The Maslov cocycle, smooth structures and real-analytic complete integrability

Abstract: This paper studies smooth obstructions to integrability and proves two main results. First, it is shown that if a smooth topological n-torus admits a real-analytically completely integrable convex hamiltonian on its cotangent bundle, then the torus is diffeomorphic to the standard n-torus. This is the first known result where the smooth structure of a manifold obstructs complete integrability. Second, it is proven that each one of the Witten-Kreck-Stolz 7-manifolds admit a real-analytically completely integrable geodesic flow on its cotangent bundle. This gives examples of topological manifolds all of whose smooth structures admit a real-analytically completely integrable convex hamiltonian on its cotangent bundle. Additional examples are provided by Eschenburgh and Aloff-Wallach spaces.


Algebraic systems of matrices and Grobner basis

Abstract: One studies a particular algebraic system where the unknowns are matrices. We solve this system according to the parameters values thanks to the theory of Grobner basis.


A Generic Approach to Searching for Jacobians

Abstract: We consider the problem of finding cryptographically suitable Jacobians. By applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family, we can search for Jacobians containing a large subgroup of prime order. For a suitable distribution of curves, the complexity is subexponential in genus 2, and O(N^{1/12}) in genus 3. We give examples of genus 2 and genus 3 hyperelliptic curves over prime fields with group orders over 180 bits in size, improving previous results. Our approach is particularly effective over low-degree extension fields, where in genus 2 we find Jacobians over F_{p^2) and trace zero varieties over F_{p^3} with near-prime orders up to 372 bits in size. For p = 2^{61}-1, the average time to find a group with 244-bit near-prime order is under an hour on a PC.


On the topology of surface singularities {z^n=f(x,y)}, for f irreducible

Abstract: The splice quotients are an interesting class of normal surface singularities with rational homology sphere links, defined by W. Neumann and J. Wahl. If Gamma is a tree of rational curves that satisfies certain combinatorial conditions, then there exist splice quotients with resolution graph Gamma. Suppose the equation z^n=f(x,y) defines a surface X_{f,n} with an isolated singularity at the origin in C^3. For f irreducible, we completely characterize, in terms of n and a variant of the Puiseux pairs of f, those X_{f,n} for which the resolution graph satisfies the combinatorial conditions that are necessary for splice quotients. This result is topological; whether or not X_{f,n} is analytically isomorphic to a splice quotient is treated separately.


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics background

Abstract: We present a convergence result for infinite products of stochastic matrices with positive diagonals. We regard infinity of the product to the left. Such a product converges partly to a fixed matrix if the minimal positive entry of each matrix does not converge too fast to zero and if either zero-entries are symmetric in each matrix or the length of subproducts which reach the maximal achievable connectivity is bounded. Variations of this result have been achieved independently in Lorenz 2005, Moreau 2005 and Hendrickx 2005. We present briefly the opinion dynamics context, discuss the relations to infinite products where infinity is to the right (inhomogeneous Markov processes) and present a small improvement and sketch another.


Curved Casimir Operators and the BGG Machinery

Abstract: We prove that the Casimir operator acting on sections of a homogeneous vector bundle over a generalized flag manifold naturally extends to an invariant differential operator on arbitrary parabolic geometries. We study some properties of the resulting invariant operators and compute their action on various special types of natural bundles. As a first application, we give a very general construction of splitting operators for parabolic geometries. Then we discuss the curved Casimir operators on differential forms with values in a tractor bundle, which nicely relates to the machinery of BGG sequences. This also gives a nice interpretation of the resolution of a finite dimensional representation by (spaces of smooth vectors in) principal series representations provided by a BGG sequence.


Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups

Abstract: We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynamics, where the semigroup $G$ of complex polynomials (under the operation of composition of functions) is such that there exists a bounded set in the plane which contains any finite critical value of any map $g \in G$. In general, the Julia set of such a semigroup $G$ may be disconnected, and each Fatou component of such $G$ is either simply connected or doubly connected (\cite{Su01,Su9}). In this paper, we show that for any two distinct Fatou components of certain types (e.g., two doubly connected components of the Fatou set), the boundaries are separated by a Cantor set of quasicircles (with uniform dilatation) inside the Julia set of $G.$ Important in this theory is the understanding of various situations which can and cannot occur with respect to how the Julia sets of the maps $g \in G$ are distributed within the Julia set of the entire semigroup $G$. We give several results in this direction and show how such results are used to generate (semi) hyperbolic semigroups possessing this postcritically boundedness condition.


Strong wavefront lemma and counting lattice points in sectors

Abstract: We compute the asymptotics of the number of integral quadratic forms with prescribed orthogonal decompositions and, more generally, the asymptotics of the number of lattice points lying in sectors of affine symmetric spaces. A new key ingredient in this article is the strong wavefront lemma, which shows that the generalized Cartan decomposition associated to a symmetric space is uniformly Lipschitz.


Detecting Infinitely Many Semisimple Representations in a Fixed Finite Dimension

Abstract: Let $n$ be a positive integer, and let $k$ be a field (of arbitrary characteristic) accessible to symbolic computation. We describe an algorithmic test for determining whether or not a finitely presented $k$-algebra $R$ has infinitely many equivalence classes of semisimple representations $R \to M_n(k')$, where $k'$ is the algebraic closure of $k$. The test reduces the problem to computational commutative algebra over $k$, via famous results of Artin, Procesi, and Shirshov. The test is illustrated by explicit examples, with $n = 3$.


Cohomology and Support Varieties for Lie Superalgebras II

Abstract: In \cite{BKN} the authors initiated a study of the representation theory of classical Lie superalgebras via a cohomological approach. Detecting subalgebras were constructed and a theory of support varieties was developed. The dimension of a detecting subalgebra coincides with the defect of the Lie superalgebra and the dimension of the support variety for a simple supermodule was conjectured to equal the atypicality of the supermodule. In this paper the authors compute the support varieties for Kac supermodules for Type I Lie superalgebras and the simple supermodules for $\mathfrak{gl}(m|n)$. The latter result verifies our earlier conjecture for $\mathfrak{gl}(m|n)$. In our investigation we also delineate several of the major differences between Type I versus Type II classical Lie superalgebras. Finally, the connection between atypicality, defect and superdimension is made more precise by using the theory of support varieties and representations of Clifford superalgebras.


Uniformly perfect analytic and conformal attractor sets

Abstract: Conditions are given which imply that analytic iterated function systems (IFS's) in the complex plane have uniformly perfect attractor sets. In particular, it is shown that the attractor set of a finitely generated conformal IFS is uniformly perfect when it contains two or more points. Also, an example of a finitely generated analytic attractor set which is not uniformly perfect is given.


Recent developments of the DDVV Conjecture

Abstract: In this paper, we give a survey of the recent develpoments of the DDVV conjecture.


Potential confinement property in the Parabolic Anderson Model

Abstract: We consider the parabolic Anderson model, the Cauchy problem for the heat equation with random potential in $Z^d$. We use i.i.d. potentials $\xi: Z^d \to \R$ in the third universality class, namely the class of almost bounded potentials, in the classification of van der Hofstad, Konig and Morters [HKM06]. This class consists of potentials whose logarithmic moment generating function is regularly varying with parameter $\gamma=1$, but do not belong to the class of so-called double-exponentially distributed potentials studied by Gartner and Molchanov (PTRF 1998). In [HKM06] the asymptotics of the expected total mass was identified in terms of a variational problem that is closely connected to the well-known logarithmic Sobolev inequality and whose solution, unique up to spatial shifts, is a perfect parabola. In the present paper we show that those potentials whose shape (after appropriate vertical shifting and spatial rescaling) is away from that parabola contribute only negligibly to the total mass. The topology used is the strong $L^1$-topology on compacts for the exponentials of the potential. In the course of the proof, we show that any sequence of approximate minimisers of the above variational formula approaches some spatial shift of the minimiser, the parabola.


Fictitious Play in 3x3 Games: the transition between periodic and chaotic behaviour

Abstract: In the 60's Shapley provided an example of a two player fictitious game with periodic behaviour. In this game, player $A$ aims to copy $B$'s behaviour and player $B$ aims to play one ahead of player $A$. In this paper we generalize Shapley's example by introducing an external parameter. We show that the periodic behaviour in Shapley's example at some critical parameter value disintegrates into unpredictable (chaotic) behaviour, with players dithering a huge number of times between different strategies. At a further critical parameter the dynamics becomes periodic again and both players aim to play one ahead of the other. We study the dynamics of a two player continuous time bimatrix fictitious play with the dynamics of a one-parameter family of $3 \times 3$ games that includes a well-known example of Shapley's as a special case. In this paper we adopt a geometric (dynamical systems) approach and study the bifurcations of simple periodic orbits. Here we concentrate on the periodic behaviour, while in a sequel we shall describe the chaotic behaviour.


