critical properties of the classical xy and classical heisenberg models a renormalization group study abstract by using two approaches of renormalization group rg mean field rg mfrg and effective field rg efrg we study the critical properties of the simple cubic lattice classical xy and classical heisenberg models the methods are illustrated by employing its simplest approximation version in which small clusters with one n 1 and two n 2 spins are used the thermal and magnetic critical exponents y t and y h and the critical parameter k c are numerically obtained and are compared with more accurate methods monte carlo series expansion and e expansion the results presented in this work are in excellent agreement with these sophisticated methods we have also shown that the exponent y h does not depend on the symmetry n of the hamiltonian hence the criteria of universality for this exponent is only a function of the dimension d

renormalisation group recursions by mean field approximations within a renormalisation group strategy different rescaled mean field approximations for the magnetisation are combined good qualitative estimates are obtained for the critical properties of classical and quantum spin systems a particularly simple realisation of the method yields recursions for arbitrary dimensionality d the critical couplings are accurate for all d and have the correct asymptotic behaviour for d approaching infinity

mean field renormalisation group transformations for geometric phase transitions it is shown that the mean field renormalisation group technique of indekeu et al 1982 may be applied directly to the self avoiding walk saw and percolation problems without reference to the n to 0 limit of interacting spin systems numerical results for the saw and bond and site percolation both directed and undirected problems on the square lattice obtained by using a variety of small cells are reported an isotropic transformation for the percolation problems on the d dimensional hypercubic lattice is also discussed

critical phenomena at surfaces in a model of nonequilibrium phase transitions critical phenomena at surfaces are studied in a simple two dimensional model of non equilibrium phase transitions belonging to the class of interacting particle systems the mean field renormalization group approach and numerical simulations are used to determine the phase diagram for this system and some of the critical exponents associated with ordinary special extraordinary and surface phase transitions

the relationship between breaking the diffie hellman protocol and computing discrete logarithms both uniform and nonuniform results concerning the security of the diffie hellman key exchange protocol are proved first it is shown that in a cyclic group g of order g prod p i e i where all the multiple prime factors of g are polynomial in log g there exists an algorithm that reduces the computation of discrete logarithms in g to breaking the diffie hellman protocol in g and has complexity sqrt max nu p i cdot log g o 1 where nu p stands for the minimum of the set of largest prime factors of all the numbers d in the interval p 2 sqrt p 1 p 2 sqrt p 1 under the unproven but plausible assumption that nu p is polynomial in log p this reduction implies that the diffie hellman problem and the discrete logarithm problem are polynomial time equivalent in g second it is proved that the diffie hellman problem and the discrete logarithm problem are equivalent in a uniform sense for groups whose orders belong to certain classes there exists a polynomial time reduction algorithm that works for all those groups moreover it is shown that breaking the diffie hellman protocol for a small but nonnegligible fraction of the instances is equally difficult as breaking it for all instances finally efficient constructions of groups are described for which the algorithm reducing the discrete logarithm problem to the diffie hellman problem is efficiently constructible

compactness by the hausdorff measure of noncompactness abstract in the present paper we establish some identities or estimates for the operator norms and the hausdorff measures of noncompactness of certain operators given by infinite matrices that map an arbitrary b k space into the sequence spaces c 0 c l and l 1 and into the matrix domains of triangles in these spaces furthermore by using the hausdorff measure of noncompactness we apply our results to characterize some classes of compact operators on the b k spaces

