the multifrontal method for sparse matrix solution theory and practice this paper presents an overview of the multifrontal method for the solution of large sparse symmetric positive definite linear systems the method is formulated in terms of frontal matrices update matrices and an assembly tree formal definitions of these notions are given based on the sparse matrix structure various advances to the basic method are surveyed they include the role of matrix reorderings the use of supernodes and other implementatjon techniques the use of the method in different computational environments is also described

an example of an indexed language of intermediate growth we give an example of a language belonging to the class of indexed languages whose growth is intermediate in particular the growth function of this language is transcendental

high order finite difference simulations of marine csem surveys using a correspondence principle for wave and diffusion fields the computer time required to solve a typical 3d marine controlled source electromagnetic surveying csem simulation can be reduced by more than one order of magnitude by transforming low frequency maxwell equations in the quasi static or diffusive limit to a hyperbolic set of partial differential equations that give a representation of electromagnetic fields in a fictitious wave domain the dispersion and stability analysis can be made equivalent to that of other types of wave simulation problems such as seismic acoustic and elastic modeling second order to eighth order spatial derivative operators are implemented for flexibility fourth order and sixth order methods are the most numerically efficient implementations for this particular scheme an implementation with high order operators requires that both electric and magnetic fields are extrapolated simultaneously into the air layer the stability condition given for high order staggered derivative operators here should be equally valid for seismic wave simulation the bandwidth of recovered fields in the diffusive domain is independent of the bandwidth of the fields in the fictitious wave domain the fields in the fictitious wave domain do not represent observable fields propagation paths and interaction reflection amplitudes are not altered by the transform from the fictitious wave domain to the diffusive frequency domain however the transform contains an exponential decay factor that damps down late arrivals in the fictitious wave domain the propagation paths that contribute most to the diffusive domain fields are airwave shallow water plus typically postcritical events such as refracted and guided waves the transform from the diffusive frequency domain to the fictitious wave domain is an ill posed problem the transform is nonunique this gives a large degree of freedom in postulating temporal waveforms for boundary conditions in the fictitious wave domain that reproduce correct diffusive frequency domain fields

new results on periodic sequences with large k error linear complexity niederreiter showed that there is a class of periodic sequences which possess large linear complexity and large k error linear complexity simultaneously this result disproved the conjecture that there exists a trade off between the linear complexity and the k error linear complexity of a periodic sequence by ding by considering the orders of the divisors of xn 1 over bbf q we obtain three main results which hold for much larger k than those of niederreiter a sequences with maximal linear complexity and almost maximal k error linear complexity with general periods b sequences with maximal linear complexity and maximal k error linear complexity with special periods c sequences with maximal linear complexity and almost maximal k error linear complexity in the asymptotic case with composite periods besides we also construct some periodic sequences with low correlation and large k error linear complexity

on the linear complexity and k error linear complexity over bbf _ p of the d ary sidel nikov sequence the d ary sidel nikov sequence s s0 s1 of period q 1 for a prime power q pm is a frequently analyzed sequence in the literature recently it turned out that the linear complexity over fp of the d ary sidel nikov sequence is considerably smaller than the period if the sequence element s q 1 2mod q 1 is chosen adequately in this paper this work is continued and tight lower bounds on the linear complexity over fp of the d ary sidel nikov sequence are given for certain cases exact values are provided finally results on the k error linear complexity over fp of the d ary sidel nikov sequence are presented

on the language of primitive words abstract a word is primitive if it is not a proper power of a shorter word we prove that the set q of primitive words over an alphabet is not an unambiguous context free language this strengthens the previous result that q cannot be deterministic context free further we show that the same holds for the set l of lyndon words we investigate the complexity of q and l we show that there are families of constant depth polynomial size unbounded fan in boolean circuits and two way deterministic pushdown automata recognizing both languages the latter result implies efficient decidability on the ram model of computation and we analyze the number of comparisons required for deciding q finally we give a new proof showing a related language not to be context free which only relies on properties of semi linear and regular sets

an algorithm for the k error linear complexity of binary sequences with period 2 sup n certain applications require pseudo random sequences which are unpredictable in the sense that recovering more of the sequence from a short segment must be computationally infeasible it is shown that linear complexity is useful in determining such sequences a generalized linear complexity that has application to the security of stream ciphers is proposed and an efficient algorithm is given for the case in which the sequence is binary with period 2 sup n this algorithm generalizes an algorithm presented by r a games and a h chan 1983

