bounds for green s functions on noncompact hyperbolic riemann orbisurfaces of finite volume in jorgenson and kramer compos math 142 679 700 2006 derived bounds for the canonical green s function and the hyperbolic green s function defined on a compact hyperbolic riemann surface in this article we extend these bounds to noncompact hyperbolic riemann orbisurfaces of finite volume and of genus greater than zero which can be realized as a quotient space of the action of a fuchsian subgroup of first kind on the hyperbolic upper half plane

a fast and accurate algorithm for solving bernstein vandermonde linear systems abstract a fast and accurate algorithm for solving a bernstein vandermonde linear system is presented the algorithm is derived by using results related to the bidiagonal decomposition of the inverse of a totally positive matrix by means of neville elimination the use of explicit expressions for the determinants involved in the process serves to make the algorithm both fast and accurate

computation of rational interpolants with prescribed poles a constructive proof for existence and unicity of the rational r belonging to gm v m 0 n 0 having prescribed n poles and interpolating m 1 hermite data is given it is based upon explicit computation of the confluent cauchy vandermonde determinant in terms of the nodes and the poles an algorithm to compute r z is presented it calculates this value from a triangular field of values of rational interpolants of lowest possible orders the algorithm is based upon a neville aitken interpolation formula and has arithmetical complexity 0 l l max m 1 n

a full rns implementation of rsa we present the first implementation of rsa in the residue number system rns which does not require any conversion either from radix to rns beforehand or rns to radix afterward our solution is based on an optimized rns version of montgomery multiplication thanks to the rns the proposed algorithms are highly parallelizable and seem then well suited to hardware implementations we give the computational procedure both parties must follow in order to recover the correct result at the end of the transaction encryption or signature

fast base extension using a redundant modulus in rns a technique to extend the base of a residue number system rns based on the chinese remainder theorem crt and the use of a redundant modulus is proposed the technique obtains the residue s of a given number in the extended moduli without resorting to the traditional mixed radix conversion mrc algorithm the base extension can be achieved in log sub 2 n table lookup cycles where n is the number of moduli in the rns the superiority of the technique compared in terms of latency and hardware requirements to the traditional szabo tanaka method is demonstrated

determinants of laplacians the determinant of the laplacian on spinor fields on a riemann surface is evaluated in terms of the value of the selberg zeta function at the middle of the critical strip a key role in deriving this relation is played by the barnes double gamma function

high speed low complexity systolic designs of novel iterative division algorithms in gf 2 sup m we extend the binary algorithm invented by stein and propose novel iterative division algorithms over gf 2 sup m for systolic vlsi realization while algorithm ebg is a basic prototype with guaranteed convergence in at most 2m 1 iterations its variants algorithms ebd and ebdf are designed for reduced complexity and fixed critical path delay respectively we show that algorithms ebd and ebdf can be mapped to parallel in parallel out systolic circuits with low area time complexities of o m sup 2 loglogm and o m sup 2 respectively compared to the systolic designs based on the extended euclid s algorithm our circuits exhibit significant speed and area advantages

explicit factorization of the vandermonde matrix the lu factorization of the vandermonde matrix is obtained using complete symmetric functions and the lower and upper triangular matrices are in turn factorized into 1 banded matrices thus expressing the vandermonde matrix as a product of 1 banded matrices

structured matrices and unconstrained rational interpolation problems we describe a fast recursive algorithm for the solution of an unconstrained rational interpolation problem by exploiting the displacement structure concept we use the interpolation data to implicitly define a convenient non hermitian structured matrix and then apply a computationally efficient procedure for its triangular factorization this leads to a transmission line interpretation that makes evident the interpolation properties we further discuss connections with the lagrange interpolating polynomial as well as questions regarding the minimality and the admissible degrees of complexity of the solutions

corner cutting algorithms for the bezier representation of free form curves abstract we give a general treatment of the problem of converting a curve expressed in a b spline expansion into its bezier form we show that corner cutting of the b spline control polygon leads to the bezier points exactly when the bezier matrix is totally positive also for spline spaces used to model space curves with curvature continuity we give an algorithm for determining the bezier representation

