an iterative algorithm for computing mean first passage times of markov chains abstract mean first passage times are an essential ingredient in both the theory and the applications of markov chains in the literature they have been expressed in elegant closed form formulas these formulas involve explicit full matrix inversions and if computed directly may incur numerical instability in this paper we present a new iterative algorithm for computing mean first passage times in a manner that does not rely on explicit full matrix inversions results regarding the convergence behavior of this algorithm are also developed

interaction of diffusion and boundary conditions on considere le systeme d equations paraboliques u t d u f u x d u n e x u 0 x ou u r n r n n 3 est un ouvert borne de frontiere lisse d diag d 1 d n e diag e 1 e n chaque d j 0 est constant e j r est continu e j 0 j 1 2 n et e 0 est constant pour f e fixe on etudie comment le flot defini ci dessus depend des parametres d

low firing rates in a compartmental model neuron a nonlinear relationship is obtained between the membrane potential of a compartmental model neuron as a function of input stimulation this is used to show how shunting effects can lead to low output firing rates in associative memory networks

local nonlinear stability of the steady state in an isothermal catalyst a first order irreversible exothermic reaction in a bounded porous catalyst is considered with smooth boundary of one two or three dimensions for small prater and nusselt numbers beta and nu and a large sherwood number sigma two isothermal models are derived an analysis of linear stability of the steady states of such models shows that oscillatory instabilities appear for appropriate values of the damkohler number if the nondimensional activation energy is larger than gamma and the lewis number is sufficiently large where gamma 4 if m nu beta sigma leqq 1 and gamma m 1 2 m if m 1 a local hopf bifurcation analysis is carried out at neutral stability points in order to ascertain whether such bifurcation is subcritical or supercritical

extension of hereditary symmetry operators two models of candidates for hereditary symmetry operators are proposed and thus many nonlinear systems of evolution equations possessing infinitely many commuting symmetries may be generated some concrete structures of hereditary symmetry operators are carefully analysed on the basis of the resulting general conditions and several corresponding nonlinear systems are explicitly given as illustrative examples

matrix analysis of a markov chain small world model abstract recently d higham showed that the small world phenomenon arising in a ring network of n nodes can be modelled by a markov chain which depends on a parameter of the form k n where k 0 and 1 the parameter can be viewed as a transitional factor interpolating between the completely local and completely global configurations of the network higham analyzed the markov chain model by a combination of matrix perturbation theory and finite difference approximations to an underlying boundary value problem using such tools he obtained asymptotic results for the limiting case when n is sufficiently large furthermore for large n higham verified the small world phenomenon on the network in the case when 3 motivated by higham s work we show that the markov chain of the small world model can be investigated more completely by direct matrix theoretic methods which produces exact results for all n and for all our results therefore allow a fuller examination of the behavior of the small world phenomenon in the markov chain model for small to moderate values of n and for all 1

a generalization of n matrices abstract ky fan defines an n matrix to be a matrix of the form a ti b b 0 t b where b is the spectral radius of b and is the maximum of the spectral radii of all principal submatrices of b of order n 1 in this paper we define the closure n 0 matrices of n matrices by letting t it is shown that if a z and a 1 a n certain inequalities of n matrices are shown to hold for n 0 matrices and a method for constructing an n matrix from an m matrix is given

an asymptotic justification of a model of isothermal catalyst an asymptotic model of isothermal catalyst is obtained rigorously from a well known model of porous catalyst for appropriate limiting values of some of the nondimensional parameters large lewis number small prater and thermal biot numbers and bounded damkohler number

nonexistence of global solutions of a nonlinear hyperbolic system consider the initial value problem utt au ivip vtt av u q x e r t t o u x 0 f x v x 0 h x ut x 0 g x vt x 0 x with 1 0 we show that there exists a bound b n oo such that if 1 pq b n all nontrivial solutions with compact support blow up in finite time

nonexistence of positive solutions to a semilinear elliptic system and blow up estimates for a reaction diffusion system abstract we get simple conditions under which the elliptic system u u p 1 v q 1 v u p 2 v q 2 in r n n 3 with p i q i 1 i 1 2 has no positive radially symmetric solutions then by using this nonexistence result we establish blow up estimates for reaction diffusion system of fujita type u t u u p 1 v q 1 v t v u p 2 v q 2 with the homogeneous dirichlet boundary value condition

