direct minimization of the least squares spectral element functional part i direct solver this paper describes an equivalent but improved least squares formulation for the numerical approximation of the solution of partial differential equations instead of using variational analysis to impose the conditions for minimizing the residual the residuals are minimized directly thus leading to a method we will denote by direct minimization dm dm circumvents setting up the normal equations which consists of matrix matrix multiplications matrix matrix multiplications are expensive may lead to loss of accuracy and destroy the sparsity pattern present in the original system the condition number of the dm formulation is the square root of the condition number which would be obtained if variational analysis was employed an element by element procedure will be presented which allows for parallelization of dm a computational comparison between dm and the conventional least squares formulation based on variational analysis will be presented

information processing in complex networks graph entropy and information functionals this paper introduces a general framework for defining the entropy of a graph our definition is based on a local information graph and on information functionals derived from the topological structure of a given graph more precisely an information functional quantifies structural information of a graph based on a derived probability distribution such a probability distribution leads directly to an entropy of a graph then the structural information content of a graph will be is interpreted and defined as the derived graph entropy another major contribution of this paper is the investigation of relationships between graph entropies in addition to this we provide numerical results demonstrating not only the feasibility of our method which has polynomial time complexity but also its usefulness with regard to practical applications aiming to an understanding of information processing in complex networks

graph characterization via ihara coefficients the novel contributions of this paper are twofold first we demonstrate how to characterize unweighted graphs in a permutation invariant manner using the polynomial coefficients from the ihara zeta function i e the ihara coefficients second we generalize the definition of the ihara coefficients to edge weighted graphs for an unweighted graph the ihara zeta function is the reciprocal of a quasi characteristic polynomial of the adjacency matrix of the associated oriented line graph since the ihara zeta function has poles that give rise to infinities the most convenient numerically stable representation is to work with the coefficients of the quasi characteristic polynomial moreover the polynomial coefficients are invariant to vertex order permutations and also convey information concerning the cycle structure of the graph to generalize the representation to edge weighted graphs we make use of the reduced bartholdi zeta function we prove that the computation of the ihara coefficients for unweighted graphs is a special case of our proposed method for unit edge weights we also present a spectral analysis of the ihara coefficients and indicate their advantages over other graph spectral methods we apply the proposed graph characterization method to capturing graph class structure and clustering graphs experimental results reveal that the ihara coefficients are more effective than methods based on laplacian spectra

diffusion wavelet embedding a multi resolution approach for graph embedding in vector space abstract in this article we propose a multiscale method of embedding a graph into a vector space using diffusion wavelets at each scale we extract a detail subspace and a corresponding lower scale approximation subspace to represent the graph representative features are then extracted at each scale to provide a scale space description of the graph the lower scale is constructed using a super node merging strategy based on nearest neighbor or maximum participation and the new adjacency matrix is generated using vertex identification this approach allows the comparison of graphs where the important structural differences may be present at varying scales additionally this method can improve the differentiating power of the embedded vectors and this property reduces the possibility of cospectrality typical in spectral methods substantially the experimental results show that augmenting the features of abstract levels to the graph features increases the graph classification accuracies in different datasets

pseudospectral solution of the two dimensional navier stokes equations in a disk an efficient and accurate algorithm for solving the two dimensional 2d incompressible navier stokes equations on a disk with no slip boundary conditions is described the vorticity stream function formulation of these equations is used and spatially the vorticity and stream functions are expressed as fourier chebyshev expansions the poisson and helmholtz equations which arise from the implicit explicit time marching scheme are solved as banded systems using a postconditioned spectral tau method the polar coordinate singularity is handled by expanding fields radially over the entire diameter using a parity modified chebyshev series and building partial regularity into the vorticity the no slip boundary condition is enforced by transferring one of the two boundary conditions imposed on the stream function onto the vorticity via a solvability constraint significant gains in run times were realized by parallelizing the code in message passage interface mpi