asymptotically fast group operations on jacobians of general curves abramovich d 1996 int math res notices p1005 aho a v 1975 design anal computer becker t 1993 graduate texts math v141 brawley j 2004 density primitixe el burgisserr p 1997 fundamentals pricnip v315 cantor dg 1987 math comput v48 p95 doi 10 2307 2007876 chow wl 1954 am j math v76 p453 doi 10 2307 2372585 coppersmith d 1990 j symb comput v9 p251 doi 10 1016 s0747 7171 08 80013 2 decker wolfram 1999 algorithmic algebra p187 eberly w 2000 j symb comput v29 p441 doi 10 1006 jsco 1999 0308 greg w 2002 adv math v172 p169 griffiths p 1994 principles algebraic hartshorne robin 1977 algebraic geometry hess f 1999 thesis technische u hess f 2002 j symb comput v33 p425 doi 10 1006 jsco 2001 0513 huang md 1994 j symb comput v18 p519 doi 10 1006 jsco 1994 1063 kemper g 2002 j symb comput v34 p229 doi 10 1006 jsco 2002 0560 kemper g 2004 math comput v73 p245 khurimakdisi k 2004 asymptotically fast lazarsfeld r 1989 sampling vector bund p500 pari group bord pari gp stein w modular forms databa volcheck e 1994 lect notes computer v877 p221

cantor versus harley optimization and analysis of explicit formulae for hyperelliptic curve cryptosystems hyperelliptic curves hec look promising for cryptographic applications because of their short operand size compared to other public key schemes the operand sizes seem well suited for small processor architectures where memory and speed are constrained however the group operation has been believed to be too complex and thus hec have not been used in this context so far in recent years a lot of effort has been made to speed up group operation of genus 2 hec in this paper we increase the efficiency of the genus 2 and genus 3 hyperelliptic curve cryptosystems hecc for certain genus 3 curves we can gain almost 80 percent performance for a group doubling this work not only improves gaudry and harley s algorithm but also improves the original algorithm introduced by cantor 1987 contrary to common belief we show that it is also practical for certain curves to use cantor s algorithm to obtain the highest efficiency for the group operation in addition we introduce a general reduction method for polynomials according to karatsuba we implemented our most efficient group operations on pentium and arm microprocessors

matrix transformations between the sequence spaces w 0 p λ v 0 p λ c 0 p λ 1 p and certain bk spaces in this paper we determine the b duals of the sets w 0 p l v 0 p l and c 0 p l for exponentially bounded email protected furthermore we characterize matrix transformations between the sequence spaces w 0 p l v 0 p l c 0 p l 1

a rapid method of evaluating the regulator and class number of a pure cubic field let 2 0 be the algebraic number field formed by adjoining 9 to the rationals 2 let r and h be respectively the regulator and class number of shanks has described a method of evaluating r for 2 vd where d is a positive integer his technique improved the speed of the usual continued fraction algorithm for finding r by allowing one to proceed almost directly from the nth to the mth step where m is approximately in in the continued fraction expansion of id this paper shows how shanks idea can be extended to the voronoi algorithm which is used to find r in cubic fields of negative discriminant it also discusses at length an algorithm for finding r and h for pure cubic fields 1 fd d an integer under a certain generalized riemann hypothesis the ideas developed here will provide a new method which will find r and h in 0 02 5 e operations when h is small this is an improvement over the 0 d h operations required by voronoi s algorithm to find r for example with d 200171999 it required only 5 minutes for an amdahl 470 vi computer to find that r 518594546 969083 and h 1 this same calculation would require about 8 days of computer time if it used only the standard voronoi algorithm a 0 and gcd w m2 m3 n2 n3 a 1 also if e m2n3 m3n2 then e a if u e we denote its conjugates by w and we also write the norm of as nia and define it as nico oco w from the simple fact that

computing in the jacobian of a hyperelliptic curve in this paper we present algorithms suitable for computer use for computation in the jacobian of a hyperelliptic curve we present a reduction algorithm which is asymptotically faster than that of gauss when the genus g is very large

