A double demonstration of a theorem of Newton, which gives a relation between the coefficient of an algebraic equation and the sums of the powers of its roots

Abstract: Translation from the Latin original, "Demonstratio gemina theorematis Neutoniani, quo traditur relatio inter coefficientes cuiusvis aequationis algebraicae et summas potestatum radicum eiusdem" (1747). E153 in the Enestrom index. In this paper Euler gives two proofs of Newton's identities, which express the sums of powers of the roots of a polynomial in terms of its coefficients. The first proof takes the derivative of a logarithm. The second proof uses induction and the fact that in a polynomial of degree $n$, the coefficient of $x^{n-k}$ is equal to the sum of the products of $k$ roots, times $(-1)^k$.


The Ring of Integers in the Canonical Structures of the Planes

Abstract: The \emph{canonical structures of the plane} are those that result, up to isomorphism, from the rings that have the form $\mathds{R}[x]/(ax^2+bx+c)$ with $a\neq 0$.That ring is isomorphic to $\mathds{R}[\theta]$, where $\theta$ is the equivalence class of x, which satisfies $\theta^2 = (-\dfrac{c}{a}) + \theta (-\dfrac{b}{a})$. On the other hand, it is known that, up to isomorphism, there are only three canonical structures: the corresponding to $\theta^2 = -1$ (the complex numbers), $\theta^2 = 1$ (the perplex or hyperbolic numbers) and $\theta^2 = 0$ (the parabolic numbers). This article copes with the algebraic structure of the rings of integers $\mathds{Z}[\theta]$ in the perplex and parabolic cases by \emph{analogy} to the complex cases: the ring of Gaussian integers. For those rings a \emph{division algorithm} is proved and it is obtained, as a consequence, the characterization of the prime and irreducible elements.


Two results from Morita theory of stable model categories

Abstract: We prove two results from Morita theory of stable model categories. Both can be regarded as topological versions of recent algebraic theorems. One is on recollements of triangulated categories, which have been studied in the algebraic case by J{\o}rgensen. We give a criterion which answers the following question: When is there a recollement for the derived category of a given symmetric ring spectrum in terms of two other symmetric ring spectra? The other result is on well generated triangulated categories in the sense of Neeman. Porta characterizes the algebraic well generated categories as localizations of derived categories of DG categories. We prove a topological analogon: a topological triangulated category is well generated if and only if it is triangulated equivalent to a localization of the derived category of a symmetric ring spectrum with several objects. Here `topological' means triangulated equivalent to the homotopy category of a spectral model category. Moreover, we show that every well generated spectral model category is Quillen equivalent to a Bousfield localization of a category of modules via a single Quillen functor.


Jacobi Forms of Critical Weight and Weil Representations

Abstract: Jacobi forms can be considered as vector valued modular forms, and Jacobi forms of critical weight correspond to vector valued modular forms of weight $\frac12$. Since the only modular forms of weight $\frac12$ on congruence subgroups of $\SL$ are theta series the theory of Jacobi forms of critical weight is intimately related to the theory of Weil representations of finite quadratic modules. This article explains this relation in detail, gives an account of various facts about Weil representations which are useful in this context, and it gives some applications of the theory developed herein by proving various vanishing theorems and by proving a conjecture on Jacobi forms of weight one on $\SL$ with character.


Beltrami operators, non--symmetric elliptic equations and quantitative Jacobian bounds

Abstract: In recent studies on the G-convergence of Beltrami operators, a number of issues arouse concerning injectivity properties of families of quasiconformal mappings. Bojarski, D'Onofrio, Iwaniec and Sbordone formulated a conjecture based on the existence of a so-called primary pair. Very recently, Bojarski proved the existence of one such pair. We provide a general, constructive, procedure for obtaining a new rich class of such primary pairs. This proof is obtained as a slight adaptation of previous work by the authors concerning the nonvanishing of the Jacobian of pairs of solutions of elliptic equations in divergence form in the plane. It is proven here that the results previously obtained when the coefficient matrix is symmetric also extend to the non-symmetric case. We also prove a much stronger result giving a quantitative bound for the Jacobian determinant of the so-called \emph{periodic} $\sigma$-harmonic sense preserving homeomorphisms of $\mathbb C$ onto itself.


Transient NN random walk on the line

Abstract: We prove strong theorems for the local time at infinity of a nearest neighbor transient random walk. First, laws of the iterated logarithm are given for the large values of the local time. Then we investigate the length of intervals over which the walk runs through (always from left to right) without ever returning.


Completely periodic directions and orbit closures of many pseudo-Anosov Teichmueller discs in Q(1,1,1,1)

Abstract: In this paper, we investigate the closure of a large class of Teichm\"uller discs in the stratum Q(1,1,1,1) or equivalently, in a GL^+_2(R)-invariant locus L of translation surfaces of genus three. We describe a systematic way to prove that the GL^+_2(R)-orbit closure of a translation surface in L is the whole of L. The strategy of the proof is an analysis of completely periodic directions on such a surface and an iterated application of Ratner's theorem to unipotent subgroups acting on an ``adequate'' splitting. This analysis applies for example to all Teichmueller discs stabilized obtained by Thurston's construction with a trace field of degree three which moreover ``obviously not Veech''. We produce an infinite series of such examples and show moreover that the favourable splitting situation does not arise everywhere on L, contrary to the situation in genus two. We also study completely periodic directions on translation surfaces in L. For instance, we prove that completely periodic directions are dense on surfaces obtained by Thurston's construction.


Nonlinear PDEs and Scale Dependence

Abstract: The properties of nonlinear PDEs that generate filtered solutions are explored with particular attention given to the constraints on the residual term. The analysis is carried out for nonlinear PDEs with an emphasis on evolution problems recast on space-time-scale. We examine the role of approximation that allow for the generation of solutions on isolated scale slices.


Local properties of J-complex curves in Lipschitz-continuous structures

Abstract: We prove the existence of primitive curves and positivity of intersections of $J$-complex curves for Lipschitz-continuous almost complex structures. These results are deduced from the Comparison Theorem for $J$-holomorphic maps in Lipschitz structures, previously known for $J$ of class $C^{1, Lip}$. We also give the optimal regularity of curves in Lipschitz structures. It occurs to be $C^{1,LnLip}$, i.e. the first derivatives of a $J$-complex curve for Lipschitz $J$ are Log-Lipschitz-continuous. A simple example that nothing better can be achieved is given. Further we prove the Genus Formula for $J$-complex curves and determine their principal Puisieux exponents (all this for Lipschitz-continuous $J$-s).


Regulators of rank one quadratic twists

Abstract: We investigate the regulators of elliptic curves with rank 1 in some families of quadratic twists of a fixed elliptic curve. In particular, we formulate some conjectures on the average size of these regulators. We also describe an efficient algorithm to compute explicitly some of the invariants of an odd quadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevich group, etc.) and we discuss the numerical data that we obtain and compare it with our predictions.


Caratheodory-Fejer interpolation and related topics in locally convex spaces

Abstract: We study Caratheodory-Herglotz functions whose values are continuous operators from a locally convex topological space which admits the factorization property into its conjugate dual space. We show how this case can be reduced to the case of functions whose values are bounded operators from a Hilbert space into itself.


Note sur les invariants du groupe affine

Abstract: In the paper, it is proved that any $C^{1}$-function on GL(n) which is locally $P$-invariant (here $P$ is the affine (sub)group of GL(n)) is locally $G$-invairant. There is also a statement for distributions (a very weak form of Baruch's results).


Non-commutative Zariski geometries and their classical limit

Abstract: We undertake a case study of two series of nonclassical Zariski geometries. We show that these geometries can be realised as representations of certain noncommutative $C^*$-algebras and introduce a natural limit construction which for each of the two series produces a classical U(1)-gauge field over a 2-dimensional Riemann surface.


Notes sur la notion d'invariant caract\'eristique

Abstract: Let $G$ be a Lie group acting on a vector space $V$. Given a set of $G$-invariants, one can ask the question : does this set of invariants characterize the group $G$ ? We recall here some known results, ask questions and state some conjectures for different choices of invariants : polynomial functions, orbits, distributions, and different types of groups.


Log-canonical threshold for curves on a smooth surface

Abstract: It is shown that the log-canonical threshold of a curve with an isolated singularity is computed by the term ideal of the curve in a suitable system of local parameters at the singularity. The proof uses the Enriques diagram of the singularity and shows that the log-canonical threshold depends only on a non-degenerate path of that diagram.


Stability of a functional equation of Deeba on semigroups

Abstract: Let $S$ be a semigroup and $X$ a Banach space. The functional equation $\phi (xyz)+ \phi (x) + \phi (y) + \phi (z) = \phi (xy) + \phi (yz) + \phi (xz)$ is said to be stable for the pair $(X, S)$ if and only if $f: S\to X$ satisfying $\| f(xyz)+f(x) + f(y) + f(z) - f(xy)- f(yz)-f(xz)\| \leq \delta $ for some positive real number $\delta$ and all $x, y, z \in S$, there is a solution $\phi : S \to X$ such that $f-\phi$ is bounded. In this paper, among others, we prove the following results: 1) this functional equation, in general, is not stable on an arbitrary semigroup; 2) this equation is stable on periodic semigroups; 3) this equation is stable on abelian semigroups; 4) any semigroup with left (or right) law of reduction can be embedded into a semigroup with left (or right) law of reduction where this equation is stable.


Representation theory, Radon transform and the heat equation on a Riemannian symmetric space

Abstract: Let X=G/K be a Riemannian symmetric space of the noncompact type. We give a short exposition of the representation theory related to X, and discuss its holomorphic extension to the complex crown, a G-invariant subdomain in the complexified symmetric space X_\C=G_\C/K_\C. Applications to the heat transform and the Radon transform for X are given.


Holomorphic geometric models for representations of $C^*$-algebras

Abstract: Representations of $C^*$-algebras are realized on section spaces of holomorphic homogeneous vector bundles. The corresponding section spaces are investigated by means of a new notion of reproducing kernel, suitable for dealing with involutive diffeomorphisms defined on the base spaces of the bundles. Applications of this technique to dilation theory of completely positive maps are explored and the critical role of complexified homogeneous spaces in connection with the Stinespring dilations is pointed out. The general results are further illustrated by a discussion of several specific topics, including similarity orbits of representations of amenable Banach algebras, similarity orbits of conditional expectations, geometric models of representations of Cuntz algebras, the relationship to endomorphisms of ${\mathcal B}({\mathcal H})$, and non-commutative stochastic analysis.


Large Deviations Principle for Self-Intersection Local Times for random walk in dimension d>4

Abstract: We obtain a large deviations principle for the self-intersection local times for a symmetric random walk in dimension d>4. As an application, we obtain moderate deviations for random walk in random sceneries in some region of parameters.


