on the uniqueness problems of entire functions and their derivatives in this paper we obtain some uniqueness theorems for entire functions and their derivatives sharing the same fixed points with the same multiplicities

graph embedding in vector spaces by node attribute statistics graph based representations are of broad use and applicability in pattern recognition they exhibit however a major drawback with regards to the processing tools that are available in their domain graph embedding into vector spaces is a growing field among the structural pattern recognition community which aims at providing a feature vector representation for every graph and thus enables classical statistical learning machinery to be used on graph based input patterns in this work we propose a novel embedding methodology for graphs with continuous node attributes and unattributed edges the approach presented in this paper is based on statistics of the node labels and the edges between them based on their similarity to a set of representatives we specifically deal with an important issue of this methodology namely the selection of a suitable set of representatives in an experimental evaluation we empirically show the advantages of this novel approach in the context of different classification problems using several databases of graphs

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

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

an entire function and its derivatives sharing a polynomial in this paper we study the growth of solutions of a first order linear differential equation and that of a second order linear differential equation from this we obtain some uniqueness theorems of a nonconstant entire function and its first derivative having the same fixed points with the same multiplicities the results in this paper also improve some known results some examples show that the results in this paper are best possible

numerical simulations of shear dependent viscoelastic flows with a combined finite element finite volume method a hybrid combined finite element finite volume method has been developed for the numerical simulation of shear dependent viscoelastic flow problems governed by a generalized oldroyd b model with a non constant viscosity function the method is applied to the 4 1 planar contraction benchmark problem to investigate the influence of the viscosity effects on the flow and results are compared with those found in the literature for creeping oldroyd b flows for a range of weissenberg numbers the method is also applied to flow in a smooth stenosed channel it is shown that the qualitative behavior of the flow is influenced by the rheological properties of the fluid namely its viscoelastic and inertial effects as well as the shear thinning viscosity these results appear in the framework of a preliminary study of the numerical simulation of steady and pulsatile blood flows in two dimensional stenotic vessels using this hybrid finite element finite volume method

weighted sharing three values and brosch s theorem this paper studies the idea of weighted sharing and proves two main theorems which generalize some results given by g brosch q zhang and other authors

graph embedding using tree edit union in this paper we address the problem of how to learn a structural prototype that can be used to represent the variations present in a set of trees the prototype serves as a pattern space representation for the set of trees to do this we construct a super tree to span the union of the set of trees this is a chicken and egg problem since before the structure can be estimated correspondences between the nodes of the super tree and the nodes of the sample tree must be to hand we demonstrate how to simultaneously estimate the structure of the super tree and recover the required correspondences by minimizing the sum of the tree edit distances over pairs of trees subject to edge consistency constraints each node of the super tree corresponds to a dimension of the pattern space and for each tree we construct a pattern vector in which the elements of the weights corresponding to each of the dimensions of the super tree we perform pattern analysis on the set of trees by performing principal components analysis on the vectors the method is illustrated on a shape analysis problem involving shock trees extracted from the skeletons of 2d objects

on the transport diffusion algorithm and its applications to the navier stokes equations this paper deals with an algorithm for the solution of diffusion and or convection equations where we mixed the method of characteristics and the finite element method globally it looks like one does one step of transport plus one step of diffusion or projection but the mathematics show that it is also an implicit time discretization of thepde in lagrangian form we give an error bound h t h h t in the interesting case that holds also for the navier stokes equations even when the reynolds number is infinite euler equation

meromorphic functions sharing three weighted values in this paper we deal with the relation between the characteristics of two meromorphic functions that share three weighted values which improve the results of q c zhang and through the results two conjectures which proposed by mues and osgood yang are solved in a weaker condition

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

unfolding kernel embeddings of graphs in this paper we investigate the use of manifold learning techniques to enhance the separation properties of standard graph kernels the idea stems from the observation that when we perform multidimensional scaling on the distance matrices extracted from the kernels the resulting data tends to be clustered along a curve that wraps around the embedding space a behavior that suggests that long range distances are not estimated accurately resulting in an increased curvature of the embedding space hence we propose to use a number of manifold learning techniques to compute a low dimensional embedding of the graphs in an attempt to unfold the embedding manifold and increase the class separation we perform an extensive experimental evaluation on a number of standard graph datasets using the shortest path borgwardt and kriegel 2005 graphlet shervashidze et al 2009 random walk kashima et al 2003 and weisfeiler lehman shervashidze et al 2011 kernels we observe the most significant improvement in the case of the graphlet kernel which fits with the observation that neglecting the locational information of the substructures leads to a stronger curvature of the embedding manifold on the other hand the weisfeiler lehman kernel partially mitigates the locality problem by using the node labels information and thus does not clearly benefit from the manifold learning interestingly our experiments also show that the unfolding of the space seems to reduce the performance gap between the examined kernels highlightswe use manifold learning techniques to enhance the performance of graph kernels we try to increase the class separation by unfolding the embedding manifold we propose a procedure to find the optimal embedding of the data the kernels which neglect the locational information show the largest improvement the unfolding of the space helps to reduce the performance gap between the kernels

