global solutions for a semilinear two dimensional klein gordon equation with exponential type nonlinearity we prove the existence and uniqueness of global solutions for a cauchy problem associated to a semilinear klein gordon equation in two space dimensions our result is based on an interpolation estimate with a sharp constant obtained by a standard variational method 2006 wiley periodicals inc

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

global well posedness and scattering for the nonstandard defocusing beam equation abstract we consider the nonstandard defocusing beam equation u t t i 1 n i 4 u u u p 1 u 0 in the subcritical case we show the global well posedness and the scattering results in low dimensions by the compactness rigidity argument of kenig and merle 2006 2008 7 3 the new ingredient is that we establish the suitable strichartz estimate to overcome the difficulty caused by the complicated dispersive estimate

long time dynamics of defocusing energy critical 3 1 dimensional wave equation with potential in the radial case using the channel of energy inequalities developed by t duyckaerts c kenig and f merle we prove that modulo a free radiation any finite energy radial solution to the defocusing energy critical wave equation with radial potential in 3 1 dimensions converges to the set of steady states as time goes to infinity for generic potentials we prove there are only finitely many steady states and in this case modulo some free radiation the solution converges to one steady state as time goes to infinity

robust self testing of the singlet in this paper we introduce a general framework to study the concept of robust self testing which can be used to self test maximally entangled pairs of qubits epr pairs and local measurement operators the result is based only on probabilities obtained from the experiment with tolerance to experimental errors in particular we show that if the results of an experiment approach the cirel son bound or approximate the mayers yao type correlations then the experiment must contain an approximate epr pair more specifically there exist local bases in which the physical state is close to an epr pair possibly encoded in a larger environment or ancilla moreover in these bases the measurements are close to the qubit operators used to achieve the cirel son bound or the mayers yao results 2012 iop publishing ltd

asymptotic behaviour and attractors for a semilinear damped wave equation with supercritical exponent we prove the existence of a global attractor for the problem where f is coercive at infinity and satisfies the growth condition while g l 2 r 3

dynamics for the energy critical nonlinear wave equation in high dimensions t duyckaerts and f merle 2008 studied the variational structure near the ground state solution w of the energy critical wave equation and classified the solutions with the threshold energy e w 0 in dimensions d 3 4 5 in this paper we extend the results to all dimensions d 6 the main issue in high dimensions is the non lipschitz continuity of the nonlinearity which we get around by making full use of the decay property of w

construction of a multisoliton blowup solution to the semilinear wave equation in one space dimension we consider the semilinear wave equation with power nonlinearity in one space dimension given a blow up solution with a characteristic point we refine the blow up behavior first derived by merle and zaag we also refine the geometry of the blow up set near a charac teristic point and show that except may be for one exceptional situation it is never symmetric with the respect to the characteristic point then we show that all blow up modalities predicted by those authors do occur more precisely given any integer k 2 and 0 we construct a blow up solution with a characteristic point a such that the asymptotic behavior of the solution near a t a shows a decoupled sum of k solitons with alternate signs whose centers in the hyperbolic geometry have 0 as a center of mass for all times

strong conical hull intersection property bounded linear regularity jameson s property g and error bounds in convex optimization the strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in euclidean space these fundamental notions occur in various branches of convex optimization constrained approximation convex feasibility problems linear inequalities for instance it is shown that the standard constraint qualification from convex analysis implies bounded linear regularity which in turn yields the strong conical hull intersection property jameson s duality for two cones which relates bounded linear regularity to property g is re derived and refined for polyhedral cones a statement dual to hoffman s error bound result is obtained a sharpening of a result on error bounds for convex inequalities by auslender and crouzeix is presented finally for two subspaces property g is quantified by the angle between the subspaces

global well posedness and scattering for the defocusing l 2 critical nonlinear schrodinger equation when d 1 in this paper we prove global well posedness and scattering for the defocusing one dimensional mass critical nonlinear schr o dinger equation we make use of a long time strichartz estimate and a frequency localized morawetz estimate this continues work begun in earlier papers for dimensions d geq 3 and d 2

scattering theory in the energy space for a class of non linear wave equations we study the asymptotic behaviour in time of the solutions and the theory of scattering in the energy space for the non linear wave equation square varphi f varphi 0 in n n 3 we prove the existence of the wave operators asymptotic completeness for small initial data and forn 4 asymptotic completeness for arbitrarily large data the assumptions onf cover the case wheref behaves slightly better than a single powerp 1 4 n 2 both near zero and at infinity see 1 5 1 6 and 1 8

global classical solutions of nonlinear wave equations is a formally selfadjoint positive elliptic operator and f is a real c2 function with non negative primitive function we treat both the homogeneous initialboundary value problem over ir x f2 for a smooth open set 2clr and the cauchy problem over ir x 1r for the initial boundary value problem we show that any strong solution of 0 1 known to exist according to heinz and yon wahl 4 if 2rn f j u2 o lul 2j j 0 1 u oo

