Diffusion covariation and co-jumps in bidimensional asset price processes with stochastic volatility and infinite activity Levy jumps

Abstract: In this paper we consider two processes driven by diffusions and jumps. The jump components are Levy processes and they can both have finite activity and infinite activity. Given discrete observations we estimate the covariation between the two diffusion parts and the co-jumps. The detection of the co-jumps allows to gain insight in the dependence structure of the jump components and has important applications in finance. Our estimators are based on a threshold principle allowing to isolate the jumps. This work follows Gobbi and Mancini (2006) where the asymptotic normality for the estimator of the covariation, with convergence speed given by the squared root of h, was obtained when the jump components have finite activity. Here we show that the speed is the squared root of h only when the activity of the jump components is moderate.


Causal inference in longitudinal studies with history-restricted marginal structural models

Abstract: A new class of Marginal Structural Models (MSMs), History-Restricted MSMs (HRMSMs), was recently introduced for longitudinal data for the purpose of defining causal parameters which may often be better suited for public health research or at least more practicable than MSMs \citejoffe,feldman. HRMSMs allow investigators to analyze the causal effect of a treatment on an outcome based on a fixed, shorter and user-specified history of exposure compared to MSMs. By default, the latter represent the treatment causal effect of interest based on a treatment history defined by the treatments assigned between the study's start and outcome collection. We lay out in this article the formal statistical framework behind HRMSMs. Beyond allowing a more flexible causal analysis, HRMSMs improve computational tractability and mitigate statistical power concerns when designing longitudinal studies. We also develop three consistent estimators of HRMSM parameters under sufficient model assumptions: the Inverse Probability of Treatment Weighted (IPTW), G-computation and Double Robust (DR) estimators. In addition, we show that the assumptions commonly adopted for identification and consistent estimation of MSM parameters (existence of counterfactuals, consistency, time-ordering and sequential randomization assumptions) also lead to identification and consistent estimation of HRMSM parameters.


On regularity of SLE_8 curves

Abstract: This paper has been withdrawn by the author due to essential mistakes in some previous versions.


A novel characterization of the Iwasawa decomposition of a simple Lie group

Abstract: It is about the uniqueness of the Iwasawa decomposition.


On Power Stable Ideals

Abstract: We define the notion of a power stable ideal in a polynomial ring $ R[X]$ over an integral domain $ R $. It is proved that a maximal ideal $\chi$ $ M $ in $ R[X]$ is power stable if and only if $ P^t $ is $ P$- primary for all $ t\geq 1 $ for the prime ideal $ P = M \cap R $. Using this we prove that for a Hilbert domain $R$ any radical ideal in $R[X]$ which is a finite intersection G-ideals is power stable. Further, we prove that if $ R $ is a Noetherian integral domain of dimension 1 then any radical ideal in $ R[X] $ is power stable. Finally, it is proved that if every ideal in $ R[X]$ is power stable then $ R $ is a field.


Uniform random sampling of planar graphs in linear time

Abstract: This article introduces new algorithms for the uniform random generation of labelled planar graphs. Its principles rely on Boltzmann samplers, as recently developed by Duchon, Flajolet, Louchard, and Schaeffer. It combines the Boltzmann framework, a suitable use of rejection, a new combinatorial bijection found by Fusy, Poulalhon and Schaeffer, as well as a precise analytic description of the generating functions counting planar graphs, which was recently obtained by Gim\'enez and Noy. This gives rise to an extremely efficient algorithm for the random generation of planar graphs. There is a preprocessing step of some fixed small cost. Then, the expected time complexity of generation is quadratic for exact-size uniform sampling and linear for approximate-size sampling. This greatly improves on the best previously known time complexity for exact-size uniform sampling of planar graphs with $n$ vertices, which was a little over $O(n^7)$.


Some result on K-algebras

Abstract: We give a new proof of the classical result due to Rodney Y. Sharp and Peter Vamos on the dimension of tensor product of a finite number of field extensions of a given field.


Domino Tiling Congruence Modulo 4

Abstract: The number of domino tilings of a region with reflective symmetry across a line is combinatorially shown to depend on the number of domino tilings of particular subregions, modulo 4. This expands upon previous congruency results for domino tilings, modulo 2, and leads to a variety of corollaries, including that the number of domino tilings of a k x 2k rectangle is congruent to 1 mod 4.


A General Fredholm Theory II: Implicit Function Theorems

Abstract: This is the revised version of the second paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. The theory will be illustrated in upcoming papers by applications to Floer Theory, Gromov-Witten Theory and Symplectic Field Theory. Some proofs have been improved and a glossary added.


Novikov algebras and Novikov structures on Lie algebras

Abstract: We study ideals of Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We present the first example of a three-step nilpotent Lie algebra which does not admit a Novikov structure. On the other hand we show that any free three-step nilpotent Lie algebra admits a Novikov structure. We study the existence question also for Lie algebras of triangular matrices. Finally we show that there are families of Lie algebras of arbitrary high solvability class which admit Novikov structures.


Mutant knots with symmetry

Abstract: Mutant knots, in the sense of Conway, are known to share the same Homfly polynomial. Their 2-string satellites also share the same Homfly polynomial, but in general their m-string satellites can have different Homfly polynomials for m>2. We show that, under conditions of extra symmetry on the constituent 2-tangles, the directed m-string satellites of mutants share the same Homfly polynomial for m<6 in general, and for all choices of m when the satellite is based on a cable knot pattern. We give examples of mutants with extra symmetry whose Homfly polynomials of some 6-string satellites are different, by comparing their quantum sl(3) invariants.


Third Order Newton's Method for Zernike Polynomial Zeros

Abstract: The Zernike radial polynomials are a system of orthogonal polynomials over the unit interval with weight x. They are used as basis functions in optics to expand fields over the cross section of circular pupils. To calculate the roots of Zernike polynomials, we optimize the generic iterative numerical Newton's Method that iterates on zeros of functions with third order convergence. The technique is based on rewriting the polynomials as Gauss hypergeometric functions, reduction of second order derivatives to first order derivatives, and evaluation of some ratios of derivatives by terminating continued fractions. A PARI program and a short table of zeros complete up to polynomials of 20th order are included.


Global Solutions to the Ultra-Relativistic Euler Equations

Abstract: We prove a global existence theorem for the $3\times 3$ system of relativistic Euler equations in one spacial dimension. It is shown that in the ultra-relativistic limit, there is a family of equations of state that satisfy the second law of thermodynamics for which solutions exist globally. With this limit and equation of state, which includes equations of state for both an ideal gas and one dominated by radiation, the relativistic Euler equations can be analyzed by a Nishida-type method leading to a large data existence theorem, including the entropy and particle number evolution, using a Glimm scheme. Our analysis uses the fact that the equations of state are of the form $p=p(n,S)$, but whose form simplifies to $p=a^{2}\rho$ when viewed as a function of $\rho$ alone.


Invertibility of the Gabor frame operator on the Wiener amalgam space

Abstract: We use a generalization of Wiener's $1/f$ theorem to prove that for a Gabor frame with the generator in the Wiener amalgam space $W(L^{\infty}, \ell^{1}_{\nu})(\mathbb{R}^{d})$, the corresponding frame operator is invertible on this space. Therefore, for such a Gabor frame, the generator of the canonical dual belongs also to $W(L^{\infty}, \ell^{1}_{\nu})(\mathbb{R}^{d}) $


On the approximate normality of eigenfunctions of the Laplacian

Abstract: The main result of this paper is a bound on the distance between the distribution of an eigenfunction of the Laplacian on a compact Riemannian manifold and the Gaussian distribution. If $X$ is a random point on a manifold $M$ and $f$ is an eigenfunction of the Laplacian with $L^2$-norm one and eigenvalue $-\mu$, then $$d_{TV}(f(X),Z)\le\frac{2}{\mu}\E\big|\|\nabla f(X)\|^2-\E\|\nabla f(X) \|^2\big|.$$ This result is applied to construct specific examples of spherical harmonics of arbitrary (odd) degree which are close to Gaussian in distribution. A second application is given to random linear combinations of eigenfunctions on flat tori.


Semisimple Algebraic Groups in Characteristic Zero

Abstract: It is shown that the classification theorems for semisimple algebraic groups in characteristic zero can be derived quite simply and naturally from the corresponding theorems for Lie algebras by using a little of the theory of tensor categories. This article is extracted from Milne 2007.


Thom polynomials and Schur functions: the singularities I_{2,2}(-)

Abstract: We give the Thom polynomials for the singularities $I_{2,2}$ associated with maps $({\bf C}^{\bullet},0) \to ({\bf C}^{\bullet+k},0)$ with parameter $k\ge 0$. Our computations combine the characterization of Thom polynomials via the ``method of restriction equations'' of Rimanyi et al. with the techniques of Schur functions.


On Two of Erd\"os's Open Problems

Abstract: In this short note we present some remarks and conjectures on two of Erd\"os's open problems in number theory.


Quantum cohomology of G/P and homology of affine Grassmannian

Abstract: Let G be a simple and simply-connected complex algebraic group, P \subset G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH^*(G/P) of a flag variety is, up to localization, a quotient of the homology H_*(Gr_G) of the affine Grassmannian \Gr_G of G. As a consequence, all three-point genus zero Gromov-Witten invariants of $G/P$ are identified with homology Schubert structure constants of H_*(Gr_G), establishing the equivalence of the quantum and homology affine Schubert calculi. For the case G = B, we use the Mihalcea's equivariant quantum Chevalley formula for QH^*(G/B), together with relationships between the quantum Bruhat graph of Brenti, Fomin and Postnikov and the Bruhat order on the affine Weyl group. As byproducts we obtain formulae for affine Schubert homology classes in terms of quantum Schubert polynomials. We give some applications in quantum cohomology. Our main results extend to the torus-equivariant setting.


The spectral shift function and spectral flow

Abstract: This paper extends Krein's spectral shift function theory to the setting of semifinite spectral triples. We define the spectral shift function under these hypotheses via Birman-Solomyak spectral averaging formula and show that it computes spectral flow.


Every compact group arises as the outer automorphism group of a II_1 factor

Abstract: We show that any compact group can be realized as the outer automorphism group of a factor of type II_1. This has been proved in the abelian case by Ioana, Peterson and Popa applying Popa's deformation/rigidity techniques to amalgamated free product von Neumann algebras. Our methods are a generalization of theirs.