approximation of time dependent viscoelastic fluid flow supg approximation in this article we consider the numerical approximation to the time dependent viscoelasticity equations with an oldroyd b constitutive equation the approximation is stabilized by using a streamline upwind petrov galerkin supg approximation for the constitutive equation we analyze both the semidiscrete and fully discrete numerical approximations for both discretizations we prove the existence of and derive a priori error estimates for the numerical approximations

finite element error estimation for quasi newtonian fluid structure interaction problems we consider a monolithic scheme for fluid structure interaction problems involving an incompressible quasi newtonian fluid the monolithic formulation is obtained using the arbitrary lagrangian eulerian ale method with matching conditions at the fluid structure interface the stability and error analysis are performed for the finite element approximation finally some numerical experiments that confirm the theoretical analysis are presented

on meromorphic functions that share three values of finite weights abstract a uniqueness theorem for two distinct non constant meromorphic functions that share three values of finite weights is proved which generalizes two previous results by h x yi and x m li and h x yi as applications of it many known results by h x yi and p li etc could be improved furthermore with the concept of finite weight sharing extensions on osgood yang s conjecture and mues conjecture and a generalization of some prevenient results by m ozawa and h ueda ect could be obtained

on the characteristics of meromorphic functions with three weighted sharing values in this paper we deal with the relation between the characteristic function of two nonconstant meromorphic functions with three weighted sharing values which improves a result given by h x yi and y h li from this we establish a theorem which improves a result given by p li and c c yang

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

on results of h ueda and g brosch concerning the unicity of meromorphic functions by emplying the main ideas from g brosch two results of r nevanlinna s four value theorem type on the unicity of meromorphic functions in the open complex plane sharing three distinct values and a fourth value or a pair of values weakly are proved which extend and generalize previous results of r nevanlinna g g gundersen h ueda and g brosch

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

an eigendecomposition approach to weighted graph matching problems an approximate solution to the weighted graph matching problem is discussed for both undirected and directed graphs the weighted graph matching problem is that of finding the optimum matching between two weighted graphs which are graphs with weights at each arc the proposed method uses an analytic instead of a combinatorial or iterative approach to the optimum matching problem using the eigendecompositions of the adjacency matrices in the case of the undirected graph matching problem or hermitian matrices derived from the adjacency matrices in the case of the directed graph matching problem a matching close to the optimum can be found efficiently when the graphs are sufficiently close to each other simulation results are given to evaluate the performance of the proposed method

a hybrid reproducing graph kernel based on information entropy abstract a number of graph kernel based methods have been developed with great success in many fields but very little research has been published that is concerned with a graph kernel in reproducing kernel hilbert space rkhs in this paper we firstly start with a derived expression for two forms of information entropy of an undirected graph they are approximated von neumann entropy and shannon entropy and depend on vertex degree statistics secondly we show the basic solution of a generalized differential operator this solution is a specific reproducing kernel called the h1 reproducing kernel in h1 space and then it is proven to satisfy the condition of mercer kernel thirdly based on the two aforementioned forms of information entropy and h1 reproducing kernel we define two reproducing graph kernels one is approximated von neumann entropy reproducing graph kernel avnergk the other is shannon entropy reproducing graph kernel sergk and then we prove that they satisfy the condition of mercer kernel finally to obtain better classification results we further propose a hybrid reproducing graph kernel hrgk based on the two reproducing graph kernels we use the hrgk as a means to establish the similarity between a pair of graphs experimental results reveal that our method gives better classification performance on graphs extracted from several graph datasets

backtrackless walks on a graph the aim of this paper is to explore the use of backtrackless walks and prime cycles for characterizing both labeled and unlabeled graphs the reason for using backtrackless walks and prime cycles is that they avoid tottering and can increase the discriminative power of the resulting graph representation however the use of such methods is limited in practice because of their computational cost in this paper we present efficient methods for computing graph kernels which are based on backtrackless walks in a labeled graph and whose worst case running time is the same as that of kernels based on random walks for clustering unlabeled graphs we construct feature vectors using ihara coefficients since these coefficients are related to the frequencies of prime cycles in the graph to efficiently compute the low order coefficients we present an o v 3 algorithm which is better than the o v 6 worst case running time of previously known algorithms in the experimental evaluation we apply the proposed method to clustering both labeled and unlabeled graphs the results show that using backtrackless walks and prime cycles instead of random walks can increase the accuracy of recognition

