Differential Equations on Complex Projective Hypersurfaces of Low Dimension

Abstract: Let $n=2,3,4,5$ and let $X$ be a smooth complex projective hypersurface of $\mathbb P^{n+1}$. In this paper we find an effective lower bound for the degree of $X$, such that every holomorphic entire curve in $X$ must satisfy an algebraic differential equation of order $k=n=\dim X$, and also similar bounds for order $k>n$. Moreover, for every integer $n\ge 2$, we show that there are no such algebraic differential equations of order $k<n$ for a smooth hypersurface in $\mathbb P^{n+1}$.


Polynomial functors and opetopes

Abstract: We give an elementary and direct combinatorial definition of opetopes in terms of trees, well-suited for graphical manipulation and explicit computation. To relate our definition to the classical definition, we recast the Baez-Dolan slice construction for operads in terms of polynomial monads: our opetopes appear naturally as types for polynomial monads obtained by iterating the Baez-Dolan construction, starting with the trivial monad. We show that our notion of opetope agrees with Leinster's. Next we observe a suspension operation for opetopes, and define a notion of stable opetopes. Stable opetopes form a least fixpoint for the Baez-Dolan construction. A final section is devoted to example computations, and indicates also how the calculus of opetopes is well-suited for machine implementation.


Infinite-dimensional diffusions as limits of random walks on partitions

Abstract: The present paper originated from our previous study of the problem of harmonic analysis on the infinite symmetric group. This problem leads to a family {P_z} of probability measures, the z-measures, which depend on the complex parameter z. The z-measures live on the Thoma simplex, an infinite-dimensional compact space which is a kind of dual object to the infinite symmetric group. The aim of the paper is to introduce stochastic dynamics related to the z-measures. Namely, we construct a family of diffusion processes in the Toma simplex indexed by the same parameter z. Our diffusions are obtained from certain Markov chains on partitions of natural numbers n in a scaling limit as n goes to infinity. These Markov chains arise in a natural way, due to the approximation of the infinite symmetric group by the increasing chain of the finite symmetric groups. Each z-measure P_z serves as a unique invariant distribution for the corresponding diffusion process, and the process is ergodic with respect to P_z. Moreover, P_z is a symmetrizing measure, so that the process is reversible. We describe the spectrum of its generator and compute the associated (pre)Dirichlet form.


Local Structure of Ideal Shapes of Knots, II, Constant Curvature Case

Abstract: The thickness, NIR(K) of a knot or link K is defined to be the radius of the largest solid tube one can put around the curve without any self intersections, which is also known as the normal injectivity radius of K. For C^{1,1} curves K, NIR(K)=min{(1/2)DCSC(K),(1/(supkappa(K))))}, where kappa(K) is the generalized curvature, and the double critical self distance DCSD(K) is the shortest length of the segments perpendicular to K at both end points. The knots and links in ideal shapes (or tight knots or links) belong to the minima of ropelength = length/thickness within a fixed isotopy class. In this article, we prove that NIR(K)=(1/2)DCSC(K), for every relative minimum K of ropelength in R^n for certain dimensions n, including n=3.


Group gradings on simple Lie algebras of type A in positive characteristic

Abstract: In this paper we consider gradings by a finite abelian group $G$ on the Lie algebra $\mathfrak{sl}_n(F)$ over an algebraically closed field $F$ of characteristic different from 2 and not dividing $n$.


Estimates of the best Sobolev constant of the embedding of $BV(\Omega)$ into $L^1(\partial\Omega)$ and related shape optimization problems

Abstract: In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality $\lambda_1(\Omega)\|u\|_{L^1(\partial\Omega)} \le \|u\|_{W^{1,1}(\Omega)}$ that are independent of $\Omega$. This estimates generalize those of \cite{BS} concerning the $p$-Laplacian to the case $p=1$. We apply our results to prove existence of an extremal for this embedding. We then study an optimal design problem related to $\lambda_1$, and eventually compute the shape derivative of the functional $\Omega\to\lambda_1(\Omega)$. As a consequence, we obtain that a ball of $\R^n$ of radius $n$ is critical for volume-preserving deformations.


Tridiagonal pairs of Krawtchouk type

Abstract: Let $K$ denote an algebraically closed field with characteristic 0 and let $V$ denote a vector space over $K$ with finite positive dimension. Let $A,A^*$ denote a tridiagonal pair on $V$ with diameter $d$. We say that $A,A^*$ has Krawtchouk type whenever the sequence $\lbrace d-2i\rbrace_{i=0}^d$ is a standard ordering of the eigenvalues of $A$ and a standard ordering of the eigenvalues of $A^*$. Assume $A,A^*$ has Krawtchouk type. We show that there exists a nondegenerate symmetric bilinear form $< , >$ on $V$ such that $<Au,v>= < u,Av>$ and $<A^*u,v >= < u,A^*v>$ for $u,v\in V$. We show that the following tridiagonal pairs are isomorphic: (i) $A,A^*$; (ii) $-A,-A^*$; (iii) $A^*,A$; (iv) $-A^*,-A$. We give a number of related results and conjectures.


Rota-Baxter Categories

Abstract: We introduce Rota-Baxter categories and construct examples of such structures.


Index reduction for Brauer classes via stable sheaves (with an appendix by Bhargav Bhatt)

Abstract: We use twisted sheaves to study the problem of index reduction for Brauer classes. In general terms, this problem may be phrased as follows: given a field $k$, a $k$-variety $X$, and a class $\alpha \in \Br(k)$, compute the index of the class $\alpha_{k(X)} \in \Br(X)$ obtained from $\alpha$ by extension of scalars to $k(X)$. We give a general method for computing index reduction which refines classical results of Schofield and van den Bergh. When $X$ is a curve of genus 1, we use Atiyah's theorem on the structure of stable vector bundles with integral slope to show that our formula simplifies dramatically, giving a complete solution to the index reduction problem in this case. Using the twisted Fourier-Mukai transform, we show that a similarly simple formula describes homogeneous index reduction on torsors under higher-dimensional abelian varieties.


Regularity of Dirichlet nearly minimizing multiple-valued functions

Abstract: In this paper, we extend the related notions of Dirichlet quasiminimizer, $\omega-$minimizer and almost minimizer to the framework of multiple-valued functions in the sense of Almgren and prove Holder regularity results. We also give examples of those minimizers with various branch sets.


Some extremal problems related to Bell-type inequalities

Abstract: The best approximation by bounded product functions is calculated for some very simple two-valued functions of two variables.


Generalizations of Khovanskii's theorems on growth of sumsets in abelian semigroups

Abstract: We show that if $P$ is a lattice polytope in the nonnegative orthant of $\R^k$ and $\chi$ is a coloring of the lattice points in the orthant such that the color $\chi(a+b)$ depends only on the colors $\chi(a)$ and $\chi(b)$, then the number of colors of the lattice points in the dilation $nP$ of $P$ is for large $n$ given by a polynomial (or, for rational $P$, by a quasipolynomial). This unifies a classical result of Ehrhart and Macdonald on lattice points in polytopes and a result of Khovanski\u\i{} on sumsets in semigroups. We also prove a strengthening of multivariate generalizations of Khovanski\u\i's theorem. Another result of Khovanski\u\i{} states that the size of the image of a finite set after $n$ applications of mappings from a finite family of mutually commuting mappings is for large $n$ a polynomial. We give a combinatorial proof of a multivariate generalization of this theorem.


Computational techniques for proving identities in symmetric compositions

Abstract: We present in this work a complete session in a Mathematica notebook. The aim of this notebook is to check identities in symmetric compositions. This notebook is a complement of our work [1] and it has all the explicit computations. We refer the reader to that paper which can be seen in http://www.uibk.ac.at/mathematik/loos/jordan/index.html. First of all we will present a few number of comands in order to simplify identities by extracting scalars, SOut. The rest of the strategy holds on the powerfull of using patterns and rules.


Characterization of the matrix whose norm is determined by its action on decreasing sequences

Abstract: Let $A=(a_{j,k})_{j,k \ge 1}$ be a non-negative matrix. In this paper, we characterize those $A$ for which $\|A\|_{E, F}$ are determined by their actions on decreasing sequences, where $E$ and $F$ are suitable normed Riesz spaces of sequences.


A slight improvement to Korenblum's constant

Abstract: Let $A^2(D)$ be the Bergman space over the open unit disk $D$ in the complex plane. Korenblum conjectured that there is an absolute constant $c \in (0,1)$ such that whenever $|f(z)|\le |g(z)|$ in the annulus $c<|z|<1$ then $||f(z)|| \le ||g(z)||$.In 2004 C.Wang gave an upper bound on $c$,that is, $c < 0.67795$, and in 2006 A.Schuster gave a lower bound ,$c > 0.21 $ .In this paper we slightly improve the upper bound for $c$.


On the threshold for k-regular subgraphs of random graphs

Abstract: The $k$-core of a graph is the largest subgraph of minimum degree at least $k$. We show that for $k$ sufficiently large, the $(k + 2)$-core of a random graph $\G(n,p)$ asymptotically almost surely has a spanning $k$-regular subgraph. Thus the threshold for the appearance of a $k$-regular subgraph of a random graph is at most the threshold for the $(k+2)$-core. In particular, this pins down the point of appearance of a $k$-regular subgraph in $\G(n,p)$ to a window for $p$ of width roughly $2/n$ for large $n$ and moderately large $k$.


Symmetric linear functions of the restricted quantum group $\bar{U}_qsl_2(\mathbb{C})$

Abstract: We determine a set of primitive idempotents and the basic algebra of the restricted quantum group $\bar{U}_qsl_2(\mathbb{C})$. As a result, we can show the dimension of the space of symmetric linear functions of $\bar{U}_qsl_2(\mathbb{C})$ is $3p-1$


Comparison Geometry for the Bakry-Emery Ricci Tensor

Abstract: For Riemannian manifolds with a measure $(M,g, e^{-f} dvol_g)$ we prove mean curvature and volume comparison results when the $\infty$-Bakry-Emery Ricci tensor is bounded from below and $f$ is bounded or $\partial_r f$ is bounded from below, generalizing the classical ones (i.e. when $f$ is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the Bakry-Emery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when $f$ is bounded. Simple examples show the bound on $f$ is necessary for these results.


