Essentialities in additive bases

Abstract: Let A be an asymptotic basis for N_0 of some order. By an essentiality of A one means a subset P such that A\P is no longer an asymptotic basis of any order and such that P is minimal among all subsets of A with this property. A finite essentiality of A is called an essential subset. In a recent paper, Deschamps and Farhi asked the following two questions : (i) does every asymptotic basis of N_0 possess some essentiality ? (ii) is the number of essential subsets of size at most k of an asymptotic basis of order h bounded by a function of k and h only (they showed the number is always finite) ? We answer the latter question in the affirmative, and the former in the negative by means of an explicit construction, for every integer h >= 2, of an asymptotic basis of order h with no essentialities.


Twisted conjugacy classes for polyfree groups

Abstract: Let $G$ be a finitely generated polyfree group. If $G$ has nonzero Euler characteristic then we show that $Aut(G)$ has a finite index subgroup in which every automorphism has infinite Reidemeister number. For certain $G$ of length 2, we show that the number of Reidemeister classes of every automorphism is infinite.


Power maps and subvarieties of the complex algebraic $n$--torus

Abstract: Given a subvariety $V$ of the complex algebraic torus ${\mathbb G}_{\rm m}^n$ defined by polynomials of total degree at most $d$ and a power map $\phi: {\mathbb G}_{\rm m}^n \to {\mathbb G}_{\rm m}^n$, the points ${\bf x}$ whose forward orbits ${\mathcal O}_\phi({\bf x})$ belong to $V$ form its {\em stable} subvariety $S(V,\phi)$. The main result of the paper provides an upper bound $T=T(n,d,\phi)$ for the number of iterations of the power map $\phi$ required to ``cut off'' the points of $V$ that do not belong to $S$.


Heat Content, Heat Trace, and Isospectrality

Abstract: We study the heat content function, the heat trace function, and questions of isospectrality for the Laplacian with Dirichlet boundary conditions on a compact manifold with smooth boundary in the context of finite coverings and warped products.


A remark on the boundedness and convergence properties of smooth sliding mode controllers

Abstract: Conventional sliding mode controllers are based on the assumption of switching control but a well-known drawback of this approach is the chattering phenomenon. To overcome the undesirable chattering effects, the discontinuity in the control law can be smoothed out in a thin boundary layer neighboring the switching surface. In this work, rigorous proofs of the boundedness and convergence properties of smooth sliding mode controllers are presented. This result corrects flawed conclusions previously reached in the literature.


Weyl groups, lattices and geometric manifolds

Abstract: By studying the action of the Weyl group of a simple Lie algebra on its root lattice, we construct torsion free subgroups of small and explicitly determined index in a large infinite class of Coxeter groups. One spin-off is the construction of hyperbolic manifolds of very small volume in up to 8 dimensions.


Lie group extensions associated to projective modules of continuous inverse algebras

Abstract: We call a unital locally convex algebra $A$ a continuous inverse algebra if its unit group $A^\times$ is open and inversion is a continuous map. For any smooth action of a, possibly infinite-dimensional, connected Lie group $G$ on a continuous inverse algebra $A$ by automorphisms and any finitely generated projective right $A$-module $E$, we construct a Lie group extension $\hat G$ of $G$ by the group $\GL_A(E)$ of automorphisms of the $A$-module $E$. This Lie group extension is a ``non-commutative'' version of the group $\Aut(\V)$ of automorphism of a vector bundle over a compact manifold $M$, which arises for $G = \Diff(M)$, $A = C^\infty(M,\C)$ and $E = \Gamma\V$. We also identify the Lie algebra $\hat\g$ of $\hat G$ and explain how it is related to connections of the $A$-module $E$.


Palindromic Saturation

Abstract: We consider two {seemingly} different definitions of infinite words which contain {the} utmost number of palindromes. We show that these two definitions coincide. {The keynote of the proof is a meticulous inspection of properties of complete return words and the application of some basic graph theory.} In fact, we provide another proof of the result announced in \cite{Zamboni}.


Central extensions of the Ptolemy-Thompson group and quantized Teichmuller theory

Abstract: The central extension of the Thompson group $T$ that arises in the quantized Teichm\"uller theory is 12 times the Euler class. This extension is obtained by taking a (partial) abelianization of the so-called braided Ptolemy-Thompson group introduced and studied in \cite{FK2}. We describe then the cyclic central extensions of $T$ by means of explicit presentations.


Spectral representation of some non stationary alpha-stable processes

Abstract: In this paper, we give a new covariation spectral representation of some non stationary symmetric $\alpha$-stable processes (S$\alpha$S). This representation is based on a weaker covariation pseudo additivity condition which is more general than the condition of independence. This work can be seen as a generalization of the covariation spectral representation of processes expressed as stochastic integrals with respect to independent increments S$\alpha$S processes (see Cambanis (1983)) or with respect to the general concept of independently scattered S$\alpha$S measures (Samorodnitsky and Taqqu 1994). Relying on this result we investigate the non stationarity structure of some harmonisable S$\alpha$S processes especially those having periodic or almost-periodic covariation functions.


The mapping class group orbit of a multicurve

Abstract: Given a set equipped with a transitive action of a group, we define the notion of an almost invariant coloring of the set. We consider the mapping class group orbit of a multicurve on a compact surface, and prove that in the case of genus at least two, no such almost invariant coloring exists. Conversely, in the case of a closed torus, one may find almost invariant colorings using arbitrarily many colors.


The operad Lie is free

Abstract: We show that the operad Lie is free as a non-symmetric operad. Then we study the generating series counting the operadic generators, finding a recursive formula for its coefficients, and showing that the asymptotic density of the operadic generators is 1/e.


Thomae's formulae for non-hyperelliptic curves and spinorial square roots of theta-constants on the moduli space of curves

Abstract: Determinantal formulae for Jacobian theta functions that go back to Klein are elaborated, via an idea due to Matone and Volpato. Also, the natural square roots of theta constants on the moduli space of curves whose existence was shown by Tsuyumine are proved to have a spinorial structure.


A remark on the constructibility of real root representations of quivers using universal extension functors

Abstract: In this paper we consider the following question: Is it possible to construct all real root representations of a given quiver Q by using universal extension functors, starting with a real Schur representation? We give a concrete example answering this question negatively.


Complexity of planar and spherical curves

Abstract: We show that the maximal number of singular moves required to pass between any two regularly homotopic planar or spherical curves with at most n crossings, grows quadratically with respect to n. Furthermore, this can be done with all curves along the way having at most n+2 crossings.


Equivalences entre conjectures de Soergel

Abstract: Soergel's category B_k(V) over a field k is defined from a Coxeter system (W,S) and a k-linear representation V of W. It's a categorification of the Hecke algebra of (W,S). In this article we prove that for some representations V and V' of W, Soergel's conjecture over B_k(V') is equivalent to that over B_k(V). In particular, when k=IR we can choose V' to be the geometric representation.


A conjectural presentation of fusion algebras

Abstract: Let g be a semisimple Lie algebra over the complex numbers. Fix a positive integer l (called the level). Let R(l,g) be the fusion algebra at level l. Then, there is an algebra homomorphism from the representation ring R(g) of g to R(l,g). We study a presentation of its kernel. The generators for the kernel were given by Gepner, Gepner-Schwimmer, Bourdeau-Mlawer-Riggs-Schnitzer for g of type A and C series. We make a conjecture for other classical groups and also for g of type G2. We also have some partial results for F4 and E series.


Nonlinear stability of stationary solutions for curvature flow with triple junction

Abstract: In this paper we analyze the motion of a network of three planar curves with a speed proportional to the curvature of the arcs, having perpendicular intersections with the outer boundary and a common intersection at a triple junction. As a main result we show that a linear stability criterion due to Ikota and Yanagida is also sufficient for nonlinear stability. We also prove local and global existence of classical smooth solutions as well as various energy estimates. Finally, we prove exponential stabilization of an evolving network starting from the vicinity of a linearly stable stationary network.


McShane's identity, using elliptic elements

Abstract: We introduce a new method to establish McShane's Identity, based upon the fact that elliptic elements of order two in the Fuchsian group uniformizing the quotient of a fixed once-punctured hyperbolic torus act so as to exclude points as being highest points of geodesics.


Commuting holonomies and rigidity of holomorphic foliations

Abstract: In this article we study deformations of a holomorphic foliation with a generic non-rational first integral in the complex plane. We consider two vanishing cycles in a regular fiber of the first integral with a non-zero self intersection and with vanishing paths which intersect each other only at their start points. It is proved that if the deformed holonomies of such vanishing cycles commute then the deformed foliation has also a first integral. Our result generalizes a similar result of Ilyashenko on the rigidity of holomorphic foliations with a persistent center singularity. The main tools of the proof are Picard-Lefschetz theory and the theory of iterated integrals for such deformations.


Estim\'ees des noyaux de Green et de la chaleur sur les espaces sym\'etriques

Abstract: We obtain an off-diagonal upper bound for Green and heat kernel of Laplace type operator on symmetric spaces.


Multiple Stratonovich integral and Hu--Meyer formula for L\'{e}vy processes

Abstract: In the framework of vector measures and the combinatorial approach to stochastic multiple integral introduced by Rota and Wallstrom [Ann. Probab. 25 (1997) 1257--1283], we present an It\^{o} multiple integral and a Stratonovich multiple integral with respect to a L\'{e}vy process with finite moments up to a convenient order. In such a framework, the Stratonovich multiple integral is an integral with respect to a product random measure whereas the It\^{o} multiple integral corresponds to integrate with respect to a random measure that gives zero mass to the diagonal sets. A general Hu--Meyer formula that gives the relationship between both integrals is proved. As particular cases, the classical Hu--Meyer formulas for the Brownian motion and for the Poisson process are deduced. Furthermore, a pathwise interpretation for the multiple integrals with respect to a subordinator is given.


Moduli of polarized Hodge structures

Abstract: Around 1970 Griffiths introduced the moduli of polarized Hodge structures/the period domain $D$ and described a dream to enlarge $D$ to a moduli space of degenerating polarized Hodge structures. Since in general $D$ is not a Hermitian symmetric domain, he asked for the existence of a certain automorphic cohomology theory for $D$, generalizing the usual notion of automorphic forms on symmetric Hermitian domains. Since then there have been many efforts in the first part of Griffith's dream but the second part still lives in darkness. The objective of the present text is two-folded. First, we give an exposition of the subject. Second, we give another formulation of the Griffiths problem, based on the classical Weierstrass uniformization theorem.


