spectra of complex networks we propose a general approach to the description of spectra of complex networks for the spectra of networks with uncorrelated vertices and a local treelike structure exact equations are derived these equations are generalized to the case of networks with correlations between neighboring vertices the tail of the density of eigenvalues ensuremath rho ensuremath lambda at large ensuremath lambda is related to the behavior of the vertex degree distribution p k at large k in particular as p k ensuremath sim k ensuremath ensuremath gamma ensuremath rho ensuremath lambda ensuremath sim ensuremath lambda 1 ensuremath 2 ensuremath gamma we propose a simple approximation which enables us to calculate spectra of various graphs analytically we analyze spectra of various complex networks and discuss the role of vertices of low degree we show that spectra of locally treelike random graphs may serve as a starting point in the analysis of spectral properties of real world networks e g of the internet

counting spanning trees in self similar networks by evaluating determinants spanning trees are relevant to various aspects of networks generally the number of spanning trees in a network can be obtained by computing a related determinant of the laplacian matrix of the network however for a large generic network evaluating the relevant determinant is computationally intractable in this paper we develop a fairly generic technique for computing determinants corresponding to self similar networks thereby providing a method to determine the numbers of spanning trees in networks exhibiting self similarity we describe the computation process with a family of networks called x y flowers which display rich behavior as observed in a large variety of real systems the enumeration of spanning trees is based on the relationship between the determinants of submatrices of the laplacian matrix corresponding to the x y flowers at different generations and is devoid of the direct laborious computation of determinants using the proposed method we derive analytically the exact number of spanning trees in the x y flowers on the basis of which we also obtain the entropies of the spanning trees in these networks moreover to illustrate the universality of our technique we apply it to some other self similar networks with distinct degree distributions and obtain explicit solutions to the numbers of spanning trees and their entropies finally we compare our results for networks with the same average degree but different structural properties such as degree distribution and fractal dimension and uncover the effect of these topological features on the number of spanning trees

on the hughes model for pedestrian flow the one dimensional case in this paper we investigate the mathematical theory of hughes model for the ow of pedestrians cf 17 consisting of a nonlinear conservation law for the density of pedestrians coupled with an eikonal equation for a potential modelling the common sense of the task we rst consider an approximation of the original model in which the eikonal equation is replaced by an elliptic approximation for such an approximated system we prove existence and uniqueness of entropy solutions in one space dimension in the sense of kru zkov 21 in which the boundary conditions are posed following the approach of bardos et al 7 we use bv estimates on the density and stability estimates on the potential in order to prove uniqueness furthermore we analyse the evolution of characteristics for the original hughes model in one space dimension and study the behaviour of simple solutions in order to reproduce interesting phenomena related to the formation of shocks and rarefaction waves the characteristic calculus is supported by numerical simulations

diameters and eigenvalues we derive a new upper bound for the diameter of a k regular graph g as a function of the eigenvalues of the adjacency matrix namely suppose the adjacency matrix of g has eigenvalues ai a2 a with ia i i vi21 ia i where ai k a ia21 then the diameter d g must satisfy d g we will consider families of graphs whose eigenvalues can be explicitly de termined these graphs are determined by sums or differences of vertex labels namely the pair i j being an edge depends only on the value i j or i j for directed graphs we will show that these graphs are expander graphs with small diameters by using an inequality on character sums which was recently proved by n m katz bell communications research 445 south street morristown new jersey 07960 this content downloaded from 207 46 13 111 on sat 21 may 2016 07 00 04 utc all use subject to http about jstor org terms

special solutions from the variable separation approach the davey stewartson equation a variable separation procedure for the davey stewartson ds equation is proposed by using a prior ansatz to its bilinear form the reduced equations for two variable separated fields have the same trilinear form although they possess different independent variables the trilinear equation can be changed to a spacetime symmetric form and can be solved by means of a boussinesq type equation system whenever a pair of solutions of the reduced fields are obtained a corresponding solution of the ds equation can be obtained algebraically the single dromion solution and some kinds of positon solutions are obtained explicitly

