analytic formulas for the regulator of a number field letr rk andw wk be the regulator and the number of roots of unity in the number fieldk we determine allk for whichr w 1 8 there are nine such fields sincew 2 we have allk withr 1 4 in particular we find the first three minima ofr over allk there are three main ingredients to the proof i a new analytic formula theorem a below ii a refinement of remak s geometric approach iii a refinement of zimmert s analytic method

classification error rate estimators evaluated by unconditional mean squared error in this article the criterion of unconditional mean squared error is used to compare four commonly used estimators of error rates in discriminant analysis the leave one out estimator which has relatively small bias is found to perform well relative to the other estimators when a large number of explanatory variables are used in the discriminant function with a small number of explanatory variables the large variance of this estimator results in poor performance we also find the estimators that assume normally distributed explanatory variables to be nonrobust when the parent distributions are skewed or have large tails

assessing the performance of an allocation rule the problem of estimating the error rates of a sample based rule on the basis of the same sample used in its construction is considered the apparent error rate is an obvious nonparametric estimate of the conditional error rate of a sample rule but unfortunately it provides too optimistic an assessment attention is focussed on the formation of improved estimates mainly through appropriate bias correction of the apparent error rate in this respect the role of the bootstrap a computer based methodology is highlighted

on effective computation of fundamental units ii introduction in part i of this paper we described the theoretical improvements of our method of computing fundamental units in algebraic number fields which we achieved in the last few years those improvements were of considerable influence on the corresponding computer program since its first 1976 2 implementation it was completely rewritten and changed in so many details that a new presentation cannot be avoided in sections 1 and 2 of this paper we describe the applications of part i 6 to the algorithm for constructing fundamental units section 3 contains a complete list of numerical examples concerning algebraic number fields of small degree and small absolute discriminant besides the fundamental units the tables contain much information about fields of degree five and six which was so far unknown the determination of those fields and their subfields is described in 3 besides the fundamental units we also listed the order of the torsion subgroup tuf of the unit group u whenever it is different from 2 it was computed by the methods of section 2 of 5 all computations were carried out on the control data cyber 76 of the computer center of the university of cologne

estimation of error rate for linear discriminant functions by resampling non gaussian populations abstract this article presents simulation results comparing various resampling estimators of classification error rate for linear discriminant type classification algorithms three non gaussian multivariate populations are studied namely exponential cauchy and uniform simulations are conducted for small sample sizes two class and three class problems and 2 d 3 d and 5 d distributions estimation procedures and sample sizes are the same as in our previous study of gaussian populations again 200 bootstrap replications are used for each simulation trial for exponential and uniform distributions the 0 632 estimator generally performs best however for cauchy distributions the convex bootstrap and the e 0 often outperform the 0 632 estimator

computing the structure of a finite abelian group we present an algorithm that computes the structure of a finite abelian group g from a generating system m the algorithm executes o m g group operations and stores o g group elements

error rate estimation in discriminant analysis a new derivation for a method to estimate error rates in discriminant analysis is presented known as the shrunken d or ds method the technique is evaluated in a sampling experiment along with seven other parametric methods as a means for estimating both the optimal and conditional error rates the results show that the best estimators are not the same for the two types of error rates and that sample size should influence the choice of an estimator

computing discrete logarithms in high genus hyperelliptic jacobians in provably subexponential time we provide a subexponential algorithm for solving the discrete logarithm problem in jacobians of high genus hyperelliptic curves over finite fields its expected running time for instances with genus g and underlying finite field fq satisfying g log q for a positive constant is given by o e f 1 3 2 3 2 o 1 g log q log g log q the algorithm works over any finite field and its running time does not rely on any unproven assumptions

an evaluation of smoothed classification error rate estimators this article examines the properties of smoothed estimators of the probabilities of misclassification in linear discriminant analysis and compares them with those of the resubstitution leave one out and bootstrap estimators smoothed estimators are found to have smaller variance than the other estimators and bias that is a function of the amount of smoothing an algorithm is presented for determining a reasonable level of smoothing as a function of the training sample sizes and the number of dimensions in the observation vector using the criterion of unconditional mean squared error this particular smoothed estimator called the ns method appears to offer a reasonable alternative to existing nonparametric estimators

error rate estimation in multiple group linear discriminant analysis a new method of error rate estimation in linear discriminant analysis appropriate to situations with several small groups is introduced it is compared with three other methods that can be generalized from the two group situation simulation studies show that in most situations it outperforms the other techniques

linearizing torsion classes in the picard group of algebraic curves over finite fields we address the problem of computing in the group of ell k torsion rational points of the jacobian variety of algebraic curves over finite fields with a view toward computing modular representations

