nearest neighbor classification with dependent training sequences the asymptotic classification risk for nearest neighbor procedures is well understood in the case of i i d training sequences in this article we generalize these results to a class of dependent models including hidden markov models in the case where the observed patterns have lebesgue densities the asymptotic risk takes the same expression as in the i i d case for discrete distributions we show that the asymptotic risk depends on the rule used for breaking ties of equal distances

gaussian heat kernel upper bounds via the phragmen lindelof theorem we prove that in presence of l gaussian estimates so called davies gaffney estimates on diagonal upper bounds imply precise off diagonal gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces

li yau gradient bounds on compact manifolds under nearly optimal curvature conditions abstract we prove li yau type gradient bounds for the heat equation either on manifolds with fixed metric or under the ricci flow in the former case the curvature condition is r i c l p for some p n 2 or sup m m r i c 2 y d 2 n x y d y where n is the dimension of the manifold in the later case one only needs scalar curvature being bounded we will explain why the conditions are nearly optimal and give an application the li yau bound for the heat equation on manifolds with fixed metric seems to be the first one allowing ricci curvature not bounded from below

weak stability of ricci expanders with positive curvature operator we prove the l infty stability of expanding gradient ricci solitons with positive curvature operator and quadratic curvature decay at infinity

a deterministic small world network created by edge iterations abstract small world networks are ubiquitous in real life systems most previous models of small world networks are stochastic the randomness makes it more difficult to gain a visual understanding on how do different nodes of networks interact with each other and is not appropriate for communication networks that have fixed interconnections here we present a model that generates a small world network in a simple deterministic way our model has a discrete exponential degree distribution we solve the main characteristics of the model

the conjugate heat equation and ancient solutions of the ricci flow abstract we prove gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the ricci flow as a consequence for dimension 4 and higher we show that the backward limit of type i solutions of the ricci flow must be a non flat gradient shrinking ricci soliton this extends perelman s previous result on backward limits of solutions in dimension 3 in which case the curvature operator is nonnegative it follows from hamilton ivey curvature pinching estimate as an application this also addresses an issue left in naber 2010 23 where naber proves the interesting result that there exists a type i dilation limit that converges to a gradient shrinking ricci soliton but that soliton might be flat the gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest

abstract hardy sobolev spaces and interpolation the purpose of this work is to describe an abstract theory of hardy sobolev spaces on doubling riemannian manifolds via an atomic decomposition we study the real interpolation of these spaces with sobolev spaces and finally give applications to riesz inequalities

invariance principle for the random conductance model with unbounded conductances we study a continuous time random walk x in an environment of i i d random conductances e 1 we obtain heat kernel bounds and prove a quenched invariance principle for x this holds even when e e

families and clustering in a natural numbers network we develop a network in which the natural numbers are the vertices the decomposition of natural numbers by prime numbers is used to establish the connections we perform data collapse and show that the degree distribution of these networks scales linearly with the number of vertices we explore the families of vertices in connection with prime numbers decomposition we compare the average distance of the network and the clustering coefficient with the distance and clustering coefficient of the corresponding random graph in case we set connections among vertices each time the numbers share a common prime number the network has properties similar to a random graph if the criterion for establishing links becomes more selective only prime numbers greater than p l are used to establish links where the network has high clustering coefficient

path integrals and the essential self adjointness of differential operators on noncompact manifolds we consider schrodinger operators on possibly noncompact riemannian manifolds acting on sections in vector bundles with locally square integrable potentials whose negative part is in the underlying kato class using path integral methods we prove that under geodesic completeness these differential operators are essentially self adjoint on mathsf c infty 0 and that the corresponding operator closures are semibounded from below these results apply to nonrelativistic pauli dirac operators that describe the energy of hydrogen type atoms on riemannian 3 manifolds

obtaining upper bounds of heat kernels from lower bounds we show that a near diagonal lower bound of the heat kernel of a dirichlet form on a metric measure space with a regular measure implies an on diagonal upper bound if in addition the dirichlet form is local and regular then we obtain a full off diagonal upper bound of the heat kernel provided the dirichlet heat kernel on any ball satisfies a near diagonal lower estimate this reveals a new phenomenon in the relationship between the lower and upper bounds of the heat kernel c 2007 wiley periodicals inc

