tight double linkage of gorenstein algebras abstract a gorenstein ideal k in a local ring r is in the class if there is a sequence of linked ideals i 0 i 1 i 2 n k with i 0 a complete intersection and i 2 i gorenstein for all i if k is in then k is linked to a complete intersection by a sequence of tight double links this means there is a sequence i 0 j 0 j 1 j 2m k such that for each p j 2 p 1 is an almost complete intersection of the form b y w and its immediate neighbors are linked to it by the similar regular sequences b y and b w respectively in this sense j 2 p and j 2 p 2 are tightly linked gorenstein ideals let k be an algebraically closed field and a k x 1 x q k with k in the k algebra a is rigid if and only if it is linked to a regular complete intersection by a sequence of semi generic tight double links it follows that if a is rigid then it is regular in codimension six

numerical chaos symplectic integrators and exponentially small splitting distances two discrete versions of duffing s equation are derived as reductions of discretizations of the nonlinear schrodinger nls equation inheriting the properties of the nsl discretizations one is shown to be integrable and using a mel nikov type analysis the other discretization is shown to be nonintegrable arguments are given why it is possible for the nonintegrable scheme to restore the homoclinic orbit at an exponentially fast rate which is faster than the order of the approximation as h 0 this suggests why there can be fundamental differences between symplectic and nonsymplectic discretizations of certain continuous hamiltonian systems

exponentially small splitting of separatrices in a weakly hyperbolic case abstract we validate the poincare melnikov method in the singular case of high frequency periodic perturbations of the hamiltonian h 0 x y 1 2 y 2 x 3 x 4 under appropriate conditions which among other things imply that we are considering the bifurcation case when the character of the fixed point changes from parabolic in the unperturbed case to hyperbolic in the perturbed one the splitting is exponentially small

algebra structures on resolutions of rings defined by grade four almost complete intersections abstract let r m k be a local ring in which 2 is a unit and let j be an indeal in r the minimal r free resolution of r j is called a dg algebra if it has the structure of an associative differential graded commutative algebra over r if j is a grade four almost complete intersection ideal in r then j is necessarily linked to a grade four gorenstein ideal i by a grade four complete intersection k it is well known that the minimal r free resolutions of r i and r k admit dg algebra structures in this paper we show that if w denotes the image of the map torr1 r k k torr1 r i k and w2 0 for example if k m i then the minimal r free resolution of s r j has a dg algebra structure in the case where the ring r is regular local and k m i we prove that the poincare series pns t of every finitely generated s module n is rational and the ring s satisfies the eisenbud conjecture

computing the dependence on a parameter of a family of unstable manifolds generalized melnikov formulas on considere la famille de diffeomorphismes de r 2 donnee par f s x f 0 x sf 1 x s s 2 f 2 x s on demontre que o n 1 f 1 x n 0 v n 1 0 2 n 1 df 1 x n 0 w n v n 1 f 2 x n 0 v n 1 ou w n y n 0

deformation and linkage of gorenstein algebras general double linkage of gorenstein algebras is defined rigidity genericity and regularity up to codimension six all pass across general double linkage rigid strongly unobstructed codimension four gorenstein algebras which lie in different herzog classes are produced introduction we begin by reviewing the classical structure theorems let p k xi x 41 a p i and f be the minimal resolution of a by free p modules f 1 if a has codimension 1 and i is unmixed then f is given by 0 p p for some fin p 2 hilbert burch 7 schaps 28 if a has codimension 2 and is cohen macaulay then f is given by x x 0 p n i ppnp where xis an n x n 1 matrix xi l 1l i x and a i x is the determinant formed by omitting the ith row from x 3 buchsbaum and eisenbud 4 if a has codimension 3 and its gorenstein then f is given by xt x x o p pn pn p where xis an n x n alternating matrix xi l i lpfi x and pfi x denotes the pfaffian of the alternating submatrix of x formed by omitting row and column i in these structure theorems as the codimension of a increases so does the number of conditions on a it is reasonable to suspect that a structure theorem for codimension four gorenstein algebras a will require that an additional hypothesis be imposed on a we observe that in each case a there is a structure theorem b all deformations are unobstructed see herzog 13 and c each algebra a is in the linkage class of a complete intersection see peskine and szpiro 27 and watanabe 30 historically these three facts were found independently with hindsight we now see that b follows from c and a follows in turn from b in his thesis received by the editors june 10 1982 and in revised form march 29 1983 some of the contents of this paper were presented to the special session on commutative algebra at the american mathematical society meeting in cincinnati on january 13 1982 1980 mathematics subject classification primary 13c05 13d25 13h10 secondary 14b07 14d115 14m05 16ao3 both authors were supported in part by the national science foundation 1984 american mathematical society 0002 9947 84 1 00 25 per page

quadratic residue codes of length 27 the structure of the supports of minimum weight words for the quadratic residue codes of length 27 is determined this does not depend on the alphabet field from the global code one can create a ternary code with the same structure

