defining functions for manifolds of matrices abstract we consider defining systems for manifolds of possibly nonsquare matrices the functions are algebraic expressions in the entries of the matrix the systems are local in the sense of being defined on the sets of a predefined cover of the manifold with open and dense sets we use bordered matrix techniques to characterize the manifold of matrices with a given rank in the case of square matrices we give conditions to express that zero is an eigenvalue with a given algebraic multiplicity and geometric multiplicity 1 in both cases the obtained systems have full linear rank hence they define the underlying manifolds in a regular way and can be used analytically to obtain local parametrizations of the manifold or numerically to set up newton systems with local quadratic convergence

structural instability analyses based on generalised path following this paper describes how quasi static conservative instability problems can be completely described using generalised path following procedures for augmented equilibrium formulations in particul

sequential testing of sorted and transformed data as an efficient way to implement long glrts it is often required to detect a long weak signal in gaussian noise and frequently the exact form of that signal is parameterized but not known a bank of matched filters provides an appropriate detector however in some practical applications there are very many matched filters and most are quite long the consequent computational needs may render the classical bank of filters approach infeasibly expensive one example and our original motivation is the detection of chirp gravitational waves by an earth based interferometer in this paper we provide a computational approach to this problem via sequential testing since the sequential tests to be used are not for constant signals we develop the theory in terms of average sample number asn for this case specifically we propose two easily calculable expressions for the asn one a bound and the other an approximation the sequential approach does yield moderate computational savings but we find that by preprocessing the data using short medium fast fourier transforms ffts and an appropriate sorting of these fft outputs such that the most informative samples are entered to a sequential test first quite high numerical efficiency can be realized the idea is simple but appears to be quite successful examples are presented in which the computational load is reduced by several orders of magnitude the fft is an example of an energy agglomerating transform but of course there are many others the point here is that the transform need not match the sought signal exactly in the sense that all energy becomes confined to a single sample it is enough that the energy becomes concentrated and the more concentrated the better

alpha trimmed means and their relationship to median filters suppose that x is a finite set of n numbers the trimmed mean of x is obtained by sorting x into ascending order removing trimming a fixed fraction alpha 0 leq alpha leq 0 5 from the high and low ends of the sorted set and computing the average of the remaining values when applied to a sliding window of length lw the trimming process is called trim filtering for 0 5 the trimmed mean of a set is the median of the set and the filtering operation is called median filtering repeated application of a median filter to the output of a previous median filter of the same length lw eventually produces a signal which is invariant to median filtering this final signal is called the root signal this paper explains the relationship between trimmed means and median filters derives a simple straightforward and fast algorithm for applying a median filter and provides a new explanation of the convergence of repeated median filtering to the root signal the latter result incorporates an approach which permits generalization of the associated concepts to a larger class of index map filters

block permutation coding of images using cosine transform we present the theory and practice of permutation coding as a new tool for very low bit rate image compression conventional source coding deals with the data information of signals while the permutation coding achieves compression through efficiently representing the positional information i e position permutation caused by ordering the data information into order statistics a set of four theorems is presented the first one reveals the information theoretic relationship between data and permutation information and the rest solves the efficient coding problem for this novel tools from finite group theory are applied to derive a compact form of representation for permutation called permutation cyclic representation pcr vectors with which various regularities and constraints in the structure of positional information are displayed whereby the coding is made very easy using a runlength and huffman method a block dct based permutation coding algorithm the bcpc is developed attempting to combine the dct s excellent features of energy packing and magnitude ordering that are found to be amenable to permutation coding this mutually beneficial characteristic significantly reduces the coding bit rate simulation results are provided for real images showing an improvement by 3 4 db in the peak snr index as compared to those representing the state of the art

information properties of order statistics and spacings we explore properties of the entropy kullback leibler information and mutual information for order statistics the probability integral transformation plays a pivotal role in developing our results we provide bounds for the entropy of order statistics and some results that relate entropy ordering of order statistics to other well known orderings of random variables we show that the discrimination information between order statistics and data distribution the discrimination information among the order statistics and the mutual information between order statistics are all distribution free and are computable using the distributions of the order statistics of the samples from the uniform distribution we also discuss information properties of spacings for uniform and exponential samples and provide a large sample distribution free result on the entropy of spacings the results show interesting symmetries of information orderings among order statistics

