Multidimensional cellular automata and generalization of Fekete's lemma

Abstract: Fekete's lemma is a well known combinatorial result on number sequences: we extend it to functions defined on $d$-tuples of integers. As an application of the new variant, we show that nonsurjective $d$-dimensional cellular automata are characterized by loss of arbitrarily much information on finite supports, at a growth rate greater than that of the support's boundary determined by the automaton's neighbourhood index.


Landen transformations and the integration of rational functions

Abstract: We present a rational version of the classical Landen transformation for elliptic integrals. This is employed to obtain explicit closed-form expressions for a large class of integrals of even rational functions and to develop an algorithm for the numerical integration of these functions.


A simple example of a new class of Landen transformation

Abstract: The rational Landen transformation is a map on the coefficients of a rational integrand that preserves the value of the integral. This is the rational analog of the classical Landen transformations for elliptic integrals that leads to the arithmetic-geometric mean of Legendre and Gauss. We present the effect of this transformation in the simplest possible case.


Smooth representations and sheaves

Abstract: The paper is concerned with `geometrization' of smooth (i.e. with open stabilizers) representations of the automorphism group of universal domains, and with the properties of `geometric' representations of such groups. As an application, we calculate the cohomology groups of several classes of smooth representations of the automorphism group of an algebraically closed extension of infinite transcendence degree of an algebraically closed field.


Covering spaces and the Kakimizu complex

Abstract: In 1992, Osamu Kakimizu defined a complex that has become known as the Kakimizu complex of a knot. Vertices correspond to isotopy classes of minimal genus Seifert surfaces of the knot. Higher dimensional simplices correspond to collections of such classes of Seifert surfaces that admit disjoint representatives. We show that this complex is simply connected.


Asymptotic expansions for functions of the increments of certain Gaussian processes

Abstract: Let $G=\{G(x),x\ge 0\}$ be a mean zero Gaussian process with stationary increments and set $\sigma^2(|x-y|)= E(G(x)-G(y))^2$. Let $f$ be a function with $Ef^{2}(\eta)<\ff$, where $\eta=N(0,1)$. When $\sigma^2$ is regularly varying at zero and \[ \lim_{h\to 0}{h^2\over \sigma^2(h)}= 0\qquad {and}\qquad \lim_{h\to 0}{\sigma^2(h)\over h}= 0 \quad {but} \quad ({d^{2}\over ds^2}\sigma^2(s))^{j_0} \] is locally integrable for some integer $j_0\ge 1$, and satisfies some additional regularity conditions, \bea && \int_a^bf(\frac{G(x+h)-G(x)}{\sigma (h)}) dx \label{abst}\nn &&\qquad = \sum_{j=0}^{j_0} (h/\sigma(h))^{j} {E(H_{j}(\eta) f(\eta))\over\sqrt {j!}} :(G')^{j}:(I_{[a,b]}) +o({h\over\sigma (h)})^{j_0}\nn \eea in $L^2$. Here $H_j$ is the $j$-th Hermite polynomial. Also $:(G')^{j}:(I_{[a,b]})$ is a $j $-th order Wick power Gaussian chaos constructed from the Gaussian field $ G'(g) $, with covariance \[ E(G'(g)G'(\wt g)) = \int \int \rho (x-y)g(x)\wt g(y) dx dy\label{3.7bqs}, \] where $ \rho(s)={1/2}{d^{2}\over ds^2}\sigma^2(s)$.


Cohomology theories for homotopy algebras and noncommutative geometry

Abstract: This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely $A_\infty, C_\infty$ and $L_\infty$-algebras. This framework is based on noncommutative geometry as expounded by Connes and Kontsevich. The developed machinery is then used to establish a general form of Hodge decomposition of Hochschild and cyclic cohomology of $C_\infty$-algebras. This generalizes and puts in a conceptual framework previous work by Loday and Gerstenhaber-Schack.


k-Disjunctive cuts and a finite cutting plane algorithm for general mixed integer linear programs

Abstract: In this paper we give a generalization of the well known split cuts of Cook, Kannan and Schrijver to cuts which are based on multi-term disjunctions. They will be called k-disjunctive cuts. The starting point is the question what kind of cuts is needed for a finite cutting plane algorithm for general mixed integer programs. We will deal with this question in detail and derive cutting planes based on k-disjunctions related to a given cut vector. Finally we will show how a finite cutting plane algorithm can be established using these cuts in combination with Gomory mixed integer cuts.


A computation of invariants of a rational self-map

Abstract: I compute the dynamical degrees in C. Voisin's example of a rational self-map of the variety of lines on a cubic fourfold.


Potential density of rational points on the variety of lines of a cubic fourfold

Abstract: We prove the potential density of rational points on the variety of lines of a sufficiently general cubic fourfold defined over a number field, where ``sufficiently general'' means that a condition of Terasoma type is satisfied. These varieties have trivial canonical bundle and have geometric Picard group equal to $\mathbb{Z}$.


Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number

Abstract: An algebraic transformation of the DeTemple-Wang half-integer approximation to the harmonic series produces the general formula and error estimate for the Ramanujan expansion for the nth harmonic number into negative powers of the nth triangular number. We also discuss the history of the Ramanujan expansion for the nth harmonic number as well as sharp estimates of its accuracy, with complete proofs, and we compare it with other approximative formulas.


Symplectic $C_\infty$-algebras

Abstract: In this paper we show that a strongly homotopy commutative (or $C_\infty$-) algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic $C_\infty$-algebra (an $\infty$-generalisation of a commutative Frobenius algebra introduced by Kontsevich). This result relies on the algebraic Hodge decomposition of the cyclic Hochschild cohomology of a $\ci$-algebra and does not generalize to algebras over other operads.


Stable modification of relative curves

Abstract: We generalize theorems of Deligne-Mumford and de Jong on semi-stable modifications of families of proper curves. The main result states that after a generically \'etale alteration of the base any (not necessarily proper) family of multipointed curves with semi-stable generic fiber admits a minimal semi-stable modification. The latter can also be characterized by the property that its geometric fibers have no certain exceptional components. The main step of our proof is uniformization of one-dimensional extensions of valued fields. Riemann-Zariski spaces are then used to obtain the result over any integral base.


On the regularity of geodesic rays associated to test configurations

Abstract: Geodesic rays of class C^{1,1} are constructed for any test configuration of a positive line bundle L on X using resolution of singularities. The construction reduces to finding a subsolution of the corresponding Monge-Ampere equation. Geometrically, this is accomplished by the use a positive line bundle on the resolution which is trivial outside of the exceptional divisor.


On the quantization of conjugacy classes

Abstract: Let G be a compact, simple, simply connected Lie group. A theorem of Freed-Hopkins-Teleman identifies the level k fusion ring R_k(G) of G with the twisted equivariant K-homology at level k+h, where h is the dual Coxeter number. In this paper, we review this result using the language of Dixmier-Douady bundles. We show that the additive generators of the group R_k(G) are obtained as K-homology push-forwards of the fundamental classes of conjugacy classes in G.


Mixed States Markov Random Fields with Symbolic Labels and Multidimensional Real Values

Abstract: New theoretical results are presented here on the recently introduced model called mixed states MRF. Such models were introduced in the context of image motion analysis and are useful to represent information which can take both discrete values accounting for symbolic states, and real values corresponding to continuous measurements. In particular, results are given when the global energy for the Gibbs formulation expressing the mixed states model, can be decomposed into one term accounting for the discrete part of the model, and a second term related to the continuous part. This decomposition theorem permits to define conditional mixed states models in a very simple way.


Regularly varying multivariate time series

Abstract: A multivariate, stationary time series is said to be jointly regularly varying if all its finite-dimensional distributions are multivariate regularly varying. This property is shown to be equivalent to weak convergence of the conditional distribution of the rescaled series given that, at a fixed time instant, its distance to the origin exceeds a threshold tending to infinity. The limit object, called the tail process, admits a decomposition in independent radial and angular components. Under an appropriate mixing condition, this tail process allows for a concise and explicit description of the limit of a sequence of point processes recording both the times and the positions of the time series when it is far away from the origin. The theory is applied to multivariate moving averages of finite order with random coefficient matrices.


Symplectic $A_\infty$-algebras and string topology operations

Abstract: In this paper we establish the existence of certain structures on the ordinary and equivariant homology of the free loop space on a manifold or, more generally, a formal Poincar\'e duality space. These structures; namely the loop product, the loop bracket and the string bracket, were introduced and studied by Chas and Sullivan under the general heading `string topology'. Our method is based on obstruction theory for $C_\infty$-algebras and rational homotopy theory. The resulting string topology operations are manifestly homotopy invariant.


Reflection Groups and Polytopes over Finite Fields, III

Abstract: When the standard representation of a crystallographic Coxeter group is reduced modulo an odd prime p, one obtains a finite group G^p acting on some orthogonal space over Z_p . If the Coxeter group has a string diagram, then G^p will often be the automorphism group of a finite abstract regular polytope. In parts I and II we established the basics of this construction and enumerated the polytopes associated to groups of rank at most 4, as well as all groups of spherical or Euclidean type. Here we extend the range of our earlier criteria for the polytopality of G^p . Building on this we investigate the class of 3-infinity groups of general rank, and then complete a survey of those locally toroidal polytopes which can be described by our construction.


Erlangen Program at Large--2: Inventing a wheel. The parabolic one

Abstract: We discuss parabolic versions of Euler's identity e^{it}=cos t + i sin t. A purely algebraic approach based on dual numbers is known to produce a very trivial relation e^{pt} = 1+pt. Therefore we use a geometric setup of parabolic rotations to recover the corresponding non-trivial algebraic framework. Our main tool is Moebius transformations which turn out to be closely related to induced representations of the group SL(2,R). Keywords: complex numbers, dual numbers, double numbers, linear algebra, invariant, computer algebra, GiNaC


Tetrahedron equations and nilpotent subalgebras of U_q(sl_n)

Abstract: A relation between q-oscillator R-matrix of the tetrahedron equation and decompositions of Poinkare-Birkhoff-Witt type bases for nilpotent subalgebras of U_q(sl_n) is observed.


Some gradient estimates for a diffusion equation on Riemannian manifolds

Abstract: In this note we present some gradient estimates for the diffusion equation $\partial_t u=\Delta u-\nabla \phi \cdot \nabla u $ on Riemannian manifolds, where $\phi $ is a C^2 function, which generalize estimates of R. Hamilton's and Qi S. Zhang's on the heat equation.


Multiplicative Order of Gauss Periods

Abstract: We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of J. von zur Gathen and I. E. Shparlinski.


