Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity

Abstract: We study the bifurcation loci of quadratic (and unicritical) polynomials and exponential maps. We outline a proof that the exponential bifurcation locus is connected; this is an analog to Douady and Hubbard's celebrated theorem that (the boundary of) the Mandelbrot set is connected. For these parameter spaces, a fundamental conjecture is that hyperbolic dynamics is dense. For quadratic polynomials, this would follow from the famous stronger conjecture that the bifurcation locus (or equivalently the Mandelbrot set) is locally connected. It turns out that a formally slightly weaker statement is sufficient, namely that every point in the bifurcation locus is the landing point of a parameter ray. For exponential maps, the bifurcation locus is not locally connected. We describe a different conjecture (triviality of fibers) which naturally generalizes the role that local connectivity has for quadratic or unicritical polynomials.


Sample Reuse Techniques of Randomized Algorithms for Control under Uncertainty

Abstract: Sample reuse techniques have significantly reduced the numerical complexity of probabilistic robustness analysis. Existing results show that for a nested collection of hyper-spheres the complexity of the problem of performing $N$ equivalent i.i.d. (identical and independent) experiments for each sphere is absolutely bounded, independent of the number of spheres and depending only on the initial and final radii. In this chapter we elevate sample reuse to a new level of generality and establish that the numerical complexity of performing $N$ equivalent i.i.d. experiments for a chain of sets is absolutely bounded if the sets are nested. Each set does not even have to be connected, as long as the nested property holds. Thus, for example, the result permits the integration of deterministic and probabilistic analysis to eliminate regions from an uncertainty set and reduce even further the complexity of some problems. With a more general view, the result enables the analysis of complex decision problems mixing real-valued and discrete-valued random variables.


Fast Parallel Frequency Sweeping Algorithms for Robust ${\cal D}$-Stability Margin

Abstract: This paper considers the robust ${\cal D}$-stability margin problem under polynomic structured real parametric uncertainty. Based on the work of De Gaston and Safonov (1988), we have developed techniques such as, a parallel frequency sweeping strategy, different domain splitting schemes, which significantly reduce the computational complexity and guarantee the convergence.


Parallel Branch and Bound Algorithm for Computing Maximal Structured Singular Value

Abstract: In this paper, we have developed a parallel branch and bound algorithm which computes the maximal structured singular value $\mu$ without tightly bounding $\mu$ for each frequency and thus significantly reduce the computational complexity.


Universal Cycles of Discrete Functions

Abstract: A connected digraph in which the in-degree of any vertex equals its out-degree is Eulerian; this baseline result is used as the basis of existence proofs for universal cycles (also known as deBruijn cycles or $U$-cycles) of several combinatorial objects. We present new results on the existence of universal cycles of certain classes of functions. These include onto functions, and 1-inequitable sequences on a binary alphabet. In each case the connectedness of the underlying graph is the non-trivial aspect to be established.


On Abstract Strichartz Estimates and the Strauss Conjecture for Nontrapping Obstacles

Abstract: The purpose of this paper is to show how local energy decay estimates for certain linear wave equations involving compact perturbations of the standard Laplacian lead to optimal global existence theorems for the corresponding small amplitude nonlinear wave equations with power nonlinearities. To achieve this goal, at least for spatial dimensions $n=3$ and 4, we shall show how the aforementioned linear decay estimates can be combined with "abstract Strichartz" estimates for the free wave equation to prove corresponding estimates for the perturbed wave equation when $n\ge3$. As we shall see, we are only partially successful in the latter endeavor when the dimension is equal to two, and therefore, at present, our applications to nonlinear wave equations in this case are limited.


Restricted Radon transforms and projections of planar sets

Abstract: We establish a mixed norm estimate for the Radon transform in the plane when the set of directions has fractional dimension. This estimate is used to prove a result about an exceptional set of directions connected with projections of planar sets. That leads to a conjecture analogous to a well-known conjecture of Furstenberg.


Geometric and spectral properties of locally tessellating planar graphs

Abstract: In this article, we derive bounds for values of the global geometry of locally tessellating planar graphs, namely, the Cheeger constant and exponential growth, in terms of combinatorial curvatures. We also discuss spectral implications for the Laplacians.


Comparison theory and smooth minimal C*-dynamics

Abstract: We prove that the C*-algebra of a minimal diffeomorphism satisfies Blackadar's Fundamental Comparability Property for positive elements. This leads to the classification, in terms of K-theory and traces, of the isomorphism classes of countably generated Hilbert modules over such algebras, and to a similar classification for the closures of unitary orbits of self-adjoint elements. We also obtain a structure theorem for the Cuntz semigroup in this setting, and prove a conjecture of Blackadar and Handelman: the lower semicontinuous dimension functions are weakly dense in the space of all dimension functions. These results continue to hold in the broader setting of unital simple ASH algebras with slow dimension growth and stable rank one. Our main tool is a sharp bound on the radius of comparison of a recursive subhomogeneous C*-algebra. This is also used to construct uncountably many non-Morita-equivalent simple separable amenable C*-algebras with the same K-theory and tracial state space, providing a C*-algebraic analogue of McDuff's uncountable family of II_1 factors. We prove in passing that the range of the radius of comparison is exhausted by simple C*-algebras.


An Asymptotic Formula for the Sequence ||exp(i n h(t))||_A

Abstract: Given a function f with an absolutely convergent Fourier series, we define the norm of f as ||f||_A = the sum of absolute values of the Fourier coefficients of f. We study the behavior of ||f^n||_A as n goes to infinity, for f of the form exp(ih(t)) where h is a real, odd and twice continuously differentiable function such that h(t + 2\pi) = h(t) + 2k\pi for some integer k. We obtain a remarkably simple asymptotic formula for the case when h'' has no zeros in (0,\pi) and satisfies an additional condition near 0 and near \pi. Corollaries of our formula are an asymptotic formula due to D.Girard, and a formula on Bessel functions, due to G.Stey.


Lattice points on the plane ax+by+cz=d and the diophantine system ax+by+cz=d ex+fy+gz=h

Abstract: The subject matter of this work are the linear, three variable diophantine equation ax+by+cz=d (1), and the diophantine system ax+by+cz=d (2) ex+fy+gz=h with the coefficients a,b,c,d,e,f,g,h being integers. Introductory number theory books, typically contain only a brief outline of how to solve equation (1). Even less or no material is offered on the system (2). The purpose of this work is to fill this gap. After some preliminary, introductory material, which includes the general solution of the two variable linear diophantine equation ax+by=c(material which we use later in the paper); we present a complete and detailed analysis of equation (1). We determine the precise conditions that the coefficients a,b,c,d must satisfy in order for integer solutions to exist. We then derive a two-parameter, parametric description of The solution set. The solution set of (1), if not empty, consists of all integer triples (x,y,z) that satisfy (1). Geometrically, this is interpreted as the set of all lattice points in 3-D space which lie on the plane with equation (1). Similarly, we offer an exhaustive analysis of the system (2) by determining the precise conditions that the coeffients must satisfy in order for integer solutions to exist. We offer seven examples with detailed solutions.


Computing stability of multi-dimensional travelling waves

Abstract: We present a numerical method for computing the pure-point spectrum associated with the linear stability of multi-dimensional travelling fronts to parabolic nonlinear systems. Our method is based on the Evans function shooting approach. Transverse to the direction of propagation we project the spectral equations onto a finite Fourier basis. This generates a large, linear, one-dimensional system of equations for the longitudinal Fourier coefficients. We construct the stable and unstable solution subspaces associated with the longitudinal far-field zero boundary conditions, retaining only the information required for matching, by integrating the Riccati equations associated with the underlying Grassmannian manifolds. The Evans function is then the matching condition measuring the linear dependence of the stable and unstable subspaces and thus determines eigenvalues. As a model application, we study the stability of two-dimensional wrinkled front solutions to a cubic autocatalysis model system. We compare our shooting approach with the continuous orthogonalization method of Humpherys and Zumbrun. We then also compare these with standard projection methods that directly project the spectral problem onto a finite multi-dimensional basis satisfying the boundary conditions.


Approximating the Value Functions of Stochastic Knapsack Problems: A Homogeneous Monge-Amp\'ere Equation and Its Stochastic Counterparts

Abstract: Stochastic knapsack problem originally was a versatile model for controls in telecommunication networks. Recently, it draws attentions of revenue management community by serving as a basic model for allocating resources over time. We develop approximation schemes for knapsack problems in this paper, a system of nonlinear but solvable partial differential equations and stochastic partial differential equation are shown to be the limit of the process that following the optimal solution of the stochastic knapsack problem.


The Minkowski question mark function: explicit series for the dyadic period function and moments

Abstract: Previously, several natural integral transforms of the Minkowski question mark function F(x) were introduced by the author. Each of them is uniquely characterized by certain regularity conditions and the functional equation, thus encoding intrinsic information about F(x). One of them - the dyadic period function G(z) - was defined as a Stieltjes transform. In this paper we introduce a family of "distributions" F_p(x) for Re p>=1, such that F_1(x) is the question mark function and F_2(x) is a discrete distribution with support on x=1. We prove that the generating function of moments of F_p(x) satisfies the three term functional equation. This has an independent interest, though our main concern is the information it provides about F(x). This approach yields the following main result: we prove that the dyadic period function is a sum of infinite series of rational functions with rational coefficients.


The ABC Theorem for Meromorphic Functions

Abstract: Using a `height-to-radical' identity, we define the archimedean contribution to the radical, $r_\arch$, and we give a new proof of the abc theorem for the field of meromorphic functions. The first step of the proof is completely formal and yields that the height is bounded by the radical, $h\leq r$, where $r=r_\na+r_\arch$ is the radical completed with the archimedean contribution. The second step is analytic in nature and uses the lemma on the logarithmic derivative to derive a bound for $r_\arch$.


Rank-level duality of conformal blocks of GL_n and SL_n

Abstract: We generalise the proof by Marian and Oprea of rank-level duality for non-abelian theta functions to the case of sections of line bundles (conformal blocks) over moduli spaces of parabolic vector bundles over a projective smooth curve. We also describe how it implies the rank-level duality between conformal blocks of Sp(2) and Sp(2n).


Geometric representation of binary codes and computation of weight enumerators

Abstract: For every linear binary code $C$, we construct a geometric triangular configuration $\Delta$ so that the weight enumerator of $C$ is obtained by a simple formula from the weight enumerator of the cycle space of $\Delta$. The triangular configuration $\Delta$ thus provides a geometric representation of $C$ which carries its weight enumerator. This is the initial step in the suggestion by M. Loebl, to extend the theory of Pfaffian orientations from graphs to general linear binary codes. Then we carry out also the second step by constructing, for every triangular configuration $\Delta$, a triangular configuration $\Delta'$ and a bijection between the cycle space of $\Delta$ and the set of the perfect matchings of $\Delta'$.