non autonomous second order hamiltonian systems abstract we study the existence of periodic solutions for a second order non autonomous dynamical system containing variable kinetic energy terms our assumptions balance the interaction between the kinetic energy and the potential energy with neither one dominating the other we study sublinear problems and the existence of non constant solutions

periodic solutions of dynamical systems by a saddle point theorem of rabinowitz on montre comment par le theoreme de point selle rabinowitz on peut obtenir des resultats nouveaux et interessants sous des hypotheses simples

transcendence of formal power series with rational coefficients we give algebraic proofs of transcendence over q x of formal power series with rational coefficients by using inter alia reduction modulo prime numbers and the christol theorem applications to generating series of languages and combinatorial objects are given

rational and transcendental growth series for the higher heisenberg groups this paper considers growth series of 2 step nilpotent groups with infinite cyclic derived subgroup every such group g has a subgroup of finite index of the form h n m where h n is the discrete heisenberg group of length 2n 1 we call n the heisenberg rank of g we show that every group of this type has some finite generating set such that the corresponding growth series is rational on the other hand we prove that if g has heisenberg rank n 2 then g possesses a finite generating set such that the corresponding growth series is a transcendental power series

fast finite difference time domain modeling for marine subsurface electromagnetic problems in the low frequency limit the displacement currents in the maxwell equations can be neglected however for numerical simulations a small displacement current should be present to achieve numerical stability this requirement leads to a large range of propagation velocities with high velocities for the high frequencies and low velocities for the low frequencies as a consequence the number of time steps may become large i show that it is possible to transform mathematically the original physical problem to one that has propagation velocities with less frequency dependence hence the number of time steps necessary for a signal to travel a certain distance with the lowest velocity is significantly reduced a typical example shows a reduction in computational time by a factor of 40 a comparison of the solutions from plane layered modeling in the frequency and wavenumber domain and the proposed method shows good agreement between the two the proposed method can also be used for other systems of diffusive equations

analytic models and ambiguity of context free languages we establish that several classical context free languages are inherently ambiguous by proving that their counting generating functions when considered as analytic functions exhibit some characteristic form of transcendental behaviour to that purpose we survey some general results on elementary analytic properties and enumerative uses of algebraic functions in relation to formal languages in particular the paper contains a general density theorem for unambiguous context free languages

periodic solutions of dynamical systems with bounded potential where ve c2 r x rn r v t t x y t x v t x jjbxiw y is bounded and v v t x 0 as 1x1 co for example we prove that if 0 d v r x m and v r x m as 1x1 co there exists at least one t periodic solution of v moreover if v j t 0 0 under suitable assumptions on v y t 0 this solution is not identically zero we also give existence results for the forced system

character sums and congruences with n we estimate character sums with n on average and individually these bounds are used to derive new results about various congruences modulo a prime p and obtain new information about the spacings between quadratic nonresidues modulo p in particular we show that there exists a positive integer n p 1 2 e such that n is a primitive root modulo p we also show that every nonzero congruence class a 0 mod p can be represented as a product of 7 factorials a n 1 n 7 mod p where max n i i 1 7 o p 11 12 e and we find the asymptotic formula for the number of such representations finally we show that products of 4 factorials n 1 n 2 n 3 n 4 with max n 1 n 2 n 3 n 4 o p 6 7 e represent almost all residue classes modulo p and that products of 3 factorials n 1 n 2 n 3 with max n 1 n 2 n 3 o p 5 6 e are uniformly distributed modulo p

on the joint subword complexity of automatic sequences let the subword complexity of a sequence u u n n 0 over a finite set s be the function m p u m where p u m denotes the number of distinct blocks u n u n m 1 of size m in u in this paper we study the complexity of u n u 1 n u r n n 0 when each u i u i n n 0 i 1 r is a q i automatic sequence over a finite set s i and q 1 q r 2 are pairwise coprime integers as an application we answer a question of allouche and shallit regarding morphic real numbers

heteroclinic solutions for the extended fisher kolmogorov equation abstract we study the extended fisher kolmogorov efk equation and its variants by variational approach we show that for every global minimum of the potential function v there is a pair of heteroclinic solutions one emanating from and the other terminating at we require neither superquadratic growth of v nor the presence of saddle focus equilibria moreover v is allowed to approach its minimal level near infinity