convergence cones near bifurcation the performance of predictor solver continuation methods near bifurcation points is considered a general theory is developed for the convergence of two commonly used solvers namely newton s method and the chord method near singular points of a nonlinear equation this theory provides a uniform treatment of continuation for simple bifurcation multiple bifurcation and multiple limit point bifurcation problems for these types of bifurcation it is shown that there are conical domains of attraction for the above iterative methods centred on solution branches which correspond to isolated roots of the algebraic bifurcation equations

breather solutions for inhomogeneous fpu models using birkhoff normal forms abstract we present results on spatially localized oscillations in some inhomogeneous nonlinear lattices of fermi pasta ulam fpu type derived from phenomenological nonlinear elastic network models proposed to study localized protein vibrations the main feature of the fpu lattices we consider is that the number of interacting neighbors varies from site to site and we see numerically that this spatial inhomogeneity leads to spatially localized normal modes in the linearized problem this property is seen in 1 d models and in a 3 d model with a geometry obtained from protein data the spectral analysis of these examples suggests some non resonance assumptions that we use to show the existence of invariant subspaces of spatially localized solutions in quartic birkhoff normal forms of the fpu systems the invariant subspaces have an additional symmetry and this fact allows us to compute periodic orbits of the quartic normal form in a relatively simple way

a geometric framework for the numerical study of singular points while bifurcation theory has developed rapidly in recent years there appears to be a need for a tighter framework for the numerical analysis of bifurcation problems this paper presents such a mathematical framework for the numerical study of the bifurcation phenomena associated with a parameter dependent equation f z lambda 0 the presentation draws from differential geometry and singularity theory and provides a basis for various numerical methods used to detect and compute certain types of bifurcation points

computation of singularities in large nonlinear systems a general approach is proposed for the numerical computation of one parameter singularities in the sense of golubitsky and schaeffer singularities and groups in bifurcation theory vol i springer verlag berlin 1985 the singularities of codimension three or less are organized in a new way our scheme is close to the numerics and in some cases gives explicit expressions for the defining equations obtained inductively where no such expressions were previously available the numerical computation by the lyapunov schmidt reduction summarizes work by many authors during the last 20 years treating all cases in a unified way a numerical example is provided where singularities with codimension up to three are detected and computed in a parameterized boundary value problem

lp estimates on functions of the laplace operator on etudie la continuite l p de certaines fonctions du laplacien sur une variete de riemann complete a geometrie bornee

numerical continuation of bifurcations of limit cycles in matlab rm cl matcont and matcont are matlab continuation packages for the interactive numerical study of a range of parameterized nonlinear dynamical systems in particular odes matcont is an interactive graphical package and rm cl matcont is a commandline version both packages allow us to compute curves of equilibria limit points hopf points limit cycles flip fold and torus bifurcation points of limit cycles we discuss computational details of the continuation of limit cycles and flip fold and torus bifurcations of limit cycles in matcont and rm cl matcont using orthogonal collocation instead of the more commonly used fully extended systems we use minimally extended systems we further describe the use of the matlab sparse matrix routines and the initialization and adaptation of the bordering vectors that are essential in minimally extended systems finally we compare the use of the minimally and the fully extended systems in the matlab environment

on the computation of multi dimensional solution manifolds of parametrized equations a new algorithm is presented for computing vertices of a simplicial triangulation of thep dimensional solution manifold of a parametrized equationf x 0 wheref is a nonlinear mapping fromr n tor m p n m 1 an essential part of the method is a constructive algorithm for computing moving frames on the manifold that is of orthonormal bases of the tangent spaces that vary smoothly with their points of contact the triangulation algorithm uses these bases together with a chord form of the gauss newton process as corrector to compute the desired vertices the jacobian matrix of the mapping is not required at all the vertices but only at the centers of certain local triangulation patches several numerical examples show that the method is very efficient in computing triangulations even around singularities such as limit points and bifurcation points this opens up new possibilities for determining the form and special features of such solution manifolds