a fast iterative solver for the variable coefficient diffusion equation on a disk we present an efficient iterative method for solving the variable coefficient diffusion equation on a unit disk the equation is written in polar coordinates and is discretized by the standard centered difference approximation under the grid arrangement by shifting half radial mesh away from the origin so that the coordinate singularity can be handled naturally without pole conditions the resultant matrix is symmetric positive definite so the preconditioned conjugate gradient pcg method can be applied different preconditioners have been tested for comparison in particular a fast direct solver derived from the equation and the semi coarsening multigrid are shown to be almost scalable with the problem size and outperform other preconditioners significantly the present elliptic solver has been applied to study the vortex dynamics of the ginzburg landau equation with a variable diffusion coefficient

hierarchical universal coding in an earlier paper we proved a strong version of the redundancy capacity converse theorem of universal coding stating that for most sources in a given class the universal coding redundancy is essentially lower bounded by the capacity of the channel induced by this class since this result holds for general classes of sources it extends rissanen s 1986 strong converse theorem for parametric families while our earlier result has established strong optimality only for mixture codes weighted by the capacity achieving prior our first result herein extends this finding to a general prior for some cases our technique also leads to a simplified proof of the above mentioned strong converse theorem the major interest in this paper however is in extending the theory of universal coding to hierarchical structures of classes where each class may have a different capacity in this setting one wishes to incur redundancy essentially as small as that corresponding to the active class and not the union of classes our main result is that the redundancy of a code based on a two stage mixture first within each class and then over the classes is no worse than that of any other code for most sources of most classes if in addition the classes can be efficiently distinguished by a certain decision rule then the best attainable redundancy is given explicitly by the capacity of the active class plus the normalized negative logarithm of the prior probability assigned to this class these results suggest some interesting guidelines as for the choice of the prior we also discuss some examples with a natural hierarchical partition into classes

universal coding of nonstationary sources we investigate the performance of the lempel ziv 1978 incremental parsing scheme on nonstationary sources we show that it achieves the best rate achievable by a finite state block coder for the nonstationary source we also show a similar result for a lossy coding scheme given by yang and kieffer see ibid vol 42 p 239 45 1996 which uses a lempel ziv scheme to perform lossy coding

diffusion kernels on statistical manifolds a family of kernels for statistical learning is introduced that exploits the geometric structure of statistical models the kernels are based on the heat equation on the riemannian manifold defined by the fisher information metric associated with a statistical family and generalize the gaussian kernel of euclidean space as an important special case kernels based on the geometry of multinomial families are derived leading to kernel based learning algorithms that apply naturally to discrete data bounds on covering numbers and rademacher averages for the kernels are proved using bounds on the eigenvalues of the laplacian on riemannian manifolds experimental results are presented for document classification for which the use of multinomial geometry is natural and well motivated and improvements are obtained over the standard use of gaussian or linear kernels which have been the standard for text classification

mass and momentum conservation of the least squares spectral collocation method for the navier stokes equations from the literature it is known that the least squares spectral element method lssem for the stationary stokes equations performs poorly with respect to mass conservation but compensates this lack by a superior conservation of momentum furthermore it is known that the least squares spectral collocation method lsscm leads to superior conservation of mass and momentum for the stationary stokes equations in the present paper we consider mass and momentum conservation of the lsscm for time dependent stokes and navier stokes equations we observe that the lsscm leads to improved conservation of mass and momentum for these problems furthermore the lsscm leads to the well known time dependent profiles for the velocity and the pressure profiles to obtain these results we use only a few elements each with high polynomial degree avoid normal equations for solving the overdetermined linear systems of equations and introduce the clenshaw curtis quadrature rule for imposing the average pressure to be zero furthermore we combined the transformation of gordon and hall transfinite mapping with the least squares spectral collocation scheme to discretize the internal flow problems

