global well posedness and scattering for the defocusing l 2 critical nonlinear schrodinger equation when d 2 in this article we prove that the defocusing cubic nonlinear schrodinger initial value problem is globally well posed and scattering for u0 l2 r2 the proof uses the bilinear estimates of planchon and vega and a frequency localized interaction morawetz estimate similar to the high frequency estimate of colliander keel staffilani takaoka and tao and especially the low frequency estimate of dodson

on some schrodinger and wave equations with time dependent potentials the existence and uniqueness of solutions in the initial value problem for schrodinger and wave equations in the presence of a large time dependent potential is studied the usual strichartz estimates for such linear evolutions are shown to hold true with optimal assumptions on the potentials as a byproduct one obtains a counterexample to the two dimensional double endpoint inhomogeneous strichartz estimate

the radial defocusing energy supercritical nls in dimension four abstract we consider the radial defocusing nonlinear schrodinger equations i u t u u p u with supercritical exponent p 4 in four space dimensions and prove that any radial solution that remains bounded in the critical sobolev space must be global and scatter

metaplectic representation on wiener amalgam spaces and applications to the schrodinger equation we study the action of metaplectic operators on wiener amalgam spaces giving upper bounds for their norms as an application we obtain new fixed time estimates in these spaces for schrodinger equations with general quadratic hamiltonians and strichartz estimates for the schrodinger equation with potentials v x x 2

the focusing energy critical nonlinear schrodinger equation in dimensions five and higher we consider the focusing energy critical nonlinear schr odinger equation iu t delta u u 4 over d 2 u in dimensions d geq 5 we prove that if a maximal lifespan solution u colon i times bbb r d to bbb c obeys sup t in i nabla u t 2 nabla w 2 then it is global and scatters both forward and backward in time here w denotes the ground state which is a stationary solution of the equation in particular if a solution has both energy and kinetic energy less than those of the ground state w at some point in time then the solution is global and scatters we also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state similar results were obtained by kenig and merle for spherically symmetric initial data and dimensions d 3 4 5

strichartz inequalities and the nonlinear schrodinger equation on compact manifolds kklwe prove strichartz estimates with fractional loss of derivatives for the schrodinger equation on any riemannian compact manifold as a consequence we infer low regularity local well posedness results in any dimension as well as global existence results for the cauchy problem of nonlinear schrodinger equations on surfaces in the case of defocusing polynomial nonlinearities and on three manifolds in the case of cubic defocusing nonlinearities we also discuss the optimality of these strichartz estimates on spheres

formulas for calculating the extremum ranks and inertias of a four term quadratic matrix valued function and their applications abstract this paper studies the quadratic matrix valued function x dxax d dxb b x d c through some expansion formulas for ranks and inertias of hermitian matrices where a b c and d are given complex matrices with a and c hermitian x is a variable matrix and denotes the conjugate transpose of a complex matrix we first introduce an algebraic linearization method for studying this matrix valued function and establish a group of explicit formulas for calculating the global maximum and minimum ranks and inertias of this matrix valued function with respect to the variable matrix x we then use these rank and inertia formulas to derive i necessary and sufficient conditions for the matrix equation x 0 to have a solution as well as the four matrix inequalities x 0 in the lowner partial ordering to be feasible respectively ii necessary and sufficient conditions for the four matrix inequalities x 0 in the lowner partial ordering to hold for all matrices x respectively iii the two matrices x and x such that the inequalities x x and x x hold for all matrices x in the lowner partial ordering respectively an application of the quadratic matrix valued function in control theory is also presented

global wellposedness of defocusing critical nonlinear schrodinger equation in the radial case is globally wellposed in time more precisely we obtain a unique solution u u ch1 0 such that for all time u t depends continuously on the data in fact the dependence is even real analytic here moreover there is scattering for t the same statement holds for radial data h s 1 and proves in particular global existence of classical solutions in the radially symmetric case also this issue was open thus this is the analogue for nls of the result for the wave equation with quintic nonlinearity obtained by m struwe str in the radial case and by m grillakis gr s s in general in the case of the wave equation the proof is based on the following two different facts

