twisted fourier mukai functors due to a theorem by orlov every exact fully faithful functor between the bounded derived categories of coherent sheaves on smooth projective varieties is of fourier mukai type we extend this result to the case of bounded derived categories of twisted coherent sheaves and at the same time we weaken the hypotheses on the functor as an application we get a complete description of the exact functors between the abelian categories of twisted coherent sheaves on smooth projective varieties

matrix decompositions using displacement rank and classes of commutative matrix algebras using the notion of displacement rank we look for a unifying approach to representations of a matrix a as sums of products of matrices belonging to commutative matrix algebras these representations are then considered in case a is the inverse of a toeplitz or a toeplitz plus hankel matrix some well known decomposition formulas for a gohberg semencul or kailath et al gader bini pan and gohberg olshevsky turn out to be special cases of the above representations new formulas for a in terms of algebras of symmetric matrices are studied and their computational aspects are discussed

a solution to certain polynomial equations with applications to nonlinear fitting we present a combinatorial method for solving a certain system of polynomial equations of vandermonde type in 2n variables by reducing it to the problem of solving two special linear systems of size n and rooting a single univariate polynomial of degree n over c all solutions can be found with fixed precision using up to polylogarithmic factors o n 2 bitwise operations in the worst case furthermore if the data is well conditioned then this can be reduced to o n bit operations up to polylogarithmic factors as an application we show how this can be used to fit data to a complex exponential sum with n terms in the same nearly optimal time

semistable bundles over an elliptic curve abstract we identify the moduli space m n d of semistable bundles of rank n and degree d on an elliptic curve c as a symmetric product of the curve its dimension being the greatest common divisor of n and d under this identification the determinant map det m n d corresponds to the abel jacobi map shc jh so that the fixed determinant moduli space is identified with a projective space we then describe the brill noether loci theta divisors and picard groups and compute the space of theta functions for both the full moduli space and the fixed determinant moduli space of any rank and any degree we extend to elliptic curves a theorem of drezet and narasimhan on the picard group of moduli spaces as well as two formuulas of beauville narasimhan and ramanan on theta functions of level one our results verify in particular for an elliptic curve a formula of e verlinde and an equality first noticed by bott and szenes relating theta functions over the full moduli space and over the fixed determinant moduli space

a fast parallel algorithm for determining all roots of a polynomial with real roots given a polynomial p z of degree n with m bit integer coefficients and an integer mu the problem of determining all its roots with error less than 2 mu is considered it is shown that this problem is in the class nc if p z has all real roots some very interesting properties of a sturm sequence of a polynomial with distinct real roots are proved and used in the design of a fast parallel algorithm for this problem using newton identities and a novel numerical integration scheme for evaluating a contour integral to high precision this algorithm determines good approximations to the linear factors of p z

matrix displacement decompositions and applications to toeplitz linear systems using the approach of bozzo di fiore and zellini new matrix displacement decomposition formulas are introduced it is shown how an arbitrary square matrix a can be expressed as sums of products of hessenberg algebra matrices and high level block matrices whose submatrices are hessenberg algebra matrices and have variable sizes in most cases these block factors are block diagonal matrices then these formulas are used in sequential and parallel solution of toeplitz systems

equivalences of derived catgories of sheaves on smooth stacks we extend orlov s representability theorem on the equivalence of derived categories of sheaves to the case of smooth stacks associated to normal projective varieties with only quotient singularities

interpolation approach to h estimation and its interconnection to loop transfer recovery abstract this paper proposes to solve the h optimal estimation problem via interpolation theory the advantages of this approach are the following i direct and simple solutions can be obtained for the optimal estimators ii frequency weightings on estimation error and disturbance can be easily incorporated and iii unnecessary high observer gains can be avoided we also show the interconnection between the h optimal estimation problem and the optimal loop transfer recovery problem the interpolation approach applies to both continuous time and discrete time systems