depth based complexity traces of graphs in this paper we aim to characterize graphs in terms of a structural measure of complexity our idea is to decompose a graph into layered substructures of increasing size and then to measure the information content of these substructures to locate dominant substructures within a graph we commence by identifying a centroid vertex which has the minimum shortest path length variance to the remaining vertices for each graph a family of centroid expansion subgraphs is derived from the centroid vertex in order to capture dominant structural characteristics of the graph since the centroid vertex is identified through a global analysis of the shortest path length distribution the expansion subgraphs provide a fine representation of a graph structure we then show how to characterize graphs using depth based complexity traces here we explore two different strategies the first strategy is to measure how the entropies on the centroid expansion subgraphs vary with the increasing size of the subgraphs the second strategy is to measure how the entropy differences vary with the increasing size of the subgraphs we perform graph classification in the principal component space of the complexity trace vectors experiments on graph datasets abstracted from some bioinformatics and computer vision databases demonstrate the effectiveness and efficiency of the proposed graph complexity traces our methods are competitive to state of the art methods highlightswe identify a centroid vertex of a graph through its shortest path matrix we establish a family of expansion subgraphs of a graph from its centroid vertex we construct depth based complexity traces of a graph on its expansion subgraphs we evaluate the performance of our complexity traces on standard graph datasets we demonstrate the effectiveness and efficiency of the graph complexity traces

thermodynamic depth of causal states objective complexity via minimal representations thermodynamic depth is an appealing but flawed structural complexity measure it depends on a set of macroscopic states for a system but neither its original introduction by lloyd and pagels nor any follow up work has considered how to select these states depth therefore is at root arbitrary computational mechanics an alternative approach to structural complexity provides a definition for a system close quote s minimal necessary causal states and a procedure for finding them we show that the rate of increase in thermodynamic depth or ital dive is the system close quote s reverse time shannon entropy rate and so depth only measures degrees of macroscopic randomness not structure to fix this we redefine the depth in terms of the causal state representation emdash epsilon machines emdash and show that this representation gives the minimum dive consistent with accurate prediction thus epsilon machines are optimally shallow copyright ital 1999 ital the american physical society

a fourier legendre spectral element method in polar coordinates in this paper a new fourier legendre spectral element method based on the galerkin formulation is proposed to solve the poisson type equations in polar coordinates the 1 r singularity at r 0 is avoided by using gauss radau type quadrature points in order to break the time step restriction in the time dependent problems the clustering of collocation points near the pole is prevented through the technique of domain decomposition in the radial direction a number of poisson type equations subject to the dirichlet or neumann boundary condition are computed and compared with the results in literature which reveals a desirable result

splitting techniques for the pseudospectral approximation of the unsteady stokes equations a pseudospectral or collocation approximation of the unsteady stokes equations is considered using the uzawa algorithm the spectral system is decoupled into a helmholtz equation and an equation with a pseudo laplace operator it is proven that the eigenvalues of the pseudo laplacian are real and negative except one zero eigenvalue belonging to the constant mode the pseudospectral method avoids spurious modes since the pressure is approximated with lower order two degrees lower polynomials than the velocity only one grid no staggered grids with the standard chebyshev gauss lobatto nodes is used for discretization for the iterative solution of the spectral system an effective preconditioner is necessary here effective finite difference preconditioners for the spectral pseudo laplacian are presented

splitting techniques with staggered grids for the navier stokes equations in the 2d case abstract a pseudo spectral approximation for the navier stokes equations in the 2d case is presented using a new splitting technique based on the uzawa algorithm the system is decoupled into helmholtz equations for the velocity and an equation with the pseudo laplacian for the pressure staggered grids with gauss and gauss lobatto nodes are employed preconditioning with finite differences is considered by extrapolation a stable second order method in time for the velocity and at least a first order method for the pressure can be achieved

improving vector space embedding of graphs through feature selection algorithms graph based pattern representation offers a versatile alternative to vectorial data structures therefore a growing interest in graphs can be observed in various fields however a serious limitation in the use of graphs is the lack of elementary mathematical operations in the graph domain actually required in many pattern recognition algorithms in order to overcome this limitation the present paper proposes an embedding of a given graph population in a vector space r n the key idea of this embedding approach is to interpret the distances of a graph g to a number of prototype graphs as numerical features of g in previous works the prototypes were selected beforehand with heuristic selection algorithms in the present paper we take a more fundamental approach and regard the problem of prototype selection as a feature selection or dimensionality reduction problem for which many methods are available with several experiments we show the feasibility of graph embedding based on prototypes obtained from such feature selection algorithms and demonstrate their potential to outperform previous approaches