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

linear algebra algorithms for divisors on an algebraic curve we use an embedding of the symmetric dth power of any algebraic curve c of genus g into a grassmannian space to give algorithms for working with divisors on c using only linear algebra in vector spaces of dimension o g and matrices of size o g2 o g when the base field k is finite or if c has a rational point over k these give algorithms for working on the jacobian of c that require o g4 field operations arising from the gaussian elimination our point of view is strongly geometric and our representation of points on the jacobian is fairly simple to deal with in particular none of our algorithms involves arithmetic with polynomials we note that our algorithms have the same asymptotic complexity for general curves as the more algebraic algorithms in florian hess 1999 ph d thesis which works with function fields as extensions of k x however for special classes of curves hess algorithms are asymptotically more efficient than ours generalizing other known efficient algorithms for special classes of curves such as hyperelliptic curves cantor 1987 superelliptic curves galbraith paulus and smart 2002 and cab curves harasawa and suzuki 2000 in all those cases one can attain a complexity of o g2

a generic approach to searching for jacobians we consider the problem of finding cryptographically suitable jacobians by applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family we can search for jacobians containing a large subgroup of prime order for a suitable distribution of curves the complexity is subexponential in genus 2 and o n1 12 in genus 3 we give examples of genus 2 and genus 3 hyperelliptic curves over prime fields with group orders over 180 bits in size improving previous results our approach is particularly effective over low degree extension fields where in genus 2 we find jacobians over fp2 and trace zero varieties over fp3 with near prime orders up to 372 bits in size for p 261 1 the average time to find a group with 244 bit near prime order is under an hour on a pc

applications of the hausdorff measure of noncompactness in some sequence spaces of weighted means in the present paper we establish some identities or estimates for the operator norms and the hausdorff measures of noncompactness of certain operators on some sequence spaces of weighted means furthermore by using the hausdorff measure of noncompactness we apply our results to characterize some classes of compact operators on those spaces

the group law on the jacobian of a curve of genus 2 an explicit description is given of the group law on the jacobian of a curve c of genus 2 the kummer surface provides a useful intermediary stage bilinear forms relating to the kummer surface imply that the global group law may be given projectively by biquadratic forms defined over the same ring as the coefficients of c it is not assumed that c has a rational weierstrass point and the theory presented applies over an arbitrary ground field

transfer matrix test of the two cell mean field renormalization group abstract a transfer matrix approach has been worked out to test the predictions of the two cell mean field renormalization group mfrg for the square ising cells with linear size up to l 11 the convergence of the finite size critical couplings towards the exact value both for mfrg and for the cluster mfa is shown and the limit of the ratio yh yt consistent with the corresponding universal value is found applications within mfa are also briefly discussed

critical properties of the ashkin teller model from the mean field renormalisation group approach by using small size clusters which carry all the basic symmetries of the ashkin teller model the authors obtain within the mean field renormalisation group approach the complete phase diagram of the model estimates of the thermal critical exponent are also obtained

mean field like variational and renormalisation group studies of 1 1 dimensional spin models two recently proposed methods the variational cluster wavefunction approximation and the mean field like renormalisation group transformation are used to describe the phase transitions in the hamiltonian version of the two dimensional z q symmetric spin models the convergence of the two schemes to the known results is analysed it is shown that in their usual form neither of these models can reproduce fine details of the phase diagram although the renormalisation group method gives in general better convergence

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

compact operators on the spaces a0r δ and acr δ abstract in c aydin f basar some new difference sequence spaces appl math comput 157 2004 677 693 aydin and basar introduced the new difference sequence spaces a c r and a 0 r and characterized the classes a c r l p and a c r c here we characterize the subclasses k x y of compact operators where x is a c r and y is one of the spaces c 0 c l l 1 b v by applying the hausdorff measure of noncompactness

two step renormalisation group approach for randomly diluted ising models a two step renormalisation group approach a migdal kadanoff or decimation transformation mk followed by a mean field renormalisation group mfrg has been applied to random bond ising models on d dimensional hypercubic lattices calculations of the critical couplings and percolation concentrations for two and three dimensions show much improvement even for small size clusters compared with the results of separate mk or mfrg calculations critical exponents nu t and nu p are also improved in some cases estimates of critical curves in the concentration temperature plane are also obtained for the two and three dimensional models

mean field renormalisation group for the spin 1 2 anisotropic heisenberg model the mean field renormalisation group approach is applied to the spin 1 2 anisotropic heisenberg model the critical temperature and the critical exponent are obtained as a function of the anisotropy for the two dimensional model on square triangular and hexagonal lattices the three dimensional model on a simple cubic lattice is also analysed

