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AbstractThe power moments of a positive measure on the real line or the circle are characterized by the non-negativity of an infinite matrix, Hankel, respectively Toeplitz, attached to the data. Except some fortunate configurations, in higher dimensions there are no non-negativity criteria for the power moments of a measure to be supported by a prescribed closed set. We combine two well studied fortunate situations, specifically a class of curves in two dimensions classified by Scheiderer and Plaumann, and compact, basic semi-algebraic sets, with the aim at enlarging the realm of geometric shapes on which the power moment problem is accessible and solvable by non-negativity certificates.

Finite rank perturbations T=N+K of a bounded normal operator N acting on a separable Hilbert space are studied thanks to a natural functional model of T; in its turn the functional model solely relies on a perturbation matrix/characteristic function previously defined by the second author. Function theoretic features of this perturbation matrix encode in a closed-form the spectral behavior of T. Under mild geometric conditions on the spectral measure of N and some smoothness constraints on K we show that the operator T admits invariant subspaces, or even it is decomposable.

The composition operators preserving total non-negativity and totalpositivity for various classes of kernels are classified, following threethemes. Letting a function act by post composition on kernels with arbitrarydomains, it is shown that such a composition operator maps the set of totallynon-negative kernels to itself if and only if the function is constant orlinear, or just linear if it preserves total positivity. Symmetric kernels arealso discussed, with a similar outcome. These classification results are abyproduct of two matrix-completion results and the second theme: an extensionof A.M. Whitney's density theorem from finite domains to subsets of the realline. This extension is derived via a discrete convolution with modulatedGaussian kernels. The third theme consists of analyzing, with tools fromharmonic analysis, the preservers of several families of totally non-negativeand totally positive kernels with additional structure: continuous Hankelkernels on an interval, P\'olya frequency functions, and P\'olya frequencysequences. The rigid structure of post-composition transforms of totallypositive kernels acting on infinite sets is obtained by combining severalspecialized situations settled in our present and earlier works.

Multivalued linear operators, also known as linear relations, are studied ona specific class of weighted, composition transforms on Fock space. Basicproperties of this class of linear relations, such as closed graph,boundedness, complex symmetry, real symmetry, or isometry are characterized insimple algebraic terms, involving their symbols.

In resonance to a recent geometric framework proposed by Douglas and Yang, afunctional model for certain linear bounded operators with rank-oneself-commutator acting on a Hilbert space is developed. By taking advantage ofthe refined existing theory of the principal function of a hyponormal operatorwe transfer the whole action outside the spectrum, on the resolvent of theunderlying operator, localized at a distinguished vector. The wholeconstruction turns out to rely on an elementary algebra body involving analyticmultipliers and Cauchy transforms. A natural field theory interpretation of theresulting resolvent functional model is proposed.

This survey contains a selection of topics unified by the concept of positivesemi-definiteness (of matrices or kernels), reflecting natural constraintsimposed on discrete data (graphs or networks) or continuous objects(probability or mass distributions). We put emphasis on entrywise operationswhich preserve positivity, in a variety of guises. Techniques from harmonicanalysis, function theory, operator theory, statistics, combinatorics, andgroup representations are invoked. Some partially forgotten classical roots inmetric geometry and distance transforms are presented with comments and fullbibliographical references. Modern applications to high-dimensional covarianceestimation and regularization are included.

We examine the structure of the Levi component MA in a minimal parabolic subgroup P= MAN of a real reductive Lie group G and work out the cases where M is metabelian, equivalently where p is solvable. When G is a linear group we verify that p is solvable if and only if M is commutative. In the general case M is abelian modulo the center ZG , we indicate the exact structure of M and P, and we work out the precise Plancherel Theorem and Fourier Inversion Formulae. This lays the groundwork for comparing tempered representations of G with those induced from generic representations of P.

We introduce the problem of transporting vector-valued distributions. In this, a salient feature is that mass may ow between vectorial entries as well as across space (discrete or continuous). The theory relies on a first step taken to define an appropriate notion of optimal transport on a graph. The corresponding distance between distributions is readily computable via convex optimization and provides a suitable generalization of Wasserstein-type metrics. Building on this, we define Wasserstein-type metrics on vector-valued distributions supported on continuous spaces as well as graphs. Motivation for developing vector-valued mass transport is provided by applications such as color image processing, multimodality imaging, polarimetric radar, as well as network problems where resources may be vectorial.

