squashed entanglement for multipartite states and entanglement measures based on the mixed convex roof new measures of multipartite entanglement are constructed based on two definitions of multipartite information and different methods of optimizing over extensions of the states one is a generalization of the squashed entanglement where one takes the mutual information of parties conditioned on the state s extension and takes the infimum over such extensions additivity of the multipartite squashed entanglement is proved for both versions of the multipartite information which turn out to be related the second one is based on taking classical extensions this scheme is generalized which enables to construct measures of entanglement based on the mixed convex roof of a quantity which in contrast to the standard convex roof method involves optimization over all decompositions of a density matrix rather than just the decompositions into pure states as one of the possible applications of these results we prove that any multipartite monotone is an upper bound on the amount of multipartite distillable key the findings are finally related to analogous results in classical key agreement

one and a half quantum de finetti theorems when n k systems of an n partite permutation invariant state are traced out the resulting state can be approximated by a convex combination of tensor product states this is the quantum de finetti theorem in this paper we show that an upper bound on the trace distance of this approximation is given by 2 frac kd 2 n where d is the dimension of the individual system thereby improving previously known bounds our result follows from a more general approximation theorem for representations of the unitary group consider a pure state that lies in the irreducible representation u mu nu subset u mu otimes u nu of the unitary group u d for highest weights and let be the state obtained by tracing out u then is close to a convex combination of the coherent states u mu g v mu rangle where g in u d and v mu rangle is the highest weight vector in u

the phase plane picture for a class of fourth order conservative differential equations we study the bounded solutions of a class of fourth order equations u u f u 0 0 we show that when is not too large then the paths in the u u plane of two bounded solutions do not cross moreover the conserved quantity associated with the equation puts an ordering on the bounded solutions in the phase plane and a continuation theorem shows that they fill up part of the phase plane we apply these results to the extended fisher kolmogorov efk equation a fourth order model equation for bi stable systems the uniqueness and ordering results imply that as long as the stable equilibrium points are real saddles the bounded solutions of the stationary efk equation correspond exactly to those of the classical second order fisher kolmogorov equation besides we establish the asymptotic stability of the heteroclinic solution of the efk equation

closed characteristics on singular energy levels of second order lagrangian systems abstract we provide a lower bound on the number of closed characteristics on singular energy levels of second order lagrangian systems in the presence of saddle focus equilibria the hypotheses on the lagrangian are mild and the bound is given in terms of the number of saddle foci and a potential function determined by the lagrangian the method of proof is variational combining techniques to minimize near a saddle focus and an analog of the method of broken geodesics

unconditional uniqueness for the cubic gross pitaevskii hierarchy via quantum de finetti we present a new simpler proof of the unconditional uniqueness of solutions to the cubic gross pitaevskii hierarchy in one of the main tools in our analysis is the quantum de finetti theorem our uniqueness result is equivalent to the one established in the celebrated works of erd s schlein and yau 2015 wiley periodicals inc

proof verification and the hardness of approximation problems we show that every language in np has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof if a string is in the language then there exists a proof such that the verifier accepts with probability 1 i e for every choice of its random string for strings not in the language the verifier rejects every provided proof with probability at least 1 2 our result builds upon and improves a recent result of arora and safra 1998 whose verifiers examine a nonconstant number of bits in the proof though this number is a very slowly growing function of the input length as a consequence we prove that no max snp hard problem has a polynomial time approximation scheme unless np p the class max snp was defined by papadimitriou and yannakakis 1991 and hard problems for this class include vertex cover maximum satisfiability maximum cut metric tsp steiner trees and shortest superstring we also improve upon the clique hardness results of feige et al 1996 and arora and safra 1998 and show that there exists a positive e such that approximating the maximum clique size in an n vertex graph to within a factor of n e is np hard

uniqueness of the stationary wave for the extended fisher kolmogorov equation abstract the extended fisher kolmogorov equation ut uxxxx uxx u u3 0 models a binary system near the lifshitz critical point and is known to exhibit a stationary heteroclinic solution joining the equilibria 1 for the classical case 0 the heteroclinic is u x tanh x 2 and is unique up to the obvious symmetries we prove the conjecture that the uniqueness persists all the way to 1 8 where the onset of spatial chaos associated with the loss of monotonicity of the stationary wave is known to occur our methods are non perturbative and employ a global cross section to the hamiltonian flow of the stationary fourth order equation on the energy level of 1 we also prove uniform a priori bounds on all bounded stationary solutions valid for any 0

