finite dimensional backward shift invariant subspaces of arveson spaces a description is given of finite dimensional backward shift invariant subspaces of arveson spaces which are certain multivariable analogues of the hardy space of the unit disk

entropy dissipation and long range interactions we study boltzmann s collision operator for long range interactions i e without grad s angular cut off assumption we establish a functional inequality showing that the entropy dissipation controls smoothness of the distribution function in a precise sense our estimate is optimal and gives a unified treatment of both the linear and the nonlinear cases we also give simple and self contained proofs of several useful results that were scattered in previous works as an application we obtain several helpful estimates for the cauchy problem and for the landau approximation in plasma physics

essential spectra of elementary operators this paper describes the essential spectrum and index function of the operator x axb where a b and x are hilbert space operators analogous results are given for the restriction of this operator to a norm ideal and partial analogues are given for sums of such operators and for the case when the operators act on a banach space

subalgebras of c algebras iii multivariable operator theory a d contraction is a d tuple t1 td of mutually commuting operators acting on a common hilbert space h such that t1 1 t2 2 td d 1 2 d for all 1 2 d h these are the higher dimensional counterparts of contractions we show that many of the operator theoretic aspects of function theory in the unit disk generalize to the unit ball bd in complex d space including von neumann s inequality and the model theory of contractions these results depend on properties of the d shift a distinguished d contraction which acts on a new h2 space associated with bd and which is the higher dimensional counterpart of the unilateral shift h 2 and the d shift are highly unique indeed by exploiting the noncommutative choquet boundary of the d shift relative to its generated c algebra we find that there is more uniqueness in dimension d 2 than there is in dimension one

smoothing estimates for boltzmann equation with full range interactions spatially homogeneous case in this work we are concerned with the regularities of the solutions to the boltzmann equation with physical collision kernels for the full range of intermolecular repulsive potentials r p 1 with p 2 we give new and constructive upper and lower bounds for the collision operator in terms of standard weighted fractional sobolev norms as an application we get the global entropy dissipation estimate which is a little stronger than that described by alexandre et al arch rational mech anal 152 4 327 355 2000 as another application we prove the smoothing effects for the strong solutions constructed by desvillettes and mouhot arch rational mech anal 193 2 227 253 2009 of the spatially homogeneous boltzmann equation with true hard potential and true moderately soft potential

regularizing effect and local existence for the non cutoff boltzmann equation the boltzmann equation without grad s angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non integrable angular singularity of the cross section however even though this has been justified satisfactorily for the spatially homogeneous boltzmann equation it is still basically unsolved for the spatially inhomogeneous boltzmann equation in this paper by sharpening the coercivity and upper bound estimates for the collision operator establishing the hypo ellipticity of the boltzmann operator based on a generalized version of the uncertainty principle and analyzing the commutators between the collision operator and some weighted pseudo differential operators we prove the regularizing effect in all time space and velocity variables on the solutions when some mild regularity is imposed on these solutions for completeness we also show that when the initial data has this mild regularity and a maxwellian type decay in the velocity variable there exists a unique local solution with the same regularity so that this solution acquires the c regularity for any positive time

forest fire model with immune trees abstract we present a generalization of the forest fire model of p bak et al by including the immunity g which is the probability that a tree is not ignited although one of its neighbors is burning when g reaches a critical value gc p which depends on the tree growth rate p the fire cannot survive any more i e a continuous phase transition takes place from a steady state with fire to a steady state without fire we present results of computer simulations and explain them by analytic calculations the fire spreading at the phase transition represents a new type of percolation which is called fluctuating site percolation

data sparse approximation to a class of operator valued functions in earlier papers we developed a method for the data sparse approximation of the solution operators for elliptic parabolic and hyperbolic pdes based on the dunford cauchy representation to the operator valued functions of interest combined with the hierarchical matrix approximation of the operator resolvents in the present paper we discuss how these techniques can be applied to approximate a hierarchy of the operator valued functions generated by an elliptic operator

field theoretic formulation of an epidemic process with immunisation the author describes a model of the spread of an epidemic on a lattice in which sites may be infected by their neighbours after which they recover and are subsequently immune it corresponds to a modification of the directed percolation problem and also to a growth model of clusters the model is analysed using a continuum field theory both within a self consistent approximation and using an epsilon expansion below six transverse dimensions

the landau equation in a periodic box the landau equation which was proposed by landau in 1936 is a fundamental equation to describe collisions among charged particles interacting with their coulombic force in this article global in time classical solutions near maxwellians are constructed for the landau equation in a periodic box our result also covers a class of generalized landau equations which describes grazing collisions in a dilute gas