enhancing the spectral gap of networks by node removal dynamics on networks are often characterized by the second smallest eigenvalue of the laplacian matrix of the network which is called the spectral gap examples include the threshold coupling strength for synchronization and the relaxation time of a random walk a large spectral gap is usually associated with high network performance such as facilitated synchronization and rapid convergence in this study we seek to enhance the spectral gap of undirected and unweighted networks by removing nodes because practically the removal of nodes often costs less than the addition of nodes addition of links and rewiring of links in particular we develop a perturbative method to achieve this goal the proposed method realizes better performance than other heuristic methods on various model and real networks the spectral gap increases as we remove up to half the nodes in most of these networks

global mean first passage times of random walks on complex networks we present a general framework applicable to a broad class of random walks on complex networks which provides a rigorous lower bound for the mean first passage time of a random walker to a target site averaged over its starting position the so called global mean first passage time gmfpt this bound is simply expressed in terms of the equilibrium distribution at the target and implies a minimal scaling of the gmfpt with the network size we show that this minimal scaling which can be arbitrarily slow is realized under the simple condition that the random walk is transient at the target site and independently of the small world scale free or fractal properties of the network last we put forward that the gmfpt to a specific target is not a representative property of the network since the target averaged gmfpt satisfies much more restrictive bounds

dissimilar behaviors of coherent exciton transport on scale free networks with identical degree sequence conventional wisdom is that the power law degree sequence has a profound impact on diverse dynamical processes taking place on scale free networks here we show that power law degree distribution alone is not sufficient to determine the behaviors of coherent exciton transport on scale free networks for this purpose we study coherent exciton transport on a class of scale free networks controlled by a tunable parameter q which have the same degree sequence independent of q but are very different in other structural properties determined by q such as average distance and fractality first through numerical determination of the time evolutions of return probabilities we compare in detail the dissimilarities in oscillation periods and oscillation amplitudes between two limiting cases of q 0 and q 1 second we demonstrate that in the network of q 0 up to the third generation the quantum transport problem between node groups partitioned by node degrees can be mapped onto a line furthermore we show that in the case of long time limit the transition probabilities in the networks of q 0 and q 1 exhibit different characteristic patterns with identical values finally we find that the degree averaged return probability decreases when the parameter q increases from 0 to 1 indicating that the stronger localization can be found in those networks with lower values of q our results suggest that it is far from enough to characterize the behaviors of coherent exciton transport on scale free networks merely by power law degree distribution

uniqueness of weak solutions to a non linear hyperbolic system in electrohydrodynamics abstract in this note we prove the uniqueness of weak solutions to a nonlinear hyperbolic system in electrohydrodynamics without the effects of a dissociation recombination process it is still open in the presence of a special dissociation recombination process although the existence of at least one weak solution was proved via the method of renormalized solutions by feireisl e feireisl weak solutions to a non linear hyperbolic system arising in the theory of dielectric liquids math methods appl sci 18 1995 1041 1052 in 1995

detecting broken pt symmetry a fundamental problem in the theory of pt invariant quantum systems is to determine whether a given system respects this symmetry or not if not the system usually develops non real eigenvalues it is shown in this contribution how to algorithmically detect the existence of complex eigenvalues for a given pt symmetric matrix the procedure uses classical results from stability theory which qualitatively locate the zeros of real polynomials in the complex plane the interest and value of the present approach lies in the fact that it avoids diagonalization of the hamiltonian at hand

