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

a spectral multiplier theorem for a sublaplacian on su 2 we prove a hormander type spectral multiplier theorem for a sublaplacian on su 2 with critical index determined by the euclidean dimension of the group this result is the analogue for su 2 of the result for the heisenberg group obtained by d muller and e m stein and by w hebisch

spanning trees on graphs and lattices in d dimensions the problem of enumerating spanning trees on graphs and lattices is considered we obtain bounds on the number of spanning trees nst and establish inequalities relating the numbers of spanning trees of different graphs or lattices a general formulation is presented for the enumeration of spanning trees on lattices in d 2 dimensions and is applied to the hypercubic body centred cubic face centred cubic and specific planar lattices including the kagome diced 4 8 8 bathroom tile union jack and 3 12 12 lattices this leads to closed form expressions for nst for these lattices of finite sizes we prove a theorem concerning the classes of graphs and lattices with the property that nst exp nz as the number of vertices n where z is a finite non zero constant this includes the bulk limit of lattices in any spatial dimension and also sections of lattices whose lengths in some dimensions go to infinity while others are finite we evaluate z exactly for the lattices we consider and discuss the dependence of z on d and the lattice coordination number we also establish a relation connecting z to the free energy of the critical ising model for planar lattices

the zeros of special functions from a fixed point method a scheme for the computation of the zeros of special functions and orthogonal polynomials is developed we study the structure of the first order difference differential equations ddes satisfied by two fundamental sets of solutions of second order odes y n prime prime x a n x y n x 0 n being the order of the solutions and an x a family of continuous functions it is proved that with a convenient normalization of the solutions t pm1 z z pm mbox rm sign d arctan y n x z y n pm 1 x z are globally convergent iterations with fixed points z x n i x n i being the zeros of yn x d is one of the coefficients in the ddes and z x is a primitive of d the structure of the ddes is also used to set global bounds on the differences between adjacent zeros of functions of consecutive orders and to find iteration steps which guarantee that all the zeros inside a given interval can be found with certainty as an illustration we describe how to implement this scheme for the calculation of the zeros of arbitrary solutions of the bessel coulomb legendre hermite and laguerre equations

mathematical models with exact renormalization for turbulent transport ii fractal interfaces non gaussian statistics and the sweeping effect this paper continues the study of a model for turbulent transport with an exact renormalization theory which has recently been proposed and developed by the authors three important topics are analyzed with complete mathematical rigor for this model 1 renormalized higher order statistics of a passively advected scalar such as the pair distance distribution and the fractal dimension of interfaces 2 the effect of non gaussian turbulent velocity statistics on renormalization theory 3 the sweeping effect of additional large scale mean velocities a special emphasis is placed on renormalization theory in the vicinity of the value of the analogue of the kolmogorov spectrum in the model in the authors earlier paper it was established that the kolmogorov value is at a phase transition boundary in the exact renormalization theory it is found here that the qualitative model despite its simplicity contains in the vicinity of the kolmogorov value a remarkable amount of the qualitative behavior of turbulent transport which has been uncovered in recent experiments and proposed in phenomenological theories in particular the richardson 4 3 law for pair dispersion and interfaces with fractal dimension defect of 2 3 occur in the model rigorously as limits when the kolmogorov spectrum is approached as a limit from one side of the phase transition boundary alternative corrections to the richardson law with the same form as those proposed heuristically in the recent literature and interfaces with fractal dimension defect 1 3 occur in the model when the kolmogorov spectrum is approached from the other side of the phase transition it is very interesting that fractal dimension defects of roughly the value either 1 3 or 2 3 for level sets and interfaces of passive scalars have been ubiquitous in recent turbulence experiments as regards non gaussian the asymptotic normality of normalized integrals b 56 corresponding to compactly supported blobs with mean zero the proof of this latter fact is done in the same way as step 2 proposition b 3 using the fact that the corresponding random processes tilde v delta s have finite domain of dependence this concludes the proof of proposition b 4

lp estimates on functions of the laplace operator on etudie la continuite l p de certaines fonctions du laplacien sur une variete de riemann complete a geometrie bornee

distribution of sizes of erased loops for loop erased random walks the loop erased random walk lerw is a simpler variant of the well known self avoiding walk saw problem which retains the no self intersection property of saws but is closer to the ordinary random walk problem in this paper we study the distribution of sizes of erased loops for lerws on arbitrary graphs and relate it to the distribution of sizes of loops formed when a bond is added to a random spanning tree on the same graph in the themodynamic limit this distribution has a power law tail and we express the exponent in terms of the fractal dimension of the chemical paths on spanning trees we show also how this relation is modified for deterministic fractals in spite of the fact that the lerw model is somewhat more tractable analytically than the saw problem the number of papers devoted to this problem has remained rather small the model was defined by lawler 1 who called it the loop erased self avoiding walk lesaw this terminology is somewhat inappropriate and

