nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

the limit behavior of solutions for the cauchy problem of the complex ginzburg landau equation this paper is devoted to the study of the limit behavior as e 0 or e 0 and a 0 for the solutions of the cauchy problem of the complex ginzburg landau equation ut e u i u a i u u 0 u 0 x u0 x 4 n 4 n 2 0 0 s 0 1 and s n 2 2 moreover the sharp convergence rate in such spaces is also given 2002 john wiley sons inc

critical exponent for evolution equations in modulation spaces abstract in this paper we propose a method to find the critical exponent for certain evolution equations in modulation spaces we define an index s q and use it to determine the critical exponent of the fractional heat equation as an example we prove that when s q is greater than the critical exponent this equation is locally well posed in the space c 0 t m p q s and when s q is less than the critical exponent this equation is ill posed in the space c 0 t m 2 q s our method may further be applied to some other evolution equations

the global cauchy problem for the nls and nlkg with small rough data abstract by using the unit cube decomposition to the frequency spaces we study the cauchy problem for the nonlinear schrodinger equation and the nonlinear klein gordon equation some global well posedness results are obtained for the small cauchy data in some modulation spaces m p 1 s

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

time frequency analysis of localization operators abstract we study a class of pseudodifferential operators known as time frequency localization operators anti wick operators gabor toeplitz operators or wave packets given a symbol a and two windows 1 2 we investigate the multilinear mapping from a 1 2 s r 2d s r d s r d to the localization operator aa 1 2 and we give sufficient and necessary conditions for aa 1 2 to be bounded or to belong to a schatten class our results are formulated in terms of time frequency analysis in particular we use modulation spaces as appropriate classes for symbols and windows

global well posedness for fractional hartree equation on modulation spaces and fourier algebra abstract we study the cauchy problem for fractional schrodinger equation with cubic convolution nonlinearity i t u 2 u k u 2 u 0 with cauchy data in the modulation spaces m p q r d for k x x 0 min d 2 we establish global well posedness results in m p q r d 1 p 2 1 q 2 d d when 2 d 1 and with radial cauchy data when d 2 2 d 2 d 1 2 similar results are proven in fourier algebra f l 1 r d l 2 r d

phase space analysis of semilinear parabolic equations abstract we present a wave packet analysis of a class of possibly degenerate parabolic equations with variable coefficients as a consequence we prove local wellposedness of the corresponding cauchy problem in spaces of low regularity namely the modulation spaces assuming a nonlinearity of analytic type as another application we deduce that the corresponding phase space flow decreases the global wave front set we also consider the action on spaces of analytic functions provided the coefficients are analytic themselves

strichartz estimates in the frame of modulation spaces abstract in this paper we study the strichartz estimates for three types of differential equations in the framework of modulation spaces the equations considered here include the wave equation the schrodinger equation and the nonelliptic schrodinger equation inhomogeneous equations of the above three types are also studied

the inclusion relation between sobolev and modulation spaces the inclusion relations between the lp sobolev spaces and the modulation spaces is determined explicitly as an application mapping properties of unimodular fourier multiplier ei d between lp sobolev spaces and modulation spaces are discussed

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

estimate for generalized unimodular multipliers on modulation spaces abstract in this paper we consider the operator e i t h d where h defined on r n is a c r n 0 positive homogeneous function with degree 0 and r r is a smooth function satisfying the following a1 there exists a constant m 1 0 such that for all n 0 n 0 r r m 1 r 1 a2 there exists a constant m 2 0 such that for all n 0 r r m 2 0 r 1 we prove the boundedness of the operator e i t h d on the modulation spaces and obtain its asymptotic estimate as t goes to infinity as applications we give the grow up rate of the solutions for the cauchy problems for the generalized wave klein gordon and schrodinger equations

banach spaces related to integrable group representations and their atomic decompositions i abstract we present a general theory of banach spaces which are invariant under the action of an integrable group representation and give their atomic decompositions with respect to coherent states i e the atoms arise from a single element under the group action several well known decomposition theories are contained as special examples and are unified under the aspect of group theory

