The Corona theorem and stable rank for the algebra $\mC +BH^\infty$

Abstract: Let $B$ be a Blaschke product. We prove in several different ways the corona theorem for the algebra $H^\infty_B:=\mC+BH^\infty$. That is, we show the equivalence of the classical {\em corona condition} on data $f_1, ..., f_n \in H^\infty_B$: \[ \forall z \in \mD, \sum_{k=1}^{n} |f_k(z)| \geq \delta >0, \] and the {\em solvability of the Bezout equation} for $g_1, ..., g_n \in H^\infty_B$: \[ \forall z\in \mD, \sum_{k=1}^n g_k (z)f_k(z)=1. \] Estimates on solutions to the Bezout equation are also obtained. We also show that the Bass stable rank of $H^\infty_B$ is 1. Let $A(\mD)_B$ be the subalgebra of all elements from $H^\infty_B$ having a continuous extension to the closed unit disk $\bar{\mD}$. Analogous results are obtained also for $A(\mD)_B$.


Kahler geometry on toric manifolds, and some other manifolds with large symmetry

Abstract: This is an expository article. Among other topics, we discuss the existence of Kahler-Ricci soliton metrics on toric Fano manifolds, and Kahler-Einstein metrics on deformations of the Mukai-Umemura 3-fold


Numerical approximations to extremal metrics on toric surfaces

Abstract: We give some detailed numerical information about extremal metrics on four different toric surfaces. These are sample of many other cases which can be treated using a computer programme outlined in the paper.


Easy Proofs of Some Borwein Algorithms for $\pi$

Abstract: In 1987 Jonathan and Peter Borwein, inspired by the works of Ramanujan, derived many efficient algorithms for computing $\pi$. We will see that by using only a formula of Gauss's and elementary algebra we are able to prove the correctness of two of them.


AB-Contexts and Stability for Gorenstein Flat Modules with Respect to Semidualizing Modules

Abstract: We investigate the properties of categories of G_C-flat R-modules where C is a semidualizing module over a commutative noetherian ring R. We prove that the category of all G_C-flat R-modules is part of a weak AB-context, in the terminology of Hashimoto. In particular, this allows us to deduce the existence of certain Auslander-Buchweitz approximations for R-modules of finite G_C-flat dimension. We also prove that two procedures for building R-modules from complete resolutions by certain subcategories of G_C-flat R-modules yield only the modules in the original subcategories.


Lipschitz perturbations of differentiable implicit functions

Abstract: Let $y=f(x)$ be a continuously differentiable implicit function solving the equation $F(x,y)=0$ with continuously differentiable $F.$ In this paper we show that if $F_\eps$ is a Lipschitz function such that the Lipschitz constant of $F_\eps-F$ goes to 0 as $\eps\to 0$ then the equation $F_\eps(x,y)=0$ has a Lipschitz solution $y=f_\eps(x)$ such that the Lipschitz constant of $f_\eps-f$ goes to 0 as $\eps\to 0$ either. As an application we evaluate the length of time intervals where the right hand parts of some nonautonomous discontinuous systems of ODEs are continuously differentiable with respect to state variables. The latter is done as a preparatory step toward generalizing the second Bogolyubov's theorem for discontinuous systems.


On Mitropol'skii Yu.A.'s Theorem on Periodic Solutions of Systems of Nonlinear Differential Equations with Non-Differentiable Right-Hand-Sides

Abstract: The smooth second Bogolyubov's theorem is generalized for Lipschitz systems.


Quantitative uniqueness for the power of Laplacian with singular coefficients

Abstract: In this paper we study the local behavior of a solution to the $l$th power of Laplacian with singular coefficients in lower order terms. We obtain a bound on the vanishing order of the nontrivial solution. Our proofs use Carleman estimates with carefully chosen weights. We will derive appropriate three-sphere inequalities and apply them to obtain doubling inequalities and the maximal vanishing order.


Positroids and Schubert matroids

Abstract: Postnikov gave a combinatorial description of the cells in a totally-nonnegative Grassmannian. These cells correspond to a special class of matroids called positroid. We prove his conjecture that a positroid is exactly an intersection of permuted Schubert matroids. This leads to a nice combinatorial description of positroids that is easily computable.


Geometric second derivative estimates in Carnot groups and convexity

Abstract: We prove some new a priori estimates for H_2-convex functions which are zero on the boundary of a bounded smooth domain \Omega in a Carnot group G. Such estimates are global and are geometric in nature as they involve the horizontal mean curvature \mathcal H of the boundary of \Omega. As a consequence of our bounds we show that if G has step two, then for any smooth $H_2$-convex function in \Omega \subset G vanishing on the boundary of \Omega one has \sum_{i,j=1}^m \int_\Omega ([X_i,X_j]u)^2 dg \leq {4/3} \int_{\partial \Omega} \mathcal H |\nabla_H u|^2 d\sigma_H .


Multiple integral representation for functionals of Dirichlet processes

Abstract: We point out that a proper use of the Hoeffding--ANOVA decomposition for symmetric statistics of finite urn sequences, previously introduced by the author, yields a decomposition of the space of square-integrable functionals of a Dirichlet--Ferguson process, written $L^2(D)$, into orthogonal subspaces of multiple integrals of increasing order. This gives an isomorphism between $L^2(D)$ and an appropriate Fock space over a class of deterministic functions. By means of a well-known result due to Blackwell and MacQueen, we show that each element of the $n$th orthogonal space of multiple integrals can be represented as the $L^2$ limit of $U$-statistics with degenerate kernel of degree $n$. General formulae for the decomposition of a given functional are provided in terms of linear combinations of conditioned expectations whose coefficients are explicitly computed. We show that, in simple cases, multiple integrals have a natural representation in terms of Jacobi polynomials. Several connections are established, in particular with Bayesian decision problems, and with some classic formulae concerning the transition densities of multiallele diffusion models, due to Littler and Fackerell, and Griffiths. Our results may also be used to calculate the best approximation of elements of $L^2(D)$ by means of $U$-statistics of finite vectors of exchangeable observations.


Gorenstein polytopes obtained from bipartite graphs

Abstract: Beck et. al. characterized the grid graphs whose perfect matching polytopes are Gorenstein and they also showed that for some parameters, perfect matching polytopes of torus graphs are Gorenstein. In this paper, we complement their result, that is, we characterize the torus graphs whose perfect matching polytopes are Gorenstein. Beck et. al. also gave a method to construct an infinite family of Gorenstein polytopes. In this paper, we introduce a new class of polytopes obtained from graphs and we extend their method to construct many more Gorenstein polytopes.


TQFT string operations in open-closed string topology

Abstract: To open-closed cobordism surfaces, open-closed string topology associates topological quantum field theory (TQFT) operations, namely string operations, which depend only on homeomorphism types of surfaces and which satisfy the sewing property. We show that most TQFT string operations vanish in open-closed string topology. We describe those open-closed cobordisms with vanishing string operations, and give a short list of open-closed cobordisms with possibly nontrivial string operations.


Open-Closed TQFT String Operations for Disc Cobordisms, Simultaneous Saddle Interactions, and Constant Homology Classes

Abstract: Previously, we showed that most of the open-closed topological quantum field theory (TQFT) string operations vanish including all the higher genus TQFT operations, and we described a small list of genus zero open-closed TQFT string operations which can be nontrivial. In this paper, we consider open-closed string operations associated to open-closed cobordisms homeomorphic to discs. These operations constitute the main part of genus zero string operations, and they include the saddle string operation of two open strings interacting at their internal points. We show not only that disc string operations are independent of their half-pair-of-pants decompositions but also that these disc string operations can be computed by \emph{simultaneous} saddle interactions of incoming open strings at the same point, and they take values in homology classes of constant open strings on some closed orientable submanifolds, which we will precisely determine. We will also discuss a role played by fundamental constant homology classes in open-closed string topology. Our main tools are saddle interaction diagrams and their deformations.


Conformal arc-length as $\frac12$ dimensional length of the set of osculating circles

Abstract: The set of osculating circles of a given curve in $\SS^3$ forms a curve in the set of oriented circles in $\SS^3$. We show that its "${\frac12}$-dimensional measure" with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a conformally invariant local quantity discovered in the first half of the last century.


Random motion with gamma-distributed alternating velocities in biological modeling

Abstract: Motivated by applications in mathematical biology concerning randomly alternating motion of micro-organisms, we analyze a generalized integrated telegraph process. The random times between consecutive velocity reversals are gamma-distributed, and perform an alternating renewal process. We obtain the probability law and the mean of the process.


Moduli spaces of reflexive sheaves of rank 2

Abstract: Let \sF be a coherent rank 2 sheaf on a scheme Y \subset \proj{n} of dimension at least two. In this paper we study the relationship between the functor which deforms a pair (\sF,\sigma), \sigma \in H^0(\sF), and the functor which deforms the corresponding pair (X,\xi) given as in the Serre correspondence. We prove that the scheme structure of e.g. the moduli scheme M_Y(P) of stable sheaves on a threefold Y at (\sF), and the scheme structure at (X) of the Hilbert scheme of curves on Y are closely related. Using this relationship we get criteria for the dimension and smoothness of M_Y(P) at (\sF), without assuming Ext^2(\sF,\sF) = 0. For reflexive sheaves on Y = \proj{3} whose deficiency module M = H_{*}^1(\sF) satisfies Ext^2(M,M) = 0 in degree zero (e.g. of diameter at most 2), we get necessary and sufficient conditions of unobstructedness which coincide in the diameter one case. The conditions are further equivalent to the vanishing of certain graded Betti numbers of the free graded minimal resolution of H_{*}^0(\sF). It follows that every irreducible component of M_{\proj{3}}(P) containing a reflexive sheaf of diameter one is reduced (generically smooth). We also determine a good lower bound for the dimension of any component of M_{\proj{3}}(P) which contains a reflexive stable sheaf with "small" deficiency module M.


Depth of segments and circles through points enclosing many points: a note

Abstract: Neumann-Lara and Urrutia showed in 1985 that in any set of n points in the plane in general positionthere is always a pair of points such that any circle through them contains at least (n-2)/60 points. In a series of papers, this result was subsequently improved till n/4.7, which is currently the best known lower bound. In this paper we propose a new approach to the problem that allows us, by using known results about j-facets of sets of points in $R^3$, to give a simple proof of a somehow stronger result: there is always a pair of points such that any circle through them has, both inside and outside, at least n/4.7 points.


A family of representations of braid groups on surfaces

Abstract: We propose a family of new representations of the braid groups on surfaces that extend linear representations of the braid groups on a disc such as the Burau representation and the Lawrence-Krammer-Bigelow representation.