hybrid calculations of the three dimensional electromagnetic response of buried conductors the hybrid method for computing the electromagnetic response of a three dimensional conductor in a layered conducting half space consists of solving a finite element problem in a localized region containing the conductor and using integral equation methods to obtain the fields outside that region the original scheme obtains the boundary values by iterating between the integral equation solution and the finite element solution after making an initial guess based on primary values from the field a two dimensional interpolation scheme is then used to speed the evaluation of the 105 to 106 green s function convolution integrals required by most problems the two algorithms presented are modifications of the original scheme both contain a search routine to identify a set of unique points where the convolution integral evaluations are required by replacing the two dimensional interpolation with a one dimensional interpolation and reading the convolution integrals from a reference table computation time

landau singularities and singularities of holonomic integrals of the ising class we consider families of multiple and simple integrals of the ising class and the linear ordinary differential equations with polynomial coefficients they are solutions of we compare the full set of singularities given by the roots of the head polynomial of these linear odes and the subset of singularities occurring in the integrals with the singularities obtained from the landau conditions for these ising class integrals we show that the landau conditions can be worked out either to give the singularities of the corresponding linear differential equation or the singularities occurring in the integral the singular behaviour of these integrals is obtained in the self dual variable w s 2 1 s2 with s sinh 2k where k j kt is the usual ising model coupling constant switching to the variable s we show that the singularities of the analytic continuation of series expansions of these integrals actually break the kramers wannier duality we revisit the singular behaviour zenine et al 2005 j phys a math gen 38 9439 74 of the third contribution to the magnetic susceptibility of ising model 3 at the points 1 3w 4w2 0 and show that 3 s is not singular at the corresponding points inside the unit circle s 1 while its analytical continuation in the variable s is actually singular at the corresponding points 2 s s2 0 outside the unit circle s 1

fast multimodel finite difference controlled source electromagnetic simulations based on a schur complement approach we have developed an efficient numerical scheme for fast multimodel 3d electromagnetic simulations by applying a schur complement approach to a frequency domain finite difference method the scheme is based on direct solvers and developed with constrained inversion algorithms in view such algorithms normally need many forward modeling jobs with different resistivities for the target zone and or background formation we geometricallydivide the computational domain intotwo subdomains an anomalous subdomain the resistivities of which were permitted to change and a background subdomain having fixed resistivities the system matrix ispartiallyfactorizedby precomputing a schur complement to eliminate unknowns associated with the background subdomain the schur complement system is then solved to compute fields inside the anomalous subdomain finally the background subdomain fields are computed using inexpensive local substitutions for each successive simulation only the relatively small schur complement system has to be solved which results in significant computational savings we applied this approach to two moderately sized 3d problems in marine controlled source electromagnetic modeling 1 a deepwater model in which the resistivities of the seawater and the air layer were kept fixed and 2 a model in which focused inversion was performed in a scenario in which the resistivities of the background formation the air layer and the seawater were known we found a significant reduction of the modeling time in inversion that depended on the relative sizes of the constrained and unconstrained volumes the smaller the unconstrained volume the larger the savings specifically for a focused inversion of the troll oil field in the north sea the gain amounted up to 80 of the total modeling time

a hybrid finite difference and integral equation method for modeling and inversion of marine controlled source electromagnetic data abstractone of the major problems in the modeling and inversion of marine controlled source electromagnetic csem data is related to the need for accurate representation of very complex geoelectrical models typical for marine environment at the same time the corresponding forward modeling algorithms should be powerful and fast enough to be suitable for repeated use in hundreds of iterations of the inversion and for multiple transmitter receiver positions to this end we have developed a novel 3d modeling and inversion approach which combines the advantages of the finite difference fd and integral equation ie methods in the framework of this approach we have solved maxwell s equations for anomalous electric fields using the fd approximation on a staggered grid once the unknown electric fields in the computation domain of the fd method are computed the electric and magnetic fields at the receivers are calculated using the ie method with the corresponding green s tensor for the background conduc

summation of power series with rational coefficients via divided differences abstract we obtain new identities involving divided differences and as an application we provide a closed form representation of power series with rational coefficients in terms of divided differences involving special functions

an efficient finite difference scheme for electromagnetic logging in 3d anisotropic inhomogeneous media we consider a problem of computing the electromagnetic field in 3d anisotropic media for electromagnetic logging the proposed finite difference scheme for maxwell equations has the following new features based on some recent and not so recent developments in numerical analysis coercivity i e the complete discrete analogy of all continuous equations in every grid cell even for nondiagonal conductivity tensors a special conductivity averaging that does not require the grid to be small compared to layering or fractures and a spectrally optimal grid refinement minimizing the error at the receiver locations and optimizing the approximation of the boundary conditions at infinity all of these features significantly reduce the grid size and accelerate the computation of electromagnetic logs in 3d geometries without sacrificing accuracy