The aim of this article is to introduce and address the problem to smoothly interpolate (empirical) probability measures. To this end, we lift the concept of a spline curve from the setting of points in a Euclidean space to that of probability measures, using the framework of optimal transport.

We investigate how a C*-algebra could consist of functions on a noncommutative set: a discretization of a C*-algebra A is a *-homomorphism A → M that factors through the canonical inclusion C(X) ⊆ ℓ∞;(X) when restricted to a commutative C*-subalgebra. Any C*-algebra admits an injective but nonfunctorial discretization, as well as a possibly noninjective functorial discretization, where M is a C*-algebra. Any subhomogenous C*-algebra admits an injective functorial discretization, where M is a W*-algebra. However, any functorial discretization, where M is an AW*-algebra, must trivialize A = B(H) for any infinite-dimensional Hilbert space H.

In a series of recent papers ([19], [20], [22], [23]) we extended the notion of square integrability, for representations of nilpotent Lie groups, to that of stepwise square integrability. There we discussed a number of applications based on the fact that nilradicals of minimal parabolic subgroups of real reductive Lie groups are stepwise square integrable. In Part I we prove stepwise square integrability for nilradicals of arbitrary parabolic subgroups of real reductive Lie groups. This is technically more delicate than the case of minimal parabolics. We further discuss applications to Plancherel formulae and Fourier inversion formulae for maximal exponential solvable subgroups of parabolics and maximal amenable subgroups of real reductive Lie groups. Finally, in Part II, we extend a number of those results to (infinite dimensional) direct limit parabolics. These extensions involve an infinite dimensional version of the Peter-Weyl Theorem, construction of a direct limit Schwartz space, and realization of that Schwartz space as a dense subspace of the corresponding L2 space.

We investigate how a C*-algebra could consist of functions on anoncommutative set: a discretization of a C*-algebra $A$ is a $*$-homomorphism$A \to M$ that factors through the canonical inclusion $C(X) \subseteq\ell^\infty(X)$ when restricted to a commutative C*-subalgebra. Any C*-algebraadmits an injective but nonfunctorial discretization, as well as a possiblynoninjective functorial discretization, where $M$ is a C*-algebra. Anysubhomogenous C*-algebra admits an injective functorial discretization, where$M$ is a W*-algebra. However, any functorial discretization, where $M$ is anAW*-algebra, must trivialize $A = B(H)$ for any infinite-dimensional Hilbertspace $H$.

In the classification theorems of Vinberg and Yakimova for commutative nilmanifolds, the relevant nilpotent groups have a very surprising analytic property. The manifolds are of the form (Formula presented.) where, in all but three cases, the nilpotent group (Formula presented.) has irreducible unitary representations whose coefficients are square integrable modulo the center (Formula presented.) of (Formula presented.). Here we show that, in those three “exceptional” cases, the group (Formula presented.) is a semidirect product (Formula presented.) or (Formula presented.) where the normal subgroup (Formula presented.) contains the center (Formula presented.) of (Formula presented.) and has irreducible unitary representations whose coefficients are square integrable modulo (Formula presented.). This leads directly to explicit harmonic analysis and Fourier inversion formulae for commutative nilmanifolds.

In this paper we prove finiteness principles for $C^{m}\left( \mathbb{R}^{n}, \mathbb{R}^{D}\right) $-selection, and for $C^{m-1,1}\left( \mathbb{R}^{n}, \mathbb{R}^{D}\right) $-selection, in particular providing a proof for a conjecture of Brudyni-Shvartsman (1994) on Lipschitz selections for the case when the domain is $X = \mathbb{R}^n$.

