spectra of digraphs in this primarily expository paper we survey classical and some more recent results on the spectra of digraphs equivalently the spectra of 0 1 matrices with emphasis on the spectral radius

nonreflecting boundary condition in finite element formulation for an elastic wave equation numerical modeling of seismic wavefields using finite difference or finite element methods requires truncation of the model to finite computational domains it is well known that the edges of such truncated models give boundary reflections on the synthetic seismograms an essential step to successful numerical modeling is to eliminate these reflections we present a simple scheme that eliminates such boundary reflections when computing synthetics we also demonstrate the efficiency and robustness of our method on a variety of geological models

initial conditions in frequency domain analysis the fem applied to the scalar wave equation the present paper describes a procedure to consider initial conditions in the frequency domain analysis of continuous media discretized by the fem the frequency domain formulation presented here is based on a standard dft procedure the fft algorithm being employed to transform from time to the frequency domain and vice versa the standard galerkin finite element method displacement model is used to replace the original differential governing equation by an integral equation amenable to numerical solution the scalar wave equation one and two dimensional is used to illustrate the proposed approach at the end of the paper examples of wave propagation for a taut string a one dimensional rod and a membrane are presented to illustrate the robustness of the formulation presented here

decrypting a class of stream ciphers using ciphertext only pseudonoise sequences generated by linear feedback shift registers 1 with some nonlinear combining function have been proposed 2 5 for cryptographic applications as running key generators in stream ciphers in this correspondence it will be shown that the number of trials to break these ciphers can be significantly reduced by using correlation methods by comparison of computer simulations and theoretical results based on a statistical model the validity of this analysis is demonstrated rubin 6 has shown that it is computationally feasible to solve a cipher proposed by pless 2 in a known plaintext attack using as few as 15 characters here the number of ciphertext symbols is determined to perform a ciphertext only attack on the pless cipher using the correlation attack our conclusion from the analysis is that the pseudonoise generator s output sequence and the sequences generated by the linear feedback shift registers should be uncorrelated this leads to constraints for the nonlinear combining function to be used

principal submatrices of co order one with the biggest perron root abstract let a be an irreducible nonnegative matrix w be any of its indices and a w be the principal submatrix of co order one obtained from a by deleting the w th column and row denote by v ext a the set of indices w such that a w has the biggest perron root among all the principal submatrices of co order one of the original matrix a we prove that exactly one jordan block corresponds to the perron root a w of a w for every w v ext a if its size is strictly greater than one for some w v ext a then the original matrix a is permutationally similar to a lower hessenberg matrix with positive entries on the superdiagonal and in the left lower corner in other words the digraph d a of a has a hamiltonian circuit and its diameter is one less than its order in the opposite case for any w v ext a there is a unique path w i i 0 p going through w in d a such that 1 a w i has the biggest perron root for i 0 p 2 a w 0 has a right positive perron eigenvector 3 a w p has a left positive perron eigenvector 4 a w i has neither a left nor a right positive perron eigenvector for i 1 p 1 thus by the spectral criterion for a nonnegative matrix to be irreducible the submatrices a w 0 a w p combined inherit the property of irreducibility we also show that a w is irreducible for every w v ext a if any of the following holds 1 a is symmetric 2 every column row of a has at least two positive nondiagonal entries 3 a has at least two columns rows all of whose entries are positive if a is an irreducible tournament matrix then either a w is also irreducible for any w v ext a or there exist exactly two indices w in and w out in v ext a such that a w in and a w out are reducible in the last case any other principal submatrix of co order one is irreducible this shows that in the general case a one vertex deleted subdigraph with the biggest perron root need not have the best connectivity properties among all one vertex deleted subdigraphs of a given strongly connected digraph d

eigenvalue multiplicities in principal submatrices let be an eigenvalue of an n by n matrix a the allowable patterns of geometric multiplicities of as an eigenvalue of a and its principal submatrices is explored a graphical hierarchy for succinctly reporting the possible patterns is defined special attention is paid to the case in which a is hermitian classical interlacing already imposes much structure on the hierarchies in the hermitian case here all the known constraints some old and some new on the geometric multiplicity hierarchies of hermitian matrices are listed some differences between allowed hierarchies for real symmetric matrices and hermitian matrices are also discussed

