spline spaces on tr meshes with hanging vertices polynomial spline spaces defined on mixed meshes consisting of triangles and rectangles are studied for the c 0 case these include triangulations with hanging vertices as well as t meshes in addition to dimension formulae explicit basis functions are constructed and their supports and stability are discussed the approximation power of the spaces is also treated

upper bounds in phase synchronous weak coherent chaotic attractors an approach is presented for coupled chaotic systems with weak coherent motion from which we estimate the upper bound value for the absolute phase difference in phase synchronous states this approach shows that synchronicity in phase implies synchronicity in the time of events a characteristic explored to derive an equation to detect phase synchronization based on the absolute difference between the time of these events we demonstrate the potential use of this approach for the phase coherent and the funnel attractor of the rossler system as well as for the spiking bursting rulkov map

dimensions of spline spaces over t meshes a t mesh is basically a rectangular grid that allows t junctions in this paper we propose a method based on bezier nets to calculate the dimension of a spline function space over a t mesh when the order of the smoothness is less than half of the degree of the spline functions a dimension formula is derived which involves only the topological quantities of the t mesh the construction of basis functions is briefly discussed furthermore the dimension formulae for t meshes after mesh operations such as edge insertion and mesh merging are also obtained

the complexity of partial derivatives abstract let l denote the nonscalar complexity in k x 1 x n we prove l x 1 x n 3l using this we determine the complexity of single power sums single elementary symmetric functions the resultant and the discriminant as root functions up to order of magnitude also we linearly reduce matrix inversion to computing the determinant

general framework for phase synchronization through localized sets we present an approach which enables one to identify phase synchronization in coupled chaotic oscillators without having to explicitly measure the phase we show that if one defines a typical event in one oscillator and then observes another one whenever this event occurs these observations give rise to a localized set our result provides a general and easy way to identify ps which can also be used to oscillators that possess multiple time scales we illustrate our approach in networks of chemically coupled neurons we show that clusters of phase synchronous neurons may emerge before the onset of phase synchronization in the whole network producing a suitable environment for information exchanging furthermore we show the relation between the localized sets and the amount of information that coupled chaotic oscillator can exchange

transmission of information in active networks shannon s capacity theorem is the main concept behind the theory of communication it says that if the amount of information contained in a signal is smaller than the channel capacity of a physical media of communication it can be transmitted with arbitrarily small probability of error this theorem is usually applicable to ideal channels of communication in which the information to be transmitted does not alter the passive characteristics of the channel that basically tries to reproduce the source of information for an active channel a network formed by elements that are dynamical systems such as neurons chaotic or periodic oscillators it is unclear if such theorem is applicable once an active channel can adapt to the input of a signal altering its capacity to shed light into this matter we show among other results how to calculate the information capacity of an active channel of communication then we show that the channel capacity depends on whether the active channel is self excitable or not and that contrary to a current belief desynchronization can provide an environment in which large amounts of information can be transmitted in a channel that is self excitable an interesting case of a self excitable active channel is a network of electrically connected hindmarsh rose chaotic neurons

algebraic complexity of computing polynomial zeros abstract using the power sum techniques of turan we evaluate all the complex zeros of an n th degree univariate polynomial with relative errors using o n 2 log n n log n log 1 arithmetic operations o n log n log 1 operations suffice to approximate with relative error a single complex zero using turan s method and all the zeros if they are real using graeffe s method in all cases n processors suffice for cn times parallel acceleration where c is a positive constant incorporating turan s techniques into another more recent algorithm gives a single complex zero with absolute error using o log 2 n log log 1 arithmetic parallel steps n 2 processors which places the problem in nc and also gives all the complex zeros in o n 2 log n log log 1 arithmetic operations where 1 is the absolutely largest zero computations with o log n 1 binary bits support the latter estimates for the arithmetic complexity which leads to a simple proof of the current best estimate for the boolean circuit complexity bit operation complexity of computing all the complex zeros of a polynomial announced by a schonhage in 1982 but not proven yet

