on bounded positive stationary solutions for a nonlocal fisher kpp equation abstract we study the existence of stationary solutions for a nonlocal version of the fisher kolmogorov petrovskii piscounov fkpp equation the main motivation is a recent study by berestycki et al 2009 where the nonlocal fkpp equation has been studied and it was shown for the spatial domain r and sufficiently small nonlocality that there are only two bounded non negative stationary solutions here we provide a similar result for r d using a completely different approach in particular an abstract perturbation argument is used in suitable weighted sobolev spaces one aim of the alternative strategy is that it can eventually be generalized to obtain persistence results for hyperbolic invariant sets for other nonlocal evolution equations on unbounded domains with small nonlocality i e to improve our understanding in applications when a small nonlocal influence alters the dynamics and when it does not

decay estimates for the wave equation in two dimensions abstract we establish strichartz estimates both reversed and some direct ones pointwise decay estimates and weighted decay estimates for the linear wave equation in dimension two with an almost scaling critical potential in the case when there is no resonance or eigenvalue at the edge of the spectrum we also prove some simple applications

self similar transport in incomplete chaos particle chaotic dynamics along a stochastic web is studied for three dimensional hamiltonian flow with hexagonal symmetry in a plane two different classes of dynamical motion obtained by different values of a control parameter and corresponding to normal and anomalous diffusion have been considered and compared it is shown that the anomalous transport can be characterized by powerlike wings of the distribution function of displacement flights which are similar to l evy flights approximate trappings of orbits near the boundary layer of islands and anomalous behavior of the moments of a distribution function considered as a function of the number of the moment the main result is related to the self similar properties of different topological and dynamical characteristics of the particle motion this self similarity appears in the weierstrass like random walk process that is responsible for the anomalous transport exponent in the mean moment dependence on t this exponent can be expressed as a ratio of fractal dimensions of space and time sets in the weierstrass like process an explicit form for the expression of the anomalous transport exponent through the local topological properties of orbits has been given

random walks on graphs ideas techniques and results random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology in recent years relevant mathematical results have been obtained in this field and new ideas have been introduced which can be fruitfully extended to different areas and disciplines here we aim at giving a brief but comprehensive perspective of these progresses with a particular emphasis on physical aspects

random walk on fractals numerical studies in two dimensions monte carlo calculations are used to investigate some statistical properties of random walks on fractal structures two kinds of lattices are used the sierpinski gasket and the infinite percolation cluster in two dimensions among other problems the authors study i the range rn of the walker number of distinct visited sites during n steps average value sn variance sigma n and asymptotic distribution ii renewal theory return to the original site probability of return p0 n mean number of returns nu n the probability distribution of the walker position p n r after n steps is discussed the asymptotic behaviour n 1 of these quantities exhibits power laws with associated exponents the numerical values of these exponents are in good agreement with recent theoretical predictions alexander and orbach 1982 rammal and toulouse 1982

quenching of flames by fluid advection we consider a simple scalar reaction advection diffusion equation with ignitiontype nonlinearity and discuss the following question what kinds of velocity profiles are capable of quenching any given flame provided the velocity s amplitude is adequately large even for shear flows the answer turns out to be surprisingly subtle if the velocity profile changes in space so that it is nowhere identically constant or if it is identically constant only in a region of small measure then the flow can quench any initial data but if the velocity profile is identically constant in a sizable region then the ensuing flow is incapable of quenching large enough flames no matter how much larger the amplitude of this velocity is the constancy region must be wider across than a couple of laminar propagating front widths the proof uses a linear pde associated to the nonlinear problem and quenching follows when the pde is hypoelliptic the techniques used allow the derivation of new nearly optimal bounds on the speed of traveling wave solutions c 2001 john wiley sons inc

mapping between hopping on hierarchical structures and diffusion on a family of fractals the authors present a simple analytical approach to studying diffusion on hierarchical and fractal structures they show that hopping on one dimensional hierarchical structures can be mapped onto diffusion on a family of loopless fractals