a precise time step integration method by step response and impulsive response matrices for dynamic problems in this paper a precise time step integration method for dynamic problems is presented the second order differential equations for dynamic problems are manipulated directly a general damping matrix is considered the transient responses are expressed in terms of the steady state responses the given initial conditions and the step response and impulsive response matrices the steady state responses for various types of excitations are readily obtainable the computation of the step response and impulsive response matrices and their time derivatives are studied in this paper a direct computation of these matrices using the taylor series solutions is not efficient when the time step size t is not small in this paper the recurrence formulae relating the response matrices at t t to those at t t 2 are constructed a recursive procedure is proposed to evaluate these matrices at t t from the matrices at t t 2m the matrices at t t 2m are obtained from the taylor series solutions to improve the computational efficiency the relations between the response matrices and their time derivatives are investigated in addition these matrices are expressed in terms of two symmetric matrices that can also be evaluated recursively besides from the physical point of view these matrices should be banded for small t both the stability and accuracy characteristics of the present algorithm are studied three numerical examples are used to illustrate the highly precise and stable algorithm 1997 john wiley sons ltd

bounds for levinger s function of nonnegative almost skew symmetric matrices the analysis of the perron eigenspace of a nonnegative matrix a whose symmetric part has rank one is continued improved bounds for the perron root of levinger s transformation 1 a a t 0 1 and its derivative are obtained the relative geometry of the corresponding left and right perron vectors is examined the results are applied to tournament matrices to obtain a comparison result for their spectral radii

linear models for keystream generators it is shown that an arbitrary binary keystream generator with m bits of memory can be linearly modeled as a non autonomous linear feedback shift register of length at most m with an additive input sequence of nonbalanced identically distributed binary random variables the sum of the squares of input correlation coefficients over all the linear models of any given length proves to be dependent on a keystream generator the minimum and maximum values of the correlation sum along with the necessary and sufficient conditions for them to be achieved are established an effective method for the linear model determination based on the linear sequential circuit approximation of autonomous finite state machines is developed linear models for clock controlled shift registers and arbitrary shift register based keystream generators are derived several examples including the basic summation generator the clock controlled cascade and the shrinking generator are presented linear models are the basis for a general structure dependent and initial state independent statistical test they may also be used for divide and conquer correlation attacks on the initial state security against the corresponding statistical attack appears hard to control in practice and generally hard to achieve with simple keystream generator schemes

eigenvalue inequalities for principal submatrices abstract for a polynomial with real roots inequalities between those roots and the roots of the derivative are demonstrated and translated into eigenvalue inequalities for a hermitian matrix and its submatrices for example given an n by n positive definite hermitian matrix with maximum eigenvalue these inequalities imply that some principal submatrix has an eigenvalue exceeding n 1 n

an upper bound on the perron value of an almost regular tournament matrix we provide an asymptotic upper bound on the perron value of an almost regular tournament matrix improving upon an existing bound due to friedland our approach employs techniques from contrained optimization

correlation immunity of nonlinear combining functions for cryptographic applications corresp pseudonoise generators for cryptographic applications consisting of several linear feedback shift registers with a nonlinear combining function have been proposed as running key generators in stream ciphers these running key generators eau sometimes be broken by ciphertext only correlation attacks on individual subsequences a new class of combining functions is presented which provides better security against such attacks the security is quantified by the smallest number m 1 of subsequences that must be simultaneously considered in a correlation attack a necessary condition for such m th order correlation immunity is proved a recursive construction is given that permits the construction of an m th order immune combining function for n subsequences for any m and n with 1 leq m finally the trade off between the length of the linear equivalent of the nonlinear generator and the order m of its immunity against correlation attacks is considered

new eigenvalue estimates for complex matrices abstract in a recent work 1 we have proved that all the eigenvalues j of a complex matrix a of order n lie in a disk with center at tr a n and radius n 1 n k 1 n k 2 tr a 2 n 1 2 in this paper we prove that the eigenvalues of a are contained in a rectangle with vertices on the boundary circle of the mentioned disk improving in this way the result in 1 we recall some inequalities which alloww us to bound the radius of the disk and the rectangle the bounds can be computed without knowing the eigenvalues of a

