a fast numerical algorithm for the composition of power series with complex coefficients abstract the composition problem for univariate complex power series p q i e the computation of the first n 1 coefficients of the composition q p is numerically solved by interpolation methods using multitape turing machines as a model of computation the composition problem of power series with integer coefficients of modulus n is possible in time o n 2 log n log where m bounds the amount of time for the multiplication of m bit numbers e g m cm log m 1 log log m 2 for multitape turing machines this algorithm is asymptotically faster than an implementation of the brent kung algorithm on a multitape turing machine the improvement is of order n 1 2 up to logarithmic terms

a multi level fast multipole bem for 3 d elastodynamics in the frequency domain to reduce computational complexity and memory requirement for 3 d elastodynamics using the boundary element method bem a multi level fast multipole bem fm bem is proposed the diagonal form for the expansion of the elastodynamic fundamental solution is used with a truncation parameter adjusted to the subdivision level a feature necessary for achieving optimal computational efficiency both the single level and mul ti level forms of the elastodynamic fm bem are considered with emphasis on the latter crucial implementation issues including the truncation of the multipole expansion the optimal number of levels the direct and inverse extrapolation steps are examined in detail with the backing of numerical experiments a complexity analysis for both the single level and multi level versions is conducted the correctness and computational performances of the proposed elastodynamic fmm are demonstrated on numerical examples featuring up to o 10 6 dofs run on a single processor pc and including the diffraction of an incident p plane wave by a semi spherical or semi ellipsoidal canyon representative of topographic site effects

implicitization of rational surfaces using toric varieties abstract a parameterized surface can be represented as a projection from a certain toric surface this generalizes the classical homogeneous and bihomogeneous parameterizations we extend to the toric case two methods for computing the implicit equation of such a rational parameterized surface the first approach uses resultant matrices and gives an exact determinantal formula for the implicit equation if the parameterization has no base points in the case the base points are isolated local complete intersections we show that the implicit equation can still be recovered by computing any non zero maximal minor of this matrix the second method is the toric extension of the method of moving surfaces and involves finding linear and quadratic relations syzygies among the input polynomials when there are no base points we show that these can be put together into a square matrix whose determinant is the implicit equation its extension to the case where there are base points is also explored

free vibrations of circular cylindrical shells with an interior plate using the receptance method this paper describes the method to analyze the free vibrations of simply supported cylindrical shells with an interior rectangular plate by using the receptance method this method is based on the ratio of a deflection or slope response to a harmonic force or moment at the joint after finding the free vibrational characteristics of the simply supported plate and shell before combination the frequency equation of the combined system is obtained by considering the continuity conditions at the joint between the plate and the shell when the line load and moment applied along the joint are assumed as the dirac delta and sinusoidal function the continuity conditions at the joint are proven to be satisfied the numerical results are compared with published works and experimental results in order to show the validity of the formulation and that the analytical results agree with those from other methods

on the spatial bandwidth of scattered fields it is shown that the scattered fields are almost space bandlimited functions the effective bandwidth w is introduced and evaluated for a very general scattering system as well as the error made using functions bandlimited to w w for representing the scattered field the effective bandwidth is very simply related to the maximum dimension of the scattering system the error drops to negligible values for modest increases of w compared to w in the case of large scatterers important consequences of the above general results are finally stressed

new resultant inequalities and complex polynomial factorization the author deduces some new probabilistic estimates on the distances between the zeros of a polynomial p x by using some properties of the discriminant of p x and applies these estimates to improve the fastest deterministic algorithm for approximating polynomial factorization over the complex field namely given a natural n positive epsilon such that log 1 epsilon o n log n and the complex coefficients of a polynomial p x sum n i 0 p ix i such that p n ne 0 sum i p i le 1 a factorization of p x within the error norm epsilon is computed as a product of factors of degrees at most n 2 by using o log 2n time and n 3 processors under the pram arithmetic model of parallel computing or by using o n 2 log 2 n arithmetic operations the algorithm is randomized of las vegas type allowing a failure with a probability at most delta for any positive delta 1 such that log 1 delta o log n except for a narrow class of polynomials p x these results can be also obtained for epsilon such that log 1 epsilon o n 2 log n

modal coupling in the vibration of fluid loaded cylindrical shells an approach is presented to evaluate the pressure field and vibratory response of a finite fluid loaded cylindrical shell with infinite rigid extensions which are connected to the shell the approach combines a generalized fourier series or in vacuo eigenfunction expansion of the velocity field of the shell with a green s function and integral equation representation of the acoustic loading on the shell although modes with different circumferential wavenumbers are decoupled all modes with identical circumferential wavenumbers are coupled via the fluid the algebraic equations which account for this coupling include both shell impedances and acoustic impedances which are mode dependent general integral expressions are presented for the acoustic impedances which include both self radiation and interaction impedances in order to illustrate the general characteristics of these acoustic impedances the impedances are investigated for a particular set of eigenfunctions low frequency and asymptotic expressi