computing discrete logarithms in real quadratic congruence function fields of large genus the discrete logarithm problem in various finite abelian groups is the basis for some well known public key cryptosystems recently real quadratic congruence function fields were used to construct a public key distribution system the security of this public key system is based on the difficulty of a discrete logarithm problem in these fields in this paper we present a probabilistic algorithm with subexponential running time that computes such discrete logarithms in real quadratic congruence function fields of sufficiently large genus this algorithm is a generalization of similar algorithms for real quadratic number fields

arithmeticity discreteness and volume we give an arithmetic criterion which is sufficient to imply the discreteness of various two generator subgroups of p sl 2 c we then examine certain two generator groups which arise as extremals in various geometric problems in the theory of kleinian groups in particular those encountered in efforts to determine the smallest co volume the margulis constant and the minimal distance between elliptic axes we establish the discreteness and arithmeticity of a number of these extremal groups the associated minimal volume arithmetic group in the commensurability class and we study whether or not the axis of a generator is simple

on the parallel generation of the residues for the continued fraction factoring algorithm etude des proprietes du developpement en fraction continue d un irrationnel quadratique application au developpement d un algorithme efficace de factorisation

computing discrete logarithms in the jacobian of high genus hyperelliptic curves over even characteristic finite fields we describe improved versions of index calculus algorithms for solving discrete logarithm problems in jacobians of high genus hyperelliptic curves de ned over even characteristic elds our rst improvement is to incorporate several ideas for the low genus case by gaudry and theriault including the large prime variant and using a smaller factor base into the large genus algorithm of enge and gaudry we extend the analysis in 24 to our new algorithm allowing us to predict accurately the number of random walk steps required to nd all relations and to select optimal degree bounds for the factor base our second improvement is the adaptation of sieving techniques from flassenberg and paulus and jacobson to our setting the new algorithms are applied to concrete problem instances arising from the weil descent attack methodology for solving the elliptic curve discrete logarithm problem demonstrating signi cant improvements in practice

smooth ideals in hyperelliptic function fields recently several algorithms have been suggested for solving the discrete logarithm problem in the jacobians of high genus hyperelliptic curves over finite fields some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are sufficiently dense we explicitly show how these density results can be derived all proofs are purely combinatorial and do not exploit analytic properties of generating functions

primitive divisors of lucas and lehmer sequences stewart reduced the problem of determining all lucas and lehmer sequences whose nth element does not have a primitive divisor to solving certain thue equations using the method of tzanakis and de weger for solving thue equations we determine such sequences for n 30 further computations lead us to conjecture that for n 30 the nth element of such sequences always has a primitive divisor

the index calculus method using non smooth polynomials we study a generalized version of the index calculus method for the discrete logarithm problem in f q when q p n p is a small prime and n the database consists of the logarithms of all irreducible polynomials of degree between given bounds the original version of the algorithm uses lower bound equal to one we show theoretically that the algorithm has the same asymptotic running time as the original version the analysis shows that the best upper limit for the interval coincides with the one for the original version the lower limit for the interval remains a free variable of the process we provide experimental results that indicate practical values for that bound we also give heuristic arguments for the running time of the waterloo variant and of the coppersmith method with our generalized database

explict bounds and heuristics on class numbers in hyperelliptic function fields in this paper we provide tight estimates for the divisor class number of hyperelliptic function fields we extend the existing methods to any hyperelliptic function field and improve the previous bounds by a factor proportional to g with the help of new results we thus obtain a faster method of computing regulators and class numbers furthermore we provide experimental data and heuristics on the distribution of the class number within the bounds on the class number these heuristics are based on recent results by katz and sarnak our numerical results and the heuristics imply that our approximation is in general far better than the bounds suggest

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

arithmetic on superelliptic curves this paper is concerned with algorithms for computing in the divisor class group of a nonsingular plane curve of the form yn c x which has only one point at infinity divisors are represented as ideals and an ideal reduction algorithm based on lattice reduction is given we obtain a unique representative for each divisor class and the algorithms for addition and reduction of divisors run in polynomial time an algorithm is also given for solving the discrete logarithm problem when the curve is defined over a finite field

hermite normal form computation using modulo determinant arithmetic this paper describes a new class of hermite normal form solution procedures which perform modulo determinant arithmetic throughout the computation this class of procedures is shown to possess a polynomial time complexity bound which is a function of the length of the input string computational results are also given

estimating the error rate of a prediction rule improvement on cross validation abstract we construct a prediction rule on the basis of some data and then wish to estimate the error rate of this rule in classifying future observations cross validation provides a nearly unbiased estimate using only the original data cross validation turns out to be related closely to the bootstrap estimate of the error rate this article has two purposes to understand better the theoretical basis of the prediction problem and to investigate some related estimators which seem to offer considerably improved estimation in small samples