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

the defocusing energy supercritical nonlinear wave equation in three space dimensions we consider the defocusing nonlinear wave equation u tt u u p u 0 in the energy supercritical regime p 4 for even values of the power p we show that blowup or failure to scatter must be accompanied by blowup of the critical sobolev norm an equivalent formulation is that solutions with bounded critical sobolev norm are global and scatter the impetus to consider this problem comes from recent work of kenig and merle who treated the case of spherically symmetric solutions

complementarity and nondegeneracy in semidefinite programming primal and dual nondegeneracy conditions are defined for semidefinite programming given the existence of primal and dual solutions it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution the converses hold if strict complementarity is assumed primal and dual nondegeneracy assumptions do not imply strict complementarity as they do in lp the primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutions x and z this is in contrast with lp where nondegeneracy assumptions exactly determine the number of variables which are zero it is shown that primal and dual nondegeneracy and strict complementarity all hold generically numerical experiments suggest probability distributions for the ranks of x and z which are consistent with the nondegeneracy conditions

nonlinear generalized dunkl wave equations and applications abstract in this paper we introduce a class of nonlinear wave equations associated with the generalized dunkl laplace operator we study local and global well posedness next we establish the linearization of bounded energy solutions in the spirit of gerard 1996 9 the proof uses strichartz type inequalities and the energy estimate

scattering for the two dimensional energy critical wave equation we investigate existence and asymptotic completeness of the wave operators for the nonlinear klein gordon equation with a defocusing exponential nonlinearity in two space dimensions a certain threshold is defined based on the value of the conserved hamiltonian below which the exponential potential energy is dominated by the kinetic energy via a trudinger moser type inequality we prove that if the energy is below or equal to the critical value then the solution approaches a free klein gordon solution at the time infinity an interesting feature in the critical case is that the strichartz estimate together with sobolev type inequalities cannot control the nonlinear term uniformly on each time interval it crucially depends on how much the energy is concentrated thus we have to trace concentration of the energy along time in order to set up favorable nonlinear estimates and only after that we can apply bourgain s induction argument or any other similar one

on the solvability of one multidimensional version of the first darboux problem for some nonlinear wave equations in the present paper for wave equations with power nonlinearity we investigate the problem of the existence or nonexistence of global solutions of a multidimensional version of the first darboux problem in the conic domain

the global cauchy problem for the non linear klein gordon equation on demontre l unicite de solutions faibles et l existence et l unicite de solutions globales fortement continues dans l espace energie

berry phase of a resonant state we derive closed analytical expressions for the complex berry phase of an open quantum system in a state which is a superposition of resonant states and evolves irreversibly due to the spontaneous decay of the metastable states the codimension of an accidental degeneracy of resonances and the geometry of the energy hypersurfaces close to a crossing of resonances differ significantly from those of bound states we discuss some of the consequences of these differences for the geometric phase factors for example instead of a diabolical point singularity there is a continuous closed line of singularities formally equivalent to a continuous distribution of magnetic charge on a diabolical circle there are different classes of topologically inequivalent non trivial closed paths in parameter space the topological invariant associated with the sum of the geometric phases dilations of the wavefunction due to the imaginary part of the berry phase and others

existence and classification of characteristic points at blow up for a semilinear wave equation in one space dimension we consider the semilinear wave equation with power nonlinearity in one space dimension we first show the existence of a blow up solution with a characteristic point then we consider an arbitrary blow up solution u x t the graph x mapsto t x of its blow up points and scr s subset bbb r the set of all characteristic points and show that scr s has an empty interior finally given x 0 in scr s we show that in selfsimilar variables the solution decomposes into a decoupled sum of at least two solitons with alternate signs and that t x forms a corner of angle pi over 2 at x 0

robust self testing of quantum systems via noncontextuality inequalities characterizing unknown quantum states and measurements is a fundamental problem in quantum information processing in this letter we provide a novel scheme to self test local quantum systems using noncontextuality inequalities our work leverages the graph theoretic framework for contextuality introduced by cabello severini and winter combined with tools from mathematical optimization that guarantee the unicity of optimal solutions as an application we show that the celebrated klyachko can binicio ifmmode breve g else u g fi lu shumovsky inequality and its generalization to contextuality scenarios with odd n cycle compatibility relations admit robust self testing

strichartz type estimates and the well posedness of an energy critical 2d wave equation in a bounded domain abstract we study the well posedness of the cauchy problem with dirichlet or neumann boundary conditions associated to an h 1 critical semilinear wave equation on a smooth bounded domain r 2 first we prove an appropriate strichartz type estimate using the l q spectral projector estimates of the laplace operator our proof follows burq lebeau and planchon 2008 4 then we show the global well posedness when the energy is below or at the threshold given by the sharp moser trudinger inequality finally in the supercritical case we prove an instability result using the finite speed of propagation and a quantitative study of the associated ode with oscillatory data

