singular hyperbolic attractors are chaotic we prove that a singular hyperbolic attractor of a 3 dimensional flow is chaotic in two different strong senses first the flow is expansive if two points remain close at all times possibly with time reparametrization then their orbits coincide second there exists a physical or sinai ruelle bowen measure supported on the attractor whose ergodic basin covers a full lebesgue volume measure subset of the topological basin of attraction moreover this measure has absolutely continuous conditional measures along the center unstable direction is a u gibbs state and is an equilibrium state for the logarithm of the jacobian of the time one map of the flow along the strong unstable direction this extends to the class of singular hyperbolic attractors the main elements of the ergodic theory of uniformly hyperbolic or axiom a attractors for flows in particular these results can be applied i to the flow defined by the lorenz equations ii to the geometric lorenz flows iii to the attractors appearing in the unfolding of certain resonant double homoclinic loops iv in the unfolding of certain singular cycles and v in some geometrical models which are singular hyperbolic but of a different topological type from the geometric lorenz models in all these cases the results show that these attractors are expansive and have physical measures which are u gibbs states

statistical properties for flows with unbounded roof function including the lorenz attractor for geometric lorenz attractors including the classical lorenz attractor we obtain a greatly simplified proof of the central limit theorem which applies also to the more general class of codimension two singular hyperbolic attractors we also obtain the functional central limit theorem and moment estimates as well as iterated versions of these results a consequence is deterministic homogenisation convergence to a stochastic differential equation for fast slow dynamical systems whenever the fast dynamics is singularly hyperbolic of codimension two

a vector valued almost sure invariance principle for hyperbolic dynamical systems we prove an almost sure invariance principle approximation by d dimensional brownian motion for vector valued holder observables of large classes of nonuniformly hyperbolic dynamical systems these systems include axiom a dieomorphisms and flows as well as systems modelled by young towers with moderate tail decay rates in particular the position variable of the planar periodic lorentz gas with finite horizon approximates a 2 dimensional brownian motion

from rates of mixing to recurrence times via large deviations abstract a classic approach in dynamical systems is to use particular geometric structures to deduce statistical properties for example the existence of invariant measures with stochastic like behaviour such as large deviations or decay of correlations such geometric structures are generally highly non trivial and thus a natural question is the extent to which this approach can be applied in this paper we show that in many cases stochastic like behaviour itself implies that the system has certain non trivial geometric properties which are therefore necessary as well as sufficient conditions for the occurrence of the statistical properties under consideration as a by product of our techniques we also obtain some new results on large deviations for certain classes of systems which include viana maps and multidimensional piecewise expanding maps

the wave equation approach for solving inverse eigenvalue problems of a multi connected region in r3 with robin conditions the asymptotic expansion for t of the trace of the wave kernel email protected t email protected u 1exp email protected 1 2 u where m u u 1 are the eigenvalues of the negative laplacian email protected email protected 3 b 1 x b 2 in the x 1 x 2 x 3 space where i 1 and

limit of the smallest eigenvalue of a large dimensional sample covariance matrix in this paper the authors show that the smallest if p leq n or the p n 1 th smallest if p n eigenvalue of a sample covariance matrix of the form 1 n xx tends almost surely to the limit 1 sqrt y 2 as n rightarrow infty and p n rightarrow y in 0 infty where x is a p times n matrix with iid entries with mean zero variance 1 and fourth moment finite also as a by product it is shown that the almost sure limit of the largest eigenvalue is 1 sqrt y 2 a known result obtained by yin bai and krishnaiah the present approach gives a unified treatment for both the extreme eigenvalues of large sample covariance matrices

stability of mixing and rapid mixing for hyperbolic flows we obtain general results on the stability of mixing and rapid mixing superpolynomial decay of correlations for hyperbolic flows amongst c r axiom a flows r 2 we show that there is a c 2 open c r dense set of flows for which each nontrivial hyperbolic basic set is rapid mixing this is the first general result on the stability of rapid mixing or even mixing for axiom a