asymptotic normality of the maximum likelihood estimator for general hidden markov models hidden markov models hmms have during the last decade become a widespread tool for modeling sequences of dependent random variables inference for such models is usually based on the maximum likelihood estimator mle and consistency of the mle for general hmms was recently proved by leroux in this paper we show that under mild conditions the mle is also asymptotically normal and prove that the observed information matrix is a consistent estimator of the fisher information

heat kernel upper bounds for jump processes and the first exit time for all t 0 x m and f l2 m the function pt x y can be considered as the transition density of the associated markov processx xt t 0 and the question of estimating of pt x y which is the main subject of this paper is closely related to the properties of x the function pt x y is also referred to as a heat kernel and this terminology goes back to the classical gauss weierstrass heat kernel associated with the heat semigroup e t 0 in r n whose markov process is brownian motion a somewhat more general but still well treated case is when m d is a riemannian metric measure space that is m is a riemannian manifold

weak type p p estimates for riesz transforms we consider the problem of l p boundedness of higher order riesz transforms associated to elliptic operators l of order 2m on as an application of the recently solved kato conjecture we show for all this generalizes the result of auscher and tchamitchian restricted to the case d 2m

point interactions in two and three dimensional riemannian manifolds we present a non perturbative renormalization of the bound state problem of n bosons interacting with finitely many dirac delta interactions on two and three dimensional riemannian manifolds using the heat kernel we formulate the problem in terms of a new operator called the principal or characteristic operator e in order to investigate the problem in more detail we then restrict the problem to one particle sector the lower bound of the ground state energy is found for a general class of manifolds e g for compact and cartan hadamard manifolds the estimate of the bound state energies in the tunneling regime is calculated by perturbation theory non degeneracy and uniqueness of the ground state is proven by the perron frobenius theorem moreover the pointwise bounds on the wave function is given and all these results are consistent with the one given in standard quantum mechanics renormalization procedure does not lead to any radical change in these cases finally renormalization group equations are derived and the function is exactly calculated this work is a natural continuation of our previous work based on a novel approach to the renormalization of point interactions developed by rajeev

hardy littlewood theory for semigroups soit t t t 0 un semi groupe symetrique sousmarkovien et soit n 2 alors on a les proprietes equivalentes a dim t t n b il existe c 2 0 s t t t f c 2 t n 2 f 1 f l 1 x t 0 en fait a b meme dans le cas n 2

nonlinear equations and weighted norm inequalities we study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities in particular we obtain explicit criteria for the solvability of the dirichlet problem u v u w u 0 on u 0 on on a regular domain in rn in the superlinear case q 1 the coefficients v w are arbitrary positive measurable functions or measures on we also consider more general nonlinear differential and integral equations and study the spaces of coefficients and solutions naturally associated with these problems as well as the corresponding capacities our characterizations of the existence of positive solutions take into account the interplay between v w and the corresponding green s kernel they are not only sufficient but also necessary and are established without any a priori regularity assumptions on v and w we also obtain sharp two sided estimates of solutions up to the boundary some of our results are new even if v 1 and is a ball or half space the corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so called 3g inequality by an elementary integration by parts argument this also gives a new unified proof for some classical inequalities including the carleson measure theorem for poisson integrals and trace inequalities for riesz potentials and green potentials

on the inequality of cover and hart in nearest neighbor discrimination when x1 1 xn n are independent identically distributed random vectors from ird x 0 1 distributed as x and when is estimated by its nearest neighbor estimate 1 then cover and hart have shown that p 1 n 2e x 1 x 2r 1 r where r is the bayes probability of error and x p 1 x x they have conditions on the distribution of x we give two proofs one due to stone and a short original one of the same result for all distributions of x if ties are carefully taken care of we also show that p 1 x1 1 xn n converges in probability to a constant for all distributions of x thereby strengthening results of wagner and fritz

