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

root refinement for real polynomials using quadratic interval refinement we consider the problem of approximating all real roots of a square free polynomial f with real coefficients given isolating intervals for the real roots and an arbitrary positive integer l the task is to approximate each root to l bits after the binary point abbott has proposed the quadratic interval refinement method qir for short which is a variant of regula falsi combining the secant method and bisection we formulate a variant of qir denoted aqir approximate qir that considers only approximations of the polynomial coefficients and chooses a suitable working precision adaptively it returns a certified result for any given real polynomial whose roots are all simple in addition we propose several techniques to improve the asymptotic complexity of qir we prove a bound on the bit complexity of our algorithm in terms of the degree of the polynomial the size and the separation of the roots that is parameters exclusively related to the geometric location of the roots for integer coefficients our variant improves in theory and practice the variant with exact integer arithmetic combining our approach with multipoint evaluation we obtain near optimal bounds that essentially match the best known theoretical bounds on root approximation as obtained by very sophisticated algorithms

numerical stability analysis for the explicit high order finite difference analysis of rotationally symmetric shells a numerical stability analysis has been formulated to accompany the already developed explicit high order finite difference analysis of rotationally symmetric shells subjected to time dependent impulsive loadings this already developed analysis utilizes a constant nodal point spacing for the spatial finite difference mesh with the governing field differential equations formulated in terms of the transverse meridional and circumferential displacements as the fundamental variables the remaining quantities which enter into the natural boundary conditions at each edge of the shell are incorporated into the complete system of equations by defining those quantities at each boundary in terms of the displacements surface loadings and inertia forces in each of the three displacement directions of the shell have been considered in the governing equations ordinary finite difference representations are used for the time derivatives all loadings and dependent variables in the circumferential direction of the shell are expressed in fourier series expansions the complete system of equations is solved implicitly for the first time increment while explicit relations are used to determine the three primary displacements within the boundary edges of the shell for the second and succeeding time increments separate implicit solutions at each boundary are then used to determine the remaining unspecified primary variables on and outside the boundaries subsequently the remaining primary variables within the boundary edges of the shell and all secondary variables are determined explicitly numerical stability or instability of numerical solutions for given choices of spatial and time increments is determined by evaluation of the eigenvalues of the explicit coefficient matrix and comparing the maximum eigenvalue with the requirements of a stability criterion developed before by the author solutions for typical shells and loadings together with results of stability analyses have been included and comparisons of the stability requirements and solutions with the requirements and solutions based upon ordinary spatial finite difference representations are included

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

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

power law corrections to exponential decay of connectivities and correlations in lattice models consider a translation invariant bond percolation model on the integer lattice which has exponential decay of connectivities that is the probability of a connection 0 x by a path of open bonds decreases like exp m x for some positive constant m which may depend on the direction x x in two and three dimensions it is shown that if the model has an appropriate mixing property and satisfies a special case of the fkg property then there is at most a power law correction to the exponential decay there exist a and c such that exp m x p 0 x a x c exp m x for all nonzero x in four or more dimensions a similar bound holds with x c replaced by exp c log x 2 in particular the power law lower bound holds for the fortuin kasteleyn random cluster model in two dimensions whenever the connectivity decays exponentially since the mixing property is known to hold in that case consequently a similar bound holds for correlations in the potts model at supercritical temperatures

a fast algorithm for reversion of power series we give an algorithm for reversion of formal power series based on an efficient way to evaluate the lagrange inversion formula our algorithm requires o n 1 2 m n mm n 1 2 operations where m n and mm n are the costs of polynomial and matrix multiplication respectively this matches an algorithm of brent and kung but we achieve a constant factor speedup whose magnitude depends on the polynomial and matrix multiplication algo rithms used benchmarks confirm that the algorithm performs well in practice

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

on weak mixing in lattice models for lattice models on d weak mixing is the property that the influence of the boundary condition on a finite decays exponentially with distance from that region for a wide class of models on 2 including all finite range models we show that weak mixing is a consequence of gibbs uniqueness exponential decay of an appropriate form of connectivity and a natural coupling property in particular on 2 the fortuin kasteleyn random cluster model is weak mixing whenever uniqueness holds and the connectivity decays exponentially and the q state potts model above the critical temperature is weak mixing whenever correlations decay exponentially a hypothesis satisfied if q is sufficiently large ratio weak mixing is the property that uniformly over events a and b occurring on subsets and respectively of the lattice p a b p a p b 1 decreases exponentially in the distance between and we show that under mild hypotheses for example finite range weak mixing implies ratio weak mixing

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

ornstein zernike theory for the bernoulli bond percolation on mathbb z d we derive a precise ornstein zernike asymptotic formula for the decay of the two point function p p 0 x of the bernoulli bond percolation on the integer lattice z d in any dimension d 2 in any direction x and for any subcritical value of p p c d

on the 2d ising wulff crystal near criticality we study the behavior of the two dimensional ising model in a finite box at temperatures that are below but very close to the critical temperature in a regime where the temperature approaches the critical point and simultaneously the size of the box grows fast enough we establish a large deviation principle that proves the appearance of a round wulff crystal

