Finite speed of propagations of the electromagnetic field in nonlinear isotropic dispersive mediums

Abstract: We propose some modification of Maxwell's equations describing mediums which electric and magnetic properties are changed essentially after interaction with outer electromagnetic field. We show for such mediums that electromagnetic waves have finite speed of propagations property for some time depending on initial energy of electromagnetic field and nonlinear parameters of the problem which are responsible for properties of medium.


Approximate and pseudo-amenability of various classes of Banach algebras

Abstract: We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors. It is shown that every boundedly approximately contractible Banach algebra has a bounded approximate identity. Among our other results, it is shown that the Fourier algebra of the free group on two generators is not approximately amenable. Further examples are obtained of ${\ell}^1$-semigroup algebras which are approximately amenable but not amenable; using these, we show that bounded approximate amenability need not imply sequential approximate amenability. Results are also given for Segal subalgebras of $L^1(G)$, where $G$ is a locally compact group, and the algebras $PF_p(\Gamma)$ of $p$-pseudofunctions on a discrete group $\Gamma$ (of which the reduced $C^*$-algebra is a special case).


Convergence of finite-dimensional laws of the weighted quadratic variations process for some fractional Brownian sheets

Abstract: In this paper we state and prove a central limit theorem for the finite-dimensional laws of the quadratic variations process of certain fractional Brownian sheets. The main tool of this article is a method developed by Nourdin and Nualart based on the Malliavin calculus.


Automorphism groups of algebraic curves with p-rank zero

Abstract: In positive characteristic, algebraic curves can have many more automorphisms than expected from the classical Hurwitz's bound. There even exist algebraic curves of arbitrary high genus g with more than 16g^4 automorphisms. It has been observed on many occasions that the most anomalous examples invariably have zero p-rank. In this paper, the K-automorphism group Aut(X) of a zero 2-rank algebraic curve X defined over an algebraically closed field K of characteristic 2 is investigated. The main result is that if the curve has genus g greater than or equal to 2, and |Aut(X)|>24g^2, then Aut(X) has a fixed point on X, apart from few exceptions. In the exceptional cases the possibilities for Aut(X) and g are determined.


The modified Calabi-Yau problems for CR-manifolds and applications

Abstract: In this paper, we derive a partial result related to a question of Yau: "Does a simply-connected complete K\"ahler manifold M with negative sectional curvature admit a bounded non-constant holomorphic function?" Main Theorem. Let $M^{2n}$ be a simply-connected complete K\"ahler manifold M with negative sectional curvature $ \le -1 $ and $S_\infty(M)$ be the sphere at infinity of $M$. Then there is an explicit {\it bounded} contact form $\beta$ defined on the entire manifold $M^{2n}$. Consequently, the sphere $S_\infty(M)$ at infinity of M admits a {\it bounded} contact structure and a bounded pseudo-Hermitian metric in the sense of Tanaka-Webster. We also discuss several open modified problems of Calabi and Yau for Alexandrov spaces and CR-manifolds.


Finding central decompositions of p-groups

Abstract: Polynomial-time algorithms are given to find a central decomposition of maximum size for a finite p-group of class 2 and for a nilpotent Lie ring of class 2. The algorithms use Las Vegas probabilistic routines to compute the structure of finite *-rings and also the Las Vegas C-MeatAxe. When p is small, the probabilistic methods can be replaced by deterministic polynomial-time algorithms. The methods introduce new group isomorphism invariants including new characteristic subgroups.


Bayesian models to adjust for response bias in survey data for estimating rape and domestic violence rates from the NCVS

Abstract: It is difficult to accurately estimate the rates of rape and domestic violence due to the sensitive nature of these crimes. There is evidence that bias in estimating the crime rates from survey data may arise because some women respondents are "gagged" in reporting some types of crimes by the use of a telephone rather than a personal interview, and by the presence of a spouse during the interview. On the other hand, as data on these crimes are collected every year, it would be more efficient in data analysis if we could identify and make use of information from previous data. In this paper we propose a model to adjust the estimates of the rates of rape and domestic violence to account for the response bias due to the "gag" factors. To estimate parameters in the model, we identify the information that is not sensitive to time and incorporate this into prior distributions. The strength of Bayesian estimators is their ability to combine information from long observational records in a sensible way. Within a Bayesian framework, we develop an Expectation-Maximization-Bayesian (EMB) algorithm for computation in analyzing contingency table and we apply the jackknife to estimate the accuracy of the estimates. Our approach is illustrated using the yearly crime data from the National Crime Victimization Survey. The illustration shows that compared with the classical method, our model leads to more efficient estimation but does not require more complicated computation.


Quenched convergence of a sequence of superprocesses in R^d among Poissonian obstacles

Abstract: We prove a convergence theorem for a sequence of super-Brownian motions moving among hard Poissonian obstacles, when the intensity of the obstacles grows to infinity but their diameters shrink to zero in an appropriate manner. The superprocesses are shown to converge in probability for the law $\mathbf{P}$ of the obstacles, and $\mathbf{P}$-almost surely for a subsequence, towards a superprocess with underlying spatial motion given by Brownian motion and (inhomogeneous) branching mechanism $\psi(u,x)$ of the form $\psi(u,x)= u^2+ \kappa(x)u$, where $\kappa(x)$ depends on the density of the obstacles. This work draws on similar questions for a single Brownian motion. In the course of the proof, we establish precise estimates for integrals of functions over the Wiener sausage, which are of independent interest.


A Characteristic Map for Symplectic Manifolds

Abstract: We construct a local characteristic map to a symplectic manifold M via certain cohomology groups of Hamiltonian vector fields. For each p in M, the Leibniz cohomology of the Hamiltonian vector fields on R^{2n} maps to the Leibniz cohomology of all Hamiltonian vector fields on M. For a particular extension g_n of the symplectic Lie algebra, the Leibniz cohomology of g_n is shown to be an exterior algebra on the canonical symplectic two-form. The Leibniz homology of g_n then maps to the Leibniz homology of Hamiltonian vector fields on R^{2n}.


Stability of bounded global solutions for Navier-Stokes equations

Abstract: In this paper some kind of asymptotic behavior of the solutions for the Navier-Stokes system on abstract Banach spaces is studied under the existence of global in time solutions. The asymptotic stability of the zero solution is also shown.


Group actions and rational ideals

Abstract: We develop the theory of rational ideals for arbitrary associative algebras R without assuming the standard finiteness conditions, noetherianness or the Goldie property. The Amitsur-Martindale ring of quotients replaces the classical ring of quotients which underlies the previous definition of rational ideals but is not available in a general setting. Our main result concerns rational actions of an affine algebraic group G on R. Working over an algebraically closed base field, we prove an existence and uniqueness result for generic rational ideals: for every G-rational ideal I of R, the closed subset of the rational spectrum Rat R that is defined by I is the closure of a unique G-orbit in Rat R. Under additional Goldie hypotheses, this was established earlier by Moeglin and Rentschler (in characteristic zero) and by Vonessen (in arbitrary characteristic), answering a question of Dixmier.


A Modified K\"ahler-Ricci Flow

Abstract: In this note, a modified K\"ahler-Ricci flow is introduced and studied. The main point is to show the flexibility of K\"ahler-Ricci flow and summarize some useful techniques.


Climbing a Legendrian mountain range without Stabilization

Abstract: We introduce a new braid-theoretic framework with which to understand the Legendrian and transversal classification of knots, namely a Legendrian Markov Theorem without Stabilization which induces an associated transversal Markov Theorem without Stabilization. We establish the existence of a nontrivial knot-type specific Legendrian and transversal MTWS by enhancing the Legendrian mountain range for the (2,3)-cable of a (2,3)-torus knot provided by Etnyre and Honda, and showing that elementary negative flypes allow us to move toward maximal tb value without having to use Legendrian stabilization. In doing so we obtain new ways to visualize convex tori and Legendrian divides and rulings, using tilings and braided rectangular diagrams.


Rayleigh's Stretched String

Abstract: We obtain rigorous a priori upper and lower bounds to the exact period of the celebrated Rayleigh stretched string differential equation.


Beyond Thresholding: Analysis and Improvements for Deterministic Parameter Estimation

Abstract: Hard-threshold estimators are popular in signal processing applications. We provide a detailed study of using hard-threshold estimators for estimating an unknown deterministic signal when additive white Gaussian noise corrupts observations. The analysis, depending heavily on Cram{\'e}r-Rao bounds, motivates piecewise-linear estimation as a simple improvement to hard thresholding. We compare the performance of two piecewise-linear estimators to a hard-threshold estimator. When either piecewise-linear estimator is optimized for the decay rate of the basis coefficients, its performance is better than the best possible with hard thresholding.


On the appearance of Eisenstein series through degeneration

Abstract: Let $\Gamma$ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane $\mathbb H$, and let $M = \Gamma \backslash \mathbb H$ be the associated finite volume hyperbolic Riemann surface. If $\gamma$ is parabolic, there is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If $\gamma$ is hyperbolic, then, following ideas due to Kudla-Millson, there is a corresponding hyperbolic Eisenstein series. In this article, we study the limiting behavior of parabolic and hyperbolic Eisenstein series on a degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. If $\gamma \in \Gamma$ corresponds to a degenerating hyperbolic element, then a multiple of the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface.


Sheaves on local Calabi-Yau varieties

Abstract: We investigate sheaves supported on the zero section of the total space of a locally-free sheaf E on a smooth, projective variety X when the top exterior power of E is isomorphic to the canonical bundle of X. We rephrase this construction using the language of A-infinity algebra and provide a simple characterisation of the case E is simply the canonical bundle itself.


Perelman's W-functional and stability of K\"ahler-Ricci flow

Abstract: In this expository note, we study the second variation of Perelman's entropy on the space of Kahler metrics at a K\"ahler-Ricci soliton. We prove that the entropy is stable in the sense of variations. In particular, Perelman's entropy is stable along the K\"ahler-Ricci flow. The Chinese version of this note has appeared in a volume in honor of professor K.C.Chang (Scientia Sinica Math., 46 (2016), 685-696).


Harmonic Analysis of Stochastic Equations and Backward Stochastic Differential Equations

Abstract: The BMO martingale theory is extensively used to study nonlinear multi-dimensional stochastic equations (SEs) in $\cR^p$ ($p\in [1, \infty)$) and backward stochastic differential equations (BSDEs) in $\cR^p\times \cH^p$ ($p\in (1, \infty)$) and in $\cR^\infty\times \bar{\cH^\infty}^{BMO}$, with the coefficients being allowed to be unbounded. In particular, the probabilistic version of Fefferman's inequality plays a crucial role in the development of our theory, which seems to be new. Several new results are consequently obtained. The particular multi-dimensional linear case for SDEs and BSDEs are separately investigated, and the existence and uniqueness of a solution is connected to the property that the elementary solutions-matrix for the associated homogeneous SDE satisfies the reverse H\"older inequality for some suitable exponent $p\ge 1$. Finally, we establish some relations between Kazamaki's quadratic critical exponent $b(M)$ of a BMO martingale $M$ and the spectral radius of the solution operator for the $M$-driven SDE, which lead to a characterization of Kazamaki's quadratic critical exponent of BMO martingales being infinite.