spectral radial basis functions for full sphere computations the singularity of cylindrical or spherical coordinate systems at the origin imposes certain regularity conditions on the spectral expansion of any infinitely differentiable function there are two efficient choices of a set of radial basis functions suitable for discretising the solution of a partial differential equation posed in either such geometry one choice is methods based on standard chebyshev polynomials although these may be efficiently computed using fast transforms differentiability to all orders of the obtained solution at the origin is not guaranteed the second is the so called one sided jacobi polynomials that explicitly satisfy the required behavioural conditions in this paper we compare these two approaches in their accuracy differentiability and computational speed we find that the most accurate and concise representation is in terms of one sided jacobi polynomials however due to the lack of a competitive fast transform chebyshev methods may be a better choice for some computationally intensive timestepping problems and indeed will yield sufficiently although not infinitely differentiable solutions provided they are adequately converged

nonextensive information theoretic kernels on measures positive definite kernels on probability measures have been recently applied to classification problems involving text images and other types of structured data some of these kernels are related to classic information theoretic quantities such as shannon s mutual information and the jensen shannon js divergence meanwhile there have been recent advances in nonextensive generalizations of shannon s information theory this paper bridges these two trends by introducing nonextensive information theoretic kernels on probability measures based on new js type divergences these new divergences result from extending the the two building blocks of the classical js divergence convexity and shannon s entropy the notion of convexity is extended to the wider concept of q convexity for which we prove a jensen q inequality based on this inequality we introduce jensen tsallis jt q differences a nonextensive generalization of the js divergence and define a k th order jt q difference between stochastic processes we then define a new family of nonextensive mutual information kernels which allow weights to be assigned to their arguments and which includes the boolean js and linear kernels as particular cases nonextensive string kernels are also defined that generalize the p spectrum kernel we illustrate the performance of these kernels on text categorization tasks in which documents are modeled both as bags of words and as sequences of characters

weisfeiler lehman graph kernels in this article we propose a family of efficient kernels for large graphs with discrete node labels key to our method is a rapid feature extraction scheme based on the weisfeiler lehman test of isomorphism on graphs it maps the original graph to a sequence of graphs whose node attributes capture topological and label information a family of kernels can be defined based on this weisfeiler lehman sequence of graphs including a highly efficient kernel comparing subtree like patterns its runtime scales only linearly in the number of edges of the graphs and the length of the weisfeiler lehman graph sequence in our experimental evaluation our kernels outperform state of the art graph kernels on several graph classification benchmark data sets in terms of accuracy and runtime our kernels open the door to large scale applications of graph kernels in various disciplines such as computational biology and social network analysis

a fast solver for elliptic boundary value problems in the square abstract we approximate the solution of advection diffusion equations by collocation at a special grid related to the differential operator and the classical legendre grid

a spectral method for polar coordinates a new set of polynomial functions that can be used in spectral expansions of c functions in polar coordinates r is defined by a singular sturm liouville equation with the use of the basis functions the spectral representations remain analytic at the pole despite the coordinate singularity because the pole condition is exactly satisfied at the origin for ag azimuthal modes not just a few of the gravest modes which is the usual case based on recurrence relations fast and stable numerical operators for 1 r r d dr the laplacian and helmholtz operators and their inverses are developed although the spacings in the azimuthal direction of the collocation points near the origin are small i e 1 m2 where m is the number of radial modes the explicit numerical method for euler s equation is not stiff at the origin namely the cfl number is o 1 where the grid size in is defined as m i e the maximum allowable timestep is proportional to 1 m not 1 m2

spectral collocation on triangular elements we consider the poisson problem on a segment of the unit disc and on triangles on the segment we transform the poisson problem by means of polar coordinates in these new coordinates we have a problem in a rectangle which can easily be mapped onto the square here standard chebyshev collocation techniques can be applied then the segment is mapped onto a triangle where the same spectral scheme may be used by numerical tests we observed the expected high spectral accuracy due to the corner singularity a singular behaviour of the solution can be expected here we improved the accuracy by auxiliary mapping techniques further it is shown that finite difference preconditioning can be successfully applied in order to construct an efficient iterative solver finally a domain decomposition technique is applied to the patching of a rectangular and a triangular element

