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

nonexistence of global solutions to critical semilinear wave equations in exterior domain in high dimensions abstract the aim of this paper is to study the long time behavior of solutions to the initial boundary value problem of critical semilinear wave equations with small data in exterior domain in high dimensions n 5 we prove that solutions cannot exist globally in time the novelty is that we first use a cutoff function to transform the exterior problem to a cauchy problem then we divide the time into two intervals in one time interval we establish an energy estimate to control the error terms while in the other interval the asymptotic behavior of two test functions is used furthermore we obtain the upper bound of the lifespan by following the idea of zhou and han 2014

lifespan of solutions to semilinear damping wave equations in de sitter spacetime abstract this paper studies blowup phenomena for two kind of initial value problem of semilinear damping wave equations in de sitter spacetime the first problem comes from the latter problem of conjecture of strauss in de sitter spacetime and the second problem arises from the latter problem of conjecture of glassey in de sitter spacetime we give blowup and lifespan of solutions on those two problems comparing with non damping case established in yan 2018 one can see that the damping term with a positive bounded time dependent coefficient does not effect blowup of solutions meanwhile we establish a new ode blowup result concerning damping term

blow up for a weakly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities abstract in this work we study the blow up of solutions of a weakly coupled system of damped semilinear wave equations in the scattering case with power nonlinearities we apply an iteration method to study both the subcritical case and the critical case in the subcritical case our approach is based on lower bounds for the space averages of the components of local solutions in the critical case we use the slicing method and a couple of auxiliary functions recently introduced by wakasa yordanov to modify the definition of the functionals with the introduction of weight terms in particular we find as critical curve for the pair p q of the exponents in the nonlinear terms the same one as for the weakly coupled system of semilinear wave equations with power nonlinearities

an example of blow up for the complex kdv equation and existence beyond the blow up the kdv equation with smooth complex initial data that blows up in a finite amount of time is studied first examples are given using the elliptic solutions of airault mckean and moser second the kdv flow of spectral coordinates is described showing that blow up occurs when eigenvalues run to infinity in finite time but eigenvalue collisions are harmless and do not cause a singularity eigenvalue collisions and blow up are studied numerically finally the solutions are continued through the singularity by continuing the coordinate flows on the appropriate riemann surface

lifespan of semilinear wave equation with scale invariant dissipation and mass and sub strauss power nonlinearity abstract in this paper we study the blow up of solutions for semilinear wave equations with scale invariant dissipation and mass in the case in which the model is somehow wave like a strauss type critical exponent is determined as the upper bound for the exponent in the nonlinearity in the main theorems two blow up results are obtained for the subcritical case and for the critical case respectively in both cases an upper bound lifespan estimate is given

improved kato s lemma on ordinary differential inequality and its application to semilinear wave equations abstract we are interested in the upper bound of the lifespan of solutions of semilinear wave equations for the sub critical case in high dimensions it has been believed that the basic tools of its analysis are kato s lemma on ordinary differential inequalities and the rescaling argument in the functional method but there is a small lack of delicate analysis and no published paper about this here we give a simple alternative proof by means of improved kato s lemma without any rescaling argument

lifespan estimates of solutions to quasilinear wave equations with scattering damping abstract the formation of singularity of solutions to the quasilinear wave equations with scattering damping and divergence form nonlinearities is investigated the upper bound lifespan estimates of solutions to the problems with variable coefficient in the critical case and constant coefficient in the sub critical case are established by applying the rescaled test function method and iterative method

weighted strichartz estimates and global existence for semilinear wave equations in this paper we prove a sharp global existence theorem in all dimensions for nonlinear wave equations with power type nonlinearities the proof is based on a weighted strichartz estimate involving powers of the lorentz distance

finite time blow up for critical wave equations in high dimensions we prove that solutions to the critical wave equation 1 1 with dimension n 4 can not be global if the initial values are positive somewhere and nonnegative this completes the solution to the famous strauss conjecture about semilinear wave equations of the form u t2u u p 0 the rest of the cases the lower dimensional case n 3 and the sub or super critical cases were settled many years earlier by the work of several authors