the entropy of ordered sequences and order statistics the entropy of a sequence of random variables under order restrictions is examined a theorem that shows the amount of entropy reduction when the sequence is ordered is presented upper and lower bounds to the entropy reduction and conditions under which they are achieved are derived some interesting properties of the entropy of the individual order statistics are also presented it is shown that the difference between the average entropy of the individual order statistics and the entropy of a member of the original independent identically distributed iid population is a constant regardless of the original distribution finally the entropies of the individual order statistics are found to be symmetric about the median when the probability density function pdf of the original iid sequence is symmetric about its mean

results on residual renyi entropy of order statistics and record values this paper explores properties of the residual renyi entropy of some ordered random variables the residual renyi entropy of the kth order statistic from a continuous distribution function is represented in terms of the residual renyi entropy of the kth order statistic from uniform distribution the monotone behavior of the residual renyi entropy of order statistic under various conditions is discussed analogues results for the residual renyi entropy of record values are also given

iterative computation of the smallest singular value and the corresponding singular vectors of a matrix abstract influenced by some techniques used for computing singular points of nonlinear equations a generalized inverse iteration method is proposed for approximating the smallest singular value n and the associated left and right singular vectors u v of a matrix a r n n in the practically relevant case n 0 the method is mathematically equivalent to inverse iteration with aa t however unlike classic inverse iteration the new method works with matrices b k r n 1 n 1 obtained by bordering a in such a way that the b k have uniformly bounded condition numbers this allows using iterative krylov type solvers for large problems if n 1 n the singular vector approximations convergence linearly with factor n n 1 k obtained as a byproduct has a relative error k n n which goes to zero r linearly with factor 2 some numerical examples confirm the theoretical results and show that the algorithm works reliable also for almost singular matrices and when using krylov solvers

btc image coding using median filter roots in this paper we source encode the truncated block used in block truncation coding it is shown that the truncated block is well approximated by wide sense markoff statistics a signal having these characteristics has a high probability of belonging to the root signal set of median filters because the root signal space is much smaller than the binary space it takes fewer bits to specify the truncated block in the root signal space obtaining in this manner rate compression using two dimensional filtering we can reduce the standard btc rate of 1 63 bits pel to 1 31 bits pel using one dimensional filtering along with a trellis encoder rates close to 1 1 bits pel are obtained with this fixed length coding method

characterization results based on non additive entropy of order statistics abstract we consider a non additive generalization of the shannon entropy measure using order statistics and show that this entropy measure characterizes the distribution function uniquely further we propose a residual non additive entropy measure for order statistics and prove a characterization result for that also

improved minimal augmentation procedure for the direct computation of critical points this paper presents a new numerical procedure for the direct computation of critical points for elastic beam structures undergoing large displacements and rotations compared to the approach described by wriggers et al comput methods appl mech engrg 70 1988 329 int j numer methods engrg 30 1990 155 two main modifications are introduced first following eriksson comput methods appl mech engrg 114 1994 77 comput methods appl mech engrg 156 1998 45 comput methods appl mech engrg 179 1999 265 int j struct stability dynam 1 1 2001 the condition of criticality is expressed by a scalar equation instead of a vectorial one next the present procedure does not use exclusively the extended system obtained from the equilibrium equations and the criticality condition but also introduces intermediate iterations based purely on equilibrium equations under load or displacement control eight numerical examples presenting bifurcation and limit points are used in order to compare the performances of this new method and the one presented in comput methods appl mech engrg 70 1988 329 int j numer methods engrg 30 1990 155

accelerated projection and iterated projection methods with applications to nonlinear integral equations a new approach is developed to numerically approximate fixed points for a class of nonlinear operators by projection and iterated projection methods it is shown that isolated solutions can be determined to any desired accuracy for a fixed choice of the projection neither the structure nor the rank of the projection is altered to achieve increased accuracy instead the fixed point is decomposed by the given projection into a finite and an infinite dimensional component successive approximations to the infinite dimensional component are employed to construct a sequence of accelerated finite dimensional systems their solutions define a sequence of higher order approximations to the finite dimensional component and consequently to the fixed point error expressions are obtained and several examples of solving nonlinear integral equations are given