Some results for the Perelman LYH-type inequality

Abstract: Let $(M,g(t))$, $0\le t\le T$, $\partial M\ne\phi$, be a compact $n$-dimensional manifold, $n\ge 2$, with metric $g(t)$ evolving by the Ricci flow such that the second fundamental form of $\partial M$ with respect to the unit outward normal of $\partial M$ is uniformly bounded below on $\partial M\times [0,T]$. We will prove a global Li-Yau gradient estimate for the solution of the generalized conjugate heat equation on $M\times [0,T]$. We will give another proof of Perelman's Li-Yau-Hamilton type inequality for the fundamental solution of the conjugate heat equation on closed manifolds without using the properties of the reduced distance. We will also prove various gradient estimates for the Dirichlet fundamental solution of the conjugate heat equation.


Admissible local systems for a class of line arrangements

Abstract: A rank one local system $\LL$ on a smooth complex algebraic variety $M$ is admissible roughly speaking if the dimension of the cohomology groups $H^m(M,\LL)$ can be computed directly from the cohomology algebra $H^*(M,\C)$. We say that a line arrangement $\A$ is of type $\CC_k$ if $k \ge 0 $ is the minimal number of lines in $\A$ containing all the points of multiplicity at least 3. We show that if $\A$ is a line arrangement in the classes $\CC_k$ for $k\leq 2$, then any rank one local system $\LL$ on the line arrangement complement $M$ is admissible. Partial results are obtained for the class $\CC_3$.


On some difficulties with a posterior probability approximation technique

Abstract: In Scott (2002) and Congdon (2006), a new method is advanced to compute posterior probabilities of models under consideration. It is based solely on MCMC outputs restricted to single models, i.e., it is bypassing reversible jump and other model exploration techniques. While it is indeed possible to approximate posterior probabilities based solely on MCMC outputs from single models, as demonstrated by Gelfand and Dey (1994) and Bartolucci et al. (2006), we show that the proposals of Scott (2002) and Congdon (2006) are biased and advance several arguments towards this thesis, the primary one being the confusion between model-based posteriors and joint pseudo-posteriors. From a practical point of view, the bias in Scott's (2002) approximation appears to be much more severe than the one in Congdon's (2006), the later being often of the same magnitude as the posterior probability it approximates, although we also exhibit an example where the divergence from the true posterior probability is extreme.


Nonlinear Schr\"odinger equation on real hyperbolic spaces

Abstract: We consider the Schr\"odinger equation with no radial assumption on real hyperbolic spaces. We obtain sharp dispersive and Strichartz estimates for a large family of admissible pairs. As a first consequence, we get strong well-posedness results for NLS. Specifically, for small intial data, we prove $L^2$ and $H^1$ global well-posedness for any subcritical nonlinearity (in contrast with the Euclidean case) and with no gauge invariance assumption on the nonlinearity $F$. On the other hand, if $F$ is gauge invariant, $L^2$ charge is conserved and hence, as in the Euclidean case, it is possible to extend local $L^2$ solutions to global ones. The corresponding argument in $H^1$ requires the conservation of energy, which holds under the stronger condition that $F$ is defocusing. Recall that global well-posedness in the gauge invariant case was already proved by Banica, Carles & Staffilani, for small radial $L^2$ data and for large radial $H^1$ data. The second important application of our global Strichartz estimates is "scattering" for NLS both in $L^2$ and in $H^1$, with no radial or gauge invariance assumption. Notice that, in the Euclidean case, this is only possible for the critical power $\gamma=1+\frac4n$ and can be false for subcritical powers while, on hyperbolic spaces, global existence and scattering of small $L^2$ solutions holds for all powers $1<\gamma\le1+\frac4n$. If we restrict to defocusing nonlinearities $F$, we can extend the $H^1$ scattering results of Banica, Carles & Staffilani to the nonradial case. Also there is no distinction anymore between short range and long range nonlinearity : the geometry of hyperbolic spaces makes every power-like nonlinearity short range.


On some smooth projective two-orbits varieties with Picard number 1

Abstract: We classify all smooth projective horospherical varieties with Picard number 1. We prove that the automorphism group of any such variety X acts with at most two orbits and that this group still acts with only two orbits on X blown up at the closed orbit. We characterize all smooth projective two-orbits varieties with Picard number 1 that satisfy this latter property.


Uniforming n-place functions on ds(alpha)

Abstract: In this paper the Erdos-Rado theorem is generalized to the class of well founded trees.


Geometric structures on loop and path spaces

Abstract: Is is known that the loop space associated to a Riemannian manifold admits a quasi-symplectic structure. This article shows that this structure is not likely to recover the underlying Riemannian metric by proving a result that is a strong indication of the "almost" independence of the quasi-symplectic structure with respect to the metric. Finally conditions to have contact structures on these spaces are studied.


Motivic and quantum invariance under stratified Mukai flops

Abstract: For stratified Mukai flops of type $A_{n,k}, D_{2k+1}$ and $E_{6,I}$, it is shown the fiber product induces isomorphisms on Chow motives. In contrast to (standard) Mukai flops, the cup product is generally not preserved. For $A_{n, 2}$, $D_5$ and $E_{6, I}$ flops, quantum corrections are found through degeneration/deformation to ordinary flops.


A statistical analysis of probabilistic counting algorithms

Abstract: This paper considers the problem of cardinality estimation in data stream applications. We present a statistical analysis of probabilistic counting algorithms, focusing on two techniques that use pseudo-random variates to form low-dimensional data sketches. We apply conventional statistical methods to compare probabilistic algorithms based on storing either selected order statistics, or random projections. We derive estimators of the cardinality in both cases, and show that the maximal-term estimator is recursively computable and has exponentially decreasing error bounds. Furthermore, we show that the estimators have comparable asymptotic efficiency, and explain this result by demonstrating an unexpected connection between the two approaches.


Majorizing measures and proportional subsets of bounded orthonormal systems

Abstract: In this article we prove that for any orthonormal system $(\vphi_j)_{j=1}^n \subset L_2$ that is bounded in $L_{\infty}$, and any $1 < k <n$, there exists a subset $I$ of cardinality greater than $n-k$ such that on $\spa\{\vphi_i\}_{i \in I}$, the $L_1$ norm and the $L_2$ norm are equivalent up to a factor $\mu (\log \mu)^{5/2}$, where $\mu = \sqrt{n/k} \sqrt{\log k}$. The proof is based on a new estimate of the supremum of an empirical process on the unit ball of a Banach space with a good modulus of convexity, via the use of majorizing measures.


Efficient l_{alpha} Distance Approximation for High Dimensional Data Using alpha-Stable Projection

Abstract: In recent years, large high-dimensional data sets have become commonplace in a wide range of applications in science and commerce. Techniques for dimension reduction are of primary concern in statistical analysis. Projection methods play an important role. We investigate the use of projection algorithms that exploit properties of the alpha-stable distributions. We show that l_{alpha} distances and quasi-distances can be recovered from random projections with full statistical efficiency by L-estimation. The computational requirements of our algorithm are modest; after a once-and-for-all calculation to determine an array of length k, the algorithm runs in O(k) time for each distance, where k is the reduced dimension of the projection.


Stability of hypersurfaces with constant $r$-th anisotropic mean curvature

Abstract: Given a positive function $F$ on $S^n$ which satisfies a convexity condition, we define the $r$-th anisotropic mean curvature function $H^F_r$ for hypersurfaces in $\mathbb{R}^{n+1}$ which is a generalization of the usual $r$-th mean curvature function. Let $X:M\to \mathbb{R}^{n+1}$ be an $n$-dimensional closed hypersurface with $H^F_{r+1}=$constant, for some $r$ with $0\leq r\leq n-1$, which is a critical point for a variational problem. We show that $X(M)$ is stable if and only if $X(M)$ is the Wulff shape.


A concentration inequality for interval maps with an indifferent fixed point

Abstract: For a map of the unit interval with an indifferent fixed point, we prove an upper bound for the variance of all observables of $n$ variables $K:[0,1]^n\to\R$ which are componentwise Lipschitz. The proof is based on coupling and decay of correlation properties of the map. We then give various applications of this inequality to the almost-sure central limit theorem, the kernel density estimation, the empirical measure and the periodogram.


Detecting rigid convexity of bivariate polynomials

Abstract: Given a polynomial $x \in {\mathbb R}^n \mapsto p(x)$ in $n=2$ variables, a symbolic-numerical algorithm is first described for detecting whether the connected component of the plane sublevel set ${\mathcal P} = \{x : p(x) \geq 0\}$ containing the origin is rigidly convex, or equivalently, whether it has a linear matrix inequality (LMI) representation, or equivalently, if polynomial $p(x)$ is hyperbolic with respect to the origin. The problem boils down to checking whether a univariate polynomial matrix is positive semidefinite, an optimization problem that can be solved with eigenvalue decomposition. When the variety ${\mathcal C} = \{x : p(x) = 0\}$ is an algebraic curve of genus zero, a second algorithm based on B\'ezoutians is proposed to detect whether $\mathcal P$ has an LMI representation and to build such a representation from a rational parametrization of $\mathcal C$. Finally, some extensions to positive genus curves and to the case $n>2$ are mentioned.


K-duality for stratified pseudomanifolds

Abstract: This paper is devoted to the study of Poincar\'e duality in K-theory for general stratified pseudomanifolds. We review the axiomatic definition of a smooth stratification $\fS$ of a topological space $X$ and we define a groupoid $T^{\fS}X$, called the $\fS$-tangent space. This groupoid is made of different pieces encoding the tangent spaces of the strata, and these pieces are glued into the smooth noncommutative groupoid $T^{\fS}X$ using the familiar procedure introduced by A. Connes for the tangent groupoid of a manifold. The main result is that $C^{*}(T^{\fS}X)$ is Poincar\'e dual to $C(X)$, in other words, the $\fS$-tangent space plays the role in $K$-theory of a tangent space for $X$.


