asymptotic stability of impulsive stochastic partial differential equations with infinite delays in this paper we study the existence and asymptotic stability in pth moment of mild solutions to nonlinear impulsive stochastic partial differential equations with infinite delay by employing a fixed point approach sufficient conditions are derived for achieving the required result these conditions do not require the monotone decreasing behaviour of the delays

fixed points and stability of neutral stochastic delay differential equations in this paper we consider a linear scalar neutral stochastic differential equation with variable delays and give conditions to ensure that the zero solution is asymptotically mean square stable by means of fixed point theory these conditions do not require the boundedness of delays nor do they ask for a fixed sign on the coefficient functions an asymptotic mean square stability theorem with a necessary and sufficient condition is proved some well known results are improved and generalized

probability inequalities for decomposition integrals recently in mathematical economics even and lehrer introduced the decomposition integral even and lehrer 2014 in this paper general versions of some well known probabilistic inequalities for the decomposition integrals and the superdecomposition integrals are discussed that are still open for research the main results of this paper generalize some previous results for particular integral inequalities obtained by several researchers in generalized probability theory

new algorithms for finding irreducible polynomials over finite fields we present a new algorithm for finding an irreducible polynomial of specified degree over a finite field our algorithm is deterministic and it runs in polynomial time for fields of small characteristic we in fact prove the stronger result that the problem of finding irreducible polynomials of specified degree over a finite field is deterministic polynomial time reducible to the problem of factoring polynomials over the prime field

simple classical groups of lie type are determined by their character degrees let g be a finite group denote by irr g the set of all irreducible complex characters of g let cd g be the set of all irreducible complex character degrees of g forgetting multiplicities that is cd g 1 irr g and let x1 g be the set of all irreducible complex character degrees of g counting multiplicities let h be a finite nonabelian simple classical group in this paper we will show that if g is a finite group and x1 g x1 h then g is isomorphic to h in particular this implies that the nonabelian simple classical groups of lie type are uniquely determined by the structure of their complex group algebras

height uniformity for integral points on elliptic curves we recall the result of d abramovich and its generalization by p pacelli on the uniformity for stably integral points on elliptic curves it says that the lang vojta conjecture on the distribution of integral points on a variety of logarithmic general type implies the uniformity for the numbers of stably integral points on elliptic curves in this paper we will investigate its analogue for their heights under the assumption of the vojta conjecture basically we will show that the vojta conjecture gives a naturally expected simple uniformity for their heights

factoring polynomials over algebraic number fields we show that if f x is a polynomial in z alpha x where alpha satisfies a monic irreducible polynomial over z then f x can be factored over q alpha x in polynomial time we also show that the splitting field of f x can be determined in time polynomial in splitting field of f x q log x

irreducible restrictions of representations of symmetric and alternating groups in small characteristics abstract building on reduction theorems and dimension bounds for symmetric groups obtained in our earlier work we classify the irreducible restrictions of representations of the symmetric and alternating groups to proper subgroups such a classification is known when the characteristic of the ground field is greater than 3 but the small characteristics cases require a substantially more delicate analysis and new ideas our results fit into the aschbacher scott program on maximal subgroups of finite classical groups

constructing nonresidues in finite fields and the extended riemann hypothesis we present a new deterministic algorithm for the problem of constructing kth power nonresidues in finite fields f p n where p is prime and k is a prime divisor of p n 1 we prove under the assumption of the extended riemann hypothesis erh that for fixed n and p our algorithm runs in polynomial time unlike other deterministic algorithms for this problem this polynomial time bound holds even if k is exponentially large more generally assuming the erh in time n log p o n we can construct a set of elements that generates the multiplicative group f p n an extended abstract of this paper appeared in proc 23rd ann acm symp on theory of computing 1991

solvability of norm equations over cyclic number fields of prime degree let l q be an abelian number field of prime degree q and let be a nonzero rational number we describe an algorithm which takes as input and the minimal polynomial of over q and determines if is a norm of an element of l we show that if we ignore the time needed to obtain a complete factorization of and a complete factorization of the discriminant of then the algorithm runs in time polynomial in the size of the input as an application we give an algorithm to test if a cyclic algebra a e over q is a division algebra

