The decomposition of the hypermetric cone into L-domains

Abstract: The hypermetric cone $\HYP_{n+1}$ is the parameter space of basic Delaunay polytopes in n-dimensional lattice. The cone $\HYP_{n+1}$ is polyhedral; one way of seeing this is that modulo image by the covariance map $\HYP_{n+1}$ is a finite union of L-domains, i.e., of parameter space of full Delaunay tessellations. In this paper, we study this partition of the hypermetric cone into L-domains. In particular, it is proved that the cone $\HYP_{n+1}$ of hypermetrics on n+1 points contains exactly {1/2}n! principal L-domains. We give a detailed description of the decomposition of $\HYP_{n+1}$ for n=2,3,4 and a computer result for n=5 (see Table \ref{TableDataHYPn}). Remarkable properties of the root system $\mathsf{D}_4$ are key for the decomposition of $\HYP_5$.


Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption

Abstract: We study the limit behaviour of solutions of a class of solutions of nonlinear parabolic equations with a degenerate strong absorption. We prove that two types of phenomena can occur: the pointwise singularity or the formation of razor blades (or persistent singularities).


Separable solutions of some quasilinear equations with source reaction

Abstract: We study the existence of singular separable solutions to a class of quasilinear equations with reaction term. In the 2-dim case, we use a dynamical system approach to construct our solutions.


A Subgroup of a Direct Product of Free Groups whose Dehn Function has a Cubic Lower Bound

Abstract: We establish a cubic lower bound on the Dehn function of a certain finitely presented subgroup of a direct product of 3 free groups.


Invariant rigid geometric structures and smooth projective factors

Abstract: We consider actions of non-compact simple Lie groups preserving an analytic rigid geometric structure of algebraic type on a compact manifold. The structure is not assumed to be unimodular, so an invariant measure may not exist. Ergodic stationary measures always exist, and when such a measure has full support, we show the following. 1) Either the manifold admits a smooth equivariant map onto a homogeneous projective variety, defined on an open dense conull invariant set, or the Lie algebra of the Zariski closure of the Gromov representation of the fundamental group contains a Lie subalgebra isomorphic to the Lie algebra of the acting group. As a corollary, a smooth non-trivial homogeneous projective factor does exist whenever the fundamental group of $M$ admits only virtually solvable linear representations, and thus in particular when $M$ is simply connected, regardless of the real rank. 2) There exist explicit examples showing that analytic rigid actions of certain simple groups (of real rank one) may indeed fail to have a smooth projective factor. 3) It is possible to generalize Gromov's theorem on the algebraic hull of the representation of the fundamental group of the manifold to the case of analytic rigid non-unimodular structures, for actions of simple groups of any real rank. An important ingredient in the proofs is a generalization of Gromov's centralizer theorem beyond the case of invariant measures.


Singular integral operators on variable Lebesgue spaces with radial oscillating weights

Abstract: We prove a Fredholm criterion for operators in the Banach algebra of singular integral operators with matrix piecewise continuous coefficients acting on a variable Lebesgue space with a radial oscillating weight over a logarithmic Carleson curve. The local spectra of these operators are massive and have a shape of spiralic horns depending on the value of the variable exponent, the spirality indices of the curve, and the Matuszewska-Orlicz indices of the weight at each point. These results extend (partially) the results of A. B\"ottcher, Yu. Karlovich, and V. Rabinovich for standard Lebesgue spaces to the case of variable Lebesgue spaces.


A weight function theory of zero order basis function interpolants and smoothers

Abstract: I develop a weight func theory of zero order basis func interpolants and smoothers.**Ch1 Basis funcs and data spaces are defined using wt funcs. Data (native)spaces are used to formulate the variational problems which define our interpolants /smoothers. Introduce the tensor prod extended B-splines.**Ch2 Prove the p'twise convergence of the minimal norm basis func interpol to its data func and obtain orders of converg. Data func spaces for the B-splines are locally Sobolev spaces.**Ch3 Another set of error estims for basis func interpol. Use distrib'n Taylor expansion of exp(i(a,x)).**Ch4 Derive local interpol errors for data funcs with bounded first derivs.**Ch5 Introduce class of tensor prod wt funcs which I call the central diff wt funcs - related to the B-splines. Apply theory to these wt funcs to obtain interpol converge results. The data func spaces are locally Sobolev spaces. **Ch6 A non-param variat smoothing problem studied with special interest in converge of smoother to its data func. This smoother is the min norm interpol stabilized by a smoothing coeff.**Ch7: A non-parametric, scalable, smoothing problem shown to converge to its data func. We discuss the software which implements these algorithms.**Ch8: Characterizes bounded linear functionals on data space.**Ch9 Bilinear form used to characterize the bounded linear functionals on the data spaces generated by the wt funcs.**Ch10 We derive an upper bound for the deriv of the 1-dim (scaled) hat basis func smoother assuming the data func has bounded derivs and large supp wrt. the data region.**Ch11: In one dim only; the local data funcs are assumed to have bounded derivs on the data region and consider a scaled hat basis func. If the basis func has large enough supp wrt. the data region then we show the order of converg of the interpol is 1. **Ch12: Exten ops derived based on Wloka; assumes rectangle condit.


A new modified Galerkin method for the two-dimensional Navier-Stokes equations

Abstract: We present a new type of modified Galerkin method. It is a construction with several (inductively defined) levels, that provides approximate solutions of increasing accuracy with every new level. These solutions are constructed as approximations of the so called induced trajectories (notion on which the definition of a class of approximate inertial manifolds used in the nonlinear and postprocessed Galerkin methods is based).


A weight function theory of positive order basis function interpolants and smoothers

Abstract: In this document I develop a weight function theory of positive order basis function interpolants and smoothers. **In Chapter 1 the basis functions and data spaces are defined directly using weight functions. The data spaces are used to formulate the variational problems which define the interpolants and smoothers discussed in later chapters. The theory is illustrated using some standard examples of radial basis functions and two classes of weight functions I will call the tensor product extended B-splines and the central difference weight functions. **In Chapter 2 I derive modified inverse-Fourier transform formulas for the basis functions and the data functions (native spaces) and to use these formulas to obtain bounds for the rates of increase of these functions and their derivatives near infinity. **Chapter 3 shows how to prove functions are basis functions without using the awkward space of test functions $S_{0,n}$ which are infinitely smooth functions of rapid decrease with several zero-valued derivatives at the origin. Worked examples include several classes of well-known radial basis functions. **In Chapter 4 we prove the existence and uniqueness of a solution to the minimal seminorm interpolation problem. We then derive orders for the pointwise convergence of the interpolant to its data function as the density of the data increases. **In Chapter 5 a well-known non-parametric variational smoothing problem will be studied with special interest in the order of pointwise convergence of the smoother to its data function. This smoothing problem is the minimal norm interpolation problem stabilized by a smoothing coefficient. **In Chapter 6 a non-parametric, scalable, variational smoothing problem will be studied, with special interest in its order of pointwise convergence to its data function. **In Chapter 7 we study the bounded linear functionals on the data spaces.


Cluster Complexes via Semi-Invariants

Abstract: We define and study virtual representation spaces having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the First Fundamental Theorem, the Saturation Theorem and the Canonical Decomposition Theorem. In the special case of Dynkin quivers with n vertices this gives the fundamental interrelationship between supports of the semi-invariants and the Tilting Triangulation of the (n-1)-sphere.


Some pseudo-Anosov maps on punctured Riemann surfaces represented by multi-twists

Abstract: This paper has been withdrawn by the author


Compositional Bernoulli numbers

Abstract: We define and study the combinatorial properties of compositional Bernoulli numbers and polynomials within the framework of rational combinatorics.


Values at s=-1 of L-functions for multi-quadratic extensions of number fields, and the fitting ideal of the tame kernel

Abstract: Fix a Galois extension E/F of totally real number fields such that the Galois group G has exponent 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in E, let S_E denote the primes of E lying above those in S, and let O_E^S denote the ring of S_E-integers of E. We then compare the Fitting ideal of K_2(O_E^S) as a Z[G]-module with a higher Stickelberger ideal. The two extend to the same ideal in the maximal order of Q[G], and hence in Z[1/2][G]. Results in Z[G] are obtained under the assumption of the Birch-Tate conjecture, especially for biquadratic extensions, where we compute the index of the higher Stickelberger ideal. We find a sufficient condition for the Fitting ideal to contain the higher Stickelberger ideal in the case where E is a biquadratic extension of F containing the first layer of the cyclotomic Z_2-extension of F, and describe a class of biquadratic extensions of F=Q that satisfy this condition.


Existence of quasi-arcs

Abstract: We show that doubling, linearly connected metric spaces are quasi-arc connected. This gives a new and short proof of a theorem of Tukia.


A separable non-remainder of H

Abstract: We prove that there is a compact separable continuum that (consistently) is not a remainder of the real line.


Triangularization of a Jordan Algebra of Schatten Operators

Abstract: We show that a Jordan algebra of compact quasinilpotent operators which contains a nonzero trace class operator has a common invariant subspace. As a consequence of this result, we obtain that a Jordan algebra of quasinilpotent Schatten operators is simultaneously triangularizable.


The Crepant Resolution Conjecture for 3-dimensional flags modulo an involution

Abstract: After fixing a non-degenerate bilinear form on a vector space V we define an involution of the manifold of flags F in V by taking a flag to its orthogonal complement. When V is of dimension 3 we check that the Crepant Resolution Conjecture of J. Bryan and T. Graber holds: the genus zero (orbifold) Gromov-Witten potential function of [F / Z_2] agrees (up to unstable terms) with the genus zero Gromov-Witten potential function of a crepant resolution Y of the quotient scheme F / Z_2, after setting a quantum parameter to -1, making a linear change of variables, and analytically continuing coefficients. The crepant resolution Y (a hypersurface in the Hilbert scheme Hilb^2 P^2) is the projectivization of a novel rank 2 vector bundle over P^2.


