isospectral problem for schrodinger operator evolutional viewpoint an isospectral transform of the schrodinger operator is considered as an evolutional problem for a transform defined by the mckean trubowitz flows associated evolutional equations are derived it is shown that for one level and two level flows these equations can be split into integrable liouville equations a relationship between the liouville equations and the darboux transforms is discussed this analysis suggests that the evolutional equations can be split into the liouville equations in the general case a hamiltonian formulation of the isospectral transform defined by the mckean trubowitz flows is presented it is shown that this transform is performed by a canonical change of variables which is related to the darboux transform

some new results on eigenvectors via dimension diameter and ricci curvature abstract we generalise for a general symmetric elliptic operator the different notions of dimension diameter and ricci curvature which coincide with the usual notions in the case of the laplace beltrami operators on riemannian manifolds if 1 denotes the spectral gap that is the first nonzero eigenvalue we investigate in this paper the best lower bound on 1 one can obtain under an upper bound on the dimension an upper bound on the diameter and a lower bound of the ricci curvature two cases are known namely if the ricci curvature is bounded below by a constant r 0 then 1 nr n 1 and this estimate is sharp for the n dimensional spheres lichnerowicz s bound if the ricci curvature is bounded below by zero then zhong yang s estimate asserts that 1 2 d 2 where d is an upper bound on the diameter this estimate is sharp for the 1 dimensional torus in the general case many interesting estimates have been obtained this paper provides a general optimal comparison result for 1 which unifies and sharpens lichnerowicz and zhong yang s estimates together with other comparison results concerning the range of the associated eigenfunctions and their derivatives

existence and uniqueness of monotone measure preserving maps introduction given a pair of borel probability measures and v on rd it is natural to ask whether v can be obtained from by redistributing its mass in a canonical way in the case of the line d 1 the answer is clear as long as both measures are free from atoms x v x 0mthere is a map y x of the line to itself for which uniquely determined almost everywhere this map may be taken to be nondecreasing by a suitable choice of y x e rw o at the remaining points interpreting and v as the initial and final distribution of a one dimensional fluid the transformation y x gives a rearrangement of fluid particles yielding final state v from the initial state this rearrangement is characterized by the fact that it preserves particle ordering obviating any need for two particles to cross although the generalization of this construction to higher dimensions is the focus of this paper the one dimensional case will be pursued slightly further when the measures are absolutely continuous with respect to lebesgue d x f x dx and dr y g y dymthen formally at least neglecting regularity issues the fundamental theorem of calculus yields

diffusion processes and heat kernels on metric spaces processes x p on any given locally compact metric space x d tx equipped with a radon measure m these processes are associated with local regular dirichlet forms which are obtained as limits of approxi mating nonlocal dirichlet forms this general method works without any restrictions on x d m and yields processes which are well defined for quasi every starting point the second main topic of this paper is to formulate and exploit the so called measure contraction property this is a condition on the original data x d m which can be regarded as a generalization of curvature bounds on the metric space x d it is a bound for distortions of the measure m under contractions of the state space x along suitable geodesics or quasi geodesics w r t the metric d in the case of rieman nian manifolds this condition is always satisfied several other examples will be discussed including uniformly elliptic operators operators with weights certain subelliptic operators manifolds with boundaries or cor ners and glueing together of manifolds the measure contraction property implies upper and lower gaussian estimates for the heat kernel and a harnack inequality for the associated harmonic functions therefore the above mentioned diffusion processes are strong feller processes and are well defined for every starting point