attenuation compensation in anisotropic least squares reverse time migration abstractanisotropic and attenuating properties of subsurface media cause amplitude loss and waveform distortion in seismic wave propagation resulting in negative influence on seismic imaging to correct the anisotropy effect and compensate amplitude attenuation a compensated amplitude vertical transverse isotropic vti least squares reverse time migration lsrtm method is adopted in this method the attenuation term of an attenuated acoustic wave equation is extended to a vti quasi differential wave equation which takes care of effects from anisotropy and attenuation the finite difference method is used to solve the equation in which attenuation terms are solved in the wavenumber domain and other terms are solved in the space or wavenumber domain stable regularization operators are derived and introduced to the equations to eliminate severe numerical noise in high frequency components during backward propagation meanwhile a demigration operator migration operator and gradient formula for att

infinitely many large amplitude homoclinic orbits for a class of autonomous hamiltonian systems abstract for a class of fourth order autonomous hamiltonian systems we give geometrical conditions that imply the existence of infinitely many homoclinic orbits to a saddle focus equilibrium our results are global in the sense that they do not rely on any transversality assumption they are applied to two example systems one arises in elasticity and in water wave theory the other describes the motion of a particle in a certain rotating potential

a most compendious and facile quantum de finetti theorem in its most basic form the finite quantum de finetti theorem states that the reduced k partite density operator of an n partite symmetric state can be approximated by a convex combination of k fold product states variations of this result include renner s exponential approximation by almost product states a theorem which deals with certain triples of representations of the unitary group and the result of d cruz et al e print quant ph 0606139 phys rev lett 98 160406 2007 for infinite dimensional systems we show how these theorems follow from a single general de finetti theorem for representations of symmetry groups each instance corresponding to a particular choice of symmetry group and representation of that group this gives some insight into the nature of the set of approximating states and leads to some new results including an exponential theorem for infinite dimensional systems

a noncommutative de finetti theorem for boolean independence abstract we introduce a family of quantum semigroups and its natural coactions on noncommutative polynomials we define three invariance conditions for the joint distribution of sequences of selfadjoint noncommutative random variables associated with these coactions for one of the invariance conditions we show that the joint distribution of an infinite sequence of noncommutative random variables satisfy it is equivalent to the fact that the sequence of the random variables is identically distributed and boolean independent with respect to the conditional expectation onto its tail algebra this is a boolean analogue of de finetti theorem on exchangeable sequences in the end of the paper we also discuss the other two invariance conditions which lead to some trivial results

closed characteristics on non compact hypersurfaces in r 2n viterbo demonstrated that any 2n 1 dimensional compact hypersur face m r 2n of contact type has at least one closed characteristic this result proved the weinstein conjecture for the standard symplectic space r 2n various extensions of this theorem have been obtained since all for compact hypersurfaces in this paper we consider non compact hypersurfaces m r 2n coming from mechanical hamiltonians and prove an analogue of viterbo s result the main result provides a strong connection between the top half homology groups hi m i n 2n 1 and the existence of closed characteristics in the non compact case including the compact case

quantum de finetti theorems under local measurements with applications quantum de finetti theorems are a useful tool in the study of correlations in quantum multipartite states in this paper we prove two new quantum de finetti theorems both showing that under tests formed by local measurements in each of the subsystems one can get an exponential improvement in the error dependence on the dimension of the subsystems we also obtain similar results for non signaling probability distributions we give several applications of the results to quantum complexity theory polynomial optimization and quantum information theory the proofs of the new quantum de finetti theorems are based on information theory in particular on the chain rule of mutual information the results constitute improvements and generalizations of a recent de finetti theorem due to brandao christandl and yard

modeling of attenuation and dispersion at the present time proper solutions for absorption modeling are based on wavefield extrapolation techniques which in some instances may be considered expensive two alternative low cost but incomplete solutions exist in the literature the first models dispersion in the frequency domain in accordance with the futterman dispersive relations but does not consider attenuation the second models both attenuation and dispersion in the time domain but assumes a digital minimum phase formulation that results in an inadequate treatment of the dispersion we show that this second solution can be adapted to perform attenuation and or dispersion modeling in agreement with the futterman attenuation dispersion relationships thus obviating the shortcoming mentioned above synthetic and real data examples are shown to illustrate the performance of the proposed algorithm

oscillating solutions of a nonlinear fourth order ordinary differential equation we study the existence of periodic solutions for a nonlinear fourth order ordinary differential equation under suitable conditions we prove the existence of at least one solution of the problem applying coincidence degree theory and the method of upper and lower solutions

deep seismic reflection data interpretation using balanced filtering method abstractinverse q filtering can perform energy compensation and phase correction of seismic reflection data but it has an instability problem due to its high pass characteristics although improved methods such as gain limited inverse q filtering and stabilized inverse q filtering overcome the instability to some extent they are not suitable for compensating deep seismic reflection events with weak energy focusing on the enhancement of deep seismic events we have developed a balanced filtering method based on the ratio of the phase compensated signal to its analytic signal counterpart the method is insensitive to the depth of seismic records and it can make shallow and deep seismic records visible simultaneously when tested on synthetic data and real seismic data compared with other methods the balanced filtering method improves the amplitude strength of the deep reflection events and the continuity of shallow and deep seismic events effectively which makes the deep reflection data easier to i