interpolation norms between row and column spaces and the norm problem for elementary operators abstract for the class of transformers acting as x a t x b t d t on the space of bounded hilbert space operators we give formulae for its norm on the hilbert schmidt class 1 x a t x b t d t b c 2 h lim n 2 n tr k 1 n a t n 1 k a s n 1 k tr k 1 n b s k b t k k 1 n d s k d t k 2 n whenever a t p b t p d t for some p 0 we also estimate from below its norm on the other schatten classes this answers a question of characterizing 1 2 interpolation norm between column and row space norm for operator valued functions with the discrete case providing the solution of the norm problem for elementary operators acting on the hilbert schmidt class

the boltzmann equation besov spaces and optimal time decay rates in rxn abstract we prove that k th order derivatives of perturbative classical solutions to the hard and soft potential boltzmann equation without the angular cut off assumption in the whole space r x n with n 3 converge in large time to the global maxwellian with the optimal decay rate of o t 1 2 k n 2 n r in the l x r l v 2 norm for any 2 r these results hold for any 0 n 2 as long as initially f 0 b 2 l v 2 in the hard potential case we prove faster decay results in the sense that if p f 0 b 2 l v 2 and i p f 0 b 2 1 l v 2 for n 2 n 2 2 then the solution decays the global maxwellian in l v 2 l x 2 with the optimal large time decay rate of o t 1 2

asymptotic analysis of the spatially homogeneous boltzmann equation grazing collisions limit in the present work we consider the asymptotic problem of the spatially homogeneous boltzmann equation when almost all collisions are grazing that is the deviation angle theta of the collision is limited near zero i e theta le epsilon we show that by taking the proper scaling to the cross section which was used in 37 that is assuming begin aligned b epsilon v v sigma 2 1 s v v gamma epsilon 3 sin 1 theta left frac theta epsilon right 1 2s mathrm 1 theta le epsilon end aligned where theta langle theta frac upsilon upsilon upsilon upsilon sigma rangle the solution f epsilon of the boltzmann equation with initial data f 0 can be globally or locally expanded in some weighted sobolev space as begin aligned f epsilon f o epsilon end aligned where the function f is the solution of landau equation which is associated with the grazing collisions limit of boltzmann equation with the same initial data f 0 this gives the rigorous justification of the landau approximation in the spatially homogeneous case in particular if taking gamma 3 and s 1 epsilon in the cross section b epsilon we show that the above asymptotic formula still holds and in this case f is the solution of landau equation with the coulomb potential going further we revisit the well posedness problem of the boltzmann equation in the limiting process we show there exists a common lifespan such that the uniform estimates of high regularities hold for each solution f epsilon thanks to the weak convergence results on the grazing collisions limit in 37 in other words we establish a unified framework to establish the well posedness results for both boltzmann and landau equations

backward shift operator and finite dimensional de branges rovnyak spaces in the ball recently bolotnikov and rodman characterized finite dimensional backward shift invariant subspaces of the arveson space in this note we study finite dimensional backward shift invariant hilbert spaces contractively included in the arveson space to that purpose we redefine in an appropriate way the backward shift operators in terms of power series expansions

on a self organized critical forest fire model the author studies a forest fire model introduced in a paper by drossel and schwabl 1992 by means of simulations with high statistics and very close to the critical point in this way he can correct some statements made in that paper in particular he finds that the critical exponents are not the classical ones proposed by drossel and schwabl in spite of scaling laws with anomalous exponents the typical states of the system are not critical in the sense of being marginally stable locally

dynamics of driven interfaces near isotropic percolation transition we consider the dynamics and kinetic roughening of interfaces embedded in uniformly random media near percolation treshold in particular we study simple discrete forest fire lattice models through monte carlo simulations in two and three spatial dimensions an interface generated in the models is found to display complex behavior away from the percolation transition the interface is self affine with asymptotic dynamics consistent with the kardar parisi zhang universality class however in the vicinity of the percolation transition there is a different behavior at earlier times by scaling arguments we show that the global scaling exponents associated with the kinetic roughening of the interface can be obtained from the properties of the underlying percolation cluster our numerical results are in good agreement with theory however we demonstrate that at the depinning transition the interface as defined in the models is no longer self affine finally we compare these results with those obtained from a more realistic reaction diffusion model of slow combustion s1063 651x 98 08708 x

fractional poincare and logarithmic sobolev inequalities for measure spaces abstract we prove generalizations of the poincare and logarithmic sobolev inequalities corresponding to the case of fractional derivatives in measure spaces with only a minimal amount of geometric structure the class of such spaces includes but is not limited to spaces of homogeneous type with doubling measures several examples and applications are given including poincare inequalities for graph laplacians fractional poincare inequalities of mouhot russ and sire 2011 17 and implications for recent work of the author and r m strain on the boltzmann collision operator gressman and strain 2010 2011 2011 9 10 11

