Proof of the BMV conjecture

We prove the BMV (Bessis, Moussa, Villani, [1]) conjecture, which states that the function ${t \mapsto \mathop{\rm Tr}\exp(A-tB)}$ , ${t \geqslant 0}$ , is the Laplace transform of a positive measure on [0,∞) if A and B are ${n \times n}$ Hermitian matrices and B is positive semidefinite. A semi-explicit representation for this measure is given.     

Average Best m-term Approximation

We introduce the concept of average best m-term approximation widths with respect to a probability measure on the unit ball or the unit sphere of $\ell_{p}^{n}$ . We estimate these quantities for the embedding $id:\ell_{p}^{n}\to\ell_{q}^{n}$ with 0<p??q??? for the normalized cone and surface measure. Furthermore, we consider certain tensor product weights and show that a typical vector with respect to such a measure exhibits a strong compressible (i.e., nearly sparse) structure. This measure may therefore be used as a random model for sparse signals.     

{SL(2, \mathbb{R})} -invariant probability measures on the moduli spaces of translation surfaces are regular

In the moduli space ${{\mathcal {H}}_g}$ of normalized translation surfaces of genus g, consider, for a small parameter ρ > 0, those translation surfaces which have two non-parallel saddle-connections of length ? ρ. We prove that this subset of ${{\mathcal {H}}_g}$ has measure o(ρ ²) w.r.t. any probability measure on ${{\mathcal {H}}_g}$ which is invariant under the natural action of ${SL(2,\mathbb{R})}$ . This implies that any such probability measure is regular, a property which is important in relation with the recent fundamental work of Eskin–Kontsevich–Zorich on the Lyapunov exponents of the KZ-cocycle.     

The type and stable type of the boundary of a Gromov hyperbolic group

Consider an ergodic non-singular action \(\Gamma \curvearrowright B\) of a countable group on a probability space. The type of this action codes the asymptotic range of the Radon–Nikodym derivative, also called the ratio set. If \(\Gamma \curvearrowright X\) is a pmp (probability-measure-preserving) action, then the ratio set of the product action \(\Gamma \curvearrowright B\times X\) is contained in the ratio set of \(\Gamma \curvearrowright B\) . So we define the stable ratio set of \(\Gamma \curvearrowright B\) to be the intersection over all pmp actions \(\Gamma \curvearrowright X\) of the ratio sets of \(\Gamma \curvearrowright B\times X\) . By analogy, there is a notion of stable type which codes the stable ratio set of \(\Gamma \curvearrowright B\) . This concept is crucially important for the identification of the limit in pointwise ergodic theorems established by the author and Amos Nevo. Here, we establish a general criteria for a nonsingular action of a countable group on a probability space to have stable type \(III_\lambda \) for some \(\lambda >0\) . This is applied to show that the action of a non-elementary Gromov hyperbolic group on its boundary with respect to a quasi-conformal measure is not type \(III_0\) and, if it is weakly mixing, then it is not stable type \(III_0\) .     

Nonuniqueness of a Gibbs measure for the Ising ball model

We study a new model, the so-called Ising ball model on a Cayley tree of order k ≥ 2. We show that there exists a critical activity \(\lambda _{cr} = \sqrt[4]{{0.064}}\) such that at least one translation-invariant Gibbs measure exists for λ ≥ λ _cr , at least three translation-invariant Gibbs measures exist for 0 < λ < λ _cr , and for some λ, there are five translation-invariant Gibbs measures and a continuum of Gibbs measures that are not translation invariant. For any normal divisor \(\hat G\) of index 2 of the group representation on the Cayley tree, we study \(\hat G\) -periodic Gibbs measures. We prove that there exists an uncountable set of \(\hat G\) -periodic (not translation invariant and “checkerboard” periodic) Gibbs measures.     

On the Number of Facets of Polytopes Representing Comparative Probability Orders

Fine and Gill (Ann Probab 4:667–673, 1976) introduced the geometric representation for those comparative probability orders on n atoms that have an underlying probability measure. In this representation every such comparative probability order is represented by a region of a certain hyperplane arrangement. Maclagan (Order 15:279–295, 1999) asked how many facets a polytope, which is the closure of such a region, might have. We prove that the maximal number of facets is at least F _n?+?1, where F _n is the nth Fibonacci number. We conjecture that this lower bound is sharp. Our proof is combinatorial and makes use of the concept of a flippable pair introduced by Maclagan. We also obtain an upper bound which is not too far from the lower bound.     

