world wide web scaling exponent from simon s 1955 model the statistical properties of the world wide web have attracted considerable attention recently since self similar regimes were first observed in the scaling of its link structure one characteristic quantity is the number of in links k that point to a particular web page its probability distribution p k shows a pronounced power law scaling p k approximately k gamma that is not readily explained by standard random graph theory here we recall a simple and elegant model for scaling phenomena in general copy and growth processes as proposed by simon in 1955 when combined with an experimental measurement of network growth in the world wide web this classical model is able to model the in link dynamics and predicts the scaling exponent gamma 2 1 in accordance with observation

recovering asymptotics of metrics from fixed energy scattering data from fixed energy scattering data is studied it is shown that if two such metrics g1 g2 have scattering data at some fixed energy which are equal up to smoothing then there exists a diffeomorphism fixing infinity such that g1 g2 is rapidly decreasing given the scattering matrix at two energies it is shown that the asymptotics of a metric and a short range potential can be determined simultaneously these results also hold for a wide class of scattering manifolds

resolvent at low energy iii the spectral measure the second and third authors were supported by australian research council discovery grants dp0771826 and dp1095448 and the second author by a future fellowship the first author was partially supported by anr grant anr 09 jcjc 0099 01 and by the pics cnrs progress in geometric analysis and applications and thanks the math department of anu for its hospitality

the schrodinger propagator for scattering metrics let g be a scattering metric on a compact manifold x with boundary i e a smooth metric giving the interior x the structure of a complete riemannian manifold with asymptotically conic ends an example is any compactly supported perturbation of the standard metric on rn consider the operator h 1 2 v where is the positive laplacian with respect to g and v is a smooth real valued function on x vanishing to second order at iox assuming that g is nontrapping we construct a global parametrix u z w t for the kernel of the schri odinger propagator u t e ith where z w io x and t 0 the parametrix is such that the difference between u and u is smooth and rapidly decreasing both as t iae 0 and as z iae iox uniformly for w on compact subsets of x let r x 1 where x is a boundary defining function for x be an asymptotic radial variable and let w t be the kernel e ir2 2tu t using the parametrix we show that w t belongs to a class of i legendre distributionsi on x i x i r 0 previously considered by hassell vasy when the metric is trapping then the parametrix construction goes through microlocally in the nontrapping part of the phase space we apply this result to determine the singularities of u t f for any tempered distribution f and for any fixed t 0 in terms of the oscillation of f near iox if the metric is nontrapping then we precisely determine the wavefront set of u t f and hence also precisely determine its singular support more generally we are able to determine the wavefront set of u t f for t 0 resp t 0 on the non backward trapped resp non forward trapped subset of the phase space this generalizes results of craig kappeler strauss and wunsch

percolation critical exponents in scale free networks we study the behavior of scale free networks having connectivity distribution p k k 2l close to the percolation threshold we show that for networks with 3 l 4 known to undergo a transition at a finite threshold of dilution the critical exponents are different than the expected mean field values of regular percolation in infinite dimensions networks with 2 l 3 possess only a percolative phase nevertheless we show that in this case percolation critical exponents are well defined near the limit of extreme dilution where all sites are removed and that also then the exponents bear a strong l dependence the regular mean field values are recovered only for l 4

morawetz estimates for the wave equation at low frequency we consider morawetz estimates for weighted energy decay of solutions to the wave equation on scattering manifolds i e those with large conic ends we show that a morawetz estimate persists for solutions that are localized at low frequencies independent of the geometry of the compact part of the manifold we further prove a new type of morawetz estimate in this context with both hypotheses and conclusion localized inside the forward light cone this result allows us to gain a 1 2 power of t decay relative to what would be dictated by energy estimates in a small part of spacetime

the quantum n body problem this selective review is written as an introduction to the mathematical theory of the schrodinger equation for n particles characteristic for these systems are the cluster properties of the potential in configuration space which are expressed in a simple geometric language the methods developed over the last 40 years to deal with this primary aspect are described by giving full proofs of a number of basic and by now classical results the central theme is the interplay between the spectral theory of n body hamiltonians and the space time and phase space analysis of bound states and scattering states