a generalized class of polynomials that are hard to factor a class of univariate polynomials is defined which make the berlekamp hensel factorization algorithm take an exponential amount of time this class contains as subclasses the swinnerton dyer polynomials discussed by berlekamp and a subset of the cyclotomic polynomials aside from shedding light on the complexity of polynomial factorization this class is also useful in testing implementations of the berlekamp hensel and related algorithms

algebra structures on some canonical resolutions the study of algebra structures on finite free resolutions of cyclic modules begins with buchsbaum and eisenbud b el the classes of cyclic modules whose minimal resolutions are known to admit an algebra structure include the residue field gu complete intersections koszul complex modules of homological dimension atmost 3 b el gorenstein of codimension four k m and herzog algebras k m23 avramov al gave examples to show that there are cyclic modules whose minimal resolutions do not admit an algebra structure let r be a noetherian local ring with maximal ideal m in this paper we construct algebra structures on the minimal resolutions of two classes of cyclic modules r i namely when z is of the form jk where j is an ideal generated by a regular sequence and when z is the ideal of maximal minors of a generic n x m matrix provided r contains the rationals in al avramov defined certain obstructions to the existance of an algebra structure on the minimal resolution of a module and then produced modules with non zero obstructions the general question is whether the vanishing of these obstructions is also sufficient for the existence of a minimal algebra resolution of a cyclic module when z is an ideal generated by a regular sequence in r and m r z for any positive integer k then these obstructions are all zero therefore avramov and schlessinger asked whether the minimal resolutions of r z admit an algebra structure corollary 3 6 of this paper provides an affirmative answer to this question

transversal homoclinic points of a class of conservative diffeomorphisms abstract we develop a global graph transformation to obtain estimates for certain invariant manifolds of a class of area preserving diffeomorphisms with symmetries the existence of homoclinic or heteroclinic points and the analyticity of the angle between the invariant manifolds at them is proved for the henon and the standard maps a lower bound of this angle is given for a large set of values of the parameters

the hardness of decoding linear codes with preprocessing the problem of maximum likelihood decoding of linear block codes is known to be hard the fact that the problem remains hard even if the code is known in advance and can be preprocessed for as long as desired in order to device a decoding algorithm is shown the hardness is based on the fact that existence of a polynomial time algorithm implies that the polynomial hierarchy collapses thus some linear block codes probably do not have an efficient decoder the proof is based on results in complexity theory that relate uniform and nonuniform complexity classes

multiplication of polynomials over finite fields the authors prove the 2 5 n o n lower bound on the number of multiplications divisions required to compute the coefficients of the product of two polynomials of degree n over a finite field by means of straight line algorithms

the minimal resolution of a codimension four almost complete intersection is a dg algebra abstract let r m k be a commutative noetherian local ring in which two is a unit we prove that if j is a five generated grade four perfect ideal in r then the minimal resolution of r j by free r modules is an associative differential graded commutative algebra this result extends and completes the work in j algebra 159 1993 1 46 where the conclusion is shown to hold provided certain technical conditions on tor are satisfied the multiplication on the resolution of r j is constructed using appropriate higher order multiplication on the resolution of r i where i is a gorenstein ideal which is linked to j

some comments on zech s logarithms some properties of zech s logarithms are stated and exploited to compute zech s logarithms from a restricted set of key elements zech s logarithms are recommended for the fast solution of quadratic cubic and quartic equations over gf 2 sup m

huneke ulrich almost complete intersections of cohen macaulay type two abstract let r m k be a commutative noetherian local ring n 2 an integer x a 2n 1 2n 1 alternating matrix with entries from m y a 1 2n 1 matrix with entries from m i the ideal i i1 yx and a the quotient ring r i assume that the grade of i is at least 2n in this case i is a perfect ideal of grade equal to 2n and i is minimally generated by 2n 1 elements we prove that the minimal resolution of a by free r modules is a dg algebra furthermore we identify the algebra torr a k and prove that if r is regular and char k 0 or n 2 2 char k then the poincare series pma z i 0 dimk torai m k zi is a rational function for every finitely generated a module m as a consequence we deduce that if the projective dimension of m is infinite then eventually the betti numbers of m form an increasing sequence with strong exponential growth

factoring polynomials over finite fields using differential equations and normal bases the deterministic factorization algorithm for polynomials over finite fields that was recently introduced by the author is based on a new type of linearization of the factorization problem the main ingredients are differential equations in rational function fields and normal bases of field extensions for finite fields of characteristic 2 it is known that this algorithm has several advantages over the classical berlekamp algorithm we develop the algorithm in a general framework and we show that it is feasible for arbitrary finite fields in the sense that the linearization can be achieved in polynomial time

