the gap lemma and geometric criteria for instability of viscous shock profiles an obstacle in the use of evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis since the evans function has in general been constructed only away from the essential spectrum a notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles here we prove a general theorem the gap lemma concerning the analytic continuation of the evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave this allows geometric stability theory to be applied in many cases where it could not be applied previously we demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves a necessary geometric condition for stability is determined in terms of the sign of a certain melnikov integral of the associated viscous profile this sign can easily be evaluated numerically we also compute it analytically for solutions of several important classes of systems in particular we show for a wide class of systems that homoclinic solitary waves are linearly unstable confirming these as the first known examples of unstable viscous shock waves we also show that strong heteroclinic undercompressive waves are sometimes unstable similar stability conditions are also derived for lax and overcompressive shocks and for nn

fixed point properties of the ising ferromagnet on the hanoi networks the ising model with ferromagnetic couplings on the hanoi networks is analyzed with an exact renormalization group in particular the fixed points are determined and the renormalization flow for certain initial conditions is analyzed hanoi networks combine a one dimensional lattice structure with a hierarchy of long range bonds to create a mix of geometric and small world properties generically those small world bonds result in nonuniversal behavior i e fixed points and scaling exponents that depend on temperature and the initial choice of coupling strengths it is shown that a diversity of different behaviors can be observed with seemingly small changes in the structure of the networks defining interpolating families of such networks we find tunable transitions between regimes with power law and certain essential singularities in the critical scaling of the correlation length these are similar to the so called inverted berezinskii kosterlitz thouless transition previously observed only in scale free or dense networks

analyzing percolation of networks inspired by the 3x 1 problem abstract in this paper we investigate percolation in a sort of networks inspired by the observation of a collatz graph cg which is the network version of the famous 3 x 1 problem in mathematics the cg consists of positive integers that are connected based on the iteration relations actually we never mean to solve the 3 x 1 problem exactly but we observe it from the viewpoint of statistical physics we focus on the so called reduced collatz graph rcg that is a subgraph with all odd numbers since even numbers can be iterated into odd ones through 3 x 1 rules considering boundary conditions we obtain a special degree distribution of rcg for finite size set of odd integers and treat the infinite case as its limit with the percolation criterion through the approach of a generating function we determine the critical condition for the network ensemble of rcg furthermore we generalize the graph model with rcg type degree distributions beyond the 3 x 1 problem get the generic criterion of percolation and phase diagram for an ensemble of positive integer networks with rcg as its extreme case

rotation invariants of vector fields from orthogonal moments abstract vector field images are a type of new multidimensional data that appear in many engineering areas although the vector fields can be visualized as images they differ from graylevel and color images in several aspects to analyze them special methods and algorithms must be originally developed or substantially adapted from the traditional image processing area in this paper we propose a method for the description and matching of vector field patterns under an unknown rotation of the field rotation of a vector field is so called total rotation where the action is applied not only on the spatial coordinates but also on the field values invariants of vector fields with respect to total rotation constructed from orthogonal gaussian hermite moments and zernike moments are introduced their numerical stability is shown to be better than that of the invariants published so far we demonstrate their usefulness in a real world template matching application of rotated vector fields

scaling and rotation invariant analysis approach to object recognition based on radon and fourier mellin transforms various types of orthogonal moments have been widely used for object recognition and classification however these moments do not natively possess scaling invariance essential image normalization and binarization prior to moments extraction will lead to error of resampling and requantifying this paper describes a new scaling and rotation invariant analysis method for object recognition in the proposed method the radon transform is utilized to project the image onto projection space to convert the rotation of the original image to a translation of the projection in the angle variable and the scaling of the original image to a scaling of the projection in the spatial variable together with an amplitude scaling of the projection and then the fourier mellin transform is applied to the result to convert the translation in the angle variable and the scaling in the spatial variable as well as the amplitude scaling of the projection to a phase shift and an amplitude scaling respectively in order to achieve a set of completely invariant descriptors a rotation and scaling invariant function is constructed a k nearest neighbors classifier is employed to implement classification theoretical and experimental results show the high classification accuracy of this approach as a result of using the rotation and scaling invariant function instead of image binarization and normalization it is also shown that this method is relatively robust in the presence of white noise