the geometrical approach to multidimensional inverse scattering we prove that in multidimensional short range potential scattering the high velocity limit of the scattering operator of an n body system determines uniquely the potential for a given long range potential the short range potential of the n body system is uniquely determined by the high velocity limit of the modified dollard scattering operator moreover we prove that any one of the dollard scattering operators determines uniquely the total potential we obtain as well a reconstruction formula with an error term our simple proof uses a geometrical time dependent method

weyl asymptotics for the laplacian on asymptotically euclidean spaces we prove a trace type formula for the laplacian on an asymptotically euclidean space and use this to obtain weyl asymptotics of the scattering phase other results include an explicit calculation of the leading order singularity of the scattering matrix and results on the behavior of the resolvent and scattering matrix near 0

regularity and decay of solutions of nonlinear harmonic oscillators abstract we prove sharp analytic regularity and decay at infinity of solutions of variable coefficients nonlinear harmonic oscillators namely we show holomorphic extension to a sector in the complex domain with a corresponding gaussian decay according to the basic properties of the hermite functions in r d our results apply in particular to nonlinear eigenvalue problems for the harmonic oscillator associated to a real analytic scattering or asymptotically conic metric in r d as well as to certain perturbations of the classical harmonic oscillator

notes on the wave equation on asymptotically euclidean manifolds abstract the primary object of this paper is to discuss the asymptotics of solutions of the wave equation on an asymptotically euclidean manifold when the initial data have compact support by going over to an appropriate conformal metric it is shown that just as for the ordinary wave equation such a solution has a forward future and a backward past radiation field the same method is then used to define end points of bicharacteristics that begin and end above the boundary the analogue of the sphere at infinity in the euclidean case and to derive a relation between the wave front sets of the two radiation fields the extension of these results to solutions with finite energy is also briefly discussed

sharp strichartz estimates on nontrapping asymptotically conic manifolds we obtain the strichartz inequalities u l q t l r x 0 1 m c u 0 l 2 m for any smooth n dimensional riemannian manifold m which is asymptotically conic at infinity with either short range or long range metric perturbation and nontrapping where u is a solution to the schrodinger equation iu t 1 2 m u 0 and 2 q r are admissible strichartz exponents 2 q n r n 2 this corresponds with the estimates available for euclidean space except for the endpoint q r 2 2 n n 2 when n 2 these estimates imply existence theorems for semi linear schrodinger equations on m by adapting arguments from cazenave and weissler and kato this result improves on our previous result which was an l 4 t x strichartz estimate in three dimensions it is closely related to results of staffilani tataru burq robbiano zuily and tataru who consider the case of asymptotically flat manifolds

time dependent method for non linear schrodinger equations in inverse scattering problems abstract we consider an inverse scattering problem for a non linear schrodinger equation with a non linearity of power type applying the time dependent technique developed by enss and weder we prove that the non linearity is uniquely determined from a scattering data defined via the scattering operator we also obtain a reconstruction formula which is valid for large scattering data

scattering metrics and geodesic flow at infinity any compact manifold with boundary admits a riemann metric on its interior taking the form x 4 dx 2 x 2 h near the boundary where x is a boundary defining function and h is a smooth symmetric 2 cotensor restricting to be positive definite and hence a metric h on the boundary the scattering theory associated to the laplacian for such a scattering metric was discussed by the first author and here it is shown as conjectured that the scattering matrix is a fourier integral operator which quantizes the geodesic flow on the boundary for the metric h at time to prove this the poisson operator of the associated generalized boundary problem is constructed as a fourier integral operator associated to a singular legendre manifold

the semiclassical resolvent and the propagator for non trapping scattering metrics abstract consider a compact manifold with boundary m with a scattering metric g or equivalently an asymptotically conic manifold m g euclidean r n with a compactly supported metric perturbation is an example of such a space let be the positive laplacian on m g and v a smooth potential on m which decays to second order at infinity in this paper we construct the kernel of the operator h 2 v 0 i 0 2 1 at a non trapping energy 0 0 uniformly for h 0 h 0 h 0 0 small within a class of legendre distributions on manifolds with codimension three corners using this we construct the kernel of the propagator e i t 2 v t 0 t 0 as a quadratic legendre distribution we also determine the global semiclassical structure of the spectral projector poisson operator and scattering matrix