The mass-critical nonlinear Schr\"odinger equation with radial data in dimensions three and higher

Abstract: We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^{4/d} u$ for large spherically symmetric L^2_x(R^d) initial data in dimensions $d\geq 3$. In the focusing case we require that the mass is strictly less than that of the ground state. As a consequence, we obtain that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time.


Embeddings of discrete groups and the speed of random walks

Abstract: For a finitely generated group G and a banach space X let \alpha^*_X(G) (respectively \alpha^#_X(G)) be the supremum over all \alpha\ge 0 such that there exists a Lipschitz mapping (respectively an equivariant mapping) f:G\to X and c>0 such that for all x,y\in G we have \|f(x)-f(y)\|\ge c\cdot d_G(x,y)^\alpha. In particular, the Hilbert compression exponent (respectively the equivariant Hilbert compression exponent) of G is \alpha^*(G)=\alpha^*_{L_2}(G) (respectively \alpha^#(G)= \alpha_{L_2}^#(G)). We show that if X has modulus of smoothness of power type p, then \alpha^#_X(G)\le \frac{1}{p\beta^*(G)}. Here \beta^*(G) is the largest \beta\ge 0 for which there exists a set of generators S of G and c>0 such that for all t\in \N we have \E\big[d_G(W_t,e)\big]\ge ct^\beta, where \{W_t\}_{t=0}^\infty is the canonical simple random walk on the Cayley graph of G determined by S, starting at the identity element. This result is sharp when X=L_p, generalizes a theorem of Guentner and Kaminker and answers a question posed by Tessera. We also show that if \alpha^*(G)\ge 1/2 then \alpha^*(G\bwr \Z)\ge \frac{2\alpha^*(G)}{2\alpha^*(G)+1}. This improves the previous bound due to Stalder and ValetteWe deduce that if we write \Z_{(1)}= \Z and \Z_{(k+1)}\coloneqq \Z_{(k)}\bwr \Z then \alpha^*(\Z_{(k)})=\frac{1}{2-2^{1-k}}, and use this result to answer a question posed by Tessera in on the relation between the Hilbert compression exponent and the isoperimetric profile of the balls in G. We also show that the cyclic lamplighter groups C_2\bwr C_n embed into L_1 with uniformly bounded distortion, answering a question posed by Lee, Naor and Peres. Finally, we use these results to show that edge Markov type need not imply Enflo type.


On the point spectrum of some perturbed differential operators with periodic coefficients

Abstract: Finiteness of the point spectrum of linear operators acting in a Banach space is investigated from point of view of perturbation theory. In the first part of the paper we present an abstract result based on analytical continuation of the resolvent function through continuous spectrum. In the second part, the abstract result is applied to differential operators which can be represented as a differential operator with periodic coefficients perturbed by an arbitrary subordinated differential operator.


The effect of memory on functional large deviations of infinite moving average processes

Abstract: The large deviations of an infinite moving average process with exponentially light tails are very similar to those of an i.i.d. sequence as long as the coefficients decay fast enough. If they do not, the large deviations change dramatically. We study this phenomenon in the context of functional large, moderate and huge deviation principles.


Asymptotic Curvature Decay and Removal of Singularities of Bach-Flat Metrics

Abstract: We prove a removal of singularities result for Bach-flat metrics in dimension 4 under the assumption of bounded L^2 norm of curvature, bounded Sobolev constant and a volume growth bound. This result extends the removal of singularities result for special classes of Bach-flat metrics obtained in \cite{TVMOD}. For the proof we analyze the decay rates of solutions to the Bach-flat equation linearized around a flat metric. This classification is used to prove that Bach-flat cones are in fact ALE of order $\tau$ for any $\tau < 2$. This result is then used to prove the removal of singularities theorem.


Eriksson's numbers game on certain edge-weighted three-node cyclic graphs

Abstract: The numbers game is a one-player game played on a finite simple graph with certain ``amplitudes'' assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. This game and its interactions with Coxeter/Weyl group theory and Lie theory have been studied by many authors. Following Eriksson, we allow the amplitudes on graph edges to be certain real numbers. Games played on such graphs are ``E-games.'' We show that for certain such three-node cyclic graphs, any numbers game will diverge when played from an initial assignment of nonnegative real numbers not all zero. This result is a key step in a Dynkin diagram classification (obtained elsewhere) of all E-game graphs which meet a certain finiteness requirement.


Volume-preserving mean curvature flow of revolution hypersurfaces in a Rotationally Symmetric Space

Abstract: In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces (tgh for short). Requiring that, for each time t, the evolving hypersurface M_t meets such tgh ortogonally, we prove that: a) the flow exists while M_t does not touch the axis of rotation; b) throughout the time interval of existence, b1) the generating curve of M_t remains a graph, and b2) the averaged mean curvature is double side bounded by positive constants; c) the singularity set (if non-empty) is finite and discrete along the axis; d) under a suitable hypothesis relating the enclosed volume to the n-volume of M, we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature.


Operad of formal homogeneous spaces and Bernoulli numbers

Abstract: It is shown that for any morphism, i: g --> h, of Lie algebras the vector space underlying the Lie algebra h is canonically a g-homogeneous formal manifold with the action of g being highly nonlinear and twisted by Bernoulli numbers. This fact is obtained from the study of a 2-coloured operad of formal homogeneous spaces and its minimal resolution, and is used to give a new conceptual explanation of both Ziv Ran's Jacobi-Bernoulli complex and Fiorenza-Manetti's L-infinity algebra structure on the mapping cone of a morphism of two Lie algebras. All these constructions are iteratively extended to the case of a morphism of arbitrary L-infinity algebras.


Uniform Sobolev inequalities for manifolds evolving by Ricci flow

Abstract: Let M be a compact n-dimensional manifold, $n\ge 2$, with metric g(t) evolving by the Ricci flow $\partial g_{ij}/\partial t=-2R_{ij}$ in (0,T) for some $T\in\Bbb{R}^+\cup\{\infty\}$ with $g(0)=g_0$. Let $\lambda_0(g_0)$ be the first eigenvalue of the operator $-\Delta_{g_0} +\frac{R(g_0)}{4}$ with respect to g_0. We extend a recent result of R. Ye and prove uniform logarithmic Sobolev inequality and uniform Sobolev inequalities along the Ricci flow for any $n\ge 2$ when either $T<\infty$ or $\lambda_0(g_0)>0$. As a consequence we extend Perelman's local $\kappa$-noncollapsing result along the Ricci flow for any $n\ge 2$ in terms of upper bound for the scalar curvature when either $T<\infty$ or $\lambda_0(g_0)>0$.


Self-similar carpets over finite fields

Abstract: Some linear dynamical systems over finite fields are studied and the self-similar character of their development is proved. Connections with aperiodic tilings, Delanoy numbers and other topics are also proved. The prime fields F_p have a canonical presentation as sets of self-similar carpets. The multiplicative inverse corresponds to mirroring.


Permanents of Circulants: a Transfer Matrix Approach (Expanded Version)

Abstract: Calculating the permanent of a (0,1) matrix is a #P-complete problem but there are some classes of structured matrices for which the permanent is calculable in polynomial time. The most well-known example is the fixed-jump (0,1) circulant matrix which, using algebraic techniques, was shown by Minc to satisfy a constant-coefficient fixed-order recurrence relation. In this note we show how, by interpreting the problem as calculating the number of cycle-covers in a directed circulant graph, it is straightforward to reprove Minc's result using combinatorial methods. This is a two step process: the first step is to show that the cycle-covers of directed circulant graphs can be evaluated using a transfer matrix argument. The second is to show that the associated transfer matrices, while very large, actually have much smaller characteristic polynomials than would a-priori be expected. An important consequence of this new viewpoint is that, in combination with a new recursive decomposition of circulant-graphs, it permits extending Minc's result to calculating the permanent of the much larger class of circulant matrices with non-fixed (but linear) jumps. It also permits us to count other types of structures in circulant graphs, e.g., Hamiltonian Cycles.


An explicit estimate on multiplicity truncation in the second main theorem for holomorphic curves encountering hypersurfaces in general position in projective space

Abstract: Yan and Chen proved a weak Cartan-type second main theorem for holomorphic curves meeting hypersurfaces in projective space that included truncated counting functions. Here we give an explicit estimate for the level of truncation.


Planar graphs and covers

Abstract: Planar locally finite graphs which are almost vertex transitive are discussed. If the graph is 3-connected and has at most one end then the group of automorphisms is a planar discontinuous group and its structure is well-known. A general result is obtained for such graphs where no restriction is put on the number of ends. It is shown that such a graph can be built up from one ended or finite planar graphs in a precise way. The results give a classification of the finitely generated groups with planar Cayley graphs.


Spherical Nilpotent Orbits in Positive Characteristic

Abstract: Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this note we classify all the spherical nilpotent G-orbits in the Lie algebra of G. The classification is the same as in the characteristic zero case obtained by D.I. Panyushev in 1994: for e a nilpotent element in the Lie algebra of G, the G-orbit G.e is spherical if and only if the height of e is at most 3.


Generalized Cherednik-Macdonald identities

Abstract: We derive generalizations of the Cherednik-Macdonald constant term identities associated to root systems which depend, besides on the usual multiplicity function, symmetrically on two quasi-periods. They are natural analogues of the Cherednik-Macdonald constant term q-identities in which the deformation parameter q is allowed to have modulus one. They unite the Cherednik-Macdonald constant term q-identities with closely related Jackson p-integral identities due to Macdonald, where the deformation parameter p is related to q by modular inversion.


An entire function with simply and multiply connected wandering domains

Abstract: We modify a construction of Kisaka and Shishikura to show that there exists an entire function which has both a simply connected and a multiply connected wandering domain. Moreover, these domains are contained in the set of fast escaping points.