on the theorem of uniform recovery of random sampling matrices we consider two theorems from the theory of compressive sensing mainly a theorem concerning uniform recovery of random sampling matrices where the number of samples needed in order to recover an s sparse signal from linear measurements with high probability is known to be m s ln s 3 lnn we present new and improved constants together with what we consider to be a more explicit proof a proof that also allows for a slightly larger class of m n matrices by considering what is called effective sparsity we also present a condition on the so called restricted isometry constants s ensuring sparse recovery via l 1 minimization we show that is sufficient and that this can be improved further to almost allow for a sufficient condition of the type

on the existence and nonexistence of global solutions for certain semilinear exterior problems with nontrivial robin boundary conditions abstract we consider three types of semilinear equations elliptic parabolic and hyperbolic posed in the n dimensional exterior domain r n d where n 2 and d is the closed unit ball in r n a nontrivial robin boundary condition is imposed on the boundary of d using a test function approach with judicious choices of the test functions we show that the considered problems share a common critical behavior we discuss separately the cases n 2 and n 3 moreover in the case n 3 the dependence of the critical exponent on initial data is discussed to the best our knowledge the study of the critical behavior in an exterior domain with a nontrivial robin boundary condition has never been studied in the literature

risk bounds of learning processes for levy processes leevy processes refer to a class of stochastic processes for example poisson processes and brownian motions and play an important role in stochastic processes and machine learning therefore it is essential to study risk bounds of the learning process for time dependent samples drawn from a leevy process or briefly called learning process for leevy process it is noteworthy that samples in this learning process are not independently and identically distributed i i d therefore results in traditional statistical learning theory are not applicable or at least cannot be applied directly because they are obtained under the sample i i d assumption in this paper we study risk bounds of the learning process for time dependent samples drawn from a leevy process and then analyze the asymptotical behavior of the learning process in particular we first develop the deviation inequalities and the symmetrization inequality for the learning process by using the resultant inequalities we then obtain the risk bounds based on the covering number finally based on the resulting risk bounds we study the asymptotic convergence and the rate of convergence of the learning process for leevy process meanwhile we also give a comparison to the related results under the sample i i d assumption

blow up and critical exponents for nonlinear hyperbolic equations we prove nonexistence results for the cauchy problem for the abstract hyperbolic equation in a banach space x utt f u t 0 u 0 u0 ut 0 u1 where f x r is a c1 function several applications to the second and higher order hyperbolic equations with local and nonlocal nonlinearities are presented we also describe an approach to kato s and john s critical exponents for the semilinear equations ut u b x t u p p 1 which are responsible for phenomena of stability unstability blow up and asymptotic behaviour

improved bounds for the nystrom method with application to kernel classification we develop two approaches for analyzing the approximation error bound for the nystrom method that approximates a positive semidefinite psd matrix by sampling a small set of columns one based on a concentration inequality for integral operators and one based on random matrix theory we show that the approximation error measured in the spectral norm can be improved from o n m to o n m1 in the case of large eigengap where n is the total number of data points m is the number of sampled data points and 0 1 2 is a positive constant that characterizes the eigengap when the eigenvalues of the kernel matrix follow a p power law our analysis based on random matrix theory further improves the bound to o n mp 1 under an incoherence assumption we present a kernel classification approach based on the nystrom method and derive its generalization performance using the improved bound we show that when the eigenvalues of the kernel matrix follow a p power law we can reduce the number of support vectors to n2p p2 1 which is sublinear in n when p 1 2 without seriously sacrificing its generalization performance