Real Zeros and Normal Distribution for statistics on Stirling permutations defined by Gessel and Stanley

Abstract: We study Stirling permutations defined by Gessel and Stanley. We prove that their generating function according to the number of descents has real roots only. We use that fact to prove that the distribution of these descents, and other, equidistributed statistics on these objects converge to a normal distribution.


Abstract: We study structures of derivation modules of Coxeter multiarrangements with quasi-constant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with quasi-constant multiplicity is combinatorially computable.


Complexity results for CR mappings between spheres

Abstract: Using elementary number theory, we prove several results about the complexity of CR mappings between spheres. It is known that CR mappings between spheres, invariant under finite groups, lead to sharp bounds for degree estimates on real polynomials constant on a hyperplane. We show here that there are infinitely many degrees for which the uniqueness of sharp examples fails. The proof uses a Pell equation and complicated explicit computations. We also show that the so-called gap phenomenon for proper mappings between balls does not occur beyond a certain target dimension. This proof uses the solution of the postage stamp problem.


The Repetition Property for Sequences on Tori Generated by Polynomials or Skew-Shifts

Abstract: The repetition property of a sequence in a metric space, a notion introduced by us in an earlier paper, is of importance in the spectral analysis of ergodic Schr\"odinger operators. It may be used to exclude eigenvalues for such operators. In this paper we study the question of when a sequence on a torus that is generated by a polynomial or a skew-shift has the repetition property. This provides classes of ergodic Schr\"odinger operators with potentials generated by skew-shifts on tori that have, contrary to earlier belief, no eigenvalues.


On the Khovanov and knot Floer homologies of quasi-alternating links

Abstract: Quasi-alternating links are a natural generalization of alternating links. In this paper, we show that quasi-alternating links are "homologically thin" for both Khovanov homology and knot Floer homology. In particular, their bigraded homology groups are determined by the signature of the link, together with the Euler characteristic of the respective homology (i.e. the Jones or the Alexander polynomial). The proofs use the exact triangles relating the homology of a link with the homologies of its two resolutions at a crossing.


Applications of Wallis Theorem

Abstract: In this paper we present theorems and applications of Wallis theorem related to trigonometric integrals.


Tong's spectrum for Rosen continued fractions

Abstract: The Rosen fractions are an infinite set of continued fraction algorithms, each giving expansions of real numbers in terms of certain algebraic integers. For each, we give a best possible upper bound for the minimum in appropriate consecutive blocks of approximation coefficients (in the sense of Diophantine approximation by continued fraction convergents). We also obtain metrical results for large blocks of ``bad'' approximations.


Asymptotically Optimal Importance Sampling for Jackson Networks with a Tree Topology

Abstract: Importance sampling (IS) is a variance reduction method for simulating rare events. A recent paper by Dupuis, Wang and Sezer (Ann. App. Probab. 17(4):1306- 1346, 2007) exploits connections between IS and stochastic games and optimal control problems to show how to design and analyze simple and efficient IS algorithms for various overflow events for tandem Jackson networks. The present paper uses the same approach to build asymptotically optimal IS schemes for stable open Jackson networks with a tree topology. Customers arrive at the single root of the tree. The rare overflow event we consider is the following: given that initially the network is empty, the system experiences a buffer overflow before returning to the empty state. Two types of buffer structures are considered: 1) A single system-wide buffer of size $n$ shared by all nodes, 2) each node $i$ has its own buffer of size $\beta_i n$, $\beta_i \in (0,1)$.


Complex Structures on Principal Bundles

Abstract: Holomorphic principal G-bundles over a complex manifold M can be studied using non-abelian cohomology groups H^1(M,G). On the other hand, if M=\Sigma is a closed Riemann surface, there is a correspondence between holomorphic principal G-bundles over \Sigma and coadjoint orbits in the dual of a central extension of the Lie algebra C^\infty(\Sigma, \g). We review these results and provide the details of an integrability condition for almost complex structures on smoothly trivial bundles. This article is a shortened version of the author's Diplom thesis.


$G$-stable pieces and Lusztig's dimension estimates

Abstract: We use $G$-stable pieces to construct some equidimensional varieties and as a consequence, obtain Lusztig's dimension estimates \cite[section 4]{L2}. This is a generalization of \cite{HL}.


Convergence of freely decomposable Kleinian groups

Abstract: We consider a compact orientable hyperbolic 3-manifold with a compressible boundary. Suppose that we are given a sequence of geometrically finite hyperbolic metrics whose conformal boundary structures at infinity diverge to a projective lamination. We prove that if this limit projective lamination is doubly incompressible, then the sequence has compact closure in the deformation space. As a consequence we generalise Thurston's double limit theorem and solve his conjecture on convergence of function groups affirmatively.


Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains

Abstract: We study Green's matrices for divergence form, second order strongly elliptic systems with bounded measurable coefficients in two dimensional domains. We establish existence, uniqueness, and pointwise estimates of the Green's matrices.


Which weakly ramified group actions admit a universal formal deformation?

Abstract: Consider a formal (mixed-characteristic) deformation functor D of a representation of a finite group G as automorphisms of a power series ring k[[t]] over a perfect field k of positive characteristic. Assume that the action of G is weakly ramified, i.e., the second ramification group is trivial. Examples of such representations are provided by a group action on an ordinary curve: the action of a ramification group on the completed local ring of any point on such a curve is weakly ramified. We prove that the only such D that are not pro-representable occur if k has characteristic two and G is of order two or isomorphic to a Klein group. Furthermore, we show that only the first of those has a non-pro-representable equicharacteristic deformation functor.


Experiments with a Positivity Preserving Operator

Abstract: We consider some multivariate rational functions which have (or are conjectured to have) only positive coefficients in their series expansion. We consider an operator that preserves positivity of series coefficients, and apply the inverse of this operator to the rational functions. We obtain new rational functions which seem to have only positive coefficients, whose positivity would imply positivity of the original series, and which, in a certain sense, cannot be improved any further.


Optimal stability estimate of the inverse boundary value problem by partial measurements

Abstract: In this work we establish log-type stability estimates for the inverse potential and conductivity problems with partial Dirichlet-to-Neumann map, where the Dirichlet data is homogeneous on the inaccessible part. This result, to some extent, improves our former result on the partial data problem in which log-log-type estimates were derived.


Cesaro's integral formula for the Bell numbers (corrected)

Abstract: M. E. Cesaro (1885) gave a quite remarkable expression for the Bell number --the number of partitions of an n-element set -- as a definite integral. This note is an exposition, correcting a typographical error in the original.


Carlitz q-Bernoulli numbers and q-Stirling numbers

Abstract: In this paper we consider carlitz q-Bernoulli numbers and q-stirling numbers of the first and the second kind. From these numbers we derive many interesting formulae associated with q-Bernoulli numbers.


A p-adic analogue of a formula of Ramanujan

Abstract: During his lifetime, Ramanujan provided many formulae relating binomial sums to special values of the Gamma function. Based on numerical computations, Van Hamme recently conjectured $p$-adic analogues to such formulae. Using a combination of ordinary and Gaussian hypergeometric series, we prove one of these conjectures.


Structure of Ann-categories and Mac Lane - Shukla cohomology

Abstract: In this paper we study the structure of a class of categories having two operations which satisfy axioms analoguos to that of rings. Such categories are called "Ann - categories". We obtain the classification theorems for regular Ann - categories and Ann - functors by using Mac Lane - Shukla cohomology of rings. These results give new interpretations of the cohomology groups and of the rings


A combinatorial realization of the Heisenberg action on the space of conformal blocks

Abstract: This paper has been withdrawn by the author. Improved versions (arXiv:1109.5548 and arXiv:0708.4190) are accepted.


Stability properties for the higher dimensional catenoid in $\rr^{n+1}$

Abstract: This paper concerns some stability properties of higher dimensional catenoids in $\rr^{n+1}$ with $n\ge 3$. We prove that higher dimensional catenoids have index one. We use $\delta$-stablity for minimal hypersurfaces and show that the catenoid is $\frac 2n$-stable and a complete $\frac 2n$-stable minimal hypersurface is a catenoid or a hyperplane provided the second fundamental form satisfies some decay conditions.


