the generalized basis reduction algorithm let fx be a convex function defined in rn which is symmetric about the origin and homogeneous of degree 1 and let l be the lattice of integers zn a definition of a reduced basis b1 bn of the lattice with respect to the distance function f is presented and we describe an algorithm which yields a reduced basis in polynomial time for fixed n in the special case in which the bodies x fx t are ellipsoids the definition of a reduced basis is identical with that given by lenstra lenstra and lovasz 1982 and the algorithm is the well known basis reduction algorithm we show that the basis vector b1 in a reduced basis is an approximation to a shortest nonzero lattice point with respect to f and relate the basis vectors bi to minkowski s successive minima the results lead to an algorithm for integer programming which executes in polynomial time for fixed n but which avoids the ellipsoidal approximations required by lenstra s algorithm we also discuss the properties of a korkine zolotarev basis for the lattice

zero crossings of a wavelet transform the completeness stability and application to pattern recognition of a multiscale representation based on zero crossings is discussed an alternative projection algorithm is described that reconstructs a signal from a zero crossing representation which is stabilized by keeping the value of the wavelet transform integral between each pair of consecutive zero crossings the reconstruction algorithm has a fast convergence and each iteration requires o n log sup 2 n computation for a signal of n samples the zero crossings of a wavelet transform define a representation which is particularly well adapted for solving pattern recognition problems as an example the implementation and results of a coarse to fine stereo matching algorithm are described

weighted complex projective 2 designs from bases optimal state determination by orthogonal measurements we introduce the problem of constructing weighted complex projective 2 designs from the union of a family of orthonormal bases if the weight remains constant across elements of the same basis then such designs can be interpreted as generalizations of complete sets of mutually unbiased bases being equivalent whenever the design is composed of d 1 bases in dimension d we show that for the purpose of quantum state determination these designs specify an optimal collection of orthogonal measurements using highly nonlinear functions on abelian groups we construct explicit examples from d 2 orthonormal bases whenever d 1 is a prime power covering dimensions d 6 10 and 12 for example where no complete sets of mutually unbiased bases have thus far been found

numerical cubature using error correcting codes we present a construction for improving numerical cubature formulas with equal weights and a convolution structure in particular equal weight product formulas using linear error correcting codes the construction is most effective in low degree with extended bch codes using it we obtain several sequences of explicit positive interior cubature formulas with good asymptotics for each fixed degree t as the dimension n to infty using a special quadrature formula for the interval g kuperberg adv in appl math 34 2005 pp 853 870 arxiv math pr 0408360 we obtain an equal weight t cubature formula on the n cube with o n lfloor t 2 rfloor points which is within a constant of the stroud lower bound we also obtain t cubature formulas on the n sphere n ball and gaussian rn with o nt 2 points when t is odd when mu is spherically symmetric and t 5 we obtain o n2 points for each t ge 4 we also obtain explicit positive interior formulas for the n simplex with o nt 1 points for t 3 we obtain o n points these constructions asymptotically improve the nonconstructive tchakaloff bound some related results were recently found independently by victoir siam j numer anal 42 2004 pp 209 227 who also noted that the basic construction more directly uses orthogonal arrays

evenly distributed unitaries on the structure of unitary designs we clarify the mathematical structure underlying unitary t designs these are sets of unitary matrices evenly distributed in the sense that the average of any tth order polynomial over the design equals the average over the entire unitary group we present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design lower bounds for the number of elements of 2 designs are derived we show how to turn mutually unbiased bases into approximate 2 designs whose cardinality is optimal in leading order designs of higher order are discussed and an example of a unitary 5 design is presented we comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups further we sketch connections to problems in linear optics and questions regarding typical entanglement

predicting nonlinear pseudorandom number generators let p be a prime and let a and b be elements of the finite field fp of p elements the inversive congruential generator icg is a sequence u n of pseudorandom numbers defined by the relation u n 1 au 1 n b mod p we show that if sufficiently many of the most significant bits of several consecutive values u n of the icg are given one can recover the initial value u 0 even in the case where the coefficients a and b are not known we also obtain similar results for the quadratic congruential generator qcg v n 1 f v n mod p where f f p x this suggests that for cryptographic applications icg and qcg should be used with great care our results are somewhat similar to those known for the linear congruential generator lcg x n 1 ax n b mod p but they apply only to much longer bit strings we also estimate limits of some heuristic approaches which still remain much weaker than those known for lcg