a theoretical analysis of the properties of median filters necessary and sufficient conditions for a signal to be invariant under a specific form of median filtering are derived these conditions state that a signal must be locally monotone to pass through a median filter unchanged it is proven that the form of successive median filtering of a signal i e the filtered output is itself again filtered eventually reduces the original signal to an invariant signal called a root signal for a signal of length l samples a maximum of frac 1 2 l 2 repeated filterings produces a root signal

some statistical properties of median filters some statistical properties of median filters are analyzed it is shown that median filters can remove impulsive plus gaussian white noise better than hanning filters when the amplitude of the impulses is large or the energy of the gaussian noise is relatively low it is also shown that unlike linear filters median filters preserve sharp changes in signals when the noise energy is not too high

numerical periodic normalization for codim 1 bifurcations of limit cycles explicit computational formulas for the coefficients of the periodic normal forms for all codim 1 bifurcations of limit cycles in generic autonomous odes are derived they include second order coefficients for the fold limit point bifurcation as well as third order coefficients for the flip period doubling and neimark sacker torus bifurcations the formulas are independent of the dimension of the phase space and involve solutions of certain boundary value problems on the interval 0 t where t is the period of the critical cycle as well as multilinear functions from the taylor expansion of the right hand sides near the cycle the formulas allow us to distinguish between sub and supercritical bifurcations in agreement with earlier asymptotic expansions of the bifurcating solutions our formulation makes it possible to use robust numerical boundary value algorithms based on orthogonal collocation rather than shooting techniques which greatly expands its applicability the actual implementation is described in detail we include three numerical examples in which codim 2 singularities are detected along branches of codim 1 bifurcations of limit cycles as zeros of the periodic normal form coefficients

on second moment properties of median filtered sequences of independent data in this paper we present the derivation of the bivariate distribution function for median filtered sequences of independent arbitrary second order random variables this result is then used to qualitatively analyze second moment properties of median filtered data the results hint towards a low pass characteristic of the median filters and a low dependency of the output spectrum on input alphabet size and distribution

a quantile approach of tsallis entropy for order statistics abstract the quantile based entropy measures possess some unique properties than its distribution function approach in this article the concept of quantile based tsallis entropy is extended for order statistics we prove that the generalized quantile information between i th order statistics and parent random variable is distribution free also the quantile version of tsallis entropy for residual lifetime has been derived and its monotonicity property is studied further some characterization results based on the tsallis quantile entropy for residual and inactivity time for the series and parallel system have been studied

a newton based method for the calculation of the distance to instability in this paper a new fast algorithm for the computation of the distance of a stable matrix to the unstable matrices is provided the method uses newton s method to find a two dimensional jordan block corresponding to a pure imaginary eigenvalue in a certain two parameter hamiltonian eigenvalue problem introduced by byers r byers a bisection method for measuring the distance of a stable matrix to the unstable matrices siam j sci statist comput 9 1988 875 881 this local method is augmented by a test step previously used by other authors to produce a global method numerical results are presented for several examples and comparison is made with the methods of boyd and balakrishnan s boyd v balakrishnan a regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its l norm systems control lett 15 1990 1 7 and he and watson c he g a watson an algorithm for computing the distance to instability siam j matrix anal appl 20 1999 101 116

state description for the root signal set of median filters median filtering is a simple digital technique for smoothing signals one main characteristic of the filter is that it maps the input signal space into a root signal space where signals invariant to median filters are called roots of the signal in this paper we develop the theory for the root signal set of median filters a tree structure for the root signal set is obtained for binary signals the number of roots r n for a signal of length n and window size filter 2s 1 is exactly represented by the difference equation r n r n 1 r n s a general solution is obtained in a z domain approach finally a method for faster one dimensional median filter operation is introduced