characteristic kernels and infinitely divisible distributions we connect shift invariant characteristic kernels to infinitely divisible distributions on rd characteristic kernels play an important role in machine learning applications with their kernel means to distinguish any two probability measures the contribution of this paper is twofold first we show using the levy khintchine formula that any shift invariant kernel given by a bounded continuous and symmetric probability density function pdf of an infinitely divisible distribution on rd is characteristic we mention some closure properties of such characteristic kernels under addition pointwise product and convolution second in developing various kernel mean algorithms it is fundamental to compute the following values i kernel mean values mp x x and ii kernel mean rkhs inner products mp mq h for probability measures p q if p q and kernel k are gaussians then the computation of i and ii results in gaussian pdfs that are tractable we generalize this gaussian combination to more general cases in the class of infinitely divisible distributions we then introduce a conjugate kernel and a convolution trick so that the above i and ii have the same pdf form expecting tractable computation at least in some cases as specific instances we explore stable distributions and a rich class of generalized hyperbolic distributions where the laplace cauchy and student s t distributions are included

a shift in the strauss exponent for semilinear wave equations with a not effective damping abstract in this note we study the global existence of small data solutions to the cauchy problem for the semilinear wave equation with a not effective scale invariant damping term namely v t t v 2 1 t v t v p v 0 x v 0 x v t 0 x v 1 x where p 1 n 2 we prove blow up in finite time in the subcritical range p 1 p 2 n and existence theorems for p p 2 n n 2 3 in this way we find the critical exponent for small data solutions to this problem our results lead to the conjecture p 2 n p 0 n 2 for n 2 where p 0 n is the strauss exponent for the classical semilinear wave equation with power nonlinearity

long time existence for the quadratic wave equation associated to the harmonic oscillator abstract we show the solution for the quadratic wave equation associated to the harmonic oscillator exists over a longer time interval than the one given by local existence theory the key point is that the nonlinearity is quadratic so that we can estimate the small divisor and perform a normal form process

lifespan of solutions to the damped wave equation with a critical nonlinearity abstract in the present paper we study a lifespan of solutions to the cauchy problem for semilinear damped wave equations dw t 2 u u t u f u t x 0 t e r n u 0 x e u 0 x x r n t u 0 x e u 1 x x r n where n 1 f u u p 1 u or u p p 1 e 0 is a small parameter and u 0 u 1 is a given initial data the main purpose of this paper is to prove that if the nonlinear term is f u u p and the nonlinear power is the fujita critical exponent p p f 1 2 n then the upper estimate to the lifespan is estimated by t e exp c e p for all e 0 1 and suitable data u 0 u 1 without any restriction on the spatial dimension our proof is based on a test function method utilized by zhang 35 we also prove a sharp lower estimate of the lifespan t e to dw in the critical case p p f

the equation utt δu u p for the critical value of p the equation u tt u u p is considered in two and three space dimensions smooth cauchy data of compact support are given at t 0 for the case of three space dimensions john has shown that solutions with sufficiently small data exist globally in time if but that small data solutions blow up in finite time if glassey has shown the two dimensional case is similar this paper shows that small data solutions blow up in finite time when p is the critical value in three dimensions and in two

concentration of norms and eigenvalues of random matrices we prove concentration results for lpn operator norms of rectangular random matrices and eigenvalues of self adjoint random matrices the random matrices we consider have bounded entries which are independent up to a possible self adjointness constraint our results are based on an isoperimetric inequality for product spaces due to talagrand

sums of random symmetric matrices and quadratic optimization under orthogonality constraints let b i be deterministic real symmetric m m matrices and i be independent random scalars with zero mean and of order of one e g xi i sim mathcal n 0 1 we are interested to know under what conditions typical norm of the random matrix s n sum i 1 n xi i b i is of order of 1 an evident necessary condition is bf e s n 2 preceq o 1 i which essentially translates to sum i 1 n b i 2 preceq i a natural conjecture is that the latter condition is sufficient as well in the paper we prove a relaxed version of this conjecture specifically that under the above condition the typical norm of s n is leq o 1 m 1 over 6 rm prob s n omega m 1 6 leq o 1 exp o 1 omega 2 for all 0 we outline some applications of this result primarily in investigating the quality of semidefinite relaxations of a general quadratic optimization problem with orthogonality constraints rm opt max limits x j in bf r m times m left f x 1 ldots x k x jx j rm t i j 1 ldots k right where f is quadratic in x x 1 x k we show that when f is convex in every one of x j a natural semidefinite relaxation of the problem is tight within a factor slowly growing with the size m of the matrices x j rm opt leq rm opt sdp leq o 1 m 1 3 ln k rm opt