order matrix transformations on some banach lattice valued sequence spaces let e and f be banach lattice with f is a dedekind complete we characterize the generalized classes x e y f of all infinite matrices of regular operators from e to f for x c and y c c 0

quantum spin two dimensional xxz model an alternative quantum renormalization group approach abstract we study the phase diagram of the two dimensional quantum spin 1 2 anisotropic heisenberg model we adapt a renormalization group approach to treat this quantum model and an analysis on low temperatures is presented and compared with the exact results in the isotropic limit obtained from the mermin wagner theorem we comment on extensions of the procedure to treat antiferromagnetic models for s 1 2 and s 1 2

gauss composition for p1 and the universal jacobian of the hurwitz space of double covers abstract in this paper we give an explicit description of the moduli space of line bundles on hyperelliptic curves including singular curves we study the universal jacobian j 2 g n of degree n line bundles over the hurwitz stack of double covers of p 1 by a curve of genus g our main results are the construction of a smooth irreducible universally closed but not separated moduli compactification j bd 2 g n of j 2 g n whose points we describe simply and explicitly as sections of certain vector bundles on p 1 a description of the global geometry and moduli properties of these stacks and a computation of the picard groups of j bd 2 g n and j 2 g n in the cases when n g is even an important ingredient of our work is the parametrization of line bundles on double covers by binary quadratic forms this parametrization generalizes the classical number theoretic correspondence between ideal classes of quadratic rings and integral binary quadratic forms which in particular gives the group law on integral binary quadratic forms first discovered by gauss

renormalization group study of the two dimensional ising model with crossing bonds abstract the simple quadratic ising model with nearest and next nearest neighbor interactions is investigated by means of the mean field renormalization group approach by conveniently choosing the size of the blocks one obtains the expected phase diagram exhibiting ferromagnetic antiferromagnetic paramagnetic and superantiferromagnetic phases the critical lines which separate the paramagnetic phase from the ordered ones have the expected asymptotic behavior for large values of the nearest neighbor coupling

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

a note on compact operators on matrix domains abstract we establish some identities or estimates for the operator norms and hausdorff measures of noncompactness of linear operators given by infinite matrices that map the matrix domains of triangles in arbitrary bk spaces with ak or in the spaces of all convergent or bounded sequences into the spaces of all null convergent or bounded sequences or of all absolutely convergent series furthermore we apply these results to the characterizations of compact operators on the matrix domains of triangles in the classical sequence spaces and on the sequence spaces studied in i djolovic compact operators on the spaces a 0 r and a c r j math anal appl 318 2 2006 658 666 i djolovic on the space of bounded euler difference sequences and some classes of compact operators appl math comput 182 2 2006 1803 1811

the parallelized pollard kangaroo method in real quadratic function fields we show how to use the parallelized kangaroo method for computing invariants in real quadratic function fields specifically we show how to apply the kangaroo method to the infrastructure in these fields we also show how to speed up the computation by using heuristics on the distribution of the divisor class number and by using the relatively inexpensive baby steps in the real quadratic model of a hyperelliptic function field furthermore we provide examples for regulators and class numbers of hyperelliptic function fields of genus 3 that are larger than those ever reported before

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

modular equations for hyperelliptic curves we dene modular equations describing the torsion subgroups of the jacobian of a hyperelliptic curve over a nite base eld we prove factorization properties that extend the well known results used in atkin s improvement of schoof s genus 1 point counting algorithm

the arithmetic of certain cubic function fields in this paper we discuss the properties of curves of the form y3 f x over a given field k of characteristic different from 3 if f x satisfies certain properties then the jacobian of such a curve is isomorphic to the ideal class group of the maximal order in the corresponding function field we seek to make this connection concrete and then use it to develop an explicit arithmetic for the jacobian of such curves from a purely mathematical perspective this provides explicit and efficient techniques for performing arithmetic in certain ideal class groups which are of fundamental interest in algebraic number theory at the same time it provides another source of groups which are suitable for diffie hellman type protocols in cryptographic applications