Penalized model-based clustering with cluster-specific diagonal covariance matrices and grouped variables

Abstract: Clustering analysis is one of the most widely used statistical tools in many emerging areas such as microarray data analysis. For microarray and other high-dimensional data, the presence of many noise variables may mask underlying clustering structures. Hence removing noise variables via variable selection is necessary. For simultaneous variable selection and parameter estimation, existing penalized likelihood approaches in model-based clustering analysis all assume a common diagonal covariance matrix across clusters, which however may not hold in practice. To analyze high-dimensional data, particularly those with relatively low sample sizes, this article introduces a novel approach that shrinks the variances together with means, in a more general situation with cluster-specific (diagonal) covariance matrices. Furthermore, selection of grouped variables via inclusion or exclusion of a group of variables altogether is permitted by a specific form of penalty, which facilitates incorporating subject-matter knowledge, such as gene functions in clustering microarray samples for disease subtype discovery. For implementation, EM algorithms are derived for parameter estimation, in which the M-steps clearly demonstrate the effects of shrinkage and thresholding. Numerical examples, including an application to acute leukemia subtype discovery with microarray gene expression data, are provided to demonstrate the utility and advantage of the proposed method.


$L^p$ Boundedness of Commutators of Riesz Transforms associated to Schr\"{o}dinger Operator

Abstract: In this paper we consider $L^p$ boundedness of some commutators of Riesz transforms associated to Schr\"{o}dinger operator $P=-\Delta+V(x)$ on $\mathbb{R}^n, n\geq 3$. We assume that $V(x)$ is non-zero, nonnegative, and belongs to $B_q$ for some $q \geq n/2$. Let $T_1=(-\Delta+V)^{-1}V,\ T_2=(-\Delta+V)^{-1/2}V^{1/2}$ and $T_3=(-\Delta+V)^{-1/2}\nabla$. We obtain that $[b,T_j] (j=1,2,3)$ are bounded operators on $L^p(\mathbb{R}^n)$ when $p$ ranges in a interval, where $b \in \mathbf{BMO}(\mathbb{R}^n)$. Note that the kernel of $T_j (j=1,2,3)$ has no smoothness.


An explicit finite difference scheme for the Camassa-Holm equation

Abstract: We put forward and analyze an explicit finite difference scheme for the Camassa-Holm shallow water equation that can handle general $H^1$ initial data and thus peakon-antipeakon interactions. Assuming a specified condition restricting the time step in terms of the spatial discretization parameter, we prove that the difference scheme converges strongly in $H^1$ towards a dissipative weak solution of Camassa-Holm equation.


Joyce invariants for K3 surfaces and mock theta functions

Abstract: We will discuss Joyce invariants of stability conditions for K3 surfaces and mock theta functions.


Convergence of weighted polynomial multiple ergodic averages

Abstract: We study here weighted polynomial multiple ergodic averages. A sequence of weights is called universally good if any polynomial multiple ergodic average with this sequence of weights converges in $L^{2}$. We find a necessary condition and show that for any bounded measurable function $\phi$ on an ergodic system, the sequence $\phi(T^{n}x)$ is universally good for almost every $x$. The linear case was understood by Host and Kra.


Testing the number of parameters with multidimensional MLP

Abstract: This work concerns testing the number of parameters in one hidden layer multilayer perceptron (MLP). For this purpose we assume that we have identifiable models, up to a finite group of transformations on the weights, this is for example the case when the number of hidden units is know. In this framework, we show that we get a simple asymptotic distribution, if we use the logarithm of the determinant of the empirical error covariance matrix as cost function.


Efficient Estimation of Multidimensional Regression Model with Multilayer Perceptron

Abstract: This work concerns estimation of multidimensional nonlinear regression models using multilayer perceptron (MLP). The main problem with such model is that we have to know the covariance matrix of the noise to get optimal estimator. however we show that, if we choose as cost function the logarithm of the determinant of the empirical error covariance matrix, we get an asymptotically optimal estimator.


Estimation of linear autoregressive models with Markov-switching, the E.M. algorithm revisited

Abstract: This work concerns estimation of linear autoregressive models with Markov-switching using expectation maximisation (E.M.) algorithm. Our method generalise the method introduced by Elliot for general hidden Markov models and avoid to use backward recursion.


The Virgin Island Model

Abstract: A continuous mass population model with local competition is constructed where every emigrant colonizes an unpopulated island. The population founded by an emigrant is modeled as excursion from zero of an one-dimensional diffusion. With this excursion measure, we construct a process which we call Virgin Island Model. Furthermore, a necessary and sufficient condition for extinction of the total population is obtained for finite initial total mass.


On the Linear Combinants of a Binary Pencil

Abstract: Let A,B denote binary forms of order d, and let C_{2r-1} = (A,B)_{2r-1} be the sequence of their linear combinants for r between 1 and (d+1)/2. It is known that C_1 and C_3 together determine the pencil generated by A and B, and hence indirectly the higher C_{2r-1}. In this paper we exhibit explicit formulae for all r>2, which allow us to recover C_{2r-1} from the knowledge of C_1 and C_3. The calculations make use of the symbolic method of classical invariant theory, as well as the quantum theory of angular momentum. Our theorem pertains to the second exterior power representation of S_d, for the group SL_2. We give an example for the group SL_3 to show that such a result may hold for other categories of representations.


Self Organizing Map algorithm and distortion measure

Abstract: We study the statistical meaning of the minimization of distortion measure and the relation between the equilibrium points of the SOM algorithm and the minima of distortion measure. If we assume that the observations and the map lie in an compact Euclidean space, we prove the strong consistency of the map which almost minimizes the empirical distortion. Moreover, after calculating the derivatives of the theoretical distortion measure, we show that the points minimizing this measure and the equilibria of the Kohonen map do not match in general. We illustrate, with a simple example, how this occurs.


Efficient Estimation of Multidimensional Regression Model using Multilayer Perceptrons

Abstract: This work concerns the estimation of multidimensional nonlinear regression models using multilayer perceptrons (MLPs). The main problem with such models is that we need to know the covariance matrix of the noise to get an optimal estimator. However, we show in this paper that if we choose as the cost function the logarithm of the determinant of the empirical error covariance matrix, then we get an asymptotically optimal estimator. Moreover, under suitable assumptions, we show that this cost function leads to a very simple asymptotic law for testing the number of parameters of an identifiable MLP. Numerical experiments confirm the theoretical results.


A note on the regularity of solutions of Hamilton-Jacobi equations with superlinear growth in the gradient variable

Abstract: We investigate the regularity of solutions of first order Hamilton-Jacobi equation with super linear growth in the gradient variable. We show that the solutions are locally H\"older continuous with H\"older exponent depending only on the growth of the Hamiltonian. The proof relies on a reverse H\"older inequality.


Scaling limits of $(1+1)$-dimensional pinning models with Laplacian interaction

Abstract: We consider a random field $\varphi:\{1,...,N\}\to \mathbb{R}$ with Laplacian interaction of the form $\sum_iV(\Delta\varphi_i)$, where $\Delta$ is the discrete Laplacian and the potential $V(\cdot)$ is symmetric and uniformly strictly convex. The pinning model is defined by giving the field a reward $\varepsilon\ge0$ each time it touches the x-axis, that plays the role of a defect line. It is known that this model exhibits a phase transition between a delocalized regime $(\varepsilon<\varepsilon_c)$ and a localized one $(\varepsilon>\varepsilon_c)$, where $0<\varepsilon_c<\infty$. In this paper we give a precise pathwise description of the transition, extracting the full scaling limits of the model. We show, in particular, that in the delocalized regime the field wanders away from the defect line at a typical distance $N^{3/2}$, while in the localized regime the distance is just $O((\log N)^2)$. A subtle scenario shows up in the critical regime $(\varepsilon=\varepsilon_c)$, where the field, suitably rescaled, converges in distribution toward the derivative of a symmetric stable L\'evy process of index 2/5. Our approach is based on Markov renewal theory.


Tridiagonal pairs of shape (1,2,1)

Abstract: Let $\mathbb F$ denote a field and let $V$ denote a vector space over $\mathbb F$ with finite positive dimension. We consider a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfies the following conditions: (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering $\lbrace V_i \rbrace_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1}+V_i+V_{i+1}$ for $0 \leq i \leq d$, where $V_{-1} = 0$ and $V_{d+1} = 0$; (iii) there exists an ordering $\lbrace V^*_i \rbrace_{i=0}^{\delta}$ of the eigenspaces of $A^*$ such that $AV^*_i \subseteq V^*_{i-1}+V^*_i+V^*_{i+1}$ for $0 \leq i \leq \delta $, where $V^*_{-1} = 0$ and $V^*_{\delta+1} = 0$; (iv) there is no subspace $W$ of $V$ such that $AW\subseteq W$, $A^*W\subseteq W$, $W \neq 0, W \neq V$. We call such a pair a {\it tridiagonal pair} on $V$. It is known that $d = \delta$ and that for $0 \leq i \leq d$ the dimensions of $V_i, V_{d-i}, V^*_i, V^*_{d-i}$ coincide; we denote this common value by $\rho_i$. The sequence $\lbrace \rho_i\rbrace_{i=0}^d$ is called the {\it shape} of the pair. In this paper we assume the shape is $(1,2,1)$ and obtain the following results. We describe six bases for $V$; one diagonalizes $A$, another diagonalizes $A^*$, and the other four underlie the split decompositions for $A,A^*$. We give the action of $A$ and $A^*$ on each basis. For each ordered pair of bases among the six, we give the transition matrix. At the end we classify the tridiagonal pairs of shape $(1,2,1)$ in terms of a sequence of scalars called the parameter array.


Decay estimates for a class of wave equations

Abstract: In this paper we use a unified way studying the decay estimate for a class of dispersive semigroup given by $e^{it\phi(\sqrt{-\Delta})}$, where $\phi: \mathbb{R}^+\to \mathbb{R}$ is smooth away from the origin. Especially, the decay estimates for the solutions of the Klein-Gordon equation and the beam equation are simplified and slightly improved.


Spectrum of the Lichnerowicz Laplacian on asymptotically hyperbolic surfaces

Abstract: We show that, on any asymptotically hyperbolic surface, the essential spectrum of the Lichnerowicz Laplacian $\Delta_L$ contains the ray $[{1/4},+\infty[$. If moreover the scalar curvature is constant then -2 and 0 are infinite dimensional eigenvalues. If, in addition, the inequality $<\Delta u, u>_{L^2}\geq \frac14||u||^2_{L^2}$ holds for all smooth compactly supported function $u$, then there is no other value in the spectrum.