mathematical analysis of a chemical reaction with lumped temperature and strong absorption abstract we consider a mathematical model for the evolution of a single reactant and of the temperature in an isothermal catalyst the temperature is assumed to be spatially homogeneous and the absorption term is supposed to be nonlipschitzian existence and uniqueness of a regular solution is proved together with some bounds the steady state problem is also investigated and information about the dead core of the reactant are given

parabolic evolution equations and nonlinear boundary conditions on etudie le probleme aux valeurs limites et initiales u 7b a t u f t u 7b b t u g t u 0 t t u 0 u 0 qui peut etre considere comme une contrepartie abstraite a des problemes paraboliques semilineaires

the role of critical exponents in blow up theorems the sequel abstract in 27 fujita showed that for positive solutions the initial value problem in r n for u t u u p with p 1 exhibited the following behavior if p p c 1 2 n then the initial value problem does not have any nontrivial non negative solution existing on r n 0 a global solution whereas if p p c there exist global small data positive solutions as well as solutions which are non global we call such a result a blow up theorem of fujita type in 50 levine discussed the various theorems of this type that had appeared in the literature prior to 1990 in this paper we revisit the literature since 1990

transition matrices for well conditioned markov chains abstract let t r n n be an irreducible stochastic matrix with stationary distribution vector set a i t and define the quantity 3 t 1 2 max j 1 n j a j 1 where aj j 1 n are the n 1 n 1 principal submatrices of a obtained by deleting the jth row and column of a results of cho and meyer and of kirkland show that 3 provides a sensitive measure of the conditioning of under perturbation of t moreover it is known that 3 t n 1 2 n in this paper we investigate the class of irreducible stochastic matrices t of order n such that 3 t n 1 2 n for such matrices correspond to markov chains with desirable conditioning properties we identify some restrictions on the zero nonzero patterns of such matrices and construct several infinite classes of matrices for which 3 is as small as possible

a hierarchy of coupled burgers systems possessing a hereditary structure a hierarchy of typical integrable coupled burgers systems is proposed by introducing a special spectral problem involving q dependent variables u0 u1 uq 1 these systems possess a hereditary structure and may reduce to a hierarchy of burgers equations under the reduction ui 0 0 or i or q 2 it is shown that their flows and lax operators commute mutually but each system is not hamiltonian

diffusivity versus absorption through the boundary for all 0 1 then we do expect to find that any solution becomes unbounded in finite time see 13 15 63 it is our goal to prove an analogous result for m f 1 for a 1 one can find that if v is a positive solution of

global existence and nonexistence for semilinear parabolic systems with nonlinear boundary conditions on etudie des problemes aux valeurs limites et initiales paraboliques semilineaires de la forme t u j a jk k u a 0 u f u dans 0 a jk j k u bu 1 u g u sur 0 u 0 u 0 sur ou est un domaine borne de r n n 1 de classe c 2

eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations abstract udy the perturbation theory of structured matrices under structured rank one perturbations and then focus on several classes of complex matrices generic jordan structures of perturbed matrices are identified it is shown that the perturbation behavior of the jordan structures in the case of singular j hamiltonian matrices is substantially different from the corresponding theory for unstructured generic rank one perturbation as it has been studied in 18 28 30 31 thus a generic structured perturbation would not be generic if considered as an unstructured perturbation in other settings of structured matrices the generic perturbation behavior of the jordan structures within the confines imposed by the structure follows the pattern of that of unstructured perturbations

on the first and second order derivatives of the perron vector abstract in a previous work 5 the authors developed formulas for the second order partial derivatives of the perron root as a function of the matrix entries at an essentially nonnegative and irreducible matrix these formulas which involve the group generalized inverse of an associated m matrix were used to investigate the concavity and convexity of the perron root as a function of the entries the authors now combine the above results together with an approach taken in an earlier joint paper 6 of the second author with l elsner and c johnson and they develop formulas for the second order derivatives of an appropriately normalized perron vector with respect to the matrix entries which again are given in terms the group generalized inverse of an associated m matrix convexity properties of the perron vector as a function of the entries of the matrix are then examined in addition formulas for the first derivative of the perron vector resulting from different normalizations of this eigenvector are also given a by product of one of these formulas yields that the group generalized inverse of a singular and irreducible m matrix can be diagonally scaled to a matrix which is entrywise column diagonally dominant