Quelques in\'egalit\'es effectives entre des fonctions arithm\'etiques usuelles

Abstract: Let us denote by $\tau(n)$ and $\si(n)$ the number and the sum of the divisors of $n$ and by $\vfi$ Euler's function. We give effective upper bounds for $\frac{n}{\vfi(n)}$ in terms of $\vfi(n)$, and for $\frac{\si(n)}{n}$ in terms of $\tau(n)$.


Single-index Regression models with right-censored responses

Abstract: In this article, we propose some new generalizations of M-estimation procedures for single-index regression models in presence of randomly right-censored responses. We derive consistency and asymptotic normality of our estimates. The results are proved in order to be adapted to a wide range of techniques used in a censored regression framework (e.g. synthetic data or weighted least squares). As in the uncensored case, the estimator of the single-index parameter is seen to have the same asymptotic behavior as in a fully parametric scheme. We compare these new estimators with those based on the average derivative technique of Burke and Lu (2005) through a simulation study.


On (twisted) Lawrence-Krammer representations

Abstract: Lawrence-Krammer representations (LK-representations for short) are linear representations of Artin-Tits groups of small type, which are of importance since they are known to be faithful when the type is spherical, or more generally when restricted to the monoid. If the construction is essentially unique for a given small and spherical type, the structure of the set of LK-representations for a given small type is not understood in general. Another important question is to ask if there exists an analogue of this construction in the non-small cases ; a first answer is given in [Digne, On the linearity of Artin Braid groups. J. Algebra 268, (2003) 39-57], where is constructed a faithful ``twisted'' LK-representation for the spherical, non-small and crystallographic types. The aim of this paper is to continue the investigations on those two topics. Regarding the first one, we classify the LK-representations of the Artin-Tits monoids and groups of small and affine type. Concerning the second one, we generalize the construction of op.cit. to any Artin-Tits monoid that appears as the submonoid of fixed points of an Artin-Tits monoid of small type under the action of graph automorphisms.


0-Cohomology of semigroups

Abstract: This article is a survey of 0-cohomology of semigroups. The main attention is devoted to applications.


The Divisor Matrix, Dirichlet Series and SL(2,Z), II

Abstract: We examine an elliptic curve constructed in an earlier paper from a certain representation of $\SL(2,\Z)$ on the space of convergent Dirichlet series. The curve is observed to be a modular curve for $\Gamma^1(15)$ and a certain orbit of modular functions is thereby associated with the Riemann zeta function. Explicit descriptions are given of these functions and of the permutation action of $\SL(2,\Z)$ on them.


A note on Larsen's conjecture and ranks of elliptic curves

Abstract: Let E be an elliptic curve defined over a number field K. Michael Larsen conjectured that for any finitely generated subgroup G of Gal(\bar K/K), the Mordell-Weil rank of E is unbounded in number fields fixed by G. We prove that the conjecture holds over K=Q for both the analytic rank and the p-infinity Selmer rank of E for every odd prime p. For arbitrary E/K, we show that Larsen's conjecture follows from the standard conjectures for ranks of elliptic curves, provided K has a real place or E has non-integral j-invariant.


Equivalence for Differential Equations

Abstract: We shall study the equivalence problem for ordinary differential equations with respect to the affine transformations group.


Plane recursive trees, Stirling permutations and an urn model

Abstract: We exploit a bijection between plane recursive trees and Stirling permutations; this yields the equivalence of some results previously proven separately by different methods for the two types of objects as well as some new results. We also prove results on the joint distribution of the numbers of ascents, descents and plateaux in a random Stirling permutation. The proof uses an interesting generalized Polya urn


Opposite relation on dual polar spaces and half-spin Grassmann spaces

Abstract: We characterize the collinearity (adjacency) relation of half-spin Grassmann spaces in terms of the relation to be opposite in the corresponding collinearity graphs. Also we show that this characterization does not hold for dual polar spaces.


On the Gorenstein locus of some punctual Hilbert schemes

Abstract: Let $k$ be an algebraically closed field and let $\Hilb_{d}^{G}(\p{N})$ be the open locus of the Hilbert scheme $\Hilb_{d}(\p{N})$ corresponding to Gorenstein subschemes. We prove that $\Hilb_{d}^{G}(\p{N})$ is irreducible for $d\le9$, we characterize geometrically its singularities for $d\le 8$ and we give some results about them when $d=9$ which give some evidence to a conjecture on the nature of the singular points in $\Hilb_{d}^{G}(\p{N})$.


A short proof of nonhomogeneity of the pseudo-circle

Abstract: The pseudo-circle is known to be nonhomogeneous. The original proofs of this fact were discovered independently by L. Fearnley and J.T. Rogers, Jr. The purpose of this paper is to provide an alternative, very short proof based on a result of D. Bellamy and W. Lewis.


A combinatorial formula for Macdonald polynomials

Abstract: In this paper we use the combinatorics of alcove walks to give a uniform combinatorial formula for Macdonald polynomials for all Lie types. These formulas are generalizations of the formulas of Haglund-Haiman-Loehr for Macdonald polynoimals of type GL(n). At q=0 these formulas specialize to the formula of Schwer for the Macdonald spherical function in terms of positively folded alcove walks and at q=t=0 these formulas specialize to the formula for the Weyl character in terms of the Littelmann path model (in the positively folded gallery form of Gaussent-Littelmann).


Regular Points of a Subcartesian Space

Abstract: We discuss properties of the regular part $S_{reg}$ of a subcartesian space $S$. We show that $S_{reg}$ is open and dense in $S$ and the restriction to $S_{reg}$ of the tangent bundle of $S$ is locally trivial.


Plactic relations for $r$-domino tableaux

Abstract: The work of C. Bonnaf{\'e}, M. Geck, L. Iancu and T. Lam \cite{Geck-Lam} shows through two conjectures that $r$-domino tableaux have an important role in Kazhdan-Lusztig theory of type $B$ with unequal parameters. In this paper we provide plactic relations on signed permutations which determine whether given two signed permutations have the same insertion $r$-domino tableaux in Garfinkle's algorithm \cite{Garfinkle1}.


Pre-Hausdorff Spaces

Abstract: This paper introduces three separation conditions for topological spaces, called T_{0,1}, T_{0,2} ("pre-Hausdorff"), and T_{1,2}. These conditions generalize the classical T_(1) and T_(2) separation axioms, and they have advantages over them topologically which we discuss. We establish several different characterizations of pre-Hausdorff spaces, and a characterization of Hausdorff spaces in terms of pre-Hausdorff. We also discuss some classical Theorems of general topology which can or cannot be generalized by replacing the Hausdorff condition by pre-Hausdorff.


On Lagrangian fibrations by Jacobians I

Abstract: Let Y->P^n be a flat family of integral Gorenstein curves, such that the compactified relative Jacobian X=\bar{J}^d(Y/P^n) is a Lagrangian fibration. We prove that the degree of the discriminant locus Delta in P^n is at least 4n+2, and we prove that X is a Beauville-Mukai integrable system if the degree of Delta is greater than 4n+20.


The Dolbeault complex with weights according to normal crossings

Abstract: In this paper, we define a Dolbeault complex with weights according to normal crossings, which is a useful tool for studying the d-bar-equation on singular complex spaces by resolution of singularities (where normal crossings appear naturally). The major difficulty is to prove that this complex is locally exact. We do that by constructing a local d-bar-solution operator which involves only Cauchy's Integral Formula (in one complex variable) and behaves well for L^p-forms with weights according to normal crossings.


The d-bar-equation on homogeneous varieties with an isolated singularity

Abstract: Let X be a regular irreducible variety in CP^{n-1}, Y the associated homogeneous variety in C^n, and N the restriction of the universal bundle of CP^{n-1} to X. In the present paper, we compute the obstructions to solving the d-bar-equation in the L^p-sense on Y for 1<=p<=\infty in terms of cohomology groups H^q(X,O(N^m)). That allows to identify obstructions explicitly if X is specified more precisely, for example if it is equivalent to CP^1 or an elliptic curve.


Stability and instability results in a model of Fermi acceleration

Abstract: We consider the static wall approximation to the dynamics of a particle bouncing on a periodically oscillating infinitely heavy plate while subject to a potential force. We assume the case of a potential given by a power of the particle's height and sinusoidal motions of the plate. We find that for powers smaller than 1 the set of escaping orbits has full Hausdorff dimension for all motions and obtain existence of elliptic island of period 2 for arbitrarily high energies for a full-measure set of motions. Moreover we obtain conditions on the potential to ensure that the total (Lebesgue) measure of elliptic islands of period 2 is either finite or infinite.


The finiteness result for Khovanov homology and localization in monoidal categories

Abstract: In the previous paper we constructed the local system of Khovanov complexes on the Vassiliev space of knots and extended it to the singular locus. In this paper we introduce the definition of the homology theory (local system) of finite type and prove the first finiteness result: the Khovanov local system restricted to the subcategory of knots of the crossing number at most n is the theory of type less or equal to n. This result can be further generalized to the categorification of Birman-Lin theorem.


Distributed Subgradient Methods and Quantization Effects

Abstract: We consider a convex unconstrained optimization problem that arises in a network of agents whose goal is to cooperatively optimize the sum of the individual agent objective functions through local computations and communications. For this problem, we use averaging algorithms to develop distributed subgradient methods that can operate over a time-varying topology. Our focus is on the convergence rate of these methods and the degradation in performance when only quantized information is available. Based on our recent results on the convergence time of distributed averaging algorithms, we derive improved upper bounds on the convergence rate of the unquantized subgradient method. We then propose a distributed subgradient method under the additional constraint that agents can only store and communicate quantized information, and we provide bounds on its convergence rate that highlight the dependence on the number of quantization levels.


Picard number of the generic fiber of an abelian fibered hyperkaehler manifold

Abstract: We shall show that the Picard number of the generic fiber, in the sense of scheme, of an abelian fibered hyperk\"ahler manifold over the projective space is always one. We then give a few applications for the Mordell-Weil group. In particular, by deforming O'Grady's 10-dimensional manifold, we construct an abelian fibered hyperk\"ahler manifold of Mordell-Weil rank 20, which is the maximum possible among all known ones.


Systems with the integer rounding property in normal monomial subrings

Abstract: Let C be a clutter and let A be its incidence matrix. If the linear system x>=0;xA<=1 has the integer rounding property, we give a description of the canonical module and the a-invariant of certain normal subrings associated to C. If the clutter is a connected graph, we describe when the aforementioned linear system has the integer rounding property in combinatorial and algebraic terms using graph theory and the theory of Rees algebras. As a consequence we show that the extended Rees algebra of the edge ideal of a bipartite graph is Gorenstein if and only if the graph is unmixed.