standard random walks and trapping on the koch network with scale free behavior and small world effect a vast variety of real life networks display the ubiquitous presence of scale free phenomenon and small world effect both of which play a significant role in the dynamical processes running on networks although various dynamical processes have been investigated in scale free small world networks analytical research about random walks on such networks is much less in this paper we will study analytically the scaling of the mean first passage time mfpt for random walks on scale free small world networks to this end we first map the classical koch fractal to a network called koch network according to this proposed mapping we present an iterative algorithm for generating the koch network based on which we derive closed form expressions for the relevant topological features such as degree distribution clustering coefficient average path length and degree correlations the obtained solutions show that the koch network exhibits scale free behavior and small world effect then we investigate the standard random walks and trapping issue on the koch network through the recurrence relations derived from the structure of the koch network we obtain the exact scaling for the mfpt we show that in the infinite network order limit the mfpt grows linearly with the number of all nodes in the network the obtained analytical results are corroborated by direct extensive numerical calculations in addition we also determine the scaling efficiency exponents characterizing random walks on the koch network

an integral equation approach to the unsteady convection diffusion equations an integral equation approach is proposed to solve the unsteady convection diffusion equations in this approach the second order adams moulton method is firstly utilized for the time discretization then by using the green s function of the laplace equation in the series form the convection diffusion equation is transformed into an integral equation that is further converted into an algebraic equation system the accuracy convergence and the stability of this integral equation approach are examined by four examples in comparison with the characteristic variational multiscale method and the finite volume element method the integral equation approach shows a higher accuracy compared with the finite volume element method the integral equation approach has a better convergence in solving the convection dominated convection diffusion problems the integral equation approach demonstrates a good stability

a stabilized superfast solver for indefinite hankel systems abstract we present a stabilized superfast solver for indefinite hankel systems whose size is a power of 2 the hankel system is transformed into a loewner system which is solved by using an inversion formula for loewner matrices this explicit formula for the inverse of a loewner matrix contains certain parameters that are computed by solving two linearized rational interpolation problems on the unit circle the heart of our hankel solver is a superfast algorithm to solve these interpolation problems this algorithm is stabilized via pivoting iterative improvement and by giving the so called difficult interpolation points an adequate treatment we have implemented our algorithm in fortran 90 numerical examples illustrate the effectiveness of our approach

potts model partition functions on two families of fractal lattices abstract the partition function of q state potts model or equivalently the tutte polynomial is computationally intractable for regular lattices the purpose of this paper is to compute partition functions of q state potts model on two families of fractal lattices based on their self similar structures and by applying the subgraph decomposition method we divide their tutte polynomials into two summands and for each summand we obtain a recursive formula involving the other summand as a result the number of spanning trees and their asymptotic growth constants and a lower bound of the number of connected spanning subgraphs or acyclic root connected orientations for each of such two lattices are obtained

a system of convection diffusion equations with small diffusion coefficient arising in semiconductor physics abstract we consider a system of two convection diffusion equations with a small diffusion parameter in one space dimension subject to dirichlet boundary conditions the system governs the evolution of the flow of electrons and holes in semiconductor devices on the dielectrical relaxation time scale the equations are coupled by nonlinear nonlocal electric field driven convection terms we prove the convergence in suitable topologies of the solutions of the diffusion convection problem to the unique solution of the convective limit problem subject to inflow boundary conditions as the diffusion coefficient tends to zero

resistance scaling and the number of spanning trees in self similar lattices we consider the problem of enumerating spanning trees in self similar lattices motivated by recent work of chang chen and yang who determined explicit formulae in the case of sierpinski graphs and some of their generalizations the aim of this note is to show that their results hold in more generality and that there is a strong relation between this enumeration problem and resistance scaling on self similar lattices

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

solving the random cauchy one dimensional advection diffusion equation numerical analysis and computing abstract in this paper a random finite difference scheme to solve numerically the random cauchy one dimensional advection diffusion partial differential equation is proposed and studied throughout our analysis both the advection and diffusion coefficients are assumed to be random variables while the deterministic initial condition is assumed to possess a discrete fourier transform for the sake of generality in our study we consider that the advection and diffusion coefficients are statistical dependent random variables under mild conditions on the data it is demonstrated that the proposed random numerical scheme is mean square consistent and stable finally the theoretical results are illustrated by means of two numerical examples