ACM bundles on cubic surfaces

Abstract: In this paper we prove that, for every $r \geq 2$, the moduli space $M^s_X(r;c_1,c_2)$ of rank $r$ stable vector bundles with Chern classes $c_1=rH$ and $c_2=(3r^2-r)/2$ on a nonsingular cubic surface $X \subset \mathbb{P}^3$ contains a nonempty smooth open subset formed by ACM bundles, i.e. vector bundles with no intermediate cohomology. The bundles we consider for this study are extremal for the number of generators of the corresponding module (these are known as Ulrich bundles), so we also prove the existence of indecomposable Ulrich bundles of arbitrarily high rank on $X$.


Escaping points of entire functions of small growth

Abstract: Let $f$ be a transcendental entire function and let $I(f)$ denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, $I(f)$ is connected. In particular, we show that $I(f)$ is connected if $f$ has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko's conjecture that $I(f)$ has no bounded components is true. We also give a new criterion related to $I(f)$ which is sufficient to ensure that $f$ has no unbounded Fatou components.


Functions of small growth with no unbounded Fatou components

Abstract: We prove a form of the $\cos \pi \rho$ theorem which gives strong estimates for the minimum modulus of a transcendental entire function of order zero. We also prove a generalisation of a result of Hinkkanen that gives a sufficient condition for a transcendental entire function to have no unbounded Fatou components. These two results enable us to show that there is a large class of entire functions of order zero which have no unbounded Fatou components. On the other hand we give examples which show that there are in fact functions of order zero which not only fail to satisfy Hinkkanen's condition but also fail to satisfy our more general condition. We also give a new regularity condition that is sufficient to ensure that a transcendental entire function of order less than 1/2 has no unbounded Fatou components. Finally, we observe that all the conditions given here which guarantee that a transcendental entire function has no unbounded Fatou components, also guarantee that the escaping set is connected, thus answering a question of Eremenko for such functions.


Tunnel effect for Kramers-Fokker-Planck type operators: return to equilibrium and applications

Abstract: In the first part of this work, we consider second order supersymmetric differential operators in the semiclassical limit, including the Kramers-Fokker-Planck operator, such that the exponent of the associated Maxwellian $\phi$ is a Morse function with two local minima and one saddle point. Under suitable additional assumptions of dynamical nature, we establish the long time convergence to the equilibrium for the associated heat semigroup, with the rate given by the first non-vanishing, exponentially small, eigenvalue. In the second part of the paper, we consider the case when the function $\phi$ has precisely one local minimum and one saddle point. We also discuss further examples of supersymmetric operators, including the Witten Laplacian and the infinitesimal generator for the time evolution of a chain of classical anharmonic oscillators.


Index theory and Groupoids

Abstract: This paper collects the notes of a serie of lectures given by the two authors during the summer school "Geometric and topological methods for Quantum Field Theory" at Villa de Leyva, Colombia, summer 2007. These lecture notes are mainly devoted to a proof using groupoids and $KK$-theory of Atiyah-Singer index theorem on compact smooth manifolds. We will present an elementary introduction to groupoids, $C^*$-algebras, $KK$-theory and pseudodifferential calculus on groupoids. We will finish by showing that the point of view adopted here generalizes to the case of conical pseudo-manifolds.


Toric complexes and Artin kernels

Abstract: A simplicial complex L on n vertices determines a subcomplex T_L of the n-torus, with fundamental group the right-angled Artin group G_L. Given an epimorphism \chi\colon G_L\to \Z, let T_L^\chi be the corresponding cover, with fundamental group the Artin kernel N_\chi. We compute the cohomology jumping loci of the toric complex T_L, as well as the homology groups of T_L^\chi with coefficients in a field \k, viewed as modules over the group algebra \k\Z. We give combinatorial conditions for H_{\le r}(T_L^\chi;\k) to have trivial \Z-action, allowing us to compute the truncated cohomology ring, H^{\le r}(T_L^\chi;\k). We also determine several Lie algebras associated to Artin kernels, under certain triviality assumptions on the monodromy \Z-action, and establish the 1-formality of these (not necessarily finitely presentable) groups.


Area, capacity and diameter versions of Schwarz's Lemma

Abstract: The now canonical proof of Schwarz's Lemma appeared in a 1907 paper of Carath\'eodory, who attributed it to Erhard Schmidt. Since then, Schwarz's Lemma has acquired considerable fame, with multiple extensions and generalizations. Much less known is that, in the same year 1907, Landau and Toeplitz obtained a similar result where the diameter of the image set takes over the role of the maximum modulus of the function. We give a new proof of this result and extend it to include bounds on the growth of the maximum modulus. We also develop a more general approach in which the size of the image is estimated in several geometric ways via notions of radius, diameter, perimeter, area, capacity, etc...


On the Representation Theory of an Algebra of Braids and Ties

Abstract: We consider the algebra ${\cal E}_n(u)$ introduced by F. Aicardi and J. Juyumaya as an abstraction of the Yokonuma-Hecke algebra. We construct a tensor space representation for ${\cal E}_n(u)$ and show that this is faithful. We use it to give a basis for ${\cal E}_n(u)$ and to classify its irreducible representations.


Asymptotic cones, bi-Lipschitz ultraflats, and the geometric rank of geodesics

Abstract: Given a geodesic inside a simply-connected, complete, non-positively curved Riemannian (NPCR) manifold M, we get an associated geodesic inside the asymptotic cone Cone(M). Under mild hypotheses, we show that if the latter is contained inside a bi-Lipschitz flat, then the original geodesic supports a non-trivial, orthogonal, parallel Jacobi field. As applications we obtain (1) constraints on the behavior of quasi-isometries between complete, simply connected, NPCR manifolds, and (2) constraints on the NPCR metrics supported by certain manifolds, and (3) a correspondence between metric splittings of complete, simply connected NPCR manifolds, and metric splittings of its asymptotic cones. Furthermore, combining our results with the Ballmann-Burns-Spatzier rigidity theorem and the classic Mostow rigidity, we also obtain (4) a new proof of Gromov's rigidity theorem for higher rank locally symmetric spaces.


The sum of irreducible fractions with consecutive denominators is never an integer in a very weak arithmetic

Abstract: Two theorems of elmentary arithmetic, one stating that the sum of the reciprocals of any number of consecutive positive integers is never an integer, and a generalization thereof by Trygve Nagell, are shown to be provable inside a very weak arithmetic, Richard Kaye's $PA^-$, in which there is no induction axiom whatsoever.


Dynatomic cycles for morphisms of projective varieties

Abstract: We prove the effectivity of the dynatomic cycles for morphisms of projective varieties. We then analyze the degrees of the dynatomic cycles and multiplicities of formal periodic points and apply these results to the existence of periodic points with arbitrarily large primitive periods.


Good Reduction of Periodic Points

Abstract: We consider the dynamical system created by iterating a morphism of a projective variety defined over the field of fractions of a discrete valuation ring. We study the primitive period of a periodic point in this field in relation to the primitive period of the reduced point in the residue field, the order of the action on the cotangent space, and the characteristic of the residue field.


All 2-dimensional links in 4-space live inside a universal 3-dimensional polyhedron

Abstract: The hexabasic book is the cone of the 1-dimensional skeleton of the union of two tetrahedra glued along a common face. The universal 3-dimensional polyhedron UP is the product of a segment and the hexabasic book. We show that any 2-dimensional link in 4-space is isotopic to a surface in UP. The proof is based on a representation of surfaces in 4-space by marked graphs, links with double intersections in 3-space. We construct a finitely presented semigroup whose central elements uniquely encode all isotopy classes of 2-dimensional links.


Finding Rational Periodic Points on Wehler K3 Surfaces

Abstract: This article examines dynamical systems on a class of K3 surfaces in $\mathbb{P}^{2} \times \mathbb{P}^{2}$ with an infinite automorphism group. In particular, this article develops an algorithm to find $\mathbb{Q}$-rational periodic points using information modulo $p$ for various primes $p$. The algorithm is applied to exhibit K3 surfaces with $\mathbb{Q}$-rational periodic points of primitive period $1,...,16$. A portion of the algorithm is then used to determine the Riemann zeta function modulo 3 of a particular K3 surface and find a family of K3 surfaces with Picard number two.


Link concordance and generalized doubling operators

Abstract: We introduce a technique for showing classical knots and links are not slice. As one application we show that the iterated Bing doubles of many algebraically slice knots are not topologically slice. Some of the proofs do not use the existence of the Cheeger-Gromov bound, a deep analytical tool used by Cochran-Teichner. We define generalized doubling operators, of which Bing doubling is an instance, and prove our nontriviality results in this more general context. Our main examples are boundary links that cannot be detected in the algebraic boundary link concordance group.


The smallest multistationary mass-preserving chemical reaction network

Abstract: Biochemical models that exhibit bistability are of interest to biologists and mathematicians alike. Chemical reaction network theory can provide sufficient conditions for the existence of bistability, and on the other hand can rule out the possibility of multiple steady states. Understanding small networks is important because the existence of multiple steady states in a subnetwork of a biochemical model can sometimes be lifted to establish multistationarity in the larger network. This paper establishes the smallest reversible, mass-preserving network that admits bistability and determines the semi-algebraic set of parameters for which more than one steady state exists.


Asymptotics of degenerating Eisenstein series

Abstract: We give some estimates for the asymptotic orders of degenerating Eisenstein series for some families of degenerating punctured Riemann surfaces, which is motivated by the question identifying $L_{2}$-cohomology of the Takhtajan-Zograf metric that is originally asked by To and Weng.


The two dimensional distribution of values of $\zeta(1+it)$

Abstract: We prove several results on the distribution function of $\zeta(1+it)$ in the complex plane, that is the joint distribution function of $\arg\zeta(1+it)$ and $|\zeta(1+it)|$. Similar results are also given for $L(1,\chi)$ (as $\chi$ varies over non-principal characters modulo a large prime $q$).


Stochastic Depletion Problems: Effective Myopic Policies for a class of Dynamic Optimization Problems

Abstract: This paper presents a general class of dynamic stochastic optimization problems we refer to as Stochastic Depletion Problems. A number of challenging dynamic optimization problems of practical interest are stochastic depletion problems. Optimal solutions for such problems are difficult to obtain, both from a pragmatic computational perspective as also from a theoretical perspective. As such, simple heuristics are highly desirable. We isolate two simple properties that, if satisfied by a problem within this class, guarantee that a myopic policy incurs a performance loss of at most 50 % relative to the optimal adaptive control policy for that problem. We are able to verify that these two properties are satisfied for several interesting families of stochastic depletion problems and as a consequence identify efficient near-optimal control policies for a number of interesting dynamic stochastic optimization problems.