the arithmetic of jacobian groups of superelliptic cubics we present two algorithms for the arithmetic of cubic curves with a totally ramified prime at infinity the first algorithm inspired by cantor s reduction for hyperelliptic curves is easily implemented with a few lines of code making use of a polynomial arithmetic package we prove explicit reducedness criteria for superelliptic curves of genus 3 and 4 which show the correctness of the algorithm the second approach quite general in nature and applicable to further classes of curves uses the fglm algorithm for switching between grobner bases for different orderings carrying out the computations symbolically we obtain explicit reduction formulae in terms of the input data

a table of fundamental pairs of units in totally real cubic fields we apply a method of cusick 5 to tabulate data on the first 250 totally real cubic fields f having discriminant d 6 885 apart from d we list the class number h and the regulator r of f also given are the integer coefficients a b c of a defining polynomial g x x3 ax2 bx c its index i and its largest zero r0 for j 1 2 we also tabulate both the integercoefficients x y z for the two units e x r r2 zj i with norm 1 forming a fundamental pair as well as the e and the integers f trace py2

resampling type error rate estimation for linear discriminant functions pearson vii distributions abstract this paper extends the work of a previous paper on error rate estimation for linear descriminant functions by considering additional non gaussian distributions from the pearson vii family the pearson vii family is an elliptically contoured family of probability densities with a parameter m that controls the tail length of the distribution for m 2 i e reasonably light tails the estimators perform similarly to previous gaussian case studies and previous simulated exponential and uniform distributions for m 1 6 i e heavy tails results similar to previous cauchy cases are obtained the 0 632 estimator generally performed best for large m but it does not perform as well as the e0 and convex bootstrap when the tails are heavy the positive bias of e0 no longer pertains when the tails are heavy

birkhoff james orthogonality and smoothness of bounded linear operators abstract we present a sufficient condition for smoothness of bounded linear operators on banach spaces for the first time let t a b x y where x is a real banach space and y is a real normed linear space we find sufficient condition for t b a t x b a x for some x s x with t x t and use it to show that t is a smooth point in b x y if t attains its norm at unique upto multiplication by scalar vector x s x tx is a smooth point of y and sup y c t y t for all closed subsets c of s x with d x c 0 for operators on a hilbert space h we show that t b a t x b a x for some x s h with t x t if and only if the norm attaining set m t x s h t x t s h 0 for some finite dimensional subspace h 0 and t h o t we also characterize smoothness of compact operators on normed spaces and bounded linear operators on hilbert spaces

on the infrastructure of the principal ideal class of an algebraic number field of unit rank one let r be the regulator and let d be the absolute value of the discriminant of an order 0 of an algebraic number field of unit rank 1 it is shown how the infrastructure idea of shanks can be used to decrease the number of binary operations needed to compute r from the best known 0 rd for most continued fraction methods to 0 rll2de these ideas can also be applied to significantly decrease the number of operations needed to determine whether or not any fractional ideal of 0 is principal

real and imaginary quadratic representations of hyperelliptic function fields a hyperelliptic function field can be always be represented as a real quadratic extension of the rational function field if at least one of the rational prime divisors is rational over the field of constants then it also can be represented as an imaginary quadratic extension of the rational function field the arithmetic in the divisor class group can be realized in the second case by cantor s algorithm we show that in the first case one can compute in the divisor class group of the function field using reduced ideals and distances of ideals in the orders involved furthermore we show how the two representations are connected and compare the computational complexity

characterization of k smooth operators between banach spaces abstract we study k smoothness of bounded linear operators defined between arbitrary banach spaces as an application we characterize k smooth operators defined from l 1 n to an arbitrary banach space we also completely characterize k smooth operators defined between arbitrary two dimensional banach spaces

smooth points in spaces of operators abstract for banach spaces x y in the space of bounded linear operators l x y we examine the relation between t l x y being a smooth point versus t l y x being a smooth point motivated by some results in the recent paper of paul et al we give some sufficient conditions for the validity of such a statement for spaces of operators

a double large prime variation for small genus hyperelliptic index calculus in this article we examine how the index calculus approach for computing discrete logarithms in small genus hyperelliptic curves can be improved by introducing a double large prime variation two algorithms are presented the first algorithm is a rather natural adaptation of the double large prime variation to the intended context on heuristic and experimental grounds it seems to perform quite well but lacks a complete and precise analysis our second algorithm is a considerably simplified variant which can be analyzed easily the resulting complexity improves on the fastest known algorithms computer experiments show that for hyperelliptic curves of genus three our first algorithm surpasses pollard s rho method even for rather small field sizes