In this paper we prove finiteness principles for $C^{m}\left( \mathbb{R}^{n}, \mathbb{R}^{D}\right) $-selection, and for $C^{m-1,1}\left( \mathbb{R}^{n}, \mathbb{R}^{D}\right) $-selection, in particular providing a proof for a conjecture of Brudyni-Shvartsman (1994) on Lipschitz selections for the case when the domain is $X = \mathbb{R}^n$. Our results raise the hope that one can start to understand constrained interpolation problems in which e.g. the interpolating function $F$ is required to be nonnegative everywhere.

Continuing our recent work in [5] we study polynomial masks of multivariate tight wavelet frames from two additional and complementary points of view: convexity and system theory. We consider such polynomial masks that are derived by means of the unitary extension principle from a single polynomial. We show that the set of such polynomials is convex and reveal its extremal points as polynomials that satisfy the quadrature mirror filter condition. Multiplicative structure of this polynomial set allows us to improve the known upper bounds on the number of frame generators derived from box splines. Moreover, in the univariate and bivariate settings, the polynomial masks of a tight wavelet frame can be interpreted as the transfer function of a conservative multivariate linear system. Recent advances in system theory enable us to develop a more effective method for tight frame constructions. Employing an example by S.W. Drury, we show that for dimension greater than 2 such transfer function representations of the corresponding polynomial masks do not always exist. However, for all wavelet masks derived from multivariate polynomials with non-negative coefficients, we determine explicit transfer function representations. We illustrate our results with several examples.

We present a particular formulation of optimal transport for matrix-valued density functions. Our aim is to devise a geometry which is suitable for comparing power spectral densities of multivariable time series. More specifically, the value of a power spectral density at a given frequency, which in the matricial case encodes power as well as directionality, is thought of as a proxy for a "matrix-valued mass density." Optimal transport aims at establishing a natural metric in the space of such matrix-valued densities which takes into account differences between power across frequencies as well as misalignment of the corresponding principle axes. Thus, our transportation cost includes a cost of transference of power between frequencies together with a cost of rotating the principle directions of matrix densities. The two endpoint matrix-valued densities can be thought of as marginals of a joint matrix-valued density on a tensor product space. This joint density, very much as in the classical Monge-Kantorovich setting, can be thought to specify the transportation plan. Contrary to the classical setting, the optimal transport plan for matrices is no longer supported on a thin zero-measure set.

We present a particular formulation of optimal transport for matrix-valued density functions. Our aim is to devise a geometry which is suitable for comparing power spectral densities of multivariable time series. More specifically, the value of a power spectral density at a given frequency, which in the matricial case encodes power as well as directionality, is thought of as a proxy for a "matrix-valued mass density." Optimal transport aims at establishing a natural metric in the space of such matrix-valued densities which takes into account differences between power across frequencies as well as misalignment of the corresponding principle axes. Thus, our transportation cost includes a cost of transference of power between frequencies together with a cost of rotating the principle directions of matrix densities. The two endpoint matrix-valued densities can be thought of as marginals of a joint matrix-valued density on a tensor product space. This joint density, very much as in the classical Monge-Kantorovich setting, can be thought to specify the transportation plan. Contrary to the classical setting, the optimal transport plan for matrices is no longer supported on a thin zero-measure set.

The increased availability of time series genetic variation data from experimental evolution studies and ancient DNA samples has created new opportunities to identify genomic regions under selective pressure and to estimate their associated fitness parameters. However, it is a challenging problem to compute the likelihood of non-neutral models for the population allele frequency dynamics, given the observed temporal DNA data. Here, we develop a novel spectral algorithm to analytically and efficiently integrate over all possible frequency trajectories between consecutive time points. This advance circumvents the limitations of existing methods which require fine-tuning the discretization of the population allele frequency space when numerically approximating requisite integrals. Furthermore, our method is flexible enough to handle general diploid models of selection where the heterozygote and homozygote fitness parameters can take any values, while previous methods focused on only a few restricted models of selection. We demonstrate the utility of our method on simulated data and also apply it to analyze ancient DNA data from genetic loci associated with coat coloration in horses. In contrast to previous studies, our exploration of the full fitness parameter space reveals that a heterozygote-advantage form of balancing selection may have been acting on these loci.