on the distribution of quadratic residues and nonresidues modulo a prime number let p be a prime number and al at be distinct integers modulo p let x be chosen at random with uniform distribution in zp and let yi x ai we prove that the joint distribution of the quadratic characters of the yi s deviates from the distribution of independent fair coins by no more than t 3 xfi p that is the probability of yi y t matching any particular quadratic character sequence of length t is in the range i t i t 3 v ii p we establish the implications of this bound on the number of occurrences of arbitrary patterns of quadratic residues and nonresidues modulo p we then explore the randomness complexity of finding these patterns in polynomial time we give exponentially low upper bounds for the probability of failure achievable in polynomial time using as a source of randomness no more than one random number modulo p

some results on boundary value problems for fuzzy differential equations with functional dependence abstract we calculate the exact solution for a class of boundary value problems for first order linear fuzzy differential equations with impulses under hukuhara differentiability by finding green s function some conclusions can be derived for impulsive fuzzy differential equations including a term with functional dependence for which existence and uniqueness results are proved

univariate polynomial factorization over finite fields abstract this paper is a tutorial introduction to univariate polynomial factorization over finite fields we recall the classical methods that induced most factorization algorithms berlekamp s and the cantor zassenhaus ones and some refinements which can be applied to these methods explicit algorithms are presented in a form suitable for almost immediate implementation we give a detailed description of an efficient implementation of the cantor zassenhaus algorithm used in the release 2 of the axiom computer algebra system

on a new factorization algorithm for polynomials over finite fields a new deterministic factorization algorithm for polynomials over finite fields was recently developed by the first author the bottleneck in this algorithm is the last stage in which the irreducible factors of the polynomial are derived from the solutions of a system of linear equations an efficient approach to the last stage was designed by the second author for the case of finite fields of characteristic 2 in this paper we describe a different approach to the last stage which works for arbitrary fields of positive characteristic in particular we obtain in this way an acceleration of the factorization algorithm of the first author which makes this algorithm polynomial time for fixed characteristic

solving equations in finite fields and some results concerning the structure of gf p sup m new methods are given to solve equations over finite fields the methods exploit the structure of gf p sup m and are first developed for fields of characteristic two then the results are generalized for fields of characteristic p 2

poincare series of modules over local rings of small embedding codepth or small linking number abstract let s n k be a commutative noetherian local ring and m be a finitely generated s module then the poincare series of m is rational if any of the following holds edim s depth s 3 edim s depth s 4 and s is gorenstein s is one link from a complete intersection s is two links from a complete intersection and s is gorenstein to prove this result we assume without loss of generality that s r i with r m regular local and the key to the argument is to produce a factorization of the canonical map r s as a composition of a complete intersection r c with a golod map c s this is accomplished by invoking a theorem of avramov and backelin once one has existence of a dg algebra structure on a minimal r free resolution of s together with detailed knowledge of the structure of the induced homology algebra tor r s k h k s linkage theory provides the main technical tool for this analysis

primitive polynomials over finite fields in this note we extend the range of previously published tables of primitive polynomials over finite fields for each p 1050 with p 97 we provide a primitive polynomial of degree n over fp moreover each polynomial has the minimal number of nonzero coefficients among all primitives of degree n over fp

formulas for the solutions of quadratic equations over gf 2 m corresp there are several known approaches to solving quadratic equations over gf 2 m here the approach to solving the equations is different from the known approaches in that the solutions are presented in terms of the parameters of the equations the formulas for solving the quadratic equations are also used to solve the equations of degrees three and four the results can be combined with other known techniques to solve equations of higher degrees

h fuzzy control for nonlinear time delay singular markovian jump systems with partly unknown transition rates abstract this paper investigates the problem of h fuzzy control for a class of nonlinear time delay singular markovian jump systems with partly unknown transition rates this class of systems under consideration is described by takagi sugeno t s fuzzy model the goal of the paper is to design fuzzy state feedback controllers such that systems with partly unknown transition rates are not only regular impulse free and stochastically stable but also satisfy a prescribed h performance for all delays no larger than a given upper bound in terms of linear matrix inequalities and systems with completely known transition rates can be viewed as a special case of the one we consider here finally numerical examples are given to illustrate the merit and usability of the approach proposed in this paper

an architecture for computing zech s logarithms in gf 2 sup m in this paper a new method for calculation of zech s logarithm in gf 2 sup m is presented for a given element the logarithm is calculated by bit operations performed on its binary representation no look up tables are used the proposed method makes feasible the implementation of an universal type device for finite field arithmetic

cryptanalytic attacks on the multiplicative knapsack cryptosystem and on shamir s fast signature scheme the basic merkle hellman additive trapdoor knapsack public key cryptosystem was recently shown to be insecure and attacks have also been developed on stronger variants of it such as the graham shamir system and the iterated knapsack cryptosystem it is shown that some simple variants of another merkle hellman system the multiplicative knapsack cryptosystem are insecure it is also shown that the shamir fast signature scheme can be broken quickly similar attacks can also be used to break the schobi massey authentication scheme these attacks have not been rigorously proved to succeed but heuristic arguments and empirical evidence indicate that they work on systems of practical size