derivation of hartree s theory for generic mean field bose systems in this paper we provide a novel strategy to prove the validity of hartree s theory for the ground state energy of bosonic quantum systems in the mean field regime for the well known case of trapped bose gases this can be shown using the strong quantum de finetti theorem which gives the structure of infinite hierarchies of k particles density matrices here we deal with the case where some particles are allowed to escape to infinity leading to a lack of compactness our approach is based on two ingredients 1 a weak version of the quantum de finetti theorem and 2 geometric techniques for many body systems our strategy does not rely on any special property of the interaction between the particles in particular our results cover those of benguria lieb and lieb yau for respectively bosonic atoms and boson stars

the pauli principle revisited by the pauli exclusion principle no quantum state can be occupied by more than one electron one can state this as a constraint on the one electron density matrix that bounds its eigenvalues by 1 shortly after its discovery the pauli principle was replaced by anti symmetry of the multi electron wave function in this paper we solve a longstanding problem about the impact of this replacement on the one electron density matrix that goes far beyond the original pauli principle our approach uses berenstein and sjamaar s theorem on the restriction of an adjoint orbit onto a subgroup and allows us to treat any type of permutational symmetry

a global condition for quasi random behavior in a class of conservative systems devaney has shown that an autonomous hamiltonian system in dimension 4 with an orbit homoclinic to a saddle focus equilibrium admits a chaotic behavior as soon as the homoclinic orbit is the transverse intersection of the stable and unstable manifolds in this paper we deal with two classes of saddle focus systems lagrangian systems defined on a two manifold in the presence of a gyroscopic force and fourth order systems arising in water wave theory we first establish by a standard variational method the existence of a homoclinic orbit then under a weak nondegeneracy condition we show that it gives rise to an infinite family of multibump homoclinic solutions and that the dynamics are chaotic our condition is much easier to check than transversality for example it is automatically satisfied for gyroscopic systems on a two torus for topological reasons 1996 john wiley sons inc

faithful squashed entanglement squashed entanglement is a measure for the entanglement of bipartite quantum states in this paper we present a lower bound for squashed entanglement in terms of a distance to the set of separable states this implies that squashed entanglement is faithful that is it is strictly positive if and only if the state is entangled we derive the lower bound on squashed entanglement from a lower bound on the quantum conditional mutual information which is used to define squashed entanglement the quantum conditional mutual information corresponds to the amount by which strong subadditivity of von neumann entropy fails to be saturated our result therefore sheds light on the structure of states that almost satisfy strong subadditivity with equality the proof is based on two recent results from quantum information theory the operational interpretation of the quantum mutual information as the optimal rate for state redistribution and the interpretation of the regularised relative entropy of entanglement as an error exponent in hypothesis testing the distance to the set of separable states is measured in terms of the locc norm an operationally motivated norm giving the optimal probability of distinguishing two bipartite quantum states each shared by two parties using any protocol formed by local quantum operations and classical communication locc between the parties a similar result for the frobenius or euclidean norm follows as an immediate consequence the result has two applications in complexity theory the first application is a quasipolynomial time algorithm solving the weak membership problem for the set of separable states in locc or euclidean norm the second application concerns quantum merlin arthur games here we show that multiple provers are not more powerful than a single prover when the verifier is restricted to locc operations thereby providing a new characterisation of the complexity class qma

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

relative entropy of entanglement and restricted measurements we introduce variants of relative entropy of entanglement based on the optimal distinguishability from unentangled states by means of restricted measurements in this way we are able to prove that the standard regularized entropy of entanglement is strictly positive for all multipartite entangled states this implies that the asymptotic creation of a multipartite entangled state by means of local operations and classical communication always requires the consumption of a nonlocal resource at a strictly positive rate

approximating the set of separable states using the positive partial transpose test the positive partial transpose test is one of the main criteria for detecting entanglement and the set of states with positive partial transpose is considered as an approximation of the set of separable states however we do not know to what extent this criterion as well as the approximation is efficient in this paper we show that the positive partial transpose test gives no bound on the distance of a density matrix from separable states more precisely we prove that as the dimension of the space tends to infinity the maximum trace distance of a positive partial transpose state from separable states tends to 1 using similar techniques we show that the same result holds for other well known separability criteria such as reduction criterion majorization criterion and symmetric extension criterion we also bring in evidence that the sets of positive partial transpose states and separable states have totally different shapes