hamiltonian theory over noncommutative rings and integrability in multidimensions a generalization of the adler gel fand dikii scheme is used to generate bi hamiltonian structures in two spatial dimensions in order to implement this scheme a hamiltonian theory is built over a noncommutative ring namely the ring of formal pseudodifferential operators bi hamiltonian structures generated in this way can be used for the kadomtsev petviashvili equation as well as other integrable equations in 2 1

nijenhuis g manifolds and lenard bicomplexes a new approach to kp systems we suggest a method to extend the theory of recursion operators to integrable hamiltonian systems in two space dimensions like kp systems the approach aims to stress the conceptual unity of the theories in one and two space dimensions a sound explanation of the appearance of bilocal operators is also given

trapping in scale free networks with hierarchical organization of modularity a wide variety of real life networks share two remarkable generic topological properties scale free behavior and modular organization and it is natural and important to study how these two features affect the dynamical processes taking place on such networks in this paper we investigate a simple stochastic process trapping problem a random walk with a perfect trap fixed at a given location performed on a family of hierarchical networks that exhibit simultaneously striking scale free and modular structure we focus on a particular case with the immobile trap positioned at the hub node having the largest degree using a method based on generating functions we determine explicitly the mean first passage time mfpt for the trapping problem which is the mean of the node to trap first passage time over the entire network the exact expression for the mfpt is calculated through the recurrence relations derived from the special construction of the hierarchical networks the obtained rigorous formula corroborated by extensive direct numerical calculations exhibits that the mfpt grows algebraically with the network order concretely the mfpt increases as a power law function of the number of nodes with the exponent much less than 1 we demonstrate that the hierarchical networks under consideration have more efficient structure for transport by diffusion in contrast with other analytically soluble media including some previously studied scale free networks we argue that the scale free and modular topologies are responsible for the high efficiency of the trapping process on the hierarchical networks

integrable coupled kdv systems we give the conditions for a system of n coupled korteweg de vries kdv type of equations to be integrable we find the recursion operators of each subclass and give all examples for n 2

global existence and blow up problems for reaction diffusion model with multiple nonlinearities abstract in this paper we study the following reaction diffusion model u t a u b x c t u g x t f u in d 0 t u n h x t r u on d 0 t u x 0 u 0 x 0 in d where d is a bounded domain in r n with smooth boundary d n 2 this paper deals with interactions among three kinds of nonlinear mechanisms nonlinear reaction nonlinear convection and nonlinear boundary flux the existence theorems of blow up positive solutions upper bound of blow up time upper estimates of blow up rate existence theorems of global positive solutions and upper estimates of global positive solutions are given under suitable assumptions on a b c f g h r and initial data u 0 x

global existence and blow up of solutions to a class of doubly degenerate parabolic equations coupled via nonlinear boundary flux the paper deals with the critical curve for a degenerate parabolic system coupled via nonlinear boundary flux by constructing the self similar super solution and subsolution we obtain the critical global existence curve and the critical curve of fujita type is conjectured with the aid of some new results an interesting feature of our results is that the critical global existence curve and the critical fujita curve are determined by a matrix and by the solution of a linear algebraic system respectively

on the theory of recursion operator the general structure and properties of recursion operators for hamiltonian systems with a finite number and with a continuum of degrees of freedom are considered weak and strong recursion operators are introduced the conditions which determine weak and strong recursion operators are found

the asymptotic justification of a nonlocal 1 d model arising in porous catalyst theory an asymptotic model of isothermal catalyst is obtained from a well known model of porous catalyst for appropriate realistic limiting values of some nondimensional parameters in this limit the original model is a singularly perturbedm d reaction diffusion system the asymptotic model consists of an ordinary differential equation coupled with a semilinear parabolic equation on a semi infinite one dimensional interval

critical exponents and blow up rate for a nonlinear diffusion equation with logarithmic boundary flux abstract in this paper we establish the critical global existence exponent and the critical fujita exponent for the nonlinear diffusion equation u t log 1 u u x x in r 0 subject to a logarithmic boundary flux log 1 u u x 0 t 1 u log p 1 u 0 t t 0 furthermore give the blow up rate for the nonglobal solutions