Thurston obstructions and Ahlfors regular conformal dimension

Abstract: Let $f: S^2 \to S^2$ be an expanding branched covering map of the sphere to itself with finite postcritical set $P_f$. Associated to $f$ is a canonical quasisymmetry class $\GGG(f)$ of Ahlfors regular metrics on the sphere in which the dynamics is (non-classically) conformal. We show \[ \inf_{X \in \GGG(f)} \hdim(X) \geq Q(f)=\inf_\Gamma \{Q \geq 2: \lambda(f_{\Gamma,Q}) \geq 1\}.\] The infimum is over all multicurves $\Gamma \subset S^2-P_f$. The map $f_{\Gamma,Q}: \R^\Gamma \to \R^\Gamma$ is defined by \[ f_{\Gamma, Q}(\gamma) =\sum_{[\gamma']\in\Gamma} \sum_{\delta \sim \gamma'} \deg(f:\delta \to \gamma)^{1-Q}[\gamma'],\] where the second sum is over all preimages $\delta$ of $\gamma$ freely homotopic to $\gamma'$ in $S^2-P_f$, and $ \lambda(f_{\Gamma,Q})$ is its Perron-Frobenius leading eigenvalue. This generalizes Thurston's observation that if $Q(f)>2$, then there is no $f$-invariant classical conformal structure.


Test ideals vs. multiplier ideals

Abstract: The generalized test ideals introduced in [HY] are related to multiplier ideals via reduction to characteristic p. In addition, they satisfy many of the subtle properties of the multiplier ideals, which in characteristic zero follow via vanishing theorems. In this note we give several examples to emphasize the different behavior of test ideals and multiplier ideals. Our main result is that every ideal in an F-finite regular local ring can be written as a generalized test ideal. We also prove the semicontinuity of $F$-pure thresholds (though the analogue of the Generic Restriction Theorem for multiplier ideals does not hold).


Some Comments around The Examples against The Generalized Jacobian Conjecture

Abstract: We have studied a faded problem, the Jacobian Conjecture ~: \noindent {\sf The Jacobian Conjecture $(JC_n)$}~: If $f_1, \cdots, f_n$ are elements in a polynomial ring $k[X_1, \cdots, X_n]$ over a field $k$ of characteristic $0$ such that the Jacobian $\det(\partial f_i/ \partial X_j) $ is a nonzero constant, then $k[f_1, \cdots, f_n] = k[X_1, \cdots, X_n]$. For this purpose, we generalize it to the following form~: \noindent {\sf The Generalized Jacobian Conjecture $(GJC)$}~: {\it Let $\varphi : S \rightarrow T$ be an unramified homomorphism of Noetherian domains with $T^\times = \varphi(S^\times)$. Assume that $T$ is a factorial domain and that $S$ is a simply connected normal domain. Then $\varphi$ is an isomorphism. } For the consistency of our discussion, we raise some serious (or idiot) questions and some comments concerning the examples appeared in the papers published by the certain excellent mathematicians (though we are unwilling to deal with them). Since the existence of such examples would be against our original target Conjecture$(GJC)$, we have to dispute their arguments about the existence of their respective (so called) counter-examples. Our conclusion is that they are not perfect counter-examples as are shown explicitly.


Additive Regression Model for Continuous Time Processes

Abstract: In the setting of additive regression model for continuous time process, we establish the optimal uniform convergence rates and optimal asymptotic quadratic error of additive regression. To build our estimate, we use the marginal integration method.


A one dimensional analysis of turbulence and its intermittence for the d-dimensional stochastic Burgers equation

Abstract: The inviscid limit of the stochastic Burgers equation is discussed in terms of the level surfaces of the minimising Hamilton-Jacobi function, the classical mechanical caustic and the Maxwell set and their algebraic pre-images under the classical mechanical flow map. The problem is analysed in terms of a reduced (one dimensional) action function. We demonstrate that the geometry of the caustic, level surfaces and Maxwell set can change infinitely rapidly causing turbulent behaviour which is stochastic in nature. The intermittence of this turbulence is demonstrated in terms of the recurrence of two processes.


Some Uniform Limit Results in Additive Regression Model

Abstract: We establish some uniform limit results in the setting of additive regression model estimation. Our results allow to give an asymptotic 100% confidence bands for these components. These results are stated in the framework of i.i.d random vectors when the marginal integration estimation method is used.


Stein's method and Poisson process approximation for a class of Wasserstein metrics

Abstract: Based on Stein's method, we derive upper bounds for Poisson process approximation in the $L_1$-Wasserstein metric $d_2^{(p)}$, which is based on a slightly adapted $L_p$-Wasserstein metric between point measures. For the case $p=1$, this construction yields the metric $d_2$ introduced in [Barbour and Brown Stochastic Process. Appl. 43 (1992) 9--31], for which Poisson process approximation is well studied in the literature. We demonstrate the usefulness of the extension to general $p$ by showing that $d_2^{(p)}$-bounds control differences between expectations of certain $p$th order average statistics of point processes. To illustrate the bounds obtained for Poisson process approximation, we consider the structure of 2-runs and the hard core model as concrete examples.


A one dimensional analysis of singularities and turbulence for the stochastic Burgers equation in d-dimensions

Abstract: The inviscid limit of the stochastic Burgers equation, with body forces white noise in time, is discussed in terms of the level surfaces of the minimising Hamilton-Jacobi function, the classical mechanical caustic and the Maxwell set and their algebraic pre-images under the classical mechanical flow map. The problem is analysed in terms of a reduced (one dimensional) action function. We give an explicit expression for an algebraic surface containing the Maxwell set and caustic in the polynomial case. Those parts of the caustic and Maxwell set which are singular are characterised. We demonstrate how the geometry of the caustic, level surfaces and Maxwell set can change infinitely rapidly causing turbulent behaviour which is stochastic in nature, and we determine its intermittence in terms of the recurrent behaviour of two processes.


Asymptotic stability of solitons of the gKdV equations with general nonlinearity

Abstract: We consider the generalized Korteweg-de Vries equation \partial_t u + \partial_x (\partial_x^2 u + f(u))=0, \quad (t,x)\in [0,T)\times \mathbb{R}, (1) with general $C^3$ nonlinearity $f$. Under an explicit condition on $f$ and $c>0$, there exists a solution in the energy space $H^1$ of (1) of the type $u(t,x)=Q_c(x-x_0-ct)$, called soliton. In this paper, under general assumptions on $f$ and $Q_c$, we prove that the family of soliton solutions around $Q_c$ is asymptotically stable in some local sense in $H^1$, i.e. if $u(t)$ is close to $Q_{c}$ (for all $t\geq 0$), then $u(t)$ locally converges in the energy space to some $Q_{c_+}$ as $t\to +\infty$. Note in particular that we do not assume the stability of $Q_{c}$. This result is based on a rigidity property of equation (1) around $Q_{c}$ in the energy space whose proof relies on the introduction of a dual problem. These results extend the main results in previous works devoted to the pure power case.


Two-sided optimal bounds for Green function of half-spaces for relativistic $\alpha$-stable process

Abstract: The purpose of this paper is to find optimal estimates for the Green function of a half-space of {\it the relativistic $\alpha$-stable process} with parameter $m$ on $\Rd$ space. This process has an infinitesimal generator of the form $mI-(m^{2/\alpha}I-\Delta)^{\alpha/2},$ where $0<\alpha<2$, $m>0$, and reduces to the isotropic $\alpha$-stable process for $m=0$. Its potential theory for open bounded sets has been well developed throughout the recent years however almost nothing was known about the behaviour of the process on unbounded sets. The present paper is intended to fill this gap and we provide two-sided sharp estimates for the Green function for a half-space. As a byproduct we obtain some improvements of the estimates known for bounded sets specially for balls. The advantage of these estimates is a clarification of the relationship between the diameter of the ball and the parameter $m$ of the process. The main result states that the Green function is comparable with the Green function for the Brownian motion if the points are away from the boundary of a half-space and their distance is greater than one. On the other hand for the remaining points the Green function is somehow related the Green function for the isotropic $\alpha$-stable process. For example, for $d\ge3$, it is comparable with the Green function for the isotropic $\alpha$-stable process, provided that the points are close enough.


Refined asymptotics around solitons for gKdV equations

Abstract: We consider the generalized Korteweg-de Vries equation $$ \partial_t u + \partial_x (\partial_x^2 u + f(u))=0, \quad (t,x)\in [0,T)\times \mathbb{R}$$ with general $C^2$ nonlinearity $f$. Under an explicit condition on $f$ and $c>0$, there exists a solution in the energy space $H^1$ of the type $u(t,x)=Q_c(x-x_0-ct)$, called soliton. Stability theory for $Q_c$ is well-known. In previous works, we have proved that for $f(u)=u^p$, $p=2,3,4$, the family of solitons is asymptotically stable in some local sense in $H^1$, i.e. if $u(t)$ is close to $Q_{c}$ (for all $t\geq 0$), then $u(t,.+\rho(t))$ locally converges in the energy space to some $Q_{c_+}$ as $t\to +\infty$, for some $c^+\sim c$. Then, the asymptotic stability result could be extended to the case of general assumptions on $f$ and $Q_c$. The objective of this paper is twofold. The main objective is to prove that in the case $f(u)=u^p$, $p=2,3,4$, $\rho(t)-c_+ t$ has limit as $t\to +\infty$ under the additional assumption $x_+ u\in L^2$. The second objective of this paper is to provide large time stability and asymptotic stability results for two soliton solutions for the case of general nonlinearity $f(u)$, when the ratio of the speeds of the solitons is small. The motivation is to accompany forthcoming works devoted to the collision of two solitons in the nonintegrable case. The arguments are refinements of previous works specialized to the case $u(t)\sim Q_{c_1}+Q_{c_2}$, for $0< c_2 \ll c_1$.


All creatures great and small

Abstract: Let kappa be an uncountable regular cardinal. Assuming 2^kappa=kappa^+, we show that the clone lattice on a set of size kappa is not dually atomic.


A Computational Approach to Essential and Nonessential Objective Functions in Linear Multicriteria Optimization

Abstract: The question of obtaining well-defined criteria for multiple criteria decision making problems is well-known. One of the approaches dealing with this question is the concept of nonessential objective function. A certain objective function is called nonessential if the set of efficient solutions is the same both with or without that objective function. In this paper we put together two methods for determining nonessential objective functions. A computational implementation is done using a computer algebra system.


Depth three towers and Jacobson-Bourbaki correspondence

Abstract: We introduce a notion of depth three tower of three rings C < B < A as a useful generalization of depth two ring extension. If A = End B_C and B | C is a Frobenius extension, this also captures the notion of depth three for a Frobenius extension in math.RA/0107064 and math.RA/0108067 such that if B | C is depth three, then A | C is depth two (cf. math.QA/0001020). If A, B and C correspond to a tower of subgroups G > H > K via the group algebra over a fixed base ring, the depth three condition is the condition that subgroup K has normal closure K^G contained in H. For a depth three tower of rings, there is a pre-Galois theory for the ring End {}_BA_C and coring (A \otimes_B A)^C involving Morita context bimodules and left coideal subrings. This is applied in the last two sections to a specialization of a Jacobson-Bourbaki correspondence theorem for augmented rings to depth two extensions with depth three intermediate division rings.