some exact results from the mean field renormalization group abstract the mean field renormalization group is applied to the study of the directed self avoiding random walk in d dimensions and to the spin ferromagnetic ising model on the bethe lattice in both cases the known exact results are obtained

behavior critical for bond diluted n vector model in the effective field theory abstract in the present work the bond diluted n vector model on the simple cubic lattice is studied by applying the effective field theory for a cluster with two spins the critical temperature and percolation concentrations for xy and heisenberg models as well the phase diagram in the t p plane respectively are obtained and the results when compared with the standard mean field approximation are an improvement even for small size clusters

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

mean field renormalisation group transformations for the triangular ising antiferromagnet the mean field renormalisation group of indekeu et al 1982 is applied to the antiferromagnetic nearest neighbour ising model on the triangular lattice the resulting phase diagram in the temperature field plane is in good agreement with other calculations while the predicted specific heat index alpha is negative and thus qualitatively wrong the study indicates that the method may be a useful approach at least for determining phase diagrams for frustrated systems and systems with competing interactions where conventional mean field theory is dubious

mean field renormalization group for the spin 1 2 xxz model the mean field renormalization group mfrg method is applied to the spin 1 2 xxz model the phase diagram is obtained for various lattice structures in two and three dimensions as well as estimates for the critical exponents the results of the original mfrg method are quite good but the more sophisticated approach that also includes surface exponents is not suitable for application to the small clusters used here

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

applications of measure of noncompactness in operators on the spaces sα s0α s c α lpα abstract in this paper we characterize some operators and matrix transformations in the sequence spaces s s 0 s c l p moreover using the hausdorff measure of noncompactness necessary and sufficient conditions are formulated for a linear operator between the mentioned spaces to be compact among other things some results of cohen and dunford are recovered

length scaling in mfrg the mean field renormalisation group mfrg has so far given poor estimates for critical indices it is argued that this can be traced to an unsatisfactory definition of the length scaling factor a new definition that improves the estimates in most cases is proposed

mean field renormalization group approach to two dimensional site directed percolation two dimensional site directed percolation is studied by the mean field renormalization group approach to bulk and surface critical properties extrapolation techniques for the exponents and the percolation threshold are described a very good estimate of the percolation threshold pc 0 7055 or 0 0001 and the first evaluation of the surface exponent yhs 0 653 or 1 are obtained

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

superintegrable systems and riemann roch theorem in algebraic geometry there is a reduction algorithm that transforms the unreduced divisor into a unique reduced divisor which existence is guaranteed by the riemann roch theorem we discuss application of this algorithm to construction of finite dimensional superintegrable systems with n degrees of freedom identifying coordinates of the reduced divisor with integrals of motion

mean field renormalisation group calculations for directed percolation on the square lattice a simple extrapolation technique for mean field renormalisation group calculations is described application of this method to large cell calculations for percolation on the directed square lattice yields pc 0 7067 or 0 0001 site problem and pc 0 6454 or 0 0002 bond problem

measure of noncompactness of matrix operators on some difference sequence spaces of weighted means for a sequence x x k we denote the difference sequence by dx x k x k 1 let u u k k 0 and v v k k 0 be the sequences of real numbers such that u k 0 v k 0 for all email protected n the difference sequence spaces of weighted means l u v d are defined as l u v d x x k w x email protected where l c c 0 and and the matrix w w n k is defined by w n k u n v k v k 1 kn in this paper we establish some identities or estimates for the operator norms and the hausdorff measures of noncompactness of certain matrix operators on l u v d further we characterize some classes of compact operators on these spaces by using the hausdorff measure of noncompactness

hausdorff measure of noncompactness of certain matrix operators on the sequence spaces of generalized means abstract in the present paper we derive some identities or estimates for the operator norms and the hausdorff measures of noncompactness of certain matrix operators on the sequence spaces of generalized means further by applying the hausdorff measure of noncompactness we obtain the necessary and sufficient conditions for such operators to be compact