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

new volume comparison results and applications to degeneration of riemannian metrics abstract we consider a condition on the ricci curvature involving vector fields which is broader than the bakry emery ricci condition under this condition volume comparison laplacian comparison isoperimetric inequality and gradient bounds are proven on the manifold specializing to the bakry emery ricci curvature condition we initiate a short cut approach to work on the original manifold which yields under a weaker than usual assumption the results mentioned above for the original manifold these results are different from most well known ones in the literature where the conclusions are made on the weighted manifold instead applications on convergence and degeneration of riemannian metrics under this curvature condition are given to this effect in particular for the bakry emery ricci curvature condition the gradient of the potential function is allowed to have singularity of order close to 1 which is broader than those allowed by the traditional weighted method in general coupling with some new techniques in constructing model functions this approach enables us to extend some of the results in the papers 14 11 36 29 and 31 the condition also covers general ricci solitons instead of just gradient ricci solitons

invariance principle for the random conductance model we study a continuous time random walk x in an environment of i i d random conductances mu e in 0 infty in mathbb z d we assume that mathbb p mu e 0 p c so that the bonds with strictly positive conductances percolate but make no other assumptions on the law of the e we prove a quenched invariance principle for x and obtain green s functions bounds and an elliptic harnack inequality

nonparametric inference for ergodic stationary time series the setting is a stationary ergodic time series the challenge is to construct a sequence of functions each based on only finite segments of the past which together provide a strongly consistent estimator for the conditional probability of the next observation given the infinite past ornstein gave such a construction for the case that the values are from a finite set and recently algoet extended the scheme to time series with coordinates in a polish space the present study relates a different solution to the challenge the algorithm is simple and its verification is fairly transparent some extensions to regression pattern recognition and on line forecasting are mentioned

riesz transforms through reverse holder and poincare inequalities we study the boundedness of riesz transforms in l p for p 2 on a doubling metric measure space endowed with a gradient operator and an injective omega accretive operator l satisfying davies gaffney estimates if l is non negative self adjoint we show that under a reverse holder inequality the riesz transform is always bounded on l p for p in some interval 2 2 epsilon and that l p gradient estimates for the semigroup imply boundedness of the riesz transform in l q for q in 2 p this improves results of auscher et al ann sci ecole norm sup 37 4 911 957 2004 and auscher and coulhon ann scuola norm sup pisa 4 531 555 2005 where the stronger assumption of a poincare inequality and the assumption e tl 1 1 were made the poincare inequality assumption is also weakened in the setting of a sectorial operator l in the last section we study elliptic perturbations of riesz transforms

a small world network of prime numbers abstract according to goldbach conjecture any even number can be broken up as the sum of two prime numbers n p q we construct a network where each node is a prime number and corresponding to every even number n we put a link between the component primes p and q in most cases an even number can be broken up in many ways and then we chose one decomposition with a probability p q through computation of average shortest distance and clustering coefficient we conclude that for 1 8 the network is of small world type and for 1 8 it is of regular type we also present a theoretical justification for such behaviour

on density estimation from ergodic processes we consider the problem of lp consistent density estimation from the initial segments of strongly dependent processes it is shown that no procedure can consistently estimate the one dimensional marginal density of every stationary ergodic process for which such a density exists a similar result is established for the problem of estimating the support of the marginal distribution of an ergodic process

fractional brownian fields over manifolds this is the publisher s final pdf the published article is copyrighted by the american mathematical society and can be found at http www ams org publications journals journalsframework tran

strichartz estimates and nonlinear wave equation on nontrapping asymptotically conic manifolds abstract we prove the global in time strichartz estimates for wave equations on the nontrapping asymptotically conic manifolds we obtain estimates for the full set of wave admissible indices including the endpoint the key points are the properties of the microlocalized spectral measure of laplacian on this setting showed in 18 and a littlewood paley squarefunction estimate as applications we prove the global existence and scattering for a family of nonlinear wave equations on this setting

