Coloured quiver mutation for higher cluster categories

Abstract: We define mutation on coloured quivers associated to tilting objects in higher cluster categories. We show that this operation is compatible with the mutation operation on the tilting objects. This gives a combinatorial approach to tilting in higher cluster categories and especially an algorithm to determine the Gabriel quivers of tilting objects in such categories.


Hopf algebras of dimension 2p^2

Abstract: Let H be a non-semisimple Hopf algebra of dimension 2p^2 over an algebraically closed field of characteristic zero, where p is an odd prime. We prove that H or H^* is pointed, which completes the classification for Hopf algebras of these dimensions.


Long cycles in graphs through fragments

Abstract: Four basic Dirac-type sufficient conditions for a graph $G$ to be hamiltonian are known involving order $n$, minimum degree $\delta$, connectivity $\kappa$ and independence number $\alpha$ of $G$: (1) $\delta \geq n/2$ (Dirac); (2) $\kappa \geq 2$ and $\delta \geq (n+\kappa)/3$ (by the author); (3) $\kappa \geq 2$ and $\delta \geq \max\lbrace (n+2)/3,\alpha \rbrace$ (Nash-Williams); (4) $\kappa \geq 3$ and $\delta \geq \max\lbrace (n+2\kappa)/4,\alpha \rbrace$ (by the author). In this paper we prove the reverse version of (4) concerning the circumference $c$ of $G$ and completing the list of reverse versions of (1)-(4): (R1) if $\kappa \geq 2$, then $c\geq\min\lbrace n,2\delta\rbrace$ (Dirac); (R2) if $\kappa \geq 3$, then $c\geq\min\lbrace n,3\delta -\kappa\rbrace$ (by the author); (R3) if $\kappa\geq 3$ and $\delta\geq \alpha$, then $c\geq\min\lbrace n,3\delta-3\rbrace$ (Voss and Zuluaga); (R4) if $\kappa\geq 4$ and $\delta\geq \alpha$, then $c\geq\min\lbrace n,4\delta-2\kappa\rbrace$. To prove (R4), we present four more general results centered around a lower bound $c\geq 4\delta-2\kappa$ under four alternative conditions in terms of fragments. A subset $X$ of $V(G)$ is called a fragment of $G$ if $N(X)$ is a minimum cut-set and $V(G)-(X\cup N(X))\neq\emptyset$.


A Fast Butterfly Algorithm for the Computation of Fourier Integral Operators

Abstract: This paper is concerned with the fast computation of Fourier integral operators of the general form $\int_{\R^d} e^{2\pi\i \Phi(x,k)} f(k) d k$, where $k$ is a frequency variable, $\Phi(x,k)$ is a phase function obeying a standard homogeneity condition, and $f$ is a given input. This is of interest for such fundamental computations are connected with the problem of finding numerical solutions to wave equations, and also frequently arise in many applications including reflection seismology, curvilinear tomography and others. In two dimensions, when the input and output are sampled on $N \times N$ Cartesian grids, a direct evaluation requires $O(N^4)$ operations, which is often times prohibitively expensive. This paper introduces a novel algorithm running in $O(N^2 \log N)$ time, i. e. with near-optimal computational complexity, and whose overall structure follows that of the butterfly algorithm [Michielssen and Boag, IEEE Trans Antennas Propagat 44 (1996), 1086-1093]. Underlying this algorithm is a mathematical insight concerning the restriction of the kernel $e^{2\pi\i \Phi(x,k)}$ to subsets of the time and frequency domains. Whenever these subsets obey a simple geometric condition, the restricted kernel has approximately low-rank; we propose constructing such low-rank approximations using a special interpolation scheme, which prefactors the oscillatory component, interpolates the remaining nonoscillatory part and, lastly, remodulates the outcome. A byproduct of this scheme is that the whole algorithm is highly efficient in terms of memory requirement. Numerical results demonstrate the performance and illustrate the empirical properties of this algorithm.


Polynomial treewidth forces a large grid-like-minor

Abstract: Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an $\ell\times\ell$ grid minor is exponential in $\ell$. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A \emph{grid-like-minor of order} $\ell$ in a graph $G$ is a set of paths in $G$ whose intersection graph is bipartite and contains a $K_{\ell}$-minor. For example, the rows and columns of the $\ell\times\ell$ grid are a grid-like-minor of order $\ell+1$. We prove that polynomial treewidth forces a large grid-like-minor. In particular, every graph with treewidth at least $c\ell^4\sqrt{\log\ell}$ has a grid-like-minor of order $\ell$. As an application of this result, we prove that the cartesian product $G\square K_2$ contains a $K_{\ell}$-minor whenever $G$ has treewidth at least $c\ell^4\sqrt{\log\ell}$.


Three geometric applications of quandle homology

Abstract: In this paper we describe three geometric applications of quandle homology. We show that it gives obstructions to tangle embeddings, provides the lower bound for the 4-move distance between links, and can be used in determining periodicity of links.


Abstract: Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by \[I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y),\] and set $M(X) = \sup I(\mu)$, where $\mu$ ranges over the collection of signed measures in $\mathcal{M}(X)$ of total mass 1. The metric space $(X, d)$ is quasihypermetric if for all $n \in \N$, all $\alpha_1, ..., \alpha_n \in \R$ satisfying $\sum_{i=1}^n \alpha_i = 0$ and all $x_1, ..., x_n \in X$, one has $\sum_{i,j=1}^n \alpha_i \alpha_j d(x_i, x_j) \leq 0$. Without the quasihypermetric property $M(X)$ is infinite, while with the property a natural semi-inner product structure becomes available on $\mathcal{M}_0(X)$, the subspace of $\mathcal{M}(X)$ of all measures of total mass 0. This paper explores: operators and functionals which provide natural links between the metric structure of $(X, d)$, the semi-inner product space structure of $\mathcal{M}_0(X)$ and the Banach space $C(X)$ of continuous real-valued functions on $X$; conditions equivalent to the quasihypermetric property; the topological properties of $\mathcal{M}_0(X)$ with the topology induced by the semi-inner product, and especially the relation of this topology to the weak-$*$ topology and the measure-norm topology on $\mathcal{M}_0(X)$; and the functional-analytic properties of $\mathcal{M}_0(X)$ as a semi-inner product space, including the question of its completeness. A later paper [Peter Nickolas and Reinhard Wolf, Distance Geometry in Quasihypermetric Spaces. II] will apply the work of this paper to a detailed analysis of the constant $M(X)$.


Derangement Polynomials and Excedances of Type B

Abstract: Adopting the definition of excedances of type B due to Brenti, we give a type B analogue of the q-derangement polynomials. The connection between q-derangement polynomials and Eulerian polynomials naturally extends to the type B case. Based on this relation, we derive some basic properties of the q-derangement polynomials of type B, including the generating function formula, the Sturm sequence property, and the asymptotic normal distribution. We also show that the q-derangement polynomials are almost symmetric in the sense that the coefficients possess the spiral property.


Liouville type of theorems for the Euler and the Navier-Stokes equations

Abstract: We prove Liouville type of theorems for weak solutions of the Navier-Stokes and the Euler equations. In particular, if the pressure satisfies $ p\in L^1 (0,T; L^1 (\Bbb R^N))$ with $\int_{\Bbb R^N} p(x,t)dx \geq 0$, then the corresponding velocity should be trivial, namely $v=0$ on $\Bbb R^N \times (0,T)$. In particular, this is the case when $p\in L^1 (0,T; \mathcal{H}^1 (\Bbb R^N))$, where $\mathcal{H}^1 (\Bbb R^N)$ the Hardy space. On the other hand, we have equipartition of energy over each component, if $p\in L^1 (0,T; L^1 (\Bbb R^N))$ with $\int_{\Bbb R^N} p(x,t)dx <0$. Similar results hold also for the magnetohydrodynamic equations.


Distance Geometry in Quasihypermetric Spaces. II

Abstract: Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by \[ I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y), \] and set $M(X) = \sup I(\mu)$, where $\mu$ ranges over the collection of signed measures in $\mathcal{M}(X)$ of total mass 1. This paper, with an earlier and a subsequent paper [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and III], investigates the geometric constant $M(X)$ and its relationship to the metric properties of $X$ and the functional-analytic properties of a certain subspace of $\mathcal{M}(X)$ when equipped with a natural semi-inner product. Using the work of the earlier paper, this paper explores measures which attain the supremum defining $M(X)$, sequences of measures which approximate the supremum when the supremum is not attained and conditions implying or equivalent to the finiteness of $M(X)$.


Distance Geometry in Quasihypermetric Spaces. III

Abstract: Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by \[ I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y), \] and set $M(X) = \sup I(\mu)$, where $\mu$ ranges over the collection of signed measures in $\mathcal{M}(X)$ of total mass 1. This paper, with two earlier papers [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and II], investigates the geometric constant $M(X)$ and its relationship to the metric properties of $X$ and the functional-analytic properties of a certain subspace of $\mathcal{M}(X)$ when equipped with a natural semi-inner product. Specifically, this paper explores links between the properties of $M(X)$ and metric embeddings of $X$, and the properties of $M(X)$ when $X$ is a finite metric space.


Group-case commutative association schemes and their character tables

Abstract: Leading towards the classification of primitive commutative association schemes as the ultimate goal, Bannai and some of his school have been trying to * identify the major sources of (primitive) commutative association schemes, * collect known group-case primitive commutative association schemes, and * compute their character tables over the last twenty years. The construction of their character tables are important first step for a systematic study of such association schemes and towards the classification of those schemes. In this talk, we briefly survey the progress made in this direction of research, and list some open problems.


A note on k[z]-automorphisms in two variables

Abstract: We prove that for a polynomial $f\in k[x,y,z]$ equivalent are: (1)$f$ is a $k[z]$-coordinate of $k[z][x,y]$, and (2) $k[x,y,z]/(f)\cong k^{[2]}$ and $f(x,y,a)$ is a coordinate in $k[x,y]$ for some $a\in k$. This solves a special case of the Abhyankar-Sathaye conjecture. As a consequence we see that a coordinate $f\in k[x,y,z]$ which is also a $k(z)$-coordinate, is a $k[z]$-coordinate. We discuss a method for constructing automorphisms of $k[x,y,z]$, and observe that the Nagata automorphism occurs naturally as the first non-trivial automorphism obtained by this method - essentially linking Nagata with a non-tame $R$-automorphism of $R[x]$, where $R=k[z]/(z^2)$.


Dynamique transverse de la lamination de Ghys-Kenyon

Abstract: Using an aperiodic and repetitive subtree of the Cayley graph of the free Abelian group with two generators, described by Kenyon, Ghys has constructed an example of minimal Riemann surface lamination having both Euclidean and hyperbolic leaves. We prove that the transverse dynamics of this lamination is represented (in a measurable way) by a 2-adic odometer. In fact, we can describe its topological transverse dynamics, and prove that the Ghys-Kenyon lamination is affable.