aspects of pairing inversion in this paper we discuss some applications of the pairing inversion problem and outline some potential approaches for solving it our analysis of these approaches gives further evidence that pairing inversion is a difficult problem

computing canonical heights using arithmetic intersection theory the canonical height on an abelian variety a defined over a global field k is an object of fundamental importance in the study of the arithmetic of a for many applications it is required to compute p for a given point p a k for instance given generators of a subgroup of the mordell weil group a k of finite index this is necessary for most known approaches to the computation of generators of the mordell weil group a k furthermore the regulator of a k which appears in the statement of the conjecture of birch and swinnerton dyer is defined in terms of the canonical height and thus we need the ability to compute canonical heights in order to gather numerical evidence for the conjecture in the case of positive rank

applications of measures of noncompactness to the infinite system of differential equations in lp spaces abstract in this paper we apply the technique of measures of noncompactness to the theory of infinite system of differential equations in the banach sequence spaces l p 1 p our aim is to present some existence results for infinite system of differential equations formulated with the help of measures of noncompactness

banach algebras of matrix transformations between some sequence spaces related to λ strong convergence and boundedness we study the spaces c 0 l c l and c l of sequences that are l strongly convergent to zero l strongly convergent and l strongly bounded and the related spaces v 0 l and v l in particular we give the characterizations of several classes of matrix transformations between those spaces furthermore we use our results to prove that the classes of matrix transformations from v l c l and c l into themselves are banach algebras as an application of our results we establish an estimate for the hausdorff measure of noncompactness of matrix operators that map c l into itself give a characterization of the subclass of compact operators and a sufficient condition for those operators to be fredholm

equivalence of the o n vector ferromagnetic and antiferromagnetic models abstract the effective field renormalization group efrg approach is used to find the neel temperature t n of the o n vector model with antiferromagnetic af interaction the efrg method is illustrated by employing approximations in which clusters with one n 1 and two n 2 spins are used the critical temperature t n is obtained as a function of component n and coordination z numbers for all values of n and z we show that t n t c where t c is the curie temperature for the ferromagnetic f case as a comparison the results of the quantum heisenberg model n 3 with f and af interactions are also presented and we find that t n t c which is different from the classical result t c t n

computing the mordell weil rank of jacobians of curves of genus two we derive the equations necessary to perform a two descent on the jacobians of curves of genus two with rational weierstrass points we compute the mordell weil rank of the jacobian of some genus two curves defined over the rationals and discuss the practicality of using this method

mixed spin 1 2 and spin 1 ising model in transverse random fields abstract we investigate the critical properties of a mixed spin 1 2 and spin 1 ising model in a transverse random field in this study we use two approaches the pair approximation and the mean field renormalization group each combined with the discretized path integral representation it is found that both methods predict quantitatively the same phase diagram critical exponents of the system are also calculated

from picard groups of hyperelliptic curves to class groups of quadratic fields let c be a hyperelliptic curve defined over mathbb q whose weierstrass points are defined over extensions of mathbb q of degree at most three and at least one of them is rational generalizing a result of r soleng in the case of elliptic curves we prove that any line bundle of degree 0 on c which is not torsion can be specialised into ideal classes of imaginary quadratic fields whose order can be made arbitrarily large this gives a positive answer for such curves to a question by agboola and pappas

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

a model for two dimensional orientational order abstract we use a generalization of kikuchi s cluster variational method cvm in the pair approximation to study anisotropic plane rotors on a square lattice as a simple model for two dimensional nematics the system was shown to exhibit nematic ordered states at low temperature disordering for all anisotropies through a continuous transition the orientational distribution functions order parameter internal energy and specific heat were calculated within the cvm and compared with the results of a recent computer simulation estimates for the non classical exponents were obtained using a mean field renormalization group analysis a spin wave argument was also used to investigate the role of long range fluctuations near the isotropic fixed point the accuracy of these different techniques is discussed within the context of our model