equidistribution for nonuniformly expanding dynamical systems and application to the almost sure invariance principle let t m mt m m be a nonuniformly expanding dynamical system such as logistic or intermittent map let v m rdv m rd be an observable and vn n 1k 0v tkvn k 0n 1v tk denote the birkhoff sums given a probability measure on m we consider vn as a discrete time random process on the probability space m m in smooth ergodic theory there are various natural choices of such as the lebesgue measure or the absolutely continuous t invariant measure they give rise to different random processes we investigate relation between such processes we show that in a large class of measures it is possible to couple redefine on a new probability space every two processes so that they are almost surely close to each other with explicit estimates of closeness the purpose of this work is to close a gap in the proof of the almost sure invariance principle for nonuniformly hyperbolic transformations by melbourne and nicol

on the θ function of a riemannian manifold with boundary let be a compact riemannian manifold of dimension n with smooth boundary let 1 2 be the eigenvalues of the laplace beltrami operator with the boundary condition n 0 the associated function t n 1 exp n t has an asymptotic expansion of the form 4 t n 2 t a 0 a 1 t 1 2 a 2 t a 3 t 3 2 a 4 t 2 the values of a 0 a 1 are well known

theory of a curved planar waveguide with robin boundary conditions a model of a thin straight strip with a uniformly curved section and with boundary requirements zeroing at the edges a linear superposition of the wave function and its normal derivative robin boundary condition is analyzed theoretically within the framework of the linear schrodinger equation and is applied to the study of the processes in the bent magnetic multilayers superconducting films and metallic ferrite filled waveguides in particular subband thresholds of the straight and curved parts of the film are calculated and analyzed as a function of the robin parameter 1 lambda with lambda being an extrapolation length entering robin boundary condition for the arbitrary robin coefficients which are equal on the opposite interfaces of the strip and for all bend parameters the lowest mode energy of the continuously curved duct is always smaller than its straight counterpart accordingly the bound state below the fundamental propagation threshold of the straight arms always exists as a result of the bend in terms of the superconductivity language it means an increased critical temperature of the curved film compared to its straight counterpart localized level dependence on the parameters of the curve is investigated with its energy decreasing with increasing bend angle and decreasing bend radius conditions of the bound state existence for the different robin parameters on the opposite edges are analyzed too in particular it is shown that the bound state below the first transverse threshold of the straight arm always exists if the inner extrapolation length is not larger than the outer one in the opposite case there is a range of the bend parameters where the curved film cannot trap the wave and form the localized mode for example for the fixed bend radius the bound state emerges from the continuum at some nonzero bend angle that depends on the difference of the two lengths lambda at the opposite interfaces various transport properties of the film such as interference blockade of the current flow at some special energies is also discussed with the special attention being paid to the transformation from the dirichlet to the neumann case as the extrapolation length lambda sweeps the positive axis

higher dimensional inverse problem of the wave equation for a general multi connected bounded domain with a finite number of smooth mixed boundary conditions the spectral function m t j 1 exp it m j 1 2 where m j j 1 are the eigenvalues of the negative laplacian d 3 u 1 3 x u 2 in the x 1 x 2 x 3 space is studied for small t for a variety of bounded domains where

robust transitivity of singular hyperbolic attractors singular hyperbolicity is a weakened form of hyperbolicity that has been introduced for vector fields in order to allow non isolated singularities inside the non wandering set a typical example of a singular hyperbolic set is the lorenz attractor however in contrast to uniform hyperbolicity singular hyperbolicity does not immediately imply robust topological properties such as the transitivity in this paper we prove that on an open and dense subset of the space of c 1 vector fields of a compact manifold any singular hyperbolic attractors is robustly transitive