inviscid limits of the complex ginzburg landau equation in the inviscid limit the generalized complex ginzburg landau equation reduces to the nonlinear schrodinger equation this limit is proved rigorously with h1 data in the whole space for the cauchy problem and in the torus with periodic boundary conditions the results are valid for nonlinearities with an arbitrary growth exponent in the defocusing case and with a subcritical or critical growth exponent at the level of l2 in the focusing case in any spatial dimension furthermore optimal convergence rates are proved the proofs are based on estimates of the schrodinger energy functional and on gagliardo nirenberg inequalities

the cauchy problem for the hartree type equation in modulation spaces we study the cauchy problem for hartree equation with cubic convolution nonlinearity f u k u 2 u under a specified condition on potential k with cauchy data in modulation spaces mp q rd we establish global well posedness results in m1 1 rd when k x x r 0 min 2 d 2 in mp q rd 1 q min p p where p is the holder conjugate of p 1 2 when k is in fourier algebra fl1 rd and local well posedness result in mp 1 rd 1 p when k m1 rd

on the hardness of approximating minimization problems we prove results indicating that it is hard to compute efficiently good approximate solutions to the graph coloring set covering and other related minimization problems specifically there is an e 0 such that graph coloring cannot be approximated with ratio n e unless p np set covering cannot be approximated with ratio c log n for any c 1 4 unless np is contained in dtime n poly log n similar results follow for related problems such as clique cover fractional chromatic number dominating set and others

frequency uniform decomposition method for the generalized bo kdv and nls equations abstract sufficient and necessary conditions for the embeddings between besov spaces b p q s 1 and modulation spaces m p q s 2 are obtained moreover using the frequency uniform decomposition method we study the cauchy problem for the generalized bo kdv and nls equations for which the global well posedness of solutions with the small rough data in certain modulation spaces m 2 1 s is shown

the global cauchy problems for the nonlinear dispersive equations on modulation spaces abstract in this paper we discuss decay estimates and strichartz estimates for dispersive equations with non homogeneous symbols on modulation spaces m p q s to obtain the global well posedness of the cauchy problems for nonlinear dispersive equations as a result we have a generalization of the result in 19 which treated the schrodinger equations with a nonlinearity of wider class

remarks on fourier multipliers and applications to the wave equation exploiting continuity properties of fourier multipliers on modulation spaces and wiener amalgam spaces we study the cauchy problem for the nlw equation local wellposedness for rough data in modulation spaces and wiener amalgam spaces is shown the results formulated in the framework of modulation spaces refine those in a benyi k a okoudjou local well posedness of nonlinear dispersive equations on modulation spaces preprint april 2007 available at arxiv 0704 0833v1 the same arguments may apply to obtain local wellposedness for the nlkg equation

lower bounds on the probability of error for classical and classical quantum channels in this paper lower bounds on error probability in coding for discrete classical and classical quantum channels are studied the contribution of the paper goes in two main directions 1 extending classical bounds of shannon to classical quantum channels and 2 proposing a new framework for lower bounding the probability of error of channels with a zero error capacity in the low rate region the relation between these two problems is revealed by showing that lovasz bound on zero error capacity emerges as a natural consequence of the sphere packing bound once we move to the more general context of classical quantum channels a variation of lovasz bound is then derived to lower bound the probability of error in the low rate region by means of auxiliary channels as a result of this study connections between the lovasz theta function the expurgated bound of gallager the cutoff rate of a classical channel and the sphere packing bound for classical quantum channels are established

functions operating on modulation spaces and nonlinear dispersive equations abstract the aim of this paper is two fold we show that if a complex function f on c operates in the modulation spaces m p 1 r n by composition then f is real analytic on r 2 c this answers negatively the open question posed in ruzhansky et al 2012 21 regarding the general power type nonlinearity of the form u u we also characterise the functions that operate in the modulation space m 1 1 r n the local well posedness of the nls nlw and nlkg equations for the real entire nonlinearities are also studied in some weighted modulation spaces m s p q r n

modulation spaces wiener amalgam spaces and brownian motions abstract we study the local in time regularity of the brownian motion with respect to localized variants of modulation spaces m s p q and wiener amalgam spaces w s p q we show that the periodic brownian motion belongs locally in time to m s p q t and w s p q t for s 1 q 1 and the condition on the indices is optimal moreover with the wiener measure on t we show that m s p q t and w s p q t form abstract wiener spaces for the same range of indices yielding large deviation estimates we also establish the endpoint regularity of the periodic brownian motion with respect to a besov type space b p s t specifically we prove that the brownian motion belongs to b p s t for s 1 p 1 and it obeys a large deviation estimate finally we revisit the regularity of brownian motion on usual local besov spaces b p q s and indicate the endpoint large deviation estimates