combined effects of two nonlinearities in lifespan of small solutions to semi linear wave equations this paper investigates the combined effects of two distinctive power type nonlinear terms with parameters p q 1 in the lifespan of small solutions to semi linear wave equations we determine the full region of p q to admit global existence of small solutions at least for spatial dimensions n 2 3 moreover for many p q when there is no global existence we obtain sharp lower bound of the lifespan which is of the same order as the upper bound of the lifespan

life span of solutions to semilinear wave equation with time dependent critical damping for specially localized initial data this paper is concerned with the blowup phenomena for initial value problem of semilinear wave equation with critical time dependent damping term dw begin aligned left begin array ll partial t 2 u x t delta u x t dfrac mu 1 t partial t u x t u x t p x t in mathbb r n times 0 t u x 0 varepsilon f x x in mathbb r n partial t u x 0 varepsilon g x x in mathbb r n end array right end aligned where n in mathbb n mu in 0 frac n 2 n 2 n 2 and varepsilon 0 is a parameter describing the smallness of initial data given data f g are compactly supported in a small area inside of the unit ball the result is the upper bound of lifespan of solution u with respects to the small parameter varepsilon when p f n le p le p 0 n mu where p f n denotes the fujita exponent for the nonlinear heat equations and p 0 n denotes the strauss exponent for nonlinear wave equation dw in n dimension with mu 0 consequently by connecting the result of d abbicco lucente reissig j differ equ 259 5040 5073 2015 our result clarifies the threshold exponent p 0 n mu for dividing blowup phenomena and global existence of small solutions when n 3 the crucial idea is to construct suitable test functions satisfying the conjugate linear equation partial t 2 phi delta phi partial t frac mu 1 t phi 0 of dw including the gauss hypergeometric functions note that the construction of test functions is different from zhou han commun partial differ equ 39 439 451 2014

nonexistence of global solutions to semilinear wave equations in high dimensions on considere le probleme de cauchy pour l equation d onde semi lineaire 2 t 2 u x t u x t p p 1 u x o f x u t x o g x avec f g c o r n

global nonexistence results for a class of hyperbolic systems abstract our concern in this paper is to prove blow up results to the non autonomous nonlinear system of wave equations u t t u a t x v p v t t v b t x u q t 0 x r n in any space dimension we show that a curve f p q 0 depending on the space dimension on the exponents p q and on the behavior of the functions a t x and b t x exists such that all nontrivial solutions to the above system blow up in a finite time whenever f p q 0 our method of proof uses some estimates developed by galaktionov and pohozaev in 11 for a single non autonomous wave equation enabling us to obtain a system of ordinary differential inequalities from which the desired result is derived our result generalizes some important results such as the ones in del santo et al 1996 12 and galaktionov and pohozaev 2003 11 the advantage here is that our result applies to a wide variety of problems

long time existence for small amplitude semilinear wave equations we prove that a certain class of semilinear wave equations has global solutions if the initial data is small these existence results rely on mixed norm angular radial space time estimates more specifically we consider power nonlinearities u f p u where f p u u p and 2 t denotes the d alembertian on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i in 1979 john investigated this equation for small initial data in dimension n 3 he proved that if p 1 2 then there is a global solution whereas for p n dimensions and the critical power p c above which global solutions exist should be the positive root of n 1 p 2 c n 1 p c 2 0 sideris proved that for all n the solutions can blow up if p p c recently zhou proved that for n 4 one has global solutions if p p c 2 here we prove that for n 8 one has global solutions if p p c furthermore we proved that for all n one has global solutions if p p c and the initial data are spherically symmetric in the radial case we use l p estimates derived from the explicit form of the fundamental solution using inequalities from classical analysis in the nonradial case we write the fundamental solution in polar coordinates and use fourier integral and special function estimates to handle the angular part