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

rigorous probabilistic analysis of equilibrium crystal shapes the rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade we review here the main results that have been obtained both in two and higher dimensions in particular we describe how the phenomenological wulff and winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases we focus on the main conceptual issues and describe the central ideas of the existing approaches

numerical analysis of rotationally symmetric shells by the modal superposition method the system of differential equations governing the analysis of rotationally symmetric shells under time dependent or static surface loadings is formulated with the transverse meridional and circumferential displacements as the dependent variables the thickness of the shell may vary and four homogeneous boundary conditions may be prescribed at each boundary edge of the shell the governing differential equations for each fourier harmonic are obtained by use of fourier series in the circumferential direction of the shell influence coefficients at each of the node points along the shell meridian are obtained for each fourier component by employing ordinary finite difference representations for the meridional co ordinate derivatives with these influence coefficients a set of homogeneous flexibility equations governing the free vibration characteristics of the shell is obtained and solved for the frequencies and mode shapes for each fourier component the solutions under time dependent or static surface loadings and due to initial displacements and velocities are then obtained by expanding the solutions in terms of the modes of free vibration of the shell for each fourier component the solution for the total shell response is obtained by fourier series summation solutions for typical shells have been found to be in excellent agreement with solutions by the method of temporal and spatial finite differences solutions for a parabolic shell are presented as an example the solution is also presented for a published cylindrical shell example and is seen to be in excellent agreement with the published finite element method results

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

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

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

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

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

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

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

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

finite difference analysis of rotationally symmetric shells using variable node point spacings the system of equations for the analysis of rotationally symmetric shells under time dependent or static surface loadings has been formulated with the transverse meridional and circumferential displacements as the dependent variables all loading functions must be continuous the thickness h of the shell may vary along the meridian four of the eight natural boundary conditions will be prescribed as time dependent boundary conditions at each boundary edge of the shell surface loadings and inertia forces in the three displacement directions of the shell have been considered fourier series are used in the circumferential direction of the shell solutions for each fourier component are found by replacing all derivatives by their finite difference equivalents and solving the resulting system of algebraic equations at successive increments of the time variable the complete system of equations is solved implicitly for the first time increment while explicit relations are used to determine the three primary displacements within the boundary edges of the shell for the second and succeeding time increments the remaining unspecified primary variables are then determined by separate implicit solutions at each boundary for the second and succeeding time increments subsequently all remaining primary and secondary variables are found explicitly a variable node point spacing may be specified over the full range of the spatial finite difference mesh a numerical stability criterion which will ensure stable solutions for selected time increments and spatial meshes has been derived and found to agree with results for typical example solutions numerical solutions obtained with the computer program which accompanies this development have been found to be stable and in agreement for a wide range of practical values of both spatial and time increments for typical shells and loadings static and dynamic solutions for a parabolic shell with fixed boundary conditions at one edge and free boundary conditions at the other edge have been obtained using both constant and variable node point spacings and presented as examples

lower bounds for boundary roughness for droplets in bernoulli percolation we consider boundary roughness for the droplet created when supercritical two dimensional bernoulli percolation is conditioned to have an open dual circuit surrounding the origin and enclosing an area at least l 2 for large l the maximum local roughness is the maximum inward deviation of the droplet boundary from the boundary of its own convex hull we show that for large l this maximum is at least of order l 1 3 logl 2 3 this complements the upper bound of order l 1 3 logl 2 3 proved in al3 for the average local roughness the exponent 1 3 on l here is in keeping with predictions from the physics literature for interfaces in two dimensions

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

mixing properties and exponential decay for lattice systems in finite volumes an infinite volume mixing or exponential decay property in a spin system or percolation model reflects the inability of the influence of the configuration in one region to propagate to distant regions but in some circumstances where such properties hold propagation can nonetheless occur in finite volumes endowed with boundary conditions we establish the absense sic of such propagation particularly in two dimensions in finite volumes which are simply connected under a variety of conditions mainly for the potts model and the fortuin kasteleyn fk random cluster model allowing external fields for example for the fk model in two dimensions we show that exponential decay of connectivity in infinite volume implies exponential decay in simply connected finite volumes uniformly over all such volumes and all boundary conditions and implies a strong mixing property for such volumes with certain types of boundary conditions for the potts model in two dimensions we show that exponential decay of correlations in infinite volume implies a strong mixing property in simply connected finite volumes which includes exponential decay of correlations in simply connected finite volumes uniformly over all such volumes and all boundary conditions

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

nearly optimal computations with structured matrices we estimate the boolean complexity of multiplication of structured matrices by a vector and the solution of nonsingular linear systems of equations with these matrices we study four basic and most popular classes that is toeplitz hankel cauchy and vandermonde matrices for which the cited computational problems are equivalent to the task of polynomial multiplication and division and polynomial and rational multipoint evaluation and interpolation the boolean cost estimates for the latter problems have been obtained by kirrinnis in 10 except for rational interpolation we supply them now as well as the boolean complexity estimates for the important problems of multiplication of transposed vandermonde matrix and its inverse by a vector all known boolean cost estimates from 10 for such problems rely on using kronecker product this implies the d fold precision increase for the d th degree output but we avoid such an increase by relying on distinct techniques based on employing fft furthermore we simplify the analysis and make it more transparent by combining the representations of our tasks and algorithms both via structured matrices and via polynomials and rational functions this also enables further extensions of our estimates to cover trummer s important problem and computations with the popular classes of structured matrices that generalize the four cited basic matrix classes as well as the transposed vandermonde matrices it is known that the solution of toeplitz hankel cauchy vandermonde and transposed vandermonde linear systems of equations is generally prone to numerical stability problems and numerical problems arise even for multiplication of cauchy vandermonde and transposed vandermonde matrices by a vector thus our fft based results on the boolean complexity of these important computations could be quite interesting because our estimates are reasonable even for more general classes of structured matrices showing rather moderate growth of the complexity as the input size increases

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

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