blow up solutions and global solutions for nonlinear parabolic problems abstract this paper deals with the blow up of positive solutions for a nonlinear parabolic equation subject to nonlinear boundary conditions we obtain the conditions under which the solutions may exist globally or blow up in a finite time by a new approach moreover upper estimates of the blow up time blow up rate and global solutions are obtained also

critical curves for fast diffusive non newtonian equations coupled via nonlinear boundary flux this paper deals with the critical curve for a nonlinear boundary value problem of a fast diffusive non newtonian system we first obtain the critical global existence curve by constructing the self similar supersolution and subsolution and then the critical fujita curve is conjectured with the aid of some new results

derivatives of the perron root at an essentially nonnegative matrix and the group inverse of an m matrix abstract the results of the work in this paper can be divided into two major parts in the first part we develop formulae using group inverses of irreducible m matrices for the second order partial derivatives with respect to the entries of the perron root r at an essentially nonnegative and irreducible matrix a we use these formulae to obtain explicit expressions for the matrices 2 r a ij 2 and 2 r a ii jj and examine these expressions at some special cases when for example in addition to a being essentially nonnegative and irreducible a is row stochastic or a is of rank 1 and so on in the second major part of this paper we investigate for a singular and irreducible m matrix q a certain matrix g which is a positive diagonal scaling of the group inverse q of q we show for instance that z gz 0 for all real vectors z and use this result and some of the formulae obtained in the first part of the paper to provide a matrix theoretic proof of a theorem due to j e cohen math proc cambridge philos soc 86 1979 343 350 which states that for an essentially nonnegative and irreducible matrix a the perron root is a convex function of the main diagonal of a

sharp bounds on the spectral radius of nonnegative matrices and digraphs abstract the spectral radius a of a matrix a is the maximum modulus of its eigenvalues we present bounds on a that are often tighter and are applicable to a larger class of nonnegative matrices than previously reported the bounds are particularly suited to matrices which are sparse we complete the paper by applying these bounds to digraphs deriving the associated equality conditions which relate to the outdegree regularity of the digraph finally we show that the equality conditions may be achieved only for very specific values of the digraph s spectral radius

blow up rate for a semilinear reaction diffusion system abstract this paper deals with the blow up rate of positive solution to semilinear reaction diffusion system u1 t u1 u2p1 un 1 t un 1 unpn 1 un u1 u1pn with null dirichlet boundary conditions the upper and lower bounds of blow up rate were obtained

blow up for quasilinear heat equations with critical fujita s exponents we consider the cauchy problem for the quasilinear heat equationwhere 0 is a fixed constant with the critical exponent in the source term c 1 2 n it is well known that if 1 c then any non negative weak solution u x t 0 blows up in a finite time for the semilinear heat equation he with 0 the above result was proved by h fujita 4 in the present paper we prove that u 0 blows up in the critical case 1 2 n with 0 a similar result is valid for the equation with gradient dependent diffusivitywith 0 and the critical exponent 1 2 n

blow up and blow up rate for a reaction diffusion model with multiple nonlinearities abstract this paper deals with interactions among three kinds of nonlinear mechanisms nonlinear diffusion nonlinear reaction and nonlinear boundary flux in a parabolic model with multiple nonlinearities the necessary and sufficient blow up conditions are established together with blow up rate estimates for the positive solutions of the problem

critical curves for degenerate parabolic equations coupled via non linear boundary flux this paper deals with the critical curve for a degenerate parabolic system coupled via non linear boundary flux by constructing the self similar supersolution and subsolution we obtain the critical global existence curve the critical curve of fujita type is conjectured with the aid of some new results the possible further extensions are also presented

the balance between nonlinear inwards and outwards boundary flux for a nonlinear heat equation abstract we study nonnegative solutions of u t u m xx x t 0 l 0 t u m x 0 t u p 0 t t 0 t u m x l t u q l t t 0 t u x 0 u 0 x x 0 l where m p q and l are positive parameters for different values of the parameters three situations may occur 1 all solutions of this problem exist for all t 0 2 for certain initial data functions the solution exists for all t 0 while for others the solution blows up as t t for some finite t 3 excepting the trivial solution when u0 0 all solutions blow up as t t for some finite t we identify in terms of the parameters which of them actually happens for solutions which blow up we find the blow up rate and the blow up set

critical curves for a degenerate parabolic equation with multiple nonlinearities this paper is concerned with the critical curves of a degenerate parabolic equation with a source and nonlinear boundary flux we obtain the critical global existence curve and critical fujita curve by constructing various self similar supersolutions and subsolutions

