Universally defined representations of Lie conformal superalgebras

Abstract: We distinguish a class of irreducible finite representations of conformal Lie (super)algebras. These representations (called universally defined) are the simplest ones from the computational point of view: a universally defined representation of a conformal Lie (super)algebra $L$ is completely determined by commutation relations of $L$ and by the requirement of associative locality of generators. We describe such representations for conformal superalgebras $W_n$, $n\ge 0$, with respect to a natural set of generators. We also consider the problem for superalgebras $K_n$. In particular, we find a universally defined representation for the Neveu--Schwartz conformal superalgebra $K_1$ and show that the analogues of this representation for $n\ge 2$ are not universally defined.


Small Deviation Probability via Chaining

Abstract: We obtain several extensions of Talagrand's lower bound for the small deviation probability using metric entropy. For Gaussian processes, our investigations are focused on processes with sub-polynomial and, respectively, exponential behaviour of covering numbers. The corresponding results are also proved for non-Gaussian symmetric stable processes, both for the cases of critically small and critically large entropy. The results extensively use the classical chaining technique; at the same time they are meant to explore the limits of this method.


Associative algebras related to conformal algebras

Abstract: In this note, we introduce a class of algebras that are in some sense related to conformal algebras. This class (called TC-algebras) includes Weyl algebras and some of their (associative and Lie) subalgebras. By a conformal algebra we generally mean what is known as $H$-pseudo-algebra over the polynomial Hopf algebra $H=\Bbbk[T_1,..., T_n]$. Some recent results in structure theory of conformal algebras are applied to get a description of TC-algebras.


On the behavior of Castelnuovo-Mumford regularity with respect to some functors

Abstract: We investigate the behavior of Castelnuovo-Mumford regularity with respect to some classical functors : Tor, the Frobenius functor in positive characteristic, taking a power or a product (on ideals). These generalizes and refines previous results on these issues by several authors. As an application we provide results on the regularity of an intersection of subschemes of a projective scheme, under appropriate geometric hypotheses. Results on the rigidity of multiple Tor modules and on the characterization of their vanishing are given, motivated by geometric applications.


Sums of $L$-functions over the rational function field

Abstract: Friedberg, Hoffstein and Lieman have constructed two related multiple Dirichlet series from quadratic and higher-order $L$-functions and Gauss sums. We compute these multiple Dirichlet series explicitly in the case of the rational function field. This is done by utilizing the functional equation of the $L$-functions and the functional equation relating the two multiple Dirichlet series. We also point out a very simple correspondence between these series and their $p$-parts.


Mapping Class Groups and Interpolating Complexes: Rank

Abstract: A family of interpolating graphs $\calC (S, \xi)$ of complexity $\xi$ is constructed for a surface $S$ and $-2 \leq \xi \leq \xi (S)$. For $\xi = -2, -1, \xi (S) -1$ these specialise to graphs quasi-isometric to the marking graph, the pants graph and the curve graph respectively. We generalise Theorems of Brock-Farb and Behrstock-Minsky to show that the rank of $\calC (S, \xi)$ is $r_\xi$, the largest number of disjoint copies of subsurfaces of complexity greater than $\xi $ that may be embedded in $S$. The interpolating graphs $\calC (S, \xi)$ interpolate between the pants graph and the curve graph.


Obtaining New Dividing Formulas n|Q(n) From the Known Ones

Abstract: In this note, we present a few methods (Theorems 1, 2, and 3) from discrete dynamical systems theory of obtaining new functions Q(n) from the known ones so that the dividing formulas n|Q(n) hold.


On certain bounds for first-crossing-time probabilities of a jump-diffusion process

Abstract: We consider the first-crossing-time problem through a constant boundary for a Wiener process perturbed by random jumps driven by a counting process. On the base of a sample-path analysis of the jump-diffusion process we obtain explicit lower bounds for the first-crossing-time density and for the first-crossing-time distribution function. In the case of the distribution function, the bound is improved by use of processes comparison based on the usual stochastic order. The special case of constant jumps driven by a Poisson process is thoroughly discussed.


Scaled entropy of filtrations of $\sigma$-fields

Abstract: We study the notion of the scaled entropy of a filtration of $\sigma$-fields (= decreasing sequence of $\sigma$-fields) introduced by the first author ({V4}). We suggest a method for computing this entropy for the sequence of $\sigma$-fields of pasts of a Markov process determined by a random walk over the trajectories of a Bernoulli action of a commutative or nilpotent countable group (Theorems~5,~6). Since the scaled entropy is a metric invariant of the filtration, it follows that the sequences of $\sigma$-fields of pasts of random walks over the trajectories of Bernoulli actions of lattices (groups ${\Bbb Z}^d$) are metrically nonisomorphic for different dimensions $d$, and for the same $d$ but different values of the entropy of the Bernoulli scheme. We give a brief survey of the metric theory of filtrations, in particular, formulate the standardness criterion and describe its connections with the scaled entropy and the notion of a tower of measures.


Embedding Theorems and Boundary-value Problems for cusp domains

Abstract: We study the Robin boundary-value problem for bounded domains with isolated singularities. Because for such domains trace spaces of space $H^1(D)$ on its boundaries are weighted Sobolev spaces $L^{2, \xi}(\partial D)$ existence and uniqueness of corresponding Robin boundary-value problems depends on properties of embedding operators $I_1: H^{1}(D)\to L^{2}(D)$ and $I_{2}:H^{1}(D)\to L^{2,\xi}(\partial D)$ i.e. on type of singularities. We obtain an exact description of the weights $\xi$ for bounded domains with 'outside peaks' on its boundaries. This result allows us to formulate correctly the corresponding Robin boundary-value problems for elliptic operators.


Near-homeomorphisms of Nobeling manifolds

Abstract: We characterize maps between $n$-dimensional N\"obeling manifolds that can be approximated by homeomorphisms.


The Ricci iteration and its applications

Abstract: In this Note we introduce and study dynamical systems related to the Ricci operator on the space of Kahler metrics as discretizations of certain geometric flows. We pose a conjecture on their convergence towards canonical Kahler metrics and study the case where the first Chern class is negative, zero or positive. This construction has several applications in Kahler geometry, among them an answer to a question of Nadel and a construction of multiplier ideal sheaves.


Chains in the noncrossing partition lattice

Abstract: We establish recursions counting various classes of chains in the noncrossing partition lattice of a finite Coxeter group. The recursions specialize a general relation which is proven uniformly (i.e. without appealing to the classification of finite Coxeter groups) using basic facts about noncrossing partitions. We solve these recursions for each finite Coxeter group in the classification. Among other results, we obtain a simpler proof of a known uniform formula for the number of maximal chains of noncrossing partitions and a new uniform formula for the number of edges in the noncrossing partition lattice. All of our results extend to the m-divisible noncrossing partition lattice.


Extremal subsets of {1,...,n} avoiding solutions to linear equations in three variables

Abstract: We refine previous results to provide examples, and in some cases precise classifications, of extremal subsets of {1,...,n} containing no solutions to a wide class of non-invariant, homogeneous linear equations in three variables, i.e.: equations of the form ax+by=cz with a+b \neq c.


Filling inequalities do not depend on topology

Abstract: Gromov's universal filling inequalities relate the filling radius and the filling volume of a Riemannian manifold to its volume. The main result of the present article is that in dimensions at least three the optimal constants in the filling inequalities depend only on dimension and orientability, not on the manifold itself. This contrasts with the analogous situation for the optimal systolic inequality, which does depend on the manifold.


A Characterization of the Morse Minimal Set up to Topological Conjugacy

Abstract: We establish necessary and sufficient conditions for a dynamical system to be topologically conjugate to the Morse minimal set, the shift orbit closure of the Morse sequence, and conditions for topological conjugacy to the closely related Teoplitz minimal set.


On the constant in the Mertens product for arithmetic progressions. I. Identities

Abstract: The aim of the paper is the proof of new identities for the constant in the Mertens product for arithmetic progressions. We deal with the problem of the numerical computation of these constants in another paper.


On the tautological ring of a Jacobian modulo rational equivalence

Abstract: We consider the Chow ring with rational coefficients of the Jacobian of a curve. Assume D is a divisor in a base point free g^r_d of the curve such that the canonical divisor K is a multiple of the divisor D. We find relations between tautological cycles. We give applications for curves having a degree d covering of P^1 whose ramification points are all of order d, and then for hyperelliptic curves.


The angel wins

Abstract: The angel-devil game is played on an infinite two-dimensional ``chessboard''. The squares of the board are all white at the beginning. The players called angel and devil take turns in their steps. When it is the devil's turn, he can turn a square black. The angel always stays on a white square, and when it is her turn she can fly at a distance of at most J steps (each of which can be horizontal, vertical or diagonal) to a new white square. Here J is a constant. The devil wins if the angel does not find any more white squares to land on. The result of the paper is that if J is sufficiently large then the angel has a strategy such that the devil will never capture her. This deceptively easy-sounding result has been a conjecture, surprisingly, for about thirty years. Several other independent solutions have appeared simultaneously, some of them prove that J=2 is sufficient (see the Wikipedia on the angel problem). Still, it is hoped that the hierarchical solution presented here may prove useful for some generalizations.


Approximate calculation of operator semigroups by perturbation of generators

Abstract: Let $\Omega$ be an operator semigroup with generator $A$ in a sequentially complete locally convex topological vector space $E$. For a semigroup with generator $A+D$, where $D$ is a bounded linear operator on $E$, two integral equations are derived. A theorem on continuous dependence of a semigroup on its generator is proved. An application to random walk on $\mathbb{Z}$ is given.


Higher Tits indices of algebraic groups

Abstract: Let G be a semisimple algebraic group over a field k. We introduce the higher Tits indices of G as the set of all Tits indices of G over all field extensions K/k. In the context of quadratic forms this notion coincides with the notion of the higher Witt indices introduced by M. Knebusch and classified by N. Karpenko and A. Vishik. We classify the higher Tits indices for exceptional algebraic groups. Our main tools involve the Chow groups and the Chow motives of projective homogeneous varieties, Steenrod operations, and the notion of the J-invariant of algebraic groups.