on the computation of manifolds of foldpoints for parameter dependent problems for parameter dependent nonlinear equations in r n a new augmentation method for computing submanifolds of foldpoints is presented the underlying theory is based on a characterization developed earlier of foldpoints on solution manifolds of parameterized equations the method allows for a locally convergent iterative computation of foldpoints of a certain class for a continuation along paths of such foldpoints and for a computation of simplicial approximations of higher dimensional submanifolds of foldpoints of the same type

heat extensions optimal atomic decompositions and sobolev embeddings in presence of symmetries on manifolds our purpose is twofold we give the heat semi group characterization of spaces of besov and triebel lizorkin types defined on riemannian manifolds of bounded geometry moreover we study boundedness and compactness of sobolev embeddings of the spaces in presence of symmetries to make it possible we first construct an optimal atomic decomposition for the above function spaces

continuation of bifurcations in periodic delay differential equations using characteristic matrices in this paper we describe a method for continuing periodic solution bifurcations in periodic delay differential equations first the notion of characteristic matrices of periodic orbits is introduced and equivalence with the monodromy operator is demonstrated an alternative formulation of the characteristic matrix is given which can be computed efficiently defining systems of bifurcations are constructed in a standard way including the characteristic matrix and its derivatives for following bifurcation curves in two parameters the pseudo arclength method is used combined with newton iteration two test examples an interrupted machining model and a traffic model with driver reaction time conclude the paper the algorithm has been implemented in the software tool pdde cont

numerical liapunov schmidt spectral methods for k determined problems we describe several generalized liapunov schmidt methods in the bordered systems formulation for operator equations and collect information concerning classification of bifurcation functions we present a short outline of spectral methods in this bifurcation context and show the convergence of the discrete defining equations and nondegeneracy conditions for spectral finite element and difference methods to the counterparts of the original operator for k determined problems this result is then applied to a model for turing patterns in a two component mixture

spectral element method in time for rapidly actuated systems in this paper the spectral element se method is applied in time to find the entire time periodic or transient solution of time dependent differential equations the time periodic solution is computed by enforcing periodicity of the element set of particular interest are periodic forcing functions possessing high frequency content to maintain the spectral accuracy for such forcing functions an h refinement scheme is employed near the semi discontinuity without increasing the number of degrees of freedom time discretization by spectral elements is applied initially to a standard form of a set of linear first order differential equations subject to harmonic excitation and an excitation admitting rapid variation other case studies include the application of the se approach to parabolic and hyperbolic partial differential equations the first order form of these equations is obtained through semi discretization using conventional finite element spectral element and finite difference schemes element clustering h refinement is applied to maintain the high accuracy and efficiency in the region of the forcing function admitting rapid variation the convergence in time of the method is demonstrated in some cases machine precision is obtained with 25 degrees of freedom per cycle finally the method is applied to a weakly nonlinear problem with time periodic solution to demonstrate its future applicability to the analysis of limit cycle oscillations in aeroelastic systems

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

a generalization of median filtering using linear combinations of order statistics we consider a class of nonlinear filters whose output is given by a linear combination of the order statistics of the input sequence assuming a constant signal in white noise the coefficients in the linear combination are chosen to minimize the output mse for several noise distributions it is shown that the optimal order statistic filter osf tends toward the median filter as the noise becomes more impulsive the optimal osf is applied to an actual noisy image and is shown to perform well combining properties of both the averaging and median filters a more general design scheme for applications involving nonconstant signals is also given

efficient numerical path following beyond critical points the paper presents a numerical pathfollowing technique based on an implicit change of parametrization using qr decomposition an automatic steplength control for tangent continuation is derived on a theoretical basis in the suggested approach turning points do not play any exceptional role for the actual computation of simple bifurcation points a special algorithm based on the augmented system of moore is developed this treatment includes imperfect bifurcations and permits an easy way of branch switching an efficient linear equation solver is worked out in connection with the qr decomposition finally numerical comparisons are included that clearly document the efficiency of the newly developed algorithms