linear complexity parallel algorithms for linear systems of equations with recursive structure abstract a method of derivation of parallel algorithms for n 1 n 1 matrices with recursive structure is presented and applied to toeplitz hankel and other toeplitz like matrices the derived algorithms executed on o n parallel processors require o n arithmetic operations per processor

real polynomial root finding by means of matrix and polynomial iterations frequently one seeks approximation to all r real roots of a polynomial of degree n with real coefficients which also has nonreal roots we split a polynomial into two factors one of which has degree r and has r real roots we approximate them at a low cost and then decrease the arithmetic time of the known algorithms for this popular problem by roughly a factor of n k if k iterations prepare splitting k is a small integer unless some nonreal roots lie close to the real axis but even if there nonreal roots near the real axis we substantially accelerate the known algorithms we also propose a dual algorithm operating with the associated structured matrices at the price of minor increase of the arithmetic time it facilitates numerical implementation our analysis and tests demonstrate the efficiency of our approach

fundamental design tradeoffs in filtering prediction and smoothing this paper examines fundamental design tradeoffs that apply to all linear filtering prediction and smoothing problems we introduce sensitivity functions that quantify filter performance and we show that their frequency response satisfies constraints imposed by unstable poles and nonminimum phase zeros of the system the constraints allow one to determine a priori whether or not a desired filter performance is attainable and how different arrangements of the measurement system influence the achievable performance

exact simulation of integrate and fire models with exponential currents neural networks can be simulated exactly using event driven strategies in which the algorithm advances directly from one spike to the next spike it applies to neuron models for which we have 1 an explicit expression for the evolution of the state variables between spikes and 2 an explicit test on the state variables that predicts whether and when a spike will be emitted in a previous work we proposed a method that allows exact simulation of an integrate and fire model with exponential conductances with the constraint of a single synaptic time constant in this note we propose a method based on polynomial root finding that applies to integrate and fire models with exponential currents with possibly many different synaptic time constants models can include biexponential synaptic currents and spike triggered adaptation currents

parameterization of linear observers and its application to observer design with the aid of the factorization approach linear observers and their estimation error dynamics are parameterized this result provides a dual representation of the well known linear controller parameterization and gives a new insight into the observer design which may be used for the observer construction and robust observer design

steady state kalman filtering with an h error bound abstract an estimator design problem is considered which involves both l 2 least squares and h worst case frequency domain aspects specifically the goal of the problem is to minimize an l 2 state estimation error criterion subject to a prespecified h constraint on the state estimation error the h estimation error constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on the l 2 state estimation error the principal result is a sufficient condition for characterizing fixed order i e full and reduced order estimators with bounded l 2 and h estimation error the sufficient condition involves a system of modified riccati equations coupled by an oblique projection i e idempotent matrix when the h constraint is absent the sufficient condition specializes to the l 2 state estimation result given in 2

displacement structure of pseudoinverses abstract a matrix or a linear operator a is said to possess an uv displacement structure if rank au va is small compared with the rank of a estimates for the rank of a v ua and more general displacements of a are presented where a is the pseudoinverse of a the general results are applied to close to toeplitz close to vandermonde and generalized cauchy matrices bezoutians toeplitz and hankel operators singular integral operators and integral operators with displacement kernel this leads to formulas for a which can be used for the fast computation of pseudosolutions for vandermonde matrices the exact displacement rank of a is evaluated it turns out that this rank is not always small