Adelic Maass spaces on U(2,2)

Abstract: Generalizing the results of Kojima, Gritsenko and Krieg, we define an adelic version of the Maass space for hermitian modular forms of weight k regarded as functions on adelic points of the quasi-split unitary group U(2,2) associated with an imaginary quadratic extension F/Q of discriminant D_F. When the class number h_F of F is odd, we show that the Maass space is invariant under the action of the local Hecke algebras of U(2,2)(Q_p) for all p not dividing D_F. As a consequence we obtain a Hecke-equivariant injective map from the Maass space to the h_F-fold direct product of the space of elliptic modular forms of weight k-1 and level D_F.


A Simple Counterexample to Havil's "Reformulation" of the Riemann Hypothesis

Abstract: This is an elementary note. It corrects a mistake in the reformulation of the Riemann Hypothesis in J. Havil's book Gamma: Exploring Euler's Constant.


Counting closed geodesics on rank one manifolds

Abstract: We establish a precise asymptotic formula for the number of homotopy classes of periodic orbits for the geodesic flow on rank one manifolds of nonpositive curvature. This extends a celebrated result of G. A. Margulis to the nonuniformly hyperbolic case and strengthens previous results by G. Knieper. We also establish some useful properties of the measure of maximal entropy.


Group Representations and High-Resolution Central Limit Theorems for Subordinated Spherical Random Fields

Abstract: We study the weak convergence (in the high-frequency limit) of the frequency components associated with Gaussian-subordinated, spherical and isotropic random fields. In particular, we provide conditions for asymptotic Gaussianity and we establish a new connection with random walks on the the dual of SO(3), which mirrors analogous results previously established for fields defined on Abelian groups (see Marinucci and Peccati (2007)). Our work is motivated by applications to cosmological data analysis, and specifically by the probabilistic modelling and the statistical analysis of the Cosmic Microwave Background radiation, which is currently at the frontier of physical research. To obtain our main results, we prove several fine estimates involving convolutions of the so-called Clebsch-Gordan coefficients (which are elements of unitary matrices connecting reducible representations of SO(3)); this allows to intepret most of our asymptotic conditions in terms of coupling of angular momenta in a quantum mechanical system. Part of the proofs are based on recently established criteria for the weak convergence of multiple Wiener-It\^o integrals.


The K\"ahler-Ricci flow with positive bisectional curvature

Abstract: We show that the K\"ahler-Ricci flow on a manifold with positive first Chern class converges to a K\"ahler-Einstein metric assuming positive bisectional curvature and certain stability conditions.


A Polynomial Invariant Of Twisted Graph Diagrams

Abstract: Twisted graph diagrams are virtual graph diagrams with bars on edges. A bijection between abstract graph diagrams and twisted graph diagrams is constructed. Then a polynomial invariant of Yamada-type is developed which provides a lower bound for the virtual crossing number of virtual graph diagrams.


Smarandache Type Function Obtained by Duality

Abstract: In this paper we extend the Smarandache function from the set $N*$ of positive integers to the set $Q$ pf rational numbers. Using the inverse formula, this function is also regarded as a generating function. We put in evidence a procedure to construct a (numerical) function starting from a given function in two particular cases. Also, connections between this function and Euler totient function as well as with Riemann zeta function are established.


Comparing composites of left and right derived functors

Abstract: We introduce a new categorical framework for studying derived functors, and in particular for comparing composites of left and right derived functors. Our central observation is that model categories are the objects of a double category whose vertical and horizontal arrows are left and right Quillen functors, respectively, and that passage to derived functors is functorial at the level of this double category. The theory of conjunctions and mates in double categories, which generalizes the theory of adjunctions and mates in 2-categories, then gives us canonical ways to compare composites of left and right derived functors. We give a number of sample applications, most of which are improvements of existing proofs in the literature.


Parametrized spaces model locally constant homotopy sheaves

Abstract: We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopy-theoretic version of the classical identification of covering spaces with locally constant sheaves. We also prove a new version of the classical result that spaces parametrized over X are equivalent to spaces with an action of the loop space of X. This gives a homotopy-theoretic version of the correspondence between covering spaces over X and sets with an action of the fundamental group of X. We then use these two equivalences to study base change functors for parametrized spaces.


On the Nonexistence of Quadratic Lyapunov Functions for Consensus Algorithms

Abstract: We provide an example proving that there exists no quadratic Lyapunov function for a certain class of linear agreement/consensus algorithms, a fact that had been numerically verified in [5]. We also briefly discuss sufficient conditions for the existence of such a Lyapunov function.


A general method for investigating the roots of all equations by approximation

Abstract: Translation of the Latin original, "Methodus generalis investigandi radices omnium aequationum per approximationem" (1776). E643 in the Enestrom index. Euler gives a series to find powers of roots of polynomials.


Dimensional reduction as a tool for mesh refinement and tracking singularities of PDEs

Abstract: We present a collection of algorithms which utilize dimensional reduction to perform mesh refinement and study possibly singular solutions of time-dependent partial differential equations. The algorithms are inspired by constructions used in statistical mechanics to evaluate the properties of a system near a critical point. The first algorithm allows the accurate determination of the time of occurrence of a possible singularity. The second algorithm is an adaptive mesh refinement scheme which can be used to approach efficiently the possible singularity. Finally, the third algorithm uses the second algorithm until the available resolution is exhausted (as we approach the possible singularity) and then switches to a dimensionally reduced model which, when accurate, can follow faithfully the solution beyond the time of occurrence of the purported singularity. An accurate dimensionally reduced model should dissipate energy at the right rate. We construct two variants of each algorithm. The first variant assumes that we have actual knowledge of the reduced model. The second variant assumes that we know the form of the reduced model, i.e. the terms appearing in the reduced model, but not necessarily their coefficients. In this case, we also provide a way of determining the coefficients. We present numerical results for the Burgers equation with zero and nonzero viscosity to illustrate the use of the algorithms.


Hecke operators in equivariant elliptic cohomology and generalized moonshine

Abstract: This paper studies connections between generalized moonshine and elliptic cohomology with a focus on the action of the Hecke correspondence and its implications for the notion of replicability.


Disconnected synchronized regions of complex dynamical networks

Abstract: This paper addresses the synchronized region problem, which is reduced to a matrix stability problem, for complex dynamical networks. For any natural number $n$, the existence of a network which has $n$ disconnected synchronized regions is theoretically demonstrated. This shows the complexity in network synchronization. Convexity characteristic of stability for matrix pencils is further discussed. Smooth and generalized smooth Chua's circuit networks are finally discussed as examples for illustration.


Analysis and control of network synchronizability

Abstract: In this paper, the investigation is first motivated by showing two examples of simple regular symmetrical graphs, which have the same structural parameters, such as average distance, degree distribution and node betweenness centrality, but have very different synchronizabilities. This demonstrates the complexity of the network synchronizability problem. For a given network with identical node dynamics, it is further shown that two key factors influencing the network synchronizability are the network inner linking matrix and the eigenvalues of the network topological matrix. Several examples are then provided to show that adding new edges to a network can either increase or decrease the network synchronizability. In searching for conditions under which the network synchronizability may be increased by adding edges, it is found that for networks with disconnected complementary graphs, adding edges never decreases their synchronizability. This implies that better understanding and careful manipulation of the complementary graphs are important and useful for enhancing the network synchronizability. Moreover, it is found that an unbounded synchronized region is always easier to analyze than a bounded synchronized region. Therefore, to effectively enhance the network synchronizability, a design method is finally presented for the inner linking matrix of rank 1 such that the resultant network has an unbounded synchronized region, for the case where the synchronous state is an equilibrium point of the network.


Complexity in linearly coupled dynamical networks: Some unusual phenomena in energy accumulation

Abstract: This paper addresses the energy accumulation problem, in terms of the $H_2$ norm, of linearly coupled dynamical networks. An interesting outer-coupling relationship is constructed, under which the $H_2$ norm of the newly constructed network with column-input and row-output shaped matrices increases exponentially fast with the node number $N$: it increases generally much faster than $2^N$ when $N$ is large while the $H_2$ norm of each node is 1. However, the $H_2$ norm of the network with a diffusive coupling is equal to $\gamma_2 N$, i.e., increasing linearly, when the network is stable, where $\gamma_2$ is the $H_2$ norm of a single node. And the $H_2$ norm of the network with antisymmetrical coupling also increases, but rather slowly, with the node number $N$. Other networks with block-diagonal-input and block-diagonal-output matrices behave similarly. It demonstrates that the changes of $H_2$ norms in different networks are very complicated, despite the fact that the networks are linear. Finally, the influence of the $H_2$ norm of the locally linearized network on the output of a network with Lur'e nodes is discussed.


The $p$-modular descent algebras

Abstract: The concept of descent algebras over a field of characteristic zero is extended to define descent algebras over a field of prime characteristic. Some basic algebraic structure of the latter, including its radical and irreducible modules, is then determined. The decomposition matrix of the descent algebras of Coxeter group types $A$, $B$, and $D$ are calculated, and used to derive a description of the decomposition matrix of an arbitrary descent algebra. The Cartan matrix of a variety of descent algebras over a finite field is then obtained.


Properties of the descent algebras of type $D$

Abstract: We establish simple combinatorial descriptions of the radical and irreducible representations specifically for the descent algebra of a Coxeter group of type $D$ over any field.


Statistical testing procedure for the interaction effects of several controllable factors in two-valued input-output systems

Abstract: Suppose several two-valued input-output systems are designed by setting the levels of several controllable factors. For this situation, Taguchi method has proposed to assign the controllable factors to the orthogonal array and use ANOVA model for the standardized SN ratio, which is a natural measure for evaluating the performance of each input-output system. Though this procedure is simple and useful in application indeed, the result can be unreliable when the estimated standard errors of the standardized SN ratios are unbalanced. In this paper, we treat the data arising from the full factorial or fractional factorial designs of several controllable factors as the frequencies of high-dimensional contingency tables, and propose a general testing procedure for the main effects or the interaction effects of the controllable factors.


Enumerative properties of Ferrers graphs

Abstract: We define a class of bipartite graphs that correspond naturally with Ferrers diagrams. We give expressions for the number of spanning trees, the number of Hamiltonian paths when applicable, the chromatic polynomial, and the chromatic symmetric function. We show that the linear coefficient of the chromatic polynomial is given by the excedance set statistic.


Tropical hyperplane arrangements and oriented matroids

Abstract: We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We show that a tropical oriented matroid determines a subdivision of a product of two simplices, and conjecture that this correspondence is a bijection.