commuting maps on some subsets of matrices that are not closed under addition abstract let m n k be the ring of all n n matrices over a field k we describe additive maps g m n k m n k such that g x x xg x for all invertible singular x m n k

dispersive estimates for schrodinger operators in dimension two we prove l1 2 l 2 for the two dimensional schrodinger operator v with the decay rate t 1 we assume that zero energy is neither an eigenvalue nor a resonance this condition is formulated as in the recent paper by jensen and nenciu on threshold expansions for the two dimensional resolvent

on strong anomalous diffusion superdiffusive behavior i e x2 t t2 with 1 2 is in general not completely characterized by a unique exponent we study some systems exhibiting strong anomalous diffusion i e x t q tq q where 2 1 2 and q q is not a linear function of q this feature is different from the weak superdiffusive regime i e q const 1 2 occurring in random shear flows strong anomalous diffusion can be generated by nontrivial chaotic dynamics e g lagrangian motion in 2d time dependent incompressible velocity fields 2d symplectic maps and 1d intermittent maps typically the function q q is piecewise linear this is due to two mechanisms a weak anomalous diffusion for the typical events and a ballistic transport for the rare excursions in order to have strong anomalous diffusion one needs a violation of the hypothesis of the central limit theorem this happens only in a very narrow region of the control parameters space in the presence of strong anomalous diffusion one does not have a unique exponent and therefore one has the failure of the usual scaling p x t t f x t of the probability density this implies that the effective equation at large scale and long time for p x t obeys neither the usual fick equation nor other linear equations involving temporal and or spatial fractional derivatives

stieltjes integral representation of effective diffusivities in time dependent flows a stieltjes integral representation for the effective diffusivity of a passive scalar in time dependent incompressible flows is developed the representation provides a summability formula for the perturbative expansion of the diffusivity in powers of the p eclet number in particular upper and lower bounds on the effective diffusivity are obtained from pad e approximants of the series

a weighted dispersive estimate for schrodinger operators in dimension two let h v where v is a real valued potential on mathbb r 2 satisfying v x lesssim langle x rangle 3 we prove that if zero is a regular point of the spectrum of h v then w 1 e ith p ac f l infty mathbb r 2 lesssim frac 1 t log 2 t w f l 1 mathbb r 2 t geq 2 with w x log 2 x 2 this decay rate was obtained by murata in the setting of weighted l 2 spaces with polynomially growing weights

l p boundedness of wave operators for two dimensional schrodinger operators let be the two dimensional schrodinger operator with the real valued potential v which satisfies the decay condition at infinity for we show that the wave operators are bounded in for any 1 p under the condition that h has no zero bound states or zero resonance extending the corresponding results for higher dimensions as w intertwine h 0 and the absolutely continuous part h p ac of h f h p ac w f h 0 w for any borel function f on 1 this reduces the various l p mapping properties of f h p ac to those of f h 0 the convolution operator by the fourier transform of the function f 2

renormalisation on sierpinski type fractals presents a family of deterministic fractals which generalise the d dimensional sierpinski gaskets and establishes their order of ramification and their fractal and spectral dimensions random walks on these fractals are renormalisable and lead to rational not necessarily polynomial mapping

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

strong commutativity preserving maps on lie ideals abstract let a be a prime ring and let r be a noncentral lie ideal of a an additive map f r a is called strong commutativity preserving scp on r if f x f y x y for all x y r in this paper we show that if f is scp on r then there exist c 2 1 and an additive map r z a such that f x x x for all x r where c is the extended centroid of a unless char a 2 and a satisfies the standard identity of degree 4

topological properties of percolation clusters percolation properties including the total cluster mass s the shell mass b and the linear geometrical size r are studied as a function of the topological chemical distance parameter l all critical exponents are shown to be related to an apparently new exponent nu defined by r approximately lv critical exponents are calculated exactly for percolation clusters on the cayley tree a model for 6d percolation for which s approximately l2 b approximately l and r2 approximately l for diffusion on such clusters one finds that l3 approximately t numerical estimates of the exponents are obtained for other dimensions a conjecture which relates nu to beta and nu is discussed