The number of open paths in an oriented $\rho$-percolation model

Abstract: We study the asymptotic properties of the number of open paths of length $n$ in an oriented $\rho$-percolation model. We show that this number is $e^{n\alpha(\rho)(1+o(1))}$ as $n \to \infty$. The exponent $\alpha$ is deterministic, it can be expressed in terms of the free energy of a polymer model, and it can be explicitely computed in some range of the parameters. Moreover, in a restricted range of the parameters, we even show that the number of such paths is $n^{-1/2} W e^{n\alpha(\rho)}(1+o(1))$ for some nondegenerate random variable $W$. We build on connections with the model of directed polymers in random environment, and we use techniques and results developed in this context.


The mapping-torus of a free group automorphism is hyperbolic relative to the canonical subgroups of polynomial growth

Abstract: We prove that the mapping torus group $\FN \rtimes_{\alpha} \Z$ of any automorphism $\alpha$ of a free group $\FN$ of finite rank $n \geq 2$ is weakly hyperbolic relative to the canonical (up to conjugation) family $\mathcal H(\alpha)$ of subgroups of $\FN$ which consists of (and contains representatives of all) conjugacy classes that grow polynomially under iteration of $\alpha$. Furthermore, we show that $\FN \rtimes_{\alpha} \Z$ is strongly hyperbolic relative to the mapping torus of the family $\mathcal H(\alpha)$. As an application, we use a result of Drutu-Sapir to deduce that $\FN \rtimes_{\alpha} \Z$ has Rapic Decay.


A Statistical Theory for the Analysis of Uncertain Systems

Abstract: This paper addresses the issues of conservativeness and computational complexity of probabilistic robustness analysis. We solve both issues by defining a new sampling strategy and robustness measure. The new measure is shown to be much less conservative than the existing one. The new sampling strategy enables the definition of efficient hierarchical sample reuse algorithms that reduce significantly the computational complexity and make it independent of the dimension of the uncertainty space. Moreover, we show that there exists a one to one correspondence between the new and the existing robustness measures and provide a computationally simple algorithm to derive one from the other.


Probabilistic Robustness Analysis -- Risks, Complexity and Algorithms

Abstract: It is becoming increasingly apparent that probabilistic approaches can overcome conservatism and computational complexity of the classical worst-case deterministic framework and may lead to designs that are actually safer. In this paper we argue that a comprehensive probabilistic robustness analysis requires a detailed evaluation of the robustness function and we show that such evaluation can be performed with essentially any desired accuracy and confidence using algorithms with complexity linear in the dimension of the uncertainty space. Moreover, we show that the average memory requirements of such algorithms are absolutely bounded and well within the capabilities of today's computers. In addition to efficiency, our approach permits control over statistical sampling error and the error due to discretization of the uncertainty radius. For a specific level of tolerance of the discretization error, our techniques provide an efficiency improvement upon conventional methods which is inversely proportional to the accuracy level; i.e., our algorithms get better as the demands for accuracy increase.


Mesures limites pour l'equation de Helmholtz dans le cas non captif

Abstract: Cet article est consacre a l'etude des mesures limites associees a la solution de l'equation de Helmholtz avec un terme source se concentrant en un point. Le potentiel est suppose regulier et l'operateur non-captif. La solution de l'equation de Schrodinger semi-classique s'ecrit alors micro-localement comme somme finie de distributions lagrangiennes. Sous une hypothese geometrique, qui generalise l'hypothese du viriel, on en deduit que la mesure limite existe et qu'elle verifie des proprietes standard. Enfin, on donne un exemple d'operateur qui ne verifie pas l'hypothese geometrique et pour lequel la mesure limite n'est pas unique. Le cas de deux termes sources est aussi traite.


Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group

Abstract: We construct a one-parameter family of properly embedded minimal annuli in the Heisenberg group Nil_3 endowed with a left-invariant Riemannian metric. These annuli are not rotationally invariant. This family gives a vertical half-space theorem and proves that each complete minimal graph in Nil_3 is entire. Also, the sister surface of an entire minimal graph in Nil_3 is an entire constant mean curvature 1/2 graph in H^2 x R, and conversely. This gives a classification of all entire constant mean curvature 1/2 graphs in H^2 x R. Finally we construct properly embedded constant mean curvature 1/2 annuli in H^2 x R.


The Euler characteristic of a category as the sum of a divergent series

Abstract: The Euler characteristic of a cell complex is often thought of as the alternating sum of the number of cells of each dimension. When the complex is infinite, the sum diverges. Nevertheless, it can sometimes be evaluated; in particular, this is possible when the complex is the nerve of a finite category. This provides an alternative definition of the Euler characteristic of a category, which is in many cases equivalent to the original one (math.CT/0610260).


Repr\'esentations de Springer pour les groupes de r\'eflexions complexes imprimitifs

Abstract: To a spetsial complex reflection group, equipped with a root lattice in the sense of Nebe, we attach a certain finite set playing a role which is analogous to the role of the set of unipotent classes of an algebraic group. In the case of imprimitive groups, we give a combinatoric parametrization of it in terms of Malle-Shoji generalized symbols. This result provides a link between the works of Shoji on Green functions for complex reflection groups and of Broue, Kim, Malle, Rouquier, et. al. on the cyclotomic Hecke algebras and their families of characters. ----- A un groupe de reflexions complexe spetsial, muni d'un reseau radiciel au sens de Nebe, nous associons un certain ensemble fini qui doit jouer un role analogue a celui de l'ensemble des classes unipotentes d'un groupe algebrique. Dans le cas des groupes imprimitifs, nous en donnons un parametrage combinatoire en termes des symboles generalises de Malle et Shoji. Ce resultat fournit un lien entre les travaux de Shoji sur les fonctions de Green pour les groupes de reflexions complexes et ceux de Broue, Kim, Malle, Rouquier, et al. sur les algebres de Hecke cyclotomiques et leurs familles de caracteres.


Explicit Formula for Constructing Binomial Confidence Interval with Guaranteed Coverage Probability

Abstract: In this paper, we derive an explicit formula for constructing the confidence interval of binomial parameter with guaranteed coverage probability. The formula overcomes the limitation of normal approximation which is asymptotic in nature and thus inevitably introduce unknown errors in applications. Moreover, the formula is very tight in comparison with classic Clopper-Pearson's approach from the perspective of interval width. Based on the rigorous formula, we also obtain approximate formulas with excellent performance of coverage probability.


Fredholm Modules on P.C.F. Self-Similar Fractals and their Conformal Geometry

Abstract: The aim of the present work is to show how, using the differential calculus associated to Dirichlet forms, it is possible to construct Fredholm modules on post critically finite fractals by regular harmonic structures. The modules are d-summable, the summability exponent d coinciding with the spectral dimension of the generalized laplacian operator associated with the regular harmonic structures. The characteristic tools of the noncommutative infinitesimal calculus allow to define a d-energy functional which is shown to be a self-similar conformal invariant.


Logarithmic limit sets of real semi-algebraic sets

Abstract: This paper is about the logarithmic limit sets of real semi-algebraic sets, and, more generally, about the logarithmic limit sets of sets definable in an o-minimal, polynomially bounded structure. We prove that most of the properties of the logarithmic limit sets of complex algebraic sets hold in the real case. This include the polyhedral structure and the relation with the theory of non-archimedean fields, tropical geometry and Maslov dequantization.


Regular and Completely Regular Differential Operators

Abstract: We define the concept of completely regular ordinary differential operators and give various criteria for operators to belong to this class. We give also criteria for Birkhof regularity of ordinary differential operators in terms of the growth of the Green function and basis property.


Spectral isolation of naturally reductive metrics on simple Lie groups

Abstract: We show that within the class of left-invariant naturally reductive metrics $\mathcal{M}_{\operatorname{Nat}}(G)$ on a compact simple Lie group $G$, every metric is spectrally isolated. We also observe that any collection of isospectral compact symmetric spaces is finite; this follows from a somewhat stronger statement involving only a finite part of the spectrum.


Data-driven efficient score tests for deconvolution problems

Abstract: We consider testing statistical hypotheses about densities of signals in deconvolution models. A new approach to this problem is proposed. We constructed score tests for the deconvolution with the known noise density and efficient score tests for the case of unknown density. The tests are incorporated with model selection rules to choose reasonable model dimensions automatically by the data. Consistency of the tests is proved.


Spectral properties of singular Sturm-Liouville operators with indefinite weight sgn x

Abstract: We consider a singular Sturm-Liouville expression with the indefinite weight sgn x. To this expression there is naturally a self-adjoint operator in some Krein space associated. We characterize the local definitizability of this operator in a neighbourhood of $\infty$. Moreover, in this situation, the point $\infty$ is a regular critical point. We construct an operator $A=(\sgn x)(-d^2/dx^2+q)$ with non-real spectrum accumulating to a real point. The obtained results are applied to several classes of Sturm-Liouville operators.


Deformation theory of representations of prop(erad)s

Abstract: We study the deformation theory of morphisms of properads and props thereby extending to a non-linear framework Quillen's deformation theory for commutative rings. The associated chain complex is endowed with a Lie algebra up to homotopy structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results. To do so, we endow the category of prop(erad)s with a model category structure. We provide a complete study of models for prop(erad)s. A new effective method to make minimal models explicit, that extends Koszul duality theory, is introduced and the associated notion is called homotopy Koszul. As a corollary, we obtain the (co)homology theories of (al)gebras over a prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex is endowed with a canonical Lie algebra up to homotopy structure in general and a Lie algebra structure only in the Koszul case. In particular, we explicit the deformation complex of morphisms from the properad of associative bialgebras. For any minimal model of this properad, the boundary map of this chain complex is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this paper provides a complete proof of the existence of a Lie algebra up to homotopy structure on the Gerstenhaber-Schack bicomplex associated to the deformations of associative bialgebras.


Brown representability for space-valued functors

Abstract: In this paper we prove two theorems which resemble the classical cohomological and homological Brown representability theorems. The main difference is that our results classify small contravariant functors from spaces to spaces up to weak equivalence of functors. In more detail, we show that every small contravariant functor from spaces to spaces which takes coproducts to products up to homotopy and takes homotopy pushouts to homotopy pullbacks is naturally weekly equivalent to a representable functor. The second representability theorem states: every contravariant continuous functor from the category of finite simplicial sets to simplicial sets taking homotopy pushouts to homotopy pullbacks is equivalent to the restriction of a representable functor. This theorem may be considered as a contravariant analog of Goodwillie's classification of linear functors.