spanning trees in a fractal scale free lattice spanning trees provide crucial insight into the origin of fractality in fractal scale free networks in this paper we present the number of spanning trees in a particular fractal scale free lattice network we first study analytically the topological characteristics of the lattice and show that it is simultaneously scale free highly clustered large world fractal and disassortative any previous model does not have all the properties as the studied one then by using the renormalization group technique we derive analytically the number of spanning trees in the network under consideration based on which we also determine the entropy for the spanning trees of the network these results shed light on understanding the structural characteristics of and dynamical processes on scale free networks with fractality moreover our method and process for employing the decimation technique to enumerate spanning trees are general and can be easily extended to other deterministic media with self similarity

maximal couplings in pt symmetric chain models with the real spectrum of energies the domain of all the coupling strengths compatible with the reality of the energies is studied for a family of non hermitian n n matrix hamiltonians h n with tridiagonal and symmetric structure at all dimensions n the coordinates are found of the extremal points at which the boundary hypersurface touches the circumscribed sphere for odd n 2m 1 or ellipsoid for even n 2k

recursive solutions for laplacian spectra and eigenvectors of a class of growing treelike networks the complete knowledge of laplacian eigenvalues and eigenvectors of complex networks plays an outstanding role in understanding various dynamical processes running on them however determining analytically laplacian eigenvalues and eigenvectors is a theoretical challenge in this paper we study the laplacian spectra and their corresponding eigenvectors of a class of deterministically growing treelike networks the two interesting quantities are determined through the recurrence relations derived from the structure of the networks beginning from the rigorous relations one can obtain the complete eigenvalues and eigenvectors for the networks of arbitrary size the analytical method opens the way to analytically compute the eigenvalues and eigenvectors of some other deterministic networks making it possible to accurately calculate their spectral characteristics

properties and applications of laplacian spectra for koch networks for a network knowledge of its laplacian eigenvalues is central to understanding its structure and dynamics in this paper we study the laplacian spectra of a family of koch networks with scale free and small world properties we derive some recursive relations between the laplacian characteristic polynomials of koch networks and their subgraphs at different iterations based on the obtained recurrence relations we determine explicitly the product of all nonzero laplacian eigenvalues as well as the sum of the reciprocals of these eigenvalues then using these results we further evaluate the number of spanning trees kirchhoff index global mean first passage time and average path length of the family of koch networks finally we determine the number of spanning forests under certain conditions we expect that our method can be adapted to other types of self similar networks

generalized symmetries and w infinity algebras in 3 dimensional toda field theory after establishing a formal theory for getting solutions of one type of high dimensional partial differential equation two sets of generalized symmetries of the 3d toda theory which arises from a particular reduction of the 4d self dual gravity equation are obtained concretely by a simple formula each set of symmetries constitutes a generalized w algebra which contains three types of the usual w algebras as special cases some open questions are discussed

spanning trees on the sierpinski gasket we present the numbers of spanning trees on the sierpinski gasket sg d n at stage n with dimension d equal to two three and four the general expression for the number of spanning trees on sg d n with arbitrary d is conjectured the numbers of spanning trees on the generalized sierpinski gasket sg d b n with d 2 and b 3 4 are also obtained

on the distributions of laplacian eigenvalues versus node degrees in complex networks abstract in this paper the important issue of laplacian eigenvalue distributions is investigated through theory guided extensive numerical simulations for four typical complex network models namely the er random graph networks ws and nw small world networks and ba scale free networks it is found that these four types of complex networks share some common features particularly similarities between the laplacian eigenvalue distributions and the node degree distributions

initial transients of solutions of the semiconductors device equations abstract carrier transport in semiconductor devices is usually described by a system of two convection diffusions equations for the carrier concentrations coupled to poisson s equation a singular perturbation analysis shows the complicated structure of solutions internal spatial layers temporal evolution on two timescales a fast timescale occurs if the initial data do not have the appropriate spatial structure we derive the initial layer equations corresponding to the fast timescale we analyse several model examples by the method of characteristics furthermore we present numerical examples