finiteness and fluctuations in growing networks we study the role of finiteness and fluctuations about average quantities for basic structural properties of growing networks we first determine the exact degree distribution of finite networks by generating function approaches the resulting distributions exhibit an unusual finite size scaling behaviour and they are also sensitive to the initial conditions we argue that fluctuations in the number of nodes of degree k become gaussian for fixed degree as the size of the network diverges we also characterize the fluctuations between different realizations of the network in terms of higher moments of the degree distribution

computing equilibria of dynamic games we develop a numerical method for computing all pure strategy subgame perfect equilibrium values of dynamic strategic games with discrete states and actions we define a monotone mapping that eliminates dominated strategies and when applied iteratively delivers an accurate approximation to the true equilibrium payoffs of the underlying game our algorithm has three parts the first provides an outer approximation to equilibrium values constructed so that any value outside of this approximation is not an equilibrium value the second provides an inner approximation any value contained within this approximation is an equilibrium value together the two approximations deliver a practical check of approximation accuracy the third part of our algorithm delivers sample equilibrium paths to illustrate our method we apply it to a dynamic oligopoly competition with endogenous production capacity

on the theory of infinitely repeated games with discounting presentation d un cadre systematique pour etudier des jeux repetes indefiniment avec actualisation en s attachant plus particulierement aux equilibres parfaits de strategie pure

inverse scattering for the magnetic schrodinger operator abstract we show that fixed energy scattering measurements for the magnetic schrodinger operator uniquely determine the magnetic field and electric potential in dimensions n 3 the magnetic potential its first derivatives and the electric potential are assumed to be exponentially decaying this improves an earlier result of eskin and ralston 1995 5 which considered potentials with many derivatives the proof is close to arguments in inverse boundary problems and is based on constructing complex geometrical optics solutions to the schrodinger equation via a pseudodifferential conjugation argument

the inverse scattering problem at fixed energy for the three dimensional schrodinger equation with an exponentially decreasing potential the relations between the faddeev functions and the functions of classical scattering theory are found in the complex domain at fixed energy for the three dimensional case without assumption of smallness of the potential it is proved that the exponentially decreasing potential is uniquely determined by its scattering amplitude at fixed energy

scattering asymptotics for riemann surfaces in this article we prove the optimal polynomial lower bound for the number of resonances of a surface with hyperbolic ends we also give weyl asymptotics for the relative scattering phase of such a surface the proofs are based on trace formulae analogous to those of the euclidean odd dimensional scattering the main technical ingredient is a new proof of the poisson formula theorem 5 7 which is applicable in the euclidean case as well our lower bound seems to be the first example of an optimal polynomial lower bound for the number of resonances holding for a general class of higher dimensional elliptic operators with no symmetries the previous general lower bounds or asymptotics were either nonoptimal 25 58 9 one dimensional or radial 65 67 and 54 41 1 or they required some degeneracy of the

scattering matrices in many body scattering in this paper we show that generalized eigenfunctions of many body hamiltonians h with short range two body interactions have distributional asymptotics at non threshold channels the leading terms of the asymptotics can be used to define a scattering matrix and we show that this is the same up to normalization as that arising from the standard wave operator approach we also prove the existence of local distributional asymptotics for locally approximate generalized eigenfunctions in the more general setting of short range perturbations of a scattering metric defined by melrose in 13

determining anisotropic real analytic conductivities by boundary measurements if an electrical potential is applied to the surface of a solid body the current flux across the surface depends on the conductivity in the interior of the body we want to consider the inverse problem of determining the conductivity by these boundary measurements

local smoothing for scattering manifolds with hyperbolic trapped sets we prove a resolvent estimate for the laplace beltrami operator on a scattering manifold with a hyperbolic trapped set and as a corollary deduce local smoothing we use a result of nonnenmacher zworski to provide an estimate near the trapped region a result of burq and cardoso vodev to provide an estimate near infinity and the microlocal calculus on scattering manifolds to combine the two