estimates on modulation spaces for schrodinger evolution operators with quadratic and sub quadratic potentials abstract in this paper we give new estimates for the solution to the schrodinger equation with quadratic and sub quadratic potentials in the framework of modulation spaces

estimates on fractional power dissipative equations in function spaces abstract we study the fractional power dissipative equations whose fundamental semigroup is given by e t with 0 by using an argument of duality and interpolation we extend space time estimates of the fractional power dissipative equations in lebesgue spaces to the hardy spaces and the modulation spaces these results are substantial extensions of some known results as applications we study both local and global well posedness of the cauchy problem for the nonlinear fractional power dissipative equation u t u u m u for initial data in the modulation spaces

strong converse to the quantum channel coding theorem a lower bound on the probability of decoding error for a quantum communication channel is presented from which the strong converse to the quantum channel coding theorem is immediately shown the results and their derivations are mostly straightforward extensions of the classical counterparts which were established by arimoto 1973 except that more careful treatment is necessary here due to the noncommutativity of operators

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

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

concavity of the auxiliary function for classical quantum channels the auxiliary function of a classical channel appears in two fundamental quantities the random coding exponent and the sphere packing exponent which yield upper and lower bounds on the error probability of decoding respectively a crucial property of the auxiliary function is its concavity and this property consequently leads to several important results in finite blocklength analysis in this paper we prove that the auxiliary function of a classical quantum channel also enjoys the same concavity property extending an earlier partial result to its full generality we also prove that the auxiliary function satisfies the data processing inequality among various other important properties furthermore we show that the concavity property of the auxiliary function enables a geometric interpretation of the random coding exponent and the sphere packing exponent of a classical quantum channel the key component in our proof is an important result from the theory of matrix geometric means

moderate deviation analysis for classical quantum channels and quantum hypothesis testing in this paper we study the tradeoffs between the error probabilities of classical quantum channels and the blocklength n when the transmission rates approach the channel capacity at a rate lower than 1 sqrt n a research topic known as moderate deviation analysis we show that the optimal error probability vanishes under this rate convergence our main technical contributions are a tight quantum sphere packing bound obtained via chaganty and sethuraman s concentration inequality in strong large deviation theory and asymptotic expansions of error exponent functions moderate deviation analysis for quantum hypothesis testing is also established the converse directly follows from our channel coding result while the achievability relies on a martingale inequality

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

well posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces abstract in this paper we prove the local and global well posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces p m q s n m q s 2 under certain conditions of s q q and

reliability function of general classical quantum channel in information theory the reliability function and its bounds describing the exponential behavior of the error probability are important quantitative characteristics of the channel performance from a more general point of view these bounds provide certain measures of distinguishability of a given set of classical states in a previous paper quantum analogs of the random coding and expurgation lower bounds for the case of pure signal states were introduced here we discuss the case of general quantum states in particular we prove the expurgation bound conjectured previously and find the quantum cutoff rate for arbitrary mixed signal states

greedy bases in weighted modulation spaces abstract in the present paper we consider weighted modulation spaces weights of exponential growth will be admissible with the molecular decomposition we shall obtain an unconditional basis as applications we obtain the greedy basis of coorbit spaces

globally well and ill posedness for non elliptic derivative schrodinger equations with small rough data abstract we show that there exists s c 0 such that the cubic quartic non elliptic derivative schrodinger equations with small data in modulation spaces m 2 1 s r n for n 3 n 2 are globally well posed if s s c and ill posed if s s c in 2d cubic case using the gabor frame we get some time global dispersive estimates for the schrodinger semi group in anisotropic lebesgue spaces which include a time global maximal function estimate in the space l x 1 2 l x 2 t by resorting to the smooth effect estimate together with the dispersive estimates in anisotropic lebesgue spaces we show that the cubic hyperbolic derivative nls in 2d has a unique global solution if the initial data in feichtinger segal algebra or in weighted sobolev spaces are sufficiently small

