on hartree equations with derivatives abstract we consider the cauchy problem of two types of hartree equations with exchange correlation correction terms i u t u v k u u in r 1 n k 1 2 u 0 in r n n 1 where v 1 u x 1 u 2 2 u 2 v 2 u x u 2 we establish the well posedness of cauchy problems and show the smoothing effect of solutions for each 0 n and 0 1

quantum monodromy and nonconcentration near a closed semi hyperbolic orbit for a large class of semiclassical operators p h z which includes schrodinger operators on manifolds with boundary we construct the quantum monodromy operator m z associated to a periodic orbit of the classical flow using estimates relating m z and p h z we prove semiclassical estimates for small complex perturbations of p h z in the case is semi hyperbolic as our main application we give logarithmic lower bounds on the mass of eigenfunctions away from semi hyperbolic orbits of the associated classical flow as a second application of the monodromy operator construction we prove if is an elliptic orbit then p h admits quasimodes which are well localized near

a new algebraic geometry algorithm for integer programming we propose a new algorithm for solving integer programming ip problems that is based on ideas from algebraic geometry the method provides a natural generalization of the farkas lemma for ip leads to a way of performing sensitivity analysis offers a systematic enumeration of all feasible solutions and gives structural information of the feasible set of a given ip we provide several examples that offer insights on the algorithm and its properties

smoothing estimates for the schrodinger equation with unbounded potentials abstract we prove a local in time smoothing estimate for a magnetic schrodinger equation with coefficients growing polynomially at spatial infinity the assumptions on the magnetic field are gauge invariant and involve only the first two derivatives the proof is based on the multiplier method and no pseudodifferential techniques are required

a strong type 2 2 estimate for a maximal operator associated to the schrodinger equation soit t f x sup t 0 t t f x ou t t f 1 exp it 2 f 1 4 on montre que etant donne un intervalle fini i i t f 2 dx c i r f x 2 dx

a smoothing property of schrodinger equations in the critical case this paper deals with the critical case of the global smoothing estimates for the schrodinger equation although such estimates fail for critical orders of weights and smoothing it is shown that they are still valid if one works with operators with symbols vanishing on a certain set the geometric meaning of this set is clarified in terms of the hamiltonian flow of the laplacian the corresponding critical case of the limiting absorption principle for the resolvent is also established obtained results are extended to dispersive equations of schrodinger type to hyperbolic equations and to equations of other orders

propagation of singularities and maximal functions in the plane in this work we mainly generalize bourgain s circular maximal function to include variable coefficient averages our techniques involve a combination of bourgain s basic ideas plus microlocal analysis in particular to see the role of curvature we rely heavily on methods used in studying propogation of singularities for hyperbolic differential equations we also show that forp 2 there is local smoothing inl p for solutions to the wave equation

regularity of solutions to nonlinear dispersive equations on considere le probleme de cauchy pour les equations dispersives non lineaires de la forme t u ip d u f u x r n t u x 0 u 0 x

local smoothing properties of dispersive equations is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data for the linear wave equation for instance the answer is no however in 10 t kato found a local smoothing property of the korteweg de vries equation the solution of the initial value problem is locally one derivative smoother than the initial datum kato s proof uses in a curcial way the algebraic properties of the symbol for the korteweg de vries equation and the fact that the underlying spatial dimension is one actually judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect one would be inclined to believe this was a special property of the k dv equation this is not however the case in this paper we attempt to describe a general local smoothing effect for dispersive equations and systems the smoothing effect is due to the dispersive nature of the linear part of the equation all the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property to mention only a few the k dv benjamin ono intermediate long wave various boussinesq and schrodinger equations are included we study thus equations and systems of the form