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

factorization method and new potentials with the oscillator spectrum a one parameter family of potentials in one dimension is constructed with the energy spectrum coinciding with that of the harmonic oscillator this is a new derivation of a class of potentials previously obtained by abraham and moses with the help of the gelfand levitan formalism

tensorization of cheeger energies the space h1 1 and the area formula for graphs abstract first we study in detail the tensorization properties of weak gradients in metric measure spaces x d m then we compare potentially different notions of the sobolev space h 1 1 x d m and of weak gradient with exponent 1 eventually we apply these results to compare the area functional 1 f w 2 d m with the perimeter of the subgraph of f in the same spirit as the classical theory

the monge problem for distance cost in geodesic spaces we address the monge problem in metric spaces with a geodesic distance x d is a polish space and dl is a geodesic borel distance which makes x dl a non branching geodesic space we show that under the assumption that geodesics are d continuous and locally compact we can reduce the transport problem to 1 dimensional transport problems along geodesics we introduce two assumptions on the transport problem which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w r t the 1 dimensional hausdorff distance induced by dl it is known that this regularity is sufficient for the construction of a transport map we study also the dynamics of transport along the geodesic the stability of our conditions and show that in this setting dl cyclical monotonicity is not sufficient for optimality

bakry emery curvature dimension condition and riemannian ricci curvature bounds the aim of the present paper is to bridge the gap between the bakry emery and the lott sturm villani approaches to provide synthetic and abstract notions of lower ricci curvature bounds we start from a strongly local dirichlet form mathcal e admitting a carre du champ gamma in a polish measure space x mathfrak m and a canonical distance mathsf d mathcal e that induces the original topology of x we first characterize the distinguished class of riemannian energy measure spaces where mathcal e coincides with the cheeger energy induced by mathsf d mathcal e and where every function f with gamma f le1 admits a continuous representative in such a class we show that if mathcal e satisfies a suitable weak form of the bakry emery curvature dimension condition mathrm be k infty then the metric measure space x mathsf d mathfrak m satisfies the riemannian ricci curvature bound mathrm rcd k infty according to duke math j 163 2014 1405 1490 thus showing the equivalence of the two notions two applications are then proved the tensorization property for riemannian energy spaces satisfying the bakry emery mathrm be k n condition and thus the corresponding one for mathrm rcd k infty spaces without assuming nonbranching and the stability of mathrm be k n with respect to sturm gromov hausdorff convergence

pilot aided ofdm channel estimation in the presence of the guard band in this letter pilot design and channel estimation are discussed for orthogonal frequency division multiplexing ofdm systems with guard subcarriers first we investigate the effects of guard band on channel estimation errors from this we propose pilot placement having a maximum distance between adjacent pilots except for the guard band and show that it achieves minimum channel estimation errors among partially equispaced pilots using equivalence of the toeplitz and circulant matrices also an efficient channel estimator is developed by introducing an extended channel and its finite impulse response fir approximation to overcome high numerical complexity caused by the presence of guard subcarriers and the use of a large number of subcarriers simulation results are presented for ofdm and orthogonal frequency division multiple access ofdma systems consistent with ieee 802 16a standards

ber minimized ofdm systems with channel independent precoders we consider the minimization of uncoded bit error rate ber for the orthogonal frequency division multiplexing ofdm system with an orthogonal precoder we analyze the ber performance of precoded ofdm systems with zero forcing and minimum mean squared error mmse receivers in the case of mmse receivers we show that for quadrature phase shift keying qpsk there exists a class of optimal precoders that are channel independent examples of this class include the discrete fourier transform dft matrix and the hadamard matrix when the precoder is the dft matrix the resulting optimal transceiver becomes the single carrier system with cyclic prefix sc cp system we also show that the worst solution corresponds to the conventional ofdm system the conventional ofdm system has the largest ber in the case of zero forcing receivers the design of optimal transceiver depends on the signal to noise ratio snr for higher snr solutions of optimal precoders are the same as those of mmse receivers

sobolev spaces on warped products abstract we study the structure of sobolev spaces on the cartesian warped products of a given metric measure space and an interval our main results are the characterization of the sobolev spaces in such products the proof that under natural assumptions the warped products possess the sobolev to lipschitz property which is key for geometric applications the results of this paper have been needed in the recent proof of the volume cone to metric cone property of rcd spaces obtained by the first author and de philippis