Higher Extension Modules and the Yoneda Product

Abstract: A chain of c submodules E =: E_0 >= E_1 >= ... >= E_c >= E_{c+1} := 0 gives rise to c composable 1-cocycles in Ext^1(E_{i-1}/E_i,E_i/E_{i+1}), i=1,...,c. In this paper we follow the converse question: When are c composable 1-cocycles induced by a module E together with a chain of submodules as above? We call such modules c-extension modules. The case c=1 is the classical correspondence between 1-extensions and 1-cocycles. For c=2 we prove an existence theorem stating that a 2-extension module exists for two composable 1-cocycles eta^M_L in Ext^1(M,L) and eta^L_N in Ext^1(L,N), if and only if their Yoneda product eta^M_L o eta^L_N in Ext^2(M,N) vanishes. We further prove a modelling theorem for c=2: In case the set of all such 2-extension modules is non-empty it is an affine space modelled over the abelian group that we call the first extension group of 1-cocycles, Ext^1(eta^M_L,eta^L_N) := Ext^1(M,N)/(Hom(M,L) o eta^L_N + eta^M_L o Hom(L,N)).


$L^2$ extension of adjoint line bundle sections

Abstract: We prove an $L^2$ extension theorem of Ohsawa-Takegoshi type for extending holomorphic sections of line bundles from a subvariety which is given as a maximal log-canonical center of a pair and is of general codimension in a projective variety. Our method of proof indicates that such a setting is the most natural one in a sense, for general $L^2$ extension of line bundle sections.


Consistance d'un estimateur de minimum de variance \'etendue

Abstract: We consider a generalization of the criterion minimized by the K-means algorithm, where a neighborhood structure is used in the calculus of the variance. Such tool is used, for example with Kohonen maps, to measure the quality of the quantification preserving the neighborhood relationships. If we assume that the parameter vector is in a compact Euclidean space and all it components are separated by a minimal distance, we show the strong consistency of the set of parameters almost realizing the minimum of the empirical extended variance.


Estimation consistante de l'architecture des perceptrons multicouches

Abstract: We consider regression models involving multilayer perceptrons (MLP) with one hidden layer and a Gaussian noise. The estimation of the parameters of the MLP can be done by maximizing the likelihood of the model. In this framework, it is difficult to determine the true number of hidden units because the information matrix of Fisher is not invertible if this number is overestimated. However, if the parameters of the MLP are in a compact set, we prove that the minimization of a suitable information criteria leads to consistent estimation of the true number of hidden units.


On Katz's bound for number of elements with given trace and norm

Abstract: In this note an improvement of the Katz's bound on the number of elements in a finite field with given trace and norm is given. The improvement is obtained by reducing the problem to estimating the number of rational points on certain toric Calabi-Yau hypersurface, and then to use detailed cohomological calculations by Rojas-Leon and the second author for such toric hypersurfaces.


A collection of sharp dilation invariant inequalities for differentiable functions

Abstract: We find best constants in several dilation invariant integral inequalities involving derivatives of functions. Some of these inequalities are new and some were known without best constants. The contents: 1. Estimate for a quadratic form of the gradient, 2. Weighted G{\aa}rding inequality for the biharmonic operator, 3.Dilation invariant Hardy's inequality with remainder term, 4. Generalized Hardy-Sobolev inequality with sharp constant, 5. Hardy's inequality with sharp Sobolev remainder term.


Laplacians on the basilica Julia set

Abstract: We consider the basilica Julia set of the polynomial $P(z)=z^{2}-1$ and construct all possible resistance (Dirichlet) forms, and the corresponding Laplacians, for which the topology in the effective resistance metric coincides with the usual topology. Then we concentrate on two particular cases. One is a self-similar harmonic structure, for which the energy renormalization factor is 2, the spectral dimension is $\log9/\log6$, and we can compute all the eigenvalues and eigenfunctions by a spectral decimation method. The other is graph-directed self-similar under the map $z\mapsto P(z)$; it has energy renormalization factor $\sqrt2$ and spectral dimension 4/3, but the exact computation of the spectrum is difficult. The latter Dirichlet form and Laplacian are in a sense conformally invariant on the basilica Julia set.


Relationship between stochastic flows and connection forms

Abstract: In this article I will prove new representation for the Levi-Civita connection in terms of the stochastic flow corresponding to Brownian motion on manifold.


Supersingular Kottwitz-Rapoport strata and Deligne-Lusztig varieties

Abstract: We investigate Siegel modular varieties in positive characteristic with Iwahori level structure. On these spaces, we have the Newton stratification, and the Kottwitz-Rapoport stratification; one would like to understand how these stratifications are related to each other. We give a simple description of all KR strata which are entirely contained in the supersingular locus as disjoint unions of Deligne-Lusztig varieties. We also give an explicit numerical description of the KR stratification in terms of abelian varieties.


Plurisubharmonic functions with weak singularities

Abstract: We study the complex Monge-Amp\`ere operator in bounded hyperconvex domains of $\C^n$. We introduce a scale of classes of weakly singular plurisubharmonic functions : these are functions of finite weighted Monge-Amp\`ere energy. They generalize the classes introduced by U.Cegrell, and give a stratification of the space of (almost) all unbounded plurisubharmonic functions. We give an interpretation of these classes in terms of the speed of decreasing of the Monge-Amp\`ere capacity of sublevel sets and solve associated complex Monge-Amp\`ere equations.


Relative radial mass and rigidity of some warped product manifolds

Abstract: We give a Riccati type formula adapted for two metrics having the same geodesics rays starting from a point or orthogonal to an hypersurface, one of these metrics being a warped product if the dimension $n$ is greater than or equal to 3. This formula has non-trivial geometric consequences such as a positive mass type theorem and other rigidity results. We also apply our result to some standard models.


The Supermagic Square in characteristic 3 and Jordan superalgebras

Abstract: Recently, the classical Freudenthal Magic Square has been extended over fields of characteristic 3 with two more rows and columns filled with (mostly simple) Lie superalgebras specific of this characteristic. This Supermagic Square will be reviewed and some of the simple Lie superalgebras that appear will be shown to be isomorphic to the Tits-Kantor-Koecher Lie superalgebras of some Jordan superalgebras.


Adaptive Confidence Sets for the Optimal Approximating Model

Abstract: In the setting of high-dimensional linear models with Gaussian noise, we investigate the possibility of confidence statements connected to model selection. Although there exist numerous procedures for adaptive point estimation, the construction of adaptive confidence regions is severely limited (cf. Li, 1989). The present paper sheds new light on this gap. We develop exact and adaptive confidence sets for the best approximating model in terms of risk. One of our constructions is based on a multiscale procedure and a particular coupling argument. Utilizing exponential inequalities for noncentral chi-squared distributions, we show that the risk and quadratic loss of all models within our confidence region are uniformly bounded by the minimal risk times a factor close to one.


Automorphic properties of generating functions for generalized rank moments and Durfee symbols

Abstract: We define two-parameter generalizations of two combinatorial constructions of Andrews: the kth symmetrized rank moment and the k-marked Durfee symbol. We prove that three specializations of the associated generating functions are so-called quasimock theta functions, while a fourth specialization gives quasimodular forms. We then define a two-parameter generalization of Andrews' smallest parts function and note that this leads to quasimock theta functions as well. The automorphic properties are deduced using q-series identities relating the relevant generating functions to known mock theta functions.


Equidistribution of expanding translates of curves and Dirichlet's theorem on Diophantine approximation

Abstract: We show that for almost all points on any analytic curve on R^{k} which is not contained in a proper affine subspace, the Dirichlet's theorem on simultaneous approximation, as well as its dual result for simultaneous approximation of linear forms, cannot be improved. The result is obtained by proving asymptotic equidistribution of evolution of a curve on a strongly unstable leaf under certain partially hyperbolic flow on the space of unimodular lattices in R^{k+1}. The proof involves ergodic properties of unipotent flows on homogeneous spaces.


Iterated Grafting and Holonomy Lifts of Teichmueller space

Abstract: Let $X$ be a closed hyperbolic surface and $\lambda, \eta$ be weighted geodesic multicurves which are short on X. We show that the iterated grafting along $\lambda$ and $\eta$ is close in the Teichmueller metric to grafting along a single multicurve which can be given explicitly in terms of $\lambda$ and $\eta$. Using this result, we study the holonomy lifts $gr_{\lambda}\rho_{X,\lambda}$ of Teichmueller geodesics $\rho_{X,\lambda}$ for integral laminations $\lambda$ and show that all of them have bounded Teichmueller distance to the geodesic $\rho_{X,\lambda}$. We obtain analogous results for grafting rays. Finally we consider the asymptotic behaviour of iterated grafting sequences $\gr_{n\lambda}X$ and show that they converge geometrically to a punctured surface.


A model of continuous time polymer on the lattice

Abstract: In this article, we try to give a rather complete picture of the behavior of the free energy for a model of directed polymer in a random environment, in which the polymer is a simple symmetric random walk on the lattice $\Z^d$, and the environment is a collection $\{W(t,x);t\ge 0, x\in \Z^d\}$ of i.i.d. Brownian motions.


De-Rham theorem and Shapiro lemma for Schwartz functions on Nash manifolds

Abstract: In this paper we continue our work on Schwartz functions and generalized Schwartz functions on Nash (i.e. smooth semi-algebraic) manifolds. Our first goal is to prove analogs of de-Rham theorem for de-Rham complexes with coefficients in Schwartz functions and generalized Schwartz functions. Using that we compute cohomologies of the Lie algebra of an algebraic group G with coefficients in the space of generalized Schwartz sections of G-equivariant bundle over a G-transitive variety M. We do it under some assumptions on topological properties of G and M. This computation for the classical case is known as Shapiro lemma.


Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case $H=1/4$

Abstract: We derive the asymptotic behavior of weighted quadratic variations of fractional Brownian motion $B$ with Hurst index $H=1/4$. This completes the only missing case in a very recent work by I. Nourdin, D. Nualart and C. A. Tudor. Moreover, as an application, we solve a recent conjecture of K. Burdzy and J. Swanson on the asymptotic behavior of the Riemann sums with alternating signs associated to $B$.


Mathematical study of resonant wind-driven oceanic motions

Abstract: We are interested here in describing the linear response of the ocean to some wind forcing, which admits fast time oscillations and may be resonant with the Coriolis force. In addition to the usual Ekman layer, we exhibit another - much larger - boundary layer, and some global vertical profile. That means in particular that the wind effect is no longer localized in the vicinity of the surface. From a mathematical point of view, the main novelty here is to introduce some systematic approach for the study of boundary effects.


Finsler conformal Lichnerowicz-Obata conjecture

Abstract: We prove the Finsler analog of the conformal Lichnerowicz-Obata conjecture showing that a complete and essential conformal vector field on a non-Riemannian Finsler manifold is a homothetic vector field of a Minkowski metric.