Counting hyperelliptic curves that admit a Koblitz model

Abstract: Let k be a finite field of odd characteristic. We find a closed formula for the number of k-isomorphism classes of pointed, and non-pointed, hyperelliptic curves of genus g over k, admitting a Koblitz model. These numbers are expressed as a polynomial in the cardinality q of k, with integer coefficients (for pointed curves) and rational coefficients (for non-pointed curves). The coefficients depend on g and the set of divisors of q-1 and q+1. These formulas show that the number of hyperelliptic curves of genus g suitable (in principle) of cryptographic applications is asymptotically (1-e^{-1})2q^{2g-1}, and not 2q^{2g-1} as it was believed. The curves of genus g=2 and g=3 are more resistant to the attacks to the DLP; for these values of g the number of curves is respectively (91/72)q^3+O(q^2) and (3641/2880)q^5+O(q^4).


Fonction Z\^eta de Hurwitz p-adique et irrationalit\'e

Abstract: The knowledge on irrationality of p-adic zeta values has recently progressed. The irrationality of zeta_2(2), \zeta_2(3) and of a few other p-adic series of Dirichlet was obtained by F. Calegari. F. Beukers gave a more elementary proof of this result. In parallel, T. Rivoal has just obtained, in the complex case, some Pade approximants of Lerch functions. It is this work which, transposed to C_p, enables us to obtain results of irrationality and linear independence.


On b-function, spectrum and multiplier ideals

Abstract: We give a survey on b-function, spectrum, and multiplier ideals together with certain interesting relations among them including the case of arbitrary subvarieties.


Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space

Abstract: We classify the entire minimal vertical graphs in the 3 dimensional Heisenberg group Nil endowed with a Riemannian left-invariant metric. This classification, which provides a solution to the Bernstein problem in Nil, is given in terms of the Abresch-Rosenberg holomorphic differential for minimal surfaces in Nil.


Contractible groups and linear dilatation structures

Abstract: A dilatation structure on a metric space, arXiv:math/0608536v4, is a notion in between a group and a differential structure, accounting for the approximate self-similarity of the metric space. The basic objects of a dilatation structure are dilatations (or contractions). The axioms of a dilatation structure set the rules of interaction between different dilatations. Linearity is also a property which can be explained with the help of a dilatation structure. In this paper we show that we can speak about two kinds of linearity: the linearity of a function between two dilatation structures and the linearity of the dilatation structure itself. Our main result here is a characterization of contractible groups in terms of dilatation structures. To a normed conical group (normed contractible group) we can naturally associate a linear dilatation structure. Conversely, any linear and strong dilatation structure comes from the dilatation structure of a normed contractible group.


Embedding Degree of Hyperelliptic Curves with Complex Multiplication

Abstract: Consider the Jacobian of a genus two curve defined over a finite field and with complex multiplication. In this paper we show that if the l-Sylow subgroup of the Jacobian is not cyclic, then the embedding degree of the Jacobian with respect to l is one.


Rational homotopy type of subspace arrangements with a geometric lattice

Abstract: Let A be a subspace arrangement with a geometric lattice such that codim(x) > 1 for every x in A. Using rational homotopy theory, we prove that the complement M(A) is rationally elliptic if and only if the sum of the orthogonal subspaces is a direct sum. The homotopy type of M(A) is also given: it is a product of odd dimensional spheres. Finally, some other equivalent conditions are given, such as Poincare duality. Those results give a complete description of arrangements (with geometric lattice and with the codimension condition on the subspaces) such that M(A) is rationally elliptic, and show that most arrangements have an hyperbolic complement.


Homotopy Lie algebra of the complements of subspace arrangements with geometric lattices

Abstract: Let A be a geometric arrangement such that codim(x) > 1 for every x in A. We prove that, if the complement space M(A) is rationally hyperbolic, then there exists an injective from a free Lie algebra L(u,v) to the homotopy Lie algebra of M(A).


A simple uniform approach to complexes arising from forests

Abstract: In this paper we present a unifying approach to study the homotopy type of several complexes arising from forests. We show that this method applies uniformly to many complexes that have been extensively studied.


Bispectrality of multivariable Racah-Wilson polynomials

Abstract: We construct a commutative algebra A_x of difference operators in R^p, depending on p+3 real parameters which is diagonalized by the multivariable Racah polynomials R_p(n;x) considered by Tratnik [27]. It is shown that for specific values of the variables x=(x_1,x_2,...,x_p) there is a hidden duality between n and x. Analytic continuation allows us to construct another commutative algebra A_n in the variables n=(n_1,n_2,...,n_p) which is also diagonalized by R_p(n;x). Thus R_p(n;x) solve a multivariable discrete bispectral problem in the sense of Duistermaat and Grunbaum [8]. Since a change of the variables and the parameters in the Racah polynomials gives the multivariable Wilson polynomials [26], this change of variables and parameters in A_x and A_n leads to bispectral commutative algebras for the multivariable Wilson polynomials.


Busemann points of Artin groups of dihedral type

Abstract: We study the horofunction boundary of an Artin group of dihedral type with its word metric coming from either the usual Artin generators or the dual generators. In both cases, we determine the horoboundary and say which points are Busemann points, that is the limits of geodesic rays. In the case of the dual generators, it turns out that all boundary points are Busemann points, but this is not true for the Artin generators. We also characterise the geodesics with respect to the dual generators, which allows us to calculate the associated geodesic growth series.


Nonorientable 3-manifolds admitting coloured triangulations with at most 30 tetrahedra

Abstract: We present the census of all non-orientable, closed, connected 3-manifolds admitting a rigid crystallization with at most 30 vertices. In order to obtain the above result, we generate, manipulate and compare, by suitable computer procedures, all rigid non-bipartite crystallizations up to 30 vertices.


Multiple Solutions for a Henon-Like Equation on the Annulus

Abstract: For the equation (-\Delta u = | |x|-2 |^\alpha u^{p-1}), (1 < |x| < 3), we prove the existence of two solutions for (\alpha) large, and of two additional solutions when (p) is close to the critical Sobolev exponent (2^*=2N/(N-2)). A symmetry--breaking phenomenon appears, showing that the least--energy solutions cannot be radial functions.


Trivialization of C(X)-algebras with strongly self-absorbing fibres

Abstract: Suppose $A$ is a separable unital $C(X)$-algebra each fibre of which is isomorphic to the same strongly self-absorbing and $K_{1}$-injective $C^{*}$-algebra $D$. We show that $A$ and $C(X) \otimes D$ are isomorphic as $C(X)$-algebras provided the compact Hausdorff space $X$ is finite-dimensional. This statement is known not to extend to the infinite-dimensional case.


Small clones and the projection property

Abstract: In 1986, the second author classified the minimal clones on a finite universe into five types. We extend this classification to infinite universes and to multiclones. We show that every non-trivial clone contains a "small" clone of one of the five types. From it we deduce, in part, an earlier result, namely that if $\mathcal C$ is a clone on a universe $A$ with at least two elements, that contains all constant operations, then all binary idempotent operations are projections and some $m$-ary idempotent operation is not a projection some $m\geq 3$ if and only if there is a Boolean group $G$ on $A$ for which $\mathcal C$ is the set of all operations $f(x_1,..., x_n)$ of the form $a+\sum_{i\in I}x_i$ for $a\in A$ and $I\subseteq \{1,..., n\}$.


Differentiable and deformation type of algebraic surfaces, real and symplectic structures

Abstract: Lecture 1: Projective and K\"ahler Manifolds, the Enriques classification, construction techniques. Lecture 2: Surfaces of general type and their Canonical models. Deformation equivalence and singularities. Lecture 3: Deformation and diffeomorphism, canonical symplectic structure for surfaces of general type. Lecture 4: Irrational pencils, orbifold fundamental groups, and surfaces isogenous to a product. Lecture 5: Lefschetz pencils, braid and mapping class groups, and diffeomorphism of ABC-surfaces. Epilogue: Deformation, diffeomorphism and symplectomorphism type of surfaces of general type.


On the distance between separatrices for the discretized logistic differential equation

Abstract: We consider the discretization y(t+\epsilon)=y(t-\epsilon)+2\epsilon\big(1-y(t)^{2}\big), $\epsilon>0$ a small parameter, of the logistic differential equation $y'=2\big(1-y^{2}\big)$, which can also be seen as a discretization of the system {y'=2\big(1-v^{2}\big), v'= 2\big(1-y^{2}\big). This system has two saddle points at $A=(1,1)$, $B=(-1, -1)$ and there exist stable and unstable manifolds. We will show that the stable manifold $W_{s}^{+}$ of the point $A=(1,1)$ and the unstable manifold $W_{i}^{-}$ of the point $B=(-1, -1)$ for the discretization do not coincide. The vertical distance between these two manifolds is exponentially small but not zero, in particular we give an asymptotic estimate of this distance. For this purpose we will use a method adapted from the paper of Sch\"afke-Volkmer \cite{SV} using formal series and accurate estimates of the coefficients.


The Rotor-Router Model on Regular Trees

Abstract: The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We show that the set of occupied sites for this model on an infinite regular tree is a perfect ball whenever it can be, provided the initial rotor configuration is acyclic (that is, no two neighboring vertices have rotors pointing to one another). This is proved by defining the rotor-router group of a graph, which we show is isomorphic to the sandpile group. We also address the question of recurrence and transience: We give two rotor configurations on the infinite ternary tree, one for which chips exactly alternate escaping to infinity with returning to the origin, and one for which every chip returns to the origin. Further, we characterize the possible "escape sequences" for the ternary tree, that is, binary words a_1 ... a_n for which there exists a rotor configuration so that the k-th chip escapes to infinity if and only if a_k=1.


On the square root of the centre of the Hecke algebra of type A

Abstract: In this paper we investigate non-central elements of the Iwahori-Hecke algebra of the symmetric group whose squares are central. In particular, we describe a commutative subalgebra generated by certain non-central square roots of central elements, and the generic existence of a rank-three submodule of the Hecke algebra contained in the square root of the centre but not in the centre. The generators for this commutative subalgebra include the longest word and elements related to trivial and sign characters of the Hecke algebra. We find elegant expressions for the squares of such generators in terms of both the minimal basis of the centre and the elementary symmetric functions of Murphy elements.


On Shalika Periods and a Theorem of Jacquet-Martin

Abstract: Let \pi be a cuspidal automorphic representation of GL_4 with central character \mu^2. It is known that \pi has Shalika period with respect to \mu if and only if the L-function L^S(s, \pi, \bigwedge^2\otimes\mu^{-1}) has a pole at s=1. Recentlt, Jacquet and Martin considered the analogous question for cuspidal representations \pi_D of the inner form GL_2(D)(\A), and obtained a partial result via the relative trace formula. In this paper, we provide a complete solution to this problem via the method of theta correspondence, and give necessary and sufficient conditions for the existence of Shalika period for \pi_D. We also resolve the analogous question in the local setting.