on the inverse problem of rotation moment invariants numerous papers have been devoted to the moment invariants mi they usually describe their derivation various invariant properties numerical behavior and their power to serve as the features in many pattern recognition tasks only few papers have dealt with the independence and completeness of the sets of mi s however both these properties are of fundamental importance from theoretical as well as practical point of view in our recent paper 1 we presented a general approach to deriving rotation moment invariants mi s were constructed as products of appropriate powers of complex moments we also proposed how to construct the basis independent and complete set of the invariants of this kind in this paper we prove stronger theorem we show the basis described in ref 1 is a basis of all possible rotation moment invariants not only of those constructed as products of moment powers in other words knowing the invariants from this basis we can calculate for instance traditional hu moment invariants 2 zernike moment invariants 3 and fourier mellin invariants 4 to name a few this theorem can be equivalently formulated as the solution of the inverse problem we show in this paper that all moments can be recovered from the invariant basis

griffiths singularities and algebraic order in the exact solution of an ising model on a fractal modular network we use an exact renormalization group transformation to study the ising model on a complex network composed of tightly knit communities nested hierarchically with the fractal scaling recently discovered in a variety of real world networks varying the ratio k j of intercommunity to intracommunity couplings we obtain an unusual phase diagram at high temperatures or small k j we have a disordered phase with a griffiths singularity in the free energy due to the presence of rare large clusters which we analyze through the yang lee zeros in the complex magnetic field plane as the temperature is lowered true long range order is not seen but there is a transition to algebraic order where pair correlations have power law decay with distance reminiscent of the xy model the transition is infinite order at small k j and becomes second order above a threshold value k j m the existence of such slowly decaying correlations is unexpected in a fat tailed scale free network where correlations longer than nearest neighbor are typically suppressed

renormalization group solution of the chutes ladder model abstract we analyze a semi infinite one dimensional random walk process with a biased motion that is incremental in one direction and long range in the other on a network with a fixed hierarchy of long range jumps we find with exact renormalization group calculations that there is a dynamical transition between a localized adsorption phase and an anomalous diffusion phase in which the mean square displacement exponent depends non universally on the bernoulli coin we relate these results to similar findings of unconventional phase behavior in hierarchical networks

quasistationary analysis of the contact process on annealed scale free networks we present an analysis of the quasistationary qs state of the contact process cp on annealed scale free networks using a mapping of the cp dynamics in a one step process and analyzing numerically and analytically the corresponding master equation the relevant qs quantities determined via the master equation exhibit an excellent agreement with direct qs stochastic simulations of the cp the high accuracy of the resulting data allows a probe of the strong corrections to scaling present in both the critical and supercritical regions corrections that mask the correct finite size scaling obtained analytically by applying an exact heterogeneous mean field approach our results represent a promising starting point for a deeper understanding of the contact process and absorbing phase transitions on real quenched complex networks

self similar disk packings as model spatial scale free networks the network of contacts in space filling disk packings such as the apollonian packing is examined these networks provide an interesting example of spatial scale free networks where the topology reflects the broad distribution of disk areas a wide variety of topological and spatial properties of these systems is characterized their potential as models for networks of connected minima on energy landscapes is discussed

rotation moment invariants for recognition of symmetric objects in this paper a new set of moment invariants with respect to rotation translation and scaling suitable for recognition of objects having n fold rotation symmetry are presented moment invariants described earlier cannot be used for this purpose because most moments of symmetric objects vanish the invariants proposed here are based on complex moments their independence and completeness are proven theoretically and their performance is demonstrated by experiments

magnetic models on apollonian networks thermodynamic and magnetic properties of ising models defined on the triangular apollonian network are investigated this and other similar networks are inspired by the problem of covering a euclidian domain with circles of maximal radii maps for the thermodynamic functions in two subsequent generations of the construction of the network are obtained by formulating the problem in terms of transfer matrices numerical iteration of this set of maps leads to very precise values for the thermodynamic properties of the model different choices for the coupling constants between only nearest neighbors along the lattice are taken into account for both ferromagnetic and antiferromagnetic constants long range magnetic ordering is obtained with exception of a size dependent effective critical behavior of the correlation length no evidence of asymptotic criticality was detected