A characterization of quadric constant mean curvature hypersurfaces of spheres

Abstract: Let $\phi:M\to\mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}$ be an immersion of a complete $n$-dimensional oriented manifold. For any $v\in\mathbb{R}^{n+2}$, let us denote by $\ell_v:M\to\mathbb{R}$ the function given by $\ell_v(x)=\phi(x),v$ and by $f_v:M\to\mathbb{R}$, the function given by $f_v(x)=\nu(x),v$, where $\nu:M\to\mathbb{S}^{n}$ is a Gauss map. We will prove that if $M$ has constant mean curvature, and, for some $v\ne{\bf 0}$ and some real number $\lambda$, we have that $\ell_v=\lambda f_v$, then, $\phi(M)$ is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that the weak stability index of any compact constant mean curvature hypersurface $M^n$ in $\mathbb{S}^{n+1}$ which is neither totally umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal to $2n+4$.


Bifurcations, Schwarzian derivatives and Feigenbaum constant revisited

Abstract: The main purpose is to show that Feigenbaum delta constant is much more universal than believed. The paper is mainly devoted to period-doubling processes in families in one parameter of endomorphisms of the interval and consider generalizations of the Feigenbaum delta constant. We formulate the so-called parenthesis permeability hypothesis, a conjecture that holds for all types of bifurcation (i.e. for flip, fold, pitchfork and transcritical bifurcations, which states that under some conditions two or three different functions may have exactly the same bifurcation points. We propose a conjecture that considerably relaxes David Singer conditions for endomorphism families to generate at most one stable orbit, showing that Feigenbaum constant appears also in some classes of functions that have more than one maximum and have positive Schwarzian in at least one sub-interval. This version contains more arguments in favor of an even greater Feiganbaum constant universality.


The subelliptic heat kernel on SU(2): Representations, Asymptotics and Gradient bounds

Abstract: The Lie group SU(2) endowed with its canonical subriemannian structure appears as a three-dimensional model of a positively curved subelliptic space. The goal of this work is to study the subelliptic heat kernel on it and some related functional inequalities.


Consistent estimation of the architecture of multilayer perceptrons

Abstract: We consider regression models involving multilayer perceptrons (MLP) with one hidden layer and a Gaussian noise. The estimation of the parameters of the MLP can be done by maximizing the likelihood of the model. In this framework, it is difficult to determine the true number of hidden units using an information criterion, like the Bayesian information criteria (BIC), because the information matrix of Fisher is not invertible if the number of hidden units is overestimated. Indeed, the classical theoretical justification of information criteria relies entirely on the invertibility of this matrix. However, using recent methodology introduced to deal with models with a loss of identifiability, we prove that suitable information criterion leads to consistent estimation of the true number of hidden units.


Generalized manifolds in products of curves

Abstract: The intent of this article is to study some special $n$-dimensional continua lying in products of $n$ curves. (The paper is an improved version of a portion of \cite{K-K-S}.) We show that if $X$ is a locally connected, so-called, quasi $n$-manifold lying in a product of $n$ curves then rank of $H^1(X)\ge n$. Moreover, if $\rank H^1(X)<2n$ then $X$ can be represented as a product of an $m$-torus and a quasi $(n-m)$-manifold, where $m\ge2n-\rank H^1(X)$. It follows that certain 2-dimensional contractible polyhedra are not embeddable in products of two curves. On the other hand, we show that any collapsible 2-dimensional polyhedron can be embedded in a product of two trees. We answer a question of R. Cauty proving that closed surfaces embeddable in products of two curves can be also embedded in products of two graphs. On the other hand, we construct an example of a 2-dimensional polyhedron which can be embedded in a product of two curves though it is not embeddable in any product of two graphes. This solves in the negative another problem of Cauty.


A change of variable formula with It\^{o} correction term

Abstract: We consider the solution $u(x,t)$ to a stochastic heat equation. For fixed $x$, the process $F(t)=u(x,t)$ has a nontrivial quartic variation. It follows that $F$ is not a semimartingale, so a stochastic integral with respect to $F$ cannot be defined in the classical It\^{o} sense. We show that for sufficiently differentiable functions $g(x,t)$, a stochastic integral $\int g(F(t),t)\,dF(t)$ exists as a limit of discrete, midpoint-style Riemann sums, where the limit is taken in distribution in the Skorokhod space of cadlag functions. Moreover, we show that this integral satisfies a change of variable formula with a correction term that is an ordinary It\^{o} integral with respect to a Brownian motion that is independent of $F$.


An abstract setting for hamiltonian actions

Abstract: In this paper we develop an abstract setup for hamiltonian group actions as follows: Starting with a continuous 2-cochain $\omega$ on a Lie algebra $h$ with values in an $h$-module $V$, we associate subalgebras $sp(h,\omega) \supeq ham(h,\omega)$ of symplectic, resp., hamiltonian elements. Then $ham(h,\omega)$ has a natural central extension which in turn is contained in a larger abelian extension of $sp(h,\omega)$. In this setting, we study linear actions of a Lie group $G$ on $V$ which are compatible with a homomorphism $g \to ham(h,\omega)$, i.e. abstract hamiltonian actions, corresponding central and abelian extensions of $G$ and momentum maps $J : g \to V$.


A remark on primality testing and decimal expansions

Abstract: We show that for any fixed base $a$, a positive proportion of primes have the property that they become composite after altering any one of their digits in the base $a$ expansion; the case $a=2$ was already established by Cohen-Selfridge and Sun, using some covering congruence ideas of Erd\H{o}s. Our method is slightly different, using a partially covering set of congruences followed by an application of the Selberg sieve upper bound. As a consequence, it is not always possible to test whether a number is prime from its base $a$ expansion without reading all of its digits. We also present some slight generalisations of these results.


Evaluation and selection of models for out-of-sample prediction when the sample size is small relative to the complexity of the data-generating process

Abstract: In regression with random design, we study the problem of selecting a model that performs well for out-of-sample prediction. We do not assume that any of the candidate models under consideration are correct. Our analysis is based on explicit finite-sample results. Our main findings differ from those of other analyses that are based on traditional large-sample limit approximations because we consider a situation where the sample size is small relative to the complexity of the data-generating process, in the sense that the number of parameters in a `good' model is of the same order as sample size. Also, we allow for the case where the number of candidate models is (much) larger than sample size.


Elliptic surfaces without 1-handles

Abstract: Harer-Kas-Kirby conjectured that every handle decomposition of the elliptic surface E(1)_{2,3} requires both 1- and 3-handles. We prove that the elliptic surface E(n)_{p,q} has a handle decomposition without 1-handles for $n\geq 1$ and (p,q)=(2,3),(2,5),(3,4),(4,5).


Edge ideals of clique clutters of comparability graphs and the normality of monomial ideals

Abstract: Let (P,<) be a finite poset and let G be its comparability graph. If cl(G) is the clutter of maximal cliques of G, we prove that cl(G) satisfies the max-flow min-cut property and that its edge ideal is normally torsion free. We prove that edge ideals of complete admissible uniform clutters are normally torsion free. The normality of a monomial ideal is expressed in terms of blocking polyhedra and the integer decomposition property. For edge ideals of clutters this property completely determine their normality


On secant varieties of Compact Hermitian Symmetric Spaces

Abstract: We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three - with one exception, the secant variety of the $21$-dimensional spinor variety in $\pp{63}$ where we show the ideal is generated in degree four. We also discuss the coordinate rings of secant varieties of compact Hermitian symmetric spaces.


Goldman flows on the Jacobian

Abstract: We show that the Goldman flows preserve the holomorphic structure on the moduli space of homomorphisms of the fundamental group of a Riemann surface into U(1), in other words the Jacobian.


The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations

Abstract: This survey article begins with a discussion of the original form of the Strominger-Yau-Zaslow conjecture, surveys the state of knowledge concering this conjecture, and explains how thinking about this conjecture naturally leads to the program initiated by the author and Bernd Siebert to study mirror symmetry via degenerations of Calabi-Yau manifolds and log structures.


Moment problems and boundaries of number triangles

Abstract: The boundary problem for graphs like Pascal's but with general multiplicities of edges is related to a `backward' problem of moments of the Hausdorff type.


Structure of T modules and restricted duals: the classical and the quantum case

Abstract: A concrete realization of Enright's $T$ modules is obtained. This is used to show their self-duality. As a consequence, the restricted duals of Verma modules are also identified.


On multiplicity problems for finite-dimensional representations of hyper loop algebras

Abstract: Given a hyper loop algebra over a non-algebraically closed field, we address multiplicity problems in the underlying abelian tensor category of finite-dimensional representations. Namely, we give formulas for the l-characters of the simple objects, the Jordan-Holder multiplicities of the Weyl modules, and the Clebsch-Gordan coefficients.


The sutured Floer homology polytope

Abstract: In this paper, we extend the theory of sutured Floer homology developed by the author. We first prove an adjunction inequality, and then define a polytope P(M,g) in H^2(M,\partial M; R) that is spanned by the Spin^c-structures which support non-zero Floer homology groups. If (M,g) --> (M',g') is a taut surface decomposition, then a natural map projects P(M',g') onto a face of P(M,g); moreover, if H_2(M) = 0, then every face of P(M,g) can be obtained in this way for some surface decomposition. We show that if (M,g) is reduced, horizontally prime, and H_2(M) = 0, then P(M,g) is maximal dimensional in H^2(M,\partial M; R). This implies that if rk(SFH(M,g)) < 2^{k+1} then (M,g) has depth at most 2k. Moreover, SFH acts as a complexity for balanced sutured manifolds. In particular, the rank of the top term of knot Floer homology bounds the topological complexity of the knot complement, in addition to simply detecting fibred knots.


A New Proof for Classification of Irreducible Modules of a Hecke Algebra of Type $ A_n$

Abstract: In this paper we give a new proof for the classification of irreducible modules of an affine Hecke algebra of type $A_n$, which was obtained by G. E. Murphy in 1995.


An operator approach to multipoint Pade approximations

Abstract: First, an abstract scheme of constructing biorthogonal rational systems related to some interpolation problems is proposed. We also present a modification of the famous step-by-step process of solving the Nevanlinna-Pick problems for Nevanlinna functions. The process in question gives rise to three-term recurrence relations with coefficients depending on the spectral parameter. These relations can be rewritten in the matrix form by means of two Jacobi matrices. As a result, a convergence theorem for multipoint Pad\'e approximants to Nevanlinna functions is proved.