On Some Discrete Differential Equations

Abstract: In this short note, we present few results on the use of the discrete Laplace transform in solving first and second order initial value problems of discrete differential equations.


Lower Bounds for the Size of Random Maximal H-Free Graphs

Abstract: We consider the next greedy randomized process for generating maximal H-free graphs: Given a fixed graph H and an integer n, start by taking a uniformly random permutation of the edges of the complete n-vertex graph. Then, construct an n-vertex graph, M_n(H), iteratively as follows. Traverse the permuted edges of the complete n-vertex graph and add each one to the (initially empty) evolving graph M_n(H) - unless its addition creates a copy of H. The result of this process is a maximal H-free graph M_n(H). The basic question we are concerned with in here is: What is the expected number of edges in M_n(H)? We give new lower bounds on the expected number of edges in M_n(H) for the case where H is a regular, strictly 2-balanced graph. In particular, we obtain new lower bounds for Turan numbers of complete balanced bipartite graphs K_{r,r}, for every fixed r > 4. This improves an old lower bound of Erdos and Spencer.


Combable functions, quasimorphisms, and the central limit theorem

Abstract: A function on a discrete group is weakly combable if its discrete derivative with respect to a combing can be calculated by a finite state automaton. A weakly combable function is bicombable if it is Lipschitz in both the left and right invariant word metrics. Examples of bicombable functions on word-hyperbolic groups include (i) homomorphisms to Z (ii) word length with respect to a finite generating set (iii) most known explicit constructions of quasimorphisms (e.g. the Epstein-Fujiwara counting quasimorphisms) We show that bicombable functions on word-hyperbolic groups satisfy a central limit theorem: if \bar{\phi}_n is the value of \phi on a random element of word length n (in a certain sense), there are E and \sigma for which there is convergence in the sense of distribution n^{-1/2}(\bar{\phi}_n - nE) \to N(0,\sigma), where N(0,\sigma) denotes the normal distribution with standard deviation \sigma. As a corollary, we show that if S_1 and S_2 are any two finite generating sets for G, there is an algebraic number lambda_{1,2} depending on S_1 and S_2 such that almost every word of length n in the S_1 metric has word length n\lambda_{1,2} in the S_2 metric, with error of size O(\sqrt{n}).


Cut ideals of K4-minor free graphs are generated by quadrics

Abstract: Cut ideals are used in algebraic statistics to study statistical models defined by graphs. Intuitively, topological restrictions on the graphs should imply structural statements about the corresponding cut ideals. Several theorems and many computer calculations support that. Sturmfels and Sullivant conjectured that the cut ideal is generated by quadrics if and only if the graph is free of K4-minors. Parts of the conjecture has been resolved by Brennan and Chen, and later by Nagel and Petrovic. We prove the full conjecture by introducing a new type of toric fiber product theorem.


Algebraic Levi-flat hypervarieties in complex projective space

Abstract: We study singular real-analytic Levi-flat hypersurfaces in complex projective space. We define the rank of an algebraic Levi-flat hypersurface and study the connections between rank, degree, and the type and size of the singularity. In particular, we study degenerate singularities of algebraic Levi-flat hypersurfaces. We then give necessary and sufficient conditions for a Levi-flat hypersurface to be a pullback of a real-analytic curve in $\mathbb{C}$ via a meromorphic function. Among other examples, we construct a nonalgebraic semianalytic Levi-flat hypersurface with compact leaves that is a perturbation of an algebraic Levi-flat variety.


Singularities on normal varieties

Abstract: In this paper we generalize the definitions of singularities of pairs and multiplier ideal sheaves to pairs on arbitrary normal varieties, without any assumption on the variety being Q-Gorenstein or the pair being log Q-Gorenstein. The main features of the theory extend to this setting in a natural way.


Panel Cointegration with Global Stochastic Trends

Abstract: This paper studies estimation of panel cointegration models with cross-sectional dependence generated by unobserved global stochastic trends. The standard least squares estimator is, in general, inconsistent owing to the spuriousness induced by the unobservable I(1) trends. We propose two iterative procedures that jointly estimate the slope parameters and the stochastic trends. The resulting estimators are referred to respectively as CupBC (continuously-updated and bias-corrected) and the CupFM (continuously-updated and fully-modified) estimators. We establish their consistency and derive their limiting distributions. Both are asymptotically unbiased and asymptotically mixed normal and permit inference to be conducted using standard test statistics. The estimators are also valid when there are mixed stationary and non-stationary factors, as well as when the factors are all stationary.


Log-Level Comparison Principle for Small Ball Probabilities

Abstract: We prove a new variant of comparison principle for logarithmic $L_2$-small ball probabilities of Gaussian processes. As an application, we obtain logarithmic small ball asymptotics for some well-known processes with smooth covariances.


Random Attractors for the Stochastic Benjamin-Bona-Mahony Equation on Unbounded Domains

Abstract: We prove the existence of a compact random attractor for the stochastic Benjamin-Bona-Mahony Equation defined on an unbounded domain. This random attractor is invariant and attracts every pulled-back tempered random set under the forward flow. The asymptotic compactness of the random dynamical system is established by a tail-estimates method, which shows that the solutions are uniformly asymptotically small when space and time variables approach infinity.


Higgs Bundles and Geometric Structures on Surfaces

Abstract: This paper concerns the relationship between locally homogeneous geometric structures on topological surfaces and the moduli of polystable Higgs bundles on Riemann surfaces, due to Hitchin and Simpson. In particular we discuss the uniformization of Riemann surfaces by hyperbolic geometry from this viewpoint, and survey more recent developments in this theory.


The cross covariogram of a pair of polygons determines both polygons, with a few exceptions

Abstract: The cross covariogram g_{K,L} of two convex sets K and L in R^n is the function which associates to each x in R^n the volume of the intersection of K and L+x. Very recently Averkov and Bianchi [AB] have confirmed Matheron's conjecture on the covariogram problem, that asserts that any planar convex body K is determined by the knowledge of g_{K,K}. The problem of determining the sets from their covariogram is relevant in probability, in statistical shape recognition and in the determination of the atomic structure of a quasicrystal from X-ray diffraction images. We prove that when K and L are convex polygons (and also when K and L are planar convex cones) g_{K,L} determines both K and L, up to a described family of exceptions. These results imply that, when K and L are in these classes, the information provided by the cross covariogram is so rich as to determine not only one unknown body, as required by Matheron's conjecture, but two bodies, with a few classified exceptions. These results are also used by Bianchi [Bia] to prove that any convex polytope P in R^3 is determined by g_{P,P}.


Vector- valued distributions and Hardy's uncertainty principle for operators

Abstract: In this paper, we generalise Hardy's uncertainty principle to vector-valued functions, and hence to operators. The principle for operators can be formulated loosely by saying that the kernel of an operator cannot be localised near the diagonal if the spectrum is also localised.


Abstract: We adapt the generalization of root systems of the second author and H. Yamane to the terminology of category theory. We introduce Cartan schemes, associated root systems and Weyl groupoids. After some preliminary general results, we completely classify all finite Weyl groupoids with at most three objects. The classification yields that there exist infinitely many standard, but only 9 exceptional cases. Key words: Nichols algebra, reflection, root system, Weyl groupoid


Canonical measures and the dynamical systems of Bergman kernels

Abstract: In this article, we construct the canonical semipositive current or the canonical measure ($=$ the potential of the canonical semipositive current) on a smooth projective variety of nonnegative Kodaira dimension in terms of a dynamical system of Bergman kernels. This current is considered to be a generalization of a K\"{a}hler-Einstein metric and coincides the one constructed independently by J. Song and G. Tian. The major difference between their work and the present article is that they found the canonical measure in terms of K\"{a}her-Ricci flows, while I found the canonical measure in terms of the dynamical system of Bergman kernels. Hence the present approach can be viewed as the discrete version of the K\"{a}hler-Ricci flow.


Complex structures on quasi-filiform nilpotent Lie algebras

Abstract: We present the classification of real nilpotent quasi-filiform Lie algebras endowed with a complex structure. A nilpotent Lie algebra g is called quasi-filiform is the nilindex is equal to dim(n)-2. We recall that the filiform case (nilindex =dim(g)-1) has already been studied.


A finiteness theorem for dual graphs of surface singularities

Abstract: Consider a fixed connected, finite graph $\Gamma$ and equip its vertices with weights $p_i$ which are non-negative integers. We show that there is a finite number of possibilities for the coefficients of the canonical cycle of a numerically Gorenstein surface singularity having $\Gamma$ as the dual graph of the minimal resolution, the weights $p_i$ of the vertices being the arithmetic genera of the corresponding irreducible components. As a consequence we get that if $\Gamma$ is not a cycle, then there is a finite number of possibilities of self-intersection numbers which one can attach to the vertices which are of valency $\geq 2$, such that one gets the dual graph of the minimal resolution of a numerically Gorenstein surface singularity. Moreover, we describe precisely the situations when there exists an infinite number of possibilities for the self-intersections of the component corresponding to some fixed vertex of $\Gamma$.


On the $L_{q,p}$-cohomology of Riemannian Manifolds with Negative Curvature

Abstract: We prove a non-vanishing result for the $L_{q,p}$-cohomology of complete simply-connected Riemannian manifolds with pinched negative curvature.


Technical report: Adaptivity and optimality of the monotone least squares estimator for four different models

Abstract: In this paper we will consider the estimation of a monotone regression (or density) function in a fixed point by the least squares (Grenander) estimator. We will show that this estimator is fully adaptive, in the sense that the attained rate is given by a functional relation using the underlying function $f_0$, and not by some smoothness parameter, and that this rate is optimal when considering the class of all monotone functions, in the sense that there exists a sequence of alternative monotone functions $f_1$, such that no other estimator can attain a better rate for both $f_0$ and $f_1$. We also show that under mild conditions the estimator attains the same rate in $L^q$ sense, and we give general conditions for which we can calculate a (non-standard) limiting distribution for the estimator.


Unconditional basic sequences in spaces of large density

Abstract: We study the problem of the existence of unconditional basic sequences in Banach spaces of high density. We show, in particular, the relative consistency with GCH of the statement that every Banach space of density $\aleph_\omega$ contains an unconditional basic sequence.