strong converse exponent for classical quantum channel coding we determine the exact strong converse exponent of classical quantum channel coding for every rate above the holevo capacity our form of the exponent is an exact analogue of arimoto s given as a transform of the renyi capacities with parameters alpha 1 it is important to note that unlike in the classical case there are many inequivalent ways to define the renyi divergence of states and hence the renyi capacities of channels our exponent is in terms of the renyi capacities corresponding to a version of the renyi divergences that has been introduced recently in muller lennert et al j math phys 54 12 122203 2013 arxiv 1306 3142 and wilde et al commun math phys 331 2 593 622 2014 arxiv 1306 1586 our result adds to the growing body of evidence that this new version is the natural definition for the purposes of strong converse problems

navier stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices abstract the cauchy problems for navier stokes equations and nonlinear heat equations are studied in modulation spaces m q s r n though the case of the derivative index s 0 has been treated in our previous work the case s 0 is also treated in this paper our aim is to reveal the conditions of s q and of m q s r n for the existence of local and global solutions for initial data u 0 m q s r n

multilinear weighted convolution of l 2 functions and applications to nonlinear dispersive equations the x s b spaces as used by beals bourgain kenig ponce vega klainerman machedon and others are fundamental tools to study the low regularity behavior of nonlinear dispersive equations it is of particular interest to obtain bilinear or multilinear estimates involving these spaces by plancherel s theorem and duality these estimates reduce to estimating a weighted convolution integral in terms of the l 2 norms of the component functions in this paper we systematically study weighted convolution estimates on l 2 as a consequence we obtain sharp bilinear estimates for the kdv wave and schrodinger x s b spaces

wave packet analysis of schrodinger equations in analytic function spaces abstract we consider a class of linear schrodinger equations in r d with analytic symbols we prove a global in time integral representation for the corresponding propagator as a generalized gabor multiplier with a window analytic and decaying exponentially at infinity which is transported by the hamiltonian flow we then provide three applications of the above result the exponential sparsity in phase space of the corresponding propagator with respect to gabor wave packets a wave packet characterization of fourier integral operators with analytic phases and symbols and the propagation of analytic singularities

continuity properties for modulation spaces with applications to pseudo differential calculus i we discuss continuity for weighted modulation spaces andprove that many such spaces can be obtained in a canonicalway from the corresponding standard modulation spaces we also discussthe trace operator a a 0 acting on modulationspaces the results are used to get inclusions betweenmodulation spaces and besov spaces and proving continuityfor pseudo differential operators and toeplitz operators

random data cauchy problem for the fourth order schrodinger equation with the second order derivative nonlinearities abstract we consider the cauchy problem of the fourth order nonlinear schrodinger equation 4nls i t e 2 u p m x u 2 x u 2 m 3 on r d d 2 with random initial data here p m is a homogeneous polynomial of degree m containing the second order derivative we show the almost sure local well posedness and small data global existence in h s r d with some range of s s c where s c is the scaling critical regularity i e s c d 2 2 m 1 our results contain the cases s s c 1 2 s c when d 3 and m 5 similar supercritical well posedness results also hold for d 2 m 4 and d 3 3 m 5

on the possibility of soft and hard turbulence in the complex ginzburg landau equation abstract we present analytical methods which predict the occurrence of both soft weak and hard strong turbulence in the complex ginzburg landau cgl equation a t ra 1 i a 1 i a a 2 on a periodic domain 0 1 d in d spatial dimensions hard turbulence is characterised by large fluctuations away from spatial and temporal averages with a cascade of energy to small scales this form of hard turbulence appears to occur not in 1d but only in 2d and 3d in parameter regions which are bounded by hyperbolic curves in the second and fourth quadrants of the planes where the system is modulationally unstable 1 2n f n n 1 a 2 n a 2n dx for n 0 for large times and large r upper bounds exist for the infinite set of fn s constructed from the hierarchy of differential inequalities fn 2nr cn a 2 fn bnf2n fn 1 for cn bn 0 f0 1 estimates for the bottom rung f2 give upper bounds for the whole ladder long time upper bounds on f2 and a 2 and hence all fn are well controlled in the soft region but become much larger in the hard region whereas spati al and temporal averages remain comparatively small when the nonlinearity is a a 2q the critical case qd 2 gives parallel results