accurate bidiagonal decomposition of totally positive cauchy vandermonde matrices and applications abstract cauchy vandermonde matrices play a fundamental role in rational interpolation theory and in other fields when all their corresponding nodes are different and positive and all poles are different and negative and follow adequate orderings these matrices are totally positive in this paper we provide fast algorithms for computing bidiagonal factorizations of these matrices and their inverses with high relative accuracy these algorithms can be used to solve with high relative accuracy other algebraic problems such as the computation of all singular values all eigenvalues or the solution of certain linear systems the error analysis of the algorithm for computing the bidiagonal factorization and the corresponding perturbation theory are also performed

a fast parallel bjorck pereyra type algorithm for solving cauchy linear equations abstract we propose a new fast o n 2 parallel algorithm for solving cauchy systems of linear equations we perform an a priori rounding error analysis and obtain componentwise bounds for the forward backward and residual errors these bounds indicate that for the class of totally positive cauchy matrices the new algorithm is forward and backward stable producing a remarkably high relative accuracy in particular hilbert linear systems often considered to be too ill conditioned to be attacked can be rapidly solved with high precision the results indicate a close resemblance between the numerical properties of cauchy matrices and the much studied vandermonde matrices in fact our proposed cauchy solver is an analog of the well known bjorck pereyra algorithm for vandermonde systems as a by product we obtain favorably backward error bounds for the original bjorck pereyra algorithm several computed examples illustrate a connection of high relative accuracy to the concepts of effective well conditioning and total positivity

eisenstein series and the selberg trace formula ii the integral int k o left g g right e left g s right dg let g sl2 r and be an arithmetic subgroup of g for which g has finite volume but is not compact the space l2 g has the spectral decomposition with respect to the casimir operator l 2 left gamma backslash g right l o 2 left gamma backslash g right oplus l sp 2 left gamma backslash g right oplus l cont 2 left gamma backslash g right where l o 2 g is the space of cusp forms and is discrete l sp 2 g is the discrete part of left l o 2 right bot given by residues of eisenstein series and l cont 2 is the continuous part of the spectrum given by integrals of eisenstein series if is a function of compact support or of sufficiently rapid decay on g then convolution with defines an endomorphism t of l2 g and the kernel function k left g g right sum limits gamma in gamma varphi left g 1 gamma g right left g g in g right 0 1

accurate bidiagonal decomposition of totally positive h bernstein vandermonde matrices and applications abstract an algorithm for the computation of the bidiagonal decomposition of strictly totally positive h bernstein vandermonde matrices is presented these matrices are collocation matrices of h bernstein bases a generalization of the bernstein basis of the space of polynomials of degree less than or equal to n the algorithm is fast its computational complexity is o n 2 and has high relative accuracy the computation of this bidiagonal decomposition is used as the initial stage for the accurate and efficient solution of several linear algebra problems with these matrices linear system solving eigenvalue computation and inverse computation numerical experiments are also included

factorizations of cauchy vandermonde matrices abstract this paper analyzes the factorization of the inverse of a cauchy vandermonde matrix as a product of bidiagonal matrices factorizations of cauchy vandermonde matrices are also considered the results are applied to derive a fast algorithm for solving linear systems whose coefficient matrices are cauchy vandermonde matrices

double basis multiplicative inversion over gf 2 sup m inversion over galois fields is much more difficult than the corresponding multiplication efficient computation of inverses in gf 2 sup m is considered by solving a set of linear equations over the ground field gf 2 the proposed algorithm uses two separate bases for the representation of its input and output elements and has low computational complexity the algorithm is also suitable for hardware implementation using vlsi technologies

arakelov self intersection numbers of minimal regular models of modular curves x_0 p 2 x0 p2 we compute an asymptotic expression for the arakelov self intersection number of the relative dualizing sheaf of edixhoven s minimal regular model for the modular curve x 0 p 2 over mathbb q the computation of the self intersection numbers are used to prove an effective version of the bogolomov conjecture for the semi stable models of modular curves x 0 p 2 and obtain a bound on the stable faltings height for those curves in a companion article banerjee and chaudhuri in isr j math 2020