the bilinear biquadratic model on the complete graph we study the spin 1 bilinear biquadratic model on the complete graph of n sites i e when each spin is interacting with every other spin with the same strength because of its complete permutation invariance this hamiltonian can be rewritten as the linear combination of the quadratic casimir operators of 3 and 2 using group representation theory we explicitly diagonalize the hamiltonian and map out the ground state phase diagram of the model furthermore the complete energy spectrum with degeneracies is obtained analytically for any number of sites

absorption compensation method by l1 norm regularization abstractabsorption in subsurface media greatly reduces the resolution of seismic data absorption not only dissipates the high frequency components of the wave but it also distorts the seismic wavelet the inverse q filtering method is an effective method to correct the attenuation and dispersion of the seismic wave we evaluated an absorption compensation method based on l1 norm regularization forward modeling in the absorption medium is described by a fredholm integral equation in the time domain and the absorption compensation comes down to solving the fredholm integral equation the solutions of the fredholm integral equation are extremely unstable generally l2 norm regularization can be used to seek stable solutions of the fredholm integral equation but it makes the solutions smooth and reduces the resolution of seismic data we used l1 norm regularization to overcome instability of solving the integral equation it makes the solutions have a higher resolution than l2 norm regularization the i

oscillatory solutions of fourth order conservative systems via the conley index in this paper we investigate periodic solutions of second order lagrangian systems which oscillate around equilibrium points of center type the main ingredients are the discretization of second order lagrangian systems that satisfy the twist property and the theory of discrete braid invariants developed by ghrist et al 2003 5 the problem with applying this topological theory directly is that the braid types in our analysis are so called improper this implies that the braid invariants do not entirely depend on the topology the relevant braid classes are non isolating neighborhoods of the flow so that their conley index is not universal in first part of this paper we develop the theory of the braid invariant for improper braid classes and in the second part this theory is applied to second order lagrangian system and in particular to the swift hohenberg equation

distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding we consider the problem of ambiguous discrimination of two quantum states when we are only allowed to perform a restricted set of measurements let the bias of a povm be defined as the total variational distance between the outcome distributions for the two states to be distinguished the performance of a set of measurements can then be defined as the ratio of the bias of this povm and the largest bias achievable by any measurements we first provide lower bounds on the performance of various povms acting on a single system such as the isotropic povm and spherical 2 and 4 designs and show how these bounds can lead to certainty relations furthermore we prove lower bounds for several interesting povms acting on multipartite systems such as the set of local povms povms which can be implemented using local operations and classical communication locc separable povms and finally povms for which every bipartition results in a measurement having positive partial transpose ppt in particular our results show that a scheme of terhal et al for hiding data against local operations and classical communication 31 has the best possible dimensional dependence

a fermionic de finetti theorem quantum versions of de finetti s theorem are powerful tools yielding conceptually important insights into the security of key distribution protocols or tomography schemes and allowing one to bound the error made by mean field approaches such theorems link the symmetry of a quantum state under the exchange of subsystems to negligible quantum correlations and are well understood and established in the context of distinguishable particles in this work we derive a de finetti theorem for finite sized majorana fermionic systems it is shown much reflecting the spirit of other quantum de finetti theorems that a state which is invariant under certain permutations of modes loses most of its anti symmetric character and is locally well described by a mode separable state we discuss the structure of the resulting mode separable states and establish in specific instances a quantitative link to the quality of the hartree fock approximation of quantum systems we hint at a link to generalized pauli principles fo

strengths and weaknesses of quantum computing recently a great deal of attention has been focused on quantum computation following a sequence of results bernstein and vazirani in proc 25th annual acm symposium theory comput 1993 pp 11 20 siam j comput 26 1997 pp 1277 1339 simon in proc 35th annual ieee symposium foundations comput sci 1994 pp 116 123 siam j comput 26 1997 pp 1340 1349 shor in proc 35th annual ieee symposium foundations comput sci 1994 pp 124 134 suggesting that quantum computers are more powerful than classical probabilistic computers following shor s result that factoring and the extraction of discrete logarithms are both solvable in quantum polynomial time it is natural to ask whether all of np can be efficiently solved in quantum polynomial time in this paper we address this question by proving that relative to an oracle chosen uniformly at random with probability 1 the class np cannot be solved on a quantum turing machine qtm in time o 2 n 2 we also show that relative to a permutation oracle chosen uniformly at random with probability 1 the class np cap conp cannot be solved on a qtm in time o 2 n 3 the former bound is tight since recent work of grover in it proc 28 th annual acm symposium theory comput 1996 shows how to accept the class np relative to any oracle on a quantum computer in time o 2 n 2