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

extremal results in sparse pseudorandom graphs abstract szemeredi s regularity lemma is a fundamental tool in extremal combinatorics however the original version is only helpful in studying dense graphs in the 1990s kohayakawa and rodl proved an analogue of szemeredi s regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs many of the key applications of szemeredi s regularity lemma use an associated counting lemma in order to prove extensions of these results which also apply to sparse graphs it remained a well known open problem to prove a counting lemma in sparse graphs the main advance of this paper lies in a new counting lemma proved following the functional approach of gowers which complements the sparse regularity lemma of kohayakawa and rodl allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph we use this to prove sparse extensions of several well known combinatorial theorems including the removal lemmas for graphs and groups the erd s stone simonovits theorem and ramsey s theorem these results extend and improve upon a substantial body of previous work

uniqueness properties of solutions of schrodinger equations under suitable assumptions on the potentials v and a we prove that if u c 0 1 h1 is a solution of the linear schrodinger equation i t x u vu a xu on rd 0 1 and if u 0 in x r 0 1 for some r 0 then u 0 in rd 0 1 as a consequence we obtain uniqueness properties of solutions of nonlinear schrodinger equations of the form i t x u g x t u u xu xu on rd 0 1 where g is a suitable nonlinear term the main ingredient in our proof is a carleman inequality of the form e x1 v lx2lt2 e x1 xv bx 2lt2 c e x1 i t x v bx1 2lt2 for any v c r h1 with v t 0 for t 0 1 in this inequality bx 2 and bx1 2 are banach spaces of functions on rd and e x1 is a suitable weight

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

complex scaling and the distribution of scattering poles the purpose of this paper is to establish sharp polynomial bounds on the number of scattering poles for a general class of compactly supported self adjoint perturbations of the laplacian in r n odd we also consider more general types of bounds that give sharper estimates in certain situations the general conclusion can be stated as follows the order of growth of the poles is the same as the order of growth of eigenvalues for corresponding compact problems from the few known cases however the exact asymptotics are expected to be different the scattering poles for compactly supported perturbations were rigorously defined by lax and phillips 13 in a more general setting they correspond to resonances the study of which has a long tradition in mathematical physics in the lax phillips theory they appear as the poles of the meromorphic continuation of the scattering matrix and coincide with the poles of the meromorphic continuation of the resolvent of the perturbed operator because of the latter characterization they can be considered as the analogue of the discrete spectral data for problems on noncompact domains the problem of estimating the counting function

the general quasilinear ultrahyperbolic schrodinger equation abstract we consider the general quasilinear schrodinger equation whose second order coefficients are given by a real symmetric non degenerate matrix we deduce conditions which guarantee that the associated initial value problem is locally well posed

an algorithmic regularity lemma for hypergraphs in this paper we will consider the problem of designing an efficient algorithm that finds an epsilon regular partition of an l uniform hypergraph

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

decay of correlations for normally hyperbolic trapping we prove that for evolution problems with normally hyperbolic trapping in phase space correlations decay exponentially in time normally hyperbolic trapping means that the trapped set is smooth and symplectic and that the flow is hyperbolic in directions transversal to it flows with this structure include contact anosov flows classical flows in molecular dynamics and null geodesic flows for black holes metrics the decay of correlations is a consequence of the existence of resonance free strips for green s functions cut off resolvents and polynomial bounds on the growth of those functions in the semiclassical parameter

smoothing effects and dispersion of singularities for the schrodinger evolution group we describe smoothing effects and dispersion of singularities for the schrodinger evolution group in the weighted sobolev spaces under a fairly general assumption on the potential it is shown that all singularities in the wavefunction vanish instantly whenever the initial state has sufficient decay we measure the regularity gained by the wavefunction by the decay property of the initial state no assumptions on the regularity of the initial state are imposed throughout the paper