commuting traces of biadditive mappings commutativity preserving mappings and lie mappings biadditive mappings b r r r where r is a prime ring with certain additional properties satisfying b x x x xb x x for all x r are characterized as an application we determine the structures of commutativity preserving mappings lie isomorphisms and lie derivations of certain prime rings

reaction spreading on graphs we study reaction diffusion processes on graphs through an extension of the standard reaction diffusion equation starting from first principles we focus on reaction spreading i e on the time evolution of the reaction product m t at variance with pure diffusive processes characterized by the spectral dimension d s the important quantity for reaction spreading is found to be the connectivity dimension d l numerical data in agreement with analytical estimates based on the features of n independent random walkers on the graph show that m t t d l in the case of erdos renyi random graphs the reaction product is characterized by an exponential growth m t e t with proportional to ln k where k is the average degree of the graph

time decay estimates for the wave equation with potential in dimension two abstract we study the wave equation with potential u t t u v u 0 in two spatial dimensions with v a real valued decaying potential with h v we study a variety of mapping estimates of the solution operators cos t h and sin t h h under the assumption that zero is a regular point of the spectrum of h we prove a dispersive estimate with a time decay rate of t 1 2 a polynomially weighted dispersive estimate which attains a faster decay rate of t 1 log t 2 for t 2 finally we prove dispersive estimates if zero is not a regular point of the spectrum of h

self organization and fractal dynamics in turbulence abstract results of analysis of the field of helicity obtained in three different turbulent laboratory flows grid flow boundary layer and jet and a simple helical fracton model has been used in order to provide a quantitative explanation of anomalous turbulent diffusion in the troposphere and in the ocean it is shown that kolmogorov turbulence is critical in respect to the localization effects of subregions with large helicity helical fractons and it breaks up into helical fractons under the condition df 2 where df 2d dw is the so called fracton dimension d is the fractal dimension of the turbulent fractal and dw is the dimension of random walks on this fractal for strictly kolmogorov turbulence d1 2 we study the internal structure of helical fractons and demonstrate that they are characterized by d f 4 3 finally we look at the influence of helical fractons on diffusion of a passive scalar in turbulence it is shown that their influence is manifested in the scaling law for the turbulent diffusivity in the form k l 8 7 in both three dimensional and quasi two dimensional situations this anomalous law is in a very good agreement with a large number of experimental data of different authors in the troposphere and in the upper ocean

on the fourth order schrodinger equation in four dimensions dispersive estimates and zero energy resonances abstract we study the fourth order schrodinger operator h 2 v for a decaying potential v in four dimensions in particular we show that the t 1 decay rate holds in the l 1 l setting if zero energy is regular furthermore if the threshold energies are regular then a faster decay rate of t 1 log t 2 is attained for large t at the cost of logarithmic spatial weights zero is not regular for the free equation hence the free evolution does not satisfy this bound due to the presence of a resonance at the zero energy we provide a full classification of the different types of zero energy resonances and study the effect of each type on the time decay in the dispersive bounds

multichannel nonlinear scattering for nonintegrable equations ii the case of anisotropic potentials and data abstract the nonlinear scattering and stability results of soffer and weinstein comm math phys 1990 are extended to the case of anisotropic potentials and data the range of nonlinearities for which the theory is shown to be valid is also extended considerably

nonlinear strong commutativity preserving maps on skew elements of prime rings with involution abstract let a be a prime ring of characteristic not 2 with involution with center z a and with skew elements k suppose that f k a is a map satisfying f x f y x y for all x y k then there exists a map k z a such that f x x x for all x k or f x x x for all x k except when a is an order in a 4 9 or 16 dimensional central simple algebra