A construction of Horikawa surface via Q-Gorenstein smoothings

Abstract: In this article we prove that Fintushel-Stern's construction of Horikawa surface, which is obtained from an elliptic surface via a rational blow-down surgery in smooth category, can be performed in complex category. The main technique involved is Q-Gorenstein smoothings.


On the moduli stack of commutative, 1-parameter formal Lie groups

Abstract: We attempt to develop a general algebro-geometric study of the moduli stack of commutative, 1-parameter formal Lie groups. We emphasize the pro-algebraic structure of this stack: it is the inverse limit, over varying n, of moduli stacks of n-buds, and these latter stacks are algebraic. Our main results pertain to aspects of the height stratification relative to fixed prime p on the stacks of buds and formal Lie groups. We conclude with a largely expository account of some foundational material on limits in bicategories.


Finite dimensional modules and perpendicular subcategories

Abstract: We explain how, under some hypotheses, one can construct a sequence of finite dimensional $kG$-modules that lie in certain prescribed additive subcategories, but whose direct limits do not. We use these to show that many of the triangulated quotients of $\Mod$ are not generated, as triangulated categories, by the corresponding quotient of $\mod$ considered as a full subcategory.


Existence and properties of geometric quotients

Abstract: In this paper, we study quotients of groupoids and coarse moduli spaces of stacks in a general setting. Geometric quotients are not always categorical, but we present a natural topological condition under which a geometric quotient is categorical. We also show the existence of geometric quotients of finite flat groupoids and give explicit local descriptions. Exploiting similar methods, we give an easy proof of the existence of quotients of flat groupoids with finite stabilizers. As the proofs do not use noetherian methods and are valid for general algebraic spaces and algebraic stacks, we obtain a slightly improved version of Keel and Mori's theorem.


Ergodic properties of a class of non-Markovian processes

Abstract: We study a fairly general class of time-homogeneous stochastic evolutions driven by noises that are not white in time. As a consequence, the resulting processes do not have the Markov property. In this setting, we obtain constructive criteria for the uniqueness of stationary solutions that are very close in spirit to the existing criteria for Markov processes. In the case of discrete time, where the driving noise consists of a stationary sequence of Gaussian random variables, we give optimal conditions on the spectral measure for our criteria to be applicable. In particular, we show that under a certain assumption on the spectral density, our assumptions can be checked in virtually the same way as one would check that the Markov process obtained by replacing the driving sequence by a sequence of independent identically distributed Gaussian random variables is strong Feller and topologically irreducible. The results of the present article are based on those obtained previously in the continuous time context of diffusions driven by fractional Brownian motion.


Lens spaces obtainable by surgery on doubly primitive knots

Abstract: In this paper, we consider which lens spaces are obtainable by Dehn surgery described by Berge on doubly primitive knots. It is given an algorithm to decide whether a given lens space is obtainable by such surgery. Also included is a complete characterization of such surgery yielding lens spaces with Klein bottles.


Some Non-Unimodal Level Algebras

Abstract: In 2005, building on his own recent work and that of F. Zanello, A. Iarrobino discovered some constructions that, he conjectured, would yield level algebras with non-unimodal Hilbert functions. This thesis provides proofs of non-unimodality for Iarrobino's level algebras, as well as for other level algebras that the author has constructed along similar lines. The key technical contribution is to extend some results published by Iarrobino in 1984. Iarrobino's results provide insight into some naturally arising vector subspaces of the vector space R_d of forms of fixed degree in a polynomial ring in several variables. In this thesis, the problem is approached by combinatorial methods and results similar to Iarrobino's are proved for a different class of vector subspaces of R_d. The combinatorial methods involve the definition of a new class of matrices called L-Matrices, which have useful properties that are inherited by their submatrices. A particular class of square L-Matrices, associated with some specialized partially ordered sets having interesting combinatorial properties, is identified. For this class of L-Matrices, necessary and sufficient conditions are given that they be nonsingular. Several larger questions are discussed whose answers are incrementally improved by the knowledge that the new non-unimodal level algebras exist.


An approximate version of the Loebl-Komlos-Sos conjecture

Abstract: Loebl, Komlos, and Sos conjectured that if at least half of the vertices of a graph G have degree at least some natural number k, then every tree with at most k edges is a subgraph of G. Our main result is an approximate version of this conjecture for large enough n=|V(G)|, assumed that n=O(k). Our result implies an asymptotic bound for the Ramsey number of trees. We prove that r(T_k,T_m)\leq k+m+o(k+m),as k+m tends to infinity.


On the approximation by weighted ridge functions

Abstract: We characterize the best $L_{2}$ approximation to a multivariate function by linear combinations of ridge functions multiplied by some fixed weight functions. In the special case when the weight functions are constants, we propose explicit formulas for both the best approximation and approximation error.


On minimal norms on $M_n$

Abstract: In this note, we show that for each minimal norm $N(\cdot)$ on the algebra $M_n$ of all $n \times n$ complex matrices, there exist norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on ${\mathbb C}^n$ such that $$N(A)=\max\{\|Ax\|_2: \|x\|_1=1, x\in {\mathbb C}^n\}$$ for all $A \in M_n$. This may be regarded as an extension of a known result on characterization of minimal algebra norms.


Minimal Homogenous Liaison and Licci Ideals

Abstract: We study the linkage classes of homogeneous ideals in polynomial rings. An ideal is said to be homogeneously licci if it can be linked to a complete intersection using only homogeneous regular sequences at each step. We ask a natural question: if $I$ is homogeneously licci, then can it be linked to a complete intersection by linking using regular sequences of forms of smallest possible degree at each step (we call such ideals minimally homogeneously licci)? In this paper we answer this question in the negative. In particular, for every $n\geq 28$ we construct a set of $n$ points in $\mathbb P^3$ which are homogeneously licci, but not minimally homogeneously licci. Moreover, we prove that one cannot distinguish between the classes of homogeneously licci and non-licci ideals based only on their Hilbert functions, nor distinguish between homogeneously licci and minimally homogeneously licci ideals based solely on the graded Betti numbers. Finally, by taking hypersurface sections, we show that the natural question has a negative answer whenever the height of the ideal is at least three.


An algebraic formulation of Thurston's characterization of rational functions

Abstract: Following Douady-Hubbard and Bartholdi-Nekrashevych, we give an algebraic formulation of Thurston's characterization of rational functions. The techniques developed are applied to the analysis of the dynamics on the set of free homotopy classes of simple closed curves induced by a rational function. The resulting finiteness results yield new information on the global dynamics of the pullback map on Teichm\"uller space used in the proof of the characterization theorem.


On the almost sure spiraling of geodesics in negatively curved manifolds

Abstract: Given a negatively curved geodesic metric space $M$, we study the statistical asymptotic penetration behavior of (locally) geodesic lines of $M$ in small neighborhoods of points, of closed geodesics, and of other compact (locally) convex subsets of $M$. We prove Khintchine-type and logarithme law-type results for the spiraling of geodesic lines around these objets. As a consequence in the tree setting, we obtain Diophantine approximation results of elements of non-archimedian local fields by quadratic irrational ones.


On the symmetric subscheme of Hilbert scheme of points

Abstract: We consider the Hilbert scheme Hilb^{d+1}(C^d) of (d+1) points in affine d-space C^d (d > 2), which includes the square of any maximal ideal. We describe equations for the most symmetric affine open subscheme of Hilb^{d+1}(C^d), in terms of Schur modules. In addition we prove that Hilb^{d+1}(C^d) is reducible for n>d>11.


Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures

Abstract: We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call semialgebras over corings). We define double-sided derived functors SemiTor and SemiExt of the functors of semitensor product and semihomomorphisms, and construct an equivalence between the exotic derived categories of semimodules and semicontramodules. Certain (co)flatness and/or (co)projectivity conditions have to be imposed on the coring and semialgebra to make the module categories abelian (and the cotensor product associative). Besides, for a number of technical reasons we mostly have to assume that the basic ring has a finite homological dimension (no such assumptions about the coring and semialgebra are made). In the final sections we construct model category structures on the categories of complexes of semi(contra)modules, and develop relative nonhomogeneous Koszul duality theory for filtered semialgebras and quasi-differential corings. Our motivating examples come from the semi-infinite cohomology theory. Comparison with the semi-infinite (co)homology of Tate Lie algebras and graded associative algebras is established in appendices, and the semi-infinite homology of a locally compact topological group relative to an open profinite subgroup is defined. An application to the correspondence between complexes of representations of an infinite-dimensional Lie algebra on the complementary central charge levels ($c$ and $26-c$ for the Virasoro) is worked out.