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

smoothing estimates for non dispersive equations this paper describes an approach to global smoothing problems for non dispersive equations based on ideas of comparison principle and canonical transformation established in authors previous paper ruzhansky and sugimoto proc lond math soc 105 393 423 2012 where dispersive equations were treated for operators a d x of order m satisfying the dispersiveness condition nabla a xi ne 0 for xi not 0 the global smoothing estimate begin aligned left left langle x right rangle s d x m 1 2 e ita d x varphi x right l 2 left mathbb r t times mathbb r n x right le c left varphi right l 2 left mathbb r n x right quad s 1 2 end aligned is well known while it is also known to fail for non dispersive operators for the case when the dispersiveness breaks we suggest the estimate in the form begin aligned left left langle x right rangle s nabla a d x 1 2 e it a d x varphi x right l 2 left mathbb r t times mathbb r n x right le c left varphi right l 2 left mathbb r n x right quad s 1 2 end aligned which is equivalent to the usual estimate in the dispersive case and is also invariant under canonical transformations for the operator a d x we show that this estimate and its variants do continue to hold for a variety of non dispersive operators a d x where nabla a xi may become zero on some set moreover other types of such estimates and the case of time dependent equations are also discussed

a fast approximation algorithm for computing thefrequencies of subgraphs in a given graph in this paper we give an algorithm which given a labeled graph on n vertices and a list of all labeled graphs on k vertices provides for each graph h of this list an approximation to the number of induced copies of h in g with total error small this algorithm has running time o n 1 over log log n cdot m n where m n is the time needed to square a n by n matrix with 0 1 entries over the integers the main tool in designing this algorithm is a variant of the regularity lemma of szemeredi

smoothing effects for schrodinger equations with electro magnetic potentials and applications to the maxwell schrodinger equations abstract we consider schrodinger equations in r 1 2 with electro magnetic potentials the potentials belong to h 1 and typically they are time independent or determined as solutions to inhomogeneous wave equations we prove kato type smoothing estimates for solutions we also apply this result to the maxwell schrodinger equations in the lorentz gauge and prove unique solvability of this system in the energy space

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

nonlocal schrodinger equations in metric measure spaces abstract in this note we consider the pointwise convergence to the initial data for the solutions of some nonlocal dyadic schrodinger equations on spaces of homogeneous type we prove the a e convergence when the initial data belongs to a dyadic version of an l 2 based besov space

near sharp strichartz estimates with loss in the presence of degenerate hyperbolic trapping we consider an n dimensional spherically symmetric asymptotically euclidean manifold with two ends and a codimension 1 trapped set which is degenerately hyperbolic by separating variables and constructing a semiclassical parametrix for a time scale polynomially beyond ehrenfest time we show that solutions to the linear schrodinger equation with initial conditions localized on a spherical harmonic satisfy strichartz estimates with a loss depending only on the dimension n and independent of the degeneracy the strichartz estimates are sharp up to an arbitrary 0 loss this is in contrast to christianson and wunsch amer j math 2013 where it is shown that solutions satisfy a sharp local smoothing estimate with loss depending only on the degeneracy of the trapped set independent of the dimension

dimension of divergence sets for the pointwise convergence of the schrodinger equation abstract in this note we consider the pointwise convergence along curves for the schrodinger equation and obtain estimates for the capacitary dimension of divergence sets which extend our previous result in 6

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

endpoint strichartz estimates for the magnetic schrodinger equation abstract we prove strichartz estimates for the schrodinger equation with an electromagnetic potential in dimension n 3 the decay and regularity assumptions on the potentials are almost critical i e close to the coulomb case in addition we require repulsivity and a nontrapping condition which are expressed as smallness of suitable components of the potentials while the potentials themselves can be large the proof is based on smoothing estimates and new sobolev embeddings for spaces associated to magnetic potentials

an algebraic approach to integer portfolio problems abstract integer variables allow the treatment of some portfolio optimization problems in a more realistic way and introduce the possibility of adding some natural features to the model we propose an algebraic approach to maximize the expected return under a given admissible level of risk measured by the covariance matrix to reach an optimal portfolio it is an essential ingredient the computation of different test sets via grobner basis of linear subproblems that are used in a dual search strategy