finding isomorphisms between finite fields we show that an isomorphism between two explicitly given finite fields of the same cardinality can be exhibited in deterministic polynomial time every finite field has cardinality p for some prime number p and some positive integer conversely if p is a prime number and a positive integer then there exists a field of cardinality p and any two fields of cardinality p are isomorphic these results are due to e h moore 1893 10 in the present paper we are interested in an algorithmic version of his theorem in particular of the uniqueness part we say that a finite field is explicitly given if for some basis of the field over its prime field we know the product of any two basis elements expressed in the same basis let more precisely p be a prime number and a positive integer then by explicit data for a finite field of cardinality p we mean a system of elements ajjk k x of the prime field fp z pz suchthat f becomes a field with the ordinary addition and multiplication by elements of fp and the multiplication determined by n

generalizations of holder s and some related inequalities in this paper we establish some new generalizations and refinements of holder s inequality and some related inequalities we also show that many existing inequalities related to the holder inequality are special cases of the inequalities presented

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

algorithms in algebraic number theory in this paper we discuss the basic problems of algorithmic algebraic number theory the emphasis is on aspects that are of interest from a purely mathematical point of view and practical issues are largely disregarded we describe what has been done and more importantly what remains to be done in the urea we hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them

classical and modular approaches to exponential diophantine equations i fibonacci and lucas perfect powers this is the first in a series of papers whereby we combine the classical approach to exponential diophantine equations linear forms in logarithms thue equations etc with a modular approach based on some of the ideas of the proof of fermat s last theorem in this paper we give new improved bounds for linear forms in three logarithms we also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the fibonacci sequence are 0 1 8 and 144 and the only perfect powers in the lucas sequence are 1 and 4

a rigorous time bound for factoring integers in this paper a probabilistic algorithm is exhibited that factors any positive integer n into prime factors in expected time at most ln 2 1 o for n oo where l a b exp b logx a loglogx l a many practical factoring algorithms including the quadratic sieve and the elliptic curve method are conjectured to have an expected running time that satisfies the same bound but this is the first algorithm for which the bound can be rigorously proved nevertheless this does not close the gap between rigorously established time bounds and merely conjectural ones for factoring algorithms this is due to the advent of a new factoring algorithm the number field sieve which is conjectured to factor any positive integer n in time ln i 0 1 the algorithm analyzed in this paper is a variant of the class group relations method which makes use of class groups of binary quadratic forms of negative discriminant this algorithm was first suggested by seysen and later improved by a k lenstra who showed that the algorithm runs in expected time at most ln 12 1 o 1 if one assumes the generalized riemann hypothesis the main device for removing the use of the generalized riemann hypothesis from the proof is the use of multipliers in addition a character sum estimate for algebraic number fields is used with an explicit dependence on possible exceptional zeros of the corresponding l functions another factoring algorithm using class groups that has been proposed is the random class groups method it is shown that there is a fairly large set of numbers that this algorithm cannot be expected to factor as efficiently as had previously been thought department of mathematics university of california berkeley california 94720 e mail address hwl math berkeley edu department of mathematics university of georgia athens georgia 30602 e mail address carl ada math uga edu this content downloaded from 157 55 39 186 on tue 12 apr 2016 08 54 17 utc all use subject to http about jstor org terms

asymptotically fast triangularization of matrices over rings this paper considers the problems of triangularizing and diagonalizing matrices over rings with particular emphasis on the integral case it begins with a description of fast algorithms for the computation of hermite and smith normal forms of integer matrices then it shows how to apply fast matrix multiplication techniques to the problem of triangularizing a matrix over a ring using elementary column operations these general results lead to an algorithm for triangularizing integer matrices that has a faster running time than the known hermite normal form algorithms the triangular matrix that is computed has small entries like the hermite normal form and will suffice for many applications

refinement of holder inequality and application to ostrowski inequality in this paper refinement of holder integral inequality is established and a necessary and sufficient condition for equality is obtained as an application we get an improved ostrowski integral inequality

