properties of the error linear complexity spectrum this paper studies the error linear complexity spectrum of binary sequences with period 2n a precise categorization of those sequences having two distinct critical points in their spectra as well as an enumeration of these sequences is given an upper bound on the maximum number of distinct critical points that the spectrum of a sequence can have is proved and a construction which yields a lower bound on this number is given in the process simpler proofs of some known results on the linear complexity and k error linear complexity of sequences with period 2n are provided

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

homotopical stable ranks for banach algebras abstract the connected stable rank and the general stable rank are homotopy invariants for banach algebras whereas the bass stable rank and the topological stable rank should be thought of as dimensional invariants this paper studies the two homotopical stable ranks viz their general properties as well as specific examples and computations the picture that emerges is that of a strong affinity between the homotopical stable ranks and a marked contrast with the dimensional ones

idempotents in nest algebras abstract the algebraic equivalence and similarity classes of idempotents within a nest algebra alg are completely characterized we obtain necessary and sufficient conditions for two idempotents to be equivalent or similar our criterion yields examples illustrating pathology and also shows that to each equivalence class of idempotents there corresponds a dimension function from into n we complete the characterization of the algebraic equivalence classes by proving that any dimension function corresponds to an equivalence class of idempotents also to each sequence of dimension functions there corresponds a commuting sequence of idempotents a criterion is obtained for when an idempotent is similar to a subidempotent of another the mapping which sends an equivalence class or idempotent to its associated dimension function plays a role in the nest algebra theory analogous to the role played by the mapping sending a projection in a type i w algebra to its center valued trace we prove that almost commuting similar idempotents are homotopic this contrasts with the situation in certain c algebras using this we show that similar simultaneously diagonalizable idempotents are homotopic and in the continuous nest case every diagonal idempotent is homotopic to a core projection

connectedness of the invertibles in certain nest algebras of order type ω we solve the connectedness problem for a class of nests of order type with finite dimensional atoms

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

computing the error linear complexity spectrum of a binary sequence of period 2 sup n binary sequences with high linear complexity are of interest in cryptography the linear complexity should remain high even when a small number of changes are made to the sequence the error linear complexity spectrum of a sequence reveals how the linear complexity of the sequence varies as an increasing number of the bits of the sequence are changed we present an algorithm which computes the error linear complexity for binary sequences of period spl lscr 2 sup n using o spl lscr log spl lscr sup 2 bit operations the algorithm generalizes both the games chan 1983 and stamp martin 1993 algorithms which compute the linear complexity and the k error linear complexity of a binary sequence of period spl lscr 2 sup n respectively we also discuss an application of an extension of our algorithm to decoding a class of linear subcodes of reed muller codes

stable ranks of split extensions of banach algebras in this paper we estimate the stable ranks of a banach algebra in terms of the stable ranks of its quotient algebra and ideal under the assumption that the quotient map splits as an application several results about the bass and the connected stable ranks of nest algebras are obtained

convergence of a finite volume scheme for coagulation fragmentation equations this paper is devoted to the analysis of a numerical scheme for the coagulation and fragmentation equation a time explicit finite volume scheme is developed based on a conservative formulation of the equation it is shown to converge under a stability condition on the time step while a first order rate of convergence is established and an explicit error estimate is given finally several numerical simulations are performed to investigate the gelation phenomenon and the long time behavior of the solution

an algorithm for the k error linear complexity of sequences over gf p m with period p n p a prime an algorithm is given for the k error linear complexity of sequences over gf pm with period pn p a prime the algorithm is derived by the generalized games chan algorithm for the linear complexity of sequences over gf pm with period pn and by using the modified cost different from that used in the stamp martin algorithm for sequences over gf 2 with period 2n a method is also given for computing an error vector which gives the k error linear complexity

cech cohomology and covering dimension for the h maximal ideal space it is proved that the n cech cohomology groups of the h maximal ideal space x h are trivial for n 2 and that single elements of h separate points from closed sets in x h these results are used to prove that the covering dimension of x h is 2

topological stable rank of nest algebras we establish a general result about extending a right invertible row over a banach algebra to an invertible matrix this is applied to the computation of right topological stable rank of a split exact sequence we also introduce a quantitative measure of stable rank these results are applied to compute the right left topological stable rank for all nest algebras this value is either 2 or infinity and rtsr t n 2 occurs only when n is of ordinal type less than 2 and the dimensions of the atoms grows sufficiently quickly we introduce general results on partial matrix algebras over a banach algebra this is used to obtain an inequality akin to rieffel s formula for matrix algebras over a banach algebra this is used to give further insight into the nest case

numerical simulation of the smoluchowski coagulation equation in this paper we develop a numerical scheme for the smoluchowski coagulation equation which relies on a conservative formulation and a finite volume approach several numerical simulations are performed to test the validity of the scheme and the expected behavior of the model in particular the gelation phenomenon and the long time behavior of the solution are numerically studied