a noncommutative de finetti theorem invariance under quantum permutations is equivalent to freeness with amalgamation we show that the classical de finetti theorem has a canonical noncommutative counterpart if we strengthen exchangeability i e invariance of the joint distribution of the random variables under the action of the permutation group to invariance under the action of the quantum permutation group more precisely for an infinite sequence of noncommutative random variables x i i in mathbb n we prove that invariance of the joint distribution of the xi s under quantum permutations is equivalent to the fact that the xi s are identically distributed and free with respect to the conditional expectation onto the tail algebra of the xi s

unknown quantum states the quantum de finetti representation we present an elementary proof of the quantum de finetti representation theorem a quantum analog of de finetti s classical theorem on exchangeable probability assignments this contrasts with the original proof of hudson and moody z wahrschein verw geb 33 343 1976 which relies on advanced mathematics and does not share the same potential for generalization the classical de finetti theorem provides an operational definition of the concept of an unknown probability in bayesian probability theory where probabilities are taken to be degrees of belief instead of objective states of nature the quantum de finetti theorem in a closely analogous fashion deals with exchangeable density operator assignments and provides an operational definition of the concept of an unknown quantum state in quantum state tomography this result is especially important for information based interpretations of quantum mechanics where quantum states like probabilities are taken to be states of knowledge rather than

symmetry and uniqueness of the solitary wave solution for the ostrovsky equation we consider herein the ostrovsky equation which arises in modeling the propagation of the surface and internal solitary waves in shallow water or the capillary waves in a plasma with the effects of rotation using the modified sliding method we prove that the solitary wave moving to the left to the ostrovsky equation is symmetric about the origin and unique up to translations we also establish the regularity and decay properties of solitary waves and obtain some results of the nonexistence of solitary wave solutions depending on the wave speed weak rotation and dispersive parameter

global branches of multi bump periodic solutions of the swift hohenberg equation in this paper we study the existence of single and multi bump periodic solutions of a class of fourth order ordinary differential equations arising in problems of pattern formation measuring the tendency to form patterns by a parameter q we view the problem as a nonlinear eigenvalue problem with the use of analytical as well as numerical methods branches of periodic solutions are investigated both locally and globally

finite de finetti theorem for conditional probability distributions describing physical theories we work in a general framework where the state of a physical system is defined by its behavior under measurement and the global state is constrained by no signaling conditions we show that the marginals of symmetric states in such theories can be approximated by convex combinations of independent and identical conditional probability distributions generalizing the classical finite de finetti theorem of diaconis and freedman our results apply to correlations obtained from quantum states even when there is no bound on the local dimension so that known quantum de finetti theorems cannot be used

su p q coherent states and a gaussian de finetti theorem we prove a generalization of the quantum de finetti theorem when the local space is an infinite dimensional fock space in particular instead of considering the action of the permutation group on n copies of that space we consider the action of the unitary group u n on the creation operators of the n modes and define a natural generalization of the symmetric subspace as the space of states invariant under unitaries in u n our first result is a complete characterization of this subspace which turns out to be spanned by a family of generalized coherent states related to the special unitary group su p q of signature p q more precisely this construction yields a unitary representation of the noncompact simple real lie group su p q we therefore find a dual unitary representation of the pair of groups u n and su p q on an n p q mode fock space the gaussian su p q coherent states resolve the identity on the symmetric subspace which implies a gaussian de finetti theorem stating that tracing over a few modes of a unitary invariant state yields a state close to a mixture of gaussian states as an application of this de finetti theorem we show that the n n upper left submatrix of an n n haar invariant unitary matrix is close in total variation distance to a matrix of independent normal variables if n3 o m we prove a generalization of the quantum de finetti theorem when the local space is an infinite dimensional fock space in particular instead of considering the action of the permutation group on n copies of that space we consider the action of the unitary group u n on the creation operators of the n modes and define a natural generalization of the symmetric subspace as the space of states invariant under unitaries in u n our first result is a complete characterization of this subspace which turns out to be spanned by a family of generalized coherent states related to the special unitary group su p q of signature p q more precisely this construction yields a unitary representation of the noncompact simple real lie group su p q we therefore find a dual unitary representation of the pair of groups u n and su p q on an n p q mode fock space the gaussian su p q coherent states resolve the identity on the symmetric subspace which implies a gaussian de finetti theorem stating that tr