The depth of a knot tunnel

Abstract: The theory of tunnel number 1 knots detailed in our previous paper, The tree of knot tunnels, provides a non-negative integer invariant called the depth of the tunnel. We give various results related to the depth invariant. Noting that it equals the minimum number of Goda-Scharlemann-Thompson tunnel moves needed to construct the tunnel, we calculate the number of distinct minimal sequences of tunnel moves that can produce a given tunnel. Next, we give a recursion that tells the minimum bridge number of a knot having a tunnel of depth D. The rate of growth of this value improves the known estimates of the growth of bridge number as a function of the Hempel distance of the associated Heegaard splitting. We also give various upper bounds for bridge number in terms of the cabling constructions needed to produce a tunnel of a knot, showing in particular that the maximum bridge number of a knot produced by N cabling constructions is the (N+2)nd Fibonacci number. Finally, we explicitly compute the slope parameters for the "short" tunnels of torus knots. In particular, we find a sequence of such tunnels for which the bridge numbers of the associated knots, as a function of the depth, achieve the minimum growth rate. The actual minimum bridge number at a given depth cannot be achieved by a torus knot.


Limit distribution theory for maximum likelihood estimation of a log-concave density

Abstract: We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form $f_0=\exp\varphi_0$ where $\varphi_0$ is a concave function on $\mathbb{R}$. The pointwise limiting distributions depend on the second and third derivatives at 0 of $H_k$, the "lower invelope" of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of $\varphi_0=\log f_0$ at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode $M(f_0)$ and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values.


Efficient computation of p-adic heights

Abstract: We analyse and drastically improve the running time of the algorithm of Mazur, Stein and Tate for computing the canonical cyclotomic p-adic height of a point on an elliptic curve E/Q, where E has good ordinary reduction at p >= 5.


Normal Hopf subalgebras in cocycle deformations of finite groups

Abstract: Let $G$ be a finite group and let $\pi: G \to G'$ be a surjective group homomorphism. Consider the cocycle deformation $L = H^{\sigma}$ of the Hopf algebra $H = k^G$ of $k$-valued linear functions on $G$, with respect to some convolution invertible 2-cocycle $\sigma$. The (normal) Hopf subalgebra $k^{G'} \subseteq k^G$ corresponds to a Hopf subalgebra $L' \subseteq L$. Our main result is an explicit necessary and sufficient condition for the normality of $L'$ in $L$.


Observability and nonlinear filtering

Abstract: This paper develops a connection between the asymptotic stability of nonlinear filters and a notion of observability. We consider a general class of hidden Markov models in continuous time with compact signal state space, and call such a model observable if no two initial measures of the signal process give rise to the same law of the observation process. We demonstrate that observability implies stability of the filter, i.e., the filtered estimates become insensitive to the initial measure at large times. For the special case where the signal is a finite-state Markov process and the observations are of the white noise type, a complete (necessary and sufficient) characterization of filter stability is obtained in terms of a slightly weaker detectability condition. In addition to observability, the role of controllability in filter stability is explored. Finally, the results are partially extended to non-compact signal state spaces.


Orbit semigroups and the representation type of quivers

Abstract: We show that a finite, connected quiver Q without oriented cycles is a Dynkin or Euclidean quiver if and only if all orbit semigroups of representations of Q are saturated.


Immersed Turnovers In Hyperbolic 3-Orbifolds

Abstract: We show that any immersion, which is not a covering of an embedded 2-orbifold, of a totally geodesic hyperbolic turnover in a complete orientable hyperbolic 3-orbifold is contained in a hyperbolic 3-suborbifold with totally geodesic boundary, called the "turnover core,'' whose volume is bounded from above by a function depending only on the area of the given turnover. Furthermore, we show that, for a given type of turnover, there are only finitely many possibilities for the turnover core. As a corollary, if the volume of a complete orientable hyperbolic 3-orbifold is at least 2\pi and if the fundamental group of the orbifold contains the fundamental group of a hyperbolic turnover (i.e., a triangle group), then the orbifold contains an embedded hyperbolic turnover.


Quiver coefficients of Dynkin type

Abstract: We study the Grothendieck classes of quiver cycles, i.e. invariant closed subvarieties of the representation space of a quiver. For quivers without oriented loops we show that the class of a quiver cycle is determined by quiver coefficients, which generalize the earlier studied quiver coefficients for equioriented quivers of type A. We conjecture that quiver coefficients satisfy positivity and finiteness properties. Our main result is a formula for the quiver coefficients for orbit closures of Dynkin type with rational singularities, which confirms the finiteness conjecture. This formula is based on Reineke's desingularization of such orbit closures. For quivers of type A3, we give positive combinatorial formulas for the quiver coefficients, which confirm the full conjecture. We also interpret quiver coefficients as formulas for degeneracy loci defined by quivers of vector bundle maps.


Brownian-time Brownian motion SIEs on $\Rp\times\Rd$: Ultra Regular direct and lattice-limits solutions and fourth order SPDEs links

Abstract: We delve deeper into the compelling regularizing effect of the Brownian-time Brownian motion density, $\KBtxy$, on the space-time-white-noise-driven stochastic integral equation we call BTBM SIE, which we recently introduced. In sharp contrast to second order heat-based SPDEs--whose real-valued mild solutions are confined to $d=1$--we prove the existence of solutions to the BTBM SIE in $d=1,2,3$ with dimension-dependent and striking Holder regularity, under both less than Lipschitz and Lipschitz conditions. In space, we show an unprecedented nearly local Lipschitz regularity for $d=1,2$--roughly, the SIE is spatially twice as regular as the Brownian sheet in these dimensions--and nearly local H\"older 1/2 regularity in d=3. In time, our solutions are locally H\"older continuous with exponent $\gamma\in(0,(4-d)/(8))$ for $1\le d\le3$. To investigate our SIE, we (a) introduce the Brownian-time random walk and we use it to formulate the spatial lattice version of the BTBM SIE; and (b) develop a delicate variant of Stroock-Varadhan martingale approach, the K-martingale approach, tailor-made for a wide variety of kernel SIEs including BTBM SIEs and the mild forms of many SPDEs of different orders on the lattice. Here, solutions types to our SIE are both direct and limits of their lattice version. The BTBM SIE is intimately connected to intriguing fourth order SPDEs in two ways. First, we show that it is connected to the diagonals of a new unconventional fourth order SPDE we call parametrized BTBM SPDE. Second, replacing $\KBtxy$ by the intimately connected kernel of our recently introduced imaginary-Brownian time-Brownian-angle process (IBTBAP), our SIE becomes the mild form of a Kuramoto-Sivashinsky SPDE with linear PDE part. Ideas developed here are adapted in separate papers to give a new approach, via our explicit IBTBAP representation, to many KS-type SPDEs in multi spatial dimensions.


A Remark on Hypercontractive Semigroups and Operator Ideals

Abstract: In this note, we answer a question raised by Johnson and Schechtman \cite{JS}, about the hypercontractive semigroup on $\{-1,1\}^{\NN}$. More generally, we prove the folllowing theorem. Let $1<p<2$. Let $(T(t))_{t>0}$ be a holomorphic semigroup on $L_p$ (relative to a probability space). Assume the following mild form of hypercontractivity: for some large enough number $s>0$, $T(s)$ is bounded from $L_p$ to $L_2$. Then for any $t>0$, $T(t)$ is in the norm closure in $B(L_p)$ (denoted by $\bar{\Gamma_2}$) of the subset (denoted by ${\Gamma_2}$) formed by the operators mapping $L_p$ to $L_2$ (a fortiori these operators factor through a Hilbert space).


A perfect stratification of M_g for g at most 5

Abstract: We find for g at most 5 a stratification of depth g-2 of the moduli space of curves M_g with the property that its strata are affine and the classes of their closures provide a Q-basis for the Chow ring of M_g. The first property confirms a conjecture of one of us. The way we establish the second property yields new (and simpler) proofs of theorems of Faber and Izadi which, taken together, amount to the statement that in this range the Chow ring is generated by the lambda-class.


Sally modules of rank one

Abstract: The structure of Sally modules of $\fkm$-primary ideals $I$ in a Cohen-Macaulay local ring $(A, \m)$ satisfying the equality $\e_1(I)=\e_0(I)-\ell_A(A/I)+1$ is explored, where $\e_0(I)$ and $\e_1(I)$ denote the first two Hilbert coefficients of $I$.


Twelve bridges from a reductive group to its Langlands dual

Abstract: These notes are based on lectures given in Wuhan (China) in July 2007. Their aim is to provide an introduction to Langlands philosophy.