free electron gas in the apollonian network multifractal energy spectrum and its thermodynamic fingerprints we study the free electron gas in an apollonian network within the tight binding framework the scale free and small world character of the underlying lattice is known to result in a quite structured energy spectrum with deltalike singularities gaps and minibands after an exact numerical diagonalization of the corresponding adjacency matrix of the network with a finite number of generations we employ a scaling analysis of the moments of the density of states to characterize its multifractality and report the associated singularity spectrum the fractal nature of the energy spectrum is also shown to be reflected in the thermodynamic behavior by logarithmic modulations on the temperature dependence of the specific heat the absence of chiral symmetry of the apollonian network leads to distinct thermodynamic behaviors due to electrons and holes thermal excitations

spectral and dynamical properties in classes of sparse networks with mesoscopic inhomogeneities we study structure eigenvalue spectra and random walk dynamics in a wide class of networks with subgraphs modules at mesoscopic scale the networks are grown within the model with three parameters controlling the number of modules their internal structure as scale free and correlated subgraphs and the topology of connecting network within the exhaustive spectral analysis for both the adjacency matrix and the normalized laplacian matrix we identify the spectral properties which characterize the mesoscopic structure of sparse cyclic graphs and trees the minimally connected nodes the clustering and the average connectivity affect the central part of the spectrum the number of distinct modules leads to an extra peak at the lower part of the laplacian spectrum in cyclic graphs such a peak does not occur in the case of topologically distinct tree subgraphs connected on a tree whereas the associated eigenvectors remain localized on the subgraphs both in trees and cyclic graphs we also find a characteristic pattern of periodic localization along the chains on the tree for the eigenvector components associated with the largest eigenvalue lambda l 2 of the laplacian further differences between the cyclic modular graphs and trees are found by the statistics of random walks return times and hitting patterns at nodes on these graphs the distribution of first return times averaged over all nodes exhibits a stretched exponential tail with the exponent sigma approximately 1 3 for trees and sigma approximately 2 3 for cyclic graphs which is independent of their mesoscopic and global structure

exact evaluation of the causal spectrum and localization properties of electronic states on a scale free network abstract a deterministic network with tree structure is considered for which the spectrum of its adjacency matrix can be exactly evaluated by a recursive renormalization approach it amounts to successively increasing number of contributions at any finite step of construction of the tree resulting in a causal chain the resulting eigenvalues can be related the full energy spectrum of a nearest neighbor tight binding model defined on this structure given this association it turns out that further properties of the eigenvectors can be evaluated like the degree of quantum localization of the tight binding eigenstates expressed by the inverse participation ratio ipr it happens that for the current model the ipr s are also suitable to be analytically expressed in terms in corresponding eigenvalue chain the resulting ipr scaling behavior is expressed by the tails of eigenvalue chains as well

patchy percolation on a hierarchical network with small world bonds the bond percolation properties of the recently introduced hanoi networks are analyzed with the renormalization group unlike scale free networks these networks are meant to provide an analytically tractable interpolation between finite dimensional lattice based models and their mean field limits in percolation the hierarchical small world bonds in the hanoi networks impose order by uniting otherwise disconnected local clusters this patchy order results in merely a finite probability to obtain a spanning cluster for certain ranges of the bond probability unlike the usual 0 1 transition found on ordinary lattices the various networks studied here exhibit a range of phase behaviors depending on the prevalence of those long range bonds fixed points in general exhibit nonuniversal behavior

analytical approach to directed sandpile models on the apollonian network we investigate a set of directed sandpile models on the apollonian network which are inspired by the work of dhar and ramaswamy phys rev lett 63 1659 1989 on euclidian lattices they are characterized by a single parameter q which restricts the number of neighbors receiving grains from a toppling node due to the geometry of the network two and three point correlation functions are amenable to exact treatment leading to analytical results for avalanche distributions in the limit of an infinite system for q 1 2 the exact recurrence expressions for the correlation functions are numerically iterated to obtain results for finite size systems when larger values of q are considered finally a detailed description of the local flux properties is provided by a multifractal scaling analysis

q state potts model on the apollonian network the q state potts model is studied on the apollonian network with monte carlo simulations and the transfer matrix method the spontaneous magnetization correlation length entropy and specific heat are analyzed as a function of temperature for different number of states q different scaling functions in temperature and q are proposed a quantitative agreement is found between results from both methods no critical behavior is observed in the thermodynamic limit for any number of states