The passage time distribution for a birth-and-death chain: Strong stationary duality gives a first stochastic proof

Abstract: A well-known theorem usually attributed to Keilson states that, for an irreducible continuous-time birth-and-death chain on the nonnegative integers and any d, the passage time from state 0 to state d is distributed as a sum of d independent exponential random variables. Until now, no probabilistic proof of the theorem has been known. In this paper we use the theory of strong stationary duality to give a stochastic proof of a similar result for discrete-time birth-and-death chains and geometric random variables, and the continuous-time result (which can also be given a direct stochastic proof) then follows immediately. In both cases we link the parameters of the distributions to eigenvalue information about the chain. We also discuss how the continuous-time result leads to a proof of the Ray-Knight theorem. Intimately related to the passage-time theorem is a theorem of Fill that any fastest strong stationary time T for an ergodic birth-and-death chain on {0, >..., d} in continuous time with generator G, started in state 0, is distributed as a sum of d independent exponential random variables whose rate parameters are the nonzero eigenvalues of the negative of G. Our approach yields the first (sample-path) construction of such a T for which individual such exponentials summing to T can be explicitly identified.


Tubular Neighborhoods of Nodal Sets and Diophantine Approximation

Abstract: We give upper and lower bounds on the volume of a tubular neighborhood of the nodal set of an eigenfunction of the Laplacian on a real analytic closed Riemannian manifold M. As an application we consider the question of approximating points on M by nodal sets, and explore analogy with approximation by rational numbers.


Clifford's theorem for coherent systems

Abstract: Final version to appear in Archiv der Mathematik.


A construction of noncontractible simply connected cell-like two dimensional Peano continua

Abstract: Using the topologist sine curve we present a new functorial construction of cone-like spaces, starting in the category of all path-connected topological spaces with a base point and continuous maps, and ending in the subcategory of all simply connected spaces. If one starts by a noncontractible n-dimensional Peano continuum for any n>0, then our construction yields a simply connected noncontractible (n + 1)-dimensional cell-like Peano continuum. In particular, starting with the circle $\mathbb{S}^1$, one gets a noncontractible simply connected cell-like 2-dimensional Peano continuum.


A universal property for the Jiang-Su algebra

Abstract: We prove that the infinite tensor power of a unital separable C*-algebra absorbs the Jiang-Su algebra Z tensorially if and only if it contains, unitally, a subhomogeneous algebra without characters. This yields a succinct universal property for Z in a category so large that there are no unital separable C*-algebras without characters known to lie outside it. This category moreover contains the vast majority of our stock-in-trade separable amenable C*-algebras, and is closed under passage to separable superalgebras and quotients, and hence to unital tensor products, unital direct limits, and crossed products by countable discrete groups. One consequence of our main result is that strongly self-absorbing ASH algebras are Z-stable, and therefore satisfy the hypotheses of a recent classification theorem of W. Winter. One concludes that Z is the only projectionless strongly self-absorbing ASH algebra, completing the classification of strongly self-absorbing ASH algebras.


Families of dendrograms

Abstract: A conceptual framework for cluster analysis from the viewpoint of p-adic geometry is introduced by describing the space of all dendrograms for n datapoints and relating it to the moduli space of p-adic Riemannian spheres with punctures using a method recently applied by Murtagh (2004b). This method embeds a dendrogram as a subtree into the Bruhat-Tits tree associated to the p-adic numbers, and goes back to Cornelissen et al. (2001) in p-adic geometry. After explaining the definitions, the concept of classifiers is discussed in the context of moduli spaces, and upper bounds for the number of hidden vertices in dendrograms are given.


1-join composition for $\alpha$-critical graphs

Abstract: Given two graphs G and H its 1-{\it join} is the graph obtained by taking the disjoint union of G and H and adding all the edges between a nonempty subset of vertices of G and a nonempty subset of vertices of H. In general, composition operations of graphs has played a fundamental role in some structural results of graph theory and in particular the 1-join composition has played an important role in decomposition theorems of several class of graphs such as the claw-free graphs, the bull-free graphs, the perfect graphs, etc. A graph G is called {\it $\alpha$-critical} if $\alpha(G\setminus e)> \alpha(G)$ for all the edges e of G, where $\alpha(G)$, the {\it stability number} of G, is equal to the maximum cardinality of a stable set of G, and a set of vertices M of G is {\it stable} if no two vertices in M are adjacent. The study $\alpha$-critical graphs is important, for instance a complete description of $\alpha$-critical graphs would yield a good characterization of the stability number of G. In this paper we give necessary and sufficient conditions that G and H must satisfy in order to its 1-join will be an $\alpha$-critical graph. Therefore we get a very useful way to construct basic $\alpha$-critical graphs using the 1-join of graphs.


On some properties of $\sigma(N)$

Abstract: We show asymptotic upper and lower bounds for the greatest common divisor of N and $\sigma(N)$. We also show that there are infinitely many integers N with fairly large g.c.d. of N and $\sigma(N)$.


Rank-based inference for bivariate extreme-value copulas

Abstract: Consider a continuous random pair $(X,Y)$ whose dependence is characterized by an extreme-value copula with Pickands dependence function $A$. When the marginal distributions of $X$ and $Y$ are known, several consistent estimators of $A$ are available. Most of them are variants of the estimators due to Pickands [Bull. Inst. Internat. Statist. 49 (1981) 859--878] and Cap\'{e}ra\`{a}, Foug\`{e}res and Genest [Biometrika 84 (1997) 567--577]. In this paper, rank-based versions of these estimators are proposed for the more common case where the margins of $X$ and $Y$ are unknown. Results on the limit behavior of a class of weighted bivariate empirical processes are used to show the consistency and asymptotic normality of these rank-based estimators. Their finite- and large-sample performance is then compared to that of their known-margin analogues, as well as with endpoint-corrected versions thereof. Explicit formulas and consistent estimates for their asymptotic variances are also given.


On the ternary Goldbach problem with primes in arithmetic progressions of a common module

Abstract: For A,epsilon>0 and any sufficiently large odd n we show that for almost all k up to n^{1/5-epsilon} there exists a representation n=p1+p2+p3 with primes in residue classes b1,b2,b3 mod k for almost all admissible triplets b1,b2,b3 of reduced residues mod k.


A deterministic version of Pollard's p-1 algorithm

Abstract: In this article we present applications of smooth numbers to the unconditional derandomization of some well-known integer factoring algorithms. We begin with Pollard's $p-1$ algorithm, which finds in random polynomial time the prime divisors $p$ of an integer $n$ such that $p-1$ is smooth. We show that these prime factors can be recovered in deterministic polynomial time. We further generalize this result to give a partial derandomization of the $k$-th cyclotomic method of factoring ($k\ge 2$) devised by Bach and Shallit. We also investigate reductions of factoring to computing Euler's totient function $\phi$. We point out some explicit sets of integers $n$ that are completely factorable in deterministic polynomial time given $\phi(n)$. These sets consist, roughly speaking, of products of primes $p$ satisfying, with the exception of at most two, certain conditions somewhat weaker than the smoothness of $p-1$. Finally, we prove that $O(\ln n)$ oracle queries for values of $\phi$ are sufficient to completely factor any integer $n$ in less than $\exp\Bigl((1+o(1))(\ln n)^{{1/3}} (\ln\ln n)^{{2/3}}\Bigr)$ deterministic time.


Modeling of Transitional Channel Flow Using Balanced Proper Orthogonal Decomposition

Abstract: We study reduced-order models of three-dimensional perturbations in linearized channel flow using balanced proper orthogonal decomposition (BPOD). The models are obtained from three-dimensional simulations in physical space as opposed to the traditional single-wavenumber approach, and are therefore better able to capture the effects of localized disturbances or localized actuators. In order to assess the performance of the models, we consider the impulse response and frequency response, and variation of the Reynolds number as a model parameter. We show that the BPOD procedure yields models that capture the transient growth well at a low order, whereas standard POD does not capture the growth unless a considerably larger number of modes is included, and even then can be inaccurate. In the case of a localized actuator, we show that POD modes which are not energetically significant can be very important for capturing the energy growth. In addition, a comparison of the subspaces resulting from the two methods suggests that the use of a non-orthogonal projection with adjoint modes is most likely the main reason for the superior performance of BPOD. We also demonstrate that for single-wavenumber perturbations, low-order BPOD models reproduce the dominant eigenvalues of the full system better than POD models of the same order. These features indicate that the simple, yet accurate BPOD models are a good candidate for developing model-based controllers for channel flow.


The Hochschild cohomology of a Poincar\'e algebra

Abstract: In this note, we define the notion of a cactus set, and show that its geometric realization is naturally an algebra over Voronov's cactus operad, which is equivalent to the framed 2-dimensional little disks operad $\mathcal{D}_2$. Using this, we show that the Hochschild cohomology of a Poincar\'e algebra A is an algebra over (the chain complexes of) $\mathcal{D}_2$.


Characterizations of probability distributions via bivariate regression of record values

Abstract: Bairamov et al. (Aust N Z J Stat 47:543-547, 2005) characterize the exponential distribution in terms of the regression of a function of a record value with its adjacent record values as covariates. We extend these results to the case of non-adjacent covariates. We also consider a more general setting involving monotone transformations. As special cases, we present characterizations involving weighted arithmetic, geometric, and harmonic means.


Vertex operator algebras associated to certain admissible modules for affine Lie algebras of type A

Abstract: Let $L(-{1/2}(l+1),0)$ be the simple vertex operator algebra associated to an affine Lie algebra of type $A_{l}^{(1)}$ with the lowest admissible half-integer level $-{1/2}(l+1)$, for even l. We study the category of weak modules for that vertex operator algebra which are in category $\cal{O}$ as modules for the associated affine Lie algebra. We classify irreducible objects in that category and prove semisimplicity of that category.


Splicing and the SL(2,C) Casson invariant

Abstract: We establish a formula for the SL(2,C) Casson invariant of spliced sums of homology spheres along knots. Along the way, we show that the SL(2,C) Casson invariant vanishes for spliced sums along knots in the 3-sphere.


A landing theorem for dynamic rays of geometrically finite entire functions

Abstract: A transcendental entire function f is called geometrically finite if the intersection of the set of singular values with the Fatou set is compact and the intersection of the postsingular set with the Julia set is finite. (In particular, this includes all entire functions with finite postsingular set.) If f is geometrically finite, then the Fatou set of f is either empty or consists of the basins of attraction of finitely many attracting or parabolic cycles. Let z_0 be a repelling or parabolic periodic point of such a map f. We show that, if f has finite order, then there exists an injective curve consisting of escaping points of f that connects z_0 to infinity. (This curve is called a dynamic ray.) In fact, the assumption of finite order can be weakened considerably; for example, it is sufficient to assume that f can be written as a finite composition of finite-order functions.