redundancy of the lempel ziv incremental parsing rule the lempel ziv codes are universal variable to fixed length codes that have become virtually standard in practical lossless data compression for any given source output string from a markov or unifilar source we upper bound the difference between the number of binary digits needed to encode the string and the self information of the string we use this result to demonstrate that for unifilar or markov sources the redundancy of encoding the first n letters of the source output with the lempel ziv incremental parsing rule lz 78 the welch modification lzw or a new variant is o ln n sup 1 and we upper bound the exact form of convergence we conclude by considering the relationship between the code length and the empirical entropy associated with a string

a direct spectral collocation poisson solver in polar and cylindrical coordinates in this paper we present a direct spectral collocation method for the solution of the poisson equation in polar and cylindrical coordinates the solver is applied to the poisson equations for several different domains including a part of a disk an annulus a unit disk and a cylinder unlike other poisson solvers for geometries such as unit disks and cylinders no pole condition is involved for the present solver the method is easy to implement fast and gives spectral accuracy we also use the weighted interpolation technique and nonclassical collocation points to improve the convergence

methodes spectrales dans une portion de disque on propose une methode spectrale pour la discretisation d equations elliptiques dans une portion de disque par passage en coordonnees polaires et approximation par des polynomes de degreeleve par rapport a chaque coordonnee on decrit la discretisation pour deux problemes modele l equation de poisson et l equation du bilaplacien on en effectue l analyse et on presente des resultats numeriques

the context tree weighting method extensions first we modify the basic binary context tree weighting method such that the past symbols x sub 1 d x sub 2 d x sub 0 are not needed by the encoder and the decoder then we describe how to make the context tree depth d infinite which results in optimal redundancy behavior for all tree sources while the number of records in the context tree is not larger than 2t 1 here t is the length of the source sequence for this extended context tree weighting algorithm we show that with probability one the compression ratio is not larger than the source entropy for source sequence length t spl rarr spl infin for stationary and ergodic sources

spectral collocation methods and polar coordinate singularities this paper considers the numerical solution of elliptic differential equations on the unit disk using polar coordinates the disk is mapped onto a rectangle the resulting transformed problem is solved by a method related to collocation since the origin is a coordinate singularity some natural trial functions are singular there and a special technique is applied to use zero as a collocation point for poisson and helmholtz equations a fast algorithm with an operation count of o n2 log n is presented numerical results show the different stability and convergence properties of the algorithms

radial collocation methods for the onset of convection in rotating spheres the viability of using collocation methods in radius and spherical harmonics in the angular variables to calculate convective flows in full spherical geometry is examined as a test problem the stability of the conductive state of a self gravitating fluid sphere subject to rotation and internal heating is considered a study of the behavior of different radial meshes previously used by several authors in polar coordinates including or not the origin is first performed the presence of spurious modes due to the treatment of the singularity at the origin to the spherical harmonics truncation and to the initialization of the eigenvalue solver is shown and ways to eliminate them are presented finally to show the usefulness of the method the neutral stability curves at very high taylor and moderate and small prandtl numbers are calculated and shown several radial collocation meshes have been tested to solve convective flows in full spherical geometry the inclusion of the origin in the mesh together with a single regularity condition is enough to obtain good results for eigenvalue problems several sources of spurious modes have been detected and methods to eliminate them are proposed the onset of convection in an internally heated sphere has been studied and asymptotic laws for high taylor and moderate prandtl numbers have been found

efficient spectral galerkin methods iii polar and cylindrical geometries we present in this paper several extremely efficient and accurate spectral galerkin methods for second and fourth order equations in polar and cylindrical geometries these methods are based on appropriate variational formulations which incorporate naturally the pole condition s in particular the computational complexities of the chebyshev galerkin method in a disk and the chebyshev legendre galerkin method in a disk or a cylinder are quasi optimal optimal up to a logarithmic term as an indication of efficiency the cpu time for the poisson solver on a disk by our chebyshev galerkin method is only about 70 of the corresponding finite difference code in fishpack