An iterative tomogravity algorithm for the estimation of network traffic

Abstract: This paper introduces an iterative tomogravity algorithm for the estimation of a network traffic matrix based on one snapshot observation of the link loads in the network. The proposed method does not require complete observation of the total load on individual edge links or proper tuning of a penalty parameter as existing methods do. Numerical results are presented to demonstrate that the iterative tomogravity method controls the estimation error well when the link data is fully observed and produces robust results with moderate amount of missing link data.


Asymptotics for Hermite-Pade rational approximants for two analytic functions with separated pairs of branch points (case of genus 0)

Abstract: We investigate the asymptotic behavior for type II Hermite-Pade approximation to two functions, where each function has two branch points and the pairs of branch points are separated. We give a classification of the cases such that the limiting counting measures for the poles of the Hermite-Pade approximants are described by an algebraic function of order 3 and genus 0. This situation gives rise to a vector-potential equilibrium problem for three measures and the poles of the common denominator are asymptotically distributed like one of these measures. We also work out the strong asymptotics for the corresponding Hermite-Pade approximants by using a 3x3 Riemann-Hilbert problem that characterizes this Hermite-Pade approximation problem.


Rational semigroup automata

Abstract: We show that for any monoid M, the family of languages accepted by M-automata (or equivalently, generated by regular valence grammars over M) is completely determined by that part of M which lies outside the maximal ideal. Hence, every such family arises as the family of languages accepted by N-automata where N is a simple or 0-simple monoid. A consequence is that every such family is either the class of regular languages, contains all the blind one-counter languages, or is the family of languages accepted by G-automata for G a non-locally-finite torsion group. We consider a natural extension of the usual definition which permits the automata to utilise more of the structure of each monoid, and also allows us to define S-automata for S an arbitrary semigroup. In the monoid case, the resulting automata are equivalent to the valence automata with rational target sets} which arise in the theory of regulated rewriting systems. We study the case that the register semigroup is completely simple or completely 0-simple, obtaining a complete characterisation of the classes of languages corresponding to such semigroups in terms of their maximal subgroups. In the process, we obtain a number of results about rational subsets of Rees matrix semigroups which may be of independent interest.


Pricing, Hedging and Optimally Designing Derivatives Via Minimization of Risk Measures

Abstract: The question of pricing and hedging a given contingent claim has a unique solution in a complete market framework. When some incompleteness is introduced, the problem becomes however more difficult. Several approaches have been adopted in the literature to provide a satisfactory answer to this problem, for a particular choice criterion. In this paper, in order to price and hedge a non-tradable contingent claim, we first start with a (standard) utility maximization problem and end up with an equivalent risk measure minimization. This hedging problem can be seen as a particular case of a more general situation of risk transfer between different agents, one of them consisting of the financial market. In order to provide constructive answers to this general optimal risk transfer problem, both static and dynamic approaches are considered. When considering a dynamic framework, our main purpose is to find a trade-off between static and very abstract risk measures as we are more interested in tractability issues and interpretations of the dynamic risk measures we obtain rather than the ultimate general results. Therefore, after introducing a general axiomatic approach to dynamic risk measures, we relate the dynamic version of convex risk measures to BSDEs.


Law of Large Numbers Limits for Many Server Queues

Abstract: This work considers a many-server queueing system in which customers with i.i.d., generally distributed service times enter service in the order of arrival. The dynamics of the system is represented in terms of a process that describes the total number of customers in the system, as well as a measure-valued process that keeps track of the ages of customers in service. Under mild assumptions on the service time distribution, as the number of servers goes to infinity, a law of large numbers (or fluid) limit is established for this pair of processes. The limit is characterised as the unique solution to a coupled pair of integral equations, which admits a fairly explicit representation. As a corollary, the fluid limits of several other functionals of interest, such as the waiting time, are also obtained. Furthermore, in the time-homogeneous setting, the fluid limit is shown to converge to its equilibrium. Along the way, some results of independent interest are obtained, including a continuous mapping result and a maximality property of the fluid limit. A motivation for studying these systems is that they arise as models of computer data systems and call centers.


Stanilov-Tsankov-Videv Theory

Abstract: We survey some recent results concerning Stanilov-Tsankov-Videv theory, conformal Osserman geometry, and Walker geometry which relate algebraic properties of the curvature operator to the underlying geometry of the manifold.


A flexible Bayesian generalized linear model for dichotomous response data with an application to text categorization

Abstract: We present a class of sparse generalized linear models that include probit and logistic regression as special cases and offer some extra flexibility. We provide an EM algorithm for learning the parameters of these models from data. We apply our method in text classification and in simulated data and show that our method outperforms the logistic and probit models and also the elastic net, in general by a substantial margin.


Overconvergence and classicality: the case of curves

Abstract: Given our set-up of a system of curves and maps between them satisfying certain assumptions, we prove a classicality criterion for overconvergent sections of line bundles over these curves. As a result, we prove such criteria for overconvergent modular forms over various Shimura curves. In particular, we provide a classicality criterion for overconvergent modular forms studied in [Kassaei: P-adic modular forms over Shimura curves over totally real fields, Compositio Math. 140 (2004), no 2, 359-395] and their higher-level generalizations.


Estimating the proportion of differentially expressed genes in comparative DNA microarray experiments

Abstract: DNA microarray experiments, a well-established experimental technique, aim at understanding the function of genes in some biological processes. One of the most common experiments in functional genomics research is to compare two groups of microarray data to determine which genes are differentially expressed. In this paper, we propose a methodology to estimate the proportion of differentially expressed genes in such experiments. We study the performance of our method in a simulation study where we compare it to other standard methods. Finally we compare the methods in real data from two toxicology experiments with mice.


A representative sampling plan for auditing health insurance claims

Abstract: A stratified sampling plan to audit health insurance claims is offered. The stratification is by dollar amount of the claim. The plan is representative in the sense that with high probability for each stratum, the difference in the average dollar amount of the claim in the sample and the average dollar amount in the population, is ``small.'' Several notions of ``small'' are presented. The plan then yields a relatively small total sample size with the property that the overall average dollar amount in the sample is close to the average dollar amount in the population. Three different estimators and corresponding lower confidence bounds for over (under) payments are studied.


Confidence distribution (CD) -- distribution estimator of a parameter

Abstract: The notion of confidence distribution (CD), an entirely frequentist concept, is in essence a Neymanian interpretation of Fisher's Fiducial distribution. It contains information related to every kind of frequentist inference. In this article, a CD is viewed as a distribution estimator of a parameter. This leads naturally to consideration of the information contained in CD, comparison of CDs and optimal CDs, and connection of the CD concept to the (profile) likelihood function. A formal development of a multiparameter CD is also presented.


Empirical Bayes methods for controlling the false discovery rate with dependent data

Abstract: False discovery rate (FDR) has been widely used as an error measure in large scale multiple testing problems, but most research in the area has been focused on procedures for controlling the FDR based on independent test statistics or the properties of such procedures for test statistics with certain types of stochastic dependence. Based on an approach proposed in Tang and Zhang (2005), we further develop in this paper empirical Bayes methods for controlling the FDR with dependent data. We implement our methodology in a time series model and report the results of a simulation study to demonstrate the advantages of the empirical Bayes approach.


A smoothing model for sample disclosure risk estimation

Abstract: When a sample frequency table is published, disclosure risk arises when some individuals can be identified on the basis of their values in certain attributes in the table called key variables, and then their values in other attributes may be inferred, and their privacy is violated. On the basis of the sample to be released, and possibly some partial knowledge of the whole population, an agency which considers releasing the sample, has to estimate the disclosure risk. Risk arises from non-empty sample cells which represent small population cells and from population uniques in particular. Therefore risk estimation requires assessing how many of the relevant population cells are likely to be small. Various methods have been proposed for this task, and we present a method in which estimation of a population cell frequency is based on smoothing using a local neighborhood of this cell, that is, cells having similar or close values in all attributes. We provide some preliminary results and experiments with this method. Comparisons are made to two other methods: 1. a log-linear models approach in which inference on a given cell is based on a ``neighborhood'' of cells determined by the log-linear model. Such neighborhoods have one or some common attributes with the cell in question, but some other attributes may differ significantly. 2 The Argus method in which inference on a given cell is based only on the sample frequency in the specific cell, on the sample design and on some known marginal distributions of the population, without learning from any type of ``neighborhood'' of the given cell, nor from any model which uses the structure of the table.


A note on the U,V method of estimation

Abstract: The U,V method of estimation provides unbiased estimators or predictors of random quantities. The method was introduced by Robbins \citer3 and subsequently studied in a series of papers by Robbins and Zhang. (See Zhang \citer5.) Practical applications of the method are featured in these papers. We demonstrate that for one U function (one for which there is an important application) the V estimator is inadmissible for a wide class of loss functions. For another important U function the V estimator is admissible for the squared error loss function.


Local polynomial regression on unknown manifolds

Abstract: We reveal the phenomenon that ``naive'' multivariate local polynomial regression can adapt to local smooth lower dimensional structure in the sense that it achieves the optimal convergence rate for nonparametric estimation of regression functions belonging to a Sobolev space when the predictor variables live on or close to a lower dimensional manifold.


Formal punctured ribbons and two-dimensional local fields

Abstract: We investigate formal ribbons on curves. Roughly speaking, formal ribbon is a family of locally linearly compact vector spaces on a curve. We establish a one-to-one correspondence between formal ribbons on curves plus some geometric data and some subspaces of two-dimensional local field.


Matrice magique associ\'ee \`a un germe de courbe plane et division par l'id\'eal Jacobien

Abstract: In the ring of holomorphic functions at the origin of C^2, we consider the equation uf'_x+vf'_y=wf where f and w are given. We introduce intersection multiplicities relative to w and f'_y along the branches of f, and we study the solutions (u,v) using these valuations. As an application, we construct an explicit functional equation satisfied by f.