Constant Weight Codes: A Geometric Approach Based on Dissections

Abstract: We present a novel technique for encoding and decoding constant weight binary codes that uses a geometric interpretation of the codebook. Our technique is based on embedding the codebook in a Euclidean space of dimension equal to the weight of the code. The encoder and decoder mappings are then interpreted as a bijection between a certain hyper-rectangle and a polytope in this Euclidean space. An inductive dissection algorithm is developed for constructing such a bijection. We prove that the algorithm is correct and then analyze its complexity. The complexity depends on the weight of the code, rather than on the block length as in other algorithms. This approach is advantageous when the weight is smaller than the square root of the block length.


A general convergence result for the Ricci flow in higher dimensions

Abstract: Let (M,g_0) be a compact Riemannian manifold of dimension n \geq 4. We show that the normalized Ricci flow deforms g_0 to a constant curvature metric provided that (M,g_0) x R has positive isotropic curvature. This condition is stronger than 2-positive flag curvature but weaker than 2-positive curvature operator.


On the cost-subdifferentials of cost-convex functions

Abstract: We are interested in the cost-convex potentials in optimal mass transport theory, and we show by direct and geometric arguments the equivalence between cost-subdifferentials and ordinary subdifferentials of cost-convex functions, under the assumptions A0, A1, A2, and A3W on cost functions introduced by Ma, Trudinger, and Wang. The connectivity of contact sets of optimal transport maps follows as a direct corollary. Our approach is quite different from the previous result of Loeper which he obtained as the first step toward his Hoelder regularity theory of potential functions, and which was based upon approximation using the regularity theory of Ma, Trudinger, and Wang. The result in this paper improves his result, by relaxing certain geometrical assumptions on the domains and targets; it also completes his Hoelder regularity theory of potential functions on the round sphere, by making it self-contained.


Convergence of iterated Aluthge transform sequence for diagonalizable matrices II: $\lambda$-Aluthge transform

Abstract: Let $\lambda \in (0,1)$ and let $T$ be a $r\times r$ complex matrix with polar decomposition $T=U|T|$. Then, the $\la$- Aluthge transform is defined by $$ \Delta_\lambda (T )= |T|^{\lambda} U |T |^{1-\lambda}. $$ Let $\Delta_\lambda^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, $n\in\mathbb{N}$. We prove that the sequence $\{\Delta_\lambda^{n}(T)\}_{n\in\mathbb{N}}$ converges for every $r\times r$ {\bf diagonalizable} matrix $T$. We show regularity results for the two parameter map $(\la, T) \mapsto \alulit{\infty}{T}$, and we study for which matrices the map $(0,1)\ni \lambda \mapsto \Delta_\lambda^{\infty}(T)$ is constant.


On The Universality Of Central Loops

Abstract: LC-loops, RC-loops and C-loops are collectively called central loops. It is shown that an LC(RC)-loop is a left(right) universal loop. But an LC(RC)-loop is a universal loop if and only if it is a right(left) universal loop. It is observed that not all RC-loops or LC-loops or C-loops are universal loops. But if an RC-loop(LC-loop, C-loop) is universal, then it is a right Bol loop(left Bol loop, Moufang loop) respectively. If a loop and its right or left isotope are commutative then the loop is a C-loop if and only if its right or left isotope is a C-loop. If a C-loop is central square and its right or left isotope is an alternative central square loop, then the latter is a C-loop. Necessary and sufficient condition for an LC-loop(RC-loop) to be a left(right)G-loop is established. Consequently, necessary and sufficient conditions for an LC-loop, and an RC-loop to be a G-loop are established. A necessary and sufficient condition for a C-loop to be a G-loop is established.


Remark on the rank of elliptic curves

Abstract: A covariant functor on the elliptic curves with complex multiplication is constructed. The functor takes values in the noncommutative tori with real multiplication. A conjecture on the rank of an elliptic curve is formulated.


Loop coproducts in string topology and triviality of higher genus TQFT operations

Abstract: Cohen and Godin constructed positive boundary topological quantum field theory (TQFT) structure on the homology of free loop spaces of oriented closed smooth manifolds by associating a certain operations called string operations to orientable surfaces with parametrized boundaries. We show that all TQFT string operations associated to surfaces of genus at least one vanish identically. This is a simple consequence of properties of the loop coproduct which will be discussed in detail. One interesting property is that the loop coproduct is nontrivial only on the degree $d$ homology group of the connected component of $LM$ consisting of contractible loops, where $d=\dim M$, with values in the degree 0 homology group of constant loops. Thus the loop coproduct behaves in a dramatically simpler way than the loop product.


A Homotopy Theoretic Proof of the BV Identity in Loop Homology

Abstract: Chas and Sullivan proved the existence of a Batalin-Vilkovisky algebra structure in the homology of free loop spaces on closed finite dimensional smooth manifolds using chains and chain homotopies. This algebraic structure involves an associative product called the loop product, a Lie bracket called the loop bracket, and a square 0 operator called the BV operator. Cohen and Jones gave a homotopy theoretic description of the loop product in terms of spectra. In this paper, we give an explicit homotopy theoretic description of the loop bracket and, using this description, we give a homological proof of the BV identity connecting the loop product, the loop bracket, and the BV operator. The proof is based on an observation that the loop bracket and the BV derivation are given by the same cycle in the free loop space, except that they differ by parametrization of loops.


Framed bicategories and monoidal fibrations

Abstract: In some bicategories, the 1-cells are `morphisms' between the 0-cells, such as functors between categories, but in others they are `objects' over the 0-cells, such as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, because the `morphisms between 0-cells', such as ring homomorphisms, are missing. We can include them by using a pseudo double category, but usually these morphisms also induce base change functors acting on the 1-cells. We avoid complicated coherence problems by describing base change `nonalgebraically', using categorical fibrations. The resulting `framed bicategories' assemble into 2-categories, with attendant notions of equivalence, adjunction, and so on which are more appropriate for our examples than are the usual bicategorical ones. We then describe two ways to construct framed bicategories. One is an analogue of rings and bimodules which starts from one framed bicategory and builds another. The other starts from a `monoidal fibration', meaning a parametrized family of monoidal categories, and produces an analogue of the framed bicategory of spans. Combining the two, we obtain a construction which includes both enriched and internal categories as special cases.


Bayesian Covariance Matrix Estimation using a Mixture of Decomposable Graphical Models

Abstract: A Bayesian approach is used to estimate the covariance matrix of Gaussian data. Ideas from Gaussian graphical models and model selection are used to construct a prior for the covariance matrix that is a mixture over all decomposable graphs. For this prior the probability of each graph size is specified by the user and graphs of equal size are assigned equal probability. Most previous approaches assume that all graphs are equally probable. We show empirically that the prior that assigns equal probability over graph sizes outperforms the prior that assigns equal probability over all graphs, both in identifying the correct decomposable graph and in more efficiently estimating the covariance matrix.


Limits of PGL(3)-translates of plane curves, I

Abstract: We classify all possible limits of families of translates of a fixed, arbitrary complex plane curve. We do this by giving a set-theoretic description of the projective normal cone (PNC) of the base scheme of a natural rational map, determined by the curve, from the $P^8$ of 3x3 matrices to the $P^N$ of plane curves of degree $d$. In a sequel to this paper we determine the multiplicities of the components of the PNC. The knowledge of the PNC as a cycle is essential in our computation of the degree of the PGL(3)-orbit closure of an arbitrary plane curve, performed in our earlier paper "Linear orbits of arbitrary plane curves".


On reconstruction formulas and algorithms for the thermoacoustic tomography

Abstract: The paper surveys recent progress in establishing uniqueness and developing inversion formulas and algorithms for the thermoacoustic tomography. In mathematical terms, one deals with a rather special inverse problem for the wave equation. In the case of constant sound speed, it can also be interpreted as a problem concerning the spherical mean transform.


Limits of PGL(3)-translates of plane curves, II

Abstract: Every complex plane curve C determines a subscheme S of the $P^8$ of 3x3 matrices, whose projective normal cone (PNC) captures subtle invariants of C. In "Limits of PGL(3)-translates of plane curves, I" we obtain a set-theoretic description of the PNC and thereby we determine all possible limits of families of plane curves whose general element is isomorphic to C. The main result of this article is the determination of the PNC as a cycle; this is an essential ingredient in our computation in "Linear orbits of arbitrary plane curves" of the degree of the PGL(3)-orbit closure of an arbitrary plane curve, an invariant of natural enumerative significance.


Compactified moduli of projective bundles

Abstract: We present a method for compactifying stacks of $\PGL_n$-torsors (Azumaya algebras) on algebraic spaces. In particular, when the ambient space is a smooth projective surface we use our methods to show that various moduli spaces are irreducible and carry natural virtual fundamental classes. We also prove a version of the Skolem-Noether theorem for certain algebra objects in the derived category, which allows us to give an explicit description of the boundary points in our compactified moduli problem.


Entrelacement d'alg\`ebres de Lie [Wreath products for Lie algebras]

Abstract: Full details are given for the definition and construction of the wreath product of two arbitrary Lie algebras, in the hope that it can lead to the definition of a suitable Lie group to be the wreath product of two given Lie groups. In the process, quite a few new notions are needed, and introduced. Such are, for example : Formal series with variables in a vector space and coefficients in some other vector space. Derivation of a formal series relative to another formal series. The Lie algebra of a vector space. Formal actions of Lie algebras over vector spaces. The basic formal action of a Lie algebra over itself (as a formal version of the analytic aspect of the infinitesimal operation law of a Lie groupuscule). More generally, the wreath product of two Lie algebras is defined, relative to a formal action of the second onto an arbitrary vector space. Main features are : A description of the triangular actions of wreath products over product vector spaces, and a Kaloujnine-Krasner type theorem : In essence, it says that all Lie extensions of a given Lie algebra by another Lie algebra are, indeed, subalgebras of their wreath product.


Non-unique ergodicity, observers' topology and the dual algebraic lamination for $\R$-trees

Abstract: We continue in this article the study of laminations dual to very small actions of a free group F on R-trees. We prove that this lamination determines completely the combinatorial structure of the R-tree (the so-called observers' topology). On the contrary the metric is not determined by the lamination, and an R-tree may be equipped with different metrics which have the same observers' topology.


The oscillation stability problem for the Urysohn sphere: A combinatorial approach

Abstract: We study the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for $\ell_2$ in the context of the Urysohn space $\Ur$. In particular, we show that this problem reduces to a purely combinatorial problem involving a family of countable ultrahomogeneous metric spaces with finitely many distances.


Hopf Algebras of Heap Ordered Trees and Permutations

Abstract: It is known that there is a Hopf algebra structure on the vector space with basis all heap-ordered trees. We give a new bialgebra structure on the space with basis all permutations and show that there is a direct bialgebra isomorphism between the Hopf algebra of heap-ordered trees and the bialgebra of permutations.