On hypercomplex pseudo-Hermitian manifolds

Abstract: The class of the hypercomplex pseudo-Hermitian manifolds is considered. The flatness of the considered manifolds with the 3 parallel complex structures is proved. Conformal transformations of the metrics are introduced. The conformal invariance and the conformal equivalence of the basic types manifolds are studied. A known example is characterized in relation to the obtained results.


Neighborliness of Marginal Polytopes

Abstract: A neighborliness property of marginal polytopes of hierarchical models, depending on the cardinality of the smallest non-face of the underlying simplicial complex, is shown. The case of binary variables is studied explicitly, then the general case is reduced to the binary case. A Markov basis for binary hierarchical models whose simplicial complexes is the complement of an interval is given.


Convergence of eigenvalues for a highly non-self-adjoint differential operator

Abstract: In this paper we study a family of operators dependent on a small parameter $\epsilon > 0$, which arise in a problem in fluid mechanics. We show that the spectra of these operators converge to N as $\epsilon \to 0$, even though, for fixed $\epsilon > 0$, the eigenvalue asymptotics are quadratic.


Even universal binary Hermitian lattices over imaginary quadratic fields

Abstract: A positive definite even Hermitian lattice is called \emph{even universal} if it represents all even positive integers. We introduce a method to get all even universal binary Hermitian lattices over imaginary quadratic fields $\Q{-m}$ for all positive square-free integers $m$ and we list optimal criterions on even universality of Hermitian lattices over $\Q{-m}$ which admits even universal binary Hermitian lattices.


Positive Legendrian regular homotopies

Abstract: In contrast with what happens for Legendrian embeddings, there always exist positive loops of Legendrian immersions.


Repr\'esentations d\'eterminantales effectives des polyn\^omes univari\'es par les matrices fl\`eches

Abstract: We first show the existence of an effective determinantal representation for any univariate polynomial with real coefficients. Then, we more precisely establish that any univariate polynomial with real coefficients has an effective determinantal representation with signature (r+s,s) if and only if it has at least r real roots with multiplicity. The effective determinantal representations we construct used arrow matrices.


Characteristic Classes on Grassmann Manifolds

Abstract: In this paper, we use characteristic classes of the canonical vector bundles and the Poincar\' {e} dualality to study the structure of the real homology and cohomology groups of oriented Grassmann manifold $G(k, n)$. Show that for $k=2$ or $n\leq 8$, the cohomology groups $H^*(G(k,n),{\bf R})$ are generated by the first Pontrjagin class, the Euler classes of the canonical vector bundles. In these cases, the Poincar\' {e} dualality: $H^q(G(k,n),{\bf R}) \to H_{k(n-k)-q}(G(k,n),{\bf R})$ can be given explicitly.


The notion of convexity and concavity on Wiener space

Abstract: We define, in the frame of an abstract Wiener space, the notions of convexity and of concavity for the equivalence classes of random variables. As application we show that some important inequalities of the finite dimensional case have their natural counterparts in this setting.


Large Deviations of Vector-valued Martingales in 2-Smooth Normed Spaces

Abstract: We derive exponential bounds on probabilities of large deviations for "light tail" martingales taking values in finite-dimensional normed spaces. Our primary emphasis is on the case where the bounds are dimension-independent or nearly so. We demonstrate that this is the case when the norm on the space can be approximated, within an absolute constant factor, by a norm which is differentiable on the unit sphere with a Lipschitz continuous gradient. We also present various examples of spaces possessing the latter property.


Nonparametric Denoising of Signals with Unknown Local Structure, I: Oracle Inequalities

Abstract: We consider the problem of pointwise estimation of multi-dimensional signals $s$, from noisy observations $(y_\tau)$ on the regular grid $\bZd$. Our focus is on the adaptive estimation in the case when the signal can be well recovered using a (hypothetical) linear filter, which can depend on the unknown signal itself. The basic setting of the problem we address here can be summarized as follows: suppose that the signal $s$ is "well-filtered", i.e. there exists an adapted time-invariant linear filter $q^*_T$ with the coefficients which vanish outside the "cube" $\{0,..., T\}^d$ which recovers $s_0$ from observations with small mean-squared error. We suppose that we do not know the filter $q^*$, although, we do know that such a filter exists. We give partial answers to the following questions: -- is it possible to construct an adaptive estimator of the value $s_0$, which relies upon observations and recovers $s_0$ with basically the same estimation error as the unknown filter $q^*_T$? -- how rich is the family of well-filtered (in the above sense) signals? We show that the answer to the first question is affirmative and provide a numerically efficient construction of a nonlinear adaptive filter. Further, we establish a simple calculus of "well-filtered" signals, and show that their family is quite large: it contains, for instance, sampled smooth signals, sampled modulated smooth signals and sampled harmonic functions.


Solving variational inequalities with Stochastic Mirror-Prox algorithm

Abstract: In this paper we consider iterative methods for stochastic variational inequalities (s.v.i.) with monotone operators. Our basic assumption is that the operator possesses both smooth and nonsmooth components. Further, only noisy observations of the problem data are available. We develop a novel Stochastic Mirror-Prox (SMP) algorithm for solving s.v.i. and show that with the convenient stepsize strategy it attains the optimal rates of convergence with respect to the problem parameters. We apply the SMP algorithm to Stochastic composite minimization and describe particular applications to Stochastic Semidefinite Feasability problem and Eigenvalue minimization.


Symmetric topological complexity of projective and lens spaces

Abstract: For real projective spaces, (a) the Euclidean immersion dimension, (b) the existence of axial maps, and (c) the topological complexity are known to be three facets of the same problem. But when it comes to embedding dimension, the classical work of Berrick, Feder and Gitler leaves a small indeterminacy when trying to identify the existence of Euclidean embeddings of these manifolds with the existence of symmetric axial maps. As an alternative we show that the symmetrized version of (c) captures, in a sharp way, the embedding problem. Extensions to the case of even torsion lens spaces and complex projective spaces are discussed.


Prehomogeneous Affine Representations and Flat Pseudo-Riemannian Manifolds

Abstract: The theory of flat Pseudo-Riemannian manifolds and flat affine manifolds is closely connected to the topic of prehomogeneous affine representations of Lie groups. In this article, we exhibit several aspects of this correspondence. At the heart of our presentation is a development of the theory of characteristic classes and characters of prehomogeneous affine representations. We give applications concerning flat affine, as well as Pseudo-Riemannian and symplectic affine flat manifolds.


Self-adjoint curl operators

Abstract: We study the exterior derivative as a symmetric unbounded operator on square integrable 1-forms on a 3D bounded domain $D$. We aim to identify boundary conditions that render this operator self-adjoint. By the symplectic version of the Glazman-Krein-Naimark theorem this amounts to identifying complete Lagrangian subspaces of the trace space of H(curl) equipped with a symplectic pairing arising from the $\wedge$-product of 1-forms on $\partial D$. Substantially generalizing earlier results, we characterize Lagrangian subspaces associated with closed and co-closed traces. In the case of non-trivial topology of the domain, different contributions from co-homology spaces also distinguish different self-adjoint extension. Finally, all self-adjoint extensions discussed in the paper are shown to possess a discrete point spectrum, and their relationship with curl curl-operators is discussed.


Deformation Spaces for Affine Crystallographic Groups

Abstract: We develop the foundations of the deformation theory of compact complete affine space forms and affine crystallographic groups. Using methods from the theory of linear algebraic groups we show that these deformation spaces inherit an algebraic structure from the space of crystallographic homomorphisms. We also study the properties of the action of the homotopy mapping class groups on deformation spaces. In our context these groups are arithmetic groups, and we construct examples of flat affine manifolds where every finite group of mapping classes admits a fixed point on the deformation space. We also show that the existence of fixed points on the deformation space is equivalent to the realisation of finite groups of homotopy equivalences by finite groups of affine diffeomorphisms. Extending ideas of Auslander we relate the deformation spaces of affine space forms with solvable fundamental group to deformation spaces of manifolds with nilpotent fundamental group. We give applications concerning the classification problem for affine space forms.


Modular forms and K3 surfaces

Abstract: For every known Hecke eigenform of weight 3 with rational eigenvalues we exhibit a K3 surface over QQ associated to the form. This answers a question asked independently by Mazur and van Straten. The proof builds on a classification of CM forms by the second author.


Cohomology of quantum groups: An analog of Kostant's Theorem

Abstract: We prove the analog of Kostant's Theorem on Lie algebra cohomology in the context of quantum groups. We prove that Kostant's cohomology formula holds for quantum groups at a generic parameter $q$, recovering an earlier result of Malikov in the case where the underlying semisimple Lie algebra $\mathfrak{g} = \mathfrak{sl}(n)$. We also show that Kostant's formula holds when $q$ is specialized to an $\ell$-th root of unity for odd $\ell \ge h-1$ (where $h$ is the Coxeter number of $\mathfrak{g}$) when the highest weight of the coefficient module lies in the lowest alcove. This can be regarded as an extension of results of Friedlander-Parshall and Polo-Tilouine on the cohomology of Lie algebras of reductive algebraic groups in prime characteristic.


mu-constancy does not imply constant bi-Lipschitz type

Abstract: We show that a family of isolated complex hypersurface singularities with constant Milnor number may fail, in the strongest sense, to have constant bi-Lipschitz type. Our example is the Briac con--Speder family $X_t:=\{(x,y,z)\in\C^3 | x^5+z^{15}+y^7z+txy^6=0 \}$ of normal complex surface germs; we show the germ $(X_0, 0)$ is not bi-Lipschitz homeomorphic with respect to the inner metric to the germ $(X_t,0)$ for $t\ne 0$.


Certain free products of graph operator algebras

Abstract: We develop a notion of a generalized Cuntz-Krieger family of projections and partial isometries where the range of the partial isometries need not have trivial intersection. We associate to these generalized Cuntz-Krieger families a directed graph, with a coloring function on the edge set. We call such a directed graph an edge-colored directed graph. We then study the $C^*$-algebras and the non-selfadjoint operator algebras associated to edge-colored directed graphs. These algebras arise as free products of directed graph algebras with amalgamation. We then determine the $C^*$-envelopes for a large class of the non-selfadjoint algebras. Finally, we relate properties of the edge-colored directed graphs to properties of the associated $C^*$-algebra, including simplicity and nuclearity. Using the free product description of these algebras we investigate the $K$-theory of these algebras.