Axiomatic theory of divergent series and cohomological equations

Abstract: A general theory of summation of divergent series based on the Hardy-Kolmogorov axioms is developed. A class of functional series is investigated by means of ergodic theory. The results are formulated in terms of solvability of some cohomological equations, all solutions to which are nonmeasurable. In particular, this realizes a construction of a nonmeasurable function as first conjectured by Kolmogorov.


A new integral basis for the centre of the Hecke algebra of type A

Abstract: We describe a recursive algorithm that produces an integral basis for the centre of the Hecke algebra of type A consisting of linear combinations of monomial symmetric polynomials of Jucys--Murphy elements. We also discuss the existence of integral bases for the centre of the Hecke algebra that consist solely of monomial symmetric polynomials of Jucys--Murphy elements. Finally, for n=3, we show that only one such basis exists for the centre of the Hecke algebra, by proving that there are exactly four bases for the centre of the corresponding symmetric group algebra which consist solely of monomial symmetric polynomials of Jucys--Murphy elements.


Bauer-Furuta invariants under Z_2-actions

Abstract: S.Bauer and M.Furuta defined a stable cohomotopy refinement of the Seiberg-Witten invariants. In this paper, we prove a vanishing theorem of Bauer-Furuta invariants for 4-manifolds with smooth Z/2-actions. As an application, we give a constraint on smooth Z/2-actions on homotopy K3#K3, and construct a nonsmoothable locally linear Z/2-action on K3#K3. We also construct a nonsmoothable locally linear Z/2-action on $K3$.


Weight 2 blocks of general linear groups and modular Alvis-Curtis duality

Abstract: We obtain the structure of weight 2 blocks and [2:1]-pairs of q-Schur algebras, and compute explicitly the modular Alvis-Curtis duality for weight 2 blocks of finite general linear groups in non-defining characteristic.


Application of Girsanov Theorem to Particle Filtering of Discretely Observed Continuous-Time Non-Linear Systems

Abstract: This article considers the application of particle filtering to continuous-discrete optimal filtering problems, where the system model is a stochastic differential equation, and noisy measurements of the system are obtained at discrete instances of time. It is shown how the Girsanov theorem can be used for evaluating the likelihood ratios needed in importance sampling. It is also shown how the methodology can be applied to a class of models, where the driving noise process is lower in the dimensionality than the state and thus the laws of state and noise are not absolutely continuous. Rao-Blackwellization of conditionally Gaussian models and unknown static parameter models is also considered.


Proof of the Double Bubble Conjecture in R^n

Abstract: The least-area hypersurface enclosing and separating two given volumes in R^n is the standard double bubble.


Perturbations of quadratic centers of genus one

Abstract: We propose a program for finding the cyclicity of period annuli of quadratic systems with centers of genus one. As a first step, we classify all such systems and determine the essential one-parameter quadratic perturbations which produce the maximal number of limit cycles. We compute the associated Poincare-Pontryagin-Melnikov functions whose zeros control the number of limit cycles. To illustrate our approach, we determine the cyclicity of the annuli of two particular reversible systems.


Estimates of the topological entropy from below for continuous self-maps on some compact manifolds

Abstract: Extending our results in "Entropy conjecture for continuous maps of nilmanifolds", to appear in Israel Jour. of Math., we confirm that Entropy Conjecture holds for every continuous self-map of a compact $K(\pi,1)$ manifold with the fundamental group $\pi$ torsion free and virtually nilpotent, in particular for every continuous map of an infra-nilmanifold. In fact we prove a stronger version, a lower estimate of the topological entropy of a map by logarithm of the spectral radius of an associated "linearization matrix" with integer entries. From this, referring to known estimates of Mahler measure of polynomials, we deduce some absolute lower bounds for the entropy.


Determining full conditional independence by low-order conditioning

Abstract: A concentration graph associated with a random vector is an undirected graph where each vertex corresponds to one random variable in the vector. The absence of an edge between any pair of vertices (or variables) is equivalent to full conditional independence between these two variables given all the other variables. In the multivariate Gaussian case, the absence of an edge corresponds to a zero coefficient in the precision matrix, which is the inverse of the covariance matrix. It is well known that this concentration graph represents some of the conditional independencies in the distribution of the associated random vector. These conditional independencies correspond to the "separations" or absence of edges in that graph. In this paper we assume that there are no other independencies present in the probability distribution than those represented by the graph. This property is called the perfect Markovianity of the probability distribution with respect to the associated concentration graph. We prove in this paper that this particular concentration graph, the one associated with a perfect Markov distribution, can be determined by only conditioning on a limited number of variables. We demonstrate that this number is equal to the maximum size of the minimal separators in the concentration graph.


Boundary Harnack inequalities for regional fractional Laplacian

Abstract: We consider boundary Harnack inequalities for regional fractional Laplacian which are generators of censored stable-like processes on G taking \kappa(x,y)/|x-y|^{n+\alpha}dxdy, x,y\in G as the jumping measure. When G is a C^{1,\beta-1} open set, 1<\alpha<\beta\leq 2 and \kappa\in C^{1}(\overline{G}\times \overline{G}) bounded between two positive numbers, we prove a boundary Harnack inequality giving dist(x,\partial G)^{\alpha-1} order decay for harmonic functions near the boundary. For a C^{1,\beta-1} open set D\subset \overline{D}\subset G, 0<\alpha\leq (1\vee\alpha)<\beta\leq 2, we prove a boundary Harnack inequality giving dist(x,\partial D)^{\alpha/2} order decay for harmonic functions near the boundary. These inequalities are generalizations of the known results for the homogeneous case on C^{1,1} open sets. We also prove the boundary Harnack inequality for regional fractional Laplacian on Lipschitz domain.


Finite Just Non-Dedekind Groups

Abstract: A group is just non-Dedekind (JND) if it is not a Dedekind group but all of whose proper homomorphic images are Dedekind groups. The aim of the paper is to classify finite JND-groups.


Deligne-Lusztig varieties and period domains over finite fields

Abstract: We prove that the Drinfeld halfspace is essentially the only Deligne-Lusztig variety which is at the same time a period domain over a finite field. This is done by comparing a cohomology vanishing theorem for DL-varieties, due to Digne, Michel, and Rouquier, with a non-vanishing theorem for PD, due to the first author. We also discuss an affineness criterion for DL-varieties.


Chen's double sieve, Goldbach's conjecture and the twin prime problem

Abstract: We give a more comrehensive treatment of Chen's double sieve and improve related constants in Goldbach's conjecture and the twin prime problem.


Gromov-Witten theory and Noether-Lefschetz theory

Abstract: Noether-Lefschetz divisors in the moduli of K3 surfaces are the loci corresponding to Picard rank at least 2. We relate the degrees of the Noether-Lefschetz divisors in 1-parameter families of K3 surfaces to the Gromov-Witten theory of the 3-fold total space. The reduced K3 theory and the Yau-Zaslow formula play an important role. We use results of Borcherds and Kudla-Millson for O(2,19) lattices to determine the Noether-Lefschetz degrees in classical families of K3 surfaces of degrees 2, 4, 6 and 8. For the quartic K3 surfaces, the Noether-Lefschetz degrees are proven to be the Fourier coefficients of an explicitly computed modular form of weight 21/2 and level 8. The interplay with mirror symmetry is discussed. We close with a conjecture on the Picard ranks of moduli spaces of K3 surfaces.


Existence of extremal Beltrami coefficients with non-constant modulus

Abstract: Suppose $[\mu]_{T(\Delta)}$ is a point of the universal Teichm\"uller space $T(\Delta)$. In 1998, it was shown by Bo\v{z}in et al. that there exists $\mu$ such that $\mu$ has non-constant modulus and is uniquely extremal in $[\mu]_{T(\Delta)}$. It is a natural problem whether there is always an extremal Beltrmai coefficient of constant modulus in $[\mu]_{T(\Delta)}$ if $[\mu]_{T(\Delta)}$ admits more than one extremal Beltrami coefficient. The purpose of this paper is to show that the answer is negative. An infinitesimal version is also obtained. Extremal sets of extremal Beltrami coefficients are considered and an open problem is proposed.


Existence results for mean field equations with turbulence

Abstract: In this paper we consider the following form of the so-called Mean field equation arising from the statistical mechanics description of two dimensional turbulence \begin{equation}\label{eq:study} - \D_g u = \rho_1 (\frac{e^{u}}{\int_\Sig e^{u} dV_g}-1)-\rho_2 (\frac{e^{-u}}{\int_\Sig e^{-u} dV_g} - 1) \end{equation} on a given closed orientable Riemannian surface ($\Sigma, g$) with volume 1, where $\rho_1, \rho_2$ are real parameters. Exploiting the variational structure of the problem and running a min-max scheme introduced by Djadli and Malchiodi, we prove that if $k$ is a positive integer, $\rho_1$ and $\rho_2$ two real numbers such that $\rho_1\in (8k\pi, 8(k+1)\pi)$ and $\rho_2<4\pi$ then $\eqref{eq:study}$ is solvable.


Generalized Brjuno functions associated to $\alpha$-continued fractions

Abstract: For \alpha in the interval [0,1], we consider the one-parameter family of \alpha-continued fraction maps, which include the Gauss map (\alpha=1) and the nearest integer (\alpha=1/2) and by-excess (\alpha=0) continued fraction maps. To each of these expansions, and to each choice of a positive function u on the interval I_\alpha=(0,max(\alpha,1-\alpha)) we associate a generalized Brjuno function B_(\alpha,u)(x). For \alpha=1/2 or \alpha=1, and u(x)=-\log(x), these functions were introduced by Yoccoz in his work on the linearization of holomorphic maps. Their regularity properties, including BMO regularity and their extension to the complex plane, have been thoroughly investigated. We compare the functions obtained with different values of \alpha and we prove that the set of (\alpha,u)-Brjuno numbers does not depend on the choice of \alpha provided that \alpha>0. We then consider the case \alpha=0, u(x)=-\log(x) and we prove that x is a Brjuno number (for \alpha> 0) if and only if both x and -x are Brjuno numbers for \alpha=0.