Cell contamination and branching process in random environment with immigration

Abstract: We consider a branching model for a population of dividing cells infected by parasites. Each cell receives parasites by inheritance from its mother cell and independent contamination from outside the population. Parasites multiply randomly inside the cell and are shared randomly between the two daughter cells when the cell divides. The law of the number of parasites which contaminate a given cell depends only on whether the cell is already infected or not. We determine the asymptotic behavior of the number of parasites in a cell line, which follows a branching process in random environment with state dependent immigration. We then derive a law of large numbers for the asymptotic proportions of cells with a given number of parasites. The main tools are branching processes in random environment and laws of large numbers for Markov tree.


A origami of genus 2 with a translation

Abstract: We study an example of a Teichmueller curve in the moduli space of algebraic curves of genus 2 coming from an origami S. It is particular in that its points admit the Klein four group as a subgroup of the automorphism group. We give an explicit description of its points in terms of affine plane curves, we show that the Teichmueller curve is a nonsingular, affine curve of genus 0 and we determine the number of cusps in the boundary of the moduli space.


Avoidance of Partially Ordered Generalized Patterns of the form $k$-$\sigma$-$k$

Abstract: Sergey Kitaev has shown that the exponential generating function for permutations avoiding the generalized pattern $\sigma$-$k$, where $\sigma$ is a pattern without dashes and $k$ is one greater than the biggest element in $\sigma$, is determined by the exponential generating function for permutations avoiding $\sigma$. We show that this also holds for permutations avoiding all the generalized patterns $\sigma_1$-$k_1$, $...$, $\sigma_n$-$k_n$, where $\sigma_1$, $...$, $\sigma_n$ are patterns without dashes and $k_i$ is one greater than the biggest element in $\sigma_i$. Similarly the exponential generating function for permutations avoiding the partially ordered generalized patterns $k_1$-$\sigma_1$-$k_1$, $...$, $k_n$-$\sigma_n$-$k_n$ can be determined from the exponential generating function for permutations avoiding the generalized patterns $\sigma_1$, $...$, $\sigma_n$, where $\sigma_1$, $...$, $\sigma_n$ are patterns without dashes and $k_i$ is one greater than the largest element in $\sigma_i$. Using this we construct a bijection between bicolored set partitions and permutations avoiding the partially ordered generalized pattern 3-12-3 (that is, permutations avoiding both the patterns 3-12-4 and 4-12-3). By using this method twice, we find a closed formula for the exponential generating function for permutations avoiding the partially ordered generalized pattern 3-121-3. Finally, we give a complete classification of when single partially ordered generalized patterns have the same set of avoiders.


The holomorphy conjecture for ideals in dimension two

Abstract: The holomorphy conjecture predicts that the local Igusa zeta function associated to a hypersurface and a character is holomorphic on $\mathbb{C}$ whenever the order of the character does not divide the order of any eigenvalue of the local monodromy of the hypersurface. In this note we propose the holomorphy conjecture for arbitrary subschemes at the level of the topological zeta function and we prove this conjecture for subschemes defined by an ideal that is generated by a finite number of complex polynomials in two variables.


Poles of the topological zeta function for plane curves and Newton polyhedra

Abstract: The local topological zeta function is a rational function associated to a germ of a complex holomorphic function. This function can be computed from an embedded resolution of singularities of the germ. For nondegenerate functions it is also possible to compute it from the Newton polyhedron. Both ways give rise to a set of candidate poles of the topological zeta function, containing all poles. For plane curves, Veys showed how to filter the actual poles out of the candidate poles induced by the resolution graph. In this note we show how to determine from the Newton polyhedron of a nondegenerate plane curve which candidate poles are actual poles.


Effective categoricity of equivalence Structures

Abstract: An equivalence structure is a set with a single binary relation, satisfying sentences stating that the relation is an equivalence relation. A computable structure A is said to be $\Delta^0_\alpha$ categorical if for any computable structure B isomorphic to A there is a $\Delta^0_\alpha$ function witnessing that the two are isomorphic. The present paper gives an exact characterization of $\Delta^0_\alpha$ equivalence structures where $\alpha = 1$ or $\alpha \geq 3$. Extensive results for $\alpha = 2$ are also given, and open cases are exhaustively described.


Effective categoricity of Abelian p-groups

Abstract: Let p be a fixed prime. An Abelian p-group is an Abelian group (not necessarily finitely generated) in which every element has for its order some power of p. The countable Abelian p-groups are classified by Ulm's theorem, and Khisamiev characterized the Abelian p-groups with computable copies. A computable structure A is said to be $\Delta^0_\alpha$ categorical if for any computable structure B isomorphic to A there is a $\Delta^0_\alpha$ function witnessing that the two are isomorphic. The present paper seeks to characterize $\Delta^0_\alpha$ categoricity for Abelian p-groups, and results of this kind are given for broad classes of Abelian p-groups and values of $\alpha$. The remaining open cases are exhaustively described.


Minimal Lagrangian diffeomorphisms between domains in the hyperbolic plane

Abstract: Let N be a complete, simply-connected surface of constant curvature \kappa \leq 0. Moreover, suppose that \Omega and \tilde{\Omega} are strictly convex domains in N with the same area. We show that there exists an area-preserving diffeomorphism from \Omega to \tilde{\Omega} whose graph is a minimal submanifold of N \times N.


On the martingale probem associated to the 2D and 3D Stochastic Navier-Stokes equations

Abstract: We consider the martingale problem associated to the Navier-Stokes in dimension 2 or 3. Existence is well known and it has been recently shown that markovian transition semi group associated to these equations can be constructed. We study the Kolmogorov operator associated to these equations. It can be defined formally as a differential operator on an infinite dimensional Hilbert space. It can be also defined in an abstract way as the infinitesimal generator of the transition semi group. We explicit cores for these abstract operators and identify them with the concrete differential operators on these cores. In dimension 2, the core is explicit and we can use a classical argument to prove uniqueness for the martingale problem. In dimension 3, we are only able to exhibit a core which is defined abstractly and does not allow to prove uniqueness for the martingale problem. Instead, we exhibit a core for a modified Kolmogorov operator which enables us to prove uniqueness for the martingale problem up to the time the solutions are regular.


On one property of distances in the infinite random quadrangulation

Abstract: We show that the Schaeffer's tree for an infinite quadrangulation only changes locally when changing the root of the quadrangulation. This follows from one property of distances in the infinite uniform random quadrangulation.


On universal Lie nilpotent associative algebras

Abstract: We study the quotient Q_i(A) of a free algebra A by the ideal M_i(A) generated by relation that the i-th commutator of any elements is zero. In particular, we completely describe such quotient for i=4 (for i<=3 this was done previously by Feigin and Shoikhet). We also study properties of the ideals M_i(A), e.g. when M_i(A)M_j(A) is contained in M_{i+j-1}(A) (by a result of Gupta and Levin, it is always contained in M_{i+j-2}(A)).


Analytification is the limit of all tropicalizations

Abstract: We introduce extended tropicalizations for closed subvarieties of toric varieties and show that the analytification of a quasprojective variety over a nonarchimedean field is naturally homeomorphic to the inverse limit of the tropicalizations of its quasiprojective embeddings.


Krull dimension for limit groups II: aligning JSJ decompositions

Abstract: This is the second paper in a sequence on Krull dimension for limit groups, answering a question of Z. Sela. In it we develop the notion of a resolution of a sequence of limit groups and show how to derive resolutions of low complexity from resolutions of high complexity.


Aggregation of weakly dependent doubly stochastic processes

Abstract: The aim of this paper is to extend the aggregation convergence results given in (Dacunha-Castelle and Fermin 2005, Dacunha-Castelle and Fermin 2008) to doubly stochastic linear and nonlinear processes with weakly dependent innovations. First, we introduce a weak dependence notion for doubly stochastic processes, based in the weak dependence definition given in (Doukhan and Louhichi 1999), and we exhibe several models satisfying this notion, such as: doubly stochastic Volterra processes and doubly stochastic Bernoulli scheme with weakly dependent innovations. Afterwards we derive a central limit theorem for the partial aggregation sequence considering weakly dependent doubly stochastic processes. Finally, show a new SLLN for the covariance function of the partial aggregation process in the case of doubly stochastic Volterra processes with interactive innovations. Keywords: Aggregation, weak dependence, doubly stochastic processes, Volterra processes, Bernoulli shift, TCL, SLLN.


A splitting theorem for holomorphic Banach bundles

Abstract: This paper is motivated by Grothendieck's splitting theorem. In the 1960s, Gohberg generalized this to a class of Banach bundles. We consider a compact complex manifold $X$ and a holomorphic Banach bundle $E \to X$ that is a compact perturbation of a trivial bundle in a sense recently introduced by Lempert. We prove that $E$ splits into the sum of a finite rank bundle and a trivial bundle, provided $H^{1}(X, \O)=0$.


Matrix subadditivity inequalities and block-matrices

Abstract: Several subadditivity results and conjectures are given for matrices (or operators), block-matrices, concave functions and norms.


Moving frames on the twistor space of self-dual positive Einstein 4-manifolds

Abstract: The twistor space of self-dual positive Einstein manifolds naturally admits two 1-parameter families of Riemannian metrics, one is the family of canonical deformation metrics and the other is the family introduced by B. Chow and D. Yang in 1989. The purpose of this paper is to compare these two families. In particular we compare the Ricci tensor and the behavior under the Ricci flow of these families. As an application, we propose a new proof to the fact that a locally irreducible self-dual positive Einstein 4-manifold is isometric to either $S^4$ with a standard metric or $\PP^2(\C)$ with a Fubini-Study metric.


Alternating, pattern-avoiding permutations

Abstract: We study the problem of counting alternating permutations avoiding collections of permutation patterns including 132. We construct a bijection between the set S_n(132) of 132-avoiding permutations and the set A_{2n + 1}(132) of alternating, 132-avoiding permutations. For every set p_1, ..., p_k of patterns and certain related patterns q_1, ..., q_k, our bijection restricts to a bijection between S_n(132, p_1, ..., p_k), the set of permutations avoiding 132 and the p_i, and A_{2n + 1}(132, q_1, ..., q_k), the set of alternating permutations avoiding 132 and the q_i. This reduces the enumeration of the latter set to that of the former.


On a set of transformations of Gaussian random functions

Abstract: We consider a set of one-dimensional transformations of Gaussian random functions. Under natural assumptions we obtain a connection between $L_2$-small ball asymptotics of the transformed function and of the original one. Also the explicit Karhunen -- Lo\'eve expansion is obtained for a proper class of Gaussian processes.