distribution of resonances for asymptotically euclidean manifolds in this paper we discuss meromorphic continuation of the resolvent and bounds on the number of resonances for scattering manifolds a class of manifolds generalizing euclidian n space subject to the basic assumption of analyticity near infinity we show that resolvent of the laplacian has a meromorphic continuation to a conic neighborhood of the continuous spectrum this involves a geometric interpretation of the complex scaling method in terms of deformations in the grauert tube of the manifold we then show that the number of resonances poles of the meromorphic continuation of the resolvent in a conic neighborhood of mathbb r of absolute value less than r 2 is mathcal o r n under the stronger assumption of global analyticity and hyperbolicity of the geodesic flow we prove a finer weyl type upper bound for the counting function for resonances in small neighborhoods of the real axis this estimate has an exponent which involves the dimension of the trapped set of the geodesic flow

symbolic functional calculus and n body resolvent estimates abstract we construct a functional calculus for symbolic functions of scattering pseudodifferential operators on manifolds with boundary this is applied to obtain mapping results on the scattering wavefront set of the b operator used by gerard et al 1994 commutator algebra and resolvent estimates advanced studies in pure mathematics vol 23 math soc japan tokyo to obtain n body resolvent estimates as an application we construct the plane waves for an n body short range schrodinger operator and obtain a scattering wavefront set estimate

high energy and smoothness asymptotic expansion of the scattering amplitude abstract we find an explicit expression for the kernel of the scattering matrix for the schrodinger operator containing at high energies all terms of power order it turns out that the same expression gives a complete description of the diagonal singularities of the kernel in the angular variables the formula obtained is in some sense universal since it applies both to short and long range electric as well as magnetic potentials

the fbi transform on compact manifolds we present a geometric theory of the fourier bros iagolnitzer transform on a compact c manifold m the fbi transform is a generalization of the classical notion of the wave packet transform we discuss the mapping properties of the fbi transform and its relationship to the calculus of pseudodifferential operators on m we also describe the microlocal properties of its range in terms of the scattering calculus of pseudodifferential operators on the noncompact manifold t m

on open scattering channels for manifolds with ends abstract in the framework of time dependent geometric scattering theory we study the existence and completeness of the wave operators for perturbations of the riemannian metric for the laplacian on a complete manifold of dimension n the smallness condition for the perturbation is expressed intrinsically and coordinate free in purely geometric terms using the harmonic radius therefore the size of the perturbation can be controlled in terms of local bounds on the injectivity radius and the ricci curvature as an application of these ideas we obtain a stability result for the scattering matrix with respect to perturbations of the riemannian metric this stability result implies that a scattering channel which interacts with other channels preserves this property under small perturbations

propagation properties of quantum scattering states the change of localization of a scattering state under the free and interacting time evolutions for quantum mechanical potential scattering is controlled if a particle of mass m is localized in a ball of radius r and has energy below mv22 then at time t the component of the state in the forbidden region x r vt decays integrably in time various sharper estimates for the free time evolution are collected in section ii the momentum support of a state in terms of its full energy support in regions where the potential is weak is controlled the potential may be fairly general of short or long range

is kinetic growth walk equivalent to canonical self avoiding walk abstract we present a monte carlo study of kinetic growth walk on square as well as triangular lattice to show that it is not equivalent to canonical self avoiding walk

highly clustered scale free networks we propose a model for growing networks based on a finite memory of the nodes the model shows stylized features of real world networks power law distribution of degree linear preferential attachment of new links and a negative correlation between the age of a node and its link attachment rate notably the degree distribution is conserved even though only the most recently grown part of the network is considered as the network grows the clustering reaches an asymptotic value larger than that for regular lattices of the same average connectivity and similar to the one observed in the networks of movie actors coauthorship in science and word synonyms these highly clustered scale free networks indicate that memory effects are crucial for a correct description of the dynamics of growing networks

spectral and scattering theory for symbolic potentials of order zero the spectral and scattering theory is investigated for a generalization to scattering metrics on two dimensional compact manifolds with boundary of the class of smooth potentials on r2 which are homogeneous of degree zero near infinity the most complete results require the additional assumption that the restriction of the potential to the circle s at infinity be morse generalized eigenfunctions associated to the essential spectrum at non critical energies are shown to originate both at minima and maxima although the latter are not germane to the l2 spectral theory asymptotic completeness is shown both in the traditional l2 sense and in the sense of tempered distributions this leads to a definition of the scattering matrix the structure of which will be described in a future publication

