cdot xi xi alpha d xi this result shows uniqueness in the setting of non negative solutions and extends some classical results for the heat equation by widder in 15 to the nonlocal diffusion framework

existence and regularity of a linear nonlocal fokker planck equation with growing drift abstract the nonlocal fokker planck equations for a class of stochastic differential equations with non gaussian stable levy motion in euclidean space are studied the existence and uniqueness of weak solution are obtained with rough drift the solution is shown to be smooth on spatial variable if all derivatives of the drift are bounded moreover the solution is jointly smooth on spatial and time variable if we assume further that the drift grows like a power of logarithm function at infinity

heights and arakelov s intersection theory introduction this paper relates the neron tate height on jacobian varieties to an intersection theory on arithmetic surfaces constructed by s arakelov a1 here i extend and prove a statement of arakelov to the effect that his intersection pairing is a neron pairing a2i i use this to prove an analogue of the hodge index theorem for arithmetic surfaces a result due independently to g faltings the results of this paper have been extracted from my thesis written under the direction of michael artin it differs from the thesis in three ways firstly it employs the language of local height functions as defined by a neron and j tate and expounded by s lang li in the thesis i used neron s language of local intersection pairings n1 the second major difference is in the proof of theorem 1 3 in my thesis i proved this result by using work of raynaud and picard functors r 1 this method was suggested by m artin in this paper i provide a simpler proof observed by s lang and independently by b gross finally the development of arakalov s theory in the context of neron functions in section 2 is due to s lang who has kindly given me leave to publish it this exposition of my results is due in considerable part to s lang it is essentially an edited transcript of a series of conversations between us which he provided it is my great pleasure to thank both him and michael artin for their help and encouragement 1 the neron pairing neron ne has given a pairing between divisors and points on abelian varieties and also on arbitrary varieties under suitable conditions we shall give a complement to the neron theory on curves let c be a curve by which we mean a complete regular curve geometrically irreducible over a field k with an absolute value v let

gaussian subordination for the beurling selberg extremal problem we determine extremal entire functions for the problem of ma jorizing minorizing and approximating the gaussian function e x 2 by en tire functions of exponential type the combination of the gaussian and a general distribution approach provides the solution of the extremal problem for a wide class of even functions that includes most of the previously known examples for instance 3 4 10 and 17 plus a variety of new interesting functions such asjxj for 1 log x 2 2 x 2 2 for 0 log x2 2 andx2n logx2 forn2 n further applications to number theory include optimal approximations of theta functions by trigonometric polynomi als and optimal bounds for certain hilbert type inequalities related to the discrete hardy littlewood sobolev inequality in dimension one

how to solve a semi infinite optimization problem abstract after an introduction to main ideas of semi infinite optimization this article surveys recent developments in theory and numerical methods for standard and generalized semi infinite optimization problems particular attention is paid to connections with mathematical programs with complementarity constraints lower level wolfe duality semi smooth approaches as well as branch and bound techniques in adaptive convexification procedures a section on recent genericity results includes a discussion of the symmetry effect in generalized semi infinite optimization

a class of extremal functions for the fourier transform we determine a class of real valued integrable functions f x and corresponding functions m x such that f x 1 and the value of mfo is miimi several applications of these functions to number theory and analysis are given

robustness in nonsmooth nonlinear multi objective programming abstract recently georgiev luc and pardalos 2013 robust aspects of solutions in deterministic multiple objective linear programming european journal of operational research 229 1 29 36 introduced the notion of robust efficient solutions for linear multi objective optimization problems in this paper we extend this notion to nonlinear case it is shown that under the compactness of the feasible set or convexity each robust efficient solution is a proper efficient solution some necessary and sufficient conditions for robustness with respect to the tangent cone and the non ascent directions are proved an optimization problem for calculating a robustness radius followed by a comparison between the newly defined robustness notion and two existing ones is presented moreover some alterations of objective functions preserving weak proper robust efficiency are studied

choosing the correct elliptic curve in the cm method we give an elementary way to distinguish between the twists of an ordinary elliptic curve e over f p in order to identify the one with p 1 2u points when p u 2 dv 2 with 2u 2v and e is constructed using the cm method for finding elliptic curves with a prescribed number of points our algorithms consist in most cases of reading off simple congruence conditions on u and v modulo 4

implementing the asymptotically fast version of the elliptic curve primality proving algorithm the elliptic curve primality proving ecpp algorithm is one of the current fastest practical algorithms for proving the primality of large numbers its running time cannot be proven rigorously but heuristic arguments show that it should run in time o log n 5 to prove the primality of n an asymptotically fast version of it attributed to j o shallit runs in time o log n 4 the aim of this article is to describe this version in more details leading to actual implementations able to handle numbers with several thousands of decimal digits

fractal hamilton jacobi kpz equations nonlinear and nonlocal evolution equations of the form u t lu u q where c is a pseudodifferential operator representing the infinitesimal generator of a levy stochastic process have been derived as models for growing interfaces in the case when the continuous brownian diffusion surface transport is augmented by a random hopping mechanism the goal of this paper is to study properties of solutions to this equation resulting from the interplay between the strengths of the diffusive linear and hyperbolic nonlinear terms posed in the whole space r n and supplemented with nonnegative bounded and sufficiently regular initial conditions

a standard branch and bound approach for nonlinear semi infinite problems this paper considers nonlinear semi infinite problems which contain at least one semi infinite constraint sic the standard branch and bound algorithm is adapted to such problems by extending usual upper and lower bounding techniques for nonlinear inequality constraints to sics this is achieved by defining the interval evaluation of parametrized functions and their generalized gradients by also adapting numerical constraint programming techniques to quantified inequalities and by introducing linear relaxations and restrictions for sics the overall efficiency of our algorithm is demonstrated on a standard set of benchmarks from the literature in comparison with the best state of the art alternative

applications of proximal calculus to fixed point theory on riemannian manifolds abstract we prove a general form of a fixed point theorem for mappings from a riemannian manifold into itself which are obtained as perturbations of a given mapping by means of general operations which in particular include the cases of sum when a lie group structure is given on the manifold and composition in order to prove our main result we develop a theory of proximal calculus in the setting of riemannian manifolds

new upper bounds on sphere packings i we continue the study of the linear programming bounds for sphere packing introduced by cohn and elkies we use theta series to give another proof of the principal theorem and present some related results and conjectures this article is in the arxiv as arxiv math mg 0110010

generalized gradients of lipschitz functionals abstract the purpose of this article is threefold i to present in a unified fashion the theory of generalized gradients whose elements are at present scattered in various sources ii to give an account of the ways in which the theory has been applied iii to prove new results concerning generalized gradients of summation functionals pointwise maxima and integral functionals on subspaces of l these last mentioned formulas are obtained with an eye to future applications in the calculus of variations and optimal control their proofs can be regarded as applications of the existing theory of subgradients of convex functionals as developed by rockafellar ioffe and levin valadier and others

holder continuous of the solutions to stochastic nonlocal heat equations abstract in this paper we aim to obtain the holder continuous of solutions to stochastic nonlocal heat equations by using campanato estimates and sobolev embedding theorem we first prove the holder continuous of the mild solution of stochastic nonlocal diffusion equations locally in r d in the sense that the solution u belongs to the space c d t l p then by using tail estimates we obtain the estimates of the mild solution in l p c d t lastly we prove the holder continuous of the mild solution on bounded domain

fast polynomial factorization and modular composition we obtain randomized algorithms for factoring degree n univariate polynomials over mathbb f q requiring o n 1 5 o 1 rm log 1 o 1 q n 1 o 1 rm log 2 o 1 q bit operations when rm log q n this is asymptotically faster than the best previous algorithms j von zur gathen and v shoup comput complexity 2 1992 pp 187 224 e kaltofen and v shoup math comp 67 1998 pp 1179 1197 for rm log q ge n it matches the asymptotic running time of the best known algorithms the improvements come from new algorithms for modular composition of degree n univariate polynomials which is the asymptotic bottleneck in fast algorithms for factoring polynomials over finite fields the best previous algorithms for modular composition use o n omega 1 2 field operations where omega is the exponent of matrix multiplication r p brent and h t kung j assoc comput mach 25 1978 pp 581 595 with a slight improvement in the exponent achieved by employing fast rectangular matrix multiplication x huang and v y pan j complexity 14 1998 pp 257 299 we show that modular composition and multipoint evaluation of multivariate polynomials are essentially equivalent in the sense that an algorithm for one achieving exponent alpha implies an algorithm for the other with exponent alpha o 1 and vice versa we then give two new algorithms that solve the problem near optimally an algebraic algorithm for fields of characteristic at most n o 1 and a nonalgebraic algorithm that works in arbitrary characteristic the latter algorithm works by lifting to characteristic 0 applying a small number of rounds of multimodular reduction and finishing with a small number of multidimensional ffts the final evaluations are reconstructed using the chinese remainder theorem as a bonus this algorithm produces a very efficient data structure supporting polynomial evaluation queries which is of independent interest our algorithms use techniques that are commonly employed in practice in contrast to all previous subquadratic algorithms for these problems which relied on fast matrix multiplication

computing canonical heights using arithmetic intersection theory the canonical height on an abelian variety a defined over a global field k is an object of fundamental importance in the study of the arithmetic of a for many applications it is required to compute p for a given point p a k for instance given generators of a subgroup of the mordell weil group a k of finite index this is necessary for most known approaches to the computation of generators of the mordell weil group a k furthermore the regulator of a k which appears in the statement of the conjecture of birch and swinnerton dyer is defined in terms of the canonical height and thus we need the ability to compute canonical heights in order to gather numerical evidence for the conjecture in the case of positive rank

non existence of positive solutions to nonlocal lane emden equations abstract this paper is concerned with non existence of positive solutions to nonlocal lane emden equations firstly we establish the fujita index for nonlocal reaction diffusion equations which is new in the blowup theory then the non existence of positive solutions to nonlocal lane emden equations is obtained because they are the limit stationary equations to the corresponding nonlocal reaction diffusion equations

on the lipschitz regularity of minimizers of anisotropic functionals abstract we prove a lipschitz regularity result for minimizers of functionals of the calculus of variations of the form f du x dx where f is a continuous convex function from r n into 0 not necessarily depending on the modulus of du

a note on the zeros of zeta and l functions let pi s t denote the argument of the riemann zeta function at the point s tfrac 1 2 it assuming the riemann hypothesis we give a new and simple proof of the sharpest known bound for s t we discuss a generalization of this bound for a large class of l functions including those which arise from cuspidal automorphic representations of gl m over a number field we also prove a number of related results including bounding the order of vanishing of an l function at the central point and bounding the height of the lowest zero of an l function

some extremal functions in fourier analysis ii we obtain extremal majorants and minorants of exponential type for a class of even functions on which includes log x and x where 1 1 we also give periodic versions of these results in which the majorants and minorants are trigonometric polynomials of bounded degree as applications we obtain optimal estimates for certain hermitian forms which include discrete analogues of the one dimensional hardy littlewood sobolev inequalities a further application provides an erdos turan type inequality that estimates the sup norm of algebraic polynomials on the unit disc in terms of power sums in the roots of the polynomials

a branch and prune algorithm for the computation of generalized aspects of parallel robots parallel robots enjoy enhanced mechanical characteristics that have to be contrasted with a more complicated design in particular they often have parallel singularities at some poses and the robots may become uncontrollable and could even be damaged in such configurations the computation of the connected components in the set of nonsingular reachable configurations called generalized aspects is therefore a key issue in their design this paper introduces a new method based on numerical constraint programming to compute a certified enclosure of the generalized aspects though this method does not allow counting their number rigorously it constructs inner approximations of the nonsingular workspace that allow commanding parallel robots safely it also provides a lower bound on the exact number of generalized aspects it is moreover the first general method able to handle any parallel robot in theory though its computational complexity currently restricts its usage to robots with three degrees of freedom finally the constraint programming paradigm it relies on makes it possible to consider various additional constraints e g collision avoidance making it suitable for practical considerations

computing hilbert class polynomials with the chinese remainder theorem we present a space efficient algorithm to compute the hilbert class polynomial h d x modulo a positive integer p based on an explicit form of the chinese remainder theorem under the generalized riemann hypothesis the algorithm uses o d 1 2 e log p space and has an expected running time of o d 1 e we describe practical optimizations that allow us to handle larger discriminants than other methods with d as large as 10 13 and h d up to 10 6 we apply these results to construct pairing friendly elliptic curves of prime order using the cm method