a multilevel preconditioner for the interior penalty discontinuous galerkin method in this article we present a multilevel preconditioner for interior penalty discontinuous galerkin discretizations of second order elliptic boundary value problems that gives rise to uniformly bounded condition numbers without any additional regularity assumptions on the solution the underlying triangulations are assumed only to be shape regular but may have hanging nodes subject to certain mild grading conditions a key role is played by certain decompositions of the discontinuous trial space into a conforming subspace and a nonconforming subspace that is controlled by the jumps across the edges

constructive recognition of classical groups in even characteristic abstract let g x gl d f be a classical group in its natural representation defined over a finite field f of even characteristic we present las vegas algorithms to construct standard generators for g which permit us to write an element of g as a straight line program in x and to construct an involution as a straight line program in x if f 4 then the algorithms run in time polynomial in the size of the input subject to the existence of a discrete logarithm oracle for f

on hermite interpolation with polynomial splines on t meshes hermite interpolation using polynomial splines on t meshes is discussed in detail leading to an error bound for interpolation of smooth functions

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

information theoretic approach to quantify complete and phase synchronization of chaos based on an information theory approach we suggest a quantitative characteristic for evaluating the degree of chaotic synchronization the proposed characteristic is tested for the cases of complete and phase synchronization of chaos it is shown that this characteristic is stable with respect to the influence of small noise and nonlinear signal distortion

construction of the jordan decomposition by means of newton s method abstract newton s method is applied to construct the semi simple part of the jordan decomposition of an algebraic element in an arbitrary algebra and to derive an efficient algorithm for its computation applications on the matrix case and on differential operators are discussed

detecting phase synchronization in noisy data from coupled chaotic oscillators two schemes are proposed to detect phase synchronization from chaotic data contaminated by noise the first is a neighborhood based method which links time delay embedding with instantaneous phase estimation the second adopts the local projection method as a preprocessing filter to noisy data both schemes utilize the state recurrence an important feature of chaotic data the proposed schemes are applied to data measured from two typical chaotic systems i e the coupled rossler systems and the coupled lorenz systems respectively the results show that phase synchronization which may be buried by noise is detected even when the noise level is high moreover the overestimation of the degree of phase synchronization which may be introduced by the hilbert transform combined with a traditional linear bandpass filter can be avoided when the data are contaminated by only measurement noise

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

bases of t meshes and the refinement of hierarchical b splines abstract in this paper we consider spaces of bivariate splines of bi degree m n with maximal order of smoothness over domains associated to a two dimensional grid we define admissible classes of domains for which suitable combinatorial technique allows us to obtain the dimension of such spline spaces and the number of tensor product b splines acting effectively on these domains following the strategy introduced recently by giannelli and juttler these results enable us to prove that under certain assumptions about the configuration of a hierarchical t mesh the hierarchical b splines form a basis of bivariate splines of bi degree m n with maximal order of smoothness over this hierarchical t mesh in addition we derive a sufficient condition about the configuration of a hierarchical t mesh that ensures a weighted partition of unity property for hierarchical b splines with only positive weights

fast algorithms for the sylvester equation ax xb t c for given matrices a fm m b fn n and c fm n over an arbitrary field f the matrix equation ax xbt c has a unique solution x fm n if and only if a and b have disjoint spectra we describe an algorithm that computes the solution x for m n n with o n logn arithmetic operations in f where 2 is such that m m matrices can be multiplied with o m arithmetic operations e g 2 376 it seems that before no better bound than o m3 n3 arithmetic operations was known the state of the art in numerical analysis is o n3 m3 flops but these algorithms due to bartels stewart and golub nash van loan involve schur decompositions i e they compute the eigenvalues of at least one of a and b and can hence not be transferred for general f

bernstein bezier finite elements on tetrahedral hexahedral pyramidal partitions abstract a construction for high order continuous finite elements on partitions consisting of tetrahedra hexahedra and pyramids based on polynomial bernstein bezier shape functions is presented along with algorithms that allow the computation of the system matrices in optimal complexity o 1 per entry