asymptotic properties of solutions to 3 particle schrodinger equations we construct a generalized fourier transformation f associated with the 3 body schrodinger operator h a v a x a and characterize all solutions of h u 0 in the agmon hormander space b as the image of f these stationary solutions admit asymptotic expansions in b in terms of spherical waves associated with scattering channels

computing supergame equilibria we present a general method for computing the set of supergame equilibria in infinitely repeated games with perfect monitoring and public randomization we present a three stage algorithm that constructs a convex set containing the set of equilibrium values constructs another convex set contained in the set of equilibrium values and produces strategies that support them we explore the properties of this algorithm by applying it to familiar games

a constructive procedure to recover asymptotics of short range or long range potentials abstract we study an inverse scattering problem for a pair of hamiltonians h h 0 on l 2 r n where h 0 and h h 0 v v being a short or long range potential by an elementary constructive method we show that the scattering operator s which is localized near a fixed energy 0 determines the asymptotics of the potential v at infinity in dimension n 3 this is done by studying the action of the scattering operator on suitable wave packets

on inverse scattering for then body schrodinger equation abstract we consider the generalized n body schrodinger operator h q x q x a a v a x a x r d x a x a r d x a x a with short range regular interactions we obtain in particular the following results 1 we give new formulas and proofs relating high energy asymptotics in a weak sense of the scattering operators with the same cluster decomposition a for and and the x ray transform pi defined on the set of all lines in x a with nonsingular directions of the effective potentials i x a x a x a these results significantly clarify some of those given in the literature 2 we describe completely ker p and give a method for reconstruction of i mod ker p from pi 3 we prove pointwise high energy asymptotics for the two cluster two cluster scattering amplitudes f with the fourier transform i in the leading term for the case when the cluster decomposition for and is the same 4 we give several additional results for the case of perturbed stratified medium when q x v a x a v b x b x a x 1 x b x 1 x d x and each v c x c rapidly decreases as x c c a b our results including proofs in some cases significantly simplify methods of high energy inverse scattering for the n body schrodinger operator given earlier by wang and by enss and weder

transparent potentials at fixed energy in dimension two fixed energy dispersion relations for the fast decaying potentials for the two dimensional schrodinger equation delta v x psi e psi x in mathbb r 2 e e fixed 0 at a fixed positive energy with a fast decaying at infinity potentialv x dispersion relations on the scattering data are given under small norm assumption using these dispersion relations we give without a complete proof of sufficiency a characterization of scattering data for the potentials from the schwartz classs c 2 for the potentials with zero scattering amplitude at a fixed energyefixed transparent potentials we give a complete proof of this characterization as a consequence we construct a family parametrized by a function of one variable of two dimensional spherically symmetric real potentials from the schwartz classs transparent at a given energy for the two dimensional case without assumption that the potential is small we show that there are no nonzero real exponentially decreasing at infinity potentials transparent at a fixed energy for any dimension greater or equal to 1 we prove that there are no nonzero real potentials with zero forward scattering amplitude at an energy interval we show that kdv type equations in dimension 2 1 related with the scattering problem the novikov veselov equations do not preserve in general these dispersion relations starting from the second one as a corollary these equations do not preserve in general the decay rate faster than x 3 for initial data from the schwartz class

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

smooth linearization near a fixed point in this paper we extend a theorem of sternberg and bi leckii we study a vector field or a diffeomorphism in the vicinity of a hyperbolic fixed point we assume that the eigenvalues of the linear part a at the fixed point satisfy qth order algebraic inequalities where q 2 2 then there is ck linearization in the vicinity of the fixed point where k is the q smoothness of a we give an explicit and simple algorithm for com puting k an illustrative example from celestial mechanics is included which shows that our main theorem is an improvement over previously known theories we also show that if the vector field or diffeomorphism depends smoothly on a parameter then the linearizing conjugation de pends smoothly on the parameter