Hyperbolic Graphs of Surface Groups

Abstract: We give a sufficient condition under which the fundamental group of a reglued graph of surfaces is hyperbolic. A reglued graph of surfaces is constructed by cutting a fixed graph of surfaces along the edge surfaces, then regluing by pseudo-Anosov homeomorphisms of the edge surfaces. By carefully choosing the regluing homeomorphism, we construct an example of such a reglued graph of surfaces, whose fundamental group is not abstractly commensurate to any surface-by-free group, i.e., which is different from all the examples given in Mosher's paper 'A hyperbolic-by-hyperbolic hyperbolic group'.


$q$-Chaos

Abstract: We consider the $L_p$ norm estimates for homogeneous polynomials of $q$-gaussian variables ($-1\leq q\leq 1$). When $-1<q<1$ the $L_p$ estimates for $1\leq p \leq 2$ are essentially the same as the free case ($q=0$), whilst the $L_p$ estimates for $2\leq p \leq \infty$ show a strong $q$-dependence. Moreover, the extremal cases $q = \pm 1$ produce decisively different formulae.


Spherical two-distance sets

Abstract: A set S of unit vectors in n-dimensional Euclidean space is called spherical two-distance set, if there are two numbers a and b, and inner products of distinct vectors of S are either a or b. The largest cardinality g(n) of spherical two-distance sets is not exceed n(n+3)/2. This upper bound is known to be tight for n=2,6,22. The set of mid-points of the edges of a regular simplex gives the lower bound L(n)=n(n+1)/2 for g(n. In this paper using the so-called polynomial method it is proved that for nonnegative a+b the largest cardinality of S is not greater than L(n). For the case a+b<0 we propose upper bounds on |S| which are based on Delsarte's method. Using this we show that g(n)=L(n) for 6<n<22, 23<n<40, and g(23)=276 or 277.


Deformations of nilpotent cones and Springer correspondences

Abstract: Let G = Sp (2n) be the symplectic group over Z. We present a certain kind of deformation of the nilpotent cone of G with G-action. This enables us to make direct links between the Springer correspondence of sp_{2n} over C, that over characteristic two, and our exotic Springer correspondence. As a by-product, we obtain a complete description of our exotic Springer correspondence.


Topology of polar weighted homogeneous hypersurfaces

Abstract: Polar weighted homogeneous polynomials are the class of special polynomials of real variables $x_i,y_i, i=1,..., n$ with $z_i=x_i+\sqrt{-1} y_i$, which enjoys a "polar action". In many aspects, their behavior looks like that of complex weighted homogeneous polynomials. We study basic properties of hypersurfaces which are defined by polar weighted homogeneous polynomials.


Construction and Uniqueness for reflected BSDE under linear increasing condition

Abstract: In this paper, we study the uniqueness of the solution of reflected BSDE with one or two barriers, under continuous and linear increasing condition of generator $g$. Before that we study the construction of solution of of reflected BSDE with one or two barriers.


Self-similar solutions and translating solitons for Lagrangian mean curvature flow

Abstract: We construct many self-similar and translating solitons for Lagrangian mean curvature flow, including self-expanders and translating solitons with arbitrarily small oscillation on the Lagrangian angle. Our translating solitons play the same role as cigar solitons in Ricci flow, and are important in studying the regularity of Lagrangian mean curvature flow. Given two transverse Lagrangian planes R^n in C^n with sum of characteristic angles less than pi, we show there exists a Lagrangian self-expander asymptotic to this pair of planes. The Maslov class of these self-expanders is zero. Thus they can serve as local models for surgeries on Lagrangian mean curvature flow. Families of self-shrinkers and self-expanders with different topologies are also constructed. This paper generalizes the work of Anciaux, Joyce, Lawlor, and Lee and Wang.


A unifying formulation of the Fokker-Planck-Kolmogorov equation for general stochastic hybrid systems

Abstract: A general formulation of the Fokker-Planck-Kolmogorov (FPK) equation for stochastic hybrid systems is presented, within the framework of Generalized Stochastic Hybrid Systems (GSHS). The FPK equation describes the time evolution of the probability law of the hybrid state. Our derivation is based on the concept of mean jump intensity, which is related to both the usual stochastic intensity (in the case of spontaneous jumps) and the notion of probability current (in the case of forced jumps). This work unifies all previously known instances of the FPK equation for stochastic hybrid systems, and provides GSHS practitioners with a tool to derive the correct evolution equation for the probability law of the state in any given example.


Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane

Abstract: We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G, then for almost every x in G with respect to the perimeter measure of E, some tangent of E at x is a vertical halfspace. This is a partial extension of a theorem of Franchi-Serapioni-Serra Cassano in step 2 Carnot groups: they have shown that, for almost every x, E has a unique tangent at x, and this tangent is a vertical halfspace.


Inverse zero-sum problems II

Abstract: Let $G$ be an additive finite abelian group. A sequence over $G$ is called a minimal zero-sum sequence if the sum of its terms is zero and no proper subsequence has this property. Davenport's constant of $G$ is the maximum of the lengths of the minimal zero-sum sequences over $G$. Its value is well-known for groups of rank two. We investigate the structure of minimal zero-sum sequences of maximal length for groups of rank two. Assuming a well-supported conjecture on this problem for groups of the form $C_m \oplus C_m$, we determine the structure of these sequences for groups of rank two. Combining our result and partial results on this conjecture, yields unconditional results for certain groups of rank two.


Bounded H_\infty-calculus for pseudodifferential Douglis-Nirenberg systems of mild regularity

Abstract: Parameter-ellipticity with respect to a closed subsector of the complex plane for pseudodifferential Douglis-Nirenberg systems is discussed and shown to imply the existence of a bounded H_\infty-calculus in suitable scales of Sobolev, Besov, and Hoelder spaces. We also admit non pseudodifferential perturbations. Applications concern systems with coefficients of mild Hoelder regularity and the generalized thermoelastic plate equations.


Higher Accuracy for Bayesian and Frequentist Inference: Large Sample Theory for Small Sample Likelihood

Abstract: Recent likelihood theory produces $p$-values that have remarkable accuracy and wide applicability. The calculations use familiar tools such as maximum likelihood values (MLEs), observed information and parameter rescaling. The usual evaluation of such $p$-values is by simulations, and such simulations do verify that the global distribution of the $p$-values is uniform(0, 1), to high accuracy in repeated sampling. The derivation of the $p$-values, however, asserts a stronger statement, that they have a uniform(0, 1) distribution conditionally, given identified precision information provided by the data. We take a simple regression example that involves exact precision information and use large sample techniques to extract highly accurate information as to the statistical position of the data point with respect to the parameter: specifically, we examine various $p$-values and Bayesian posterior survivor $s$-values for validity. With observed data we numerically evaluate the various $p$-values and $s$-values, and we also record the related general formulas. We then assess the numerical values for accuracy using Markov chain Monte Carlo (McMC) methods. We also propose some third-order likelihood-based procedures for obtaining means and variances of Bayesian posterior distributions, again followed by McMC assessment. Finally we propose some adaptive McMC methods to improve the simulation acceptance rates. All these methods are based on asymptotic analysis that derives from the effect of additional data. And the methods use simple calculations based on familiar maximizing values and related informations. The example illustrates the general formulas and the ease of calculations, while the McMC assessments demonstrate the numerical validity of the $p$-values as percentage position of a data point. The example, however, is very simple and transparent, and thus gives little indication that in a wide generality of models the formulas do accurately separate information for almost any parameter of interest, and then do give accurate $p$-value determinations from that information. As illustration an enigmatic problem in the literature is discussed and simulations are recorded; various examples in the literature are cited.


Representation of nonnegative convex polynomials

Abstract: We provide a specific representation of convex polynomials nonnegative on a convex (not necessarily compact) basic closed semi-algebraic subset K of Rn. Namely, they belong to a specific subset of the quadratic module generated by the concave polynomials that define K.


Generalized iteration, catastrophes and generalized Sharkovsky's ordering

Abstract: We define iteration of functions that map n-dimensional vector spaces into m-dimensional vector spaces (m at most equal to n). It happens that usual iteration and Fibonacci iterative methods become special cases of this generalized iteration. Mathematical objects such as orbits, bifurcations, chaos, Feigenbaum constant, (generalized) Sharkovsky ordering, (generalized) Julia and Mandelbrot sets and a new kind of catastrophe can be found and studied in this enlarged context.


Counting cluster-tilted algebras of type $A_n$

Abstract: The purpose of this paper is to give an explicit formula for the number of non-isomorphic cluster-tilted algebras of type $A_n$, by counting the mutation class of any quiver with underlying graph $A_n$. It will also follow that if $T$ and $T'$ are cluster-tilting objects in a cluster category $\mathcal{C}$, then $\End_{\mathcal{C}}(T)$ is isomorphic to $\End_{\mathcal{C}}(T')$ if and only if $T=\tau^i T'$.


On the Well-possedness of the Problem of Reconstruction of Non-separate Boundary Conditions

Abstract: We consider an inverse spectral problem with the third-order differential equation and the non-separated boundary conditions. Two theorems on the uniqueness of the solution of this problem are proved, and a method for establishing the unknown conditions is obtained, using 19 eigenvalues. The method of approximate calculation of unknown boundary conditions is explained, with the help of an example.


Maximal Crossed Product Orders over Discrete Valuation Rings

Abstract: The problem of determining when a (classical) crossed product $T=S^f*G$ of a finite group $G$ over a discrete valuation ring $S$ is a maximal order, was answered in the 1960's for the case where $S$ is tamely ramified over the subring of invariants $S^G$. The answer was given in terms of the conductor subgroup (with respect to $f$) of the inertia. In this paper we solve this problem in general when $S/S^G$ is residually separable. We show that the maximal order property entails a restrictive structure on the sub-crossed product graded by the inertia subgroup. In particular, the inertia is abelian. Using this structure, one is able to extend the notion of the conductor. As in the tame case, the order of the conductor is equal to the number of maximal two sided ideals of $T$ and hence to the number of maximal orders containing $T$ in its quotient ring. Consequently, $T$ is a maximal order if and only if the conductor subgroup is trivial.


Stable laws and products of positive random matrices

Abstract: Let $S$ be the multiplicative semigroup of $q\times q$ matrices with positive entries such that every row and every column contains a strictly positive element. Denote by $(X_n)_{n\geq1}$ a sequence of independent identically distributed random variables in $S$ and by $X^{(n)} = X_n ... X_1$, $ n\geq 1$, the associated left random walk on $S$. We assume that $(X_n)_{n\geq1}$ verifies the contraction property $\P(\bigcup_{n\geq1}[X^{(n)} \in S^\circ])>0$, where $S^\circ $ is the subset of all matrices which have strictly positive entries. We state conditions on the distribution of the random matrix $X_1$ which ensure that the logarithms of the entries, of the norm, and of the spectral radius of the products $X^{(n)}$, $n\ge 1$, are in the domain of attraction of a stable law.