Toric dynamical systems

Abstract: Toric dynamical systems are known as complex balancing mass action systems in the mathematical chemistry literature, where many of their remarkable properties have been established. They include as special cases all deficiency zero systems and all detailed balancing systems. One feature is that the steady state locus of a toric dynamical system is a toric variety, which has a unique point within each invariant polyhedron. We develop the basic theory of toric dynamical systems in the context of computational algebraic geometry and show that the associated moduli space is also a toric variety. It is conjectured that the complex balancing state is a global attractor. We prove this for detailed balancing systems whose invariant polyhedron is two-dimensional and bounded.


Some counterexamples in dynamics of rational semigroups

Abstract: We give an example of two rational functions with non-equal Julia sets that generate a rational semigroup whose completely invariant Julia set is a closed line segment. We also give an example of polynomials with unequal Julia sets that generate a non nearly Abelian polynomial semigroup with the property that the Julia set of one generator is equal to the Julia set of the semigroup. These examples show that certain conjectures in the field of dynamics of rational semigroups do not hold as stated and therefore require the allowance of certain exceptional cases.


On the Dreaded Right Bousfield Localization

Abstract: I verify the existence of right Bousfield localizations of right semimodel categories, and I apply this to construct a model of the homotopy limit of a left Quillen presheaf as a right semimodel category.


Generalized Cauchy identities, trees and multidimensional Brownian motions. Part II: Combinatorial differential calculus

Abstract: We present an analogue of the differential calculus in which the role of polynomials is played by certain ordered sets and trees. Our combinatorial calculus has all nice features of the usual calculus and has an advantage that the elements of the considered ordered sets might carry some additional information. In this way an analytic proof of generalized Cauchy identities from the previous work of the second author can be directly reformulated in our new language of the combinatorial calculus; furthermore the additional information carried by vertices determines uniquely the bijections presented in Part I of this series.


The special cuts of 600-cell

Abstract: A polytope is called {\em regular-faced} if every one of its facets is a regular polytope. The 4-dimensional regular-faced polytopes were determined by G. Blind and R. Blind \cite{BlBl2,roswitha,roswitha2}. The last class of such polytopes is the one which consists of polytopes obtained by removing a set of non-adjacent vertices (an independent set) of the 600-cell. These independent sets are enumerated up to isomorphism and it is determined that the number of polytopes in this last class is $314,248,344$.


On Local Behavior of Holomorphic Functions Along Complex Submanifolds of C^N

Abstract: In this paper we establish some general results on local behavior of holomorphic functions along complex submanifolds of $\Co^{N}$. As a corollary, we present multi-dimensional generalizations of an important result of Coman and Poletsky on Bernstein type inequalities on transcendental curves in $\Co^{2}$.


Diophantine subsets of function fields of curves

Abstract: We consider diophantine subsets of function fields of curves and show, roughly speaking, that they are either very small or very large. In particular, this implies that the ring of polynomials $k[t]$ is a not a diophantine subset of the field of rational functions $k(t)$ for many fields $k$.


Asymptotic behavior of the rate of adaptation

Abstract: We consider the accumulation of beneficial and deleterious mutations in large asexual populations. The rate of adaptation is affected by the total mutation rate, proportion of beneficial mutations and population size $N$. We show that regardless of mutation rates, as long as the proportion of beneficial mutations is strictly positive, the adaptation rate is at least $\mathcal{O}(\log^{1-\delta}N)$ where $\delta$ can be any small positive number, if the population size is sufficiently large. This shows that if the genome is modeled as continuous, there is no limit to natural selection, that is, the rate of adaptation grows in $N$ without bound.


On the Theory of Surfaces in the Four-dimensional Euclidean Space

Abstract: For a two-dimensional surface in the four-dimensional Euclidean space we introduce an invariant linear map of Weingarten type in the tangent space of the surface, which generates two invariants k and kappa. The condition k = kappa = 0 characterizes the surfaces consisting of flat points. The minimal surfaces are characterized by the equality kappa^2=k. The class of the surfaces with flat normal connection is characterized by the condition kappa = 0. For the surfaces of general type we obtain a geometrically determined orthonormal frame field at each point and derive Frenet-type derivative formulas. We apply our theory to the class of the rotational surfaces, which prove to be surfaces with flat normal connection, and describe the rotational surfaces with constant invariants.


On topological spaces possessing uniformly distributed sequences

Abstract: Two classes of topological spaces are introduced on which every probability Radon measure possesses a uniformly distributed sequence or a uniformly tight uniformly distributed sequence. It is shown that these classes are stable under multiplication by completely regular Souslin spaces


Extrema of low eigenvalues of the Dirichlet-Neumann Laplacian on a disk

Abstract: We study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet-Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact 1-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.


On the theorem of M.Golomb

Abstract: Let $X_{1},...,X_{n}$ be compact spaces and $X=X_{1}\times ... \times X_{n}.$ Consider the approximation of a function $f\in C(X)$ by sums $g_{1}(x_{1})+... g_{n}(x_{n}),$ where $g_{i}\in C(X_{i}),$ $i=1,...,n.$ In [8], M.Golomb obtained a formula for the error of this approximation in terms of measures constructed on special points of $X$, called "projection cycles". However, his proof had a gap, which was pointed out by Marshall and O'Farrell [15]. But the question if the formula was correct, remained open. The purpose of the paper is to prove that Golomb's formula holds in a stronger form.


Characteristic 2 approach to bivariate interpolation problems

Abstract: We investigate bivariate interpolation problems in characteristic 2. Given a nonnegative integer $t$, we describe all the sub-linear systems generated by monomials, in which there is no curve passing through a general point with multiplicity at least $2^t$. As an application, we show that a certain linear system of plane curves with 10 base points is non-special.


Intransitive geometries and fused amalgams

Abstract: We study geometries that arise from the natural $G_2(K)$ action on the geometry of one-dimensional subspaces, of nonsingular two-dimensional subspaces, and of nonsingular three-dimensional subspaces of the building geometry of type $C_3(K)$ where $K$ is a perfect field of characteristic 2. One of these geometries is intransitive in such a way that the non-standard geometric covering theory by the first and the last author is not applicable. In this paper we introduce the concept of fused amalgams in order to extend the geometric covering theory so that it applies to that geometry. This yields an interesting new amalgamation result for the group $G_2(K)$.


Sparse inverse covariance estimation with the lasso

Abstract: We consider the problem of estimating sparse graphs by a lasso penalty applied to the inverse covariance matrix. Using a coordinate descent procedure for the lasso, we develop a simple algorithm that is remarkably fast: in the worst cases, it solves a 1000 node problem (~500,000 parameters) in about a minute, and is 50 to 2000 times faster than competing methods. It also provides a conceptual link between the exact problem and the approximation suggested by Meinhausen and Buhlmann (2006). We illustrate the method on some cell-signaling data from proteomics.


A novel operation associated with Gauss' arithmetic-geometric means

Abstract: The arithmetic mean is the mean for addition and the geometric mean is that for multiplication. Then what kind of binary operation is associated with the arithmetic-geometric mean (AGM) due to C. F. Gauss? If it is possible to construct an arithmetic operation such that AGM is the mean for this operation, it can be regarded as an intermediate operation between addition and multiplication in view of the meaning of AGM. In this paper such an operation is introduced and several of its algebraic properties are proved.


Stability and convergence of the Method of Fundamental Solutions for Helmholtz problems on analytic domains

Abstract: The Method of Fundamental Solutions (MFS) is a popular tool to solve Laplace and Helmholtz boundary value problems. Its main drawback is that it often leads to ill-conditioned systems of equations. In this paper we investigate for the interior Helmholtz problem on analytic domains how the singularities (charge points) of the MFS basis functions have to be chosen such that approximate solutions can be represented by the MFS basis in a numerically stable way. For Helmholtz problems on the unit disc we give a full analysis which includes the high frequency (short wavelength) limit. For more difficult and nonconvex domains such as crescents we demonstrate how the right choice of charge points is connected to how far into the complex plane the solution of the boundary value problem can be analytically continued, which in turn depends on both domain shape and boundary data. Using this we develop a recipe for locating charge points which allows us to reach error norms of typically 10^{-11} on a wide variety of analytic domains. At high frequencies of order only 3 points per wavelength are needed, which compares very favorably to boundary integral methods.