analysis of bifurcations of limit cycles with lyapunov exponents and numerical normal forms in this paper we focus on the combination of normal form and lyapunov exponent computations in the numerical study of the three codim 2 bifurcations of limit cycles with dimension of the center manifold equal to 4 or to 5 in generic autonomous odes the normal form formulas are independent of the dimension of the phase space and involve solutions of certain linear boundary value problems the formulas allow one to distinguish between the complicated bifurcation scenarios which can happen near these codim 2 bifurcations where 3 tori and 4 tori can be present we apply our techniques to the study of a known laser model a novel model from population biology and a model of mechanical vibrations these models exhibit limit point neimark sacker period doubling neimark sacker and double neimark sacker bifurcations lyapunov exponents are computed to numerically confirm the results of the normal form analysis in particular with respect to the existence of stable invariant tori of various dimensions conversely the normal forms are essential to understand the significance of the lyapunov exponents

equilibrium subsets for multi parametric structural analysis the paper discusses multi parametric formulations for structural equilibrium problems by seeking subsets of equilibrium states which also fulfil extra conditions more detailed information on structural behaviour can be obtained these conditions can be related to analyses of e g critical behaviour imperfections or optimisation the paper shows a general problem setting and some interesting augmentations a solution method for the special case of one dimensional equilibrium subsets is described the algorithm is a generalisation of a path following algorithm where new methods are developed for the evaluation of the path tangent and for the isolation of special solution points examples are used to show the properties and the possibilities of the ideas

on solving nonlinear equations with simple singularities or nearly singular solutions the paper begins with a review of the convergence theory for newton s method near simple and other regular singularities followed by a brief discussion of the inherent difficulty of singular problems and the effect of rounding errors on the achievable solution accuracy then several ways to accelerate the slow convergence of newton s method are described in some detail these are richardson extrapolation overrelaxation bordering by a singularity condition and appending the usual linear model by a quadratic term the last two approaches require limited second derivative information which can be obtained analytically by differencing or through secant updating numerical comparisons reported in the final sections lead to the conclusion that the collection of the additional information pays off in terms of speed and accuracy

an algebraic geometric method to calculate branches near a bifurcation point abstract systems of nonlinear algebraic equations with a parameter arises in many branches of mathematics and engineering computation of the solution curve is usually hindered by the existence of a bifurcation point where the rank of the corresponding matrix is not full and usually there is another branch curve passing it we introduce a method from algebraic geometry to find the branch curve a computational example is provided

optimum quantization of order statistics we present necessary conditions for minimum mean square error mmse quantization of order statistics these conditions are obtained by observing that in a quantize sort cascade system operations of quantization and sorting are interchangeable we consider mmse quantization of a single order statistic as well as of a group of order statistics and highlight several interesting properties numerical characteristics of order statistic quantizers are presented for gaussian and uniform distributions and detailed comparisons are made

characterization and computation of singular points with maximum rank deficiency characterizations are given of points at which a mapping has a rank drop possibly greater than one multiple bifurcation points can then be computed without needing a full unfolding of the singularity nondegeneracy conditions are studied in detail

the approximate solution of defining equations for generalized turning points a set of defining equations for turning points is shown to produce a test function that can be computed iteratively to double accuracy an inexact newton method is then derived that gives a method for the efficient calculation of turning points applications are given to discretized boundary value problems using conjugate gradients

floquet theory as a computational tool we describe how classical floquet theory may be utilized in a continuation framework to construct an efficient fourier spectral algorithm for approximating periodic orbits at each continuation step only a single square matrix whose size equals the dimension of the phase space needs to be factorized the rest of the required numerical linear algebra just consists of back substitutions with this matrix the eigenvalues of this key matrix are the floquet exponents whose crossing of the imaginary axis indicates bifurcation and change in stability hence we also describe how the new periodic orbits created at a period doubling bifurcation point may be efficiently computed using our approach

an edge gradient enhancing adaptive order statistic filter we consider an adaptive order statistic os filter that can enhance gradients of edges while suppressing impulsive and some nonimpulsive noise components in particular comparison and selection cs filters are introduced in which the output is determined by comparing the sample mean and median values inside each window it is shown theoretically that cs filters can enhance gradients of a variety of edges including noisy blurred ones some root properties of cs filters are also shown finally experimental results are presented to illustrate the performance characteristics of these filters