strichartz estimates and local smoothing estimates for asymptotically flat schrodinger equations abstract in this article we study global in time strichartz estimates for the schrodinger evolution corresponding to long range perturbations of the euclidean laplacian this is a natural continuation of a recent article d tataru parametrices and dispersive estimates for schrodinger operators with variable coefficients amer j math 130 2008 571 634 of the third author where it is proved that local smoothing estimates imply strichartz estimates by d tataru parametrices and dispersive estimates for schrodinger operators with variable coefficients amer j math 130 2008 571 634 the local smoothing estimates are known to hold for small perturbations of the laplacian here we consider the case of large perturbations in three increasingly favorable scenarios i without non trapping assumptions we prove estimates outside a compact set modulo a lower order spatially localized error term ii with non trapping assumptions we prove global estimates modulo a lower order spatially localized error term and iii for time independent operators with no resonance or eigenvalue at the bottom of the spectrum we prove global estimates for the projection onto the continuous spectrum

a geometric buchberger algorithm for integer programming let ip a c denote the family of integer programs of the form min cx ax b x nn obtained by varying the right hand side vector b but keeping a and c fixed a test set for ipa c is a set of vectors in zn such that for each nonoptimal solution to a program in this family there is at least one element g in this set such that g has an improved cost value as compared to we describe a unique minimal test set for this family called the reduced grobner basis of ip a c an algorithm for its construction is presented which we call a geometric buchberger algorithm for integer programming and we show how an integer program may be solved using this test set the reduced grobner basis is then compared with some other known test sets from the literature we also indicate an easy procedure to construct test sets with respect to all cost functions for a matrix a zn 2 n of full row rank

quintic nls in the exterior of a strictly convex obstacle we consider the defocusing energy critical nonlinear schr odinger equation in the exterior of a smooth compact strictly convex obstacle in three dimensions for the initial value problem with dirichlet boundary condition we prove global well posedness and scattering for all initial data in the energy space

quasimodes and resonances sharp lower bounds we prove that asymptotically any cluster of quasimodes close to each other approximates at least the same number of resonances counting multiplicities as a consequence we get that the counting function of the number of resonances close to the real axis is bounded from below essentially by the counting function of the quasimodes

approximate hypergraph partitioning and applications szemeredi s regularity lemma is a cornerstone result in extremal combinatorics it roughly asserts that any dense graph is composed of a finite number of pseudorandom graphs the regularity lemma has found many applications in theoretical computer science and thus a lot of attention was given to designing algorithmic versions of this lemma our main results in this paper are the following i we introduce a new approach to the problem of constructing regular partitions of graphs which results in a surprisingly simple o n time algorithmic version of the regularity lemma thus improving over the previous o n 2 time algorithms furthermore unlike all the previous approaches for this problem see n alon and a naor siam j comput 35 2006 pp 787 803 r a duke h lefmann and v rodl siam j comput 24 1995 pp 598 620 a frieze and r kannan electron j combin 6 1999 article 17 a frieze and r kannan the regularity lemma and approximation schemes for dense problems in proceedings of the 37th annual symposium on foundations of computer science burlington vt 1996 ieee computer society press los alamitos ca 1996 pp 12 20 and y kohayakawa v rodl and l thoma siam j comput 32 2003 pp 1210 1235 which only guaranteed to find tower size partitions our algorithm will find a small regular partition if one exists in the graph ii for any constant r geq3 we give an o n time randomized algorithm for constructing regular partitions of r uniform hypergraphs thus improving the previous o n 2r 1 time deterministic algorithms a czygrinow and v rodl siam j comput 30 2000 pp 1041 1066 a frieze and r kannan the regularity lemma and approximation schemes for dense problems in proceedings of the 37th annual symposium on foundations of computer science burlington vt 1996 ieee computer society press los alamitos ca 1996 pp 12 20 these two results are obtained as an application of an efficient algorithm for approximating partition problems of hypergraphs which we obtain here given a directed hypergraph with bounded edge arities a set of constraints on the set sizes and densities of a possible partition of its vertex set and an approximation parameter we provide in o n time a partition approximating the constraints if a partition satisfying them exists we can also test in o 1 time for the existence of such a partition given the approximation parameter this algorithm extends the result of goldreich goldwasser and ron for graph partition problems o goldreich s goldwasser and d ron j acm 45 1998 pp 653 750 and encompasses more recent hypergraph related results such as the maximal constraint satisfaction approximation of g andersson and l engebretsen random structures algorithms 21 2002 pp 14 32