Confidence regions for the multinomial parameter with small sample size

Abstract: Consider the observation of n iid realizations of an experiment with d>1 possible outcomes, which corresponds to a single observation of a multinomial distribution M(n,p) where p is an unknown discrete distribution on {1,...,d}. In many applications, the construction of a confidence region for p when n is small is crucial. This concrete challenging problem has a long history. It is well known that the confidence regions built from asymptotic statistics do not have good coverage when n is small. On the other hand, most available methods providing non-asymptotic regions with controlled coverage are limited to the binomial case d=2. In the present work, we propose a new method valid for any d>1. This method provides confidence regions with controlled coverage and small volume, and consists of the inversion of the "covering collection"' associated with level-sets of the likelihood. The behavior when d/n tends to infinity remains an interesting open problem beyond the scope of this work.


On the Rankin-Selberg problem: higher power moments of the Riesz mean error term

Abstract: Let $\Delta_1(x;\phi)$ be the error term of the first Riesz means of the Rankin-Selberg zeta function. We study the higher power moments of $\Delta_1(x;\phi)$ and derive an asymptotic formula for 3-rd, 4-th and 5-th power moments by using Ivi\'c 's large value arguments.


On Berry--Esseen bounds for non-instantaneous filters of linear processes

Abstract: Let $X_n=\sum_{i=1}^{\infty}a_i\epsilon_{n-i}$, where the $\epsilon_i$ are i.i.d. with mean 0 and at least finite second moment, and the $a_i$ are assumed to satisfy $|a_i|=O(i^{-\beta})$ with $\beta >1/2$. When $1/2<\beta<1$, $X_n$ is usually called a long-range dependent or long-memory process. For a certain class of Borel functions $K(x_1,...,x_{d+1})$, $d\ge0$, from ${\mathcal{R}}^{d+1}$ to $\mathcal{R}$, which includes indicator functions and polynomials, the stationary sequence $K(X_n,X_{n+1},...,X_{n+d})$ is considered. By developing a finite orthogonal expansion of $K(X_n,...,X_{n+d})$, the Berry--Esseen type bounds for the normalized sum $Q_N/\sqrt{N},Q_N=\sum_{n=1}^N(K(X_ n,...,X_{n+d})-\mathrm{E}K(X_n,...,X_{n+d}))$ are obtained when $Q_N/\sqrt{N}$ obeys the central limit theorem with positive limiting variance.


Number of Least Area Planes in Gromov Hyperbolic 3-Spaces

Abstract: We show that for a generic simple closed curve C in the asymptotic boundary of a Gromov hyperbolic 3-space with cocompact metric X, there exist a unique least area plane P in X with asymptotic boundary C. This result has interesting topological applications for constructions of canonical 2-dimensional objects in 3-manifolds.


The dbar steepest descent method for orthogonal polynomials on the real line with varying weights

Abstract: We obtain Plancherel-Rotach type asymptotics valid in all regions of the complex plane for orthogonal polynomials with varying weights of the form $e^{-NV(x)}$ on the real line, assuming that $V$ has only two Lipschitz continuous derivatives and that the corresponding equilibrium measure has typical support properties. As an application we extend the universality class for bulk and edge asymptotics of eigenvalue statistics in unitary invariant Hermitian random matrix theory. Our methodology involves developing a new technique of asymptotic analysis for matrix Riemann-Hilbert problems with nonanalytic jump matrices suitable for analyzing such problems even near transition points where the solution changes from oscillatory to exponential behavior.


On the symmetry of arithmetical functions in almost all short intervals,IV

Abstract: We study the arithmetic (real) function, with f 'essentially bounded'. In particular, we obtain non-trivial bounds, through f 'correlations', for the 'Selberg integral' and the 'symmetry integral' of f in almost all short intervals $[x-h,x+h]$, $N\le x\le 2N$, beyond the 'classical' level, up to a very high level of distribution (for $h$ not too small). This time we go beyond Large Sieve inequality. Precisely, our method applies Weil bound for Kloosterman sums.


Structure of the spaces of matrix monotone functions and of matrix convex functions and Jensen's type inequality for operators

Abstract: Let $n \in \N$ and $M_n$ be the algebra of $n \times n$ matrices. We call a function $f$ matrix monotone of order $n$ or $n$-monotone in short whenever the inequality $f(a) \leq f(b)$ holds for every pair of selfadjoint matrices $a, b \in M_n$ such that $a \leq b$ and all eigenvalues of $a$ and $b$ are contained in $I$. Matrix convex (concave) functions on $I$ are similarily defined. The spaces for $n$-monotone functions and $n$-convex functions are written as $P_n(I)$ and $K_n(I)$. In this note we discuss several assertions at each leven $n$ for which we regard themas the problems of double piling structure of those sequences $\{P_n(I)\}_{n\in\N}$ and $\{K_n(I)\}_{n\in\N}$. In order to see clear insight of the aspect of the problems, however, we choose the following three main assertions among them and discuss their mutual dependence: \begin{enumerate} \item[(i)] $f(0)\leq 0$ and $f$ is $n$-convex in $[0,\alpha)$, \item[(ii)] For each matrix $a$ with its spectrum in $[0,\alpha)$ and a contraction $c$ in the matrix algebra $M_n$, \[ f(c^{\star}a c)\leq c^{\star}f(a)c, \] \item[(iii)] The functon $g(t)/t$ is $n$-monotone in $(0,\alpha)$. \end{enumerate} In particular, we show that for any $n \in \N$ two conditions $(ii)$ and $(iii)$ are equivalent.


Stochastic analysis on Gaussian space applied to drift estimation

Abstract: In this paper we consider the nonparametric functional estimation of the drift of Gaussian processes using Paley-Wiener and Karhunen-Lo\`eve expansions. We construct efficient estimators for the drift of such processes, and prove their minimaxity using Bayes estimators. We also construct superefficient estimators of Stein type for such drifts using the Malliavin integration by parts formula and stochastic analysis on Gaussian space, in which superharmonic functionals of the process paths play a particular role. Our results are illustrated by numerical simulations and extend the construction of James-Stein type estimators for Gaussian processes by Berger and Wolper.


Currents and Flat Chains Associated to Varifolds, with an Application to Mean Curvature Flow

Abstract: We prove under suitable hypotheses that convergence of integral varifolds implies convergence of associated mod 2 flat chains and subsequential convergence of associated integer-multiplicity rectifiable currents. The convergence results imply restrictions on the kinds of singularities that can occur in mean curvature flow.


The Kolmogorov operator associated to a Burgers SPDE in spaces of continuous functions

Abstract: We are concerned with a viscous Burgers equation forced by a perturbation of white noise type. We study the corresponding transition semigroup in a space of continuous functions weighted by a proper potential, and we show that the infinitesimal generator is the closure (with respect to a suitable topology) of the Kolmogorov operator associated to the stochastic equation. In the last part of the paper we use this result to solve the corresponding Fokker-Planck equation.


Equiangular tight frames from complex Seidel matrices containing cube roots of unity

Abstract: We derive easily verifiable conditions which characterize when complex Seidel matrices containing cube roots of unity have exactly two eigenvalues. The existence of such matrices is equivalent to the existence of equiangular tight frames for which the inner product between any two frame vectors is always a common multiple of the cube roots of unity. We also exhibit a relationship between these equiangular tight frames, complex Seidel matrices, and highly regular, directed graphs. We construct examples of such frames with arbitrarily many vectors.


A symmetry property of some harmonic algebraic curves

Abstract: The aim of this note is to give a surprising symmetry property of some harmonic algebraic curves: when all the roots $z_i$ of a complex polynomial $P$ lie on the unit circle $\U$, the points of $\U$ different from the $z_i$, and such that $\Arg(P(z))=\theta$, form a regular $n$-gon, where $n$ is the degree of $P$.


The growth of a C_0-semigroup characterised by its cogenerator

Abstract: We characterise contractivity, boundedness and polynomial boundedness for a C_0-semigroup on a Banach space in terms of its cogenerator V (or the Cayley transform of the generator) or its resolvent. In particular, we extend results of Gomilko and Brenner, Thomee and show that polynomial boundedness of a semigroup implies polynomial boundedness of its cogenerator. As is shown by an example, the result is optimal. For analytic semigroups we show that the converse holds, i.e., polynomial boundedness of the cogenerators implies polynomial boundedness of the semigroup. In addition, we show by simple examples in (C^2,\|\cdot\|_p), p \neq 2, that our results on the characterisation of contractivity are sharp. These examples also show that the famous Foias-Sz.-Nagy theorem on cogenerators of contraction semigroups on Hilbert spaces fails in (C^2,\|\cdot\|_p) for p\neq 2.


Operators of Harmonic Analysis in Weighted Spaces with Non-standard Growth

Abstract: Last years there was increasing an interest to the so called function spaces with non-standard growth, known also as variable exponent Lebesgue spaces. For weighted such spaces on homogeneous spaces, we develop a certain variant of Rubio de Francia's extrapolation theorem. This extrapolation theorem is applied to obtain the boundedness in such spaces of various operators of harmonic analysis, such as maximal and singular operators, potential operators, Fourier multipliers, dominants of partial sums of trigonometric Fourier series and others, in weighted Lebesgue spaces with variable exponent. There are also given their vector-valued analogues.


Codings of separable compact subsets of the first Baire class

Abstract: Let $X$ be a Polish space and $K$ a separable compact subset of the first Baire class on $X$. For every sequence $\bs$ dense in $\kk$, the descriptive set-theoretic properties of the set \[ \lbf=\{L\in[\nn]: (f_n)_{n\in L} \text{is pointwise convergent}\} \] are analyzed. It is shown that if $K$ is not first countable, then $\lbf$ is $\PB^1_1$-complete. This can also happen even if $K$ is a pre-metric compactum of degree at most two, in the sense of S. Todorcevic. However, if $K$ is of degree exactly two, then $\lbf$ is always Borel. A deep result of G. Debs implies that $\lbf$ contains a Borel cofinal set and this gives a tree-representation of $\kk$. We show that classical ordinal assignments of Baire-1 functions are actually $\PB^1_1$-ranks on $\kk$. We also provide an example of a $\SB^1_1$ Ramsey-null subset $A$ of $[\nn]$ for which there does not exist a Borel set $B\supseteq A$ such that the difference $B\setminus A$ is Ramsey-null.