characterizing graph symmetries through quantum jensen shannon divergence in this paper we investigate the connection between quantum walks and graph symmetries we begin by designing an experiment that allows us to analyze the behavior of the quantum walks on the graph without causing the wave function collapse to achieve this we base our analysis on the recently introduced quantum jensen shannon divergence in particular we show that the quantum jensen shannon divergence between the evolution of two quantum walks with suitably defined initial states is maximum when the graph presents symmetries hence we assign to each pair of nodes of the graph a value of the divergence and we average over all pairs of nodes to characterize the degree of symmetry possessed by a graph

on the characteristics of meromorphic functions that share three values cm in this paper by studying the counting functions of the common 1 points of meromorphic functions a more precise relation between the characteristics of meromorphic functions that share three values cm has been obtained as applications of this many known results can be improved

fuzzy multilevel graph embedding structural pattern recognition approaches offer the most expressive convenient powerful but computational expensive representations of underlying relational information to benefit from mature less expensive and efficient state of the art machine learning models of statistical pattern recognition they must be mapped to a low dimensional vector space our method of explicit graph embedding bridges the gap between structural and statistical pattern recognition we extract the topological structural and attribute information from a graph and encode numeric details by fuzzy histograms and symbolic details by crisp histograms the histograms are concatenated to achieve a simple and straightforward embedding of graph into a low dimensional numeric feature vector experimentation on standard public graph datasets shows that our method outperforms the state of the art methods of graph embedding for richly attributed graphs highlights we propose an explicit graph embedding method we perform multilevel analysis of graph to extract global topological structural and attribute information we use homogeneity of subgraphs in graph for extracting topological structural details we encode numeric information by fuzzy histograms and symbolic information by crisp histograms our method outperforms graph embedding methods for richly attributed graphs

graph characteristics from the heat kernel trace graph structures have been proved important in high level vision since they can be used to represent structural and relational arrangements of objects in a scene one of the problems that arises in the analysis of structural abstractions of objects is graph clustering in this paper we explore how permutation invariants computed from the trace of the heat kernel can be used to characterize graphs for the purposes of measuring similarity and clustering the heat kernel is the solution of the heat equation and is a compact representation of the path length distribution on a graph the trace of the heat kernel is given by the sum of the laplacian eigenvalues exponentiated with time we explore three different approaches to characterizing the heat kernel trace as a function of time our first characterization is based on the zeta function which from the mellin transform is the moment generating function of the heat kernel trace our second characterization is unary and is found by computing the derivative of the zeta function at the origin the third characterization is derived from the heat content i e the sum of the elements of the heat kernel we show how the heat content can be expanded as a power series in time and the coefficients of the series can be computed using the laplacian spectrum we explore the use of these characterizations as a means of representing graph structure for the purposes of clustering and compare them with the use of the laplacian spectrum experiments with the synthetic and real world databases reveal that each of the three proposed invariants is effective and outperforms the traditional laplacian spectrum moreover the heat content invariants appear to consistently give the best results in both synthetic sensitivity studies and on real world object recognition problems

numerical study of shear dependent non newtonian fluids in compliant vessels the aim of this contribution is to present recent results on numerical modelling of non newtonian flow in compliant stenosed vessels with application in hemodynamics we consider two models of shear thinning non newtonian fluids and compare them with the newtonian model for the structure problem the generalized string equation for radial symmetric tubes is used and extended to a stenosed vessel the global iterative approach to approximate the fluid structure interaction is used finally we present numerical experiments for some non newtonian models comparisons with the newtonian model and the results for hemodynamic wall parameters such as the wall shear stress and the oscillatory shear index

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

indexing hierarchical structures using graph spectra hierarchical image structures are abundant in computer vision and have been used to encode part structure scale spaces and a variety of multiresolution features in this paper we describe a framework for indexing such representations that embeds the topological structure of a directed acyclic graph dag into a low dimensional vector space based on a novel spectral characterization of a dag this topological signature allows us to efficiently retrieve a promising set of candidates from a database of models using a simple nearest neighbor search we establish the insensitivity of the signature to minor perturbation of graph structure due to noise occlusion or node split merge to accommodate large scale occlusion the dag rooted at each nonleaf node of the query votes for model objects that share that part effectively accumulating local evidence in a model dag s topological subspaces we demonstrate the approach with a series of indexing experiments in the domain of view based 3d object recognition using shock graphs