Hilbert's Nullstellensatz and an Algorithm for Proving Combinatorial Infeasibility

Abstract: Systems of polynomial equations over an algebraically-closed field K can be used to concisely model many combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution over K. In this paper, we investigate an algorithm aimed at proving combinatorial infeasibility based on the observed low degree of Hilbert's Nullstellensatz certificates for polynomial systems arising in combinatorics and on large-scale linear-algebra computations over K. We report on experiments based on the problem of proving the non-3-colorability of graphs. We successfully solved graph problem instances having thousands of nodes and tens of thousands of edges.


Inverse Zero-Sum Problems III

Abstract: Let $G$ be a finite abeilian group. A sequence $S$ with terms from $G$ is zero-sum if the sum of terms in $S$ equals zero. It is a minimal zero-sum sequence if no proper, nontrivial subsequence is zero-sum. The maximal length of a minimal zero-sum subsequence in $G$ is the Davenport constant, denoted $D(G)$. For a rank 2 group $G=C_n \oplus C_n$, it is known that $D(G)=2n-1$. However, the structure of all maximal length minimal zero-sum sequences remains open. If every such sequence contains a term with multiplicity $n-1$, then $C_n \oplus C_n$ is said to have Property B, and it is conjectured that this is true for all rank 2 groups $C_n \oplus C_n$. In this paper, we show that Property B is multiplicative, namely, if $G=C_n \oplus C_n$ and $G=C_m \oplus C_m$ both satisfy Property B, with $m, n\geq 3$ odd and $mn>9$, then $C_{mn}\oplus C_{mn}$ satisfies Property B also. Combined with previous work in the literature, this reduces the question of establishing Property B to the prime cases, and in such case the complete structural description of the sequence follows.


Novel Bounds on Marginal Probabilities

Abstract: We derive two related novel bounds on single-variable marginal probability distributions in factor graphs with discrete variables. The first method propagates bounds over a subtree of the factor graph rooted in the variable, and the second method propagates bounds over the self-avoiding walk tree starting at the variable. By construction, both methods not only bound the exact marginal probability distribution of a variable, but also its approximate Belief Propagation marginal (``belief''). Thus, apart from providing a practical means to calculate bounds on marginals, our contribution also lies in an increased understanding of the error made by Belief Propagation. Empirically, we show that our bounds often outperform existing bounds in terms of accuracy and/or computation time. We also show that our bounds can yield nontrivial results for medical diagnosis inference problems.


An existence result for the sandpile problem on flat tables with walls

Abstract: We derive an existence result for solutions of a differential system which characterizes the equilibria of a particular model in granular matter theory, the so-called partially open table problem for growing sandpiles. Such result generalizes a recent theorem of Cannarsa and Cardaliaguet established for the totally open table problem. Here, due to the presence of walls at the boundary, the surface flow density at the equilibrium may result no more continuous nor bounded, and its explicit mathematical characterization is obtained by domain decomposition techniques. At the same time we show how these solutions can be numerically computed as stationary solutions of a dynamical two-layer model for growing sandpiles and we present the results of some simulations.


Vari\'et\'es homog\`enes sous $\PGL_n$

Abstract: Let $A$ be an Azumaya algebra over a field. If $G$ is the group of automorphisms of $A$ and $X$ denotes a projective homogeneous variety under $G$, we construct in a very explicit way and under suitable hypotheses a bundle $\mathcal{V}$ on $S$, where $S$ is a (generalized) Severi-Brauer variety associated to $A$, and a canonical isomorphism between $X$ and a flag bundle on $\mathcal{V}$. This allows to explicitely compute Chow groups of $X$ in terms of the Chow groups of $S$.


Frame potential and finite abelian groups

Abstract: This article continues a prior investigation of the authors with the goal of extending characterization results of convolutional tight frames from the context of cyclic groups to general finite abelian groups. The collections studied are formed by translating a number of \emph{generators} by elements of a fixed subgroup and it is shown, under certain norm conditions, that tight frames with this structure are characterized as local minimizers of the frame potential. Natural analogs to the downsampling and upsampling operators of finite cyclic groups are studied for arbitrary subgroups of finite abelian groups. Directions of further study are also proposed.


On cusps and flat tops

Abstract: We develop non-invertible Pesin theory for a new class of maps called cusp maps. These maps may have unbounded derivative, but nevertheless verify a property analogous to $C^{1+\epsilon}$. We do not require the critical points to verify a non-flatness condition, so the results are applicable to $C^{1+\epsilon}$ maps with flat critical points. If the critical points are too flat, then no absolutely continuous invariant probability measure can exist. This generalises a result of Benedicks and Misiurewicz.


Algorithm for solving optimization problems with Interval Valued Probability Measure

Abstract: We are concerned with three types of uncertainties: probabilistic, possibilitistic and interval. By using possibility and necessity measures as an Interval Valued Probability Measure (IVPM), we present IVPM's interval expected values whose possibility distributions are in the form of polynomials. By working with interval expected values of independent uncertainty coefficients in a linear optimization problem together with operations suggested in Lodwick and Jamison (2007), the problem after applying these operations becomes a linear programming problem with constant coefficients. This is achieved by the application of two functions. The first is applied to the interval coefficients, v: I -> R^k, where I= {[a,b] | a <= b}. The second is u: R^k -> R, applied to the product we got from a previous function. Similar concepts hold for any types of optimization problems with linear constraints. Moreover, it implied that optimization problems containing all three types of uncertainties in one problem can be solved as ordinary optimization problems.


Symmetry of Reidemeister torsion on $SU_2$-representation spaces of knots

Abstract: We study two sorts of actions on the space of conjugacy classes of irreducible $SU_2$-representations of a knot group. One of them is an involution which comes from the algebraic structure of $SU_2$ and the other is the action by the outer automorphism group of the knot group. In particular, we consider them on an 1-dimensional smooth part of the space, which is canonically oriented and metrized via a Reidemeister torsion volume form. As an application we show that the Reidemeister torsion function on the 1-dimensional subspace has symmetry about the metrization.


Normal Toric Ideals of Low Codimension

Abstract: Every normal toric ideal of codimension two is minimally generated by a Grobner basis with squarefree initial monomials. A polynomial time algorithm is presented for checking whether a toric ideal of fixed codimension is normal.


Large p-groups actions with a p-elementary abelian second ramification group

Abstract: Let $k$ be an algebraically closed field of characteristic $p>0$ and $C$ a connected nonsingular projective curve over $k$ with genus $g \geq 2$. Let $(C,G)$ be a "big action", i.e. a pair $(C,G)$ where $G$ is a $p$-subgroup of the $k$-automorphism group of $C$ such that$\frac{|G|}{g} >\frac{2 p}{p-1}$. We denote by $G_2$ the second ramification group of $G$ at the unique ramification point of the cover $C \to C/G$. The aim of this paper is to describe the big actions whose $G_2$ is $p$-elementary abelian. In particular, we obtain a structure theorem by considering the $k$-algebra generated by the additive polynomials. We more specifically explore the case where there is a maximal number of jumps in the ramification filtration of $G_2$. In this case, we display some universal families.


Computing Arakelov class groups

Abstract: Shanks's infrastructure algorithm and Buchmann's algorithm for computing class groups and unit groups of rings of integers of algebraic number fields are most naturally viewed as computations inside Arakelov class groups. In this paper we discuss the basic properties of Arakelov class groups and of the set of reduced Arakelov divisors. As an application we describe Buchmann's algorithm in this context.


Pseudodifferential multi-product representation of the solution operator of a parabolic equation

Abstract: By using a time slicing procedure, we represent the solution operator of a second-order parabolic pseudodifferential equation on $\R^n$ as an infinite product of zero-order pseudodifferential operators. A similar representation formula is proven for parabolic differential equations on a compact Riemannian manifold. Each operator in the multi-product is given by a simple explicit Ansatz. The proof is based on an effective use of the Weyl calculus and the Fefferman-Phong inequality.


Four primality testing algorithms

Abstract: In this expository paper we describe four primality tests. The first test is very efficient, but is only capable of proving that a given number is either composite or 'very probably' prime. The second test is a deterministic polynomial time algorithm to prove that a given numer is either prime or composite. The third and fourth primality tests are at present most widely used in practice. Both tests are capable of proving that a given number is prime or composite, but neither algorithm is deterministic. The third algorithm exploits the arithmetic of cyclotomic fields. Its running time is almost, but not quite polynomial time. The fourth algorithm exploits elliptic curves. Its running time is difficult to estimate, but it behaves well in practice.


The Classifying Space of a Topological 2-Group

Abstract: Categorifying the concept of topological group, one obtains the notion of a 'topological 2-group'. This in turn allows a theory of 'principal 2-bundles' generalizing the usual theory of principal bundles. It is well-known that under mild conditions on a topological group G and a space M, principal G-bundles over M are classified by either the first Cech cohomology of M with coefficients in G, or the set of homotopy classes [M,BG], where BG is the classifying space of G. Here we review work by Bartels, Jurco, Baas-Bokstedt-Kro, and others generalizing this result to topological 2-groups and even topological 2-categories. We explain various viewpoints on topological 2-groups and Cech cohomology with coefficients in a topological 2-group C, also known as 'nonabelian cohomology'. Then we give an elementary proof that under mild conditions on M and C there is a bijection between the first Cech cohomology of M with coefficients in C and [M,B|C|] where B|C| is the classifying space of the geometric realization of the nerve of C. Applying this result to the 'string 2-group' String(G) of a simply-connected compact simple Lie group G, it follows that principal String(G)-2-bundles have rational characteristic classes coming from elements of the rational cohomology of BG modulo the ideal generated by c, where c is any nonzero element in the 4th cohomology of BG.


Graphic Bernstein Results in Curved Pseudo-Riemannian Manifolds

Abstract: We generalize a Bernstein-type result due to Albujer and Al\'ias, for maximal surfaces in a curved Lorentzian product 3-manifold of the form $\Sigma_1\times \mathbb{R}$, to higher dimension and codimension. We consider $M$ a complete spacelike graphic submanifold with parallel mean curvature, defined by a map $f: \Sigma_1\to \Sigma_2$ between two Riemannian manifolds $(\Sigma_1^m, g_1)$ and $(\Sigma^n_2, g_2)$ of sectional curvatures $K_1$ and $K_2$, respectively. We take on $\Sigma_1\times \Sigma_2$ the pseudo-Riemannian product metric $g_1-g_2$. Under the curvature conditions, $\mathrm{Ricci}_1 \geq 0$ and $K_1\geq K_2$, we prove that, if the second fundamental form of $M$ satisfies an integrability condition, then $M$ is totally geodesic, and it is a slice if $\mathrm{Ricci}_1(p)>0$ at some point. For bounded $K_1$, $K_2$ and hyperbolic angle $\theta$, we conclude $M$ must be maximal. If $M$ is a maximal surface and $K_1\geq K_2^+$, we show $M$ is totally geodesic with no need for further assumptions. Furthermore, $M$ is a slice if at some point $p\in \Sigma_1$, $K_1(p)> 0$, and if $\Sigma_1$ is flat and $K_2<0$ at some point $f(p)$, then the image of $f$ lies on a geodesic of $\Sigma_2$.