Subelliptic Spin_C Dirac operators, I

Abstract: We consider modifications of the classical dbar-Neumann conditions that define Fredholm problems for the Spin_C Dirac operator. In part II, we use boundary layer methods to obtain subelliptic estimates for these boundary value problems. Using these results, we obtain an expression for the finite part of the holomorphic Euler characteristic of a strictly pseudoconvex manifold as the index of a Spin_C-Dirac operator with a subelliptic boundary condition. We also prove an analogue of the Agranovich-Dynin formula expressing the change in the index in terms of a relative index on the boundary. If X is a complex manifold partitioned by a strictly pseudoconvex hypersurface, then we obtain formulae for the holomorphic Euler characteristic of X as sums of indices of Spin_C-Dirac operators on the components. This is a subelliptic analogue of Bojarski's formula in the elliptic case.


Hives and the fibres of the convolution morphism

Abstract: By the geometric Satake correspondence, the number of components of certain fibres of the affine Grassmannian convolution morphism equals the tensor product multiplicity for representations of the Langlands dual group. On the other hand, in the case of GL_n, combinatorial objects called hives also count tensor product multiplicities. The purpose of this paper is to give a simple bijection between hives and the components of these fibres. In particular, we give a description of the individual components. We also describe a conjectural generalization involving the octahedron recurrence.


Subelliptic Spin_C Dirac operators, II Basic Estimates

Abstract: We assume that the manifold with boundary, X, has a Spin_C-structure with spinor bundle S. Along the boundary, this structure agrees with the structure defined by an infinite order integrable almost complex structure and the metric is Kahler. The induced CR-structure on bX is integrable and either strictly pseudoconvex or strictly pseudoconcave. We assume that E->X is a complex vector bundle, which has an infinite order integrable complex structure along bX, compatible with that defined along bX. In this paper use boundary layer methods to prove subelliptic estimates for the twisted Spin_C- Dirac operator acting on sections on S\otimes E. We use boundary conditions that are modifications of the classical dbar-Neumann condition. These results are proved by using the extended Heisenberg calculus.


Universality results for largest eigenvalues of some sample covariance matrix ensembles

Abstract: For sample covariance matrices with iid entries with sub-Gaussian tails, when both the number of samples and the number of variables become large and the ratio approaches to one, it is a well-known result of A. Soshnikov that the limiting distribution of the largest eigenvalue is same as the of Gaussian samples. In this paper, we extend this result to two cases. The first case is when the ratio approaches to an arbitrary finite value. The second case is when the ratio becomes infinity or arbitrarily small.


Cobordism, Relative Indices and Stein Fillings

Abstract: In this paper we build on the framework developed in "Subelliptic Boundary Value Problems for the Spin_C Dirac Operator, I, II, III" to obtain a more complete understanding of the gluing properties for indices of boundary value problems for the SpinC-Dirac operator with sub-elliptic boundary conditions. We extend our analytic results for sub-elliptic boundary value problems for the SpinC-Dirac operator, and gluing results for the indices of these boundary problems to SpinC-manifolds with several pseudoconvex (pseudoconcave) boundary components. These results are applied to study Stein fillability for compact, 3-dimensional, contact manifolds.


Slicing, skinning, and grafting

Abstract: We prove that a Bers slice is never algebraic, meaning that its Zariski closure in the character variety has strictly larger dimension. A corollary is that skinning maps are never constant. The proof uses grafting and the theory of complex projective structures.


A homotopy method for finding eigenvalues and eigenvectors

Abstract: Suppose we want to find the eigenvalues and eigenvectors for the linear operator L, and suppose that we have solved this problem for some other "nearby" operator K. In this paper we show how to represent the eigenvalues and eigenvectors of L in terms of the corresponding properties of K.


Differential operators on toric varieties and Fourier transform

Abstract: We show that Fourier transforms on the Weyl algebras have a geometric counterpart in the framework of toric varieties, namely they induce isomorphisms between twisted rings of differential operators on regular toric varieties, whose fans are related to each other by reflections of one-dimensional cones. The simplest class of examples is provided by the toric varieties related by such reflections to projective spaces. It includes the blow-up at a point in affine space and resolution of singularities of varieties appearing in the study of the minimal orbit of sl(n+1).


Selfsimilar Equivalence of Porous Medium and p-Laplacian Flows

Abstract: We demonstrate the equivalence between the two popular models of nonlinear diffusion, the porous medium equation and the p-Laplacian equation. The equivalence is shown at the level of selfsimilar solutions.


Holomorphic fiber bundle with Stein base and Stein fibers

Abstract: In this article, we prove that if $\Pi: X\to \Omega$ is a surjective holomorphic map, with $\Omega$ a Stein space and $X$ a complex manifold of dimension $n\geq 3,$ and if, for every $x\in \Omega$ there exists an open neighborhood $U$ such that $\Pi^{-1}(U)$ is Stein, then $X$ is Stein


On Vojta's $1+\epsilon$ Conjecture

Abstract: I gave a geometric proof of Vojta's 1 + epsilon conjecture. Some gaps in the published paper were spotted and kindly pointed out to me by Paul Vojta. These were addressed in "Erratum".


Fibers of tropicalization

Abstract: We use functoriality of tropicalization and the geometry of projections of subvarieties of tori to show that the fibers of the tropicalization map are dense in the Zariski topology. For subvarieties of tori over fields of generalized power series, points in each tropical fiber are obtained "constructively" using Kedlaya's transfinite version of Newton's method.


The Large Sieve Inequality for Quadratic Polynomial Amplitudes

Abstract: We provide here a modest improvement upon a large sieve inequality for quadratic polynomial amplitudes orginally due to Liangyi Zhao.


Nonlinarity of Boolean functions and hyperelliptic curves

Abstract: We study the nonlinearity of functions defined on a finite field with 2^m elements which are the trace of a polynomial of degree 7 or more general polynomials. We show that for m odd such functions have rather good nonlinearity properties. We use for that recent results of Maisner and Nart about zeta functions of supersingular curves of genus 2. We give some criterion for a vectorial function not to be almost perfect nonlinear.


Recursive Parameter Estimation: Convergence

Abstract: We consider estimation procedures which are recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment. We propose a wide class of recursive estimation procedures for the general statistical model and study convergence.


Rate of Convergence in Recursive Parameter Estimation procedures

Abstract: We consider estimation procedures which are recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment. We study rate of convergence of recursive estimation procedures for the general statistical model.


The rate of convergence of Euler approximations for solutions of stochastic differential equations driven by fractional Brownian motion

Abstract: The paper focuses on discrete-type approximations of solutions to non-homogeneous stochastic differential equations (SDEs) involving fractional Brownian motion (fBm). We prove that the rate of convergence for Euler approximations of solutions of pathwise SDEs driven by fBm with Hurst index $H>1/2$ can be estimated by $O(\delta^{2H-1})$ ($\delta$ is the diameter of partition). For discrete-time approximations of Skorohod-type quasilinear equation driven by fBm we prove that the rate of convergence is $O(\delta^H)$.


Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian

Abstract: We investigate integrable second order equations of the form F(u_{xx}, u_{xy}, u_{yy}, u_{xt}, u_{yt}, u_{tt})=0. Familiar examples include the Boyer-Finley equation, the potential form of the dispersionless Kadomtsev-Petviashvili equation, the dispersionless Hirota equation, etc. The integrability is understood as the existence of infinitely many hydrodynamic reductions. We demonstrate that the natural equivalence group of the problem is isomorphic to Sp(6), revealing a remarkable correspondence between differential equations of the above type and hypersurfaces of the Lagrangian Grassmannian. We prove that the moduli space of integrable equations of the dispersionless Hirota type is 21-dimensional, and the action of the equivalence group Sp(6) on the moduli space has an open orbit.


Recursive Parameter Estimation: Asymptotic expansion

Abstract: We consider estimation procedures which are recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment. The model considered in the paper is very general as we do not impose any preliminary restrictions on the probabilistic nature of the observation process and cover a wide class of nonlinear recursive procedures. In this paper we study asymptotic behaviour of the recursive estimators. The results of the paper can be used to determine the form of a recursive procedure which is expected to have the same asymptotic properties as the corresponding non-recursive one defined as a solution of the corresponding estimating equation.


Riemann surfaces, ribbon graphs and combinatorial classes

Abstract: This survey paper begins with the description of the duality between arc systems and ribbon graphs embedded in a punctured surface. Then we explain how to cellularize the moduli space of curves in two different ways: using Jenkins-Strebel differentials and using hyperbolic geometry. We also briefly discuss how these two methods are related. Next, we recall the definition of Witten cycles and we illustrate their connection with tautological classes and Weil-Petersson geometry. Finally, we exhibit a simple direct argument to prove that Witten classes are stable.


Semimartingale Stochastic Approximation Procedures and Recursive Estimation

Abstract: The semimartingale stochastic approximation procedure, namely, the Robbins-Monro type SDE is introduced which naturally includes both generalized stochastic approximation algorithms with martingale noises and recursive parameter estimation procedures for statistical models associated with semimartingales. General results concerning the asymptotic behaviour of the solution are presented. In particular, the conditions ensuring the convergence, rate of convergence and asymptotic expansion are established. The results concerning the Polyak weighted averaging procedure are also presented.


Subjective Questions and Answers for a Mathematics Instructor of Higher Education

Abstract: This article of mathematical education reflects author's experience with job applications and teaching methods and procedures to employ in the American Higher Education. It is organized as a standard questionnaire.


Transversals in trees

Abstract: A transversal in a rooted tree is any set of nodes that meets every path from the root to a leaf. We let c(T,k) denote the number of transversals of size k in a rooted tree T. We define a partial order on the set of all rooted trees with n nodes by saying that a tree T succeeds a tree T' if c(T,k) is at least c(T',k) for all k and strictly greater than c(T',k) for at least one k. We prove that, for every choice of positive integers d and n, the set of all rooted trees on n nodes where each node has at most d children has a unique minimal element with respect to this partial order and we describe this tree.


Computing the core of ideals in arbitrary characteristic

Abstract: Let $R$ be a local Gorenstein ring with infinite residue field of arbitrary characteristic. Let $I$ be an $R$--ideal with $g=\height I >0$, analytic spread $\ell$, and let $J$ be a minimal reduction of $I$. We further assume that $I$ satisfies $G_{\ell}$ and ${\depth} R/I^j \geq \dim R/I -j+1$ for $1 \leq j \leq \ell-g$. The question we are interested in is whether $\core{I}=J^{n+1}:\ds \sum_{b \in I} (J,b)^n$ for $n \gg 0$. In the case of analytic spread one Polini and Ulrich show that this is true with even weaker assumptions (\cite[Theorem 3.4]{PU}). We give a negative answer to this question for higher analytic spreads and suggest a formula for the core of such ideals.