Non-annulation effective et positivit\'e locale des fibr\'es en droites amples adjoints

Abstract: We prove that Seshadri constants of some ample divisors are bigger than 1 on smooth threefolds whose anticanonical bundle is nef or on Fano varieties of small coindice. The main tools are (some known cases of) the Kawamata's effective non-vanishing conjecture and the adjunction theory. We prove the non-vanishing conjecture in dimension 3 in the case of line bundles of "high" volume using Kawamata's subadjunction formula.


Resolution of symplectic cyclic orbifold singularities

Abstract: In this paper we present a method to obtain resolutions of symplectic orbifolds arising from symplectic reduction of a Hamiltonian S^1-manifold at a regular value. As an application, we show that all isolated cyclic singularities of a symplectic orbifold admit a resolution and that pre-quantisations of symplectic orbifolds are symplectically fillable by a smooth manifold.


Test configurations and Geodesic rays

Abstract: For smooth test configurations, there always exist C^{1,1} geodesic rays in Kahler metric space parallel to the algebraic ray. The $\yen$ invariant agrees with Futaki invariant, at least under nice assumptions. Explicit examples in Toric cases are calculated. On simple test configurations, Donaldson's correspondence between HCMA solution and holomorphic disc family is extended.


Density theorems for bipartite graphs and related Ramsey-type results

Abstract: In this paper, we present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density. Our results imply several new bounds for classical problems in graph Ramsey theory and improve and generalize earlier results of various researchers. The proofs combine probabilistic arguments with some combinatorial ideas. In addition, these techniques can be used to study properties of graphs with a forbidden induced subgraph, edge intersection patterns in topological graphs, and to obtain several other Ramsey-type statements.


Finite vertex algebras and nilpotence

Abstract: I show that simple finite vertex algebras are commutative, and that the Lie conformal algebra structure underlying a reduced (i.e., without nilpotent elements) finite vertex algebra is nilpotent.


On Hadamard matrices at roots of unity

Abstract: This paper has been withdrawn by the authors, the main result being known since Lam and Leung, J. Algebra 2000.


Restriction of characters and products of characters

Abstract: Let G be a finite p-group, for some prime p, and $\psi, \theta \in \Irr(G)$ be irreducible complex characters of G. It has been proved that if, in addition, $\psi,\theta$ are faithful characters, then the product $\psi\theta$ is a multiple of an irreducible or it is the nontrivial linear combination of at least $\frac{p+1}{2}$ distinct irreducible characters of G. We show that if we do not require the characters to be faithful, then given any integer k>0, we can always find a p-group G and irreducible characters $\Psi$ and $\Theta$ such that $\Psi\Theta$ is the nontrivial combination of exactly k distinct irreducible characters. We do this by translating examples of decompositions of restrictions of characters into decompositions of products of characters.


A short Brownian motion proof of the Riemann hypothesis

Abstract: We give a short probabilistic (a Brownian motion) proof of the Riemann hypothesis based on some surprising, unexpected and deep algebraic conjecture (MAC in short) concerning the relation between the Riemann zeta $\xi$ and a trivial zeta $\zeta_{t}$. That algebraic conjecture was firstly discovered and formulated in [MA]


Ascent of module structures, vanishing of Ext, and extended modules

Abstract: Let $(R,\m)$ and $(S,\n)$ be commutative Noetherian local rings, and let $\phi:R\to S$ be a flat local homomorphism such that $\m S = \n$ and the induced map on residue fields $R/\m \to S/\n$ is an isomorphism. Given a finitely generated $R$-module $M$, we show that $M$ has an $S$-module structure compatible with the given $R$-module structure if and only if $\Ext^i_R(S,M)=0$ for each $i\ge 1$. We say that an $S$-module $N$ is {\it extended} if there is a finitely generated $R$-module $M$ such that $N\cong S\otimes_RM$. Given a short exact sequence $0 \to N_1\to N \to N_2\to 0$ of finitely generated $S$-modules, with two of the three modules $N_1,N,N_2$ extended, we obtain conditions forcing the third module to be extended. We show that every finitely generated module over the Henselization of $R$ is a direct summand of an extended module, but that the analogous result fails for the $\m$-adic completion.


Noncommutative geometry and lower dimensional volumes in Riemannian geometry

Abstract: In this paper we explain how to define "lower dimensional'' volumes of any compact Riemannian manifold as the integrals of local Riemannian invariants. For instance we give sense to the area and the length of such a manifold in any dimension. Our reasoning is motivated by an idea of Connes and involves in an essential way noncommutative geometry and the analysis of Dirac operators on spin manifolds. However, the ultimate definitions of the lower dimensional volumes don't involve noncommutative geometry or spin structures at all.


Uniform Uncertainty Principle and signal recovery via Regularized Orthogonal Matching Pursuit

Abstract: This paper seeks to bridge the two major algorithmic approaches to sparse signal recovery from an incomplete set of linear measurements -- L_1-minimization methods and iterative methods (Matching Pursuits). We find a simple regularized version of the Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and the strong uniform guarantees of the L_1-minimization. Our algorithm ROMP reconstructs a sparse signal in a number of iterations linear in the sparsity (in practice even logarithmic), and the reconstruction is exact provided the linear measurements satisfy the Uniform Uncertainty Principle.


Symbolic Models for Nonlinear Control Systems: Alternating Approximate Bisimulations

Abstract: Symbolic models are abstract descriptions of continuous systems in which symbols represent aggregates of continuous states. In the last few years there has been a growing interest in the use of symbolic models as a tool for mitigating complexity in control design. In fact, symbolic models enable the use of well known algorithms in the context of supervisory control and algorithmic game theory, for controller synthesis. Since the 1990's many researchers faced the problem of identifying classes of dynamical and control systems that admit symbolic models. In this paper we make a further progress along this research line by focusing on control systems affected by disturbances. Our main contribution is to show that incrementally globally asymptotically stable nonlinear control systems with disturbances admit symbolic models. When specializing these results to linear systems, we show that these symbolic models can be easily constructed.


Sampling Lissajous and Fourier knots

Abstract: A Lissajous knot is one that can be parameterized by a single cosine function in each coordinate. Lissajous knots are highly symmetric, and for this reason, not all knots are Lissajous. We prove several theorems which allow us to place bounds on the number of Lissajous knot types with given frequencies and to efficiently sample all possible Lissajous knots with a given set of frequencies. In particular, we systematically tabulate all Lissajous knots with small frequencies and as a result substantially enlarge the tables of known Lissajous knots. A Fourier (i, j, k) knot is similar to a Lissajous knot except that each coordinate is now described by a finite sum of i, j, and k cosine functions respectively. According to Lamm, every knot is a Fourier-(1,1,k) knot for some k. By randomly searching the set of Fourier-(1,1,2) knots we find that all 2-bridge knots up to 14 crossings are either Lissajous or Fourier-(1,1,2) knots. We show that all twist knots are Fourier-(1,1,2) knots and give evidence suggesting that all torus knots are Fourier-(1,1,2) knots. As a result of our computer search, several knots with relatively small crossing numbers are identified as potential counterexamples to interesting conjectures.


A Batalin-Vilkovisky Algebra structure on the Hochschild Cohomology of Truncated Polynomials

Abstract: The main result of this paper is to calculate the Batalin-Vilkovisky structure of $HH^*(C^*(\mathbf{K}P^n;R);C^*(\mathbf{K}P^n;R))$ for $ \mathbf{K}=\mathbb{C}$ and $\mathbb{H}$, and $R=\mathbb{Z}$ and any field; and shows that in the special case when $M=\mathbb{C}P^1=S^2$, and $R=\mathbb{Z}$, this structure can not be identified with the BV-structure of $\mathbb{H}_*(LS^2;\mathbb{Z})$ computed by Luc Memichi in \cite{menichi2}. However, the induced Gerstenhaber structures are still identified in this case. Moreover, according to a recent work of Y.Felix and J.Thomas \cite{felix--thomas}, the main result of the present paper eventually calculates the BV-structure of the rational loop homology, $\mathbb{H}_*(L\mathbb{C}P^n;\mathbb{Q})$ and $\mathbb{H}_*(L\mathbb{H}P^n;\mathbb{Q})$, of projective spaces.


Ergodic BSDEs and Optimal Ergodic Control in Banach Spaces

Abstract: In this paper we introduce a new kind of Backward Stochastic Differential Equations, called ergodic BSDEs, which arise naturally in the study of optimal ergodic control. We study the existence, uniqueness and regularity of solution to ergodic BSDEs. Then we apply these results to the optimal ergodic control of a Banach valued stochastic state equation. We also establish the link between the ergodic BSDEs and the associated Hamilton-Jacobi-Bellman equation. Applications are given to ergodic control of stochastic partial differential equations.


Orbit inequivalent actions of non-amenable groups

Abstract: Consider two free measure preserving group actions $\Gamma \actson (X, \mu), \Delta \actson (X, \mu)$, and a measure preserving action $\Delta \actson^a (Z, \nu)$ where $(X, \mu), (Z, \nu)$ are standard probability spaces. We show how to construct free measure preserving actions $\Gamma \actson^c (Y, m)$, $\Delta \actson^d (Y, m)$ on a standard probability space such that $E_{\Delta}^d \subset E_{\Gamma}^c$ and $d$ has $a$ as a factor. This generalizes the standard notion of co-induction of actions of groups from actions of subgroups. We then use this construction to show that if $\Gamma$ is a countable non-amenable group, then $\Gamma$ admits continuum many orbit inequivalent free, measure preserving, ergodic actions on a standard probability space.


Real embeddings, eta invariant and Chern-Simons current

Abstract: We present an alternate proof of the Bismut-Zhang localization formula for $\eta$-invariants without using the analytic techniques developed by Bismut-Lebeau. A Riemann-Roch property for Chern-Simons currents, which is of independent interest, is established in due course.


Energy of harmonic functions and Gromov's proof of Stallings' theorem

Abstract: We provide the details for Gromov's proof of Stallings' theorem on groups with infinitely many ends using harmonic functions. The main technical result of the paper is a compactness theorem for a certain family of harmonic functions.


Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property

Abstract: In this paper we set up a representation theorem for tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property in terms of Ky Fan norms. Examples of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property include unitarily invariant norms on finite factors (type ${\rm II}\sb 1$ factors and $M_n(\cc)$) and symmetric gauge norms on $L^\infty[0,1]$ and $\cc^n$. As the first application, we obtain that the class of unitarily invariant norms on a type ${\rm II}\sb 1$ factor coincides with the class of symmetric gauge norms on $L^\infty[0,1]$ and von Neumann's classical result \cite{vN} on unitarily invariant norms on $M_n(\cc)$. As the second application, Ky Fan's dominance theorem \cite{Fan} is obtained for finite von Neumann algebras satisfying the weak Dixmier property. As the third application, some classical results in non-commutative $L^p$-theory (e.g., non-commutative H$\ddot{\text{o}}$lder's inequality, duality and reflexivity of non-commutative $L^p$-spaces) are obtained for general unitarily invariant norms on finite factors. We also investigate the extreme points of $\NN(\M)$, the convex compact set (in the pointwise weak topology) of normalized unitarily invariant norms (the norm of the identity operator is 1) on a finite factor $\M$. We obtain all extreme points of $\NN(M_2(\cc))$ and many extreme points of $\NN(M_n(\cc))$ ($n\geq 3$). For a type ${\rm II}\sb 1$ factor $\M$, we prove that if $t$ ($0\leq t\leq 1$) is a rational number then the Ky Fan $t$-th norm is an extreme point of $\NN(\M)$.


Unitarily invariant norms related to factors

Abstract: Let $\M$ be a semi-finite factor and let $\J(\M)$ be the set of operators $T$ in $\M$ such that $T=ETE$ for some finite projection $E$. In this paper we obtain a representation theorem for unitarily invariant norms on $\J(\M)$ in terms of Ky Fan norms. As an application, we prove that the class of unitarily invariant norms on $\J(\M)$ coincides with the class of symmetric gauge norms on a classical abelian algebra, which generalizes von Neumann's classical result \cite{vN} on unitarily invariant norms on $M_n(\cc)$. As another application, Ky Fan's dominance theorem \cite{Fan} is obtained for semi-finite factors. Some classical results in non-commutative $L^p$-theory (e.g., non-commutative H$\ddot{\text{o}}$lder's inequality, duality and reflexivity of non-commutative $L^p$-spaces) are extended to general unitarily invariant norms related to semi-finite factors. We also prove that up to a scale the operator norm is the unique unitarily invariant norm associated to a type ${\rm III}$ factor.


Importance Tempering

Abstract: Simulated tempering (ST) is an established Markov chain Monte Carlo (MCMC) method for sampling from a multimodal density $\pi(\theta)$. Typically, ST involves introducing an auxiliary variable $k$ taking values in a finite subset of $[0,1]$ and indexing a set of tempered distributions, say $\pi_k(\theta) \propto \pi(\theta)^k$. In this case, small values of $k$ encourage better mixing, but samples from $\pi$ are only obtained when the joint chain for $(\theta,k)$ reaches $k=1$. However, the entire chain can be used to estimate expectations under $\pi$ of functions of interest, provided that importance sampling (IS) weights are calculated. Unfortunately this method, which we call importance tempering (IT), can disappoint. This is partly because the most immediately obvious implementation is na\"ive and can lead to high variance estimators. We derive a new optimal method for combining multiple IS estimators and prove that the resulting estimator has a highly desirable property related to the notion of effective sample size. We briefly report on the success of the optimal combination in two modelling scenarios requiring reversible-jump MCMC, where the na\"ive approach fails.


On the Brauer groups of quasilocal fields and the norm groups of their finite Galois extensions

Abstract: This paper shows that divisible abelian torsion groups are realizable as Brauer groups of quasilocal fields. It describes the isomorphism classes of Brauer groups of primarily quasilocal fields and solves the analogous problem concerning the reduced components of the Brauer groups of two basic types of Henselian valued absolutely stable fields. For a quasilocal field E and a finite separable extension R/E, we find two sufficient conditions for validity of the norm group equality $N(R/E) = N(R_{0}/E)$, where R_{0} is the maximal abelian extension of E in R. This is used for deriving information on the arising specific relations between Galois groups and norm groups of finite Galois extensions of E.


Jet Geometrical Objects Produced by Linear ODEs Systems and Superior Order ODEs

Abstract: The aim of this paper is to construct a Riemann-Lagrange geometry on 1-jet spaces, in the sense of d-connections, d-torsions, d-curvatures, electromagnetic d-field and geometric electromagnetic Yang-Mills energy, starting from a given linear ODEs system or a given superior order ODE. The case of a non-homogenous linear ODE of superior order is disscused.


Rubbling and Optimal Rubbling of Graphs

Abstract: A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move one pebble is removed at vertices v and w adjacent to a vertex u and an extra pebble is added at vertex u. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using rubbling moves. The rubbling number of a graph is the smallest number m needed to guarantee that any vertex is reachable from any pebble distribution of m pebbles. The optimal rubbling number is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. We determine the rubbling and optimal rubbling number of some families of graphs including cycles.


Congruence obstructions to pseudomodularity of Fricke groups

Abstract: A pseudomodular group is a finite coarea nonarithmetic Fuchsian group whose cusp set is exactly $\mathbb{P}^1(\mathbb{Q})$. Long and Reid constructed finitely many of these by considering Fricke groups, i.e., those that uniformize one-cusped tori. We prove that a zonal Fricke group with rational cusps is pseudomodular if and only if its cusp set is dense in the finite adeles of $\mathbb{Q}$. We then deduce that infinitely many such Fricke groups are not pseudomodular.


Traces on pseudodifferential operators and sums of commutators

Abstract: The aim of this paper is to show that various known characterizations of traces on classical pseudodifferentials operators (PsiDOs) can actually be obtained by very elementary considerations on PsiDOs, using only basic properties of these operators. Thereby, we give a unified treatment of the determinations of the space of traces (i) on PsiDOs of noninteger orders or of regular parity-class, (ii) on integer order PsiDOs, (iii) on nonpositive order PsiDOs in dimension greather than or equal to 2, and (iv) on nonpositive order PsiDOs in dimension 1.


An Explicit Construction of Type A Demazure Atoms

Abstract: Demazure characters of type A, which are equivalent to key polynomials, have been decomposed by Lascoux and Sch\"{u}tzenberger into standard bases. We prove that the resulting polynomials, which we call Demazure atoms, can be obtained from a certain specialization of nonsymmetric Macdonald polynomials. This combinatorial interpretation for Demazure atoms accelerates the computation of the right key associated to a semi-standard Young tableau. Utilizing a related construction, we provide a new combinatorial description for the key polynomials.


Structure and randomness in combinatorics

Abstract: Combinatorics, like computer science, often has to deal with large objects of unspecified (or unusable) structure. One powerful way to deal with such an arbitrary object is to decompose it into more usable components. In particular, it has proven profitable to decompose such objects into a \emph{structured} component, a \emph{pseudo-random} component, and a \emph{small} component (i.e. an error term); in many cases it is the structured component which then dominates. We illustrate this philosophy in a number of model cases.


A convexity property of expectations under exponential weights

Abstract: Take a random variable X with some finite exponential moments. Define an exponentially weighted expectation by E^t(f) = E(e^{tX}f)/E(e^{tX}) for admissible values of the parameter t. Denote the weighted expectation of X itself by r(t) = E^t(X), with inverse function t(r). We prove that for a convex function f the expectation E^{t(r)}(f) is a convex function of the parameter r. Along the way we develop correlation inequalities for convex functions. Motivation for this result comes from equilibrium investigations of some stochastic interacting systems with stationary product distributions. In particular, convexity of the hydrodynamic flux function follows in some cases.


Sums of the error term function in the mean square for $\zeta(s)$

Abstract: Sums of the form $\sum_{n\le x}E^k(n) (k\in{\bf N}$ fixed) are investigated, where $$ E(T) = \int_0^T|\zeta(1/2+it)|^2 dt - T\Bigl(\log {T\over2\pi} + 2\gamma -1\Bigr)$$ is the error term in the mean square formula for $|\zeta(1/2+it)|$. The emphasis is on the case k=1, which is more difficult than the corresponding sum for the divisor problem. The analysis requires bounds for the irrationality measure of ${\rm e}^{2\pi m}$ and for the partial quotients in its continued fraction expansion.


Effective Iitaka fibrations

Abstract: We show that the M-canonical map of an n-dimensional complex projective manifold X of Kodaira dimension two is birational to an Iitaka fibration for a computable positive integer M. M depends on the index b of a general fibre F of the Iitaka fibration and on the Betti number of the canonical covering of F, In particular, M is a universal constant if the dimension n is smaller than or equal to 4.


On deformations of maps and curve singularities

Abstract: We study several deformation functors associated to the normalization of a reduced curve singularity $(X,0) \subset (\c^n,0)$. The main new results are explicit formulas, in terms of classical invariants of (X,0), for the cotangent cohomology groups $T^i, i = 0,1,2,$ of these functors. Thus we obtain precise statements about smoothness and dimension of the corresponding local moduli spaces. We apply the results to obtain explicit formulas resp. estimates for the $\hoa{A}_e$-codimension of a parametrized curve singularity, where $\hoa{A}_e$ denotes the Mather-Wall group of left-right equivalence.


Neumann Heat kernel monotonicity

Abstract: We prove that the diagonal of the transition probabilities for the d-dimensional Bessel processes on (0, 1], reflected at 1, which we denote by $p_R^N(t, r,r)$, is an increasing function of r for d>2 and that this is false for d=2.


Volume and homology of one-cusped hyperbolic 3-manifolds

Abstract: Let M be a complete, finite-volume, orientable hyperbolic manifold having exactly one cusp. If we assume that pi_1(M) has no subgroup isomorphic to a genus-2 surface group, and that either (a) H_1(M;Z_p) has dimension at least 5 for some prime p, or (b) H_1(M;Z_2) has dimension at least 4, and the subspace of H^2(M;Z_2) spanned by the image of the cup product has dimension at most 1, then vol M > 5.06 If we assume that H_1(M;Z_2) has dimension at least 7, and that the compact core of M does not contain a genus-2 closed incompressible surface, then vol M > 5.06.


On elements of prime order in the plane Cremona group over a perfect field

Abstract: We show that the plane Cremona group over a perfect field $k$ of characteristic $p \ge 0$ contains an element of prime order $\ell\ge 7$ not equal to $p$ if and only if there exists a 2-dimensional algebraic torus $T$ over $k$ such that $T(k)$ contains an element of order $\ell$. If $p = 0$ and $k$ does not contain a primitive $\ell$-th root of unity, we show that there are no elements of prime order $\ell > 7$ in $\Cr_2(k)$ and all elements of order 7 are conjugate.