construction of class regions by a randomized algorithm a randomized subclass method a randomized algorithm is proposed for solving the problem of finding hyper rectangles sufficiently approximating the true region in each class this method yields a suboptimal solution but is more efficient than previous methods the performance is analysed based on a criterion of pac probably approximately correct learning experimental results show that the proposed method can solve large problems which were not able to be solved previously

an efficient construction and application usefulness of rectangle greedy covers we develop efficient construction methods of a rectangle greedy cover rgc and evaluate its usefulness in applications an rgc is a greedy cover of the set of given positive instances by exclusive axis parallel hyperrectangles namely axis parallel hyperrectangles that exclude all the given negative instances an rgc is expected to be a compact classification rule with high readability because the number of its component rectangles is expected to be small and it can be seen as a disjunctive normal form which is one of the most readable representations for us we propose two approaches of rgc construction enumeration approach and direct approach in enumeration approach the maximal exclusive positive subsets mepss are enumerated first and then an ordinary greedy set covering is done using the enumerated mepss we make clear the relation between enumeration of the maximal frequent itemsets and enumeration of the mepss and convert an efficient enumeration algorithm lcmmax 1 of maximal frequent itemsets to an enumeration algorithm lcmmax rnaive of mepss we also develop a more efficient version of lcmmax rnaive or lcmmax r by incorporating effective dynamic reordering of instances using excluded frequency and bit parallel exclusiveness check in direct approach each component meps of an rgc is searched not from enumerated mepss but directly from the dataset that consists of the remaining uncovered positive instances and the whole negative instances we developed an algorithm called mrf that efficiently finds an maximum sized meps for given positive and negative instances mrf is made from lcmmax r by modifying it so as to find a maximum sized meps only an rgc is constructed by mrf repetition that is by repeatedly executing mrf using the remaining uncovered positive instances according to our experimental evaluation using uci repository datasets lcmmax r was about 5 11 times faster than lcmmax rnaive which indicates effectiveness of the introduced two improvements mrf repetition however was significantly faster than lcmmax r and it was fast enough for practical usage the experimental results using uci repository datasets also showed that accuracy of a nearest rectangle classifier using an rgc is close to that using the hyperrectangles output by the randomized subclass method rsm 2 though the number of component rectangles of an rgc is significantly smaller than the number of the hyperrectangles output by rsm the performance of rgc was also shown to be comparable to that of the six popular classifiers including logistic regression and support vector machine the disjunctive normal form representation of the classification rules obtained by rgc was demonstrated to be simpler and more readable for us than that obtained by rsm and c4 5 highlightsefficient construction methods of a rectangle greedy cover rgc are developed the relation between frequent itemsets and exclusive positive subsets is made clear direct approach is faster than enumeration approach to construct an rgc empirical classification accuracy of an rgc is comparable to that of svm the dnf rules obtained by an rgc are shown to be more readable than those by c4 5

isometric decomposition operators function spaces ep qλ and applications to nonlinear evolution equations abstract by using the isometric decomposition to the frequency spaces we will introduce a new class of function spaces e p q which is a subspace of gevrey 1 class g 1 r n c r n for 0 and we will study the cauchy problem for the nonlinear schrodinger equation the complex ginzburg landau equation and the navier stokes equation some well posed results are obtained for the cauchy data in e 2 1 0 and the regularity behavior in e 2 1 ct g 1 r n for the complex ginzburg landau equation and the navier stokes equation is also obtained as time t 0

the inclusion relations between α modulation spaces and lp sobolev spaces or local hardy spaces abstract in this paper we first discuss equivalent norms for modulation spaces which are composed from decomposition with non compact frequency support then we determine sharp inclusion relations between modulation spaces and l p sobolev spaces and between modulation spaces and local hardy spaces

nonlinear schrodinger equations in the sobolev space of critical order the cauchy problem for the nonlinear schrodinger equations is considered in the sobolev spacehn 2 rn of critical ordern 2 where the embedding intol rn breaks down and any power behavior of interaction works as a subcritical nonlinearity under the interaction of exponential type the existence and uniqueness is proved for globalhn 2 solutions with small cauchy data