on almost regular tournament matrices abstract spectral and determinantal properties of a special class m n of 2n 2n almost regular tournament matrices are studied in particular the maximum perron value of the matrices in this class is determined and shown to be achieved by the brualdi li matrix which has been conjectured to have the largest perron value among all tournament matrices of even order we also establish some determinantal inequalities for matrices in m n and describe the structure of their associated walk spaces

an efficient time frequency domain algorithm for modal analysis of non linear models discretized by the fem abstract the present paper describes an efficient time frequency domain approach for modal analysis of non linear models discretized by the fem an efficient recurrence relationship for displacement and velocity based on the green s function of the model and its time derivative is presented in both a time and a frequency domain context in the time domain procedure the green s function related terms are analytical in the frequency domain procedure the fundamental terms are indirectly evaluated taking into account steady state responses of the system the pseudo force method is adopted in order to deal with the model non linearity results for the time frequency domain approaches discussed here showed the same level of accuracy as those obtained by the newmark modified newton algorithm at a lower computational cost

properties of the brualdi li tournament matrix the brualdi li tournament matrix is conjectured to have the largest spectral radius among all tournament matrices of even order in this paper two forms of the characteristic polynomial of the brualdi li tournament matrix are found using the first form it is shown that the roots of the characteristic polynomial are simple and that the brualdi li tournament matrix is diagonalizable using the second form an expression is found for the coefficients of the powers of the variable in the characteristic polynomial these coefficients give information about the cycle structure of the cycles of length 1 5 of the directed graph associated with the brualdi li tournament matrix

on the perron roots of principal submatrices of co order one of irreducible nonnegative matrices let a be an irreducible nonnegative matrix and a be the perron root spectral radius of a denote by min a the minimum of the perron roots of all the principal submatrices of co order one it is well known that the interval min a a does not contain any eigenvalues of a consider any principal submatrix a v of co order one whose perron root is equal to min a we show that the jordan structure of min a as an eigenvalue of a is obtained from that of the perron root of a v as follows one largest jordan block disappears and the others remain the same so if only one jordan block corresponds to the perron root of the submatrix then min a is not an eigenvalue of a by schneider s theorem this holds if and only if there is a hamiltonian chain in the singular digraph of a v in the general case the jordan structure for the perron root of the submatrix a v and therefore that for the eigenvalue min a of a can be arbitrary but if the perron root a w of a principal submatrix a w of co order one is strictly greater than min a then a w is a simple eigenvalue of a w we also obtain different representations for the generalized eigenvectors corresponding to the eigenvalues of a contained in the annulus min a a

explicit time integration algorithms for structural dynamics with optimal numerical dissipation abstract a new predictor corrector explicit time integration algorithm is presented for solving structural dynamics problems the basis of the algorithm is the implicit generalized method developed by the authors like its implicit parent the explicit generalized method is a one parameter family of algorithms in which the parameter defines the high frequency numerical dissipation for a given value of high frequency dissipation the explicit generalized method minimizes low frequency dissipation the algorithm can be utilized effectively for structural dynamics calculations in which numerical dissipation is needed to reduce spurious oscillations parameter values are given that enable physical damping to be treated explicitly while maintaining second order accuracy without the need for an expensive second corrector pass

a time domain fem approach based on implicit green s functions for non linear dynamic analysis the present paper describes an unconditionally stable algorithm to integrate the equations of motion in time the standard fem displacement model is employed to perform space discretization and the time marching process is carried out through an algorithm based on the green s function of the mechanical system in nodal co ordinates in the present implicit green s function approach imga mechanical system green s functions are not explicitly computed rather they are implicitly considered through an iterative pseudo forces process under certain simplifying hypothesis iterations are not necessary and the imga becomes cheaper than standard newmark newton raphson algorithm at the end of the paper numerical examples are presented in order to illustrate the accuracy of the present approach

on the maximal spectral radius of even tournament matrices and the spectrum of almost skew symmetric compact operators abstract we study asymptotic expansions of the spectral radii of certain tournament matrices in particular the sequence of brualdi li tournament matrices we also discuss the location of the spectra of certain almost skew symmetric compact operators