the complexity of partial derivatives abstract let l denote the nonscalar complexity in k x 1 x n we prove l x 1 x n 3l using this we determine the complexity of single power sums single elementary symmetric functions the resultant and the discriminant as root functions up to order of magnitude also we linearly reduce matrix inversion to computing the determinant

lower bounds for special cases of syntactic multilinear abps abstract algebraic branching programs abps are standard models for computing polynomials syntactic multilinear abps smabps are restrictions of abps where every variable is allowed to occur at most once in every path from the source to the sink node proving super polynomial lower bounds for syntactic multilinear abps remains a challenging open question in algebraic complexity theory the current best lower bound for smabps is only quadratic in the number of variables 1 in this article we develop a new approach for proving syntactic multilinear branching program lower bounds convert the smabp into an equivalent multilinear formula with a super polynomial blow up in size and then exploit the structural limitations of the resulting formula to obtain an upper bound on the rank of partial derivative matrix of the polynomial computed by the smabp using this approach we prove exponential lower bounds for special cases of smabps namely sum of read once oblivious smabps multilinear r pass abps and set multilinear abps en route we also prove an exponential lower bound for a special class of syntactic multilinear arithmetic circuits using a similar approach

rigorous sensitivity analysis for systems of linear and nonlinear equations methods are presented for performing a rigorous sensitivity analysis of numerical problems with independent noncorrelated data for general systems of linear and nonlinear equations the methods may serve for the following two purposes first to bound the dependency of the solution on changes in the input data in contrast to condition numbers a componentwise sensitivity analysis of the solution vector is performed second to estimate the true solution set for problems the input data of which are afflicted with tolerances the methods presented are very effective with the additional property that due to an automatic error control mechanism every computed result is guaranteed to be correct examples are given for linear systems demonstrating that the computed bounds are in general very sharp interesting comparisons to traditional condition numbers are given

interpolation of polynomials given by straight line programs we give an algorithm for the interpolation of a polynomial a given by a straight line program its complexity is polynomial in t log d l n where t is an input bound on the number of terms in a d is a bound on its partial degree in all variables l is the length of the given straight line program and n is the number of variables

asymptotic fluid structure interaction theories for acoustic radiation prediction it is well known that at extremely high or low frequencies the fluid structure interaction effects can be represented asymptotically by simple equations thus it appears that an optimum computation scheme for predicting acoustic pressure field radiated from a submerged elastic structure could be a combination of various asymptotic theories and the exact formulation this paper explores the ranges of applicability of some asymptotic theories using the problem of radiation from a spherical elastic shell as the bench mark it is found that the added mass and plane wave approximations are quite adequate respectively for ka 5 where ka is the ratio of the circumference of the spherical shell to the acoustic wavelength for the intermediate frequency range a theory termed second order doubly asymptotic approximation daa2 is suitable at intermediate frequencies near resonances however the exact formulation is needed

complex power reciprocity and radiation modes for submerged bodies this study of the surface interaction between a submerged body and the surrounding fluid begins by developing reciprocity relations between alternative pressure and normal velocity distributions on the wetted surface a corollary of these principles is proof of the symmetry of the matrix representing the acoustic contribution to the structural impedance even in situations where the acoustic relation between surface pressure and normal velocity is not symmetric the reciprocity properties lead to two eigenvalue problems whose solution yields velocity and pressure radiation modes each of which decouples the complex surface acoustic power the matrices required to obtain the eigensolutions are shown to arise in the ordinary course of modeling fluid structure interaction further analysis reveals that the velocity and pressure modes occur in a one to one correspondence with a relative phase angle that decreases monotonically as the radiated power increases relative to the reactive power using the radiati

acoustic radiations for submerged elastic structures using natural mode expansions in conjunction with radiation modes approach this work formulates submerged elastic structures using in vacuo vibrational mode expansions with which the acoustic impedance loading is derived based on radiation mode theory the displacement of natural modes on the normal direction is expanded as linear combinations of a set of velocity radiation modes that the expansion coefficients characterize as the radiation characteristics of each vibration mode this type of expansion allows one to represent the surface pressure by the corresponding set of pressure radiation modes thus a symmetric impedance matrix associated with the natural vibration mode expansions is derived when a variational increment is applied to the virtual work done by the surface pressure against the normal displacement the equation of the submerged structures is obtained according to hamilton s principles by incorporating the description of radiation modes this equation of natural mode expansions is used to study the coupling among vibration modal amplitudes due to the modal cross impedances and the convergence of near and farfield solutions in addition a slender submerged spheroidal shell vibrating axisymmetrically serves as a numerical example to demonstrate the effectiveness of the analysis procedure this numerical example reveals that the acoustic impedances decrease with ascending mode numbers causing the high order vibration modes to react independently moreover the convergence of the surface pressure and normal velocity is examined on the basis of independent reaction of the vibration modes accurately predicting farfield solutions depends only on the convergence of the surface quantities whose components pertain to strong radiation modes the numerical example indicates that the number of vibration modes used in the expansion for predicting farfield solutions is less than the modes required for the surface solution