Prime numbers with Beatty sequences

Abstract: A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes of the form $p=2[\alpha n]+1$, where $1<\alpha<2$ is a fixed irrational number. An argument of P. Ribenboim coupled with classical results about the distribution of fractional parts of irrational multiples of primes in an arithmetic progression immediately imply that this conjecture holds in a much more precise asymptotic form. Motivated by this observation, we give an asymptotic formula for the number of primes $p=q[\alpha n+\beta]+a$ with $n\le N$, where $\alpha,\beta$ are real numbers such that $\alpha$ is positive and irrational of finite type (which is true for almost all $\alpha$) and $a,q$ are integers with $0\le a<q\le N^\kappa$ and $\gcd(a,q)=1$, where $\kappa>0$ depends only on $\alpha$. We also prove a similar result for primes $p=[\alpha n+\beta]$ such that $p\equiv a\pmod q$.


On the injectivity of the X-ray transform for Anosov thermostats

Abstract: We consider Anosov thermostats on a closed surface and the X-ray transform on functions which are up to degree two in the velocities. We show that the subspace where the X-ray transform fails to be s-injective is finite dimensional. Furthermore, if the surface is negatively curved and the thermostat is pure Gaussian (i.e. no magnetic field is present), then the X-ray transform is s-injective.


On deciding stability of multiclass queueing networks under buffer priority scheduling policies

Abstract: One of the basic properties of a queueing network is stability. Roughly speaking, it is the property that the total number of jobs in the network remains bounded as a function of time. One of the key questions related to the stability issue is how to determine the exact conditions under which a given queueing network operating under a given scheduling policy remains stable. While there was much initial progress in addressing this question, most of the results obtained were partial at best and so the complete characterization of stable queueing networks is still lacking. In this paper, we resolve this open problem, albeit in a somewhat unexpected way. We show that characterizing stable queueing networks is an algorithmically undecidable problem for the case of nonpreemptive static buffer priority scheduling policies and deterministic interarrival and service times. Thus, no constructive characterization of stable queueing networks operating under this class of policies is possible. The result is established for queueing networks with finite and infinite buffer sizes and possibly zero service times, although we conjecture that it also holds in the case of models with only infinite buffers and nonzero service times. Our approach extends an earlier related work [Math. Oper. Res. 27 (2002) 272--293] and uses the so-called counter machine device as a reduction tool.


Special K\"ahler-Ricci potentials and Ricci solitons

Abstract: On a manifold of dimension at least six, let $(g,\tau)$ be a pair consisting of a K\"ahler metric g which is locally K\"ahler irreducible, and a nonconstant smooth function $\tau$. Off the zero set of $\tau$, if the metric $\hat{g}=g/\tau^2$ is a gradient Ricci soliton which has soliton function $1/\tau$, we show that $\hat{g}$ is K\"ahler with respect to another complex structure, and locally of a type first described by Koiso. Moreover, $\tau$ is a special K\"ahler-Ricci potential, a notion defined in earlier works of Derdzinski and Maschler. The result extends to dimension four with additional assumptions. We also discuss a Ricci-Hessian equation, which is a generalization of the soliton equation, and observe that the set of pairs $(g,\tau)$ satisfying a Ricci-Hessian equation is invariant, in a suitable sense, under the map $(g,\tau)\to (\hat{g},1/\tau)$.


Singular and tangent slit solutions to the Loewner equation

Abstract: We consider the Loewner differential equation generating univalent maps of the unit disk (or of the upper half-plane) onto itself minus a single slit. We prove that the circular slits, tangent to the real axis are generated by H\"older continuous driving terms with exponent 1/3 in the Loewner equation. Singular solutions are described, and the critical value of the norm of driving terms generating quasisymmetric slits in the disk is obtained.


Deconvolution by simulation

Abstract: Given samples (x_1,...,x_m) and (z_1,...,z_n) which we believe are independent realizations of random variables X and Z respectively, where we further believe that Z=X+Y with Y independent of X, the problem is to estimate the distribution of Y. We present a new method for doing this, involving simulation. Experiments suggest that the method provides useful estimates.


Shape restricted regression with random Bernstein polynomials

Abstract: Shape restricted regressions, including isotonic regression and concave regression as special cases, are studied using priors on Bernstein polynomials and Markov chain Monte Carlo methods. These priors have large supports, select only smooth functions, can easily incorporate geometric information into the prior, and can be generated without computational difficulty. Algorithms generating priors and posteriors are proposed, and simulation studies are conducted to illustrate the performance of this approach. Comparisons with the density-regression method of Dette et al. (2006) are included.


Non- and semi-parametric analysis of failure time data with missing failure indicators

Abstract: A class of estimating functions is introduced for the regression parameter of the Cox proportional hazards model to allow unknown failure statuses on some study subjects. The consistency and asymptotic normality of the resulting estimators are established under mild conditions. An adaptive estimator which achieves the minimum variance-covariance bound of the class is constructed. Numerical studies demonstrate that the asymptotic approximations are adequate for practical use and that the efficiency gain of the adaptive estimator over the complete-case analysis can be quite substantial. Similar methods are also developed for the nonparametric estimation of the survival function of a homogeneous population and for the estimation of the cumulative baseline hazard function under the Cox model.


Nonparametric estimation of a distribution function under biased sampling and censoring

Abstract: This paper derives the nonparametric maximum likelihood estimator (NPMLE) of a distribution function from observations which are subject to both bias and censoring. The NPMLE is obtained by a simple EM algorithm which is an extension of the algorithm suggested by Vardi (Biometrika, 1989) for size biased data. Application of the algorithm to many models is discussed and a simulation study compares the estimator's performance to that of the product-limit estimator (PLE). An example demonstrates the utility of the NPMLE to data where the PLE is inappropriate.


Estimating a Polya frequency function_2

Abstract: We consider the non-parametric maximum likelihood estimation in the class of Polya frequency functions of order two, viz. the densities with a concave logarithm. This is a subclass of unimodal densities and fairly rich in general. The NPMLE is shown to be the solution to a convex programming problem in the Euclidean space and an algorithm is devised similar to the iterative convex minorant algorithm by Jongbleod (1999). The estimator achieves Hellinger consistency when the true density is a PFF_2 itself.


A super Frobenius formula for the characters of Iwahori-Hecke algebras

Abstract: We establish a super Frobenius formula for the characters of Iwahori-Hecke algebras. We show that the Hall-Littlewood sypersymmetric function, up to constant, generates the values of the irreducible characters of Iwahori-Hecke algebras at the elements corresponding to cycle permutations. Our formula in this article includes both the ordinary quantum case and the classical super case.


A comparison of the accuracy of saddlepoint conditional cumulative distribution function approximations

Abstract: Consider a model parameterized by a scalar parameter of interest and a nuisance parameter vector. Inference about the parameter of interest may be based on the signed root of the likelihood ratio statistic R. The standard normal approximation to the conditional distribution of R typically has error of order O(n^{-1/2}), where n is the sample size. There are several modifications for R, which reduce the order of error in the approximations. In this paper, we mainly investigate Barndorff-Nielsen's modified directed likelihood ratio statistic, Severini's empirical adjustment, and DiCiccio and Martin's two modifications, involving the Bayesian approach and the conditional likelihood ratio statistic. For each modification, two formats were employed to approximate the conditional cumulative distribution function; these are Barndorff-Nielson formats and the Lugannani and Rice formats. All approximations were applied to inference on the ratio of means for two independent exponential random variables. We constructed one and two-sided hypotheses tests and used the actual sizes of the tests as the measurements of accuracy to compare those approximations.


Multivariate medians and measure-symmetrization

Abstract: We discuss two research areas dealing respectively with (1) a class of multivariate medians and (2) a symmetrization algorithm for probability measures.


Statistical thinking: From Tukey to Vardi and beyond

Abstract: Data miners (minors?) and neural networkers tend to eschew modelling, misled perhaps by misinterpretation of strongly expressed views of John Tukey. I discuss Vardi's views of these issues as well as other aspects of Vardi's work in emision tomography and in sampling bias.


Errors Theory using Dirichlet Forms, Linear Partial Differential Equations and Wavelets

Abstract: We present an application of error theory using Dirichlet Forms in linear partial differential equations (LPDE). We study the transmission of an uncertainty on the terminal condition to the solution of the LPDE thanks to the decomposition of the solution on a wavelets basis. We analyze the basic properties and a particular class of LPDE where the wavelets bases show their powerful, the combination of error theory and wavelets basis justifies some hypotheses, helpful to simplify the computation.


Statistical inverse problems in active network tomography

Abstract: The analysis of computer and communication networks gives rise to some interesting inverse problems. This paper is concerned with active network tomography where the goal is to recover information about quality-of-service (QoS) parameters at the link level from aggregate data measured on end-to-end network paths. The estimation and monitoring of QoS parameters, such as loss rates and delays, are of considerable interest to network engineers and Internet service providers. The paper provides a review of the inverse problems and recent research on inference for loss rates and delay distributions. Some new results on parametric inference for delay distributions are also developed. In addition, a real application on Internet telephony is discussed.


Densities for Ornstein-Uhlenbeck processes with jumps

Abstract: We consider an Ornstein-Uhlenbeck process with values in R^n driven by a L\'evy process (Z_t) taking values in R^d with d possibly smaller than n. The L\'evy noise can have a degenerate or even vanishing Gaussian component. Under a controllability condition and an assumption on the L\'evy measure of (Z_t), we prove that the law of the Ornstein-Uhlenbeck process at any time t>0 has a density on R^n. Moreover, when the L\'evy process is of $\alpha$-stable type, $\alpha \in (0,2)$, we show that such density is a $C^{\infty}$-function.


Using data network metrics, graphics, and topology to explore network characteristics

Abstract: Yehuda Vardi introduced the term network tomography and was the first to propose and study how statistical inverse methods could be adapted to attack important network problems (Vardi, 1996). More recently, in one of his final papers, Vardi proposed notions of metrics on networks to define and measure distances between a network's links, its paths, and also between different networks (Vardi, 2004). In this paper, we apply Vardi's general approach for network metrics to a real data network by using data obtained from special data network tools and testing procedures presented here. We illustrate how the metrics help explicate interesting features of the traffic characteristics on the network. We also adapt the metrics in order to condition on traffic passing through a portion of the network, such as a router or pair of routers, and show further how this approach helps to discover and explain interesting network characteristics.