a strategy for the solution of soil dynamic problems involving plasticity by transform a closed solution for dynamic problems with the transform technique is possible for linear behaviour only therefore a strategy for the solution involving nonlinear material behaviour has been developed herewith an iterative equalization is made possible by the repeated transform of the differential equations given through the initial strain method

an efficient time domain fem bem coupling approach based on fem implicit green s functions and truncation of bem time convolution process abstract this paper describes an efficient scheme for coupling the fe and the be methods the domain of the original problem is subdivided into two sub domains which are independently modelled by the fem and the bem both sub domains are analyzed separately and the coupling is enforced at their interface the fem time marching process is carried out through an algorithm based on implicit green s functions of the related mechanical system and the convolution process of the bem is compressed by proper interpolation of the time generated matrices the efficiency of the proposed scheme is improved by a special procedure to deal with different time step choices in each sub domain at the end of the paper numerical examples are presented in order to illustrate the accuracy and efficiency of the present approach

a note on the sequence of brualdi li matrices abstract we consider the sequence 2n n 1 2 2n where 2n is the perron value of the brualdi li matrix of order 2n we prove that the sequence converges find its limit and show that asymptotically the sequence is monotonically decreasing in particular this addresses some problems raised by friedland and katz

bounds for the perron root using max eigenvalues abstract using the techniques of max algebra a new proof of al pin s lower and upper bounds for the perron root of a nonnegative matrix is given the bounds depend on the row sums of the matrix and its directed graph if the matrix has zero main diagonal entries then these bounds may improve the classical row sum bounds this is illustrated by a generalized tournament matrix

the perron eigenspace of nonnegative almost skew symmetric matrices and levinger s transformation let a be a nonnegative square matrix whose symmetric part has rank one tournament matrices are of this type up to a positive shift by 1 2i when the symmetric part of a is irreducible the perron value and the left and right perron vectors of l a 1 a at are studied and compared as functions of 0 1 2 in particular upper bounds are obtained for both the perron value and its derivative as functions of the parameter via the notion of the q numerical range

on the location of the spectrum of hypertournament matrices inclusion regions for the spectrum of a hypertournament matrix a are obtained based on a complex curve that relates the real and imaginary parts of the eigenvalues these results generalize and in certain cases improve the work of s kirkland linear and multilinear algebra 30 1991 261 the bounds obtained depend on the variance of the score vector their tightness is investigated using the notion of numerical range 2001 elsevier science inc all rights reserved

extremal digraphs with given clique number abstract we classify those digraphs with a given number of vertices and a given clique number that maximize the perron root of the adjacency matrix

a frequency domain fem approach based on implicit green s functions for non linear dynamic analysis abstract the present paper describes an efficient algorithm to integrate the equations of motion implicitly in the frequency domain the standard fem displacement model galerkin formulation is employed to perform space discretization and the time marching process is carried out through an algorithm based on the green s function of the mechanical system in nodal coordinates in the present formulation mechanical system green s functions are implicitly calculated in the frequency domain once the green s functions related matrices are computed a time integration procedure which demands low computational effort when applied to non linear mechanical systems becomes available at the end of the paper numerical examples are presented in order to illustrate the accuracy of the present approach

a new family of explicit time integration methods for linear and non linear structural dynamics a new family of explicit single step time integration methods with controllable high frequency dissipation is presented for linear and non linear structural dynamic analyses the proposed methods are second order accurate and completely explicit with a diagonal mass matrix even when the damping matrix is not diagonal in the linear structural dynamics or the internal force vector is a function of velocities in the non linear structural dynamics stability and accuracy of the new explicit methods are analysed for the linear undamped damped cases furthermore the new methods are compared with other explicit methods

normal matrices and their principal submatrices of co order one abstract let a be a normal matrix v be any of its indices a v be the matrix obtained from a by deleting the v th row and column and be an eigenvalue of a v in our paper we construct the eigenspace of a associated with from that of a v in particular it is shown that if there is a unique jordan block of size strictly greater than one in the part of the jordan form of a v corresponding to then the geometric multiplicity of decreases by one under the transition from a v to a in other words the typical change of the spectral properties holds for the results obtained are applied to circulant matrices moreover in appendix to our paper we consider almost regular tournament matrices as principal submatrices of co order one of regular tournament matrices in particular it is observed that the brualdi li tournament matrix b 2 n of order 2n is permutationally similar to a principal submatrix of co order one of the circulant matrix of order 2n 1 with the first row 0 1 1 n 0 0 n as a consequence of this fact the weak brualdi li conjecture is formulated for principal submatrices of co order one of the adjacency matrices of cayley tournaments