A spectral sequence to compute L2-Betti numbers of groups and groupoids

Abstract: We construct a spectral sequence for L2-type cohomology groups of discrete measured groupoids. Based on the spectral sequence, we prove the Hopf-Singer conjecture for aspherical manifolds with poly-surface fundamental groups. More generally, we obtain a permanence result for the Hopf-Singer conjecture under taking fiber bundles whose base space is an aspherical manifold with poly-surface fundamental group. As further sample applications of the spectral sequence, we obtain new vanishing theorems and explicit computations of L2-Betti numbers of groups and manifolds and obstructions to the existence of normal subrelations in measured equivalence relations.


The concrete theory of numbers : Problem of simplicity of Fermat number-twins

Abstract: The problem of simplicity of Fermat number-twins $f_{n}^{\pm}=2^{2^n}\pm3$ is studied. The question for what $n$ numbers $f_{n}^{\pm}$ are composite is investigated. The factor-identities for numbers of a kind $x^2 \pm k $ are found.


A vanishing theorem for sheaves of small differential operators in positive characteristic

Abstract: Let $X$ be a smooth variety over an algebraically closed field $k$ of positive characteristic, ${\rm D}_X$ the sheaf of PD-differential operators, and ${\bar D}_X$ its central reduction, the sheaf of small differential operators. In this paper we show that if $X$ is a line-hyperplane incidence variety (a partial flag variety of type $(1,n,n+1)$) or a quadric of arbitrary dimension (in this case the characteristic is supposed to be odd) then ${\rm H}^{i}(X,{\bar D}_X)=0$ for $i>0$. Using this vanishing result and the derived localization theorem for crystalline differential operators (\cite{BMR}) we show that the Frobenius pushforward of the structure sheaf is a tilting bundle on these varieties, provided that $p>h$, the Coxeter number of the corresponding group.


D-affinity and Frobenius morphism on quadrics

Abstract: We compute decomposition of Frobenius push-forwards of line bundles on quadrics into a direct sum of line bundles and spinor bundles. As an application we show when the Frobenius push-forward gives a tilting bundle and we apply it to study D-modules on quadrics.


Toroidal embeddings and polyhedral divisors

Abstract: Given an effective action of an (n-1)-dimensional torus on an n-dimensional normal affine variety, Mumford constructs a toroidal embedding, while Altmann and Hausen give a description in terms of a polyhedral divisor on a curve. We compare the fan of the toroidal embedding with this polyhedral divisor.


The braid groups of the projective plane and the Fadell-Neuwirth short exact sequence

Abstract: We study the pure braid groups $P_n(RP^2)$ of the real projective plane $RP^2$, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence $1 \to P_m(RP^2 {x_1,...,x_n} \to P_{n+m}(RP^2) \stackrel{p_{\ast}}{\to} P_n(RP^2) \to 1$, where $n\geq 2$ and $m\geq 1$, and $p_{\ast}$ is the homomorphism which corresponds geometrically to forgetting the last $m$ strings. This problem is equivalent to that of the existence of a section for the associated fibration $p: F_{n+m}(RP^2) \to F_n(RP^2)$ of configuration spaces. Van Buskirk proved in 1966 that $p$ and $p_{\ast}$ admit a section if $n=2$ and $m=1$. Our main result in this paper is to prove that there is no section if $n\geq 3$. As a corollary, it follows that $n=2$ and $m=1$ are the only values for which a section exists. As part of the proof, we derive a presentation of $P_n(RP^2)$: this appears to be the first time that such a presentation has been given in the literature.


Extension of automorphisms to C*-crossed products with non-trivial centre

Abstract: Given a quasi-special endomorphism $\rho$ of a C*-algebra A with nontrivial center, we study an extension problem for automorphisms of A to a minimal cross-product B of A by $\rho$. Exploiting some aspects of the underlying generalized Doplicher-Roberts duality theory based on Pimsner algebras, an obstruction to the existence of such extensions is found and described in terms of sections of a suitable group bundle.


The intrinsic torsion of almost quaternion-Hermitian manifolds

Abstract: We study the intrinsic torsion of almost quaternion-Hermitian manifolds via the exterior algebra. In particular, we show how it is determined by particular three-forms formed from simple combinations of the exterior derivatives of the local Kaehler forms. This gives a practical method to compute the intrinsic torsion and is applied in a number of examples. In addition we find simple characterisations of HKT and QKT geometries entirely in the exterior algebra and compute how the intrinsic torsion changes under a twist construction.


Sobolev regularity of solutions of the cohomological equation

Abstract: We refine the theory of the cohomological equation for translation flows on higher genus surfaces with the goal of proving optimal results on the Sobolev regularity of solutions and of distributional obstructions. For typical translation surfaces our results are sharp and we find the expected relation between the regularity of the distributional obstructions and the Lyapunov exponents of the Kontsevich-Zorich renormalization cocycle. As a consequence we exactly determine the dimension of the space of obstructions in each Sobolev regularity class in terms of the Kontsevich-Zorich exponents. For a fixed arbitrary translation surface and a typical direction, our results are probably not optimal but are the best which can be achieved with the available harmonic analysis techniques we have introduced in an earlier paper.


Weighted lattice polynomials of independent random variables

Abstract: We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions of independent random variables. Since weighted lattice polynomial functions include ordinary lattice polynomial functions and, particularly, order statistics, our results encompass the corresponding formulas for these particular functions. We also provide an application to the reliability analysis of coherent systems.


Indecomposable representations of quivers on infinite-dimensional Hilbert spaces

Abstract: We study indecomposable representations of quivers on separable infinite-dimensional Hilbert spaces by bounded operators. We consider a complement of Gabriel's theorem for these representations. Let $\Gamma$ be a finite, connected quiver. If its underlying undirected graph contains one of extended Dynkin diagrams $\tilde{A_n} (n \geq 0)$, $\tilde{D_n} (n \geq 4)$, $\tilde{E_6}$,$\tilde{E_7}$ and $\tilde{E_8}$, then there exists an indecomposable representation of $\Gamma$ on separable infinite-dimensional Hilbert spaces.


Non-abelian free groups admit non-essentially free actions on rooted trees

Abstract: We show that every countable non-abelian free group $\Gamma $ admits a spherically transitive action on a rooted tree $T$ such that the action of $\Gamma $ on the boundary of $T$ is not essentially free. This reproves a result of Bergeron and Gaboriau. The existence of such an action answers a question of Grigorchuk, Nekrashevich and Sushchanskii.


Scalar extension of bicoalgebroids

Abstract: After recalling the definition of a bicoalgebroid, we define comodules and modules over a bicoalgebroid. We construct the monoidal category of comodules, and define Yetter--Drinfel'd modules over a bicoalgebroid. It is proved that the Yetter--Drinfel'd category is monoidal and pre--braided just as in the case of bialgebroids, and is embedded into the one--sided center of the comodule category. We proceed to define Braided Cocommutative Coalgebras (BCC) over a bicoalgebroid, and dualize the scalar extension construction of Brzezinski and Militaru [2] and Balint and Slachanyi [1], originally applied to bialgebras and bialgebroids, to bicoalgebroids. A few classical examples of this construction are given. Identifying the comodule category over a bicoalgebroid with the category of coalgebras of the associated comonad, we obtain a comonadic (weakened) version of Schauenburg's theorem. Finally, we take a look at the scalar extension and braided cocommutative coalgebras from a (co--)monadic point of view.


Multiplicity of solutions of a zero mass nonlinear equation on a Riemannian manifold

Abstract: The relation between the number of solutions of a nonlinear equation on a Riemannian manifold and the topology of the manifold itself is studied. The technique is based on Ljusternik-Schnirelmann category and Morse theory.


Analog of the Skewes number for twin primes

Abstract: The results of the computer investigation of the sign changes of the difference between the number of twin primes $\pi_2(x)$ and the Hardy--Littlewood conjecture $c_2\Li_2(x)$ are reported. It turns out that $\pi_2(x) - c_2\Li_2(x)$ changes the sign at unexpectedly low values of $x$ and for $x<2^{42}$ there are over 90000 sign changes of this difference. It is conjectured that the number of sign changes of $\pi_2(x) - c_2\Li_2(x)$ for $x\in (1, T)$ is given by $\sqrt T/\log(T)$.


Moduli spaces of coherent systems of small slope on algebraic curves

Abstract: Let $C$ be an algebraic curve of genus $g\ge2$. A coherent system on $C$ consists of a pair $(E,V)$, where $E$ is an algebraic vector bundle over $C$ of rank $n$ and degree $d$ and $V$ is a subspace of dimension $k$ of the space of sections of $E$. The stability of the coherent system depends on a parameter $\alpha$. We study the geometry of the moduli space of coherent systems for $0<d\le2n$. We show that these spaces are irreducible whenever they are non-empty and obtain necessary and sufficient conditions for non-emptiness.


The integral of the supremum process of Brownian motion

Abstract: In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area) A(T), covered by the process in the time interval [0,T]. The Laplace transform of A(T) follows as a consequence. The main proof involves a double Laplace transform of A(T) and is based on excursion theory and local time for Brownian motion.


Tail estimates for the Brownian excursion area and other Brownian areas

Abstract: Several Brownian areas are considered in this paper: the Brownian excursion area, the Brownian bridge area, the Brownian motion area, the Brownian meander area, the Brownian double meander area, the positive part of Brownian bridge area, the positive part of Brownian motion area. We are interested in the asymptotics of the right tail of their density function. Inverting a double Laplace transform, we can derive, in a mechanical way, all terms of an asymptotic expansion. We illustrate our technique with the computation of the first four terms. We also obtain asymptotics for the right tail of the distribution function and for the moments. Our main tool is the two-dimensional saddle point method.


Internal sets and internal functions in Colombeau theory

Abstract: Inspired by nonstandard analysis, we define and study internal subsets and internal functions in algebras of Colombeau generalized functions. We prove a saturation principle for internal sets and provide applications to Colombeau algebras.


A vanishing theorem for a class of logarithmic D-modules

Abstract: Let $O_X$ (resp. $D_X$) be the sheaf of holomorphic functions (resp. the sheaf of linear differential operators with holomorphic coefficients) on $X$ (=the complex affine n-space). Let $Y$ be a locally weakly quasi-homogeneous free divisor defined by a polynomial $f$. In this paper we prove that, locally, the annihilating ideal of $1/f^k$ over $D_X$ is generated by linear differential operators of order 1 (for $k$ big enough). For this purpose we prove a vanishing theorem for the extension groups of a certain logarithmic $D_X$--module with $O_X$. The logarithmic $D_X$--module is naturally associated with $Y$. This result is related to the so called Logarithmic Comparison Theorem.


The low-dimensional homotopy of the stable mapping class group

Abstract: Due to the deep work of Tillmann, Madsen, Weiss and Galatius, the cohomology of the stable mapping class group $\gaminf$ is known with rational or finite field coefficients. Little is known about the integral cohomology. In this paper, we study the first four cohomology groups. Also, we compute the first few steps of the Postnikov tower of $B \gaminf^+$, the Quillen plus construction applied to $B \gaminf$. Our method relies on the Madsen-Weiss theorem, a few known computations of stable homotopy groups of spheres and projective spaces and on a certain action of the binary icosahedral group on a surface. Using the latter, we can also describe an explicit geometric generator of the third homotopy group $\pi_3 (B \gaminf)$.