Legendrian ribbons in overtwisted contact structures

Abstract: We show that a null-homologous transverse knot K in the complement of an overtwisted disk in a contact 3-manifold is the boundary of a Legendrian ribbon if and only if it possesses a Seifert surface S such that the self-linking number of K with respect to S satisfies $\sel(K,S)=-\chi(S)$. In particular, every null-homologous topological knot type in an overtwisted contact manifold can be represented by the boundary of a Legendrian ribbon. Finally, we show that a contact structure is tight if and only if every Legendrian ribbon minimizes genus in its relative homology class.


Semiclassical Limits of Quantum Affine Spaces

Abstract: Semiclassical limits of generic multiparameter quantized coordinate rings A = O_q(k^n) of affine spaces are constructed and related to A, for k an algebraically closed field of characteristic zero and q a multiplicatively antisymmetric matrix whose entries generate a torsionfree subgroup of k*. A semiclassical limit of A is a Poisson algebra structure on the corresponding classical coordinate ring R = O(k^n), and results of Oh, Park, Shin and the authors are used to construct homeomorphisms from the Poisson prime and Poisson primitive spectra of R onto the prime and primitive spectra of A. The Poisson primitive spectrum of R is then identified with the space of symplectic cores in k^n in the sense of Brown and Gordon, and an example is presented (over the complex numbers) for which the Poisson primitive spectrum of R is not homeomorphic to the space of symplectic leaves in k^n. Finally, these results are extended from quantum affine spaces to quantum affine toric varieties.


Pseudo-Riemannian Jacobi-Videv Manifolds

Abstract: We exhibit several families of Jacobi-Videv pseudo-Riemannian manifolds which are not Einstein. We also exhibit Jacobi-Videv algebraic curvature tensors where the Ricci operator defines an almost complex structure.


Differential geometry of curves in Lagrange Grassmannians with given Young diagram

Abstract: Curves in Lagrange Grassmannians appear naturally in the intrinsic study of geometric structures on manifolds. By a smooth geometric structure on a manifold we mean any submanifold of its tangent bundle, transversal to the fibers. One can consider the time-optimal problem naturally associate with a geometric structure. The Pontryagin extremals of this optimal problem are integral curves of certain Hamiltonian system in the cotangent bundle. The dynamics of the fibers of the cotangent bundle w.r.t. this system along an extremal is described by certain curve in a Lagrange Grassmannian, called Jacobi curve of the extremal. Any symplectic invariant of the Jacobi curves produces the invariant of the original geometric structure. The basic characteristic of a curve in a Lagrange Grassmannian is its Young diagram. The number of boxes in its kth column is equal to the rank of the kth derivative of the curve (which is an appropriately defined linear mapping) at a generic point. We will describe the construction of the complete system of symplectic invariants for parameterized curves in a Lagrange Grassmannian with given Young diagram. It allows to develop in a unified way local differential geometry of very wide classes of geometric structures on manifolds, including both classical geometric structures such as Riemannian and Finslerian structures and less classical ones such as sub-Riemannian and sub-Finslerian structures, defined on nonholonomic distributions.


Spectral geometry, homogeneous spaces, and differential forms with finite Fourier series

Abstract: Let G be a compact Lie group acting transitively on Riemannian manifolds M and N. Let p be a G equivariant Riemannian submersion from M to N. We show that a smooth differential form on N has finite Fourier series if and only if the pull back has finite Fourier series on M


Additivity of Spin^c Quantization under Cutting

Abstract: A G-equivariant spin^c structure on a manifold gives rise to a virtual representation of the group G, called the spin^c quantization of the manifold. We present a cutting construction for S^1-equivariant spin^c manifolds, and show that the quantization of the original manifold is isomorphic to the direct sum of the quantizations of the cut spaces. Our proof uses Kostant-type formulas, which express the quantization in terms of local data around the fixed point set of the S^1-action.


Functional analysis via extensions of the band depth

Abstract: The notion of data depth has long been in use to obtain robust location and scale estimates in a multivariate setting. The depth of an observation is a measure of its centrality, with respect to a data set or a distribution. The data depths of a set of multivariate observations translates to a center-outward ordering of the data. Thus, data depth provides a generalization of the median to a multivariate setting (the deepest observation), and can also be used to screen for extreme observations or outliers (the observations with low data depth). Data depth has been used in the development of a wide range of robust and non-parametric methods for multivariate data, such as non-parametric tests of location and scale [Li and Liu (2004)], multivariate rank-tests [Liu and Singh (1993)], non-parametric classification and clustering [Jornsten (2004)], and robust regression [Rousseeuw and Hubert (1999)]. Many different notions of data depth have been developed for multivariate data. In contrast, data depth measures for functional data have only recently been proposed [Fraiman and Muniz (1999), L\'{o}pez-Pintado and Romo (2006a)]. While the definitions of both of these data depth measures are motivated by the functional aspect of the data, the measures themselves are in fact invariant with respect to permutations of the domain (i.e. the compact interval on which the functions are defined). Thus, these measures are equally applicable to multivariate data where there is no explicit ordering of the data dimensions. In this paper we explore some extensions of functional data depths, so as to take the ordering of the data dimensions into account.


The hook fusion procedure and its generalisations

Abstract: The fusion procedure provides a way to construct new solutions to the Yang-Baxter equation. In the case of the symmetric group the fusion procedure has been used to construct diagonal matrix elements using a decomposition of the Young diagram into its rows or columns. We present a new construction which decomposes the diagram into hooks, the great advantage of this is that it minimises the number of auxiliary parameters needed in the procedure. We go on to use the hook fusion procedure to find diagonal matrix elements computationally and calculate supporting evidence to a previous conjecture. We are motivated by the construction of certain elements that allow us to generate representations of the symmetric group and single out particular irreducible components. In this way we may construct higher representations of the symmetric group from elementary ones. We go some way to generalising the hook fusion procedure by considering other decompositions of Young diagrams, specifically into ribbons. Finally, we adapt our construction to the quantum deformation of the symmetric group algebra known as the Hecke algebra.


Distribution of periodic torus orbits and Duke's theorem for cubic fields

Abstract: We study periodic torus orbits on spaces of lattices. Using the action of the group of adelic points of the underlying tori, we define a natural equivalence relation on these orbits, and show that the equivalence classes become uniformly distributed. This is a cubic analogue of Duke's theorem about the distribution of closed geodesics on the modular surface: suitably interpreted, the ideal classes of a cubic totally real field are equidistributed in the modular 5-fold SL_3(Z)\SL_3(R)/SO_3(R). In particular, this proves (a stronger form of) the folklore conjecture that the collection of maximal compact flats in SL_3(Z)\SL_3(R)/SO_3(R) of volume less than V becomes equidistributed as V goes to infinity. The proof combines subconvexity estimates, measure classification, and local harmonic analysis.


Remark on fundamental groups and effective Diophantine methods for hyperbolic curves

Abstract: We discuss how non-commutative fundamental groups could eventually contribute to algorithms for finding rational points on hyperbolic curves.


Complex Datasets and Inverse Problems. Tomography, Networks and Beyond

Abstract: This book is a collection of papers dedicated to the memory of Yehuda Vardi. Yehuda was the chair of the Department of Statistics of Rutgers University when he passed away unexpectedly on January 13, 2005. On October 21--22, 2005, some 150 leading scholars from many different fields, including statistics, telecommunications, biomedical engineering, bioinformatics, biostatistics and epidemiology, gathered at Rutgers in a conference in his honor. This conference was on ``Complex Datasets and Inverse Problems: Tomography, Networks, and Beyond,'' and was organized by the editors. The present collection includes research work presented at the conference, as well as contributions from Yehuda's colleagues. The theme of the conference was networks and other important and emerging areas of research involving incomplete data and statistical inverse problems. Networks are abundant around us: communication, computer, traffic, social and energy are just a few examples. As enormous amounts of network data are collected in this information age, the field has attracted a great amount of attention from researchers in statistics and computer engineering as well as telecommunication providers and various government agencies. However, few statistical tools have been developed for analyzing network data as they are typically governed by time-varying and mutually dependent communication protocols sitting on complicated graph-structured network topologies. Many prototypical applications in these and other important technologies can be viewed as statistical inverse problems with complex, massive, high-dimensional and possibly biased/incomplete data. This unifying theme of inverse problems is particularly appropriate for a conference and volume dedicated to the memory of Yehuda. Indeed he made influential contributions to these fields, especially in medical tomography, biased data, statistical inverse problems, and network tomography.


Automorphisms and derivations of free Poisson algebras in two variables

Abstract: Let P be a free Poisson algebra in two variables over a field of characteristic zero. We prove that the automorphisms of P are tame and that the locally nilpotent derivations of P are triangulable.


"Boundary blowup" type sub-solutions to semilinear elliptic equations with Hardy potential

Abstract: Semilinear elliptic equations which give rise to solutions blowing up at the boundary are perturbed by a Hardy potential. The size of this potential effects the existence of a certain type of solutions (large solutions): if the potential is too small, then no large solution exists. The presence of the Hardy potential requires a new definition of large solutions, following the pattern of the associated linear problem. Nonexistence and existence results for different types of solutions will be given. Our considerations are based on a Phragmen-Lindelof type theorem which enables us to classify the solutions and sub-solutions according to their behavior near the boundary. Nonexistence follows from this principle together with the Keller-Osserman upper bound. The existence proofs rely on sub- and super-solution techniques and on estimates for the Hardy constant derived in Marcus, Mizel and Pinchover.


Quaternion CR-submanifolds of a Locally Conformal Quaternion Kaehler Manifold

Abstract: The purpose of the present paper is to study the differential geometric properties of a quaternion CR-submanifold in a locally conformal quaternion Kaehler manifold.