parallel complexity of computations with general and toeplitz like matrices filled with integers and extensions computations with toeplitz and toeplitz like matrices are fundamental for many areas of algebraic and numerical computing the list of computational problems reducible to toeplitz and toeplitz like computations includes in particular the evaluation of the greatest common divisor gcd the least common multiple lcm and the resultant of two polynomials computing pade approximation and the berlekamp massey recurrence coefficients as well as numerous problems reducible to these transition to toeplitz and toeplitz like computations is currently the basis for the design of the parallel randomized nc rnc algorithms for these computational problems our main result is in constructing nearly optimal randomized parallel algorithms for toeplitz and toeplitz like computations and consequently for numerous related computational problems including the computational problems listed above where all the input values are integers and all the output values are computed exactly this includes randomized parallel algorithms for computing the rank the determinant and a basis for the null space of an n n toeplitz or toeplitz like matrix a filled with integers as well as a solution x to a linear system a x f if the system is consistent our algorithms use o log n log n log a parallel time and o n log n processors each capable of performing in unit time an arithmetic operation a comparision or a rounding of a rational number to a closest integer the cost bounds cover the cost of the verification of the correctness of the output the computations by these algorithms can be performed with the precision of o n log a bits which matches the precision required in order to represent the output except for the rank computation where the precision of the computation decreases the algorithms involve either a single random parameter or at most 2n 1 parameters the cited processor bounds are less by roughly factor n than ones supported by the known algorithms that run in polylogarithmic arithmetic time and do not use rounding to the closest integers technically we first devise new algorithms supporting our old nearly optimal complexity estimates for parallel computations with general matrices filled with integers then we decrease dramatically by roughly factor n1 376 the processor bounds required in these algorithms in the case where the input matrix is toeplitz like our algorithms exploit and combine some new techniques which may be of independent interest e g in the study of parallel and sequential computation of recursive factorization of integer matrices as well as our earlier techniques of variable diagonal relating to each other several known algebraic and numerical methods stream contraction and the truncation of displacement generators in toeplitz like computations our development and application of these techniques may be of independent

complexity of computations with matrices and polynomials this paper reviews the complexity of polynomial and matrix computations as well as their various correlations to each other and some major techniques for the design of algebraic and numerical algorithms

observer parameterization for simultaneous observation the stable inverse approach is used to obtain the observers for the simultaneous observation of a given set of plants a parameterization in terms of a stable inverse and a stable null space is proposed for all simultaneous observers to verify the effectiveness of the proposed method a design example is also given

unknown inputs functional observers designs for delay descriptor systems new time and frequency domain designs of unknown inputs functional observers uifo for a class of descriptor systems with a constant time delay are investigated in this paper the unknown inputs are present in both the state and the measurement equations the time procedure design is based on the unbiasedness of the estimation error and the lyapunov krasovskii stability theory after given the existence condition of such observers the gain implemented in the functional observer with internal delay design is obtained in terms of linear matrix inequalities the particular cases where the observer is independent of internal delay and where it is independent of delay are also investigated a design algorithm that summarises the different steps of the uifo design is proposed then the frequency procedure design is derived from time domain results by applying the factorisation approach where we define some useful matrix fraction descriptions note that the order of this unknown input observers is equal to

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

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

efficient algorithms for the evaluation of the eigenvalues of block banded toeplitz matrices on presente des algorithmes de calcul des valeurs propres des matrices bandes blocs de toeplitz

parallel computation of polynomial gcd and some related parallel computations over abstract fields several fundamental problems of computations with polynomials and structured matrices are well known for their resistance to effective parallel solution this includes computing the gcd the 1cm and the resultant of two polynomials as well as any selected entry of either the extended euclidean scheme for these polynomials or the pade approximation table associated to a fixed power series the solution of the berlekamp massey problem of recovering the n coefficients of a linear recurrence sequence from its first 2n terms the solution of a toeplitz linear system of equations computing the ranks and the null spaces of toeplitz matrices and several other computations with toeplitz hankel vandermonde and cauchy generalized hilbert matrices and with matrices having similar structures our algorithms enable us to reach new record estimates for randomized parallel arithmetic complexity of all these computations over any field of constants that is o log n h time and o n2 log log n log n h g arithmetic processors where n is the input size g varies from 1 to 2 and h varies from 2 to 4 depending on the computational problem and on the fixed field of constants the estimates include the cost of correctness tests for the solution the results have further applications to many related computations over abstract fields and rely on some new techniques in particular on recursive pade approximation and regularization of pade approximation via randomization which might be of some independent technical interest in particular an auxiliary algorithm computes over any field the coefficients of a polynomial from the power sums of its zeros which is an important problem of algebraic coding