entangled games are hard to approximate we establish the first hardness results for the problem of computing the value of one round games played by a verifier and a team of provers who can share quantum entanglement in particular we show that it is np hard to approximate within an inverse polynomial the value of a one round game with i a quantum verifier and two entangled provers or ii a classical verifier and three entangled provers previously it was not even known if computing the value exactly is np hard we also describe a mathematical conjecture which if true would imply hardness of approximation of entangled prover games to within a constant using our techniques we also show that every language in pspace has a two prover one round interactive proof system with perfect completeness and soundness 1 1 mathrm poly even against entangled provers we start our proof by describing two ways to modify classical multiprover games to make them resistant to entangled provers we then show that a strategy for the modified game that uses entanglement can be rounded to one that does not the results then follow from classical inapproximability bounds our work implies that unless mathrm p mathrm np the values of entangled prover games cannot be computed by semidefinite programs that are polynomial in the size of the verifier s system a method that has been successful for more restricted quantum games

de finetti representation theorem for infinite dimensional quantum systems and applications to quantum cryptography we show that the quantum de finetti theorem holds for states on infinite dimensional systems provided they satisfy certain experimentally verifiable conditions this result can be applied to prove the security of quantum key distribution based on weak coherent states or other continuous variable states against general attacks

homoclinic solutions for swift hohenberg and suspension bridge type equations we establish the existence of homoclinic solutions for a class of fourth order equations which includes the swift hohenberg model and the suspension bridge equation in the first case the nonlinearity has three zeros corresponding to a double well potential while in the second case the nonlinearity is asymptotically constant on one side the swift hohenberg model is a higher order extension of the classical fisher kolmogorov model its more complicated dynamics give rise to further possibilities of pattern formation the suspension bridge equation as studied by chen and mckenna j differential equations 136 1997 325 355 we give a positive answer to an open question raised by the authors c 2002 elsevier science usa

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

de finetti theorems for braided parafermions the classical de finetti theorem in probability theory relates symmetry under the permutation group with the independence of random variables this result has application in quantum information here we study states that are invariant with respect to a natural action of the braid group and we emphasize the pictorial formulation and interpretation of our results we prove a new type of de finetti theorem for the four string double braid group acting on the parafermion algebra to braid qudits a natural symmetry in the quon language for quantum information we prove that a braid invariant state is extremal if and only if it is a product state furthermore we provide an explicit characterization of braid invariant states on the parafermion algebra including finding a distinction that depends on whether the order of the parafermion algebra is square free we characterize the extremal nature of product states an inverse de finetti theorem

second order lagrangian twist systems simple closed characteristics we consider a special class of lagrangians that play a fundamental role in the theory of second order lagrangian systems twist systems this subclass of lagrangian systems is defined via a convenient monotonicity property that such systems share this monotonicity property twist property allows a finite dimensional reduction of the variational principle for finding closed characteristics in fixed energy levels this reduction has some similarities with the method of broken geodesics for the geodesic variational problem on riemannian manifolds on the other hand the monotonicity property can be related to the existence of local twist maps in the associated hamiltonian flow the finite dimensional reduction gives rise to a second order monotone recurrence relation we study these recurrence relations to find simple closed characteristics for the lagrangian system more complicated closed characteristics will be dealt with in future work furthermore we give conditions on the lagrangian that guarantee the twist property

a de finetti representation for finite symmetric quantum states consider a symmetric quantum state on an n fold product space that is the state is invariant under permutations of the n subsystems we show that conditioned on the outcomes of an informationally complete measurement applied to a number of subsystems the state in the remaining subsystems is close to having product form this immediately generalizes the so called de finetti representation to the case of finite symmetric quantum states

quantum de finetti theorem under fully one way adaptive measurements we prove a version of the quantum de finetti theorem permutation invariant quantum states are well approximated as a probabilistic mixture of multifold product states the approximation is measured by distinguishability under measurements that are implementable by fully one way local operations and classical communication locc our result strengthens brandao and harrow s de finetti theorem where a kind of partially one way locc measurements was used for measuring the approximation with essentially the same error bound as main applications we show i a quasipolynomial time algorithm which detects multipartite entanglement with an amount larger than an arbitrarily small constant measured with a variant of the relative entropy of entanglement and ii a proof that in quantum merlin arthur proof systems polynomially many provers are not more powerful than a single prover when the verifier is restricted to one way locc operations

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

de finetti theorem on the car algebra the symmetric states on a quasi local c algebra on the infinite set of indices j are those invariant under the action of the group of the permutations moving only a finite but arbitrary number of elements of j the celebrated de finetti theorem describes the structure of the symmetric states i e exchangeable probability measures in classical probability in the present paper we extend the de finetti theorem to the case of the car algebra that is for physical systems describing fermions namely after showing that a symmetric state is automatically even under the natural action of the parity automorphism we prove that the compact convex set of such states is a choquet simplex whose extremal i e ergodic w r t the action of the group of permutations previously described are precisely the product states in the sense of araki moriya in order to do that we also prove some ergodic properties naturally enjoyed by the symmetric states which have a self containing interest