Mackey functors on compact closed categories

Abstract: We develop and extend the theory of Mackey functors as an application of enriched category theory. We define Mackey functors on a lextensive category $\E$ and investigate the properties of the category of Mackey functors on $\E$. We show that it is a monoidal category and the monoids are Green functors. Mackey functors are seen as providing a setting in which mere numerical equations occurring in the theory of groups can be given a structural foundation. We obtain an explicit description of the objects of the Cauchy completion of a monoidal functor and apply this to examine Morita equivalence of Green functors.


On the stabilization of the elasticity system by the boundary

Abstract: We obtain free of resonances regions for the elasticity system in the exterior of a strictly convex body with dissipative boundary conditions under some natural assumptions on the behaviour of the geodesics on the boundary.


Thue equations and torsion groups of elliptic curves

Abstract: A new characterization of rational torsion subgroups of elliptic curves is found, for points of order greater than 4, through the existence of solution for systems of Thue equations.


On the shadow boundary of a centrally symmetric convex body

Abstract: We discuss the concept of the shadow boundary of a centrally symmetric convex ball $K$ (actually being the unit ball of a Minkowski normed space) with respect to a direction ${\bf x}$ of the Euclidean n-space $R^n$. We introduce the concept of general parameter spheres of $K$ corresponding to this direction and prove that the shadow boundary is a topological manifold if all of the non-degenerated general parameter spheres are, too. In this case, using the approximation theorem of cell-like maps we get that they are homeomorphic to the $(n-2)$-dimensional sphere $S^{(n-2)}$. We also prove that the bisector (equidistant set of the corresponding normed space) in the direction ${\bf x}$ is homeomorphic to $R^{(n-1)}$ iff all of the non-degenerated general parameter spheres are $(n-2)$-manifolds implying that if the bisector is a homeomorphic copy of $R^{(n-1)}$ then the corresponding shadow boundary is a topological $(n-2)$-sphere.


On Scattering for CMV Matrices

Abstract: Adamjan-Arov (Lax--Phillips) model space is considered as a scattering representation space for a CMV matrix in context of an extended Marchenko--Faddeev scattering theory. That is, there exists a basis in which the multiplication by independent variable is a CMV matrix. This basis as well as Verblunski coefficients are computed explicitly in terms of Nehari interpolation. Asymptotically the Verblynski coefficients go to zero. Moreover, relations between the basis and wandering subspaces are established. Transformation from scattering representation to spectral representation is given.


Rational formality of function spaces

Abstract: Let $X$ be a nilpotent space such that there exists $N\geq 1$ with $H^N(X,\mathbb Q) \ne 0$ and $H^n(X,\mathbb Q)=0$ if $n>N$. Let $Y$ be a m-connected space with $m\geq N+1$ and $H^*(Y,\mathbb Q)$ is finitely generated as algebra. We assume that the odd part of the rational Hurewicz homomorphism: $\pi_{odd}(X)\otimes \mathbb Q\to H_{odd}(X,\mathbb Q)$ is non-zero. We prove that if the space $\mathcal F(X,Y)$ of continuous maps from $X$ to $Y$ is rationally formal, then $Y$ has the rational homotopy type of a finite product of Eilenberg Mac Lane spaces. At the opposite, we exhibit an example of a rationally formal space $\mathcal F(S^2,Y)$ where $Y$ is not rationally equivalent to a product of Eilenberg Mac Lane spaces.


Moduli space of Brody curves, energy and mean dimension

Abstract: We study the mean dimension of the moduli space of Brody curves. We introduce the notion of "mean energy" and show that this can be used to estimate the mean dimension.


Explicit birational geometry of threefolds of general type

Abstract: Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}(V)>0$ and $P_{24}(V)>1$ (which answers an open problem of J. Kollar and S. Mori). We also prove that the canonical volume has an universal lower bound $\text{Vol}(V) \geq 1/2660$ and that the pluri-canonical map $\Phi_m$ is birational onto its image for all $m\geq 77$. As an application of our method, we prove Fletcher's conjecture on weighted hyper-surface 3-folds with terminal quotient singularities. Another featured result is the optimal lower bound $\text{Vol}(V)\geq {1/420}$ among all those 3-folds $V$ with $\chi({\mathcal O}_V)\leq 1$.


Injectivity and Projectivity in Analysis and Topology

Abstract: We give new proofs for many injectivity results in analysis that make more careful use of the duality between unital abelian C*-algebras and compact Hausdorff spaces. We then extend many of these results to incorporate group actions. Our approach uses only elementary topological constructions and eliminates the need for results from the theory of Boolean algebras and AW*-algebras that were used in earlier proofs.


The forgotten monoid

Abstract: We study properties of the forgotten monoid which appeared in work of Lascoux and Schutzenberger and recently resurfaced in the construction of dual equivalence graphs by Assaf. In particular, we provide an explicit characterization of the forgotten classes in terms of inversion numbers and show that there are n^2-3n+4 forgotten classes in the symmetric group S_n. Each forgotten class contains a canonical element that can be characterized by pattern avoidance. We also show that the sum of Gessel's quasi-symmetric functions over a forgotten class is a 0-1 sum of ribbon-Schur functions.


Connections between real polynomial solutions of hypergeometric-type differential equations with Rodrigues formula

Abstract: Starting from the Rodrigues representation of polynomial solutions of the general hypergeometric-type differential equation complementary polynomials are constructed using a natural method. Among the key results is a generating function in closed form leading to short and transparent derivations of recursion relations and an addition theorem. The complementary polynomials satisfy a hypergeometric-type differential equation themselves, have a three-term recursion among others and obey Rodrigues formulas. Applications to the classical polynomials are given.


Recollement of Deformed Preprojective Algebras and the Calogero-Moser Correspondence

Abstract: The aim of this paper is to clarify the relation between the following objects: $ (a) $ rank 1 projective modules (ideals) over the first Weyl algebra $ A_1(\C)$; $ (b) $ simple modules over deformed preprojective algebras $ \Pi_{\lambda}(Q) $ introduced by Crawley-Boevey and Holland; and $ (c) $ simple modules over the rational Cherednik algebras $ H_{0,c}(S_n) $ associated to symmetric groups. The isomorphism classes of each type of these objects can be parametrized geometrically by the same space (namely, the Calogero-Moser algebraic varieties); however, no natural functors between the corresponding module categories seem to be known. We construct such functors by translating our earlier results on $\A$-modules over $ A_1 $ to a more familiar setting of representation theory. In the last section we extend our construction to the case of Kleinian singularities $ \C^2/\Gamma $, where $ \Gamma $ is a finite cyclic subgroup of $ \SL(2, \C) $.


Quasi-invariance properties of a class of subordinators

Abstract: We study absolute-continuity properties of a class of stochastic processes, including the gamma and the Dirichlet processes. We prove that the laws of a general class of non-linear transformations of such processes are locally equivalent to the law of the original process and we compute explicitly the associated Radon-Nikodym densities. This work unifies and generalizes to random non-linear transformations several previous results on quasi-invariance of gamma and Dirichlet processes.


Independence property and hyperbolic groups

Abstract: We prove that existentially closed $CSA$-groups have the independence property. This is done by showing that there exist words having the independence property relatively to the class of torsion-free hyperbolic groups.


On a generalization of Dehn's algorithm

Abstract: Viewing Dehn's algorithm as a rewriting system, we generalise to allow an alphabet containing letters which do not necessarily represent group elements. This extends the class of groups for which the algorithm solves the word problem to include nilpotent groups, many relatively hyperbolic groups including geometrically finite groups and fundamental groups of certain geometrically decomposable manifolds. The class has several nice closure properties. We also show that if a group has an infinite subgroup and one of exponential growth, and they commute, then it does not admit such an algorithm. We dub these Cannon's algorithms.


On the discreteness and rationality of F-jumping coefficients

Abstract: This paper studies the jumping coefficients of principal ideals of regular local rings. Recently M. Blickle, M. Mustata and K. Smith showed that, when $R$ is of essentially finite type over a field and $F$-finite, bounded intervals contain finitely many jumping coefficients and that those are rational. In a later paper they extended these results to principal ideals of $F$-finite complete regular local rings. The aim of this paper is to extend these results on the discreteness and rationality of jumping coefficients to principal ideals of arbitrary (i.e. not necessarily $F$-finite) excellent regular local rings containing fields of positive characteristic. Our proof uses a very different method: we do not use $D$-modules and instead we analyze the modules of nilpotents elements in the injective hull or $R$ under some non-standard Frobenius actions. This new method undoubtedly holds a potential for more applications.


Using integral transforms to estimate higher order derivatives

Abstract: Integral transformations are used to estimate high order derivatives of various special functions. Applications are given to numerical integration, where estimates of high order derivatives of the integrand are needed to achieve bounds on the error. The main idea is to find a suitable integral representation of the function whose derivatives are to be estimated, differentiate repeatedly under the integral sign, and estimate the resulting integral.


Duality of antidiagonals and pipe dreams

Abstract: Weighted enumeration of reduced pipe dreams (or rc-graphs) results in a combinatorial expression for Schubert polynomials. The duality between the set of reduced pipe dreams and certain antidiagonals has important geometric implications [A. Knutson and E. Miller, Gr\"obner geometry of Schubert polynomials, Ann. Math. 161, 1245-1318]. The original proof of the duality was roundabout, relying on the algebra of certain monomial ideals and a recursive characterization of reduced pipe dreams. This paper provides a direct combinatorial proof.


Implicit function density computation

Abstract: If two random variables X and A are functionally related via f(X)=A for some strictly monotone continuously differentiable function f:R->R, the distribution of X may easily be computed from the distribution of A.


On McQuillan's "tautological inequality" and the Weyl-Ahlfors theory of associated curves

Abstract: In 1941, L. Ahlfors gave another proof of a 1933 theorem of H. Cartan on approximation to hyperplanes of holomorphic curves in P^n. Ahlfors' proof built on earlier work of H. and J. Weyl (1938), and proved Cartan's theorem by studying the associated curves of the holomorphic curve. This work has subsequently been reworked by H.-H. Wu in 1970, using differential geometry, M. Cowen and P. A. Griffiths in 1976, further emphasizing curvature, and by Y.-T. Siu in 1987 and 1990, emphasizing meromorphic connections. This paper gives another variation of the proof, motivated by successive minima as in the proof of Schmidt's Subspace Theorem, and using McQuillan's "tautological inequality." In this proof, essentially all of the analysis is encapsulated within a modified McQuillan-like inequality, so that most of the proof primarily uses methods of algebraic geometry, in particular flag varieties. A diophantine conjecture based on McQuillan's inequality is also posed.