On the expansion of the resolvent for elliptic boundary contact problems

Abstract: Let $A$ be an elliptic operator on a compact manifold with boundary $M$, and let $\wp : \partial\M \to Y$ be a covering map, where $Y$ is a closed manifold. Let $A_C$ be a realization of $A$ subject to a coupling condition $C$ that is elliptic with parameter in the sector $\Lambda$. By a coupling condition we mean a nonlocal boundary condition that respects the covering structure of the boundary. We prove that the resolvent trace $\Tr_{L^2} (A_C-\lambda)^{-N}$ for $N$ sufficiently large has a complete asymptotic expansion as $|\lambda| \to \infty$, $\lambda \in \Lambda$. In particular, the heat trace $\Tr_{L^2}e^{-tA_C}$ has a complete asymptotic expansion as $t \to 0^+$, and the $\zeta$-function has a meromorphic extension to $\C$.


Long cycles in fullerene graphs

Abstract: It is conjectured that every fullerene graph is hamiltonian. Jendrol' and Owens proved [J. Math. Chem. 18 (1995), pp. 83--90] that every fullerene graph on n vertices has a cycle of length at least 4n/5. In this paper, we improve this bound to 5n/6-2/3.


The Range of Approximate Unitary Equivalence Classes of Homomorphisms from AH-algebras

Abstract: Let $C$ be a unital AH-algebra and $A$ be a unital simple C*-algebra with tracial rank zero. It has been shown that two unital monomorphisms $\phi, \psi: C\to A$ are approximately unitarily equivalent if and only if $$ [\phi]=[\psi] {\rm in} KL(C,A) and \tau\circ \phi=\tau\circ \psi \tforal \tau\in T(A), $$ where $T(A)$ is the tracial state space of $A.$ In this paper we prove the following: Given $\kappa\in KL(C,A)$ with $\kappa(K_0(C)_+\setminus \{0\})\subset K_0(A)_+\setminus \{0\}$ and with $\kappa([1_C])=[1_A]$ and a continuous affine map $\lambda: T(A)\to T_{\mathtt{f}}(C)$ which is compatible with $\kappa,$ where $T_{\mathtt{f}}(C)$ is the convex set of all faithful tracial states, there exists a unital monomorphism $\phi: C\to A$ such that $$ [\phi]=\kappa\andeqn \tau\circ \phi(c)=\lambda(\tau)(c) $$ for all $c\in C_{s.a.}$ and $\tau\in T(A).$ Denote by ${\rm Mon}_{au}^e(C,A)$ the set of approximate unitary equivalence classes of unital monomorphisms. We provide a bijective map $$ \Lambda: {\rm Mon}_{au}^e (C,A)\to KLT(C,A)^{++}, $$ where $KLT(C,A)^{++}$ is the set of compatible pairs of elements in $KL(C,A)^{++}$ and continuous affine maps from $T(A)$ to $T_{\mathtt{f}}(C).$ Moreover, we realized that there are compact metric spaces $X$, unital simple AF-algebras $A$ and $\kappa\in KL(C(X), A)$ with $\kappa(K_0(C(X))_+\setminus\{0\})\subset K_0(A)_+\setminus \{0\}$ for which there is no \hm $h: C(X)\to A$ so that $[h]=\kappa.$


Simultaneous preconditioning and symmetrization of non-symmetric linear systems

Abstract: Motivated by the theory of self-duality which provides a variational formulation and resolution for non self-adjoint partial differential equations \cite{G1, G2}, we propose new templates for solving large non-symmetric linear systems. The method consists of combining a new scheme that simultaneously preconditions and symmetrizes the problem, with various well known iterative methods for solving linear and symmetric problems. The approach seems to be efficient when dealing with certain ill-conditioned, and highly non-symmetric systems.


Infinite Dimensional Multiplicity Free Spaces II: Limits of Commutative Nilmanifolds

Abstract: We study direct limits $(G,K) = \varinjlim (G_n,K_n)$ of Gelfand pairs of the form $G_n = N_n\rtimes K_n$ with $N_n$ nilpotent, in other words pairs $(G_n,K_n)$ for which $G_n/K_n$ is a commutative nilmanifold. First, we extend the criterion of \cite{W3} for a direct limit representation to be multiplicity free. Then we study direct limits $G/K = \varinjlim G_n/K_n$ of commutative nilmanifolds and look to see when the regular representation of $G = \varinjlim G_n$ on an appropriate Hilbert space $\varinjlim L^2(G_n/K_n)$ is multiplicity free. One knows that the $N_n$ are commutative or 2--step nilpotent. In many cases where the derived algebras $[\gn_n,\gn_n]$ are of bounded dimension we construct $G_n$--equivariant isometric maps $\zeta_n : L^2(G_n/K_n) \to L^2(G_{n+1}/K_{n+1})$ and prove that the left regular representation of $G$ on the Hilbert space $L^2(G/K) := \varinjlim \{L^2(G_n/K_n),\zeta_n\}$ is a multiplicity free direct integral of irreducible unitary representations. The direct integral and its irreducible constituents are described explicitly. One constituent of our argument is an extension of the classical Peter--Weyl Theorem to parabolic direct limits of compact groups.


Infinite Dimensional Multiplicity Free Spaces I: Limits of Compact Commutative Spaces

Abstract: We study direct limits $(G,K) = \varinjlim (G_n,K_n)$ of compact Gelfand pairs. First, we develop a criterion for a direct limit representation to be a multiplicity--free discrete direct sum of irreducible representations. Then we look at direct limits $G/K = \varinjlim G_n/K_n$ of compact riemannian symmetric spaces, where we combine our criterion with the Cartan--Helgason Theorem to show in general that the regular representation of $G = \varinjlim G_n$ on a certain function space $\varinjlim L^2(G_n/K_n)$ is multiplicity free. That method is not applicable for direct limits of nonsymmetric Gelfand pairs, so we introduce two other methods. The first, based on ``parabolic direct limits'' and ``defining representations'', extends the method used in the symmetric space case. The second uses some (new) branching rules from finite dimensional representation theory. In both cases we define function spaces $\cA(G/K)$, $\cC(G/K)$ and $L^2(G/K)$ to which our multiplicity--free criterion applies.


2-Cocycles of Original Deformative Schr\"{o}dinger-Virasoro Algebras

Abstract: Both original and twisted Schr\"{o}dinger-Virasoro algebras also their deformations were introduced and investigated in a series of papers by Henkel, Roger and Unterberger. In the present paper we aim to determine the 2-cocycles of original deformative Schr\"{o}dinger-Virasoro algebras.


Smooth Solutions of Non-linear Stochastic Partial Differential Equations

Abstract: In this paper, we study the regularities of solutions of nonlinear stochastic partial differential equations in the framework of Hilbert scales. Then we apply our general result to several typical nonlinear SPDEs such as stochastic Burgers and Ginzburg-Landau's equations on the real line, stochastic 2D Navier-Stokes equations in the whole space and a stochastic tamed 3D Navier-Stokes equation in the whole space, and obtain the existence of their respectively smooth solutions.


Properties of Nested Sampling

Abstract: Nested sampling is a simulation method for approximating marginal likelihoods proposed by Skilling (2006). We establish that nested sampling has an approximation error that vanishes at the standard Monte Carlo rate and that this error is asymptotically Gaussian. We show that the asymptotic variance of the nested sampling approximation typically grows linearly with the dimension of the parameter. We discuss the applicability and efficiency of nested sampling in realistic problems, and we compare it with two current methods for computing marginal likelihood. We propose an extension that avoids resorting to Markov chain Monte Carlo to obtain the simulated points.


An LQ problem for the heat equation on the halfline with Dirichlet boundary control and noise

Abstract: We study a linear quadratic problem for a system governed by the heat equation on a halfline with Dirichlet boundary control and Dirichlet boundary noise. We show that this problem can be reformulated as a stochastic evolution equation in a certain weighted L2 space. An appropriate choice of weight allows us to prove a stronger regularity for the boundary terms appearing in the infinite dimensional state equation. The direct solution of the Riccati equation related to the associated non-stochastic problem is used to find the solution of the problem in feedback form and to write the value function of the problem.


A Unified Quantum SO(3) Invariant for Rational Homology 3-Spheres

Abstract: Given a rational homology 3-sphere M with the first integral homology of rank b and a link L inside M, colored by odd numbers, we construct a unified invariant I_{M,L} belonging to a modification of the Habiro ring where b is inverted. Our unified invariant dominates the whole set of the SO(3) Witten-Reshetikhin-Turaev invariants of the pair (M,L). If b=1 and L is empty, I_M coincides with Habiro's invariant of integral homology 3-spheres. For b>1, the unified invariant defined by the third author is determined by I_M. One of the applications are the new Ohtsuki series (perturbative expansions of I_M at roots of unity) dominating all quantum SO(3) invariants.


Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise

Abstract: We consider two exit problems for the Korteweg-de Vries equation perturbed by an additive white in time and colored in space noise of amplitude a. The initial datum gives rise to a soliton when a=0. It has been proved recently that the solution remains in a neighborhood of a randomly modulated soliton for times at least of the order of a^{-2}. We prove exponential upper and lower bounds for the small noise limit of the probability that the exit time from a neighborhood of this randomly modulated soliton is less than T, of the same order in a and T. We obtain that the time scale is exactly the right one. We also study the similar probability for the exit from a neighborhood of the deterministic soliton solution. We are able to quantify the gain of eliminating the secular modes to better describe the persistence of the soliton.


On finite generation of symbolic Rees rings of space monomial curves and existence of negative curves

Abstract: In this paper, we shall study finite generation of symbolic Rees rings of the defining ideal of the space monomial curves $(t^a, t^b, t^c)$ for pairwise coprime integers $a$, $b$, $c$ such that $(a,b,c) \neq (1,1,1)$. If such a ring is not finitely generated over a base field, then it is a counterexample to the Hilbert's fourteenth problem. Finite generation of such rings is deeply related to existence of negative curves on certain normal projective surfaces. We study a sufficient condition (Definition 3.6) for existence of a negative curve. Using it, we prove that, in the case of $(a+b+c)^2 > abc$, a negative curve exists. Using a computer, we shall show that there exist examples in which this sufficient condition is not satisfied.