We study n by n symmetric random matrices H, possibly discrete, with iidabove-diagonal entries. We show that H is singular with probability at mostexp(-n^c), and the spectral norm of the inverse of H is O(sqrt{n}).Furthermore, the spectrum of H is delocalized on the optimal scale o(n^{-1/2}).These results improve upon a polynomial singularity bound due to Costello, Taoand Vu, and they generalize, up to constant factors, results of Tao and Vu, andErdos, Schlein and Yau.

Given a subset K of the unit Euclidean sphere, we estimate the minimal numberm = m(K) of hyperplanes that generate a uniform tessellation of K, in the sensethat the fraction of the hyperplanes separating any pair x, y in K is nearlyproportional to the Euclidean distance between x and y. Random hyperplanesprove to be almost ideal for this problem; they achieve the almost optimalbound m = O(w(K)^2) where w(K) is the Gaussian mean width of K. Using the mapthat sends x in K to the sign vector with respect to the hyperplanes, weconclude that every bounded subset K of R^n embeds into the Hamming cube {-1,1}^m with a small distortion in the Gromov-Haussdorf metric. Since for manysets K one has m = m(K) << n, this yields a new discrete mechanism of dimensionreduction for sets in Euclidean spaces.

We consider finitely generated shift-invariant spaces (SIS) with additional invariance in $L^2(\R^d)$. We prove that if the generators and their translates form a frame, then they must satisfy some stringent restrictions on their behavior at infinity. Part of this work (non-trivially) generalizes recent results obtained in the special case of a principal shift-invariant spaces in $L^2(\R)$ whose generator and its translates form a Riesz basis.

The massless Nelson model describes non-relativistic, spinless quantum particles interacting with a relativistic, massless, scalar quantum field. The interaction is linear in the field. We analyze the one particle sector. First, we construct the renormalized mass shell of the non-relativistic particle for an arbitrarily small infrared cut-off that turns off the interaction with the low energy modes of the field. No ultraviolet cut-off is imposed. Second, we implement a suitable Bogolyubov transformation of the Hamiltonian in the infrared regime. This transformation depends on the total momentum of the system and is non-unitary as the infrared cut-off is removed. For the transformed Hamiltonian we construct the mass shell in the limit where both the ultraviolet and the infrared cut-off are removed. Our approach is constructive and leads to explicit expansion formulae which are amenable to rigorously control the S-matrix elements.

We show that there are no non-trivial linear dependencies among p-norms of vectors in finite dimensions that hold for all p. The proof is by complex analytic continuation.

Spectral clustering is one of the most widely used techniques for extracting the underlying global structure of a data set. Compressed sensing and matrix completion have emerged as prevailing methods for efficiently recovering sparse and partially observed signals respectively. We combine the distance preserving measurements of compressed sensing and matrix completion with the power of robust spectral clustering. Our analysis provides rigorous bounds on how small errors in the affinity matrix can affect the spectral coordinates and clusterability. This work generalizes the current perturbation results of two-class spectral clustering to incorporate multi-class clustering with k eigenvectors. We thoroughly track how small perturbation from using compressed sensing and matrix completion affect the affinity matrix and in succession the spectral coordinates. These perturbation results for multi-class clustering require an eigengap between the kth and (k+1)th eigenvalues of the affinity matrix, which naturally occurs in data with k well-defined clusters. Our theoretical guarantees are complemented with numerical results along with a number of examples of the unsupervised organization and clustering of image data.

In this article, we study the geodesic problem in a generalized metric space, in which the distance function satisfies a relaxed triangle inequality $d(x,y)\leq \sigma (d(x,z)+d(z,y))$ for some constant $\sigma \geq 1$, rather than the usual triangle inequality. Such a space is called a quasimetric space. We show that many well-known results in metric spaces (e.g. Ascoli-Arzel\`{a} theorem) still hold in quasimetric spaces. Moreover, we explore conditions under which a quasimetric will induce an intrinsic metric. As an example, we introduce a family of quasimetrics on the space of atomic probability measures. The associated intrinsic metrics induced by these quasimetrics coincide with the $d_{\alpha}$ metric studied early in the study of branching structures arisen in ramified optimal transportation. An optimal transport path between two atomic probability measures typically has a "tree shaped" branching structure. Here, we show that these optimal transport paths turn out to be geodesics in these intrinsic metric spaces.