local smoothing for the quantum liouville equation abstract we analyze in this work the regularity properties of the density operator solution to the quantum liouville equation as was recently done for the strichartz inequalities we extend to systems of orthonormal functions the local smoothing estimates satisfied by the solutions to the schrodinger equation we show in particular that the local density associated to the solution to the free linear quantum liouville equation admits locally fractional derivatives of given order provided the data belong to some schatten spaces

strichartz estimates via the schrodinger maximal operator we consider the schrodinger operator e it delta acting on initial data f in dot h s we show that an affirmative answer to a question of carleson concerning the sharp range of s for which lim t to 0 e it delta f x f x a e x in mathbb r n would imply an affirmative answer to a question of planchon concerning the sharp range of q and r for which e it delta is bounded in l x q mathbb r n l r t mathbb r when n 2 we unconditionally improve the range for which the mixed norm estimates hold

local smoothing property and strichartz inequality for schrodinger equations with potentials superquadratic at infinity abstract we study smoothing properties for time dependent schrodinger equations i u t 1 2 u v x u u 0 l 2 r n with potentials which satisfy v x o x m at infinity m 2 we show that the solution u t x is 1 m times differentiable with respect to x at almost all t r and explain that this is the result of the fact that the sojourn time of classical particles with energy in arbitrary compact set is less than ct 1 m during 0 t when is very large we also show strichartz s inequality with derivative loss for such potentials and give its application to nonlinear schrodinger equations

the cauchy problem for quasi linear schrodinger equations with x rn t r with x1 xn and summation convention one may think of the equation in 1 1 as a non linear schrodinger equation where the operator modeling the dispersion relation is non isotropic and depends also on the unknown function its conjugate and their gradients in the space variables under appropriate assumptions on the function f the method presented here extends to fully non linear schrodinger equations of the form

magnetic virial identities weak dispersion and strichartz inequalities we show a family of virial type identities for the schrodinger and wave equations with electromagnetic potentials as a consequence some weak dispersive inequalities in space dimension n 3 involving morawetz and smoothing estimates are proved finally we apply them to prove strichartz inequalities for the wave equation with a non trapping electromagnetic potential with almost coulomb decay

extremizability of fourier restriction to the paraboloid abstract in this article we prove that nearly all valid scale invariant fourier restriction inequalities for the paraboloid in r 1 d have extremizers and that l p normalized extremizing sequences are precompact modulo symmetries this result had previously been established for the case q 2 in the range where the boundedness of the restriction operator is still an open question our result is conditional on improvements toward the restriction conjecture

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

dispersive smoothing effects for kdv type equations in this paper we study the smoothness properties of solutions of some nonlinear equations of korteweg de vries kdv type which are of the form tu a x t u3 f u2 u1 u x t 1 wherex r uj jxu andkandjare nonnegative integers our principal condition is thata x t be positive and bounded so that the dispersion is dominant it is shown under certain additional conditions onaandfthatc solutionsu x t are obtained fort 0 if the initial datau x 0 decays faster than it does polynomially onr and has certain initial sobolev regularity a quantitative relationship between the rate of decay and the amount of gain of smoothness is given lets0be the sobolev index if ru2 x 0 1 x m dx 2 for an integerm 0 and the solution obeys u hs0 for an existence time 0 t t thenu x t hmloc r for all 0 t t andu x t l1 0 t h m 1 loc r our method can also be extended to address the fully nonlinear dispersive equations related to 1