On a Smale Theorem and Nonhomogeneous Equilibria in Cooperative Systems

Abstract: A standard result by Smale states that n dimensional strongly cooperative dynamical systems can have arbitrary dynamics when restricted to unordered invariant hyperspaces. In this paper this result is extended to the case when all solutions of the strongly cooperative system are bounded and converge towards one of only two equilibria outside of the hyperplane. An application is given in the context of strongly cooperative systems of reaction diffusion equations. It is shown that such a system can have a continuum of spatially inhomogeneous steady states, even when all solutions of the underlying reaction system converge to one of only three equilibria.


An explicit construction of the Quillen homotopical category of dg Lie algebras

Abstract: Let $\g_1$ and $\g_2$ be two dg Lie algebras, then it is well-known that the $L_\infty$ morphisms from $\g_1$ to $\g_2$ are in 1-1 correspondence to the solutions of the Maurer-Cartan equation in some dg Lie algebra $\Bbbk(\g_1,\g_2)$. Then the gauge action by exponents of the zero degree component $\Bbbk(\g_1,\g_2)^0$ on $MC\subset\Bbbk(\g_1,\g_2)^1$ gives an explicit "homotopy relation" between two $L_\infty$ morphisms. We prove that the quotient category by this relation (that is, the category whose objects are $L_\infty$ algebras and morphisms are $L_\infty$ morphisms modulo the gauge relation) is well-defined, and is a localization of the category of dg Lie algebras and dg Lie maps by quasi-isomorphisms. As localization is unique up to an equivalence, it is equivalent to the Quillen-Hinich homotopical category of dg Lie algebras [Q1,2], [H1,2]. Moreover, we prove that the Quillen's concept of a homotopy coincides with ours. The last result was conjectured by V.Dolgushev [D].


A note on Poisson homogeneous spaces

Abstract: We identify the cotangent bundle Lie algebroid of a Poisson homogeneous space G/H of a Poisson Lie group G as a quotient of a transformation Lie algebroid over G. As applications, we describe the modular vector fields of G/H, and we identify the Poisson cohomology of G/H with coefficients in powers of its canonical line bundle with relative Lie algebra cohomology of the Drinfeld Lie algebra associated to G/H. We also construct a Poisson groupoid over G/H which is symplectic near the identity section. This note serves as preparation for forthcoming papers, in which we will compute explicitly the Poisson cohomology and study their symplectic groupoids for certain examples of Poisson homogeneous spaces related to semi-simple Lie groups.


Verification theorem and construction of $\epsilon$-optimal controls for control of abstract evolution equations

Abstract: We study several aspects of the dynamic programming approach to optimal control of abstract evolution equations, including a class of semilinear partial differential equations. We introduce and prove a verification theorem which provides a sufficient condition for optimality. Moreover we prove sub- and superoptimality principles of dynamic programming and give an explicit construction of $\epsilon$-optimal controls.


The rational homotopy type of a blow-up in the stable case

Abstract: Suppose that f:V->W is an embedding of closed oriented manifolds whose normal bundle has the structure of a complex vector bundle. It is well known in both complex and symplectic geometry that one can then construct a manifold W' which is the blow-up of W along V. Assume that dim(W)>2.dim(V)+2 and that H^1(f) is injective. We construct an algebraic model of the rational homotopy type of the blow-up W' from an algebraic model of the embedding and the Chern classes of the normal bundle. This implies that if the space W is simply connected then the rational homotopy type of W' depends only on the rational homotopy class of f and on the Chern classes of the normal bundle.


On Zermelo'-like problems: a Gauss-Bonnet inequality and a E. Hopf theorem

Abstract: The goal of this paper is to describe Zermelo's navigation problem on Riemannian manifolds as a time-optimal control problem and give an efficient method in order to evaluate its control curvature. We will show that up to change the Riemannian metric on the manifold the control curvature of Zermelo's problem has a simple to handle expression which naturally leads to a generalization of the classical Gauss-Bonnet formula in an inequality. This Gauss-Bonnet inequality enables to generalize for Zermelo's problems the E. Hopf theorem on flatness of Riemannian tori without conjugate points.


On the generalization of the Costas property in the continuum

Abstract: We extend the definition of the Costas property to functions in the continuum, namely on intervals of the reals or the rationals, and argue that such functions can be used in the same applications as discrete Costas arrays. We construct Costas bijections in the real continuum within the class of piecewise continuously differentiable functions, but our attempts to construct a fractal-like Costas bijection there are successful only under slight but necessary deviations from the usual arithmetic laws. Furthermore, we are able, contingent on the validity of Artin's conjecture, to set up a limiting process according to which sequences of Welch Costas arrays converge to smooth Costas bijections over the reals. The situation over the rationals is different: there, we propose an algorithm of great generality and flexibility for the construction of a Costas fractal bijection. Its success, though, relies heavily on the enumerability of the rationals, and therefore it cannot be generalized over the reals in an obvious way.


A geometric invariant theory construction of moduli spaces of stable maps

Abstract: We construct the moduli spaces of stable maps, \bar M_g,n(P^r,d), via geometric invariant theory (GIT). This construction is only valid over Spec C, but a special case is a GIT presentation of the moduli space of stable curves of genus g with n marked points, \bar M_g,n; this is valid over Spec Z. Our method follows that used in the case n=0 by Gieseker to construct \bar M_g, though our proof that the semistable set is nonempty is entirely different.


Fixed point theorem for discontinuous mappings on PN spaces

Abstract: We present a study on strong t-continuity and measure of discontinuity on PN spaces. As an application, we prove a fixed point theorem for a self mapping on PN spaces by means of measure of discontinuity.


On (g, \phi)-contraction in fuzzy metric spaces

Abstract: In this paper, we give a generalization of Hicks type contractions and Golet type contractions on fuzzy metric spaces. We prove some fixed point theorems for this new type contractions mappings on fuzzy metric spaces.


Wall-Crossing Morphisms in Khovanov-Rozansky Homology

Abstract: We define a wall-crossing morphism for Khovanov-Rozansky homology; that is, a map between the KR homology of knots related by a crossing change. Using this map, we extend KR homology to an invariant of singular knots categorifying the Vasilliev derivative of the HOMFLY polynomial, and of $\mathfrak{sl}_n$ quantum invariants.


A universal property of the monoidal 2-category of cospans of finite linear orders and surjections

Abstract: We prove that the monoidal 2-category of cospans of finite linear orders and surjections is the universal monoidal category with an object X with a semigroup and a cosemigroup structures, where the two structures satisfy a certain 2-dimensional separable algebra condition.


A universal enveloping for L-infinity algebras

Abstract: For any L-infinity algebra L, we construct an A-infinity structure on the space of symmetric tensors Sym*(L), which generalizes the classical universal enveloping for Lie algebras. Our construction is based on an invariant homotopy on a cobar construction of the symmetric coalgebra, which is obtained through its relation with permutahedra and Young tableaux.


BGG correspondence for toric complete intersections

Abstract: We generalize the classical Bernstein-Gelfand-Gelfand correspondence to complete intersections in toric varieties.


Controlling for individual heterogeneity in longitudinal models, with applications to student achievement

Abstract: Longitudinal data tracking repeated measurements on individuals are highly valued for research because they offer controls for unmeasured individual heterogeneity that might otherwise bias results. Random effects or mixed models approaches, which treat individual heterogeneity as part of the model error term and use generalized least squares to estimate model parameters, are often criticized because correlation between unobserved individual effects and other model variables can lead to biased and inconsistent parameter estimates. Starting with an examination of the relationship between random effects and fixed effects estimators in the standard unobserved effects model, this article demonstrates through analysis and simulation that the mixed model approach has a ``bias compression'' property under a general model for individual heterogeneity that can mitigate bias due to uncontrolled differences among individuals. The general model is motivated by the complexities of longitudinal student achievement measures, but the results have broad applicability to longitudinal modeling.


Rate of Convergence of Space Time Approximations for stochastic evolution equations

Abstract: Stochastic evolution equations in Banach spaces with unbounded nonlinear drift and diffusion operators driven by a finite dimensional Brownian motion are considered. Under some regularity condition assumed for the solution, the rate of convergence of various numerical approximations are estimated under strong monotonicity and Lipschitz conditions. The abstract setting involves general consistency conditions and is then applied to a class of quasilinear stochastic PDEs of parabolic type.


The absolute order on the symmetric group, constructible partially ordered sets and Cohen-Macaulay complexes

Abstract: The absolute order is a natural partial order on a Coxeter group W. It can be viewed as an analogue of the weak order on W in which the role of the generating set of simple reflections in W is played by the set of all reflections in W. By use of a notion of constructibility for partially ordered sets, it is proved that the absolute order on the symmetric group is homotopy Cohen-Macaulay. This answers in part a question raised by V. Reiner and the first author. The Euler characteristic of the order complex of the proper part of the absolute order on the symmetric group is also computed.


Jordan structures and non-associative geometry

Abstract: In this survey paper we give an overview over constructions of geometries associated to Jordan structures (algebras, triple systems and pairs), featuring analogs of these constructions with the Lie functor on the one hand and with the approach of non-commutative geometry on the other hand.


Absolute continuity of the measures of the Dunkl intetwining operator and its dualand applications

Abstract: In this paper we consider the representing measures $\mu_{x},x\in \QTR{Bbb}{R}^{d}$, and $\nu_{y},y\in \QTR{Bbb}{R}^{d}$, of the Dunkl intertwining operator and of its dual. When the multiplicity function is positive, we prove that for all $x\in \QTR{Bbb}{R}_{\QTO{mbox}{reg}}^{d}$ we have $d\mu_{x}(y)=\QTR{cal}{K}(x,y)dy$ and for almost all $y\in \QTR{Bbb}{R}^{d}$ we have $d\nu_{y}(x)=\QTR{cal}{K}(x,y)\omega_{k}(x)dx,$ where $\QTR{cal}{K}(x,.)$ is a positive integrable function on $\QTR{Bbb}{R}^{d}$ with support in $\{y\in \QTR{Bbb}{R}^{d}/\Vert y\Vert \leq \Vert x\Vert \}$ and the function $\QTR{cal}{K}(.,y)$ is locally integrable on $\QTR{Bbb}{R}^{d}$ with respect to the measure $\omega_{k}(x)dx$ and with support in $\{x\in \QTR{Bbb}{R}^{d}/\Vert x\Vert \geq \Vert y\Vert \}$. Next we present some applications of this result.