Orbit structure of a distinguished Stein invariant domain in the complexification of a Hermitian symmetric space

Recently, Bruinier and Ono proved that the coefficients of certain weight \(-1/2\) harmonic weak Maaß forms are given as “traces” of singular moduli for harmonic weak Maaß forms. Here, we prove that similar results hold for the coefficients of harmonic weak Maaß forms of weight \(3/2+k\) , \(k\) even, and weight \(1/2-k\) , \(k\) odd, by extending the theta lift of Bruinier–Funke and Bruinier–Ono. Moreover, we generalize these results to include twisted traces of singular moduli using earlier work of the author and Ehlen on the twisted Bruinier–Funke-lift. Employing a general duality result between weight \(k\) and \(2-k\) , we obtain formulas for all half-integral weights. We also show that the non-holomorphic part of the theta lift in weight \(1/2-k\) , \(k\) odd, is connected to the vanishing of the special value of the \(L\) -function of a certain derivative of the lifted function.     

Coinvariants and the regular representation of a cyclic P-group

We consider an indecomposable representation of a cyclic p-group ${Z_{p^r}}$ over a field of characteristic p. We show that the top degree of the corresponding ring of coinvariants is less than ${\frac{(r^2+3r)p^r}{2}}$ . This bound also applies to the degrees of the generators for the invariant ring of the regular representation.     

Characteristic functions as distributional boundary values of analytic functions in tube domains

We propose necessary and sufficient conditions for a complex-valued function f on \( {{\mathbb{R}}^n} \) to be a characteristic function of a probability measure. Certain analytic extensions of f to tubular domains in \( {{\mathbb{C}}^n} \) are studied. In order to extend the class of functions under study, we also consider the case where f is a generalized function (distribution). The main result is given in terms of completely monotonic functions on convex cones in \( {{\mathbb{R}}^n} \) .     

A structural characterization of numéraires of convex sets of nonnegative random variables

We introduce the concept of numéraire s of convex sets in ${L^0_{+}}$ , the nonnegative orthant of the topological vector space L ⁰ of all random variables built over a probability space. A necessary and sufficient condition for an element of a convex set ${\mathcal{C} \subseteq L^0_{+}}$ to be a numéraire of ${\mathcal{C}}$ is given, inspired from ideas in financial mathematics.     

Homeomorphisms of the annulus with a transitive lift

Let f be a homeomorphism of the closed annulus A that preserves the orientation, the boundary components and that has a lift ${\tilde{f}}$ to the infinite strip à which is transitive. We show that, if the rotation numbers of both boundary components of A are strictly positive, then there exists a closed nonempty unbounded set ${B^{-} \subset \tilde{A}}$ such that B ^? is bounded to the right, the projection of B ^? to A is dense, ${B^{-}-(1, 0) \subset B^{-}}$ and ${\tilde{f}(B^{-}) \subset B^{-}}$ . Moreover, if p ₁ is the projection on the first coordinate of Ã, then there exists d > 0 such that, for any ${\tilde z \in B^{-}}$ , $$\limsup_{n\to\infty}\frac{p_1(\tilde f^n(\tilde z))-p_1(\tilde z)}{n}<-d.$$ In particular, using a result of Franks, we show that the rotation set of any homeomorphism of the annulus that preserves orientation, boundary components, which has a transitive lift without fixed points in the boundary is an interval with 0 in its interior.     

Computable representations for convex hulls of low-dimensional quadratic forms

Let ${\mathcal{C}}$ be the convex hull of points ${{\{{1 \choose x}{1 \choose x}^T \,|\, x\in \mathcal{F}\subset \Re^n\}}}$ . Representing or approximating ${\mathcal{C}}$ is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables. We show that if n ≤ 4 and ${\mathcal{F}}$ is a simplex, then ${\mathcal{C}}$ has a computable representation in terms of matrices X that are doubly nonnegative (positive semidefinite and componentwise nonnegative). We also prove that if n = 2 and ${\mathcal{F}}$ is a box, then ${\mathcal{C}}$ has a representation that combines semidefiniteness with constraints on product terms obtained from the reformulation-linearization technique (RLT). The simplex result generalizes known representations for the convex hull of ${{\{(x_1, x_2, x_1x_2)\,|\, x\in\mathcal{F}\}}}$ when ${\mathcal{F}\subset\Re^2}$ is a triangle, while the result for box constraints generalizes the well-known fact that in this case the RLT constraints generate the convex hull of ${{\{(x_1, x_2, x_1x_2)\,|\, x\in\mathcal{F}\}}}$ . When n = 3 and ${\mathcal{F}}$ is a box, we show that a representation for ${\mathcal{C}}$ can be obtained by utilizing the simplex result for n = 4 in conjunction with a triangulation of the 3-cube.     