decay of correlations for slowly mixing flows we show that polynomial decay of correlations is prevalent for a class of nonuniformly hyperbolic flows these flows are the continuous time analogue of a class of nonuniformly hyperbolic dieomorphisms for which young proved polynomial decay of correlations roughly speaking in situations where the decay rate o 1 n has previously been proved for dieomorphisms we establish the decay rate o 1 t for flows applications include certain classes of semidispersing billiards as well as dispersing billiards with vanishing curvature in addition we obtain results for suspension flows with unbounded roof functions in particular the classical planar lorentz flow with a doubly periodic array of circular scatterers has decay rate 1 t as anticipated by physicists

generalized bounded variation and applications to piecewise monotonic transformations we prove the quasi compactness of the perron frobenius operator of piecewise monotonic transformations when the inverse of the derivative is holder continuous or more generally of bounded p variation

rapid mixing for the lorenz attractor and statistical limit laws for their time 1 maps we prove that every geometric lorenz attractor satisfying a strong dissipativity condition has superpolynomial decay of correlations with respect to the unique sinai ruelle bowen measure moreover we prove the central limit theorem and almost sure invariance principle for the time 1 map of the flow of such attractors in particular our results apply to the classical lorenz attractor

eigenvalues of the laplacian for the third boundary values problem an extension to higher dimensions soit r a r b 0 0 2 une coque spherique on suppose connu exactement le spectre de valeurs propres 0 1 2 3 j quand j pour le probleme 3 u 0 dans u r 1 u r a 0 u r 2 u r b 0 ou 1 et 2 sont des constantes positives on determine la geometrie de et les impedances 1 et 2

corrigendum in just relax convex programming methods for identifying sparse signals in noise mar 06 1030 1051 this correspondence closes a gap in the proof of the main lemma from the paper just relax convex programming methods for identifying sparse signals in noise 2006

on the singular values of gaussian random matrices this short note is about the singular value distribution of gaussian random matrices i e gaussian ensemble or ge of size n we present a new approach for deriving the p d f of the singular values directly from the singular value decomposition svd form which also takes advantage of the rotational invariance of ge and the lie algebra of the orthogonal group our method is direct and more general than the conventional approach that relies on the wishart ensemble and the combination of qr and cholesky decomposition directly based on this p d f and its interpretation by statistical mechanics we give the physics proof that in the thermodynamic limit n the singular value distribution satisfies the quadrant law similar to the celebrated semi circle lawestablished by wigner more than 40 years ago for the spectral distribution of gaussian orthogonal or unitary ensembles this quadrant law was also proved earlier and mathematically more rigorously by some authors based on probabilistic estimations and the moment method but not directly from the p d f formula 2001 elsevier science inc all rights reserved

an almost sure invariance principle for hilbert space valued martingales we obtain an almost sure approximation of a martingale with values in a real separable hilbert space h by a suitable h valued brownian motion

can one hear the dimension of a fractal we consider the spectrum of the laplacian in a bounded open domain of n with a rough boundary i e with possibly non integer dimension and we discuss a conjecture by m v berry generalizing weyl s conjecture then using ideas mark kac developed in his famous study of the drum we give upper and lower bounds for the second term of the expansion of the partition function the main thesis of the paper is to show that the relevant measure of the roughness of the boundary should be based on minkowski dimensions and on minkowski measures rather than on haussdorff ones

why simple shrinkage is still relevant for redundant representations shrinkage is a well known and appealing denoising technique introduced originally by donoho and johnstone in 1994 the use of shrinkage for denoising is known to be optimal for gaussian white noise provided that the sparsity on the signal s representation is enforced using a unitary transform still shrinkage is also practiced with nonunitary and even redundant representations typically leading to very satisfactory results in this correspondence we shed some light on this behavior the main argument in this work is that such simple shrinkage could be interpreted as the first iteration of an algorithm that solves the basis pursuit denoising bpdn problem while the desired solution of bpdn is hard to obtain in general we develop a simple iterative procedure for the bpdn minimization that amounts to stepwise shrinkage we demonstrate how the simple shrinkage emerges as the first iteration of this novel algorithm furthermore we show how shrinkage can be iterated turning into an effective algorithm that minimizes the bpdn via simple shrinkage steps in order to further strengthen the denoising effect