Lieb-Thirring inequalities with improved constants

Abstract: Following Eden and Foias we obtain a matrix version of a generalised Sobolev inequality in one-dimension. This allow us to improve on the known estimates of best constants in Lieb-Thirring inequalities for the sum of the negative eigenvalues for multi-dimensional Schroedinger operators.


On the spinor structure of the homogeneous and isotropic universe in closed model

Abstract: The closed homogeneous and isotropic universe is considered. The bundles of Weyl and Dirac spinors for this universe are explicitly described. Some explicit formulas for the basic fields and for the connection components in stereographic and in spherical coordinates are presented.


On some lower bounds on the number of bicliques needed to cover a bipartite graph

Abstract: The biclique covering number of a bipartite graph G is the minimum number of complete bipartite subgraphs (bicliques) whose union contains every edge of G. In this little note we compare three lower bounds on the biclique covering number: A bound jk(G) proposed by Jukna & Kulikov (Discrete Math. 2009); the well-known fooling set bound fool(G); the "tensor-power" fooling set bound fool^\infty(G). We show jk \le fool le fool^\infty \le min_Q (rk Q)^2, where the minimum is taken over all matrices with a certain zero/nonzero-pattern. Only the first inequality is really novel, the third one generalizes a result of Dietzfelbinger, Hromkovi\v{c}, Schnitger (1994). We also give examples for which fool \ge (rk)^{log_4 6} improving on Dietzfelbinger et al.


The periodic table of n-categories for low dimensions I: degenerate categories and degenerate bicategories

Abstract: We examine the periodic table of weak n-categories for the low-dimensional cases. It is widely understood that degenerate categories give rise to monoids, doubly degenerate bicategories to commutative monoids, and degenerate bicategories to monoidal categories; however, to understand this correspondence fully we examine the totalities of such structures together with maps between them and higher maps between those. Categories naturally form a 2-category {\bfseries Cat} so we take the full sub-2-category of this whose 0-cells are the degenerate categories. Monoids naturally form a category, but we regard this as a discrete 2-category to make the comparison. We show that this construction does not yield a biequivalence; to get an equivalence we ignore the natural transformations and consider only the {\it category} of degenerate categories. A similar situation occurs for degenerate bicategories. The tricategory of such does not yield an equivalence with monoidal categories; we must consider only the categories of such structures. For doubly degenerate bicategories the tricategory of such is not naturally triequivalent to the category of commutative monoids (regarded as a tricategory). However in this case considering just the categories does not give an equivalence either; to get an equivalence we consider the {\it bicategory} of doubly degenerate bicategories. We conclude with a hypothesis about how the above cases might generalise for n-fold degenerate n-categories.


Trace fields and commensurability of link complements

Abstract: This paper investigates the strength of the trace field as a commensurability invariant of hyperbolic 3-manifolds. We construct an infinite family of two-component hyperbolic link complements which are pairwise incommensurable and have the same trace field, and infinitely many 1-cusped finite volume hyperbolic 3-manifolds with the same property. We also show that the two-component link complements above have integral traces, but each has a mutant with a nonintegral trace.


The hardness of computing an eigenform

Abstract: In this article, we give evidence that computing Fourier coefficients of the Hecke eigenforms for composite indices is no easier than factoring integers. In particular, we show that the existence of a polynomial time algorithm that, given n, computes the n-th Fourier coefficient of a (fixed) Hecke eigenform implies that we can factor most RSA moduli (numbers that are products of two distinct primes) in polynomial time.


A Selberg integral for the Lie algebra A_n

Abstract: A new q-binomial theorem for Macdonald polynomials is employed to prove an A_n analogue of the celebrated Selberg integral. This confirms the g=A_n case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for every simple Lie algebra g.


An Algebraic Chain Model of String Topology

Abstract: A chain complex model for the free loop space of a connected, closed and oriented manifold is presented, and on its homology, the Gerstenhaber and Batalin-Vilkovisky algebra structures are defined and identified with the string topology structures. The gravity algebra on the equivariant homology of the free loop space is also modeled. The construction includes non simply-connected case, and therefore gives an algebraic and chain level model of Chas-Sullivan's String Topology.


The symplectic structure of curves in three dimensional spaces of constant curvature and the equations of mathematical physics

Abstract: This paper defines a symplectic form on the infinite dimensional Fr\'echet manifold of framed curves of fixed length over a simply connected Riemannian manifold of constant curvature. The paper then considers Hamiltonian systems generated by the geometric invariants of the curves on the base manifold and relates them to the equations of mathematical physics.


Projectively Osserman manifolds

Abstract: One says that a smooth manifold M is a pseudo-Riemannian manifold of signature (p,q) if the tangent bundle TM is equipped with a smooth non-degenerate symmetric inner product g of signature (p,q). Similarly one says that M is an affine manifold if TM is equipped with a torsion free connection. One says g is Osserman if the eigenvalues of the Jacobi operator are constant on the pseudo-sphere bundles of unit timelike and spacelike vectors. We extend this concept from the pseudo-Riemannian to the affine setting to define the notion of a projectively Osserman manifold. This notion is the focus of the paper. We establish some basic results concerning projectively Osserman manifolds and exhibit examples of this structure which arise in several different geometrical contexts.


On the collisions of singular points of complex algebraic plane curves

Abstract: We study the "generic" degenerations of curves with two singular points when the points merge. First, the notion of generic degeneration is defined precisely. Then a method to classify the possible results of generic degenerations is proposed in the case of linear singularity types. We discuss possible bounds on the singularity invariants of the resulting type in terms of the initial types. In particular the strict upper bound on the resulting multiplicity is proved and a sufficient condition for $\delta=const$ collision is given.


On the enumeration of complex plane curves with two singular points

Abstract: We study equi-singular strata of plane curves with two singular points of prescribed types. The method of the previous work [Kerner06] is generalized to this case. In particular we consider the enumerative problem for plane curves with two singular points of linear singularity types. First the problem for two ordinary multiple points of fixed multiplicities is solved. Then the enumeration for arbitrary linear types is reduced to the case of ordinary multiple points and to the understanding of "merging" of singular points. Many examples and numerical answers are given.


Random permutations and unique fully supported ergodicity for the Euler adic transformation

Abstract: There is only one fully supported ergodic invariant probability measure for the adic transformation on the space of infinite paths in the graph that underlies the Eulerian numbers. This result may partially justify a frequent assumption about the equidistribution of random permutations.


Several Dirac Operator in parabolic geometry

Abstract: In this thesis, we show the existence of a sequence of differential operators starting with with the Dirac operator in k Clifford variables, $D=(D_1,..., D_k)$, where $D_i=\sum_j e_j\cdot \partial_{ij}: C^\infty((\R^n)^k,\S)\to C^\infty((\R^n)^k,\S)$ ($\S$ is the spinor module). This operator is the Cauchy-Riemann operator for n=2 and its resolution is the Dolbeault complex. For higher n, the resolution of D is not known in general. While this problem was treated many times in the language of Clifford analysis and some partial results are known, we give a description of this operator in Parabolic geometry, which is a special type of Cartan geometry modeled on $G/P$, where P is a Parabolic subgroup of G. We construct sequences of invariant differential operators starting with the Dirac operator in several variables and assume that these sequences coinside in some cases with the resolution. We describe the structure of these sequences precisely in case the dimension $n$ is odd and give a conjecture that these sequences have similar structure for n even, $k\leq n/2$ (the s.c. {\it stable range}). We also give some information about these sequences in case n even, k>n/2. In the last chapter, explicite formulas for the operators are derived for the case k=2.


Affineness of Deligne-Lusztig varieties for minimal length elements

Abstract: We prove that the Deligne-Lusztig varieties associated to elements of the Weyl group which are of minimal length in their twisted class are affine. Our proof differs from the proof of He and Orlik-Rapoport and it is inspired by the case of regular elements, which correspond to the varieties involved in Brou\'e's conjectures.


Smooth toric DM stacks

Abstract: We give a new definition of smooth toric DM stacks in the same spirit of toric varieties. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a stacky fan. We also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks.


On the number of stable quiver representations over finite fields

Abstract: We prove a new formula for the generating function of polynomials counting absolutely stable representations of quivers over finite fields. The case of irreducible representations is studied in more detail.


Generic Continuous Spectrum for Ergodic Schr"odinger Operators

Abstract: We consider discrete Schr"odinger operators on the line with potentials generated by a minimal homeomorphism on a compact metric space and a continuous sampling function. We introduce the concepts of topological and metric repetition property. Assuming that the underlying dynamical system satisfies one of these repetition properties, we show using Gordon's Lemma that for a generic continuous sampling function, the associated Schr"odinger operators have no eigenvalues in a topological or metric sense, respectively. We present a number of applications, particularly to shifts and skew-shifts on the torus.


Borel subalgebras of root-reductive Lie algebras

Abstract: This paper generalizes the classification in a paper of Dimitrov and Penkov of Borel subalgebras of gl_infty. Root-reductive Lie algebras are direct limits of finite-dimensional reductive Lie algebras along inclusions preserving the root spaces with respect to nested Cartan subalgebras. A Borel subalgebra of a root-reductive Lie algebra is by definition a maximal locally solvable subalgebra. The main general result of this paper is that a Borel subalgebra of an infinite-dimensional indecomposable root-reductive Lie algebra is the simultaneous stabilizer of a certain type of generalized flag in each of the standard representations. For the three infinite-dimensional simple root-reductive Lie algebras more precise results are obtained. The map sending a maximal closed (isotropic) generalized flag in the standard representation to its stabilizer hits Borel subalgebras, yielding a bijection in the cases of sl_infty and sp_infty; in the case of so_infty the fibers are of size one and two. A description is given of a nice class of toral subalgebras contained in any Borel subalgebra. Finally, certain Borel subalgebras of a general root-reductive Lie algebra are seen to correspond bijectively with Borel subalgebras of the commutator subalgebra, which are understood in terms of the special cases.