On the representation of tight functionals as integrals

We consider a vector lattice $\mathcal L $ of bounded real continuous functions on a topological space $X$ that separates the points of $X$ and contains the constant functions. A notion of tightness for linear functionals is defined, and by an elementary argument we prove with the aid of the classical Riesz representation theorem that every tight continuous linear functional on $\mathcal L $ can be represented by integration with respect to a Radon measure. This result leads incidentally to an simple proof of Prokhorov’s existence theorem for the limit of a projective system of Radon measures.     

Donsker-type theorems for nonparametric maximum likelihood estimators

Let ${\mathcal{P}}$ be a nonparametric probability model consisting of smooth probability densities and let ${\hat{p}_{n}}$ be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law ${\mathbb{P}}$ . With $\hat{\mathbb{P}}_{n}$ denoting the measure induced by the density ${\hat{p}_{n}}$ , define the stochastic process ${\hat{\nu}}_{n}: f\longmapsto \sqrt{n} \int fd({\hat{\mathbb{P}}}_{n} -\mathbb{P})$ where f ranges over some function class ${\mathcal{F}}$ . We give a general condition for Donsker classes ${\mathcal{F}}$ implying that the stochastic process $\hat{\nu}_{n}$ is asymptotically equivalent to the empirical process in the space ${\ell ^{\infty }(\mathcal{F})}$ of bounded functions on ${ \mathcal{F}}$ . This implies in particular that $\hat{\nu}_{n}$ converges in law in ${\ell ^{\infty }(\mathcal{F})}$ to a mean zero Gaussian process. We verify the general condition for a large family of Donsker classes ${\mathcal{ F}}$ . We give a number of applications: convergence of the probability measure ${\hat{\mathbb{P}}_{n}}$ to ${\mathbb{P}}$ at rate ${\sqrt{n}}$ in certain metrics metrizing the topology of weak(-star) convergence; a unified treatment of convergence rates of the MLE in a continuous scale of Sobolev-norms; ${\sqrt{n}}$ -efficient estimation of nonlinear functionals defined on ${\mathcal{P}}$ ; limit theorems at rate ${\sqrt{n}}$ for the maximum likelihood estimator of the convolution product ${\mathbb{P\ast P}}$ .     

Martin representation and Relative Fatou Theorem for fractional Laplacian with a gradient perturbation

Let $L=\Delta ^{\alpha /2}+ b\cdot \nabla $ with $\alpha \in (1,2)$ . We prove the Martin representation and the Relative Fatou Theorem for non-negative singular L-harmonic functions on $\mathcal{C }^{1,1}$ bounded open sets.     

Comparison of Different Definitions of Traces for a Class of Ramified Domains with Self-Similar Fractal Boundaries

We consider a class of ramified bidimensional domains Ω with a self-similar fractal boundary Γ^?∞?, which is supplied with a probability measure μ called the self-similar measure. Emphasis is put on the case when the domain is not a ε???δ domain as defined by Jones and the fractal set is not totally disconnected. We compare two notions of trace on Γ^?∞? for functions in W ^1,q (Ω): the classical one, see for instance the book by Jonnson and Wallin, 1984, using the strict definition of a function at a point of $\overline{\Omega}$ , and another one proposed in 2007 and heavily relying on self-similarity. We prove that the two traces coincide μ-almost everywhere on Γ^?∞?. As a corollary, we characterize the critical number $\bar q$ for which for all $q<\bar q$ (resp. $q > \bar q$ ) there is a (resp. no) continuous extension operator from W ^1,q (Ω) to W ^1,q (?²).     

Cabling procedure for the colored HOMFLY polynomials

We discuss using the cabling procedure to calculate colored HOMFLY polynomials. We describe how it can be used and how the projectors and $\mathcal{R}$ -matrices needed for this procedure can be found. The constructed matrix expressions for the projectors and $\mathcal{R}$ -matrices in the fundamental representation allow calculating the HOMFLY polynomial in an arbitrary representation for an arbitrary knot. The computational algorithm can be used for the knots and links with ¦Q¦m ≤ 12, where m is the number of strands in a braid representation of the knot and ¦Q¦ is the number of boxes in the Young diagram of the representation. We also discuss the justification of the cabling procedure from the group theory standpoint, deriving expressions for the fundamental $\mathcal{R}$ -matrices and clarifying some conjectures formulated in previous papers.     

The 0-Homogenous Complete Lift Metric

The complete lift of a Riemannian metric g on a differentiable manifold M is not 0-homogeneous on the fibers of the tangent bundle TM. In this paper we introduce a new kind of lift G of g, which is 0-homogeneous. It determines a pseudo-Riemannian metric on ${\widetilde {TM}}$ , which depends only on the metric g. We obtain the Levi-Civita connection of this metric and study conformal vector fields on ( ${\widetilde {TM},G}$ ). Finally, we introduce the almost product and complex structures which preserve homogeneity and study certain geometrical properties of these structures.