theory of resistor networks the two point resistance the resistance between two arbitrary nodes in a resistor network is obtained in terms of the eigenvalues and eigenfunctions of the laplacian matrix associated with the network explicit formulae for two point resistances are deduced for regular lattices in one two and three dimensions under various boundary conditions including that of a mobius strip and a klein bottle the emphasis is on lattices of finite sizes we also deduce summation and product identities which can be used to analyse large size expansions in two and higher dimensions

explicit determination of mean first passage time for random walks on deterministic uniform recursive trees the determination of mean first passage time mfpt for random walks in networks is a theoretical challenge and is a topic of considerable recent interest within the physics community in this paper according to the known connections between mfpt effective resistance and the eigenvalues of graph laplacian we first study analytically the mfpt between all node pairs of a class of growing treelike networks which we term deterministic uniform recursive trees durts since one of its particular cases is a deterministic version of the famous uniform recursive tree the interesting quantity is determined exactly through the recursive relation of the laplacian spectra obtained from the special construction of durts the analytical result shows that the mfpt between all couples of nodes in durts varies as n text text ln text n for large networks with node number n second we study trapping on a particular network of durts focusing on a special case with the immobile trap positioned at a node having largest degree we determine exactly the average trapping time att that is defined as the average of fpt from all nodes to the trap in contrast to the scaling of the mfpt the leading behavior of att is a linear function of n interestingly we show that the behavior for att of the trapping problem is related to the trapping location which is in comparison with the phenomenon of trapping on fractal t graph although both networks exhibit tree structure finally we believe that the methods could open the way to exactly calculate the mfpt and att in a wide range of deterministic media

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

geometry of minimum spanning trees on scale free networks abstract the minimum spanning trees on scale free graphs are shown to be scale free as well in the presence of random edge weights the probability distribution of the weights on the tree differs from regular lattices reflecting the typically short distances small world property we consider also the trees in the absence of such randomness and the ensuing massive degeneracy which is analyzed with graph theoretical arguments

eigenvalue spectrum of transition matrix of dual sierpinski gaskets and its applications the eigenvalue spectrum of the transition matrix of a network contains much information about its structural properties and is related to the behavior of various dynamical processes performed on it in this paper we study the eigenvalues of the transition matrix of the dual sierpinski gaskets embedded in d dimensional euclidean spaces we obtain all the eigenvalues as well as their corresponding degeneracies by making use of the spectral decimation technique we then apply the obtained eigenvalues to determine the exact number of spanning trees in the studied fractals and derive an explicit formula of the eigentime identity for random walks taking place on them

bounding the gap between extremal laplacian eigenvalues of graphs abstract let g be a graph whose laplacian eigenvalues are 0 1 2 n we investigate the gap expressed either as a difference or as a ratio between the extremal non trivial laplacian eigenvalues of a connected graph that is n and 2 this gap is closely related to the average density of cuts in a graph we focus here on the problem of bounding the gap from below

initial time layer problem for quantum drift diffusion model we study the initial time layer problem to the quantum drift diffusion model the limit of vanishing debye length is established by entropy method and compactness argument as a byproduct the global weak solution is also obtained for this model

pseudofractal scale free web we find that scale free random networks are excellently modeled by simple deterministic graphs our graph has a discrete degree distribution degree is the number of connections of a vertex which is characterized by a power law with exponent gamma 1 ln 3 ln 2 properties of this compact structure are surprisingly close to those of growing random scale free networks with gamma in the most interesting region between 2 and 3 we succeed to find exactly and numerically with high precision all main characteristics of the graph in particular we obtain the exact shortest path length distribution for a large network ln n 1 the distribution tends to a gaussian of width approximately sqrt ln n centered at l approximately ln n we show that the eigenvalue spectrum of the adjacency matrix of the graph has a power law tail with exponent 2 gamma

a singular perturbation analysis of the transient semiconductor device equations this paper is concerned with the asymptotic analysis of the transient semiconductor device equations by appropriately scaling the equations the semiconductor device problem is reformulated as a si