spectral properties of the apollonian network abstract a first characterization of the spectral properties of the apollonian network is presented results are based on the numerically evaluated eigenvalue spectra of adjacency matrices for the first nine generations of the network the main features refer to the presence of several gaps on the eigenvalue axis and a very large number of degenerated eigenvalues the increase in the degeneracy of a family of eigenvalues is found to obey a very simple empirical relation

critical properties of contact process on the apollonian network we investigate an epidemic spreading process by means of a computational simulation on the apollonian network which is simultaneously small world scale free euclidean space filling and matching graphs an analysis of the critical behavior of the contact process cp is presented using a monte carlo method our model shows a competition between healthy and infected individuals in a given biological or technological system leading to a continuous phase transition between the active and inactive states whose critical exponents and 1 are calculated employing a finite size scaling analysis we show that the continuous phase transition belongs to the mean field directed percolation universality class in regular lattices

nonequilibrium model on apollonian networks we investigate the majority vote model with two states ensuremath 1 1 and a noise parameter q on apollonian networks the main result found here is the presence of the phase transition as a function of the noise parameter q previous results on the ising model in apollonian networks have reported no presence of a phase transition we also studied the effect of redirecting a fraction p of the links of the network by means of monte carlo simulations we obtained the exponent ratio ensuremath gamma ensuremath nu ensuremath beta ensuremath nu and 1 ensuremath nu for several values of rewiring probability p the critical noise q c and u were also calculated therefore the results presented here demonstrate that the majority vote model belongs to a different universality class than equilibrium ising model on apollonian network

implicit moment invariants the use of traditional moment invariants in object recognition is limited to simple geometric transforms such as rotation scaling and affine transformation of the image this paper introduces so called implicit moment invariants implicit invariants measure the similarity between two images factorized by admissible image deformations for many types of image deformations traditional invariants do not exist but implicit invariants can be used as features for object recognition in the paper we present implicit moment invariants with respect to polynomial transform of spatial coordinates describe their stable and efficient implementation by means of orthogonal moments and demonstrate their performance in artificial as well as real experiments

on the spectrum of the normalized laplacian of iterated triangulations of graphs the eigenvalues of the normalized laplacian of a graph provide information on its topological and structural characteristics and also on some relevant dynamical aspects specifically in relation to random walks in this paper we determine the spectra of the normalized laplacian of iterated triangulations of a generic simple connected graph as an application we also find closed forms for their multiplicative degree kirchhoff index kemeny s constant and number of spanning trees

on the independence of rotation moment invariants abstract the problem of the independence and completeness of rotation moment invariants is addressed in this paper first a general method for constructing invariants of arbitrary orders by means of complex moments is described as a major contribution of the paper it is shown that for any set of invariants there exists a relatively small basis by means of which all other invariants can be generated the method how to construct such a basis and how to prove its independence and completeness is presented some practical impacts of the new results are mentioned at the end of the paper

scaling of clusters near discontinuous percolation transitions in hyperbolic networks we investigate the onset of the discontinuous percolation transition in small world hyperbolic networks by studying the systems size scaling of the typical largest cluster approaching the transition p p c to this end we determine the average size of the largest cluster s max n p in the thermodynamic limit using real space renormalization of cluster generating functions for bond and site percolation in several models of hyperbolic networks that provide exact results we determine that all our models conform to the recently predicted behavior regarding the growth of the largest cluster which found diverging albeit subextensive clusters spanning the system with finite probability well below p c and at most quadratic corrections to unity in p for p p c our study suggests a large universality in the cluster formation on small world hyperbolic networks and the potential for an alternative mechanism in the cluster formation dynamics at the onset of discontinuous percolation transitions

hierarchical regular small world networks two new networks are introduced that resemble small world properties these networks are recursively constructed but retain a fixed regular degree they possess a unique one dimensional lattice backbone overlaid by a hierarchical sequence of long distance links mixing real space and small world features both networks one 3 regular and the other 4 regular lead to distinct behaviors as revealed by renormalization group studies the 3 regular network is planar has a diameter growing as with system size n and leads to super diffusion with an exact anomalous exponent dw 1 306 but possesses only a trivial fixed point tc 0 for the ising ferromagnet in turn the 4 regular network is non planar has a diameter growing as exhibits ballistic diffusion dw 1 and a non trivial ferromagnetic transition tc 0 it suggests that the 3 regular network is still quite geometric while the 4 regular network qualifies as a true small world with mean field properties as an engineering application we discuss synchronization of processors on these networks