Let A be a matrix whose entries are real i.i.d. centered random variables with unit variance and suitable moment assumptions. Then the smallest singular value of A is of order n^{-1/2} with high probability. The lower estimate of this type was proved recently by the authors; in this note we establish the matching upper estimate.

We prove two basic conjectures on the distribution of the smallest singular value of random n times n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n^{-1/2}, which is optimal for Gaussian matrices. Moreover, we give a optimal estimate on the tail probability. This comes as a consequence of a new and essentially sharp estimate in the Littlewood-Offord problem: for i.i.d. random variables X_k and real numbers a_k, determine the probability P that the sum of a_k X_k lies near some number v. For arbitrary coefficients a_k of the same order of magnitude, we show that they essentially lie in an arithmetic progression of length 1/p.

We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product $v(K) = (\Vol K)(\Vol K^\circ)$ of the volume of a symmetric convex body $K \in \R^n$ and its polar body $K^\circ$. The Mahler conjecture asserts that the Mahler volume $v(K)$ is minimized (non-uniquely) when $K$ is an $n$-cube. The bottleneck conjecture (in its least general form) asserts that the volume of a certain domain $K^\diamond \subseteq K \times K^\circ$ is minimized when $K$ is an ellipsoid. It implies the Mahler conjecture up to a factor of $(\pi/4)^n \gamma_n$, where $\gamma_n$ is a monotonic factor that begins at $4/\pi$ and converges to $\sqrt{2}$. This strengthens a result of Bourgain and Milman, who showed that there is a constant $c$ such that the Mahler conjecture is true up to a factor of $c^n$. The proof uses a version of the Gauss linking integral to obtain a constant lower bound on $\Vol K^\diamond$, with equality when $K$ is an ellipsoid. It applies to a more general conjecture concerning the join of any two necks of the pseudospheres of an indefinite inner product space. Because the calculations are similar, we will also analyze traditional Gauss linking integrals in the sphere $S^{n-1}$ and in hyperbolic space $H^{n-1}$.

The smoothed analysis of algorithms is concerned with the expected running time of an algorithm under slight random perturbations of arbitrary inputs. Spielman and Teng proved that the shadow-vertex simplex method has polynomial smoothed complexity. On a slight random perturbation of an arbitrary linear program, the simplex method finds the solution after a walk on polytope(s) with expected length polynomial in the number of constraints n, the number of variables d and the inverse standard deviation of the perturbation 1/sigma. We show that the length of walk in the simplex method is actually polylogarithmic in the number of constraints n. Spielman-Teng's bound on the walk was O(n^{86} d^{55} sigma^{-30}), up to logarithmic factors. We improve this to O(log^7 n (d^9 + d^3 \s^{-4})). This shows that the tight Hirsch conjecture n-d on the length of walk on polytopes is not a limitation for the smoothed Linear Programming. Random perturbations create short paths between vertices. We propose a randomized phase-I for solving arbitrary linear programs, which is of independent interest. Instead of finding a vertex of a feasible set, we add a vertex at random to the feasible set. This does not affect the solution of the linear program with constant probability. This overcomes one of the major difficulties of smoothed analysis of the simplex method -- one can now statistically decouple the walk from the smoothed linear program. This yields a much better reduction of the smoothed complexity to a geometric quantity -- the size of planar sections of random polytopes. We also improve upon the known estimates for that size, showing that it is polylogarithmic in the number of vertices.

We investigate pseudodifferential operators on arbitrary locally compact abelian groups. As symbol classes for the Kohn-Nirenberg calculus we introduce a version of Sjoestrand's class. Pseudodifferential operators with such symbols form a Banach algebra that is closed under inversion. Since "hard analysis" techniques are not available on locally compact abelian groups, a new time-frequency approach is used with the emphasis on modulation spaces, Gabor frames, and Banach algebras of matrices. Sjoestrand's original results are thus understood as a phenomenon of abstract harmonic analysis rather than "hard analysis" and are proved in their natural context and generality.