a nearly optimal algorithm for deciding connectivity queries in smooth and bounded real algebraic sets a roadmap for a semi algebraic set s is a curve which has a non empty and connected intersection with all connected components of s hence this kind of object introduced by canny can be used to answer connectivity queries with applications for instance to motion planning but has also become of central importance in effective real algebraic geometry since it is used in higher level algorithms in this article we provide a probabilistic algorithm which computes roadmaps for smooth and bounded real algebraic sets its output size and running time are polynomial in nd nlog d where d is the maximum of the degrees of the input polynomials d is the dimension of the set under consideration and n is the number of variables more precisely the running time of the algorithm is essentially subquadratic in the output size even under our assumptions it is the first roadmap algorithm with output size and running time polynomial in nd nlog d

experimental investigation of the wave propagation on a point driven submerged capped cylinder using k space analysis classically sparse accelerometer measurements are made on vibrating structures to help in deducing the physics of the vibration however the experimenter now has at his command more sophisticated measurement tools such as nearfield acoustical holography or laser doppler velocimetry which provide much more data often complete mapping of the acceleration or displacements on the surface of these structures can be acquired as a result more sophisticated analysis tools must be developed such a tool useful for vibrating structures which are basically cylindrical or planar is presented herein the measured spatial velocity is decomposed at fixed temporal frequencies into its helical wave spectrum also known as the k space spectrum this decomposition contains a great deal of information about the physics of the vibrator for example it provides the dominant wavenumbers the dispersion diagram of free waves that exist on the shell these free waves reach maximum amplitudes when close to a resonance fr

a nonlinear lower bound for constant depth arithmetical circuits via the discrete uncertainty principle we prove a superlinear lower bound on the size of a bounded depth bilinear arithmetical circuit computing cyclic convolution our proof uses the strengthening of the donoho stark uncertainty principle d l donoho p b stark uncertainty principles and signal recovery siam journal of applied mathematics 49 1989 906 931 given by tao t tao an uncertainty principle for cyclic groups of prime order mathematical research letters 12 2005 121 127 and a combinatorial lemma by raz and shpilka r raz a shpilka lower bounds for matrix product in arbitrary circuits with bounded gates siam journal of computing 32 2003 488 513 this combination and an observation on ranks of circulant matrices which we use to give a much shorter proof of the donoho stark principle may have other applications

subquadratic time factoring of polynomials over finite fields new probabilistic algorithms are presented for factoring univariate polynomials over finite fields the algorithms factor a polynomial of degree n over a finite field of constant cardinality in time o n 1 815 previous algorithms required time n 2 o 1 the new algorithms rely on fast matrix multiplication techniques more generally to factor a polynomial of degree n over the finite field f q with q elements the algorithms use o n 1 815 log q arithmetic operations in f q the new baby step giant step techniques used in our algorithms also yield new fast practical algorithms at super quadratic asymptotic running time and subquadratic time methods for manipulating normal bases of finite fields

hardness randomness tradeoffs for bounded depth arithmetic circuits in this paper we show that lower bounds for bounded depth arithmetic circuits imply derandomization of polynomial identity testing for bounded depth arithmetic circuits more formally if there exists an explicit polynomial f that cannot be computed by a depth d arithmetic circuit of small size then there exists an efficient deterministic black box algorithm to test whether a given depth d 5 circuit that computes a polynomial of relatively small individual degrees is identically zero or not in particular if we are guaranteed that the tested circuit computes a multilinear polynomial then we can perform the identity test efficiently to the best of our knowledge this is the first hardness randomness tradeoff for bounded depth arithmetic circuits the above results are obtained using the arithmetic nisan wigderson generator of kabanets and impagliazzo together with a new theorem on bounded depth circuits which is the main technical contribution of our work this theorem deals with polynomial equations of the form p x 1 dots x n y equiv0 and shows that if p has a circuit of depth d and size s and if the polynomial f x 1 dots x n satisfies p x 1 dots x n f equiv0 then f has a circuit of depth d 3 and size mathrm poly s m r where m is the total degree of f and r is the degree of y in p this circuit for f can be found probabilistically in time mathrm poly s m r in the other direction we observe that the methods of kabanets and impagliazzo can be used to show that derandomizing identity testing for bounded depth circuits implies lower bounds for the same class of circuits more formally if we can derandomize polynomial identity testing for bounded depth circuits then nexp does not have bounded depth arithmetic circuits that is either mathrm nexp not subseteq p mathrm poly or the permanent is not computable by polynomial size bounded depth arithmetic circuits

on a problem posed by steve smale the 17th of the problems proposed by steve smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of n complex polynomials in n unknowns in time polynomial on the average in the size n of the input system a partial solution to this problem was given by carlos beltr an and luis miguel pardo who exhibited a randomized algorithm doing so in this paper we further extend this result in several directions firstly we exhibit a linear homotopy algorithm that eciently implements a nonconstructive idea of