Sensitivity of principal Hessian direction analysis

Abstract: We provide sensitivity comparisons for two competing versions of the dimension reduction method principal Hessian directions (pHd). These comparisons consider the effects of small perturbations on the estimation of the dimension reduction subspace via the influence function. We show that the two versions of pHd can behave completely differently in the presence of certain observational types. Our results also provide evidence that outliers in the traditional sense may or may not be highly influential in practice. Since influential observations may lurk within otherwise typical data, we consider the influence function in the empirical setting for the efficient detection of influential observations in practice.


Generalized Banach contraction in probabilistic metric/normed spaces

Abstract: In this paper, we present the generalization of B-contraction and C-contraction due to Sehgal and Hicks respectively. We also study some properties of C-contraction in probabilistic metric space.


Regularization by free additive convolution, square and rectangular cases

Abstract: The free convolution is the binary operation on the set of probability measures on the real line which allows to deduce, from the individual spectral distributions, the spectral distribution of a sum of independent unitarily invariant square random matrices or of a sum of free operators in a non commutative probability space. In the same way, the rectangular free convolution allows to deduce, from the individual singular distributions, the singular distribution of a sum of independent unitarily invariant rectangular random matrices. In this paper, we consider the regularization properties of these free convolutions on the whole real line. More specifically, we try to find continuous semigroups $(\mu_t)$ of probability measures such that $\mu_0$ is the Dirac mass at zero and such that for all positive $t$ and all probability measure $\nu$, the free convolution of $\mu_t$ with $\nu$ (or, in the rectangular context, the rectangular free convolution of $\mu_t$ with $\nu$) is absolutely continuous with respect to the Lebesgue measure, with a positive analytic density on the whole real line. In the square case, we prove that in semigroups satisfying this property, no measure can have a finite second moment, and we give a sufficient condition on semigroups to satisfy this property, with examples. In the rectangular case, we prove that in most cases, for $\mu$ in a continuous rectangular-convolution-semigroup, the rectangular convolution of $\mu$ with $\nu$ either has an atom at the origin or doesn't put any mass in a neighborhood of the origin, thus the expected property does not hold. However, we give sufficient conditions for analyticity of the density of the rectangular convolution of $\mu$ with $\nu$ except on a negligible set of points, as well as existence and continuity of a density everywhere.


Inverse Conductivity Problem for a Parabolic Equation using a Carlemen Estimate with one Observation

Abstract: For the heat equation in a bounded domain we give a stability result for a smooth diffusion coefficient. The key ingredients are a global Carleman-type estimate, a Poincar\'e-type estimate and an energy estimate with a single observation acting on a part of the boundary.


3-quasi-Sasakian manifolds

Abstract: In the present paper we carry on a systematic study of 3-quasi-Sasakian manifolds. In particular we prove that the three Reeb vector fields generate an involutive distribution determining a canonical totally geodesic and Riemannian foliation. Locally, the leaves of this foliation turn out to be Lie groups: either the orthogonal group or an abelian one. We show that 3-quasi-Sasakian manifolds have a well-defined rank, obtaining a rank-based classification. Furthermore, we prove a splitting theorem for these manifolds assuming the integrability of one of the almost product structures. Finally, we show that the vertical distribution is a minimum of the corrected energy.


Representations of tame quivers and affine canonical bases

Abstract: An integral PBW-basis of type $A_1^{(1)}$ has been constructed by Zhang [Z] and Chen [C] using the Auslander-Reiten quiver of the Kronecker quiver. We associate a geometric order to elements in this basis following an idea of Lusztig [L1] in the case of finite type. This leads to an algebraic realization of a bar-invariant basis of $\uq2$. For any affine symmetric type, we obtain an integral PBW-basis of the generic composition algebra, by using an algebraic construction of the integral basis for a tube in [DDX], an embedding of the module category of the Kronecker quiver into the module category of the tame quiver, and a list of the root vectors of indecomposable modules according to the preprojective, regular, and preinjective components of the Auslander-Reiten quiver of the tame quiver. When the basis elements are ordered to be compatible with the geometric order given by the dimensions of the orbit varieties and the extension varieties, we can show that the transition matrix between the PBW-basis and a monomial basis is triangular with diagonal entries equal to 1. Therefore we obtain a bar-invariant basis. By a orthogonalization for the PBW-basis with the inner product, we finally give an algebraic way to realize the canonical bases of the quantized enveloping algebras of all symmetric affine Kac-Moody Lie algebras.


Rational points on certain hyperelliptic curves over finite fields

Abstract: Let $K$ be a field, $a, b\in K$ and $ab\neq 0$. Let us consider the polynomials $g_{1}(x)=x^n+ax+b, g_{2}(x)=x^n+ax^2+bx$, where $n$ is a fixed positive integer. In this paper we show that for each $k\geq 2$ the hypersurface given by the equation \begin{equation*} S_{k}^{i}: u^2=\prod_{j=1}^{k}g_{i}(x_{j}),\quad i=1, 2. \end{equation*} contains a rational curve. Using the above and Woestijne's recent results \cite{Woe} we show how one can construct a rational point different from the point at infinity on the curves $C_{i}:y^2=g_{i}(x), (i=1, 2)$ defined over a finite field, in polynomial time.


The absolute Galois group acts faithfully on the connected components of the moduli space of surfaces of general type

Abstract: We show that the Galois group $Gal(\bar{\Q} /\Q)$ operates faithfully on the set of connected components of the moduli spaces of surfaces of general type, and also that for each element $\sigma \in Gal(\bar{\Q} /\Q)$ different from the identity and from complex conjugation, there is a surface of general type such that $X$ and the Galois conjugate variety $X^{\sigma}$ have nonisomorphic fundamental groups. The result was announced by the second author at the Alghero Conference 'Topology of algebraic varieties' in september 2006. Before the present paper was actually written, we received a very interesting preprint by Robert Easton and Ravi Vakil (\cite{e-v}), where it is proven, with a completely different type of examples, that the Galois group $Gal(\bar{\Q} /\Q)$ operates faithfully on the set of irreducible components of the moduli spaces of surfaces of general type. We also give other simpler examples of surfaces with nonisomorphic fundamental groups which are Galois conjugate, hence have isomorphic algebraic fundamental groups.


Singular unitarity in "quantization commutes with reduction"

Abstract: Let $M$ be a connected compact quantizable K\"ahler manifold equipped with a Hamiltonian action of a connected compact Lie group $G$. Let $M//G=\phi^{-1}(0)/G=M_0$ be the symplectic quotient at value 0 of the moment map $\phi$. The space $M_0$ may in general not be smooth. It is known that, as vector spaces, there is a natural isomorphism between the quantum Hilbert space over $M_0$ and the $G$-invariant subspace of the quantum Hilbert space over $M$. In this paper, without any regularity assumption on the quotient $M_0$, we discuss the relation between the inner products of these two quantum Hilbert spaces under the above natural isomorphism; we establish asymptotic unitarity to leading order in Planck's constant of a modified map of the above isomorphism under a ``metaplectic correction'' of the two quantum Hilbert spaces.


On Lesieur's Measured Quantum Groupoids

Abstract: In his thesis ([L1]), which is published in an expended and revised version ([L2]), Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras, using intensively the notion of pseudo-multiplicative unitary, which had been introduced in a previous article of the author, in collaboration with Jean-Michel Vallin [EV]. In [L2], the axioms given are very complicated and are here simplified.


Jacobi-Nijenhuis algebroids and their modular classes

Abstract: Jacobi-Nijenhuis algebroids are defined as a natural generalization of Poisson-Nijenhuis algebroids, in the case where there exists a Nijenhuis operator on a Jacobi algebroid which is compatible with it. We study modular classes of Jacobi and Jacobi-Nijenhuis algebroids.


Some necessary and sufficient conditions for parastrophic invariance of the associative law in quasigroups

Abstract: Every quasigroup $(S,\cdot)$ belongs to a set of 6 quasigroups, called parastrophes denoted by $(S,\pi_i)$, $i\in \{1,2,3,4,5,6\}$. It is shown that isotopy-isomorphy is a necessary and sufficient condition for any two distinct quasigroups $(S,\pi_i)$ and $(S,\pi_j)$, $i,j\in \{1,2,3,4,5,6\}$ to be parastrophic invariant relative to the associative law. In addition, a necessary and sufficient condition for any two distinct quasigroups $(S,\pi_i)$ and $(S,\pi_j)$, $i,j\in \{1,2,3,4,5,6\}$ to be parastrophic invariance under the associative law is either if the $\pi_i$-parastrophe of $H$ is equivalent to the $\pi_i$-parastrophe of the holomorph of the $\pi_i$-parastrophe of $S$ or if the $\pi_i$-parastrophe of $H$ is equivalent to the $\pi_k$-parastrophe of the $\pi_i$-parastrophe of the holomorph of the $\pi_i$-parastrophe of $S$, for a particular $k\in \{1,2,3,4,5,6\}$.


Weak Inverse Property Loops and Some Isotopy-Isomorphy Properties

Abstract: Two distinct isotopy-isomorphy conditions, different from those of J. M. Osborn and Wilson's condition, for a weak inverse property loop(WIPL) are shown. Only one of them characterizes isotopy-isomorphy in WIPLs while the other is just a sufficient condition for isotopy-isomorphy. Under the sufficient condition called the ${\cal T}$ condition, Artzy's result that isotopic cross inverse property loops(CIPL) are isomorphic is proved for WIP loops.


On a Pair of Universal Weak Inverse Property Loops

Abstract: A new condition called ${\cal T}$ condition is introduced for the first time and used to study a pair of isotopic loops. Under this condition, a loop in the pair is a WIPL if and only if the other loop is a WIPL. Furthermore, such WIPLs are isomorphic. The translation elements $f$ and $g$ of a CIPL with the ${\cal T}$ condition(such that its $f,g$-isotope is an automorphic inverse property loop) are found to be alternative, flexible, centrum and equal elements. A necessary and sufficient condition for a pair WIPLs with a weak ${\cal T}$ condition to be isomorphic is shown. A CIPL and an isomorph are observed to have this weak ${\cal T}$ condition.


Unitary Representations of Wavelet Groups and Encoding of Iterated Function Systems in Solenoids

Abstract: For points in $d$ real dimensions, we introduce a geometry for general digit sets. We introduce a positional number system where the basis for our representation is a fixed $d$ by $d$ matrix over $\bz$. Our starting point is a given pair $(A, \mathcal D)$ with the matrix $A$ assumed expansive, and $\mathcal D$ a chosen complete digit set, i.e., in bijective correspondence with the points in $\bz^d/A^T\bz^d$. We give an explicit geometric representation and encoding with infinite words in letters from $\mathcal D$. We show that the attractor $X(A^T,\mathcal D)$ for an affine Iterated Function System (IFS) based on $(A,\mathcal D)$ is a set of fractions for our digital representation of points in $\br^d$. Moreover our positional "number representation" is spelled out in the form of an explicit IFS-encoding of a compact solenoid $\sa$ associated with the pair $(A,\mathcal D)$. The intricate part (Theorem \ref{thenccycl}) is played by the cycles in $\bz^d$ for the initial $(A,\mathcal D)$-IFS. Using these cycles we are able to write down formulas for the two maps which do the encoding as well as the decoding in our positional $\mathcal D$-representation. We show how some wavelet representations can be realized on the solenoid, and on symbolic spaces.


