on a class of degenerate parabolic equations with dynamic boundary conditions we consider a quasi linear parabolic possibly degenerate equation with nonlinear dynamic boundary conditions the corresponding class of initial and boundary value problems has already been studied previously proving well posedness of weak solutions and the existence of the global attractor assuming that the nonlinearities are subcritical to a given exponent the goal of this article is to show that the previous analysis can be redone for supercritical nonlinearities by proving an additional l estimate on the solutions in particular we derive new conditions which reflect an exact balance between the internal and the boundary mechanisms involved even when both the nonlinear sources contribute in opposite directions then we show how to construct a trajectory attractor for the weak solutions of the associated parabolic system and prove that any solution belonging to the attractor is bounded which implies uniqueness finally we also prove for the semilinear reaction diffusion equation with nonlinear dynamic boundary conditions that the fractal dimension of the global attractor is of the order n 1 as diffusion 0 in any space dimension n 2 improving some recent results in gal 2012 23

blow up for some quasilinear equations with dynamical boundary conditions of parabolic type sufficient conditions for existence of blow up solutions to parabolic and elliptic equations with dynamical boundary conditions having positive or vanishing initial energy are obtained estimates for the time interval of solution existence are given we also discuss the blow up rates and blow up sets by means of numerical experiments

a note on blow up of solutions of a quasilinear heat equation with vanishing initial energy abstract in this work we consider an initial boundary value problem related to the equation u t div u m 2 u f u and prove under suitable conditions on f a blow up result for solutions with vanishing or negative initial energy

consistency property of extended least squares parameter estimation for stochastic diffusion equation the extended least squares parameter estimate for stochastic heat diffusion equations is considered the unknown parameter is a heat diffusion coefficient which is a function of a spatial variable almost sure convergence for the estimated parameter is proved a numerical example is demonstrated for supporting the theoretical results developed here

random sequential renormalization of networks application to critical trees we introduce the concept of random sequential renormalization rsr for arbitrary networks rsr is a graph renormalization procedure that locally aggregates nodes to produce a coarse grained network it is analogous to the quasi parallel renormalization schemes introduced by c song et al c song et al nature london 433 392 2005 and studied by f radicchi et al f radicchi et al phys rev lett 101 148701 2008 but much simpler and easier to implement here we apply rsr to critical trees and derive analytical results consistent with numerical simulations critical trees exhibit three regimes in their evolution under rsr i for n 0 ensuremath nu ensuremath lesssim nl n 0 where n is the number of nodes at some step in the renormalization and n 0 is the initial size of the tree rsr is described by a mean field theory and fluctuations from one realization to another are small the exponent ensuremath nu 1 2 is derived using random walk and other arguments the degree distribution becomes broader under successive steps reaching a power law p k 1 k ensuremath gamma with ensuremath gamma 2 and a variance that diverges as n 0 1 2 at the end of this regime both of these latter results are obtained from a scaling theory ii for n 0 ensuremath nu mathrm star ensuremath lesssim n ensuremath lesssim n 0 1 2 with ensuremath nu mathrm star ensuremath approx 1 4 hubs develop and fluctuations between different realizations of the rsr are large trees are short and fat with an average radius that is mathcal o 1 crossover functions exhibiting finite size scaling in the critical region n n 0 1 2 ensuremath rightarrow ensuremath infty connect the behaviors in the first two regimes iii for n ensuremath lesssim n 0 ensuremath nu mathrm star star configurations appear with a central hub surrounded by many leaves the distribution of stars is broadly distributed over this range the scaling behaviors found under rsr are identified with a continuous transition in a process called agglomerative percolation ap with the coarse grained nodes in rsr corresponding to clusters in ap that grow by simultaneously attaching to all their neighboring clusters

agglomerative percolation on the bethe lattice and the triangular cactus agglomerative percolation ap on the bethe lattice and the triangular cactus is studied to establish the exact mean field theory for ap using the self consistent simulation method based on the exact self consistent equations the order parameter p and the average cluster size s are measured from the measured p and s the critical exponents k and k for k 2 and 3 are evaluated here k and k are the critical exponents for p and s when the growth of clusters spontaneously breaks the zk symmetry of the k partite graph the obtained values are 2 1 79 3 2 0 88 1 3 1 35 5 and 3 0 94 2 by comparing these exponents with those for ordinary percolation 1 and 1 we also find 3 2 these results quantitatively verify the conjecture that the ap model belongs to a new universality class if the zk symmetry is broken spontaneously and the new universality class depends on k

parameter identification for hyperbolic stochastic systems parameter identification is studied for hyperbolic stochastic systems explicit sufficient conditions ensuring consistency of the maximum likelihood estimate are given