connecting invertible analytic toeplitz operators in g t p abstract we prove that there exists an orthonormal basis f for classical hardy space h 2 such that each invertible analytic toeplitz operator could be connected to the identity via a norm continuous path of invertible elements of the lower triangular operators with respect to f

on the computation of the linear complexity and the k error linear complexity of binary sequences with period a power of two the linear games chan algorithm for computing the linear complexity c s of a binary sequence s of period lscr 2n requires the knowledge of the full sequence while the quadratic berlekamp massey algorithm requires knowledge of only 2c s terms we show that we can modify the games chan algorithm so that it computes the complexity in linear time knowing only 2c s terms the algorithms of stamp martin and lauder paterson can also be modified without loss of efficiency to compute analogs of the k error linear complexity for finite binary sequences viewed as initial segments of infinite sequences with period a power of two we also develop an algorithm which given a constant c and an infinite binary sequence s with period lscr 2n computes the minimum number k of errors and an associated error sequence needed over a period of s for bringing the linear complexity of s below c the algorithm has a time and space bit complexity of o lscr we apply our algorithm to decoding and encoding binary repeated root cyclic codes of length lscr in linear o lscr time and space a previous decoding algorithm proposed by lauder and paterson has o lscr loglscr 2 complexity

dimension and stable rank in the k theory of c algebras in topological k theory which can be viewed as the algebraic side of the theory of vector bundles some of the interesting properties which one investigates are for example the conditions under which bundles must possess trivial direct summands or the extent to which the cancellation property for direct whitney sums of bundles holds such properties turn out to be controlled in part by the dimension of the base space and results describing the nature of this control are among what are frequently called stability results 17 during the past few years there has been a sudden flowering of a theory for c algebras see surveys 35 36 37 since c algebras are profitably thought of as non commutative locally compact spaces with the finitely generated projective modules being the appropriate generalization of vector bundles according to swan s theorem 40 it would be natural to look for stability results for c algebras but until now the theory has been focussed on the c groups of c algebras and there has been little discussion of stability properties presumably in part for lack of an appropriate concept of dimension for c algebras one of the main objectives of this paper is to introduce a concept of dimension for c algebras which directly generalizes the classical concept of dimension for compact spaces 16 20 22 and to develop some techniques for calculating it notably for c algebras which are obtained as crossed product algebras for an action of the group of integers in algebraic k theory which is the generalization of topological c theory to rings a substantial number of stability properties have been discovered 2 39 42 48 within that theory the concept which has played a role most analogous to that of dimension is the bass stable rank but despite its many successes the bass stable rank has often been difficult to calculate even for relatively uncomplicated rings another main objective of the present paper is to make accessible for application to c algebras some of the stability results from algebraic c theory this comes about from the fact that the concept of dimension which we introduce dominates the bass stable rank i was led to seek stability results for c algebras by my desire to determine whether the cancellation property for projective modules holds for the irrational rotation c algebras discussed in 30 which as elliott 11 has suggested can appropriately be called 2 dimensional non commutative tori in more concrete terms the question is whether two projections in such an algebra which have the same trace must be unitarily equivalent the present paper provides the general theory which is needed for the affirmative answer which i obtained in a subsequent paper i will derive the

the arveson frequency response and systems theory a frequency domain theory which is applicable to all linear systems and a class of non linear systems is formulated in a hilbert resolution space setting our approach is predicated on an analytic characterization of certain non selfadjoint operator algebras due to arveson although originally used as an analytic tool arveson in fact formulated an operator valued frequency response function for the elements of these algebras which in particular include the causal operators defined on a hilbert resolution space the resultant arveson frequency response afr is well defined for all linear and non linear systems on a hilbert resolution space and in the linear case it is characterized by an analyticity theory which is identical to that of the classical time invariant frequency response concept moreover it yields computationally viable formulae for the most widely studied classes of linear systems including the time varying abcd systems finally the classical frequency response concepts and their ti

from the discrete to the continuous coagulation fragmentation equations the connection between the discrete and the continuous coagulation fragmentation models is investigated a weak stability principle relying on a priori estimates and weak compactness in l 1 is developed for the continuous model we approximate the continuous model by a sequence of discrete models and writing the discrete models as modified continuous ones we prove the convergence of the latter towards the former with the help of the above mentioned stability principle another application of this stability principle is the convergence of an explicit time and size discretization of the continuous coagulation fragmentation model

the discrete coagulation fragmentation equations existence uniqueness and density conservation the discrete coagulation fragmentation equation describes the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones these models have many applications in pure and applied science ranging from cluster formation in galaxies to the kinetics of phase transformations in binary alloys our results relate to existence uniqueness density conservation and continuous dependence and they generalise the corresponding results in ref 2 for the becker doring equations for which the processes are restricted to clusters gaining or shedding one particle examples are given which illustrate the role of the assumptions on the kinetic coefficients and show the rich set of analytic phenomena supported by the general discrete coagulation fragmentation equations

a fast algorithm for determining the complexity of a binary sequence with period 2 n corresp the complexity of a periodic sequence s is defined to be the least number of stages in a linear feedback shift register that generates s