semiclassical non concentration near hyperbolic orbits for a large class of semiclassical pseudodifferential operators including schrodinger operators p h h2 g v x on compact riemannian manifolds we give logarithmic lower bounds on the mass of eigenfunctions outside neighbourhoods of generic closed hyperbolic orbits more precisely we show that if a is a pseudodifferential operator which is microlocally equal to the identity near the hyperbolic orbit and microlocally zero away from the orbit then u c log 1 h h p h u clog 1 h i a u this generalizes earlier estimates of colin de verdiere and parisse y colin de verdiere b parisse equilibre instable en regime semi classique i concentration microlocale comm partial differential equations 19 1994 1535 1563 equilibre instable en regime semi classique ii conditions de bohr sommerfeld ann inst h poincare phys theor 61 1994 347 367 obtained for a special case and of burq and zworski n burq m zworski geometric control in the presence of a black box j amer math soc 17 2004 443 471 for real hyperbolic orbits

smoothing properties for the higher order nonlinear schrodinger equation with constant coefficients abstract we study local and global existence and smoothing properties for the initial value problem associated to a higher order nonlinear schrodinger equation with constant coefficients which appears as a model for propagation of pulse in an optical fiber

local well posedness for the modified kdv equation in almost critical widehat h r_s spaces we study the cauchy problem for the modified kdv equation u t u xxx u 3 x 0 u 0 u 0 for data u 0 in the space h r s defined by the norm u 0 h r s s u 0 lr local well posedness of this problem is established in the parameter range 2 r 1 s 1 2 1 2r so the case s r 0 1 which is critical in view of scaling considerations is almost reached to show this result we use an appropriate variant of the fourier restriction norm method as well as bi and trilinear estimates for solutions of the airy equation

effective limiting absorption principles and applications the limiting absorption principle asserts that if h is a suitable schrodinger operator and f lives in a suitable weighted l2 space namely h 0 1 2 sigma for some sigma 0 then the functions r lambda i varepsilon f h lambda i varepsilon 1 f converge in another weighted l2 space h 0 1 2 sigma to the unique solution u of the helmholtz equation h lambda u f which obeys the sommerfeld outgoing radiation condition in this paper we investigate more quantitative or effective versions of this principle for the schrodinger operator on asymptotically conic manifolds with short range potentials and in particular consider estimates of the form r lambda i varepsilon f h 0 1 2 sigma leq c lambda h f h 0 1 2 sigma we are particularly interested in the exact nature of the dependence of the constants c lambda h on both lambda and h it turns out that the answer to this question is quite subtle with distinctions being made between low energies lambda ll 1 medium energies lambda sim 1 and large energies lambda gg 1 and there is also a non trivial distinction between qualitative estimates on a single operator h possibly obeying some spectral condition such as non resonance or a geometric condition such as non trapping and quantitative estimates which hold uniformly for all operators h in a certain class using elementary methods integration by parts and ode techniques we give some sharp answers to these questions as applications of these estimates we present a global in time local smoothing estimate and pointwise decay estimates for the associated time dependent schrodinger equation as well as an integrated local energy decay estimate and pointwise decay estimates for solutions of the corresponding wave equation under some additional assumptions on the operator h

regularity of solutions to the schrodinger equation soit f dans l espace de schwartz s r n et soit s t f x u x t rn exp ix exp it 2 f d x r n t r f est la transformee de fourier de f on considere u solution de l equation de schrodinger u i u t on pose s f x sup 0 t 1 s t f x x r n on introduit des espaces de sobolev h s en posant h s f s f hd s s r ou f hd s rn 1 2 s f 2 d 1 2 on etudie pour quelles valeurs de s on a b s f x 2 dx 1 2 c b f hd s f s r n pour toutes les boucles b de r n