accurate eigenvalues of certain sign regular matrices we present a new o n 3 algorithm for computing all eigenvalues of certain sign regular matrices to high relative accuracy in floating point arithmetic the accuracy and cost are unaffected by the conventional eigenvalue condition numbers a matrix is called sign regular when the signs of its nonzero minors depend only of the order of the minors the sign regular matrices we consider are the ones which are nonsingular and whose kth order nonzero minors are of sign 1 k k 1 2 for all k this class of matrices can also be characterized as nonsingular totally nonnegative matrices with columns in reverse order we exploit a characterization of these particular sign regular matrices as products of nonnegative bidiagonals and the reverse identity we arrange the computations in such a way that no subtractive cancellation is encountered thus guaranteeing high relative forward accuracy

a hardware algorithm for modular multiplication division a mixed radix 4 2 algorithm for modular multiplication division suitable for vlsi implementation is proposed the algorithm is based on montgomery method for modular multiplication and on the extended binary gcd algorithm for modular division both algorithms are modified and combined into the proposed algorithm so that almost all the hardware components are shared the new algorithm carries out both calculations using simple operations such as shifts additions and subtractions the radix 2 signed digit representation is used to avoid carry propagation in all additions and subtractions a modular multiplier divider based on the algorithm performs an n bit modular multiplication division in o n clock cycles where the length of the clock cycle is constant and independent of n the modular multiplier divider has a linear array structure with a bit slice feature and can be implemented with much smaller hardware than that necessary to implement both multiplier and divider separately

pivoting and backward stability of fast algorithms for solving cauchy linear equations abstract three fast o n2 algorithms for solving cauchy linear systems of equations are proposed a rounding error analysis indicates that the backward stability of these new cauchy solvers is similar to that of gaussian elimination thus suggesting to employ various pivoting techniques to achieve a favorable backward stability it is shown that cauchy structure allows one to achieve in o n2 operations partial pivoting ordering of the rows and several other judicious orderings in advance without actually performing the elimination the analysis also shows that for the important class of totally positive cauchy matrices it is advantageous to avoid pivoting which yields a remarkable backward stability of the suggested algorithms it is shown that vandermonde and chebyshev vandermonde matrices can be efficiently transformed into cauchy matrices using discrete fourier cosine or sine transforms this allows us to use the proposed algorithms for cauchy matrices for rapid and accurate solution of vandermonde and chebyshev vandermonde linear systems the analytical results are illustrated by computed examples

new efficient structure for a modular multiplier for rns modular multiplication is a very important arithmetic operation in residue based real time computing systems in realizing these multipliers rom based structures are more efficient for small moduli due to the exponential growth of rom sizes implementations with arithmetic components are more suitable for medium and large moduli this paper presents a new modular multiplier that can deal efficiently with medium and large size moduli the design of this modular multiplier that multiplies two n bit residue digits consists basically of a n spl times n binary multiplier a n 1 k spl times k binary multiplier k n three n bit adders and a small size combinational circuit when compared with the most competitive published work the new multiplier reduces significantly both time delay and hardware requirements the design is very suitable for vlsi realization

points de petite hauteur sur les courbes modulaires x0 n let n be a square free integer prime to 6 let the imbeding of x 0 n in its jacobian relative to the point we show that the set is finite and that is infinite this explicit form of the bogomolov conjecture is obtained by an estimation of the self intersection of the dualizing sheaf in the sense of arakelov theory of modular curves this result is obtained by estimating several quantities attached to the arakelov metric on x 0 n starting with petersson s trace formula

an rns montgomery modular multiplication algorithm we present a new rns modular multiplication for very large operands the algorithm is based on montgomery s method adapted to mixed radix and is performed using a residue number system by choosing the moduli of the rns system reasonably large and implementing the system on a ring of fairly simple processors an effect corresponding to a redundant high radix implementation is achieved the algorithm can be implemented to run in o n time on o n processors where n is the number of moduli in the rns system and the unit of time is a simple residue operation possibly by table look up two different implementations are proposed one based on processors attached to a broadcast bus another on an oriented ring structure