Index theory for linear self-adjoint operator equations and nontrivial solutions for asymptotically linear operator equations

Abstract: We will first establish an index theory for linear self-adjoint operator equations. And then with the help of this index theory we will discuss existence and multiplicity of solutions for asymptotically linear operator equations by making use of the dual variational methods and Morse theory. Finally, some interesting examples concerning second order Hamiltonian systems, first order Hamiltonian systems and elliptical partial differential equations will be presented to illustrate our results.


The Cauchy Operator for Basic Hypergeometric Series

Abstract: We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's ${}_2\phi_1$ transformation formula and Sears' ${}_3\phi_2$ transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can be considered as a generalization of the operator $T(bD_q)$. Using this operator, we obtain extensions of the Askey-Wilson integral, the Askey-Roy integral, Sears' two-term summation formula, as well as the $q$-analogues of Barnes' lemmas. Finally, we find that the Cauchy operator is also suitable for the study of the bivariate Rogers-Szeg\"o polynomials, or the continuous big $q$-Hermite polynomials.


Periodic Orbits of Twisted Geodesic Flows and The Weinstein-Moser Theorem

Abstract: In this paper, we establish the existence of periodic orbits of a twisted geodesic flow on all low energy levels and in all dimensions whenever the magnetic field form is symplectic and spherically rational. This is a consequence of a more general theorem concerning periodic orbits of autonomous Hamiltonian flows near Morse-Bott non-degenerate, symplectic extrema. Namely, we show that all energy levels near such extrema carry periodic orbits, provided that the ambient manifold meets certain topological requirements. This result is a partial generalization of the Weinstein-Moser theorem. The proof of the generalized Weinstein-Moser theorem is a combination of a Sturm-theoretic argument and a Floer homology calculation.


Anticipated backward stochastic differential equations

Abstract: In this paper we discuss new types of differential equations which we call anticipated backward stochastic differential equations (anticipated BSDEs). In these equations the generator includes not only the values of solutions of the present but also the future. We show that these anticipated BSDEs have unique solutions, a comparison theorem for their solutions, and a duality between them and stochastic differential delay equations.


Free Boolean algebras over unions of two well orderings

Abstract: Given a partially ordered set $P$ there exists the most general Boolean algebra $F(P)$ which contains $P$ as a generating set, called the {\it free Boolean algebra} over $P$. We study free Boolean algebras over posets of the form $P=P_0\cup P_1$, where $P_0,P_1$ are well orderings. We call them {\it nearly ordinal algebras}. Answering a question of Maurice Pouzet, we show that for every uncountable cardinal $\kappa$ there are $2^\kappa$ pairwise non-isomorphic nearly ordinal algebras of cardinality $\kappa$. Topologically, free Boolean algebras over posets correspond to compact 0-dimensional distributive lattices. In this context, we classify all closed sublattices of the product $(\omega_1+1)\times(\omega_1+1)$, thus showing that there are only $\aleph_1$ many of them. In contrast with the last result, we show that there are $2^{\aleph_1}$ topological types of closed subsets of the Tikhonov plank $(\omega_1+1)\times(\omega+1)$.


Radon transform on real symmetric varieties: kernel and cokernel

Abstract: We define and study the (minimal) Radon transform on a real symmetric variety.


Some Blow-Up Problems for a Semilinear Parabolic Equation with a Potential

Abstract: The blow-up rate estimate for the solution to a semilinear parabolic equation $u_t=\Delta u+V(x) |u|^{p-1}u$ in $\Omega \times (0,T)$ with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data $u(x,0)=M\vf (x)$ as $M$ goes to infinity, which have been found in \cite{cer}, are improved under some reasonable and weaker conditions compared with \cite{cer}.


Remark on the Garnier system in two variables

Abstract: We remark on the Garnier system in two variables.


Decomposition and Enumeration of Triangulated Surfaces

Abstract: We describe some theoretical results on triangulations of surfaces and we develop a theory on roots, decompositions and genus-surfaces. We apply this theory to describe an algorithm to list all triangulations of closed surfaces with at most a fixed number of vertices. We specialize the theory for the case where the number of vertices is at most 11 and we get theoretical restrictions on genus-surfaces allowing us to get the list of triangulations of closed surfaces with at most 11 vertices.


A rigidity theorem for the mapping class group action on the space of unmeasured foliations on a surface

Abstract: Let $S$ be a surface of finite type which is not a sphere with at most four punctures, a torus with at most two punctures, or a closed surface of genus two. Let $\mathcal{MF}$ be the space of equivalence classes of measured foliations of compact support on $S$ and let $\mathcal{UMF}$ be the quotient space of $\mathcal{MF}$ obtained by identifying two equivalence classes whenever they can be represented by topologically equivalent foliations, that is, forgetting the transverse measure. The extended mapping class group $\Gamma^*$ of $S$ acts as by homeomorphisms of $\mathcal{UMF}$. We show that the restriction of the action of the whole homeomorphism group of $\mathcal{UMF}$ on some dense subset of $\mathcal{UMF}$ coincides with the action of $\Gamma^*$ on that subset. More precisely, let $\mathcal{D}$ be the natural image in $\mathcal{UMF}$ of the set of homotopy classes of not necessarily connected essential disjoint and pairwise nonhomotopic simple closed curves on $S$. The set $\mathcal{D}$ is dense in $\mathcal{UMF}$, it is invariant by the action of $\Gamma^*$ on $\mathcal{UMF}$ and the restriction of the action of $\Gamma^*$ on $\mathcal{D}$ is faithful. We prove that the restriction of the action on $\mathcal{D}$ of the group $\mathrm{Homeo}(\mathcal{UMF})$ coincides with the action of $\Gamma^*(S)$ on that subspace.


Cohen-Macaulay multigraded modules

Abstract: Let S be a standard N^r-graded algebra over a local ring A, and let M be a finitely generated Z^r-graded S-module. We characterize the Cohen-Macaulayness of M in terms of the vanishing of certain sheaf cohomology modules. As a consequence, we apply our result to study the Cohen-Macaulayness of multi-Rees modules (also called Rees modification). Our work extends previous studies on the Cohen-Macaulayness of multi-Rees algebras.


When does elementary bi-embeddability imply isomorphism?

Abstract: A first-order theory has the Schroder-Bernstein property if any two of its models that are elementarily bi-embeddable are isomorphic. We prove that if a countable theory T has the Schroder-Bernstein property then it is classifiable (it is superstable and has NDOP and NOTOP) and satisfies a slightly stronger condition than nonmultidimensionality, namely: there cannot be a model M of T, a type p over M, and an automorphism f of M such that for every two distinct natural numbers i and j, f^i(p) is orthogonal to f^j(p). We also make some conjectures about how the class of theories with the Schroder-Bernstein property can be characterized.


The Schroder-Bernstein property for theories of abelian groups

Abstract: A first-order theory has the Schroder-Bernstein property if any two of its models that are elementarily bi-embeddable are isomorphic. We prove that if G is an abelian group, then the follwing are equivalent: 1. Th(G, +) has the Schroder-Bernstein property; 2. Th(G, +) is omega-stable; 3. G is the direct sum of a divisible group and a torsion group of bounded exponent; 4. Th(G, +) is superstable, and if (H, +) is a saturated elementary extension of (G,+), every map in Aut(H/H^0) is unipotent.


The Riemann Mapping Theorem for semianalytic domains and o-minimality

Abstract: We consider the Riemann Mapping Theorem in the case of a bounded simply connected and semianalytic domain. We show that the germ at 0 of the Riemann map (i.e. biholomorphic map) from the upper half plane to such a domain can be realized in a certain quasianalytic class if the angle of the boundary at the point to which 0 is mapped, is greater than 0. This quasianalytic class was introduced and used by Ilyashenko in his work on Hilbert's 16th problem. With this result we can prove that the Riemann map from a bounded simply connected semianalytic domain onto the unit ball is definable in an o-minimal structure, provided that at singular boundary points the angles of the boundary are irrational multiples of $\pi$.


Revesibility of chordal SLE

Abstract: We prove that the chordal SLE$_\kappa$ trace is reversible for $\kappa\in(0,4]$.


2-Frame flow dynamics and hyperbolic rank rigidity in nonpositive curvature

Abstract: This paper presents hyperbolic rank rigidity results for rank 1, nonpositively curved spaces. Let $M$ be a compact, rank 1 manifold with nonpositive sectional curvature and suppose that along every geodesic in $M$ there is a parallel vector field making curvature $-a^2$ with the geodesic direction. We prove that $M$ has constant curvature equal to $-a^2$ if $M$ is odd dimensional, or if $M$ is even dimensional and has sectional curvature pinched as follows: $-\Lambda^2 < K < -\lambda^2$ where $\lambda/\Lambda > >.93$. When $-a^2$ is the upper curvature bound this gives a shorter proof of the hyperbolic rank rigidity theorem of Hamenst\"{a}dt, subject to the pinching condition in even dimension; in all other cases it is a new result. We also present a rigidity result using only an assumption on maximal Lyapunov exponents in direct analogy with work done by Connell. The proof of the main theorem is simplified considerably by assuming strict negative curvature; in fact, in all dimensions but 7 and 8 it is then an immediate consequence of ergodicity of the $(dim(M)-1)$-frame flow. In these exceptional dimensions, recourse to the dynamics of the 2-frame flow must be made and the scheme of proof developed there can be generalized to deal with rank 1, nonpositively curved spaces.


On the Riemann zeta-function, Part I: Outline

Abstract: Results of a multipart work are outlined. Use is made therein of the conjunction of the Riemann hypothesis, RH, and hypotheses advanced by the author. Let z(n) be the nth nonreal zero of the Riemann zeta-function with positive imaginary part in order of magnitude thereof. A relation is obtained, of the pair z(n) and the first derivative thereat of the zeta-function, to the preceding such pairs and the values of zeta at the points one-half plus a nonnegative multiple of four. That relation is derived from two forms of the density of the Laplace representation, on a certain vertical strip, of a meromorphic function constructed from zeta. Specific functions which play a central role therein are proven to have analytic extensions to the entire complex plane. It is established that the Laplace density is positive. That positivity implies RH and that each nonreal zero of zeta is simple. A metric geometry expression of the positivity of the density is derived. An analogous context is delineated relative to Dirichlet L-functions and the Ramanujan tau Dirichlet function.


On the Green's matrices of strongly parabolic systems of second order

Abstract: We establish existence and various estimates of fundamental matrices and Green's matrices for divergence form, second order strongly parabolic systems in arbitrary cylindrical domains under the assumption that solutions of the systems satisfy an interior H\"{o}lder continuity estimate. We present a unified approach valid for both the scalar and the vectorial cases.