seismic absorption compensation a least squares inverse scheme the problem of instability plagues conventional inverse q filtering we formulate the deabsorption problem as an inverse problem in terms of least squares and impose regularization by means of bayes theorem the solution is iterative and nonparametric and returns a reflectivity that has been constrained to be sparse the inverse scheme is tested on both synthetic and real data and the results obtained demonstrate the viability of the approach

testing product states quantum merlin arthur games and tensor optimization we give a test that can distinguish efficiently between product states of n quantum systems and states that are far from product if applied to a state whose maximum overlap with a product state is 1 e the test passes with probability 1 e regardless of n or the local dimensions of the individual systems the test uses two copies of we prove correctness of this test as a special case of a more general result regarding stability of maximum output purity of the depolarizing channel a key application of the test is to quantum merlin arthur games with multiple merlins where we obtain several structural results that had been previously conjectured including the fact that efficient soundness amplification is possible and that two merlins can simulate many merlins qma k qma 2 for k 2 building on a previous result of aaronson et al this implies that there is an efficient quantum algorithm to verify 3 sat with constant soundness given two unentangled proofs of o n qubits we also show how qma 2 with log sized proofs is equivalent to a large number of problems some related to quantum information such as testing separability of mixed states as well as problems without any apparent connection to quantum mechanics such as computing injective tensor norms of 3 index tensors as a consequence we obtain many hardness of approximation results as well as potential algorithmic applications of methods for approximating qma 2 acceptance probabilities finally our test can also be used to construct an efficient test for determining whether a unitary operator is a tensor product which is a generalization of classical linearity testing

stable patterns for fourth order parabolic equations we consider fourth order parabolic equations of gradient type for the sake of simplicity the analysis is carried out for the specific equation u

product state approximations to quantum states we show that for any many body quantum state there exists an unentangled quantum state such that most of the two body reduced density matrices are close to those of the original state this is a statement about the monogamy of entanglement which cannot be shared without limit in the same way as classical correlation our main application is to hamiltonians that are sums of two body terms for such hamiltonians we show that there exist product states with energy that is close to the ground state energy whenever the interaction graph of the hamiltonian has high degree this proves the validity of mean field theory and gives an explicitly bounded approximation error if we allow states that are entangled within small clusters of systems but product across clusters then good approximations exist when the hamiltonian satisfies one or more of the following properties 1 high degree 2 small expansion or 3 a ground state where the blocks in the partition have sublinear entanglement previously this was known only in the case of small expansion or in the regime where the entanglement was close to zero our approximations allow an extensive error in energy which is the scale considered by the quantum pcp probabilistically checkable proof and nlts no low energy trivial state conjectures thus our results put restrictions on the possible hamiltonians that could be used for a possible proof of the qpcp or nlts conjectures by contrast the classical pcp constructions are often based on constraint graphs with high degree likewise we show that the parallel repetition that is possible with classical constraint satisfaction problems cannot also be possible for quantum hamiltonians unless qpcp is false the main technical tool behind our results is a collection of new classical and quantum de finetti theorems which do not make any symmetry assumptions on the underlying states

positivity and the attractor dimension in a fourth order reaction diffusion equation in this paper we investigate the semilinear partial differential equation ut fuxxxx uxx u 1 u2 with a view particularly to obtaining some insight into how one might establish positivity preservation results for equations containing fourth order spatial derivatives the maximum principle cannot be applied to such equations however progress can be made by employing some very recent best possible interpolation inequalities due to the third named author in which the interpolation constants are both explicitly known and sharp these are used to estimate the lx distance between u and 1 during the evolution a positivity preservation result can be obtained under certain restrictions on the initial datum we also establish an explicit two sided estimate for the fractal dimension of the attractor which is sharp in terms of the physical parameters

central limit theorem for mixing quantum systems and the ccr algebra of fluctuations we analyse macroscopic fluctuations of an infinite quantum system and introduce the ccr c algebra of normal fluctuations a non commutative central limit theorem for mixing quantum systems is proved

homotopy classes for stable connections between hamiltonian saddle focus equilibria for a class of hamiltonian systems in 4 the set of homoclinic and heteroclinic orbits which connect saddle focus equilibria is studied using a variational approach the oscillatory properties of a saddle focus equilibrium and the variational nature of the problem give rise to connections in many homotopy classes of the configuration plane punctured at the saddle foci this variational approach does not require any assumptions on the intersections of stable and unstable manifolds such as transversality moreover these connections are shown to be local minimizers of an associated action functional this result has applications to spatial pattern formation in a class of fourth order bistable evolution equations

existence and numerical approximations of periodic solutions of semilinear fourth order differential equations abstract a multiplicity result of existence of periodic solutions with prescribed wavelength for a class of fourth order nonautonomous differential equations related either to the extended fisher kolmogorov or to the swift hohenberg equation is proved variational approach is used some numerical solutions are calculated via the finite element method