weighted sobolev spaces on metric measure spaces abstract we investigate weighted sobolev spaces on metric measure spaces x d m denoting by the weight function we compare the space w x d m which always concides with the closure h x d m of lipschitz functions with the weighted sobolev spaces w 1 p x d m and h 1 p x d m defined as in the euclidean theory of weighted sobolev spaces under mild assumptions on the metric measure structure and on the weight we show that w x d m h x d m we also adapt the results proved by muckenhoupt and the ones proved by zhikov to the metric measure setting considering appropriate conditions on that ensure the equality w 1 p x d m h x d m

calculus and heat flow in metric measure spaces and applications to spaces with ricci bounds from below this paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces x mathsf d mathfrak m our main results are a general study of the relations between the hopf lax semigroup and hamilton jacobi equation in metric spaces x d the equivalence of the heat flow in l 2 x mathfrak m generated by a suitable dirichlet energy and the wasserstein gradient flow of the relative entropy functional mathrm ent mathfrak m in the space of probability measures open image in new window the proof of density in energy of lipschitz functions in the sobolev space w 1 2 x mathsf d mathfrak m a fine and very general analysis of the differentiability properties of a large class of kantorovich potentials in connection with the optimal transport problem is the fourth achievement of the paper our results apply in particular to spaces satisfying ricci curvature bounds in the sense of lott and villani ann math 169 903 991 2009 and sturm acta math 196 65 131 2006 and acta math 196 133 177 2006 and require neither the doubling property nor the validity of the local poincare inequality

metric measure spaces with riemannian ricci curvature bounded from below in this paper we introduce a synthetic notion of riemannian ricci bounds from below for metric measure spaces x d m which is stable under measured gromov hausdorff convergence and rules out finsler geometries it can be given in terms of an enforcement of the lott sturm and villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow besides stability it enjoys the same tensorization global to local and local to global properties in these spaces which we call rcd k spaces we prove that the heat flow which can be equivalently characterized either as the flow associated to the dirichlet form or as the wasserstein gradient flow of the entropy satisfies wasserstein contraction estimates and several regularity properties in particular bakry emery estimates and the l lip feller regularization we also prove that the distance induced by the dirichlet form coincides with d that the local energy measure has density given by the square of cheeger s relaxed slope and as a consequence that the underlying brownian motion has continuous paths all these results are obtained independently of poincare and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact as the infinite dimensional ones

equivalent definitions of bv space and of total variation on metric measure spaces abstract in this paper we introduce a new definition of bv based on measure upper gradients and prove the equivalence of this definition and the coincidence of the corresponding notions of total variation with the definitions based on relaxation of l 1 norm of the slope of lipschitz functions or upper gradients as in the previous work by the first author with gigli and savare in the sobolev case the proof requires neither local compactness nor doubling and poincare

a riemannian interpolation inequality a la borell brascamp and lieb a concavity estimate is derived for interpolations between l 1 m mass densities on a riemannian manifold the inequality sheds new light on the theorems of prekopa leindler borell brascamp and lieb that it generalizes from euclidean space due to the curvature of the manifold the new riemannian versions of these theorems incorporate a volume distortion factor which can however be controlled via lower bounds on ricci curvature the method uses optimal mappings from mass transportation theory along the way several new properties are established for optimal mass transport and interpolating maps on a riemannian manifold

collapsing of riemannian manifolds and eigenvalues of laplace operator on etudie la perturbation des valeurs propres de l operateur de laplace sous des deformations des varietes qui ne conservent pas les dimensions