on the asymptotic behavior of counting functions associated to degenerating hyperbolic riemann surfaces abstract in this article we will study what we call weighted counting functions on hyperbolic riemann surfaces of finite volume ifmis compact then we define the weighted counting function forw 0 to be n m w t n t t n w where n is the set of eigenvalues of the laplacian which acts on the space of smooth functions onm ifmis non compact then we define the weighted counting functionnm w t via the inverse laplace transform from which one can express the weighted counting function in terms of spectral data associated tom see proposition 5 1 and remark 5 2 using the convergence results from jl2 concerning the regularized heat trace on finite volume hyperbolic riemann surfaces we shall prove the following results t heorem let mldenote a degenerating family of compact or non compact hyperbolic riemann surfaces of finite volume which converges to the non compact hyperbolic surface m0 let s ldenote the set of lengths of pinching geodesics on the family of surfaces ml for w 0and t 1 4 let g l w t 2 w 1 16 1 2 n 1 l k s 1 l k sinfh nl k 2 t 1 4 nl k 2 2 w 1 2 2 j w 1 2 nl k t 1 4 where js x is the j bessel function and x is the standard gamma function for0 t 3 2andt 0 we have n m l w t g l w t n m 0 w t o 1 as l 0 b for w 0andt 0 we have n m l w t g l w t o l k s l log 1 l k as l 0 c for w 0andt 1 4 we have g l w t 2 w 1 t 1 4 w 1 2 4 1 2 w 3 2 l k s l log 1 l k o 1 as l 0 observe that by takingt

a lagrange matrices approach to confluent cauchy matrices we present a new approach to study inversion and factorization properties of confluent cauchy matrices we consider a class of generalized vandermonde matrices called lagrange matrices that are connected in a simple way with the cauchy matrices we apply the methods introduced in adv appl math 10 1989 348 to obtain inversion and factorization theorems for lagrange matrices and their confluent forms and then derive corresponding results for cauchy matrices a lagrange matrix v is associated with a pair of vectors x0 x1 xn and y0 y1 yn v is the matrix representation of the substitution operator that sends the vector p x0 p x1 p xn t to the vector p y0 p y1 p yn t for any polynomial p of degree at most n

accurate computations with said ball vandermonde matrices abstract a generalization of the vandermonde matrices which arise when the power basis is replaced by the said ball basis is considered when the nodes are inside the interval 0 1 then those matrices are strictly totally positive an algorithm for computing the bidiagonal decomposition of those said ball vandermonde matrices is presented which allows us to use known algorithms for totally positive matrices represented by their bidiagonal decomposition the algorithm is shown to be fast and to guarantee high relative accuracy some numerical experiments which illustrate the good behaviour of the algorithm are included

hauteur de faltings de quotients de j 0 n discriminants d algebres de hecke et congruences entre formes modulaires in this paper we compute the faltings height of the jacobian of the modular curve j 0 n we explain how it is related to the discriminant of the hecke algebra this discriminant measures congruences between modular forms as a corollary we obtain a lower bound for this discriminant the faltings height of a quotient a of j 0 n is also computed

total positivity and neville elimination abstract neville elimination is described in terms of schur complements of matrices and used to improve some well known characterizations of totally positive and strictly totally positive matrices

an inversion formula and fast algorithms for cauchy vandermonde matrices abstract matrices of composed type consisting of a vandermonde and a cauchy part and their connection to rational interpolation are investigated an inversion formula is presented and fast solution algorithms with o n log2 n complexity for cauchy vandermonde systems are developed

new systolic architectures for inversion and division in gf 2 sup m we present two systolic architectures for inversion and division in gf 2 sup m based on a modified extended euclidean algorithm our architectures are similar to those proposed by others in that they consist of two dimensional arrays of computing cells and control cells with only local intercell connections and have o m sup 2 area time product however in comparison to similar architectures both our architectures have critical path delays that are smaller gate counts that range from being considerably smaller to only slightly larger and latencies that are identical for inversion but somewhat larger for division one architecture uses an adder or an m l bit ring counter inside each control cell while the other architecture distributes the ring counters into the computing cells thereby reducing each control cell to just two gates

symmetric functions and the vandermonde matrix this work deduces the lower and the upper triangular factors of the inverse of the vandermonde matrix using symmetric functions and combinatorial identities the l and u matrices are in turn factored as bidiagonal matrices the elements of the upper triangular matrices in both the vandermonde matrix and its inverse are obtained recursively the particular value x i 1 q qi 1 in the indeterminates of the vandermonde matrix is investigated and it leads to q binomial and q stirling matrices it is also shown that q stirling matrices may be obtained from the pascal matrix