Studies on the Chazy equations

Abstract: In this paper, we study the Chazy III,IX and X equations. For the Chazy III equation, by making the birational transformations the Chazy III equation is transformed into a third-order ordinary differential equation of rational type. For this equation, we find its meromorphic solutions, whose free parameters are essentially two. We also show that the system associated with this equation admits new special solutions solved by $tanh(t)$. For the Chazy IX equation, we transform the Chazy IX equation to a system of the first-order ordinary differential equations by birational transformations. For this system, we give two new birational B{\"a}cklund transformations. We also give the holomorphy condition of this system. Thanks to this holomorphy condition, we obtain a new partial differential system in two variables involving the Chazy IX equation, This system satisfies the compatibility condition, and admits a travelling wave solution. For the Chazy X equation, we transform the Chazy X equation to a system of the first-order ordinary differential equations by birational transformations. For this system, we give two birational B{\"a}cklund transformations. One of them is new. We also give the holomorphy condition of this system. Thanks to this holomorphy condition, we can recover this system.


An EL-labeling of the subgroup lattice

Abstract: In a 2001 paper, Shareshian conjectured that the subgroup lattice of a finite, solvable group has an EL-labeling. We construct such a labeling, and verify that our labeling has the expected properties.


On the endomorphism algebra of modular Gelfand-Graev representations

Abstract: We study the endomorphism algebras of a modular Gelfand-Graev representation of a finite reductive group by investigating modular properties of homomorphisms constructed by Curtis and Curtis-Shoji.


Degenerations of quadratic differentials on CP1

Abstract: We describe the connected components of the complement of a natural "diagonal" of real codimension 1 in a stratum of quadratic differentials on CP1. We establish a natural bijection between the set of these connected components and the set of generic configurations that appear on such "flat spheres". We also prove that the stratum has only one topological end. Finally, we elaborate a necessary toolkit destined to evaluation of the Siegel-Veech constants.


Applied statistics: A review

Abstract: The main phases of applied statistical work are discussed in general terms. The account starts with the clarification of objectives and proceeds through study design, measurement and analysis to interpretation. An attempt is made to extract some general notions.


An order result for the exponential divisor function

Abstract: The integer $d=\prod_{i=1}^s p_i^{b_i}$ is called an exponential divisor of $n=\prod_{i=1}^s p_i^{a_i}>1$ if $b_i \mid a_i$ for every $i\in \{1,2,...,s\}$. Let $\tau^{(e)}(n)$ denote the number of exponential divisors of $n$, where $\tau^{(e)}(1)=1$ by convention. The aim of the present paper is to establish an asymptotic formula with remainder term for the $r$-th power of the function $\tau^{(e)}$, where $r\ge 1$ is an integer. This improves an earlier result of {\sc M. V. Subbarao} [5].


Generalization error for multi-class margin classification

Abstract: In this article, we study rates of convergence of the generalization error of multi-class margin classifiers. In particular, we develop an upper bound theory quantifying the generalization error of various large margin classifiers. The theory permits a treatment of general margin losses, convex or nonconvex, in presence or absence of a dominating class. Three main results are established. First, for any fixed margin loss, there may be a trade-off between the ideal and actual generalization performances with respect to the choice of the class of candidate decision functions, which is governed by the trade-off between the approximation and estimation errors. In fact, different margin losses lead to different ideal or actual performances in specific cases. Second, we demonstrate, in a problem of linear learning, that the convergence rate can be arbitrarily fast in the sample size $n$ depending on the joint distribution of the input/output pair. This goes beyond the anticipated rate $O(n^{-1})$. Third, we establish rates of convergence of several margin classifiers in feature selection with the number of candidate variables $p$ allowed to greatly exceed the sample size $n$ but no faster than $\exp(n)$.


On certain arithmetic functions involving exponential divisors, II

Abstract: Asymptotic properties of certain arithmetic functions involving exponential divisors are investigated.


First Passage Densities and Boundary Crossing Probabilities for Diffusion Processes

Abstract: We consider the boundary crossing problem for time-homogeneous diffusions and general curvilinear boundaries. Bounds are derived for the approximation error of the one-sided (upper) boundary crossing probability when replacing the original boundary by a different one. In doing so we establish the existence of the first-passage time density and provide an upper bound for this function. In the case of processes with diffusion interval equal to whole real line this is extended to a lower bound, as well as bounds for the first crossing time of a lower boundary. An extension to some time-inhomogeneous diffusions is given. These results are illustrated by numerical examples.


Canonical Hilbert-Burch matrices for ideals of $k[x,y]$

Abstract: An Artinian ideal $I$ of $k[x,y]$ has many Hilbert-Burch matrices. We show that there is a canonical choice. As an application, we determine the dimension of certain affine Gr\"obner cells and their Betti strata recovering results of Ellingsrud and Str{\o}mme, G\"ottsche and Iarrobino.


Relative Hyperbolicity, Trees of Spaces and Cannon-Thurston Maps

Abstract: We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result for inclusion of vertex (or edge) subgroups in finite graphs of (strongly) relatively hyperbolic groups. This generalises a result of Bowditch for punctured surfaces in 3 manifolds and a result of Mitra for trees of hyperbolic metric spaces.


Hyper-atoms and the Kemperman's critical pair Theory

Abstract: In the present work, we introduce the notion of a hyper-atom and prove their main structure theorem. We then apply the global isoperimetric methodology to give a new proof for Kemperman's structure Theory and a slight improvement.


Defining Relations of Low Degree of Invariants of Two $4 \times 4$ Matrices

Abstract: Over a field K of characteristic 0, we study the algebra of invariants of the general linear group GL(4,K) acting by simultaneous conjugation on two matrices of order 4. It coincides with the trace algebra generated by all traces of products of two generic matrices of order 4. It is known that the minimal degree of the defining relations of any homogeneous minimal generating set of this algebra is equal to 12. Starting with the generating set given recently by Drensky and Sadikova, we have determined all relations of degree < 15. For this purpose we have developed further algorithms based on representation theory of the general linear group and easy computer calculations with standard functions of Maple.


Globular realization and cubical underlying homotopy type of time flow of process algebra

Abstract: We construct a small realization as flow of every precubical set (modeling for example a process algebra). The realization is small in the sense that the construction does not make use of any cofibrant replacement functor and of any transfinite construction. In particular, if the precubical set is finite, then the corresponding flow has a finite globular decomposition. Two applications are given. The first one presents a realization functor from precubical sets to globular complexes which is characterized up to a natural S-homotopy. The second one proves that, for such flows, the underlying homotopy type is naturally isomorphic to the homotopy type of the standard cubical complex associated with the precubical set.


Fixed points of smooth varieties with Kodaira dimension zero

Abstract: In this paper, we study the growth of the number of fixed points from iterating an endomorphism of an abelian variety. Using the estimates obtained on an abelian variety, we are able to extend the results to endomorphisms on varieties of Kodaira dimension zero and more generally their periodic subvarieties.


Torus knots are Fourier-(1,1,2) knots

Abstract: Every torus knot can be represented as a Fourier-(1,1,2) knot which is the simplest possible Fourier representation for such a knot. This answers a question of Kauffman and confirms the conjecture made by Boocher, Daigle, Hoste and Zheng. In particular, the torus knot T(p,q) can be parameterized as x(t)=cos(pt), y(t)=cos(qt+pi/(2p)), and z(t)=cos(pt+pi/2)\cos((q-p)t+pi/(2p)-pi/(4q)).


Non commutative functional calculus: bounded operators

Abstract: In this paper we develop a functional calculus for bounded operators defined on quaternionic Banach spaces. This calculus is based on the notion of slice-regularity, see \cite{gs}, and the key tools are a new resolvent operator and a new eigenvalue problem.


Non commutative functional calculus: unbounded operators

Abstract: In a recent work, \cite{cgss}, we developed a functional calculus for bounded operators defined on quaternionic Banach spaces. In this paper we show how the results from \cite{cgss} can be extended to the unbounded case, and we highlight the crucial differences between the two cases. In particular, we deduce a new eigenvalue equation, suitable for the construction of a functional calculus for operators whose spectrum is not necessarily real.


A new functional calculus for non-commuting operators

Abstract: In this paper we use the notion of slice monogenic functions \cite{slicecss} to define a new functional calculus for an $n$-tuple $T$ of not necessarily commuting operators. This calculus is different from the one discussed in \cite{jefferies} and it allows the explicit construction of the eigenvalue equation for the $n$-tuple $T$ based on a new notion of spectrum for $T$. Our functional calculus is consistent with the Riesz-Dunford calculus in the case of a single operator.