Frames of subspaces and operators

Abstract: We study the relationship between operators, orthonormal basis of subspaces and frames of subspaces (also called fusion frames) for a separable Hilbert space $\mathcal{H}$. We get sufficient conditions on an orthonormal basis of subspaces $\mathcal{E} = \{E_i \}_{i\in I}$ of a Hilbert space $\mathcal{K}$ and a surjective $T\in L(\mathcal{K}, \mathcal{H})$ in order that $\{T(E_i)\}_{i\in I}$ is a frame of subspaces with respect to a computable sequence of weights. We also obtain generalizations of results in [J. A. Antezana, G. Corach, M. Ruiz and D. Stojanoff, Oblique projections and frames. Proc. Amer. Math. Soc. 134 (2006), 1031-1037], which related frames of subspaces (including the computation of their weights) and oblique projections. The notion of refinament of a fusion frame is defined and used to obtain results about the excess of such frames. We study the set of admissible weights for a generating sequence of subspaces. Several examples are given.


Bootstrapping confidence intervals for the change-point of time series

Abstract: We study an AMOC time series model with an abrupt change in the mean and dependent errors that fulfill certain mixing conditions. We obtain confidence intervals for the unknown change-point via bootstrapping methods. Precisely we use a block bootstrap of the estimated centered error sequence. Then we reconstruct a sequence with a change in the mean using the same estimators as before. The difference between the change-point estimator of the resampled sequence and the one for the original sequence can be use as an approximation of the difference between the real change-point and its estimator. This enables us to construct confidence intervals using the empirical distribution of the resampled time series. A simulation study shows that the resampled confidence intervals are usually closer to their target levels and at the same time smaller than the asymptotic intervals.


Power-free values, repulsion between points, differing beliefs and the existence of error

Abstract: Let f be a cubic polynomial. Then there are infinitely many primes p such that f(p) is square-free.


On the derived category of 1-motives, I

Abstract: We consider the category of Deligne 1-motives over a perfect field k of exponential characteristic p and its derived category for a suitable exact structure after inverting p. As a first result, we provide a fully faithful embedding into an etale version of Voevodsky's triangulated category of geometric motives. Our second main result is that this full embedding "almost" has a left adjoint, that we call \LAlb. Applied to the motive of a variety we thus get a bounded complex of 1-motives, that we compute fully for smooth varieties and partly for singular varieties. As an application we give motivic proofs of Roitman type theorems (in characteristic 0).


Elementary Maps on Triangular Algebras

Abstract: In this note we prove that elementary maps on triangular algebras are automically additive.


Additivity of Maps on Triangular Algebras

Abstract: We prove that every multiplicative bijective map, Jordan bijective map, and Jordan triple bijective map from a triangular algebra onto any ring is automatically additive.


The law of the supremum of a stable L\'{e}vy process with no negative jumps

Abstract: Let $X=(X_t)_{t\ge0}$ be a stable L\'{e}vy process of index $\alpha \in(1,2)$ with no negative jumps and let $S_t=\sup_{0\le s\le t}X_s$ denote its running supremum for $t>0$. We show that the density function $f_t$ of $S_t$ can be characterized as the unique solution to a weakly singular Volterra integral equation of the first kind or, equivalently, as the unique solution to a first-order Riemann--Liouville fractional differential equation satisfying a boundary condition at zero. This yields an explicit series representation for $f_t$. Recalling the familiar relation between $S_t$ and the first entry time $\tau_x$ of $X$ into $[x,\infty)$, this further translates into an explicit series representation for the density function of $\tau_x$.


Local stability of ergodic averages

Abstract: The mean ergodic theorem is equivalent to the assertion that for every function K and every epsilon, there is an n with the property that the ergodic averages A_m f are stable to within epsilon on the interval [n,K(n)]. We show that even though it is not generally possible to compute a bound on the rate of convergence of a sequence of ergodic averages, one can give explicit bounds on n in terms of K and || f || / epsilon. This tells us how far one has to search to find an n so that the ergodic averages are "locally stable" on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov. Finally, we explain how our positive results can be viewed as an application of a body of general proof-theoretic methods falling under the heading of "proof mining."


On dynamical bit sequences

Abstract: Let X^{(k)}(t) = (X_1(t), ..., X_k(t)) denote a k-vector of i.i.d. random variables, each taking the values 1 or 0 with respective probabilities p and 1-p. As a process indexed by non-negative t, $X^{(k)}(t)$ is constructed--following Benjamini, Haggstrom, Peres, and Steif (2003)--so that it is strong Markov with invariant measure ((1-p)\delta_0+p\delta_1)^k. We derive sharp estimates for the probability that ``X_1(t)+...+X_k(t)=k-\ell for some t in F,'' where F \subset [0,1] is nonrandom and compact. We do this in two very different settings: (i) Where \ell is a constant; and (ii) Where \ell=k/2, k is even, and p=q=1/2. We prove that the probability is described by the Kolmogorov capacitance of F for case (i) and Howroyd's 1/2-dimensional box-dimension profiles for case (ii). We also present sample-path consequences, and a connection to capacities that answers a question of Benjamini et. al. (2003)


Thom polynomials for maps of curves with isolated singularities

Abstract: Thom (residual) polynomials in characteristic classes are used in the analysis of geometry of functional spaces. They serve as a tool in description of classes Poincar\'e dual to subvarieties of functions of prescribed types. We give explicit universal expressions for residual polynomials in spaces of functions on complex curves having isolated singularities and multisingularities, in terms of few characteristic classes. These expressions lead to a partial explicit description of a stratification of Hurwitz spaces.


Helix, shadow boundary and minimal submanifolds

Abstract: Inspired by a Blaschke's work about analytic convex surfaces, we study {\em shadow boundaries} of Riemannian submanifolds $M$, which are defined by a parallel vector field along $M$. Since a shadow boundary is just a closed subset of $M$, first, we will give a condition that guarantee its smoothness. It depends on the second fundamental form of the submanifold. It is natural to search for what kind of properties might have such submanifolds of $M$? Could they be totally geodesic or minimal? Answers to these and related questions are given in this work.


On Colorings of Squares of Outerplanar Graphs

Abstract: We study vertex colorings of the square $G^2$ of an outerplanar graph $G$. We find the optimal bound of the inductiveness, chromatic number and the clique number of $G^2$ as a function of the maximum degree $\Delta$ of $G$ for all $\Delta\in \nats$. As a bonus, we obtain the optimal bound of the choosability (or the list-chromatic number) of $G^2$ when $\Delta \geq 7$. In the case of chordal outerplanar graphs, we classify exactly which graphs have parameters exceeding the absolute minimum.


On multipartite posets

Abstract: A poset $\mathbf{P} = (X,\preceq)$ is {\em $m$-partite} if $X$ has a partition $X = X_1 \cup ... \cup X_m$ such that (1) each $X_i$ forms an antichain in $\mathbf{P}$, and (2) $x\prec y$ implies $x\in X_i$ and $y\in X_j$ where $i<j$. In this article we derive a tight asymptotic upper bound on the order dimension of $m$-partite posets in terms of $m$ and their bipartite sub-posets in a constructive and elementary way.


Randomly coloring planar graphs with fewer colors than the maximum degree

Abstract: We study Markov chains for randomly sampling $k$-colorings of a graph with maximum degree $\Delta$. Our main result is a polynomial upper bound on the mixing time of the single-site update chain known as the Glauber dynamics for planar graphs when $k=\Omega(\Delta/\log{\Delta})$. Our results can be partially extended to the more general case where the maximum eigenvalue of the adjacency matrix of the graph is at most $\Delta^{1-\eps}$, for fixed $\eps > 0$. The main challenge when $k \le \Delta + 1$ is the possibility of "frozen" vertices, that is, vertices for which only one color is possible, conditioned on the colors of its neighbors. Indeed, when $\Delta = O(1)$, even a typical coloring can have a constant fraction of the vertices frozen. Our proofs rely on recent advances in techniques for bounding mixing time using "local uniformity" properties.


Vertex coloring acyclic digraphs and their corresponding hypergraphs

Abstract: We consider vertex coloring of an acyclic digraph $\Gdag$ in such a way that two vertices which have a common ancestor in $\Gdag$ receive distinct colors. Such colorings arise in a natural way when bounding space for various genetic data for efficient analysis. We discuss the corresponding {\em down-chromatic number} and derive an upper bound as a function of $D(\Gdag)$, the maximum number of descendants of a given vertex, and the degeneracy of the corresponding hypergraph. Finally we determine an asymptotically tight upper bound of the down-chromatic number in terms of the number of vertices of $\Gdag$ and $D(\Gdag)$.


On the dynamics of codimension one holomorphic foliations with ample normal bundle

Abstract: We investigate the accumulation to singular points of leaves of codimension one foliations whose normal bundle is ample, with emphasis on the nonexistence of Levi-flat hypersurfaces.


Murre's conjectures and explicit Chow--Kuenneth projectors for some varieties

Abstract: In this paper, we investigate Murre's conjectures on the structure of rational Chow groups and exhibit explicit Chow--Kuenneth projectors for some examples. More precisely, the examples we study are the varieties which have a nef tangent bundle. For surfaces and threefolds which have a nef tangent bundle explicit Chow--Kuenneth projectors are obtained which satisfy Murre's conjectures and the motivic Hard Lefschetz theorem is verified.


Equivalence of sparse circulants: the bipartite \'Ad\'am problem

Abstract: We consider n-by-n circulant matrices having entries 0 and 1. Such matrices can be identified with sets of residues mod n, corresponding to the columns in which the top row contains an entry 1. Let A and B be two such matrices, and suppose that the corresponding residue sets S_A and S_B have size at most 3. We prove that the following are equivalent: (1) there are integers u,v mod n, with u a unit, such that S_A = uS_B + v; (2) there are permutation matrices P,Q such that A=PBQ. Our proof relies on some new results about vanishing sums of roots of unity. We give examples showing this result is not always true for denser circulants, as well as results showing it continues to hold in some situations. We also explain how our problem relates to the Adam problem on isomorphisms of circulant directed graphs.