Sequences of knots and their limits

Abstract: Hyperfinite knots, or limits of equivalence classes of knots induced by a knot invariant taking values in a metric space, were introduced in a previous article by the author. In this article, we present new examples of hyperfinite knots stemming from sequences of torus knots.


A Tribute to Ingram Olkin

Abstract: It is with pleasure and pride that I introduce this special section in honor of Ingram Olkin. This tribute is especially fitting because, among the many profound and far-reaching contributions that he has made to our profession, Ingram Olkin was the key force behind the genesis of Statistical Science. As put so eloquently by Morrie DeGroot [1], the founding Executive Editor of Statistical Science.


Classification of certain cellular classes of chain complexes

Abstract: Let (R, m) be a local commutative ring. Suppose that m is principal and that m^2 = 0. We give a complete description of the cellular lattice of perfect chain complexes of modules over this ring.


The derivations, central extensions and automorphism group of the Lie algebra $W$

Abstract: In this paper, we study the derivations, central extensions and the automorphisms of the infinite-dimensional Lie algebra W which appeared in [8] and Dong-Zhang's recent work [22] on the classification of some simple vertex operator algebras.


On the universal enveloping algebra of a Lie-Rinehart algebra

Abstract: We review the extent to which the universal enveloping algebra of a Lie-Rinehart algebra resembles a Hopf algebra, and refer to this structure as a Rinehart bialgebra. We then prove a Cartier-Milnor-Moore type theorem for such Rinehart bialgebras.


The countable Telescope Conjecture for module categories

Abstract: By the Telescope Conjecture for Module Categories, we mean the following claim: "Let R be any ring and (A, B) be a hereditary cotorsion pair in Mod-R with A and B closed under direct limits. Then (A, B) is of finite type." We prove a modification of this conjecture with the word 'finite' replaced by 'countable'. We show that a hereditary cotorsion pair (A, B) of modules over an arbitrary ring R is generated by a set of strongly countably presented modules provided that B is closed under unions of well-ordered chains. We also characterize the modules in B and the countably presented modules in A in terms of morphisms between finitely presented modules, and show that (A, B) is cogenerated by a single pure-injective module provided that A is closed under direct limits. Then we move our attention to strong analogies between cotorsion pairs in module categories and localizing pairs in compactly generated triangulated categories.


Special values of L-functions and false Tate curve extensions II

Abstract: In this paper we show how one can combine the p-adic Rankin-Selberg product construction of Hida with freeness results of Hecke modules of Wiles to establish interesting congruences between special values of L-functions. These congruences is a part of some deep conjectural congruences that follow from the work of Kato on the non-commutative Iwasawa theory of the false Tate curve extension.


An algebraic characterization of simple closed curves on surfaces with boundary

Abstract: We characterize in terms of the Goldman Lie algebra which conjugacy classes in the fundamental group of a surface with non empty boundary are represented by simple closed curves. We prove the following: A non power conjugacy class X contains an embedded representative if and only if the Goldman Lie bracket of X with the third power of X is zero. The proof uses combinatorial group theory and Chas' combinatorial description of the bracket recast here in terms of an exposition of the Cohen-Lustig algorithm. Using results of Ivanov, Korkmaz and Luo there are corollaries characterizing which permutations of conjugacy classes are related to diffeomorphisms of the surfaces. The problem is motivated by a group theoretical statement from the sixties equivalent to the Poincare conjecture due to Jaco and Stallings and by a question of Turaev from the eighties. Our main theorem actually counts the minimal possible number of self-intersection points of representatives of a conjugacy class X in terms of the bracket of X with the third power of X.


A refined version of the Lang-Trotter Conjecture

Abstract: Let $E$ be an elliptic curve defined over the rational numbers and $r$ a fixed integer. Using a probabilistic model consistent with the Chebotarev theorem for the division fields of $E$ and the Sato-Tate distribution, Lang and Trotter conjectured an asymptotic formula for the number of primes up to $x$ which have Frobenius trace equal to $r$, where $r$ is a {\it fixed} integer. However, as shown in this note, this asymptotic estimate cannot hold for {\it all} $r$ in the interval $|r|\le 2\sqrt{x}$ with a uniform bound for the error term, because an estimate of this kind would contradict the Chebotarev density theorem as well as the Sato-Tate conjecture. The purpose of this note is to refine the Lang-Trotter conjecture, by taking into account the "semicircular law", to an asymptotic formula that conjecturally holds for arbitrary integers $r$ in the interval $|r|\le 2\sqrt{x}$, with a uniform error term. We demonstrate consistency of our refinement with the Chebotarev theorem for a fixed division field, and with the Sato-Tate conjecture. We also present numerical evidence for the refined conjecture.


Symbolic dynamics for the geodesic flow on Hecke surfaces

Abstract: In this paper we discuss a coding and the associated symbolic dynamics for the geodesic flow on Hecke triangle surfaces. We construct an explicit cross section for which the first return map factors through a simple (explicit) map given in terms of the generating map of a particular continued fraction expansion closely related to the Hecke triangle groups. We also obtain explicit expressions for the associated first return times.


Semisimple torsion in groups of finite Morley rank

Abstract: We prove several results about groups of finite Morley rank without unipotent p-torsion: p-torsion always occurs inside tori, Sylow p-subgroups are conjugate, and p is not the minimal prime divisor of our approximation to the ``Weyl group.'' These results are quickly finding extensive applications within the classification project.


A generation theorem for groups of finite Morley rank

Abstract: We deal with two forms of the "uniqueness cases" in the classification of large simple $K^*$-groups of finite Morley rank of odd type, where large means the $m_2(G)$ at least three. This substantially extends results known for even larger groups having \Prufer 2-rank at least three, to cover the two groups $\PSp_4$ and $\G_2$. With an eye towards distant developments, we carry out this analysis for $L^*$-groups which is substantially broader than the $K^*$ setting.


Linear groups of finite Morley rank

Abstract: We show that a non-algebraic simple group of finite Morley rank with a definable representation over a field has no involutions, and otherwise resembles a bad group. In particular, the modern form of the Cherlin-Zilber alebaricity conjecture hold for such groups.


The Hausdorff dimension of the set of dissipative points for a Cantor-like model set for singly cusped parabolic dynamics

Abstract: In this paper we introduce and study a certain intricate Cantor-like set $C$ contained in unit interval. Our main result is to show that the set $C$ itself, as well as the set of dissipative points within $C$, both have Hausdorff dimension equal to 1. The proof uses the transience of a certain non-symmetric Cauchy-type random walk.


Cluster multiplication in regular components via generalized Chebyshev polynomials

Abstract: We introduce a multivariate generalization of normalized Chebyshev polynomials of the second kind. We prove that these polynomials arise in the context of cluster characters associated to Dynkin quivers of type $\mathbb A$ and representation-infinite quivers. This allows to obtain a simple combinatorial description of cluster algebras of type $\mathbb A$. We also provide explicit multiplication formulas for cluster characters associated to regular modules over the path algebra of any representation-infinite quiver.


Stability conditions and Stokes factors

Abstract: Let A be the category of modules over a complex, finite-dimensional algebra. We show that the space of stability conditions on A parametrises an isomonodromic family of irregular connections on P^1 with values in the Hall algebra of A. The residues of these connections are given by the holomorphic generating function for counting invariants in A constructed by D. Joyce.


Multiple disjunction for spaces of Poincare embeddings

Abstract: We obtain multirelative connectivity statements about spaces of Poincare embeddings, as precursors to analogous statements about spaces of smooth embeddings. The latter are the key to convergence results in the functor calculus approach to spaces of embeddings.


A Volume Product Representation and its Ramifications in lp,

Abstract: We represent the volume product for the unit p-ball in a a form free of its gamma symbolism;this will enable us to confirm Mahler's lower bound and Santalo's upper bound by the use of basic only gamma function theory and moderately advanced classical analysis.


Continuous Dependence for Backward Parabolic Operators with Log-Lipschitz Coefficients

Abstract: We prove continuous dependence on Cauchy data for a backward parabolic operator whose coefficients are Log-Lipschitz continuous in time.


Elliptic fibrations and symplectic automorphisms on K3 surfaces

Abstract: Nikulin has classified all finite abelian groups acting symplectically on a K3 surface and he has shown that the induced action on the K3 lattice $U^3\oplus E_8(-1)^2$ depends only on the group but not on the K3 surface. For all the groups in the list of Nikulin we compute the invariant sublattice and its orthogonal complement by using some special elliptic K3 surfaces.


Cox rings and combinatorics II

Abstract: We study varieties with a finitely generated Cox ring. In a first part, we generalize a combinatorial approach developed in earlier work for varieties with a torsion free divisor class group to the case of torsion. Then we turn to modifications, e.g., blow ups, and the question how the Cox ring changes under such maps. We answer this question for a certain class of modifications induced from modifications of ambient toric varieties. Moreover, we show that every variety with finitely generated Cox ring can be explicitly constructed in a finite series of toric ambient modifications from a combinatorially minimal one.


Automorphism groups of generalized Reed-Solomon codes

Abstract: We look at AG codes associated to the projective line, re-examining the problem of determining their automorphism groups (originally investigated by Duer in 1987 using combinatorial techniques) using recent methods from algebraic geometry. We (re)classify those finite groups that can arise as the automorphism group of an AG code for the projective line and give an explicit description of how these groups appear. We also give examples of generalized Reed-Solomon codes with large automorphism groups G, such as G=PSL(2,q), and explicitly describe their G-module structure.


A computation in Khovanov-Rozansky Homology

Abstract: We investigate the Khovanov-Rozansky invariant of a certain tangle and its compositions. Surprisingly the complexes we encounter reduce to ones that are very simple. Furthermore, we discuss a "local" algorithm for computing Khovanov-Rozansky homology and compare our results with those for the "foam" version of sl_3-homology.


Higher-dimensional linking integrals

Abstract: We derive an integral formula for the linking number of two submanifolds of the n-sphere S^n, of the product S^n x R^m, and of other manifolds which appear as "nice" hypersurfaces in Euclidean space. The formulas are geometrically meaningful in that they are invariant under the action of the special orthogonal group on the ambient space.


Quantization of Projective Homogeneous Spaces and Duality Principle

Abstract: We introduce a general recipe to construct quantum projective homogeneous spaces, with a particular interest for the examples of the quantum Grassmannians and the quantum generalized flag varieties. Using this construction, we extend the quantum duality principle to quantum projective homogeneous spaces.