On level crossings for a general class of piecewise-deterministic Markov processes

Abstract: We consider a piecewise-deterministic Markov process governed by a jump intensity function, a rate function that determines the behaviour between jumps, and a stochastic kernel describing the conditional distribution of jump sizes. We study the point process of upcrossings of a level $b$ by the Markov process. Our main result shows that, under a suitable scaling $\nu(b)$, the point process converges, as $b$ tends to infinity, weakly to a geometrically compound Poisson process. We also prove a version of Rice's formula relating the stationary density of the process to level crossing intensities. This formula provides an interpretation of the scaling factor $\nu(b)$. While our proof of the limit theorem requires additional assumptions, Rice's formula holds whenever the (stationary) overall intensity of jumps is finite.


On the degree of Polar Transformations -- An approach through Logarithmic Foliations

Abstract: We investigate the degree of the polar transformations associated to a certain class of multi-valued homogeneous functions. In particular we prove that the degree of the pre-image of generic linear spaces by a polar transformation associated to a homogeneous polynomial $F$ is determined by the zero locus of $F$. For zero dimensional-dimensional linear spaces this was conjecture by Dolgachev and proved by Dimca-Papadima using topological arguments. Our methods are algebro-geometric and rely on the study of the Gauss map of naturally associated logarithmic foliations.


A criterion for transience of multidimensional branching random walk in random environment

Abstract: We develop a criterion for transience for a general model of branching Markov chains. In the case of multi-dimensional branching random walk in random environment (BRWRE) this criterion becomes explicit. In particular, we show that \emph{Condition L} of Comets and Popov is necessary and sufficient for transience as conjectured. Furthermore, the criterion applies to two important classes of branching random walks and implies that the critical branching random walk is transient resp. dies out locally.


On Mordell-Weil groups of elliptic curves induced by Diophantine triples

Abstract: We study the possible structure of the groups of rational points on elliptic curves of the form y^2=(ax+1)(bx+1)(cx+1), where a,b,c are non-zero rationals such that the product of any two of them is one less than a square.


Set theoretic solutions of the Yang-Baxter equation, graphs and computations

Abstract: We extend our recent work on set-theoretic solutions of the Yang-Baxter or braid relations with new results about their automorphism groups, strong twisted unions of solutions and multipermutation solutions. We introduce and study graphs of solutions and use our graphical methods for the computation of solutions of finite order and their automorphisms. Results include a detailed study of solutions of multipermutation level 2.


Self-duality of Selmer groups

Abstract: The first part of the paper gives a new proof of self-duality for Selmer groups: if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the Q_pG-representation naturally associated to the p-infinity Selmer group of A/F is self-dual. The second part describes a method for obtaining information about parities of Selmer ranks from the local Tamagawa numbers of A in intermediate extensions of F/K.


On the invertibility of "rectangular" bi-infinite matrices and applications in time--frequency analysis

Abstract: Finite dimensional matrices having more columns than rows have no left inverses while those having more rows than columns have no right inverses. We give generalizations of these simple facts to bi--infinite matrices and use those to obtain density results for $p$--frames of time--frequency molecules in modulation spaces and identifiability results for operators with bandlimited Kohn--Nirenberg symbols.


Necessary Conditions for Geometric Realizability of Simplicial Complexes

Abstract: We associate with any simplicial complex $\K$ and any integer $m$ a system of linear equations and inequalities. If $\K$ has a simplicial embedding in $\R^m$ then the system has an integer solution. This result extends the work of I. Novik (2000).


Measurement of time--varying Multiple--Input Multiple--Output Channels

Abstract: We derive a criterion on the measurability / identifiability of Multiple--Input Multiple--Output (MIMO) channels based on the size of the so-called spreading support of its subchannels. Novel MIMO transmission techniques provide high-capacity communication channels in time-varying environments and exact knowledge of the transmission channel operator is of key importance when trying to transmit information at a rate close to channel capacity.


The Dirichlet problem in the plane with semianalytic raw data, quasianalyticity and o-minimal structures

Abstract: We investigate the Dirichlet solution for a semianalytic continuous function on the boundary of a semianalytic bounded domain in the plane. We show that the germ of the Dirichlet solution at a boundary point with angle greater than 0 lies in a certain quasianalytic class used by Ilyashenko in his work on Hilbert's 16th problem. With this result we can prove that the Dirichlet solution is definable in an o-minimal structure if the angle at a singular boundary point of the domain is an irrational multiple of $\pi$.


Linear Prediction of Long-Memory Processes: Asymptotic Results on Mean-squared Errors

Abstract: We present two approaches for linear prediction of long-memory time series. The first approach consists in truncating the Wiener-Kolmogorov predictor by restricting the observations to the last $k$ terms, which are the only available values in practice. We derive the asymptotic behaviour of the mean-squared error as $k$ tends to $ + \infty$. By contrast, the second approach is non-parametric. An AR($k$) model is fitted to the long-memory time series and we study the error that arises in this misspecified model.


Ideals of varieties parameterized by certain symmetric tensors

Abstract: The ideal of a Segre variety is generated by the 2-minors of a generic hypermatrix of indeterminates. We extend this result to the case of Segre-Veronese varieties. The main tool is the concept of weak generic hypermatrix which allows us to treat also the case of projection of Veronese surfaces from a set of generic points and of Veronese varieties from a Cohen-Macaulay subvariety of codimension 2.


Riesz and Szeg\"o type factorizations for noncommutative Hardy spaces

Abstract: Let $\A$ be a finite subdiagonal algebra in Arveson's sense. Let $H^p(\A)$ be the associated noncommutative Hardy spaces, $0<p\le\8$. We extend to the case of all positive indices most recent results about these spaces, which include notably the Riesz, Szeg\"o and inner-outer type factorizations. One new tool of the paper is the contractivity of the underlying conditional expectation on $H^p(\A)$ for $p<1$.


BMO functions and Carleson measures with values in uniformly convex spaces

Abstract: This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbb T$, respectively. For $1< q<\infty$ and a Banach space $B$ we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1-|z|)^{q-1}\|\nabla f(z)\|^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\T}\|f(z)-f(z_0)\|^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ iff $B$ admits an equivalent norm which is $q$-uniformly convex, where $$P_{z_0}(z)=\frac{1-|z_0|^2}{|1-\bar{z_0}z|^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm.


Sums of lens spaces bounding rational balls

Abstract: We classify connected sums of three-dimensional lens spaces which smoothly bound rational homology balls. We use this result to determine the order of each lens space in the group of rational homology 3-spheres up to rational homology cobordisms, and to determine the concordance order of each 2-bridge knot.


Noncommutative Burkholder/Rosenthal inequalities II: applications

Abstract: We show norm estimates for the sum of independent random variables in noncommutative $L_p$-spaces for $1<p<\infty$ following our previous work. These estimates generalize the classical Rosenthal inequality in the commutative case. Among applications, we derive an equivalence for the $p$-norm of the singular values of a random matrix with independent entries, and characterize those symmetric subspaces and unitary ideals which can be realized as subspaces of a noncommutative $L_p$ for $2<p<\infty$.


Intersections of polynomial orbits, and a dynamical Mordell-Lang conjecture

Abstract: We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the Mordell-Lang conjecture.


$C^*$-alg\`ebres gradu\'ees par un semi-treillis

Abstract: Graded $C^*$-algebras by a semi lattice were introduced and studied by Anne Boutet de Monvel, Vladimir Georgescu and their collaborators in relation with the quantum N body problem. This thesis is devoted to a systematic study of these algebras and their properties. In particular, we show that they are completely described by their homogenous subalgebras and that they are invariant under several operations such as tensor products, crossed products (by actions of locally compact groups that respect the graduation). We give the $K$-theory of graded $C^*$-algebras and finally, we study some examples of such algebras.


Realizing Kasparov's KK-theory groups as the homotopy classes of maps of a Quillen model category

Abstract: In this article we build a Quillen model category structure on the category of sequentially complete l.m.c.-C*-algebras such that the corresponding homotopy classes of maps Ho(A,B) for separable C*-algebras A and B coincide with the Kasparov groups KK(A,B). This answers an open question posed by Mark Hovey about the possibility of describing KK-theory for C*-algebras using the language of Quillen model categories.


Amenable groups and Hadamard spaces with a totally disconnected isometry group

Abstract: Let $X$ be a locally compact Hadamard space and $G$ be a totally disconnected group acting continuously, properly and cocompactly on $X$. We show that a closed subgroup of $G$ is amenable if and only if it is (topologically locally finite)-by-(virtually abelian). We are led to consider a set $\bdfine X$ which is a refinement of the visual boundary $\bd X$. For each $x \in \bdfine X$, the stabilizer $G_x$ is amenable.


Integral group ring of the Mathieu simple group M24

Abstract: We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic group $M_{24}$. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.


Energy of zeros of random sections on Riemann Surface

Abstract: The purpose of this paper is to determine the asymptotic of the average energy of a configuration of N zeros of system of random polynomials of degree N as N tends to infinity and more generally the zeros of random holomorphic sections of a line bundle L over any Riemann surface M. And we compare our results to the well-known minimum of energies.


Intrinsically Linked Graphs with Knotted Components

Abstract: We construct a graph G such that any embedding of G into R^{3} contains a nonsplit link of two components, where at least one of the components is a nontrivial knot. Further, for any m < n we produce a graph H so that every embedding of H contains a nonsplit n component link, where at least m of the components are nontrivial knots. We then turn our attention to complete graphs and show that for any given n, every embedding of a large enough complete graph contains a two component link whose linking number is a nonzero multiple of n.


Multivariable generalizations of the Schur class: positive kernel characterization and transfer function realization

Abstract: The operator-valued Schur-class is defined to be the set of holomorphic functions $S$ mapping the unit disk into the space of contraction operators between two Hilbert spaces. There are a number of alternate characterizations: the operator of multiplication by $S$ defines a contraction operator between two Hardy Hilbert spaces, $S$ satisfies a von Neumann inequality, a certain operator-valued kernel associated with $S$ is positive-definite, and $S$ can be realized as the transfer function of a dissipative (or even conservative) discrete-time linear input/state/output linear system. Various multivariable generalizations of this class have appeared recently,one of the most encompassing being that of Muhly and Solel where the unit disk is replaced by the strict unit ball of the elements of a dual correspondence $E^{\sigma}$ associated with a $W^{*}$-correspondence $E$ over a $W^{*}$-algebra $\cA$ together with a $*$-representation $\sigma$ of $\cA$. The main new point which we add here is the introduction of the notion of reproducing kernel Hilbert correspondence and identification of the Muhly-Solel Hardy spaces as reproducing kernel Hilbert correspondences associated with a completely positive analogue of the classical Szeg\"o kernel. In this way we are able to make the analogy between the Muhly-Solel Schur class and the classical Schur class more complete. We also illustrate the theory by specializing it to some well-studied special cases; in some instances there result new kinds of realization theorems.


