a constructive proof of the fefferman stein decomposition of bmo rn let s vec r left left f r 1 f cdots r n f right f in h 1 left text r n text r right right subset h 1 left text r n text r n 1 right 22 1 begin gathered s vec r left vec g left g j right j 0 cdots n right in text bmo left text r n text r n 1 right sum limits j 0 n vec r j g j 0 text in bmo left text r n text r n 1 right left text r right end gathered 22 2 j 0

weak type estimates for a singular convolution operator on the heisenberg group on the heisenberg group ehn with coordinates z t e cn x r define the distribution k z t l z 6 t where l z is a homogeneous distribution on cn of degree 2n smooth away from the origin and 3 t is the dirac mass in the t variable we prove that the operator given by convolution with k maps hl h n to weak l1 hn

acceleration of euclidean algorithm and rational number reconstruction we accelerate the known algorithms for computing a selected entry of the extended euclidean algorithm for integers and consequently for the modular and numerical rational number reconstruction problems the acceleration is from quadratic to nearly linear time matching the known complexity bound for the integer gcd which our algorithm computes as a special case

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

a probabilistic factorization algorithm with quadratic forms of negative discriminant we propose a probabilistic algorithm for factorization of an integer n with run time expvlog n log log n v54 1 asymptotically our algorithm will be as fast as the wellknown factorization algorithm of morrison and brillhart the latter algorithm will fail in several cases and heuristic assumptions are needed for its run time analysis our new algorithm will be analyzed under the assumption of the extended riemann hypothesis and it will be of las vegas type on input n the new algorithm will factor n with probability 12 in case of prime n the algorithm will prove the primality of n with probability 2 introduction until the last decade the centuries old problem of factoring integers was mainly a problem for specialists worldwide interest in factoring integers increased dramatically in 1978 when rivest shamir and adleman 32 published their public key cryptosystem whose security relies on the fact that some large integers are hard to factor gauss 9 already discovered a close connection between the factorization of a natural number n and the theory of quadratic forms of discriminant 4n now quadratic forms are one of the most important tools for factoring integers examples for efficient factorization algorithms are among others the algorithms of morrison and brillhart 26 and lenstra and schnorr 22 the former works with the continued fraction expansion of fn which is closely related to the theory of quadratic forms of discriminant 4n see 20 while the latter works with quadratic forms of discriminant 4n at present the most efficient factorization algorithm is the quadratic sieve algorithm see 29 which can also be expressed in terms of quadratic forms for an overview of modem factorization algorithms we refer to the papers of guy 10 monier 25 and pomerance 29 a deeper understanding of the theory of quadratic forms is of great importance for the analysis of modem factorization algorithms in this paper we shall only deal with the theory of quadratic forms of negative discriminant which is considerably more simple than the theory of forms of positive discriminants using this theory we obtain a probabilistic factorization algorithm with run time exp vlog n log log n v4 in this paper we denote by log x the natural logarithm of x we have 5 4 1 118 received may 17 1985 revised june 12 1986 1980 mathematics subject classification primary 10a30 68c25 1987 american mathematical society 0025 5718 87 1 00 25 per page

defect free global minima in thomson s problem of charges on a sphere given n unit points charges on the surface of a unit conducting sphere what configuration of charges minimizes the coulombic energy ensuremath sum igj 1 n 1 r ij due to an exponential rise in good local minima finding global minima for this problem or even approaches to do so has proven extremely difficult for n 10 h 2 hk k 2 2 recent theoretical work based on elasticity theory and subsequent numerical work has shown that for n ensuremath gtrsim 500 char21 1000 adding dislocation defects to a symmetric icosadeltahedral lattice lowers the energy here we show that in fact this approach holds for all n and we give a complete or near complete catalogue of defect free global minima

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

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 l 1 3 algorithm for ideal class group and regulator computation in certain number fields we analyse the complexity of the computation of the class group structure regulator and a system of fundamental units of a certain class of number fields our approach differs from buchmann s who proved a complexity bound of l 1 2 o 1 when the discriminant tends to infinity with fixed degree we achieve a subexponential complexity in o l 1 3 o 1 when both the discriminant and the degree of the extension tend to infinity by using techniques due to enge and gaudry in the context of algebraic curves over finite fields

random walks on discrete groups of polynomial volume growth let mu be a probability measure with finite support on a discrete group gamma of polynomial volume growth the main purpose of this paper is to study the asymptotic behavior of the convolution powers mu n of mu if mu is centered then we prove upper and lower gaussian estimates we prove a central limit theorem and we give a generalization of the berry esseen theorem these results also extend to noncentered probability measures we study the associated riesz transform operators the main tool is a parabolic harnack inequality for centered probability measures which is proved by using ideas from homogenization theory and by adapting the method of krylov and safonov this inequality implies that the positive mu harmonic functions are constant finally we give a characterization of the mu harmonic functions which grow polynomially