A stochastic-Lagrangian particle system for the Navier-Stokes equations

Abstract: This paper is based on a formulation of the Navier-Stokes equations developed by P. Constantin and the first author (\texttt{arxiv:math.PR/0511067}, to appear), where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. In this paper, we take $N$ copies of the above process (each based on independent Wiener processes), and replace the expected value with $\frac{1}{N}$ times the sum over these $N$ copies. (We remark that our formulation requires one to keep track of $N$ stochastic flows of diffeomorphisms, and not just the motion of $N$ particles.) We prove that in two dimensions, this system of interacting diffeomorphisms has (time) global solutions with initial data in the space $\holderspace{1}{\alpha}$ which consists of differentiable functions whose first derivative is $\alpha$ H\"older continuous (see Section \ref{sGexist} for the precise definition). Further, we show that as $N \to \infty$ the system converges to the solution of Navier-Stokes equations on any finite interval $[0,T]$. However for fixed $N$, we prove that this system retains roughly $O(\frac{1}{N})$ times its original energy as $t \to \infty$. Hence the limit $N \to \infty$ and $T\to \infty$ do not commute. For general flows, we only provide a lower bound to this effect. In the special case of shear flows, we compute the behaviour as $t \to \infty$ explicitly.


Parameter Collapse due to the Zeros in the Inverse Condition

Abstract: Helton, Lasserre, and Putinar (2008, Ann. Probability; arXiv:math/0702314) expose the relationship between three properties of a measure: the conditional triangularity property of the associated orthogonal polynomials, the zeros in the inverse condition of the truncated moment matrix, and conditional independence. The purpose of this article is to provide examples of parameter collapse to product structure given that the zeros in the inverse condition holds up to some degree d. Specifically, start with a parameterized family of probability density functions; require that the zeros in the inverse condition up to degree d holds; and validate that imposing this restriction on the parameterized family results in a measure with product structure, or at least that conditional independence holds. Algorithms related to parameter collapse are supplied, including the computation of the zeros in the inverse condition up to degree d.


A new class of transport distances

Abstract: We introduce a new class of distances between nonnegative Radon measures in Euclidean spaces. They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances proposed by Benamou-Brenier and provide a wide family interpolating between the Wasserstein and the homogeneous (dual) Sobolev distances. From the point of view of optimal transport theory, these distances minimize a dynamical cost to move a given initial distribution of mass to a final configuration. An important difference with the classical setting in mass transport theory is that the cost not only depends on the velocity of the moving particles but also on the densities of the intermediate configurations with respect to a given reference measure. We study the topological and geometric properties of these new distances, comparing them with the notion of weak convergence of measures and the well established Kantorovich-Rubinstein-Wasserstein theory. An example of possible applications to the geometric theory of gradient flows is also given.


On the Hard Lefschetz property of stringy Hodge numbers

Abstract: For projective varieties with a certain class of 'mild' isolated singularities and for projective threefolds with arbitrary Gorenstein canonical singularities, we show that the stringy Hodge numbers satisfy the Hard Lefschetz property. This result fits nicely with a 6-dimensional counterexample of Mustata and Payne for the Hard Lefschetz property for stringy Hodge numbers in general. We also give such an example, ours is a hypersurface singularity.


Moments of Two-Variable Functions and the Uniqueness of Graph Limits

Abstract: For a symmetric bounded measurable function W on [0,1]^2, "moments" of W can be defined as values t(F,W) indexed by simple graphs. We prove that every such function is determined by its moments up to a measure preserving transformation of the variables. This implies that the limit of a convergent dense graph sequence is unique up to measure preserving transformation.


Testing properties of graphs and functions

Abstract: We define an analytic version of the graph property testing problem, which can be formulated as studying an unknown 2-variable symmetric function through sampling from its domain and studying the random graph obtained when using the function values as edge probabilities. We give a characterization of properties testable this way, and extend a number of results about ``large graphs'' to this setting. These results can be applied to the original graph-theoretic property testing.


Bergman approximations of harmonic maps into the space of Kahler metrics on toric varieties

Abstract: We generalize the results of Song-Zelditch on geodesics in spaces of Kahler metrics on toric varieties to harmonic maps of any compact Riemannian manifold with boundary into the space of Kahler metrics on a toric variety. We show that the harmonic map equation can always be solved and that such maps may be approximated in the C^2 topology by harmonic maps into the spaces of Bergman metrics. In particular, WZW maps, or equivalently solutions of a homogeneous Monge-Ampere equation on the product of the manifold with a Riemann surface with S^1 boundary admit such approximations. We also show that the Eells-Sampson flow on the space of Kahler potentials is transformed to the usual heat flow on the space of symplectic potentials under the Legendre transform.


The Three Gap Theorem and Riemannian Geometry

Abstract: The classical Three Gap Theorem asserts that for a natural number n and a real number p, there are at most three distinct distances between consecutive elements in the subset of [0,1) consisting of the reductions modulo 1 of the first n multiples of p. Regarding it as a statement about rotations of the circle, we find results in a similar spirit pertaining to isometries of compact Riemannian manifolds and the distribution of points along their geodesics.


Contact surgeries and the transverse invariant in knot Floer homology

Abstract: We study naturality properties of the transverse invariant in knot Floer homology under contact (+1)-surgery. This can be used as a calculational tool for the transverse invariant. As a consequence, we show that the Eliashberg-Chekanov twist knots E_n are not transversely simple for n odd and n>3.


Geometry and Rank of Fibered Hyperbolic 3-Manifolds

Abstract: Assume that M is a closed hyperbolic 3-manifold fibering over the circle with fiber a closed orientable surface of genus g. We show that if M has large diameter and its injectivity radius is bounded below, then the rank of the fundamental group of M is 2g+1.


Two paradigms for topological quantum computation

Abstract: We present two paradigms relating algebraic, topological and quantum computational statistics for the topological model for quantum computation. In particular we suggest correspondences between the computational power of topological quantum computers, computational complexity of link invariants and images of braid group representations. While at least parts of these paradigms are well-known to experts, we provide supporting evidence for them in terms of recent results. We give a fairly comprehensive list of known examples and formulate two conjectures that would further support the paradigms.


On approximation of continuous functions by entire functions on subsets of the real line

Abstract: We generalize the classical Bernstein theorem concerning the constructive description of classes of functions uniformly continuous on the real line. The approximation of continuous bounded functions by entire functions of exponential type on unbounded closed proper subsets of the real line is studded.


On the two dimensional Bilinear Hilbert Transform

Abstract: We investigate the Bilinear Hilbert Transform in the plane and the pointwise convergence of bilinear averages in Ergodic theory, arising from $\Z^2$ actions. Our techniques combine novel one and a half dimensional phase-space analysis with more standard one dimensional theory.


Symmetries and the Riemann Hypothesis

Abstract: Associated to classical semi-simple groups and their maximal parabolics are genuine zeta functions. Naturally related to Riemann's zeta and governed by symmetries, including that of Weyl, these zetas are expected to satisfy the Riemann hypothesis.


Repeat distributions from unequal crossovers

Abstract: It is a well-known fact that genetic sequences may contain sections with repeated units, called repeats, that differ in length over a population, with a length distribution of geometric type. A simple class of recombination models with single crossovers is analysed that result in equilibrium distributions of this type. Due to the nonlinear and infinite-dimensional nature of these models, their analysis requires some nontrivial tools from measure theory and functional analysis, which makes them interesting also from a mathematical point of view. In particular, they can be viewed as quadratic, hence nonlinear, analogues of Markov chains.


An extended class of minimax generalized Bayes estimators of regression coefficients

Abstract: We derive minimax generalized Bayes estimators of regression coefficients in the general linear model with spherically symmetric errors under invariant quadratic loss for the case of unknown scale. The class of estimators generalizes the class considered in Maruyama and Strawderman (2005) to include non-monotone shrinkage functions.


Nevanlinna-Pick interpolation for $C+BH^\infty$

Abstract: Given an inner function $B$ we classify the invariant subspaces of the algebra $H^\infty_B:=\mathbb{C}+BH^\infty$. We derive a formula in terms of these invariant subspaces for the distance of an element in $L^\infty$ to a certain weak*-closed ideal in $H^\infty_B$ and use this to prove an analogue of the Nevanlinna-Pick interpolation theorem.


Chernoff and Trotter type product formulas

Abstract: We consider the abstract Cauchy problem x'=Ax, x(0)=x_0\in D(A) for linear operators A on a Banach space X. We prove uniqueness of the (local) solution of this problem for a natural class of operators A. Moreover, we establish that the solution x(\cdot) can be represented as a limit of sequence F(t/n)^{n} as n\to\infty in the weak operator topology, where a function F:[0,\infty)\to L(X) satisfies F'(0)y=Ay, y\in D(A). As a consequence, we deduce necessary and sufficient conditions that a linear operator C is closable and its closure is a generator of C_0-semigroup. We also obtain some criteria for the sum of two generators of C_0-semigroups to be a generator of C_0-semigroup such that the Trotter formula is valid.


Isoparametric hypersurfaces with four principal curvatures revisited

Abstract: The classification of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial characterization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersurface of the type constructed by Ferus, Karcher and M\"{u}nzner. The proof of the characterization in [2] is an extremely long calculation by exterior derivatives with remarkable cancellations, which is motivated by the idea that an isoparametric hypersurface is defined by an over-determined system of partial differential equations. Therefore, exterior differentiating sufficiently many times should gather us enough information for the conclusion. In spite of its elementary nature, the magnitude of the calculation and the surprisingly pleasant cancellations make it desirable to understand the underlying geometric principles. In this paper, we give a conceptual, and considerably shorter, proof of the characterization based on Ozeki and Takeuchi's expansion formula for the Cartan-M\"{u}nzner polynomial. Along the way the geometric meaning of these four sets of equations also becomes clear.


Relative regular modules. Applications to von Neumann regular rings

Abstract: We use the concept of a regular object with respect to another object in an arbitrary category, defined in \cite{dntd}, in order to obtain the transfer of regularity in the sense of Zelmanowitz between the categories $R-$mod and $S-$mod, when $S$ is an excellent extension of the ring $R$. Consequently, we obtain a result of \cite{ps}: if $S$ is an excellent extension of the ring $R$, then $S$ is von Neumann regular ring if and only if $R$ is also von Neumann regular ring. In the second part, using relative regular modules, we give a new proof of a classical result: the von Neumann regular property of a ring is Morita invariant. Finally, the von Neumann regularity of the Morita ring is investigated.