a unified definition and computation of laplacian spectral distances abstract laplacian spectral kernels and distances e g biharmonic heat diffusion wave kernel distances are easily defined through a filtering of the laplacian eigenpairs they play a central role in several applications such as dimensionality reduction with spectral embeddings diffusion geometry image smoothing geometric characterisations and embeddings of graphs extending the results recently derived in the discrete setting 38 39 to the continuous case we propose a novel definition of the laplacian spectral kernels and distances whose approximation requires the solution of a set of inhomogeneous laplace equations their discrete counterparts are equivalent to a set of sparse symmetric and well conditioned linear systems which are efficiently solved with iterative methods finally we discuss the optimality of the laplacian spectrum for the approximation of the spectral kernels the relation between the spectral and green kernels and the stability of the spectral distances with respect to the evaluation of the laplacian spectrum and to multiple laplacian eigenvalues

finite volume and weno scheme in one dimensional vascular system modelling a finite volume and weno model is developed validated and applied to a one dimensional numerical investigation of the vascular system suitable balance laws are spatially integrated using a finite volume weno approach time integration is performed using a five steps fourth order accurate runge kutta strong stability preserving scheme a detailed investigation of different sets of reflecting and non reflecting boundary conditions is performed to achieve a correct representation of limited portions of the vascular system moreover special attention is devoted to the numerical discretisation of the source term finally as an application of the whole model the effect of an abdominal aorta aneurysm on the blood flow is evaluated

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

on a result of g brosch abstract we prove a uniqueness theorem for non constant meromorphic functions f g which share three values 0 1 and f a g b share the value 0 for a b 0 1 our theorem improves a result of g brosch

a quantum jensen shannon graph kernel for unattributed graphs in this paper we use the quantum jensen shannon divergence as a means of measuring the information theoretic dissimilarity of graphs and thus develop a novel graph kernel in quantum mechanics the quantum jensen shannon divergence can be used to measure the dissimilarity of quantum systems specified in terms of their density matrices we commence by computing the density matrix associated with a continuous time quantum walk over each graph being compared in particular we adopt the closed form solution of the density matrix introduced in rossi et al 2013 27 28 to reduce the computational complexity and to avoid the cumbersome task of simulating the quantum walk evolution explicitly next we compare the mixed states represented by the density matrices using the quantum jensen shannon divergence with the quantum states for a pair of graphs described by their density matrices to hand the quantum graph kernel between the pair of graphs is defined using the quantum jensen shannon divergence between the graph density matrices we evaluate the performance of our kernel on several standard graph datasets from both bioinformatics and computer vision the experimental results demonstrate the effectiveness of the proposed quantum graph kernel highlightswe compute a density matrix for a graph using the continuous time quantum walk we compute the quantum jensen shannon divergence between graph density matrixes we define a quantum jensen shannon graph kernel using the quantum divergence we evaluate the performance of our quantum kernel on standard graph datasets we demonstrate the effectiveness of the proposed quantum kernel

small functions and uniqueness of meromorphic functions we discuss the uniqueness of meromorphic functions sharing three weighted values and provide a complete answer to a question of t c alzahary

an improvement of the nevanlinna gundersen theorem a well known result of nevanlinna states that for two nonconstant meromorphic functions f and g on the complex plane c and for four distinct values aj c if f aj g aj for all 1 j 4 then g is a mobius transformation of f in 1983 gundersen generalized the result of nevanlinna to the case where the above condition is replaced by min f aj 1 min g aj 1 for j 1 2 and f aj g aj for j 3 4 in this paper we prove that the theorem of gundersen remains valid to the case where min f aj 1 min g aj 1 for j 1 2 and min f aj 2 min g aj 2 for j 3 4 furthermore we work on the case where aj are small functions

some np complete problems similar to graph isomorphism the graph isomorphism problem has so far resisted attempts at determining its complexity status it has not been shown to be np complete nor in p in this paper several altered or generalized versions of the isomorphism problem are presented and shown to be np complete one of these is the problem of determining whether a given graph has a fixed point free automorphism some speculation is made on the possible implications of these results on deciding the complexity status of isomorphism various classes and hierarchies of problems in np are discussed

numerical analysis of a fem for a transient viscoelastic flow abstract we present the numerical analysis of a coupled method for the numerical simulation of transient viscoelastic flows obeying a differential constitutive equation with a newtonian viscosity the scheme used is based on euler implicit method in time and maintain at each time step a coupling between the velocity u and the viscoelastic part of the stress approximation in space is made by finite element method fe the approximate stress velocity and pressure are respectively p 1 discontinuous p 2 continuous and p 1 continuous upwinding needed for convection of is made by a discontinuous galerkin fe method