global existence for semilinear wave equations with the critical blow up term in high dimensions abstract we are interested in almost global existence cases in the general theory for nonlinear wave equations which are caused by critical exponents of nonlinear terms such situations can be found in only three cases in the theory cubic terms in two space dimensions quadratic terms in three space dimensions and quadratic terms including a square of unknown functions itself in four space dimensions except for the last case criteria to classify nonlinear terms into the almost global or global existence case are well studied and known to be so called null condition and non positive condition our motivation of this work is to find such a kind of the criterion in four space dimensions in our previous paper an example of the non single term for the almost global existence case is introduced in this paper we show an example of the global existence case these two examples have nonlinear integral terms which are closely related to derivative loss due to high dimensions but it may help us to describe the final form of the criterion

the sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions the final open part of strauss conjecture on semilinear wave equations was the blow up theorem for the critical case in high dimensions this problem was solved by yordanov and zhang 2006 18 or zhou 2007 21 independently but the estimate for the lifespan the maximal existence time of solutions was not clarified in both papers in this paper we refine their theorems and introduce a new iteration argument to get the sharp upper bound of the lifespan as a result with the sharp lower bound by li and zhou 1995 10 the lifespan t e of solutions of utt u u2 in r4 0 with the initial data u x 0 ef x ut x 0 eg x of a small parameter e 0 compactly supported smooth functions f and g has an estimate exp ce 2 t e exp ce 2 where c and c are positive constants depending only on f and g this upper bound has been known to be the last open optimality of the general theory for fully nonlinear wave equations

finite time blow up to critical semilinear wave equation outside the ball in 3 d abstract in this paper we study the initial boundary value problem of critical semilinear wave equation with small initial data in 3 d blow up result will be established no matter how small the data are we find a radial and positive solution of the corresponding linear problem then our blow up result follows from the idea of zhou and han 2014 what is more we get the upper bound of the lifespan estimate

finite time blowup of solutions to semilinear wave equation in an exterior domain abstract we consider the initial boundary value problem of semilinear wave equation with nonlinearity u p in exterior domain in r n n 3 especially the lifespan of blowup solutions with small initial data are studied the result gives upper bounds of lifespan which is essentially the same as the cauchy problem in r n at least in the case n 4 their estimates are sharp in view of the work by zha zhou 21 the idea of the proof is to use special solutions to linear wave equation with dirichlet boundary condition which are constructed via an argument based on wakasa yordanov 15

remarks on a weighted energy estimate and its application to nonlinear wave equations in one space dimension abstract a weighted energy estimate with tangential derivatives on the light cone is applied for the cauchy problem of semilinear wave equations with the null conditions in one space dimension the well posedness and lifespan of the solutions are considered based on the vector field method

moment inequalities for sums of random matrices and their applications in optimization in this paper we consider various moment inequalities for sums of random matrices which are well studied in the functional analysis and probability theory literature and demonstrate how they can be used to obtain the best known performance guarantees for several problems in optimization first we show that the validity of a recent conjecture of nemirovski is actually a direct consequence of the so called non commutative khintchine s inequality in functional analysis using this result we show that an sdp based algorithm of nemirovski which is developed for solving a class of quadratic optimization problems with orthogonality constraints has a logarithmic approximation guarantee this improves upon the polynomial approximation guarantee established earlier by nemirovski furthermore we obtain improved safe tractable approximations of a certain class of chance constrained linear matrix inequalities secondly we consider a recent result of delage and ye on the so called data driven distributionally robust stochastic programming problem one of the assumptions in the delage ye result is that the underlying probability distribution has bounded support however using a suitable moment inequality we show that the result in fact holds for a much larger class of probability distributions given the close connection between the behavior of sums of random matrices and the theoretical properties of various optimization problems we expect that the moment inequalities discussed in this paper will find further applications in optimization