weight functions for chebyshev quadrature la fonction poids 1 x 2 1 2 r x ou r x est une fonction rationnelle d ordre 1 1 est etudiee il s agit de determiner si elle admet des quadratures de tchebychev

averaging sets a generalization of mean values and spherical designs 0 1 r one cannot of course expect there to be a single point x at which every one assumes its mean value but there is an easy generalization in terms of weighted averages there exist at most m points xi and positive weights oi summing to 1 so that aifj xi i fi dt for j 1 2 m

the multicomponent am fm image representation we compute am fm representations for multicomponent nonstationary images using a statistical component model components are isolated with a filterbank comprising frequency and orientation selective channels the modulating functions for each component are estimated from the channel responses using localized nonlinear operators followed by optimal mmse estimators we also demonstrate reconstruction from the representation

optimal routing in double loop networks in this paper we study the problem of finding the shortest path in circulant graphs with an arbitrary number of jumps we provide algorithms specifically tailored for weighted undirected and directed circulant graphs with two jumps which compute the shortest path our method only requires o logn arithmetic operations and the total bit complexity is o log 2nloglognlogloglogn where n is the number of the graph s vertices this elementary and efficient shortest path algorithm has been derived from the closest vector problem cvp of lattices in dimension two and with an 1 norm

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

a complementary survey on double loop networks we give a survey on double loop networks with emphasis on new development since the surveys in 1986 1991 and 1995

doubly linked ring networks we consider networks of processors where each processor either has one in link and one out link or two in links and two out links we study three properties of such networks 1 diameter 2 connectivity and 3 the ring property we propose a class of networks which seem to achieve the optimum as far as these three properties are concerned

optimal distance networks of low degree for parallel computers the authors introduce and study a family of interconnection schemes the midimew networks based on circulant graphs of degree 4 a family of such circulants is determined and shown to be optimal with respect to two distance parameters simultaneously namely maximum distance and average distance among all circulants of degree 4 these graphs are regular point symmetric and maximally connected and one such optimal graph exists for any given number of nodes the proposed interconnection schemes consist of mesh connected networks with wrap around links and are isomorphic to the optimal distance circulants previously considered ways to construct one such network for any number of nodes are shown their good properties to build interconnection schemes for multicomputers are examined and some interesting particular cases are discussed the problem of routing is also addressed and a basic algorithm is provided which is adequate for implementing the routing policy required to convey messages traversing shortest paths between nodes

a characterization of positive quadrature formulae a positive quadrature formula with n nodes which is exact for polynomials of degree in r 1 0 r is based on the zeros of certain quasi orthogonal polynomials of degree n we show that the quasi orthogonal polynomials that lead to the positive quadrature formulae can all be expressed as characteristic polynomials of a symmetric tridiagonal matrix with positive subdiagonal entries as a consequence for a fixed n every positive quadrature formula is a gaussian quadrature formula for some nonnegative measure

unsupervised texture segmentation using gabor filters abstract this paper presents a texture segmentation algorithm inspired by the multi channel filtering theory for visual information processing in the early stages of human visual system the channels are characterized by a bank of gabor filters that nearly uniformly covers the spatial frequency domain and a systematic filter selection scheme is proposed which is based on reconstruction of the input image from the filtered images texture features are obtained by subjecting each selected filtered image to a nonlinear transformation and computing a measure of energy in a window around each pixel a square error clustering algorithm is then used to integrate the feature images and produce a segmentation a simple procedure to incorporate spatial information in the clustering process is proposed a relative index is used to estimate the true number of texture categories

on weight functions admitting chebyshev quadrature in this paper we prove the existence of chebyshev quadrature for three new weight functions which are quite different from the two known examples given by ullman 15 and byrd and stalla 2 in particular we indicate a simple method to construct weight functions for which there exist infinitely many chebyshev quadrature rules

chebyshev quadrature rules for a new class of weight functions proof is given that the weight functions w x p i fr p x x x i x on 0 1 admit chebyshev quadratures for any fixed p 1 and every n for the particular cases when p i and p 2 the nodes are tabulated to ten decimal places for n point rules with n 2 4 6 8 and 12 numerical tables are also given for a coefficient in the expression of the error term