a generalization of quantum stein s lemma given many independent and identically distributed i i d copies of a quantum system described either by the state or called null and alternative hypotheses respectively what is the optimal measurement to learn the identity of the true state in asymmetric hypothesis testing one is interested in minimizing the probability of mistakenly identifying instead of while requiring that the probability that is identified in the place of is bounded by a small fixed number quantum stein s lemma identifies the asymptotic exponential rate at which the specified error probability tends to zero as the quantum relative entropy of and

inverse q filtering by fourier transform although some attention has been paid to the idea that seismic migration is equivalent to a type of deconvolution of the spatial wavelet less thought has been given to the opposite perspective that deconvolution of the earth q filter might itself be equivalent to a form of migration the key point raised in this paper is that a dispersive 1 d backward propagation can form the basis of a number of different algorithms for inverse q filtering each of which is akin to a particular migration algorithm an especially efficient algorithm can be derived by means of a coordinate transformation equivalent to that in the stolt frequency wavenumber migration this fast algorithm valid for q constant with depth can be extended to accommodate depth variable q by cascading a series of constant q compensations as in cascaded migration by combining a cascaded phase compensation with a windowed approach to amplitude compensation we obtain an algorithm that is sufficiently efficient to be used routinely for pre

weak decoupling duality and quantum identification if a quantum system is subject to noise it is possible to perform quantum error correction reversing the action of the noise if and only if no information about the system s quantum state leaks to the environment in this paper we develop an analogous duality in the case that the environment approximately forgets the identity of the quantum state a weaker condition satisfied by randomizing maps and approximate unitary designs specifically we show that the environment approximately forgets quantum states if and only if the original channel approximately preserves pairwise fidelities of pure inputs an observation we call weak decoupling duality using this tool we then go on to study the task of using the output of a channel to simulate restricted classes of measurements on a space of input states the case of simulating measurements that test whether the input state is an arbitrary pure state is known as equality testing or quantum identification an immediate consequence of weak decoupling duality is that the ability to perform quantum identification cannot be cloned we furthermore establish that the optimal amortized rate at which quantum states can be identified through a noisy quantum channel is equal to the entanglement assisted classical capacity of the channel despite the fact that the task is quantum not classical and entanglement assistance is not allowed in particular this rate is strictly positive for every nonconstant quantum channel including classical channels

homotopy classes for stable periodic and chaotic patterns in fourth order hamiltonian systems we investigate periodic and chaotic solutions of hamiltonian systems in 4 which arise in the study of stationary solutions of a class of bistable evolution equations under very mild hypotheses variational techniques are used to show that in the presence of two saddle focus equilibria minimizing solutions respect the topology of the configuration plane punctured at these points by considering curves in appropriate covering spaces of this doubly punctured plane we prove that minimizers of every homotopy type exist and characterize their topological properties

a noncommutative extended de finetti theorem abstract the extended de finetti theorem characterizes exchangeable infinite sequences of random variables as conditionally i i d and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability our main result is a noncommutative version of this theorem in contrast to the classical result of ryll nardzewski exchangeability turns out to be stronger than spreadability for infinite sequences of noncommutative random variables out of our investigations emerges noncommutative conditional independence in terms of a von neumann algebraic structure closely related to popa s notion of commuting squares and kummerer s generalized bernoulli shifts our main result is applicable to classical probability quantum probability in particular free probability braid group representations and jones subfactors

de finetti theorems for easy quantum groups we study sequences of noncommutative random variables which are invariant under quantum transformations coming from an orthogonal quantum group satisfying the easiness condition axiomatized in our previous paper for 10 easy quantum groups we obtain de finetti type theorems characterizing the joint distribution of any infinite quantum invariant sequence in particular we give a new and unified proof of the classical results of de finetti and freedman for the easy groups sn on which is based on the combinatorial theory of cumulants we also recover the free de finetti theorem of kostler and speicher and the characterization of operator valued free semicircular families due to curran we consider also finite sequences and prove an approximation result in the spirit of diaconis and freedman

unique games with entangled provers are easy we consider one round games between a classical verifier and two provers who share entanglement we show that when the constraints enforced by the verifier are unique constraints i e permutations the value of the game can be well approximated by a semidefinite program sdp essentially the only algorithm known previously was for the special case of binary answers as follows from the work of tsirelson in 1980 among other things our result implies that the variant of the unique games conjecture where we allow the provers to share entanglement is false our proof is based on a novel quantum rounding technique showing how to take a solution to an sdp and transform it into a strategy for entangled provers using our approximation by an sdp we also show a parallel repetition theorem for unique entangled games