lower bounds on the arithmetic self intersection number of the relative dualizing sheaf on arithmetic surfaces we give an explicitly computable lower bound for the arithmetic self intersection number of the dualizing sheaf on a large class of arithmetic surfaces if some technical conditions are satisfied then this lower bound is positive in particular these technical conditions are always satisfied for minimal arithmetic surfaces with simple multiplicities and at least one reducible fiber but we have also used our techniques to obtain lower bounds for some arithmetic surfaces with non reduced fibers

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

new explicit formulas for faltings delta invariant in this paper we give new explicit formulas for faltings delta invariant in terms of integrals of theta functions and we deduce an explicit lower bound for delta only in terms of the genus and an explicit upper bound for the arakelov green function in terms of delta furthermore we give a canonical extension of delta and the zhang kawazumi invariant varphi to the moduli space of indecomposable principally polarised complex abelian varieties

calculus on arithmetic surfaces in a2 and a3 arakelov introduces an intersection calculus for arithmetic surfaces that is for stable models of curves over a number field in this paper we intend to show that his intersection product has a lot of useful properties more precisely we show that the following properties from the theory of algebraic surfaces have an analogue in our situation

bounding sup norms of cusp forms of large level let f be an l2 normalized weight zero hecke maas cusp form of square free level n character and laplacian eigenvalue 1 4 it is shown that f n 1 37 from which the hybrid bound f 1 4 n for some 0 is derived the first bound holds also for f yk 2f where f is a holomorphic cusp form of weight k with the implied constant now depending on k

binary ternary plus minus modular inversion in rns a fast rns modular inversion for finite fields arithmetic has been published at ches 2013 conference it is based on the binary version of the plus minus euclidean algorithm in the context of elliptic curve cryptography i e 160 550 bits finite fields it significantly speeds up modular inversions in this paper we propose an improved version based on both radix 2 and radix 3 this new algorithm leads to 30 percent speed up for a maximal area overhead about 4 percent on virtex 5 fpgas

systolic vlsi arrays for polynomial gcd computation the problem of finding a greatest common divisor gcd of any two nonzero polynomials is fundamental to algebraic and symbolic computations as well as to the decoder implementation for a variety of error correcting codes this paper describes new systolic arrays that can lead to efricient vlsi solutions to both the gcd problem and the extended gcd problem

polynomial least squares fitting in the bernstein basis the problem of polynomial least squares fitting in which the usual monomial basis is replaced by the bernstein basis is considered the coefficient matrix of the overdetermined system to be solved in the least squares sense is then a rectangular bernstein vandermonde matrix in order to use the method based on the qr decomposition of a the first stage consists of computing the bidiagonal decomposition of the coefficient matrix a starting from that bidiagonal decomposition an algorithm for obtaining the qr decomposition of a is the applied finally a triangular system is solved by using the bidiagonal decomposition of the r factor of a some numerical experiments showing the behavior of this approach are included

on computing multiplicative inverses in gf 2 sup m the design of a modular standard basis inversion for galois fields gf 2 sup m based on euclid s algorithm for computing the greatest common divisor of two polynomials is presented the asymptotic complexity is linear with m both in computation time and area requirement thus resulting in an at complexity of o m sup 2 this is a significant improvement over the best previous proposal which achieves at complexity of only o m sup 3

a test for strict sign regularity a characterization of strict sign regular matrices is obtained it is given in an algorithmic form which allows the easy construction of a test to check if a matrix is strictly sign regular the test is based on the neville elimination of several submatrices of a and requires less computational work than those which can be derived from other previously known characterizations of these matrices

bernstein bezoutian matrices and curve implicitization a new application of bernstein bezoutian matrices a type of resultant matrices constructed when the polynomials are given in the bernstein basis is presented in particular the approach to curve implicitization through sylvester and bezout resultant matrices and bivariate interpolation in the usual power basis is extended to the case in which the polynomials appearing in the rational parametric equations of the curve are expressed in the bernstein basis avoiding the basis conversion from the bernstein to the power basis the coefficients of the implicit equation are computed in the bivariate tensor product bernstein basis and their computation involves the bidiagonal factorization of the inverses of certain totally positive matrices