spherical designs discrepancy and numerical integration a spherical design is a point configuration on the sphere which yields exact equal weight quadrature formulae for polynomials up to a given de gree until now only very specific constructions for spherical designs are known we establish connections to spherical cap discrepancy and show some general discrepancy bounds furthermore we reformulate the problem of construct ing designs as an optimization problem and develop an algorithm for finding practical designs

chebyshev type quadrature for jacobi weight functions a chebyshev type quadrature formula is an integration formula with equal coefficients we denote by n n w the minimal number of nodes for which a chebyshev type formula for the weight function w exists having degree of exactness n the asymptotic behavior of n n w is determined for jacobi weight functions with nonnegative parameters also some numerical results are given

existence of solutions to systems of underdetermined equations and spherical designs this paper is concerned with proving the existence of solutions to an underdetermined system of equations and with the application to existence of spherical t designs with t 1 2 points on the unit sphere s 2 in r 3 we show that the construction of spherical designs is equivalent to solution of underdetermined equations a new verification method for underdetermined equations is derived using brouwer s fixed point theorem application of the method provides spherical t designs which are close to extremal maximum determinant points and have the optimal order o t 2 for the number of points an error bound for the computed spherical designs is provided

the art of signaling fifty years of coding theory in 1948 shannon developed fundamental limits on the efficiency of communication over noisy channels the coding theorem asserts that there are block codes with code rates arbitrarily close to channel capacity and probabilities of error arbitrarily close to zero fifty years later codes for the gaussian channel have been discovered that come close to these fundamental limits there is now a substantial algebraic theory of error correcting codes with as many connections to mathematics as to engineering practice and the last 20 years have seen the construction of algebraic geometry codes that can be encoded and decoded in polynomial time and that beat the gilbert varshamov bound given the size of coding theory as a subject this review is of necessity a personal perspective and the focus is reliable communication and not source coding or cryptography the emphasis is on connecting coding theories for hamming and euclidean space and on future challenges specifically in data networking wireless communication and quantum information theory

well conditioned spherical designs for integration and interpolation on the two sphere a set mathcal x n of n points on the unit sphere is a spherical t design if the average value of any polynomial of degree at most t over mathcal x n is equal to the average value of the polynomial over the sphere this paper considers the characterization and computation of spherical t designs on the unit sphere mathbb s 2 subset mathbb r 3 when n geq t 1 2 the dimension of the space mathbb p t of spherical polynomials of degree at most t we show how to construct well conditioned spherical designs with n geq t 1 2 points by maximizing the determinant of a matrix while satisfying a system of nonlinear constraints interval methods are then used to prove the existence of a true spherical t design very close to the calculated points and to provide a guaranteed interval containing the determinant the resulting spherical designs have good geometrical properties separation and mesh norm we discuss the usefulness of the points for both equal weight numerical integration and polynomial interpolation on the sphere and give an example

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

qmc designs optimal order quasi monte carlo integration schemes on the sphere we study equal weight numerical integration or quasi monte carlo qmc rules for functions in a sobolev space hs sd with smoothness parameter s d 2 defined over the unit sphere sd in rd 1 focusing on n point configurations that achieve optimal order qmc error bounds as is the case for efficient spherical designs we are led to introduce the concept of qmc designs these are sequences of n point configurations xn on s d such that the worst case error satisfies sup f h s f hs 1 1 n x xn f x sd f x d d x o n s d n with an implied constant that depends on the hs sd norm but is independent of n here d is the normalized surface measure on s d we provide methods for generation and numerical testing of qmc designs an essential tool is an expression for the worst case error in terms of a reproducing kernel for the space hs sd with s d 2 as a consequence of this and a recent result of bondarenko et al on the existence of spherical designs with appropriate number of points we show that minimizers of the n point energy for this kernel form a sequence of qmc designs for hs sd furthermore without appealing to the bondarenko et al result we prove that point sets that maximize the sum of suitable powers of the euclidean distance between pairs of points form a sequence of qmc designs for hs sd with s in the interval d 2 d 2 1 for such spaces there exist reproducing kernels with simple closed forms that are useful for numerical testing of optimal order quasi monte carlo integration numerical experiments suggest that many familiar sequences of point sets on the sphere equal area points spiral points minimal coulomb or logarithmic energy points and fekete points are qmc designs for appropriate values of s for comparison purposes we show that configurations of random points that are independently and uniformly distributed on the sphere do not constitute qmc designs for any s d 2 if xn is a sequence of qmc designs for h s sd we prove that it is also a sequence of qmc designs for hs sd for all s d 2 s this leads to the question of determining the supremum of such s here called the qmc strength of the sequence for which we provide estimates based on computations for the aforementioned sequences received by the editor august 15 2012 and in revised form february 26 2013 2010 mathematics subject classification primary 65d30 65d32 secondary 11k38 41a55