the homotopy theory of dg categories and derived morita theory the main purpose of this work is to study the homotopy theory of dg categories up to quasi equivalences our main result is a description of the mapping spaces between two dg categories c and d in terms of the nerve of a certain category of c d bimodules we also prove that the homotopy category ho dg cat possesses internal hom s relative to the derived tensor product of dg categories we use these two results in order to prove a derived version of morita theory describing the morphisms between dg categories of modules over two dg categories c and d as the dg category of c d bi modules finally we give three applications of our results the first one expresses hochschild cohomology as endomorphisms of the identity functor as well as higher homotopy groups of the classifying space of dg categories i e the nerve of the category of dg categories and quasi equivalences between them the second application is the existence of a good theory of localization for dg categories defined in terms of a natural universal property our last application states that the dg category of continuous morphisms between the dg categories of quasi coherent resp perfect complexes on two schemes resp smooth and proper schemes is quasi equivalent to the dg category of quasi coherent resp perfect complexes on their product

symmetry stabilization for fast discrete monomial transforms and polynomial evaluation abstract we discuss and explain the issues present in implementing a fast algorithm for discrete monomial transforms dmts in the context of certain applications in mri and data analysis the dmt of a sequence f 0 m with respect to sample points z0 zm c m 1 is the sequence of sums f 0 f m where f j mk 0 k zjk for different sets of sample points the dmt is an important component of many applications driscoll healy and rockmore described a fast o m log2m operations algorithm via matrix factorization for computing the dmt while the asymptotic complexity of the algorithm is guaranteed for any choice of the sample points the numerical stability is not in this paper we present a heuristic for stabilizing the fast dmt which makes its implementation feasible numerically for certain sets of sample points while retaining its computational advantages even for moderate problem sizes where it is shown to be more efficient than implementations using horner s rule

a fast algorithm for the multiplication of generalized hilbert matrices with vectors on presente un algorithme rapide de complexite logarithmique pour la multiplication des matrices de hilbert generalisees par des vecteurs

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

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

a fast algorithm for computing the determinants of banded circulant matrices let c n be a k 1 diagonal complex circulant matrix of order n k 1 and let detc n be the determinant of c n an algorithm for computing detc n is presented with the cost of oklog 2k log 2n k 4 multiplication and an asymptotic formula for detc n is obtained moreover a result on symmetric circulant matrices with integer entries is also given using mathematica in a personal computer we give some numerical examples which illustrate that the algorithm is very efficient and the asymptotic formula is accurate enough when the order n of the circulant matrix is sufficiently large

non uniqueness of fourier mukai kernels we prove that the kernels of fourier mukai functors are not unique in general on the other hand we show that the cohomology sheaves of those kernels are unique we also discuss several properties of the functor sending an object in the derived category of the product of two smooth projective schemes to the corresponding fourier mukai functor

improved parallel polynomial division the authors compute the first n coefficients of the reciprocal r x r x p x 1 bmod x n given a natural n and a polynomial p x p 0 ne 0 by using o h log n arithmetic steps and o n h 1 2 h log h n processors for any h h 1 2 ldots log n under the pram arithmetic models provided that o log m steps and m processors suffice to perform discrete fourier transforms on m points and that log 0 n n log h n log 2 log h 1 n h 1 ldots log n log n max h log h n 0 the same estimates apply to some other computations such as the division with a remainder of two polynomials of degrees o n and the inversion of an n times n triangular toeplitz matrix this improves the known estimates of reif tate and georgiev the presented techniques are extended to parallel implementation of other recursive processes such as the evaluation modulo x n of the mth root p x 1 m of p x