On the location and classification of all prime numbers

Abstract: We will describe an algorithm to arrange all the positive and negative integer numbers. This array of numbers permits grouping them in six different Classes, $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$, and $\zeta$. Particularly, numbers belong to Class $\alpha$ are defined as $\alpha=1+6 n$, and those of Class $\beta$, as $\beta=5+6n$, where $n=0,\pm1,\pm2,\pm3,\pm4,...$ These two Classes $\alpha$ and $\beta$,contain: i) all prime numbers, except + 2, -2 and $\pm$3, which belong to $\epsilon$, $\delta$, and $\gamma$ Classes, respectively, and ii) all the other odd numbers, except those that are multiple of $\pm$3, according to the sequence $\pm$9, $\pm$15, $\pm$21, $\pm$27, ... Besides, products between numbers of the Class $\alpha$, and also those between numbers of the Class $\beta$, generates numbers belonging to the Class $\alpha$. On the other side, products between numbers of Class $\alpha$ with numbers of Class $\beta$, result in numbers of Class $\beta$. Then, both Classes $\alpha$ and $\beta$ include: i) all the prime numbers except $\pm$2 and $\pm$3, and ii) all the products between $\alpha$ numbers, as $\alpha\cdot\alpha^{\prime}$; all the products between $\beta$ numbers, as $\beta\cdot\beta^{\prime}$; and also all the products between numbers of Classes $\alpha$ and $\beta$, as $\alpha\cdot\beta$, which necessarily are composite numbers, whose factorization is completely determined.


Self-dual polygons and self-dual curves

Abstract: We study projectively self-dual polygons and curves in the projective plane. Our results provide a partial answer to problem No 1994-17 in the book of Arnold's problems.


Compositions inside a rectangle and unimodality

Abstract: Let c^{k,l}(n) be the number of compositions (ordered partitions) of the integer n whose Ferrers diagram fits inside a k-by-l rectangle. The purpose of this note is to give a simple, algebraic proof of a conjecture of Vatter that the sequence c^{k,l}(0), c^{k,l}(1), ..., c^{k,l}(kl) is unimodal. The problem of giving a combinatorial proof of this fact is discussed, but is still open.


Hyperbolic geometry and moduli of real cubic surfaces

Abstract: Let M_0^R be the moduli space of smooth real cubic surfaces. We show that each of its components admits a real hyperbolic structure. More precisely, one can remove some lower-dimensional geodesic subspaces from a real hyperbolic space H^4 and form the quotient by an arithmetic group to obtain an orbifold isomorphic to a component of the moduli space. There are five components. For each we describe the corresponding lattices in PO(4,1). We also derive several new and several old results on the topology of M_0^R. Let M_s^R be the moduli space of real cubic surfaces that are stable in the sense of geometric invariant theory. We show that this space carries a hyperbolic structure whose restriction to M_0^R is that just mentioned. The corresponding lattice in PO(4,1), for which we find an explicit fundamental domain, is nonarithmetic.


A note on Reeb dynamics on the tight 3-sphere

Abstract: We show that a nondegenerate tight contact form on the 3-sphere has exactly two simple closed Reeb orbits if and only if the differential in linearized contact homology vanishes. Moreover, in this case the Floquet multipliers and Conley-Zehnder indices of the two Reeb orbits agree with those of a suitable irrational ellipsoid in 4-space.


Coloring and The Lonely Graph

Abstract: We improve upper bounds on the chromatic number proven independently in \cite{reedNote} and \cite{ingo}. Our main lemma gives a sufficient condition for two paths in graph to be completely joined. Using this, we prove that if a graph has an optimal coloring with more than $\frac{\omega}{2}$ singleton color classes, then it satisfies $\chi \leq \frac{\omega + \Delta + 1}{2}$. It follows that a graph satisfying $n - \Delta < \alpha + \frac{\omega - 1}{2}$ must also satisfy $\chi \leq \frac{\omega + \Delta + 1}{2}$, improving the bounds in \cite{reedNote} and \cite{ingo}. We then give a simple argument showing that if a graph satisfies $\chi > \frac{n + 3 - \alpha}{2}$, then it also satisfies $\chi(G) \leq \left\lceil\frac{\omega(G) + \Delta(G) + 1}{2}\right\rceil$. From this it follows that a graph satisfying $n - \Delta < \alpha + \omega$ also satisfies $\chi(G) \leq \left\lceil\frac{\omega(G) + \Delta(G) + 1}{2}\right\rceil$ improving the bounds in \cite{reedNote} and \cite{ingo} even further at the cost of a ceiling. In the next sections, we generalize our main lemma to constrained colorings (e.g. r-bounded colorings). We present a generalization of Reed's conjecture to r-bounded colorings and prove the conjecture for graphs with maximal degree close to their order. Finally, we outline some applications (in \cite{BorodinKostochka} and \cite{ColoringWithDoublyCriticalEdge}) of the theory presented here to the Borodin-Kostochka conjecture and coloring graphs containing a doubly critical edge.


A few remarks concerning the geometry of the moduli spaces of semistable sheaves supported on plane curves of multiplicity four

Abstract: This paper has been withdrawn by the author due to a crucial error in the proofs. The error has been corrected and the paper has been expanded in arXiv:0910.5327


On the variety of two dimensional real associative algebras

Abstract: This paper consists of a description of the variety of two dimensional associative algebras within the framework of Nonstandard Analysis. By decomposing each algebra in A^2 as sum of a Jordan algebra and a Lie algebra, we calculate the isomorphism classes of two dimensional real associative algebras over the field of real numbers and determine the components and the contractions of the variety.


Entire spacelike radial graphs in the Minkowski space, asymptotic to the light-cone, with prescribed scalar curvature

Abstract: Existence and uniqueness in ${\Bbb R}^{n,1}$ of entire spacelike hypersurfaces contained in the future of the origin $O$ and asymptotic to the light-cone, with scalar curvature prescribed at their generic point $M$ as a negative function of the unit vector $\overrightarrow{Om}$ pointing in the direction of $\overrightarrow{OM}$, divided by the square of the norm of $\overrightarrow{OM}$ (a dilation invariant problem). The solutions are seeked as graphs over the future unit-hyperboloid emanating from $O$ (the hyperbolic space); radial upper and lower solutions are constructed which, relying on a previous result in the Cartesian setting, imply their existence.


On the Calabi-Yau problem for maximal surfaces in L^3

Abstract: In this paper we construct an example of a weakly complete maximal surface in the Lorentz-Minkowski space L^3, which is bounded by a hyperboloid. Moreover, all the singularities of our example are of lightlike type.


Nonexistence of permutation binomials of certain shapes

Abstract: Suppose x^m + c*x^n is a permutation polynomial over GF(p), where p>5 is prime, m>n>0, and c is in GF(p)^*. We prove that gcd(m-n,p-1) is not 2 or 4. In the special case that either (p-1)/2 or (p-1)/4 is prime, this was conjectured in a recent paper by Masuda, Panario and Wang.


Hilbert $\widetilde{\C}$-modules: structural properties and applications to variational problems

Abstract: We develop a theory of Hilbert $\widetilde{\C}$-modules by investigating their structural and functional analytic properties. Particular attention is given to finitely generated submodules, projection operators, representation theorems for $\widetilde{\C}$-linear functionals and $\widetilde{\C}$-sesquilinear forms. By making use of a generalized Lax-Milgram theorem, we provide some existence and uniqueness theorems for variational problems involving a generalized bilinear or sesquilinear form.


Permutation binomials over finite fields

Abstract: We prove that if x^m + c*x^n permutes the prime field GF(p), where m>n>0 and c is in GF(p)^*, then gcd(m-n,p-1) > sqrt{p} - 1. Conversely, we prove that if q>=4 and m>n>0 are fixed and satisfy gcd(m-n,q-1) > 2q*(log log q)/(log q), then there exist permutation binomials over GF(q) of the form x^m + c*x^n if and only if gcd(m,n,q-1) = 1.


Boundary of the braid groups and Markov--Ivanovsky normal form

Abstract: We describe random walk boundaries (in particular, the Poisson--Furstenberg, or PF-boundary) for a vast family of groups in terms of the hyperbolic boundary of a special free subgroup. We prove that almost all trajectories of the random walk (with respect to an arbitrary nondegenerate measure on the group) converge to points of that boundary. This implies the stability (in the sense of \cite{Ver}) of the so-called Markov--Ivanovsky normal form for braids.


On some permutation polynomials over F_q of the form x^r*h(x^((q-1)/d)))

Abstract: In a recent paper, Akbary and Wang gave a sufficient condition for x^u + x^r to permute GF(q), in terms of the period of a certain sequence involving sums of cosines. As an application they gave necessary and sufficient conditions in case u,r,q satisfy certain special properties. We show that the Akbary-Wang sufficient condition follows from a more general sufficient condition which does not involve sums of cosines. This leads to vastly simpler proofs of the Akbary-Wang results, as well as generalizations to polynomials of the form x^r*h(x^{(q-1)/d}).


Some families of permutation polynomials over finite fields

Abstract: We give necessary and sufficient conditions for a polynomial of the form x^r*(1+x^v+x^(2v)+...+x^(kv))^t to permute the elements of the finite field GF(q). Our results yield especially simple criteria in case (q-1)/gcd(q-1,v) is a small prime.


Norm convergence of multiple ergodic averages for commuting transformations

Abstract: Let $T_1, ..., T_l: X \to X$ be commuting measure-preserving transformations on a probability space $(X, \X, \mu)$. We show that the multiple ergodic averages $\frac{1}{N} \sum_{n=0}^{N-1} f_1(T_1^n x) ... f_l(T_l^n x)$ are convergent in $L^2(X,\X,\mu)$ as $N \to \infty$ for all $f_1,...,f_l \in L^\infty(X,\X,\mu)$; this was previously established for $l=2$ by Conze and Lesigne and for general $l$ assuming some additional ergodicity hypotheses on the maps $T_i$ and $T_i T_j^{-1}$ by Frantzikinakis and Kra (with the $l=3$ case of this result established earlier by Zhang). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal lemmas, although we will not need the full strength of those lemmas. In particular, the $l=2$ case of our arguments are a finitary analogue of those of Conze and Lesigne.


Weyl closure of hypergeometric systems

Abstract: We show that A-hypergeometric systems and Horn hypergeometric systems are Weyl closed for very generic parameters.