Noetherian types of homogeneous compacta and dyadic compacta

Abstract: The Noetherian type of a space is the least $\kappa$ such that it has a base that is $\kappa$-like with respect to containment. Just as all known homogeneous compacta have cellularity at most $2^\omega$, they satisfy similar upper bounds in terms of Noetherian type and related cardinal functions. We prove these and many other results about these cardinal functions. For example, every homogeneous dyadic compactum has Noetherian type $\omega$. Assuming GCH, every point in a homogeneous compactum $X$ has a local base that is $c(X)$-like with respect to containment. If every point in a compactum has a well-quasiordered local base, then some point has a countable local $\pi$-base.


The Maximal Integral Domain Generated By A Commutative Ring

Abstract: In this paper, we exhibit the creation of the maximal integral domain mid(R) generated by a nonzero commutative ring R.


Drinfeld-Hecke algebras over cocommutative algebras

Abstract: If A is a cocommutative algebra with coproduct, then so is the smash product algebra of a symmetric algebra Sym(V) with A, where V is an A-module. Such smash product algebras, with A a group ring or a Lie algebra, have families of deformations that have been studied widely in the literature; examples include symplectic reflection algebras and infinitesimal Hecke algebras. We introduce a family of deformations of these smash product algebras for general A, and characterize the PBW property. We then characterize the Jacobi identity for "grouplike" algebras (that include group rings and the nilCoxeter algebra), and precisely identify the PBW deformations in the example where A is the nilCoxeter algebra. We end with the more prominent case - where A is a Hopf algebra. We show the equivalence of several versions of the "deformed" relations in the smash product, and identify the PBW deformations which are Hopf algebras as well.


Tensor Product of the Fundamental Representations for the Quantum Loop Algebras of Type A at Roots of Unity

Abstract: In this paper, we consider the necessary and sufficient conditions for the tensor product of the fundamental representations for the restricted quantum loop algebras of type A at roots of unity to be irreducible.


Gluing pseudoholomorphic curves along branched covered cylinders II

Abstract: This paper and its prequel ("Part I") prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves U_+ and U_- in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit gamma, the total multiplicity of the negative ends of U_+ at covers of gamma agrees with the total multiplicity of the positive ends of U_- at covers of gamma. However, unlike in the usual gluing story, here the individual multiplicities are allowed to differ. In this situation, one can often glue U_+ and U_- to an index 2 curve by inserting genus zero branched covers of R-invariant cylinders between them. This paper shows that the signed count of such gluings equals a signed count of zeroes of a certain section of an obstruction bundle over the moduli space of branched covers of the cylinder. Part I obtained a combinatorial formula for the latter count and, assuming the result of the present paper, deduced that the differential d in embedded contact homology satisfies d^2=0. The present paper completes all of the analysis that was needed in Part I. The gluing technique explained here is in principle applicable to more gluing problems. We also prove some lemmas concerning the generic behavior of pseudoholomorphic curves in symplectizations, which may be of independent interest.


Vanishing and injectivity theorems for LMMP

Abstract: This preprint has been withdrawn. It is because I will never publish this preprint since everything has been contained in my new preprint: arXiv:0907.1506. Please refer to arXiv:0907.1506. Please do not cite this preprint any more.


Notes on the log minimal model program


The abelianization of a symmetric mapping class group

Abstract: We determine the abelianization of the symmetric mapping class group of a double unbranched cover using the Riemann theta constant, Schottky theta constant, and the theta multiplier. We also give lower bounds of the abelianizations of some finite index subgroups of the mapping class group.


Mirzakharni's recursion formula is equivalent to the Witten-Kontsevich theorem

Abstract: In this paper, we give a proof of Mirzakhani's recursion formula of Weil-Petersson volumes of moduli spaces of curves using the Witten-Kontsevich theorem. We also describe properties of intersections numbers involving higher degree $\kappa$ classes.


A Blichfeldt-type inequality for the surface area

Abstract: In 1921 Blichfeldt gave an upper bound on the number of integral points contained in a convex body in terms of the volume of the body. More precisely, he showed that $#(K\cap\Z^n)\leq n! \vol(K)+n$, whenever $K\subset\R^n$ is a convex body containing $n+1$ affinely independent integral points. Here we prove an analogous inequality with respect to the surface area $\F(K)$, namely $ #(K\cap\Z^n) < \vol(K) + ((\sqrt{n}+1)/2) (n-1)! \F(K)$. The proof is based on a slight improvement of Blichfeldt's bound in the case when $K$ is a non-lattice translate of a lattice polytope, i.e., $K=t+P$, where $t\in\R^n\setminus\Z^n$ and $P$ is an $n$-dimensional polytope with integral vertices. Then we have $#((t+P)\cap\Z^n)\leq n! \vol(P)$. Moreover, in the 3-dimensional case we prove a stronger inequality, namely $#(K\cap\Z^n) < \vol(K) + 2 \F(K)$.


Another property of minimal surfaces in Euclidean space

Abstract: The new property of minimal surfaces is obtained in this article.


Smooth multiparameter perturbation of polynomials and operators

Abstract: This paper has been withdrawn by the authors due to a gap in the proof of the main result (in 5.3).


Pr\"ufer's Ideal Numbers as Gelfand's maximal Ideals

Abstract: Polyadic arithmetics is a branch of mathematics related to $p$--adic theory. The aim of the present paper is to show that there are very close relations between polyadic arithmetics and the classic theory of commutative Banach algebras. Namely, let $\ms A$ be the algebra consisting of all complex periodic functions on $\Z$ with the uniform norm. Then the polyadic topological ring can be defined as the ring of all characters $\ms A\to\C$ with convolution operations and the Gelfand topology.


Casimir operators, abelian subspaces and u-cohomology

Abstract: This survey paper is an exposition of old and recent results of Kostant and al. on the relationships between the exterior algebra of a simple Lie algebra and the action of the Casimir operator on it. Our exposition relies on u-cohomology and it is basically self-contained.


The General Definition of the Complex Monge-Amp\`ere Operator on Compact K\"ahler Manifolds

Abstract: We introduce a wide subclass ${\cal F}(X,\omega)$ of quasi-plurisubharmonic functions in a compact K\"ahler manifold, on which the complex Monge-Amp\`ere operator is well-defined and the convergence theorem is valid. We also prove that ${\cal F}(X,\omega)$ is a convex cone and includes all quasi-plurisubharmonic functions which are in the Cegrell class.


Determination of the body force of a two-dimensional isotropic elastic body

Abstract: Let $\Omega$ represent a two$-$dimensional isotropic elastic body. We consider the problem of determining the body force $F$ whose form $\phi(t)(f_1(x),f_2(x))$ with $\phi$ be given inexactly. The problem is nonlinear and ill-posed. Using the Fourier transform, the methods of Tikhonov's regularization and truncated integration, we construct a regularized solution from the data given inexactly and derive the explicitly error estimate. Numerical part is given


Discontinuity and Involutions on Countable Sets

Abstract: For any infinite subset $X$ of the rationals and a subset $F \subseteq X$ which has no isolated points in $X$ we construct a function $f: X \to X$ such that $f(f(x))=x$ for each $x\in X$ and $F $ is the set of discontinuity points of $f$.


TYZ expansion for the Kepler manifold

Abstract: The main goal of the paper is to address the issue of the existence of Kempf's distortion function and the Tian-Yau-Zelditch (TYZ) asymptotic expansion for the Kepler manifold - an important example of non compact manfold. Motivated by the recent results for compact manifolds we construct Kempf's distortion function and derive a precise TYZ asymptotic expansion for the Kepler manifold. We get an exact formula: finite asymptotic expansion of $n-1$ terms and exponentially small error terms uniformly with respect to the discrete quantization parameter $m\to \infty $ and $\rho \to \infty$, $\rho$ being the polar radius in $\C^n$. Moreover, the coefficents are calculated explicitly and they turned out to be homogeneous functions with respect to the polar radius in the Kepler manifold. We also prove and derive an asymptotic expansion of the obtstruction term with the coefficients being defined by geometrical quantities. We show that our estimates are sharp by analyzing the nonharmonic behaviour of $T_m$ and the error term of the approximation of the Fubini--Study metric by $m\omega$ for $m\to +\infty$. The arguments of the proofs combine geometrical methods, quantization tools and functional analytic techniques for investigating asymptotic expansions in the framework of analytic-Gevrey spaces.


Extremal metrics on Hartogs domains

Abstract: An $n$-dimensional Hartogs domain $D_F$ with strongly pseudoconvex boundary can be equipped with a natural \K metric $g_F$. In this paper we prove that if $g_F$ is an extremal \K metric then $(D_F, g_F)$ is biholomorphically isometric to the $n$-dimensional complex hyperbolic space.


On the complexity of solving ordinary differential equations in terms of Puiseux series

Abstract: We prove that the binary complexity of solving ordinary polynomial differential equations in terms of Puiseux series is single exponential in the number of terms in the series. Such a bound was given by Grigoriev [10] for Riccatti differential polynomials associated to ordinary linear differential operators. In this paper, we get the same bound for arbitrary differential polynomials. The algorithm is based on a differential version of the Newton-Puiseux procedure for algebraic equations.


A tree approach to $p$-variation and to integration

Abstract: We consider a real-valued path; it is possible to associate a tree to this path, and we explore the relations between the tree, the properties of $p$-variation of the path, and integration with respect to the path. In particular, the fractal dimension of the tree is estimated from the variations of the path, and Young integrals with respect to the path, as well as integrals from the rough paths theory, are written as integrals on the tree. Examples include some stochastic paths such as martingales, L\'evy processes and fractional Brownian motions (for which an estimator of the Hurst parameter is given).


Predictability, entropy and information of infinite transformations

Abstract: We show that a certain type of quasi finite, conservative, ergodic, measure preserving transformation always has a maximal zero entropy factor, generated by predictable sets. We also construct a conservative, ergodic, measure preserving transformation which is not quasi finite; and consider distribution asymptotics of information showing that e.g. for Boole's transformation, information is asymptotically mod-normal with square root normalization. Lastly we see that certain ergodic, probability preserving transformations with zero entropy have analogous properties and consequently entropy dimension of at most 1/2.