hearing the shape of a general convex domain abstract the spectral function t j 1 exp t j where j j 1 are the eigenvalues of the laplace operator i 1 2 x i 2 in the x 1 x 2 plane is studied for a general convex domain together with an impedance condition on a part of its boundary and another impedance condition on the remaining part of that boundary

an inverse problem for a general annular bounded domain in r2 with mixed boundary conditions and its physical applications this paper deals with the very interesting problem about the influence of the boundary conditions on the distribution of the eigenvalues of the negative laplacian in r 2 the asymptotic expansion of the trace of the heat kernel q t u 1 exp t m n where m n n 1 are the eigenvalues of the negative laplacian d 2 k 1 2 x k 2 in the x 1 x 2 plane is studied for short time t of a general annular bounded domain w in r 2 together with its smooth inner boundary w 1 and its smooth outer boundary w 2 where a finite number of dirichlet neumann and robin boundary conditions on the piecewise smooth components c i i 1 m of w 1 and on the piecewise smooth components c i i m 1 n of w 2 such that w 1 i 1 m c i and w 2 i m 1 n c i are considered in this paper one may extract information on the geometry of w by analyzing the asymptotic expansions of q t for short time t some applications of q t for an ideal gas enclosed in w are given thermodynamic quantities of an ideal gas enclosed in w are examined we use an asymptotic expansion for high temperatures to obtain the partition function of an ideal gas showing the leading corrections to the internal energy due to a finite container we show that the ideal gas cannot feel the shape of its container although it can feel some geometrical properties of it

on the error of estimating the sparsest solution of underdetermined linear systems let a be an n m matrix with m n and suppose that the underdetermined linear system as x admits a sparse solution s0 for which s0 0 1 2 spark a such a sparse solution is unique due to a well known uniqueness theorem suppose now that we have somehow a solution s as an estimation of s0 and suppose that s is only approximately sparse that is many of its components are very small and nearly zero but not mathematically equal to zero is such a solution necessarily close to the true sparsest solution more generally is it possible to construct an upper bound on the estimation error s s0 2 without knowing s0 the answer is positive and in this paper we construct such a bound based on minimal singular values of submatrices of a we will also state a tight bound which is more complicated but besides being tight enables us to study the case of random dictionaries and obtain probabilistic upper bounds we will also study the noisy case that is where x as n moreover we will see that where s0 0 grows to obtain a predetermined guaranty on the maximum of s s0 2 s is needed to be sparse with a better approximation this can be seen as an explanation to the fact that the estimation quality of sparse recovery algorithms degrades where s0 0 grows

on hearing the shape of rectilinear regions from complete knowledge of the eigenvalues of the negative laplacian on a bounded domain one may extract information on the geometry and the boundary conditions by analyzing the asymptotic expansion of a spectral function explicit calculations are performed for an equilateral triangular domain with dirichlet or neumann boundary conditions yielding in particular the corner angle terms in three dimensions some applications to eigenvalue problems for an equilateral triangular prism are dealt with including the solid vertex terms

an inverse problem of the wave equation for a general doubly connected region in r2 with a finite number of piecewise smooth robin boundary conditions the spectral distribution m t w 1 exp ite w 1 2 where e w w 1 are the eigenvalues of the negative laplacian d k 1 2 x k 2in the x 1 x 2 plane is studied for a variety of domains where 0

mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation abstract over the last 10 years or so advanced statistical properties including exponential decay of correlations have been established for certain classes of singular hyperbolic flows in three dimensions the results apply in particular to the classical lorenz attractor however many of the proofs rely heavily on the smoothness of the stable foliation for the flow in this paper we show that many statistical properties hold for singular hyperbolic flows with no smoothness assumption on the stable foliation these properties include existence of srb measures central limit theorems and associated invariance principles as well as results on mixing and rates of mixing the properties hold equally for singular hyperbolic flows in higher dimensions provided the center unstable subspaces are two dimensional