computing the sign or the value of the determinant of an integer matrix a complexity survey computation of the sign of the determinant of a matrix and the determinant itself is a challenge for both numerical and exact methods we survey the complexity of existing methods to solve these problems when the input is an n n matrix a with integer entries we study the bit complexities of the algorithms asymptotically in n and the norm of a existing approaches rely on numerical approximate computations on exact computations or on both types of arithmetic in combination

circuit complexity proof complexity and polynomial identity testing the ideal proof system we introduce a new and natural algebraic proof system whose complexity measure is essentially the algebraic circuit size of nullstellensatz certificates this enables us to exhibit close connections between effective nullstellensatze proof complexity and algebraic circuit complexity in particular we show that any super polynomial lower bound on any boolean tautology in our proof system implies that the permanent does not have polynomial size algebraic circuits vnp vp we also show that super polynomial lower bounds on the number of lines in polynomial calculus proofs imply the permanent versus determinant conjecture note that there was no proof system prior to ours for which lower bounds on an arbitrary tautology implied any complexity class lower bound our proof system helps clarify the relationships between previous algebraic proof systems in doing so we highlight the importance of polynomial identity testing pit in proof complexity in particular we use pit to illuminate ac0 p frege lower bounds which have been open for nearly 30 years with no satisfactory explanation as to their apparent difficulty finally we explain the obstacles that must be overcome in any attempt to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity using the algebraic structure of our proof system we propose a novel route to such lower bounds although such lower bounds remain elusive this proposal should be contrasted with the difficulty of extending ac0 p circuit lower bounds to ac0 p frege lower bounds

acoustic radiation from a finite length shell with non axisymmetric substructures using a surface variational principle abstract a modal based method is developed to analyze the acoustic radiation of axisymmetric submerged shells of finite length with non axisymmetric internal substructures subjected to time harmonic loads in this method a variational principle is used to derive impedance relations between the surface pressure and the surface velocity of the shell these impedance relations are combined with the structural equations of motions based on a lagrange energy formulation to form a complete set of equations for the fluid structure interaction problem fourier series expansions are used to represent the circumferential dependence of the surface pressure and velocity the method is demonstrated for two different configurations of substructures circular ribs supporting length wise beams and a spatially and modally dense array of oscillators since the substructures couple the circumferential modes a large system of equations must be inverted a matrix decomposition technique is used to reduce the size of the system of equations of the first example for the second example the asymptotic limit for an infinite number of substructures is developed using an integral form for the substructure impedance it is shown that the monte carlo simulations for the oscillator substructures converge to the asymptotic results it is also shown that below the ring frequency oscillators can induce damping that is more effective in reducing far field radiation than high loss factors in the shell

an extension of strassen s degree bound with every set p 1 cdots p m of multivariate polynomials we associate in a natural way several algebraic varieties i e irreducible zariski closed sets the degree of each of these closed sets can be nicely bounded in terms of the number l ns p 1 cdots p m of nonscalar operations which are necessary to evaluate p 1 cdots p m we establish lower bounds l ns p geqq omega k lg n for single specific polynomials p of degree n depending on o k variables with 0 1 coefficients typical examples are l ns sum i 1 k x i n y i geqq frac 1 2 k lg n l ns sum i 1 k x 1 x 2 cdots x i n y i frac 1 2 k lg n provided k omega k lg n for almost all polynomials p of degree n depending on k variables k ll n these lower bounds hold for any field characteristic

complexity of parallel matrix computations abstract we estimate parallel complexity of several matrix computations under both boolean and arithmetic machine models using deterministic and probabilistic approaches those computations include the evaluation of the inverse the determinant and the characteristic polynomial of a matrix recently processor efficiency of the previous parallel algorithms for numerical matrix inversion has been substantially improved in pan and reif 1987 reaching optimum estimates up to within a logarithmic factor that work however applies neither to the evaluation of the determinant and the characteristic polynomial nor to exact matrix inversion nor to the numerical inversion of ill conditioned matrices we present four new approaches to the solution of those latter problems having several applications to combinatorial computations in order to extend the suboptimum time and processor bounds of pan and reif 1987 to the case of computing the inverse determinant and characteristic polynomial of an arbitrary integer input matrix in addition processor efficient algorithms using polylogarithmic parallel time are devised for some other matrix computations such as triangular and qr factorizations of a matrix and its reduction to hessenberg form

rigid continuation paths i quasilinear average complexity for solving polynomial systems how many operations do we need on the average to compute an approximate root of a random gaussian polynomial system beyond smale s 17th problem that asked whether a polynomial bound is possible we prove a quasi optimal bound text input size 1 o 1 this improves upon the previously known text input size frac32 o 1 bound the new algorithm relies on numerical continuation along rigid continuation paths the central idea is to consider rigid motions of the equations rather than line segments in the linear space of all polynomial systems this leads to a better average condition number and allows for bigger steps we show that on the average we can compute one approximate root of a random gaussian polynomial system of n equations of degree at most d in n 1 homogeneous variables with o n 5 d 2 continuation steps this is a decisive improvement over previous bounds that prove no better than sqrt 2 min n d continuation steps on the average