Khovanov-Rozansky homology and the braid index of a knot

Abstract: We prove the existence of a knot whose braid index the Morton-Franks-Williams inequality fails to detect but a related inequality (KR-MFW inequality), which uses new information of Khovanov-Rozansky homology, detects. We also prove, by examples, that there exists infinitely many knots for which the KR-MFW inequality fails to detect the braid indices.


Stochastic Differential Games with Reflection and Related Obstacle Problems for Isaacs Equations

Abstract: In this paper we first investigate zero-sum two-player stochastic differential games with reflection with the help of theory of Reflected Backward Stochastic Differential Equations (RBSDEs). We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straight-forward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions of the associated upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations with obstacles, respectively. The method differs heavily from those used for control problems with reflection, it has its own techniques and its own interest. On the other hand, we also prove a new estimate for RBSDEs being sharper than that in El Karoui, Kapoudjian, Pardoux, Peng and Quenez [7], which turns out to be very useful because it allows to estimate the $L^p$-distance of the solutions of two different RBSDEs by the $p$-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution of the approximating Isaacs equation which is constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.


Linearity Defect and Regularity over a Koszul Algebra

Abstract: Let A be a Koszul algebra, and $mod A$ the category of finitely generated graded left A-modules. The "linearity defect" ld_A(M) of $M \in mod A$ is an invariant defined by Herzog and Iyengar. An exterior algebra E is a Koszul algebra which is the Koszul dual S^! of a polynomial ring S. Eisenbud et al. showed that $ld_E(M) < \infty$ for all $M \in mod E$. Improving their result, we show the following (and many other facts): (*) If A is a Koszul complete intersection, then $reg_{A^!} (M) < \infty$ and $ld_{A^!} (M) < \infty$ for all $M \in mod A^!$. (**) There is a uniform bound of $ld(J)$, where J is a graded ideal of E.


On multipliers of Hilbert modules over locally C*-algebras

Abstract: In this paper, we investigate the structure of the multiplier module of a Hilbert module over a locally C*-algebra and the relationship between the set of all adjointable operators from a Hilbert A-module E to a Hilbert A-module F and the set of all adjointable operators from the multiplier module M(E)of E to the multiplier module M(F) of F.


Selmer groups for elliptic curves in Z_l^d-extensions of function fields of characteristic p

Abstract: Let $F$ be a function field of characteristic $p>0$, $\F/F$ a Galois extension with $Gal(\F/F)\simeq \Z_l^d$ (for some prime $l\neq p$) and $E/F$ a non-isotrivial elliptic curve. We study the behaviour of Selmer groups $Sel_E(L)_r$ ($r$ any prime) as $L$ varies through the subextensions of $\F$ via appropriate versions of Mazur's Control Theorem. As a consequence we prove that $Sel_E(\F)_r$ is a cofinitely generated (in some cases cotorsion) $\Z_r[[Gal(\F/F)]]$-module.


Surface subgroups of right-angled Artin groups

Abstract: We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group $A(K)$ has such a subgroup if its defining graph $K$ contains an $n$-hole (i.e. an induced cycle of length $n$) with $n\geq 5$. We construct another eight "forbidden" graphs and show that every graph $K$ on $\le 8$ vertices either contains one of our examples, or contains a hole of length $\ge 5$, or has the property that $A(K)$ does not contain hyperbolic closed surface subgroups. We also provide several sufficient conditions for a \RAAG to contain no hyperbolic surface subgroups. We prove that for one of these "forbidden" subgraphs $P_2(6)$, the right angled Artin group $A(P_2(6))$ is a subgroup of a (right angled Artin) diagram group. Thus we show that a diagram group can contain a non-free hyperbolic subgroup answering a question of Guba and Sapir. We also show that fundamental groups of non-orientable surfaces can be subgroups of diagram groups. Thus the first integral homology of a subgroup of a diagram group can have torsion (all homology groups of all diagram groups are free Abelian by a result of Guba and Sapir).


Realisability and Localisation

Abstract: Let $A$ be a differential graded algebra with cohomology ring $H^*A$. A graded module over $H^*A$ is called \emph{realisable} if it is (up to direct summands) of the form $H^*M$ for some differential graded $A$-module $M$. Benson, Krause and Schwede have stated a local and a global obstruction for realisability. The global obstruction is given by the Hochschild class determined by the secondary multiplication of the $A_{\infty}$-algebra structure of $H^*A$. In this thesis we mainly consider differential graded algebras $A$ with graded-commutative cohomology ring. We show that a finitely presented graded $H^*A$-module $X$ is realisable if and only if its $\mathfrak{p}$-localisation $X_{\mathfrak{p}}$ is realisable for all graded prime ideals $\mathfrak{p}$ of $H^*A$. In order to obtain such a local-global principle also for the global obstruction, we define the \emph{localisation of a differential graded algebra $A$ at a graded prime $\mathfrak{p}$ of $H^*A$}, denoted by $A_{\mathfrak{p}}$, and show the existence of a morphism of differential graded algebras inducing the canonical map $H^*A \to (H^*A)_{\mathfrak{p}}$ in cohomology. The latter result actually holds in a much more general setting: we prove that every smashing localisation on the derived category of a differential graded algebra is induced by a morphism of differential graded algebras. Finally we discuss the relation between realisability of modules over the group cohomology ring and the Tate cohomology ring.


Natural Moves for Knots and Links

Abstract: We propose some natural generalizations of Reidemeister moves that do not increase the number of crossings in the generated diagrams. Experimentations make us conjecture that this class of monotonic moves is complete for computing canonical forms and then deciding isotopy.


On the skein exact squence for knot Floer homology

Abstract: The aim of this paper is to study the skein exact sequence for knot Floer homology. We prove precise graded version of this sequence, and also one using $\HFm$. Moreover, a complete argument is also given purely within the realm of grid diagrams.


Banach-like metrics and metrics of compact sets

Abstract: We present and study a family of metrics on the space of compact subsets of $R^N$ (that we call ``shapes''). These metrics are ``geometric'', that is, they are independent of rotation and translation; and these metrics enjoy many interesting properties, as, for example, the existence of minimal geodesics. We view our space of shapes as a subset of Banach (or Hilbert) manifolds: so we can define a ``tangent manifold'' to shapes, and (in a very weak form) talk of a ``Riemannian Geometry'' of shapes. Some of the metrics that we propose are topologically equivalent to the Hausdorff metric; but at the same time, they are more ``regular'', since we can hope for a local uniqueness of minimal geodesics. We also study properties of the metrics obtained by isometrically identifying a generic metric space with a subset of a Banach space to obtain a rigidity result.


Green's formula with $\bbc^{*}$-action and Caldero-Keller's formula for cluster algebras

Abstract: It is known that Green's formula over finite fields gives rise to the comultiplications of Ringel-Hall algebras and quantum groups (see\cite{Green}, also see \cite{Lusztig}). In this paper, we deduce the projective version of Green's formula in a geometric way. Then following the method of Hubery in \cite{Hubery2005}, we apply this formula to proving Caldero-Keller's multiplication formula for acyclic cluster algebras of arbitrary type.


Unit distance graphs with ambiguous chromatic number

Abstract: First Laszlo Szekely and more recently Saharon Shelah and Alexander Soifer have presented examples of infinite graphs whose chromatic numbers depend on the axioms chosen for set theory. The existence of such graphs may be relevant to the Chromatic Number of the Plane problem. In this paper we construct a new class of graphs with ambiguous chromatic number. They are unit distance graphs with vertex set R^n, and hence may be seen as further evidence that the chromatic number of the plane might depend on set theory.


Sharp Examples for Planar Quasiconformal Distortion of Hausdorff Measures and Removability

Abstract: In his celebrated paper on area distortion for quasiconformal mappings, Astala showed optimal area distortion bounds and dimension distortion estimates for planar quasiconformal mappings. He asked (Question 4.4) whether a finer result held, namely absolute continuity of Hausdorff measures under push-forward by quasiconformal mappings. This was proved in one particular case relevant for removability questions, in joint work of Astala, Clop, Mateu, Orobitg and the author ("Distortion of Hausdorff measures and improved Painlev'e removability for bounded quasiregular mappings", Duke Math J., to appear [ACMOU]) (Theorem 1.1), the other cases remaining open. A related question that we left open in [ACMOU] (Question 4.2) (which was asked by Astala to the author before [ACMOU] in an equivalent form in a personal communication) is whether BMO removability for K-quasiregular mappings and ($L^{\infty}$) removability for K-quasiregular mappings are indeed different problems. In this paper we give a series of examples answering in the positive Question 4.2 in [ACMOU], at the same time proving sharpness in two different senses of Theorem 1.1 in [ACMOU], and also giving examples that would yield sharpness in those two different senses as well for the absolute continuity of Hausdorff measures under push-forward by quasiconformal mappings, were it to be proved.


Global parametrices and dispersive estimates for variable coefficient wave equations

Abstract: In this article we consider variable coefficient, time-dependent wave equations. Using phase space methods we construct outgoing parametrices and prove Strichartz-type estimates globally in time. This is done in the context of C^2 metrics which satisfy a weak aymptotic flatness condition at infinity.


Nonlinear Dynamics of the 3D Pendulum

Abstract: A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom. The pendulum is acted on by a gravitational force. Symmetry assumptions are shown to lead to the planar 1D pendulum and to the spherical 2D pendulum models as special cases. The case where the rigid body is asymmetric and the center of mass is distinct from the pivot location leads to the 3D pendulum. Full and reduced 3D pendulum models are introduced and used to study important features of the nonlinear dynamics: conserved quantities, equilibria, invariant manifolds, local dynamics near equilibria and invariant manifolds, and the presence of chaotic motions. These results demonstrate the rich and complex dynamics of the 3D pendulum.


Eigenfunctions of transfer operators and cohomology

Abstract: The eigenfunctions with eigenvalues 1 or -1 of the transfer operator of Mayer are in bijective correspondence with the eigenfunctions with eigenvalue 1 of a transfer operator connected to the nearest integer continued fraction algorithm. This is shown by relating these eigenspaces of these operators to cohomology groups for the modular group with coefficients in certain principal series representations.


Fa\`a di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations

Abstract: We consider the combinatorial Dyson-Schwinger equation X=B^+(P(X)) in the non-commutative Connes-KreimerHopf algebra of planar rooted trees H, where B^+ is the operator of grafting on a root, and P a formal series. The unique solution X of this equation generates a graded subalgebra A_P of\H. We describe all the formal series P such that A_P is a Hopf subalgebra. We obtain in this way a 2-parameters family of Hopf subalgebras of H, organized into three isomorphism classes: a first one, restricted to a olynomial ring in one variable; a second one, restricted to the Hopf subalgebra of ladders, isomorphic to the Hopf algebra of quasi-symmetric functions; a last (infinite) one, which gives a non-commutative version of the Fa\`a di Bruno Hopf algebra. By taking the quotient, the last classe gives an infinite set of embeddings of the Fa\`a di Bruno algebra into the Connes-Kreimer Hopf algebra of rooted trees. Moreover, we give an embedding of the free Fa\`a di Bruno Hopf algebra on D variables into a Hopf algebra of decorated rooted trees, togetherwith a non commutative version of this embedding.