almost sure invariance principle for dynamical systems by spectral methods we prove the almost sure invariance principle for stationary d valued random processes with very precise dimension independent error terms solely under a strong assumption concerning the characteristic functions of these processes this assumption is easy to check for large classes of dynamical systems or markov chains using strong or weak spectral perturbation arguments

on the limit of the largest eigenvalue of the large dimensional sample covariance matrix in this paper the authors show that the largest eigenvalue of the sample covariance matrix tends to a limit under certain conditions when both the number of variables and the sample size tend to infinity the above result is proved under the mild restriction that the fourth moment of the elements of the sample sums of squares and cross products sp matrix exist

on decay of correlations in anosov flows there is some disagreement about the meaning of the phrase chaotic flow however there is no doubt that mixing anosov flows provides an example of such systems anosov systems were introduced and extensively studied in his classical memoir a among other things he proved the following fact known now as anosov alternative for flows either every strong stable and strong unstable manifold is everywhere dense or the flow gt is a suspension over an anosov diffeomorphism by a constant roof function if the first alternative holds gt is mixing with respect to every gibbs measure see pp2 now the natural question is what is the estimated rate of mixing this is certainly one of the simplest questions concerning correlation decay in continuous time systems nevertheless the only results obtained until recently dealt with the case when the system discussed had an additional algebraic structure the easier case of anosov diffeomorphisms can be treated by the methods of thermodynamic formalism of sinai ruelle and bowen b2 namely one uses markov partitions to construct an isomorphism between the diffeomorphism and a subshift of finite type and then proves that all such subshifts are exponentially mixing this method would succeed also for flows if any suspension over a subshift of finite type had exponentially decaying correlations however the simplest example suspensions with locally constant roof functions never have such a property ri one can use the above observation to produce examples of axiom a flows with arbitrary slow correlation decay it became clear therefore that some additional geometric properties should be taken into account in recent work chernov chl ch2 has employed a uniform nonintegrability condition to get a subexponential estimate for correlation functions for geodesic flows on surfaces of variable negative curvature his method relies on the technique of markov approximation developed in chl the aim of this paper is to combine geometric considerations of chernov with the thermodynamic formalism approach the later method seems to be more appropriate than markov approximations since it gives simpler

an inverse eigenvalue problem for a general convex domain the underlying problem is to deduce the precise shape of a membrane from the complete knowledge of the eigenvalues i for the lapace operator a cf a ax in the x1x2 plane let d g r be a simply connected bounded domain with a smooth boundary ad let us suppose that the a are the eigenvalues of the impedance problem au k o in d

a fast approach for overcomplete sparse decomposition based on smoothed ell 0 norm in this paper a fast algorithm for overcomplete sparse decomposition called sl0 is proposed the algorithm is essentially a method for obtaining sparse solutions of underdetermined systems of linear equations and its applications include underdetermined sparse component analysis sca atomic decomposition on overcomplete dictionaries compressed sensing and decoding real field codes contrary to previous methods which usually solve this problem by minimizing the l 1 norm using linear programming lp techniques our algorithm tries to directly minimize the l 1 norm it is experimentally shown that the proposed algorithm is about two to three orders of magnitude faster than the state of the art interior point lp solvers while providing the same or better accuracy

rapid decay of correlations for nonuniformly hyperbolic flows we show that superpolynomial decay of correlations rapid mixing is prevalent for a class of nonuniformly hyperbolic flows these flows are the continuous time analogue of the class of nonuniformly hyperbolic maps for which young proved exponential decay of correlations the proof combines techniques of dolgopyat and operator renewal theory it follows from our results that planar periodic lorentz flows with finite horizons and flows near homoclinic tangencies are typically rapid mixing