asymptotic expansions in time for solutions of schrodinger type equations let ia i p d v be a dissipative operator in l2 rn where p d is an elliptic differential operator of order m with real constant coefficients and v is a compact operator from the weighted sobolev space hm s rn to h0 p s rn s r for some m 1 let r z be the resolvent of a then an asymptotic expansion of r z as z approaches a critical value of the polynomial p is given the coefficient operators in the expansion are computed explicitly by using the resolvent expansion and the results of m murata 9 an asymptotic expansion of e ita as t is given

strong commutativity preserving maps in prime rings with involution abstract let a be a prime ring of characteristic not 2 with center z a and with involution let s be the set of symmetric elements of a suppose that f s a is an additive map such that f x f y x y for all x y s then unless a is an order in a 4 dimensional central simple algebra there exists an additive map s z a such that f x x x for all x s or f x x x for all x s

invariant manifolds for a class of dispersive hamiltonian partial differential equations we construct an invariant manifold of periodic orbits for a class of non linear schrodinger equations using standard ideas of the theory of center manifolds we rederive the results of soffer and weinstein comm math phys 133 119 146 1997 j differential equations98 376 390 1992 on the large time asymptotics of small solutions scattering theory

strong commutativity preserving maps on subsets of matrices that are not closed under addition abstract let m n d be the ring of all n n matrices over a division ring d where n 2 is an integer and let gl n d be the set of all invertible matrices in m n d we describe maps f gl n d m n d such that f x f y x y for all x y gl n d the analogous result for singular matrices is also obtained

predictability in spatially extended systems the predictability problem is discussed in turbulent fluids and in spatially extended systems we perform a numerical analysis of the spatial propagation of a small perturbation initially localized at a point of a one dimensional lattice of coupled maps where the interactions can be either local or non local in space in both cases the nonlinear terms in the dynamics of the perturbation cannot be neglected and the predictability problem cannot be reduced to the linearized evolution equations only for non local interactions is the predictability time proportional to the inverse lyapunov exponent lambda as in dynamical systems with a few degrees of freedom for local interactions it is proportional to the size of the system and is practically independent of the chaoticity degree lambda

diffusion on random clusters and the parasite problem in the parasite problem a particle ant diffuses randomly on a random percolation cluster in the limit of concentration to 0 lattice animal monte carlo simulations and scaling arguments show that for large animals the distance r travelled by this parasite increases as t 1 za with time t the authors find za approximately 3 4 on the simple cubic lattice and za approximately 2 6 on the square lattice this anomalous diffusion is in rough agreement with a generalisation of a suggestion by alexander and orbach 1982 heuristic arguments in favour of this suggestion are given also they look at corrections to scaling for concentrations equal to the percolation threshold

a counterexample to dispersive estimates for schrodinger operators in higher dimensions in dimension n 3 we show the existence of a compactly supported potential in the differentiability class c alpha alpha frac n 3 2 for which the solutions to the linear schrodinger equation in mathbb r n i partial t u delta u vu quad u 0 f fail to satisfy an evolution estimate of the form u t cdot infty leq c t frac n 2 u 0 cdot 1 this contrasts with known results in dimensions n 3 where a pointwise decay condition on v is generally sufficient to imply dispersive bounds

the influence of fractional diffusion in fisher kpp equations we study the fisher kpp equation where the laplacian is replaced by the generator of a feller semigroup with power decaying kernel an important example being the fractional laplacian in contrast with the case of the standard laplacian where the stable state invades the unstable one at constant speed we prove that with fractional diffusion generated for instance by a stable levy process the front position is exponential in time our results provide a mathematically rigorous justification of numerous heuristics about this model

description of diffusive and propagative behavior on fractals the known properties of diffusion on fractals are reviewed in order to give a general outlook of these dynamic processes after that we propose a description developed in the context of the intrinsic metric of fractals which leads us to a differential equation able to describe diffusion in real fractals in the asymptotic regime we show that our approach has a stronger physical justification than previous works on this field the most important result we present is the introduction of a dependence on time and space for the conductivity in fractals which is deduced by scaling arguments and supported by computer simulations finally the diffusion equation is used to introduce the possibility of reaction diffusion processes on fractals and analyze their properties specifically an analytic expression for the speed of the corresponding travelling fronts which can be of great interest for application purposes is derived