computation of approximate polynomial gcds and an extension computation of approximate polynomial greatest common divisors gcds is important both theoretically and due to its applications to control linear systems network theory and computer aided design we study two approaches to the solution so far omitted by the researchers despite intensive recent work in this area correlation to numerical pade approximation enabled us to improve computations for both problems gcds and pade reduction to the approximation of polynomial zeros enabled us to obtain a new insight into the gcd problem and to devise effective solution algorithms in particular unlike the known algorithms we estimate the degree of approximate gcds at a low computational cost and this enables us to obtain certified correct solution for a large class of input polynomials we also restate the problem in terms of the norm of the perturbation of the zeros rather than the coefficients of the input polynomials which leads us to the fast certified solution for any pair of input polynomials via the computation of their roots and the maximum matchings or connected components in the associated bipartite graph

parallel algorithms for algebraic problems fast parallel algorithms are presented for the following problems in symbolic manipulation of univariate polynomials computing all entries of the extended euclidean scheme of two polynomials over an arbitrary field gcd and 1cm of many polynomials factoring polynomials over finite fields and the squarefree decomposition of polynomials over fields of characteristic zero and over finite fields for the following estimates assume that the input polynomials have degree at most n and the finite field has p d elements the euclidean algorithm is deterministic and runs in parallel time o log 2 n all the other algorithms are probabilistic las vegas in the general case but when applicable to bf q or bf r they can be implemented deterministically over these fields the algorithms for gcd and lcm use parallel time o log 2 n the factoring algorithm runs in parallel time o log 2 n log 2 d 1 log p the algorithm for squarefree decomposition runs in parallel time o log 2 n

a new grobner basis conversion method based on stabilization techniques we propose a new method for converting a grobner basis w r t one term order into a grobner basis w r t another term order by using the algorithm stabilization techniques proposed by shirayanagi and sweedler first we guess the support of the desired grobner basis from a floating point grobner basis by exploiting the supportwise convergence property of the stabilized buchberger s algorithm next assuming this support to be correct we use linear algebra namely the method of indeterminate coefficients to determine the exact values for the coefficients related work includes the fglm algorithm and its modular version our method is new in the sense that it can be thought of as a floating point approach to the linear algebra method the results of maple computing experiments indicate that our method can be very effective in the case of non rational coefficients especially the ones including transcendental constants

parametrization of newton s iteration for computations with structured matrices and applications abstract we apply a new parametrized version of newton s iteration in order to compute over any field f of constants the solution or at least squares solution to linear system b x v with an n n toeplitz or toeplitz like matrix b as well as the determinant of b and the coefficients of its characteristic polynomial det i b dramatically improving the processor efficiency of the known fast parallel algorithms our algorithms together with some previously known and some recent results of 1 5 as well as with our new techniques for computing polynomial god s and lcm s imply respective improvement of the known estimates for parallel arithmetic complexity of several fundamental computations with polynomials and with both structured and general matrices

parallel solution of certain toeplitz linear systems using the concept of approximate algorithm it is shown that 6 log n 6 parallel steps and 2n processors suffice to approximate with any precision the solution of a linear system with an n times n triangular toeplitz matrix a moreover 7 log n 7 steps are sufficient for an exact computation whereas the number of processors is increased to frac 5 2 n 2 if a is also banded and k is its bandwidth the number of processors is reduced to frac 5 2 n k 1 two applications are shown it is proved that if b is any matrix belonging to the algebra generated over the complex field by a given n times n matrix then the system bx b can be solved with no more than 9 log n 4 steps with o n 2 processors it is proved that given a toeplitz matrix a a i j such that a i j 0 if i j k or j i h a k 1 ne 0 then 13 log n o log 2 k steps and max frac 5 2 n k h n n 1 2 processors are sufficient to solve the system ax