an inverse problem for a general vibrating annular membrane in r3 with its physical applications further results this paper deals with the very interesting problem about the influence of the boundary conditions on the distribution of the eigenvalues of the negative laplacian in r 3 the trace of the heat semigroup q t email protected v 1 exp email protected v where m v v 1 are the eigenvalues of the negative laplacian email protected email protected b 1 email protected x b 2 in the x 1 x 2 x 3 space is studied for short time t for a general vibrating annular membrane w in r 3 together with its smooth inner bounding surface s 1 and its smooth outer bounding surface s 2 where a finite number of dirichlet neumann and robin boundary conditions on the piecewise smooth components s i i 1 m of s 1 and on the piecewise smooth components s i i m 1 n of s 2 is considered such that s email protected i 1 ms i and s email protected i m 1 ns i in this paper one may extract information on the geometry of w by analyzing the asymptotic expansions of q t for short time t some applications of q t for an ideal gas enclosed in the general vibrating annular membrane w are given we show that the asymptotic expansion of q t for short time t plays an important role in investigating the influence of the annular region w on the thermodynamic quantities of an ideal gas some applications of q t for an ideal gas enclosed in a compact n dimensional riemannian manifold are also given we show that the ideal gas cannot feel the shape of its container although it can feel some geometrical properties of it

higher dimensional inverse problem for a multi connected bounded domain with piecewise smooth robin boundary conditions and its physical applications the asymptotic expansions of the trace of the heat kernel q t email protected j 1 exp email protected j for short time t have been derived for a variety of bounded domains where l j j 1 are the eigenvalues of the negative laplace operator email protected email protected i 1 3 x i 2 in the x 1 x 2 x 3 space the dependence of q t on the connectivity of bounded domains and the boundary conditions is analyzed particular attention is given for an arbitrary multiply connected bounded domain w in r 3 together with piecewise smooth robin boundary conditions where the coefficients in these conditions are assumed to be piecewise smooth positive functions some applications of an ideal gas enclosed in the multiply connected bounded domain w are given we show that the ideal gas cannot feel the shape of its container although it can feel some geometrical properties of it

an inverse problem for the three dimensional multi connected vibrating membrane with robin boundary conditions this paper deals with the very interesting problem about the influence of the boundary conditions on the distribution of the eigenvalues of the negative laplacian in r 3 the trace of the heat semigroup q t n 1exp t m n where m n n 1 are the eigenvalues of the negative laplacian 2 3 b 1 x b 2 in the x 1 x 2 x 3 space is studied for a general multiply connected bounded domain w in r 3 surrounded by simply connected bounded domains w j with smooth bounding surfaces s j j 1 n where a finite number of piecewise smooth robin boundary conditions on the piecewise smooth components s i i 1 k j 1 k j of the bounding surfaces s j are considered such that s j k j i 1 k j 1s i where k 0 0 some applications of q t for an ideal gas enclosed in the multiply connected bounded container w with robin boundary conditions are given we show that the asymptotic expansion of q t for short time t plays an important role in investigating the influence of the finite container w on the thermodynamic quantities of an ideal gas

an inverse eigenvalue problem of the wave equation for a multi connected region in r2 together with three different types of boundary conditions the trace of the wave kernel m t email protected w 1exp ite 1 2 w where e w w 1 are the eigenvalues of the negative laplacian email protected email protected 2 k 1 email protected x k 2 in the x 1 x 2 plane is studied for a variety of bounded domains where

on selecting a maximum volume sub matrix of a matrix and related problems given a matrix email protected r m x n n vectors in m dimensions we consider the problem of selecting a subset of its columns such that its elements are as linearly independent as possible this notion turned out to be important in low rank approximations to matrices and rank revealing qr factorizations which have been investigated in the linear algebra community and can be quantified in a few different ways in this paper from a complexity theoretic point of view we propose four related problems in which we try to find a sub matrix email protected r m x k of a given matrix email protected r m x n such that i s m a x c the largest singular value of c is minimum ii s m i n c the smallest singular value of c is maximum iii k c email protected m a x c s m i n c the condition number of c is minimum and iv the volume of the parallelepiped defined by the column vectors of c is maximum we establish the np hardness of these problems and further show that they do not admit ptas we then study a natural greedy heuristic for the maximum volume problem and show that it has approximation ratio 2 o k l o g k our analysis of the greedy heuristic is tight to within a logarithmic factor in the exponent which we show by explicitly constructing an instance for which the greedy heuristic is 2 w k from optimal when a has unit norm columns a related problem is to select the maximum number of vectors with a given volume we show that if the optimal solution selects k columns then greedy will select w k logk columns providing a logk approximation