path regularity for feller semigroups via gaussian kernel estimates and generalizations to arbitrary semigroups on c0 given 1 1 we present a dyadic growth condition open image in new window on the finite dimensional distributions of operator semigroups on c0 e which for 0 and feller semigroups assures that the corresponding feller process has paths in local holder spaces and in weighted besov spaces of order we show that for operator semigroups satisfying gaussian kernel estimates of order m 1 condition open image in new window holds for all open image in new window and even for all open image in new window in the case of feller semigroups such gaussian kernel estimates are typical for feller semigroups on fractals of walk dimension m and for semigroups generated by elliptic operators on d of order m d

on solutions to the heat equation with the initial condition in the orlicz slobodetskii space we study the boundary value problem t x u x t u x 0 u x where x2 t2 0 t r n 1 is a bounded lipschitz boundary domain u belongs to certain orlicz slobodetskii space y r r under certain assumptions on the orlicz function r we prove that the solution u belongs to orlicz sobolev space w 1 r 0 t

riesz transform gaussian bounds and the method of wave equation for an abstract self adjoint operator l and a local operator a we study the boundedness of the riesz transform al on lp for some 0 a very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation we also discuss the relation between the gaussian bounds and the finite speed propagation property using the wave equation methods we obtain a new natural form of the gaussian bounds for the heat kernels for a large class of the generating operators we describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature

gaussian bounds and parabolic harnack inequality on locally irregular graphs a well known theorem of delmotte is that gaussian bounds parabolic harnack inequality and the combination of volume doubling and poincare inequality are equivalent for graphs in this paper we consider graphs for which these conditions hold but only for sufficiently large balls and prove a similar equivalence

sharp two sided heat kernel estimates of twisted tubes and applications we prove on diagonal bounds for the heat kernel of the dirichlet laplacian delta d omega in locally twisted three dimensional tubes in particular we show that for any fixed x the heat kernel decays for large times as rm e e 1t t 3 2 where e1 is the fundamental eigenvalue of the dirichlet laplacian on the cross section of the tube this shows that any suitably regular local twisting speeds up the decay of the heat kernel with respect to the case of straight untwisted tubes moreover the above large time decay is valid for a wide class of subcritical operators defined on a straight tube we also discuss some applications of this result such as sobolev inequalities and spectral estimates for schrodinger operators delta d omega v

comportement des noyaux de la chaleur des operateurs de schrodinger et applications a certaines equations paraboliques semi lineaires abstract we consider general schrodinger operators on domains of riemannian manifolds with possibly exponential volume growth we prove sharp large time gaussian upper bounds these bounds are then used to prove new l p l p estimates for the corresponding semigroups applications to semi linear parabolic equations are given

the hodge de rham laplacian and lp boundedness of riesz transforms on non compact manifolds let m be a complete non compact riemannian manifold satisfying the volume doubling property as well as a gaussian upper bound for the corresponding heat kernel we study the boundedness of the riesz transform d 12 on both hardy spaces hp and lebesgue spaces lp under two different conditions on the negative part r of the ricci curvature first we prove that if r is subcritical for some 0 1 then the riesz transform d 12 on differential 1 forms is bounded from the associated hardy space h p 1t m to lp m for all p 1 2 as a consequence d 12 is bounded on lp for all p 1 p0 where p0 2 depends on and the constant appearing in the doubling property second we prove that if 01 r 12v t 1p1 p1dtt 1 r 12v t 1p2 p2dtt for some p1 2 and p2 3 then the riesz transform d 12 is bounded on lp for all 1 2 under conditions on r and the potential v we prove both positive and negative results on the boundedness of da 12 on lp

the speed of propagation for kpp type problems ii general domains this paper is devoted to nonlinear propagation phenomena in general unbounded domains of rn for reaction diffusion equations with kolmogorov petrovsky piskunov kpp type nonlinearities this article is the second in a series of two and it is the follow up of the paper 7 which dealt which the case of periodic domains this paper is concerned with general domains and we give various definitions of the spreading speeds at large times for solutions with compactly supported initial data we study the relationships between these new notions and analyze their dependence on the geometry of the domain and on the initial condition some a priori bounds are proved for large classes of domains the case of exterior domains is also discussed in detail lastly some domains which are very thin at infinity and for which the spreading speeds are infinite are exhibited the construction is based on some new heat kernel estimates in such domains