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

global existence for 2d nonlinear schrodinger equations via high low frequency decomposition method abstract we study global existence of solutions for the cauchy problem of the nonlinear schrodinger equation i u t u u 2 m u in the 2 dimension case where m is a positive integer m 2 using the high low frequency decomposition method we prove that if 10 m 6 10 m 5 s 1 then for any initial value h s r 2 the cauchy problem has a global solution in c r h s r 2 and it can be split into u t e i t y t with y c r h 1 r 2 satisfying y t h 1 1 t 2 1 s 10 m 5 s 10 m 6 where is an arbitrary sufficiently small positive number

bilinear identities involving the k plane transform and fourier extension operators we prove certain l2 n bilinear estimates for fourier extension operators associated to spheres and hyperboloids under the action of the k plane transform as the estimates are l2 based they follow from bilinear identities in particular these are the analogues of a known identity for paraboloids and may be seen as higher dimensional versions of the classical l2 2 bilinear identity for fourier extension operators associated to curves in 2

linear matrix inequalities riccati equations and indefinite stochastic linear quadratic controls this paper deals with an optimal stochastic linear quadratic lq control problem in infinite time horizon where the diffusion term in dynamics depends on both the state and the control variables in contrast to the deterministic case we allow the control and state weighting matrices in the cost functional to be indefinite this leads to an indefinite lq problem which may still be well posed due to the deep nature of uncertainty involved the problem gives rise to a stochastic algebraic riccati equation sare which is however fundamentally different from the classical algebraic riccati equation as a result of the indefinite nature of the lq problem to analyze the sare we introduce linear matrix inequalities lmis whose feasibility is shown to be equivalent to the solvability of the sare moreover we develop a computational approach to the sare via a semi definite programming associated with the lmis finally numerical experiments are reported to illustrate the proposed approach

performance bounds for linear stochastic control we develop computational bounds on performance for causal state feedback stochastic control with linear dynamics arbitrary noise distribution and arbitrary input constraint set this can be very useful as a comparison with the performance of suboptimal control policies which we can evaluate using monte carlo simulation our method involves solving a semidefinite program a linear optimization problem with linear matrix inequality constraints a convex optimization problem which can be efficiently solved numerical experiments show that the lower bound obtained by our method is often close to the performance achieved by several widely used suboptimal control policies which shows that both are nearly optimal as a by product our performance bound yields approximate value functions that can be used as control lyapunov functions for suboptimal control policies

the lyapunov order for real matrices abstract the real lyapunov order in the set of real n n matrices is a relation defined as follows a b if for every real symmetric matrix s sb b t s is positive semidefinite whenever sa a t s is positive semidefinite we describe the main properties of the lyapunov order in terms of linear systems theory nevenlinna pick interpolation and convexity

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

semilinear schrodinger flows on hyperbolic spaces scattering in h 1 we prove global well posedness and scattering in h1 for the defocusing nonlinear schrodinger equations left begin array ll i partial t delta g u u u 2 sigma u 0 phi end array right on the hyperbolic spaces mathbb h d d 2 for exponents sigma in 0 2 d 2 the main unexpected conclusion is scattering to linear solutions in the case of small exponents for comparison on euclidean spaces scattering in h1 is not known for any exponent sigma in 1 d 2 d and is known to fail for sigma in 0 1 d our main ingredients are certain noneuclidean global in time strichartz estimates and noneuclidean morawetz inequalities

some remarks on the schrodinger equation with a potential in lrtlsx we study the dispersive properties of the linear schrodinger equation with a time dependent potential v t x we show that an appropriate integrability condition in space and time on v i e the boundedness of a suitable lrtlsx norm is sufficient to prove the full set of strichartz estimates we also construct several counterexamples which show that our assumptions are optimal both for local and for global strichartz estimates in the class of large unsigned potentials v lrtlsx

some new strichartz estimates for the schrodinger equation abstract we deal with fixed time and strichartz estimates for the schrodinger propagator as an operator on wiener amalgam spaces we discuss the sharpness of the known estimates and we provide some new estimates which generalize the classical ones as an application we present a result on the wellposedness of the linear schrodinger equation with a rough time dependent potential

nonlinear profile decomposition for the h 12 h 12 rd energy subcritical wave equation abstract we consider the nonlinear wave equation u t t u u u 4 d 1 0 in dimensions d 2 this equation is critical in h 1 2 h 1 2 r d and subcritical with respect to the energy we prove the nonlinear profile decomposition the proof must address the lack of compactness due to the lorentz symmetry

homogeneous riccati equations versus quadratic inequalities it is proved that under certain hypotheses the existence of hermitian solutions of a homogenized algebraic riccati equation of continuous type is guaranteed if the corresponding quadratic inequality admits a hermitian solution both finite dimensional matrices and infinite dimensional hilbert space operators riccati equations are considered

regular articlegeneralized strichartz inequalities for the wave equation we make a synthetic exposition of the generalized strichartz inequalities for the wave equation obtained in 6 together with the limiting cases recently obtained in 13 with as simple proofs as possible the proofs combine stationary phase estimates dyadic decompositions the hardy littlewood inequality or the holder inequality in time and abstract duality and interpolation arguments