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

a classification of matrices of class z abstract we generalize the classes n 0 and f 0 studied by k fan g johnson and r smith schur complements and lattices are examined for matrices in these classes

a nonlinear diffusion system coupled via nonlinear boundary flux abstract in this paper we deal with the global existence and nonexistence of solutions to a nonlinear diffusion system coupled via nonlinear boundary flux by constructing various kinds of sub and super solutions and using the basic properties of m matrix we give the necessary and sufficient conditions for global existence of nonnegative solutions the critical curve of fujita type is conjectured with the aid of some new results which extend the recent results of wang et al nonlinear anal 71 2009 2134 2140 and li et al j math anal appl 340 2008 876 883 to more general equations

lu decomposition of m matrices by elimination without pivoting abstract it is shown that if a or a is a singular m matrix satisfying the generalized diagonal dominance condition y t a 0 for some vector y 0 then a can be factored into a lu by a certain elimination algorithm where l is a lower triangular m matrix with unit diagonal and u is an upper triangular m matrix the existence of lu decomposition of symmetric permutations of a and for irreducible m matrices and symmetric m matrices follow as colollaries this work is motivated by applications to the solution of homogeneous systems of linear equations ax 0 where a or a is an m matrix these applications arise e g in the analysis of markov chains input output economic models and compartmental systems a converse of the theorem metioned above can be established by considering the reduced normal form of a

critical exponents for a fast diffusive polytropic filtration equation with nonlinear boundary flux abstract in this paper we deal with a fast diffusive polytropic filtration equation u t x u m x p 2 u m x 1 p 1 1 m in r 0 subject to a nonlinear boundary flux u m x p 2 u m x 0 t u q 0 t t 0 we first get the behavior of the solution at infinity and establish the critical global existence exponent and critical fujita exponent for the fast diffusive polytropic filtration equation furthermore give the blow up set and upper bound of the blow up rate for the nonglobal solutions

coupled kdv equations with multi hamiltonian structures abstract we present a large class of systems of n equations which possess n 1 purely differential compatible hamiltonian structures our generic equations are isospectral to n 1 0 e i 2 i 2 n 1 0 i 2 i 2 n but our class also includes degenerate nondispersive systems which are unrelated to this linear problem embedded in this class for each n there are n distinct coupled kdv systems when n 2 our class includes 3 known equations dispersive water waves ito s equation and reduced benney s equations these equations are thus tri hamiltonian we also present 2 examples of 3 component quadri hamiltonian systems which generalise the above mentioned dispersive and nondispersive water waves

perron complement and perron root for a nonnegative irreducible matrix a this paper is concerned with the estimation and determination of the unique perron root or spectral radius of a we present a new method that utilizes the relation between perron roots of the nonnegative matrix and its generalized perron complement several numerical examples are given to show that our method is effective at least for some classes of nonnegative matrices

two types of multicomponent matrix loop algebras and its application two types of multicomponent matrix lie algebras are constructed which are devoted to obtain two types of new loop algebras am 1 by making use of tu scheme integrable multicomponent levi hierarchy and multicomponent ablowitz kaup newell segur akns hierarchy are generated which contain an arbitrary positive integer m furthermore we expanded the multicomponent matrix loop algebra into a large one and work out integrable coupling of levi hierarchy and akns hierarchy this method proposed in this paper can be used generally