the vlasov poisson boltzmann system without angular cutoff this paper is concerned with the vlasov poisson boltzmann system for plasma particles of two species in three space dimensions the boltzmann collision kernel is assumed to be angular non cutoff with 3 2s and 1 2 s 1 where s are two parameters describing the kinetic and angular singularities respectively we establish the global existence and convergence rates of classical solutions to the cauchy problem when initial data is near maxwellians this extends the results in duan et al j diff eqs 252 12 6356 6386 2012 math models methods appl sci 23 6 927 2013 for the cutoff kernel with 2 1 to the case 3 2 as long as the angular singularity exists instead and is strong enough i e s is close to 1 the proof is based on the time weighted energy method building also upon the recent studies of the non cutoff boltzmann equation in gressman and strain j amer math soc 24 3 771 847 2011 and the vlasov poisson landau system in guo j amer math soc 25 759 812 2012

formation of space time structure in a forest fire model we present a general stochastic forest fire model which shows a variety of different structures depending on the parameter values the model contains three possible states per site tree burning tree empty site and three parameters tree growth probability p lightning probability f and immunity g we review analytic and computer simulation results for a quasideterministic state with spiral shaped fire fronts for a percolation like phase transition and a self organized critical state possible applications to excitable systems are discussed

spreading analysis and finite size scaling study of the critical behavior of a forest fire model with immune trees abstract in a forest fire model with immune trees ffmit recently proposed by drossel and schwabl physica a 199 1993 183 the sites of the lattice are either empty a tree or a burning tree and at each time step the system is updated in parallel according to the following rules i a burning tree becomes an empty site ii trees grow with probability p from empty sites and iii a green tree becomes a burning tree with probability 1 g if at least one next neighbor is burning fixing an arbitrary grow probability p and starting with a small immunity g increments of g causes the fire density of the steady state to decrease until the fire becomes irreversibly extinguished at a certain critical point of coordinates pc gc the set of critical points defines a critical curve gc p which in the l limit divides the p g plane in two regions a steady state with fire fronts for g

caratheodory fejer interpolation in the ball with mixed derivatives we use reproducing kernel hilbert space methods to solve a caratheodory fejer type interpolation problem with mixed derivatives corresponding to a so called lower inclusive set in the class of schur multipliers

singular perturbation theory of traveling waves in excitable media a review abstract waves of chemical or electrical activity traveling through space have been observed in several contexts chemical reaction mixtures in non convecting liquid phase cell suspensions nerve axons and neuromuscular tissues typically wave supporting preparations are excitable that is they respond and sensitivity to perturbations are rapidly damped out but suprathreshold disturbances trigger an abrupt and substantial response the abruptness of the response can be exploited by the methods of singular perturbation theory to obtain a mathematical description of wave propagation in spatially distributed excitable media singular perturbation analysis of propagating waves in one spatial dimension is straightforward and uncontentious but the analysis of propagating waves in one spatial dimension is straightforward and several fundamentally different ways we compare and contrast the approaches taken by greenberg zykov fife krinskii and his collaborators and ourselves with particular emphasis on the case of rotating spiral waves our intention is to bring some order to the important but difficult theory of propagating waves in two dimensional excitable media in conclusion we discuss briefly some possible extensions of the singular perturbation approach to propagating wave surfaces in three dimensional space

upper maxwellian bounds for the spatially homogeneous boltzmann equation for the spatially homogeneous boltzmann equation with cutoff hard potentials it is shown that solutions remain bounded from above uniformly in time by a maxwellian distribution provided the initial data have a maxwellian upper bound the main technique is based on a comparison principle that uses a certain dissipative property of the linear boltzmann equation implications of the technique to propagation of upper maxwellian bounds in the spatially inhomogeneous case are discussed

on a forest fire model with supposed self organized criticality we study a stochastic forest fire model introduced by p baket al as a model showing self organized criticality this model involves a growth parameterp and the criticality is supposed to show up in the limitp 0 by simulating the model on much larger lattices and with much smaller values ofp we find that the correlations with longest range do not show a nontrivial critical phenomenon in this limit though we cannot rule out percolation like critical behavior on a smaller but still divergent length scale in contrast the model shows nontrivialdeterministic evolution over time scales 1 p in the limitp 0