uniqueness for critical nonlinear wave equations and wave maps via the energy inequality we show uniqueness of sufficiently regular solutions to critical semilinear wave equations and wave maps in the a priori much larger class of distribution solutions with finite energy assuming only that the energy is nonincreasing in time 1999 john wiley sons inc

on the solvability of the cauchy characteristic problem for a nonlinear equation with iterated wave operator in the principal part we consider one multidimensional version of the cauchy characteristic problem in the light cone of the future for a hyperbolic equation with power nonlinearity with iterated wave operator in the principal part depending on the exponent of nonlinearity and spatial dimension of equation we investigate the problem on the nonexistence of global solutions of the cauchy characteristic problem the question on the local solvability of that problem is also considered

the global cauchy problem for the critical non linear wave equation abstract we study the global cauchy problem for the non linear wave equation p 1 0 for the critical value p n 2 n 2 in space dimension n 3 we identify a weak space time integrability property stip of the solutions and prove that it is sufficient to ensure the uniqueness of weak solutions the global existence of finite energy solutions with the naturally associated stip and the global existence of regular solutions with some n dependent restrictions on the regularity for spherically symmetric solutions we prove that the previous crucial stip follows from the morawetz inequality actually in a much stronger form than necessary thereby proving that all the previous results hold in the spherically symmetric case

the role of the geometric phases in adiabatic population tracking for non hermitian hamiltonians the definition of instantaneous eigenstate populations for a dynamical non self adjoint system is not obvious the nadirect extension of the definition used for the self adjoint case leads to inconsistencies the resulting artifacts can induce a false inversion of population or a false adiabaticity we show that the inconsistency can be avoided by introducing geometric phases in another possible definition of populations an example is given which demonstrates both the anomalous effects and their removal by our approach

bounded solutions to an energy subcritical non linear wave equation on r3 abstract in this work we consider an energy subcritical semi linear wave equation 3 p 5 t 2 u u x u p 1 u x t r 3 r u t 0 u 0 h s p r 3 t u t 0 u 1 h s p 1 r 3 where s p 3 2 2 p 1 and the function r 3 1 1 is a radial continuous function with a limit at infinity we prove that unless the elliptic equation w x w p 1 w has a nonzero radial solution w c 2 r 3 h s p r 3 any radial solution u with a finite uniform upper bound on the critical sobolev norm u t t u t h s p h s p r 3 for all t in the maximal lifespan must be a global solution in time and scatter

nondispersive radial solutions to energy supercritical non linear wave equations with applications in this paper we establish optimal pointwise decay estimates for non dispersive compact radial solutions to non linear wave equations in 3 dimensions in the energy supercritical range as an application we show for the full energy supercritical range in the defocusing case that if the scale invariant sobolev norm of a radial solution remains bounded in its maximal interval of existence then the solution must exist for all times and scatter

distinguishing separable and entangled states we show how to design families of operational criteria that distinguish entangled from separable quantum states the simplest of these tests corresponds to the well known peres horodecki positive partial transpose ppt criterion and the more complicated tests are strictly stronger the new criteria are tractable due to powerful computational and theoretical methods for the class of convex optimization problems known as semidefinite programs we successfully applied the results to many low dimensional states from the literature where the ppt test fails as a by product of the criteria we provide an explicit construction of the corresponding entanglement witnesses

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

inward outward energy theory of non radial solutions to 3d semi linear wave equation abstract the topic of this paper is a semi linear energy sub critical defocusing wave equation t 2 u u u p 1 u in the 3 dimensional space with 3 p 5 we generalize inward outward energy theory and weighted morawetz estimates for radial solutions to the non radial case as an application we show that if 3 p 5 and 5 p 2 then the solution scatters as long as the initial data u 0 u 1 satisfy r 3 x 1 1 2 u 0 2 1 2 u 1 2 1 p 1 u 0 p 1 d x if p 3 we can also prove the scattering result if initial data u 0 u 1 are contained in the critical sobolev space and satisfy the inequality r 3 x 1 2 u 0 2 1 2 u 1 2 1 4 u 0 p 1 d x these assumptions on the decay rate of initial data as x are weaker than previously known scattering results in the non radial case

optimal classical simulation of state independent quantum contextuality simulating quantum contextuality with classical systems requires memory a fundamental yet open question is what is the minimum memory needed and therefore the precise sense in which quantum systems outperform classical ones here we make rigorous the notion of classically simulating quantum state independent contextuality qsic in the case of a single quantum system submitted to an infinite sequence of measurements randomly chosen from a finite qsic set we obtain the minimum memory needed to simulate arbitrary qsic sets via classical systems under the assumption that the simulation should not contain any oracular information in particular we show that while classically simulating two qubits tested with the peres mermin set requires log 2 24 4 585 bits simulating a single qutrit tested with the yu oh set requires at least 5 740 bits