On Khintchine exponents and Lyapunov exponents of continued fractions

Abstract: Assume that $x\in [0,1) $ admits its continued fraction expansion $x=[a_1(x), a_2(x),...]$. The Khintchine exponent $\gamma(x)$ of $x$ is defined by $\gamma(x):=\lim\limits_{n\to \infty}\frac{1}{n}\sum_{j=1}^n \log a_j(x)$ when the limit exists. Khintchine spectrum $\dim E_\xi$ is fully studied, where $ E_{\xi}:=\{x\in [0,1):\gamma(x)=\xi\} (\xi \geq 0)$ and $\dim$ denotes the Hausdorff dimension. In particular, we prove the remarkable fact that the Khintchine spectrum $\dim E_{\xi}$, as function of $\xi \in [0, +\infty)$, is neither concave nor convex. This is a new phenomenon from the usual point of view of multifractal analysis. Fast Khintchine exponents defined by $\gamma^{\phi}(x):=\lim\limits_{n\to\infty}\frac{1}{\phi(n)} \sum_{j=1}^n \log a_j(x)$ are also studied, where $\phi (n)$ tends to the infinity faster than $n$ does. Under some regular conditions on $\phi$, it is proved that the fast Khintchine spectrum $\dim (\{x\in [0,1]: \gamma^{\phi}(x)= \xi \}) $ is a constant function. Our method also works for other spectra like the Lyapunov spectrum and the fast Lyapunov spectrum.


Generic points in systems of specification and Banach valued Birkhoff ergodic average

Abstract: We prove that systems satisfying the specification property are saturated in the sense that the topological entropy of the set of generic points of any invariant measure is equal to the measure-theoretic entropy of the measure. We study Banach valued Birkhoff ergodic averages and obtain a variational principle for its topological entropy spectrum. As application, we examine a particular example concerning with the set of real numbers for which the frequencies of occurrences in their dyadic expansions of infinitely many words are prescribed. This relies on our explicit determination of a maximal entropy measure.


Quotients of fake projective planes

Abstract: Recently, Prasad and Yeung classified all possible fundamental groups of fake projective planes. According to their result, many fake projective planes admit a nontrivial group of automorphisms, and in that case it is isomorphic to $\bbZ/3\bbZ$, $\bbZ/7\bbZ$, $7:3$, or $(\bbZ/3\bbZ)^2$, where $7:3$ is the unique non-abelian group of order 21. Let $G$ be a group of automorphisms of a fake projective plane $X$. In this paper we classify all possible structures of the quotient surface $X/G$ and its minimal resolution.


On a nonhierarchical version of the Generalized Random Energy Model. II. Ultrametricity

Abstract: We study the Gibbs measure of the nonhierarchical versions of the Generalized Random Energy Models introduced in previous work. We prove that the ultrametricity holds only provided some nondegeneracy conditions on the hamiltonian are met.


On peak phenomena for non-commutative $H^\infty$

Abstract: A non-commutative extension of Amar and Lederer's peak set result is given. As its simple applications it is shown that any non-commutative $H^\infty$-algebra $H^\infty(M,\tau)$ has unique predual,and moreover some restriction in some of the results of Blecher and Labuschagne are removed, making them hold in full generality.


H\"ormander type pseudodifferential calculus on homogeneous groups

Abstract: We produce, on general homogeneous groups, an analogue of the usual H\"ormander pseudodifferential calculus on Euclidean space, at least as far as products and adjoints are concerned. In contrast to earlier works, we do not limit ourselves to analogues of classical symbols, nor to the Heisenberg group. The key technique is to understand ``multipliers'' of any given order j, and the operators of convolution with their inverse Fourier transforms, which we here call convolution operators of order j. (Here a ``multiplier'' is an analogue of a H\"ormander-type symbol a(x,\xi), which is independent of x.) Specifically, we characterize the space of inverse Fourier transforms of multipliers of any order j, and use this characterization to show that the composition of convolution operators of order j_1 and j_2 is a convolution operator of order j_1+j_2.


A Truncation Approach for Fast Computation of Distribution Functions

Abstract: In this paper, we propose a general approach for improving the efficiency of computing distribution functions. The idea is to truncate the domain of summation or integration.


Confidence Interval for the Mean of a Bounded Random Variable and Its Applications in Point Estimation

Abstract: In this article, we derive an explicit formula for computing confidence interval for the mean of a bounded random variable. Moreover, we have developed multistage point estimation methods for estimating the mean value with prescribed precision and confidence level based on the proposed confidence interval.


Estimating Traffic Parameters with Rigorous Error Control

Abstract: To perform a queuing analysis or design in a communications context, we need to estimate the values of the input parameters, specifically the mean of the arrival rate and service time. In this paper, we propose an approach for estimating the arrival rate of Poisson processes and the average service time for servers under the assumption that the service time is exponential. In particular, we derive sample size (i.e., the number of i.i.d. observations) required to obtain an estimate satisfying a pre-specified relative accuracy with a given confidence level. A remarkable feature of this approach is that no a priori information about the parameter is needed. In contrast to conventional methods such as, standard error estimation and confidence interval construction, which only provides post-experimental evaluations of the estimate, this approach allows experimenters to rigorously control the error of estimation.


Infinite class field towers

Abstract: This paper studies infinite class field towers of number fields $K$ that are ramified over $\Q$ only at one finite prime. In particular, we show the existence of such towers for a general family of primes including $p=2$, 3 and 5.


Asymptotic Normality of the Additive Regression Components for Continuous Time Processes

Abstract: In multivariate regression estimation, the rate of convergence depends on the dimension of the regressor. This fact, known as the curse of the dimensionality, motivated several works. The additive model, introduced by Stone (10), offers an efficient response to this problem. In the setting of continuous time processes, using the marginal integration method, we obtain the quadratic convergence rate and the asymptotic normality of the components of the additive model.


Reconstruction of Random Colourings

Abstract: Reconstruction problems have been studied in a number of contexts including biology, information theory and and statistical physics. We consider the reconstruction problem for random $k$-colourings on the $\Delta$-ary tree for large $k$. Bhatnagar et. al. showed non-reconstruction when $\Delta \leq \frac12 k\log k - o(k\log k)$ and reconstruction when $\Delta \geq k\log k + o(k\log k)$. We tighten this result and show non-reconstruction when $\Delta \leq k[\log k + \log \log k + 1 - \ln 2 -o(1)]$ and reconstruction when $\Delta \geq k[\log k + \log \log k + 1+o(1)]$.


Classification of Quiver Hopf Algebras and Pointed Hopf Algebras of Type One

Abstract: The quiver Hopf algebras are classified by means of ramification systems with irreducible representations. This leads to the classification of Nichols algebras over group algebras and pointed Hopf algebras of type one.


Efficient Counting and Asymptotics of $k$-noncrossing tangled-diagrams

Abstract: In this paper we enumerate $k$-noncrossing tangled-diagrams. A tangled-diagram is a labeled graph whose vertices are $1,...,n$ have degree $\le 2$, and are arranged in increasing order in a horizontal line. Its arcs are drawn in the upper halfplane with a particular notion of crossings and nestings. Our main result is the asymptotic formula for the number of $k$-noncrossing tangled-diagrams $T_{k}(n) \sim c_k n^{-((k-1)^2+(k-1)/2)} (4(k-1)^2+2(k-1)+1)^n$ for some $c_k>0$.


Mapping properties of fundamental operators in harmonic analysis related to Bessel operators

Abstract: We prove sharp power-weighted strong type, weak type and restricted weak type inequalities for the heat and Poisson integral maximal operators, Riesz transform and a Littlewood-Paley type square function, emerging naturally in the harmonic analysis related to Bessel operators.


Quasi-nearly subharmonicity and separately quasi-nearly subharmonic functions

Abstract: Wiegerinck has shown that a separately subharmonic function need not be subharmonic. Improving previous results of Lelong, of Avanissian, of Arsove and of us, Armitage and Gardiner gave an almost sharp integrability condition which ensures a separately subharmonic function to be subharmonic. Completing now our recent counterparts to the cited results of Lelong, Avanissian and Arsove for so called quasi-nearly subharmonic functions, we present a counterpart to the cited result of Armitage and Gardiner for separately quasi-nearly subharmonic functions. This counterpart enables us to slightly improve Armitage's and Gardiner's original result, too. The method we use is a rather straightforward and technical, but still by no means easy, modification of Armitage's and Gardiner's argument combined with an old argument of Domar.


Feynman diagrams and minimal models for operadic algebras

Abstract: We construct an explicit minimal model for an algebra over the cobar-construction of a differential graded operad. The structure maps of this minimal model are expressed in terms of sums over decorated trees. We introduce the appropriate notion of a homotopy equivalence of operadic algebras and show that our minimal model is homotopy equivalent to the original algebra. All this generalizes and gives a conceptual explanation of well-known results for A-infinity algebras. Further, we show that these results carry over to the case of algebras over modular operads; the sums over trees get replaced by sums over general Feynman graphs. As a by-product of our work we prove gauge-independence of Kontsevich's `dual construction' producing graph cohomology classes from contractible differential graded Frobenius algebras.


Coherent presentations of structure monoids and the Higman-Thompson groups

Abstract: Structure monoids and groups are algebraic invariants of equational varieties. We show how to construct presentations of these objects from coherent categorifications of equational varieties, generalising several results of Dehornoy. We subsequently realise the higher Thompson groups $F_{n,1}$ and the Higman-Thompson groups $G_{n,1}$ as structure groups. We go on to obtain presentations of these groups via coherent categorifications of the varieties of higher-order associativity and of higher-order associativity and commutativity, respectively. These categorifications generalise Mac Lane's pentagon and hexagon conditions for coherently associative and commutative bifunctors.


Maximum vertex occupation time and inert fugitive: recontamination does help

Abstract: Given a simple graph $G$, we consider the node search problem with inert fugitive. We are interested in minimizing the maximum vertex occupation time, i.e. the maximum number of steps in which a vertex is occupied by a searcher during a search of $G$. We prove that a search program which does not allow a recontamination may not find an optimal solution to this problem, and the difference between the maximum vertex occupation time computed by a monotone search program and a program without such restriction may be arbitrarily large.


On the locality of the Pr\"ufer code

Abstract: The Pr\"ufer code is a bijection between trees on the vertex set $[n]$ and strings on the set $[n]$ of length $n-2$ (Pr\"ufer strings of order $n$). In this paper we examine the `locality' properties of the Pr\"ufer code, i.e. the effect of changing an element of the Pr\"ufer string on the structure of the corresponding tree. Our measure for the distance between two trees $T,T^*$ is $\Delta(T,T^*)=n-1-| E(T)\cap E(T^*)|$. We randomly mutate the $\mu$th element of the Pr\"ufer string of the tree $T$, changing it to the tree $T^*$, and we asymptotically estimate the probability that this results in a change of $\ell$ edges, i.e. $P(\Delta=\ell | \mu).$ We find that P(\Delta=\ell | \mu)$ is on the order of $ n^{-1/3+o(1)}$ for any integer $\ell>1,$ and that $P(\Delta=1 | \mu)=(1-\mu/n)^2+o(1).$ This result implies that the probability of a `perfect' mutation in the Pr\"ufer code (one for which $\Delta(T,T^*)=1$) is $1/3.$


Moufang symmetry II. Moufang-Mal'tsev pairs and triality

Abstract: A concept of the Moufang-Malt'tsev pair is elaborated. This concept is based on the generalized Maurer-Cartan equations of a local analytic Moufang loop. Triality can be seen as a fundamental property of such pairs. Based on triality, the Yamagutian is constructed. Properties of the Yamagutian are studied.