prediction of the vibroacoustic behavior of a submerged shell with non axisymmetric internal substructures by a condensed transfer function method the vibroacoustic behavior of axisymmetric stiffened shells immersed in water has been intensively studied in the past on the contrary little attention has been paid to the modeling of these shells coupled to non axisymmetric internal frames indeed breaking the axisymmetry couples the circumferential orders of the fourier series and considerably increases the computational costs in order to tackle this issue we propose a sub structuring approach called the condensed transfer function ctf method that will allow assembling a model of axisymmetric stiffened shell with models of non axisymmetric internal frames the ctf method is developed in the general case of mechanical subsystems coupled along curves a set of orthonormal functions called condensation functions which depend on the curvilinear abscissa along the coupling line is considered this set is then used as a basis for approximating and decomposing the displacements and the applied forces at the line junctions thanks to the definition and calculation of condensed transfer functions for each uncoupled subsystem and by using the superposition principle for passive linear systems the behavior of the coupled subsystems can be deduced a plane plate is considered as a test case to study the convergence of the method with respect to the type and the number of condensation functions taken into account the ctf method is then applied to couple a submerged non periodically stiffened shell described using the circumferential admittance approach caa with internal substructures described by finite element method fem the influence of non axisymmetric internal substructures can finally be studied and it is shown that it tends to increase the radiation efficiency of the shell and can modify the vibrational and acoustic energy distribution

algorithmic applications of baur strassen s theorem shortest cycles diameter and matchings consider a directed or an undirected graph with integral edge weights from the set w w that does not contain negative weight cycles in this article we introduce a general framework for solving problems on such graphs using matrix multiplication the framework is based on the usage of baur strassen s theorem and of strojohann s determinant algorithm it allows us to give new and simple solutions to the following problems finding shortest cycles we give a simple o wn time algorithm for finding shortest cycles in undirected and directed graphs for directed graphs and undirected graphs with nonnegative weights this matches the time bounds obtained in 2011 by roditty and williams on the other hand no algorithm working in o wn time was previously known for undirected graphs with negative weights furthermore our algorithm for a given directed or undirected graph detects whether it contains a negative weight cycle within the same running time computing diameter and radius we give a simple o wn time algorithm for computing a diameter and radius of an undirected or directed graphs to the best of our knowledge no algorithm with this running time was known for undirected graphs with negative weights finding minimum weight perfect matchings we present an o wn time algorithm for finding minimum weight perfect matchings in undirected graphs this resolves an open problem posted by sankowski 2009 who presented such an algorithm but only in the case of bipartite graphs these three problems that are solved in the full generality demonstrate the utility of this framework hence we believe that it can find applications for solving larger spectra of related problems as an illustrative example we apply it to the problem of computing a set of vertices that lie on cycles of length at most t for some given t we give a simple o wn time algorithm for this problem that improves over the o wn t time algorithm given by yuster in 2011 besides giving this flexible framework the other main contribution of this article is the development of a novel combinatorial interpretation of the dual solution for the minimum weight perfect matching problem despite the long history of the matching problem such a combinatorial interpretation was not known previously this result sheds a new light on the problem as there exist many structural theorems about unweighted matchings but almost no results that could cope with the weighted case

polar varieties and efficient real elimination let s 0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f 1 ldots f p this paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of s 0 for this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of s 0 this leads to a procedure which solves our algorithmic problem in time that is polynomial in the extrinsic description length of the input equations f 1 ldots f p and in a suitably introduced intrinsic geometric parameter called the degree of the real interpretation of the given equation system f 1 ldots f p

algebraic and numerical techniques for the computation of matrix determinants we review modify and combine together several numerical and algebraic techniques in order to compute the determinant of a matrix or the sign of such a determinant the resulting algorithms enable us to obtain the solution by using a lower precision of computations and relatively few arithmetic operations the problem has important applications to computational geometry

error minimization of multipole expansion in this paper we focus on the truncation error of the multipole expansion for the fast multipole method and the multilevel fast multipole algorithm when the buffer size is large enough the error can be controlled and minimized by using the conventional selection rules on the other hand if the buffer size is small the conventional selection rules no longer hold and the new approach which we have recently proposed is needed however this method is still not sufficient to minimize the error for small buffer cases we clarify this fact and show that the information about the placement of true worst case interaction is needed a novel algorithm to minimize the truncation error is proposed

fast computation of a rational point of a variety over a finite field we exhibit a probabilistic algorithm which computes a rational point of an absolutely irreducible variety over a finite field defined by a reduced regular sequence its time space complexity is roughly quadratic in the logarithm of the cardinality of the field and a geometric invariant of the input system this invariant called the degree is bounded by the bezout number of the system our algorithm works for fields of any characteristic but requires the cardinality of the field to be greater than a quantity which is roughly the fourth power of the degree of the input variety

tensor rank and lower bounds for arithmetic formulas we show that any explicit example for a tensor a n r f with tensor rank nr 1 o 1 where r r n log n log log n is super constant implies an explicit super polynomial lower bound for the size of general arithmetic formulas over f this shows that strong enough lower bounds for the size of arithmetic formulas of depth 3 imply super polynomial lower bounds for the size of general arithmetic formulas one component of our proof is a new approach for homogenization and multilinearization of arithmetic formulas that gives the following results we show that for any n variate homogeneous polynomial f of degree r if there exists a fanin 2 formula of size s and depth d for f then there exists a homogeneous formula of size o d r 1 r s for f in particular for any r o log n if there exists a polynomial size formula for f then there exists a polynomial size homogeneous formula for f this refutes a conjecture of nisan and wigderson 1996 and shows that super polynomial lower bounds for homogeneous formulas for polynomials of small degree imply super polynomial lower bounds for general formulas we show that for any n variate set multilinear polynomial f of degree r if there exists a fanin 2 formula of size s and depth d for f then there exists a set multilinear formula of size o d 2 r s for f in particular for any r o log n log log n if there exists a polynomial size formula for f then there exists a polynomial size set multilinear formula for f this shows that super polynomial lower bounds for set multilinear formulas for polynomials of small degree imply super polynomial lower bounds for general formulas

greatest common divisors of polynomials given by straight line programs algorithms on multivariate polynomials represented by straight line programs are developed first it is shown that most algebraic algorithms can be probabilistically applied to data that are given by a straight line computation testing such rational numeric data for zero for instance is facilitated by random evaluations modulo random prime numbers then auxiliary algorithms that determine the coefficients of a multivariate polynomial in a single variable are constructed the first main result is an algorithm that produces the greatest common divisor of the input polynomials all in straight line representation the second result shows how to find a straight line program for the reduced numerator and denominator from one for the corresponding rational function both the algorithm for that construction and the greatest common divisor algorithm are in random polynomial time for the usual coefficient fields and output a straight line program which with controllably high probability correctly determines the requested answer the running times are polynomial functions in the binary input size the input degrees as unary numbers and the logarithm of the inverse of the failure probability the algorithm for straight line programs for the numerators and denominators of rational functions implies that every degree bounded rational function can be computed fast in parallel that is in polynomial size and polylogarithmic depth

the techniques of trilinear aggregating and the recent progress in the asymptotic acceleration of matrix operations abstract the recent progress in the asymptotic acceleration of matrix multiplication and of related matrix operations is surveyed the techniques of trilinear aggregating and their applications to the above problems and to the disproval of the direct sum conjecture over some rings of constants are presented in some detail

black box learning of multigrid parameters abstract this paper studies the optimality of the restriction and prolongation operators in the geometric multigrid method gmg gmg is used in solving discretized partial differential equation pde and it relies greatly on the restriction and prolongation operators many methods to find these operators were proposed but most of them have limited optimality proofs to study their optimality we introduce stochastic convergence functional which estimates the spectral radius of the iteration matrix for given gmg parameters we implement the gmg method in a modern machine learning framework that can automatically compute the gradients of the introduced convergence functional with respect to restriction and prolongation operators therefore we can minimize the proposed functional starting from some initial parameters and get better ones after some iterations of stochastic gradient descent to illustrate the performance of the proposed approach we carry out experiments on the discretized poisson equation helmholtz equation and singularly perturbed convection diffusion equation and demonstrate that proposed approach gives operators which lead to faster convergence

fast computation of zeros of polynomial systems with bounded degree under finite precision a solution for smale s 17th problem for the case of systems with bounded degree was recently given this solution an algorithm computing approximate zeros of complex polynomial systems in average polynomial time assumed infinite precision in this paper we describe a finite precision version of this algorithm our main result shows that this version works within the same time bounds and requires a precision which on the average amounts to a polynomial amount of bits in the mantissa of the intervening floating point numbers

lower bounds for the approximative complexity abstract we generalize several methods for obtaining lower bounds for the complexity of polynomials such as those based on substitution transcendence degree or algebraic geometric degree to the approximative model

random walks on polytopes and an affine interior point method for linear programming let k be a polytope in rn defined by m linear inequalities we give a new markov chain algorithm to draw a nearly uniform sample from k the underlying markov chain is the first to have a mixing time that is strongly polynomial when started from a central point we use this result to design an affine interior point algorithm that does a single random walk to solve linear programs approximately

automatic computation of partial derivatives and rounding error estimates with applications to large scale systems of nonlinear equations abstract recently a new approach has been proposed to efficiently compute the accurate values of partial derivatives of a function or functions and simultaneously to estimate the rounding errors in the computed function values in this paper the use of the method in the solution of nonlinear equations is investigated the method makes use of the computational graph and when applied to the evaluation of a function traverses it from the top the function node down to the bottom the input variable nodes a remarkable analogy is observed between the partial derivatives and the shortest paths on the computational graph the top down traversing on the computational graph has the following advantages over the existing algorithms using the bottom up traversing 1 the gradient of a function can be computed within the same complexity as that of the evaluation of the function alone the complexity being independent of the number of input variables 2 a fairly sharp estimate of the rounding error in the function evaluation is obtained on the basis of which a computationally meaningful norm may be introduced in the space of residuals to afford a convergence criterion for an iterative method of solving the system of nonlinear equations as an example a system of nonlinear equations with 108 variables for a distillation tower of a chemical plant is numerically analyzed in detail it is shown that by the use of the proposed method we could satisfactorily resolve two main problems encountered in computing a numerical solution of the system of nonlinear equations i e how to compute the accurate jacobian matrix and when we should stop the iteration

arithmetic cryptography we study the possibility of computing cryptographic primitives in a fully black box arithmetic model over a finite field f in this model the input to a cryptographic primitive e g encryption scheme is given as a sequence of field elements the honest parties are implemented by arithmetic circuits that make only a black box use of the underlying field and the adversary has a full non black box access to the field this model captures many standard information theoretic constructions we prove several positive and negative results in this model for various cryptographic tasks on the positive side we show that under coding related intractability assumptions computational primitives like commitment schemes public key encryption oblivious transfer and general secure two party computation can be implemented in this model on the negative side we prove that garbled circuits additively homomorphic encryption and secure computation with low online complexity cannot be achieved in this model our results reveal a qualitative difference between the standard boolean model and the arithmetic model and explain in retrospect some of the limitations of previous constructions

numerical accuracy of multipole expansion for 2d mlfma a numerical study of the multipole expansion for the multilevel fast multipole algorithm mlfma is presented in the numerical implementation of mlfma the error comes from three sources the truncation of the addition theorem the approximation of the integration the aggregation and disaggregation process these errors are due to the factorization of the green s function which is the mathematical core of the algorithm among the three error sources we focus on the truncation error and a new approach of selecting truncation numbers for the addition theorem is proposed using this approach the error prediction and control can be improved for the small buffer sizes and high accuracy requirements

on fixed polynomial size circuit lower bounds for uniform polynomials in the sense of valiant we consider the problem of fixed polynomial lower bounds on the size of arithmetic circuits computing uniform families of polynomials assuming the generalised riemann hypothesis grh we show that for all k there exist polynomials with coefficients in ma having no arithmetic circuits of size o n k over c allowing any complex constant we also build a family of polynomials that can be evaluated in am having no arithmetic circuits of size o n k then we investigate the link between fixed polynomial size circuit bounds in the boolean and arithmetic settings in characteristic zero it is proved that np size n k or ma size n k or np ma imply lower bounds on the circuit size of uniform polynomials in n variables from the class vnp over c assuming grh in positive characteristic p uniform polynomials in vnp have circuits of fixed polynomial size if and only if both vp vnp over f p and mod p p has circuits of fixed polynomial size

locally decodable codes with two queries and polynomial identity testing for depth 3 circuits in this work we study two seemingly unrelated notions locally decodable codes ldcs are codes that allow the recovery of each message bit from a constant number of entries of the codeword polynomial identity testing pit is one of the fundamental problems of algebraic complexity we are given a circuit computing a multivariate polynomial and we have to determine whether the polynomial is identically zero we improve known results on ldcs and on polynomial identity testing and show a relation between the two notions in particular we obtain the following results 1 we show that if e mathbb f n mapsto mathbb f m is a linear ldc with two queries then m exp omega n previously this was known only for fields of size ll 2 n o goldreich e comput complexity 15 2006 pp 263 296 2 we show that from every depth 3 arithmetic circuit sigma pi sigma circuit cal c with a bounded constant top fan in that computes the zero polynomial one can construct an ldc more formally assume that cal c is minimal no subset of the multiplication gates sums to zero and simple no linear function appears in all the multiplication gates denote by d the degree of the polynomial computed by cal c and by r the rank of the linear functions appearing in cal c then we can construct a linear ldc with two queries that encodes messages of length r operatorname polylog d by codewords of length o d 3 we prove a structural theorem for sigma pi sigma circuits with a bounded top fan in that compute the zero polynomial in particular we show that if such a circuit is simple minimal and of polynomial size then its rank r is only polylogarithmic in the number of variables a priori it could have been linear 4 we give new pit algorithms for sigma pi sigma circuits with a bounded top fan in a a deterministic algorithm that runs in quasipolynomial time and b a randomized algorithm that runs in polynomial time and uses only a polylogarithmic number of random bits moreover when the circuit is multilinear our deterministic algorithm runs in polynomial time previously deterministic subexponential time algorithms for pit in bounded depth circuits were known only for depth 2 circuits in the black box model d grigoriev m karpinski and m f singer siam j comput 19 1990 pp 1059 1063 m ben or and p tiwari proceedings of the 20th annual acm symposium on theory of computing acm press new york 1988 pp 301 309 a r klivans and d spielman proceedings of the 33rd annual acm symposium on theory of computing acm press new york 2001 pp 216 223 in particular for the special case of depth 3 circuits with three multiplication gates our result resolves an open question asked by klivans and spielman

truncation error analysis of multipole expansion the multilevel fast multipole algorithm is based on the multipole expansion which has numerical error sources such as truncation of the addition theorem numerical integration and interpolation anterpolation of these we focus on the truncation error and discuss its control precisely the conventional selection rule fails when the buffer size is small compared to the desired numerical accuracy we propose a new approach and show that the truncation error can be controlled and predicted regardless of the number of buffer sizes

lower bounds on the bounded coefficient complexity of bilinear maps we prove lower bounds of order n log n for both the problem of multiplying polynomials of degree n and of dividing polynomials with remainder in the model of bounded coefficient arithmetic circuits over the complex numbers these lower bounds are optimal up to order of magnitude the proof uses a recent idea of r raz proc 34th stoc 2002 proposed for matrix multiplication it reduces the linear problem of multiplying a random circulant matrix with a vector to the bilinear problem of cyclic convolution we treat the arising linear problem by extending j morgenstern s bound j acm 20 pp 305 306 1973 in a unitarily invariant way this establishes a new lower bound on the bounded coefficient complexity of linear forms in terms of the singular values of the corresponding matrix in addition we extend these lower bounds for linear and bilinear maps to a model of circuits that allows a restricted number of unbounded scalar multiplications

change of order for regular chains in positive dimension we discuss changing the variable order for a regular chain in positive dimension this quite general question has applications going from implicitization problems to the symbolic resolution of some systems of differential algebraic equations we propose a modular method reducing the problem to computations in dimension zero and one the problems raised by the choice of the specialization points and the lack of the crucial information of what are the free and algebraic variables for the new order are discussed strong but not unusual hypotheses for the initial regular chain are required the main required subroutines are change of order in dimension zero and a formal newton iteration

a minimal code list it is proved that a certain code list for a particular function has minimal possible length this result is of interest in search of optimal automatic differentiation methods

definability and fast quantifier elimination in algebraically closed fields abstract the bezout inequality an affine version not including multiplicities of the classical bezout theorem is derived for applications in algebraic complexity theory upper bounds for the cardinality and number of sets definable by first order formulas over algebraically closed fields are given this is used for fast quantifier elimination in algebraically closed fields

efficient vlsi fault simulation let c be an acyclic boolean circuit with n gates and n inputs a circuit manufacture error may result in a stuck at s a fault in a circuit identical to c except a gate v only outputs a fixed boolean value the s a fault simulation problem for c is to determine all possible s a faults which can be detected i e faults circuit and c would give distinct outputs by a given test pattern input we consider the case where c is a tree i e has fan out 1 we give a practical algorithm for fault simulation which simultaneously determines all detectable s a faults for every gate in the circuit tree c our algorithm required only the evaluation of a circuit fs c which has 7n gates and has depth 3 d 1 when d is the depth of c thus the sequential time of our algorithm is 7n and the parallel time is 3 d 1 furthermore fs c requires only a small constant factor more vlsi area than does the original circuit c we also extend our results to get efficient methods for fault simulation of oblivious vlsi circuits with feedback lines

rapid solution of integral equations of scattering theory in two dimensions abstract the present paper describes an algorithm for rapid solution of boundary value problems for the helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory cpu time requirements of previously published algorithms of this type are of the order n 2 where n is the number of nodes in the discretization of the boundary of the scatterer the cpu time requirements of the algorithm of the present paper are n 4 3 and can be further reduced making it considerably more practical for large scale problems

the bit complexity of matrix multiplication and of related computations in linear algebra the segmented λ algorithms abstract the numbers of bit operations bt required for matrix multiplication mm matrix inversion mi the evaluation of the determinant of a matrix det and the solution of a system of linear equations sle are estimated from above and below for sle the estimates are nearly sharp the bit complexity classes turn out to be different from the arithmetical complexity classes for those problems for instance bt mm o bt det while mm and det require the same number of arithmetical operations within a constant factor the bit complexity gives a more refined measure of stability of algorithms than their condition in particular the study shows that the large condition of the so called apa algorithms of small degrees for mm causes only reasonable growth of their bit time which suggests that fast apa algorithms are practically efficient a new extension of the class of apa algorithms is introduced

permanent does not have succinct polynomial size arithmetic circuits of constant depth we show that over fields of characteristic zero there does not exist a polynomial p n and a constant free succinct arithmetic circuit family f n using division by constants where f n has size at most p n and depth o 1 such that f n computes the nxn permanent a circuit family f n is succinct if there exists a nonuniform boolean circuit family c n with o logn many inputs and size n o 1 such that c n can correctly answer direct connection language queries about f n succinctness is a relaxation of uniformity to obtain this result we develop a novel technique that further strengthens the connection between black box derandomization of polynomial identity testing and lower bounds for arithmetic circuits from this we obtain the lower bound by giving an explicit construction computable in the polynomial hierarchy of a hitting set for arithmetic circuits