an efficient hybrid time laplace domain method for elastodynamic analysis based on the explicit green s approach abstract this work aims at presenting a novel hybrid method applied to the numerical solution of elastodynamic problems the formulation is based on the so called explicit green s approach exga recently developed by mansur et al mansur w j loureiro f s soares jr d dors c 2007 explicit time domain approaches based on numerical green s functions computed by finite differences the exga family journal of computational physics 227 851 870 in which a recursive time integration procedure based on green s function is adopted here the finite element method is employed to calculate the green s function in the laplace domain afterwards the zakian s laplace inversion algorithm is employed to compute numerically the green s function in the time domain only at time t t time integration is then carried out by employing the standard exga procedure and another where the convolution integral is replaced by a particular solution that is directly related to the applied load finally some numerical problems are solved to demonstrate the computational efficiency and reliability of the proposed hybrid method

iterative optimum symbol by symbol decoding and fast correlation attacks the relation between fast correlation attacks on stream ciphers based on linear feedback shift registers lfsrs and the hartmann rudolph 1976 algorithm for optimal symbol by symbol decoding of linear block codes is theoretically investigated by iterating approximate versions of the hr algorithm novel iterative probabilistic decoding algorithms for fast correlation attacks are developed in particular these algorithms can be applied when a large number of nonorthogonal parity checks are used a number of properties of the proposed iterated versions of the hr algorithm including convergence conditions and critical bit error rates are derived

exact probabilistic analysis of memoryless combiners exact analytical expressions for various probability distributions in a type of stream ciphers known as memoryless combiners are derived they relate to arbitrary linear functions and subvectors of output bits as well as to arbitrary linear functions and subvectors of input bits when the output sequence is assumed to be given practical approximations to the exact expressions to be used in one step and iterative fast correlation attacks and statistical distinguishers are proposed more generally the results also apply to arbitrary binary linear codes that are nonlinearly transformed by time variant boolean functions for comparison the corresponding probability distributions for linear block codes on a memoryless time variant binary symmetric channel are also determined and some practical consequences pointed out

perron frobenius type results on the numerical range we present results connecting the shape of the numerical range to intrinsic properties of a matrix a w hena is a nonnegative matrix these results are to a large extent analogous to the perron frobenius theory especially as it pertains to irreducibility and cyclicity in the combinatorial sense special attention is given to polygonal circular and elliptic numerical ranges the main vehicles for obtaining these results are the hermitian and skew hermitian parts of a as well as levinger s transformation aa 1 a a 2002 elsevier science

explicit time domain approaches based on numerical green s functions computed by finite differences the exga family the present paper describes a new family of time stepping methods to integrate dynamic equations of motion the scalar wave equation is considered here however the method can be applied to time domain analyses of other hyperbolic e g elastodynamics or parabolic e g transient diffusion problems the algorithms presented require the knowledge of the green s function of mechanical systems in nodal coordinates the finite difference method is used here to compute numerically the problem green s function however any other numerical method can be employed e g finite elements finite volumes etc the green s matrix and its time derivative are computed explicitly through the range 0 dt with either the fourth order runge kutta algorithm or the central difference scheme in order to improve the stability of the algorithm based on central differences an additional matrix called step response is also calculated the new methods become more stable and accurate when a sub stepping procedure is adopted to obtain the green s and step response matrices and their time derivatives at the end of the time step three numerical examples are presented to illustrate the high precision of the present approach

a note on perron vectors for almost regular tournament matrices abstract let t be an almost regular tournament matrix of order n with right perron vector w we show that if the i th row sum of t is n 2 2 and the j th row sum is n 2 then w i w j thus in the round robin competition corresponding to t the ranking schemes of kendall and wei and of ramanujacharyula agree with the ranking generated by the row sums of t