on the selection of the redundant carrier positions in uw ofdm unique word uw ofdm is a new multicarrier technique that was recently proposed in m huemer c hofbauer and j b huber the potential of unique words in ofdm proc inowo hamburg germany sep 2010 pp 140 144 in this technique the guard interval that contains a priori known samples is part of the fft interval to be able to construct the uw ofdm signal some carriers must be sacrificed to transmit redundant information these carriers are called the redundant carriers it turns out that the average redundant energy needed strongly depends on the positions of the redundant carriers in this paper we look for the redundant carrier placement that minimizes the average redundant energy however this optimization is a np hard problem therefore we reformulate the optimization problem and implement a branch and bound b b algorithm to find the optimum the results of the b b search algorithm are compared to a theoretical lower bound and with the results for two low cost redundant carrier placements i e the quasi uniform qu distribution h steendam the quasi uniform redundant carrier placement for uw ofdm vehicular technology conf fall 2012 quebec canada sep 2012 for the case where no guard bands are present and the maximum distance md distribution in case forbidden guard bands are present we show in this paper that when the full band is available for redundant carriers both the redundant carrier placement found with the b b algorithm and the qu distribution result in a redundant energy that is very close to the theoretical lower bound although the b b slightly outperforms the qu distribution this implies that the qu distribution is a good low cost solution to the redundant carrier placement problem in the case of no guard bands however when there are forbidden guard bands in which the redundant carriers cannot be placed the qu distribution can no longer be used in that case the results of the b b algorithm are compared to that of the md distribution we show in this paper that the proposed b b algorithm outperforms the md distribution in the sense that it needs noticeably lower redundant energy

strict and pointwise convergence of bv functions in metric spaces abstract consider a metric space x that is equipped with a doubling measure and supports a poincare inequality we show that if u i u strictly in bv x i e if u i u in l 1 x and d u i x d u x then for a subsequence not relabeled we have u i x u x for h almost every x x s u

transport inequalities gradient estimates entropy and ricci curvature we present various characterizations of uniform lower bounds for the ricci curvature of a smooth riemannian manifold m in terms of convexity properties of the entropy considered as a function on the space of probability measures on m as well as in terms of transportation inequalities for volume measures heat kernels and brownian motions and in terms of gradient estimates for the heat semigroup 2004 wiley periodicals inc

the spectral class of the quantum mechanical harmonic oscillator the purpose of this paper is to study the so calledspectral classq of anharmonic oscillatorsq d2 q having the same spectrum n 2n n 0 as the harmonic oscillatorq0 d2 x2 1 thenorming constants t n mathop lim limits x uparrow infty ell g 1 n e n x mathord left vphantom e n x e n right kern nulldelimiterspace e n x of the eigenfunctions ofq form a complete set of coordinates inq in terms of which the potential may be expressed asq x2 1 2d2 lg with theta det left delta ij e ti 1 int limits x infty e i 0 e j 0 0 leqq i j infty right e n 0 being thenth eigenfunctionq0 the spectrum and norming constants are canonically conjugate relative to the bracket f g fd gdx to wit i j 0 ti 2 j 1 or 0 according to whetheri j or not and ti tj 0 this prompts an investigation of the symplectic geometry ofq the function is related to the theta function of a singular algebraic curve numerical results are also presented

monge kantorovitch measure transportation and monge ampere equation on wiener space let w h be an abstract wiener space assume two i i 1 2 probabilities on w w we give some conditions for the wasserstein distance between 1 and 2 with respect to the cameron martin space to be finite where the infimum is taken on the set of probability measures on w w whose first and second marginals are 1 and 2 in this case we prove the existence of a unique cyclically monotone map t iw with w h such that t maps 1 to 2 moreover if 2 then t is stochastically invertible i e there exists s w w such that s t iw 1 a s and t s iw 2 a s if in addition 1 then there exists a 1 convex function in the gaussian sobolev space such that these results imply that the quasi invariant transformations of the wiener space with finite wasserstein distance from can be written as the composition of a transport map t and a rotation i e a measure preserving map we give also 1 convex sub solutions and ito type solutions of the monge ampere equation on w

perimeter as relaxed minkowski content in metric measure spaces abstract in this note we prove that on general metric measure spaces the perimeter is equal to the relaxation of the minkowski contents w r t convergence in measure

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

on the nonlinear fock description of quantum systems with quadratic spectra the nonlinear version of the fock approach developed previously for systems with extended equidistant spectra eleonsky v m and korolev v g 1995 j phys a math gen 28 4973 is also shown to describe systems with quadratic spectra some general properties of shift operators for such spectra are studied it is found in particular that families of shift operators related to the generators of the algebra su 1 1 as well as supersymmetry ladder operators arise naturally in that approach in a unified way