The time constant vanishes only on the percolation cone in directed first passage percolation

Abstract: We consider the directed first passage percolation model on ${\bf Z}^2$. In this model, we assign independently to each edge $e$ a passage time $t(e)$ with a common distribution $F$. We denote by $\vec{T}({\bf 0}, (r,\theta))$ the passage time from the origin to $(r, \theta)$ by a northeast path for $(r, \theta)\in {\bf R}^+\times [0,\pi/2]$. It is known that $\vec{T}({\bf 0}, (r, \theta))/r$ converges to a time constant $\vec{\mu}_F (\theta)$. Let $\vec{p}_c$ denote the critical probability for oriented percolation. In this paper, we show that the time constant has a phase transition divided by $\vec{p}_c$, as follows: (1) If $F(0) < \vec{p}_c$, then $\vec{\mu}_F(\theta) >0$ for all $0\leq \theta\leq \pi/2$. (2) If $F(0) = \vec{p}_c$, then $\vec{\mu}_F(\theta) >0$ if and only if $\theta\neq \pi/4$. (3) If $F(0)=p > \vec{p}_c$, then there exists a percolation cone between $\theta_p^-$ and $\theta_p^+$ for $0\leq \theta^-_p< \theta^+_p \leq \pi/2$ such that $\vec{\mu} (\theta) >0$ if and only if $\theta\not\in [\theta_p^-, \theta^+_p]$. Furthermore, all the moments of $\vec{T}({\bf 0}, (r, \theta))$ converge whenever $\theta\in [\theta_p^-, \theta^+_p]$. As applications, we describe the shape of the directed growth model on the distribution of $F$. We give a phase transition for the shape divided by $\vec{p}_c$.


Complex interpolation of compact operators mapping into lattice couples

Abstract: Suppose that (A_0,A_1) and (B_0,B_1) are Banach couples, and that T is a linear operator which maps A_0 compactly into B_0 and A_1 boundedly (or even compactly) into B_1. Does this imply that T maps [A_0,A_1]_s to [B_0,B_1]_s compactly for 0<s<1 ? (Here, as usual, [A_0,A_1]_s denotes the complex interpolation space of Alberto Calderon.) This question has been open for 44 years. Affirmative answers are known for it in many special cases. We answer it affirmatively in the case where (A_0,A_1) is arbitrary and (B_0,B_1) is a couple of Banach lattices having absolutely continuous norms or the Fatou property. Our result has some overlap with a recent result by Evgeniy Pustylnik.


On linear versions of some addition theorems

Abstract: Let K \subset L be a field extension. Given K-subspaces A,B of L, we study the subspace spanned by the product set AB = {ab | a \in A, b \in B}. We obtain some lower bounds on the dimension of this subspace and on dim B^n in terms of dim A, dim B and n. This is achieved by establishing linear versions of constructions and results in additive number theory mainly due to Kemperman and Olson.


On contracting hyperplane elements from a 3-connected matroid

Abstract: Let $\tilde{K}_{3,n}$, $n\geq 3$, be the simple graph obtained from $K_{3,n}$ by adding three edges to a vertex part of size three. We prove that if $H$ is a hyperplane of a 3-connected matroid $M$ and $M \not\cong M^*(\tilde{K}_{3,n})$, then there is an element $x$ in $H$ such that the simple matroid associated with $M/x$ is 3-connected.


A Remark on Triangle-Critical Graphs

Abstract: A connected $k$-chromatic graph $G$ with $k \geq 3$ is said to be triangle-critical, if every edge of $G$ is contained in an induced triangle of $G$ and the removal of any triangle from $G$ decreases the chromatic number of $G$ by three. B. Toft posed the problem of showing that the complete graphs on more than two vertices are the only triangle-critical graphs. By applying a method of M. Stiebitz [Discrete Math. 64 (1987), 91--93], we answer the problem affirmatively for triangle-critical $k$-chromatic graphs with $k \leq 6$.


Asymptotically Conical Associative 3-folds

Abstract: Given an associative 3-fold in R^7 which is asymptotically conical with generic rate less than 1, we show that its moduli space of deformations is locally homeomorphic to the kernel of a smooth map between smooth manifolds. Moreover, the virtual dimension of the moduli space is computed and shown to be non-negative for rates greater than -1, whereas the associative 3-fold is expected to be isolated for rates less than or equal to -1.


Interval Estimation of Bounded Variable Means via Inverse Sampling

Abstract: In this paper, we develop interval estimation methods for means of bounded random variables based on a sequential procedure such that the sampling is continued until the sample sum is no less than a prescribed threshold.


Algorithmic and combinatorial methods for enumerating the relators of a group presentation

Abstract: The main achievement of this thesis is an algorithm which given a finite group presentation and natural numbers n and k, computes all the relators of length and area up to n and k respectively. The complexity of this algorithm is better by a factor which is over-exponential than that of classical methods using van Kampen diagrams.


On $l^2$ norms of some weighted mean matrices

Abstract: We give another proof of a result of Bennett on the $l^{p}$ operator norms of some weighted mean matrices for the case $p=2$ and we also present some related results.


Almost Periodic Szeg\H{o} Cocycles with Uniformly Positive Lyapunov Exponents

Abstract: We exhibit examples of almost periodic Verblunsky coefficients for which Herman's subharmonicity argument applies and yields that the associated Lyapunov exponents are uniformly bounded away from zero.


Type II$_1$ von Neumann algebra representations of Hecke operators on Maass forms and the Ramanujan-Petersson conjectures

Abstract: Classical Hecke operators on Maass forms are unitarely equivalent, up to a commuting phase, to completely positive maps on II$_1$ factors, associated to a pair of isomorphic subfactors, and an intertwining unitary. This representation is obtained through a quantized representation of the Hecke operators. in this representation, the Hecke operators act on the Berezin's quantization, deformation algebra of the fundamental domain of $\PSL(2,\Z)$ in the upper halfplane. The Hecke operators are inheriting from the ambient, non-commutative algebra on which they act, a rich structure of matrix inequalities. Using this construction we obtain that, for every prime $p$, the essential spectrum of the classical Hecke operator $T_p$ is contained in the interval $[-2\sqrt p, 2\sqrt p]$, predicted by the Ramanujan Petersson conjectures. In particular, given an open interval containing $[-2\sqrt p, 2\sqrt p]$, there are at most a finite number of possible exceptional eigenvalues lying outside this interval. The main tool for obtaining this representation of the Hecke operators (unitarely equivalent to the classical representation, up to commuting phase) is a Schurr type, positive "square root" of the state on $\PGL(2,\Q)$, measuring the displacement of fundamental domain by translations in $\PGL(2,\Q)$. The "square root" is obtained from the matrix coefficients of the discrete series representations of $\PSL(2,\R)$ restricted to $\PGL(2, \Q)$. The methods in this paper may also be applied to any finite index, modular subgroup $\Gamma_0(p^n)$, $n\geq 1$, of $\PSL(2,\Z)$. In this case the essential norm of the Hecke operator is equal to the norm of the corresponding convolution operator on the cosets Hilbert space $\ell^2((\Gamma_0(p^n))\backslash \PSL(2,\Z[1/p])$.


Directional isoperimetric inequalities and rational homotopy invariants

Abstract: We estimate the second order linking invariants of Lipschitz maps from an n-dimensional ellipse. The estimate uses a new directionally-dependent version of the isoperimetric inequality for cycles inside the ellipse. Using this work, we prove new lower bounds for the k-dilation of maps from one ellipse to another.


Isoperimetric inequalities and rational homotopy invariants

Abstract: We estimate the linear isoperimetric constants of an n-dimensional ellipse. Using these estimates and a technique of Gromov, we estimate the Hopf and linking invariants of Lipschitz maps from ellipses to round spheres. Using these estimates, we give a lower bound for the k-dilation of degree non-zero maps between ellipses.


A natural extension for the greedy beta-transformation with three deleted digits

Abstract: We give an explicit expression for the invariant measure, absolutely continuous with respect to the Lebesgue measure, of the greedy beta-transformation with three deleted digits. We define a version of the natural extension of the transformation to obtain this expression. We get that the transformation is exact and weakly Bernoulli.


Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations

Abstract: In this work we continue the study of the Weyl asymptotics of the distribution of eigenvalues of non-self-adjoint (pseudo)differential operators with small random perturbations, by treating the case of multiplicative perturbations in arbitrary dimension. We were led to quite essential improvements of many of the probabilistic aspects.


A Systematic Study of Frame Sequence Operators and their Pseudoinverses

Abstract: In this note we investigate the operators associated with frame sequences in a Hilbert space $H$, i.e., the synthesis operator $T:\ell ^{2}(\mathbb{N}) \to H$, the analysis operator $T^{\ast}:H\to $ $% \ell ^{2}(\mathbb{N}) $ and the associated frame operator $S=TT^{\ast}$ as operators defined on (or to) the whole space rather than on subspaces. Furthermore, the projection $P$ onto the range of $T$, the projection $Q$ onto the range of $T^{\ast}$ and the Gram matrix $G=T^{\ast}T$ are investigated. For all these operators, we investigate their pseudoinverses, how they interact with each other, as well as possible classification of frame sequences with them. For a tight frame sequence, we show that some of these operators are connected in a simple way.


Moufang symmetry III. Integrability of generalized Lie equations

Abstract: Integrability of generalized Lie equations of a local analytic Moufang loop is inquired.


Strong solutions for stochastic porous media equations with jumps

Abstract: We prove global well-posedness in the strong sense for stochastic generalized porous media equations driven by square integrable martingales with stationary independent increments.


A Schwartz type algebra for the Tangent Groupoid

Abstract: We construct an algebra of smooth functions over the tangent groupoid associated to any Lie groupoid. This algebra is a field of algebras over the closed interval [0, 1] which fiber at zero is the algebra of Schwartz functions over the Lie algebroid, whereas any fiber out of zero is the convolution algebra of the initial groupoid. Our motivation comes from index theory for Lie groupoids. In fact, our construction gives an intermediate algebra between the enveloping C-algebra and the convolution algebra of compactly supported functions of the tangent groupoid; and it will allows us, in a further work, to define other analytic index morphisms as a sort of deformations.