Inverse scattering on conformally compact manifolds

Abstract: We study inverse scattering for $\Delta_g+V$ on $(X,g)$ a conformally compact manifold with metric $g,$ with variable sectional curvature $-\alf^2(y)$ at the boundary and $V\in C^\infty(X)$ not vanishing at the boundary. We prove that the scattering matrix at a fixed energies $(\lambda_1,$ $\lambda_2)$ in a suitable subset of $\mc$, determines $\alf,$ and the Taylor series of both the potential and the metric at the boundary.


Rational structure on algebraic tangles and closed incompressible surfaces in the complements of algebraically alternating knots and links

Abstract: Let $F$ be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle $(B,T)$. Then $F$ separates the strings of $T$ in $B$ and the boundary slope of $F$ is uniquely determined by $(B,T)$ and hence we can define the slope of the algebraic tangle. In addition to the Conway's tangle sum, we define a natural product of two tangles. The slopes and binary operation on algebraic tangles lead an algebraic structure which is isomorphic to the rational numbers. We introduce a new knot and link class, algebraically alternating knots and links, roughly speaking which are constructed from alternating knots and links by replacing some crossings with algebraic tangles. We give a necessary and sufficient condition for a closed surface to be incompressible and meridionally incompressible in the complement of an algebraically alternating knot or link $K$, in particular we show that if $K$ is a knot, then the complement of $K$ does not contain such a surface.


Euler-Hurwitz series and non-linear Euler sums

Abstract: In this paper we derive two expressions for the Hurwitz zeta function involving the complete Bell polynomials in the restricted case where q is a positive integer greater than 1. The arguments of the complete Bell polynomials comprise the generalised harmonic number functions. These in turn give rise to Euler-Hurwitz series which may then be used to determine identities for combinations of both linear and non-linear Euler sums.


Note sur les corps 2-rationnels

Abstract: We compute the Galois group of the maximal 2-ramified pro-2-extension of a 2-rational number field


A proof by calibration of an isoperimetric inequality in the Heisenberg group H^n

Abstract: Let $D$ be a closed disk centered at the origin in the horizontal hyperplane $\{t=0\}$ of the sub-Riemannian Heisenberg group $\hh^n$, and $C$ the vertical cylinder over $D$. We prove that any finite perimeter set $E$ such that $D\subset E\subset C$ has perimeter larger than or equal to the one of the rotationally symmetric sphere with constant mean curvature of the same volume, and that equality holds only for the spheres using a recent result by Monti and Vittone [12].


Examples of area-minimizing surfaces in the subriemannian Heisenberg group H^1 with low regularity

Abstract: We give new examples of entire area-minimizing t-graphs in the subriemannian Heisenberg group H^1. Most of the examples are locally lipschitz in Euclidean sense. Some regular examples have prescribed singular set consisting of either a horizontal line or a finite number of horizontal halflines extending from a given point. Amongst them, a large family of area-minimizing cones is obtained.


Simplices and spectra of graphs

Abstract: In this note we show the n-2-dimensional volumes of codimension 2 faces of an n-dimensional simplex are algebraically independent functions of the lengths of edges. In order to prove this we compute the complete spectrum of a combinatorially interesting graph.


Global regularity for a modified critical dissipative quasi-geostrophic equation

Abstract: In this paper, we consider the modified quasi-geostrophic equation \begin{gather*} \del_t \theta + (u \cdot \grad) \theta + \kappa \Lambda^\alpha \theta = 0 u = \Lambda^{\alpha - 1} R^{\perp}\theta. \end{gather*} with $\kappa > 0$, $\alpha \in (0,1]$ and $\theta_0 \in \lp{2}(\R^2)$. We remark that the extra $\Lambda^{\alpha - 1}$ is introduced in order to make the scaling invariance of this system similar to the scaling invariance of the critical quasi-geostrophic equations. In this paper, we use Besov space techniques to prove global existence and regularity of strong solutions to this system.


Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology

Abstract: We associate to each infinite primitive Lie pseudogroup a Hopf algebra of `transverse symmetries', by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a `quantum group' counterpart of the infinite-dimensional primitive Lie algebra of the pseudogroup. It is first constructed via its action on the \'etale groupoid associated to the pseudogroup, and then realized as a bicrossed product of a universal enveloping algebra by a Hopf algebra of regular functions on a formal group. The bicrossed product structure allows to express its Hopf cyclic cohomology in terms of a bicocyclic bicomplex analogous to the Chevalley-Eilenberg complex. As an application, we compute the relative Hopf cyclic cohomology modulo the linear isotropy for the Hopf algebra of the general pseudogroup, and find explicit cocycle representatives for the universal Chern classes in Hopf cyclic cohomology. As another application, we determine all Hopf cyclic cohomology groups for the Hopf algebra associated to the pseudogroup of local diffeomorphisms of the line.


A complex surface of general type with p_g=0, K^2=3 and H_1=Z/2Z

Abstract: This paper is an addendum to [4], in which the authors constructed a simply connected minimal complex surface of general type with p_g=0 and K^2=3. In this paper we construct a new non-simply connected minimal surface of general type with p_g=0, K^2=3 and H_1=Z/2Z using a rational blow-down surgery and Q-Gorenstein smoothing theory.


Covariants and the no-name lemma

Abstract: A close connection between the no-name lemma (concerning algebraic groups acting on vector bundles) and the existence of sufficiently many independent rational covariants is pointed out. In particular, this leads to a new natural proof of the no-name lemma. For linearly reductive groups, the approach has a refined variant based on integral covariants. This fits into the usual context of invariant theory, and yields a version of the no-name lemma that has a constructive nature.


Quivers with potentials associated to triangulated surfaces

Abstract: We attempt to relate two recent developments: cluster algebras associated to triangulations of surfaces by Fomin-Shapiro-Thurston, and quivers with potentials and their mutations introduced by Derksen-Weyman-Zelevinsky. To each ideal triangulation of a bordered surface with marked points we associate a quiver with potential, in such a way that whenever two ideal triangulations are related by a flip of an arc, the respective quivers with potentials are related by a mutation with respect to the flipped arc. We prove that if the surface has non-empty boundary, then the quivers with potentials associated to its triangulations are rigid and hence non-degenerate.


Arithmetic Groups Have Rational Representation Growth

Abstract: Let G be an arithmetic lattice in a semisimple algebraic group over a number field. We show that if G has the congruence subgroup property, then the number of n-dimensional irreducible representations of G grows like n^a, where a is a rational number.


Cohen-Macaulay admissible clutters

Abstract: There is a one-to-one correspondence between square-free monomial ideals and clutters, which are also known as simple hypergraphs. It was conjectured that unmixed admissible clutters are Cohen-Macaulay. We prove the conjecture for uniform admissible clutters of heights 2 and 3. For admissible clutters of greater heights, we give a family of examples to show that the conjecture may fail. When the height is 4, we give an additional condition under which unmixed admissible clutters are Cohen-Macaulay.


Minimality of the well-rounded retract

Abstract: We prove that the well-rounded retract of SO_n\SL_n(R) is a minimal SL_n(Z)-invariant spine.


Primitive Central Idempotents of the Group Algebra

Abstract: An approach to representations of finite groups is presented without recourse to character theory. Considering the group algebra C[G] as an algebra of linear maps on C[G] (by left multiplication), we derive the primitive central idempotents as a simultaneous eigenbasis of the centre, Z(C[G]). We apply this framework to obtain the irreducible representations of a class of finite meta-abelian groups. In particular, we give a general construction of the isomorphism between simple blocks of C[G] and the corresponding matrix algebra where G can be any finite group.


Generating function for GL_n-invariant differential operators in the skew Capelli identity

Abstract: Let Alt_n be the vector space of all alternating n-by-n complex matrices, on which the complex general linear group GL_n acts by $x \mapsto g x g^t$. The aim of this paper is to show that Pfaffian of a certain matrix whose entries are multiplication operators or derivations acting on polynomials on Alt_n provides a generating function for the GL_n-invariant differential operators that play a role in the skew Capelli identity, with coefficients the Hermite polynomials.


Wild Harmonic Bundles and Wild Pure Twistor D-modules

Abstract: We study (i) asymptotic behaviour of wild harmonic bundles, (ii) the relation between semisimple meromorphic flat connections and wild harmonic bundles, (iii) the relation between wild harmonic bundles and polarized wild pure twistor $D$-modules. As an application, we show the hard Lefschetz theorem for algebraic semisimple holonomic $D$-modules, conjectured by M. Kashiwara.


Good formal structure for meromorphic flat connections on smooth projective surfaces

Abstract: We prove the algebraic version of a conjecture of C. Sabbah on the existence of the good formal structure for meromorphic flat connections on surfaces after some blow up.


A report on "Regulators of canonical extension are torsion; the smooth divisor case"

Abstract: In this note, we report on a work jointly done with C. Simpson on a generalization of Reznikov's theorem which says that the Chern-Simons classes and in particular the Deligne Chern classes (in degrees $> 1$) are torsion, of a flat vector bundle on a smooth complex projective variety. We consider the case of a smooth quasi--projective variety with an irreducible smooth divisor at infinity. We define the Chern-Simons classes of the Deligne's \textit{canonical extension} of a flat vector bundle with unipotent monodromy at infinity, which lift the Deligne Chern classes and prove that these classes are torsion. The details of the proof can be found in arxiv:0707.0372 [math.AG].


On flows associated to Sobolev vector fields in Wiener spaces: an approach \`a la DiPerna-Lions

Abstract: In this paper we extend the DiPerna-Lions theory on ODEs with Sobolev vector fields to the setting of abstract Wiener spaces.


Dilational Hilbert Scales and Deconvolutional Sharpening

Abstract: Operationally, index functions of variable Hilbert scales can be viewed as generators for families of spaces and norms. Using a one parameter family of index functions based on the dilations of a given index function, a new class of scales (dilational Hilbert scales (DHS)) is derived which generates new interpolatory inequalities (dilational interpolatory inequalities (DII)) which have the ordinary Hilbert scales (OHS) interpolatory inequalities as special cases. They therefore represent a one-parameter family generalization of OHS, and are a precise and concise subset of VHS approriate for deriving error estimates for deconvolution. The role of the Hilbert scales in deriving error estimates for the approximate solution of inverse problems is discussed along with an application of DHS to deconvolution sharpening.