Cohomological characterizations of projective spaces and hyperquadrics

Abstract: We confirm Beauville's conjecture that claims that if the p-th exterior power of the tangent bundle of a smooth projective variety contains the p-th power of an ample line bundle, then the variety is either the projective space or the p-dimensional quadric hypersurface.


Trace Estimates for Stable Processes

Abstract: In this paper we study the behaviour in time of the trace (the partition function) of the heat semigroup associated with symmetric stable processes in domains of $\Rd$. In particular, we show that for domains with the so called {\it{$R$-smoothness}} property the second terms in the asymptotic as $t\to 0$ involves the surface area of the domain, just as in the case of Brownian motion.


All CAT(0) Boundaries of a Group of the Form HxK are CE Equivalent

Abstract: M. Bestvina has shown that for any given torsion-free CAT(0) group G, all of its boundaries are shape equivalent. He then posed the question of whether they satisfy the stronger condition of being cell-like equivalent. In this article we prove that the answer is "Yes" in the situation where the group in question splits as a direct product with infinite factors. We accomplish this by proving an interesting theorem in shape theory.


Optimalit\'e, congruences et calculs d'invariants des vari\'et\'es symplectiques r\'eelles de dimension quatre

Abstract: This paper follows a previous one in which were introduced deformation invariants $\chi^d_r$, $d \in H_2 (X ; \Z)$, $r \in \N$, of closed real symplectic four-manifolds $(X, \omega, c_X)$, invariants which produced lower bounds in real enumerative geometry. We prove here using methods of symplectic field theory that the lower bounds are sharp when $r \leq 1$ and the real locus of the manifold contains a sphere, torus or real projective plane (under stronger assumptions in this last case). We also prove that a big power of two divides $\chi^d_r$ as soon as r is not too big and when the real locus contains a sphere or real projective plane (under the same stronger assumptions in this last case). We finally present some explicit computations in the case of the projective plane or quadric ellipsoid surface as well as the general formulas used to get them, formulas which involve some relative invariants that we first define.


Infinite topology of curve complexes and non-Poincare duality of Teichmueller modular groups

Abstract: In this note, we fill in a gap in the literature by proving that the Teichmueller modular groups (mapping class groups) are not Poincare duality groups and the complexes of curves of surfaces have infinite homotopy type (i.e. are not homotopy equivalent to a finite CW-complex).


Multiple extensions of a finite Euler's pentagonal number theorem and the Lucas formulas

Abstract: Motivated by the resemblance of a multivariate series identity and a finite analogue of Euler's pentagonal number theorem, we study multiple extensions of the latter formula. In a different direction we derive a common extension of this multivariate series identity and two formulas of Lucas. Finally we give a combinatorial proof of Lucas' formulas.


Cobordisms of maps with singularities of a given class

Abstract: Let P be a connected smooth p-manifold. We describe the group of all cobordism classes of smooth maps of n-manifolds to P with singularities of a given $cal K$-invariant class in terms of certain stable homotopy groups by applying the relative homotopy principle on the existence level. We also deal with the oriented version and construct a classifying space to which this oriented cobordism group is represented as the set of homotopy classes of P in the codimension n<p and n\geqq p\geqq 2.


A new twist on Lorenz links

Abstract: Twisted torus links are given by twisting a subset of strands on a closed braid representative of a torus link. T--links are a natural generalization, given by repeated positive twisting. We establish a one-to-one correspondence between positive braid representatives of Lorenz links and T--links, so Lorenz links and T--links coincide. Using this correspondence, we identify over half of the simplest hyperbolic knots as Lorenz knots. We show that both hyperbolic volume and the Mahler measure of Jones polynomials are bounded for infinite collections of hyperbolic Lorenz links. The correspondence provides unexpected symmetries for both Lorenz links and T-links, and establishes many new results for T-links, including new braid index formulas.


The mapping class group and the Meyer function for plane curves

Abstract: For each d>=2, the mapping class group for plane curves of degree d will be defined and it is proved that there exists uniquely the Meyer function on this group. In the case of d=4, using our Meyer function, we can define the local signature for 4-dimensional fiber spaces whose general fibers are non-hyperelliptic compact Riemann surfaces of genus 3. Some computations of our local signature will be given.


Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

Abstract: We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in $\R^\ell$, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in $\ell$. More precisely, we prove the following. Let $\R$ be a real closed field and let \[ {\mathcal P} = \{P_1,...,P_m\} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k], \] with ${\rm deg}_Y(P_i) \leq 2, {\rm deg}_X(P_i) \leq d, 1 \leq i \leq m$. Let $S \subset \R^{\ell+k}$ be a semi-algebraic set, defined by a Boolean formula without negations, whose atoms are of the form, $P \geq 0, P\leq 0, P \in {\mathcal P}$. Let $\pi: \R^{\ell+k} \to \R^k$ be the projection on the last k co-ordinates. Then, the number of stable homotopy types amongst the fibers $S_{\x} = \pi^{-1}(\x) \cap S$ is bounded by \[ (2^m\ell k d)^{O(mk)}. \]


Classification of free actions on complete intersections of four quadrics

Abstract: In this paper we classify all free actions of finite groups on Calabi-Yau complete intersection of 4 quadrics in $\PP^7$, up to projective equivalence. We get some examples of smooth Calabi-Yau threefolds with large nonabelian fundamental groups. We also observe the relation between some of these examples and moduli of polarized abelian surfaces.


Holomorphic Motions, Fatou Linearization, and Quasiconformal Rigidity for Parabolic Germs

Abstract: By applying holomorphic motions, we prove that a parabolic germ is quasiconformally rigid, that is, any two topologically conjugate parabolic germs are quasiconformally conjugate and the conjugacy can be chosen to be more and more near conformal as long as we consider these germs defined on smaller and smaller neighborhoods. Before proving this theorem, we use the idea of holomorphic motions to give a conceptual proof of the Fatou linearization theorem. As a by-product, we also prove that any finite number of analytic germs at different points in the Riemann sphere can be extended to a quasiconformal homeomorphism which can be more and more near conformal as as long as we consider these germs defined on smaller and smaller neighborhoods of these points.


Moderate deviations and laws of the iterated logarithm for the local times of additive L\'{e}vy processes and additive random walks

Abstract: We study the upper tail behaviors of the local times of the additive L\'{e}vy processes and additive random walks. The limit forms we establish are the moderate deviations and the laws of the iterated logarithm for the L_2-norms of the local times and for the local times at a fixed site.


Exact Hausdorff measure on the boundary of a Galton--Watson tree

Abstract: A necessary and sufficient condition for the almost sure existence of an absolutely continuous (with respect to the branching measure) exact Hausdorff measure on the boundary of a Galton--Watson tree is obtained. In the case where the absolutely continuous exact Hausdorff measure does not exist almost surely, a criterion which classifies gauge functions $\phi$ according to whether $\phi$-Hausdorff measure of the boundary minus a certain exceptional set is zero or infinity is given. Important examples are discussed in four additional theorems. In particular, Hawkes's conjecture in 1981 is solved. Problems of determining the exact local dimension of the branching measure at a typical point of the boundary are also solved.


Guaranteed Accuracy for Conic Programming Problems in Vector Lattices

Abstract: This paper presents rigorous forward error bounds for linear conic optimization problems. The error bounds are formulated in a quite general framework; the underlying vector spaces are not required to be finite-dimensional, and the convex cones defining the partial ordering are not required to be polyhedral. In the case of linear programming, second order cone programming, and semidefinite programming specialized formulas are deduced yielding guaranteed accuracy. All computed bounds are completely rigorous because all rounding errors due to floating point arithmetic are taken into account. Numerical results, applications and software for linear and semidefinite programming problems are described.


Denjoy constructions for fibred homeomorphisms of the torus

Abstract: We construct different types of quasiperiodically forced circle homeomorphisms with transitive but non-minimal dynamics. Concerning the recent Poincar\'e-like classification for this class of maps of Jaeger-Stark, we demonstrate that transitive but non-minimal behaviour can occur in each of the different cases. This closes one of the last gaps in the topological classification. Actually, we are able to get some transitive quasiperiodically forced circle homeomorphisms with rather complicated minimal sets. For example, we show that, in some of the examples we construct, the unique minimal set is a Cantor set and its intersection with each vertical fibre is uncountable and nowhere dense (but may contain isolated points). We also prove that minimal sets of the later kind cannot occur when the dynamics are given by the projective action of a quasiperiodic SL(2,R)-cocycle. More precisely, we show that, for a quasiperiodic SL(2,R)-cocycle, any minimal strict subset of the torus either is a union of finitely many continuous curves, or contains at most two points on generic fibres.


Plurisubharmonic functions on the octonionic plane and Spin(9)-invariant valuations on convex sets

Abstract: A new class of plurisubharmonic functions on the octonionic plane O^2= R^{16} is introduced. An octonionic version of theorems of A.D. Aleksandrov and Chern- Levine-Nirenberg, and Blocki are proved. These results are used to construct new examples of continuous translation invariant valuations on convex subsets of O^2=R^{16}. In particular a new example of Spin(9)-invariant valuation on R^{16} is given.


Nonlinear Dirac equations on Riemann surfaces

Abstract: We develop analytical methods for nonlinear Dirac equations. Examples of such equations include Dirac-harmonic maps with curvature term and the equations describing the generalized Weierstrass representation of surfaces in three-manifolds. We provide the key analytical steps, i.e., small energy regularity and removable singularity theorems and energy identities for solutions.


Backward stochastic differential equations with random stopping time and singular final condition

Abstract: In this paper we are concerned with one-dimensional backward stochastic differential equations (BSDE in short) of the following type: \[Y_t=\xi -\int_{t\wedge \tau}^{\tau}Y_r|Y_r|^q dr-\int_{t\wedge \tau}^{\tau}Z_r dB_r,\qquad t\geq 0,\] where $\tau$ is a stopping time, $q$ is a positive constant and $\xi$ is a $\mathcal{F}_{\tau}$-measurable random variable such that $\mathbf{P}(\xi =+\infty)>0$. We study the link between these BSDE and the Dirichlet problem on a domain $D\subset \mathbb{R}^d$ and with boundary condition $g$, with $g=+\infty$ on a set of positive Lebesgue measure. We also extend our results for more general BSDE.