an efficient parity detection technique using the two moduli set 2h 1 2h 1 lu and chiang used both the table lookup and fractional number approaches to discover the parity of an rns number to eliminate the need for table space and time for computing fractions a two moduli set 2 h 1 2 h 1 is used to speed up the technique proposed by lu and chiang based on this modified two moduli set it is found that the parity of an rns number x x 1 x 2 is x 1 x 2mod42 if x 1 x 2 on the contrary if x 1

an iterative modular multiplication algorithm in rns most current cryptosystems need to compute modular multiplication with large numbers modular multiplication is a time consuming operation and thus many different techniques have been proposed for the acceleration a novel approach residue number system rns which has the advantages of parallel carry free and high speed arithmetic is usually used for large number computations however division and the magnitude comparison which most modular multiplication algorithms involve are difficult to be processed in rns in this paper we present an iterative modular multiplication algorithm in rns a subtle iterative model eliminating division and the magnitude comparison in modular multiplications proposed by chiou and yang and improved further by leong et al can be used to achieve our purpose our new algorithm has the property of easy parallelization and is more efficient than other iterative modular multiplication algorithms proposed previously

hardware complexity of modular multiplication and exponentiation large integer modular multiplication mm and modular exponentiation me are the foundation of most public key cryptosystems specifically rsa diffie helleman eigamal and the elliptic curve cryptosystems thus mm algorithms have been studied widely and extensively most of the work is based on the well known montgomery multiplication method and its variants which require standard multiplication operations despite their better complexity orders karatsuba and fft algorithms seem to rarely be used for hardware implementation in this paper we review their hardware complexity and propose original implementations of mm and me that become useful for 24 bit operators karatsuba algorithm or 373 bit operators fft algorithm

bounds on faltings s delta function through covers let x be a compact riemann surface of genus gx 1 in 1984 g faltings introduced a new invariant fal x associated to x in this paper we give explicit bounds for fal x in terms of fundamental dierential geometric invariants arising from x when gx 1 as an application we are able to give bounds for faltings s delta function for the family of modular curves x0 n in terms of the genus only in combination with work of a abbes p michel and e ullmo this leads to an asymptotic formula for the faltings height of the jacobian j0 n associated to x0 n

novel rns parameter selection for fast modular multiplication the parameter selection of residue number systems rns has a great impact on its computational efficiency this paper shows that a base extension the most costly operation in rns montgomery multiplication can be more efficient when the intervals between the rns moduli are small we propose a systematic rns parameter selection procedure and two methods to select rns moduli that lead to a reduced complexity our experimental results confirm the advantages of the selected moduli

asymptotic behavior of the selberg zeta functions for degenerating families of hyperbolic manifolds let mk be a degenerating sequence of finite volume hyperbolic manifolds of dimension d with d 2 or d 3 with finite volume limit m let z m k s be the associated sequence of selberg zeta functions and let mathcal z k s be the product of local factors in the euler product expansion of z m k s corresponding to the pinching geodesics on mk the main result in this article is to prove that z m k s mathcal z k s converges to z m infty s for all s in mathbf c with re s d 1 2 the significant feature of our analysis is that the convergence of z m k s mathcal z k s to z m infty s is obtained up to the critical line including the right half of the critical strip a region where the euler product definition of the selberg zeta function does not converge in the case d 2 our result reproves by different means the main theorem in schulze j funct anal 236 120 160 2006

factorizations of confluent cauchy vandermonde matrices abstract the confluent cauchy vandermonde matrices are considered which were studied earlier by various authors in different way in this paper we analyze the factorization of the inverse of a special type of confluent cauchy vandermonde matrix as a product of block bidiagonal matrices by using block neville elimination factorizations of confluent cauchy vandermonde matrices are also considered which generalize the results of j j martinez j m pena factorizations of cauchy vandermonde matrices linear algebra appl 284 1998 229 237

a matricial description of neville elimination with applications to total positivity abstract the neville elimination process used by the authors in some previous papers in connection with totally positive matrices is studied in detail in the case of nonsingular matrices a wide class of matrices is found where neville elimination has a lower computational cost than gauss elimination finally some new characterizations are obtained for strictly totally positive and nonsingular totally positive matrices in terms of their neville elimination and that of their inverses