The Frobenius action on rank 2 vector bundles over curves in small genus and small characteristic

Abstract: Let X be a general proper and smooth curve of genus 2 (resp. of genus 3) defined over an algebraically closed field of characteristic p. When 3\leq p \leq 7, the action of Frobenius on rank 2 semi-stable vector bundles with trivial determinant is completely determined by its restrictions to the 30 lines (resp. the 126 Kummer surfaces) that are invariant under the action of some order 2 line bundle over X. Those lines (resp. those Kummer surfaces) are closely related to the elliptic curves (resp. the abelian varieties of dimension 2) that appear as the Prym varieties associated to double \'etale coverings of X. We are therefore able to compute explicit equations of this action in these cases. We perform some of these computations and draw some consequences.


Rigidity results for some boundary quasilinear phase transitions

Abstract: We consider a quasilinear equation given in the half-space, i.e. a so called boundary reaction problem. Our concerns are a geometric Poincar\'e inequality and, as a byproduct of this inequality, a result on the symmetry of low-dimensional bounded stable solutions, under some suitable assumptions on the nonlinearities. More precisely, we analyze the following boundary problem $$ \left\{\begin{matrix} -{\rm div} (a(x,|\nabla u|)\nabla u)+g(x,u)=0 \qquad {on $\R^n\times(0,+\infty)$} -a(x,|\nabla u|)u_x = f(u) \qquad{\mbox{on $\R^n\times\{0\}$}}\end{matrix} \right.$$ under some natural assumptions on the diffusion coefficient $a(x,|\nabla u|)$ and the nonlinearities $f$ and $g$. Here, $u=u(y,x)$, with $y\in\R^n$ and $x\in(0,+\infty)$. This type of PDE can be seen as a nonlocal problem on the boundary $\partial \R^{n+1}_+$. The assumptions on $a(x,|\nabla u|)$ allow to treat in a unified way the $p-$laplacian and the minimal surface operators.


Quasi-Minimal, Pseudo-Minimal Systems and Dense Orbits

Abstract: We give a short discussion about a weaker form of minimality (called quasi-minimality). We call a system quasi-minimal if all dense orbits form an open set. It is hard to find examples which are not already minimal. Since elliptic behaviour makes them minimal, these systems are regarded as parabolic systems. Indeed, we show that a quasi-minimal homeomorphism on a manifold is not expansive (hyperbolic).


Tate modules of universal p-divisible groups

Abstract: A p-divisible group over a complete local domain determines a Galois representation on the Tate module of its generic fibre. We determine the image of this representation for the universal deformation in mixed characteristic of a bi-infinitesimal group and for the p-rank strata of the universal deformation in positive characteristic of an infinitesimal group. The method is a reduction to the known case of one-dimensional groups by a deformation argument based on properties of the stratification by Newton polygons.


Exponential sums and rank of triple persymmetric matrices over F_2

Abstract: We obtain using exponential quadratic sums, explicit expressions for the number of triple persymmetric matrices over F_2 of given rank. (A matrix [a(i,j)] is persymmetric if a(i,j) = a(r,s) for i+j = r+s)


Pseudoknot RNA structures with arc-length $\ge 4$

Abstract: In this paper we study $k$-noncrossing RNA structures with minimum arc-length 4 and at most $k-1$ mutually crossing bonds. Let ${\sf T}_{k}^{[4]}(n)$ denote the number of $k$-noncrossing RNA structures with arc-length $\ge 4$ over $n$ vertices. We prove (a) a functional equation for the generating function $\sum_{n\ge 0}{\sf T}_{k}^{[4]}(n)z^n$ and (b) derive for $k\le 9$ the asymptotic formula ${\sf T}_{k}^{[4]}(n)\sim c_k n^{-((k-1)^2+(k-1)/2)} \gamma_k^{-n}$. Furthermore we explicitly compute the exponential growth rates $\gamma_k^{-1}$ and asymptotic formulas for $4\le k\le 9$.


On some transformations of bilateral birth-and-death processes with applications

Abstract: A method yielding simple relationships among bilateral birth-and-death processes is outlined. This allows one to relate birth and death rates of two processes in such a way that their transition probabilities, first-passage-time densities and ultimate crossing probabilities are mutually related by some product-form expressions.


Infinitesimal or cocommutative dipterous bialgebras and good triples of operads

Abstract: The works of Poincare, Birkhoff, Witt and Cartier, Milnor, Moore on the connected cocommutative Hopf algebras translated in the language of operads means that the triple of operads (Com, As, Lie) endowed with the Hopf compatiblity relation is good. In this paper, we focus on left dipterous (resp. right dipterous) algebras which are associative algebras with an extra left (resp. right) module on themselves and look for good triples were $As$ is replaced by the dipterous operad Dipt. Since the work of Loday and Ronco, the triple of operads (As, Dipt, B_\infty) endowed with the semi-Hopf compatibility relations is known to be good. In this paper, we prove that the triple of operads (As, Dipt, Grove) endowed with the so-called (nonunital) semi-infinitesimal compatibility relations is good. For that, explicit constructions of the free dipterous algebra and the free grove-algebra over a K-vector space V are given. These constructions turn out to be related to rooted planar trees and the little an large Schroeder numbers. Many examples of dipterous algebras are given, notably the free L-dipterous algebras. As a corollary of our results, we also recover that the triple of operads (2As, Dipt, Vect) endowed both with the unital semi-Hopf and with the unital semi-infinitesimal compatibility relations is good, where 2As denotes the operad of 2-associative algebras. We also open this paper on a good triple, related to the Connes-Kreimer Hopf algebra in quantum field theory, (Com, Dipt, Prim_{Com} Dipt) endowed with the Hopf compatibility relations and also present a general theorem giving good triples from entangled dipterous like operads named associative molecules.


Bases in Lie and Quantum Algebras

Abstract: Applications of algebras in physics are related to the connection of measurable observables to relevant elements of the algebras, usually the generators. However, in the determination of the generators in Lie algebras there is place for some arbitrary conventions. The situation is much more involved in the context of quantum algebras, where inside the quantum universal enveloping algebra, we have not enough primitive elements that allow for a privileged set of generators and all basic sets are equivalent. In this paper we discuss how the Drinfeld double structure underlying every simple Lie bialgebra characterizes uniquely a particular basis without any freedom, completing the Cartan program on simple algebras. By means of a perturbative construction, a distinguished deformed basis (we call it the analytical basis) is obtained for every quantum group as the analytical prolongation of the above defined Lie basis of the corresponding Lie bialgebra. It turns out that the whole construction is unique, so to each quantum universal enveloping algebra is associated one and only one bialgebra. In this way the problem of the classification of quantum algebras is moved to the classification of bialgebras. In order to make this procedure more clear, we discuss in detail the simple cases of su(2) and su_q(2).


A Note on Coseparable Coalgebras

Abstract: Given a coalgebra $C$ over a commutative ring $R,$ we show that $C$ can be considered as a (not necessarily counital) $C^{\ast op}$-coring. Moreover, we show that this coring has a left (right) counity if and only if $C$ is coseparable as an $R$-coalgebra.


Mass transport generated by a flow of Gauss maps

Abstract: Let $A \subset \mathbb{R}^d$, $d\ge 2$, be a compact convex set and let $\mu = \varrho_0 dx$ be a probability measure on $A$ equivalent to the restriction of Lebesgue measure. Let $\nu = \varrho_1 dx$ be a probability measure on $B_r := \{x\colon |x| \le r\}$ equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping $T$ such that $\nu = \mu \circ T^{-1}$ and $T = \phi \cdot {\rm n}$, where $\phi\colon A \to [0,r]$ is a continuous potential with convex sub-level sets and ${\rm n}$ is the Gauss map of the corresponding level sets of $\phi$. Moreover, $T$ is invertible and essentially unique. Our proof employs the optimal transportation techniques. We show that in the case of smooth $\phi$ the level sets of $\phi$ are driven by the Gauss curvature flow $\dot{x}(s) = -s^{d-1} \frac{\varrho_1(s {\rm n})}{\varrho_0(x)} K(x) \cdot {\rm n}(x)$, where $K$ is the Gauss curvature. As a by-product one can reprove the existence of weak solutions of the classical Gauss curvature flow starting from a convex hypersurface.


Estimation of Wiener--Ito integrals and polynomials of independent Gaussian random variables

Abstract: In this paper I prove good estimates on the moments and tail distribution of $k$-fold Wiener--It\^o integrals and also present their natural counterpart for polynomials of independent Gaussian random variables. The proof is based on the so-called diagram formula for Wiener--It\^o integrals which yields a good representation for their products as a sum of such integrals. I intend to show in a subsequent paper that this method also yields good estimates for degenerate $U$-statistics. The main result of this paper is a generalization of the estimates of Hanson and Wright about bilinear forms of independent standard normal random variables. On the other hand, it is a weaker estimate than the main result of a paper of Lata{\l}a [6]. But that paper contains an error, and it is not clear whether its result is true. This question is also discussed here.


Acyclic edge coloring

Abstract: This paper has been withdrawn by the author due to an error in the proof.


Search for primes of the form $m^2+1$

Abstract: The results of the computer hunt for the primes of the form $q = m^2+1$ up to $10^{20}$ are reported. The number of sign changes of the difference $\pi_q(x) - \frac{C_q}{2}\int_2^x{du \over \sqrt{u}\log(u)}$ and the error term for this difference is investigated. The analogs of the Brun's constant and the Skewes number are calculated. An analog of the B conjecture of Hardy--Littlewood is formulated. It is argued that there is no Chebyshev bias for primes of the form $q=m^2+1$. All encountered integrals we were able to express by the logarithmic integral.


Optimization of periodic composite structures for sub-wavelength focusing

Abstract: Recently, there has been plenty of work in designing and fabricating materials with an effective negative refractive index. Veselago realized that a slab of material with a refractive index of -1 would act as a lens. Pendry suggested that the Veselago lens would act as a superlens, providing a perfect image of an object in contrast to conventional lenses which are only able to focus a point source to an image having a diameter of the order of the wavelength of the incident field. Recent work has shown that similar focusing effects can be obtained with certain slabs of ``conventional'' periodic composite materials: photonic crystals. The present work seeks to answer the question of what periodic dielectric composite medium (described by dielectric coefficient with positive real part) gives an optimal image of a point source. An optimization problem is formulated and it is shown that a solution exists provided the medium has small absorption. Solutions are characterized by an adjoint-state gradient condition, and several numerical examples illustrate both the plausibility of this design approach, and the possibility of obtaining smaller image spot sizes than with typical photonic crystals.