Weighted Boundedness of the Maximal, Singular and Potential Operators in Variable Exponent Spaces

Abstract: We present a brief survey of recent results on boundedness of some classical operators within the frameworks of weighted spaces $L^{p(\cdot)}(\varrho)$ with variable exponent $p(x)$, mainly in the Euclidean setting and dwell on a new result of the boundedness of the Hardy-Littlewood maximal operator in the space $L^{p(\cdot)}(X,\varrho)$ over a metric measure space $X$ satisfying the doubling condition. In the case where $X$ is bounded, the weight function satisfies a certain version of a general Muckenhoupt-type condition For a bounded or unbounded $X$ we also consider a class of weights of the form $\varrho(x)=[1+d(x_0,x)]^{\bt_\infty}\prod_{k=1}^m w_k(d(x,x_k))$, $x_k\in X$, where the functions $w_k(r)$ have finite upper and lower indices $m(w_k)$ and $M(w_k)$. Some of the results are new even in the case of constant $p$.


Sample autocovariances of long-memory time series

Abstract: We find the asymptotic distribution of the sample autocovariances of long-memory processes in cases of finite and infinite fourth moment. Depending on the interplay of assumptions on moments and the intensity of dependence, there are three types of convergence rates and limit distributions. In particular, a normal approximation with the standard rate does not always hold in practically relevant cases.


On filling families of finite subsets of the Cantor set

Abstract: Let $\ee>0$ and $\fff$ be a family of finite subsets of the Cantor set $\ccc$. Following D. H. Fremlin, we say that $\fff$ is $\ee$-filling over $\ccc$ if $\fff$ is hereditary and for every $F\subseteq\ccc$ finite there exists $G\subseteq F$ such that $G\in\fff$ and $|G|\geq\ee |F|$. We show that if $\fff$ is $\ee$-filling over $\ccc$ and $C$-measurable in $[\ccc]^{<\omega}$, then for every $P\subseteq\ccc$ perfect there exists $Q\subseteq P$ perfect with $[Q]^{<\omega}\subseteq\fff$. A similar result for weaker versions of density is also obtained.


A classification of separable Rosenthal compacta and its applications

Abstract: The present work consists of three parts. In the first one we determine the prototypes of separable Rosenthal compacta and we provide a classification theorem. The second part concerns an extension of a theorem of S. Todorcevic. The last one is devoted to applications.


On pairs of definable orthogonal families

Abstract: We introduce the notion of an M-family of infinite subsets of $\nn$ which is implicitly contained in the work of A. R. D. Mathias. We study the structure of a pair of orthogonal hereditary families $\aaa$ and $\bbb$, where $\aaa$ is analytic and $\bbb$ is $C$-measurable and an M-family.


A strong boundedness result for separable Rosenthal compacta

Abstract: It is proved that the class of separable Rosenthal compacta on the Cantor set having a uniformly bounded dense sequence of continuous functions, is strongly bounded.


Definability under duality

Abstract: It is shown that if $A$ is an analytic class of separable Banach spaces with separable dual, then the set $A^*=\{Y:\exists X\in A \text{with} Y\cong X^*\}$ is analytic. The corresponding result for pre-duals is false.


On antichains of spreading models of Banach spaces

Abstract: We show that for every separable Banach space $X$, either $\spw(X)$ (the set of all spreading models of $X$ generated by weakly-null sequences in $X$, modulo equivalence) is countable, or $\spw(X)$ contains an antichain of the size of the continuum. This answers a question of S. J. Dilworth, E. Odell and B. Sari.


Limit distributions for the problem of collecting pairs

Abstract: Let $N_n=\{1,2,...,n\}$. Elements are drawn from the set $N_n$ with replacement, assuming that each element has probability $1/n$ of being drawn. We determine the limiting distributions for the waiting time until the given portion of pairs $jj$, $j\in N_n$, is sampled. Exact distributions of some related random variables and their characteristics are also obtained.


Braid ordering and knot genus

Abstract: The genus of knots is a one of the fundamental invariant and can be seen as a complexity of knots. In this paper, we give a lower bound of genus using Dehornoy floor, which is a measure of complexity of braids in terms of braid ordering.


On classes of Banach spaces admitting "small" universal spaces

Abstract: We characterize those classes $\ccc$ of separable Banach spaces admitting a separable universal space $Y$ (that is, a space $Y$ containing, up to isomorphism, all members of $\ccc$) which is not universal for all separable Banach spaces. The characterization is a byproduct of the fact, proved in the paper, that the class $\mathrm{NU}$ of non-universal separable Banach spaces is strongly bounded. This settles in the affirmative the main conjecture form \cite{AD}. Our approach is based, among others, on a construction of $\llll_\infty$-spaces, due to J. Bourgain and G. Pisier. As a consequence we show that there exists a family $\{Y_\xi:\xi<\omega_1\}$ of separable, non-universal, $\llll_\infty$-spaces which uniformly exhausts all separable Banach spaces. A number of other natural classes of separable Banach spaces are shown to be strongly bounded as well.


On the elicitation of continuous, symmetric, unimodal distributions

Abstract: In this brief note, we highlight some difficulties that can arise when fitting a continuous, symmetric, unimodal distribution to a set of expert's judgements. A simple analysis shows it is possible to fit a Cauchy distribution to an expert's beliefs when their beliefs actually follow a normal distribution. This example stresses the need for careful distribution fitting and for feedback to the expert about what the fitted distribution implies about their beliefs.


On unconditionally saturated Banach spaces

Abstract: We prove a structural property of the class of unconditionally saturated separable Banach spaces. We show, in particular, that for every analytic set $\aaa$, in the Effros-Borel space of subspaces of $C[0,1]$, of unconditionally saturated separable Banach spaces, there exists an unconditionally saturated Banach space $Y$, with a Schauder basis, that contains isomorphic copies of every space $X$ in the class $\aaa$.


On the number of unique expansions in non-integer bases

Abstract: Let $q > 1$ be a real number and let $m=m(q)$ be the largest integer smaller than $q$. It is well known that each number $x \in J_q:=[0, \sum_{i=1}^{\infty} m q^{-i}]$ can be written as $x=\sum_{i=1}^{\infty}{c_i}q^{-i}$ with integer coefficients $0 \le c_i < q$. If $q$ is a non-integer, then almost every $x \in J_q$ has continuum many expansions of this form. In this note we consider some properties of the set $\mathcal{U}_q$ consisting of numbers $x \in J_q$ having a unique representation of this form. More specifically, we compare the size of the sets $\mathcal{U}_q$ and $\mathcal{U}_r$ for values $q$ and $r$ satisfying $1< q < r$ and $m(q)=m(r)$.


Absolute Chow-Kuenneth decomposition for rational homogeneous bundles and for log homogeneous varieties

Abstract: In this paper, we investigate Murre's conjecture on the existence of a Chow--Kuenneth decomposition for a rational homogeneous bundle $Z\to S$ over a smooth variety, defined over complex numbers. Chow-K\"unneth decomposition is exhibited for $Z$ whenever $S$ has a Chow--Kuenneth decomposition. The same conclusion holds for a class of log homogeneous varieties, studied by M. Brion.


Asymptotic Nets and Discrete Affine Surfaces with Indefinite Metric

Abstract: Asymptotic net is an important concept in discrete differential geometry. In this paper, we show that we can associate affine discrete geometric concepts to an arbitrary non-degenerate asymptotic net. These concepts include discrete affine area, mean curvature, normal and co-normal vector fields and cubic form, and they are related by structural and compatibility equations. We consider also the particular cases of affine minimal surfaces and affine spheres.


KW-sections for exceptional type Vinberg's $\theta$-groups

Abstract: Let $k$ be an algebraically closed field of characteristic not equal to 2 or 3, let $G$ be an almost simple algebraic group of type $F_4$, $G_2$ or $D_4$ and let $\theta$ be an automorphism of $G$ of finite order, coprime to the characteristic. In this paper we consider the $\theta$-group (in the sense of Vinberg) associated to these choices; we classify the positive rank automorphisms and give their Kac diagrams and we describe the little Weyl group in each case. As a result we show that all such $\theta$-groups have KW-sections, confirming a conjecture of Popov in these cases.


A 3-Variable Bracket

Abstract: Kauffman's bracket is an invariant of regular isotopy of knots and links which since its discovery in 1985 it has been used in many different directions: (a) it implies an easy proof of the invariance of (in fact, it is equivalent to) the Jones polynomial; (b) it is the basic ingredient in a completely combinatorial construction for quantum 3-manifold invariants; (c) by its fundamental character it plays an important role in some theories in Physics; it has been used in the context of virtual links; it has connections with many objects other objects in Mathematics and Physics. I show in this note that, surprisingly enough, the same idea that produces the bracket can be slightly modified to produce algebraically stronger regular isotopy and ambient isotopy invariants living in the quotient ring $R/I$, where the ring $R$ and the ideal $I$ are: \begin{center} $R=\Z[\alpha,\beta,\delta]$, $I=< p_1, p_2 >$, with $p_1=\alpha^2 \delta + 2 \alpha \beta \delta ^2 -\delta ^2+\beta ^2 \delta, p_2=\alpha \beta \delta ^3+\alpha ^2 \delta ^2+\beta ^2 \delta ^2+\alpha \beta \delta -\delta.$ \end{center} It is easy to prove that any pair of links distinguished by the usual bracket is also distinguishable by the new invariant. The contrary is not necessarily true. However, a explicit example of a pair of knots not distinguished by the bracket and distinguished by this new invariant is an open problem.


Manifolds covered by lines and extremal rays

Abstract: Let $X$ be a smooth complex projective variety and let $H \in \pic(X)$ be an ample line bundle. Assume that $X$ is covered by rational curves with degree one with respect to $H$ and with anticanonical degree greater than or equal to $(\dim X -1)/2$. We prove that there is a covering family of such curves whose numerical class spans an extremal ray in the cone of curves $\cone(X)$.


Estimation of the Brownian dimension of a continuous It\^{o} process

Abstract: In this paper, we consider a $d$-dimensional continuous It\^{o} process which is observed at $n$ regularly spaced times on a given time interval $[0,T]$. This process is driven by a multidimensional Wiener process and our aim is to provide asymptotic statistical procedures which give the minimal dimension of the driving Wiener process, which is between 0 (a pure drift) and $d$. We exhibit several different procedures, all similar to asymptotic testing hypotheses.