power commuting additive maps on invertible or singular matrices abstract let f be a field and let m n f be the ring of all n n matrices over f where n 2 is an integer we characterize additive maps f m n f m n f satisfying f x x m x m f x for all invertible singular x m n f where m 2 is an integer

dispersive estimates for schrodinger equations with threshold resonance and eigenvalue let h v x be a three dimensional schrodinger operator we study the time decay in lp spaces of scattering solutions e ithpcu where pc is the orthogonal projection onto the continuous spectral subspace of l2 r3 for h under suitable decay assumptions on v x it is shown that they satisfy the so called lp lq estimates e ithpcu p 4 t 3 1 2 1 p u q for all 1 q 2 p with 1 p 1 q 1 if h has no threshold resonance and eigenvalue and for all 3 2 q 2 p 3 if otherwise

reaction diffusion on random spatial networks with scale free jumping rates via effective medium theory we study epidemic processes using a metapopulation approach on the line featuring random transport rates between arbitrarily distant sites an average transport network is found using a recently developed variant of the effective medium approximation ema that is capable of dealing with these long range connections using a feynman kac argument in the effective medium we derive an estimate on the size of the infected domain and reproduce the known result of its exponential growth in time we hereby demonstrate the applicability of long range ema to dynamical processes on networks more intricate than simple diffusion

turing pattern formation in the brusselator model with superdiffusion the effect of superdiffusion on pattern formation and pattern selection in the brusselator model is studied our linear stability analysis shows in particular that unlike the case of normal diffusion the turing instability can occur even when diffusion of the inhibitor is slower than that of the initiator a weakly nonlinear analysis yields a system of amplitude equations analysis of which predicts parameter regimes where hexagons stripes and their coexistence are expected numerical computations of the original brusselator model near the stability boundaries confirm the results of the analysis in addition further from the stability boundaries we find a regime of self replicating spots

front propagation in reactive systems with anomalous diffusion abstract we study front propagation of reactive fields in systems whose diffusive behavior is anomalous both superdiffusive and subdiffusive the features of the front propagation depend not only on the scaling exponent x t 2 t 2 but also on the detailed shape of the probability distribution of the diffusive process from the analysis of different systems we have three possible behavior of front propagation the usual fisher kolmogorov like scenario i e the field has a spatial exponential tail moving with constant speed v f and thickness the field has a spatial exponential tail but v f and change in time as a power law and finally the field has a spatial power law tail and v f increases exponentially in time a linear analysis of the front tail is in quantitative agreement with the numerical simulations it is remarkable the fact that anomalous diffusion is neither necessary nor sufficient condition for the linear front propagation moreover if the probability distribution of the transport process follows the scaling relation given by the flory argument the front propagation is standard fisher kolmogorov like even in presence of super or sub diffusion

multiplicity results for a non local problem with concave and convex nonlinearities abstract we shall prove a multiplicity result for a non local problem with a super critical nonlinearity of the form 1 s u u u p 2 u u q 2 in u 0 on r n where n is a bounded domain 0 s 1 and 1 q 2 p as a consequence of our results for each p 2 there exists 0 such that for each 0 problem 1 has a sequence of solutions with a negative energy this result is new for super critical values of p we shall also explore the existence of symmetric solutions for symmetric domains our methodology is based on a variational principle established by one of the authors that allows one to deal with super critical problems when the standard euler lagrange functional is restricted to certain convex sets

existence and uniqueness of weak solutions for nonlocal parabolic problems via the galerkin method abstract fractional differential equations are becoming increasingly popular as a modeling tool to describe a wide range of non classical phenomena with spatial heterogeneities throughout the applied science and engineering a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal operators which include fractional laplacians on bounded domains in r n we develop the galerkin method to prove existence and uniqueness of weak solutions to nonlocal parabolic problems moreover we study the existence of orthonormal basis of eigenvectors associated to these nonlocal operators