quantum sphere packing bounds with polynomial prefactors we study lower bounds on the optimal error probability in classical coding over classical quantum channels at rates below the capacity commonly termed quantum sphere packing bounds winter and dalai have derived such bounds for classical quantum channels however the exponents in their bounds only coincide when the channel is classical in this paper we show that these two exponents admit a variational representation and are related by the golden thompson inequality reaffirming that dalai s expression is stronger in general classical quantum channels second we establish a finite blocklength sphere packing bound for classical quantum channels which significantly improves dalai s prefactor from the order of subexponential to polynomial furthermore the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of o log n n indicating our sphere packing bound is almost exact in the high rate regime finally for a special class of symmetric classical quantum channels we can completely characterize its optimal error probability without the constant composition code assumption the main technical contributions are two converse hoeffding bounds for quantum hypothesis testing and the saddle point properties of error exponent functions

sharp estimates of unimodular multipliers on frequency decomposition spaces abstract this paper is devoted to give a general method for the study of unimodular multipliers on certain function spaces defined by decomposition on the frequency plane these spaces include modulation spaces modulation spaces and homogeneous and inhomogeneous besov spaces we give a complete characterization of the fourier multipliers on these function spaces and also give a characterization of unimodular multipliers under some mild assumptions as applications we obtain some sharp boundedness properties of unimodular fourier multipliers between these function spaces we also obtain the asymptotic estimates for certain free dispersive semigroups

monotonicity of a relative renyi entropy we show that a recent definition of relative renyi entropy is monotone under completely positive trace preserving maps this proves a recent conjecture of muller lennert et al on quantum renyi entropies a new definition some properties j math phys 54 122203 2013 e print arxiv 1306 3142v1 see also e print arxiv 1306 3142

unimodular fourier multipliers for modulation spaces abstract we investigate the boundedness of unimodular fourier multipliers on modulation spaces surprisingly the multipliers with general symbol e i where 0 2 are bounded on all modulation spaces but in general fail to be bounded on the usual l p spaces as a consequence the phase space concentration of the solutions to the free schrodinger and wave equations are preserved as a byproduct we also obtain boundedness results on modulation spaces for singular multipliers sin for 0

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

modulation spaces and non linear hartree type equations abstract we study the cauchy problem for hartree equation with cubic convolution nonlinearity f u k u 2 u under a specified condition on potential k with cauchy data in modulation spaces m p q r n we establish global well posedness results in m p p r n with 1 p 2 n n when k x x r 0 m i n 2 n 2 in m p q r n with 1 q p 2 when k m 1 r n

concavity of certain matrix trace and norm functions ii abstract we refine epstein s method to prove joint concavity convexity of matrix trace functions of lieb type tr f a p 1 2 b q a p 1 2 and symmetric anti norm functions of the form f a p b q where and are positive linear maps is an operator mean and f x with a certain power is an operator monotone function on 0 moreover the variational method of carlen frank and lieb is extended to general non decreasing convex concave functions on 0 so that we prove joint concavity convexity of more trace functions of lieb type

α modulation spaces ii derivative nls abstract the current paper is a continuation of our early work 11 we use the techniques and properties of modulation spaces developed in 11 to study the cauchy problem for the derivative nonlinear schrodinger equations by applying some new type of inhomogeneous linear estimate to control the high low frequency interaction appearing from the nonlinearity we obtain the critical regularity index for the global well posedness with small initial data in modulation spaces for one type of derivative nonlinear schrodinger equations we also study the generalized derivative schrodinger equations

ill posedness for the navier stokes equations in critical besov spaces b q 1 abstract we study the cauchy problem for the incompressible navier stokes equations in two and higher spatial dimensions 0 1 u t u u u p 0 div u 0 u 0 x u 0 for arbitrarily small 0 we show that the solution map u 0 u in critical besov spaces b q 1 q 1 2 is discontinuous at origin it is known that the navier stokes equation is globally well posed for small data in bmo 1 20 taking notice of the embedding b q 1 bmo 1 q 2 we see that for sufficiently small 0 u 0 b q 1 q 2 can guarantee that 0 1 has a unique global solution in bmo 1 however this solution is instable in b q 1 and the solution can have an inflation in b q 1 for certain initial data