A primer on the (2+1) Einstein universe

Abstract: The Einstein universe is the conformal compactification of Minkowski space. It also arises as the ideal boundary of anti-de Sitter space. The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of its homogeneous submanifolds and causal structure, with particular emphasis on dimension $2 + 1$, in which there is a rich interplay with symplectic geometry.


Extreme values for Benedicks-Carleson quadratic maps

Abstract: We consider the quadratic family of maps given by $f_{a}(x)=1-a x^2$ with $x\in [-1,1]$, where $a$ is a Benedicks-Carleson parameter. For each of these chaotic dynamical systems we study the extreme value distribution of the stationary stochastic processes $X_0,X_1,...$, given by $X_{n}=f_a^n$, for every integer $n\geq0$, where each random variable $X_n$ is distributed according to the unique absolutely continuous, invariant probability of $f_a$. Using techniques developed by Benedicks and Carleson, we show that the limiting distribution of $M_n=\max\{X_0,...,X_{n-1}\}$ is the same as that which would apply if the sequence $X_0,X_1,...$ was independent and identically distributed. This result allows us to conclude that the asymptotic distribution of $M_n$ is of Type III (Weibull).


Strong Uniqueness of the Ricci Flow

Abstract: In this paper, we derive some local a priori estimates for Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let $g(t)$ be a smooth complete solution to the Ricci flow on $\mathbb{R}^{3}$, with the canonical Euclidean metric $E$ as initial data, then $g(t)$ is trivial, i.e. $g(t)\equiv E$.


Configurations of Extremal Even Unimodular Lattices

Abstract: We extend the results of Ozeki on the configurations of extremal even unimodular lattices. Specifically, we show that if L is such a lattice of rank 56, 72, or 96, then L is generated by its minimal-norm vectors.


Estimates of Gromov's box distance

Abstract: In 1999, M. Gromov introduced the box distance function $\sikaku$ on the space of all mm-spaces. In this paper, by using the method of T. H. Colding (cf. \cite[Lemma 5.10]{Colding}), we estimate $\sikaku(\mathbb{S}^n,\mathbb{S}^m)$ and $\sikaku (\mathbb{C}P^n, \mathbb{C}P^m)$, where $\mathbb{S}^n$ is the $n$-dimensional unit sphere in $\mathbb{R}^{n+1}$ and $\mathbb{C}P^n$ is the $n$-dimensional complex projective space equipped with the Fubini-Study metric. In paticular, we give the complete answer to an Exercise of Gromov's Green book (cf. \cite[Section $3{1/2}.18$]{gromov}). We also estimate $\sikaku \big(SO(n), SO(m)\big)$ from below, where SO(n) is the special orthogonal group.


Stark units and main conjectures for totally real fields

Abstract: Main theorem of [Buyukboduk, arXiv:0706.0377v1] suggests that it should be possible to lift the Kolyvagin systems of Stark units constructed in [Buyukboduk, arXiv:math/0703426v1] to a Kolyvagin system over the cyclotomic Iwasawa algebra. This is what we prove in this paper. This construction gives the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank is greater than one. As a result of this construction, we reduce the main conjectures of Iwasawa theory for totally real fields to a statement of local Iwasawa theory. This statement, however, turns out to be interesting in its own right as it suggests a relation between solutions to $p$-adic and complex Stark conjectures.


On the well-posedness of the Cauchy problem for the generalized Korteweg-de Vries-Burgers equation

Abstract: Considered is the generalized Korteweg-de Vries-Burgers equation $$ u_{t}+u_{xxx}+uu_{x}+|D_{x}|^{2\alpha}u=0,\quad t\in \mathbb{R}^{+}, x\in \mathbb{R}, $$ with $0\leq \alpha\le 1$. We prove a sharp results on the associated Cauchy problem in the Sobolev space $ {H}^s(\mathbb{R})$. For $s>-\min\{\frac {3+2\alpha}4, 1\}$ we give the well-posedness of solutions of the Cauchy problem, while for $\frac 12\le\alpha\le 1$ and for $s<-\min\{\frac {3+2\alpha}4, 1\}$ we show some ill-posedness issues.


On $(2k)$-Minimal Submanifolds

Abstract: Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature is zero. It is classical that minimal submanifolds are the critical points of the volume function. In this paper, we examine the critical points of the total $(2k)$-th Gauss-Bonnet curvature function, called $(2k)$-minimal submanifolds. We prove that they are characterized by the vanishing of a higher mean curvature, namely the $(2k+1)$-Gauss-Bonnet curvature. Furthermore, we show that several properties of usual minimal submanifolds can be naturally generalized to $(2k)$-minimal submanifolds.


Spinorial Characterization of Surfaces into 3-dimensional homogeneous Manifolds

Abstract: We give a spinorial characterization of isometrically immersed surfaces into 3-dimensional homogeneous manifolds with 4-dimensional isometry group in terms of the existence of a particular spinor, called generalized Killing spinor. This generalizes results by T. Friedrich for $\R^3$ and B. Morel for $\Ss^3$ and $\HH^3$. The main argument is the interpretation of the energy-momentum tensor of a genralized Killing spinor as the second fondamental form up to a tensor depending on the structure of the ambient space


Polynomial rate convergence to an invariant measure for the continuum time limit of the Minority Game

Abstract: In this paper we show that the continuum time version of the Minority Game satisfies the criteria for the application of a theorem on the existence of an invariant measure. We consider the special case of a game with "sufficiently" asymmetric initial condition where the number of possible choices for each individual is S=2 and $\Gamma<+\infty$. An upper bound for the asymptotic behavior, as the number of agents grows to infinity, of the waiting time for reaching the stationary state is then obtained.


Finite jet determination of CR mappings

Abstract: We prove the following finite jet determination result for CR mappings: Given a smooth generic submanifold M of C^N, N >= 2, which is essentially finite and of finite type at each of its points, for every point p on M there exists an integer l(p), depending upper-semicontinuously on p, such that for every smooth generic submanifold M' of C^N of the same dimension as M, if h_1 and h_2: (M,p)->M' are two germs of smooth finite CR mappings with the same l(p) jet at p, then necessarily their k-jets agree for all positive integers k. In the hypersurface case, this result provides several new unique jet determination properties for holomorphic mappings at the boundary in the real-analytic case; in particular, it provides the finite jet determination of arbitrary real-analytic CR mappings between real-analytic hypersurfaces in C^N of D'Angelo finite type. It also yields a new boundary version of H. Cartan's uniqueness theorem: if Omega and Omega' are two bounded domains in C^N with smooth real-analytic boundary, then there exists an integer k, depending only on the boundary of Omega, such that if H_1 and H_2: Omega -> Omega' are two proper holomorphic mappings extending smoothly up to the boundary of Omega near some point boundary point p and agreeing up to order k at p, then necessarily H_1=H_2.


Multi-triangulations as complexes of star polygons

Abstract: Maximal $(k+1)$-crossing-free graphs on a planar point set in convex position, that is, $k$-triangulations, have received attention in recent literature, with motivation coming from several interpretations of them. We introduce a new way of looking at $k$-triangulations, namely as complexes of star polygons. With this tool we give new, direct, proofs of the fundamental properties of $k$-triangulations, as well as some new results. This interpretation also opens-up new avenues of research, that we briefly explore in the last section.


A nilpotent quotient algorithm for L-presented groups

Abstract: The main part of this paper contains a description of a nilpotent quotient algorithm for L-presented groups and a report on applications of its implementation in the computer algebra system GAP. The appendix introduces two new infinite series of L-presented groups. Apart from being of interest in their own right, these new L-presented groups serve as examples for applications of the nilpotent quotient algorithm.


Higher-Order Calculus of Variations on Time Scales

Abstract: We prove a version of the Euler-Lagrange equations for certain problems of the calculus of variations on time scales with higher-order delta derivatives.


Remarks on the calculus of variations on time scales

Abstract: The calculus of variations is a classical subject which has gain throughout the last three hundred years a level of rigor and elegance that only time can give. In this note we show that, contrary to the classical field, available formulations and results on the recent calculus of variations on time scales are still at the heuristic level.


Connections between Romanovski and other polynomials

Abstract: A connection between Romanovski polynomials and those polynomials that solve the one-dimensional Schr\"odinger equation with the trigonometric Rosen-Morse and hyperbolic Scarf potential is established. The map is constructed by reworking the Rodrigues formula in an elementary and natural way. The generating function is summed in closed form from which recursion relations and addition theorems follow. Relations to some classical polynomials are also given.


On contact tops and integrable tops

Abstract: In this paper, we introduce a geometric structure called top, which is a trivialized bundle of plane pencils over a Riemannian 3-manifold, defined as the set of kernels of a circle of 1-forms (e.g. of contact and integrable forms) with particular properties with respect to the metric. We classify the manifolds which admit tops and we describe the associated metrics.


Remarks on the density of the law of the occupation time for Bessel bridges and stable excursions

Abstract: Smoothness and asymptotic behaviors are studied for the densities of the law of the occupation time on the positive line for Bessel bridges and the normalized excursion of strictly stable processes. The key role is played by these properties for functions defined by Riemann--Liouville fractional integrals.


Verhulst's logistic curve

Abstract: We observe that the elementary logistic differential equation dP/dt=(1-P/M)kP may be solved by first changing the variable to R=(M-P)/P. This reduces the logistic differential equation to the simple linear differential equation dR/dt=-kR, which can be solved without using the customary but slightly more elaborate methods applied to the original logistic DE. The resulting solution in terms of R can be converted by simple algebra to the familiar sigmoid expression involving P. A biological argument is given for introducing logistic growth via the simpler DE for R. It is also shown that the sigmoid P may be written in terms of the hyperbolic tangent by a simple translation that is also motivated by a biological argument.


Representing a product system representation as a contractive semigroup and applications to regular isometric dilations

Abstract: In this paper we propose a new technical tool for analyzing representations of Hilbert $C^*$-product systems. Using this tool, we give a new proof that every doubly commuting representation over $\mathbb{N}^k$ has a regular isometric dilation, and we also prove sufficient conditions for the existence of a regular isometric dilation of representations over more general subsemigroups of $\mathbb{R}_+^k$.