scattering theory for riemannian laplacians we introduce a notion of scattering theory for the laplace beltrami operator on non compact connected and complete riemannian manifolds a principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity another condition is certain bounds of derivatives up to order one of the trace of this quantity these conditions are shown to be optimal for existence and completeness of a wave operator our theory does not involve prescribed asymptotic behavior of the metric at infinity like asymptotic euclidean or hyperbolic metrics studied previously in the literature a consequence of the theory is spectral theory for the laplace beltrami operator including identification of the continuous spectrum and absence of singular continuous spectrum

boundary control wave field continuation and inverse problems for the wave equation abstract we consider the problem of the wave field continuation and recovering of coefficients for the wave equation in a bounded domain in r n n 1 the inverse data is a response operator mapping neumann boundary data into dirichlet ones the reconstruction procedure is local this means that observing boundary response for larger times we may recover coefficients deeper in the domain the approach is based upon ideas and results of the boundary control theory yielding some natural multidimensional analogs of the classical gel fand levitan krein equations

approximation and equidistribution of phase shifts spherical symmetry consider a semiclassical hamiltonian h v h h 2 delta v e where h 0 is a semiclassical parameter is the positive laplacian on mathbb r d v is a smooth compactly supported central potential function and e 0 is an energy level in this setting the scattering matrix sh e is a unitary operator on l 2 mathbb s d 1 hence with spectrum lying on the unit circle moreover the spectrum is discrete except at 1

scattering poles for asymptotically hyperbolic manifolds for a class of manifolds x that includes quotients of real hyperbolic n 1 dimensional space by a convex co compact discrete group we show that the resonances of the meromorphically continued resolvent kernel for the laplacian on x coincide with multiplicities with the poles of the meromorphically continued scattering operator for x in order to carry out the proof we use shmuel agmon s perturbation theory of resonances to show that both resolvent resonances and scattering poles are simple for generic potential perturbations

subgame perfect equilibria in discounted stochastic games abstract this paper considers policies and payoffs corresponding to subgame perfect equilibrium strategies in discounted stochastic games with finitely many states it is shown that a policy is induced by an equilibrium strategy if and only if it can be supported with the threat of reverting to the induced policy that gives the least equilibrium payoff for the deviator it follows that the correspondence of subgame perfect equilibrium payoffs is the largest fixed point of a correspondence valued operator defined by the players incentive compatibility conditions moreover the fixed point iteration converges to the equilibrium payoff correspondence

semiclassical estimates in asymptotically euclidean scattering we consider long range semiclassical perturbations of the laplacian on asymptotically euclidean manifolds we obtain precise resolvent estimates under non trapping assumptions the novelty lies in a systematic use of geometric microlocal methods

the wave equation on asymptotically de sitter like spaces in this paper we obtain the asymptotic behavior of solutions of the klein gordon equation on lorentzian manifolds x g which are de sitter like at infinity such manifolds are lorentzian analogues of the so called riemannian conformally compact or asymptotically hyperbolic spaces under global assumptions on the null bicharacteristic flow namely that the boundary of the compactification x is a union of two disjoint manifolds y and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to and to the other manifold as the parameter goes to we also define the scattering operator and show that it is a fourier integral operator associated to the bicharacteristic flow from y to y

le saut en zero de la fonction de decalage spectral abstract we give a formula for the jump at zero of the spectral shift function associated with schrodinger operator on manifolds with conical ends we show that according to its decay a zero energy resonant state has a non integer contribution

infinite order percolation and giant fluctuations in a protein interaction network we investigate a model protein interaction network whose links represent interactions between individual proteins this network evolves by the functional duplication of proteins supplemented by random link addition to account for mutations when link addition is dominant an infinite order percolation transition arises as a function of the addition rate in the opposite limit of high duplication rate the network exhibits giant structural fluctuations in different realizations for biologically relevant growth rates the node degree distribution has an algebraic tail with a peculiar rate dependence for the associated exponent