the dynamics of the 3d radial nls with the combined terms in this paper we show the scattering and blow up result of the radial solution with the energy below the threshold for the nonlinear schrodinger equation nls with the combined terms iu t delta u u 4 u u 2 u qquad qquad qquad qquad rm cnls in the energy space h 1 mathbb r 3 the threshold is given by the ground state w for the energy critical nls iu t u u 4 u this problem was proposed by tao visan and zhang in comm pdes 32 1281 1343 2007 the main difficulty is lack of the scaling invariance illuminated by ibrahim et al in analysis pde 4 3 405 460 2011 we need to give the new radial profile decomposition with the scaling parameter then apply it to the scattering theory our result shows that the defocusing dot h 1 subcritical perturbation u 2 u does not affect the determination of the threshold of the scattering solution of cnls in the energy space

nonlinear scattering theory for a class of wave equations in hs abstract for the semi linear higher order wave equation and the nonlinear higher order schrodinger equation we show that the scattering operators map a band in h s into h s if the nonlinearities have sub critical powers in h s the smoothness of the scattering operators and the uniform boundedness of strong solutions for the defocusing nls equation are also shown provided that the nonlinearities have subcritical growth in h 1 moreover the spatial decaying behavior of solutions in energy space for the defocusing nls equation are obtained

solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method abstract we establish in this paper a group of closed form formulas for calculating the global maximum and minimum ranks and inertias of the quadratic hermitian matrix function x q x p x with respect to the variable matrix x by using a linearization method and some known formulas for extremum ranks and inertias of linear hermitian matrix functions where both p and q are complex hermitian matrices and x is the conjugate transpose of x we then derive the global maximum and minimum ranks and inertias of the two quadratic hermitian matrix functions 1 x q 1 x p 1 x and 2 x q 2 x p 2 x subject to a consistent matrix equation a x b respectively by using some pure algebraic operations of matrices and their generalized inverses as consequences we establish necessary and sufficient conditions for the solutions of the matrix equation a x b to satisfy the quadratic hermitian matrix equalities x p 1 x q 1 and x p 2 x q 2 respectively and for the quadratic matrix inequalities x p 1 x q 1 and x p 2 x q 2 in the lowner partial ordering to hold respectively in addition we give complete solutions to four lowner partial ordering optimization problems on the matrix functions 1 x and 2 x subject to a x b examples are also presented to illustrative applications of the equality constrained quadratic optimizations in some matrix completion problems

remarks on the inhomogeneous fractional nonlinear schrodinger equation using a sharp gagliardo nirenberg type inequality well posedness issues of the initial value problem for a fractional inhomogeneous schrodinger equation are investigated

optimal comparison strategies in ulam s searching game with two errors abstract suppose x is an n bit integer by a comparison question we mean a question of the form does x satisfy either condition a x b or c x d we describe strategies to find x using the smallest possible number q n of comparison questions and allowing up to two of the answers to be erroneous as proved in this self contained paper with the exception of n 2 q n is the smallest number q satisfying berlekamp s inequality 2 q 2 n q 2 q 1 this result would disappear if we only allowed questions of the form does x satisfy the condition a x b since no strategy can find the unknown x 0 1 2n 1 with less than q n questions our result provides extremely simple optimal searching strategies for ulam s game with two lies the game of twenty questions where up to two of the answers may be erroneous

matrix riccati inequality existence of solutions abstract we consider the various criteria including hamiltonian and frequency domain ones for solvability of the matrix riccati inequality arising from control problems some criteria are given in terms of symplectic algebra which is also used as an important technical tool

on the multi interval ulam renyi game for 3 lies 4 intervals suffice abstract we study the problem of identifying an initially unknown m bit number by using yes no questions when up to a fixed number e of the answers can be erroneous in the variant we consider here questions are restricted to be the union of up to a fixed number of intervals for any e 1 let k e be the minimum k such that for all sufficiently large m there exists a strategy matching the information theoretic lower bound and only using k interval questions it is known that k e o e 2 and it has been conjectured that the k e e this linearity conjecture is supported by the known results for small values of e as for e 2 we have k e e we focus on the case e 3 and show k 3 4 improving upon the previously known bound k 3 10