using number fields to compute logarithms in finite fields we describe an adaptation of the number field sieve to the problem of computing logarithms in a finite field we conjecture that the running time of the algorithm when restricted to finite fields of an arbitrary but fixed degree is l q 1 3 64 9 1 3 o 1 where q is the cardinality of the field l q s c exp c log q s log log q 1 s and the o 1 is for q the number field sieve factoring algorithm is conjectured to factor a number the size of q in the same amount of time

a polynomial bound for the orders of primitive solvable groups the structure of primitive solvable permutation groups has bee r thoroughly investigated since galois relying on the known structure theorems we determine the largest order a n of a primitive solvable group of degree ii theorem 3 in the course of the proof we obtain the largest order a n of a transitive solvable group of degree n as well theorem 2 although the formula describing the exact value of i is quite complicated we are able to give an asymptotic evaluation it turns out thar the order of a primitive solvable group of degree n is bounded by 24 12 for a constant c 3 24399 theorem i a similar asymptotic estimate obtained recentiy by babai cameron and pilfy 111 shows that the orders of those primitive permutation groups of degree iz in which all nonabelian simple sections have orders less than a iixed integer k are bounded by n where c c k depends on k only related estimates under the assumption of the classification of finite simple groups are given by cameron 2 strong motivation for such asymptotic estimates comes from recent results of luks 151 on the complexity of graph isomorphism testing the results of the present paper have immediate application to graphs of vaience k p we mean that the field multiplication can be defined on the k dimensional vector space over gf p so that this equ ality holds 127 002 8693 82 g g i27 i so2 gg g

finite time stability of a class of fuzzy cellular neural networks with multi proportional delays abstract this paper is concerned with finite time stability for a class of fuzzy cellular neural networks with multi proportional delays based on the finite time stability theory we establish a novel result to ensure the finite time stability of the addressed system by employing the differential inequality techniques we also use numerical simulations to demonstrate our theoretical result

recognizing units in number fields we present a deterministic polynomial time algorithm that decides whether a power product ilki yn is a unit in the ring of integers of k where k is a number field yi are nonzero elements of k and ni are rational integers the main algorithm is based on the factor refinement method for ideals which might be of independent interest

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

factoring multivariate polynomials over algebraic number fields we present an algorithm to factor multivariate polynomials over algebraic number fields that is polynomial time in the degrees of the polynomial to be factored the algorithm is an immediate generalization of the polynomial time algorithm to factor univariate polynomials with rational coefficients

finiteness results for modular curves of genus at least 2 a curve x over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is modular if it is dominated by x 1 n for some n if in addition the image of its jacobian in j 1 n is contained in the new subvariety of j 1 n then x is called a new modular curve we prove that for each g 2 the set of new modular curves over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i of genus g is finite and computable for the computability result we prove an algorithmic version of the de franchis severi theorem similar finiteness results are proved for new modular curves of bounded gonality for new modular curves whose jacobian is a quotient of j 0 n new with n divisible by a prescribed prime and for modular curves new or not with levels in a restricted set we study new modular hyperelliptic curves in detail in particular we find all new modular curves of genus 2 explicitly and construct what might be the complete list of all new modular hyperelliptic curves of all genera finally we prove that for each field k of characteristic zero and g 2 the set of genus g curves over k dominated by a fermat curve is finite and computable

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

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

algorithmic enumeration of ideal classes for quaternion orders we provide algorithms to count and enumerate representatives of the right ideal classes of an eichler order in a quaternion algebra defined over a number field we analyze the run time of these algorithms and consider several related problems including the computation of two sided ideal classes isomorphism classes of orders connecting ideals for orders and ideal principalization we conclude by giving the complete list of definite eichler orders with class number at most 2

representations of the general linear groups which are irreducible over subgroups we classify all triples g v h such that sl n q leq g leq gl n q v is a representation of g of dimension greater than one over an algebraically closed field bbb f of characteristic prime to q and h is a proper subgroup of g such that the restriction v downarrow h is irreducible this problem is a natural part of the aschbacher scott program on maximal subgroups in finite classical groups