computation of hopf bifurcation with bordered matrices hopf bifurcation for dynamical systems dot x f x lambda is characterized by rank deficiency two of a 2 omega 2 i where a f x x lambda in mathbb r n times n is the jacobian at the hopf point and pm i omega are the hopf eigenvalues replacing w 2 by nu and allowing nu leq 0 we introduce real and imaginary hopf matrices a with hopf numbers nu defining a codimension 2 manifold mathcal m h nu subset mathbb r n times n times mathbb r which can be described by two scalar equations alpha a nu 0 beta a nu 0 given by the solution of certain linear bordered systems these two scalar equations form a suitable augmentation of f x lambda 0 for the numerical computation of branches of hopf points in two parameter systems in contrast to chu govaerts and spence siam j numer anal 31 1994 pp 524 539 we suggest an a dependent bordering having certain advantages we construct a scalar test function for hopf points in the

efficient finite differences for parameter dependent singularities in two point boundary value problems we will consider the implementation of finite differences to the numerical computation of simple turning points cubic turning points simple bifurcation points and bifurcation points with corank 2 in b v ps for ordinary differential equations the singularity will be characterized by an equation or a set of equations which will be augmented with the original system to produce a regular problem block elimination will be employed for saving on the amount of work while richardson extrapolation will be used to produce accurate results numerical examples will be presented to show the efficiency of the above described algorithm

characterization and computation of generalized turning points a minimal set of augmenting equations is given so that general parameter dependent problems together with these equations will be nonsingular at points of simple rank deficiency applications to de

the approximation of generalized turn ng points by projections methods with superconvergence to the critical parameter a procedure is given that generates characterizations of singular manifolds for mildly nonlinear mappings between banach spaces this characterization is used to develop a method for determining generalized turning points by using projection methods as a discretization applications are given to parameter dependent two point boundary value problems in particular collocation at gauss points is shown to achieve superconvergence in approximating the parameter at simple turning points

the approximation of generalized turning points newline by krylov subspace methods certain types of singular solutions of nonlinear parameter dependent operator equations were characterized by griewank and reddien 5 6 as regular solutions of suitable augmented systems for their numerical approximation an approach based on the use of krylov subspaces is here presented the application to boundary value problems is illustrated by numerical examples

superlinearly convergent algorithms for solving singular equations and smooth reformulations of complementarity problems we propose a new algorithm for solving smooth nonlinear equations in the case where their solutions can be singular compared to other techniques for computing singular solutions a distinctive feature of our approach is that we do not employ second derivatives of the equation mapping in the algorithm and we do not assume their existence in the convergence analysis important examples of once but not twice differentiable equations whose solutions are inherently singular are smooth equation based reformulations of the nonlinear complementarity problems reformulations of complementarity problems serve both as illustration of and motivation for our approach and one of them we consider in detail we show that the proposed method possesses local superlinear quadratic convergence under reasonable assumptions we further demonstrate that these assumptions are in general not weaker and not stronger than regularity conditions employed in the context of other superlinearly convergent newton type algorithms for solving complementarity problems which are typically based on nonsmooth reformulations therefore our approach appears to be an interesting complement to the existing ones

the entropy of consecutive order statistics calculation of the entropy of a set of consecutive order statistics is relatively more complicated than that of the entropy of the individual order statistic which has been studied by wong and chan 1990 we provide some fundamental relations occurring in the entropy of consecutive order statistics which are very useful for computations we first consider the decomposition of the entropy of order statistics and derive some recurrence relations in the first r order statistics we also establish a dual principle for the entropy of order statistics which yields a dual relation from a given relation in the entropy of order statistics

computation of periodic solution bifurcations in odes using bordered systems we consider numerical methods for the computation and continuation of the three generic secondary periodic solution bifurcations in autonomous odes namely the fold the period doubling or flip bifurcation and the torus or neimark sacker bifurcation in the fold and flip cases we append one scalar equation to the standard periodic bvp that defines the periodic solution in the torus case four scalar equations are appended evaluation of these scalar equations and their derivatives requires the solution of linear bvps whose sparsity structure after discretization is identical to that of the linearization of the periodic bvp therefore the calculations can be done using existing numerical linear algebra techniques such as those implemented in the software auto and colsys