Slice monogenic functions

Abstract: In this paper we offer a definition of monogenicity for functions defined on $\rr^{n+1}$ with values in the Clifford algebra $\rr_n$ following an idea inspired by the recent papers \cite{gs}, \cite{advances}. This new class of monogenic functions contains the polynomials (and, more in general, power series) with coefficients in the Clifford algebra $\rr_n$. We will prove a Cauchy integral formula as well as some of its consequences. Finally, we deal with the zeroes of some polynomials and power series.


A correlated topic model of Science

Abstract: Topic models, such as latent Dirichlet allocation (LDA), can be useful tools for the statistical analysis of document collections and other discrete data. The LDA model assumes that the words of each document arise from a mixture of topics, each of which is a distribution over the vocabulary. A limitation of LDA is the inability to model topic correlation even though, for example, a document about genetics is more likely to also be about disease than X-ray astronomy. This limitation stems from the use of the Dirichlet distribution to model the variability among the topic proportions. In this paper we develop the correlated topic model (CTM), where the topic proportions exhibit correlation via the logistic normal distribution [J. Roy. Statist. Soc. Ser. B 44 (1982) 139--177]. We derive a fast variational inference algorithm for approximate posterior inference in this model, which is complicated by the fact that the logistic normal is not conjugate to the multinomial. We apply the CTM to the articles from Science published from 1990--1999, a data set that comprises 57M words. The CTM gives a better fit of the data than LDA, and we demonstrate its use as an exploratory tool of large document collections.


Continuous functions taking every value a given number of times

Abstract: We give necessary and sufficient conditions on a function $f:[0,1]\to {0,1,2,...,\omega,\continuum}$ under which there exists a continuous function $F:[0,1]\to [0,1]$ such that for every $y\in[0,1]$ we have $|F^{-1}(y)|=f(y)$.


Thompson's Group F

Abstract: We introduce forest diagrams and strand diagrams for elements of Thompson's group F. A forest diagram is a pair of infinite, bounded binary forests together with an order-preserving bijection of the leaves. Using forest diagrams, we derive a simple length formula for elements of F, and we discuss applications to the geometry of the Cayley graph, including a new upper bound on the isoperimetric constant (a.k.a. Cheeger constant) of F. Strand diagrams are similar to tree diagrams, but they can be concatenated like braids. Motivated by the fact that configuration spaces are classifying spaces for braid groups, we present a classifying space for F that is the ``configuration space'' of finitely many points on a line, with the points allowed to split and merge in pairs. Strand diagrams are related to a description of F as a groupoid, which we use to derive presentations for F, T, V, and the braided Thompson group BV. In addition to the new results, we include a thorough exposition of the basic theory of the group F. Highlights include a simplified proof that the commutator subgroup of F is simple, a discussion of open problems (with a focus on amenability), and a simplified derivation of the standard presentation and normal forms for F using forest diagrams.


Explicit Evaluation of Certain Exponential Sums of Quadratic Functions over $\Bbb F_{p^n}$, $p$ Odd

Abstract: Let $p$ be an odd prime and let $f(x)=\sum_{i=1}^ka_ix^{p^{\alpha_i}+1}\in\Bbb F_{p^n}[x]$, where $0\le \alpha_1<...<\alpha_k$. We consider the exponential sum $S(f,n)=\sum_{x\in\Bbb F_{p^n}}e_n(f(x))$, where $e_n(y)=e^{2\pi i\text{Tr}_n(y)/p}$, $y\in\Bbb F_{p^n}$, $\text{Tr}_n=\text{Tr}_{\Bbb F_{p^n}/\Bbb F_p}$. There is an effective way to compute the nullity of the quadratic form $\text{Tr}_{mn}(f(x))$ for all integer $m>0$. Assuming that all such nullities are known, we find relative formulas for $S(f,mn)$ in terms of $S(f,n)$ when $\nu_p(m) \le \min\{\nu_p(\alpha_i):1\le i\le k\}$, where $\nu_p$ is the $p$-adic order. We also find an explicit formula for $S(f,n)$ when $\nu_2(\alpha_1)=...= \nu_2(\alpha_k)<\nu_2(n)$. These results generalize those by Carlitz and by Baumert and McEliece. Parallel results with $p=2$ were obtained in a previous paper by the second author.


The $\mathcal{Q}_p$ Carleson Measure Problem

Abstract: Let $\mu$ be a nonnegative Borel measure on the open unit disk $\mathbb{D}\subset\mathbb{C}$. This note shows how to decide that the M\"obius invariant space $\mathcal{Q}_p$, covering $\mathcal{BMOA}$ and $\mathcal{B}$, is boundedly (resp., compactly) embedded in the quadratic tent-type space $T^\infty_p(\mu)$. Interestingly, the embedding result can be used to determine the boundedness (resp., the compactness) of the Volterra-type and multiplication operators on $\mathcal{Q}_p$.


Derangements and Relative Derangements of Type $B$

Abstract: By introducing the notion of relative derangements of type $B$, also called signed relative derangements, which are defined in terms of signed permutations, we obtain a type $B$ analogue of the well-known relation between relative derangements and the classical derangements. While this fact can be proved by using the principle of inclusion and exclusion, we present a combinatorial interpretation with the aid of the intermediate structure of signed skew derangements.


Prediction of Fractional Processes with Long-range Dependence

Abstract: We introduce a class of Gaussian processes with stationary increments which exhibit long-range dependence. The class includes fractional Brownian motion with Hurst parameter H>1/2 as a typical example. We establish infinite and finite past prediction formulas for the processes in which the predictor coefficients are given explicitly in terms of the MA and AR coefficients.


Relationship Between Bicomplex Generalized Analytic Functions and Solutions of the Complexified Schr\"odinger Equation

Abstract: Using three different representations of the bicomplex numbers $T\cong Cl_{C}(1,0) \cong Cl_{C}(0,1)$, which is a commutative ring with zero divisors defined by $T={w_0+w_1 {i_1}+w_2{i_2}+w_3 {j} | w_0,w_1,w_2,w_3 \in{R}}$ where ${i_1^{2}}=-1, {i_2^{2}}=-1, {j^{2}}=1 and {i_1}{i_2}={j}={i_2}{i_1}$, we construct three classes of bicomplex pseudoanalytic functions. In particular, we obtain some specific systems of Vekua equations of two complex variables and we established some connections between one of these systems and the classical Vekua equations. We consider also the complexification of the real stationary two-dimensional Schr{\"o}dinger equation. With the aid of any of its particular solutions, we construct a specific bicomplex Vekua equation possessing the following special property. The scalar parts of its solutions are solutions of the original complexified Schr{\"o}dinger equation and the vectorial parts are solutions of another complexified Schr{\"o}dinger equation.


Proof of Lemma 6.3 in ``The crossing number of $K_{4,n}$ on the torus and the Klein bottle"

Abstract: We found that Lemma 6.3 in the paper ``The crossing number of $K_{4,n}$ on the torus and the Klein bottle" is wrong.


Gradient regularity for elliptic equations in the Heisenberg Group

Abstract: We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised by Manfredi & Mingione (Math. Ann. 2007) where only dimension dependent bounds for the growth exponent are given. We also obtain explicit a priori local regularity estimates, and cover the case of the horizontal p-Laplacean operator, extending some regularity proven by Domokos & Manfredi (Cont. Math. 2005). In turn, the a priori estimates found are shown to imply the suitable local Calderon-Zygmund theory for the related class of non-homogeneous, possibly degenerate equations involving discontinuous coefficients. These last results extend to the sub-elliptic setting a few classical non-linear Euclidean results of Iwaniec and Dibenedetto & Manfredi, and to the non-linear case estimates of the same nature that were available in the sub-elliptic setting only for solutions to linear equations.


On Kalai's conjectures concerning centrally symmetric polytopes

Abstract: In 1989 Kalai stated the three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture, A, became known as the ``$3^d$-conjecture''. It is well-known that the three conjectures hold in dimensions d \leq 3. We show that in dimension 4 only conjectures A and B are valid, while conjecture C fails. Furthermore, we show that both conjectures B and C fail in all dimensions d \geq 5.


On a conjecture of Hacon and McKernan in dimension three

Abstract: We prove that there exists a universal constant $r_3$ such that if $X$ is a smooth projective threefold over $\mathbb{C}$ with non-negative Kodaira dimension, then the linear system $|r K_X|$ admits a fibration that is birational to the Iitaka fibration as soon as $r \geq r_3$ and sufficiently divisible. This gives an affirmative answer to a conjecture of Hacon and McKernan in the case of threefolds. Viehweg and Zhang have posted a stronger result along these lines using different methods.