ising model on the apollonian network with node dependent interactions this work considers an ising model on the apollonian network where the exchange constant j i j approximately 1 k i k j mu between two neighboring spins i j is a function of the degree k of both spins using the exact geometrical construction rule for the network the thermodynamical and magnetic properties are evaluated by iterating a system of discrete maps that allows for very precise results in the thermodynamic limit the results can be compared to the predictions of a general framework for spin models on scale free networks where the node distribution p k approximately k gamma with node dependent interacting constants we observe that by increasing mu the critical behavior of the model changes from a phase transition at t infinity for a uniform system mu 0 to a t 0 phase transition when mu 1 in the thermodynamic limit the system shows no true critical behavior at a finite temperature for the whole mu or 0 interval the magnetization and magnetic susceptibility are found to present noncritical scaling properties

competitive cluster growth in complex networks in this work we propose an idealized model for competitive cluster growth in complex networks each cluster can be thought of as a fraction of a community that shares some common opinion our results show that the cluster size distribution depends on the particular choice for the topology of the network of contacts among the agents as an application we show that the cluster size distributions obtained when the growth process is performed on hierarchical networks e g the apollonian network have a scaling form similar to what has been observed for the distribution of a number of votes in an electoral process we suggest that this similarity may be due to the fact that social networks involved in the electoral process may also possess an underlining hierarchical structure

inverted berezinskii kosterlitz thouless singularity and high temperature algebraic order in an ising model on a scale free hierarchical lattice small world network we have obtained exact results for the ising model on a hierarchical lattice incorporating three key features characterizing many real world networks a scale free degree distribution a high clustering coefficient and the small world effect by varying the probability p of long range bonds the entire spectrum from an unclustered non small world network to a highly clustered small world system is studied using the self similar structure of the network we obtain analytic expressions for the degree distribution p k and clustering coefficient c for all p as well as the average path length ensuremath ell for p 0 and 1 the ferromagnetic ising model on this network is studied through an exact renormalization group transformation of the quenched bond probability distribution using up to 562 phantom rule 0 2em 0ex 500 renormalized probability bins to represent the distribution for pl0 494 we find power law critical behavior of the magnetization and susceptibility with critical exponents continuously varying with p and exponential decay of correlations away from t c for p ensuremath ge 0 494 in fact where the network exhibits small world character the critical behavior radically changes we find a highly unusual phase transition namely an inverted berezinskii kosterlitz thouless singularity between a low temperature phase with nonzero magnetization and finite correlation length and a high temperature phase with zero magnetization and infinite correlation length with power law decay of correlations throughout the phase approaching t c from below the magnetization and the susceptibility respectively exhibit the singularities of mathrm exp ensuremath c sqrt t c ensuremath t and mathrm exp d sqrt t c ensuremath t with c and d positive constants with long range bond strengths decaying with distance we see a phase transition with power law critical singularities for all p and evaluate an unusually narrow critical region and important corrections to power law behavior that depend on the exponent characterizing the decay of long range interactions

stability of large amplitude viscous shock profiles of hyperbolic parabolic systems we establish nonlinear l1 h3 lp orbital stability 2 p with sharp rates of decay of large amplitude lax type shock profiles for a class of symmetric hyperbolic parabolic systems including compressible gas dynamics and magnetohydrodynamics mhd under the necessary conditions of strong spectral stability i e a stable point spectrum of the linearized operator about the wave transversality of the profile and hyperbolic stability of the associated ideal shock this yields in particular together with the spectral stability results of 50 the nonlinear stability of arbitrarily large amplitude shock profiles of isentropic navier stokes equations for a gamma law gas as 1 the first complete large amplitude stability result for a shock profile of a system with real i e partial viscosity a corresponding small amplitude result was established in 53 54 for general systems of kawashima class by a combination of kawashima type energy estimates and pointwise green function bounds where the small amplitude assumption was used only to close the energy estimates here under the mild additional assumption that hyperbolic characteristic speeds relative to the shock are not only nonzero but of a common sign we close the estimates instead by use of a goodman type weighted norm 25 26 designed to control estimates in the crucial hyperbolic modes