Modeling and Optimal Control of Networks of Pipes and Canals

Abstract: This paper deals with the optimal control of systems governed by nonlinear systems of conservation laws at junctions. The applications considered range from gas compressors in pipelines to open channels management. The existence of an optimal control is proved. From the analytical point of view, these results are based on the well posedness of a suitable initial boundary value problem and on techniques for quasidifferential equations in a metric space.


Switching to nonhyperbolic cycles from codim 2 bifurcations of equilibria in ODEs

Abstract: The paper provides full algorithmic details on switching to the continuation of all possible codim 1 cycle bifurcations from generic codim 2 equilibrium bifurcation points in n-dimensional ODEs. We discuss the implementation and the performance of the algorithm in several examples, including an extended Lorenz-84 model and a laser system.


A combinatorial approach to surgery formulas in Heegaard Floer homology

Abstract: Using the combinatorial approach to Heegaard Floer homology we obtain a relatively easy formula for computation of hat Heegaard Floer homology for the three-manifold obtained by rational surgery on a knot K inside a homology sphere Y.


Moufang symmetry IV. Reductivity and hidden associativity

Abstract: It is shown how integrability of the generalized Lie equations of a local analytic Moufang loop is related to the reductivity conditions and Sagle-Yamaguti identity.


Veraverbeke's theorem at large - On the maximum of some processes with negative drift and heavy tail innovations

Abstract: Veraverbeke's (1977) theorem relates the tail of the distribution of the supremum of a random walk with negative drift to the tail of the distribution of its increments, or equivalently, the probability that a centered random walk with heavy-tail increments hits a moving linear boundary. We study similar problems for more general processes. In particular, we derive an analogue of Veraverbeke's theorem for fractional integrated ARMA models without prehistoric influence, when the innovations have regularly varying tails. Furthermore, we prove some limit theorems for the trajectory of the process, conditionally on a large maximum. Those results are obtained by using a general scheme of proof which we present in some detail and should be of value in other related problems.


Strongly-Representable Operators

Abstract: Recently in [1] a new class of maximal monotone operators has been introduced. In this note we study domain range properties as well as connections with other classes and calculus rules for these operators we called strongly-representable. While not every maximal monotone operator is strongly-representable, every maximal monotone NI operator is strongly-representable, and every strongly representable operator is locally maximal monotone, maximal monotone locally, and ANA. As a consequence the conjugate of the Fitzpatrick function of a maximal monotone operator is not necessarily a representative function.


Symplectic Calabi-Yau manifolds, minimal surfaces and the hyperbolic geometry of the conifold

Abstract: Given an SO(3)-bundle with connection, the associated two-sphere bundle carries a natural closed 2-form. Asking that this be symplectic gives a curvature inequality first considered by Reznikov. We study this inequality in the case when the base has dimension four, with three main aims. Firstly, we use this approach to construct symplectic six-manifolds with c_1=0 which are never Kahler; e.g., we produce such manifolds with b_1=0=b_3 and also with c_2.omega <0, answering questions posed by Smith-Thomas-Yau. Examples come from Riemannian geometry, via the Levi-Civita connection on Lambda^+. The underlying six-manifold is then the twistor space and often the symplectic structure tames the Eells-Salamon twistor almost complex structure. Our second aim is to exploit this to deduce new results about minimal surfaces: if a certain curvature inequality holds, it follows that the space of minimal surfaces (with fixed topological invariants) is compactifiable; the minimal surfaces must also satisfy an adjunction inequality, unifying and generalising results of Chen--Tian. One metric satisfying the curvature inequality is hyperbolic four-space H^4. Our final aim is to show that the corresponding symplectic manifold is symplectomorphic to the small resolution of the conifold xw-yz=0 in C^4. We explain how this fits into a hyperbolic description of the conifold transition, with isometries of H^4 acting symplectomorphically on the resolution and isometries of H^3 acting biholomorphically on the smoothing.


Presque $\mathbf{C}_p$-repr\'esentations et $(\phi,\Gamma)$-modules

Abstract: We associate two almost $C_p$-representations to a $(\phi,\Gamma)$-module, and we compute their dimensions and heights. As a corollary, we get a full faithfulness result for $B_e$-representations.


Cohomology of diagrams of algebras

Abstract: We consider cohomology of diagrams of algebras by Beck's approach, using comonads. We then apply this theory to computing the cohomology of $\Psi$-rings. Our main result is that there is a spectral sequence connecting the cohomology of the diagram of an algebra to the cohomology of the underlying algebra.


Poincare duality complexes in dimension four

Abstract: We describe an algebraic structure on chain complexes yielding algebraic models which classify homotopy types of Poincare duality complexes of dimension 4. Generalizing Turaev's fundamental triples of Poincare duality complexes of dimension 3, we introduce fundamental triples for Poincare duality complexes of dimension n > 2 and show that two Poincare duality complexes are orientedly homotopy equivalent if and only if their fundamental triples are isomorphic. As applications we establish a conjecture of Turaev and obtain a criterion for the existence of degree 1 maps between n-dimensional manifolds.


A Note on Walking Versus Waiting

Abstract: This mathematical recreation extends the analysis of a recent paper, asking when a traveller at a bus stop and not knowing the time of the next bus is best advised to wait or to start walking toward the destination. A detailed analysis and solution is provided for a very general class of probability distributions of bus arrival time, and the solution characterised in terms of a function analogous to the hazard rate in reliability theory. The note also considers the question of intermediate stops. It is found that the optimal strategy is not always the laziest, even when headways are not excessively long. For the common special case where one knows the (uniform) headway but not the exact timetable, it is shown that one should wait if the headway is less than the walking time (less bus travel time), and walk if the headway is more than twice this much. In between it may be better to wait or to walk, depending on one's confidence in being able to catch up to a passing bus.


Random walk on a discrete torus and random interlacements

Abstract: We investigate the relation between the local picture left by the trajectory of a simple random walk on the torus (Z/NZ)^d, d >= 3, until u N^d time steps, u > 0, and the model of random interlacements recently introduced by Sznitman. In particular, we show that for large N, the joint distribution of the local pictures in the neighborhoods of finitely many distant points left by the walk up to time u N^d converges to independent copies of the random interlacement at level u.


Curves over higher local fields

Abstract: In this work, we prove the vanishing of the two cohomological group of the higher local field. This generalize the well-known propriety of finite field and one dimensional local field. We apply this result to study the arithmetic of curve defined over higher local field.


Extension of C*-bundles

Abstract: Different (fibrewise) amalgamated products of continuous C*-bundles have been studied over the last years, one of the main question being to know when these amalgamated products are continuous C*-bundles. In order to gather these approaches in a joint framework, we first recall a few definitions from the theory of deformations of C*-algebras and we fix several notations that will be used in the sequel. Then we characterise the continuity properties of different amalgamated products of (continuous) C(X)-algebras.


Coarse embeddability into Banach spaces

Abstract: The main purposes of this paper are (1) To survey the area of coarse embeddability of metric spaces into Banach spaces, and, in particular, coarse embeddability of different Banach spaces into each other; (2) To present new results on the problems: (a) Whether coarse non-embeddability into $\ell_2$ implies presence of expander-like structures? (b) To what extent $\ell_2$ is the most difficult space to embed into?


A cascade of determinantal Calabi--Yau threefolds

Abstract: We study Kustin--Miller unprojections of Calabi--Yau threefolds. As an application we work out the geometric properties of Calabi--Yau threefolds defined as linear sections of determinantal varieties. We compute their Hodge numbers and describe the morphisms corresponding to the faces of their K\"{a}hler--Mori cone.


A Cyclic Operad in the Category of Artin Stacks and Gravitational Correlators

Abstract: We define an Artin stack which may be considered as a substitute for the non-existing (or empty) moduli space of stable two-pointed curves of genus zero. We show that this Artin stack can be viewed as the first term of a cyclic operad in the category of stacks. Applying the homology functor we obtain a linear cyclic operad. We formulate conjectures which assert that cohomology of a smooth projective variety has the structure of an algebra over this homology operad and that gravitational quantum cohomology can naturally expressed in terms of this algebra. As a test for these conjectures we show how certain well-known relations between gravitational correlators can be deduced from them.


Cohomological support loci for Abel-Prym curves

Abstract: For an Abel-Prym curve contained in a Prym variety, we determine the cohomological support loci of its twisted ideal sheaves and the dimension of its theta-dual.


On variance stabilisation by double Rao-Blackwellisation

Abstract: Population Monte Carlo has been introduced as a sequential importance sampling technique to overcome poor fit of the importance function. In this paper, we compare the performances of the original Population Monte Carlo algorithm with a modified version that eliminates the influence of the transition particle via a double Rao-Blackwellisation. This modification is shown to improve the exploration of the modes through an large simulation experiment on posterior distributions of mean mixtures of distributions.


A Characterization of Jacobians by the Existence of Picard Bundles

Abstract: Based on the Matsusaka-Ran criterion we give a criterion to characterize when a principal polarized abelian variety is a Jacobian by the existence of Picard bundles.


A Torelli theorem for curves over finite fields

Abstract: We study hyperbolic curves and their Jacobians over finite fields in the context of anabelian geometry.


Loop groups and string topology

Abstract: Survey article on loop groups and their representations, following a course of three lectures held at the summer school "algebraic groups" at the Georg-August-Universitaet zu Goettingen, June 27--July 13, 2005. We discuss loop groups, their central extensions, and positive energy representations.


Some arithmetic properties of matroidal ideals

Abstract: In this paper, we study various properties of matroidal ideals.


Heterodimensional tangencies on cycles leading to strange attractors

Abstract: In this paper, we study heterodimensional cycles of two-parameter families of 3-dimensional diffeomorphisms some element of which contains nondegenerate heterodimensional tangencies of the stable and unstable manifolds of two saddle points with different indexes, and prove that such diffeomorphisms can be well approximated by another element which has a quadratic homoclinic tangency associated to one of these saddle points. Moreover, it is shown that the tangency unfolds generically with respect to the family. This result together with some theorem in Viana, we detect strange attractors appeared arbitrarily close to the original element with the heterodimensional cycle.


Lower bound theorem for normal pseudomanifolds

Abstract: In this paper we present a self-contained combinatorial proof of the lower bound theorem for normal pseudomanifolds, including a treatment of the cases of equality in this theorem. We also discuss McMullen and Walkup's generalised lower bound conjecture for triangulated spheres in the context of the lower bound theorem. Finally, we pose a new lower bound conjecture for non-simply connected triangulated manifolds.