blow up theorem for semilinear wave equations with non zero initial position abstract one of the features of solutions of semilinear wave equations can be found in blow up results for non compactly supported data in spite of finite propagation speed of the linear wave we have no global in time solution for any power nonlinearity if the spatial decay of the initial data is weak this was first observed by asakura 1986 2 finding out a critical decay to ensure the global existence of the solution but the blow up result is available only for zero initial position having positive speed in this paper the blow up theorem for non zero initial position by uesaka 2009 22 is extended to higher dimensional case and the assumption on the nonlinear term is relaxed to include an example u p 1 u moreover the critical decay of the initial position is clarified by example

commuting linear differential expressions we consider the question when do two ordinary linear differential expressions commute it turns out that the set of all expressions which commute with a given one form a commutative ring here we study the algebraic structure of these rings as an application a complete characterization of normal differential expressions is obtained

characterisation of the factors of quasi differential expressions a necessary and sufficient condition for a general scalar quasi differential expression of order n to be factorisable into a product of expressions of order n k and k for any 0 k n is given the factors are characterised completely in terms of elements of the null space of the expression and its adjoint the results obtained extend existing results due to both polya and zettl from the case of classical linear differential expressions to quasi differential expressions

from random matrix theory to coding theory volume of a metric ball in unitary group volume estimates of metric balls in manifolds find diverse applications in information and coding theory in this paper new results for the volume of a metric ball in unitary group are derived via tools from random matrix theory the first result is an integral representation of the exact volume which involves a toeplitz determinant of bessel functions a simple but accurate limiting volume formula is then obtained by invoking szeg s strong limit theorem for large toeplitz matrices the derived asymptotic volume formula enables analytical evaluation of some coding theoretic bounds of unitary codes in particular the gilbert varshamov lower bound and the hamming upper bound on the cardinality as well as the resulting bounds on code rate and minimum distance are derived moreover bounds on the scaling law of code rate are found finally a closed form bound on the diversity sum relevant to unitary space time codes is obtained which was only computed numerically in the literature

matrix concentration inequalities via the method of exchangeable pairs this paper derives exponential concentration inequalities and polynomial moment inequalities for the spectral norm of a random matrix the analysis requires a matrix extension of the scalar concentration theory developed by sourav chatterjee using stein s method of exchangeable pairs when applied to a sum of independent random matrices this approach yields matrix generalizations of the classical inequalities due to hoeffding bernstein khintchine and rosenthal the same technique delivers bounds for sums of dependent random matrices and more general matrix valued functions of dependent random variables

almost global solutions of semilinear wave equations with the critical exponent in high dimensions abstract we are interested in the almost global in time existence of classical solutions in the general theory for nonlinear wave equations all the three such cases are known to be sharp due to blow up results in the critical case for model equations however it is known that we have a possibility to get the global in time existence for two of them in low space dimensions if the nonlinear term is of derivatives of the unknown function and satisfies the so called null condition or non positive condition but another one for the quadratic term in four space dimensions is out of the case as the nonlinear term should include a square of the unknown function itself in this paper we get one more example guaranteeing the sharpness of the almost global in time existence in four space dimensions it is also the first example of the blow up of classical solutions for non single and indefinitely signed term in high dimensions such a term arises from the neglect of derivative loss factors in duhamel s formula for positive and single nonlinear term this fact may help us to describe a criterion to get the global in time existence in this critical situation