Infinite smooth Lyndon words

Abstract: In a recent paper, Brlek, Jamet and Paquin showed that some extremal infinite smooth words are also infinite Lyndon words. This result raises a natural question: are they the only ones? If no, what do the infinite smooth words that are also Lyndon words look like? In this paper, we give the answer, proving that the only infinite smooth Lyndon words are $m_{\{a<b\}}$, with $a,b$ even, $m_{\{1<b\}}$ and $\Delta^{-1}_1(m_{\{1<b\}})$, with $b$ odd, where $m_\A$ is the minimal infinite smooth word with respect to the lexicographic order over a numerical alphabet $\A$ and $\Delta$ is the run-length encoding function.


Stochastic calculus for convoluted L\'{e}vy processes

Abstract: We develop a stochastic calculus for processes which are built by convoluting a pure jump, zero expectation L\'{e}vy process with a Volterra-type kernel. This class of processes contains, for example, fractional L\'{e}vy processes as studied by Marquardt [Bernoulli 12 (2006) 1090--1126.] The integral which we introduce is a Skorokhod integral. Nonetheless, we avoid the technicalities from Malliavin calculus and white noise analysis and give an elementary definition based on expectations under change of measure. As a main result, we derive an It\^{o} formula which separates the different contributions from the memory due to the convolution and from the jumps.


Fully closed maps and non-metrizable higher-dimensional Anderson-Choquet continua

Abstract: Fedorchuk's fully closed (continuous) maps and resolutions are applied in constructions of non-metrizable higher-dimensional analogues of Anderson, Choquet, and Cook's continua. Certain theorems on dimension-lowering maps are proved for inductive dimensions and fully closed maps from spaces that need not be hereditarily normal, and some examples of continua have non-coinciding dimensions.


On the Form of Odd Perfect Gaussian Integers

Abstract: We extend the sum-of-divisors function to the complex plane via the Gaussian integers. Then we prove a modified form of Euler's classification of odd perfect numbers.


New examples of p-adically rigid automorphic forms

Abstract: We give examples of cohomological automorphic forms for unitary groups which are $p$-adically rigid.


Invariant measures for interval maps with critical points and singularities

Abstract: We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure.


Asymptotic Behavior of Systems involving Delays: Preliminary Results

Abstract: This paper investigates the relations between the particular eigensolutions of a limiting functional differential equation of any order, which is the nominal (unperturbed) linear autonomous differential equations, and the associate ones of the corresponding perturbed functional differential equation. Both differential equations involve point and distributed delayed dynamics. The proofs are based on a Perron type theorem for functional equations so that the comparison is governed by the real part of a dominant zero of the characteristic equation of the nominal differential equation. The obtained results are also applied to investigate the global stability of the perturbed equation based on that of its corresponding limiting equation.


Quadrature formulas for integrals transforms generated by orthogonal polynomials

Abstract: By using the three-term recurrence equation satisfied by a family of orthogonal polynomials, the Christoffel-Darboux-type bilinear generating function and their asymptotic expressions, we obtain quadrature formulas for integral transforms generated by the classical orthogonal polynomials. These integral transforms, related to the so-called Poisson integrals, correspond to a modified Fourier Transform in the case of the Hermite polynomials, a Bessel Transform in the case of the Laguerre polynomials and to an Appell Transform in the case of the Jacobi polynomials.


On special values of spinor L-functions of Siegel cusp eigenforms of genus 3

Abstract: We compute the special values for the spinor L-function L(s,F12) in the critical strip s={12,...,19}, where F12 is the unique (up to a scalar) Siegel cusp form of degree 3 and weight 12, which was constructed by Miyawaki. These values are proportional to the product of Petersson inner products of Ramanujan's Delta by itself and the cusp form of weight 20 for SL(2,Z) by itself by a rational number and some power of Pi. We also verify this result numerically using Dokchitser's ComputeL PARI package. To our knowledge this is the first example of a spinor L-function of Siegel cusp forms of degree 3, when the special values can be computed explicitly.


Eight Hateful Sequences

Abstract: In his July 1974 Scientific American column, Martin Gardner mentioned the Handbook of Integer Sequences, which then contained 2372 sequences. Today the On-Line Encyclopedia of Integer Sequences (the OEIS) contains 140000 sequences. This paper discusses eight of them, suggested by the theme of the Eighth Gathering For Gardner: they are all infinite, and all 'ateful in one way or another. Each one is connected with an unsolved problem. The sequences are related to: hateful numbers, Angelini's 1995 puzzle, the persistence of a number, Alekseyev's 123 sequence, the curling number conjecture, Quet's prime-generating recurrence, the traveling salesman's problem, and the Riemann Hypothesis.


Functoriality in Morse theory on closed manifolds

Abstract: We develop functoriality for Morse theory, namely, to a pair of Morse-Smale systems and a generic smooth map between the underlying manifolds we associate a chain map between the corresponding Morse complexes, which descends to the correct map on homology. This association does not in general respect composition. We give sufficient conditions under which composition is preserved. As an application we provide a new proof that the cup product as defined in Morse theory on the chain level agrees with the cup product in singular cohomology. In appendices we present a proof (due to Paul Biran) that the unstable manifolds of a Morse-Smale system are the open cells of a CW structure on the underlying manifold, and also we show that the Morse complex of the triple is canonically isomorphic to the cellular complex of the CW structure. This gives a new proof that the Morse complex is actually a complex and that it computes the homology of the manifold.


Spectral gaps of the one-dimensional Schr\"odinger operators with singular periodic potentials

Abstract: The behaviour of the lengths of spectral gaps $\{\gamma_{n}(q)\}_{n=1}^{\infty}$ of the Hill-Schr\"odinger operators S(q)u=-u''+q(x)u,\quad u\in \mathrm{Dom}(S(q)) with real-valued 1-periodic distributional potentials $q(x)\in H_{1{-}per}^{-1}(\mathbb{R})$ is studied. We show that they exhibit the same behaviour as the Fourier coefficients $\{\widehat{q}(n)\}_{n=-\infty}^{\infty}$ of the potentials $q(x)$ with respect to the weighted sequence spaces $h^{s,\varphi}$, $s>-1$, $\varphi\in \mathrm{SV}$. The case $q(x)\in L_{1{-}per}^{2}(\mathbb{R})$, $s\in \mathbb{Z}_{+}$, $\varphi\equiv 1$ corresponds to the Marchenko-Ostrovskii Theorem.


An Improvement on Olson's Constant for Z_p+Z_p

Abstract: For a prime number p greater than 6000, the Olson's constant for the group Z_p+Z_p is given by Ol(Z_p+Z_p)=p-1+Ol(Z_p).


On the adjoint quotient of Chevalley groups over arbitrary base schemes

Abstract: For a split semisimple Chevalley group scheme G with Lie algebra g over an arbitrary base scheme S, we consider the quotient of g by the adjoint action of G. We study in detail the structure of g over S. Given a maximal torus T with Lie algebra t and associated Weyl group W, we show that the Chevalley morphism t/W -> g/G is an isomorphism except for the group Sp_{2n} over a base with 2-torsion. In this case this morphism is only dominant and we compute it explicitly. We compute the adjoint quotient in some other classical cases, yielding examples where the formation of the quotient g -> g/G commutes, or does not commute, with base change on S.


Modular forms on noncongruence subgroups and Atkin-Swinnerton-Dyer relations

Abstract: We give new examples of weight three cusp forms on noncongruence subgroups of SL(2, Z) whose Scholl representation is modular and which satisfy three term Atkin-Swinnerton-Dyer relations.


Orderings of the rationals and dynamical systems

Abstract: This paper is devoted to a systematic study of a class of binary trees encoding the structure of rational numbers both from arithmetic and dynamical point of view. The paper is divided into two parts. The first one is a critical review of rather standard topics such as Stern-Brocot and Farey trees and their connections with continued fraction expansion and the question mark function. In the second part we introduce a class of one-dimensional maps which can be used to generate the binary trees in different ways and study their ergodic properties. This also leads us to study some random processes (Markov chains and martingales) arising in a natural way in this context.


Yoneda representations of flat functors and classifying toposes

Abstract: In this paper, we first introduce a technique that we call "Yoneda representation of flat functors", based on ideas from indexed category theory; then we provide applications of this technique to the theory of classifying toposes. Specifically, we obtain results characterizing the models of a theory classified by a topos of the form Sh(C,J) in terms of the models of a theory classified by the topos [C^op, Set].


On the moduli space of Donaldson-Thomas instantons

Abstract: In alignment with a programme by Donaldson and Thomas [DT], Thomas [Th] constructed a deformation invariant for smooth projective Calabi-Yau threefolds, which is now called the Donaldson-Thomas invariant, from the moduli space of (semi-)stable sheaves by using algebraic geometry techniques. In the same paper [Th], Thomas noted that certain perturbed Hermitian-Einstein equations might possibly produce an analytic theory of the invariant. This article sets up the equations on symplectic 6-manifolds, and gives the local model and structures of the moduli space coming from the equations. We then describe a Hitchin-Kobayashi style correspondence for the equations on compact K\"ahler threefolds, which turns out to be a special case of results by Alvarez-Consul and Garcia-Prada [AG].


A weak compactness theorem of the Donaldson-Thomas instantons on compact K\"ahler threefolds

Abstract: In arXiv:0805.2192, we set up a gauge-theoretic equation on symplectic 6-manifolds, which is a version of the Hermitian-Einstein equation perturbed by Higgs fields, and call Donaldson-Thomas equation, to analytically approach the Donaldson-Thomas invariants. In this article, we consider the equation on compact K\"ahler threefolds, and study some of analytic properties of solutions to them, using analytic methods in higher-dimensional Yang-Mills theory developed by Nakajima and Tian with some additional arguments concerning an extra non-linear term coming from the Higgs fields. We prove that a sequence of solutions to the Donaldson-Thomas equation of a unitary vector bundle over a compact K\"ahler threefold has a converging subsequence outside a closed subset whose real 2-dimensional Hausdorff measure is finite, provided that the L^2-norms of the Higgs fields are uniformly bounded. We also prove an n/2-compactness theorem of solutions to the equations on compact K\"ahler threefolds.


The Donaldson-Thomas instantons on compact Kahler threefolds and a convergence

Abstract: The contents of this article are now presented in the appendix of arXiv:0805.2195v2.


Mass formula for self-orthogonal codes over Z_{p^2}

Abstract: In this note, we establish a mass formula for self-orthogonal codes over Z_{p^2}, where p is a prime. As a consequence, an alternative proof of the known mass formulas for self-dual codes over Z_{p^2} is obtained. We also establish a mass formula for even quaternary codes, which includes a mass formula for Type II quaternary codes as a special case.