the keller segel model with logistic sensitivity function and small diffusivity the keller segel model is the classical model for chemotaxis of cell populations it consists of a drift diffusion equation for the cell density coupled to an equation for the chemoattractant here a variant of this model is studied in one dimensional position space where the chemotactic drift is turned off for a limiting cell density by a logistic term and where the chemoattractant density solves an elliptic equation modeling a quasi stationary balance of reaction and diffusion with production of the chemoattractant by the cells the case of small cell diffusivity is studied by asymptotic and numerical methods on a time scale characteristic for the convective effects convergence of solutions to weak entropy solutions of the limiting nonlinear hyperbolic conservation law is proven numerical and analytic evidence indicates that solutions of this problem converge to irregular patterns of cell aggregates separated by entropic shocks from vacuum regions as time tends to infinity close to each of these p

an algorithmic test for diagonalizability of finite dimensional pt invariant systems a non hermitian operator does not necessarily have a complete set of eigenstates contrary to a hermitian one an algorithm is presented which allows one to decide whether the eigenstates of a given pt invariant operator on a finite dimensional space are complete or not in other words the algorithm checks whether a given pt symmetric matrix is diagonalizable the procedure neither requires to calculate any single eigenvalue nor any numerical approximation

an analytic family of cal p cal t symmetric hamiltonians with real eigenvalues we study nonlinear perturbative expansions for symmetric local schrodinger operators the schrodinger operator is a sum of the harmonic oscillator hamiltonian and a local symmetric potential depending in general nonlinearly on the perturbation parameter a specific class of models having real spectrum for any value of the parameter is proposed

analysis of a finite element method for the drift diffusion semiconductor device equations the multidimensional case an explicit finite element method for numerically solving the drift diffusion semiconductor device equations in two space dimensions is analyzed the method is based on the use of a mixed finite element method for the approximation of the electric field and a discontinuous upwinding finite element method for the approximation of the electron and hole concentrations the mixed method gives an approximate electric field in the precise form needed by the discontinuous method which is trivially conservative and fully parallelizable it is proven that the method produces uniformly bounded concentrations and electric fields and that it converges to the exact solution provided there is a convergent subsequence of the electron concentrations numerical simulations are presented that display the performance of the method and indicate the behavior of the solution

convergence of a finite element method for the drift diffusion semiconductor device equations the zero diffusion case in this paper a new explicit finite element method for numerically solving the drift diffusion semiconductor device equations is introduced and analyzed the method uses a mixed finite element method for the approximation of the electric field a finite element method using discontinuous finite elements is used to approximate the concentrations which may display strong gradients the use of discontinuous finite elements renders the scheme for the concentrations trivially conservative and fully parallelizable in this paper we carry out the analysis of the model method which employs a continuous piecewise linear approximation to the electric field and a piecewise constant approximation to the electron concentration in a model problem namely the so called unipolar case with the diffusion terms neglected the resulting system of equations is equivalent to a conservation law whose flux the electric field depends globally on the solution the concentration of electrons by exploiting the similarities of this system with classical scalar conservation laws the techniques to analyze the monotone schemes for conservation laws are adapted to the analysis of the new scheme the scheme considered as a scheme for the electron concentration is shown to satisfy a maximum principle and to be total variation bounded its convergence to the unique weak solution is proven numerical experiments displaying the performance of the scheme are shown

generalisations and randomisation of the plane koch curve the koch curve evolves from a base equilateral triangle by the trisection of each side and the replication of the original triangle on the mid section the process being repeated ad infinitum by the addition of sets of successively smaller triangles the process is generalised to replace the trisectioning by 2k 1 sectioning it is shown that a square is the only other regular polygon on which the 2k 1 sectioning procedure can be implemented the koch curves thus generated are strictly self similar their fractal dimensions being similarity dimensions and enclose simply connected areas randomisation of the generating procedure is also discussed