the number and degree distribution of spanning trees in the tower of hanoi graph the number of spanning trees of a graph is an important invariant related to topological and dynamic properties of the graph such as its reliability communication aspects synchronization and so on however the practical enumeration of spanning trees and the study of their properties remain a challenge particularly for large networks in this paper we study the number and degree distribution of the spanning trees in the hanoi graph we first establish recursion relations between the number of spanning trees and other spanning subgraphs of the hanoi graph from which we find an exact analytical expression for the number of spanning trees of the n disc hanoi graph this result allows the calculation of the spanning tree entropy which is then compared with those for other graphs with the same average degree then we introduce a vertex labeling which allows to find for each vertex of the graph its degree distribution among all possible spanning trees

front propagation into unstable states universal algebraic convergence towards uniformly translating pulled fronts fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes pulled fronts and pushed fronts the term pulled front expresses that these fronts are pulled along by the spreading of linear perturbations about the unstable state accordingly their asymptotic speedv equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state the central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towardsv we show that when such fronts evolve from sufficiently steep initial conditions which initially decay faster than e x forx 1 t hey have auniversal relaxation behavioras timet 1 the velocity of a pulled front always relaxes algebraically like v t d v 3 2t c 3 p d d 2 t 3 2 c o 1 t 2 the parametersv and d are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state this front velocity is independent of the precise value of the front amplitude which one tracks to measure the front position the interior of the front is essentially slaved to the leading edge and develops universally as x t d8v t x rt dt 0 v t 0 c o 1 t 2 where8v x vt is a uniformly translating front solution with velocity v v our result which can be viewed as a general center manifold result for pulled front propagation is derived in detail for the well known nonlinear diffusion equation of type t d 2 x c 3 where the invaded unstable state is d 0 even for this simple case the subdominant t 3 2 term extends an earlier result of bramson our analysis is then generalized to more general sets of partial differential equations with higher spatial or temporal derivatives to pdes with memory kernels and also to difference equations such as those that occur in numerical finite difference codes our universal result for pulled fronts thus implies independence i of the level curve which is used to track the front position ii of the precise nonlinearities iii of the precise form of the linear operators in the dynamical equation and iv of the precise initial conditions as long as they are sufficiently steep the only remainders of the explicit form of the dynamical equation are the nonlinear solutions 8v and the three saddle point parameters v and d as our simulations confirm all our analytical predictions in every detail it can be concluded that we have a complete analytical understanding of the propagation mechanism and relaxation behavior of pulled fronts if they are uniformly translating for t 1 an immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation breaks down for weakly curved pulled fronts in two or three dimensions in addition to our main result for pulled fronts we also discuss the propagation and convergence of fronts emerging from

zeros of the macdonald function of complex order the z zeros of the modified bessel function of the third kind k n z also known as modified hankel function or macdonald function are considered for arbitrary complex values of the order n approximate expressions for the zeros applicable in the cases of very small or very large n are given the behaviour of the zeros for varying n or arg n obtained numerically is illustrated by means of some graphics

on the convergence of riesz means on compact manifolds the purpose of this paper is to study the lp mapping properties of the riesz means of eigenfunction expansions associated to elliptic differential operators on compact manifolds in order to be more specific let us first introduce some notation let m be a compact connected c manifold of dimension n 2 and p an elliptic differential operator on m with c00 coefficients we shall assume that p is self adjoint with respect to some positive c00 density dx kept fixed throughout that is pu v u pv where as usual

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

some two vertex resistances of the three towers hanoi graph formed by a fractal graph the resistance distance between two vertices of a simple connected graph g is equal to the resistance between two equivalent points on an electrical network constructed to correspond to g with each edge being replaced by a unit resistor in this study we apply the principle of substitution to three towers hanoi graph to procure an equivalent network that enables us to only use the series and parallel principles and the delta wye transformation to acquire the some two vertex resistance distances of three towers hanoi graph

functional calculus for infinitesimal generators of holomorphic semigroups we give a functional calculus formula for infinitesimal generators of holomorphic semigroups of operators on banach spaces which involves the bochner riesz kernels of such generators the rate of smoothness of operating functions is related to the exponent of the growth on vertical lines of the operator norm of the semigroup the strength of the formula is tested on poisson and gauss semigroups inl1 rn andl1 g for a stratified lie groupg we give also a self contained theory of smooth absolutely continuous functions on the half line 0