the bit complexity of discrete solutions of partial differential equations compact multigrid abstract the topic of partial differential equations pdes is an interesting area where the techniques of discrete mathematics and numerical algorithms can be brought together to solve problems that would normally be considered more properly in the domain of continuous mathematics we investigate the bit complexity of discrete solutions to linear pdes which is a realistic measure for such computers as the connection machine cm 1 and maspar we show that for a large class of linear pdes satisfying some routine assumptions of the multigrid methods the n point discretization of their solution can be compressed to only a constant number of bits per discretization point without loss of information or introducing errors beyond the order of the discretization error more specifically we show that the bit complexity of the compressed solution is o n for both storage space and the total number of operations we also compute the compressed solution by a parallel algorithm using o log n time and n log n bit serial processors the best previous bounds on the bit complexity for both sequential time and storage space were at least n log n furthermore the order of n log n bit serial processors were required to support the o log n parallel time in the known algorithms we believe this is the first case where a linear or algebraic system can be provably compressed i e the bit complexity of storage of the compressed solution is less than the solution size and also the first case where the use of data compression provably speeds up the time to solve the system in the compressed form

filtering and smoothing in an h sup infinity setting the problems of filtering and smoothing are considered for linear systems in an h sup infinity setting i e the plant and measurement noises have bounded energies are in l sub 2 but are otherwise arbitrary two distinct situations for the initial condition of the system are considered the initial condition is assumed known in one case while in the other the initial condition is not known but the initial condition the plant and measurement noise are in some weighted ball of r sup n xl sub 2 finite horizon and infinite horizon cases are considered necessary and sufficient conditions are presented for the existence of estimators both filters and smoothers that achieve a prescribed performance bound and algorithms that result in performance within the bounds are developed in case of smoothers the optimal smoother is also presented the approach uses basic quadratic optimization theory in a time domain setting as a consequence of which both linear time varying and time invariant systems can be considered with equal ease in the smoothing problem for linear time varying systems one considers only the finite horizon case

uniqueness of enhancement for triangulated categories the paper contains general results on the uniqueness of a dg enhancement for trian gulated categories as a consequence we obtain such uniqueness for the unbounded categories of quasi coherent sheaves for the triangulated categories of perfect complexes and for the bounded de rived categories of coherent sheaves on quasi projective schemes if a scheme is projective then we also prove a strong uniqueness for the triangulated category of perfect complexes and for the bounded de rived categories of coherent sheaves these results directly imply that fully faithful functors from the bounded derived categories of coherent sheaves and the triangulated categories of perfect complexes on projective schemes can be represented by objects on the product

fast and efficient parallel solution of sparse linear systems this paper presents a parallel algorithm for the solution of a linear system a bf x bf b with a sparse n times n symmetric positive definite matrix a associated with the graph g a that has n vertices and has an edge for each nonzero entry of a if g a has an s n separator family and a known s n separator tree then the algorithm requires only o log 3 n time and e m s n log n processors for the evaluation of the solution vector bf x a 1 bf b where e is the number of edges in g a and m n is the number of processors sufficient for multiplying two n times n rational matrices in time o log n furthermore for this computational cost the algorithm computes a recursive factorization of a such that the solution of any other linear system a bf x bf b with the same matrix a requires only o log 2 n time and e log n s n 2 processors

schur aggregation for linear systems and determinants we apply our recent preconditioning techniques to the solution of linear systems of equations and computing determinants we combine these techniques with the sherman morrison woodbury formula its new variations aggregation iterative refinement and advanced algorithms that rapidly compute sums and products either error free or with the desired high accuracy our theoretical and experimental study shows the power of this approach

fault detection via factorization approach abstract problems of designing fault detection and identification filters in the frequency domain are formulated and solved using the factorization approach a characterization of all fault detection filters is derived this enables the derivation of necessary and sufficient conditions for the existence of fault identification as well as detection and isolation filters it is shown that these conditions are a generalization of existing results the formulas of constructing the filters are also derived in comparison with the algorithms given in previous work they are computationally straightforward and simple finally the proposed method for designing fault identification filters is extended so that more practical cases can be handled