Mirror symmetry and T-duality in the complement of an anticanonical divisor

Abstract: We study the geometry of complexified moduli spaces of special Lagrangian submanifolds in the complement of an anticanonical divisor in a compact Kahler manifold. In particular, we explore the connections between T-duality and mirror symmetry in concrete examples, and show how quantum corrections arise in this context.


Convolution estimates and model surfaces of low codimension

Abstract: We give examples of measures on certain k-surfaces in R^d. These measures satisfy convolution estimates which are nearly optimal.


Computing word length in alternate presentations of Thompson's group F

Abstract: We introduce a new method for computing the word length of an element of Thompson's group F with respect to a "consecutive" generating set of the form X_n={x_0,x_1,...,x_n}, which is a subset of the standard infinite generating set for F. We use this method to show that (F,X_n) is not almost convex, and has pockets of increasing, though bounded, depth dependent on n.


Curvature line parametrization from circle patterns

Abstract: We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by respective discrete nets (circular nets and planar quadrilateral nets) with infinitesimal quads. It is shown that choosing the points of discrete nets on the smooth surface one can obtain second-order approximation globally. Also a simple geometric construction for approximate determination of principal directions of smooth surfaces is given.


Smooth maps of a foliated manifold in a symplectic manifold

Abstract: The immersions of a smooth manifold $M$ in a symplectic manifold $(N,\sigma)$ inducing a given closed form $\omega$ on $M$ satisfy the $C^0$-dense $h$-principle in the space of all continuous maps which pull back the deRham cohomology class of $\sigma$ onto that of $\omega$. In this paper we prove a foliated version of this result due to Gromov.


Convex Hull Realizations of the Multiplihedra

Abstract: We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. We use this realization to unite the approach to A_n-maps of Iwase and Mimura to that of Boardman and Vogt. We include a review of the appearance of the nth multiplihedron for various n in the studies of higher homotopy commutativity, (weak) n-categories, A_infinity-categories, deformation theory, and moduli spaces. We also include suggestions for the use of our realizations in some of these areas as well as in related studies, including enriched category theory and the graph associahedra.


Stringy product on twisted orbifold K-theory for abelian quotients

Abstract: In this paper we present a model to calculate the stringy product on twisted orbifold K-theory of Adem-Ruan-Zhang for abelian complex orbifolds. In the first part we consider the non-twisted case on an orbifold presented as the quotient of a manifold acted by a compact abelian Lie group. We give an explicit description of the obstruction bundle, we explain the relation with the product defined by Jarvis-Kaufmann-Kimura and, via a Chern character map, with the Chen-Ruan cohomology, and we explicitely calculate the stringy product for a weighted projective orbifold. In the second part we consider orbifolds presented as the quotient of a manifold acted by a finite abelian group and twistings coming from the group cohomology. We show a decomposition formula for twisted orbifold K-theory that is suited to calculate the stringy product and we use this formula to calculate two examples when the group is $(\integer/2)^3$.


On finite groups whose derived subgroup has bounded rank

Abstract: Let $G$ be a finite group with derived subgroup of rank $r$. We prove that $\gzz\leq |G'|^{2r}$. Motivated by the results of I. M. Isaacs in \cite{isa} we show that if $G$ is capable then $\gz\leq |G'|^{4r}$. This answers a question of L. Pyber. We prove that if $G$ is a capable $p$-group then the rank of $G/\mathbf{Z}(G)$ is bounded above in terms of the rank of $G'$.


Multiplicity free expansions of Schur $P$-functions

Abstract: After deriving inequalities on coefficients arising in the expansion of a Schur $P$-function in terms of Schur functions we give criteria for when such expansions are multiplicity free. From here we study the multiplicity of an irreducible spin character of the twisted symmetric group in the product of a basic spin character with an irreducible character of the symmetric group, and determine when it is multiplicity free.


Compact symmetric solutions to the postage stamp problem

Abstract: We derive lower and upper bounds on possible growth rates of certain sets of positive integers $A_k=\{1= a_1 < a_2 < ... < a_{k}\}$ such that all integers $n\in \{0, 1, 2, ..., ka_{k}\}$ can be represented as a sum of no more than $k$ elements of $A_k$, with repetition.


On tensor products of polynomial representations

Abstract: We determine the necessary and sufficient combinatorial conditions for which the tensor product of two irreducible polynomial representations of $GL(n,\mathbb{C})$ is isomorphic to another. As a consequence we discover families of Littlewood-Richardson coefficients that are non-zero, and a condition on Schur non-negativity.


Schur positivity of skew Schur function differences and applications to ribbons and Schubert classes

Abstract: Some new relations on skew Schur function differences are established both combinatorially using Sch\"utzenberger's jeu de taquin, and algebraically using Jacobi-Trudi determinants. These relations lead to the conclusion that certain differences of skew Schur functions are Schur positive. Applying these results to a basis of symmetric functions involving ribbon Schur functions confirms the validity of a Schur positivity conjecture due to McNamara. A further application reveals that certain differences of products of Schubert classes are Schubert positive.


Zeta functions and topological entropy of the Markov-Dyck shifts

Abstract: The Markov-Dyck shifts arise from finite directed graphs. An expression for the zeta function of a Markov-Dyck shift is given. The derivation of this expression is based on a formula in Keller (G. Keller, {\it Circular codes, loop counting, and zeta-functions}, J. Combinatorial Theory {\bf 56} (1991), pp. 75--83). For a class of examples that includes the Fibonacci-Dyck shift the zeta functions and topological entropy ae determined.


A Blass-Sagan bijection on Eulerian equivalence classes

Abstract: Following the treatment of Blass and Sagan, we present an algorithmic bijection between the Eulerian equivalence classes of totally cyclic orientations and the spanning trees without internal activity edges for a given graph.


Orientations, lattice polytopes, and group arrangements I: Chromatic and tension polynomials of graphs

Abstract: This is the first one of a series of papers on association of orientations, lattice polytopes, and abelian group arrangements to graphs. The purpose is to interpret the integral and modular tension polynomials of graphs at zero and negative integers. The whole exposition is put under the framework of subgroup arrangements and the application of Ehrhart polynomials. Such viewpoint leads to the following main results of the paper: (i) the reciprocity law for integral tension polynomials; (ii) the reciprocity law for modular tension polynomials; and (iii) a new interpretation for the value of the Tutte polynomial $T(G;x,y)$ of a graph $G$ at $(1,0)$ as the number of cut-equivalence classes of acyclic orientations on $G$.


Braided Hom-Lie bialgebras

Abstract: We introduce the new concept of braided Hom-Lie bialgebras which is a generalization of Sommerh\"{a}user-Majid's braided Lie bialgebras and Yau's Hom-Lie bialgebras. Using this concept we give the unified product construction for Hom-Lie bialgebras which can be seen as a Hom-Lie version of Bespalov-Drabant's cocycle cross product bialgebras. Some special cases of unified products such as crossed product and matched pair of braided Hom-Lie bialgebras are investigated. As an application, we solve the Agore-Militaru extending problem for Hom-Lie bialgebras by using some non-abelian cohomology theory. Furthermore, one dimensional flag extending structures for Hom-Lie bialgebras are also investigated.


The role of string topology in symplectic field theory

Abstract: We outline a program for incorporating holomorphic curves with Lagrangian boundary conditions into symplectic field theory, with an emphasis on ideas, geometric intuition, and a description of the resulting algebraic structures.


On geodesics of Berger tangent sphere bundle of Hermitian locally symmetric manifold

Abstract: We propose a special deformation of the Sasaki metric on tangent and unit tangent bundle of a Hermitian locally symmetric manifold. Geodesics of this deformed metric have different projections on a base manifold for tangent or unit tangent bundle cases in contrast to usual Sasaki metric. Nevertheless, the projections of geodesics of the unit tangent bundle still preserve the property to have all geodesic curvatures constant.


Relaxation of the flow of triods by Curve Shortening Flow via the vector-valued parabolic Allen-Cahn equation

Abstract: In this paper we find solutions $u_\epsilon$ to a certain class of vector-valued parabolic Allen-Cahn equation that as $\epsilon \to 0$ develops as interface a given triod evolving under curve shortening flow.


Parity properties of Costas arrays defined via finite fields

Abstract: A Costas array of order $n$ is an arrangement of dots and blanks into $n$ rows and $n$ columns, with exactly one dot in each row and each column, the arrangement satisfying certain specified conditions. A dot occurring in such an array is even/even if it occurs in the $i$-th row and $j$-th column, where $i$ and $j$ are both even integers, and there are similar definitions of odd/odd, even/odd and odd/even dots. Two types of Costas arrays, known as Golomb-Costas and Welch-Costas arrays, can be defined using finite fields. When $q$ is a power of an odd prime, we enumerate the number of even/even odd/odd, even/odd and odd/even dots in a Golomb-Costas array. We show that three of these numbers are equal and they differ by $\pm 1$ from the fourth. For a Welch-Costas array of order $p-1$, where $p$ is an odd prime, the four numbers above are all equal to $(p-1)/4$ when $p\equiv 1\pmod{4}$, but when $p\equiv 3\pmod{4}$, we show that the four numbers are defined in terms of the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$, and thus behave in a much less predictable manner.


Invariance of Gamma-dimension for projective families

Abstract: This article is withdrawn because of a mistake in the main result of the paper.


Radius and profile of random planar maps with faces of arbitrary degrees

Abstract: We prove some asymptotic results for the radius and the profile of large random rooted planar maps with faces of arbitrary degrees. Using a bijection due to Bouttier, Di Francesco and Guitter between rooted planar maps and certain four-type trees with positive labels, we derive our results from a conditional limit theorem for four-type spatial Galton-Watson trees.


Filtering and estimation in stochastic volatility models with rationally distributed disturbances

Abstract: This paper deals with the filtering problem for a class of discrete time stochastic volatility models in which the disturbances have rational probability density functions. This includes the Cauchy distributions and Student t-distributions with odd number of degrees of freedom. Using state space realizations to represent the rational probability density functions we are able to solve the filtering problem exactly. However the size of the involved state space matrices grows exponentially with each time step of the filter. Therefore we use stochastically balanced truncation techniques to approximate the high order rational functions involved. In a simulation study we show the applicability of this approach. In addition a simple method of moments estimator is derived.