Small-time expansions for the transition distributions of L\'evy processes

Abstract: Let $X$ be a L\'evy process with absolutely continuous L\'evy measure $\nu$. Small time polynomial expansions of order $n$ in $t$ are obtained for the tails $P(X_{t}\geq{}y)$ of the process, assuming smoothness conditions on the L\'evy density away from the origin. By imposing additional regularity conditions on the transition density $p_{t}$ of $X_{t}$, an explicit expression for the remainder of the approximation is also given. As a byproduct, polynomial expansions of order $n$ in $t$ are derived for the transition densities of the process. The conditions imposed on $p_{t}$ require that its derivatives remain uniformly bounded away from the origin, as $t\to{}0$; such conditions are shown to be satisfied for symmetric stable L\'evy processes as well as for other related L\'evy processes of relevance in mathematical finance. The expansions seem to correct asymptotics previously reported in the literature.


Non-Walker Self-Dual Neutral Einstein Four-Manifolds of Petrov Type III

Abstract: The local structure of the manifolds named in the title is described. Although curvature homogeneous, they are not, in general, locally homogeneous. Not all of them are Ricci-flat, which answers an existence question about type III Jordan-Osserman metrics, raised by Diaz-Ramos, Garcia-Rio and Vazquez-Lorenzo (2006).


A Morse theoretic description of string topology

Abstract: Let M be a closed, oriented, n-dimensional manifold. In this paper we give a Morse theoretic description of the string topology operations introduced by Chas and Sullivan, and extended by the first author, Jones, Godin, and others. We do this by studying maps from surfaces with cylindrical ends to M, such that on the cylinders, they satisfy the gradient flow equation of a Morse function on the loop space, LM. We then give Morse theoretic descriptions of related constructions, such as the Thom and Euler classes of a vector bundle, as well as the shriek, or unkehr homomorphism.


Coniveau 2 complete intersections and effective cones

Abstract: The goal of this paper is first of all to propose a strategy to attack the generalized Hodge conjecture for coniveau 2 complete intersections, and secondly to state a conjecture concerning the cones of effective cycle classes in intermediate dimensions. Our main results show that the generalized Hodge conjecture for coniveau 2 complete intersections would follow from a particular case of this effectiveness conjecture.


$\LE$-diagrams and totally positive bases inside the nonnegative Grassmannian

Abstract: There is a cell decomposition of the nonnegative Grassmannian. For each cell, totally positive bases(TP-bases) is defined as the minimal set of Pl\"ucker variables such that all other nonzero Pl\"ucker variables in the cell can be expressed in those variables in a subtraction-free rational function. This is the generalization of the TP-bases defined for nonnegative part of $GL_k$ defined in \cite{FZ5}. For each cell, we have a $\LE$-diagram and a natural way to label the dots inside the diagram with Pl\"ucker variables. Those set of Pl\"ucker variables form a TP-bases of the cell. Using mutations coming from 3-term Pl\"ucker relation, we conjecture that they can be mutated to a special set of Pl\"ucker variable $\S$. All other nonzero Pl\"ucker variables in the cell will be expressed as a subtraction-free Laurent polynomial in variables of $\S$. We define TP-diagrams to express the transformation procedure in terms of moves on a diagram. We will prove the conjecture for certain class of cells called weakly-connected cells. Then we will study the connection with cluster algebras through lattice-path-matroid cells.


The Seventeen Elements of Pythagorean Triangles

Abstract: This is an exhaustive study of the seventeen elements of Pythagorean triangles, from the point of view of when such an element is an irrational number, a rational number, or an integer. For each of these 17 elements,precice conditions for their integrality,rationality, or irrationality; are given. These 17 elements are:The radius of the incircle, the radius of the circumscribed circle,the three radii of the three exterior circles tangential to the three lines containing the triangles'sides, the lengths of the three heights, the lenghts of the three internal angle bisectors, the lengthe of the three external angle bisectors, and the lengths of the three medians.


The holomorphic Gauss Parametrization

Abstract: We give a local parametric description of all holomorphic hypersurfaces in complex Euclidean and projective spaces with constant index of relative nullity, together with applications. This is a complex analogue to the parametrization for real hypersurfaces in Euclidean space known as the Gauss parametrization.


Birational rigidity of Fano varieties and field extensions

Abstract: The aim of this note is to settle some foundational questions about the behavior of birational rigidity in extensions of algebraically closed fields.


Nonabelian harmonic analysis and functional equations on compact groups

Abstract: Making use of nonabelian harmonic analysis and representation theory, we solve the functional equation $$f_1(xy)+f_2(yx)+f_3(xy^{-1})+f_4(y^{-1}x)=f_5(x)f_6(y)$$ on arbitrary compact groups. The structure of its general solution is completely described. Consequently, several special cases of the above equation, in particular, the Wilson equation and the d'Alembert long equation, are solved on compact groups.


Multi-product splitting and Runge-Kutta-Nystrom integrators

Abstract: The splitting of $\e^{h(A+B)}$ into a single product of $\e^{h A}$ and $\e^{hB}$ results in symplectic integrators when $A$ and $B$ are classical Lie operators. However, at high orders, a single product splitting, with exponentially growing number of operators, is very difficult to derive. This work shows that, if the splitting is generalized to a sum of products, then a simple choice of the basis product reduces the problem to that of extrapolation, with analytically known coefficients and only quadratically growing number of operators. When a multi-product splitting is applied to classical Hamiltonian systems, the resulting algorithm is no longer symplectic but is of the Runge-Kutta-Nystr\"om (RKN) type. Multi-product splitting, in conjunction with a special force-reduction process,explains why at orders $p=4$ and 6, RKN integrators only need $p-1$ force evaluations.


Edge-Graph Diameter Bounds for Convex Polytopes with Few Facets

Abstract: We show that the edge graph of a 6-dimensional polytope with 12 facets has diameter at most 6, thus verifying the d-step conjecture of Klee and Walkup in the case of d=6. This implies that for all pairs (d,n) with n-d \leq 6 the diameter of the edge graph of a d-polytope with n facets is bounded by 6, which proves the Hirsch conjecture for all n-d \leq 6. We show this result by showing this bound for a more general structure -- so-called matroid polytopes -- by reduction to a small number of satisfiability problems.


Convexit\'e holomorphe du rev\^etement de Malcev d'apr\`es S. Leroy

Abstract: In these notes, we present the strategy employed by S. Leroy to attack the nilpotent case of the Shafarevich conjecture. The projective case was treated in a paper of L. Katzarkov but Leroy's proof is largely independant and works in the K\"ahler setting as well. It is entirely based on the higher Albanese manifolds constructed by R. Hain.


Pseudodifferential operator calculus for generalized Q-rank 1 locally symmetric spaces, I

Abstract: This paper is the first of two papers constructing a calculus of pseudodifferential operators suitable for doing analysis on Q-rank 1 locally symmetric spaces and Riemannian manifolds generalizing these. This generalization is the interior of a manifold with boundary, where the boundary has the structure of a tower of fibre bundles. The class of operators we consider on such a space includes those arising naturally from metrics which degenerate to various orders at the boundary, in directions given by the tower of fibrations. As well as Q-rank 1 locally symmetric spaces, examples include Ricci-flat metrics on the complement of a divisor in a smooth variety constructed by Tian and Yau. In this first part of the calculus construction, parametrices are found for "fully elliptic differential \bfa-operators", which are uniformly elliptic operators on these manifolds that satisfy an additional invertibility condition at infinity. In the second part we will consider operators that do not satisfy this condition.


Triangulations of the sphere and degenerations of K3 surfaces

Abstract: W. Thurston proved that to a triangulation of the sphere of non-negative combinatorial curvature, one can associate an element in a certain lattice over the Eisenstein integers such that its orbit is a complete invariant of the triangulation. In this paper, we show that this association can be obtained naturally by using Type III degenerations of K3 surfaces.


Finite index subgroups of fully residually free groups

Abstract: Using graph-theoretic techniques for f.g. subgroups of $F^{\mathbb{Z}[t]}$ we provide a criterion for a f.g. subgroup of a f.g. fully residually free group to be of finite index. Moreover, we show that this criterion can be checked effectively. Also we obtain an analogue of Greenberg-Stallings Theorem for f.g. fully residually free groups, and prove that a f.g. non-abelian subgroup of a f.g. fully residually free group is of finite index in its commensurator.


The modified K\"ahler-Ricci flow and solitons

Abstract: We investigate the K\"ahler-Ricci flow modified by a holomorphic vector field. We find equivalent analytic criteria for the convergence of the flow to a K\"ahler-Ricci soliton. In addition, we relate the asymptotic behavior of the scalar curvature along the flow to the lower boundedness of the modified Mabuchi energy.


On principal fibrations associated with one algebra

Abstract: In this paper we study two types of fibrations associated with a 3-dimensional unital associative irreducible algebra and their basic properties. We investigate trivial principal fibrations of degenerate semi-Euclidean sphere and their semi-conformal and projective models. We use Norden normalization method for constructing second model.


Solving the conjugacy problem in Garside groups by cyclic sliding

Abstract: We present a solution to the conjugacy decision problem and the conjugacy search problem in Garside groups, which is theoretically simpler than the usual one, with no loss of efficiency. This is done by replacing the well known cycling and decycling operations by a new one, called cyclic sliding, which appears to be a more natural choice. We give an analysis of the complexity of our algorithm in terms of fundamental operations with simple elements, so our analysis is valid for every Garside group. This paper intends to be self-contained, not requiring any previous knowledge of prior algorithms, and includes all the details for the algorithm to be implemented on a computer.


Connected Components of Hurwitz Schemes and Malle's Conjecture

Abstract: Let Z(X) be the number of degree-d extensions of F_q(t) with bounded discriminant and some specified Galois group. The problem of computing Z(X) can be related to a problem of counting F_q-rational points on certain Hurwitz spaces. Ellenberg and Venkatesh used this idea to develop a heuristic for the asymptotic behavior of Z'(X), the number of -geometrically connected- extensions, and showed that this agrees with the conjectures of Malle for function fields. We extend Ellenberg-Venkatesh's argument to handle the more complicated case of covers of P^1 which may not be geometrically connected, and show thatthe resulting heuristic suggests a natural modification to Malle's conjecture which avoids the counterexamples, due to Kl\"uners, to the original conjecture.


Counting Multisections in Conic Bundles over a Curve defined over F_q

Abstract: For a given conic bundle X over a curve C defined over F_q, we count irreducible branch covers of C in X of degree d and height e>>1. As a special case, we get the number of algebraic numbers of degree d and height e over the function field F_q (C).


Splitting of Sharply 2-Transitive Groups of Characteristic 3

Abstract: We give a group theoretic proof of the splitting of sharply 2-transitive groups of characteristic 3.


On NP complete problems I

Abstract: We study the quadratic residue problem known as an NP complete problem by way of the prime number and show that a nondeterministic polynomial process does not belong to the class P because of a random distribution of solutions for the quadratic residue problem.