pfaffian orientations and perfect matchings of scale free networks counting perfect matchings is of great interest in theoretical computer science mathematics among other disciplines however explicitly determining the number of perfect matchings of general graphs is a challenge in this paper we present a first analytical study of the enumeration problem of perfect matchings on scale free networks which are ubiquitous in real life systems we focus on two iteratively growing scale free networks one is fractal while the other is non fractal for each network we construct an orientation and prove that it is pfaffian then based on the connection between the number of perfect matchings and determinant of the skew adjacency matrix corresponding to a pfaffian orientation we derive the recursive expressions for the number of perfect matchings of the networks at two successive iterations furthermore using the recursive relations we show that for both scale free networks their entropy is zero which is in sharp contrast to the previously obtained results for other networks without scale free behavior but having the same average degree as the studied networks such as square lattice and honeycomb lattice whose entropies are larger than zero our results indicate from another angle that scale free networks have a distinct architecture in comparison with those lattices lacking the power law property

laplace s method on a computer algebra system with an application to the real valued modified bessel functions we examine a maple implementation of two distinct approaches to laplace s method used to obtain asymptotic expansions of laplace type integrals one algorithm uses power series reversion whereas the other expands all quantities in taylor or puiseux series these algorithms are used to derive asymptotic expansions for the real valued modified bessel functions of pure imaginary order and real argument that mimic the well known corresponding expansions for the unmodified bessel functions

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

summability of hermite expansions ii we study the summability of n dimensional hermite expansions where n 1 we prove that the critical index for the riesz summability is n 1 2 we also prove analogues of the fejer lebesgue theorem and riemann s localisation principle when the index of the riesz means is 3n 2 6

l p eigenfunction bounds for the hermite operator author s koch herbert tataru daniel abstract we obtain l p eigenfunction bounds for the harmonic oscillator in r n and for other related operators improving earlier results of thangavelu and karadzhov we also construct suitable counterexamples which show that our estimates are sharp

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

bochner riesz profile of anharmonic oscillator l d2dx2 x we investigate spectral multipliers bochner riesz means and the convergence of eigenfunction expansion corresponding to the schrodinger operator with anharmonic potential l d2dx2 x we show that the bochner riesz profile of the operator l completely coincides with such profile of the harmonic oscillator h d2dx2 x2 it is especially surprising because the bochner riesz profile of the one dimensional standard laplace operator is known to be essentially different and the case of operators h and l resembles more the profile of multidimensional laplace operators another surprising element of the main obtained result is the fact that the proof is not based on restriction type estimates and instead an entirely new perspective has to be developed to obtain the critical exponent for bochner riesz means convergence

local characteristics entropy and limit theorems for spanning trees and domino tilings via transfer impedances let g be a finite graph or an infinite graph on which mathbb z d acts with finite fundamental domain if g is finite let mathbf t be a random spanning tree chosen uniformly from all spanning trees of g if g is infinite methods from pemantle show that this still makes sense producing a random essential spanning forest of g a method for calculating local characteristics i e finite dimensional marginals of mathbf t from the transfer impedance matrix is presented this differs from the classical matrix tree theorem in that only small pieces of the matrix n dimensional minors are needed to compute small n dimensional marginals calculation of the matrix entries relies on the calculation of the green s function for g which is not a local calculation however it is shown how the calculation of the green s function may be reduced to a finite computation in the case when g is an infinite graph admitting a z d action with finite quotient the same computation also gives the entropy of the law of mathbf t these results are applied to the problem of tiling certain lattices by dominos the so called dimer problem another application of these results is to prove modified versions of conjectures of aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity included is a generalization of moments to tree valued random variables and criteria for these generalized moments to determine a distribution