the topology of the adiabatic passage process for molecular photodissociative dynamics we study population transfer processes for photodissociation as described within the optical potential model in the context of adiabatic or quasi adiabatic dynamics by a reformulation of the adiabatic passage theory in terms of the fibre bundle topology we extend the analysis made by yatsenko et al 2002 phys rev a 65 043407 from conservative to dissipative systems we show that this topology is associated with the problem of eigenvalue labelling and hence with the continuous eigenvalue following process in the control parameter space problems involving adiabatic passage the direct chirping process and non adiabatic transitions are studied with particular regard to the presence of resonance states we also discuss the role played by the molecular continua

blow up of the radially symmetric solutions for the quadratic nonlinear schrodinger system without mass resonance abstract we consider the quadratic nonlinear schrodinger system i t u u v u i t v v u 2 on i r d where 1 d 6 and 0 in the lower dimensional case d 1 2 3 it is known that the h 1 solution is global in time on the other hand there are finite time blow up solutions when d 4 5 6 and 1 2 the condition of 1 2 is called mass resonance in this paper we prove finite time blow up under radially symmetric assumption when d 5 6 and 1 2 and we show blow up or grow up when d 4

uniform decay estimates and the lorentz invariance of the classical wave equation on etudie le comportement asymptotique des solutions de u 0 ou t 2 1 2 n 2 pour des conditions initiales u 0 u t g x en t 0 avec g reguliere a support compact dans r n

existence and universality of the blow up profile for the semilinear wave equation in one space dimension in this paper we consider the semilinear wave equation with a power nonlinearity in one space dimension we exhibit a universal one parameter family of functions which stand for the blow up profile in self similar variables at a non characteristic point for general initial data the proof is done in self similar variables we first characterize all the solutions of the associated stationary problem as a one parameter family then we use energy arguments coupled with dispersive estimates to show that the solution approaches this family in the energy norm in the non characteristic case and to a finite decoupled sum of such a solution in the characteristic case finally in the case where this sum is reduced to one element which is the case for noncharacteristic points we use modulation theory coupled with a nonlinear argument to show the exponential convergence in the self similar time variable of the various parameters and conclude the proof this step provides us with a result of independent interest the trapping of the solution in self similar variables near the set of stationary solutions valid also for non characteristic points the proof of these results is based on a new analysis in the self similar variable 2007 elsevier inc all rights reserved

a blowing up wave equation with exponential type nonlinearity and arbitrary positive energy abstract we consider in two space dimensions the nonexistence of global solutions to the class of klein gordon equations t t u u u f a u t x 0 t we give sufficient conditions on the data such that the solution of the above klein gordon equation blows up in finite time

coupling of eigenvalues of complex matrices at diabolic and exceptional points the paper presents a general theory of coupling of eigenvalues of complex matrices of an arbitrary dimension depending on real parameters the cases of weak and strong coupling are distinguished and their geometric interpretation in two and three dimensional spaces is given general asymptotic formulae for eigenvalue surfaces near diabolic and exceptional points are presented demonstrating crossing and avoided crossing scenarios two physical examples illustrate effectiveness and accuracy of the presented theory

almost quantum adiabatic dynamics and generalized time dependent wave operators we consider quantum dynamics for which the strict adiabatic approximation fails but which do not escape too far from the adiabatic limit to treat these systems we introduce a generalization of the time dependent wave operator theory which is usually used to treat dynamics which do not escape too far from an initial subspace called the active space our generalization is based on a time dependent adiabatic deformation of the active space the geometric phases associated with the almost adiabatic representation are also derived we use this formalism to study the adiabaticity of a dynamics surrounding an exceptional point of a non hermitian hamiltonian we show that the generalized time dependent wave operator can be used to correct easily the adiabatic approximation which is very imperfect in this situation

error bounds for linear matrix inequalities for iterative sequences that converge to the solution set of a linear matrix inequality we show that the distance of the iterates to the solution set is at most o epsilon 2 d the nonnegative integer d is the so called degree of singularity of the linear matrix inequality and epsilon denotes the amount of constraint violation in the iterate for infeasible linear matrix inequalities we show that the minimal norm of epsilon approximate primal solutions is at least 1 o epsilon 1 2 d 1 and the minimal norm of epsilon approximate farkas type dual solutions is at most o 1 epsilon 2 d 1 as an application of these error bounds we show that for any bounded sequence of epsilon approximate solutions to a semidefinite programming problem the distance to the optimal solution set is at most o epsilon 2 k where k is the degree of singularity of the optimal solution set

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