a generalized cameron kantor theorem abstract in this paper we prove that the non trivial unipotent characters of psl n q are reducible over any proper subgroup with the only exception of the unipotent character of psl 4 2 of degree 14 this result can be viewed as a generalization of a theorem of cameron and kantor

generalization of a sharp holder s inequality and its application abstract in this paper we generalize hu ke s sharpness of holder s inequality as application the obtained result is used to improve the well known opial olech inequality

a new sharpened and generalized version of holder s inequality and its applications abstract in this paper we show a new sharpened and generalized version of holder s inequality as applications the result is used to deal with an open problem proposed by k l chung an improvement of k l chung s result is obtained

a note on locality of algebras in 15 the second author stated two conjectures characterizing algebraic commutative local algebras with unit over a field f in this paper we settle conjecture 1 from 15 in positive and conjecture 2 in positive when charf 0 and in negative otherwise we also discuss locality of arbitrary commutative algebras with unit in this context as an application we propose deterministic and randomized polynomial time algorithms that test locality of finite dimensional algebras over algebraic number fields

fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays the fixed point theory is first used to consider the stability for stochastic partial differential equations with delays some conditions for the exponential stability in pth mean as well as in sample path of mild solutions are given these conditions do not require the monotone decreasing behavior of the delays which is necessary in t caraballo k liu exponential stability of mild solutions of stochastic partial differential equations with delays stoch anal appl 17 1999 743 763 ruhollan jahanipur stability of stochastic delay evolution equations with monotone nonlinearity stoch anal appl 21 2003 161 181 even in this special case our results also improve the results in t caraballo k liu exponential stability of mild solutions of stochastic partial differential equations with delays stoch anal appl 17 1999 743 763

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

fixed points and exponential stability for stochastic volterra levin equations in this paper we study a stochastic volterra levin equation by using fixed point theory we give some conditions for ensuring that this equation is exponentially stable in mean square and is also almost surely exponentially stable our result generalizes and improves on the results in 14 1 30

exponential stability of a class of nonlinear systems via fixed point theory abstract in this paper the problem of globally stochastically exponential stability in the p th moment for a class of t s fuzzy stochastic impulsive genetic regulatory networks with random discrete delays distributed delays and parameter uncertainties is discussed by utilizing the theory of stochastic analysis semigroup theory and fixed point theory a novel sufficient condition to guarantee the globally stochastically exponential stability in the p th moment of the considered genetic regulatory networks is derived eventually an example is given to demonstrate that the obtained result is effective

factoring multivariate integral polynomials an algorithm is presented to factorize polynomials in several variables with integral coefficients that is polynomial time in the degrees of the polynomial to be factored for any fixed number of variables the algorithm generalizes the algorithm presented by a k lenstra et al to factorize integral polynomials in one variable

a rigorous subexponential algorithm for computation of class groups let c d denote the gauss class group of quadratic forms of a negative discriminant d or equivalently the class group of the imaginary quadratic field q a we give a rigorous proof that there exists a las ve gas algorithm that will compute the structure of c d with an expected running time of l d vf2 1 bit operations where l d exp vlog d log log d thus of course also includes the computation of the class number h d the cardinality of c d ibm research division almaden research center k53 802 650 harry road san jose california 95 120 6099 current address k mccurley organization 1423 sandia national laboratories albuquerque new mexico 87185 this content downloaded from 207 46 13 180 on thu 08 sep 2016 05 47 29 utc all use subject to http about jstor org terms

degrees of rational characters of finite groups abstract a classical theorem of john thompson on character degrees states that if the degree of any complex irreducible character of a finite group g is 1 or divisible by a prime p then g has a normal p complement in this paper we consider fields of values of characters and prove some improvements of this result

on the computation of the class number of an algebraic number field it is shown how the analytic class number formula can be used to produce an algorithm which efficiently computes the class number h of an algebraic number field f the method assumes the truth of the generalized riemann hypothesis in order to estimate the residue of the dedekind zeta function of f at s 1 sufficiently well that h can be determined unambiguously

on a conjecture of g e wall abstract we prove that if g is a finite almost simple group having socle of lie type of rank r then the number of maximal subgroups of g is at most c r 2 3 g where c is an absolute constant this verifies a conjecture of wall for groups of sufficiently large rank using this we prove that any finite group g has at most 2 c g 3 2 maximal subgroups

rational points of bounded height on fano varieties a prime pe7z let v be an algebraic variety defined over f and li a metrized line bundle on v i e a system l consisting of a line bundle l and a family of banach v adic metrics on l f for all v s satisfying well known conditions cf 1 chap vi then we can define a height function hl x f i r by the formula h x is x 1 where s is a local section of l non vanishing