spreading dynamics in heterogeneous graphs beyond the assortativity coefficient abstract we study spreading dynamics of a reaction diffusion process in a special class of heterogeneous graphs with poissonian degree distribution and composed of both local and long range links the behavior of the spreading dynamics on such networks are investigated by relating them to the topological features of graphs we find that the degree of assortativity can give just some indication about the large scale behavior of the spreading dynamics while a detailed description of the process can be addressed by introducing new more appropriate topological quantities linked to the distance between nodes

an integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection diffusion by a steady incompressible velocity field are developed here under these conditions a rigorous stieltjes integral representation for effective diffusivity in turbulent transport is derived this representation is valid for all peclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the peclet number one consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all peclet numbers a nonlocal variational principle for effective diffusivity is developed along with applications to advection diffusion by random arrays of vortices a new class of rigorous examples is introduced these examples have an explicit stieltjes measure for the effective diffusivity furthermore the effective diffusivity behaves likek0 pe 1 2 in the limit of large peclet numbers wherek0 is the molecular diffusivity formal analogies with the theory of composite materials are exploited systematically

q exponential weibull and q weibull distributions an empirical analysis abstract in a comparative study the q exponential and weibull distributions are employed to investigate frequency distributions of basketball baskets cyclone victims brand name drugs by retail sales and highway length in order to analyze the intermediate cases a distribution the q weibull one which interpolates the q exponential and weibull ones is introduced it is verified that the basketball baskets distribution is well described by a q exponential whereas the cyclone victims and brand name drugs by retail sales ones are better adjusted by a weibull distribution on the other hand for highway length the q exponential and weibull distributions do not give satisfactory adjustment being necessary to employ the q weibull distribution furthermore the introduction of this interpolating distribution gives an illumination from the point of view of the stretched exponential against inverse power law q exponential with q 1 controversy

a remark on lp boundedness of wave operators for two dimensional schrodinger operators let h v be a two dimensional schrodinger operator with a real potential v x satisfying the decay condition 6 let h0 we show that the wave operators are bounded in lp r2 under the condition that h has no zero resonances or bound states in this paper the condition imposed in a previous paper k yajima commun math phys 208 125 152 1999 is removed

dispersive estimates for schrodinger operators in dimensions one and three we consider l 1 l estimates for the time evolution of hamiltonians h v in dimensions d 1 and d 3 with bound t 2 we require decay of the potentials but no regularity in d 1 the decay assumption is 1 x v x dx whereas in d 3 it is v x c 1 x 3

resolvent expansion and time decay of the wave functions for two dimensional magnetic schrodinger operators we consider two dimensional schrodinger operators h b v given by eq 1 1 below we prove that under certain regularity and decay assumptions on b and v the character of the expansion for the resolvent h b v 1 as 0 is determined by the flux of the magnetic field b through mathbb r 2 subsequently we derive the leading term of the asymptotic expansion of the unitary group e ith b v as t and show how the magnetic field improves its decay in t with respect to the decay of the unitary group e ith 0 v

commutativity preserving maps we briefly describe several techniques that have been developed to solve the problem of characterizing commutativity preserving maps we prove a new result describing the general form of injective continuous commutativity preserving maps on full matrix algebras

center manifold for nonintegrable nonlinear schrodinger equations on the line in this paper we study the following nonlinear schrodinger equation on the line where f is real valued and it satisfies suitable conditions on regularity on growth as a function of u and on decay as x the generic potential v is real valued and it is chosen so that the spectrum of consists of one simple negative eigenvalue and absolutely continuous spectrum filling 0 the solutions to this equation have in general a localized and a dispersive component the nonlinear bound states that bifurcate from the zero solution at the energy of the eigenvalue of h define an invariant center manifold that consists of the orbits of time periodic localized solutions we prove that all small solutions approach a particular periodic orbit in the center manifold as t in general the periodic orbits are different for t our result implies also that the nonlinear bound states are asymptotically stable in the sense that each solution with initial data near a nonlinear bound state is asymptotic as t to the periodic orbits of nearby nonlinear bound states that are in general different for t