robust transitive singular sets for 3 flows are partially hyperbolic attractors or repellers inspired by lorenz remarkable chaotic flow we describe in this paper the structure of all c 1 robust transitive sets with singularities for flows on closed 3 manifolds they are partially hyperbolic with volume expanding central direction and are either attractors or repellers in particular any c 1 robust attractor with singularities for flows on closed 3 manifolds always has an invariant foliation whose leaves are forward contracted by the flow and has positive lyapunov exponent at every orbit showing that any c 1 robust attractor resembles a geometric lorenz attractor

the wave equation approach to an inverse problem for a general multi connected domain in r2 with mixed boundary conditions the spectral function email protected t email protected j 1 exp email protected j 1 2 where m j j 1 are the eigenvalues of the negative laplacian email protected email protected u 1 email protected x u 2 in r 2 is studied for small t for a variety of domains where

the lorenz attractor is mixing we study a class of geometric lorenz flows introduced independently by afraimovic bykov sil nikov and by guckenheimer williams and give a verifiable condition for such flows to be mixing as a consequence we show that the classical lorenz attractor is mixing

restricted isometry constants where ell p sparse recovery can fail for 0 p leq 1 this paper investigates conditions under which the solution of an underdetermined linear system with minimal lscrp norm 0 p les 1 is guaranteed to be also the sparsest one matrices are constructed with restricted isometry constants ric delta2m arbitrarily close to 1 radic2 ap 0 707 where sparse recovery with p 1 fails for at least one m sparse vector as well as matrices with delta2m arbitrarily close to one where lscr1 minimization succeeds for any m sparse vector this highlights the pessimism of sparse recovery prediction based on the ric and indicates that there is limited room for improving over the best known positive results of foucart and lai which guarantee that lscr1 minimization recovers all m sparse vectors for any matrix with delta2m 2 3 radic2 7 ap 0 4531 these constructions are a by product of tight conditions for lscrp recovery 0 les p les 1 with matrices of unit spectral norm which are expressed in terms of the minimal singular values of 2m column submatrices compared to lscr1 minimization lscrp minimization recovery failure is shown to be only slightly delayed in terms of the ric values furthermore in this case the minimization is nonconvex and it is important to consider the specific minimization algorithm being used it is shown that when lscrp optimization is attempted using an iterative reweighted lscr1 scheme failure can still occur for delta2m arbitrarily close to 1 radic2

hearing the shape of a general doubly connected domain in r3 with impedance boundary conditions the basic problem in this paper is that of determining the geometry of a general doubly connected domain in r3 together with an impedance condition on its inner bounding surface and another impedance condition on its outer bounding surface from the complete knowledge of the eigenvalues j j 1 for the three dimensional laplacian using the asymptotic expansion of the spectral function t j 1 exp t j for small positive t

on hearing the shape of a general multi connected vibrating membrane in r2 with piecewise smooth positive functions in the robin boundary conditions the asymptotic expansion of the trace of the heat kernel q t email protected j 1 exp email protected j as t 0 has been derived for a variety of domains where l j are the eigenvalues of the negative laplace email protected email protected i 1 email protected email protected x i 2in the x 1 x 2 plane the dependence of q t on the connectivity of domains and the boundary conditions are analyzed particular attention is given for a general multiply connected bounded domain in r 2 together with a finite number of piecewise robin boundary conditions where the coefficients in the boundary conditions are piecewise smooth positive functions some applications of an ideal gas enclosed in the general multiply connected domain are given