Lyapunov exponents for the one-dimensional parabolic Anderson model with drift

Abstract: We consider the solution $u$ to the one-dimensional parabolic Anderson model with homogeneous initial condition $u(0, \cdot) \equiv 1$, arbitrary drift and a time-independent potential bounded from above. Under ergodicity and independence conditions we derive representations for both the quenched Lyapunov exponent and, more importantly, the $p$-th annealed Lyapunov exponents for {\it all} $p \in (0, \infty).$ These results enable us to prove the heuristically plausible fact that the $p$-th annealed Lyapunov exponent converges to the quenched Lyapunov exponent as $p \downarrow 0.$ Furthermore, we show that $u$ is $p$-intermittent for $p$ large enough. As a byproduct, we compute the optimal quenched speed of the random walk appearing in the Feynman-Kac representation of $u$ under the corresponding Gibbs measure. In this context, depending on the negativity of the potential, a phase transition from zero speed to positive speed appears.


The similarity problem for $J$-nonnegative Sturm-Liouville operators

Abstract: Sufficient conditions for the similarity of the operator $A := 1/r(x) (-d^2/dx^2 +q(x))$ with an indefinite weight $r(x)=(\sgn x)|r(x)|$ are obtained. These conditions are formulated in terms of Titchmarsh-Weyl $m$-coefficients. Sufficient conditions for the regularity of the critical points 0 and $\infty$ of $J$-nonnegative Sturm-Liouville operators are also obtained. This result is exploited to prove the regularity of 0 for various classes of Sturm-Liouville operators. This implies the similarity of the considered operators to self-adjoint ones. In particular, in the case $r(x)=\sgn x$ and $q\in L^1(R, (1+|x|)dx)$, we prove that $A$ is similar to a self-adjoint operator if and only if $A$ is $J$-nonnegative. The latter condition on $q$ is sharp, i.e., we construct $q\in \cap_{\gamma <1} L^1(R, (1+|x|)^\gamma dx)$ such that $A$ is $J$-nonnegative with the singular critical point 0. Hence $A$ is not similar to a self-adjoint operator. For periodic and infinite-zone potentials, we show that $J$-positivity is sufficient for the similarity of $A$ to a self-adjoint operator. In the case $q\equiv 0$, we prove the regularity of the critical point 0 for a wide class of weights $r$. This yields new results for "forward-backward" diffusion equations.


Cyclic Approximation to K-Stasis

Abstract: If a linear combination of k smooth vector fields is zero at a point, then, generically, near this point there are small cycles comprised of segments from the flow of each vector field. This answers a question posed in arXiv:math/0504365.


Recurrence relations for characters of affine Lie algebra $A_{\ell}^{(1)}$

Abstract: By using the known description of combinatorial bases for Feigin-Stoyanovsky's type subspaces of standard modules for affine Lie algebra $\mathfrak{sl}(l+1,\mathbb{C})^{\widetilde{}}$, as well as certain intertwining operators between standard modules, we obtain exact sequences of Feigin-Stoyanovsky's type subspaces at fixed level $k$. This directly leads to systems of recurrence relations for formal characters of those subspaces.


Discrete Affine Minimal Surfaces with Indefinite Metric

Abstract: Inspired by the Weierstrass representation of smooth affine minimal surfaces with indefinite metric, we propose a constructive process producing a large class of discrete surfaces that we call discrete affine minimal surfaces. We show that they are critical points of an affine area functional defined on the space of quadrangular discrete surfaces. The construction makes use of asymptotic coordinates and allows defining the discrete analogs of some differential geometric objects, such as the normal and co normal vector fields, the cubic form and the compatibility equations.


Global Symplectic Uncertainty Propagation on SO(3)

Abstract: This paper introduces a global uncertainty propagation scheme for rigid body dynamics, through a combination of numerical parametric uncertainty techniques, noncommutative harmonic analysis, and geometric numerical integration. This method is distinguished from prior approaches, as it allows one to consider probability densities that are global, and are not supported on only a single coordinate chart on the manifold. The use of Lie group variational integrators, that are symplectic and stay on the Lie group, as the underlying numerical propagator ensures that the advected probability densities respect the geometric properties of uncertainty propagation in Hamiltonian systems, which arise as consequence of the Gromov nonsqueezing theorem from symplectic geometry. We also describe how the global uncertainty propagation scheme can be applied to the problem of global attitude estimation.


Derived Arithmetic Fuchsian Groups of Genus Two

Abstract: We classify all torsion-free derived arithmetic Fuchsian groups of genus two by commensurability class. In particular, we show that there exist no such groups arising from quaternion algebras over number fields of degree greater than 5. We also prove some results on the existence and form of maximal orders for a class of quaternion algebras related to these groups. Using these results in conjunction with a computer program, one can determine an explicit set of generators for each derived arithmetic Fuchsian group containing a torsion-free subgroup of genus two. We show this for a number of examples.


Birkhoff spectra for one-dimensional maps with some hyperbolicity

Abstract: We study the multifractal analysis for smooth dynamical systems in dimension one. It is characterized the Hausdorff dimension of the level set obtained from the Birkhoff averages of a continuous function by the local dimensions of hyperbolic measures for a topologically mixing $C^2$ map modelled by an abstract dynamical system. A characterization which corresponds to above is also given for the ergodic basins of invariant probability measures. And it is shown that the complement of the set of quasi-regular points carries full Hausdorff dimension.


Equality of pressures for diffeomorphisms preserving hyperbolic measures

Abstract: For a diffeomorphism which preserves a hyperbolic measure the potential $\phi^u=-\log|{\rm Jac} df|_{E^u}|$ is studied. Various types of pressure of $\phi^u$ are introduced. It is shown that these pressures satisfy a corresponding variational principle.


Mukai duality for gerbes with connection

Abstract: We study gerbes with connection over an etale stack via noncommutative algebras of differential forms on a groupoid presenting the stack. We then describe a dg-category of modules over any such algebra, which we claim represents a dg-enhancement of the derived category of coherent analytic sheaves on the gerbe in question. This category can be used to phrase and prove Fourier-Mukai type dualities between gerbes and other noncommutative spaces. As an application of the theory, we show that a gerbe with flat connection on a torus is dual (in a sense analogous to Fourier-Mukai duality or T-duality) to a noncommutative holomorphic dual torus.


Equations of 2-linear ideals and arithmetical rank

Abstract: In this paper we consider reduced homogeneous ideals $\Jcal\subset S$ of a polynomial ring $S$, having a 2-linear resolution. 1. We study systems of generators of $\Jcal\subset S$. 2. We compute the arithmetical rank for a large class of projective curves having a 2-linear resolution. 3. We show that the fiber cone $\proj \Fcal(I_{\Lcal})$ of a lattice ideal $I_{\Lcal}$ of codimension two is a set theoretical complete intersection.


The Hochschild cohomology ring of a class of special biserial algebras

Abstract: We consider a class of self-injective special biserial algebras $\Lambda_N$ over a field $K$ and show that the Hochschild cohomology ring of $\Lambda_N$ is a finitely generated $K$-algebra. Moreover the Hochschild cohomology ring of $\Lambda_N$ modulo nilpotence is a finitely generated commutative $K$-algebra of Krull dimension two. As a consequence the conjecture of Snashall-Solberg \cite{SS}, concerning the Hochschild cohomology ring modulo nilpotence, holds for this class of algebras.


On the Gromov hyperbolicity of the Kobayashi metric on strictly pseudoconvex regions in the almost complex case

Abstract: We prove that every bounded strictly $J$-convex region equipped with the Kobayashi metric is hyperbolic in the sense of Gromov. We apply this result to the study of the dynamics of pseudo-holomorphic maps.


Bijections for Baxter Families and Related Objects

Abstract: The Baxter number can be written as $B_n = \sum_0^n \Theta_{k,n-k-1}$. These numbers have first appeared in the enumeration of so-called Baxter permutations; $B_n$ is the number of Baxter permutations of size $n$, and $\Theta_{k,l}$ is the number of Baxter permutations with $k$ descents and $l$ rises. With a series of bijections we identify several families of combinatorial objects counted by the numbers $\Theta_{k,l}$. Apart from Baxter permutations, these include plane bipolar orientations with $k+2$ vertices and $l+2$ faces, 2-orientations of planar quadrangulations with $k+2$ white and $l+2$ black vertices, certain pairs of binary trees with $k+1$ left and $l+1$ right leaves, and a family of triples of non-intersecting lattice paths. This last family allows us to determine the value of $\Theta_{k,l}$ as an application of the lemma of Gessel and Viennot. The approach also allows us to count certain other subfamilies, e.g., alternating Baxter permutations, objects with symmetries and, via a bijection with a class of plan bipolar orientations also Schnyder woods of triangulations, which are known to be in bijection with 3-orientations.


A sharp Wirtinger inequality and some related functional spaces

Abstract: We consider the generalized Wirtinger inequality \[ (\int_{0}^{T} a |u|^q )^{1/q} \le C \biggm(\int_{0}^{T} a^{1-p} |u'|^{p}\biggm)^{1/p}, \] with $p,q>1$, $T>0$, $a\in L^1[0,T]$, $a\ge0$, $a\not\equiv0$ and where $u$ is a $T$-periodic function satisfying the constraint \[ \int_{0}^{T} a |u|^{q-2}u =0. \] We provide the best constant $C>0$ as well as all extremals. Furthermore, we characterize the natural functional space where the inequality is defined.


Nonlinear optimal control synthesis via occupation measures

Abstract: We consider nonlinear optimal control problems (OCPs) for which all problem data are polynomial. In the first part of the paper, we review how occupation measures can be used to approximate pointwise the optimal value function of a given OCP, using a hierarchy of linear matrix inequality (LMI) relaxations. In the second part, we extend the methodology to approximate the optimal value function on a given set and we use such a function to constructively and computationally derive an almost optimal control law. Numerical examples show the effectiveness of the approach.


A new test procedure of independence in copula models via chi-square-divergence

Abstract: We introduce a new test procedure of independence in the framework of parametric copulas with unknown marginals. The method is based essentially on the dual representation of $\chi^2$-divergence on signed finite measures. The asymptotic properties of the proposed estimate and the test statistic are studied under the null and alternative hypotheses, with simple and standard limit distributions both when the parameter is an interior point or not.