some inequalities for unitarily invariant norm abstract we shall prove the inequalities a b a b aa bb 2 ab a b a b 4 ab for all n n complex matrices a b and all unitarily invariant norms if further a b are positive definite it is proved that j 1 k j a b j 1 k j a 1 b 1 k n 0 1 where denotes the operator means considered by kubo and ando and j x 1 j n denote the eigenvalues of x arranged in the decreasing order whenever these all are real a number of inequalities are obtained as applications

the cauchy problem in local spaces for the complex ginzburg landau equation i compactness methods we continue the study of the initial value problem for the complex ginzburg landau equation with a 0 b 0 g 0 in initiated in a previous paper i we treat the case where the initial data and the solutions belong to local uniform spaces more precisely to spaces of functions satisfying local regularity conditions and uniform bounds in local norms but no decay conditions or arbitrarily weak decay conditions at infinity in in i we used compactness methods and an extended version of recent local estimates 3 and proved in particular the existence of solutions globally defined in time with local regularity of the initial data corresponding to the spaces l r for r 2 or h 1 here we treat the same problem by contraction methods this allows us in particular to prove that the solutions obtained in i are unique under suitable subcriticality conditions and to obtain for them additional regularity properties and uniform bounds the method extends some of those previously applied to the nonlinear heat equation in global spaces to the framework of local uniform spaces

gevrey regularity of mild solutions to the parabolic elliptic system of drift diffusion type in critical besov spaces abstract this paper is devoted to studying analyticity of mild solutions to the parabolic elliptic system of drift diffusion type with small initial data in critical besov spaces more precisely by using multilinear singular integrals and fourier localization argument we show that global in time mild solutions are gevrey regular i e e t 1 v e t 1 w l t b p r 2 n p r n l t 1 b p r n p r n for all 1 p 2 n and 1 r where 1 is the fourier multiplier whose symbol is given by 1 i 1 n i as a corollary we obtain decay estimates of mild solutions in critical besov spaces without using cumbersome recursive estimation of higher order derivatives

concavity of the auxiliary function appearing in quantum reliability function reliability functions characterize the asymptotic behavior of the error probability for transmission of data on a channel holevo introduced the quantum channel and gave an expression for a random coding lower bound involving an auxiliary function holevo ogawa and nagaoka conjectured that this auxiliary function is concave here we give a proof of this conjecture

the cauchy problem in local spaces for the complex ginzburg landau equation ii contraction methods we continue the study of the initial value problem for the complex ginzburg landau equation with a 0 b 0 g 0 in initiated in a previous paper i we treat the case where the initial data and the solutions belong to local uniform spaces more precisely to spaces of functions satisfying local regularity conditions and uniform bounds in local norms but no decay conditions or arbitrarily weak decay conditions at infinity in in i we used compactness methods and an extended version of recent local estimates 3 and proved in particular the existence of solutions globally defined in time with local regularity of the initial data corresponding to the spaces l r for r 2 or h 1 here we treat the same problem by contraction methods this allows us in particular to prove that the solutions obtained in i are unique under suitable subcriticality conditions and to obtain for them additional regularity properties and uniform bounds the method extends some of those previously applied to the nonlinear heat equation in global spaces to the framework of local uniform spaces

a characterization of modulation spaces by symplectic rotations abstract this note contains a new characterization of modulation spaces m m p r n 1 p by symplectic rotations precisely instead to measure the time frequency content of a function by using translations and modulations of a fixed window as building blocks we use translations and metaplectic operators corresponding to symplectic rotations technically this amounts to replace in the computation of the m m p r n norm the integral in the time frequency plane with an integral on r n u 2 n r with respect to a suitable measure u 2 n r being the group of symplectic rotations more conceptually we are considering a sort of polar coordinates in the time frequency plane to have invariance under symplectic rotations we choose a gaussian as suitable window function we also provide a similar and easier characterization with the group u 2 n r being reduced to the n dimensional torus t n

gabor representations of evolution operators we perform a time frequency analysis of fourier multipliers and more generally pseudodierential operators with symbols of gevrey analytic and ultra analytic type as an application we show that gabor frames which provide optimally sparse decompositions for schrodinger type propagators 14 surprisingly reveal to be an equally ecient tool for representing solutions to hyperbolic and parabolic type dierential equations with constant coecients in fact the gabor matrix represen tation of the corresponding propagator displays super exponential decay away from the diagonal