Corps de nombres peu ramifies et formes automorphes autoduales

Abstract: Let S be a finite set of primes, p in S, and Q_S a maximal algebraic extension of Q unramified outside S and infinity. Assume that |S|>=2. We show that the natural maps Gal(Q_p^bar/Q_p) --> Gal(Q_S/Q) are injective. Much of the paper is devoted to the problem of constructing selfdual automorphic cuspidal representations of GL(2n,A_Q) with prescribed properties at all places, that we study via the twisted trace formula of J. Arthur. The techniques we develop shed also some lights on the orthogonal/symplectic alternative for selfdual representations of GL(2n).


On the residual finiteness and other properties of (relative) one-relator groups

Abstract: A relative one-relator presentation has the form P = < X,H ; R > where X is a set, H is a group, and R is a group word on X and H. We show that if the group word on X obtained from R by deleting all the terms from H has what we call the unique max-min property, then the group defined by P is residually finite if and only if H is residually finite (Theorem 1). We apply this to obtain new results concerning the residual finiteness of (ordinary) one-relator groups (Theorem 4). We also obtain results concerning the conjugacy problem for one-relator groups (Theorem 5), and results concerning the relative asphericity of presentations of the form P (Theorem 6).


Flocking in noisy environments

Abstract: We consider a perturbed version of the dynamics of a flock introduced by Cucker and Smale ("Emergent behaviour in flocks") and prove, under similar conditions, that nearly-alignment (a concept that is precised in the text) is achieved with a certain probability, bounded from below.


Cyclic projectors and separation theorems in idempotent convex geometry

Abstract: Semimodules over idempotent semirings like the max-plus or tropical semiring have much in common with convex cones. This analogy is particularly apparent in the case of subsemimodules of the n-fold cartesian product of the max-plus semiring it is known that one can separate a vector from a closed subsemimodule that does not contain it. We establish here a more general separation theorem, which applies to any finite collection of closed semimodules with a trivial intersection. In order to prove this theorem, we investigate the spectral properties of certain nonlinear operators called here idempotent cyclic projectors. These are idempotent analogues of the cyclic nearest-point projections known in convex analysis. The spectrum of idempotent cyclic projectors is characterized in terms of a suitable extension of Hilbert's projective metric. We deduce as a corollary of our main results the idempotent analogue of Helly's theorem.


Probabilistic Representations of Solutions of the Forward Equations

Abstract: In this paper we prove a stochastic representation for solutions of the evolution equation $ \partial_t \psi_t = {1/2}L^*\psi_t $ where $ L^* $ is the formal adjoint of an elliptic second order differential operator with smooth coefficients corresponding to the infinitesimal generator of a finite dimensional diffusion $ (X_t).$ Given $ \psi_0 = \psi $, a distribution with compact support, this representation has the form $ \psi_t = E(Y_t(\psi))$ where the process $ (Y_t(\psi))$ is the solution of a stochastic partial differential equation connected with the stochastic differential equation for $ (X_t) $ via Ito's formula.


Automorphisms of Generalized Down-Up Algebras

Abstract: A generalization of down-up algebras was introduced by Cassidy and Shelton (J. Algebra 279 (2004), no. 1), the so-called generalized down-up algebras. We describe the automorphism group of conformal Noetherian generalized down-up algebras L(f,r,s,\gamma) such that r is not a root of unity, listing explicitly the elements of the group. In the last section we apply these results to Noetherian down-up algebras, thus obtaining a characterization of the automorphism group of Noetherian down-up algebras A(\alpha, \beta, \gamma) for which the roots of the polynomial X^2-\alpha X-\beta are not both roots of unity.


Three experimental pearls in Costas arrays

Abstract: The results of 3 experiments in Costas arrays are presented, for which theoretical explanation is still not available: the number of dots on the main diagonal of exponential Welch arrays, the parity populations of Golomb arrays generated in fields of characteristic 2, and the maximal cross-correlation between pairs of Welch or Golomb arrays generated in fields of size equal to a Sophie Germain prime.


Integral degree of a ring and reduction numbers

Abstract: The supremum of reduction numbers of ideals having principal reductions is expressed in terms of the integral degree, a new invariant of the ring, which is finite provided the ring has finite integral closure. As a consequence, one obtains bounds for the Castelnuovo-Mumford regularity of the Rees algebra and for the Artin-Rees numbers.


Characterization of the Radon-Nikodym Property in terms of inverse limits

Abstract: We clarify the relation between inverse systems, the Radon-Nikodym property, the Asymptotic Norming Property of James-Ho, and the GFDA spaces introduced in our earlier paper on differentiability of Lipschitz maps into Banach spaces.


Solving SPDEs driven by colored noise: a chaos approach

Abstract: An Ito-Skorokhod bi-linear equation driven by infinitely many independent colored noises is considered in a normal triple of Hilbert spaces. The special feature of the equation is the appearance of the Wick product in the definition of the Ito-Skorokhod integral, requiring innovative approaches to computing the solution. A chaos expansion of the solution is derived and several truncations of this expansion are studied. A recursive approximation of the solution is suggested and the corresponding approximation error bound is computed.


The slice-ribbon conjecture for 3-stranded pretzel knots

Abstract: We determine the smooth concordance order of the 3-stranded pretzel knots P(p,q,r) with p,q,r odd. We show that each one of finite order is, in fact, ribbon, thereby proving the slice-ribbon conjecture for this family of knots. As corollaries we give new proofs of results first obtained by Fintushel-Stern and Casson-Gordon.


On Point Coverings of Boxes in $\mathbb R^d$

Abstract: Families of boxes in $\mathbb R^d$ are considered. In the paper an upper bound on the size of a minimum transversal in terms of the space dimension and the independence number of the given family was improved.


Spectral stability of noncharacteristic isentropic Navier-Stokes boundary layers

Abstract: Building on work of Barker, Humpherys, Lafitte, Rudd, and Zumbrun in the shock wave case, we study stability of compressive, or "shock-like", boundary layers of the isentropic compressible Navier-Stokes equations with gamma-law pressure by a combination of asymptotic ODE estimates and numerical Evans function computations. Our results indicate stability for gamma in the interval [1, 3] for all compressive boundary-layers, independent of amplitude, save for inflow layers in the characteristic limit (not treated). Expansive inflow boundary-layers have been shown to be stable for all amplitudes by Matsumura and Nishihara using energy estimates. Besides the parameter of amplitude appearing in the shock case, the boundary-layer case features an additional parameter measuring displacement of the background profile, which greatly complicates the resulting case structure. Moreover, inflow boundary layers turn out to have quite delicate stability in both large-displacement and large-amplitude limits, necessitating the additional use of a mod-two stability index studied earlier by Serre and Zumbrun in order to decide stability.


Twisted conjugacy classes in nilpotent groups

Abstract: A group is said to have the $R_\infty$ property if every automorphism has an infinite number of twisted conjugacy classes. We study the question whether $G$ has the $R_\infty$ property when $G$ is a finitely generated torsion-free nilpotent group. As a consequence, we show that for every positive integer $n\ge 5$, there is a compact nilmanifold of dimension $n$ on which every homeomorphism is isotopic to a fixed point free homeomorphism. As a by-product, we give a purely group theoretic proof that the free group on two generators has the $R_\infty$ property. The $R_{\infty}$ property for virtually abelian and for $\mathcal C$-nilpotent groups are also discussed.


Separating populations with wide data: A spectral analysis

Abstract: In this paper, we consider the problem of partitioning a small data sample drawn from a mixture of $k$ product distributions. We are interested in the case that individual features are of low average quality $\gamma$, and we want to use as few of them as possible to correctly partition the sample. We analyze a spectral technique that is able to approximately optimize the total data size--the product of number of data points $n$ and the number of features $K$--needed to correctly perform this partitioning as a function of $1/\gamma$ for $K>n$. Our goal is motivated by an application in clustering individuals according to their population of origin using markers, when the divergence between any two of the populations is small.


Undercomplete Blind Subspace Deconvolution via Linear Prediction

Abstract: We present a novel solution technique for the blind subspace deconvolution (BSSD) problem, where temporal convolution of multidimensional hidden independent components is observed and the task is to uncover the hidden components using the observation only. We carry out this task for the undercomplete case (uBSSD): we reduce the original uBSSD task via linear prediction to independent subspace analysis (ISA), which we can solve. As it has been shown recently, applying temporal concatenation can also reduce uBSSD to ISA, but the associated ISA problem can easily become `high dimensional' [1]. The new reduction method circumvents this dimensionality problem. We perform detailed studies on the efficiency of the proposed technique by means of numerical simulations. We have found several advantages: our method can achieve high quality estimations for smaller number of samples and it can cope with deeper temporal convolutions.


Non-archimedean analytification of algebraic spaces

Abstract: It is now a classical result that an algebraic space locally of finite type over $\mathbf{C}$ is analytifiable if and only if it is locally separated. In this paper we study non-archimedean analytifications of algebraic spaces. We construct a quotient for any etale non-archimedean analytic equivalence relation whose diagonal is a closed immersion, and deduce that any separated algebraic space locally of finite type over any non-archimedean field $k$ is analytifiable in both the category of rigid spaces and the category of analytic spaces over $k$. Also, though local separatedness remains a necessary condition for analytifiability in either of these categories, we present many surprising examples of non-analytifiable locally separated smooth algebraic spaces over $k$ that can even be defined over the prime field.


The SSM Toolbox for Matlab

Abstract: State Space Models (SSM) is a MATLAB 7.0 software toolbox for doing time series analysis by state space methods. The software features fully interactive construction and combination of models, with support for univariate and multivariate models, complex time-varying (dynamic) models, non-Gaussian models, and various standard models such as ARIMA and structural time-series models. The software includes standard functions for Kalman filtering and smoothing, simulation smoothing, likelihood evaluation, parameter estimation, signal extraction and forecasting, with incorporation of exact initialization for filters and smoothers, and support for missing observations and multiple time series input with common analysis structure. The software also includes implementations of TRAMO model selection and Hillmer-Tiao decomposition for ARIMA models. The software will provide a general toolbox for doing time series analysis on the MATLAB platform, allowing users to take advantage of its readily available graph plotting and general matrix computation capabilities.