propagation of singularities in many body scattering in the presence of bound states abstract in this paper we describe the propagation of singularities of tempered distributional solutions u s of h u 0 where h is a many body hamiltonian h v 0 v a v a and is not a threshold of h under the assumption that the inter particle e g two body interactions v a are real valued polyhomogeneous symbols of order 1 e g coulomb type with the singularity at the origin removed here the term singularity refers to a microlocal description of the lack of decay at infinity thus we prove that the set of singularities of u is a union of maximally extended broken bicharacteristics of h these are curves in the characteristic variety of h which can be quite complicated due to the existence of bound states we use this result to describe the wave front relation of the s matrices we also analyze lagrangian properties of this relation which shows that the relation is not too large in terms of its dimension

relative time delay for perturbations of elliptic operators and semiclassical asymptotics the main goal of this paper is to compare two long range perturbations of constant coefficient operators on rn such that their difference is short range typical examples are schrodinger hamiltonians hj h2 vj with vj x o x v2 x v1 x o x with 0 n we can also consider perturbations by magnetic fields and cases where is the laplace beltrami operator for asymptotically flat metrics on rn for the scattering pair h2 h1 the average time delay d h depending on the energy and the parameter h is well defined it is related to the spectral shift function and also to the scattering phase under suitable assumptions we prove asymptotic results on d as high energy regime and as h 0 semi classical regime

inverse backscattering in two dimensions this article extends the authors previous results commun math phys 124 169 215 1989 to inverse scattering in two space dimensions the new problem in two dimensions is the behavior of the backscattering amplitude near zero energy generically this has the form a xi mathord left vphantom xi left xi right right kern nulldelimiterspace left xi right xi mathord left vphantom xi left xi right left xi right right kern nulldelimiterspace left xi right left xi right 2 pi 2 pi beta ln left xi right 1 b xi whereb 0 0 andb is holder continuous in order to work in weighted holder spaces as before the constant and the functionb must now be interpreted as coordinates on the space of backscattering data in this setting the mapping to backscattering data is again a local diffeomorphism at a dense open set in the real valued potentials

discounted stochastic games we give an alternative proof to a result of mertens and parthasarathy stating that every n player discounted stochastic game with general setup and with a norm continuous transition has a subgame perfect equilibrium

an approximation formula in the inverse scattering problem an approximate formula is given by which the potential q x can be recovered with any accuracy if only the scattering data of the schrodinger operator q x is known for some sufficiently high energy value k thus the scattering data for all k around k is not needed in order to get a good approximate value for the potential the main tools for the proof are an asymptotic formula for the s matrix y sait j math phys 25 3105 1984 and the spectral decomposition theorem for the schrodinger operator q x based on the limiting absorption principle

asymptotic behavior of generalized eigenfunctions inn body scattering abstract in this paper an asymptotic expansion is proved for locally at infinity outgoing functions on asymptotically euclidian spaces this is applied to n body scattering where the two body interactions are one step polyhomogeneous symbols of order 1 or 2 hence long range and short range respectively the asymptotic behavior of the n body resolvent applied to schwartz functions is thereby deduced away from the singular set where some of the potentials do not decay at infinity

global uniqueness for a two dimensional inverse boundary value problem we show that the coefficient y x of the elliptic equation vie qyvu 0 in a two dimensional domain is uniquely determined by the corresponding dirichlet to neumann map on the boundary and give a reconstruction pro

subgame perfect equilibria for stochastic games for an n person stochastic game with borel state space s and compact metric action sets a1 a2 an sufficient conditions are given for the existence of subgame perfect equilibria one result is that such equilibria exist if the law of motion q s a is for fixed s continuous in a a1 a2 an for the total variation norm and the payoff functions f1 f2 fn are bounded borel measurable functions of the sequence of states s1 s2 sn and in addition are continuous when sn is given the product of discrete topologies on s

absence of embedded eigenvalues for riemannian laplacians in this paper we study absence of embedded eigenvalues for schrodinger operators on non compact connected riemannian manifolds a principal example is given by a manifold with an end possibly more than one in which geodesic coordinates are naturally defined in this case one of our geometric conditions is a positive lower bound of the second fundamental form of angular submanifolds at infinity inside the end another condition is an upper bound of the trace of this quantity while a third one is a bound of the derivatives of part of the trace some oscillatory behaviour of the trace is allowed in addition to geometric bounds we need conditions on the potential a regularity property of the domain of the schrodinger operator and the unique continuation property examples include ends endowed with asymptotic euclidean or hyperbolic metrics