on the coefficients that arise from laplace s method laplace s method is one of the best known techniques in the asymptotic approximation of integrals the salient step in the technique s historical development was erdelyi s use of watson s lemma to obtain an infinite asymptotic expansion valid for any laplace type integral published in 1956 erdelyi s expansion contains coefficients cs that must be calculated in each application of laplace s method a tedious process that has traditionally involved the reversion of a series this paper shows that the coefficients cs in fact have a very simple general form in effect we extend erdelyi s theorem our results greatly simplify calculation of the cs in any particular application and clarify the theoretical basis of erdelyi s expansion it turns out that faa di bruno s formula has always played a central role in it we prove or derive the following the correct dimensionless groups erdelyi s expansion is properly expressed in terms of scaled coefficientscs two explicit expressions for cs in terms of combinatorial objects called partial ordinary bell polynomials this form is probably computationally optimal and makes checking for correctness a relatively straightforward process a recursive expression for cs each coefficient cs can be expressed as a polynomial in s where and are quantities in erdelyi s formulation the main insight that emerges is that the traditional approach to laplace s method involving reversion of a series is less efficient and need only be invoked if one is interested in the role of the reversion coefficients in erdelyi s expansion a point which erdelyi himself alluded to we consider as an example an integral that occurs in a variational approach to finding the binding energy of helium dimers we also present a three line computer code to generate the coefficients cs exactly in the general case in a sequel paper to be published in siam review a new representation for the gamma function is obtained and the link with faa di bruno s formula is explained

on the fkpp equation with gaussian shear advection abstract a boundary value problem for the fisher kolmogorov petrovskii piskunov fkpp equation with shear random advection is investigated it is demonstrated that the upper bounds for front propagation in the fkpp equation with gaussian advection obtained by an analysis of its linear ensemble averaged upper solution can give too rough predictions it is shown that the unboundedness of the gaussian advection affects ensemble averaged solutions of linear and nonlinear problems in a different way this analytical prediction is confirmed by direct numerical simulations the phenomenon of propagation failure due to zero boundary conditions is studied and the critical conditions are found numerical procedure for the analysis of random wave propagation is suggested by introduction of mean front position and its variance some numerical experiments are presented

the weak type l1 convergence of eigenfunction expansions for pseudodifferential operators soit m une variete compacte connexe lisse sans bord de dimension n 2 soit p d un operateur pseudodifferentiel elliptique d ordre 1 sur m qui est auto adjoint par rapport a une densite positive lisse dx on demontre la convergence faible de type l 1 pour les developpements en fonctions propres

bounds for spectral multipliers on nilpotent groups a criterion is given for the lp boundedness of a class of spec tral multiplier operators associated to left invariant homogeneous subelliptic second order differential operators on nilpotent lie groups generalizing a theo rem of hormander for radial fourier multipliers on euclidean space the order of differentiability required is half the homogeneous dimension of the group improving previous results in the same direction

synchronization is optimal in nondiagonalizable networks we consider maximization of the synchronizability of oscillator networks by assigning weights and directions to the links of a given interaction topology by extending the master stability formalism to all possible network structures we show that unless some oscillator is linked to all the others maximally synchronizable networks are necessarily nondiagonalizable and can always be obtained by imposing unidirectional information flow with normalized input strengths the results provide insights into hierarchical structures observed in complex networks in which synchronization is important

maximum matchings in scale free networks with identical degree distribution the size and number of maximum matchings in a network have found a large variety of applications in many fields as a ubiquitous property of diverse real systems power law degree distribution was shown to have a profound influence on size of maximum matchings in scale free networks where the size of maximum matchings is small and a perfect matching often does not exist in this paper we study analytically the maximum matchings in two scale free networks with identical degree sequence and show that the first network has no perfect matchings while the second one has many for the first network we determine explicitly the size of maximum matchings and provide an exact recursive solution for the number of maximum matchings for the second one we design an orientation and prove that it is pfaffian based on which we derive a closed form expression for the number of perfect matchings moreover we demonstrate that the entropy for perfect matchings is equal to that corresponding to the extended sierpiski graph with the same average degree as both studied scale free networks our results indicate that power law degree distribution alone is not sufficient to characterize the size and number of maximum matchings in scale free networks

farey graphs as models for complex networks farey sequences of irreducible fractions between 0 and 1 can be related to graph constructions known as farey graphs these graphs were first introduced by matula and kornerup in 1979 and further studied by colbourn in 1982 and they have many interesting properties they are minimally 3 colorable uniquely hamiltonian maximally outerplanar and perfect in this paper we introduce a simple generation method for a farey graph family and we study analytically relevant topological properties order size degree distribution and correlation clustering transitivity diameter and average distance we show that the graphs are a good model for networks associated with some complex systems

plancherel type estimates and sharp spectral multipliers abstract we study general spectral multiplier theorems for self adjoint positive definite operators on l2 x where x is any open subset of a space of homogeneous type we show that the sharp hormander type spectral multiplier theorems follow from the appropriate estimates of the l2 norm of the kernel of spectral multipliers and the gaussian bounds for the corresponding heat kernel the sharp hormander type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the riesz means summability which we also study here we discuss several examples which include sharp spectral multiplier theorems for a class of scattering operators on r3 and new spectral multiplier theorems for the laguerre and hermite expansions