Fonction constante et d\'eriv\'ee nulle : un r\'esultat si trivial..

Abstract: We study various proofs of the caracterization of constant functions, more precisely of the theorem: a derivable function, defined on a real interval, is constant if, and only if, its derivative is null. Our aim is to study the relationships of these proofs with the mathematical curriculum of secondary schools and the begining of undergraduate studies in France, from various point of views (epistemological, historical, didactical).


Least Squares and Shrinkage Estimation under Bimonotonicity Constraints

Abstract: In this paper we describe active set type algorithms for minimization of a smooth function under general order constraints, an important case being functions on the set of bimonotone r-by-s matrices. These algorithms can be used, for instance, to estimate a bimonotone regression function via least squares or (a smooth approximation of) least absolute deviations. Another application is shrinkage estimation in image denoising or, more generally, regression problems with two ordinal factors after representing the data in a suitable basis which is indexed by pairs (i,j) in {1,...,r}x{1,...,s}. Various numerical examples illustrate our methods.


Rational representations of $GL_2$

Abstract: Let $F$ be an algebraically closed field of characteristic $p$. We fashion an infinite dimensional basic algebra $\underleftarrow{\mathcal{C}}_p(F)$, with a transparent combinatorial structure, which we expect to control the rational representation theory of $GL_2(F)$.


Sudden extinction of a critical branching process in random environment

Abstract: Let $T$ be the extinction moment of a critical branching process $Z=(Z_{n},n\geq 0) $ in a random environment specified by iid probability generating functions. We study the asymptotic behavior of the probability of extinction of the process $Z$ at moment $n\to \infty$, and show that if the logarithm of the (random) expectation of the offspring number belongs to the domain of attraction of a non-gaussian stable law then the extinction occurs owing to very unfavorable environment forcing the process, having at moment $T-1$ exponentially large population, to die out. We also give an interpretation of the obtained results in terms of random walks in random environment.


Homotopy, homology, and $GL_2$

Abstract: We define weak 2-categories of finite dimensional algebras with bimodules, along with collections of operators $\mathbb{O}_{(c,x)}$ on these 2-categories. We prove that special examples $\mathbb{O}_p$ of these operators control all homological aspects of the rational representation theory of the algebraic group $GL_2$, over a field of positive characteristic. We prove that when $x$ is a Rickard tilting complex, the operators $\mathbb{O}_{(c,x)}$ honour derived equivalences, in a differential graded setting. We give a number of representation theoretic corollaries, such as the existence of tight $\mathbb{Z}_+$-gradings on Schur algebras $S(2,r)$, and the existence of braid group actions on the derived categories of blocks of these Schur algebras.


Universal classes for algebraic groups

Abstract: We exhibit cocycles representing certain classes in the rational cohomology of of the general linear group with coefficients in the divided powers of a Frobenius twist of the adjoint representation. These classes' existence was anticipated by van der Kallen, and they intervene in the proof that reductive linear algebraic groups have finitely generated cohomology algebras.


A phase transition for non-intersecting Brownian motions, and the Painleve II equation

Abstract: We consider n non-intersecting Brownian motions with two fixed starting positions and two fixed ending positions in the large n limit. We show that in case of 'large separation' between the endpoints, the particles are asymptotically distributed in two separate groups, with no interaction between them, as one would intuitively expect. We give a rigorous proof using the Riemann-Hilbert formalism. In the case of 'critical separation' between the endpoints we are led to a model Riemann-Hilbert problem associated to the Hastings-McLeod solution of the Painleve II equation. We show that the Painleve II equation also appears in the large n asymptotics of the recurrence coefficients of the multiple Hermite polynomials that are associated with the Riemann-Hilbert problem.


Bifunctor cohomology and Cohomological finite generation for reductive groups

Abstract: Let G be a reductive linear algebraic group over a field k. Let A be a finitely generated commutative k-algebra on which G acts rationally by k-algebra automorphisms. Invariant theory tells that the ring of invariants A^G=H^0(G,A) is finitely generated. We show that in fact the full cohomology ring H^*(G,A) is finitely generated. The proof is based on the strict polynomial bifunctor cohomology classes constructed by the junior author. We also continue the study of bifunctor cohomology of the divided powers of a Frobenius twist of the adjoint representation.


Kobayashi geodesics in A_g

Abstract: We consider Kobayashi geodesics in the moduli space of abelian varieties A_g that is, algebraic curves that are totally geodesic submanifolds for the Kobayashi metric. We show that Kobayashi geodesics can be characterized as those curves whose logarithmic tangent bundle splits as a subbundle of the logarithmic tangent bundle of A_g. Both Shimura curves and Teichmueller curves are examples of Kobayashi geodesics, but there are other examples. We show moreover that non-compact Kobayashi geodesics always map to the locus of real multiplication and that the Q-irreducibility of the induced variation of Hodge structures implies that they are defined over a number field.


A simulation study comparing likelihood and non-likelihood approaches in analyzing overdispersed count data

Abstract: Overdispersed count data are modelled with likelihood and non-likelihood approaches. Likelihood approaches include the Poisson mixtures with three distributions, the gamma, the lognormal, and the inverse Gaussian distributions. Non-likelihood approaches include the robust sandwich estimator and quasilikelihood. In this simulation study, overdispersed count data were simulated under the Poisson mixtures with the gamma, the lognormal and the inverse Gaussian distributions, then analyzed with the five likelihood and non-likelihood approaches. Our results indicated that 1) when the count data are mildly overdispersed, there are virtually no differences in type I error rate, standard error of the main effect, and empirical power among the five methods; 2) when the count data are very overdispersed, none of these five approaches is robust to model misspecification as evaluated by type I error rate, standard error of the main effect, and empirical power. This simulation study raises caution on using non-likelihood method for analyzing very overdispered count data because of likely higher type I error and inappropriate power levels. Unlike non-likelihood approaches, likelihood approaches allow for statistical tests based on likelihood ratios and for checking model fit and provide basis for power and sample size calculations. When likelihood approaches are used, we suggest comparing likelihood values to select the appropriate parametric method for analyzing very overdispersed count data.


On the frequentist coverage of Bayesian credible intervals for lower bounded means

Abstract: For estimating a lower bounded location or mean parameter for a symmetric and logconcave density, we investigate the frequentist performance of the $100(1-\alpha)%$ Bayesian HPD credible set associated with priors which are truncations of flat priors onto the restricted parameter space. Various new properties are obtained. Namely, we identify precisely where the minimum coverage is obtained and we show that this minimum coverage is bounded between $1-\frac{3\alpha}{2}$ and $1-\frac{3\alpha}{2}+\frac{\alpha^2}{1+\alpha}$; with the lower bound $1-\frac{3\alpha}{2}$ improving (for $\alpha \leq 1/3$) on the previously established ([9]; [8]) lower bound $\frac{1-\alpha}{1+\alpha}$. Several illustrative examples are given.


On a $p$-adic extension of the Jacquet-Langlands correspondence to weight 1

Abstract: We consider a novel version of the classical Jacquet-Langlands {correspondence}, explore a $p$-adic extension of the Jacquet-Langlands correspondence, and as an explicit example we find an overconvergent automorphic form of weight~1 which corresponds to a classical modular form of weight~1, using both experimental and theoretical methods.


Terwilliger Algebras of Wreath Powers of One-Class Association Schemes

Abstract: In this paper, we study the subconstituent algebras, also called as Terwilliger algebras, of association schemes that are obtained as the wreath product of one-class association schemes $K_n=H(1, n)$ for $n\ge 2$. We find that the $d$-class association scheme $K_{n_{1}}\wr K_{n_{2}} \wr ... \wr K_{n_{d}}$ formed by taking the wreath product of $K_{n_{i}}$ has the triple-regularity property. We determine the dimension of the Terwilliger algebra for the association scheme $K_{n_{1}}\wr K_{n_{2}}\wr ... \wr K_ {n_{d}}$. We give a description of the structure of the Terwilliger algebra for the wreath power $(K_n)^{\wr d}$ for $n \geq 2$ by studying its irreducible modules. In particular, we show that the Terwilliger algebra of $(K_n)^{\wr d}$ is isomorphic to $M_{d+1}(\mathbb{C})\oplus M_1(\mathbb{C})^{\oplus \frac12d(d+1)}$ for $n\ge3$, and $M_{d+1}(\mathbb{C})\oplus M_1(\mathbb{C})^{\oplus \frac12d(d-1)}$ for $n=2$.


On the Schrodinger equation outside strictly convex obstacles

Abstract: We prove sharp Strichartz estimates for the semi-classical Schrodinger equation on a compact manifold with smooth, strictly geodesically concave boundary. We deduce sharp (classical) Strichartz estimates for the Schrodinger equation outside a strictly convex obstacle, local existence for the H^1-critical (quintic) Schrodinger equation and scattering for the sub-critical Schrodinger equation in 3D.


The Julia sets of basic uniCremer polynomials of arbitrary degree

Abstract: Let $P$ be a polynomial of degree $d$ with a Cremer point $p$ and no repelling or parabolic periodic bi-accessible points. We show that there are two types of such Julia sets $J_P$. The \emph{red dwarf} $J_P$ are nowhere connected im kleinen and such that the intersection of all impressions of external angles is a continuum containing $p$ and the orbits of all critical images. The \emph{solar} $J_P$ are such that every angle with dense orbit has a degenerate impression disjoint from other impressions and $J_P$ is connected im kleinen at its landing point. We study bi-accessible points and locally connected models of $J_P$ and show that such sets $J_P$ appear through polynomial-like maps for generic polynomials with Cremer points.


Localized factorizations of integers

Abstract: We determine the order of magnitude of H^{(k+1)}(x,\vec{y},2\vec{y}), the number of integers up to x that are divisible by a product d_1...d_k with y_i<d_i\le 2y_i, when the numbers \log y_1,...,\log y_k have the same order of magnitude and k\ge 2. This generalizes a result by K. Ford when k=1. As a corollary of these bounds, we determine the number of elements up to multiplicative constants that appear in a (k+1)-dimensional multiplication table as well as how many distinct sums of k+1 Farey fractions there are modulo 1.


Multifractal analysis for multimodal maps

Abstract: Given a multimodal interval map $f:I \to I$ and a H\"older potential $\phi:I \to \mathbb{R}$, we study the dimension spectrum for equilibrium states of $\phi$. The main tool here is inducing schemes, used to overcome the presence of critical points. The key issue is to show that enough points are `seen' by a class of inducing schemes. We also compute the Lyapunov spectrum. We obtain the strongest results when $f$ is a Collet-Eckmann map, but our analysis also holds for maps satisfying much weaker growth conditions.