depth based hypergraph complexity traces from directed line graphs in this paper we aim to characterize the structure of hypergraphs in terms of structural complexity measure measuring the complexity of a hypergraph in a straightforward way tends to be elusive since the hyperedges of a hypergraph may exhibit varying relational orders we thus transform a hypergraph into a line graph which not only accurately reflects the multiple relationships exhibited by the hyperedges but is also easier to manipulate for complexity analysis to locate the dominant substructure within a line graph we identify a centroid vertex by computing the minimum variance of its shortest path lengths a family of centroid expansion subgraphs of the line graph is then derived from the centroid vertex we compute the depth based complexity traces for the hypergraph by measuring either the directed or undirected entropies of its centroid expansion subgraphs the resulting complexity traces provide a flexible framework that can be applied to both hypergraphs and graphs we perform hyper graph classification in the principal component space of the complexity trace vectors experiments on hyper graph datasets abstracted from bioinformatics and computer vision data demonstrate the effectiveness and efficiency of the complexity traces highlightswe aim to characterize hypergraphs in terms of structural complexity measures straightforwardly measuring the complexity of a hypergraph tends to be elusive we transform a hypergraph into a line graph for measuring the complexities we compute depth based complexity traces for the hypergraph on line graphs the complexity traces provide a flexible framework for hypergraphs and graphs

spectral schemes on triangular elements abstract the poisson problem with homogeneous dirichlet boundary conditions is considered on a triangle the transformation from square to triangle is realized by mapping an edge of the square onto a corner of the triangle then standard chebyshev collocation techniques can be implemented numerical experiments demonstrate the expected high spectral accuracy for smooth solutions furthermore it is shown that finite difference preconditioning can be successfully employed to construct an efficient iterative solver then the convection diffusion equation is considered here finite difference preconditioning with central differences leads to instability however using the first order upstream scheme we obtain a stable method finally a domain decomposition technique is applied to the patching of rectangular and triangular elements

finite state adaptive block to variable length noiseless coding of a nonstationary information source the author associates with a given nonstationary finite alphabet information source a certain class of stationary processes that he terms the stationary hull of the given source he shows that the optimum average rate at which the given source can be noiselessly coded by means of a finite state adaptive block to variable length coding schemes is the largest entropy rate among those processes in the stationary hull he explains what he means by an adaptive block to variable length coding scheme

conservation of mass and momentum of the least squares spectral collocation scheme for the stokes problem from the literature it is known that spectral least squares schemes perform poorly with respect to mass conservation and compensate this lack by a superior conservation of momentum this should be revised since the here presented new least squares spectral collocation scheme leads to an outstanding performance with respect to conservation of momentum and mass the reasons can be found in using only a few elements each with high polynomial degree avoiding normal equations for solving the overdetermined linear systems of equations and by introducing the clenshaw curtis quadrature rule for imposing the average pressure to be zero furthermore we combined the transformation of gordon and hall transfinite mapping with our least squares spectral collocation scheme to discretize the internal flow problems

splitting techniques for the unsteady stokes equations a pseudospectral or collocation approximation of the unsteady stokes equations is presented using the uzawa algorithm the spectral system is decoupled into helmholtz equations for the velocity components and an equation with the pseudo laplacian for the pressure both the velocity components and the pressure are approximated by polynomials of the same degree n compatible velocity and pressure approximations are constructed to ensure that the discrete problem is well posed the resulting discrete velocity field is identically divergence free only one grid no staggered grids with the standard chebyshev gauss lobatto nodes is used here we further compare our treatment with an inhomogeneous neumann condition for the pressure this scheme was proposed in g karniadakis m israeli and s a orszag j comp phys 97 1991 pp 414 443 the comparison shows that the neumann condition yields somewhat better approximations for the pressure in the time discretization a high order backward differentiation scheme for the intermediate velocity is combined with a high order extrapolant for the pressure it is numerically shown that a stable third order method in time can be achieved in the case of periodic boundary conditions this result is confirmed by fourier analysis