continuous bordering of matrices and continuous matrix decompositions let cal m n be the set of all real n times n matrices of rank ge n 1 we prove that for n ge 2 there are no continuous vector fields l r cal m n rightarrow bbb r n such that the bordered matrix is regular for all a in cal m n this result has some relevance for the numerical analysis of steady state bifurcation as a by product we show that there is no nonvanishing continuous vector field u cal m n n 1 rightarrow bbb r n with au a 0 for all a in cal m n n 1 where cal m n n 1 subset cal m n is the set of all matrices of rank deficiency one this implies that there is no singular value decomposition of a depending continuously on a in any matrix set which contains cal m n n 1 as another application we prove that in general there is no global analytic singular value decomposition for analytic matrix valued functions of more than one real variable

matrix manifolds and the jordan structure of the bialternate matrix product the bialternate product of matrices was introduced at the end of the 19th century and recently revived as a computational tool in problems where real matrices with conjugate pairs of pure imaginary eigenvalues are important i e in stability theory and hopf bifurcation problems we give a complete description of the jordan structure of the bialternate product 2a in of an n n matrix a thus extending several partial results in the literature we use these results to obtain regular local defining systems for some manifolds of matrices which occur naturally in applications in particular for manifolds with resonant conjugate pairs of pure imaginary eigenvalues such defining systems can be used analytically to obtain local parameterizations of the manifolds or numerically to set up newton systems with local quadratic convergence we give references to explicit numerical applications and implementations in software we expect that the analysis provided in this paper can be used to further improve such implementations

a neural mass model with direct and indirect excitatory feedback loops identification of bifurcations and temporal dynamics neural mass modeling is a part of computational neuroscience that was developed to study the general behavior of a neuronal population this type of mesoscopic model is able to generate output signals that are comparable to experimental data such as electroencephalograms classically neural mass models consider two interconnected populations excitatory pyramidal cells and inhibitory interneurons however many authors have included an excitatory feedback on the pyramidal cell population two distinct approaches have been developed a direct feedback on the main pyramidal cell population and an indirect feedback via a secondary pyramidal cell population in this letter we propose a new neural mass model that couples these two approaches we perform a detailed bifurcation analysis and present a glossary of dynamical behaviors and associated time series our study reveals that the model is able to generate particular realistic time series that were never pointed out in either simulated or experimental data finally we aim to evaluate the effect of balance between both excitatory feedbacks on the dynamical behavior of the model for this purpose we compute the codimension 2 bifurcation diagrams of the system to establish a map of the repartition of dynamical behaviors in a direct versus indirect feedback parameter space a perspective of this work is from a given temporal series to estimate the parameter value range especially in terms of direct versus indirect excitatory feedback

computation of cusp singularities for operator equations and their discretizations we discuss the direct calculation of cusp singularities as solutions of a minimally augmented defining system which is nonsingular under the canonical cusp conditions the underlying operator equation may have any finite fredholm index and bounds on the discretization error are derived for the case of projection methods

on uncertainty and information properties of ranked set samples ranked set sampling is a sampling design which has a wide range of applications in industrial statistics economics and environmental and ecological studies etc it is well known that ranked set samples provide more fisher information than simple random samples of the same size about the unknown parameters of the underlying distribution in parametric inferences in this paper we consider the uncertainty and information content of ranked set samples in both perfect and imperfect ranking scenarios in terms of shannon entropy renyi and kullback leibler kl information measures it is proved that under these information measures ranked set sampling design performs better than its simple random sampling counterpart of the same size the information content is also a monotone function of the set size in ranked set sampling moreover the effect of ranking error on the information content of the data is investigated

source coding with a permutation based reversible memory binding transform for data compression in categorical data domains a general purpose reversible memory binding transform mbt is developed which uses a permutation transform technique to bind memory information to a transformed signal alphabet the algorithm performs well in conjunction with a huffman coder for both ordered sources such as pixel intensities and categorical sources such as vector quantized codebook indices