global existence and decay of solutions for soft potentials to the fokker planck boltzmann equation without cut off abstract this paper is concerned with the large time behavior of classical solutions near a given global maxwellian to the cauchy problem of the fokker planck boltzmann equation for non cutoff soft potentials our analysis is based on the coercivity of the fokker planck operator and elementary weighted energy method

multipliers of elementary operators and comparison of row and column space schatten p norms abstract sharp upper estimates for the norm of the weighted elementary operator of the form n 1 c n z n a n b n w n d n acting from one symmetrically normed ideal of compact hilbert space operators to another are given particularly we relate the norm of n 1 c n z n a n a n w n c n with norms of n 1 a n a n and n 1 c n c n on the appropriate domains and co domains

the spectral picture of la rb abstract let a and b be commuting n tuples of operators on a hilbert space h and let l a r b be the 2 n tuple of left and right multiplications induced by a and b on l h we show that t l a r b l h t a h t b h te l a r b l h te a h t b h t a h te b h and ind l a r b 1 n ind a ind b whenever a and b are fredholm similar results hold for the restrictions of l a r b to norm ideals in l h giving in particular the spectral picture of a i i b on h h a s fainstein working independently has announced versions of these results for banach spaces

fredholm and invertible tuples of operators the deformation problem using j l taylor s definition of joint spectrum we study fredholm and invertible tuples of operators on a hilbert space we give the foundations for a several variables theory including a natural generalization of atkinson s theorem and an index which well behaves we obtain a characterization of joint invertibility in terms of a single operator and study the main examples at length we then consider the deformation problem and solve it for the class of almost doubly commuting fredholm pairs with a semi fredholm coordinate 1 introduction 1 let t be a bounded linear operator on a banach space t is said to be invertible if there exists an operator s on such that ts st 1 the identity operator on 9c by the open mapping theorem this is equivalent to ker t 0 and r t range of t the last formulation does not rely upon the existence of an inverse for t but rather on the action of the operator t when t is replaced by an tuple of commuting operators several definitions of nonsingular ity exist j l taylor 19 has obtained one which reflects the actions of the operators by considering the koszul complex associated with the tuple 2 in this paper we develop a general several variables theory on the basis of taylor s work and study commuting and almost commuting commuting mod ulo the compacts tuples of operators on a hilbert space we obtain a characterization of joint invertibility in terms of the invertibility of a single operator which is essential for our approach from that we get a number of corollaries which generalize nicely the known elementary results in one variable at the same time the referred characterization allows us to define a continuous invariant under compact perturbations integer valued index on the class of fred holm tuples those almost commuting tuples which are invertible in the calkin algebra this index extends the classical one for fredholm operators we prove that an almost commuting tuple of essentially normal operators with all commu tators in trace class has index zero 2 and that a natural generalization of atkinson s theorem holds for tuples

the threshold of coexistence and critical behaviour of a predator prey cellular automaton we study a probabilistic cellular automaton to describe two population biology problems the threshold of species coexistence in a predator prey system and the spreading of an epidemic in a population by carrying out mean field approximations and numerical simulations we obtain the phase boundaries thresholds related to the transition between an active state where prey and predators present a stable coexistence and a prey absorbing state the numerical estimates for the critical exponents show that the transition belongs to the directed percolation universality class in the limit where the cellular automaton maps into a model for the spreading of an epidemic with immunization we observe a crossover from directed percolation class to the dynamic percolation class patterns of growing clusters related to species coexistence and spreading of epidemic are shown and discussed

rational hyperholomorphic functions in r4 we introduce the notion of rationality for hyperholomorphic functions functions in the kernel of the cauchy fueter operator following the case of one complex variable we give three equivalent definitions the first in terms of cauchy kovalevskaya quotients of polynomials the second in terms of realizations and the third in terms of backward shift invariance also introduced and studied are the counterparts of the arveson space and blaschke factors

exact and approximate solutions of some operator equations based on the cayley transform abstract we consider the operator equation sx j 1 m u j xv j y where u j v j are some communicative sets of operators but in general u j need not compute with v j particular cases of this equation are the sylvester and ljapunov equations we give a new representation and an approximation of the solution which is suitable to perform it algorithmically error estimates are given which show exponential covergence for bounded operators and polynomial convergence for unbounded ones based on these considerations we construct an iterative process and give an existence theorem for the operator equation z 2 a 1 z a 2 0 arising for example when solving an abstract second order differential equation with non commutative coefficients

the boltzmann equation with frictional force for soft potentials in the whole space abstract this paper is concerned with the cauchy problem of the boltzmann equation with frictional force for both cutoff and non cutoff soft potentials in the whole space and our main purpose is to establish its global solvability result near a given global maxwellian and to deduce the temporal convergence rates of such a global solution toward the global maxwellian when initial perturbation is sufficiently small the analysis is based on the time weighted energy method building also upon the recent studies of the cutoff vlasov poisson boltzmann system 14 15 the non cutoff vlasov poisson boltzmann system 8 and non cutoff vlasov maxwell boltzmann system 9