on the regularity of inverses of singular integral operators sur r d on considere un operateur de convolution sf f k ou k est une distribution homogene de degre d on suppose que s s etend a un operateur borne et inversible sur l 2 r d l inverse de s est donne par convolution avec une distribution l aussi homogene de degre d on montre qu une regularite possedee par k est partagee par l

an octahedral equal area partition of the sphere and near optimal configurations of points we construct a new area preserving map from the unit sphere to the regular octahedron both centered at the origin its inverse map allows the construction of uniform and refinable grids on a sphere starting from any triangular uniform and refinable grid on the faces of the octahedron we prove that our new grids are diameter bounded and then for the resulting configurations of points we calculate some riesz s energies and we compare them with the optimal ones for some configurations we also calculate the point energies and we list the minimum and maximum values concluding that these values are very close finally we show how we can map a hemisphere of the earth onto a square using our new area preserving projection the simplicity and the symmetry of our formulas lead to fast computations

dependency of units in number fields we develop a method for validating the indepencence of units in algebraic number fields in case that a given system of units has a dependency we compute a certificate for this

bmol hn spaces and carleson measures for schrodinger operators abstract let l h n v be a schrodinger operator on the heisenberg group h n where h n is the sub laplacian and the nonnegative potential v belongs to the reverse holder class b q 2 here q is the homogeneous dimension of h n in this article we investigate the dual space of the hardy type space h l 1 h n associated with the schrodinger operator l which is a kind of bmo type space bmo l h n defined by means of a revised sharp function related to the potential v we give the fefferman stein type decomposition of bmo l functions with respect to the adjoint riesz transforms r j l for l and characterize bmo l h n in terms of the carleson measure we also establish the bmo l boundedness of some operators such as the adjoint riesz transforms r j l the littlewood paley function s q l the lusin area integral s q l the hardy littlewood maximal function and the semigroup maximal function all results hold for stratified groups as well

energies and spacings of point charges on a sphere numerical methods are in general required for the determination of the stable configurations of n point charges on a sphere the stable configurations for n up to 50 have previously been ascertained and the authors extend the calculations for values up to 101 they report for the first time some remarkable global features of these configurations they show that the minimum energy accurately follows a simple half integral power law in 1 n over the full range they have investigated this power law is explicable in terms of the idealization of mapping a planar wigner lattice onto the surface of the unit sphere the pair distribution functions of the larger n configurations indicate predominant hexagonal coordination the coefficients of the observed power law are closely straddled by values calculated on the basis of hexagonal and square wigner lattices this highly accurate description of the energy permits them to remark on the detailed deviations of the individual structures from the general trend for n 30 they note that structures with n prime are relatively less stable while structures with n equal to 6 12 32 44 48 and 60 seem more stable

hardy space estimates of hausdorff operators on the heisenberg group abstract this is a continuation of a series of works about hausdorff operator on hardy spaces in the setting of the heisenberg group in this paper we obtain the boundedness from lebesgue spaces to hardy spaces for fractional hausdorff operators and their compositions with riesz transforms we also establish the boundedness for two kinds of special hausdorff operators the hausdorff poisson operator and the hausdorff gauss operator on hardy spaces

computing residue class rings and picard groups of orders abstract let k be a global field and o be an order of k we develop algorithms for the computation of the unit group of residue class rings for ideals in o as an application we show how to compute the unit group and the picard group of o provided that we are able to compute the unit group and class group of the maximal order o of k

asymptotics for minimal discrete energy on the sphere we investigate the energy of arrangements of n points on the surface of the unit sphere sd in rd 1 that interact through a power law potential v 1 rs where s 0 and r is euclidean distance with ed s n denoting the minimal energy for such n point arrangements we obtain bounds valid for all n for ed s n in the cases when 0 s d and 2 d s for s d we determine the precise asymptotic behavior of ed d n as n as a corollary lower bounds are given for the separation of any pair of points in an n point minimal energy configuration when s d 2 for the unit sphere in r3 d 2 we present two conjectures concerning the asymptotic expansion of e2 s n that relate to the zeta function l s for a hexagonal lattice in the plane we prove an asymptotic upper bound that supports the first of these conjectures of related interest we derive an asymptotic formula for the partial sums of l s when 0 s 2 the divergent case