fast depth based subgraph kernels for unattributed graphs in this paper we investigate two fast subgraph kernels based on a depth based representation of graph structure both methods gauge depth information through a family of k layer expansion subgraphs rooted at a vertex 1 the first method commences by computing a centroid based complexity trace for each graph using a depth based representation rooted at the centroid vertex that has minimum shortest path length variance to the remaining vertices 2 this subgraph kernel is computed by measuring the jensen shannon divergence between centroid based complexity entropy traces the second method on the other hand computes a depth based representation around each vertex in turn the corresponding subgraph kernel is computed using isomorphisms tests to compare the depth based representation rooted at each vertex in turn for graphs with n vertices the time complexities for the two new kernels are o n 2 and o n 3 in contrast to o n 6 for the classic gartner graph kernel 3 key to achieving this efficiency is that we compute the required shannon entropy of the random walk for our kernels with o n 2 operations this computational strategy enables our subgraph kernels to easily scale up to graphs of reasonably large sizes and thus overcome the size limits arising in state of the art graph kernels experiments on standard bioinformatics and computer vision graph datasets demonstrate the effectiveness and efficiency of our new subgraph kernels highlightswe investigate two fast subgraph kernels based on a depth based representation our kernels gauge depth information rooted at a vertex for a graph the time complexities for the two kernels are o n 2 and o n 3 we evaluate the performance of our subgraph kernels on standard graph datasets we demonstrate the effectiveness of the proposed subgraph kernels

a 3d pseudospectral method for cylindrical coordinates application to the simulations of rotating cavity flows the present work proposes a collocation spectral method for solving the three dimensional navier stokes equations using cylindrical coordinates the whole diameter r

simple universal lossy data compression schemes derived from the lempel ziv algorithm two universal lossy data compression schemes one with fixed rate and the other with fixed distortion are presented based on the well known lempel ziv algorithm in the case of fixed rate r the universal lossy data compression scheme works as follows first pick a codebook b sub n consisting of all reproduction sequences of length n whose lempel ziv codeword length is spl les nr and then use b sub n to encode the entire source sequence n block by n block this fixed rate data compression scheme is universal in the sense that for any stationary ergodic source or for any individual sequence the sample distortion performance as n spl rarr spl infin is given almost surely by the distortion rate function a similar result is shown in the context of fixed distortion lossy source coding

pattern vectors from algebraic graph theory graph structures have proven computationally cumbersome for pattern analysis the reason for this is that before graphs can be converted to pattern vectors correspondences must be established between the nodes of structures which are potentially of different size to overcome this problem in this paper we turn to the spectral decomposition of the laplacian matrix we show how the elements of the spectral matrix for the laplacian can be used to construct symmetric polynomials that are permutation invariants the coefficients of these polynomials can be used as graph features which can be encoded in a vectorial manner we extend this representation to graphs in which there are unary attributes on the nodes and binary attributes on the edges by using the spectral decomposition of a hermitian property matrix that can be viewed as a complex analogue of the laplacian to embed the graphs in a pattern space we explore whether the vectors of invariants can be embedded in a low dimensional space using a number of alternative strategies including principal components analysis pca multidimensional scaling mds and locality preserving projection lpp experimentally we demonstrate that the embeddings result in well defined graph clusters our experiments with the spectral representation involve both synthetic and real world data the experiments with synthetic data demonstrate that the distances between spectral feature vectors can be used to discriminate between graphs on the basis of their structure the real world experiments show that the method can be used to locate clusters of graphs

a general formula for channel capacity a formula for the capacity of arbitrary single user channels without feedback not necessarily information stable stationary etc is proved capacity is shown to equal the supremum over all input processes of the input output inf information rate defined as the liminf in probability of the normalized information density the key to this result is a new converse approach based on a simple new lower bound on the error probability of m ary hypothesis tests among equiprobable hypotheses a necessary and sufficient condition for the validity of the strong converse is given as well as general expressions for spl epsiv capacity