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

a formula for obtaining new hereditary symmetries and new integrable equations in this paper we give a simple formula to obtain hereditary symmetries related to the isospectral eigenvalue problem of integrable equations using this formula i we prove that eigenfunctions of hereditary strong symmetry are symmetries for the whole hierarchy which improves the result of fokas and anderson j math phys 23 1066 1982 ii we find new integrable equations and iii we give strong symmetries of these new integrable equations and various equations for eigenfunctions of the isospectral eigenvalue problems

on the multi component extensions of the burgers equation we construct a multi component burgers equation by means of a hereditary symmetry this n component system has n parameters

the algebraic structure of zero curvature representations and application to coupled kdv systems the author first establishes an algebraic structure related to zero curvature representations and propose a new approach for calculating symmetry algebras of integrable systems then he deduces a hierarchy of nonisospectral flows associated with coupled kdv systems from a spectral problem with the laurent polynomial dependent form of the spectral parameter furthermore the commutator relations of lax operators corresponding to isospectral and nonisospectral flows are worked out according to this algebraic structure and thus a symmetry algebra for coupled kdv systems is engendered from this general theory

recursion operators and bi hamiltonian structures in multidimensions ii the algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied in particular the factorization of certain operators called recursion operators establishes the bi hamiltonian nature of all these equations recently we have presented the recursion operator and the bi hamiltonian formulation of the kadomtsev petviashvili equation a two spatial dimensional analogue of the korteweg devries equation here we present the general theory associated with recursion operators for bi hamiltonian equations in two spatial and one temporal dimensions as an application we show that general classes of equations which include the kadomtsev petviashvili and the davey stewartson equations possess infinitely many commuting symmetries and infinitely many constants of motion in involution under two distinct poisson brackets furthermore we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems

on thelu factorization ofm matrices in this paper we give in theorem 1 a characterization based on graph theory of when anm matrixa admits anlu factorization intom matrices wherel is a nonsingular lower triangularm matrix andu is an upper triangularm matrix this result generalizes earlier factorization results of fiedler and ptak 1962 and kuo 1977 as a consequence of theorem 1 we show in theorem 3 that the conditionx t a 0 t for somex 0 for anm matrixa is both necessary and sufficient forpap t to admit such anlu factorization for everyn n permutation matrixp this latter result extends recent work of funderlic and plemmons 1981 finally theorem 1 is extended in theorem 5 to give a characterization similarly based on graph theory of when anm matrixa admits anlu factorization intom matrices

k symmetries and tau symmetries of evolution equations and their lie algebras on the basis of tu s work 1951 the author establishes a more general skeleton on k symmetries and tau symmetries of evolution equations and their lie algebraic structures and discuss carefully the corresponding symmetries and lie algebras for kdv and jaulent miodek hierarchies

distinct scalings for mean first passage time of random walks on scale free networks with the same degree sequence in general the power law degree distribution has profound influence on various dynamical processes defined on scale free networks in this paper we will show that power law degree distribution alone does not suffice to characterize the behavior of trapping problems on scale free networks which is an integral major theme of interest for random walks in the presence of an immobile perfect absorber in order to achieve this goal we study random walks on a family of one parameter denoted by q scale free networks with identical degree sequence for the full range of parameter q in which a trap is located at a fixed site we obtain analytically or numerically the mean first passage time mfpt for the trapping issue in the limit of large network order number of nodes for the whole class of networks the mfpt increases asymptotically as a power law function of network order with the exponent obviously different for different parameter q which suggests that power law degree distribution itself is not sufficient to characterize the scaling behavior of mfpt for random walks at least trapping problem performed on scale free networks

a model of porous catalyst accounting for incipiently non isothermal effects abstract an approximate model accounting for incipiently non isothermal effects is derived from a well known model of porous catalyst for appropriate realistic limiting values of the parameters in this limit the original model is a singularly perturbed m d reaction diffusion system and the approximate model is given by the m d heat equation with nonlinear boundary condition coupled with infinitely many ifm 2 1 d semilinear parabolic equations one for each point of the boundary of the spatial domain some limiting cases are still considered in the approximate model that lead to further simplifications

global solvability for the porous medium equation with boundary flux governed by nonlinear memory abstract we introduce the study of global existence and blow up in finite time for nonlinear diffusion equations with flux at the boundary governed by memory via a simple transformation the memory term arises out of a corresponding model introduced in previous studies of tumor induced angiogenesis the present study is also in the spirit of extending work on models of the heat equation with local nonlocal and delay nonlinearities present in the boundary flux specifically we establish an identical set of necessary and sufficient conditions for blow up in finite time as previously established in the case of local flux conditions at the boundary