Lifting of Characters for Nonlinear Simply Laced Groups

Abstract: One aspect of the Langlands program for linear groups is lifting of characters, which relates virtual representations on a group $G$ with those on an endoscopic group for $G$. The goal of this paper is to extend this theory to nonlinear two-fold covers of real groups in the simply laced case. Suppose $\tG$ is a two-fold cover of a real reductive group $G$. The main result is that there is an operation, denoted $\Lift_G^{\tG}$, taking a stable virtual character of $G$ to 0 or a virtual genuine character of $\tG$, and $\Lift_G^{\tG}(\Theta_\pi)$ may be explicitly computed if $\pi$ is a stable sum of standard modules.


Discrete Fourier analysis on fundamental domain of $A_d$ lattice and on simplex in $d$-variables

Abstract: A discrete Fourier analysis on the fundamental domain $\Omega_d$ of the $d$-dimensional lattice of type $A_d$ is studied, where $\Omega_2$ is the regular hexagon and $\Omega_3$ is the rhombic dodecahedron, and analogous results on $d$-dimensional simplex are derived by considering invariant and anti-invariant elements. Our main results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the simplex is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^d$. The basic trigonometric functions on the simplex can be identified with Chebyshev polynomials in several variables already appeared in literature. We study common zeros of these polynomials and show that they are nodes for a family of Gaussian cubature formulas, which provides only the second known example of such formulas.


Concordance invariants from higher order covers

Abstract: We generalize the Manolescu-Owens smooth concordance invariant delta(K) of knots K in the 3-sphere to invariants delta_{p^n}(K) obtained by considering covers of order p^n, with p prime. Our main result shows that for any odd prime p, the direct sum of delta_{p^n} as n ranges through the natural numbers, yields a homomorphism of infinite rank from the smooth concordance group to Z^\infty. We also show that unlike delta, these new invariants typically are not multiples of the knot signature, even for alternating knots. A significant portion of the article is devoted to exploring examples.


Well-posedness for the 1D Zakharov-Rubenchik system

Abstract: Local and global well-posedness results are established for the initial value problem associated to the 1D Zakharov-Rubenchik system. We show that our results are sharp in some situations by proving Ill-posedness results otherwise. The global results allow us to study the norm growth of solutions corresponding to the Schrodinger equation term. We use ideas recently introduced to study the classical Zakharov systems.


The topology of Birkhoff varieties

Abstract: Our main theorem is that the inclusion of a Birkhoff variety in the affine Grassmannian is a homotopy equivalence. We also construct analogues of tubular neighborhoods for Birkhoff and Schubert varieties. We include some observations on torus-equivariant cohomology.


Tropical descendant Gromov-Witten invariants

Abstract: We define tropical Psi-classes on the moduli space of rational tropical curves in R^2 and consider intersection products of Psi-classes and pull-backs of evaluations on this space. We show a certain WDVV equation which is sufficient to prove that tropical numbers of curves satisfying certain Psi- and evaluation conditions are equal to the corresponding classical numbers. We present an algorithm that generalizes Mikhalkin's lattice path algorithm and counts rational plane tropical curves satisfying certain Psi- and evaluation conditions.


Measurability of optimal transportation and strong coupling of martingale measures

Abstract: We consider the optimal mass transportation problem in $\RR^d$ with measurably parameterized marginals, for general cost functions and under conditions ensuring the existence of a unique optimal transport map. We prove a joint measurability result for this map, with respect to the space variable and to the parameter. The proof needs to establish the measurability of some set-valued mappings, related to the support of the optimal transference plans, which we use to perform a suitable discrete approximation procedure. A motivation is the construction of a strong coupling between orthogonal martingale measures. By this we mean that, given a martingale measure, we construct in the same probability space a second one with specified covariance measure. This is done by pushing forward one martingale measure through a predictable version of the optimal transport map between the covariance measures. This coupling allows us to obtain quantitative estimates in terms of the Wasserstein distance between those covariance measures.


An example concerning the Theory of Levels for codimension-one foliations

Abstract: We give an example of a codimension-one foliation which is transversely of class C^1 and which does not satisfy the "Local Minimal Set" property.


beta-family congruences and the f-invariant

Abstract: In previous work, the authors have each introduced methods for studying the 2-line of the p-local Adams-Novikov spectral sequence in terms of the arithmetic of modular forms. We give the precise relationship between the congruences of modular forms introduced by the first author with the Q-spectrum and the f-invariant of the second author. This relationship enables us to refine the target group of the f-invariant in a way which makes it more manageable for computations.


On the generalised Ritt problem as a computational problem

Abstract: The Ritt problem asks if there is an algorithm that tells whether one prime differential ideal is contained in another one if both are given by their characteristic sets. We give several equivalent formulations of this problem. In particular, we show that it is equivalent to testing if a differential polynomial is a zero divisor modulo a radical differential ideal. The technique used in the proof of equivalence yields algorithms for computing a canonical decomposition of a radical differential ideal into prime components and a canonical generating set of a radical differential ideal. Both proposed representations of a radical differential ideal are independent of the given set of generators and can be made independent of the ranking.


Continuous extension of arithmetic volumes

Abstract: This paper is the sequel of the paper "Continuity of volumes on arithmetic varieties", in which we established the arithmetic volume function of smooth hermitian Q-invertible sheaves and proved its continuity. The continuity of the volume function has a lot of applications as treated in the paper as above. In this paper, we would like to consider its continuous extension over R.


On adaptive stratification

Abstract: This paper investigates the use of stratified sampling as a variance reduction technique for approximating integrals over large dimensional spaces. The accuracy of this method critically depends on the choice of the space partition, the strata, which should be ideally fitted to thesubsets where the functions to integrate is nearly constant, and on the allocation of the number of samples within each strata. When the dimension is large and the function to integrate is complex, finding such partitions and allocating the sample is a highly non-trivial problem. In this work, we investigate a novel method to improve the efficiency of the estimator "on the fly", by jointly sampling and adapting the strata and the allocation within the strata. The accuracy of estimators when this method is used is examined in detail, in the so-called asymptotic regime (i.e. when both the number of samples and the number of strata are large). We illustrate the use of the method for the computation of the price of path-dependent options in models with both constant and stochastic volatility. The use of this adaptive technique yields variance reduction by factors sometimes larger than 1000 compared to classical Monte Carlo estimators.


On the equivariant cohomology of subvarieties of a B-regular variety

Abstract: By a $B$-regular variety, we mean a smooth projective variety over $C$ admitting an algebraic action of the upper triangular Borel subgroup $B \subset SL_2(C)$ such that the unipotent radical in $B$ has a unique fixed point. A result of M. Brion and the first author describes the equivariant cohomology algebra (over $C$) of a $B$-regular variety $X$ as the coordinate ring of a remarkable affine curve in $X \times P^1$. The main result of this paper uses this fact to classify the $B$-invariant subvarieties $Y$ of a $B$-regular variety $X$ for which the restriction map $i_Y:H^*(X) \to H^*(Y)$ is surjective.


Vertex Degree of Random Intersection Graph

Abstract: A random intersection graph is constructed by independently assigning a subset of a given set of objects $W,$ to each vertex of the vertex set $V$ of a simple graph $G.$ There is an edge between two vertices of $V,$ iff their respective subsets(in $W$,) have at least one common element. The strong threshold for the connectivity between any two arbitrary vertices of vertex set $V,$ is derived. Also we determine the almost sure probability bounds for the vertex degree of a typical vertex of graph $G.$


A Strong threshold for the size of random caps to cover a sphere

Abstract: In this article, we consider `$N$'spherical caps of area $4\pi p$ were uniformly distributed over the surface of a unit sphere. We are giving the strong threshold function for the size of random caps to cover the surface of a unit sphere. We have shown that for large $N,$ if $\frac{Np}{\log\:N} > 1/2$ the surface of sphere is completely covered by the $N$ caps almost surely, and if $\frac{Np}{\log\:N} \leq 1/2$ a partition of the surface of sphere is remains uncovered by the $N$ caps almost surely.


Number of Edges in Random Intersection Graph on Surface of a Sphere

Abstract: In this article, we consider `$N$'spherical caps of area $4\pi p$ were uniformly distributed over the surface of a unit sphere. We study the random intersection graph $G_N$ constructed by these caps. We prove that for $p = \frac{c}{N^{\al}},\:c >0$ and $\al >2,$ the number of edges in graph $G_N$ follow the Poisson distribution. Also we derive the strong law results for the number of isolated vertices in $G_N$: for $p = \frac{c}{N^{\al}},\:c >0$ for $\al < 1,$ there is no isolated vertex in $G_N$ almost surely i.e., there are atleast $N/2$ edges in $G_N$ and for $\al >3,$ every vertex in $G_N$ is isolated i.e., there is no edge in edge set $\cE_N.$


Bialgebra structures of 2-associative algebras

Abstract: This work is devoted to study new bialgebra structures related to 2-associative algebras. A 2-associative algebra is a vector space equipped with two associative multiplications. We discuss the notions of 2-associative bialgebras, 2-bialgebras and 2-2-bialgebras. The first structure was revealed by J.-L. Loday and M. Ronco in an analogue of a Cartier-Milnor-Moore theorem, the second was suggested by Loday and the third is a variation of the second one. The main results of this paper are the construction of 2-associative bialgebras, 2-bialgebras and 2-2-bialgebras starting from an associative algebra and the classification of these structures in low dimensions.


$L^p-L^q$ estimates on the solutions to $u_{tt}-u_{x_1x_1}=\triangle u_t$

Abstract: This paper focuses the study on the $L^p-L^q$ estimates on the solutions to an asymmetric wave equation with dissipation which arises in the study for the magneto-hydrodynamics by using the method of Green function.


On degeneration of surface in Fitting compactification of moduli of stable vector bundles

Abstract: The new compactification of moduli scheme of Gieseker-stable vector bundles with the given Hilbert polynomial on a smooth projective polarized surface (S;H), over the field k = \bar k of zero characteristic, is constructed in previous papers of the author. Families of locally free sheaves on the surface S are completed by the locally free sheaves on the schemes which are certain modifications of S. We obtain the class of modified surfaces to appear in the construction. Keywords: moduli space, semistable coherent sheaves, blowup algebra, algebraic surface.


Combinatorial differential geometry and ideal Bianchi-Ricci identities

Abstract: We apply the graph complex method to vector fields depending naturally on a set of vector fields and a linear symmetric connection. We characterize all possible systems of generators for such vector-field valued operators including the classical ones given by normal tensors and covariant derivatives. We also describe the size of the space of such operators and prove the existence of an `ideal' basis consisting of operators with given leading terms which satisfy the (generalized) Bianchi-Ricci identities without the correction terms.