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

stability analysis of uncertain genetic sum regulatory networks this paper addresses the problem of establishing robust stability of uncertain genetic networks with sum regulatory functions specifically we first consider uncertain genetic networks where the regulation occurs at the transcriptional level and we derive a sufficient condition for robust stability by introducing a bounding set of the uncertain nonlinearity we hence show that this condition can be formulated as a convex optimization through polynomial lyapunov functions and polynomial descriptions of the bounding set by exploiting the square matricial representation smr of polynomials which allows to establish whether a polynomial is a sum of squares sos via a linear matrix inequality lmi then we propose a method for computing a family of bounding sets by means of convex optimizations it is worthwhile to remark that these results are derived in spite of the fact that the variable equilibrium point cannot be computed as being the solution of a system of parameter dependent nonlinear equations and is hence unknown lastly the proposed approach is extended to models where the regulation occurs at different levels and both mrna and protein dynamics are nonlinear

complexity of counting csp with complex weights we give a complexity dichotomy theorem for the counting constraint satisfaction problem csp in short with algebraic complex weights to this end we give three conditions for its tractability let f be any finite set of algebraic complex valued functions defined on an arbitrary finite domain we show that csp f is solvable in polynomial time if all three conditions are satisfied and is p hard otherwise our dichotomy theorem generalizes a long series of important results on counting problems and reaches a natural culmination a the problem of counting graph homomorphisms is the special case when f has a single symmetric binary function dyer and greenhill 2000 bulatov and grohe 2005 goldberg et al 2010 cai et al 2013 b the problem of counting directed graph homomorphisms is the special case when f has a single but not necessarily symmetric binary function dyer et al 2007 cai and chen 2010 c the unweighted form of csp is when all functions in f take values in 0 1 bulatov 2008 dyer and richerby 2013

parallel machine scheduling with high multiplicity in high multiplicity scheduling problems identical jobs are encoded in the efficient format of describing one of the jobs and the number of identical jobs similarly identical machines are efficiently encoded in the same manner we investigate parallel machine high multiplicity problems where there are three possible machine speed structures identical proportional or unrelated for the objectives of minimizing the sum of job completion times and minimizing the makespan we consider both nonpreemptive and preemptive problems for some problems we develop polynomial time algorithms for several problems we demonstrate that the recognition versions can be solved in polynomial time while the optimization versions require pseudo polynomial time we also show that changing from standard binary encoding to high multiplicity encoding does not affect the complexity class of np complete problems

black box galois representations abstract we develop methods to study 2 dimensional 2 adic galois representations of the absolute galois group of a number field k unramified outside a known finite set of primes s of k which are presented as black box representations where we only have access to the characteristic polynomials of frobenius automorphisms at a finite set of primes using suitable finite test sets of primes depending only on k and s we show how to determine the determinant det whether or not is residually reducible and further information about the size of the isogeny graph of whose vertices are homothety classes of stable lattices the methods are illustrated with examples for k q and for k imaginary quadratic being the representation attached to a bianchi modular form these results form part of the first author s thesis 2

rational points on symmetric products of a curve of course faltings theorem 3 says that the left hand side of 1 is actually bounded independently of x however mumford s result has certain uniformity properties which when combined with faltings theorem help one to bound the number of points in c k see 15 this idea is due to parshin in this paper we will give estimates similar to 1 for certain higher dimensional varieties for each integer d 1 let cd be the d fold symmetric product of the curve c thus c d is a smooth projective variety of dimension d as above let