local smoothing and convergence properties of schrodinger type equations abstract the initial value problem for the schrodinger type equation u t ip d u u 0 x u 0 x x r n is considered here p is a principal type polynomial of order m i e p c 1 m 1 r or a certain type of other real symbol such as p m m 1 the following results are proved a if u 0 l 2 r n then for any fixed r 0 write u 0 u 1 u 2 where u 1 and u 2 are square integrable and the fourier transform of u 1 vanishes for r and that of u 2 vanishes for r if u 1 t x e itp d u 1 x u 2 t x e itp d u 2 x then u 1 is real analytic in r r n and satisfies for every multi index sup t x d u 1 t x c u 0 l 2 while 1 x 2 1 2 u 2 t x l 2 r t h m 1 2 r x n b if u 0 l 2 s r n s 0 then for every t 0 u t h loc m 1 s r n c if u 0 h s r n s 1 2 then for a e x r n u t x u 0 x as t 0

almost sure local well posedness and scattering for the 4d cubic nonlinear schrodinger equation abstract we consider the cauchy problem for the defocusing cubic nonlinear schrodinger equation in four space dimensions and establish almost sure local well posedness and conditional almost sure scattering for random initial data in h x s r 4 with 1 3 s 1 the main ingredient in the proofs is the introduction of a functional framework for the study of the associated forced cubic nonlinear schrodinger equation which is inspired by certain function spaces used in the study of the schrodinger maps problem and is based on strichartz spaces as well as variants of local smoothing inhomogeneous local smoothing and maximal function spaces additionally we prove an almost sure scattering result for randomized radially symmetric initial data in h x s r 4 with 1 2 s 1

systematic identification of statistically significant network measures we present a graph embedding space i e a set of measures on graphs for performing statistical analyses of networks key improvements over existing approaches include discovery of motif hubs multiple overlapping significant subgraphs computational efficiency relative to subgraph census and flexibility the method is easily generalizable to weighted and signed graphs the embedding space is based on scalars functionals of the adjacency matrix representing the network scalars are global involving all nodes although they can be related to subgraph enumeration there is not a one to one mapping between scalars and subgraphs improvements in network randomization and significance testing we learn the distribution rather than assuming gaussianity are also presented the resulting algorithm establishes a systematic approach to the identification of the most significant scalars and suggests machine learning techniques for network classification

dispersive analysis of charge transfer models we prove dispersive estimates for the time dependent schrodinger equation with a charge transfer hamiltonian as a by product we also obtain another proof of asymptotic completeness of the wave operators for a charge transfer model established earlier by k yajima and j m graf we also consider a more general matrix non self adjoint charge transfer problem this model appears naturally in the study of nonlinear multisoliton systems and is specifically motivated by the problem of asymptotic stability of multisoliton states of a nonlinear schrodinger equation 2004 wiley periodicals inc

on a class of solutions to the generalized derivative schrodinger equations ii abstract in this note we shall continue our study on the initial value problem associated for the generalized derivative schrodinger gdnls equation t u i x 2 u u x u x t r 0 1 and 1 inspiring by cazenave naumkin s works we shall establish the local well posedness for a class of data of arbitrary size in an appropriate weighted sobolev space thus removing the size restriction on the data required in our previous work the main new tool in the proof is the homogeneous and inhomogeneous versions of the kato smoothing effect for the linear schrodinger equation with lower order variable coefficients established by kenig ponce vega

microlocal dispersive smoothing for the schrodinger equation this paper establishes a connection between the microlocal smoothness of solutions of the initial value problem for schrodinger s equation and the global behavior of bicharacteristics of the principal symbol in particular localized initial data gives rise to solutions which are microlocally smooth at all points which are not trapped backwards by the bicharacteristic flow the origin of the phenomenon is in the dispersive nature of the equation the results imply microlocal regularity properties of the fundamental solution 1995 john wiley sons inc