volume and time doubling of graphs and random walks the strongly recurrent case this paper proves upper and lower off diagonal sub gaussian transition probability estimates for strongly recurrent random walks under sufficient and necessary conditions besides the known conditions volume doubling and the elliptic harnack inequality a new property is introduced time doubling 2001 john wiley sons inc

heat semigroup and singular pdes we provide in this work a semigroup approach to the study of singular pdes in the line of the paracontrolled approach developed recently by gubinelli imkeller and perkowski starting from a heat semigroup we develop a functional calculus and introduce a paraproduct based on the semigroup for which commutator estimates and schauder estimates are proved together with their paracontrolled extensions this machinery allows us to investigate singular pdes in potentially unbounded riemannian manifolds under mild geometric conditions as an illustration we study the generalized parabolic anderson model equation and prove under mild geometric conditions its well posed character in small time on a potentially unbounded 2 dimensional riemannian manifold for an equation driven by a coloured noise and for all times for the linear parabolic anderson model equation in 2 dimensional unbounded manifolds this machinery can be extended to an even more singular setting and deal with sobolev spaces rather than holder spaces

regularity of kahler ricci flows on fano manifolds in this paper we will establish a regularity theory for the kahler ricci flow on fano n manifolds with ricci curvature bounded in l p norm for some p n using this regularity theory we will also solve a long standing conjecture for dimension 3 as an application we give a new proof of the yau tian donaldson conjecture for fano 3 manifolds the results have been announced in 45

from u bounds to isoperimetry with applications to h type groups abstract in this paper we study u bounds in relation to l 1 type coercive inequalities and isoperimetric problems for a class of probability measures on a general metric space r n d we prove the equivalence of an isoperimetric inequality with several other coercive inequalities in this general framework the usefulness of our approach is illustrated by an application to the setting of h type groups and an extension to infinite dimensions

analysis on lie groups theorie locale diffusion de la chaleur sur des groupes de lie connexes la theorie globale les inegalites de sobolev bases graduees sur des corps lisses construction des bases graduees estimation locale du noyau diffusion de la chaleur demonstration de l inegalite de sobolev locale formes de dirichlet sur des groupes le semi groupe de convolution les conditions de sobolev modifiees

analysis on graphs and noncommutative geometry abstract we study the form of the continuous time heat kernel for a second order discrete laplacian on a weighted graph the analysis is shown to be closely related to the theory of symmetric markov semigroups on noncommutative l p spaces and to the noncommutative geometry of connes the paper obtains better pointwise upper bounds on the heat kernels than those previously known by the use of a novel metric on the graph in certain cases it is shown that the new estimates are optimal of their type the metric is investigated at some length and compared with other known metrics on graphs

surgery of the faber krahn inequality and applications to heat kernel bounds 2 cutting faber krahn inequalities 3 2 1 fk functions 3 2 2 cutting a manifold 6 2 3 non parabolic case 7 2 4 fk functions in balls 9 2 5 relative fk function 16

a small world network derived from the deterministic uniform recursive tree by line graph operation the deterministic uniform recursive tree is one of the deterministic versions of the uniform recursive tree zhang et al 2008 eur phys j b 63 507 13 studied the properties of durt including its topological characteristics and spectral properties although durt shows a logarithmic scaling with the size of the network durt is not a small world network since its clustering coefficient is zero lu et al 2012 physica a 391 87 92 proposed a deterministic small world network by adding some edges with a simple rule in each durt iteration in this paper we intoduce a method for constructing a new deterministic small world network by the line graph operation in each durt iteration the line graph operation brings about cliques at each node of the previous given graph and the resulting line graph possesses larger clustering coefficients on the other hand this operation can decrease the diameter at almost one then giving the analytic solutions to several topological characteristics of the model proposed