convergence results for continuous time adaptive stochastic filtering algorithms abstract the adaptive stochastic filtering problem for gaussian processes is considered the self tuning synthesis procedure is used to derive two algorithms for this problem almost sure convergence for the parameter estimate and the filtering error will be established the convergence analysis is based on an almost supermartingale convergence lemma that allows a stochastic lyapunov like approach

regularized maximum likelihood estimate for an infinite dimensional parameter in stochastic parabolic systems the purpose of this paper is to study the identification problem of an infinite dimensional parameter more precisely a spatially varying parameter in stochastic diffusion equations in a previous study s i aihara and y sunahara siam j control optim 26 1988 pp 1062 1075 some explicit conditions for the consistency property of the maximum likelihood estimate mle is explored here an algorithm for generating the mle is developed with the aid of the regularization technique proposed by c kravaris and j h seinfeld siam j control optim 23 1985 pp 217 241 after the consistency property of the mle by a regularization is proved necessary conditions for the regularized mle rmle are derived proposed is an iterative algorithm for computing one of the solutions of the necessary conditions derived the convergence property of the sequence generated by the proposed algorithm is also shown finally numerical examples are presented

renormalization flows in complex networks complex networks have acquired a great popularity in recent years since the graph representation of many natural social and technological systems is often very helpful to characterize and model their phenomenology additionally the mathematical tools of statistical physics have proven to be particularly suitable for studying and understanding complex networks nevertheless an important obstacle to this theoretical approach is still represented by the difficulties to draw parallelisms between network science and more traditional aspects of statistical physics in this paper we explore the relation between complex networks and a well known topic of statistical physics renormalization a general method to analyze renormalization flows of complex networks is introduced the method can be applied to study any suitable renormalization transformation finite size scaling can be performed on computer generated networks in order to classify them in universality classes we also present applications of the method on real networks

agglomerative percolation on bipartite networks nonuniversal behavior due to spontaneous symmetry breaking at the percolation threshold ordinary bond percolation op can be viewed as a process where clusters grow by joining them pairwise adding links chosen randomly one by one from a set of predefined virtual links in contrast in agglomerative percolation ap clusters grow by choosing randomly a target cluster and joining it with all its neighbors as defined by the same set of virtual links previous studies showed that ap is in different universality classes from op for several types of virtual networks linear chains trees erdos renyi networks but most surprising were the results for two dimensional 2d lattices while ap on the triangular lattice was found to be in the op universality class it behaved completely differently on the square lattice in the present paper we explain this striking violation of universality by invoking bipartivity while the square lattice is a bipartite graph the triangular lattice is not in conformity with this we show that ap on the honeycomb and simple cubic 3d lattices both of which are bipartite are also not in the op universality classes more precisely we claim that this violation of universality is basically due to a z 2 symmetry that is spontaneously broken at the percolation threshold we also discuss ap on bipartite random networks and suitable generalizations of ap on k partite graphs

complex networks renormalization flows and fixed points recently it has been claimed that some complex networks are self similar under a convenient renormalization procedure we present a general method to study renormalization flows in graphs we find that the behavior of some variables under renormalization such as the maximum number of connections of a node obeys simple scaling laws characterized by critical exponents this is true for any class of graphs from random to scale free networks from lattices to hierarchical graphs therefore renormalization flows for graphs are similar as in the renormalization of spin systems an analysis of classic renormalization for percolation and the ising model on the lattice confirms this analogy critical exponents and scaling functions can be used to classify graphs in universality classes and to uncover similarities between graphs that are inaccessible to a standard analysis

epidemic dynamics and endemic states in complex networks we study by analytical methods and large scale simulations a dynamical model for the spreading of epidemics in complex networks in networks with exponentially bounded connectivity we recover the usual epidemic behavior with a threshold defining a critical point below that the infection prevalence is null on the contrary on a wide range of scale free networks we observe the absence of an epidemic threshold and its associated critical behavior this implies that scale free networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents might possess these results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks

continuity of the explosive percolation transition the explosive percolation problem on the complete graph is investigated via extensive numerical simulations we obtain the cluster size distribution at the moment when the cluster size heterogeneity becomes maximum the distribution is found to be well described by the power law form with the decay exponent ensuremath tau 2 06 2 followed by a hump we then use the finite size scaling method to make all the distributions at various system sizes up to n 2 37 collapse perfectly onto a scaling curve characterized solely by the single exponent ensuremath tau we also observe that the instant of that collapse converges to a well defined percolation threshold from below as n ensuremath rightarrow ensuremath infty based on these observations we show that the explosive percolation transition in the model should be continuous contrary to the widely spread belief of its discontinuity