on the asymptotic stability of bound states in 2d cubic schrodinger equation we consider the cubic nonlinear schrodinger equation in two space dimensions with an attractive potential we study the asymptotic stability of the nonlinear bound states i e periodic in time localized in space solutions our result shows that all solutions with small localized in space initial data converge to the set of bound states therefore the center manifold in this problem is a global attractor the proof hinges on dispersive estimates that we obtain for the non autonomous non hamiltonian linearized dynamics around the bound states

on the propagator of sierpinski gaskets the authors present numerical calculations for the propagator p r t the probability to reach a distance r at time t having started at the origin at t 0 on sierpinski gaskets the results are confronted with approximate analytical expressions it is shown that p r t approximately t ds 2 pi xi where xi is the scaling variable xi r t1 dw in the short xi regime the scaling function follows the form pi xi approximately exp c1 xi dw while for large xi pi xi is given asymptotically by pi xi approximately xi alpha exp c2 xi nu with alpha df dw 2 dw 1 and nu dw dw 1 this result extends the previously derived expressions the numerically observed oscillations which are superimposed on the power law decay of the autocorrelation function p r 0 t are analysed in terms of typical residence times on hierarchical substructures

analytical solutions for diffusion on fractal objects a generalization of the euclidean diffusion equation is proposed for diffusion on fractals on the basis of a scaling argument a renormalization group theory for the green s function and numerical tests conjectures on the applicability to natural fractals such as porous media are presented

time decay for solutions of schrodinger equations with rough and time dependent potentials in this paper we establish dispersive estimates for solutions to the linear schrodinger equation in three dimensions frac 1 i partial t psi triangle psi v psi 0 qquad psi s f 0 1 where v t x is a time dependent potential that satisfies the conditions sup t v t cdot l frac 3 2 mathbb r 3 sup x in mathbb r 3 int mathbb r 3 int infty infty frac v hat tau x x y d tau dy c 0 here c0 is some small constant and v hat tau x denotes the fourier transform with respect to the first variable we show that under these conditions 0 1 admits solutions l t l2x 3 l2t l6x 3 for any f l2 3 satisfying the dispersive inequality psi t infty le c t s frac32 f 1 text for all times t s 0 2 for the case of time independent potentials v x 0 2 remains true if int mathbb r 6 frac v x v y x y 2 dxdy 4 pi 2 text and v mathcal k sup x in mathbb r 3 int mathbb r 3 frac v y x y dy 4 pi we also establish the dispersive estimate with an e loss for large energies provided v mathcal k v 2 infty

weighted dispersive estimates for two dimensional schrodinger operators with aharonov bohm magnetic field abstract we consider two dimensional schrodinger operators h with an aharonov bohm magnetic field and an additional electric potential we obtain an explicit leading term of the asymptotic expansion of the unitary group e i t h for t in weighted l 2 spaces in particular we show that the magnetic field improves the decay of e i t h with respect to the unitary group of non magnetic schrodinger operators and that the decay rate in time is determined by the magnetic flux

dispersive estimates for the three dimensional schrodinger equation with rough potentials the three dimensional schrodinger propogator e ith h v is a bounded map from l 1 to l with norm controlled by t 3 2 provided the potential satisfies two conditions an integrability condition limiting the singularities and decay of v and a zero energy spectral condition on h this is shown by expressing the spectral measure of h in terms of its resolvents and proving a family of l p mapping estimates for the resolvents previous results in this direction had required v to satisfy explicit pointwise bounds

dynamical features of reaction diffusion fronts in fractals the speed of front propagation in fractals is studied by using i the reduction of the reaction transport equation into a hamilton jacobi equation and ii the local equilibrium approach different equations proposed for describing transport in fractal media together with logistic reaction kinetics are considered finally we analyze the main features of wave fronts resulting from this dynamic process i e why they are accelerated and what is the exact form of this acceleration