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

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

the solutions for nonlinear schrodinger equations abstract we consider the nonlinear schrodinger equation i u t u f u 0 f u u 2 m u n m n n x t r n r u 0 x u 0 x we give a formal solution of the cauchy problem by this formal solution we obtain that there exists a constant b independent of p n such that for any initial data u 0 h s b 0 s n 2 2 m n 1 4 n 2 s the schrodinger equation has a unique global solution u l r b p 2 s moreover when m 0 we give a unique weak solution of the cauchy problem in d distribution

ulam s searching game with a fixed number of lies abstract paul tries to find an unknown x from l to n by asking q yes no questions in response carole may lie up to k times for k fixed and n q sufficiently large necessary and sufficient conditions are given for paul to win

how to use expert advice we analyze algorithms that predict a binary value by combining the predictions of several prediction strategies called experts our analysis is for worst case situations i e we make no assumptions about the way the sequence of bits to be predicted is generated we measure the performance of the algorithm by the difference between the expected number of mistakes it makes on the bit sequence and the expected number of mistakes made by the best expert on this sequence where the expectation is taken with respect to the randomization in the predictins we show that the minimum achievable difference is on the order of the square root of the number of mistakes of the best expert and we give efficient algorithms that achieve this our upper and lower bounds have matching leading constants in most cases we then show how this leads to certain kinds of pattern recognition learning algorithms with performance bounds that improve on the best results currently know in this context we also compare our analysis to the case in which log loss is used instead of the expected number of mistakes

minimal initial data for potential navier stokes singularities abstract assuming some initial data u 0 h 1 2 r 3 lead to a singularity for the 3d navier stokes equations we show that there are also initial data with the minimal h 1 2 norm which will produce a singularity

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3

on a quadratic matrix inequality and the corresponding algebraic riccati equation abstract the purpose of this paper is to report some new results concerning the geometry of the set x1d4ab 0 of all solutions of a quadratic matrix inequality qmi arising in stochastic realization theory and the subset x1d4ab 0 of all solutions of the corresponding algebraic riccati equation are the boundary interior and extreme points of x1d4ab are studied some new results linking x1d4ab 0 to x1d4ab are presented a complete characterization of some subsets of x1d4ab 0 and x1d4ab are given additional properties of the feedback matrix corresponding to any element in x1d4ab 0 are obtained

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

strichartz type estimates for fractional heat equations abstract we obtain strichartz estimates for the fractional heat equations by using both the abstract strichartz estimates of keel tao and the hardy littlewood sobolev inequality we also prove an endpoint homogeneous strichartz estimate via replacing l x r n by bmo x r n and a parabolic homogeneous strichartz estimate meanwhile we generalize the strichartz estimates by replacing the lebesgue spaces with either besov spaces or sobolev spaces moreover we establish the strichartz estimates for the fractional heat equations with a time dependent potential of an appropriate integrability as an application we prove the global existence and uniqueness of regular solutions in spatial variables for the generalized navier stokes system with l r r n data

two batch search with lie cost we consider the problem of searching for an unknown number in the search space u 0 m 1 q ary questions can be asked and some of the answers may be wrong an arbitrary integer weighted bipartite graph gamma is given stipulating the cost gamma i j of each answer jnei when the correct answer is i i e the cost of a wrong answer correct answers are supposed to be cost less it is assumed that a maximum cost e for the sum of the cost of all wrong answers can be afforded by the responder during the whole search we provide tight upper and lower bounds for the largest size m m q e gamma n for which it is possible to find an unknown number x isinu with n q ary questions and maximum lie cost e our results improve the bounds of cicalese et al 2004 and ahlswede et al 2008 the questions in our strategies can be asked in two batches of nonadaptive questions finally we remark that our results can be further generalized to a wider class of error models including also unidirectional errors

global well posedness and scattering for the defocusing mass critical nonlinear schrodinger equation for radial data in high dimensions we establish global well posedness and scattering for solutions to the defocusing mass critical pseudoconformal nonlinear schrodinger equation iut u u 4 nu for large spherically symmetric lx2 rn initial data in dimensions n 3 after using the concentration compactness reductions in 32 to reduce to eliminating blow up solutions that are almost periodic modulo scaling we obtain a frequency localized morawetz estimate and exclude a mass evacuation scenario somewhat analogously to 10 23 36 in order to conclude the argument

energy concentration of the focusing energy critical fnls abstract we consider the fractional nonlinear schrodinger equation fnls with non local dispersion and focusing energy critical hartree type nonlinearity x 2 u 2 u we consider the energy concentration phenomena of radial blowup solutions near the maximal existence time we use the concentration compactness approach of 19 for confined case and kinetic energy trapping approach of 21 for unconfined case

global well posedness and scattering for the defocusing energy supercritical cubic nonlinear wave equation abstract in this paper we consider the defocusing cubic nonlinear wave equation u t t u u 2 u 0 in the energy supercritical regime in dimensions d 6 with no radial assumption on the initial data we prove that if a solution satisfies an a priori bound in the critical homogeneous sobolev space throughout its maximal interval of existence that is u l t h x s c h x s c 1 then the solution is global and it scatters our analysis is based on the methods of the recent works of kenig and merle 2008 21 and killip and visan 2010 26 27 treating the energy supercritical nonlinear schrodinger and wave equations