Distributions vectorielles homog\`enes sur une alg\`ebre de Jordan

Abstract: We study distributions on a Euclidean Jordan algebra V with values in a finite dimensional representation space for the identity component G of the structure group of V and homogeneous equivariance condition. We show that such distributions exist if and only if the representation is spherical, and that then the dimension of the space of these distributions is r+1 (where r is the rank of V). We give also construction of these distributions and of those that are invariant under the semi-simple part of G.


Gerby Localization, Z_3-Hodge Integrals and the GW Theory of C^3/Z_3

Abstract: We exhibit a set of recursive relations that completely determine all equivariant Gromov-Witten invariants of the quotient orbifold C^3/Z_3. We interpret such invariants as G-Hodge Integrals, and produce relations among them via Atiyah-Bott localization on moduli spaces of twisted stable maps to gerbes over the projective line.


Quantum cohomology of [C^N/\mu_r]

Abstract: We give a construction of the moduli space of stable maps to the classifying stack B\mu_r of a cyclic group by a sequence of r-th root constructions on M_{0, n}. We prove a closed formula for the total Chern class of \mu_r-eigenspaces of the Hodge bundle, and thus of the obstruction bundle of the genus zero Gromov-Witten theory of stacks of the form [C^N/\mu_r]. We deduce linear recursions for all genus-zero Gromov-Witten invariants.


Local dynamics for fibered holomorphic transformations

Abstract: Fibered holomorphic dynamics are skew-product transformations over an irrational rotation, whose fibers are holomorphic functions. In this paper we study such a dynamics on a neighborhood of an invariant curve. We obtain some results analogous to the results in the non fibered case.


Multiplicity one Conjectures

Abstract: In the first part, in the local non archimedean case, we consider distributions on GL(n+1) which are invariant under the adjoint action of GL(n). We conjecture that such distributions are invariant by transposition. This would imply multiplicity at most one for restrictions from GL(n+1) to GL(n). We reduce ourselves to distributions with "singular" support and then finish the proof for n< 9. In the second part we show that similar Theorems for orthogonal or unitary groups follow from the case of GL(n)


Limits of Hypergraphs, Removal and Regularity Lemmas. A Non-standard Approach

Abstract: We study the integral and measure theory of the ultraproduct of finite sets. As a main application we construct limit objects for hypergraph sequences. We give a new proof for the Hypergraph Removal Lemma and the Hypergraph Regularity Lemma.


Rational functions with linear relations

Abstract: We find all polynomials f,g,h over a field K such that g and h are linear and f(g(x))=h(f(x)). We also solve the same problem for rational functions f,g,h, in case the field K is algebraically closed.


A remarkable moduli space of rank 6 vector bundles related to cubic surfaces

Abstract: We study the moduli space $\fM^s(6;3,6,4)$ of simple rank 6 vector bundles $\E$ on $\PP^3$ with Chern polynomial $1+3t+6t^2+4t^3$ and properties of these bundles, especially we prove some partial results concerning their stability. We first recall how these bundles are related to the construction of sextic nodal surfaces in $\PP^3$ having an even set of 56 nodes (cf. \cite{CaTo}). We prove that there is an open set, corresponding to the simple bundles with minimal cohomology, which is irreducible of dimension 19 and bimeromorphic to an open set $\fA^0$ of the G.I.T. quotient space of the projective space $\fB:=\{B\in \PP(U^\vee\otimes W\otimes V^\vee)\}$ of triple tensors of type $(3,3,4)$ by the natural action of $SL(W)\times SL(U)$. We give several constructions for these bundles, which relate them to cubic surfaces in 3-space $\PP^3$ and to cubic surfaces in the dual space $(\PP^3)^{\vee}$. One of these constructions, suggested by Igor Dolgachev, generalizes to other types of tensors. Moreover, we relate the socalled {\em cross-product involution} for $(3,3,4)$-tensors, introduced in \cite{CaTo}, with the Schur quadric associated to a cubic surface in $\PP^3$ and study further properties of this involution.


The Gorenstein Colength of an Artinian Local Ring

Abstract: In this paper, we make the notion of approximating an Artinian local ring by a Gorenstein Artin local ring precise using the concept of Gorenstein colength. We also answer the question as to when the Gorenstein colength is at most two.


Multiresolution wavelet analysis of integer scale Bessel functions

Abstract: We identify multiresolution subspaces giving rise via Hankel transforms to Bessel functions. They emerge as orthogonal systems derived from geometric Hilbert-space considerations, the same way the wavelet functions from a multiresolution scaling wavelet construction arise from a scale of Hilbert spaces. We study the theory of representations of the $C^{\ast}$-algebra $% O_{\nu +1}$ arising from this multiresolution analysis. A connection with Markov chains and representations of $O_{\nu +1}$ is found. Projection valued measures arising from the multiresolution analysis give rise to a Markov trace for quantum groups $SO_q$.


Combinatorial Hopf algebras and K-homology of Grassmanians

Abstract: Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, we study six combinatorial Hopf algebras. These Hopf algebras can be thought of as K-theoretic analogues of the by now classical ``square'' of Hopf algebras consisting of symmetric functions, quasisymmetric functions, noncommutative symmetric functions and the Malvenuto-Reutenauer Hopf algebra of permutations. In addition, we develop a theory of set-valued P-partitions and study three new families of symmetric functions which are weight generating functions of reverse plane partitions, weak set-valued tableaux and valued-set tableaux.


Bagging multiple comparisons from microarray data

Abstract: The problem of large-scale simultaneous hypothesis testing is re-visited. Bagging and subagging procedures are put forth with the purpose of improving the discovery power of the tests. The procedures are implemented in both simulated and real data. It is shown that bagging and subagging significantly improve power at the cost of a small increase in false discovery rate with the proposed `maximum contrast' subagging having an edge over bagging, i.e., yielding similar power but significantly smaller false discovery rates.


Real interpolation of Sobolev spaces

Abstract: We prove that $W^{1}_{p}$ is an interpolation space between $W^{1}_{p_{1}}$ and $W^{1}_{p_{2}}$ for $p>q_{0}$ and $1\leq p_{1}<p<p_{2}\leq \infty$ on some classes of manifolds and general metric spaces, where $q_{0}$ depends on our hypotheses.


Dye's theorem in the almost continuous category

Abstract: We prove an almost continuous version of Dye's theorem: any two non-atomic probability measure preserving homeomorphisms of Polish spaces are almost continuously orbit equivalent. More precisely they are orbit equivalent by a map which is defined and continuous on a Polish subset of full measure with an inverse satisfying the same conditions. This result includes all of the recent results on almost continuous orbit equivalence. We also deal with the case of infinite invariant measures.


On the convergence to equilibrium of Kac's random walk on matrices

Abstract: We consider Kac's random walk on $n$-dimensional rotation matrices, where each step is a random rotation in the plane generated by two randomly picked coordinates. We show that this process converges to the Haar measure on $\mathit{SO}(n)$ in the $L^2$ transportation cost (Wasserstein) metric in $O(n^2\ln n)$ steps. We also prove that our bound is at most a $O(\ln n)$ factor away from optimal. Previous bounds, due to Diaconis/Saloff-Coste and Pak/Sidenko, had extra powers of $n$ and held only for $L^1$ transportation cost. Our proof method includes a general result of independent interest, akin to the path coupling method of Bubley and Dyer. Suppose that $P$ is a Markov chain on a Polish length space $(M,d)$ and that for all $x,y\in M$ with $d(x,y)\ll1$ there is a coupling $(X,Y)$ of one step of $P$ from $x$ and $y$ (resp.) that contracts distances by a $(\xi+o(1))$ factor on average. Then the map $\mu\mapsto\mu P$ is $\xi$-contracting in the transportation cost metric.


Variance reduction for particle filters of systems with time-scale separation

Abstract: We present a particle filter construction for a system that exhibits time-scale separation. The separation of time-scales allows two simplifications that we exploit: i) The use of the averaging principle for the dimensional reduction of the system needed to solve for each particle and ii) the factorization of the transition probability which allows the Rao-Blackwellization of the filtering step. Both simplifications can be implemented using the coarse projective integration framework. The resulting particle filter is faster and has smaller variance than the particle filter based on the original system. The method is tested on a multiscale stochastic differential equation and on a multiscale pure jump diffusion motivated by chemical reactions.


A note on De Concini and Procesi's curious identity

Abstract: We give a short, case-free and combinatorial proof of de Concini and Procesi's formula for the volume of the simplicial cone spanned by the simple roots of any finite root system. The argument presented here also extends their formula to include the non-crystallographic root systems.


Double Shuffle Relations of Euler Sums

Abstract: In this paper we shall develop a theory of (extended) double shuffle relations of Euler sums which generalizes that of multiple zeta values (see Ihara, Kaneko and Zagier, \emph{Derivation and double shuffle relations for multiple zeta values}. Compos. Math. \textbf{142} (2)(2006), 307--338). After setting up the general framework we provide some numerical evidence for our two main conjectures. At the end we shall prove the following long standing conjecture: for every positive integer n $$\zeta(\{3\}^n)=8^n\zeta(\{\ol2,1\}^n).$$ The main idea is to use the double shuffle relations and the distribution relation. This particular distribution relation doesn't follow from the double shuffle relation in general. But we believe it does follow from the extended double shuffle relations.


Real interpoaltion of Sobolev spaces associated to a weight

Abstract: We study the interpolation property of Sobolev spaces of order 1 denoted by $W^{1}_{p,V}$, arising from Schr\"{o}dinger operators with positive potential. We show that for $1\leq p_1<p<p_2<q_{0}$ with $p>s_0$, $W^{1}_{p,V}$ is a real interpolation space between $W_{p_1,V}^{1}$ and $W_{p_2,V}^{1}$ on some classes of manifolds and Lie groups. The constants $s_{0}, q_{0}$ depend on our hypotheses.


The weighted complexity and the determinant functions of graphs

Abstract: The complexity of a graph can be obtained as a derivative of a variation of the zeta function or a partial derivative of its generalized characteristic polynomial evaluated at a point [\textit{J. Combin. Theory Ser. B}, 74 (1998), pp. 408--410]. A similar result for the weighted complexity of weighted graphs was found using a determinant function [\textit{J. Combin. Theory Ser. B}, 89 (2003), pp. 17--26]. In this paper, we consider the determinant function of two variables and discover a condition that the weighted complexity of a weighted graph is a partial derivative of the determinant function evaluated at a point. Consequently, we simply obtain the previous results and disclose a new formula for the Bartholdi zeta function. We also consider a new weighted complexity, for which the weights of spanning trees are taken as the sum of weights of edges in the tree, and find a similar formula for this new weighted complexity. As an application, we compute the weighted complexities of the product of the complete graphs.