La $\mathrm{Z}_l$-cohomologie du mod\`ele de Deligne-Carayol est sans torsion

Abstract: This article is the $\mathrm{Z}_l$-version of my paper "Monodromie du faisceau pervers des cycles \'evanescents de quelques vari\'et\'es de Shimura simples" in Invent. Math. 2009 vol 177 pp. 239-280, where we study the vanishing cycles of some unitary Shimura variety. The aim is to prove that the cohomology sheaves of this complexe are free so that, thanks to the main theorem of Berkovich on vanishing cycles, we can deduce that the $\mathrm{Z}_l$-cohomology of the model of Deligne-Carayol is free. There will be a second article which will be the $\mathrm{Z}_l$ version of my paper "Conjecture de monodromie-poids pour quelques vari\'t\'es de Shimura unitaires" in Compositio vol 146 part 2, pp. 367-403. The aim of this second article will be to study the torsion of the cohomology groups of these Shimura varieties.


Enumerating the Saneblidze-Umble diagonal terms

Abstract: The author presents a computer implementation, calculating the terms of the Saneblidze-Umble diagonals on the permutahedron and the associahedron. The code is analyzed for correctness and presented in the paper, the source code of which simultaneously represents both the paper and the program.


Backlund Transformations and Darboux Integrability for Nonlinear Wave Equations

Abstract: We prove that second-order hyperbolic Monge-Ampere equations for one function of two variables are connected to the wave equation by a Backlund transformation if and only if they are integrable by the method of Darboux at second order. One direction of proof, proving Darboux integrability, follows the implications of the wave equation for the invariants of the G-structure associated to the Backlund transformation. The other direction constructs Backlund transformations for Darboux integrable equations as solutions of an involutive exterior differential system. Explicit transformations are given for several equations on the Goursat-Vessiot list of Darboux-integrable equations.


On Shokurov-type b-divisorial algebras of higher rank

Abstract: The purpose of this paper is to lay the foundations for the theory of higher rank b-divisorial algebras of Shokurov type. We develop techniques to deal with such objects and propose two natural conjectures regarding Shokurov algebras and adjoint algebras. We confirm these conjectures in the case of affine curves.


An application of linear programming duality to discrete Fourier analysis and additive problems

Abstract: Suppose that f is a function from Z_p -> [0,1] (Z_p is my notation for the integers mod p, not the p-adics), and suppose that a_1,...,a_k are some places in Z_p. In some additive number theory applications it would be nice to perturb f slightly so that Fourier transform f^ vanishes at a_1,...,a_k, while additive properties are left intact. In the present paper, we show that even if we are unsuccessful in this, we can at least say something interesting by using the principle of the separating hyperplane, a basic ingredient in linear programming duality.


On the existence of A-loops with some commutative inner mappings and others of order 2

Abstract: The existence of A$_\rho$-loops, A$_\lambda$-loops and A$_\mu$-loops that are neither extra loops nor CC-loops such that any two of their inner mappings $R(x,y),L(x,y)$ and $T(x)$ commute while the other one is of order 2 is shown.


Sur la d\'efinissabilit\'e existentielle de la non-nullit\'e dans les anneaux

Abstract: We investigate the rings in which the set of nonzero elements is positive-existential (i.e. a finite union of projections of "algebraic" sets). In the case of Noetherian domains, we prove in particular that this condition is satisfied whenever the ring in question is not local Henselian, while it is not satisfied for any excellent local Henselian domain which is not a field. As a byproduct, we obtain an answer to a question of Popescu on strong approximation for Henselian pairs.


Free resolutions over short local rings

Abstract: The structure of minimal free resolutions of finite modules M over commutative local rings (R,m,k) with m^3=0 and rank_k(m^2) < rank_k(m/m^2)is studied. It is proved that over generic R every M has a Koszul syzygy module. Explicit families of Koszul modules are identified. When R is Gorenstein the non-Koszul modules are classified. Structure theorems are established for the graded k-algebra Ext_R(k,k) and its graded module Ext_R(M,k).


A combinatorial basis for the free Lie algebra of the labelled rooted trees

Abstract: The pre-Lie operad can be realized as a space T of labelled rooted trees. A result of F. Chapoton shows that the pre-Lie operad is a free twisted Lie algebra. That is, the S-module T is obtained as the plethysm of the S-module Lie with an S-module F. In the context of species, we construct an explicit basis of F. This allows us to give a new proof of Chapoton's results. Moreover it permits us to show that F forms a sub nonsymmetric operad of the pre-Lie operad T.


Weak Convergence in the Prokhorov Metric of Methods for Stochastic Differential Equations

Abstract: We consider the weak convergence of numerical methods for stochastic differential equations (SDEs). Weak convergence is usually expressed in terms of the convergence of expected values of test functions of the trajectories. Here we present an alternative formulation of weak convergence in terms of the well-known Prokhorov metric on spaces of random variables. For a general class of methods, we establish bounds on the rates of convergence in terms of the Prokhorov metric. In doing so, we revisit the original proofs of weak convergence and show explicitly how the bounds on the error depend on the smoothness of the test functions. As an application of our result, we use the Strassen - Dudley theorem to show that the numerical approximation and the true solution to the system of SDEs can be re-embedded in a probability space in such a way that the method converges there in a strong sense. One corollary of this last result is that the method converges in the Wasserstein distance, another metric on spaces of random variables. Another corollary establishes rates of convergence for expected values of test functions assuming only local Lipschitz continuity. We conclude with a review of the existing results for pathwise convergence of weakly converging methods and the corresponding strong results available under re-embedding.


Note Integer Factoring Methods III

Abstract: The best deterministic unconditionally proven integer factorization algorithms have exponential running time complexities of O(N^(1/4)) arithmetic operations, and conditional on the Riemann hypothesis, there is a deterministic algorithm of exponential running time complexity O(N^(1/5)). This note proposes a new deterministic integer factorization algorithm of deterministic exponential time complexity O(N^(1/6)). Furthermore, an algorithm for decomposing composite integers that have factor differences of the form q - p = (r - 1)N^(1/2) + u, where r > 1 is a fixed parameter, and | u | < N^(1/3), in deterministic logarithmic time and various other results are included.


Large deviations for occupation times of Markov processes with $L_{\mathbf{2}}$ semigroups

Abstract: Our aim is to unify and extend the large deviation upper and lower bounds for the occupation times of a Markov process with $L_2$ semigroups under minimal conditions on the state space and the process trajectories; for example, no strong Markov property is needed. The methods used here apply in both continuous and discrete time. We present the proofs for continuous time only because of the inherent technical difficulties in that situation; the proofs can be adapted for discrete time in a straightforward manner.


A signed analog of Euler's reduction formula for the double zeta function

Abstract: The double zeta function is a function of two arguments defined by a double Dirichlet series, and was first studied by Euler in response to a letter from Goldbach in 1742. By calculating many examples, Euler inferred a closed form evaluation of the double zeta function in terms of values of the Riemann zeta function, in the case when the two arguments are positive integers with opposite parity. Here, we consider a signed analog of Euler's evaluation: namely a reduction formula for the signed double zeta function that reduces to Euler's evaluation when the signs are specialized to 1. This formula was first stated in a 1997 paper by Borwein, Bradley and Broadhurst and was subsequently proved by Flajolet and Salvy using contour integration. The purpose here is to give an elementary proof based on a partial fraction identity.


Expected Utility Optimization - Calculus of Variations Approach

Abstract: In this paper, I'll derive the Hamilton-Jacobi (HJ) equation for Merton's problem in Utility Optimization Theory using a Calculus of Variations (CoV) Approach. For stochastic control problems, Dynamic Programming (DP) has been used as a standard method. To the best of my knowledge, no one has used CoV for this problem. In addition, while the DP approach cannot guarantee that the optimum satisfies the HJ equation, the CoV approach does. Be aware that this is the first draft of this paper and many flaws might be introduced.


On tight contact structures with negative maximal twisting number on small Seifert manifolds

Abstract: We study some properties of transverse contact structures on small Seifert manifolds, and we apply them to the classification of tight contact structures on a family of small Seifert manifolds.


On Quasiminimal Excellent Classes

Abstract: A careful exposition of Zilber's quasiminimal excellent classes and their categoricity is given, leading to two new results: the L_w1,w(Q)-definability assumption may be dropped, and each class is determined by its model of dimension aleph_0.


A spectral condition for odd cycles in graphs

Abstract: We give a sharp spectral condition for the existence of odd cycles in a graph of given order. We also prove a related stability result.


The Analytic Classification of Plane Branches

Abstract: In this paper we give a solution to Zariski's problem of analytic classification of plane branches.


Criteria for virtual fibering

Abstract: We prove that an irreducible 3-manifold whose fundamental group satisfies a certain group-theoretic property called RFRS is virtually fibered. As a corollary, we show that 3-dimensional reflection orbifolds and arithmetic hyperbolic orbifolds defined by a quadratic form virtually fiber. These include the Seifert Weber dodecahedral space and the Bianchi orbifolds. Moreover, we show that a taut sutured compression body has a finite-sheeted cover with a depth one taut-oriented foliation.


Analytic classification of plane branches up to multiplicity 4

Abstract: We perform the analytic classification of plane branches of multiplicity less or equal than four.


Gr\"obner bases and Betti numbers of monoidal complexes

Abstract: In this note we consider monoidal complexes and their associated algebras, called toric face rings. These rings generalize Stanley-Reisner rings and affine monoid algebras. We compute initial ideals of the presentation ideal of a toric face ring, and determine its graded Betti numbers. Our results generalize celebrated theorems of Hochster in combinatorial commutative algebra.


Integration over complex manifolds via Hochschild homology

Abstract: Given a holomorphic vector bundle $\cale$ on a connected compact complex manifold X, [FLS] construct a $\compl$-linear functional $I_{\cale}$ on $\hh{2n}{\compl}$. This is done by constructing a linear functional on the 0-th completed Hochschild homology $\choch{0}{(\dif(\cale))}$ of the sheaf of holomorphic differential operators on $\cale$ using topological quantum mechanics. They show that this functional is $\int_X$ if $\cale$ has non zero Euler characteristic. They conjecture that this functional is $\int_X$ for all $\cale$. A subsequent work [Ram] by the author proved that the linear functional $I_{\cale}$ is independent of the vector bundle $\cale$. This note builds upon the work in [Ram] to prove that $I_{\cale}=\int_X$ for an arbitrary holomorphic vector bundle $\cale$ on an arbitrary connected compact complex manifold X. This is done using an argument that is very natural from the geometric point of view. This argument enables us to extend the construction in [FLS] to a construction of a linear functional $I_{\cale}$ on $\text{H}^{2n}_{c}(Y,\compl)$ for an arbitrary holomorphic vector bundle $\cale$ on an arbitrary connected complex manifold Y and prove that $I_{\cale} = \int_Y$. We also generalize a result of [Ram] pertaining to "cyclic homology analogs" of $I_{\cale}$.