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

tight informationally complete quantum measurements we introduce a class of informationally complete positive operator valued measures which are in analogy with a tight frame as close as possible to orthonormal bases for the space of quantum states these measures are distinguished by an exceptionally simple state reconstruction formula which allows painless quantum state tomography complete sets of mutually unbiased bases and symmetric informationally complete positive operator valued measures are both members of this class the latter being the unique minimal rank one members recast as ensembles of pure quantum states the rank one members are in fact equivalent to weighted 2 designs in complex projective space these measures are shown to be optimal for quantum cloning and linear quantum state tomography

optimizing quantum process tomography with unitary 2 designs we show that weighted unitary 2 designs define optimal measurements on the system ancilla output state for ancilla assisted process tomography of unital quantum channels examples include complete sets of mutually unbiased unitary operator bases each of these specifies a minimal series of optimal orthogonal measurements general quantum channels are also considered

on the explicit constructions of certain unitary t designs unitary t designs are good finite subsets of the unitary group ug d that approximate the whole unitary group ug d well unitary t designs have been applied in randomized benchmarking tomography quantum cryptography and many other areas of quantum information science if a unitary t design itself is a group then it is called a unitary t group although it is known that unitary t designs in ug d exist for any t and d the unitary t groups do not exist for t geq 4 if d geq 3 as it is shown by guralnick tiep 2005 and bannai navarro rizo tiep bnrt 2018 explicit constructions of exact unitary t designs in ug d are not easy in general in particular explicit constructions of unitary 4 designs in ug 4 have been an open problem in quantum information theory we prove that some exact unitary t 1 designs in the unitary group ug d are constructed from unitary t groups in ug d that satisfy certain specific conditions based on this result we specifically construct exact unitary 3 designs in ug 3 from the unitary 2 group sl 3 2 in ug 3 and also unitary 4 designs in ug 4 from the unitary 3 group sp 4 3 in ug 4 numerically we also discuss some related problems and a few applications to physics problems

composite parameterization and haar measure for all unitary and special unitary groups we adopt the concept of the composite parameterization of the unitary group u d to the special unitary group su d furthermore we also consider the haar measure in terms of the introduced parameters we show that the well defined structure of the parameterization leads to a concise formula for the normalized haar measure on u d and su d with regard to possible applications of our results we consider the computation of high order integrals over unitary groups

an effective and fast iris recognition system based on a combined multiscale feature extraction technique the randomness of iris pattern makes it one of the most reliable biometric traits on the other hand the complex iris image structure and the various sources of intra class variations result in the difficulty of iris representation although a number of iris recognition methods have been proposed it has been found that several accurate iris recognition algorithms use multiscale techniques which provide a well suited representation for iris recognition in this paper and after a thorough analysis and summarization a multiscale edge detection approach has been employed as a pre processing step to efficiently localize the iris followed by a new feature extraction technique which is based on a combination of some multiscale feature extraction techniques this combination uses special gabor filters and wavelet maxima components finally a promising feature vector representation using moment invariants is proposed this has resulted in a compact and efficient feature vector in addition a fast matching scheme based on exclusive or operation to compute bits similarity is proposed where the result experimentation was carryout out using casia database the experimental results have shown that the proposed system yields attractive performances and could be used for personal identification in an efficient and effective manner and comparable to the best iris recognition algorithm found in the current literature

optimal fault tolerant routing algorithm and fault tolerant diameter in directed double loop networks this paper addresses the reliability of directed double loop networks g n r s and studies the problems about optimal fault tolerant routing fault tolerant diameter etc in g n r s firstly we study the shapes of the l shaped tiles which are the minimum distance diagram of directed double loop networks we divide them into four types there are different parameters for each type according to the solutions to the congruence equation then we study the distributions of the optimal equivalent nodes for different destination nodes and present an optimal fault tolerant routing algorithm and a formula computing the fault tolerant diameter finally we prove that there is a lower bound for fault tolerant diameter and show that there may be lots of double optimal directed double loop networks in some infinite clusters of g n r s according to the proposed fault tolerant routing algorithm the reliability and transmission performance will be optimal when some faults occur in g n r s