This note deals with a problem of the probabilistic Ramsey theory in functional analysis. Given a linear operator $T$ on a Hilbert space with an orthogonal basis, we define the isomorphic structure $\Sigma(T)$ as the family of all subsets of the basis so that $T$ restricted to their span is a nice isomorphism. Our main result is a dimension-free optimal estimate of the size of $\Sigma(T)$. It improves and extends in several ways the principle of restricted invertibility due to Bourgain and Tzafriri. With an appropriate notion of randomness, we obtain a randomized principle of restricted invertibility.

We study random submatrices of a large matrix A. We show how to approximately compute A from its random submatrix of the smallest possible size O(r log r) with a small error in the spectral norm, where r = ||A||_F^2 / ||A||_2^2 is the numerical rank of A. The numerical rank is always bounded by, and is a stable relaxation of, the rank of A. This yields an asymptotically optimal guarantee in an algorithm for computing low-rank approximations of A. We also prove asymptotically optimal estimates on the spectral norm and the cut-norm of random submatrices of A. The result for the cut-norm yields a slight improvement on the best known sample complexity for an approximation algorithm for MAX-2CSP problems. We use methods of Probability in Banach spaces, in particular the law of large numbers for operator-valued random variables.

We develop an approach through geometric functional analysis to error correcting codes and to reconstruction of signals from few linear measurements. An error correcting code encodes an n-letter word x into an m-letter word y in such a way that x can be decoded correctly when any r letters of y are corrupted. We prove that most linear orthogonal transformations Q from R^n into R^m form efficient and robust robust error correcting codes over reals. The decoder (which corrects the corrupted components of y) is the metric projection onto the range of Q in the L_1 norm. An equivalent problem arises in signal processing: how to reconstruct a signal that belongs to a small class from few linear measurements? We prove that for most sets of Gaussian measurements, all signals of small support can be exactly reconstructed by the L_1 norm minimization. This is a substantial improvement of recent results of Donoho and of Candes and Tao. An equivalent problem in combinatorial geometry is the existence of a polytope with fixed number of facets and maximal number of lower-dimensional facets. We prove that most sections of the cube form such polytopes.

Large deviation estimates are by now a standard tool inthe Asymptotic Convex Geometry, contrary to small deviationresults. In this note we present a novel application of a smalldeviations inequality to a problem related to the diameters of random sections of high dimensional convex bodies. Our results imply an unexpected distinction between the lower and the upper inclusions in Dvoretzky Theorem.

In modern communication systems such as the Internet, random losses of information can be mitigated by oversampling the source. This is equivalent to expanding the source using overcomplete systems of vectors (frames), as opposed to the traditional basis expansions. Dependencies among the coefficients in frame expansions often allow for better performance comparing to bases under random losses of coefficients. We show that for any n-dimensional frame, any source can be linearly reconstructed from only (n log n) randomly chosen frame coefficients, with a small error and with high probability. Thus every frame expansion withstands random losses better (for worst case sources) than the orthogonal basis expansion, for which the (n log n) bound is attained. The proof reduces to M.Rudelson's selection theorem on random vectors in the isotropic position, which is based on the non-commutative Khinchine's inequality.

In the paper "Isoperimetry of waists and local versus global asymptotic convex geometries", it was proved that the existence of nicely bounded sections of two symmetric convex bodies K and L implies that the intersection of randomly rotated K and L is nicely bounded. In this appendix, we achieve a polynomial bound on the diameter of that intersection (in the ratio of the dimensions of the sections).

Existence of nicely bounded sections of two symmetric convex bodies K and L implies that the intersection of random rotations of K and L is nicely bounded. For L = subspace, this main result immediately yields the unexpected phenomenon: "If K has one nicely bounded section, then most sections of K are nicely bounded". This 'existence implies randomness' consequence was proved independently in [Giannopoulos, Milman and Tsolomitis]. The main result represents a new connection between the local asymptotic convex geometry (study of sections of convex bodies) and the global asymptotic convex geometry (study of convex bodies as a whole). The method relies on the new 'isoperimetry of waists' on the sphere due to Gromov.