modular multiplication without trial division let n 1 we present a method for multiplying two integers called n residues modulo n while avoiding division by n n residues are represented in a nonstandard way so this method is useful only if several computations are done modulo one n the addition and subtraction algorithms are unchanged 1 description some algorithms 1 2 4 5 require extensive modular arith metic we propose a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms other recent algorithms for modular arithmetic appear in 3 6 fix n 1 define an a residue to be a residue class modulo n select a radix r coprime to n possibly the machine word size or a power thereof such that r n and such that computations modulo r are inexpensive to process let r l and n be integers satisfying 0 n then return t n else return t to validate redc observe mn tn n tmod r so t is an integer also tr tmod n so t tr x mod n thirdly 0 t mn rn rn so 0 t 2n if r and n are large then t mn may exceed the largest double precision value one can circumvent this by adjusting m so r m 0 given two numbers x and y between 0 and n 1 inclusive let z redc xy then z xy r x mod n so xr l yr x zrx mod n also 0 z n so z is the product of x and y in this representation other algorithms for operating on n residues in this representation can be derived from the algorithms normally used the addition algorithm is unchanged since xr x yr x zr x mod n if and only if x y z mod n also unchanged are

an algorithmic and architectural study on montgomery exponentiation in rns the modular exponentiation on large numbers is computationally intensive an effective way for performing this operation consists in using montgomery exponentiation in the residue number system rns this paper presents an algorithmic and architectural study of such exponentiation approach from the algorithmic point of view new and state of the art opportunities that come from the reorganization of operations and precomputations are considered from the architectural perspective the design opportunities offered by well known computer arithmetic techniques are studied with the aim of developing an efficient arithmetic cell architecture furthermore since the use of efficient rns bases with a low hamming weight are being considered with ever more interest four additional cell architectures specifically tailored to these bases are developed and the tradeoff between benefits and drawbacks is carefully explored an overall comparison among all the considered algorithmic approaches and cell architectures is presented with the aim of providing the reader with an extensive overview of the montgomery exponentiation opportunities in rns

an effective bound for the huber constant for cofinite fuchsian groups let be a cofinite fuchsian group acting on hyperbolic two space let m be the corresponding quotient space for a closed geodesic of m let l denote its length the prime geodesic counting function m u is defined as the number of inconjugate primitive closed geodesics such that e l 1 m u 0 m j 1 4 we call the absolute constant c m the huber constant the objective of this paper is to give an effectively computable upper bound of c m for an arbitrary cofinite fuchsian group as a corollary we bound the huber constant for psl 2 z showing that c m 16 607 349 020 658 exp 30 44086643

lax integrability and the peakon problem for the modified camassa holm equation peakons are special weak solutions of a class of nonlinear partial differential equations modelling non linear phenomena such as the breakdown of regularity and the onset of shocks we show that the natural concept of weak solutions in the case of the modified camassa holm equation studied in this paper is dictated by the distributional compatibility of its lax pair and as a result it differs from the one proposed and used in the literature based on the concept of weak solutions used for equations of the burgers type subsequently we give a complete construction of peakon solutions satisfying the modified camassa holm equation in the sense of distributions our approach is based on solving certain inverse boundary value problem the solution of which hinges on a combination of classical techniques of analysis involving stieltjes continued fractions and multi point pade approximations we propose sufficient conditions needed to ensure the global existence of peakon solutions and analyze the large time asymptotic behaviour whose special features include a formation of pairs of peakons that share asymptotic speeds as well as toda like sorting property

bidiagonal decomposition of rectangular totally positive said ball vandermonde matrices error analysis perturbation theory and applications abstract an algorithm for computing the bidiagonal decomposition of said ball vandermonde matrices in the rectangular case is presented which allows one to use known algorithms for totally positive matrices represented by their bidiagonal decomposition the algorithm is fast and possesses high relative accuracy the error analysis of the algorithm is also addressed along with the perturbation theory for the bidiagonal factorization of totally positive said ball vandermonde matrices some numerical experiments showing the good behavior of the algorithm are included