the lifespan of solutions to nonlinear systems of a high dimensional wave equation abstract in this work we study the lifespan of solutions to a p q system in the higher dimensional case n 4 a suitable local existence curve in the p q plane is found the curve characterizes the local solutions in sobolev space h s with s 0 further some lower and upper bounds of the lifespan of classical solutions are found too this work is an extension of work geometric optics and related topics progress in nonlinear differential equations and their applications vol 32 birkhauser basel 1997 pp 117 140 where a suitable global existence small data curve is studied in the subcritical case we give almost precise results for the lower bounds of the lifespan by using a suitable weighted strichartz estimate for the higher dimensional wave equation

new blow up results for nonlinear boundary value problems in exterior domains abstract we consider exterior boundary value problems with nontrivial boundary conditions including three types of semilinear equations heat equation wave equation and damped wave equation using a test function approach with a choice of a suitable family of test functions we obtain the critical exponents in the sense of fujita for the considered problems our obtained results close some open questions from the literature

almost global existence for semilinear wave equations with mixed nonlinearities in four space dimensions abstract we study the long time existence for the small amplitude semilinear wave equations with mixed nonlinearities of the form c 1 u t p c 2 u 2 with p 2 when the spatial dimension is four by exploiting the local energy estimates and their recent variants we prove the almost global existence up to exp c e 2 which is sharp in general for the case p 2 3 due to technical reason we need to assume the initial data to be radial

blow up of solutions to critical semilinear wave equations with variable coefficients abstract we verify the critical case p p 0 n of strauss conjecture 30 concerning the blow up of solutions to semilinear wave equations with variable coefficients in r n where n 2 the perturbations of laplace operator are assumed to be smooth and decay exponentially fast at infinity we also obtain a sharp lifespan upper bound for solutions with compactly supported data when p p 0 n the unified approach to blow up problems in all dimensions combines several classical ideas in order to generalize and simplify the method of zhou 43 and zhou han 45 exponential eigenfunctions of the laplacian 37 are used to construct the test function q for linear wave equation with variable coefficients and john s method of iterations 13 is augmented with the slicing method of agemi kurokawa and takamura 1 for lower bounds in the critical case

matrix infinitely divisible series tail inequalities and their applications in this paper we study tail inequalities of the largest eigenvalue of a matrix infinitely divisible i d series which is a finite sum of fixed matrices weighted by i d random variables we obtain several types of tail inequalities including bennett type and bernstein type inequalities this allows us to further bound the expectation of the spectral norm of a matrix i d series moreover by developing a new lower bound function for q s s 1 log s 1 s that appears in the bennett type inequality we derive a tighter tail inequality of the largest eigenvalue of the matrix i d series than the bernstein type inequality when the matrix dimension is high the resulting lower bound function is of independent interest and can improve any bennett type concentration inequality that involves the function q s the class of i d probability distributions is large and includes gaussian and poisson distributions among many others therefore our results encompass the existing work on matrix gaussian series as a special case lastly we show that the tail inequalities of a matrix i d series have applications in several optimization problems including the chance constrained optimization problem and the quadratic optimization problem with orthogonality constraints in addition we also use the resulting tail bounds to show that random matrices constructed from i d random variables satisfy the restricted isometry property rip when it acts as a measurement matrix in compressed sensing

strong converse for identification via quantum channels we present a simple proof of the strong converse for identification via discrete memoryless quantum channels based on a novel covering lemma the new method is a generalization to quantum communication channels of ahlswede s 1979 1992 approach to classical channels it involves a development of explicit large deviation estimates to the case of random variables taking values in self adjoint operators on a hilbert space this theory is presented separately in an appendix and we illustrate it by showing its application to quantum generalizations of classical hypergraph covering problems

robust estimates of covariance matrices in the large dimensional regime this paper studies the limiting behavior of a class of robust population covariance matrix estimators originally due to maronna in 1976 in the regime where both the number of available samples and the population size grow large using tools from random matrix theory we prove that for sample vectors made of independent entries having some moment conditions the difference between the sample covariance matrix and a scaled version of such robust estimator tends to zero in spectral norm almost surely this result can be applied to various statistical methods arising from random matrix theory that can be made robust without altering their first order behavior