Products of Brauer Severi surfaces

Abstract: Let $\{P_i\}_{1 \leq i \leq r}$ and $\{Q_i\}_{1 \leq i \leq r}$ be two collections of Brauer Severi surfaces (resp. conics) over a field $k$. We show that the subgroup generated by the $P_i's$ in $Br(k)$ is the same as the subgroup generated by the $Q_i's$ \iff $\Pi P_i $ is birational to $\Pi Q_i$. Moreover in this case $\Pi P_i$ and $\Pi Q_i$ represent the same class in $M(k)$, the Grothendieck ring of $k$-varieties. The converse holds if $char(k)=0$. Some of the above implications also hold over a general noetherian base scheme.


Complexity test modules

Abstract: A method is provided for computing an upper bound of the complexity of a module over a local ring, in terms of vanishing of certain cohomology modules. We then specialize to complete intersections, which are precisely the rings over which all modules have finite complexity.


Stability of Hodge bundles and a numerical characterization of Shimura varieties

Abstract: Consider a family f:A --> U of g-dimensional abelian varieties over a quasiprojective manifold U. Suppose that the induced map from U to the moduli scheme of polarized abelian varieties is generically finite and that there is a projective manifold Y, containing U as the complement of a normal crossing divisor S, such that the sheaf of logarithmic one forms is nef and that its determinant is ample with respect to U. We characterize whether $U$ is a Shimura variety by numerical data attached to the variation of Hodge structures, rather than by properties of the map from U to the moduli scheme or by the existence of CM points. More precisely, we show that U is a Shimura variety, if and only if two conditions hold. First, each irreducible local subsystem V of the complex weight one variation of Hodge structures is either unitary or satisfies the Arakelov equality. Secondly, for each factor M in the universal cover of U whose tangent bundle behaves like the one of a complex ball, an iterated Kodaira-Spencer map associated with V has minimal possible length in the direction of M.


Classification of hyperfinite factors up to completely bounded isomorphism of their preduals

Abstract: In this paper we consider the following problem: When are the preduals of two hyperfinite (=injective) factors $\M$ and $\N$ (on separable Hilbert spaces) cb-isomorphic (i.e., isomorphic as operator spaces)? We show that if $\M$ is semifinite and $\N$ is type III, then their preduals are not cb-isomorphic. Moreover, we construct a one-parameter family of hyperfinite type III$_0$-factors with mutually non cb-isomorphic preduals, and we give a characterization of those hyperfinite factors $\M$ whose preduals are cb-isomorphic to the predual of the unique hyperfinite type III$_1$-factor. In contrast, Christensen and Sinclair proved in 1989 that all infinite dimensional hyperfinite factors with separable preduals are cb-isomorphic. More recently Rosenthal, Sukochev and the first-named author proved that all hyperfinite type III$_\lambda$-factors, where $0< \lambda\leq 1$, have cb-isomorphic preduals.


Stationarity and Self-similarity Characterization of the Set-indexed Fractional Brownian Motion

Abstract: The set-indexed fractional Brownian motion (sifBm) has been defined by Herbin-Merzbach (2006) for indices that are subsets of a metric measure space. In this paper, the sifBm is proved to statisfy a strenghtened definition of increment stationarity. This new definition for stationarity property allows to get a complete characterization of this process by its fractal properties: The sifBm is the only set-indexed Gaussian process which is self-similar and has stationary increments. Using the fact that the sifBm is the only set-indexed process whose projection on any increasing path is a one-dimensional fractional Brownian motion, the limitation of its definition for a self-similarity parameter 0<H<1/2 is studied, as illustrated by some examples. When the indexing collection is totally ordered, the sifBm can be defined for 0<H<1.


An exact geometric mass formula

Abstract: We show an exact geometric mass formula for superspecial points in the reduction of any quaternionic Shimura variety modulo at a good prime $p$.


Baxter operator and Archimedean Hecke algebra

Abstract: In this paper we introduce Baxter integral Q-operators for finite-dimensional Lie algebras gl(n+1) and so(2n+1). Whittaker functions corresponding to these algebras are eigenfunctions of the Q-operators with the eigenvalues expressed in terms of Gamma-functions. The appearance of the Gamma-functions is one of the manifestations of an interesting connection between Mellin-Barnes and Givental integral representations of Whittaker functions, which are in a sense dual to each other. We define a dual Baxter operator and derive a family of mixed Mellin-Barnes-Givental integral representations. Givental and Mellin-Barnes integral representations are used to provide a short proof of the Friedberg-Bump and Bump conjectures for G=GL(n+1) proved earlier by Stade. We also identify eigenvalues of the Baxter Q-operator acting on Whittaker functions with local Archimedean L-factors. The Baxter Q-operator introduced in this paper is then described as a particular realization of the explicitly defined universal Baxter operator in the spherical Hecke algebra H(G(R),K), K being a maximal compact subgroup of G. Finally we stress an analogy between Q-operators and certain elements of the non-Archimedean Hecke algebra H(G(Q_p),G(Z_p)).


Relations between tautological cycles on Jacobians

Abstract: We study tautological cycle classes on the Jacobian of a curve. We prove a new result about the ring of tautological classes on a general curve that allows, among other things, easy dimension calculations and leads to some general results about the structure of this ring. Next we obtain a vanishing result for some of the generating classes p_i; this gives an improvement of an earlier result of Herbaut. Finally we lift a result of Herbaut and van der Geer-Kouvidakis to the Chow ring (as opposed to its quotient modulo algebraic equivalence) and we give a method to obtain further explicit cycle relations. As an ingredient for this we prove a theorem about how Polishchuk's operator D lifts to the tautological subalgebra of Chow(J).


The size of isoperimetric surfaces in 3-manifolds and a rigidity result for the upper hemisphere

Abstract: We characterize the standard $\mathbb{S}^3$ as the closed Ricci-positive 3-manifold with scalar curvature at least 6 having isoperimetric surfaces of largest area: $4\pi$. As a corollary we answer in the affirmative an interesting special case of a conjecture of Min-Oo's on the scalar curvature rigidity of the upper hemisphere..


Peak Quasisymmetric Functions and Eulerian Enumeration

Abstract: Via duality of Hopf algebras, there is a direct association between peak quasisymmetric functions and enumeration of chains in Eulerian posets. We study this association explicitly, showing that the notion of $\cd$-index, long studied in the context of convex polytopes and Eulerian posets, arises as the dual basis to a natural basis of peak quasisymmetric functions introduced by Stembridge. Thus Eulerian posets having a nonnegative $\cd$-index (for example, face lattices of convex polytopes) correspond to peak quasisymmetric functions having a nonnegative representation in terms of this basis. We diagonalize the operator that associates the basis of descent sets for all quasisymmetric functions to that of peak sets for the algebra of peak functions, and study the $g$-polynomial for Eulerian posets as an algebra homomorphism.


Metric Embedding for Nearest Neighbor Classification

Abstract: The distance metric plays an important role in nearest neighbor (NN) classification. Usually the Euclidean distance metric is assumed or a Mahalanobis distance metric is optimized to improve the NN performance. In this paper, we study the problem of embedding arbitrary metric spaces into a Euclidean space with the goal to improve the accuracy of the NN classifier. We propose a solution by appealing to the framework of regularization in a reproducing kernel Hilbert space and prove a representer-like theorem for NN classification. The embedding function is then determined by solving a semidefinite program which has an interesting connection to the soft-margin linear binary support vector machine classifier. Although the main focus of this paper is to present a general, theoretical framework for metric embedding in a NN setting, we demonstrate the performance of the proposed method on some benchmark datasets and show that it performs better than the Mahalanobis metric learning algorithm in terms of leave-one-out and generalization errors.


A symmetric Finsler space with Chern connection

Abstract: We define a symmetry for a Finsler space with Chern connection and investigate its implementation and properties and find a relation between them and flag curvature.


On elliptic differential operators with shifts

Abstract: We give an index formula for elliptic differential operators whose coefficients include shifts forming an infinite group.


Homogeneous geodesics in homogeneous Finsler spaces

Abstract: In this paper, we study homogeneous geodesics in homogeneous Finsler spaces. We first give a simple criterion that characterizes geodesic vectors. We show that the geodesics on a Lie group, relative to a bi-invariant Finsler metric, are the cosets of the one-parameter subgroups. The existence of infinitely many homogeneous geodesics on compact semi-simple Lie group is established. We introduce the notion of naturally reductive homogeneous Finsler space. As a special case, we study homogeneous geodesics in homogeneous Randers spaces. Finally, we study some curvature properties of homogeneous geodesics. In particular, we prove that the S-curvature vanishes along the homogeneous geodesics.


Wedderburn polynomials over division rings, II

Abstract: A polynomial $f(t)$ in an Ore extension $K[t;S,D]$ over a division ring $K$ is a Wedderburn polynomial if $f(t)$ is monic and is the minimal polynomial of an algebraic subset of $K$. These polynomials have been studied in "Wedderburn polynomials over division rings,I (Journal of Pure and Applied Algebra, Vol. 186, (2004), 43-76). In this paper, we continue this study and give some applications to triangulation, diagonalization and eigenvalues of matrices over a division ring in the general setting of $(S,D)$-pseudo-linear transformations. In the last section we introduce and study the notion of $G$-algebraic sets which, in particular, permits generalization of Wedderburn's theorem relative to factorization of central polynomials.


On probabilities for separating sets of order statistics

Abstract: Consider a set of order statistics that arise from sorting samples from two different populations, each with their own, possibly different distribution function. The probability that these order statistics fall in disjoint, ordered intervals, and that of the smallest statistics, a certain number come from the first populations, are given in terms of the two distribution functions. The result is applied to computing the joint probability of the number of rejections and the number of false rejections for the Benjamini-Hochberg false discovery rate procedure.


Avoidable Sets in The Bicyclic Inverse Semigroup

Abstract: A subset $U$ of a set $S$ with a binary operation is called {\it avoidable} if $S$ can be partitioned into two subsets $A$ and $B$ such that no element of $U$ can be written as a product of two distinct elements of $A$ or as the product of two distinct elements of $B$. The avoidable sets of the bicyclic inverse semigroup are classified.


The diameter of random Cayley digraphs of given degree

Abstract: We consider random Cayley digraphs of order $n$ with uniformly distributed generating set of size $k$. Specifically, we are interested in the asymptotics of the probability such a Cayley digraph has diameter two as $n\to\infty$ and $k=f(n)$. We find a sharp phase transition from 0 to 1 at around $k = \sqrt{n \log n}$. In particular, if $f(n)$ is asymptotically linear in $n$, the probability converges exponentially fast to 1.