Integrable systems and complex geometry

Abstract: In this paper, we discuss an interaction between complex geometry and integrable systems. Section 1 reviews the classical results on integrable systems. New examples of integrable systems, which have been discovered, are based on the Lax representation of the equations of motion. These systems can be realized as straight line motions on a Jacobi variety of a so-called spectral curve. In section 2, we study a Lie algebra theoretical method leading to integrable systems and we apply the method to several problems. In section 3, we discuss the concept of the algebraic complete integrability (a.c.i.) of hamiltonian systems. Algebraic integrability means that the system is completely integrable in the sens of the phase space being folited by tori, which in addition are real parts of a complex algebraic tori (abelian varieties). The method is devoted to illustrate how to decide about the a.c.i. of hamiltonian systems and is applied to some examples. Finally, in section 4 we study an a.c.i. in the generalized sense which appears as covering of a.c.i. system. The manifold invariant by the complex flow is covering of abelian variety.


Examples of Non-Rigid CAT(0) Groups from the Category of Knot Groups

Abstract: C Croke and B Kleiner have constructed an example of a CAT(0) group with more than one visual boundary. J Wilson has proven that this same group has uncountably many distinct boundaries. In this article we prove that the knot group of any connected sum of two non-trivial torus knots also has uncountably many distinct CAT(0) boundaries.


A method for the resolution of the Jacobi equation Y''+RY = 0 on the manifold Sp(2)/SU(2)

Abstract: In this paper a method for the resolution of the differential equation of the Jacobi vector fields in the manifold V1 = Sp(2)/SU(2) is exposed. These results are applied to determine areas and volumes of geodesic spheres and balls.


Kolmogorov condition near hyperbolic singularities of integrable Hamiltonian systems

Abstract: In this paper we show that, if an integrable Hamiltonian system admits a nondegenerate hyperbolic singularity then it will satisfy the Kolmogorov condegeneracy condition near that singularity (under a mild additional condition, which is trivial if the singularity contains a fixed point)


A uniform Sobolev inequality under Ricci flow

Abstract: Let ${\bf M}$ be a compact Riemannian manifold and the metrics $g=g(t)$ evolve by the Ricci flow. We prove the following result. The Sobolev imbedding by Aubin or Hebey, perturbed by a scalar curvature term and modulo sharpness of constants, holds uniformly for $({\bf M}, g(t))$ for all time if the Ricci flow exists for all time; and if the Ricci flow develops a singularity in finite time, then the same Sobolev imbedding holds uniformly after a standard normalization. As a consequence, long time non-collapsing results are derived, which improve Perelman's local non-collapsing results. An application to 3-d Ricci flow with surgery is also presented.


The vanishing of the contact invariant in the presence of torsion

Abstract: We prove that the Ozsvath-Szabo contact invariant of a closed contact 3-manifold with positive Giroux torsion vanishes.


Partitions with independent iterates in random dynamical systems

Abstract: Consider an invertible measure-preserving transformation of a probability space. A finite partition of the space is called weakly independent if there are infinitely many images of this partition under powers of the transformation that are jointly independent. Krengel proved that a transformation is weakly mixing if and only if weakly independent partitions of the underlying space are dense among all finite partitions. Using the tools developed in the later papers of del Junco-Reinhold-Weiss and del Junco-Begun we obtain Krengel- type results for weakly mixing random dynamical systems (or equivalently, skew products that are relatively weakly mixing).


Effective equidistribution of S-integral points on symmetric varieties

Abstract: Let K be a global field of characteristic not 2. Let Z be a symmetric variety defined over K and S a finite set of places of K. We obtain counting and equidistribution results for the S-integral points of Z. Our results are effective when K is a number field.


Small values of Lusternik-Schnirelmann and systolic categories for manifolds

Abstract: We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna, by generalizing their result in dimension 3, to all higher dimensions. We examine its ramifications in systolic topology, and provide a sufficient condition for ensuring a lower bound of 3 for systolic category.


Critical Delays and Polynomial Eigenvalue Problems

Abstract: In this work we present a new method to compute the delays of delay differential equations (DDEs), such that the DDE has a purely imaginary eigenvalue. For delay differential equations with multiple delays, the critical curves or critical surfaces in delay space (that is, the set of delays where the DDE has a purely imaginary eigenvalue) are parameterized. We show how the method is related to other works in the field by treating the case where the delays are integer multiples of some delay value, i.e., commensurate delays. The parametrization is done by solving a {\em quadratic eigenvalue problem} which is constructed from the vectorization of a matrix equation and hence typically of large size. For commensurate delay differential equations, the corresponding equation is a polynomial eigenvalue problem. As a special case of the proposed method, we find a closed form for a parameterization of the critical surface for the scalar case. We provide several examples with visualizations where the computation is done with some exploitation of the structure of eigenvalue problems.


t-Wise Independence with Local Dependencies

Abstract: In this note we prove a large deviation bound on the sum of random variables with the following dependency structure: there is a dependency graph $G$ with a bounded chromatic number, in which each vertex represents a random variable. Variables that are represented by neighboring vertices may be arbitrarily dependent, but collections of variables that form an independent set in $G$ are $t$-wise independent.


Right and Left Joint System Representation of a Rational Matrix Function in General Position (System Representation Theory for Dummies)

Abstract: For a rational matrix function R of one variable in general position, the matrix functions R(x)/R(y) and R(y)\R(x) of two variables are considered. For these matrix functions of two variables, representations which are analogous to the system representations (or realizations) of a rational matrix function of one variable are constructed. This representation is called the joint right [the joint left] system representation.


Pascal's triangle and word bases for blob algebra ideals

Abstract: We give an elementary proof of isomorphism of the blob (diagram) algebra and the corresponding extended Temperley-Lieb algebra (defined by presentation).


Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies

Abstract: Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy 0 if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy is 0 whenever the repetitivity function satisfies a certain growth restriction.


On fundamental groups related to the Hirzebruch surface F_1

Abstract: Given a projective surface and a generic projection to the plane, the braid monodromy factorization (and thus, the braid monodromy type) of the complement of its branch curve is one of the most important topological invariants, stable on deformations. From this factorization, one can compute the fundamental group of the complement of the branch curve, either in C^2 or in CP^2. In this article, we show that these groups, for the Hirzebruch surface F_{1,(a,b)}, are almost-solvable. That is - they are an extension of a solvable group, which strengthen the conjecture on degeneratable surfaces.


A classification of CO spaces which are continuous images of compact ordered spaces

Abstract: A compact Hausdorff space X is called a CO space, if every closed subset of X is homeomorphic to an open subset of X. Every successor ordinal with its order topology is a CO space. We find an explicit characterization of the class K of CO spaces which are a continuous image of a Dedkind complete totally ordered set. (The topology of a totally ordered set is taken to be its order topology). We show that every member of K can be described as a finite disjoint sum of very simple spaces. Every summand has either form: (1) mu + 1 + nu^*, where mu and nu are cardinals, and nu^* is the reverse order of nu; or (2) the summand is the 1-point-compactification of a discrete space with cardinality aleph_1.


Continuation of connecting orbits in 3D-ODEs: (I) Point-to-cycle connections

Abstract: We propose new methods for the numerical continuation of point-to-cycle connecting orbits in 3-dimensional autonomous ODE's using projection boundary conditions. In our approach, the projection boundary conditions near the cycle are formulated using an eigenfunction of the associated adjoint variational equation, avoiding costly and numerically unstable computations of the monodromy matrix. The equations for the eigenfunction are included in the defining boundary-value problem, allowing a straightforward implementation in AUTO, in which only the standard features of the software are employed. Homotopy methods to find connecting orbits are discussed in general and illustrated with several examples, including the Lorenz equations. Complete AUTO demos, which can be easily adapted to any autonomous 3-dimensional ODE system, are freely available.


The Explicit Chaotic Representation of the powers of increments of Levy Processes

Abstract: An explicit formula for the chaotic representation of the powers of increments, (X_{t+t_0}-X_{t_0})^n, of a Levy process is presented. There are two different chaos expansions of a square integrable functional of a Levy process: one with respect to the compensated Poisson random measure and the other with respect to the orthogonal compensated powers of the jumps of the Levy process. Computationally explicit formulae for both of these chaos expansions of (X_{t+t_0}-X_{t_0})^n are given in this paper. Simulation results verify that the representation is satisfactory. The CRP of a number of financial derivatives can be found by expressing them in terms of (X_{t+t_0}-X_{t_0})^n using Taylor's expansion.


Constraints, MMSNP and expander relational structures

Abstract: We give a poly-time construction for a combinatorial classic known as Sparse Incomparability Lemma, studied by Erdos, Lovasz, Nesetril, Rodl and others: We show that every Constraint Satisfaction Problem is poly-time equivalent to its restriction to structures with large girth. This implies that the complexity classes CSP and Monotone Monadic Strict NP introduced by Feder and Vardi are computationally equivalent. The technical novelty of the paper is a concept of expander relations and a new type of product for relational structures: a generalization of the zig-zag product, the twisted product.


Forbidden lists (NP and CSP for combinatorialists)

Abstract: We present a definition of the class NP in combinatorial context as the set of languages of structures defined by finitely many forbidden lifted substructures. We apply this to special syntactically defined subclasses and show how they correspond to naturally defined (and intensively studied) combinatorial problems. We show that some types of combinatorial problems like edge colorings and graph decompositions express the full computational power of the class NP. We then characterize Constraint Satisfaction Problems (i.e. H-coloring problems) which are expressible by finitely many forbidden lifted substructures. This greatly simplifies and generalizes the earlier attempts to characterize this problem. As a corollary of this approach we perhaps find a proper setting of Feder and Vardi analysis of CSP languages within the class MMSNP.


Correction of paper published in J. Combinatorial Theory 21, 1976: On the Existence of Hadamard Matrices

Abstract: In the paper On the Existence of Hadamard Matrices in J. Combinatorial Theory 21, 1976, it is shown that for a natural number q > 3, we can construct an Hadamard Matrix of order 2^s q for s \geq t where t = [2 log_2(q-3)]. I will show that this bound is not a consequence of the proof given in the paper and explain the error in the argumentation.


The two possible values of the chromatic number of a random graph

Abstract: Given d \in (0,infty) let k_d be the smallest integer k such that d < 2k\log k. We prove that the chromatic number of a random graph G(n,d/n) is either k_d or k_d+1 almost surely.


Lambda Mu Calculus and Duality: Call-by-Name and Call-by-Value

Abstract: Under the extension of Curry-Howard's correspondence to classical logic, Gentzen's NK and LK systems can be seen as syntax-directed systems of simple types respectively for Parigot's Lambda Mu Calculus and Curien-Herbelin's Lambda Bar Mu Mu Tidle Calculus. We aim at showing their computational equivalence. We define translations between these calculi. We prove simulation theorems for an undirected evaluation as well as for call-by-name and call-by-value evaluations.