competition of synchronization domains in arrays of chaotic homoclinic systems we investigate the response of an open chain of bidirectionally coupled chaotic homoclinic systems to external periodic stimuli when one end of the chain is driven by a periodic signal the system propagates a phase synchronization state in a certain range of coupling strengths and external frequencies when two simultaneous forcings are applied at different points of the array a rich phenomenology of stable competitive states is observed including temporal alternation and spatial coexistence of synchronization domains

construction of the irreducible modular representations of a finite group abstract a complete procedure is described for constructing the irreducible kg modules and their brauer characters where k is a finite field of characteristic p and g is a finite permutation or matrix group the central idea is to construct a sequence s 1 s n of kg modules each having relatively small dimension such that each s i has one or more irreducible constituents that are not constituents of s 1 s i 1 the meataxe used in conjunction with condensation is used to extract the new irreducibles from each s i the algorithm has been implemented in magma and is capable of constructing irreducibles of dimension over 200 000

non transitive maps in phase synchronization abstract concepts from ergodic theory are used to describe the existence of special non transitive maps in attractors of phase synchronous chaotic oscillators in particular it is shown that for a class of phase coherent oscillators these special maps imply phase synchronization we illustrate these ideas in the sinusoidally forced chua s circuit and two coupled rossler oscillators furthermore these results are extended to other coupled chaotic systems in addition a phase for a chaotic attractor is defined from the tangent vector of the flow finally it is discussed how these maps can be used for the real time detection of phase synchronization in experimental systems

adaptive isogeometric analysis by local h refinement with t splines isogeometric analysis based on non uniform rational b splines nurbs as basis functions preserves the exact geometry but suffers from the drawback of a rectangular grid of control points in the parameter space which renders a purely local refinement impossible this paper demonstrates how this difficulty can be overcome by using t splines instead t splines allow the introduction of so called t junctions which are related to hanging nodes in the standard fem obeying a few straightforward rules rectangular patches in the parameter space of the t splines can be subdivided and thus a local refinement becomes feasible while still preserving the exact geometry furthermore it is shown how state of the art a posteriori error estimation techniques can be combined with refinement by t splines numerical examples underline the potential of isogeometric analysis with t splines and give hints for further developments

phase synchronization in the perturbed chua circuit we show experimental and numerical results of phase synchronization between the chaotic chua circuit and a small sinusoidal perturbation experimental real time phase synchronized states can be detected with oscilloscope visualization of the attractor using specific sampling rates arnold tongues demonstrate robust phase synchronized states for perturbation frequencies close to the characteristic frequency of the unperturbed chua

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

efficient computation of the characteristic polynomial of a threshold graph an efficient algorithm is presented to compute the characteristic polynomial of a threshold graph threshold graphs were introduced by chvatal and hammer as well as by henderson and zalcstein in 1977 a threshold graph is obtained from a one vertex graph by repeatedly adding either an isolated vertex or a dominating vertex which is a vertex adjacent to all the other vertices threshold graphs are special kinds of cographs which themselves are special kinds of graphs of clique width 2 we obtain a running time of o n log 2 n for computing the characteristic polynomial while the previously fastest algorithm ran in quadratic time we improve the running time drastically in the case where there is a small number of alternations between 0 s and 1 s in the sequence defining a threshold graph a simple recurrence equation for the determinant of a weighted threshold graph matrix is presented the characteristic polynomial is computed in o n log 2 n arithmetic operations the algorithm is more efficient if the alternations in the defining sequence are o n as the numbers may be of length n log n the bit complexity is investigated too

dynamic matrix rank we consider maintaining information about the rank of a matrix under changes of the entries for nxn matrices we show an upper bound of o n 1 5 7 5 arithmetic operations and a lower bound of w n arithmetic operations per element change the upper bound is valid when changing up to o n 0 5 7 5 entries in a single column of the matrix we also give an algorithm that maintains the rank using o n 2 arithmetic operations per rank one update these bounds appear to be the first nontrivial bounds for the problem the upper bounds are valid for arbitrary fields whereas the lower bound is valid for algebraically closed fields the upper bound for element updates uses fast rectangular matrix multiplication and the lower bound involves further development of an earlier technique for proving lower bounds for dynamic computation of rational functions