multiple critical points for indefinite functionals and applications abstract in this paper we prove a multiplicity result for critical points of an indefinite functional on a manifold by using a generalization of the ljusternik schnirelman category the abstract result is applied to prove the existence of multiple timelike trajectories for lorentzian manifolds of splitting type

hilbert series of fixed free algebras and noncommutative classical invariant theory we compute an asymptotic formula for the number of invariants of a given degree which compares nicely with the corresponding result in the classical commutative case in section 6 we show that if g is an infinite cyclic group generated by a unipotent matrix then

on the lower bound of the linear complexity over f sub p of sidelnikov sequences for a sidelnikov sequence of period p sup m 1 tight lower bounds are obtained on its linear complexity l over f sub p in particular these bounds imply that uniformly over all p and m l is close to its largest possible value p sup m 1

3d integral equation modeling with a linear dependence on dimensions we have improved the integral equation method for modeling 3d electromagnetic fields by using the separability of its inherent 3 3 dyadic green s tensors conventional integral equation approaches exhibit a quadratic dependence on model size at least for the vertical dimension in contrast our approach has a linear dependence on all three dimensions we tested our method on an example of induction logging in deviated boreholes

conservative modeling of 3 d electromagnetic fields part ii biconjugate gradient solution and an accelerator the preceding paper derives a staggered grid finite difference approximation applicable to electromagnetic induction in the earth the staggered grid finite difference approximation results in a linear system of equations ax b where a is symmetric but not hermitian this is solved using the biconjugate gradient method preconditioned with a modified partial cholesky decomposition of a this method takes advantage of the sparsity of a and converges much more quickly than methods used previously to solve the 3 d induction problem when simulating a conductivity model at a number of frequencies the rate of convergence slows as frequency approaches 0 the convergence rate at low frequencies can be improved by an order of magnitude by alternating the incomplete cholesky preconditioned biconjugate gradient method with a procedure designed to make the approximate solutions conserve current

3d finite difference frequency domain modeling of controlled source electromagnetic data direct solution and optimization for high accuracy three dimensional modeling of marine controlled source electromagnetic csem data is vital to improve the understanding of electromagnetic em responses collected in increasingly complex geologic settings a modeling tool for simulating 3d marine csem surveys based on a finite difference discretization of the helmholtz equation for the electric fields has been developed optimizations for csem simulations include the use of a frequency domain technique a staggering scheme that reduces inaccuracies especially for horizontal electric dipole sources located near the seafloor and a new interpolation technique that provides highly accurate em field values for receivers located in the immediate vicinity of the seafloor source singularities are eliminated through a secondary field approach in which the primary fields are computed analytically for a homogeneous or a 1d layered background the secondary fields are computed using the finite difference technique exploiting recent advances in computer technology and algorithmic developments the system of finite difference equations is solved using the mumps direct matrix solver in combination with the other optimizations this allows accurate em field computations for moderately sized models on small computer clusters the explicit availability of matrix factorizations is advantageous for multisource modeling and makes the algorithm well suited for future use within an inversion scheme comparisons of simulated data for 1 1d models to data generated using a 1d reflectivity technique and 2 3d models to published 3d data demonstrate the accuracy and benefits of the approach

on the transcendence of certain series r p stanley in s observed that the series f x c c xn is transcendental over c x for even t 1 and stated that it is unknown for odd t 1 whether or not it is transcendental we show here that the reduction of f is algebraic over any field of positive characteristic p and we then deduce from the explicit equations obtained that f is transcendental over any field of characteristic zero for any integer r 1 we also give a generalisation of this result in the case of multinomial coefficients note that if t 1 then f is algebraic of degree at most 2 over any field in fact f 1 4x and that for all t 2 1 f 1 over any field of characteristic 2 so that we may suppose that in the case of positive characteristic pi p 2 if k is a field k x will denote the field of formal power series with coefficients in k i e se k x if f c k a x a e k ken by an algebraic function over k we mean an element of k x which is algebraic over the field of rational functions k x an element which is not algebraic is called transcendental we intend to prove the following theorem