Gaussian Approximations of Multiple Integrals

Abstract: Fix an integer k, and let I(l), l=1,2,..., be a sequence of k-dimensional vectors of multiple Wiener-It\^o integrals with respect to a general Gaussian process. We establish necessary and sufficient conditions to have that, as l diverges, the law of I(l) is asymptotically close (for example, in the sense of Prokhorov's distance) to the law of a k-dimensional Gaussian vector having the same covariance matrix as I(l). The main feature of our results is that they require minimal assumptions (basically, boundedness of variances) on the asymptotic behaviour of the variances and covariances of the elements of I(l). In particular, we will not assume that the covariance matrix of I(l) is convergent. This generalizes the results proved in Nualart and Peccati (2005), Peccati and Tudor (2005) and Nualart and Ortiz-Latorre (2007). As shown in Marinucci and Peccati (2007b), the criteria established in this paper are crucial in the study of the high-frequency behaviour of stationary fields defined on homogeneous spaces.


An Infinite Family of Quadratic Quadrinomial APN Functions

Abstract: We present an infinite family of quadratic APN functions on a finite field of dimension over GF(2) divisible by 3.


Ubiquity and a general logarithm law for geodesics

Abstract: There are two fundamental results in the classical theory of metric Diophantine approximation: Khintchine's theorem and Jarnik's theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We start by discussing these theorems and show that they are both in fact a simple consequence of the notion of `local ubiquity'. The local ubiquity framework introduced here is a much simplified and more transparent version of that in \cite{memoirs}. Furthermore, it leads to a single local ubiquity theorem that unifies the Lebesgue and Hausdorff theories. As an application of our framework we consider the theory of metric Diophantine approximation on limit sets of Kleinian groups. In particular, we obtain a general Hausdorff measure version of Sullivan's logarithm law for geodesics -- an aspect overlooked in \cite{memoirs}.


Euler Scheme and Tempered Distributuions

Abstract: Given a smooth R^d-valued diffusion, we study how fast the Euler scheme with time step 1/n converges in law. To be precise, we look for which class of test functions f the approximate expectation E[f(X^{n,x}_1)] converges with speed 1/n to E[f(X^x_1)]. If X is uniformly elliptic, we show that this class contains all tempered distributions, and all measurable functions with exponential growth. We give applications to option pricing and hedging, proving numerical convergence rates for prices, deltas and gammas.


The geometry of the critical set of nonlinear periodic Sturm-Liouville operators

Abstract: We study the critical set C of the nonlinear differential operator F(u) = -u" + f(u) defined on a Sobolev space of periodic functions H^p(S^1), p >= 1. Let R^2_{xy} \subset R^3 be the plane z = 0 and, for n > 0, let cone_n be the cone x^2 + y^2 = tan^2 z, |z - 2 pi n| < pi/2; also set Sigma = R^2_{xy} U U_{n > 0} cone_n. For a generic smooth nonlinearity f: R -> R with surjective derivative, we show that there is a diffeomorphism between the pairs (H^p(S^1), C) and (R^3, Sigma) x H where H is a real separable infinite dimensional Hilbert space.


Affine systems: asymptotics at infinity for fractal measures

Abstract: We study measures on $\mathbb{R}^d$ which are induced by a class of infinite and recursive iterations in symbolic dynamics. Beginning with a finite set of data, we analyze prescribed recursive iteration systems, each involving subdivisions. The construction includes measures arising from affine and contractive iterated function systems with and without overlap (IFSs), i.e., limit measures $\mu$ induced by a finite family of affine mappings in $\mathbb{R}^d$ (the focus of our paper), as well as equilibrium measures in complex dynamics. By a systematic analysis of the Fourier transform of the measure $\mu$ at hand (frequency domain), we identify asymptotic laws, spectral types, dichotomy, and chaos laws. In particular we show that the cases when $\mu$ is singular carry a gradation, ranging from Cantor-like fractal measures to measures exhibiting chaos, i.e., a situation when small changes in the initial data produce large fluctuations in the outcome, or rather, the iteration limit (in this case the measures). Our method depends on asymptotic estimates on the Fourier transform of $\mu$ for paths at infinity in $\mathbb{R}^d$. We show how properties of $\mu$ depend on perturbations of the initial data, e.g., variations in a prescribed finite set of affine mappings in $\mathbb{R}^d$, in parameters of a rational function in one complex variable (Julia sets and equilibrium measures), or in the entries of a given infinite positive definite matrix.


Some intersection numbers of divisors on toroidal compactifications of A_g

Abstract: We study the top intersection numbers of the boundary and Hodge class divisors on toroidal compactifications of the moduli space $A_g$ of principally polarized abelian varieties and compute those numbers that live away from the stratum which lies over the closure of $A_{g-3}$ in the Satake compactification.


Knotted Polyhedral Tori

Abstract: For every knot K with stick number k there is a knotted polyhedral torus of knot type K with 3k vertices. We prove that at least 3k-2 vertices are necessary.


Monge-Ampere equations and moduli spaces of manifolds of circular type

Abstract: A (bounded) manifold of circular type is a complex manifold M of dimension n admitting a (bounded) exhaustive real function u, defined on M minus a point x_o, so that: a) it is a smooth solution on $M\setminus {x_o}$ to the Monge-Amp\`ere equation $(d d^c u)^n = 0$; b) x_o is a singular point for u of logarithmic type and e^u extends smoothly on the blow up of M at x_o; c) $d d^c (e^u) >0$ at any point of $M\setminus {x_o}$. This class of manifolds naturally includes all smoothly bounded, strictly linearly convex domains and all smoothly bounded, strongly pseudoconvex circular domains of $\bC^n$. The moduli spaces of bounded manifolds of circular type are studied. In particular, for each biholomorphic equivalence class of them it is proved the existence of an essentially unique manifold in normal form. It is also shown that the class of normalizing maps for an n-dimensional manifold M is a new holomorphic invariant with the following property: it is parameterized by the points of a finite dimensional real manifold of dimension n^2 when M is a (non-convex) circular domain while it is of dimension $n^2 + 2 n$ when M is a strictly convex domain. New characterizations of the circular domains and of the unit ball are also obtained.


Conformal quaternionic contact curvature and the local sphere theorem

Abstract: A tensor invariant is defined on a quaternionic contact manifold in terms of the curvature and torsion of the Biquard connection involving derivatives up to third order of the contact form. This tensor, called quaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry and to the Chern-Moser tensor in CR geometry. It is shown that a quaternionic contact manifold is locally quaternionic contact conformal to the standard flat quaternionic contact structure on the quaternionic Heisenberg group, or equivalently, to the standard 3-sasakian structure on the sphere iff the quaternionic contact conformal curvature vanishes.


Smoothness Theorem for Differential BV Algebras

Abstract: Associated to a differential BV algebra are two differential graded Lie algebras: we call one classical and the other, which contains a formal h-bar parameter, quantum. The classical dgLa is always smooth formal. In this paper, we give necessary and sufficient conditions for the quantum dgLa to be smooth formal. These conditions are equivalent to the degeneration of a version of the noncommutative Hodge to de Rham spectral sequence. References added.


Poisson transform for higher-rank graph algebras and its applications

Abstract: Higher-rank graph generalisations of the Popescu-Poisson transform are constructed, allowing us to develop a dilation theory for higher rank operator tuples. These dilations are joint dilations of the families of operators satisfying relations encoded by the graph structure which we call $\Lambda$-contractions or $\Lambda$-coisometries. Besides commutant lifting results and characterisations of pure states on higher rank graph algebras several applications to the structure theory of non-selfadjoint graph operator algebras are presented generalising recent results in special cases.


Harmonic morphisms and hyperelliptic graphs

Abstract: We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical Riemann-Hurwitz formula, study the functorial maps on Jacobians and harmonic 1-forms induced by a harmonic morphism, and present a discrete analogue of the canonical map from a Riemann surface to projective space. We also discuss several equivalent formulations of the notion of a hyperelliptic graph, all motivated by the classical theory of Riemann surfaces. As an application of our results, we show that for a 2-edge-connected graph G which is not a cycle, there is at most one involution $\iota$ on G for which the quotient $G/\iota$ is a tree. We also show that the number of spanning trees in a graph G is even if and only if G admits a non-constant harmonic morphism to the graph B_2 consisting of 2 vertices connected by 2 edges. Finally, we use the Riemann-Hurwitz formula and our results on hyperelliptic graphs to classify all hyperelliptic graphs having no Weierstrass points.


Multiplicity Bounds for Quadratic Monomial Ideals

Abstract: We prove the multiplicity bounds conjectured by Herzog-Huneke-Srinivasan and Herzog-Srinivasan in the following cases: the strong conjecture for edge ideals of bipartite graphs, and the weaker Taylor bound conjecture for all quadratic monomial ideals. We attach a directed graph to a bipartite graph with perfect matching, and describe operations on the directed graph that would reduce the problem to a Cohen-Macaulay bipartite graph. We determine when equality holds in the conjectured bound for edge ideals of bipartite graphs, and verify that when equality holds, the resolution is pure. We characterize bipartite graphs that have Cohen-Macaulay edge ideals and quasi-pure resolutions.


A dichotomy characterizing analytic digraphs of uncountable Borel chromatic number in any dimension

Abstract: We study the extension of the Kechris-Solecki-Todorcevic dichotomy on analytic graphs to dimensions higher than 2. We prove that the extension is possible in any dimension, finite or infinite. The original proof works in the case of the finite dimension. We first prove that the natural extension does not work in the case of the infinite dimension, for the notion of continuous homomorphism used in the original theorem. Then we solve the problem in the case of the infinite dimension. Finally, we prove that the natural extension works in the case of the infinite dimension, but for the notion of Baire-measurable homomorphism.


Surfaces from Circles

Abstract: In the search for appropriate discretizations of surface theory it is crucial to preserve such fundamental properties of surfaces as their invariance with respect to transformation groups. We discuss discretizations based on M\"obius invariant building blocks such as circles and spheres. Concrete problems considered in these lectures include the Willmore energy as well as conformal and curvature line parametrizations of surfaces. In particular we discuss geometric properties of a recently found discrete Willmore energy. The convergence to the smooth Willmore functional is shown for special refinements of triangulations originating from a curvature line parametrization of a surface. Further we treat special classes of discrete surfaces such as isothermic and minimal. The construction of these surfaces is based on the theory of circle patterns, in particular on their variational description.