On the paper ``Weak convergence of some classes of martingales with jumps''

Abstract: This note extends some results of Nishiyama [Ann. Probab. 28 (2000) 685--712]. A maximal inequality for stochastic integrals with respect to integer-valued random measures which may have infinitely many jumps on compact time intervals is given. By using it, a tightness criterion is obtained; if the so-called quadratic modulus is bounded in probability and if a certain entropy condition on the parameter space is satisfied, then the tightness follows. Our approach is based on the entropy techniques developed in the modern theory of empirical processes.


Structural properties of proportional fairness: stability and insensitivity

Abstract: In this article we provide a novel characterization of the proportionally fair bandwidth allocation of network capacities, in terms of the Fenchel--Legendre transform of the network capacity region. We use this characterization to prove stability (i.e., ergodicity) of network dynamics under proportionally fair sharing, by exhibiting a suitable Lyapunov function. Our stability result extends previously known results to a more general model including Markovian users routing. In particular, it implies that the stability condition previously known under exponential service time distributions remains valid under so-called phase-type service time distributions. We then exhibit a modification of proportional fairness, which coincides with it in some asymptotic sense, is reversible (and thus insensitive), and has explicit stationary distribution. Finally we show that the stationary distributions under modified proportional fairness and balanced fairness, a sharing criterion proposed because of its insensitivity properties, admit the same large deviations characteristics. These results show that proportional fairness is an attractive bandwidth allocation criterion, combining the desirable properties of ease of implementation with performance and insensitivity.


Good rough path sequences and applications to anticipating stochastic calculus

Abstract: We consider anticipative Stratonovich stochastic differential equations driven by some stochastic process lifted to a rough path. Neither adaptedness of initial point and vector fields nor commuting conditions between vector field is assumed. Under a simple condition on the stochastic process, we show that the unique solution of the above SDE understood in the rough path sense is actually a Stratonovich solution. We then show that this condition is satisfied by the Brownian motion. As application, we obtain rather flexible results such as support theorems, large deviation principles and Wong--Zakai approximations for SDEs driven by Brownian motion along anticipating vectorfields. In particular, this unifies many results on anticipative SDEs.


Central limit theorem and almost sure central limit theorem for the product of some partial sums

Abstract: In this paper, we give the central limit theorem and almost sure central limit theorem for products of some partial sums of independent identically distributed random variables.


Stationary distributions of a model of sympatric speciation

Abstract: This paper deals with a model of sympatric speciation, that is, speciation in the absence of geographical separation, originally proposed by U. Dieckmann and M. Doebeli in 1999. We modify their original model to obtain a Fleming--Viot type model and study its stationary distribution. We show that speciation may occur, that is, the stationary distribution puts most of the mass on a configuration that does not concentrate on the phenotype with maximum carrying capacity, if competition between phenotypes is intense enough. Conversely, if competition between phenotypes is not intense, then speciation will not occur and most of the population will have the phenotype with the highest carrying capacity. The length of time it takes speciation to occur also has a delicate dependence on the mutation parameter, and the exact shape of the carrying capacity function and the competition kernel.


Open Problems in Algebraic Statistics

Abstract: Algebraic statistics is concerned with the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry. This article presents a list of open mathematical problems in this emerging field, with main emphasis on graphical models with hidden variables, maximum likelihood estimation, and multivariate Gaussian distributions. This article is based on a lecture presented at the IMA in Minneapolis during the 2006/07 program on Applications of Algebraic Geometry.


A boundedness result for toric log Del Pezzo surfaces

Abstract: In this paper we give an upper bound for the Picard number of the rational surfaces which resolve minimally the singularities of toric log Del Pezzo surfaces of given index $\ell$. This upper bound turns out to be a quadratic polynomial in the variable $\ell$.


Heat kernel estimates for the Grusin operator

Abstract: We study the geometry associated to the Grusin operator G=\Delta_{x}+|x|^{2}\partial_{u}^{2} on \mathbb{R}_{x}^{n}\times\mathbb{R}_{u}, to obtain heat kernel estimates for this operator. The main work is to find the shortest geodesics connecting two given points in $\mathbb{R}^{n+1}$. This gives the Carnot-Caratheodory distance d_{CC}, associated to this operator. The main result in the second part is to give Gaussian bounds for the heat kernel K_{t} in terms of the Carnot-Caratheodory distance. In particular we obtain the following estimate |k_{t}(\zeta,\eta)|\leq C t^{-\frac{n}{2}-1}\min(1+\frac{d_{CC}(\zeta,\eta)} {|x+\xi|},1+\frac{d_{CC}(\zeta,\eta)^{2}}{4t})^{\alpha}e^{-\frac{1}{4t}d_{CC} (\zeta,\eta)^{2}} for all $\zeta=(x,u_{1}), \eta=(\xi,u)\in\mathbb{R}^{n+1}$, where $\alpha = \max{\frac{n}{2}-1,0}$. Here the homogeneous dimension is q=n+2, so that $\frac{n}{2}-1=\frac{q-4}{2}$. This shows that our result for $n\geq2$ corresponds with the result on the Heisenberg group, which was given by Beals, Gaveau, Greiner in [1].


Probabilistic validation of homology computations for nodal domains

Abstract: Homology has long been accepted as an important computable tool for quantifying complex structures. In many applications, these structures arise as nodal domains of real-valued functions and are therefore amenable only to a numerical study based on suitable discretizations. Such an approach immediately raises the question of how accurate the resulting homology computations are. In this paper, we present a probabilistic approach to quantifying the validity of homology computations for nodal domains of random fields in one and two space dimensions, which furnishes explicit probabilistic a priori bounds for the suitability of certain discretization sizes. We illustrate our results for the special cases of random periodic fields and random trigonometric polynomials.


Einstein solvmanifolds with a simple Einstein derivation

Abstract: The structure of a solvable Lie groups admitting an Einstein left-invariant metric is, in a sense, completely determined by the nilradical of its Lie algebra. We give an easy-to-check necessary and sufficient condition for a nilpotent algebra to be an Einstein nilradical whose Einstein derivation has simple eigenvalues. As an application, we classify filiform Einstein nilradicals (modulo known classification results on filiform graded Lie algebras).


Uniform convergence of exact large deviations for renewal reward processes

Abstract: Let (X_n,Y_n) be i.i.d. random vectors. Let W(x) be the partial sum of Y_n just before that of X_n exceeds x>0. Motivated by stochastic models for neural activity, uniform convergence of the form $\sup_{c\in I}|a(c,x)\operatorname {Pr}\{W(x)\gecx\}-1|=o(1)$, $x\to\infty$, is established for probabilities of large deviations, with a(c,x) a deterministic function and I an open interval. To obtain this uniform exact large deviations principle (LDP), we first establish the exponentially fast uniform convergence of a family of renewal measures and then apply it to appropriately tilted distributions of X_n and the moment generating function of W(x). The uniform exact LDP is obtained for cases where X_n has a subcomponent with a smooth density and Y_n is not a linear transform of X_n. An extension is also made to the partial sum at the first exceedance time.


Heavy traffic limit for a processor sharing queue with soft deadlines

Abstract: This paper considers a GI/GI/1 processor sharing queue in which jobs have soft deadlines. At each point in time, the collection of residual service times and deadlines is modeled using a random counting measure on the right half-plane. The limit of this measure valued process is obtained under diffusion scaling and heavy traffic conditions and is characterized as a deterministic function of the limiting queue length process. As special cases, one obtains diffusion approximations for the lead time profile and the profile of times in queue. One also obtains a snapshot principle for sojourn times.


Geometry of the theta divisor of a compactified jacobian

Abstract: We study the theta divisor of the compactified jacobian of a nodal, possibly reducible, curve. We compute its irreducible components and give it a geometric interpretation consistent with the classical Brill-Noether theory of smooth curves. Some applications on hyperelliptic stable curves are appended.


Teaching the Kepler laws for freshmen

Abstract: We present a natural proof of Kepler's law of ellipses in the spirit of Euclidean geometry. Moreover we discuss two existing Euclidean geometric proofs, one by Feynman in hist Lost Lecture from 1964 and the other by Newton in the Principia of 1687.


LBB Stability of a Mixed Discontinuous/Continuous Galerkin Finite Element Pair

Abstract: We introduce a new mixed discontinuous/continuous Galerkin finite element for solving the 2- and 3-dimensional wave equations and equations of incompressible flow. The element, which we refer to as P1dg-P2, uses discontinuous piecewise linear functions for velocity and continuous piecewise quadratic functions for pressure. The aim of introducing the mixed formulation is to produce a new flexible element choice for triangular and tetrahedral meshes which satisfies the LBB stability condition and hence has no spurious zero-energy modes. We illustrate this property with numerical integrations of the wave equation in two dimensions, an analysis of the resultant discrete Laplace operator in two and three dimensions, and a normal mode analysis of the semi-discrete wave equation in one dimension.


Estimates for the maximal singular integral in terms of the singular integral:the case of even kernels

Abstract: The purpose of this paper is to describe the smooth homogeneous Calderon-Zygmund operators for which the maximal singular integral T*f may be controlled by the singular integral Tf. We consider two types of control. The first is the L2 estimate of T*f by Tf, namely the estimate of the L2 norm of T*f by a constant times the L2 norm of Tf. The second is the pointwise estimate of T*f(x) by a constant times M(Tf)(x), where M denotes the Hardy-Littlewood maximal operator. Notice that this is an improved variant of Cotlar's inequality, because the term Mf(x) is missing on the right hand side. Our main result states that, for even operators, both are equivalent to a purely algebraic condition formulated in terms of the expansion of the kernel in spherical harmonics. The condition holds by higher order Riesz transforms, which then satisfy an improved version of Cotlar's inequality


Integrality of instanton numbers

Abstract: We prove the results announced in a joint paper (arXiv:hep-th/0603106) with Maxim Kontsevich and Albert Schwarz.