Cohen-Macaulayness and computation of Newton graded toric rings

Abstract: Let $H$ be a positive semigroup in $\mathbb{Z}^d$ generated by $A$, and let $K[H]$ be the associated semigroup ring over a field $K$. We investigate heredity of the Cohen-Macaulay property from $K[H]$ to both its $A$-Newton graded ring and to its face rings. We show by example that neither one inherits in general the Cohen-Macaulay property. On the positive side we show that for every $H$ there exist generating sets $A$ for which the Newton graduation preserves Cohen-Macaulayness. This gives an elementary proof for an important vanishing result on $A$-hypergeometric Euler-Koszul homology. As a tool for our investigations we develop an algorithm to compute algorithmically the Newton filtration on a toric ring.


Crossed products of k-graph C*-algebras by Z^l

Abstract: An action of Z^l by automorphisms of a k-graph induces an action of Z^l by automorphisms of the corresponding k-graph C*-algebra. We show how to construct a (k+l)-graph whose C*-algebra coincides with the crossed product of the original k-graph algebra by Z^l. We then investigate the structure of the crossed-product C*-algebra.


The mean curvature flow for isoparametric submanifolds

Abstract: A submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant. We study the mean curvature flow with initial data an isoparametric submanifold in Euclidean space and sphere. We show that the mean curvature flow preserves the isoparametric condition, develops singularities in finite time, and converges in finite time to a smooth submanifold of lower dimension. We also give a precise description of the collapsing.


A formula for the hypergeometric function of type $BC_n$

Abstract: Formulae of Berezin and Karpelevic for the radial parts of invariant differential operators and the spherical function on a complex Grassmann manifold are generalized to the hypergeometric functions associated with root system of type $BC_n$ under condition that the multiplicity of the middle roots is zero or one.


A phase transition behavior for Brownian motions interacting through their ranks

Abstract: Consider a time-varying collection of n points on the positive real axis, modeled as exponentials of n Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. If at each time point we divide the points by their sum, under suitable assumptions the rescaled point process converges to a stationary distribution (depending on n and the vector of drifts) as time goes to infinity. This stationary distribution can be exactly computed using a recent result of Pal and Pitman. The model and the rescaled point process are both central objects of study in models of equity markets introduced by Banner, Fernholz, and Karatzas. In this paper, we look at the behavior of this point process under the stationary measure as $n$ tends to infinity. Under a certain `continuity at the edge' condition on the drifts, we show that one of the following must happen: either (i) all points converge to zero, or (ii) the maximum goes to one and the rest go to zero, or (iii) the processes converge in law to a non-trivial Poisson-Dirichlet distribution. The proof employs, among other things, techniques from Talagrand's analysis of the low temperature phase of Derrida's Random Energy Model of spin glasses. The main result establishes a universality property for the BFK models and aids in explicit asymptotic computations using known results about the Poisson-Dirichlet law.


Reparametrizations of Continuous Paths

Abstract: A reparametrization (of a continuous path) is given by a surjective weakly increasing self-map of the unit interval. We show that the monoid of reparametrizations (with respect to compositions) can be understood via ``stop-maps'' that allow to investigate compositions and factorizations, and we compare it to the distributive lattice of countable subsets of the unit interval. The results obtained are used to analyse the space of traces in a topological space, i.e., the space of continuous paths up to reparametrization equivalence. This space is shown to be homeomorphic to the space of regular paths (without stops) up to increasing reparametrizations. Directed versions of the results are important in directed homotopy theory.


Algebraic definition of Holonomy on Poisson Manifold

Abstract: We give an algebraic construction of connection on the symplectic leaves of Poisson manifold, introduced in \cite{Ginzburg}. This construction is suitable for the definition of the linearized holonomy on a regular symplectic foliation.


An explicit stationary phase formula for the local formal Fourier-Laplace transform

Abstract: We give an explicit formula (i.e., a formal stationary phase formula) for the local Fourier-Laplace transform of a formal germ of meromorphic connection of one complex variable with a possibly irregular singularity. This is a complex analogue of the formulas in the preprint math/0702436v1.


Cusps of Hilbert modular varieties

Abstract: Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that every Sol 3-manifold is diffeomorphic to a cusp cross-section of a (generalized) Hilbert modular surface. We also deduce an obstruction to geometric bounding in this setting. Consequently, there exist Sol 3-manifolds that cannot arise as a cusp cross-section of a 1-cusped nonsingular Hilbert modular surface.


Bohr and Rogosinski abscissas for ordinary Dirichlet series

Abstract: We prove that the abscissas of Bohr and Rogosinski for ordinary Dirichlet series, mapping the right half-plane into the bounded convex domain $G\subset \mathbb{C} $ are independent of the domain $G$. Furthermore, we obtain new estimates about these abscissas.


An introduction to Seiberg-Witten theory on closed 3-manifolds

Abstract: This is a version of the author's diploma thesis written at the University of Cologne in 2002/03. The topic is the construction of Seiberg-Witten invariants of closed 3-manifolds. In analogy to the four dimensional case, the structure of the moduli space is investigated. The Seiberg-Witten invariants are defined and their behaviour under deformation of the Riemannian metric is analyzed. Since it is essentially an exposition of results which were already known during the time of writing, the thesis has not been published. In particular, the author does not claim any originality concerning the results. Moreover, new developments of the theory are not included. However, the detailed account--together with the appendices on the required functional analytic and geometric background--might be of interest for people starting to work in the area of gauge field theory.


Projective structures and projective bundles over compact Riemann surfaces

Abstract: A projective structure on a compact Riemann surface X of genus g is given by an atlas with transition functions in PGL(2,C). Equivalently, a projective structure is given by a projective sl(2,C)-bundle over X equipped with a section s and a foliation F which is both transversal to the fibers and the section s. From this latter geometric bundle picture, we survey on classical problems and results on projective structures. We will give a complete description of projective (actually affine) structures on the torus with an explicit versal family of foliated bundle picture.


Low Regularity local well-posedness for the 1+3 dimensional Dirac-Klein-Gordon system

Abstract: We prove that the Cauchy problem for the Dirac-Klein-Gordon system of equations in 1+3 dimensions is locally well-posed in a range of Sobolev spaces for the Dirac spinor and the meson field. The result contains and extends the earlier known results for the same problem. Our proof relies on the null structure in the system, and bilinear spacetime estimates of Klainerman-Machedon type.


Almost Everywhere Convergence of Inverse Dunkl Transform on the Real Line

Abstract: In this paper, we will first show that the maximal operator $S_*^\alpha$ of spherical partial sums $S_R^\alpha$, associated to Dunkl transform on $\mathbb{R}$ is bounded on $L^p(\mathbb{R}, |x|^{2\alpha+1} dx)$ functions when $\frac{4(\alpha+1)}{2\alpha+3}<p<\frac{4(\alpha+1)}{2\alpha+1}$, and it implies that, for every $L^p(\mathbb{R}, |x|^{2\alpha+1} dx)$ function $f(x)$, $S_R^\alpha f(x)$ converges to $f(x)$ almost everywhere as $R\to \infty$. On the other hand we obtain a sharp version by showing that $S_*^\alpha$ is bounded from the Lorentz space $L^{p_i,1}(\mathbb{R}, |x|^{2\alpha+1})$ into $L^{p_i,\infty}(\mathbb{R}, |x|^{2\alpha+1}),\quad i=0,1$ where $p_0=\frac{4(\alpha+1)}{2\alpha+3}$ and $p_1=\frac{4(\alpha+1)}{2\alpha+1}$.


Hypergroups with Unique Alpha-Means

Abstract: Let $K$ be a commutative hypergroup and $\alpha\in \hat{K}$. We show that $K$ is $\alpha$-amenable with the unique $\alpha$-mean $m_\alpha$ if and only if $m_\alpha\in L^1(K)\cap L^2(K)$ and $\alpha$ is isolated in $\hat{K}$. In contrast to the case of amenable noncompact locally compact groups, examples of polynomial hypergroups with unique $\alpha$-means ($\alpha\not=1$) are given. Further examples emphasize that the $\alpha$-amenability of hypergroups depends heavily on the asymptotic behavior of Haar measures and characters.


On a class of $\mathrm{II}_1$ factors with at most one Cartan subalgebra

Abstract: We prove that the normalizer of any diffuse amenable subalgebra of a free group factor $L(\Bbb F_r)$ generates an amenable von Neumann subalgebra. Moreover, any II$_1$ factor of the form $Q \vt L(\Bbb F_r) $, with $Q$ an arbitrary subfactor of a tensor product of free group factors, has no Cartan subalgebras. We also prove that if a free ergodic measure preserving action of a free group $\Bbb F_r$, $2\leq r \leq \infty$, on a probability space $(X,\mu)$ is profinite then the group measure space factor $L^\infty(X)\rtimes \Bbb F_r$ has unique Cartan subalgebra, up to unitary conjugacy.


Familles fuchsiennes d'\'equations aux (q-)diff\'erences et confluence

Abstract: In a first part, we give a method for solving a family of fuchsian systems of operators of pseudo-derivations associated to a family of homographies with two parameters which unify and generalize the differential, the difference and the $q$-difference cases. In a second part, we study the problems of confluence related to these families.


Orbit-counting for nilpotent group shifts

Abstract: We study the asymptotic behaviour of the orbit-counting function and a dynamical Mertens' theorem for the full $G$-shift for a finitely-generated torsion-free nilpotent group $G$. Using bounds for the M{\"o}bius function on the lattice of subgroups of finite index and known subgroup growth estimates, we find a single asymptotic of the shape \[ \sum_{|\tau|\le N}\frac{1}{e^{h|\tau|}}\sim CN^{\alpha}(\log N)^{\beta} \] where $|\tau|$ is the cardinality of the finite orbit $\tau$. For the usual orbit-counting function we find upper and lower bounds together with numerical evidence to suggest that for actions of non-cyclic groups there is no single asymptotic in terms of elementary functions.


Cohomology of Congruence Subgroups of SL(4,\Z) II

Abstract: In a previous paper [3] we computed cohomology groups H^5 (Gamma_0 (N), \C), where Gamma_0 (N) is a certain congruence subgroup of SL (4, \Z), for a range of levels N. In this note we update this earlier work by extending the range of levels and describe cuspidal cohomology classes and additional boundary phenomena found since the publication of [3]. The cuspidal cohomology classes in this paper are the first cuspforms for GL(4) concretely constructed in terms of Betti cohomology.