a singularity theory approach to a semilinear boundary value problem where a b 3 are real parameters with 0 a b 1 this problem arises in population genetics with the unknown function u corresponding to the relative frequency of a gene the parameter a is given by a combination of the death rates of the various genotypes involved b gives the frequency of the gene on the boundary of the region and 1 is the reciprocal of the diffusion rate of the gene see aronson and weinberger 1 for further details solutions of 1 1 correspond to possible steady states of the gene populations we are interested mainly in how the bifurcation diagram changes as a and b vary and we investigate this question by exploiting singularity theory for bifurcation problems as developed by golubitsky and schaeffer c51 equation 1 1 is a simple example of a semilinear elliptic boundary value problem and a standard method of investigating the bifurcation of

percolation in media with columnar disorder we study a generalization of site percolation on a simple cubic lattice where not only single sites are removed randomly but also entire parallel columns of sites we show that typical clusters near the percolation transition are very anisotropic with different scaling exponents for the sizes parallel and perpendicular to the columns below the critical point there is a griffiths phase where cluster size distributions and spanning probabilities in the direction parallel to the columns have power law tails with continuously varying non universal powers this region is very similar to the griffiths phase in subcritical directed percolation with frozen disorder in the preferred direction and the proof follows essentially the same arguments as in that case but in contrast to directed percolation in disordered media the number of active growth sites in a growing cluster at criticality shows a power law while the probability of a cluster to continue to grow shows logarithmic behavior

exact closed form of the return probability on the bethe lattice an exact closed form solution for the return probability of a random walk on the bethe lattice is given the long time asymptotic form confirms a previously known expression it is however shown that this exact result reduces to the proper expression when the bethe lattice degenerates on a line unlike the asymptotic result which is singular this is shown to be an artefact of the asymptotic expansion the density of states is also calculated

fractality in complex networks critical and supercritical skeletons fractal scaling char22 a power law behavior of the number of boxes needed to tile a given network with respect to the lateral size of the box char22 is studied we introduce a box covering algorithm that is a modified version of the original algorithm introduced by song et al nature london 433 392 2005 this algorithm enables easy implementation fractal networks are viewed as comprising a skeleton and shortcuts the skeleton embedded underneath the original network is a special type of spanning tree based on the edge betweenness centrality it provides a scaffold for the fractality of the network when the skeleton is regarded as a branching tree it exhibits a plateau in the mean branching number as a function of the distance from a root for nonfractal networks on the other hand the mean branching number decays to zero without forming a plateau based on these observations we construct a fractal network model by combining a random branching tree and local shortcuts the scaffold branching tree can be either critical or supercritical depending on the small worldness of a given network for the network constructed from the critical supercritical branching tree the average number of vertices within a given box grows with the lateral size of the box according to a power law an exponential form in the cluster growing method the critical and supercritical skeletons are observed in protein interaction networks and the world wide web respectively the distribution of box masses i e the number of vertices within each box follows a power law p m m ensuremath sim m ensuremath ensuremath eta the exponent ensuremath eta depends on the box lateral size ensuremath ell b for small values of ensuremath ell b ensuremath eta is equal to the degree exponent ensuremath gamma of a given scale free network whereas ensuremath eta approaches the exponent ensuremath tau ensuremath gamma ensuremath gamma ensuremath 1 as ensuremath ell b increases which is the exponent of the cluster size distribution of the random branching tree finally we study the perimeter h ensuremath alpha of a given box ensuremath alpha i e the number of edges connected to different boxes from a given box ensuremath alpha as a function of the box mass m b ensuremath alpha it is obtained that the average perimeter over the boxes with box mass m b is likely to scale as h m b ensuremath sim m b irrespective of the box size ensuremath ell b

small world to fractal transition in complex networks a renormalization group approach we show that renormalization group rg theory applied to complex networks is useful to classify network topologies into universality classes in the space of configurations the rg flow readily identifies a small world fractal transition by finding i a trivial stable fixed point of a complete graph ii a nontrivial point of a pure fractal topology that is stable or unstable according to the amount of long range links in the network and iii another stable point of a fractal with shortcuts that exist exactly at the small world fractal transition as a collateral the rg technique explains the coexistence of the seemingly contradicting fractal and small world phases and allows us to extract information on the distribution of shortcuts in real world networks a problem of importance for information flow in the system

adaptive control via a simple switching algorithm in this paper we present an adaptive stabilization control for systems with unknown constant parameters and stochastic disturbances which may be neither open loop stable nor minimum phase the ideas come from previous works j f zhang and h f chen adaptive stabilization under the weakest condition proc 31st control and design conference december 14 18 1992 pp 3620 3621 and h f chen continuous time stochastic adaptive control stabilizing the system and minimizing the quadratic loss function tech report institute of systems science academia sinica beijing 1992 but here we not only simplify the construction procedure of an adaptive control but also reduce the computational load significantly so that the adaptive control in this paper is more practical furthermore parameter estimation is carried out in only a finite time period and unlike previous work the parameter estimates are generated by ordinary differential equations rather than stochastic differential equations