Pre-quantization of the Moduli Space of Flat G-Bundles over a Surface

Abstract: For a simply connected, compact, simple Lie group G, the moduli space of flat G-bundles over a closed surface is known to be pre-quantizable at integer levels. For non-simply connected G, however, integrality of the level is not sufficient for pre-quantization, and this paper determines the obstruction -- namely a certain cohomology class in H^3(G^2;Z) -- that places further restrictions on the underlying level. The levels that admit a pre-quantization of the moduli space are determined explicitly for all non-simply connected, compact, simple Lie groups G.


Generalisations of the Tits representation

Abstract: We construct a group K_n with properties similar to infinite Coxeter groups. In particular, it has a geometric representation featuring hyperplanes and simplicial chambers. The generators of K_n are given by 2-element subsets of {0, .., n}. We give some easy combinatorial results on the finite residues of K_n.


Calibrated associative and Cayley embeddings

Abstract: Using the Cartan-Kahler theory, and results on real algebraic structures, we prove two embedding theorems. First, the interior of a smooth, compact 3-manifold may be isometrically embedded into a G_2-manifold as an associative submanifold. Second, the interior of a smooth, compact 4-manifold K, whose double has a trivial bundle of self-dual 2-forms, may be isometrically embedded into a Spin(7)-manifold as a Cayley submanifold. Along the way, we also show that Bochner's Theorem on real analytic approximation of smooth differential forms, can be obtained using real algebraic tools developed by Akbulut and King.


Universal derived equivalences of posets of tilting modules

Abstract: We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their tilting modules are related by a simple combinatorial construction, which we call flip-flop. We deduce that the posets of tilting modules of derived equivalent path algebras of quivers without oriented cycles are universally derived equivalent.


On Almost Representations of Property (T) Groups

Abstract: Property (T) for groups means a dichotomy: a representation either has an invariant vector or all vectors are far from being invariant. We show that, under a stronger condition of A.Zuk, a similar dichotomy holds for almost representations as well.


Correction to: HZ-algebra spectra are differential graded algebras

Abstract: This correction article is actually unnecessary. The proof of Theorem 1.2, concerning commutative HQ-algebra spectra and commutative differential graded algebras, in the author's paper [American Journal of Mathematics vol. 129 (2007) 351-379 (arxiv:math/0209215v4)] is correct as originally stated. Neil Strickland carefully proved that D is symmetric monoidal; so Proposition 4.7 and hence also Theorem 1.2 hold as stated. Strickland's proof will appear in joint work with Stefan Schwede; see related work in Strickland's [arxiv:0810.1747]. Note here D is defined as a colimit of chain complexes; in contrast, non-symmetric monoidal functors analogous to D are defined as homotopy colimits of spaces in previous work of the author.


Parametrization of the regular equivalences of the canonical controller and its applications

Abstract: We study control problems for linear systems in the behavioral framework. Our focus is a class of regular controllers that are equivalent to the canonical controller. The canonical controller is a particular controller that is guaranteed to be a solution whenever a solution exists. However, it has been shown that in most cases, the canonical controller is not regular. The main result of the paper is a parametrization of all regular controllers that are equivalent to the canonical controller. The parametrization is then used to solve two control problems. The first problem is related to designing a regular controller that uses as few control channels as possible. The second problem is to design a regular controller that satisfies a predefined input-output partitioning constraint. In both problems, based on the parametrization, we present algorithms that does the controller design.


Graphical methods for efficient likelihood inference in Gaussian covariance models

Abstract: In graphical modelling, a bi-directed graph encodes marginal independences among random variables that are identified with the vertices of the graph. We show how to transform a bi-directed graph into a maximal ancestral graph that (i) represents the same independence structure as the original bi-directed graph, and (ii) minimizes the number of arrowheads among all ancestral graphs satisfying (i). Here the number of arrowheads of an ancestral graph is the number of directed edges plus twice the number of bi-directed edges. In Gaussian models, this construction can be used for more efficient iterative maximization of the likelihood function and to determine when maximum likelihood estimates are equal to empirical counterparts.


Limited scope adic transformations

Abstract: We introduce a family of adic transformations on diagrams that are nonstationary and nonsimple. This family includes some previously studied adic transformations. We relate the dimension group of each these diagrams to the dynamical system determined by the adic transformation on the infinite edge paths, and we explicitly compute the dimension group for a subfamily. We also determine the ergodic adic invariant probability measures for this subfamily, and show that each system of the subfamily is loosely Bernoulli. We also give examples of particular adic transformations with roots of unity as well as one which is totally ergodic called the Euler adic. We also show that the Euler adic is loosely Bernoulli.


Explicit gradient estimates for minimal Lagrangian surfaces of dimension two

Abstract: We derive explicit, uniform, a priori interior Hessian and gradient estimates for special Lagrangian equations of all phases in dimension two.


A Proof that Thompson's Groups have Infinitely Many Relative Ends

Abstract: We show that each of Thompson's groups F, T, and V have infinitely many ends relative to certain subgroups. We go on to show that T and V both have Serre's property FA, i.e., any action of T or V on a tree will have a fixed point. (The proof of the latter statement was originally due to Ken Brown, and our proof is based on his notes.)


On Gr-Functors between Gr-Categories: Obstruction theory for Gr-Functors of the type $(\varphi,f)$

Abstract: Each Gr-functor of the type $(\varphi,f)$ of a Gr-category of the type $(\Pi,\C)$ has the obstruction be an element $\overline{k}\in H^3(\Pi,\C).$ When this obstruction vanishes, there exists a bijection between congruence classes of Gr-functors of the type $(\varphi,f)$ and the cohomology group $H^2(\Pi,\C).$ Then the relation of Gr-category theory and the group extension problem can be established and used to prove that each Gr-category is Gr-equivalent to a strict one.


The marginalization paradox and the formal Bayes' law

Abstract: It has recently been shown that the marginalization paradox (MP) can be resolved by interpreting improper inferences as probability limits. The key to the resolution is that probability limits need not satisfy the formal Bayes' law, which is used in the MP to deduce an inconsistency. In this paper, I explore the differences between probability limits and the more familiar pointwise limits, which do imply the formal Bayes' law, and show how these differences underlie some key differences in the interpretation of the MP.


The theory of the exponential differential equations of semiabelian varieties

Abstract: The complete first order theories of the exponential differential equations of semiabelian varieties are given. It is shown that these theories also arises from an amalgamation-with-predimension construction in the style of Hrushovski. The theory includes necessary and sufficient conditions for a system of equations to have a solution. The necessary condition generalizes Ax's differential fields version of Schanuel's conjecture to semiabelian varieties. There is a purely algebraic corollary, the "Weak CIT" for semiabelian varieties, which concerns the intersections of algebraic subgroups with algebraic varieties.


New equivalences for pattern avoiding involutions

Abstract: We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard's conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of S_5, S_6, and S_7 for both permutations and involutions.


On Rational Nilpotent Orbits of $SL_{n}$ and $Sp_{2n}$ over a Local Non-Archimedean Field

Abstract: We relate the partition-type parametrization of rational (arithmetic) nilpotent adjoint orbits of the split classical groups $SL_n$ and $Sp_{2n}$ over local non-Archimedean fields with a parametrization, introduced by DeBacker in 2002, which uses the associated Bruhat-Tits building to relate the question to one over the residue field.


Homological algebra for affine Hecke algebras

Abstract: In this paper we study homological properties of modules over an affine Hecke algebra H. In particular we prove a comparison result for higher extensions of tempered modules when passing to the Schwartz algebra S, a certain topological completion of the affine Hecke algebra. The proof is self-contained and based on a direct construction of a bounded contraction of certain standard resolutions of H-modules. This construction applies for all positive parameters of the affine Hecke algebra. This is an important feature since it is an ingredient to analyse how the irreducible discrete series representations of H arise in generic families over the parameter space of H. For irreducible non-simply laced affine Hecke algebras this will enable us to give a complete classification of the discrete series characters for all positive parameters (we will report on this application in a separate article).


Isogonal Conjugacy and Fermat Problems

Abstract: We consider three types of isogonal conjugacy of two points with respect to a given triangle and characterize any of these types by a geometric equality. As an application to the Fermat problem with positive weights, we prove that in the general case the given weights determine uniquely a point X and the solution to the Fermat problem is the point Y, which is isogonally conjugate of type I to the point X. We obtain a similar characterization of the solution to the Fermat problem in the case of mixed weights as well.


Remarks on $\eta$-Einstein unit tangent bundles

Abstract: We study the geometric properties of the base manifold for the unit tangent bundle satisfying the $\eta$-Einstein condition with the standard contact metric structure. One of the main theorems is that the unit tangent bundle of 4-dimensional Einstein manifold, equipped with the canonical contact metric structure, is $\eta$-Einstein manifold if and only if base manifold is the space of constant sectional curvature 1 or 2.


Data selection and confounding in the court case of Lucia de Berk

Abstract: The nurse Lucia de Berk was convicted by the Dutch courts as a serial killer with 7 murders and 3 attempts at murder in three hospitals where she worked. The nurse however always professed her innocence and indeed was never observed in such an act of murder. The courts based their decision on circumstantial evidence and upon the use of statistics. In the appeal court, the use of statistical calculations was repealed but the use of "data" and "statistical insights" were not excluded. The trial hinged importantly on the role of statistics and data gathering. It appears that data selection and confounding feature strongly in this case. The notion of "nominal correlation" can be used to highlight those two features. This suggests a mistrial with the conviction of an innocent person.


Four-dimensional almost Hermitian manifolds with vanishing Tricerri-Vanhecke Bochner curvature tensor

Abstract: We study curvature properties of four-dimensional almost Hermitian manifolds with vanishing Bochner curvature tensor as defined by Tricerri and Vanhecke. We give local structure theorems for such Kaehler manifolds, and find out several examples related to the theorems.


Which canonical algebras are derived equivalent to incidence algebras of posets?

Abstract: We give a full description of all the canonical algebras over an algebraically closed field that are derived equivalent to incidence algebras of finite posets. These are the canonical algebras whose number of weights is either 2 or 3.