Global well-posedness for dissipative Korteweg-de Vries equations

Abstract: This paper is devoted to the well-posedness for dissipative KdV equations $u_t+u_{xxx}+|D_x|^{2\alpha}u+uu_x=0$, $0<\alpha\leq 1$. An optimal bilinear estimate is obtained in Bourgain's type spaces, which provides global well-posedness in $H^s(\R)$, $s>-3/4$ for $\alpha\leq1/2$ and $s>-3/(5-2\alpha)$ for $\alpha>1/2$.


Permutations with Extremal number of Fixed Points

Abstract: We extend Stanley's work on alternating permutations with extremal number of fixed points in two directions: first, alternating permutations are replaced by permutations with a prescribed descent set; second, instead of simply counting permutations we study their generating polynomials by number of excedances. Several techniques are used: Desarmenien's desarrangement combinatorics, Gessel's hook-factorization and the analytical properties of two new permutation statistics "DEZ" and "lec". Explicit formulas for the maximal case are derived by using symmetric function tools.


On path factors of (3,4)-biregular bigraphs

Abstract: A (3,4)-biregular bigraph G is a bipartite graph where all vertices in one part have degree 3 and all vertices in the other part have degree 4. A path factor of G is a spanning subgraph whose components are nontrivial paths. We prove that a simple (3,4)-biregular bigraph always has a path factor such that the endpoints of each path have degree three. Moreover we suggest a polynomial algorithm for the construction of such a path factor.


On a complex differential Riccati equation

Abstract: We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schrodinger equation and enjoys many properties similar to those of the ordinary differential Riccati equation as, e.g., the famous Euler theorems, the Picard theorem and others. Besides these generalizations of the classical "one-dimensional" results we discuss new features of the considered equation like, e.g., an analogue of the Cauchy integral theorem.


Noether Symmetries and Conservations Laws For Non-critical Kohn-Laplace Equations on Three-Dimensional Heisenberg Group

Abstract: We show which Lie point symmetries of non-critical semilinear Kohn-Laplace equations on the Heisenberg group $H^1$ are Noether symmetries and we establish their respectives conservations laws.


On the Linearization of the Painleve' III-VI Equations and Reductions of the Three-Wave Resonant System

Abstract: We extend similarity reductions of the coupled (2+1)-dimensional three-wave resonant interaction system to its Lax pair. Thus we obtain new 3x3 matrix Fuchs--Garnier pairs for the third and fifth Painleve' equations, together with the previously known Fuchs--Garnier pair for the fourth and sixth Painleve' equations. These Fuchs--Garnier pairs have an important feature: they are linear with respect to the spectral parameter. Therefore we can apply the Laplace transform to study these pairs. In this way we found reductions of all pairs to the standard 2x2 matrix Fuchs--Garnier pairs obtained by M. Jimbo and T. Miwa. As an application of the 3x3 matrix pairs, we found an integral auto-transformation for the standard Fuchs--Garnier pair for the fifth Painleve' equation. It generates an Okamoto-like B\"acklund transformation for the fifth Painleve' equation. Another application is an integral transformation relating two different 2x2 matrix Fuchs--Garnier pairs for the third Painleve' equation.


Partial Inertial Manifolds for infinite-dimensional dynamical systems: Example for P.D.E.s with a state-dependent delay

Abstract: We propose a new notion of Partial Inertial Manifold to study the long-time asymptotic behavior of dissipative differential equations. As shown on an example, such manifolds may exist in the cases when the classical Inertial manifold does not exist (or not known to exist).


Concentration of the Spectral Measure for Large Random Matrices with Stable Entries

Abstract: We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these random matrices, such as the largest eigenvalue or the largest singular value.


A discrete mean value of the derivative of the Riemann zeta function

Abstract: In this article we compute a discrete mean value of the derivative of the Riemann zeta function. This mean value will be important for several applications concerning the size of $\zeta'(\rho)$ where $\zeta(s)$ is the Riemann zeta function and $\rho$ is a non-trivial zero of the Riemann zeta function.


Extreme values of zeta prime rho

Abstract: In this article we exhibit small and large values of $\zeta'(\rho)$ by applying Soundararajan's resonance method. Our results assume the Riemann hypothesis.


The fundamental progroupoid of a general topos

Abstract: It is well known that the category of covering projections (that is, locally constant objects) of a locally connected topos is equivalent to the classifying topos of a strict progroupoid (or, equivalently, a localic prodiscrete groupoid), the \emph{fundamental progroupoid}, and that this progroupoid represents first degree cohomology. In this paper we generalize these results to an arbitrary topos. The fundamental progroupoid is now a localic progroupoid, and can not be replaced by a localic groupoid. The classifying topos in not any more a Galois topos. Not all locally constant objects can be considered as covering projections. The key contribution of this paper is a novel definition of covering projection for a general topos, which coincides with the usual definition when the topos is locally connected. The results in this paper were presented in a talk at the Category Theory Conference, Vancouver July 2004.


The Stokes phenomenon in the confluence of the hypergeometric equation using Riccati equation

Abstract: In this paper we study the confluence of two regular singular points of the hypergeometric equation into an irregular one. We study the consequence of the divergence of solutions at the irregular singular point for the unfolded system. Our study covers a full neighborhood of the origin in the confluence parameter space. In particular, we show how the divergence of solutions at the irregular singular point explains the presence of logarithmic terms in the solutions at a regular singular point of the unfolded system. For this study, we consider values of the confluence parameter taken in two sectors covering the complex plane. In each sector, we study the monodromy of a first integral of a Riccati system related to the hypergeometric equation. Then, on each sector, we include the presence of logarithmic terms into a continuous phenomenon and view a Stokes multiplier related to a 1-summable solution as the limit of an obstruction that prevents a pair of eigenvectors of the monodromy operators, one at each singular point, to coincide.


Coherence and phase synchronization: generalization to pairs of multivariate time series, and removal of zero-lag contributions

Abstract: Coherence and phase synchronization between time series corresponding to different spatial locations are usually interpreted as indicators of the connectivity between locations. In neurophysiology, time series of electric neuronal activity are essential for studying brain interconnectivity. Such signals can either be invasively measured from depth electrodes, or computed from very high time resolution, non-invasive, extracranial recordings of scalp electric potential differences (EEG: electroencephalogram) and magnetic fields (MEG: magnetoencephalogram) by means of a tomography such as sLORETA (standardized low resolution brain electromagnetic tomography). There are two problems in this case. First, in the usual situation of unknown cortical geometry, the estimated signal at each brain location is a vector with three components (i.e. a current density vector), which means that coherence and phase synchronization must be generalized to pairs of multivariate time series. Second, the inherent low spatial resolution of the EEG/MEG tomography introduces artificially high zero-lag coherence and phase synchronization. In this report, solutions to both problems are presented. Two additional generalizations are briefly mentioned: (1) conditional coherence and phase synchronization; and (2) non-stationary time-frequency analysis. Finally, a non-parametric randomization method for connectivity significance testing is outlined. The new connectivity measures proposed here can be applied to pairs of univariate EEG/MEG signals, as is traditional in the published literature. However, these calculations cannot be interpreted as connectivity, since it is in general incorrect to associate an extracranial electrode or sensor to the underlying cortex.


Surfaces with K^2<3\chi and finite fundamental group

Abstract: In this paper we continue the study of algebraic fundamentale group of minimal surfaces of general type S satisfying K_S^2<3\chi(S). We show that, if K_S^2= 3\chi(S)-1 and the algebraic fundamental group of S has order 8, then S is a Campedelli surface. In view of the results of math.AG/0512483 and math.AG/0605733, this implies that the fundamental group of a surface with K^2<3\chi that has no irregular etale cover has order at most 9, and if it has order 8 or 9, then S is a Campedelli surface. To obtain this result we establish some classification results for minimal surfaces of general type such that K^2=3p_g-5 and such that the canonical map is a birational morphism. We also study rational surfaces with a Z_2^3-action.


On existence of log minimal models

Abstract: In this paper, we prove that the log minimal model program in dimension $d-1$ implies the existence of log minimal models for effective lc pairs (eg of nonnegative Kodaira dimension) in dimension $d$. In fact, we prove that the same conclusion follows from a weaker assumption, namely, the log minimal model program with scaling in dimension $d-1$. This enables us to prove that effective lc pairs in dimension five have log minimal models. We also give new proofs of the existence of log minimal models for effective lc pairs in dimension four and the Shokurov reduction theorem. Other applications appear in a paper of Birkar-Paun.


Birational geometry

Abstract: This article contains the notes of a graduate course on birational geometry focusing on the minimal model program. Topics covered include singularities, vanishing, nonvanishing, cone and contraction, base point freeness, finite generation, flips, termination, minimal models and Mori fibre spaces.


Necessary Conditions for Schur-Positivity

Abstract: In recent years, there has been considerable interest in showing that certain conditions on skew shapes A and B are sufficient for the difference s_A - s_B of their skew Schur functions to be Schur-positive. We determine necessary conditions for the difference to be Schur-positive. Our conditions are motivated by those of Reiner, Shaw and van Willigenburg that are necessary for s_A = s_B, and we deduce a strengthening of their result as a special case.


On the asymptotic behavior of Faber polynomials for domains with piecewise analytic boundary

Abstract: For a function g(w) analytic and univalent in {w:1<|w|<\infty} with a simple pole at \infty and a continuous extension to {w:|w|\geq 1}, we consider the Faber polynomials F_n(z), n=0,1,2,..., associated to g(w) via their generating function g'(w)/(g(w)-z)=\sum_{n=0}^\infty F_n(z)w^{-(n+1)}. Assuming that g(w) maps the unit circle T onto a piecewise analytic curve L whose exterior domain has no outward-pointing cusps, and under an additional assumption concerning the "Lehman expansion" of g(w) about those points of T mapped onto corners of L, we obtain asymptotic formulas for F_n(z) that yield fine results on the location, limiting distribution and accumulation points of the zeros of the Faber polynomials. The asymptotic formulas are shown to hold uniformly and the exact rate of decay of the error terms involved is provided.


Generalised knot groups distinguish the square and granny knots (with an appendix by David Savitt)

Abstract: Given a knot K we may construct a group G_n(K) from the fundamental group of K by adjoining an nth root of the meridian that commutes with the corresponding longitude. These "generalised knot groups" were introduced independently by Wada and Kelly, and contain the fundamental group as a subgroup. The square knot SK and the granny knot GK are a well known example of a pair of distinct knots with isomorphic fundamental groups. We show that G_n(SK) and G_n(GK) are non-isomorphic for all n>1. This confirms a conjecture of Lin and Nelson, and shows that the isomorphism type of G_n(K), n>1, carries more information about K than the isomorphism type of the fundamental group. An appendix by David Savitt contains some results on representations of the trefoil group in PSL(2,p) that are needed for the proof.