Augmented GARCH sequences: Dependence structure and asymptotics

Abstract: The augmented GARCH model is a unification of numerous extensions of the popular and widely used ARCH process. It was introduced by Duan and besides ordinary (linear) GARCH processes, it contains exponential GARCH, power GARCH, threshold GARCH, asymmetric GARCH, etc. In this paper, we study the probabilistic structure of augmented $\mathrm {GARCH}(1,1)$ sequences and the asymptotic distribution of various functionals of the process occurring in problems of statistical inference. Instead of using the Markov structure of the model and implied mixing properties, we utilize independence properties of perturbed GARCH sequences to directly reduce their asymptotic behavior to the case of independent random variables. This method applies for a very large class of functionals and eliminates the fairly restrictive moment and smoothness conditions assumed in the earlier theory. In particular, we derive functional CLTs for powers of the augmented GARCH variables, derive the error rate in the CLT and obtain asymptotic results for their empirical processes under nearly optimal conditions.


Density estimation with heteroscedastic error

Abstract: It is common, in deconvolution problems, to assume that the measurement errors are identically distributed. In many real-life applications, however, this condition is not satisfied and the deconvolution estimators developed for homoscedastic errors become inconsistent. In this paper, we introduce a kernel estimator of a density in the case of heteroscedastic contamination. We establish consistency of the estimator and show that it achieves optimal rates of convergence under quite general conditions. We study the limits of application of the procedure in some extreme situations, where we show that, in some cases, our estimator is consistent, even when the scaling parameter of the error is unbounded. We suggest a modified estimator for the problem where the distribution of the errors is unknown, but replicated observations are available. Finally, an adaptive procedure for selecting the smoothing parameter is proposed and its finite-sample properties are investigated on simulated examples.


Consistency of the $\alpha$-trimming of a probability. Applications to central regions

Abstract: The sequence of $\alpha$-trimmings of empirical probabilities is shown to converge, in the Painlev\'{e}--Kuratowski sense, on the class of probability measures endowed with the weak topology, to the $\alpha$-trimming of the population probability. Such a result is applied to the study of the asymptotic behaviour of central regions based on the trimming of a probability.


A New Differential Test for Series of Positive Terms

Abstract: A new differential test for series of positive terms is proved. Let f(x) be a positive continuous function corresponded to a series of positive terms f(k), and g(x) is a derivative of reciprocal of f(x). Then, the convergence and divergence of the series may be determined from a value of fgx for enough large x. The rest may make the limit form, and is universal and complete.


Analytic perturbations and systematic bias in statistical modeling and inference

Abstract: In this paper we provide a comprehensive study of statistical inference in linear and allied models which exhibit some analytic perturbations in their design and covariance matrices. We also indicate a few potential applications. In the theory of perturbations of linear operators it has been known for a long time that the so-called ``singular perturbations'' can have a big impact on solutions of equations involving these operators even when their size is small. It appears that so far the question of whether such undesirable phenomena can also occur in statistical models and their solutions has not been formally studied. The models considered in this article arise in the context of nonlinear models where a single parameter accounts for the nonlinearity.


Pranab Kumar Sen: Life and works

Abstract: In this article, we describe briefly the highlights and various accomplishments in the personal as well as the academic life of Professor Pranab Kumar Sen.


Smooth estimation of mean residual life under random censoring

Abstract: We propose here a smooth estimator of the mean residual life function based on randomly censored data. This is derived by smoothing the product-limit estimator using the Chaubey-Sen technique (Chaubey and Sen (1998)). The resulting estimator does not suffer from boundary bias as is the case with standard kernel smoothing. The asymptotic properties of the estimator are investigated. We establish strong uniform consistency and asymptotic normality. This complements the work of Chaubey and Sen (1999) which considered a similar estimation procedure in the case of complete data. It is seen that the properties are similar, though technically more difficult to prove, to those in the complete data case with appropriate modifications due to censoring.


Conformal Metrics

Abstract: This paper surveys some selected topics in the theory of conformal metrics and their connections to complex analysis, partial differential equations and conformal differential geometry.


Order restricted inference for comparing the cumulative incidence of a competing risk over several populations

Abstract: There is a substantial literature on testing for the equality of the cumulative incidence functions associated with one specific cause in a competing risks setting across several populations against specific or all alternatives. In this paper we propose an asymptotically distribution-free test when the alternative is that the incidence functions are linearly ordered, but not equal. The motivation stems from the fact that in many examples such a linear ordering seems reasonable intuitively and is borne out generally from empirical observations. These tests are more powerful when the ordering is justified. We also provide estimators of the incidence functions under this ordering constraint, derive their asymptotic properties for statistical inference purposes, and show improvements over the unrestricted estimators when the order restriction holds.


Statistical inference under order restrictions on both rows and columns of a matrix, with an application in toxicology

Abstract: We present a general methodology for performing statistical inference on the components of a real-valued matrix parameter for which rows and columns are subject to order restrictions. The proposed estimation procedure is based on an iterative algorithm developed by Dykstra and Robertson (1982) for simple order restriction on rows and columns of a matrix. For any order restrictions on rows and columns of a matrix, sufficient conditions are derived for the algorithm to converge in a single application of row and column operations. The new algorithm is applicable to a broad collection of order restrictions. In practice, it is easy to design a study such that the sufficient conditions derived in this paper are satisfied. For instance, the sufficient conditions are satisfied in a balanced design. Using the estimation procedure developed in this article, a bootstrap test for order restrictions on rows and columns of a matrix is proposed. Computer simulations for ordinal data were performed to compare the proposed test with some existing test procedures in terms of size and power. The new methodology is illustrated by applying it to a set of ordinal data obtained from a toxicological study.


Totally free arrangements of hyperplanes

Abstract: A central arrangement $\A$ of hyperplanes in an $\ell$-dimensional vector space $V$ is said to be {\it totally free} if a multiarrangement $(\A, m)$ is free for any multiplicity $ m : \A\to \Z_{> 0}$. It has been known that $\A$ is totally free whenever $\ell \le 2$. In this article, we will prove that there does not exist any totally free arrangement other than the obvious ones, that is, a product of one-dimensional arrangements and two-dimensional ones.


On the structure of a family of probability generating functions induced by shock models

Abstract: We explore conditions for a class of functions defined via an integral representation to be a probability generating function of some positive integer valued random variable. Interest in and research on this question is motivated by an apparently surprising connection between a family of classic shock models due to Esary et. al. (1973) and the negatively aging nonparametric notion of ``strongly decreasing failure rate'' (SDFR) introduced by Bhattacharjee (2005). A counterexample shows that there exist probability generating functions with our integral representation which are not discrete SDFR, but when used as shock resistance probabilities can give rise to a SDFR survival distribution in continuous time.


Adaptive approximate Bayesian computation

Abstract: Sequential techniques can enhance the efficiency of the approximate Bayesian computation algorithm, as in Sisson et al.'s (2007) partial rejection control version. While this method is based upon the theoretical works of Del Moral et al. (2006), the application to approximate Bayesian computation results in a bias in the approximation to the posterior. An alternative version based on genuine importance sampling arguments bypasses this difficulty, in connection with the population Monte Carlo method of Cappe et al. (2004), and it includes an automatic scaling of the forward kernel. When applied to a population genetics example, it compares favourably with two other versions of the approximate algorithm.


A Bayesian test for excess zeros in a zero-inflated power series distribution

Abstract: Power series distributions form a useful subclass of one-parameter discrete exponential families suitable for modeling count data. A zero-inflated power series distribution is a mixture of a power series distribution and a degenerate distribution at zero, with a mixing probability $p$ for the degenerate distribution. This distribution is useful for modeling count data that may have extra zeros. One question is whether the mixture model can be reduced to the power series portion, corresponding to $p=0$, or whether there are so many zeros in the data that zero inflation relative to the pure power series distribution must be included in the model i.e., $p\geq0$. The problem is difficult partially because $p=0$ is a boundary point. Here, we present a Bayesian test for this problem based on recognizing that the parameter space can be expanded to allow $p$ to be negative. Negative values of $p$ are inconsistent with the interpretation of $p$ as a mixing probability, however, they index distributions that are physically and probabilistically meaningful. We compare our Bayesian solution to two standard frequentist testing procedures and find that using a posterior probability as a test statistic has slightly higher power on the most important ranges of the sample size $n$ and parameter values than the score test and likelihood ratio test in simulations. Our method also performs well on three real data sets.


Posterior consistency of Dirichlet mixtures of beta densities in estimating positive false discovery rates

Abstract: In recent years, multiple hypothesis testing has come to the forefront of statistical research, ostensibly in relation to applications in genomics and some other emerging fields. The false discovery rate (FDR) and its variants provide very important notions of errors in this context comparable to the role of error probabilities in classical testing problems. Accurate estimation of positive FDR (pFDR), a variant of the FDR, is essential in assessing and controlling this measure. In a recent paper, the authors proposed a model-based nonparametric Bayesian method of estimation of the pFDR function. In particular, the density of p-values was modeled as a mixture of decreasing beta densities and an appropriate Dirichlet process was considered as a prior on the mixing measure. The resulting procedure was shown to work well in simulations. In this paper, we provide some theoretical results in support of the beta mixture model for the density of p-values, and show that, under appropriate conditions, the resulting posterior is consistent as the number of hypotheses grows to infinity.


On hereditarily indecomposable compacta and factorization of maps

Abstract: We prove a general factorization theorem for maps with hereditarily indecomposable fibers and apply it to reprove a theorem of Mackoviak on the existence of universal hereditarily indecomposable continua.


Robust estimation in finite population sampling

Abstract: The paper proposes some robust estimators of the finite population mean. Such estimators are particularly suitable in the presence of some outlying observations. Included as special cases of our general result are robust versions of the ratio estimator and the Horvitz-Thompson estimator. The robust estimators are derived on the basis of certain predictive influence functions.


Estimation of population-level summaries in general semiparametric repeated measures regression models

Abstract: This paper considers a wide family of semiparametric repeated measures regression models, in which the main interest is on estimating population-level quantities such as mean, variance, probabilities etc. Examples of our framework include generalized linear models for clustered/longitudinal data, among many others. We derive plug-in kernel-based estimators of the population level quantities and derive their asymptotic distribution. An example involving estimation of the survival function of hemoglobin measures in the Kenya hemoglobin study data is presented to demonstrate our methodology.


Fundamental groups of symmetric sextics. II

Abstract: We study the moduli spaces and compute the fundamental groups of plane sextics of torus type with the set of inner singularities $2\bold{A}_8$ or $\bold{A}_{17}$. We also compute the fundamental groups of a number of other sextics, both of and not of torus type. The groups found are simplest possible, i.e., $\Bbb{Z}_2*\Bbb{Z}_3$ and $\Bbb{Z}_6$, respectively.