entropy dissipation estimates for the landau equation in the coulomb case and applications abstract we present in this paper an estimate which bounds from below the entropy dissipation d f of the landau operator with coulomb interaction by a weighted h 1 norm of the square root of f as a consequence we get a weighted l t 1 l v 3 estimate for the solutions of the spatially homogeneous landau equation with coulomb interaction and the propagation of l 1 moments of any order for this equation we also present an application of our estimate to the landau equation with moderately soft potentials providing thus a new proof of some recent results of 30

global existence and full regularity of the boltzmann equation without angular cutoff we prove the global existence and uniqueness of classical solutions around an equilibrium to the boltzmann equation without angular cutoff in some sobolev spaces in addition the solutions thus obtained are shown to be non negative and c in all variables for any positive time in this paper we study the maxwellian molecule type collision operator with mild singularity one of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator this norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated h theorem

complete nevanlinna pick kernels abstract we give a new treatment of quiggin s and mccullough s characterization of complete nevanlinna pick kernels we show that a kernel has the matrix valued nevanlinna pick property if and only if it has the vector valued nevanlinna pick property we give a representation of all complete nevanlinna pick kernels and show that they are all restrictions of a universal complete nevanlinna pick kernel

self organized critical forest fire model a forest fire model is introduced which contains a lightning probability f this leads to a self organized critical state in the limit f ensuremath rightarrow 0 provided that the time scales of tree growth and burning down of forest clusters are separated we derive scaling laws and calculate all critical exponents the values of the critical exponents are confirmed by computer simulations for a two dimensional system we show that the forest density in the critical state assumes its minimum possible value i e that energy dissipation is maximum

positive extension problems for a class of structured matrices abstract we consider positive definite semidefinite extension problems for matrices with structure determined via a stein equation some related extremal problems maximal and minimal rank extensions maximal determinant extension are also considered connections with interpolation problems for a certain class of analytic contractive valued functions on the unit ball of c d are discussed

computer simulations of the forest fire model abstract we present results of simulations of the forest fire model proposed by p bak et al containing a tree growth rate p and fire spreading to nearest neighbors the space time structure of the fire and the scaling properties of the forest clusters show that the model cannot be critical in the limit p 0 instead we observe regular and quasideterministic spiral shaped fire fronts we present a deterministic forest fire model that helps understand the origin of these spirals

an operator valued poisson kernel abstract versions of the cauchy and poisson formulas in the unit ball of c n are defined by means of appropriate operator valued kernels connections with some problems of dilation theory are then made

well posedness of spatially homogeneous boltzmann equation with full range interaction in the present work we give the coercivity estimates for the boltzmann collision operator q without angular cut off to clarify in which case the functional langle q g f f rangle will become the truly sub elliptic based on this observation and commutator estimates in alexandre et al arch rat mech anal 198 39 123 2010 the upper bound estimates for the collision operator in chen and he arch rat mech anal 201 2 501 548 2011 and the stability results in desvillettes and mouhot arch rat mech anal 193 2 227 253 2009 in the function space l 1 q cap h n we establish global well posedness or local well posedness for the spatially homogeneous boltzmann equation with full range interaction covering most of physical collision kernels

self organized criticality in forest fire models we review properties of the self organized critical soc forest fire model ffm self organized critical systems drive themselves into a critical state without fine tuning of parameters after an introduction the rules of the model and the conditions for spiral shaped and soc large scale structures are given for the soc state critical exponents and scaling relations are introduced the existence of an upper critical dimension and the universal behavior of the model are discussed the relations and differences between ffm and percolation systems are outlined considering an extension of the ffm into the regions beyond the critical point phase transitions and the various structures found in these regions are illustrated

sharp anisotropic estimates for the boltzmann collision operator and its entropy production abstract this article provides sharp constructive upper and lower bound estimates for the boltzmann collision operator with the full range of physical non cut off collision kernels n and s 0 1 in the trilinear l 2 r n energy q g f f these new estimates prove that for a very general class of g v the global diffusive behavior on f in the energy space is that of the geometric fractional derivative semi norm identified in the linearized context in our earlier works gressman and strain 2010 15 2011 16 we further prove new global entropy production estimates with the same anisotropic semi norm this resolves the longstanding widespread heuristic conjecture about the sharp diffusive nature of the non cut off boltzmann collision operator in the energy space l 2 r n