We find a sharp combinatorial bound for the metric entropy of sets in R^n and general classes of functions. This solves two basic combinatorial conjectures on the empirical processes. 1. A class of functions satisfies the uniform Central Limit Theorem if the square root of its combinatorial dimension is integrable. 2. The uniform entropy is equivalent to the combinatorial dimension under minimal regularity. Our method also constructs a nicely bounded coordinate section of a symmetric convex body in R^n. In the operator theory, this essentially proves for all normed spaces the restricted invertibility principle of Bourgain and Tzafriri.

This is an informal discussion on one of the basic problems in the theory of empirical processes, addressed in our preprint "Combinatorics of random processes and sections of convex bodies", which is available at ArXiV and from our web pages.

Every convex body K in R^n has a coordinate projection PK that contains at least vol(0.1 K) cells of the integer lattice PZ^n, provided this volume is at least one. Our proof of this counterpart of Minkowski's theorem is based on an extension of the combinatorial density theorem of Sauer, Shelah and Vapnik-Chervonenkis to Z^n. This leads to a new approach to sections of convex bodies. In particular, fundamental results of the asymptotic convex geometry such as the Volume Ratio Theorem and Milman's duality of the diameters admit natural versions for coordinate sections.

Given a JBW*-triple Z and a normal contractive projection P : Z → Z, we show that the (Murray-von Neumann) type of each summand of P(Z) is dominated by the type of Z.

We establish an asymptotic formula for determinants of truncated Wiener-Hopf+Hankel operators with symbol equal to the exponential of a constant times the characteristic function of an interval. This is done by reducing it to the corresponding (known) asymptotics for truncated Toeplitz+Hankel operators. The determinants in question arise in random matrix theory in determining the limiting distribution for the number of eigenvalues in an interval for a scaled Laguerre ensemble of positive Hermitian matrices.

We study infinite order differential operators acting in the spaces of exponential type entire functions. We derive conditions under which such operators preserve the set of Laguerre entire functions which consists of the polynomials possessing real nonpositive zeros only and of their uniform limits on compact subsets of the complex plane. We obtain integral representations of some particular cases of these operators and apply these results to obtain explicit solutions to some Cauchy problems for diffusion equations with nonconstant drift term.

For a given class ${\cal F}$ of uniform frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation $|< f_k,f_l >|$ among all frames $\{f_k\}_{k \in {\cal I}} \in {\cal F}$. We first analyze finite-dimensional Grassmannian frames. Using links to packings in Grassmannian spaces and antipodal spherical codes we derive bounds on the minimal achievable correlation for Grassmannian frames. These bounds yield a simple condition under which Grassmannian frames coincide with uniform tight frames. We exploit connections to graph theory, equiangular line sets, and coding theory in order to derive explicit constructions of Grassmannian frames. Our findings extend recent results on uniform tight frames. We then introduce infinite-dimensional Grassmannian frames and analyze their connection to uniform tight frames for frames which are generated by group-like unitary systems. We derive an example of a Grassmannian Gabor frame by using connections to sphere packing theory. Finally we discuss the application of Grassmannian frames to wireless communication and to multiple description coding.

We show that $(g_2,a,b)$ is a Gabor frame when $a>0, b>0, ab0$. This is accomplished by expressing the Zak transform of $g_2$ in terms of the Zak transform of the Gaussian $g_1(t)=(2\gamma)^{{1/4}} \exp (-\pi \gamma t^2)$, together with an appropriate use of the Ron-Shen criterion for being a Gabor frame. As a side result it follows that the windows, generating tight Gabor frames, that are canonically associated to $g_2$ and $g_1$ are the same at critical density $a=b=1$. Also, we display the ``singular'' dual function corresponding to the hyperbolic secant at critical density.

The Vapnik-Chervonenkis dimension of a set K in R^n is the maximal dimensionof the coordinate cube of a given size, which can be found in coordinateprojections of K. We show that the VC dimension of a convex body governs itsentropy. This has a number of consequences, including the optimal Elton'stheorem and a uniform central limit theorem in the real valued case.