hybrid finite difference integral equation solver for 3d frequency domain anisotropic electromagnetic problems the modeling of the controlled source electromagnetic csem and single well and crosswell electromagnetic em configurations requires fine gridding to take into account the 3d nature of the geometries encountered in these applications that include geological structures with complicated shapes and exhibiting large variations in conductivities such as the seafloor bathymetry the land topography and targets with complex geometries and large contrasts in conductivities such problems significantly increase the computational cost of the conventional finite difference fd approaches mainly due to the large condition numbers of the corresponding linear systems to handle these problems we employ a volume integral equation ie approach to arrive at an effective preconditioning operator for our fd solver we refer to this new hybrid algorithm as the finite difference integral equation method fdie this fdie preconditioning operator is divergence free and is based on a magnetic field formulation similar to the lippman schwinger ie method this scheme allows us to use a background elimination approach to reduce the computational domain resulting in a smaller size stiffness matrix furthermore it yields a linear system whose condition number is close to that of the conventional lippman schwinger ie approach significantly reducing the condition number of the stiffness matrix of the fd solver moreover the fd framework allows us to substitute convolution operations by the inversion of banded matrices which significantly reduces the computational cost per iteration of the hybrid method compared to the standard ie approaches also well established fd homogenization and optimal gridding algorithms make the fdie more appropriate for the discretization of strongly inhomogeneous media some numerical studies are presented to illustrate the accuracy and effectiveness of the presented solver for csem single well and crosswell em applications

subharmonic solutions for nonautonomous sublinear second order hamiltonian systems some existence theorems are obtained for subharmonic solutions of nonautonomous second order hamiltonian systems by the minimax methods in critical point theory

linear complexity over f sub p of sidel nikov sequences helleseth kim and no 2003 described the linear complexity over f sub p of sidel nikov sequences of length p sup m 1 for p 3 5 and 7 in this correspondence the result is generalized to all odd primes

transcendence of binomial and lucas formal power series abstract the formal power series formula is transcendental over q x whentis an integer 2 this is due to stanley forteven and independently to flajolet and to woodcock and sharif for the general case while stanley and flajolet used analytic methods and studied the asymptotics of the coefficients of this series woodcock and sharif gave a purely algebraic proof their basic idea is to reduce this series modulo prime numbersp and to use thep lucas property ifn nipiis the basepexpansion of the integern then equation the series reduced modulopis then proved algebraic over f p x the field of rational functions over the galois field f p but its degree is not a bounded function ofp we generalize this method to characterize all formal power series that have thep lucas property for many prime numbersp and that are furthermore algebraic over q x

cranks and t cores new statistics on partitions calledcranks are defined which combinatorially prove ramanujan s congruences for the partition function modulo 5 7 11 and 25 explicit bijections are given for the equinumerous crank classes the cranks are closely related to thet core of a partition usingq series some explicit formulas are given for the number of partitions which aret cores some related questions for self conjugate and distinct partitions are discussed

canonical positions for the factors in paperfolding sequences it has been proved by the first author that the number of factors of length k in any paperfolding sequence is equal to 4k once k 7 we prove here that for every integer k there exists a set of integers pk of cardinality 4k for k 7 such that the factors of length k of any paperfolding sequence are exactly the factors beginning at the places indexed by the integers in pk this gives a bijective proof of the result mentioned above then we give an explicit uniform linear upper bound for the recurrence function of paperfolding sequences finally we study the set of all factors of all paperfolding sequences evaluating their number for a given length and studying the language of these words

on the subword complexity of thue morse polynomial extractions let the subword complexity of a sequence u u n n 0 over a finite set s be the function m p u m where p u m is the number of distinct blocks of length m in u let t t n n 0 denote the thue morse sequence in this paper we study the complexity of the sequences t h t h n n 0 when h n q n is a polynomial with h n n in particular we solve an open problem of allouche and shallit regarding t n 2 n 0 we also study the vector space over z 2z spanned by the sequences t h

a domain decomposition finite difference method for parallel numerical implementation of time dependent maxwell s equations a domain decomposition technique together with an implicit finite difference scheme is used to design a parallel algorithm to solve for electromagnetic scattering in the time domain by an infinite square metallic cylinder the implicit difference scheme yields second order accuracy unconditional stability and at each time step a large system of linear equations the domain decomposition technique reduces the solution of this large system to that of many independent smaller subsystems the present algorithm can be easily implemented on coarse grain parallel vector supercomputers to obtain a speedup close to the number of available central processing units cpus