Betti numbers of transversal monomial ideals

Abstract: In this paper, by a modification of a previously constructed minimal free resolution for a transversal monomial ideal, the Betti numbers of this ideal is explicitly computed. For convenient characteristics of the ground field, up to a change of coordinates, the ideal of $t$-minors of a generic pluri-circulant matrix is a transversal monomial ideal . Using a Gr\"obner basis for this ideal, it is shown that the initial ideal of a generic pluri-circulant matrix is a stable monomial ideal when the matrix has two square blocks. By means of the Eliahou-Kervair resolution, the Betti numbers of this initial ideal is computed and it is proved that, for some significant values of $t$, this ideal has the same Betti numbers as the corresponding transversal monomial ideal. The ideals treated in this paper, naturally arise in the study of generic singularities of algebraic varieties.


Global Well-posedness of the 1D Dirac-Klein-Gordon system in Sobolev spaces of negative index

Abstract: We prove that the Cauchy problem for the Dirac-Klein-Gordon system of equations in 1D is globally well-posed in a range of Sobolev spaces of negative index for the Dirac spinor and positive index for the scalar field. The main ingredient in the proof is the theory of almost conservation law and I-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao. Our proof also relies on the null structure in the system, and bilinear spacetime estimates of Klainerman-Machedon type.


On the essential spectrum of Nadirashvili-Martin-Morales minimal surfaces

Abstract: We show that the spectrum of a complete submanifold properly immersed into a ball of a Riemannian manifold is discrete, provided the norm of the mean curvature vector is sufficiently small. In particular, the spectrum of a complete minimal surface properly immersed into a ball of $\mathbb{R}^{3}$ is discrete. This gives a positive answer to a question of Yau.


On the connected components of the conjugacy class of projectors on $ \ell_p\oplus\ell_q $

Abstract: We characterize the projectors $ P $ on a Banach space $ E $ having the property of being connected to all the others projectors obtained as a conjugation of $ P $. Using this characterization we show an example of Banach space where the conjugacy class of a projector splits into several path-connected components, and describe the conjugacy classes of projectors onto subspaces of $ \ell_p\oplus\ell_q $ with $ p\neq q $.


The group of symmetries of the Tower of Hanoi graph

Abstract: I prove that the group of symmetries of the Tower of Hanoi graph with k pegs and n disks, denoted H_n^k, is isomorphic to the group of permutations of k elements, S_k, for all k greater than or equal to 3 and positive n.


Left invariant complex structures on U(2) and SU(2)xSU(2) revisited

Abstract: We compute the torsion-free linear maps from the Lie algebra su(2) into itself, deduce a new determination of the integrable complex structures and their equivalence classes under the action of the automorphism group for u(2) and su(2)xsu(2), and prove that in both cases the set of complex structures is a differentiable manifold. u(2)x u(2), su(2)^N and u(2)^N are also considered. Extensions of complex structures from u(2) to su(2)xsu(2) are studied, local holomorphic charts given, and attention is paid to what representations of u(2) we can get from a substitute to the regular representation on a space of holomorphic functions for the complex structure.


Survey on Affine Spheres

Abstract: We give a survey of the theory of affine spheres, emphasizing the convex cases and relationsships to Monge-Ampere equations and geometric structures on manifolds.


Representation theory of mv-algebras

Abstract: In this paper we develop a general representation theory for mv-algebras. We furnish the appropriate categorical background to study this problem. Our guide line is the theory of classifying topoi of coherent extensions of universal algebra theories. Our main result corresponds, in the case of mv-algebras and mv-chains, to the representation of commutative rings with unit as rings of global sections of sheaves of local rings. \emph{We prove that any mv-algebra is isomorphic to the mv-algebra of all global sections of a sheaf of mv-chains on a compact topological space}. This result is intimately related to McNaughton's theorem, and we explain why our representation theorem can be viewed as a vast generalization of McNaughton's. On spite of the language utilized in this abstract, we wrote this paper in a way that, we hope, could be read without much acquaintance with either sheaf theory or mv-algebra theory.


Monotone images of Cremer Julia sets

Abstract: We show that if $P$ is a quadratic polynomial with a fixed Cremer point and Julia set $J$, then for any monotone map $\ph:J\to A$ from $J$ onto a locally connected continuum $A$, $A$ is a single point.


K-theoretic exceptional collections at roots of unity

Abstract: Using cyclotomic specializations of the equivariant $K$-theory with respect to a torus action we derive congruences for discrete invariants of exceptional objects in derived categories of coherent sheaves on a class of varieties that includes Grassmannians and smooth quadrics. For example, we prove that if $X={\Bbb P}^{n_1-1}\times...\times{\Bbb P}^{n_k-1}$, where $n_i$'s are powers of a fixed prime number $p$, then the rank of an exceptional object on $X$ is congruent to $\pm 1$ modulo $p$.


Compactification for essentially finite-type maps

Abstract: We show that any separated essentially finite-type map $f$ of noetherian schemes globally factors as $f = hi$ where $i$ is an injective localization map and $h$ a separated finite-type map. In particular, via Nagata's compactification theorem, $h$ can be chosen to be proper. We apply these results to Grothendieck duality. We also obtain other factorization results and provide essentialized versions of many general results such as Zariski's Main Theorem, Chow's Lemma, and blow-up descriptions of birational maps.


Proving a manifold to be hyperbolic once it has been approximated to be so

Abstract: The computer program SnapPea can approximate whether or not a three manifold whose boundary consists of tori has a complete hyperbolic structure, but it can not prove conclusively that this is so. This article provides a method for proving that such a manifold has a complete hyperbolic structure based on the approximations of SNAP, a program that includes the functionality of SnapPea plus other features. The approximation is done by triangulating the manifold, identifying consistency and completeness equations with respect to this triangulation, and then trying to solve the system of equations using Newton's Method. This produces an approximate, not actual solution. The method developed here uses Kantorovich's theorem to prove that an actual solution exists, thereby assuring that the manifold has a complete hyperbolic structure. Using this, we can definitively prove that every manifold in the SnapPea cusped census has a complete hyperbolic structure.


Linear algebra meets Lie algebra: the Kostant-Wallach theory

Abstract: In two languages, Linear Algebra and Lie Algebra, we describe the results of Kostant and Wallach on the fibre of matrices with prescribed eigenvalues of all leading principal submatrices. In addition, we present a brief introduction to basic notions in Algebraic Geometry, Integrable Systems, and Lie Algebra aimed at specialists in Linear Algebra.


Schatten-von Neumann properties in the Weyl calculus

Abstract: Let $\Op_t(a)$, for $t\in \mathbf R$, be the pseudo-differential operator $$ f(x) \mapsto (2\pi)^{-n}\iint a((1-t)x+ty,\xi)f(y)e^{i\scal {x-y}\xi} dyd\xi $$ and let $\mathscr I_p$ be the set of Schatten-von Neumann operators of order $p\in [1,\infty ]$ on $L^2$. We are especially concerned with the Weyl case (i.{}e. when $t=1/2$). We prove that if $m$ and $g$ are appropriate metrics and weight functions respectively, $h_g$ is the Planck's function, $h_g^{k/2}m\in L^p$ for some $k\ge 0$ and $a\in S(m,g)$, then $\Op_t(a)\in \mathscr I_p$, iff $a\in L^p$. Consequently, if $0\le \delta <\rho \le 1$ and $a\in S^r_{\rho ,\delta}$, then $\Op_t(a)$ is bounded on $L^2$, iff $a\in L^\infty$.


Large deviations for random walk in a random environment

Abstract: In this work, we study the large deviation properties of random walk in a random environment on $\mathbb{Z}^d$ with $d\geq1$. We start with the quenched case, take the point of view of the particle, and prove the large deviation principle (LDP) for the pair empirical measure of the environment Markov chain. By an appropriate contraction, we deduce the quenched LDP for the mean velocity of the particle and obtain a variational formula for the corresponding rate function $I_q$. We propose an Ansatz for the minimizer of this formula. This Ansatz is easily verified when $d=1$. In his 2003 paper, Varadhan proves the averaged LDP for the mean velocity and gives a variational formula for the corresponding rate function $I_a$. Under the non-nestling assumption (resp. Kalikow's condition), we show that $I_a$ is strictly convex and analytic on a non-empty open set $\mathcal{A}$, and that the true velocity $\xi_o$ is an element (resp. in the closure) of $\mathcal{A}$. We then identify the minimizer of Varadhan's variational formula at any $\xi\in\mathcal{A}$. For walks in high dimension, we believe that $I_a$ and $I_q$ agree on a set with non-empty interior. We prove this for space-time walks when the dimension is at least 3+1. In the latter case, we show that the cheapest way to condition the asymptotic mean velocity of the particle to be equal to any $\xi$ close to $\xi_o$ is to tilt the transition kernel of the environment Markov chain via a Doob $h$-transform.


On the notion of Cohen-Macaulayness for non Noetherian rings

Abstract: There exist many characterizations of Noetherian Cohen-Macaulay rings in the literature. These characterizations do not remain equivalent if we drop the Noetherian assumption. The aim of this paper is to provide some comparisons between some of these characterizations in non Noetherian case. Toward solving a conjecture posed by Glaz, we give a generalization of the Hochster-Eagon result on Cohen-Macaulayness of invariant rings, in the context of non Noetherian rings.


Multicomplexes and spectral sequences

Abstract: In this note we present some algebraic examples of multicomplexes whose differentials differ from those in the spectral sequences associated to the multicomplexes. The motivation for constructing examples showing the algebraic distinction between a multicomplex and its associated spectral sequence comes from the author's work on Morse-Bott homology with A. Banyaga.


The ring of projective invariants of eight points on the line via representation theory

Abstract: The ring of projective invariants of eight ordered points on the line is a quotient of the polynomial ring on V, where V is a fourteen-dimensional representation of S_8, by an ideal I_8, so the modular fivefold (P^1)^8 // GL(2) is Proj(Sym* (V)/I_8). We show that there is a unique cubic hypersurface S in PV whose equation s is skew-invariant, and that the singular locus of S is the modular fivefold. In particular, over Z[1/3], the modular fivefold is cut out by the 14 partial derivatives of s. Better: these equations generate I_8. In characteristic 3, the cubic s is needed to generate the ideal. The existence of such a cubic was predicted by Dolgachev. Over Q, we recover the 14 quadrics found by computer calculation by Koike, and our approach yields a conceptual representation-theoretic description of the presentation. Additionally we find the graded Betti numbers of a minimal free resolution in any characteristic. The proof over Q is by pure thought, using Lie theory and commutative algebra. Over Z, the assistance of a computer was necessary. This result will be used as the base case describing the equations of the moduli space of an arbitrary number of points on P^1, with arbitrary weighting, in a later paper, completing the program of our previous paper (Duke Math. J.). The modular fivefold, and corresponding ring, are known to have a number of special incarnations, due to Deligne-Mostow, Kondo, and Freitag-Salvati Manni, for example as ball quotients or ring of modular forms respectively.