Fiber products of elliptic surfaces with section and associated Kummer fibrations

Abstract: We investigate Calabi--Yau three folds which are small resolutions of fiber products of elliptic surfaces with section admitting reduced fibers. We start by the classification of all fibers that can appear on such varieties. Then, we find formulas to compute the Hodge numbers of obtained three folds in terms of the types of singular fibers of the elliptic surfaces. Next we study Kummer fibrations associated to these fiber products.


Asymptotically optimal quantization schemes for Gaussian processes

Abstract: We describe quantization designs which lead to asymptotically and order optimal functional quantizers. Regular variation of the eigenvalues of the covariance operator plays a crucial role to achieve these rates. For the development of a constructive quantization scheme we rely on the knowledge of the eigenvectors of the covariance operator in order to transform the problem into a finite dimensional quantization problem of normal distributions. Furthermore we derive a high-resolution formula for the $L^2$-quantization errors of Riemann-Liouville processes.


Correspondences between modular Calabi--Yau fiber products

Abstract: We describe two ways to construct finite rational morphisms between fiber products of rational elliptic surfaces with section and some Calabi--Yau manifolds. We use them to construct correspondences between such fiber products that admit at most five singular fibers and rigid Calabi--Yau threefolds.


An algorithmic implementation of the Pi function based on a new sieve

Abstract: In this paper we propose an algorithm that correctly discards a set of numbers (from a previously defined sieve) with an interval of integers. Leopoldo's Theorem states that the remaining integer numbers will generate and count the complete list of primes of absolute value greater than 3 in the interval of interest. This algorithm avoids the problem of generating large lists of numbers, and can be used to compute (even in parallel) the prime counting function $\pi(h)$.


The 2/3 - convergence rate for the Poisson bracket

Abstract: In this paper we introduce a new method for approaching the C^0 - rigidity results for the Poisson bracket. Using this method, we provide a different proof for the lower semi-continuity under C^0 perturbations, for the uniform norm of the Poisson bracket. We find the precise rate for the modulus of the semi-continuity. This extends the previous results of Cardin-Viterbo, Zapolsky, Entov and Polterovich. Using our method, we prove a C^0 - rigidity result in the spirit of the work of Humiliere. We also discuss a general question of the C^0 - rigidity for multilinear differential operators.


Connectedness in graph limits

Abstract: We define direct sums and a corresponding notion of connectedness for graph limits. Every graph limit has a unique decomposition as a direct sum of connected components. As is well-known, graph limits may be represented by symmetric functions on a probability space; there are natural definitions of direct sums and connectedness for such functions, and there is a perfect correspondence with the corresponding properties of the graph limit. Similarly, every graph limit determines an infinite random graph, which is a.s. connected if and only if the graph limit is connected. There are also characterizations in terms of the asymptotic size of the largest component in the corresponding finite random graphs, and of minimal cuts in sequences of graphs converging to a given limit.


Moufang symmetry XI. Integrability of generalized Lie equations of continuous Moufang transformations

Abstract: Integrability of generalized Lie equations of continuous Moufang transformations is inquired.


Moufang symmetry XII. Reductivity and hidden associativity of infinitesimal Moufang transformations

Abstract: It is shown how integrability of the generalized Lie equations of continous Moufang transformatiosn is related to the reductivity conditions and Sagle-Yamaguti identity.


On the flat remainder in normal forms of families of analytic planar saddles

Abstract: We give an explicit expression for the (finitely) flat remainder after analytic normal form reduction of a family of planar saddles of diffeomorphisms or vector fields. We distinguish between a rational or irrational ratio of the moduli of the eigenvalues at the saddle for a certain value of the parameter.


Representability of Hilbert schemes and Hilbert stacks of points

Abstract: We show that the Hilbert functor of points on an arbitrary separated algebraic stack is an algebraic space. We also show the algebraicity of the Hilbert stack of points on an algebraic stack and the algebraicity of the Weil restriction of an algebraic stack along a finite flat morphism. For the latter two results, no separation assumptions are necessary.


A Weak Chevalley-Warning Theorem for Quasi-finite Fields

Abstract: There exists a function f: N -> N such that for every positive integer d, every quasi-finite field K and every projective hypersurface X of degree d and dimension at least f(d), the set X(K) is non-empty. This is a special case of a more general result about intersections of hypersurfaces of fixed degree in projective spaces of sufficiently high dimension over fields with finitely generated Galois groups.


Clones from ideals

Abstract: On an infinite base set X, every ideal of subsets of X can be associated with the clone of those operations on X which map small sets to small sets. We continue earlier investigations on the position of such clones in the clone lattice.


Pro-p groups of positive deficiency

Abstract: Let \Gamma be a finitely presentable pro-p group with a nontrivial finitely generated closed normal subgroup N of infinite index. Then def(\Gamma)\leq 1, and if def(\Gamma)=1 then \Gamma is a pro-p duality group of dimension 2, N is a free pro-p group and \Gamma/N is virtually free. In particular, if the centre of \Gamma is nontrivial and def(\Gamma)\geq 1, then def(\Gamma)=1, cd G \leq 2 and \Gamma is virtually a direct product F \times Z_p, with F a finitely generated free pro-p group.


Searching for Strange Hypergeometric Identities By Sheer Brute Force

Abstract: We describe a systematic search for all strange hypergeometric identities up to a certain complexity with sheer brute force that lead us to the discovery of two new infinite families of closed-form evaluations.


Khovanov homology and tight contact structures

Abstract: Using the relation between Khovanov homology and the Heegaard Floer homology of branched double covers, we show how Khovanov homology can be used to establish tightness of branched double covers of certain transverse knots. We give examples of several infinite families of knots whose branched covers are tight for Khovanov-homological reasons, and show that some of these branched covers are not Stein fillable.


The universal Hopf algebra associated with a Hopf-Lie-Rinehart algebra

Abstract: We introduce a notion of Hopf-Lie-Rinehart algebra and show that the universal algebra of a Hopf-Lie-Rinehart algebra acquires an ordinary Hopf algebra structure.


Quadratic equations over free groups are NP-complete

Abstract: We prove that the problems of deciding whether a quadratic equation over a free group has a solution is NP-complete.


A note on three dimensional good sets

Abstract: We show that as in the case of n- fold Cartesian product for n greater than or equal to 4, even in 3-fold Cartesian product, a related component of a good set need not be a full component.


Automatic Transversality and Orbifolds of Punctured Holomorphic Curves in Dimension Four

Abstract: We derive a numerical criterion for J-holomorphic curves in 4-dimensional symplectic cobordisms to achieve transversality without any genericity assumption. This generalizes results of Hofer-Lizan-Sikorav and Ivashkovich-Shevchishin to allow punctured curves with boundary that generally need not be somewhere injective or immersed. As an application, we combine this with the intersection theory of punctured holomorphic curves to prove that certain geometrically natural moduli spaces are globally smooth orbifolds, consisting generically of embedded curves, plus unbranched multiple covers that form isolated orbifold singularities.


Computational class field theory

Abstract: Class field theory furnishes an intrinsic description of the abelian extensions of a number field that is in many cases not of an immediate algorithmic nature. We outline the algorithms available for the explicit computation of such extensions.


A homotopical algebra of graphs related to zeta series

Abstract: The purpose of this paper is to develop a homotopical algebra for graphs, relevant to zeta series and spectra of finite graphs. More precisely, we define a Quillen model structure in a category of graphs (directed and possibly infinite, with loops and multiple arcs allowed). The weak equivalences for this model structure are the Acyclics (graph morphisms which preserve cycles). The cofibrations and fibrations for the model are determined from the class of Whiskerings (graph morphisms produced by grafting trees). Our model structure seems to fit well with the importance of acyclic directed graphs in many applications. In addition to the weak factorization systems which form this model structure, we also describe two Freyd-Kelly factorization systems based on Folding, Injecting, and Covering graph morphisms.


The open mapping theorem for regular quaternionic functions

Abstract: The basic results of a new theory of regular functions of a quaternionic variable have been recently stated, following an idea of Cullen. In this paper we prove the minimum modulus principle and the open mapping theorem for regular functions. The proofs involve some peculiar geometric properties of such functions which are of independent interest.


Generalization of One of Lie's Theorems

Abstract: We prove a generalization of one of Lie's Theorems in the context of Lie-like algebras$^{2-nd}$.


Gradient NLW on curved background in 4+1 dimensions

Abstract: We obtain a sharp local well-posedness result for the Gradient Nonlinear Wave Equation on a nonsmooth curved background. In the process we introduce variable coefficient versions of Bourgain's $X^{s,b}$ spaces, and use a trilinear multiscale wave packet decomposition in order to prove a key trilinear estimate.


On the endomorphisms of Weyl modules over affine Kac-Moody algebras at the critical level

Abstract: We present an independent short proof of the main result of arXiv:0706.3725 that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to the algebra of functions on the space of monodromy-free opers on the disc with regular singularity and residue determined by the highest weight of the Weyl module. We derive this from the results of arXiv:0712.1183 about the shift of argument subalgebras.


On low rank perturbation of matrices

Abstract: The article is devoted to different aspects of the question "What can be done with a matrix by low rank perturbation?" It is proved that one can change a geometrically simple spectrum drastically by a rank 1 permutation, but the situation is quite different if one restricts oneself to normal matrices. Also, the Jordan normal form of a perturbed matrix is considered. It is proved that with respect to rank as a distance all almost unitary matrices are near unitary.


Polygones de Newton de certaines sommes de caract\`eres et s\'eries de Poincar\'e

Abstract: In this paper, we shall precise the asymptotic behaviour of Newton polygons of $L$ functions associated to character sums, coming from some $n$ variable Laurent polynomials. In order to do this, we use the free sum on convex polytopes. This operation allows the determination of the limit of generic Newton polygons for the sum $\Delta=\Delta_1\oplus \Delta_2$ when we know the limit of generic Newton polygons for each factor. To our knowledge, these are the first results concerning the asymptotic behaviour of Newton polygons for multivariable polynomials.


PGA Tour Scores as a Gaussian Random Variable

Abstract: In this paper it is demonstrated that the scoring at each PGA Tour stroke play event can be reasonably modeled as a Gaussian random variable. All 46 stroke play events in the 2007 season are analyzed. The distributions of scores are favorably compared with a Gaussian distribution using the Kolmogorov-Smirnov test. This observation suggests performance tracking on the PGA tour should be done in terms of the z-score, calculated by subtracting the mean from the raw score and dividing by the standard deviation. This methodology measures performance relative to the field of competitors, independent of the venue, and in terms of a statistic that has quantitative meaning. Several examples of the use of this scoring methodology are provided, including a calculation of the probability that Tiger Woods will break Byron Nelson's record of eleven consecutive PGA Tour victories.