Semiparametrically efficient rank-based inference for shape I. optimal rank-based tests for sphericity

Abstract: We propose a class of rank-based procedures for testing that the shape matrix $\mathbf{V}$ of an elliptical distribution (with unspecified center of symmetry, scale and radial density) has some fixed value ${\mathbf{V}}_0$; this includes, for ${\mathbf{V}}_0={\mathbf{I}}_k$, the problem of testing for sphericity as an important particular case. The proposed tests are invariant under translations, monotone radial transformations, rotations and reflections with respect to the estimated center of symmetry. They are valid without any moment assumption. For adequately chosen scores, they are locally asymptotically maximin (in the Le Cam sense) at given radial densities. They are strictly distribution-free when the center of symmetry is specified, and asymptotically so when it must be estimated. The multivariate ranks used throughout are those of the distances--in the metric associated with the null value ${\mathbf{V}}_0$ of the shape matrix--between the observations and the (estimated) center of the distribution. Local powers (against elliptical alternatives) and asymptotic relative efficiencies (AREs) are derived with respect to the adjusted Mauchly test (a modified version of the Gaussian likelihood ratio procedure proposed by Muirhead and Waternaux [Biometrika 67 (1980) 31--43]) or, equivalently, with respect to (an extension of) the test for sphericity introduced by John [Biometrika 58 (1971) 169--174]. For Gaussian scores, these AREs are uniformly larger than one, irrespective of the actual radial density. Necessary and/or sufficient conditions for consistency under nonlocal, possibly nonelliptical alternatives are given. Finite sample performances are investigated via a Monte Carlo study.


Forbidden patterns and shift systems

Abstract: The scope of this paper is two-fold. First, to present to the researchers in combinatorics an interesting implementation of permutations avoiding generalized patterns in the framework of discrete-time dynamical systems. Indeed, the orbits generated by piecewise monotone maps on one-dimensional intervals have forbidden order patterns, i.e., order patterns that do not occur in any orbit. The allowed patterns are then those patterns avoiding the so-called forbidden root patterns and their shifted patterns. The second scope is to study forbidden patterns in shift systems, which are universal models in information theory, dynamical systems and stochastic processes. Due to its simple structure, shift systems are accessible to a more detailed analysis and, at the same time, exhibit all important properties of low-dimensional chaotic dynamical systems (e.g., sensitivity to initial conditions, strong mixing and a dense set of periodic points), allowing to export the results to other dynamical systems via order-isomorphisms.


Generating trees for permutations avoiding generalized patterns

Abstract: We construct generating trees with one, two, and three labels for some classes of permutations avoiding generalized patterns of length 3 and 4. These trees are built by adding at each level an entry to the right end of the permutation, which allows us to incorporate the adjacency condition about some entries in an occurrence of a generalized pattern. We use these trees to find functional equations for the generating functions enumerating these classes of permutations with respect to different parameters. In several cases we solve them using the kernel method and some ideas of Bousquet-M\'elou. We obtain refinements of known enumerative results and find new ones.


Special Fuzzy Matrices for Social Scientists

Abstract: This book introduces special classes of Fuzzy and Neutrosophic matrices. These special classes of matrices are used in the construction of multi-expert special fuzzy models using FCM, FRM and FRE and their Neutorosophic analogues (simultaneous or otherwise, according to ones need). Using the six basic models, we have constructed a multi-expert multi-model called the Super Special Hexagonal Fuzzy and Neutrosophic model. Given any special input vector, these models can give the resultant using special operations. When coupled with computer programming, these operations can give the solution within a reasonable time period. Such multi-expert multi-model systems are not only a boon to social scientists, but also to anyone who wants to use Fuzzy or Neutrosophic models.


Active Set and EM Algorithms for Log-Concave Densities Based on Complete and Censored Data

Abstract: We develop an active set algorithm for the maximum likelihood estimation of a log-concave density based on complete data. Building on this fast algorithm, we indidate an EM algorithm to treat arbitrarily censored or binned data.


On admissible rank one local systems

Abstract: A rank one local system $\LL$ on a smooth complex algebraic variety $M$ is 1-admissible if the dimension of the first cohomology group $H^1(M,\LL)$ can be computed from the cohomology algebra $H^*(M,\C)$ in degrees $\leq 2$. Under the assumption that $M$ is 1-formal, we show that all local systems, except finitely many, on a non-translated irreducible component $W$ of the first characteristic variety $\V_1(M)$ are 1-admissible, see Proposition 3.1. The same result holds for local systems on a translated component $W$, but now $H^*(M,\C)$ should be replaced by $H^*(M_0,\C)$, where $M_0$ is a Zariski open subset obtained from $M$ by deleting some hypersurfaces determined by the translated component $W$, see Theorem 4.3.


On the space of morphisms between \'etale groupoids

Abstract: Given two \'etale groupoids $\Cal G$ and $\Cal G'$, we consider the set of pointed morphisms from $\Cal G$ to $\Cal G'$. Under suitable hypothesis we introduce on this set a structure of Banach manifold which can be considered as the space of objects of an \'etale groupoid whose space of orbits is the space of morphisms from $\Cal G$ to $\Cal G'$.


q-Terms, singularities and the extended Bloch group

Abstract: Our paper originated from a generalization of the Volume Conjecture to multisums of $q$-hypergeometric terms. This generalization was sketched by Kontsevich in a problem list in Aarhus University in 2006; \cite{Ko}. We introduce the notion of a $q$-hypergeometric term (in short, $q$-term). The latter is a product of ratios of $q$-factorials in linear forms in several variables. In the first part of the paper, we show how to construct elements of the Bloch group (and its extended version) given a \qterm. Their image under the Bloch-Wigner map or the Rogers dilogarithm is a finite set of periods of weight 2, in the sense of Kontsevich-Zagier. In the second part of the paper we introduce the notion of a special $q$-term, its corresponding sequence of polynomials, and its generating series. Examples of special $q$-terms come naturally from Quantum Topology, and in particular from planar projections of knots. The two parts are tied together by a conjecture that relates the singularities of the generating series of a special $q$-term with the periods of the corresponding elements of the extended Bloch group. In some cases (such as the $4_1$ knot), the conjecture is known.


Semigroups of valuations on local rings

Abstract: In this paper the question of which semigroups are realizable as the semigroup of values attained on a Noetherian local ring which is dominated by a valuation is considered. We give some striking examples, indicating that there may be no constraints on the semigroup beyond those known classically.


A solid angle theory for real polytopes

Abstract: We extend many theorems from the context of solid angle sums over rational polytopes to the context of solid angle sums over real polytopes. Moreover, we consider any real dilation parameter, as opposed to the traditional integer dilation parameters. One of the main results is an extension of Macdonald's solid angle quasipolynomial for rational polytopes to a real analytic function of the dilation parameter, for any real convex polytope.


A renormalization approach to irrational rotations

Abstract: We introduce a renormalization procedure which allows us to study in a unified and concise way different properties of the irrational rotations on the unit circle $\beta \mapsto \set{\alpha+\beta}$, $\alpha \in \R\setminus \Q$. In particular we obtain sharp results for the diffusion of the walk on $\Z$ generated by the location of points of the sequence $\{n\alpha +\beta\}$ on a binary partition of the unit interval. Finally we give some applications of our method.


Abstract: We characterize Banach spaces with polynomial numerical index 1 when they have the Radon-Nikod\'ym property. The holomorphic numerical index is introduced and the characterization of the Banach space with holomorphic numerical index 1 is obtained when it has the Radon-Nikod\'ym property.


Cross Validation for Comparing Multiple Density Estimation Procedures

Abstract: We demonstrate the consistency of cross validation for comparing multiple density estimators using simple inequalities on the likelihood ratio. In nonparametric problems, the splitting of data does not require the domination of test data over the training/estimation data, contrary to Shao (1993). The result is complementary to that of Yang (2005) and Yang (2006).


Symmetries of spatial graphs and Simon invariants

Abstract: An ordered and oriented 2-component link L in the 3-sphere is said to be achiral if it is ambient isotopic to its mirror image ignoring the orientation and ordering of the components. Kirk-Livingston showed that if L is achiral then the linking number of L is not congruent to 2 modulo 4. In this paper we study orientation-preserving or reversing symmetries of 2-component links, spatial complete graphs on 5 vertices and spatial complete bipartite graphs on 3+3 vertices in detail, and determine the necessary conditions on linking numbers and Simon invariants for such links and spatial graphs to be symmetric.


Rational solutions of the Noumi and Yamada system of type $A_4^{(1)}$

Abstract: In this paper, we completely classify the raional solutions of the Noumi and Yamada system of type A_4^{(1)}, which is a generalization of the forth Painlev\'e equation. The rational solutions are classified to three types by the B\"acklund transformation group.


Semiparametrically efficient rank-based inference for shape II. Optimal R-estimation of shape

Abstract: A class of R-estimators based on the concepts of multivariate signed ranks and the optimal rank-based tests developed in Hallin and Paindaveine [Ann. Statist. 34 (2006)] is proposed for the estimation of the shape matrix of an elliptical distribution. These R-estimators are root-n consistent under any radial density g, without any moment assumptions, and semiparametrically efficient at some prespecified density f. When based on normal scores, they are uniformly more efficient than the traditional normal-theory estimator based on empirical covariance matrices (the asymptotic normality of which, moreover, requires finite moments of order four), irrespective of the actual underlying elliptical density. They rely on an original rank-based version of Le Cam's one-step methodology which avoids the unpleasant nonparametric estimation of cross-information quantities that is generally required in the context of R-estimation. Although they are not strictly affine-equivariant, they are shown to be equivariant in a weak asymptotic sense. Simulations confirm their feasibility and excellent finite-sample performances.


Farey Statistics in Time n^{2/3} and Counting Primitive Lattice Points in Polygons

Abstract: We present algorithms for computing ranks and order statistics in the Farey sequence, taking time O (n^{2/3}). This improves on the recent algorithms of Pawlewicz [European Symp. Alg. 2007], running in time O (n^{3/4}). We also initiate the study of a more general algorithmic problem: counting primitive lattice points in planar shapes.


2004 IMS Medallion Lecture: Local Rademacher complexities and oracle inequalities in risk minimization

Abstract: Let $\mathcal{F}$ be a class of measurable functions $f:S\mapsto [0,1]$ defined on a probability space $(S,\mathcal{A},P)$. Given a sample (X_1,...,X_n) of i.i.d. random variables taking values in S with common distribution P, let P_n denote the empirical measure based on (X_1,...,X_n). We study an empirical risk minimization problem $P_nf\to \min$, $f\in \mathcal{F}$. Given a solution $\hat{f}_n$ of this problem, the goal is to obtain very general upper bounds on its excess risk \[\mathcal{E}_P(\hat{f}_n):=P\hat{f}_n-\inf_{f\in \mathcal{F}}Pf,\] expressed in terms of relevant geometric parameters of the class $\mathcal{F}$. Using concentration inequalities and other empirical processes tools, we obtain both distribution-dependent and data-dependent upper bounds on the excess risk that are of asymptotically correct order in many examples. The bounds involve localized sup-norms of empirical and Rademacher processes indexed by functions from the class. We use these bounds to develop model selection techniques in abstract risk minimization problems that can be applied to more specialized frameworks of regression and classification.