Cartan and Berwald connections in the pullback formalism

Abstract: Adopting the pullback approach to global Finsler geometry, the aim of the present paper is to provide new intrinsic (coordinate-free) proofs of intrinsic versions of the existence and uniqueness theorems for the Cartan and Berwald connections on a Finsler manifold. To accomplish this, the notions of semispray and nonlinear connection associated with a given regular connection, in the pullback bundle, is introduced and investigated. Moreover, it is shown that for the Cartan and Berwald connections, the associated semispray coincides with the canonical spray and the associated nonlinear connection coincides with the Barthel connection. An explicit intrinsic expression relating both connections is deduced.


Theory of dimension for large discrete sets and applications

Abstract: We define two notions of discrete dimension based on the Minkowski and Hausdorff dimensions in the continuous setting. After proving some basic results illustrating these definitions, we apply this machinery to the study of connections between the Erdos and Falconer distance problems in geometric combinatorics and geometric measure theory, respectively.


The Norm Index Theorem (An Analytic Proof)

Abstract: We give an analytic proof of the norm index theorem $[I_:K^* N(I_L)] =[L:K]$ for cyclic extensions of number fields using spectral theory of the idele class group.


New Generalizations of Poisson Algebras

Abstract: We introduce many new generalizations of Poisson algebras which can be constructed inside the associative algebra of linear transformations over a vector space.


Proving the triviality of rational points on Atkin-Lehner quotients of Shimura curves

Abstract: In this paper we give a method for studying global rational points on certain quotients of Shimura curves by Atkin-Lehner involutions. We obtain explicit conditions on such quotients for rational points to be ``trivial'' (coming from CM points only) and exhibit an explicit infinite family of such quotients satisfying these conditions.


Functoriality and special values of L-functions

Abstract: This is a semi-expository article concerning Langlands functoriality and Deligne's conjecture on the special values of $L$-functions. The emphasis is on symmetric power $L$-functions associated to a holomorphic cusp form, while appealing to a recent work of Mahnkopf on the special values of automorphic $L$-functions.


l-Adic representations associated to modular forms over imaginary quadratic fields

Abstract: Let pi be a regular algebraic cuspidal automorphic representation of GL(2) over an imaginary quadratic number field K such that the central character of pi is invariant under the non-trivial automorphism of K. We show that pi is associated with an l-adic Galois representation rho over K such that at each prime of K outside an explicit finite set the Frobenius polynomial of rho agrees with the Hecke polynomial of pi.


Piecewise principal comodule algebras

Abstract: A comodule algebra P over a Hopf algebra H with bijective antipode is called principal if the coaction of H is Galois and P is H-equivariantly projective (faithfully flat) over the coaction-invariant subalgebra B. We prove that principality is a piecewise property: given N comodule-algebra surjections P->Pi whose kernels intersect to zero, P is principal if and only if all Pi's are principal. Furthermore, assuming the principality of P, we show that the lattice these kernels generate is distributive if and only if so is the lattice obtained by intersection with B. Finally, assuming the above distributivity property, we obtain a flabby sheaf of principal comodule algebras over a certain space that is universal for all such N-families of surjections P->Pi and such that the comodule algebra of global sections is P.


Mixed Lefschetz Theorems and Hodge-Riemann Bilinear Relations

Abstract: Statements analogous to the Hard Lefschetz Theorem (HLT) and the Hodge-Riemann bilinear relations (HRR) hold in a variety of contexts: they impose restrictions on the cohomology algebra of a smooth compact K\"ahler manifold or on the intersection cohomology of a projective toric variety; they restrict the local monodromy of a polarized variation of Hodge structure; they impose conditions on the possible $f$-vectors of convex polytopes. While the statements of these theorems depend on the choice of a K\"ahler class, or its analog, there is usually a cone of possible K\"ahler classes. It is then natural to ask whether the HLT and HRR remain true in a mixed context. In this note we present a unified approach to proving the mixed HLT and HRR, generalizing the previously known results, and proving it in new cases such as the intersection cohomology of non-rational polytopes.


Poincare duality and Periodicity

Abstract: We construct periodic families of Poincare complexes, partially solving a question of Hodgson that was posed in the proceedings of the 1982 Northwestern homotopy theory conference. We also construct infinite families of Poincare complexes whose top cell falls off after one suspension but which fail to embed in a sphere of codimension one. We give a homotopy theoretic description of the four-fold periodicity in knot cobordism.


Graphs of functions and vanishing free entropy

Abstract: Suppose X is an n-tuple of selfadjoint elements in a tracial von Neumann algebra M. If z is a selfadjoint element in M and for some selfadjoint element y in the von Neumann algebra generated by X $\delta_0(y, z) < \delta_0(y) + \delta_0(z)$, then $\chi(X \cup \{z\}) = -\infty$ (here $\chi$ and $\delta_0$ denote the microstates free entropy and free entropy dimension, respectively). In particular, if z lies in the von Neumann algebra generated by X, then $\chi(X \cup \{z\}) = -\infty$. The statement and its proof are motivated by geometric-measure-theoretic results on graphs of functions. A similar statement for the nonmicrostates free entropy is obtained under the much stronger hypothesis that z lies in the algebra generated by X.


A generalization of the Shestakov-Umirbaev inequality

Abstract: We give a generalization of the Shestakov-Umirbaev inequality which plays an important role in their solution of the tame generators conjecture.


Pareto Optima of Multicriteria Integer Linear Programs

Abstract: We settle the computational complexity of fundamental questions related to multicriteria integer linear programs, when the dimensions of the strategy space and of the outcome space are considered fixed constants. In particular we construct: 1. polynomial-time algorithms to exactly determine the number of Pareto optima and Pareto strategies; 2. a polynomial-space polynomial-delay prescribed-order enumeration algorithm for arbitrary projections of the Pareto set; 3. an algorithm to minimize the distance of a Pareto optimum from a prescribed comparison point with respect to arbitrary polyhedral norms; 4. a fully polynomial-time approximation scheme for the problem of minimizing the distance of a Pareto optimum from a prescribed comparison point with respect to the Euclidean norm.


Conditions for Generic Initial Ideals to be Almost Reverse Lexicographic

Abstract: Let $I$ be a homogeneous Artinian ideal in a polynomial ring $R=k[x_1,...,x_n]$ over a field $k$ of characteristic 0. We study an equivalent condition for the generic initial ideal $\gin(I)$ with respect to reverse lexicographic order to be almost reverse lexicographic. As a result, we show that Moreno-Socias conjecture implies Fr\"{o}berg conjecture. And for the case $\Codim I \le 3$, we show that $R/I$ has the strong Lefschetz property if and only if $\gin(I)$ is almost reverse lexicographic. Finally for a monomial complete intersection Artinian ideal $I=(x_1^{d_1},...,x_n^{d_n})$, we prove that $\gin(I)$ is almost reverse lexicographic if $d_i > \sum_{j=1}^{i-1} d_j - i + 1$ for each $i \ge 4$. Using this, we give a positive partial answer to Moreno-Socias conjecture, and to Fr\"{o}berg conjecture.


Generalized Bounded Variation and Inserting point masses

Abstract: Let $d\mu$ be a probability measure on the unit circle and $d\nu$ be the measure formed by adding a pure point to $d\mu$. We give a simple formula for the Verblunsky coefficients of $d\nu$ based on a result of Simon. Then we consider $d\mu_0$, a probability measure on the unit circle with $\ell^2$ Verblunsky coefficients $(\alpha_n (d\mu_0))_{n=0}^{\infty}$ of bounded variation. We insert $m$ pure points to $d\mu$, rescale, and form the probability measure $d\mu_m$. We use the formula above to prove that the Verblunsky coefficients of $d\mu_m$ are in the form $\alpha_n(d\mu_0) + \sum_{j=1}^m \frac{\ol{z_j}^{n} c_j}{n} + E_n$, where the $c_j$'s are constants of norm 1 independent of the weights of the pure points and independent of $n$; the error term $E_n$ is in the order of $o(1/n)$. Furthermore, we prove that $d\mu_m$ is of $(m+1)$-generalized bounded variation - a notion that we shall introduce in the paper. Then we use this fact to prove that $\lim_{n \to \infty} \vp_n^*(z, d\mu_m)$ is continuous and is equal to $D(z, d\mu_m)^{-1}$ away from the pure points.


Szego kernels, Toeplitz operators, and equivariant fixed point formulae

Abstract: Let $\gamma$ be an automorphism of a polarized complex projective manifold $(M,L)$. Then $\gamma$ induces an automorphism $\gamma_k$ of the space of global holomorphic sections of the $k$-th tensor power of $L$, for every $k=1,2,...$; for $k\gg 0$, the Lefschetz fixed point formula expresses the trace of $\gamma_k$ in terms of fixed point data. More generally, one may consider the composition of $\gamma_k$ with the Toeplitz operator associated to some smooth function on $M$. Still more generally, in the presence of the compatible action of a compact and connected Lie group preserving $(M,L,\gamma)$, one may consider induced linear maps on the equivariant summands associated to the irreducible representations of $G$. In this paper, under familiar assumptions in the theory of symplectic reductions, we show that the traces of these maps admit an asymptotic expansion as $k\to +\infty$, and compute its leading term.


The characteristic quasi-polynomials of the arrangements of root systems and mid-hyperplane arrangements

Abstract: Let $q$ be a positive integer. In our recent paper, we proved that the cardinality of the complement of an integral arrangement, after the modulo $q$ reduction, is a quasi-polynomial of $q$, which we call the characteristic quasi-polynomial. In this paper, we study general properties of the characteristic quasi-polynomial as well as discuss two important examples: the arrangements of reflecting hyperplanes arising from irreducible root systems and the mid-hyperplane arrangements. In the root system case, we present a beautiful formula for the generating function of the characteristic quasi-polynomial which has been essentially obtained by Ch. Athanasiadis and by A. Blass and B. Sagan. On the other hand, it is hard to find the generating function of the characteristic quasi-polynomial in the mid-hyperplane arrangement case. We determine them when the dimension is less than six.


Asymptotically Optimal Estimator of the Parameter of Semi-Linear Autoregression

Abstract: The difference equations $\xi_{k}=af(\xi_{k-1})+\epsilon_{k}$, where $(\epsilon_k)$ is a square integrable difference martingale, and the differential equation ${\rm d}\xi=-af(\xi){\rm d}t+{\rm d}\eta$, where $\eta$ is a square integrable martingale, are considered. A family of estimators depending, besides the sample size $n$ (or the observation period, if time is continuous) on some random Lipschitz functions is constructed. Asymptotic optimality of this estimators is investigated.


Equisingularity classes of birational projections of normal singularities to a plane

Abstract: Given a birational normal extension S of a two-dimensional local regular ring R, we describe all the equisingularity types of the complete ideals J in R whose blowing-up has some point at which the local ring is analytically isomorphic to S. The problem of classifying the germs of such normal surface singularities was already posed by Spivakovsky (Ann. of Math. 1990). This problem has two parts: discrete and continous. The continous part is to some extent equivalent to the problem of the moduli of plane curve singularities, while the main result of this paper solves completely the discrete part.