perron vector bounds for a tournament matrix with applications to a conjecture of brualdi and li abstract for a primitive generalized tournament matrix we present upper and lower bounds on an entry in its perron vector in terms of the corresponding row sum of the matrix the bounds are then used to help prove that if n is even and sufficiently large any tournament matrix of order n which maximizes the perron value must be almost regular throughout we use both analytic and combinatorial techniques

solution of the conjecture of brualdi and li abstract we establish the conjecture of brualdi and li on the maximal perron root of a tournament matrix of even order

on the use of pseudo forces to consider initial conditions in 3d time and frequency domain acoustic analysis this work is mainly concerned with a general strategy based on well known concepts of classical mechanics for taking into account initial conditions in frequency domain fd and time domain td analyses a general approach extended here to three dimensional applications is presented special problems associated with analyses through discrete fourier transform dft algorithms as those occurring in consequence of a non correct choice of extended period or those connected with aliasing phenomenon are also discussed furthermore an alternative starting procedure for time marching schemes in td analyses is proposed at the end of the paper to validate the proposed techniques and to demonstrate their generality two and three dimensional problems with non homogeneous initial conditions are solved through frequency and time domain approaches by employing the finite element method fem numerical results are compared with existing analytical solutions

tournament matrices with extremal spectral properties abstract for a tournament matrix m of order n we define its walk space wm to be span m j 1 j 0 n 1 where 1 is the all ones vector we show that the dimension of wm equals the number of eigenvalues of m whose real parts are greater than 1 2 we then focus on tournament matrices whose walk space has particularly simple structure and characterize them in terms of their spectra specifically we characterize those tournament matrices such that mj1 is an eigenvector of m for some j 0 we also characterize the tournament matrices m such that jn 2m is a skew hadamard matrix throughout we illustrate our results with examples

statistical distinguishers for irregularly decimated linear recurring sequences a clock controlled linear feedback shift register lfsr is a simple keystream generator consisting of an lfsr whose clock is controlled by another lfsr so that the output is an irregularly decimated lfsr sequence statistical distinguishers for keystream generators are algorithms whose objective is to distinguish the keystream sequence from a purely random sequence previously proposed statistical distinguishers for clock controlled lfsrs are based on detecting binary linear relations in the keystream sequence that hold with a probability sufficiently different from one half in this paper a novel approach based on the correlation analysis of clock controlled lfsrs is introduced it significantly reduces the required computation time and essentially consists of a probabilistic reconstruction of bits in the regularly clocked lfsr sequence that satisfy the lfsr recurrence or any linear recurrence derived from low weight multiples of the lfsr characteristic polynomial the keystream sequence length and the computation time required for a reliable statistical distinction are analyzed both theoretically and experimentally the new approach shows that there is no need to deal with the majority bits of blocks of lfsr bits as proposed recently if individual bits or blocks of bits are considered instead then the underlying probabilities can be computed theoretically instead of heuristically which in turn allows a simple and general theoretical analysis of the keystream sequence length required

pick s inequality and tournaments abstract let t be a tournament on n nodes and let a be its adjacency matrix a brauer and i gentry observed that an inequality due to g pick implies that im 1 2 cot 2n for all eigenvalues of a we say that t is a pick or p tournament if equality holds for some we determine when equality holds in pick s inequality for arbitrary real matrices and use this to show that the p tournaments can all be constructed from the transitive tournament m by reversing the arcs between the sets of certain node partitions or cuts u the cuts are specified by 1 n vectors u such that utw 0 where w 1 n 1 t e i n this links the cuts to cyclotomic polynomials there is at least one p tournament on n nodes if and only if n 2k k 1 up to isomorphism there is precisely one p tournament on n nodes if and only if n p or n 2p for some odd prime p for odd n up toisomorphism the only regular p tournament on n nodes has matrix zn circ 0 1 1 0 0 a composition rule is used on the matrices zp to form all p tournaments on n 2kpl nodes l 1

numerical range of matrices and levinger s theorem abstract this paper is in a sense a continuation of the author s previous paper on the numerical range of matrices in particular its connection to levinger s theorem for the case of nonnegative matrices is extended for general complex matrices an elementary proof of levinger s theorem is also included