small designs for path connected spaces and path connected homogeneous spaces author s kane dm abstract 2015 american mathematical society we prove the existence of designs of small size in a number of contexts in particular our techniques can be applied to prove the existence of n designs on sd of size od nd logd 1 n

optimal asymptotic bounds for spherical designs in this paper we prove the conjecture of korevaar and meyers for each n cdt d there exists a spherical t design in the sphere s d consisting of n points where cd is a constant depending only on d

embedding meshes and tori on double loop networks of the same size double loop networks are extensions of ring networks and are widely used in the design and implementation of local area networks and parallel processing architectures however embedding of other types of networks on double loop networks has not been well studied due to the topological complexity of double loop networks the traditional l shape designed to compute the diameter of double loop networks is not effective to solve the embedding problem we propose a novel tessellation approach to partition the geometric plane of double loop networks into a set of parallelogram tiles called p shape based on the characteristics of p shape we design a simple embedding scheme namely p shape embedding that embeds meshes and tori on double loop networks in a systematic way under p shape embedding we evaluate the embedding metrics of dilation average dilation and congestion which depend heavily on the parameters of p shape a main merit of p shape embedding is that a large fraction of embedded mesh torus edges have edge dilation 1 resulting in a low average dilation these are the first results to our knowledge for embedding meshes and tori on double loop networks which is of great significance due to the popularity of these architectures our p shape construction bridges between regular graphs and double loop networks and provides a powerful tool for studying double loop networks

quadrature rules with neighborhood of spherical designs on the two sphere abstract in this paper we concentrate on quadrature rules with their point sets located on a neighborhood of a spherical design we show that any point set in a small enough neighborhood of a fundamental spherical design can establish a positive quadrature rule a preliminary bound of the neighborhood radius is given to guarantee this property the perturbation range of the weights is asymptotically linearly dependent on the radius of the neighborhood numerical experiments are proposed to test the asymptotic sharpness of the theoretical results

minkowski s convex body theorem and integer programming the paper presents an algorithm for solving integer programming problems whose running time depends on the number n of variables as non this is done by reducing an n variable problem to 2n5i 2 problems in n i variables for some i greater than zero chosen by the algorithm the factor of on5 2 per variable improves the best previously known factor which is exponential in n minkowski s convex body theorem and other results from geometry of numbers play a crucial role in the algorithm several related algorithms for lattice problems are presented the complexity of these problems with respect to polynomial time reducibilities is studied

chebyshev type quadrature and zeros of faber polynomials abstract with any probability measure on 1 1 we associate a sequence of polynomials fn z which are faber polynomials of a univalent function f z on z 1 if the zeros of fn z are in the open unit disk then there exists a chebyshev type quadrature formula for with n nodes which is exact for all polynomials f t up to degree n 1 for the normalized jacobi measures d t c 1 t 1 2 1 t 1 2 d t with 1 2 the function f z can be expressed in terms of hypergeometric functions using this expression it is proved that the zeros of the associated faber polynomials are in the open unit disk in case 0 0 for some 0 0 this result solves to a large extent a problem of forster

isoentangled mutually unbiased bases symmetric quantum measurements and mixed state designs discrete structures in hilbert space play a crucial role in finding optimal schemes for quantum measurements we solve the problem of whether a complete set of five isoentangled mutually unbiased bases exists in dimension four providing an explicit analytical construction the reduced density matrices of these 20 pure states forming this generalized quantum measurement form a regular dodecahedron inscribed in a sphere of radius sqrt 3 20 located inside the bloch ball of radius 1 2 such a set forms a mixed state 2 design a discrete set of quantum states with the property that the mean value of any quadratic function of density matrices is equal to the integral over the entire set of mixed states with respect to the flat hilbert schmidt measure we establish necessary and sufficient conditions mixed state designs need to satisfy and present general methods to construct them furthermore it is shown that partial traces of a projective design in a composite hilbert space form a mixed state design while decoherence of elements of a projective design yields a design in the classical probability simplex we identify a distinguished two qubit orthogonal basis such that four reduced states are evenly distributed inside the bloch ball and form a mixed state 2 design