The AS-Cohen-Macaulay property for quantum flag manifolds of minuscule weight

Abstract: It is shown that quantum homogeneous coordinate rings of generalised flag manifolds corresponding to minuscule weights, their Schubert varieties, big cells, and determinantal varieties are AS-Cohen-Macaulay. The main ingredient in the proof is the notion of a quantum graded algebra with a straightening law, introduced by T.H. Lenagan and L. Rigal [J. Algebra 301 (2006), 670-702]. Using Stanley's Theorem it is moreover shown that quantum generalised flag manifolds of minuscule weight and their big cells are AS-Gorenstein.


Probability Measures and Effective Randomness

Abstract: We study the question, ``For which reals $x$ does there exist a measure $\mu$ such that $x$ is random relative to $\mu$?'' We show that for every nonrecursive $x$, there is a measure which makes $x$ random without concentrating on $x$. We give several conditions on $x$ equivalent to there being continuous measure which makes $x$ random. We show that for all but countably many reals $x$ these conditions apply, so there is a continuous measure which makes $x$ random. There is a meta-mathematical aspect of this investigation. As one requires higher arithmetic levels in the degree of randomness, one must make use of more iterates of the power set of the continuum to show that for all but countably many $x$'s there is a continuous $\mu$ which makes $x$ random to that degree.


Continuous first-passage percolation and continuous greedy paths model: linear growth

Abstract: We study a random growth model on $\R^d$ introduced by Deijfen. This is a continuous first-passage percolation model. The growth occurs by means of spherical outbursts with random radii in the infected region. We aim at finding conditions on the distribution of the random radii to determine whether the growth of the process is linear or not. To do so, we compare this model with a continuous analogue of the greedy lattice paths model and transpose results in the lattice setting to the continuous setting.


Lower bounds for sup + inf and sup * inf and an Extension of Chen-Lin result in dimension 3

Abstract: We give two results about Harnack type inequalities. First, on compact smooth Riemannian surface without boundary, we have an estimate of the type $\sup +\inf$. The second result concerns the solutions of prescribed scalar curvature equation on the unit ball of ${\mathbb R}^n$ with Dirichlet condition. Next, we give an inequality of the type $(\sup_K u)^{2s-1} \times \inf_{\Omega} u \leq c$ for positive solutions of $\Delta u=Vu^5$ on $\Omega \subset {\mathbb R}^3$, where $K$ is a compact set of $\Omega$ and $V$ is $s-$ h\"olderian, $s\in ]-1/2,1]$. For the case $s=1/2$, we prove that if $\min_{\Omega} u>m>0$ and the h\"olderian constant $A$ of $V$ is small enough (in certain meaning), we have the uniform boundedness of the supremum of the solutions of the previous equation on any compact set of $\Omega$. ----- Nous donnons quelques estimations des solutions d'equations elliptiques sur les surfaces de Riemann et sur des ouverts en dimension n> 2. Nous traitons le cas holderien pour l'equation de la courbure scalaire prescrite en dimension 3.


Corings with exact rational functors and injective objects

Abstract: We describe how some aspects of abstract localization on module categories have applications to the study of injective comodules over some special types of corings. We specialize the general results to the case of Doi-Koppinen modules, generalizing previous results in this setting.


Uniqueness at infinity in time for the Maxwell-Schr"odinger system with arbitrarily large asymptotic data

Abstract: We prove the uniqueness of solutions of the Maxwell-Schr"odinger system with given asymptotic behaviour at infinity in time. The assumptions include suitable restrictions on the growth of solutions for large time and on the accuracy of their asymptotics, but no restriction on their size. The result applies to the solutions with prescribed asymptotics constructed in a previous paper.


Module Shifts and Measure Rigidity in Linear Cellular Automata

Abstract: Suppose R is a finite commutative ring of prime characteristic, A is a finite R-module, M:=Z^D x N^E, and F is an R-linear cellular automaton on A^M. If mu is an F-invariant measure which is multiply shift-mixing in a certain way, then we show that mu must be the Haar measure on a coset of some submodule shift of A^M. Under certain conditions, this means mu must be the uniform Bernoulli measure on A^M.


Constructing Simplicial Branched Covers

Abstract: Branched covers are applied frequently in topology - most prominently in the construction of closed oriented PL d-manifolds. In particular, strong bounds for the number of sheets and the topology of the branching set are known for dimension d<=4. On the other hand, Izmestiev and Joswig described how to obtain a simplicial covering space (the partial unfolding) of a given simplicial complex, thus obtaining a simplicial branched cover [Adv. Geom. 3(2):191-255, 2003]. We present a large class of branched covers which can be constructed via the partial unfolding. In particular, for d<=4 every closed oriented PL d-manifold is the partial unfolding of some polytopal d-sphere.


On natural and conformally equivariant quantizations

Abstract: The concept of conformally equivariant quantizations was introduced by Duval, Lecomte and Ovsienko in \cite{DLO} for manifolds endowed with flat conformal structures. They obtained results of existence and uniqueness (up to normalization) of such a quantization procedure. A natural generalization of this concept is to seek for a quantization procedure, over a manifold $M$, that depends on a pseudo-Riemannian metric, is natural and is invariant with respect to a conformal change of the metric. The existence of such a procedure was conjectured by P. Lecomte in \cite{Leconj} and proved by C. Duval and V. Ovsienko in \cite{DO1} for symbols of degree at most 2 and by S. Loubon Djounga in \cite{Loubon} for symbols of degree 3. In two recent papers \cite{MR,MR1}, we investigated the question of existence of projectively equivariant quantizations using the framework of Cartan connections. Here we will show how the formalism developed in these works adapts in order to deal with the conformally equivariant quantization for symbols of degree at most 3. This will allow us to easily recover the results of \cite{DO1} and \cite{Loubon}. We will then show how it can be modified in order to prove the existence of conformally equivariant quantizations for symbols of degree 4.


Constructing Combinatorial 4-Manifolds

Abstract: Every closed oriented PL 4-manifold is a branched cover of the 4-sphere branched over a PL-surface with finitely many singularities by Piergallini [Topology 34(3):497-508, 1995]. This generalizes a long standing result by Hilden and Montesinos to dimension four. Izmestiev and Joswig [Adv. Geom. 3(2):191-225, 2003] gave a combinatorial equivalent of the Hilden and Montesinos result, constructing closed oriented combinatorial 3-manifolds as simplicial branched covers of combinatorial 3-spheres. The construction of Izmestiev and Joswig is generalized and applied to the result of Piergallini, obtaining closed oriented combinatorial 4-manifolds as simplicial branched covers of simplicial 4-spheres.


Characteristic Functions of Liftings

Abstract: We introduce characteristic functions for certain contractive liftings of row contractions. These are multi-analytic operators which classify the liftings up to unitary equivalence and provide a kind of functional model. The most important cases are subisometric and coisometric liftings. We also identify the most general setting which we call reduced liftings. We derive properties of these new characteristic functions and discuss the relation to Popescu's definition for completely non-coisometric row contractions. Finally we apply our theory to completely positive maps and prove a one-to-one correspondence between the fixed point sets of completely positive maps related to each other by a subisometric lifting.


On the transformation semitopological semigroup

Abstract: In this paper we introduce the notion of weighted (weakly) almost periodic compactifcation of a semitopological semigroup and generalize this notion to corresponding notion for transformation semigroup.The inclusion relation and equality of some well known function spaces on a weighted transformation semigroup is also investigated.


Parastrophic invariance of Smarandache quasigroups

Abstract: Every quasigroup $(L,\cdot)$ belongs to a set of 6 quasigroups, called parastrophes denoted by $(L,\pi_i)$, $i\in \{1,2,3,4,5,6\}$. It is shown that $(L,\pi_i)$ is a Smarandache quasigroup with associative subquasigroup $(S,\pi_i) \forall i\in \{1,2,3,4,5,6\}$ if and only if for any of some four $j\in \{1,2,3,4,5,6\}$, $(S,\pi_j)$ is an isotope of $(S,\pi_i)$ or $(S,\pi_k)$ for one $k\in \{1,2,3,4,5,6\}$ such that $i\ne j\ne k$. Hence, $(L,\pi_i)$ is a Smarandache quasigroup with associative subquasigroup $(S,\pi_i) \forall i\in \{1,2,3,4,5,6\}$ if and only if any of the six Khalil conditions is true for any of some four of $(S,\pi_i)$.


Smarandache Isotopy Theory Of Smarandache: Quasigroups And Loops

Abstract: The concept of Smarandache isotopy is introduced and its study is explored for Smarandache: groupoids, quasigroups and loops just like the study of isotopy theory was carried out for groupoids, quasigroups and loops. The exploration includes: Smarandache; isotopy and isomorphy classes, Smarandache $f,g$ principal isotopes and G-Smarandache loops.


Tangential projections and secant defective varieties

Abstract: Going one step further in Zak's classification of Scorza varieties with secant defect equal to one, we characterize the Veronese embedding of $\P^n$ given by the complete linear system of quadrics and its smooth projections from a point as the only smooth irreducible complex and non-degenerate projective subvarieties of $\P^N$ that can be projected isomorphically into $\P^{2n}$ when $N\geq\binom{n+2}{2}-2$.


On the Derivatives of Central Loops

Abstract: The right(left) derivative, $a^{-1},e-$ and $e,a^{-1}-$ isotopes of a C-loop are shown to be C-loops. Furthermore, for a central loop $(L,F)$, it is shown that $\big\{F,F^{a^{-1}},F_{a^{-1},e}\big\}$ and $\big\{F,F_{a^{-1}},F_{e,a^{-1}}\big\}$ are systems of isotopic C-loops that obey a form of generalized distributive law. Quasigroup isotopes $(L,\otimes)$ and $(L,\ominus)$ of a loop $(L,\theta)$ and its parastrophe $(L,\theta ^*)$ respectively are proved to be isotopic if either $(L,\otimes)$ or $(L,\ominus )$ is commutative. If $(L,\theta)$ is a C-loop, then it is shown that $\big\{(L,\theta),(L,\theta ^*),(L,\otimes),(L,\oplus)\big\}$ is a system of isotopic C-quasigroup under the above mentioned condition. It is shown that C-loops are isotopic to some finite indecomposable groups of the classes ${\cal D}_i,i=1,2,3,4,5$ and that the center of such C-loops have a rank of 1,2 or 3.


On Isotopic Characterization of Central Loops

Abstract: The representation sets of central loops are investigated and the results obtained are used to construct a finite C-loop. It is shown that for certain types of isotopisms, the central identities are isotopic invariant.