Smooth dependence on parameters of solution of cohomology equations over Anosov systems and applications to cohomology equations on diffeomorphism groups

Abstract: We consider the dependence on parameters of the solutions of cohomology equations over Anosov diffeomorphisms. We show that the solutions depend on parameters as smoothly as the data. As a consequence we prove optimal regularity results for the solutions of equations taking value in diffeomorphism groups. These results are motivated by applications to rigidity theory, dynamical systems, and geometry. In particular, in the context of diffeomorphism groups we show: Let $f$ be a transitive Anosov diffeomorphism of a compact manifold $M$. Suppose that $\eta \in C^{\reg}(M,\Diff^r(N))$ for a compact manifold $N$, $k,r \in \N$, $r \geq 1$, and $0 < \alpha \leq \Lip$. We show that if there exists a $\varphi\in C^{\reg}(M,\Diff^1(N))$ solving \begin{equation*} \varphi_{f(x)} = \eta_x \circ \varphi_x \end{equation*} then in fact $\varphi \in C^{\reg}(M,\Diff^r(N))$.


The de Rham comparison theorem for Deligne-Mumford stacks

Abstract: The de Rham comparison theorem for varieties, first proved by Faltings, gives the de Rham cohomology of a variety in terms of its p-adic etale cohomology. We extend this theorem to proper, smooth Deligne-Mumford stacks. Two approaches are given, which both in the end reduce the problem to the already known comparison theorem for varieties. The first approach employs the formalism of Weil cohomologies. Unfortunately, this does not result in a complete proof of the comparison theorem, as the author was unable to prove the required compatibility between intersections and cup products. Nevertheless, the author thought the results that are obtained and the method that is suggested interesting enough to include them. The second approach uses simplicial methods and is based on versions of the comparison theorem by Kisin and Tsuji. The latter approach does result in a complete proof of the extended comparison theorem.


About the choice of a basis in Kedlaya's algorithm

Abstract: Kedlaya's algorithm (Kedlaya, J. Ramanujan Math. Soc 16, 2001) can be used to count the points of arbitrary hyperelliptic curves over finite fields of characteristic p, where p is an odd prime. The algorithm uses the cohomology of a p-adic lift of the curve. The Frobenius morphism of the curve induces an automorphism of this cohomological space. The key step of the algorithm is to determine this automorphism with a sufficiently high p-adic precision: it is given in the form of a matrix with respect to a certain basis. Edixhoven has found a basis that has the property that the coefficients of the matrix are p-adically integral. This allows a smaller required precision, because a (semi-linear) power of this matrix must be computed up to some given precision. This text describes Edixhoven's basis and provides a proof of the fact that the basis is suitable.


Obstructing Sliceness in a Family of Montesinos Knots

Abstract: Using Gauge theoretical techniques employed by Lisca for 2-bridge knots and by Greene-Jabuka for 3-stranded pretzel knots, we show that no member of the family of Montesinos knots M(0;[m_1+1,n_1+2],[m_2+1,n_2+2],q), with certain restrictions on m_i, n_i, and q, can be (smoothly) slice. Our techniques use Donaldson's diagonalization theorem and the fact that the 2-fold covers of Montisinos knots bound plumbing 4-manifolds, many of which are negative definite. Some of our examples include knots with signature 0 and square determinant.


Weak Approximation for General Degree Two del Pezzo Surfaces

Abstract: We address weak approximation for certain del Pezzo surfaces defined over the function field of a curve. We study the rational connectivity of the smooth locus of degree two del Pezzo surfaces with two A1 singularities in order to prove weak approximation for degree two del Pezzo surfaces with square-free discriminant.


Necessary and Sufficient Conditions for Success of the Nuclear Norm Heuristic for Rank Minimization

Abstract: Minimizing the rank of a matrix subject to constraints is a challenging problem that arises in many applications in control theory, machine learning, and discrete geometry. This class of optimization problems, known as rank minimization, is NP-HARD, and for most practical problems there are no efficient algorithms that yield exact solutions. A popular heuristic algorithm replaces the rank function with the nuclear norm--equal to the sum of the singular values--of the decision variable. In this paper, we provide a necessary and sufficient condition that quantifies when this heuristic successfully finds the minimum rank solution of a linear constraint set. We additionally provide a probability distribution over instances of the affine rank minimization problem such that instances sampled from this distribution satisfy our conditions for success with overwhelming probability provided the number of constraints is appropriately large. Finally, we give empirical evidence that these probabilistic bounds provide accurate predictions of the heuristic's performance in non-asymptotic scenarios.


Constructing geometrically infinite groups on boundaries of deformation spaces

Abstract: Consider a geometrically finite Kleinian group $G$ without parabolic or elliptic elements, with its Kleinian manifold $M=(\H^3\cup \Omega_G)/G$. Suppose that for each boundary component of $M$, either a maximal and connected measured lamination in the Masur domain or a marked conformal structure is given. In this setting, we shall prove that there is an algebraic limit $\Gamma$ of quasi-conformal deformations of $G$ such that there is a homeomorphism $h$ from $\mathrm{Int} M$ to $\H^3/\Gamma$ compatible with the natural isomorphism from $G$ to $\Gamma$, the given laminations are unrealisable in $\H^3/\Gamma$, and the given conformal structures are pushed forward by $h$ to those of $\H^3/\Gamma$. Based on this theorem and its proof, in the subsequent paper, the Bers-Thurston conjecture, saying that every finitely generated Kleinian group is an algebraic limit of quasi-conformal deformations of minimally parabolic geometrically finite group, is proved using recent solutions of Marden's conjecture by Agol, Calegari-Gabai, and the ending lamination conjecture by Minsky collaborating with Brock, Canary and Masur.


Combinatorics and topology of straightening maps I: compactness and bijectivity

Abstract: We study the parameter space structure of degree $d \ge 3$ one complex variable polynomials as dynamical systems acting on $\C$. We introduce and study {\it straightening maps}. These maps are a natural higher degree generalization of the ones introduced by Douady and Hubbard to prove the existence of small copies of the Mandelbrot set inside itself. We establish that straightening maps are always injective and that their image contains all the corresponding hyperbolic systems. Also, we characterize straightening maps with compact domain. Moreover, we give two classes of bijective straightening maps. The first produces an infinite collection of embedded copies of the $(d-1)$-fold product of the Mandelbrot set in the connectedness locus of degree $d \ge 3$. The second produces an infinite collection of full families of quadratic connected filled Julia sets in the cubic connectedness locus, such that each filled Julia set is quasiconformally embedded.


Appell Polynomials and Their Zero Attractors

Abstract: A polynomial family $\{p_n(x)\}$ is Appell if it is given by $\frac{e^{xt}}{g(t)} = \sum_{n=0}^\infty p_n(x)t^n$ or, equivalently, $p_n'(x) = p_{n-1}(x)$. If $g(t)$ is an entire function, $g(0)\neq 0$, with at least one zero, the asymptotics of linearly scaled polynomials $\{p_n(nx)\}$ are described by means of finitely zeros of $g$, including those of minimal modulus. As a consequence, we determine the limiting behavior of their zeros as well as their density. The techniques and results extend our earlier work on Euler polynomials.


The Atiyah--Segal completion theorem in twisted K-theory

Abstract: In this note we prove the analogue of the Atiyah-Segal completion theorem for equivariant twisted K-theory in the setting of an arbitrary compact Lie group G and an arbitrary twisting of the usually considered type. The theorem generalizes a result by C. Dwyer, who has proven the theorem for finite G and twistings of a more restricted type. While versions of the general result have been known to experts, to our knowledge no proof appears in the current literature. Our goal is to fill in this gap. The proof we give proceeds in two stages. We first prove the theorem in the case of a twisting arising from a graded central extension of G, following the Adams-Haeberly-Jackowski-May proof of the classical Atiyah-Segal completion theorem. After establishing that the theorem holds for this special class of twistings, we then deduce the general theorem by a Mayer-Vietoris argument.


MultiResolution Anomaly Detection Method for Long Range Dependent Time Series

Abstract: Driven by network intrusion detection, we propose a MultiResolution Anomaly Detection (MRAD) method, which effectively utilizes the multiscale properties of Internet features and network anomalies. In this paper, several theoretical properties of the MRAD method are explored. A major new result is the mathematical formulation of the notion that a two-scaled MRAD method has larger power than the average power of the detection method based on the given two scales. Test threshold is also developed. Comparisons between MRAD method and other classical outlier detectors in time series are reported as well.


Strong uniqueness for a class of singular SDEs for catalytic branching diffusions

Abstract: A new result for the strong uniqueness for catalytic branching diffusions is established, which improves the work of Dawson, D.A.; Fleischmann, K.; Xiong, J.[Strong uniqueness for cyclically symbiotic branching diffusions. Statist. Probab. Lett. 73, no. 3, 251--257 (2005)].


Submanifolds with Biharmonic Gauss Map

Abstract: We generalize the Ruh-Vilms problem by characterizing the submanifolds in Euclidean spaces with proper biharmonic Gauss map and we construct examples of such hypersurfaces.


Semistar-Krull and Valuative Dimension of Integral Domains

Abstract: Given a stable semistar operation of finite type $\star$ on an integral domain $D$, we show that it is possible to define in a canonical way a stable semistar operation of finite type $\star[X]$ on the polynomial ring $D[X]$, such that, if $n:=\star$-$\dim(D)$, then $n+1\leq \star[X]\text{-}\dim(D[X])\leq 2n+1$. We also establish that if $D$ is a $\star$-Noetherian domain or is a Pr\"{u}fer $\star$-multiplication domain, then $\star[X]\text{-}\dim(D[X])=\star\text{-}\dim(D)+1$. Moreover we define the semistar valuative dimension of the domain $D$, denoted by $\star$-$\dim_v(D)$, to be the maximal rank of the $\star$-valuation overrings of $D$. We show that $\star$-$\dim_v(D)=n$ if and only if $\star[X_1,...,X_n]$-$\dim_v(D[X_1,...,X_n])=2n$, and that if $\star$-$\dim_v(D)<\infty$ then $\star[X]$-$\dim_v(D[X])=\star$-$\dim_v(D)+1$. In general $\star$-$\dim(D)\leq\star$-$\dim_v(D)$ and equality holds if $D$ is a $\star$-Noetherian domain or is a Pr\"{u}fer $\star$-multiplication domain. We define the $\star$-Jaffard domains as domains $D$ such that $\star$-$\dim(D)<\infty$ and $\star$-$\dim(D)=\star$-$\dim_v(D)$. As an application, $\star$-quasi-Pr\"{u}fer domains are characterized as domains $D$ such that each $(\star,\star')$-linked overring $T$ of $D$, is a $\star'$-Jaffard domain, where $\star'$ is a stable semistar operation of finite type on $T$. As a consequence of this result we obtain that a Krull domain $D$, must be a $w_D$-Jaffard domain.