meromorphic functions that share four small functions in this paper we prove that if two nonconstant meromorphic functions f and g share two small functions cm and share other two small functions im then f must be a quasi mobius transformation of g this result is a generalization of several results obtained by g g gundersen and li yang

spectral embedding of graphs in this paper we explore how to embed symbolic relational graphs with unweighted edges in a pattern space we adopt a graph spectral approach we use the leading eigenvectors of the graph adjacency matrix to define eigenmodes of the adjacency matrix for each eigenmode we compute vectors of spectral properties these include the eigenmode perimeter eigenmode volume cheeger number inter mode adjacency matrices and intermode edge distance we embed these vectors in a pattern space using two contrasting approaches the first of these involves performing principal or independent components analysis on the covariance matrix for the spectral pattern vectors the second approach involves performing multidimensional scaling on the l2 norm for pairs of pattern vectors we illustrate the utility of the embedding methods on neighbourhood graphs representing the arrangement of corner features in 2d images of 3d polyhedral objects two problems are investigated the first of these is the clustering of graphs representing distinct objects viewed from different directions the second is the identification of characteristic views of single objects these two studies reveal that both embedding methods result in well structured view spaces for graph data extracted from 2d views of 3d objects

a study of graph spectra for comparing graphs and trees the spectrum of a graph has been widely used in graph theory to characterise the properties of a graph and extract information from its structure it has also been employed as a graph representation for pattern matching since it is invariant to the labelling of the graph there are however a number of potential drawbacks in using the spectrum as a representation of a graph firstly more than one graph may share the same spectrum it is well known for example that very few trees can be uniquely specified by their spectrum secondly the spectrum may change dramatically with a small change structure there are a wide variety of graph matrix representations from which the spectrum can be extracted among these are the adjacency matrix combinatorial laplacian normalised laplacian and unsigned laplacian spectra can also be derived from the heat kernel matrix and path length distribution matrix the choice of matrix representation clearly has a large effect on the suitability of spectrum in a number of pattern recognition tasks in this paper we investigate the performance of the spectra as a graph representation in a variety of situations firstly we investigate the cospectrality of the various matrix representations over large graph and tree sets extending the work of previous authors we then show that the euclidean distance between spectra tracks the edit distance between graphs over a wide range of edit costs and we analyse the accuracy of this relationship we then use the spectra to both cluster and classify the graphs and demonstrate the effect of the graph matrix formulation on error rates these results are produced using both synthetic graphs and trees and graphs derived from shape and image data

edit distance based kernel functions for structural pattern classification a common approach in structural pattern classification is to define a dissimilarity measure on patterns and apply a distance based nearest neighbor classifier in this paper we introduce an alternative method for classification using kernel functions based on edit distance the proposed approach is applicable to both string and graph representations of patterns by means of the kernel functions introduced in this paper string and graph classification can be performed in an implicit vector space using powerful statistical algorithms the validity of the kernel method cannot be established for edit distance in general however by evaluating theoretical criteria we show that the kernel functions are nevertheless suitable for classification and experiments on various string and graph datasets clearly demonstrate that nearest neighbor classifiers can be outperformed by support vector machines using the proposed kernel functions

coined quantum walks lift the cospectrality of graphs and trees in this paper we explore how a spectral technique suggested by coined quantum walks can be used to distinguish between graphs that are cospectral with respect to standard matrix representations the algorithm runs in polynomial time and moreover can distinguish many graphs for which there is no subexponential time algorithm that is proven to be able to distinguish between them in the paper we give a description of the coined quantum walk from the field of quantum computing the evolution of the walk is governed by a unitary matrix we show how the spectrum of this matrix is related to the spectrum of the transition matrix of the classical random walk however despite this relationship the behaviour of the quantum walk is vastly different from the classical walk this leads us to define a new matrix based on the amplitudes of paths of the walk whose spectrum we use to characterise graphs we carry out three sets of experiments using this matrix representation firstly we test the ability of the spectrum to distinguish between sets of graphs that are cospectral with respect to standard matrix representation these include strongly regular graphs and incidence graphs of balanced incomplete block designs bibds secondly we test our method on all regular graphs on up to 14 vertices and all trees on up to 24 vertices this demonstrates that the problem of cospectrality is often encountered with conventional algorithms and tests the ability of our method to resolve this problem thirdly we use distances obtained from the spectra of s u 3 to cluster graphs derived from real world image data and these are qualitatively better than those obtained with the spectra of the adjacency matrix thus we provide a spectral representation of graphs that can be used in place of standard spectral representations far less prone to the problems of cospectrality