semiclassical resonances generated by a closed trajectory of hyperbolic type we determine all the resonances in certain rectangular regions of the complex plane of the schrodinger operator h2 v whenh 0 under the assumption that the set of trapped points of energy 0 for the classical flow form a closed trajectory and that the corresponding poincare map is hyperbolic

best constants in some operator smoothness estimates abstract we provide a short proof of the inequality cf ben artzi and klainerman regularity and decay of evolution equations preprint and kato and yajima rev math phys 1 1989 481 496 1 x 2 1 2 1 1 4 e it u 2 dt c u 2 with explicit essentially exact values for c

on the dimension of divergence sets of dispersive equations we refine results of carleson sjogren and sjolin regarding the pointwise convergence to the initial data of solutions to the schrodinger equation we bound the hausdorff dimension of the sets on which convergence fails for example with initial data in h 1 mathbb r 3 the sets of divergence have dimension at most one

an optimal algorithm for checking regularity we present a deterministic algorithm cal a that in o m2 time verifies whether a given m by m bipartite graph g is regular in the sense of szemeredi regular partitions of graphs in problemes combinatoires et theorie des graphes orsay 1976 colloques internationaux cnrs 260 cnrs paris 1978 pp 399 401 in the case in which g is not regular enough our algorithm outputs a witness to this irregularity algorithm cal a may be used as a subroutine in an algorithm that finds an varepsilon regular partition of a given n vertex graph gamma in time o n2 this time complexity is optimal up to a constant factor and improves upon the bound o m n proved by alon et al the algorithmic aspects of the regularity lemma j algorithms 16 1994 pp 80 109 where m n o n2 376 is the time required to square a 0 1 matrix over the integers our approach is elementary except that it makes use of linear sized expanders to accomplish a suitable form of deterministic sampling

an endline bilinear cone restriction estimate for mixed norms we prove an l 2 times l 2 rightarrow l t ql x p bilinear fourier extension estimate for the cone when p q are on the critical line 1 q frac n 1 2 1 1 p this extends previous results by wolff tao and lee vargas

the selberg zeta function for convex co compact schottky groups we give a new upper bound on the selberg zeta function for a convex co compact schottky group acting on the hyperbolic space n 1 in strips parallel to the imaginary axis the zeta function is bounded by exp c s where is the dimension of the limit set of the group this bound is more precise than the optimal global bound exp c s n 1 and it gives new bounds on the number of resonances scattering poles of n 1 the proof of this result is based on the application of holomorphic l2 techniques to the study of the determinants of the ruelle transfer operators and on the quasi self similarity of limit sets we also study this problem numerically and provide evidence that the bound may be optimal our motivation comes from molecular dynamics and we consider n 1 as the simplest model of quantum chaotic scattering

local smoothing for the schrodinger equation with a prescribed loss we consider a family of surfaces of revolution each with a single periodic geodesic which is degenerately unstable we prove a local smoothing estimate for solutions to the linear schrodinger equation with a loss that depends on the degeneracy and we construct explicit examples to show our estimate is saturated on a weak semiclassical time scale as a byproduct of our proof we obtain a cutoff resolvent estimate with a sharp polynomial loss

an algebraic geometry algorithm for scheduling in presence of setups and correlated demands we study here a problem of schedulingn job types onm parallel machines when setups are required and the demands for the products are correlated random variables we model this problem as a chance constrained integer program

well posedness of cauchy problem for the fourth order nonlinear schrodinger equations in multi dimensional spaces abstract we study the well posedness of cauchy problem for the fourth order nonlinear schrodinger equations i t u e u 2 u p x u 2 x u 2 t r x r n where e 1 0 1 n 2 denotes the spatial dimension and p is a polynomial excluding constant and linear terms