scalar extensions of derived categories and non fourier mukai functors abstract orlov s famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a fourier mukai functor this result has been extended by lunts and orlov to include functors from perfect complexes to quasi coherent complexes in this paper we show that the latter extension is false without the full faithfulness hypothesis our results are based on the properties of scalar extensions of derived categories whose investigation was started by pawel sosna and the first author

fast and efficient parallel solution of dense linear systems the most efficient previously known parallel algorithms for the inversion of a nonsingular n n matrix a or solving a linear system ax b over the rational numbers require o log2n time and m n n processors provided that m n processors suffice in order to multiply two n n rational matrices in time o logn furthermore the known polylog arithmetic time algorithms for those problems are numerically unstable in this paper we apply newton s iteration and initially choose an approximate inverse matrix by following ben israel this quadratically convergent and numerically stable iterative method takes o log2n parallel time using m n processors to compute the inverse within the relative precision 2 nc for a positive constant c of an n n rational matrix a with the condition number at most nd for a constant d this is the optimum processor bound and by a factor of n improvement of the previously known processor bounds for polylogarithmic time matrix inversion the algorithm does not require to precompute the condition number of the input matrix but it just converges slower for ill conditioned input matrices

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

very fast parallel polynomial arithmetic parallel algorithms for polynomial arithmetic multiplication of n polynomials of degree m polynomial division with remainder and polynomial interpolation are presented these algorithms can be implemented using polynomial time constructible families of boolean circuits of polynomial size and optimal order depth or log space constructible families of polynomial size and near optimal depth for computations over mathbb z mathbb q finite fields and several other domains arithmetic circuits of polynomial size and optimal order depth are obtained for polynomial arithmetic over arbitrary fields

sequential and parallel complexity of approximate evaluation of polynomial zeros our new sequential and parallel algorithms establish new record upper bounds on both arithmetic and boolean complexity of approximating to complex polynomial zeros o n2 log b log n arithmetic operations or o n log n log bn parallel steps and n log b log bn processors suffice in order to approximate with absolute errors 2m b to all the complex zeros of an nth degree polynomial p x whose coefficients have mod 2m if we only need such an approximation to a single zero of p x then o n log b log n arithmetic operations or o log2 n log bn steps and n log n log b log bn processors suffice which places the latter problem in nc that is in the class of problems that can be solved using polylogarithmic parallel time and a polynomial number of processors those estimates are reached in computations with o bn binary bits where the polynomial has integer coefficients we also reach the sequential boolean time bounds o bn3 log bn log log bn for approximating to all the zeros very minor improvement of the bound announced in 1982 by schonhage and o bn2log log n log bn log log bn for approximating to a single zero among further implications are the improvements of the known algorithms and complexity estimates for computing matrix eigenvalues for polynomial factorization over the field of complex numbers and for solving systems of polynomial equations the computations rely on recursive application of turan s proximity test of 1968 on its more recent extensions to root radii computations on contour integration via fast fourier transform fft within geometric constructions for search and exclusion and for the final minor improvements of the complexity bounds on the recursive factorization of p x over discs on the complex plane via numerical integration and newton s iterations

on the existence of fourier mukai functors a theorem by orlov states that any equivalence f d b mathrm coh x rightarrow d b mathrm coh y between the bounded derived categories of coherent sheaves of two smooth projective varieties x and y is isomorphic to a fourier mukai transform phi e r pi 2 e mathop otimes limits l l pi 1 where the kernel e is in d b mathrm coh x times y in the case of an exact functor which is not necessarily fully faithful we compute some sheaves that play the role of the cohomology sheaves of the kernel and that are isomorphic to the latter whenever an isomorphism f cong phi e exists we then exhibit a class of functors that are not full or faithful and still satisfy the above result