Minimal Prime Ideals and Semistar Operations

Abstract: Let $R$ be a commutative integral domain and let $\star$ be a semistar operation of finite type on $R$, and $I$ be a quasi-$\star$-ideal of $R$. We show that, if every minimal prime ideal of $I$ is the radical of a $\star$-finite ideal, then the set $\Min(I)$ of minimal prime ideals over $I$ is finite.


Graph-theoretic criteria for injectivity and unique equilibria in general chemical reaction systems

Abstract: In this paper we discuss the question of how to decide when a general chemical reaction system is incapable of admitting multiple equilibria, regardless of parameter values such as reaction rate constants, and regardless of the type of chemical kinetics, such as mass-action kinetics, Michaelis-Menten kinetics, etc. Our results relate previously described linear algebraic and graph-theoretic conditions for injectivity of chemical reaction systems. After developing a translation between the two formalisms, we show that a graph-theoretic test developed earlier in the context of systems with mass action kinetics, can be applied to reaction systems with arbitrary kinetics. The test, which is easy to implement algorithmically, and can often be decided without the need for any computation, rules out the possibility of multiple equilibria for the systems in question.


Well-posedness of the transport equation by stochastic perturbation

Abstract: We consider the linear transport equation with a globally Holder continuous and bounded vector field. While this deterministic PDE may not be well-posed, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of partial differential equation that become well-posed under the influece of noise. The key tool is a differentiable stochastic flow constructed and analysed by means of a special transformation of the drift of Ito-Tanaka type.


A class of iterative methods for solving nonlinear operator equations

Abstract: The article deals with gradient-like iterative methods for solving nonlinear operator equations on Hilbert and Banach spaces. The authors formulate a general principle of studying such methods. This principle allows to formulate simple conditions of convergence of the method under consideration, to estimate the rate of this convergence and to give effective a priori and aposteriori error estimates in terms of a scalar function that is constructed on the base of estimates for properties of invertibility and smoothness of linearizations of the left-hand side of the equations under study. The principle is applicable for analysis of such classical methods as method of minimal residuals, method of steepest descent, method of minimal errors and others. The main results are obtained for operator equations on Hilbert spaces and Banach spaces with a special property, that is called Bynum property.


Totally geodesic submanifolds of the exceptional Riemannian symmetric spaces of rank 2

Abstract: The present article is the final part of a series on the classification of the totally geodesic submanifolds of the irreducible Riemannian symmetric spaces of rank 2. After this problem has been solved for the 2-Grassmannians in my previous papers cited in the present paper as [K1] and [K2], and for the space SU(3)/SO(3) in [K3], Section 6, we now solve the classification for the remaining irreducible Riemannian symmetric spaces of rank 2 and compact type: SU(6)/Sp(3), SO(10)/U(5), E6/(U(1)*Spin(10)), E6/F4, G2/SO(4), SU(3), Sp(2) and G2. Similarly as for the spaces already investigated in the earlier papers, it turns out that for many of the spaces investigated here, the earlier classification of the maximal totally geodesic submanifolds of Riemannian symmetric spaces by Chen and Nagano ([CN], Paragraph 9) is incomplete. In particular, in the spaces Sp(2), G2/SO(4) and G2, there exist maximal totally geodesic submanifolds, isometric to 2- or 3-dimensional spheres, which have a "skew" position in the ambient space in the sense that their geodesic diameter is strictly larger than the geodesic diameter of the ambient space. They are all missing from [CN].


A complete lift for semisprays

Abstract: In this paper, we define a complete lift for semisprays. If $S$ is a semispray on a manifold $M$, its complete lift is a new semispray $S^c$ on $TM$. The motivation for this lift is two-fold: First, geodesics for $S^c$ correspond to the Jacobi fields for $S$, and second, this complete lift generalizes and unifies previously known complete lifts for Riemannian metrics, affine connections, and regular Lagrangians. When $S$ is a spray, we prove that the projective geometry of $S^c$ uniquely determines $S$. We also study how symmetries and constants of motions for $S$ lift into symmetries and constants of motions for $S^c$.


Reliability analysis of semicoherent systems through their lattice polynomial descriptions

Abstract: A semicoherent system can be described by its structure function or, equivalently, by a lattice polynomial function expressing the system lifetime in terms of the component lifetimes. In this paper we point out the parallelism between the two descriptions and use the natural connection of lattice polynomial functions and relevant random events to collect exact formulas for the system reliability. We also discuss the equivalence between calculating the reliability of semicoherent systems and calculating the distribution function of a lattice polynomial function of random variables.


The warping degree of a knot diagram

Abstract: For an oriented knot diagram D, the warping degree d(D) is the smallest number of crossing changes which are needed to obtain the monotone diagram from D in the usual way. We show that d(D) + d(-D) + 1 is less than or equal to the crossing number of D. Moreover the equality holds if and only if D is an alternating diagram.


Remarques sur un article r'ecent de B. Poonen

Abstract: B. Poonen recently produced smooth threefolds over a number field which do not have a rational point but have no Brauer-Manin obstruction even after descent to a finite 'etale cover. In this note I show that the varieties he produces have zero-cycles of degree 1. ----- B. Poonen a r'ecemment exhib'e des exemples de vari'et'es projectives et lisses de dimension 3 sur un corps de nombres qui n'ont pas de point rationnel et pour lesquelles il n'y a pas d'obstruction de Brauer-Manin apr`es rev^etement fini 'etale. Dans cette note, je montre que les vari'et'es qu'il construit poss`edent des z'ero-cycles de degr'e 1.


On the flux of pseudo-Anosov homeomorphisms

Abstract: We exhibit a pseudo-Anosov homeomorphism of a surface S which acts trivially on the first homology group of S and whose flux is non zero


Proteus mirabilis swarm-colony development with drift

Abstract: We prove a global existence result for a model describing the swarming phenomenon of the bacterium Proteus mirabilis. The model consists of an ordinary differential equation coupled with an age-structured equation involving nonlinear degenerate diffusion and an additional drift term.


General Existence Results for Reflected BSDE and BSDE

Abstract: In this paper, we are concerned with the problem of existence of solutions for generalized reflected backward stochastic differential equations (GRBSDEs for short) and generalized backward stochastic differential equations (GBSDEs for short) when the generator $fds + gdA_s$ is continuous with general growth with respect to the variable $y$ and stochastic quadratic growth with respect to the variable $z$. We deal with the case of a bounded terminal condition $\xi$ and a bounded barrier $L$ as well as the case of unbounded ones. This is done by using the notion of generalized BSDEs with two reflecting barriers studied in \cite{EH}. The work is suggested by the interest the results might have in finance, control and game theory.


Hyperbolicity of geometric orbifolds

Abstract: We study complex hyperbolicity in the setting of geometric orbifolds introduced by F. Campana. Generalizing classical methods to this context, we obtain degeneracy statements for entire curves with ramification in situations where no Second Main Theorem is known from value distribution theory.


Kuelshammer ideals and the scalar problem for blocks with dihedral defect groups

Abstract: In by now classical work, K. Erdmann classified blocks of finite groups with dihedral defect groups (and more generally algebras of dihedral type) up to Morita equivalence. In the explicit description by quivers and relations of such algebras with two simple modules, several subtle problems about scalars occurring in relations remained unresolved. In particular, for the dihedral case it is a longstanding open question whether blocks of finite groups can occur for both possible scalars 0 and 1. In this article, using Kuelshammer ideals (a.k.a. generalized Reynolds ideals), we provide the first examples of blocks where the scalar is 1, thus answering the above question to the affirmative. Our examples are the principal blocks of PGL_2(F_q), the projective general linear group of 2x2-matrices with entries in the finite field F_q, where q=p^n\equiv \pm 1 mod 8, with p an odd prime number.


Some notes on trees and paths

Abstract: These notes cover background material on trees which are used in the paper `On uniqueness of the signature of a path of variation and the reduced path group'.


Desperately seeking mathematical truth

Abstract: This article discusses epistemological problems in the philosophy of mathematics and issues concerning the reliability of the mathematical literature.


Euclidean components for a class of self-injective algebras

Abstract: We determine the length of composition series of projective modules of G-transitive algebras with an Auslander-Reiten component of Euclidean tree class. Furthermore we show that modules with certain length of composition series are periodic. We apply these results to G-transitive blocks of the Universal enveloping of restricted p-Lie algebras and prove that G-transitive principal blocks only allow components with Euclidean tree class if p=2. Finally we deduce conditions for a smash product of a local basic algebra with a commutative semi-simple group algebra to have components with Euclidean tree class, depending on the components of the Auslander-Reiten quiver of the basic algebra. We also develop some properties of the bilinear form dim Hom_A(-,-) for the representation ring G(A) of any finite-dimensional algebra A.


Vari'et'es presque rationnelles, leurs points rationnels et leurs d'eg'en'erescences

Abstract: This survey, which contains very few proofs, addresses the general question: Over a given type of field, is there a natural class of varieties which automatically have a rational point? Fields under consideration here include: finite fields, p-adic fields, function fields in one or two variables over an algebraically closed field. Classical answers are given by the Chevalley-Warning theorem and by Tsen's theorem. More general answers were provided by a theorem of Graber, Harris and Starr and by a theorem of Esnault. The latter results apply to rationally connected varieties. We discuss these varieties from various angles : weak approximation, R-equivalence, Chow group of zero-cycles. Ongoing work on `rationally simply connected' varieties over function fields in two variables is also mentioned. A common thread in this report is the study of the special fibre of a scheme over a discrete valuation ring: if the generic fibre has a simple geometry, what does it imply for the special fibre?


Integral models of unitary representations of current groups with values in semidirect products

Abstract: We describe a general construction of irreducible unitary representations of the group of currents with values in the semidirect product of a locally compact subgroup $P_0$ and a one-parameter group ${\mathbb R {}}^*_+=\{r:r>0\}$ of automorphisms of $P_0$. This construction is determined by a a faithful unitary representation of $P_0$ (canonical representation) whose images under the action of the group of automorphisms tend to the identity representation as $r\to 0$. We apply this construction to the groups of currents of the maximal parabolic subgroups of the groups of motions of the $n$-dimensional real and complex Lobachevsky spaces. The obtained representations of the groups of parabolic currents can be uniquely extended to the groups of currents with values in the semisimple groups O(n,1) and U(n,1). This gives a new description of the representations of the groups of currents of these groups constructed in the 70s and realized in the Fock space. The key role in our construction is played by the so-called special representation of the parabolic subgroup $P$ and the remarkable $\sigma$-finite measure (Lebesgue measure) $\mathcal L$ in the space of distributions.