Smoothing-inspired lack-of-fit tests based on ranks

Abstract: A rank-based test of the null hypothesis that a regressor has no effect on a response variable is proposed and analyzed. This test is identical in structure to the order selection test but with the raw data replaced by ranks. The test is nonparametric in that it is consistent against virtually any smooth alternative, and is completely distribution free for all sample sizes. The asymptotic distribution of the rank-based order selection statistic is obtained and seen to be the same as that of its raw data counterpart. Exact small sample critical values of the test statistic are provided as well. It is shown that the Pitman-Noether efficiency of the proposed rank test compares very favorably with that of the order selection test. In fact, their asymptotic relative efficiency is identical to that of the Wilcoxon signed rank and $t$-tests. An example involving microarray data illustrates the usefulness of the rank test in practice.


A nonparametric control chart based on the Mann-Whitney statistic

Abstract: Nonparametric or distribution-free charts can be useful in statistical process control problems when there is limited or lack of knowledge about the underlying process distribution. In this paper, a phase II Shewhart-type chart is considered for location, based on reference data from phase I analysis and the well-known Mann-Whitney statistic. Control limits are computed using Lugannani-Rice-saddlepoint, Edgeworth, and other approximations along with Monte Carlo estimation. The derivations take account of estimation and the dependence from the use of a reference sample. An illustrative numerical example is presented. The in-control performance of the proposed chart is shown to be much superior to the classical Shewhart $\bar{X}$ chart. Further comparisons on the basis of some percentiles of the out-of-control conditional run length distribution and the unconditional out-of-control ARL show that the proposed chart is almost as good as the Shewhart $\bar{X}$ chart for the normal distribution, but is more powerful for a heavy-tailed distribution such as the Laplace, or for a skewed distribution such as the Gamma. Interactive software, enabling a complete implementation of the chart, is made available on a website.


On the length of lemniscates

Abstract: We show that for a monic polynomial p of degree d, the length of the level set {z: |p(z)|=1} is at most 9.2 d, which improves an earlier estimate due to P. Borwein. For d=2 we show that the extremal level set is the Bernoullis' Lemniscate. One ingredient of our proofs is the fact that for an extremal polynomial this level set is connected.


Regression rank scores in nonlinear models

Abstract: Consider the nonlinear regression model $Y_i=g({\bf x}_i,\boldmath $\theta$)+e_i,\quad i=1,...,n$(1) with ${\bf x}_i\in \mathbb{R}^k,$ $\boldmath{\theta}=(\theta_0,\theta_1,...,\theta_p)^{\prime}\in \boldmath $\Theta$$ (compact in $\mathbb{R}^{p+1}$), where $g({\bf x},\boldmath $\theta$)=\theta_0+\tilde{g}({\bf x},\theta_1,...,\theta_p)$ is continuous, twice differentiable in $\boldmath $\theta$$ and monotone in components of $\boldmath $\theta$$. Following Gutenbrunner and Jure\v{c}kov\'{a} (1992) and Jure\v{c}kov\'{a} and Proch\'{a}zka (1994), we introduce regression rank scores for model (1), and prove their asymptotic properties under some regularity conditions. As an application, we propose some tests in nonlinear regression models with nuisance parameters.


Chernoff-Savage and Hodges-Lehmann results for Wilks' test of multivariate independence

Abstract: We extend to rank-based tests of multivariate independence the Chernoff-Savage and Hodges-Lehmann classical univariate results. More precisely, we show that the Taskinen, Kankainen and Oja (2004) normal-score rank test for multivariate independence uniformly dominates -- in the Pitman sense -- the classical Wilks (1935) test, which establishes the Pitman non-admissibility of the latter, and provide, for any fixed space dimensions $p,q$ of the marginals, the lower bound for the asymptotic relative efficiency, still with respect to Wilks' test, of the Wilcoxon version of the same.


Stanley depth of complete intersection monomial ideals

Abstract: We compute the Stanley depth of irreducible monomial ideals and we show that the Stanley depth of a monomial complete intersection ideal is the same as the Stanley depth of it's radical. Also, we give some bounds for the Stanley depth of a monomial complete intersection ideal.


Surgery presentations of coloured knots and of their covering links

Abstract: We consider knots equipped with a representation of their knot groups onto a dihedral group D_{2n} (where n is odd). To each such knot there corresponds a closed 3-manifold, the (irregular) dihedral branched covering space, with the branching set over the knot forming a link in it. We report a variety of results relating to the problem of passing from the initial data of a D_{2n}-coloured knot to a surgery presentation of the corresponding branched covering space and covering link. In particular, we describe effective algorithms for constructing such presentations. A by-product of these investigations is a proof of the conjecture that two D_{2n}-coloured knots are related by a sequence of surgeries along unit-framed unknots in the kernel of the representation if and only if they have the same coloured untying invariant (a Z_{n}-valued algebraic invariant of D_{2n}-coloured knots).


Ext^1-quivers for the Witt algebra W(1,1)

Abstract: Let g be the finite dimensional Witt algebra W(1,1) over an algebraically closed field of characteristic p > 3. It is well known that all simple W(1,1)-modules are finite dimensional. Each simple module admits a character \chi in g^*. Given such a \chi one can form the (finite dimensional) reduced enveloping algebra u(g,\chi). The simple modules for u(g,\chi) are precisely those simple W(1,1)-modules admitting the character \chi. In this paper the authors compute Ext^1 between pairs of simple modules for u(g,\chi).


Long-time extinction of solutions of some semilinear parabolic equations

Abstract: We study the long time behaviour of solutions of semi-linear parabolic equation of the following type $\partial_t u-\Delta u+a_0(x)u^q=0$ where $a_0(x) \geq d_0 \exp(\frac{\omega(|x|)}{|x|^2})$, $d_0>0$, $1>q>0$ and $\omega$ a positive continuous radial function. We give a Dini-like condition on the function $\omega$ by two different method which implies that any solution of the above equation vanishes in a finite time. The first one is a variant of a local energy method and the second one is derived from semi-classical limits of some Schr\"odinger operators.


$U$-tests for variance components in one-way random effects models

Abstract: We consider a test for the hypothesis that the within-treatment variance component in a one-way random effects model is null. This test is based on a decomposition of a $U$-statistic. Its asymptotic null distribution is derived under the mild regularity condition that the second moment of the random effects and the fourth moment of the within-treatment errors are finite. Under the additional assumption that the fourth moment of the random effect is finite, we also derive the distribution of the proposed $U$-test statistic under a sequence of local alternative hypotheses. We report the results of a simulation study conducted to compare the performance of the $U$-test with that of the usual $F$-test. The main conclusions of the simulation study are that (i) under normality or under moderate degrees of imbalance in the design, the $F$-test behaves well when compared to the $U$-test, and (ii) when the distribution of the random effects and within-treatment errors are nonnormal, the $U$-test is preferable even when the number of treatments is small.


On the Simes inequality and its generalization

Abstract: The Simes inequality has received considerable attention recently because of its close connection to some important multiple hypothesis testing procedures. We revisit in this article an old result on this inequality to clarify and strengthen it and a recently proposed generalization of it to offer an alternative simpler proof.


Composition-Diamond lemma for differential algebras

Abstract: In this paper, we establish the Composition-Diamond lemma for free differential algebras. As applications, we give Groebner-Shirshov bases for free Lie-differential algebra and free commutative-differential algebra, respectively.


Multiple testing procedures under confounding

Abstract: While multiple testing procedures have been the focus of much statistical research, an important facet of the problem is how to deal with possible confounding. Procedures have been developed by authors in genetics and statistics. In this chapter, we relate these proposals. We propose two new multiple testing approaches within this framework. The first combines sensitivity analysis methods with false discovery rate estimation procedures. The second involves construction of shrinkage estimators that utilize the mixture model for multiple testing. The procedures are illustrated with applications to a gene expression profiling experiment in prostate cancer.


New HKT manifolds arising from quaternionic representations

Abstract: We give a procedure for constructing an $8n$-dimensional HKT Lie algebra starting from a $4n$-dimensional one by using a quaternionic representation of the latter. The strong (respectively, weak, hyper-K\"ahler, balanced) condition is preserved by our construction. As an application of our results we obtain a new compact HKT manifold with holonomy in $SL(n,\Bbb H)$ which is not a nilmanifold. We find in addition new compact strong HKT manifolds. We also show that every K\"ahler Lie algebra equipped with a flat, torsion-free complex connection gives rise to an HKT Lie algebra. We apply this method to two distinguished 4-dimensional K\"ahler Lie algebras, thereby obtaining two conformally balanced HKT metrics in dimension 8. Both techniques prove to be an effective tool for giving the explicit expression of the corresponding HKT metrics.


Stochastic expansions and Hopf algebras

Abstract: We study solutions to nonlinear stochastic differential systems driven by a multi-dimensional Wiener process. A useful algorithm for strongly simulating such stochastic systems is the Castell--Gaines method, which is based on the exponential Lie series. When the diffusion vector fields commute, it has been proved that at low orders this method is more accurate in the mean-square error than corresponding stochastic Taylor methods. However it has also been shown that when the diffusion vector fields do not commute, this is not true for strong order one methods. Here we prove that when there is no drift, and the diffusion vector fields do not commute, the exponential Lie series is usurped by the sinh-log series. In other words, the mean-square error associated with a numerical method based on the sinh-log series, is always smaller than the corresponding stochastic Taylor error, in fact to all orders. Our proof utilizes the underlying Hopf algebra structure of these series, and a two-alphabet associative algebra of shuffle and concatenation operations. We illustrate the benefits of the proposed series in numerical studies.


A Fast Algorithm for Stallings' Folding Process

Abstract: We show that for a fixed free group F and an arbitrary finitely generated subgroup H (as given above) we can perform the Stalling's folding process in time O(N log^*(N)), where N is the sum of the word lengths of the given generators of H.


Compact anti-self-dual orbifolds with torus actions

Abstract: We give a classification of toric anti-self-dual conformal structures on compact 4-orbifolds with positive Euler characteristic. Our proof is twistor theoretic: the interaction between the complex torus orbits in the twistor space and the twistor lines induces meromorphic data, which we use to recover the conformal structure. A compact anti-self-dual orbifold can also be constructed by adding a point at infinity to an asymptotically locally Euclidean (ALE) scalar-flat K\"ahler orbifold. We use this observation to classify ALE scalar-flat K\"ahler 4-orbifolds whose isometry group contain a 2-torus.