a sharpening of the welch bounds and the existence of real and complex spherical t designs the welch bounds for a finite set of unit vectors are a family of inequalities indexed by t 1 2 ldots which describe how evenly spread the vectors are they have important applications in signal analysis where sequences giving equality in the first welch bound are known as welch bound equality sequences or as unit norm tight frames here we consider sequences of vectors giving equality in the higher order welch bounds these are seen to correspond to tight frames for the complex symmetric t tensors which we prove always exist we show that for t 1 the welch bounds can be sharpened for real vectors and again vectors giving equality always exist we give a unified treatment of various conditions for equality in both the real and complex cases in particular we give an explicit description of the corresponding cubature rules t designs our results set up a framework for the construction and classification several configurations of vectors of recent interest these include mutually unbiased bases complex equiangular lines spherical half designs projective t designs and minimisers of the higher order frame potential one interesting consequence is a construction of sets of complex equiangular lines which were previously unknown

some constructions of spherical 5 designs abstract spherical designs were introduced by delsarte goethals and seidel in 1977 a spherical t design in rn is a finite set x sn 1 with the property that for every polynomial p with degree t the average value of p on x equals the average value of p on sn 1 this paper contains some existence and nonexistence results mainly for spherical 5 designs in r3 delsarte goethals and seidel proved that if x is a spherical 5 design in r3 then x 12 and if xz sfnc 12 then x consists of the vertices of a regular icosahedron we show that such designs exist with cardinality 16 18 20 22 24 and every integer 26 if x is a spherical 5 design in rn then x n n 1 if x n n 1 then x has been called tight tight spherical 5 designs in rn are known to exist only for n 2 3 7 23 and possibly n u2 2 for odd u 7 any tight spherical 5 design in rn must consist of n n 1 2 antipodal pairs of points we show that for n 3 there are no spherical 5 designs in rn consisting of n n 1 2 1 antipodal pairs of points

computational existence proofs for spherical t designs spherical t designs provide quadrature rules for the sphere which are exact for polynomials up to degree t in this paper we propose a computational algorithm based on interval arithmetic which for given t upon successful completion will have proved the existence of a t design with t 1 2 nodes on the unit sphere s 2 subset mathbb r 3 and will have computed narrow interval enclosures which are known to contain these nodes with mathematical certainty since there is no theoretical result which proves the existence of a t design with t 1 2 nodes for arbitrary t our method contributes to the theory because it was tested successfully for t 1 2 100 the t design is usually not unique our method aims at finding a well conditioned one the method relies on computing an interval enclosure for the zero of a highly nonlinear system of dimension t 1 2 we therefore develop several special approaches which allow us to use interval arithmetic efficiently in this particular situation the computations were all done using the matlab toolbox intlab

chebyshev type quadrature and partial sums of the exponential series chebyshev type quadrature for the weight functions 1 ai wa t l r l 1 a 1 7tvl 12 is related to a problem concerning partial sums of the exponential series namely the problem to extend the nth partial sum to a polynomial of degree 27v having all zeros on the circle z a n using this connection we show that the minimal number n of nodes needed for chebyshev type quadrature of degree n for wait satisfies an inequality c n n c2n with positive constants q c2 as an application we prove that the minimal number n of nodes for chebyshev type quadrature of degree n on a torus embedded in r3 satisfies an inequality c1 2 n cjn2

attacking the pollard generator let p be a prime and let c be an integer modulo p the pollard generator is a sequence un of pseudorandom numbers defined by the relation un 1equivun 2 c mod p it is shown that if c and 9 14 of the most significant bits of two consecutive values un un 1 of the pollard generator are given one can recover in polynomial time the initial value u0 with a probabilistic algorithm this result is an improvement of a theorem in a recent paper which requires that 2 3 of the most significant bits be known

analysis of the structured perturbation for the bsccb linear system abstract in this paper based on block style spectral decomposition of the block skew circulant with circulant blocks bsccb matrix the structure perturbation is discussed which includes the condition number and relative error of the bsccb linear system then the optimal backward perturbation bound of the bsccb linear system is analyzed simultaneously the algorithm for the optimal backward perturbation bound is presented at the end of the paper a numerical example is provided to verify the effectiveness of the algorithm