on and off diagonal heat kernel behaviors on certain infinite dimensional local dirichlet spaces this paper studies on diagonal and off diagonal bounds for symmetric diffusion semi groups that admit a continuous kernel in the case where the underlying space is typically infinite dimensional for invariant diffusions on the infinite dimensional torus the equivalence between a certain on diagonal estimate and a certain off diagonal behavior is proved this gives a necessary and sufficient condition in terms of the associated infinite symmetric matrix for an elliptic harnack inequality to be satisfied as a consequence elliptic and parabolic harnack inequalities are in fact equivalent properties in this setting in terms of potential theory this gives a necessary and sufficient condition for the associated harmonic sheaf to be a brelot harmonic sheaf a new distance associated to certain dirichlet spaces is also introduced it plays a crucial role in relating on diagonal behavior to off diagonal behavior in cases where the intrinsic distance is infinite almost everywhere

on the asymptotic probability of error in nonparametric discrimination let x y x 1 y 1 cdots x n y n be independent identically distributed random vectors from r d times 0 1 and let hat y be the k nearest neighbor estimate of y from x and the x i y i s we show that for all distributions of x y the limit of l n p hat y neq y exists and satisfies lim n rightarrow infty l n leq 1 a k r ast a k frac alpha sqrt k k 3 25 big 1 frac beta sqrt k 3 big k text odd k geq 5 where r ast is the bayes probability of error and alpha 0 3399 cdots and beta 0 9749 cdots are universal constants this bound is shown to be best possible in a certain sense

a new approach to pointwise heat kernel upper bounds on doubling metric measure spaces abstract on doubling metric measure spaces endowed with a strongly local regular dirichlet form we show some characterisations of pointwise upper bounds of the heat kernel in terms of global scale invariant inequalities that correspond respectively to the nash inequality and to a gagliardo nirenberg type inequality when the volume growth is polynomial this yields a new proof and a generalisation of the well known equivalence between classical heat kernel upper bounds and relative faber krahn inequalities or localised sobolev or nash inequalities we are able to treat more general pointwise estimates where the heat kernel rate of decay is not necessarily governed by the volume growth a crucial role is played by the finite propagation speed property for the associated wave equation and our main result holds for an abstract semigroup of operators satisfying the davies gaffney estimates

explicit constants for gaussian upper bounds on heat kernels on utilise des inegalites logarithmiques de sobolev pour obtenir des bornes superieures gaussiennes sur les noyaux de la chaleur associes a des operateurs d ordre 2 elliptiques et hypoelliptiques

geometric fractal growth model for scale free networks we introduce a deterministic model for scale free networks whose degree distribution follows a power law with the exponent ensuremath gamma at each time step each vertex generates its offspring whose number is proportional to the degree of that vertex with proportionality constant m ensuremath 1 mg1 we consider the two cases first each offspring is connected to its parent vertex only forming a tree structure second it is connected to both its parent and grandparent vertices forming a loop structure we find that both models exhibit power law behaviors in their degree distributions with the exponent ensuremath gamma 1 mathrm ln 2m ensuremath 1 mathrm ln m thus by tuning m the degree exponent can be adjusted in the range 2l ensuremath gamma l3 we also solve analytically a mean shortest path distance d between two vertices for the tree structure showing the small world behavior that is d ensuremath sim mathrm ln n mathrm ln overline k where n is system size and overline k is the mean degree finally we consider the case that the number of offspring is the same for all vertices and find that the degree distribution exhibits an exponential decay behavior

a small world network derived from the deterministic uniform recursive tree abstract as the deterministic version of the uniform recursive tree urt the deterministic uniform recursive tree durt has been intensively studied by zhang et al 2008 21 they gave several important properties of durt including its topological characteristics and spectral properties although durt shows a logarithmic scaling with the size of the network durt is not a small world network since its clustering coefficient is zero in this paper we propose a new deterministic small world network by adding some edges with a simple rule in each durt iteration and then give the analytic solutions to several topological characteristics of the model proposed

lp self improvement of generalized poincare inequalities in spaces of homogeneous type abstract in this paper we study self improving properties in the scale of lebesgue spaces of generalized poincare inequalities in spaces of homogeneous type in contrast with the classical situation the oscillations involve approximation of the identities or semigroups whose kernels decay fast enough and the resulting estimates take into account their lack of localization the techniques used do not involve any classical poincare or sobolev poincare inequalities and therefore they can be used in general settings where these estimates do not hold or are unknown we apply our results to the case of riemannian manifolds with doubling volume form and assuming gaussian upper bounds for the heat kernel of the semigroup e t with being the laplace beltrami operator we obtain generalized poincare inequalities with oscillations that involve the semigroup e t and with right hand sides containing either or 1 2