Symplectic surgeries and normal surface singularities

Abstract: We show that every negative definite configuration of symplectic surfaces in a symplectic 4--manifold has a strongly symplectically convex neighborhood. We use this to show that, if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration, and if the complex singularity with resolution diffeomorphic to a neighborhood of the configuration has a smoothing, then the configuration can be symplectically replaced by the smoothing of the singularity. This generalizes the symplectic rational blowdown procedure used in recent constructions of small exotic 4--manifolds.


On Local Integrability Conditions Of Jet Groupoids

Abstract: A Jet groupoid R_q over a manifold X is a special Lie groupoid consisting of q-jets of local diffeomorphisms from X to X. As a subbundle of the q-th order jet bundle of the trivial bundle X times X, a jet groupoid can be considered as a nonlinear system of partial differential equations (PDE). This leads to the concept of formal integrability. On the other hand, each jet groupoid is the symmetry groupoid of a geometric object, modelled as a section of a natural bundle. Using the jet groupoids, we give a local characterisation of formal integrability for transitive jet groupoids in terms of their corresponding geometric objects.


A property of cyclotomic polynomials

Abstract: Given two cyclotomic polynomials $\Phi_n(x)$ and $\Phi_m(x)$, $n\not= m$, we determine the minimal natural number k such that we can write $$k=a(x)\Phi_n (x)+b(x)\Phi_m(x),$$ with a(x) and b(x) integer polynomials.


Qualitative properties of coupled parabolic systems of evolution equations

Abstract: We apply functional analytical and variational methods in order to study well-posedness and qualitative properties of evolution equations on product Hilbert spaces. To this aim we introduce an algebraic formalism for matrices of sesquilinear mappings. We apply our results to parabolic problems of different nature: a coupled diffusive system arising in neurobiology, a strongly damped wave equation, a heat equation with dynamic boundary conditions, and a general semilinear Hodgkin--Huxley sytem.


Determinants related to Dirichlet characters modulo 2, 4 and 8 of binomial coefficients and the algebra of recurrence matrices

Abstract: Using recurrence matrices, defined and described with some details, we study a few determinants related to evaluations of binomial coefficients on Dirichlet characters modulo 2, 4 and 8.


Gradient estimates for a degenerate parabolic equation with gradient absorption and applications

Abstract: Qualitative properties of non-negative solutions to a quasilinear degenerate parabolic equation with an absorption term depending solely on the gradient are shown, providing information on the competition between the nonlinear diffusion and the nonlinear absorption. In particular, the limit as time goes to infinity of the mass of integrable solutions is identified, together with the rate of expansion of the support for compactly supported initial data. The persistence of dead cores is also shown. The proof of these results strongly relies on gradient estimates which are first established.


Twisted cohomology of the Hilbert schemes of points on surfaces

Abstract: We calculate the cohomology spaces of the Hilbert schemes of points on surfaces with values in locally constant systems. For that purpose, we generalise I. Grojnoswki's and H. Nakajima's description of the ordinary cohomology in terms of a Fock space representation to the twisted case. We further generalise M. Lehn's work on the action of the Virasoro algebra to the twisted case. Building on work by M. Lehn and Ch. Sorger, we then give an explicit description of the cup-product in the twisted case whenever the surface has a numerically trivial canonical divisor. We formulate our results in a way that they apply to the projective and non-projective case in equal measure. As an application of our methods, we give explicit models for the cohomology rings of the generalised Kummer varieties and of a series of certain even dimensional Calabi--Yau manifolds.


A note on the lambda-structure on the Burnside ring

Abstract: Let G be a finite group and let S be a G-set. The Burnside ring of G has a natural structure of a lambda-ring. However, a priori the images of S under the lambda-operations can only be computed implicitly. In this paper we establish an explicit formula, expressing these images as linear combinations of classes of G-sets.


Stanley Decompositions, Pretty Clean Filtrations and Reductions Modulo Regular Elements

Abstract: We study the behavior of Stanley decompositions and of pretty clean filtrations under reduction modulo a regular element.


Pathwise coordinate optimization

Abstract: We consider ``one-at-a-time'' coordinate-wise descent algorithms for a class of convex optimization problems. An algorithm of this kind has been proposed for the $L_1$-penalized regression (lasso) in the literature, but it seems to have been largely ignored. Indeed, it seems that coordinate-wise algorithms are not often used in convex optimization. We show that this algorithm is very competitive with the well-known LARS (or homotopy) procedure in large lasso problems, and that it can be applied to related methods such as the garotte and elastic net. It turns out that coordinate-wise descent does not work in the ``fused lasso,'' however, so we derive a generalized algorithm that yields the solution in much less time that a standard convex optimizer. Finally, we generalize the procedure to the two-dimensional fused lasso, and demonstrate its performance on some image smoothing problems.


Quasisymmetric and unipotent tensor categories

Abstract: We classify braided tensor categories over C of exponential growth which are quasisymmetric, i.e., the squared braiding is the identity on the product of any two simple objects. This generalizes the classification results of Deligne on symmetric categories of exponential growth, and of Drinfeld on quasitriangular quasi-Hopf algebras. In particular, we classify braided categories of exponential growth which are unipotent, i.e., those whose only simple object is the unit object. We also classify fiber functors on such categories. Finally, using the Etingof-Kazhdan quantization theory of Poisson algebraic groups, we give a classification of coconnected Hopf algebras, i.e. of unipotent categories of exponential growth with a fiber functor.


The reverse engineering problem with probabilities and sequential behavior: Probabilistic Sequential Networks

Abstract: The reverse engineering problem with probabilities and sequential behavior is introducing here, using the expression of an algorithm. The solution is partially founded, because we solve the problem only if we have a Probabilistic Sequential Network. Therefore the probabilistic structure on sequential dynamical systems is introduced here, the new model will be called Probabilistic Sequential Network, PSN. The morphisms of Probabilistic Sequential Networks are defined using two algebraic conditions, whose imply that the distribution of probabilities in the systems are close. It is proved here that two homomorphic Probabilistic Sequential Networks have the same equilibrium or steady state probabilities. Additionally, the proof of the set of PSN with its morphisms form the category PSN, having the category of sequential dynamical systems SDS, as a full subcategory is given. Several examples of morphisms, subsystems and simulations are given.


Sch'nol's Theorem For Strongly Local Forms

Abstract: We prove a variant of Sch'nol's theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with $\delta$- or Kirchhoff boundary conditions.


Rankin-Cohen Deformations and Representation Theory

Abstract: In this paper, we use the unitary representation theory of $SL_2(\mathbb R)$ to understand the Rankin-Cohen brackets for modular forms. Then we use this interpretation to study the corresponding deformation problems that Paula Cohen, Yuri Manin and Don Zagier initiated. Two uniqueness results are established.


Liouville-type theorems for foliations with complex leaves

Abstract: We discuss various problems regarding the structure of the foliation of some foliated submanifolds S of C^n, in particular Levi flat ones. As a general scheme, we suppose that S is bounded along a coordinate (or a subset of coordinates), and prove that the leaves of its foliation are complex planes.


Floer Homology for Symplectomorphism

Abstract: Let (M,\omega) be a compact symplectic manifold, and \phi be a symplectic diffeomorphism on M, we define a Floer-type homology FH_*(\phi) which is a gen- eralization of Floer homology for symplectic fixed points defined by Dostoglou and Salamon for monotone symplectic manifolds. These homology groups are modules over a suitable Novikov ring and depend only on \phi up to a Hamiltonian isotopy.


Seip's differentiability concepts as a particular case of the Bertram--Gloeckner--Neeb construction

Abstract: From the point of view of unification of differentiation theory, it is of interest to note that the general construction principle of Bertram, Gloeckner and Neeb leading to a C^k differentiability concept from a given C^0 one, besides subsuming the Keller--Bastiani C_c^k differentiabilities on real Hausdorff locally convex spaces, also does the same to the "arc-generated" interpretation of the Lipschitz theory of differentiation by Frolicher and Kriegl, and likewise to the "compactly generated" theory of Seip's continuous differentiabilities. In this article, we give the details of the proof for the assertion concerning Seip's theory. We also give an example indicating that the premises in Seip's various inverse and implicit function theorems may be too strong in order for these theorems to have much practical value. Also included is a presentation of the BGN--setting reformulated so as to be consistent with the Kelley--Morse--Godel--Bernays--von Neumann type approach to set theory, as well as a treatment of the function space constructions and development of their basic properties needed in the proof of the main result.


Defining amalgams of compact Lie groups

Abstract: For $n \geq 2$ let $\Delta$ be a Dynkin diagram of rank $n$ and let $I = {1, >..., n}$ be the set of labels of $\Delta$. A group $G$ admits a weak Phan system of type $\Delta$ over $\mathbb{C}$ if $G$ is generated by subgroups $U_i$, $i \in I$, which are central quotients of simply connected compact semisimple Lie groups of rank one, and contains subgroups $U_{i,j} = \gen{U_i ,U_j}$, $i \neq j \in I$, which are central quotients of simply connected compact semisimple Lie groups of rank two such that $U_i$ and $U_j$ are rank one subgroups of $U_{i,j}$ corresponding to a choice of a maximal torus and a fundamental system of roots for $U_{i,j}$. It is shown in this article that $G$ then is a central quotient of the simply connected compact semisimple Lie group whose complexification is the simply connected complex semisimple Lie group of type $\Delta$.


Products in Residue Classes

Abstract: We consider a problem of P. Erdos, A. M. Odlyzko and A. Sarkozy about the representation of residue classes modulo m by products of two not too large primes. While it seems that even the Extended Riemann Hypothesis is not powerful enough to achieve the expected results, here we obtain some unconditional results ``on average'' over moduli m and residue classes modulo m and somewhat stronger results when the average is restricted to prime moduli m = p. We also consider the analogous question wherein the primes are replaced by easier sequences so, quite naturally, we obtain much stronger results.