Images directes et fonctions L en cohomologie rigide

Abstract: Let $k$ be a perfect field of characteristic $p>0$, $\mathcal{V}$ a complete discrete valuation ring with residue field $k$ and field of fractions $K$ of characteristic 0, and $S$ a separated $k$-scheme of finite type. When $S$ is smooth over $k$, we partially prove here a conjecture of Berthelot about the overconvergence of the higher direct images of the structure sheaf under a proper smooth morphism $f:X\to S$; when $k$ is perfect and $\mathcal{V}$ is tamely ramified such direct images are always convergent, not only for the structure sheaf but also for (almost) every convergent $F$-isocrystals. More generally, we prove this overconvergence when $f$ is liftable over $\mathcal{V}$, or when $X$ is a relative complete intersection in some projective spaces over $S$, and taking as coefficients any overconvergent isocrystals. We then apply these results to $L$-functions with coefficients such direct images with Frobenius structure: we derive rationality or meromorphy for these $L$-functions (Dwork's conjecture), and we study their $p$-adic unit zeroes and poles (Katz's conjecture) ; and explicit case concerns the ordinary abelian schemes. A more precise presentation of results by chapters is given in the introduction.


A class of statistical models to weaken independence in two-way contingency tables

Abstract: In this paper we study a new class of statistical models for contingency tables. We define this class of models through a subset of the binomial equations of the classical independence model. We use some notions from Algebraic Statistics to compute their sufficient statistic, and to prove that they are log-linear. Moreover, we show how to compute maximum likelihood estimates and to perform exact inference through the Diaconis-Sturmfels algorithm. Examples show that these models can be useful in a wide range of applications.


Distribution of Angles in Hyperbolic Lattices

Abstract: We prove an effective equidistribution result about angles in a hyperbolic lattice. We use this to generalize a result due to F. P. Boca.


On some generalized reinforced random walks on integers

Abstract: We consider Reinforced Random Walks where transition probabilities are a function of the proportion of times the walk has traversed an edge. We give conditions for recurrence or transience. A phase transition is observed, similar to Pemantle \cite{Pem000} on trees.


On the non--existence of certain hyperovals in dual Andr\'e planes of order $2^{2h}$

Abstract: No regular hyperoval of the Desarguesian affine plane $AG(2,2^{2h})$, with $h>1$, is inherited by a dual Andr\'e plane of order $2^{2h}$ with dimension 2 over its centre.


Symmetric Numerical Semigroups Generated by Fibonacci and Lucas Triples

Abstract: The symmetric numerical semigroups S(F_a,F_b,F_c) and S(L_k,L_m,L_n) generated by three Fibonacci (F_a,F_b,F_c) and Lucas (L_k,L_m,L_n) numbers are considered. Based on divisibility properties of the Fibonacci and Lucas numbers we establish necessary and sufficient conditions for both semigroups to be symmetric and calculate their Hilbert generating series, Frobenius numbers and genera.


Solution to the Burnside Problem

Abstract: The Burnside Problem asks whether a finitely generated group of exponent n is finite. We present a solution for 2-generator groups of prime power exponent. Results of P. Hall and G. Higman extends the finiteness conclusion to groups having composite exponents. Our main result, called the Generalized Burnside Theorem, is a solvability theorem that applies to a family of groups called GB (Generalized Burnside) groups that contain infinite as well as finite groups. The final section discusses the extension to k-generator groups although details are left for another time.


The Kahler-Ricci flow and K-stability

Abstract: We consider the K\"ahler-Ricci flow on a Fano manifold. We show that if the curvature remains uniformly bounded along the flow, the Mabuchi energy is bounded below, and the manifold is K-polystable, then the manifold admits a K\"ahler-Einstein metric. The main ingredient is a result that says that a sufficiently small perturbation of a cscK manifold admits a cscK metric if it is K-polystable.


On 0-homology of categorical at zero semigroups

Abstract: The isomorphism of 0-homology groups of a categorical at zero semigroup and homology groups of its 0-reflector is proved. Some applications of 0-homology to Eilenberg-MacLane homology of semigroups are given.


Monodromy of a family of hypersurfaces

Abstract: Let $Y$ be an $(m+1)$-dimensional irreducible smooth complex projective variety embedded in a projective space. Let $Z$ be a closed subscheme of $Y$, and $\delta$ be a positive integer such that $\mathcal I_{Z,Y}(\delta)$ is generated by global sections. Fix an integer $d\geq \delta +1$, and assume the general divisor $X \in |H^0(Y,\ic_{Z,Y}(d))|$ is smooth. Denote by $H^m(X;\mathbb Q)_{\perp Z}^{\text{van}}$ the quotient of $H^m(X;\mathbb Q)$ by the cohomology of $Y$ and also by the cycle classes of the irreducible components of dimension $m$ of $Z$. In the present paper we prove that the monodromy representation on $H^m(X;\mathbb Q)_{\perp Z}^{\text{van}}$ for the family of smooth divisors $X \in |H^0(Y,\ic_{Z,Y}(d))|$ is irreducible.


Component models for large networks

Abstract: Being among the easiest ways to find meaningful structure from discrete data, Latent Dirichlet Allocation (LDA) and related component models have been applied widely. They are simple, computationally fast and scalable, interpretable, and admit nonparametric priors. In the currently popular field of network modeling, relatively little work has taken uncertainty of data seriously in the Bayesian sense, and component models have been introduced to the field only recently, by treating each node as a bag of out-going links. We introduce an alternative, interaction component model for communities (ICMc), where the whole network is a bag of links, stemming from different components. The former finds both disassortative and assortative structure, while the alternative assumes assortativity and finds community-like structures like the earlier methods motivated by physics. With Dirichlet Process priors and an efficient implementation the models are highly scalable, as demonstrated with a social network from the Last.fm web site, with 670,000 nodes and 1.89 million links.


Poisson (co)homology of polynomial Poisson algebras in dimension four : Sklyanin's case

Abstract: In this paper, we compute the Poisson (co)homology of a polynomial Poisson structure given by two Casimir polynomial functions which define a complete intersection with an isolated singularity.


Large induced trees in K_r-free graphs

Abstract: For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of G that is a tree. In this paper, we study the problem of bounding t(G) for graphs which do not contain a complete graph K_r on r vertices. This problem was posed twenty years ago by Erdos, Saks, and Sos. Substantially improving earlier results of various researchers, we prove that every connected triangle-free graph on n vertices contains an induced tree of order \sqrt{n}. When r >= 4, we also show that t(G) >= (\log n)/(4 \log r) for every connected K_r-free graph G of order n. Both of these bounds are tight up to small multiplicative constants, and the first one disproves a recent conjecture of Matousek and Samal.


Algebraic K-theory over the infinite dihedral group: an algebraic approach

Abstract: We prove that the Waldhausen nilpotent class group of an injective index 2 amalgamated free product is isomorphic to the Farrell-Bass nilpotent class group of a twisted polynomial extension. As an application, we show that the Farrell-Jones Conjecture in algebraic K-theory can be sharpened from the family of virtually cyclic subgroups to the family of finite-by-cyclic subgroups.


Some remarks on Nil groups in algebraic K-theory

Abstract: This note explains consequences of recent work of Frank Quinn for computations of Nil groups in algebraic K-theory, in particular the Nil groups occurring in the K-theory of polynomial rings, Laurent polynomial rings, and the group ring of the infinite dihedral group.


Categorifying Coloring Numbers

Abstract: Coloring numbers are one of the simplest combinatorial invariants of knots and links to describe. And with Joyce's introduction of quandles, we can understand them more algebraically. But can we extend these invariants to tangles -- knots and links with free ends? Indeed we can, once we categorify. Starting from the definition of coloring numbers, we will categorify them and establish this extension to tangles. Then, decategorifying will leave us with matrix representations of the monoidal category of tangles.


Betti numbers of graded modules and the Multiplicity Conjecture in the non-Cohen-Macaulay case

Abstract: We use the results by Eisenbud and Schreyer to prove that any Betti diagram of a graded module over a standard graded polynomial ring is a positive linear combination Betti diagrams of modules with a pure resolution. This implies the Multiplicity Conjecture of Herzog, Huneke and Srinivasan for modules that are not necessarily Cohen-Macaulay. We give a combinatorial proof of the convexity of the simplicial fan spanned by the pure diagrams.


Logics preserving degrees of truth from varieties of residuated lattices

Abstract: Let K be a variety of (commutative, integral) residuated lattices. The substructural logic usually associated with K is an algebraizable logic that has K as its equivalent algebraic semantics, and is a logic that preserves truth, i.e., 1 is the only truth value preserved by the inferences of the logic. In this paper we introduce another logic associated with K, namely the logic that preserves degrees of truth, in the sense that it preserves lower bounds of truth values in inferences. We study this second logic mainly from the point of view of abstract algebraic logic. We determine its algebraic models and we classify it in the Leibniz and the Frege hierarchies: we show that it is always fully selfextensional, that for most varieties K it is non-protoalgebraic, and that it is algebraizable if and only K is a variety of generalized Heyting algebras, in which case it coincides with the logic that preserves truth. We also characterize the new logic in three ways: by a Hilbert style axiomatic system, by a Gentzen style sequent calculus, and by a set of conditions on its closure operator. Concerning the relation between the two logics, we prove that the truth preserving logic is the purely inferential extension of the one that preserves degrees of truth with either the rule of Modus Ponens or the rule of Adjunction for the fusion connective.


Spherical Spectral Synthesis and Two-Radius Theorems on Damek-Ricci Spaces

Abstract: We prove that spherical spectral analysis and synthesis hold in Damek-Ricci spaces and derive two-radius theorems.


Weighted enumeration of spanning subgraphs with degree constraints

Abstract: The Heilmann-Lieb Theorem on (univariate) matching polynomials states that the polynomial $\sum_k m_k(G) y^k$ has only real nonpositive zeros, in which $m_k(G)$ is the number of $k$-edge matchings of a graph $G$. There is a stronger multivariate version of this theorem. We provide a general method by which ``theorems of Heilmann-Lieb type'' can be proved for a wide variety of polynomials attached to the graph $G$. These polynomials are multivariate generating functions for spanning subgraphs of $G$ with certain weights and constraints imposed, and the theorems specify regions in which these polynomials are nonvanishing. Such theorems have consequences for the absence of phase transitions in certain probabilistic models for spanning subgraphs of $G$.


Dynamics of the Nearly Parametric Pendulum

Abstract: Dynamically stable periodic rotations of a driven pendulum provide a unique mechanism for generating a uniform rotation from bounded excitations. This paper studies the effects of a small ellipticity of the driving, perturbing the classical parametric pendulum. The first finding is that the region in the parameter plane of amplitude and frequency of excitation where rotations are possible increases with the ellipticity. Second, the resonance tongues, which are the most characteristic feature of the classical bifurcation scenario of a parametrically driven pendulum, merge into a single region of instability.


A note on multi-type cookie random walk on integers

Abstract: We consider a random walk on integers where at the first visits to a site the walker gets a positive drift, but where after a certain number of visits the walker gets a negative drift. We prove that the walker is almost surely transient to the left with positive speed.