a history of graph entropy measures this survey seeks to describe methods for measuring the entropy of graphs and to demonstrate the wide applicability of entropy measures setting the scene with a review of classical measures for determining the structural information content of graphs we discuss graph entropy measures which play an important role in a variety of problem areas including biology chemistry and sociology in addition we examine relationships between selected entropy measures illustrating differences quantitatively with concrete examples

pole condition for singular problems the pseudospectral approximation abstract this paper deals with the pseudospectral solution of differential equations with coordinate singularities such as those which describe situations in spherical or cylindrical geometries we use the differential equation together with a smoothness assumption on the solution to construct pole conditions the pole conditions which are straightforward and easily implemented serve as numerical boundary conditions at the coordinate singularity standard pseudospectral methods including fast transformation techniques can then be applied to the singular problem the method is illustrated using the eigenvalue problem of bessel s equation and a poisson equation on the unit disk numerical results show that spectral convergence is achieved

shannon and von neumann entropy of random networks with heterogeneous expected degree entropic measures of complexity are able to quantify the information encoded in complex network structures several entropic measures have been proposed in this respect here we study the relation between the shannon entropy and the von neumann entropy of networks with given expected degree sequence we find in different examples of network topologies that when the degree distribution contains some heterogeneity an intriguing correlation emerges between the two entropic quantities this results seems to suggest that heterogeneity in the expected degree distribution is implying an equivalence between a quantum and a classical description of networks which respectively corresponds to the von neumann and the shannon entropy

fixed rate encoding of nonstationary information sources an expression is obtained for the optimum distortion theoretically attainable when an information source with finite alphabet is encoded at a fixed rate with respect to a single letter fidelity criterion the expression is demonstrated by means of an appropriate coding theorem and converse this new result generalizes the coding theorem of shannon for stationary ergodic sources the coding theorem of gray davisson for stationary nonergodic sources that of gray saadat for asymptotically mean stationary sources and that of ziv for an individual sequence

a pseudospectral approach for polar and spherical geometries the singularities that arise in polar and spherical coordinates have in the past caused significant difficulties in obtaining accurate solutions to convection diffusion type problems in many fields astrophysics geophysics meteorology etc viewing pseudospectral methods as a limiting case of finite difference methods rather than as based on expansions in terms of orthogonal functions leads naturally to very simple yet highly effective fast fourier transform fft based pseudospectral methods in such geometries

coding for a binary independent piecewise identically distributed source two weighting procedures are presented for compaction of output sequences generated by binary independent sources whose unknown parameter may occasionally change the resulting codes need no knowledge of the sequence length t i e they are strongly sequential and also the number of parameter changes is unrestricted the additional transition redundancy of the first method was shown to achieve the merhav lower bound i e log t bits per transition for the second method we could prove that additional transition redundancy is not more than 3 2 log t bits per transition which is more than the merhav bound however the storage and computational complexity of this method are also more interesting than those of the first method simulations show that the difference in redundancy performance between the two methods is negligible

a sparse mesh for compact finite difference fourier solvers with radius dependent spectral resolution in circular domains this paper presents a new method for the resolution of elliptic and parabolic equations in circular domains it can be trivially extended to cylindrical domains the algorithm uses a mixed fourier compact finite difference method the main advantage of the method is achieved by a new concept of mesh the topology of the new grid keeps constant the aspect ratio of the cells avoiding the typical clustering for radial structured meshes at the center the reduction of the number of nodes has as a consequence the reduction in memory consumption in the case of fluid mechanics problems this technique also increases the time step for a constant courant number several examples are given in the paper which show the potential of the method

spectral collocation schemes on the unit disc two spectral collocation schemes on the unit disc are presented the first one is based on the mapping of gordon and hall here the unit square is directly mapped onto the unit disc by means of an interpolation technique unlike other poisson solvers on the unit disc no polar coordinates are involved hence the usual problems with the singularity of polar coordinates are avoided this is also shown for more complex geometries the second method is based on a diameter approach where the collocation nodes are no more clustering in the center numerical results are presented which demonstrate the high accuracy of our new spectral collocation schemes