On the infimum convolution inequality

Abstract: In the paper we study the infimum convolution inequalites. Such an inequality was first introduced by B. Maurey to give the optimal concentration of measure behaviour for the product exponential measure. We show how IC-inequalities are tied to concentration and study the optimal cost functions for an arbitrary probability measure. In particular, we show the optimal IC-inequality for product log-concave measures and for uniform measures on the l_p^n balls. Such an optimal inequality implies, for a given measure, in particular the Central Limit Theorem of Klartag and the tail estimates of Paouris.


Decompositions and statistics for beta(1,0)-trees and nonseparable permutations

Abstract: The subject of pattern avoiding permutations has its roots in computer science, namely in the problem of sorting a permutation through a stack. A formula for the number of permutations of length n that can be sorted by passing it twice through a stack (where the letters on the stack have to be in increasing order) was conjectured by West, and later proved by Zeilberger. Goulden and West found a bijection from such permutations to nonseparable planar maps, and later, Jacquard and Schaeffer presented a bijection from these planar maps to certain labeled plane trees, called beta(1,0)-trees. Using generating trees, Dulucq, Gire and West showed that nonseparable planar maps are equinumerous with permutations avoiding the (classical) pattern 2413 and the barred pattern 41\bar{3}52; they called these permutations nonseparable. We give a new bijection between beta(1,0)-trees and permutations avoiding the dashed patterns 3-1-4-2 and 2-41-3. These permutations can be seen to be exactly the reverse of nonseparable permutations. Our bijection is built using decompositions of the permutations and the trees, and it translates seven statistics on the trees into statistics on the permutations. Among the statistics involved are ascents, left-to-right minima and right-to-left maxima for the permutations, and leaves and the rightmost and leftmost paths for the trees. In connection with this we give a nontrivial involution on the beta(1,0)-trees, which specializes to an involution on unlabeled rooted plane trees, where it yields interesting results. Lastly, we conjecture the existence of a bijection between nonseparable permutations and two-stack sortable permutations preserving at least four permutation statistics.


Carry Propagation in Multiplication by Constants

Abstract: Suppose that a random n-bit number V is multiplied by an odd constant M, greater than or equal to 3, by adding shifted versions of the number V corresponding to the 1s in the binary representation of the constant M. Suppose further that the additions are performed by carry-save adders until the number of summands is reduced to two, at which time the final addition is performed by a carry-propagate adder. We show that in this situation the distribution of the length of the longest carry-propagation chain in the final addition is the same (up to terms tending to 0 as n tends to infinity) as when two independent n-bit numbers are added, and in particular the mean and variance are the same (again up to terms tending to 0). This result applies to all possible orders of performing the carry-save additions.


On the solvability of systems of pseudodifferential operators

Abstract: The paper studies the solvability for square systems of pseudodifferential operators. We assume that the system is of principal type, i.e., the principal symbol vanishes of first order on the kernel. We shall also assume that the eigenvalues of the principal symbol close to zero have constant multiplicity. We prove that local solvability for the system is equivalent to condition (PSI) on the eigenvalues of the principal symbol. This condition rules out any sign changes from - to + of the imaginary part of the eigenvalue when going in the positive direction on the bicharacteristics of the real part. Thus we need no conditions on the lower order terms. We obtain local solvability by proving a localizable a priori estimate for the adjoint operator with a loss of 3/2 derivatives (compared with the elliptic case).


About the logic of the prime number distribution

Abstract: There are two basic number sequences which play a major role in the prime number distribution. The first Number Sequence SQ1 contains all prime numbers of the form 6n+5 and the second Number Sequence SQ2 contains all prime numbers of the form 6n+1. All existing prime numbers seem to be contained in these two number sequences, except of the prime numbers 2 and 3. Riemanns Zeta Function also seems to indicate, that there is a logical connection between the mentioned number sequences and the distribution of prime numbers. This connection is indicated by lines in the diagram of the Zeta Function, which are formed by the points s where the Zeta Function is real. Another key role in the distribution of the prime numbers plays the number 5 and its periodic occurrence in the two number sequences SQ1 and SQ2. All non-prime numbers in SQ1 and SQ2 are caused by recurrences of these two number sequences with increasing wave-lengths in themselves, in a similar fashion as Overtones (harmonics) or Undertones derive from a fundamental frequency. On the contrary prime numbers represent spots in these two basic Number Sequences SQ1 and SQ2 where there is no interference caused by these recurring number sequences. The distribution of the non-prime numbers and prime numbers can be described in a graphical way with a -Wave Model- (or Interference Model) -- see Table 2.


Weak Hopf monoids in braided monoidal categories

Abstract: We develop the theory of weak bimonoids in braided monoidal categories and show them to be quantum categories in a certain sense. Weak Hopf monoids are shown to be quantum groupoids. Each separable Frobenius monoid R leads to a weak Hopf monoid R \otimes R.


The morphology of infinite tournaments. Application to the growth of their profile

Abstract: A tournament is \emph{acyclically indecomposable} if no acyclic autonomous set of vertices has more than one element. We identify twelve infinite acyclically indecomposable tournaments and prove that every infinite acyclically indecomposable tournament contains a subtournament isomorphic to one of these tournaments. The {\it profile} of a tournament $T$ is the function $\phi_T$ which counts for each integer $n$ the number $\phi_T(n)$ of tournaments induced by $T$ on the $n$-element subsets of $T$, isomorphic tournaments being identified. As a corollary of the result above we deduce that the growth of $\phi_T$ is either polynomial, in which case $\phi_T(n)\simeq an^k$, for some positive real $a$, some non-negative integer $k$, or as fast as some exponential.


A Simple Solution to a Major Problem: Proof of the Riemann Hypothesis

Abstract: Starting from the symmetrical reflection functional equation of the zeta function, we have found that the sigma values satisfying zeta(s) = 0 must also satisfy both |zeta(s)| = |zeta(1 - s)| and |gamma(s/2)zeta(s)| = |gamma((1 - s)/2)zeta(1 - s)|. We have shown that sigma = 1/2 is the only numeric solution that satisfies this requirement.


Exceptional symmetric domains

Abstract: We give the presentation of exceptional bounded symmetric domains using the Albert algebra and exceptional Jordan triple systems. The first chapter is devoted to Cayley-Graves algebras, the second to exceptional Jordan triple systems. In the third chapter, we give a geometric description of the two exceptional bounded symmetric domains, their boundaries and their compactification.


Local estimates and global continuities in Lebesgue spaces for bilinear operators

Abstract: In this paper, we first prove some local estimates for bilinear operators (closely related to the bilinear Hilbert transform and similar singular operators) with truncated symbol. Such estimates, in accordance with the Heisenberg uncertainty principle correspond to a description of ``off-diagonal'' decay. In addition they allow us to prove global continuities in Lebesgue spaces for bilinear operators with spatial dependent symbol.


A compactly supported formula for equivariant localization, and, simplicial complexes of Bialynicki-Birula decompositions

Abstract: Let X be a projective scheme carrying a circle action S with isolated fixed points. We associate a simplicial complex Delta(X,S) of "closure chains" using a refinement of its Morse/Bialynicki-Birula decomposition. If this decomposition is a stratification (e.g. when X is a flag manifold), then Delta(X,S) is just the order complex of the poset of fixed points. For X a toric variety, Delta(X,S) is a triangulation of the moment polytope. We compute some other examples, including a Bott-Samelson manifold and the punctual Hilbert scheme of 4 points in the plane. Summing over the facets of Delta(X,S), we obtain a positive formula for the Duistermaat-Heckman measure on the moment polytope of X, defined for any torus action extending S. We explain how, through brutal use of partial fractions, this can be extended to an AB/BV-type formula for integrating general classes. Throughout we work with equivariant Chow groups, and do not make any smoothness requirements on X.


A sampling inequality for fractional order Sobolev semi-norms using arbitrary order data

Abstract: To improve convergence results obtained using a framework for unsymmetric meshless methods due to Schaback (Preprint G\"ottingen 2006), we extend, in two directions, the Sobolev bound due to Arcang\'eli et al. (Numer Math 107, 181-211, 2007), which itself extends two others due to Wendland and Rieger (Numer Math 101, 643-662, 2005) and Madych (J. Approx Theory 142, 116-128, 2006). The first is to incorporate discrete samples of arbitrary order derivatives into the bound, which are used to obtain higher order convergence in higher order Sobolev norms. The second is to optimally bound fractional order Sobolev semi-norms, which are used to obtain more optimal convergence rates when solving problems requiring fractional order Sobolev spaces, notably inhomogeneous boundary value problems.


Extensions of Lie-Rinehart algebras and cotangent bundle reduction

Abstract: Let Q denote a smooth manifold acted upon smoothly by a Lie group G. The G-action lifts to an action on the total space T of the cotangent bundle of Q and hence on the standard symplectic Poisson algebra of smooth functions on T. The Poisson algebra of G-invariant functions on T yields a Poisson structure on the space T/G of G-orbits. We relate this Poisson algebra with extensions of Lie-Rinehart algebras and derive an explicit formula for this Poisson structure in terms of differentials. We then show, for the particular case where the G-action on Q is principal, how an explicit description of the Poisson algebra derived in the literature by an ad hoc construction is essentially a special case of the formula for the corresponding extension of Lie-Rinehart algebras. By means of various examples, we also show that this kind of description breaks down when the G-action does not define a principal bundle.


On a formula for the spectral flow and its applications

Abstract: We consider a continuous path of bounded symmetric Fredholm bilinear forms with arbitrary endpoints on a real Hilbert space, and we prove a formula that gives the spectral flow of the path in terms of the spectral flow of the restriction to a finite codimensional closed subspace. We also discuss the case of restrictions to a continuous path of finite codimensional closed subspaces. As an application of the formula, we introduce the notion of spectral flow for a periodic semi-Riemannian geodesic, and we compute its value in terms of the Maslov index.


Note on Frobenius monoidal functors

Abstract: It is well known that strong monoidal functors preserve duals. In this short note we show that a slightly weaker version of functor, which we call "Frobenius monoidal", is sufficient.


Geometric Properties of Assur Graphs

Abstract: In our previous paper, we presented the combinatorial theory for minimal isostatic pinned frameworks - Assur graphs - which arise in the analysis of mechanical linkages. In this paper we further explore the geometric properties of Assur graphs, with a focus on singular realizations which have static self-stresses. We provide a new geometric characterization of Assur graphs, based on special singular realizations. These singular positions are then related to dead-end positions in which an associated mechanism with an inserted driver will stop or jam.