Cobordism of fold maps, stably framed manifolds and immersions

Abstract: We give complete geometric invariants of cobordisms of fold maps with oriented singular set and cobordisms of even codimensional fold maps. These invariants are given in terms of cobordisms of stably framed manifolds and cobordisms of immersions with prescribed normal bundles defined by the author in his earlier works.


Geodesic Equations on Diffeomorphism Groups

Abstract: We bring together those systems of hydrodynamical type that can be written as geodesic equations on diffeomorphism groups or on extensions of diffeomorphism groups with right invariant $L^2$ or $H^1$ metrics. We present their formal derivation starting from Euler's equation, the first order equation satisfied by the right logarithmic derivative of a geodesic in Lie groups with right invariant metrics.


On Bass' question for finitely generated algebras over large fields

Abstract: Recently Cortinas-Haesemayer-Walker-Weibel gave affirmative answer to Bass' 1972 question on NK-groups for algebras of essentially finite type over large fields of characteristic 0. Here we give an alternative short proof of this result for algebras of finite type over such fields. Our approach is based on classical techniques in higher K-theory of rings and a direct K_i-analogue of an old observation of Murthy-Pedrini, dating back from the same 1972.


An open image theorem for a general class of abelian varieties

Abstract: Let K be a number field and A/K be a polarized abelian variety with absolutely trivial endomorphism ring. We show that if the Neron model of A/K has at least one fiber with potential toric dimension one, then for almost all rational primes ell, the Galois group of the splitting field of the ell-torsion of A is GSp_{2g}(Z/ell).


Ordinary differential equations in Banach spaces and the spectral flow

Abstract: We give a definition of the spectral flow for continuous paths in the space of bounded and essentially hyperbolic operators. We provide a homotopical characterization of the spectral flow in terms of a group homomorphism of the fundamental group of the projectors of the Calkin algebra with the infinite cyclic group Z. This characterization helps us to exhibit examples of infinite-dimensional Banach spaces where the spectral flow is not injective nor surjective. We prove that a path with spectral flow equal to an integer m exists if and only if there exists a projector P connected by an arc to a projector Q such that Range(Q) has co-dimension m in Range(P). We prove that if A is an asymptotically hyperbolic and essentially splitting path the differential operator F(u) = du/dt - Au is Fredholm. Moreover if A is also essentially hyperbolic the Fredholm index coincides with minus the spectral flow of A.


Abstract: We treat two different topics on the log minimal model program, especially for four-dimensional log canonical pairs. (a) Finite generation of the log canonical ring in dimension four. (b) Abundance theorem for irregular fourfolds. We obtain (a) as a direct consequence of the existence of four-dimensional log minimal models by using Fukuda's theorem on the four-dimensional log abundance conjecture. We can prove (b) only by using traditional arguments. More precisely, we prove the abundance conjecture for irregular $(n+1)$-folds on the assumption that the minimal model conjecture and the abundance conjecture hold in dimension $\leq n$.


Markov convexity and local rigidity of distorted metrics

Abstract: It is shown that a Banach space admits an equivalent norm whose modulus of uniform convexity has power-type p if and only if it is Markov p-convex. Counterexamples are constructed to natural questions related to isomorphic uniform convexity of metric spaces, showing in particular that tree metrics fail to have the dichotomy property.


Random fields of multivariate test statistics, with applications to shape analysis

Abstract: Our data are random fields of multivariate Gaussian observations, and we fit a multivariate linear model with common design matrix at each point. We are interested in detecting those points where some of the coefficients are nonzero using classical multivariate statistics evaluated at each point. The problem is to find the $P$-value of the maximum of such a random field of test statistics. We approximate this by the expected Euler characteristic of the excursion set. Our main result is a very simple method for calculating this, which not only gives us the previous result of Cao and Worsley [Ann. Statist. 27 (1999) 925--942] for Hotelling's $T^2$, but also random fields of Roy's maximum root, maximum canonical correlations [Ann. Appl. Probab. 9 (1999) 1021--1057], multilinear forms [Ann. Statist. 29 (2001) 328--371], $\bar{\chi}^2$ [Statist. Probab. Lett 32 (1997) 367--376, Ann. Statist. 25 (1997) 2368--2387] and $\chi^2$ scale space [Adv. in Appl. Probab. 33 (2001) 773--793]. The trick involves approaching the problem from the point of view of Roy's union-intersection principle. The results are applied to a problem in shape analysis where we look for brain damage due to nonmissile trauma.


Rodeo: Sparse, greedy nonparametric regression

Abstract: We present a greedy method for simultaneously performing local bandwidth selection and variable selection in nonparametric regression. The method starts with a local linear estimator with large bandwidths, and incrementally decreases the bandwidth of variables for which the gradient of the estimator with respect to bandwidth is large. The method--called rodeo (regularization of derivative expectation operator)--conducts a sequence of hypothesis tests to threshold derivatives, and is easy to implement. Under certain assumptions on the regression function and sampling density, it is shown that the rodeo applied to local linear smoothing avoids the curse of dimensionality, achieving near optimal minimax rates of convergence in the number of relevant variables, as if these variables were isolated in advance.


Disproof of the Continuum Hypothesis and Determination of the Cardinality of Continuum by Approximations of Sets

Abstract: A set theory is developed based on the approximations of sets and denoted by AS. In AS the set of all sets exists but the argument for Russell's and Cantor's paradox fail. The Axioms of Separation, Replacement and Foundation are not valid. All the other axioms of ZF are valid and all the basic sets, such as complement, intersection and cartesian product, exist although complement is not quite the same set as in ZF. The set of all sets can be equipped with the topology of approximations (Ta). Every set is closed and every function is continuous in Ta. This implies that the Continuum Hypothesis is false. The sets containing a subset which is perfect in Ta are of the greatest cardinality. A simple observation shows that the concept of well-ordering must be defined in a slightly different way than in ZF. We prove that a set can be well-ordered if and only if it has no perfect subset. Therefore the cardinalities of arbitrary sets are always comparable without assuming the Axiom of Choice. The cardinals following the smallest infinite cardinal w are 2w, 3w, ..., w^2, 2w^2, 3w^2, >..., w^3.... each being of greater cardinality than the previous one, which is not the case in ZF. Immediately after these cardinals does not follow w^w which is not a well-orderable set but some well-ordered cardinal k, and this one is followed by the cardinals 2k, 3k, ..., k^2, 2k^2, 3k^2, ..., k^3...., etc. The greatest cardinal is P(w) and is not a well-orderable set. The cofinality of a well-ordered set is either 2 or w. The only regular cardinals are 0, 1, 2, w and P(w). All other cardinals are singular. The only strong limit cardinal is w. The only inaccessible cardinal is P(w). Strongly inaccessible cardinals do not exist.


Approximation and learning by greedy algorithms

Abstract: We consider the problem of approximating a given element $f$ from a Hilbert space $\mathcal{H}$ by means of greedy algorithms and the application of such procedures to the regression problem in statistical learning theory. We improve on the existing theory of convergence rates for both the orthogonal greedy algorithm and the relaxed greedy algorithm, as well as for the forward stepwise projection algorithm. For all these algorithms, we prove convergence results for a variety of function classes and not simply those that are related to the convex hull of the dictionary. We then show how these bounds for convergence rates lead to a new theory for the performance of greedy algorithms in learning. In particular, we build upon the results in [IEEE Trans. Inform. Theory 42 (1996) 2118--2132] to construct learning algorithms based on greedy approximations which are universally consistent and provide provable convergence rates for large classes of functions. The use of greedy algorithms in the context of learning is very appealing since it greatly reduces the computational burden when compared with standard model selection using general dictionaries.


On the LMO conjecture

Abstract: We give a proof of the LMO conjecture which say that for any simply connectd simple Lie group $G$, the LMO invariant of rational homology 3-spheres recovers the perturvative invariant $\tau^{PG}$. By Habiro-Le theorem, this implies that the LMO invariant is the universal quantum invariant of integral homology 3-spheres.


High breakdown point robust regression with censored data

Abstract: In this paper, we propose a class of high breakdown point estimators for the linear regression model when the response variable contains censored observations. These estimators are robust against high-leverage outliers and they generalize the LMS (least median of squares), S, MM and $\tau$-estimators for linear regression. An important contribution of this paper is that we can define consistent estimators using a bounded loss function (or equivalently, a redescending score function). Since the calculation of these estimators can be computationally costly, we propose an efficient algorithm to compute them. We illustrate their use on an example and present simulation studies that show that these estimators also have good finite sample properties.


Primes in the form $[\alpha p+\beta]$

Abstract: Let \beta be a real number. Then for almost all irrational \alpha>0 (in the sense of Lebesgue measure) \limsup_{x\to\infty}\pi_{\alpha,\beta}^*(x)(\log x)^2/x>=1, where \pi_{\alpha,\beta}^*(x)={p<=x: both p and [\alpha p+\beta] are primes}.


Poincare series of filtrations corresponding to ideals on surfaces

Abstract: Earlier the authors considered and, in some cases, computed Poincare series of two sorts of multi-index filtrations on the ring of germs of functions on a complex (normal) surface singularity (in particular on the complex plane). A filtration from the first class was defined by a curve (with several branches) on the surface singularity. The other one (so called divisorial filtration) was defined by a set of components of the exceptional divisor of a modification of the surface singularity. Here we define a filtration corresponding to an ideal or to a set of ideals in the ring of germs of functions on a surface singularity and compute the corresponding Poincare series in some cases. For the complex plane this notion unites the two classes of filtrations described above.


Equivalences between fusion systems of finite groups of Lie type

Abstract: We prove, for certain pairs G,G of finite groups of Lie type, that the p-fusion systems for G and G' are equivalent. In other words, there is an isomorphism between a Sylow p-subgroup of G and one of G' which preserves p-fusion. This occurs, for example, when G=H(q) and G'=H(q') for a simple Lie type H, and q and q' are prime powers, both prime to p, which generate the same closed subgroup of the p-adic units. Our proof uses homotopy theoretic properties of the p-completed classifying spaces of G and G', and we know of no purely algebraic proof of this result.


An isomorphism between the completion of an Algebra and its Caratheodory Extension

Abstract: Let $\Omega$ denote an algebra of sets and $\mu$ a $\sigma$-finite measure. We then prove that the completion of $\Omega$ under the pseudometric $d(A,B)$ = $\mu^{\ast}(A \triangle B)$ is $\sigma$-algebra isomorphic and isometric to the Caratheodory Extension of $\Omega$ under the equivalence relation $\sim$.