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

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

scaling and percolation in the small world network model in this paper we study the small world network model of watts and strogatz which mimics some aspects of the structure of networks of social interactions we argue that there is one nontrivial length scale in the model analogous to the correlation length in other systems which is well defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero giving a normal critical point in this limit this length scale governs the crossover from large to small world behavior in the model as well as the number of vertices in a neighborhood of given radius on the network we derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex vertex distance on the network and using series expansion and pad e approximants find an approximate analytic form for the scaling function we calculate the effective dimension of small world graphs and show that this dimension varies as a function of the length scale on which it is measured in a manner reminiscent of multifractals we also study the problem of site percolation on small world networks as a simple model of disease propagation and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms in epidemiological terms the point at which an epidemic occurs the typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph all our analytic results are confirmed by extensive numerical simulations of the model

anchored nash inequalities and heat kernel bounds for static and dynamic degenerate environments abstract we introduce anchored versions of the nash inequality they allow to control the l 2 norm of a function by dirichlet forms that are not uniformly elliptic we then use them to provide heat kernel upper bounds for diffusions in degenerate static and dynamic random environments as an example we apply our results to the case of a random walk with degenerate jump rates that depend on an underlying exclusion process at equilibrium

approximate solutions to second order parabolic equations i analytic estimates we establish a new type of local asymptotic formula for the green s function gt x y of a uniformly parabolic linear operator t l with nonconstant coefficients using dilations and taylor expansions at a point z z x y for a function z with bounded derivatives such that z x x x rn our method is based on dilation at z dyson and taylor series expansions we use the baker campbell hausdorff commutator formula to explicitly compute the terms in the dyson series our procedure leads to an explicit elementary algorithmic construction of approximate solutions to parabolic equations that are accurate to arbitrarity prescribed order in the short time limit we establish mapping properties and precise error estimates in the exponentially weighed lp type sobolev spaces was p rn that appear in practice

on the strong universal consistency of nearest neighbor regression function estimates two results are presented concerning the consistency of the k nearest neighbor regression estimate we show that all modes of convergence in l 1 in probability almost sure complete are equivalent if the regression variable is bounded under the additional condition k log n we also obtain the strong universal consistency of the estimate

logarithmic sobolev and multilinear holder s inequalities via heat flow monotonicity formulas abstract heat flow monotonicity formulas have evolved in recent years as a powerful tool in deriving functional and geometric inequalities which are in turn useful in mathematical analysis and applications this paper aims mainly at proving logarithmic sobolev and multilinear holder s inequalities through the heat flow method precisely two entropy monotonicity formulas are constructed via the heat flow it is shown that the first entropy monotonicity formula is intimately related to the concavity of the power of shannon entropy and fisher information from which the associated logarithmic sobolev inequality for probability measure in euclidean setting is recovered the second monotonicity formula combines very well with convolution and diffusion semigroup properties of the heat kernel to establish the proof of the multilinear holder inequalities

weighted norm inequalities off diagonal estimates and elliptic operators part iv riesz transforms on manifolds and weights this is the fourth article of our series here we study weighted norm inequalities for the riesz transform of the laplace beltrami operator on riemannian manifolds and of subelliptic sum of squares on lie groups under the doubling volume property and gaussian upper bounds

towards a nonperturbative renormalization group analysis we prove that a certain class of convex gradient models in high dimensional spaces without the presence of a small parameter renormalizes to a free field as a consequence we establish a certain asymptotic formula for the partition function in some ways this is a realization of gawedzki and kupiainen s idea to use correlation inequalities to augment the rigorous renormalization group methods we use the more particular suggestion of spencer to use certain inequalities of brascamp and lieb and also the formulation of the correlation functions in terms of the solutions to some partial differential equations given by helffer and sjostrand