sharp spectral multipliers for operators satisfying generalized gaussian estimates abstract let l be a non negative self adjoint operator acting on l 2 x where x is a space of homogeneous type assume that l generates a holomorphic semigroup e t l which satisfies generalized m th order gaussian estimates in this article we study singular and dyadically supported spectral multipliers for abstract self adjoint operators we show that in this setting sharp spectral multiplier results follow from plancherel or stein tomas type estimates these results are applicable to spectral multipliers for a large class of operators including m th order elliptic differential operators with constant coefficients biharmonic operators with rough potentials and laplace type operators acting on fractals

on the reversion of an asymptotic expansion and the zeros of the airy functions the general theories of the derivation of inverses of functions from their power series and asymptotic expansions are discussed and compared the asymptotic theory is applied to obtain asymptotic expansions of the zeros of the airy functions and their derivatives and also of the associated values of the functions or derivatives a maple code is constructed to generate exactly the coefficients in these expansions the only limits on the number of coefficients are those imposed by the capacity of the computer being used and the execution time that is available the sign patterns of the coefficients suggest open problems pertaining to error bounds for the asymptotic expansions of the zeros and stationary values of the airy functions

scaling and renormalization for the kolmogorov petrovskii piskunov equationwith turbulent convection the problem of determining the upper bounds for the ensemble averaged reaction front position and speed in a fully developed three dimensional turbulent flow has been examined in which the reaction is of kolmogorov petrovskii piskunov type and turbulent velocity is a gaussian random field exhibiting long range correlations and infrared divergence in the limit of large reynolds number an asymptotic method has been developed that gives the general formalism for determining the upper bounds for reaction front in the long time large distance limit two anomalous scaling regimes and corresponding scaling functions have been determined by the use of exact renormalization procedure

mathematical models with exact renormalization for turbulent transport the advection diffusion of a passive scalar by incompressible velocity fields which admit a statistical description and involve a continuous range of excited spatial and or temporal scales is very important in applications ranging from fully developed turbulence to the diffusion of tracers in heterogeneous porous media a variety of renormalization theories which typically utilize partial resummation of divergent perturbation series according to various recipes have been applied to this problem in various contexts in this paper a simple model problem for the advection diffusion of a passive scalar is introduced and the complete renormalization theory is developed with full mathematical rigor explicit formulas for the anomalous time scaling in various regimes as well as the green s function for the large scale long time ensemble average are developed here formulas for the renormalized higher order statistics are also developed the simple form of the model problem is deceptive the renormalization theory for this problem exhibits a remarkable range of different renormalization phenomena as parameters in the velocity statistics are varied these phenomena include the existence of several distinct anomalous scaling regimes as the spectral parameter tilde varepsilon is varied as well as explicit regimes in tilde varepsilon where the effective equation for the ensemble average is not a simple diffusion equation but instead involves an explicit random nonlocal eddy diffusivity we use fourier analysis and the feynman kac formula as main tools in the explicit exact renormalization theory developed here

counting spanning trees in a small world farey graph the problem of spanning trees is closely related to various interesting problems in the area of statistical physics but determining the number of spanning trees in general networks is computationally intractable in this paper we perform a study on the enumeration of spanning trees in a specific small world network with an exponential distribution of vertex degrees which is called a farey graph since it is associated with the famous farey sequence according to the particular network structure we provide some recursive relations governing the laplacian characteristic polynomials of a farey graph and its subgraphs then making use of these relations obtained here we derive the exact number of spanning trees in the farey graph as well as an approximate numerical solution for the asymptotic growth constant characterizing the network finally we compare our results with those of different types of networks previously investigated

a general method for computing tutte polynomials of self similar graphs abstract self similar graphs were widely studied in both combinatorics and statistical physics motivated by the construction of the well known 3 dimensional sierpinski gasket graphs in this paper we introduce a family of recursively constructed self similar graphs whose inner duals are of the self similar property by combining the dual property of the tutte polynomial and the subgraph decomposition trick we show that the tutte polynomial of this family of graphs can be computed in an iterative way and in particular the exact expression of the formula of the number of their spanning trees is derived furthermore we show our method is a general one that is easily extended to compute tutte polynomials for other families of self similar graphs such as farey graphs 2 dimensional sierpinski gasket graphs hanoi graphs modified koch graphs apollonian graphs pseudofractal scale free web fractal scale free network etc