fredholm theory for elliptic operators on quasi asymptotically conical spaces we consider the mapping properties of generalized laplace type operators l r on the class of quasi asymptotically conical qac spaces which provide a riemannian generalization of the qale manifolds considered by joyce joy00 our main result gives conditions under which such operators are fredholm when between

statistical ensemble of scale free random graphs a thorough discussion of the statistical ensemble of scale free connected random tree graphs is presented methods borrowed from field theory are used to define the ensemble and to study analytically its properties the ensemble is characterized by two global parameters the fractal and the spectral dimensions which are explicitly calculated it is discussed in detail how the geometry of the graphs varies when the weights of the nodes are modified the stability of the scale free regime is also considered when it breaks down either a scale is spontaneously generated or else a singular node appears and the graphs become crumpled a new computer algorithm to generate these random graphs is proposed possible generalizations are also discussed in particular more general ensembles are defined along the same lines and the computer algorithm is extended to arbitrary degenerate scale free random graphs

correlated random networks we develop a statistical theory of networks a network is a set of vertices and links given by its adjacency matrix c and the relevant statistical ensembles are defined in terms of a partition function z summation operator exp betah c the simplest cases are uncorrelated random networks such as the well known erdos renyi graphs here we study more general interactions h c which lead to correlations for example between the connectivities of adjacent vertices in particular such correlations occur in optimized networks described by partition functions in the limit beta infinity they are argued to be a crucial signature of evolutionary design in biological networks

multidimensional inverse scattering for the nonlinear klein gordon equation with a potential in this paper we solve the multidimensional inverse scattering problem for the nonlinear klein gordon equation on rn n 2 2 t2u x t u x t u x t v0 x u x t j 1 vj x u 2 j0 j u x t 0 we prove that the small amplitude limit of the scattering operator determines uniquely all the vj j 0 1 our proof gives as well a method for the reconstruction of the vj j 0 1

tree networks with causal structure a geometry of networks endowed with a causal structure is discussed using the conventional framework of the equilibrium statistical mechanics the popular growing network models appear as particular causal models we focus on a class of tree graphs an analytically solvable case general formulas are derived describing the degree distribution the ancestor descendant correlation and the probability that a randomly chosen node lives at a given geodesic distance from the root it is shown that the hausdorff dimension d h of the causal networks is generically infinite in contrast to the maximally random trees where it is generically finite

recovering asymptotics of short range potentials any compact smooth manifold with boundary admits a riemann metric of the form near the boundary where x is the boundary defining function and h restricts to a riemannian metric h on the boundary melrose has associated a scattering matrix to such a metric which was shown by he and zworski to be a fourier integral operator it is shown here that the principal symbol of the difference of the scattering matrices for two potentials at fixed energy determines a weighted integral of the lead term of v 1 v 2 over all geodesics on the boundary this is used to prove that the entire taylor series of the potential at the boundary is determined by the scattering matrix at a non zero fixed energy for certain manifolds including euclidean space

existence of four dimensional polymer collapse i kinetic growth trails abstract we present the results of simulations of kinetic growth trails kgt bond avoiding walks in four dimensions we use a mapping from a kinetic growth model to a static model of self interacting trails isat at a particular temperature to argue that this temperature is precisely the collapse temperature of four dimensional interacting trails to do this we show that the kinetic growth trails behave neither like static non interacting trails which should behave as excluded volume dominated four dimensional polymers that is self avoiding walks or collapsed four dimensional polymers but rather show an intermediate behaviour this is the first indication of collapse in any four dimensional lattice polymer model and so may be helpful in deciding which of the competing models of polymers is a good model in lower dimensions we have calculated various exponents of the kgt model and identified them with certain critical exponents of the static isat problem

a parametrix for the fundamental solution of the klein gordon equation on asymptotically de sitter spaces abstract in this paper we construct a parametrix for the forward fundamental solution of the wave and klein gordon equations on asymptotically de sitter spaces without caustics we use this parametrix to obtain asymptotic expansions for solutions of u f and to obtain a uniform l p estimate for a family of bump functions traveling to infinity

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved