knottedness is in np modulo grh abstract given a tame knot k presented in the form of a knot diagram we show that the problem of determining whether k is knotted is in the complexity class np assuming the generalized riemann hypothesis grh in other words there exists a polynomial length certificate that can be verified in polynomial time to prove that k is non trivial grh is not needed to believe the certificate but only to find a short certificate this result complements the result of hass lagarias and pippenger that unknottedness is in np our proof is a corollary of major results of others in algebraic geometry and geometric topology

ranks of elliptic curves this paper gives a general survey of ranks of elliptic curves over the field of rational numbers the rank is a measure of the size of the set of rational points the paper includes discussions of the birch and swinnertondyer conjecture the parity conjecture ranks in families of quadratic twists and ways to search for elliptic curves of large rank

constructing hyperelliptic curves of genus 2 suitable for cryptography in this article we show how to generalize the cm method for elliptic curves to genus two we describe the algorithm in detail and discuss the results of our implementation

checking the odd goldbach conjecture up to 10 20 vinogradov s theorem states that any sufficiently large odd integer is the sum of three prime numbers this theorem allows us to suppose the conjecture that this is true for all odd integers in this paper we describe the implementation of an algorithm which allowed us to check this conjecture up to 10 20

the period functionsʼ higher order derivatives abstract we prove a formula for the n th derivative of the period function t in a period annulus of a planar differential system for n 1 we obtain freire gasull and guillamon formula for the period s first derivative chicone and dumortier 1993 17 we apply such a result to hamiltonian systems with separable variables and other systems we give some sufficient conditions for the period function of conservative second order ode s to be convex

the primality of r1031 on decrit une technique permettant de demontrer que le nombre r1031 10 1031 1 9 compose uniquement de 1 est premier

a study on nonnegativity preservation in finite element approximation of nagumo type nonlinear differential equations preservation of nonnengativity and boundedness in the finite element solution of nagumo type equations with general anisotropic diffusion is studied linear finite elements and the backward euler scheme are used for the spatial and temporal discretization respectively an explicit an implicit and two hybrid explicitimplicit treatments for the nonlinear reaction term are considered conditions for the mesh and the time step size are developed for the numerical solution to preserve nonnegativity and boundedness the effects of lumping of the mass matrix and the reaction term are also discussed the analysis shows that the nonlinear reaction term has significant effects on the conditions for both the mesh and the time step size numerical examples are given to demonstrate the theoretical findings

on the computation of a table of complex cubic fields with discriminant d 10 6 a method for finding all the nonisomorphic complex cubic fields with discriminant d 106 is described three different methods were used to find the class number of each of these fields the speed of these techniques is discussed and several tables illustrating the computational results are presented these include tables of the distribution of the fields and the class numbers and the class group structures of these fields

effective primality tests for some integers of the forms 5ⁿ 1 and 7ⁿ 1 etablissement de tests de caractere premier deterministes et rapides fonctionnant en temps polynomial pour des nombres de la forme ar n 1 r 5 7 a r n

length dependence of solutions of fitzhugh nagumo equations a l aide de constructions et de methodes de semi groupe on obtient des resultats de decroissance sur les normes des differences entre la solution pour l infini et la solution l grand mais fini

the finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes we construct a new nonlinear finite volume scheme for diffusion equation on polygonal meshes and prove that the scheme satisfies the discrete extremum principle our scheme is locally conservative and has only cell centered unknowns numerical results are presented to show how our scheme works for preserving discrete extremum principle and positivity on various distorted meshes

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

primality testing and jacobi sums we present a theoretically and algorithmically simplified version of a primality testing algorithm that was recently invented by adleman and rumely the new algorithm performs well in practice it is the first pnmality test in existence that can routinely handle numbers of hundreds of decimal digits

a performance analysis of a simple prime testing algorithm this paper gives an empirical performance analysis of a prime proving program designed and implemented by the author and j l selfridge in 1974 the algorithm has been commonly referred to as the down algorithm because of its recursive characteristics it is shown among other things that of the 2270 primes tested 94 of them were proved in less than 3 seconds of ibm 360 67 time per number introduction there is a variety of algorithms available for using a computer to prove numbers prime see for example h c williams 6 the most straightforward method is to divide the number by each prime which is less than its square root if none of the remainders is zero the number is a prime this method has the advantage of being very easy to program but it uses a prohibitive amount of computer time for numbers exceeding 10 digits in length also the proof is nonconstructive in that the only way to verify the correctness of the proof is to repeat the calculation with another program perhaps on another computer recently michael 0 rabin 3 and r solovay and v strassen 5 have published probabilistic algorithms for testing primality these are quick relatively easy to program tests which can assert positively that a number n is composite but when it asserts the primality of n it does so with a very small probability of error that is n is asserted to be prime by a procedure that on the average will make no more than one mistake in 21 applications while these procedures can be useful in producing large collections of essentially composite free numbers they are not able to provide the mathematician with a conclusive proof of the primality of a single value of n in 1975 j l selfridge and the author 4 designed and implemented a program for prime proving which is not probabilistic and is effective for numbers between 10 and 35 decimal digits in length in this note we give a performance analysis of this algorithm based on the results of using the program on a collection of 2270 numbers in this range the algorithm for a detailed description of the program and the algorithm the reader should consult 4 but for completeness we will state the two relevant theorems their proofs can be found in 1 theorem 1 proth pocklington lehmer if n 1 2vp lpa2 pakc1 where cl has no prime factors less than b and 2vpalpa2 pakb nk then n is a prime if the following two conditions hold for any choices of bi received april 20 1980 revised february 8 1982 1980 mathematics subject classification primary 10a25 the research for this paper has been supported in part by the air force office of scientific research 1983 american mathematical society 0025 5718 82 oo0 0990 02 25

the monotonicity of period function for codimension four quadratic system q4 abstract in this paper we study the codimension four quadratic system q 4 z iz 4 z 2 2 z 2 b ic z 2 where b and c are real constants i 2 1 z x iy b ic 2 it is proved that the period function of periodic trajectories for q 4 is monotone the content of this paper can be viewed as a contribution to the proof of chicone s conjecture cf mr 94h 58072

construction of cm picard curves in this article we generalize the cm method for elliptic and hyperelliptic curves to picard curves we describe the algorithm in detail and discuss the results of our implementation

the factorization of the ninth fermat number in this paper we exhibit the full prime factorization of the ninth fermat number f9 2 512 1 it is the product of three prime factors that have 7 49 and 99 decimal digits we found the two largest prime factors by means of the number field sieve which is a factoring algorithm that depends on arithmetic in an algebraic number field in the present case the number field used was q fifth root 2 the calculations were done on approximately 700 workstations scattered around the world and in one of the final stages a supercomputer was used the entire factorization took four months

generalised mersenne numbers revisited generalised mersenne numbers gmns were defined by solinas in 1999 and feature in the nist fips 186 2 and secg standards for use in elliptic curve cryptography their form is such that modular reduction is extremely efficient thus making them an attractive choice for modular multiplication implementation however the issue of residue multiplication efficiency seems to have been overlooked asymptotically using a cyclic rather than a linear convolution residue multiplication modulo a mersenne number is twice as fast as integer multiplication this property does not hold for prime gmns unless they are of mersenne s form in this work we exploit an alternative generalisation of mersenne numbers for which an analogue of the above property and hence the same efficiency ratio holds even at bitlengths for which schoolbook multiplication is optimal while also maintaining very efficient reduction moreover our proposed primes are abundant at any bitlength whereas gmns are extremely rare our multiplication and reduction algorithms can also be easily parallelised making our arithmetic particularly suitable for hardware implementation furthermore the field representation we propose also naturally protects against side channel attacks including timing attacks simple power analysis and differential power analysis which is essential in many cryptographic scenarios in constrast to gmns

the structure factor of primes although the prime numbers are deterministic they can be viewed by some measures as pseudo random numbers in this article we numerically study the pair statistics of the primes using statistical mechanical methods particularly the structure factor in an interval with m large and l m smaller than unity we show that the structure factor of the prime number configurations in such intervals exhibits well defined bragg like peaks along with a small diffuse contribution this indicates that primes are appreciably more correlated and ordered than previously thought our numerical results definitively suggest an explicit formula for the locations and heights of the peaks this formula predicts infinitely many peaks in any non zero interval similar to the behavior of quasicrystals however primes differ from quasicrystals in that the ratio between the location of any two predicted peaks is rational we also show numerically that the diffuse part decays slowly as m and l increases this suggests that the diffuse part vanishes in an appropriate infinite system size limit

cnn model for studying dynamics and travelling wave solutions of fitzhugh nagumo equation in this paper a cellular neural network cnn model of fitzhugh nagumo equation is introduced dynamical behavior of this model is investigated using harmonic balance method for the cnn model of fitzhugh nagumo equation propagation of solitary waves have been proved

properties of the period function for some hamiltonian systems and homogeneous solutions of a semilinear elliptic equation abstract in this work we study the period function t of solutions to the conservative equation x t f x t 0 we present conditions on f that imply the monotonicity and convexity of t as a consequence we obtain the criterium established by c chicone and find conditions easier to apply we also get a condition obtained by cima gasull and manosas about monotonicity and following some of their calculations present results on the period function of hamiltonian systems where h x y f x n 1 y n using the monotonicity of t we count the homogeneous solutions to the semilinear elliptic equation u u 1 in two dimensions

abelian integrals and bifurcation theory abstract conditions are given for uniqueness of limit cycles for autonomous equations in the plane the results are applicable to codimension two bifurcations near equilibrium points for vector fields

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

ramanujan s class invariants and their use in elliptic curve cryptography the complex multiplication cm method is a method frequently used for the generation of elliptic curves ecs over a prime field f p the most demanding and complex step of this method is the computation of the roots of a special type of class polynomials called hilbert polynomials however there are several polynomials called class polynomials which can be used in the cm method having much smaller coefficients and fulfilling the prerequisite that their roots can be easily transformed to the roots of the corresponding hilbert polynomials in this paper we propose the use of a new class of polynomials which are derived from ramanujan s class invariants t n we explicitly describe the algorithm for the construction of the new polynomials and give the necessary transformation of their roots to the roots of the corresponding hilbert polynomials we provide a theoretical asymptotic bound for the bit precision requirements of all class polynomials and also using extensive experimental assessments we compare the efficiency of using the new polynomials against the use of the other class polynomials our comparison shows that the new class of polynomials clearly surpass all of the previously used polynomials when they are used in the generation of prime order elliptic curves

finding suitable curves for the elliptic curve method of factorization using the parametrizations of kubert we show how to produce in nite families of elliptic curves which have prescribed nontrivial torsion over q and rank at least one these curves can be used to speed up the ecm factorization algorithm of lenstra we also brie y discuss curves with complex multiplication in this context

explicit primality criteria for p 1 p n 1 deterministic polynomial time primality criteria for 2 n 1 have been known since the work of lucas in 1876 1878 little is known however about the existence of deterministic polynomial time primality tests for numbers of the more general form n n p 1 p n 1 where p is any fixed prime when n p 1 2 we show that it is always possible to produce a lucas like deterministic test for the primality of n n which requires that only o q p log q p 3 log n n modular multiplications be performed modulo n n as long as we can find a prime q of the form 1 kp such that n n k 1 is not divisible by q we also show that for all p with 3 p 10 7 such a q can be found very readily and that the most difficult case in which to find a q appears somewhat surprisingly to be that for p 3 some explanation is provided as to why this case is so difficult

speeding the pollard and elliptic curve methods of factorization since 1974 several algorithms have been developed that attempt to factor a large number n by doing extensive computations module n and occasionally taking gcds with n these began with pollard s p 1 and monte carlo methods more recently williams published a p 1 method and lenstra discovered an elliptic curve method ecm we present ways to speed all of these one improvement uses two tables during the second phases of p 1 and ecm looking for a match polynomial preconditioning lets us search a fixed table of size n with n 2 o n multiplications a parametrization of elliptic curves lets step 1 of ecm compute the x coordinate of np from that of p in about 9 3 1og2 n multiplications for arbitrary p

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

table errata new primality criteria and factorizations of 2 1 math comp 29 1975 620 647 mr 52 5546 by the author d h lehmer and j l selfridge

improved computation of square roots in specific finite fields in this paper we study exponentiation in the specific finite fields f with very special exponents such as those that occur in algorithms for computing square roots here q is a prime power q pk where k 1 and k is odd our algorithmic approach improves the corresponding exponentiation resulted from the better rewritten exponent to the best of our knowledge it is the first major improvement to the tonelli shanks algorithm for example the number of multiplications can be reduced to at least 60 percent on the average when p 1 mod 16 several numerical examples are given that show the speedup of the proposed methods

on the number of critical points of the period in the study of subharmonic bifurcations from a planar hamiltonian system it is often assumed that the period of a one parameter family of periodic orbits of the unperturbed system is a monotone function of the parameter see e g hale and taboas s chow and hale es however this hypothesis is in many instances difficult to verify for example it is well known that every orbit in the first quadrant of the vollterra lotka equation is periodic it is only in recent years that the monotonicity of the periods of these periodic orbits has been shown hsu 9 and waldvogel 151 such problems also occur in the study of bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation u f u o uer 1 1 on some interval in 14 smoller and wasserman showed that if f u u u u u cc

factorization of the tenth fermat number we describe the complete factorization of the tenth fermat number f10 by the elliptic curve method ecm f10 is a product of four prime factors with 8 10 40 and 252 decimal digits the 40 digit factor was found after about 140 mflop years of computation we also discuss the complete factorization of other fermat numbers by ecm and summarize the factorizations

the period function for quadratic integrable systems with cubic orbits this paper is concerned with the monotonicity of the period function for families of quadratic systems with centers whose orbits lie on cubic planar curves it is proved that each such system has a period function with at most one critical point

on the number of elliptic pseudoprimes for an elliptic curve e with complex multiplication by an order in k q v d a point p of infinite order on e and any prime p with d i p 1 we have that p 1 p o mod p where 0 is the point at infinity and calculations are done using the addition law for e any composite number which satisfies these conditions is called an elliptic pseudoprime in this paper it is shown that assuming the generalized riemann hypothesis elliptic pseudoprimes are less numerous than primes in particular on the grh the number of elliptic pseudoprimes less than x is o x log log x log2 x for certain curves it is shown that infinitely many elliptic pseudoprimes exist

finiteness for critical periods of planar analytic vector fields limburgs univ centrum dept math b 3610 diepenbeek belgium chicone c univ missouri dept math columbia mo 65211

turning points and traveling waves in fitzhugh nagumo type equations abstract consider the following fitzhugh nagumo type equation u t u x x f u w w t g u w where f u w u u a w 1 u for some smooth function a w and g u w u w by allowing a w to cross zero and one the corresponding traveling wave equation possesses special turning points which result in very rich dynamics in this work we examine the existence of fronts backs and pulses solutions in particular the co existence of different fronts will be discussed

physical bound preserving finite volume methods for the nagumo equation on distorted meshes abstract in this paper we present a boundedness preserving finite volume scheme for the nagumo equation in this method we use the implicit euler method for the time discretization and construct a maximum principle preserving discrete normal flux for the diffusion term for the nonlinear reaction term we design a type of picard iteration to ensure that at each iterative step it keeps physical boundedness moreover we prove that the numerical solution of the resulting scheme can preserve the bound of the solution for the nagumo equation on distorted meshes some numerical results are presented to verify the theoretical analysis

elliptic curves over finite fields and the computation of square roots mod p in this paper we present a deterministic algorithm to compute the number of f points of an elliptic curve that is defined over a finite field fv and which is given by a weierstrass equation the algorithm takes 0 log9 q elementary operations as an application wc give an algorithm to compute square roots mod p for fixed i e z it takes 0 log9p elementary operations to compute fx mod p 1 introduction in this paper we present an algorithm to compute the number of f points of an elliptic curve defined over a finite field f which is given by a weierstrass equation we restrict ourselves to the case where the characteristic of f is not 2 or 3 the algorithm is deterministic does not depend on any unproved hypotheses and takes 0 log9 0 if one applies fast multiplication techniques the algorithm will take 0 x 1 2log p 6 f elementary operations for any e 0 let be an elliptic curve defined over the prime field fp and let an affine model of it be given by a weierstrass equation y2 x3 ax b a befp an explicit formula for the number of f points on is given by

positivity preserving finite volume scheme for the nagumo type equations on distorted meshes abstract in this paper we present a nonlinear positivity preserving finite volume scheme for the nagumo type equations with anisotropic tensor diffusion coefficient for the diffusion term we use the positivity preserving finite volume scheme for the time direction we use the backward euler approximation we deal with nonlinear reaction term implicitly and decompose nonlinear reaction coefficient into two nonnegative functions thus we get a system of nonlinear algebraic equations the advantages of our scheme are that it can be applied to distorted meshes and has no severe constraint on the time step the numerical results verify the theoretical result

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

primality test for numbers m with a large power of 5 dividing m 4 1 the quintic reciprocity law is used to produce an algorithm that runs in polynomial time and that determines the primality of numbers m such that m4 1 is divisible by a power of 5 which is larger that m provided that a small prime p p 1 mod 5 is given such that m is not a fifth power modulo p the same test equations are used for all such m a sufficiency test together with its probability of succeeding in determining primality is given when the condition on m modulo p is omitted

stability of the travelling wave solution of the fitzhugh nagumo system on demontre la stabilite relative au systeme complet d equations aux derivees partielles de ces ondes de propagation

modularity of galois traces of class invariants zagier showed that the galois traces of the values of j invariant at cm points are fourier coefficients of a weakly holomorphic modular form of weight 3 2 and bruinier funke expanded his result to the sums of the values of arbitrary modular functions at heegner points in this paper we identify the galois traces of real valued class invariants with modular traces of the values of certain modular functions at heegner points so that they are fourier coefficients of weight 3 2 weakly holomorphic modular forms

implementation of a new primality test an implementation of the cohen lenstra version see ibid vol 42 p 297 330 1984 of the adleman pomerance rumely primality test see l m adleman c pomerance and r s rumely ann of math vol 117 p 173 206 1983 is presented primality of prime numbers of up to 213 decimal digits can now routinely be proved within approximately ten minutes

computational modeling of electrochemical coupling a novel finite element approach towards ionic models for cardiac electrophysiology we propose a novel efficient finite element solution technique to simulate the electrochemical response of excitable cardiac tissue we apply a global local split in which the membrane potential of the electrical problem is introduced globally as a nodal degree of freedom while the state variables of the chemical problem are treated locally as internal variables on the integration point level this particular discretization is efficient and highly modular since different cardiac cell models can be incorporated in a straightforward way through only minor local modifications on the constitutive level here we derive the underlying algorithmic framework for a recently proposed ionic model for human ventricular cardiomyocytes and demonstrate its integration into an existing nonlinear finite element infrastructure to ensure unconditional algorithmic stability we apply an implicit backward euler scheme to discretize the evolution equations for both the electrical potential and the chemical state variables in time to increase robustness and guarantee optimal quadratic convergence we suggest an incremental iterative newton raphson scheme and illustrate the consistent linearization of the weak form of the excitation problem this particular solution strategy allows us to apply an adaptive time stepping scheme which automatically generates small time steps during the rapid upstroke and large time steps during the plateau the repolarization and the resting phases we demonstrate that solving an entire cardiac cycle for a real patient specific geometry characterized through a transmembrane potential four ion concentrations thirteen gating variables and fifteen ionic currents requires computation times of less than ten minutes on a standard desktop computer

the critical points of the period function of x x2 x α x 1 0 0 α 1 on etudie les points critiques de la fonction periode des solutions periodiques de l equation x x 2 x x 1 0 0 2 1 on montre que le nombre de ces points critiques est borne

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