Commutator maps, measure preservation, and T-systems

Abstract: Let G be a finite simple group. We show that the commutator map $a : G \times G \to G$ is almost equidistributed as the order of G goes to infinity. This somewhat surprising result has many applications. It shows that for a subset X of G we have $a^{-1}(X)/|G|^2 = |X|/|G| + o(1)$, namely $a$ is almost measure preserving. From this we deduce that almost all elements $g \in G$ can be expressed as commutators $g = [x,y]$ where x,y generate G. This enables us to solve some open problems regarding T-systems and the Product Replacement Algorithm (PRA) graph. We show that the number of T-systems in G with two generators tends to infinity as the order of G goes to infinity. This settles a conjecture of Guralnick and Pak. A similar result follows for the number of connected components of the PRA graph of G with two generators. Some of our results apply for more general finite groups, and more general word maps. Our methods are based on representation theory, combining classical character theory with recent results on character degrees and values in finite simple groups. In particular the so called Witten zeta function plays a key role in the proofs.


A note on the convergence of renewal and regenerative processes to a Brownian bridge

Abstract: The standard functional central limit theorem for a renewal process with finite mean and variance, results in a Brownian motion limit. This note shows how to obtain a Brownian bridge process by a direct procedure that does not involve conditioning. Several examples are also considered.


A simple proof of the algebraic version of a conjecture by Vogan

Abstract: In a recent manuscript, D.Vogan conjectures that four canonical globalizations of Harish-Chandra modules commute with certain n-cohomology groups. In this article we prove that Vogan's conjecture holds for one of the globalizations if and only if it holds for the dual. Using a previously published result of one of the authors, which establishes the conjecture for the minimal globalization, we can therefore deduce Vogan's conjecture for the maximal globalization.


On the Cauchy problem for higher-order nonlinear dispersive equations

Abstract: We study a class of higher-order KdV equations. We show that the associated initial value problem is well posed in weighted Besov and Sobolev spaces for small initial data. We also prove ill-posedness results when in H^s(\R), for any real s.


On representations of certain pseudo-Anosov maps of Riemann surfaces with punctures

Abstract: Let $S$ be a Riemann surface of type $(p,n)$ with $3p+n>4$ and $n\geq 1$. Let $\alpha_1,\alpha_2\subset S$ be two simple closed geodesics such that $\{\alpha_1, \alpha_2\}$ fills $S$. It was shown by Thurston that most maps obtained through Dehn twists along $\alpha_1$ and $\alpha_2$ are pseudo-Anosov. Let $a$ be a puncture. In this paper, we study the family $\mathcal{F}(S,a)$ of pseudo-Anosov maps on $S$ that projects to the trivial map as $a$ is filled in, and show that there are infinitely many elements in $\mathcal{F}(S,a)$ that cannot be obtained from Dehn twists along two filling geodesics. We further characterize all elements in $\mathcal{F}(S,a)$ that can be constructed by two filling geodesics. Finally, for any point $b\in S$, we obtain a family $\mathcal{H}$ of pseudo-Anosov maps on $S\backslash \{b\}$ that is not obtained from Thurston's construction and projects to an element $\chi\in \mathcal{F}(S,a)$ as $b$ is filled in, some properties of elements in $\mathcal{H}$ are also discussed.


Arithmetic structures in smooth subsets of F_p

Abstract: Fix integers a_1,...,a_d satisfying a_1 + ... + a_d = 0. Suppose that f : Z_N -> [0,1], where N is prime. We show that if f is ``smooth enough'' then we can bound from below the sum of f(x_1)...f(x_d) over all solutions (x_1,...,x_d) in Z_N to a_1 x_1 + ... + a_d x_d == 0 (mod N). Note that d = 3 and a_1 = a_2 = 1 and a_3 = -2 is the case where x_1,x_2,x_3 are in arithmetic progression. By ``smooth enough'' we mean that the sum of squares of the lower order Fourier coefficients of f is ``small'', a property shared by many naturally-occurring functions, among them certain ones supported on sumsets and on certain types of pseudoprimes. The paper can be thought of as a generalization of another result of the author, which dealt with a F_p^n analogue of the problem. It appears that the method in that paper, and to a more limited extent the present paper, uses ideas similar to those of B. Green's ``arithmetic regularity lemma'', as we explain in the paper.


An Ergodic Result

Abstract: A rather general ergodic type scheme is presented on arbitrary sets X, as they are generated by arbitrary mappings T : X \longrightarrow X. The structures considered on X are given by suitable subsets of the set of all of its finite partitions. Ergodicity is studied not with respect to subsets of X, but with the {\it inverse limits} of families of finite partitions.


Homotopy characterization of ANR mapping spaces

Abstract: Let Y be an absolute neighborhood retract (ANR) for the class of metric spaces and let X be a Hausdorff space. Let map(X,Y) denote the space of continuous maps from X to Y with the compact open topology. It is shown that if X is a CW complex then map(X,Y) is an ANR for the class of metric spaces if and only if map(X,Y) is metrizable and has the homotopy type of a CW complex. The same holds also when X is a compactly generated hemicompact space (metrizability assumption is void in this case).


About Goldbach strong conjecture

Abstract: In this work we use the number classification in families of the form 6n+1, and 6n+5 with n integer (Such families contain all odd prime numbers greater than 3 and other compound numbers related with primes). We will use this kind of classification in order to attempt an approach to Goldbach strong conjecture. By means of a geometric method of binary bands of numbers we conceive a new form of study of the stated problem.


An elementary sieve

Abstract: In this paper we review the properties of families of numbers of the form $6n\pm1$, with $n$ integer (in which there are all prime numbers greater than 3 and other compound numbers with particular properties) to later use them in a new sieve that allows the separation of numbers $n$ that generate primes from those that only generate compounds. In principle, this can be used to find the amount of prime numbers up to a given number $h$; this means, $\pi(h)$.


Zero product preservers of C*-algebras

Abstract: Let T be be a zero-product preserving bounded linear map between C*-algebras A and B. Here neither A nor B is necessarily unital. In this note, we investigate when T gives rise to a Jordan homomorphism. In particular, we show that A and B are isomorphic as Jordan algebras if T is bijective and sends zero products of self-adjoint elements to zero products. They are isomorphic as C*-algebras if T is bijective and preserves the full zero product structure.


Dirac concentrations in Lotka-Volterra parabolic PDEs

Abstract: We consider parabolic partial differential equations of Lotka-Volterra type, with a non-local nonlinear term. This models, at the population level, the darwinian evolution of a population; the Laplace term represents mutations and the nonlinear birth/death term represents competition leading to selection. Once rescaled with a small diffusion, we prove that the solutions converge to a moving Dirac mass. The velocity and weights cannot be obtained by a simple expression, e.g., an ordinary differential equation. We show that they are given by a constrained Hamilton-Jacobi equation. This extends several earlier results to the parabolic case and to general nonlinearities. Technical new ingredients are a $BV$ estimate in time on the non-local nonlinearity, a characterization of the concentration point (in a monomorphic situation) and, surprisingly, some counter-examples showing that jumps on the Dirac locations are indeed possible.


Galois cohomology of completed link groups

Abstract: In this paper we compute the Galois cohomology of the pro-p completion of primitive link groups. Here, a primitive link group is the fundamental group of a tame link in the 3-sphere whose linking number diagram is irreducible modulo p (e.g. none of the linking numbers is divisible by p). The result is that (with Z/pZ-coefficients) the Galois cohomology is naturally isomorphic to the Z/pZ-cohomology of the discrete link group. The main application of this result is that for such groups the Baum-Connes conjecture or the Atiyah conjecture are true for every finite extension (or even every elementary amenable extension), if they are true for the group itself.


Densities for Rough Differential Equations under Hoermander's Condition

Abstract: We consider stochastic differential equations dY=V(Y)dX driven by a multidimensional Gaussian process X in the rough path sense. Using Malliavin Calculus we show that Y(t) admits a density for t in (0,T] provided (i) the vector fields V=(V_1,...,V_d) satisfy Hoermander's condition and (ii) the Gaussian driving signal X satisfies certain conditions. Examples of driving signals include fractional Brownian motion with Hurst parameter H>1/4, the Brownian Bridge returning to zero after time T and the Ornstein-Uhlenbeck process.


A convergent finite difference method for a nonlinear variational wave equation

Abstract: We establish rigorously convergence of a semi-discrete upwind scheme for the nonlinear variational wave equation $u_{tt} - c(u)(c(u) u_x)_x = 0$ with $u|_{t=0}=u_0$ and $u_t|_{t=0}=v_0$. Introducing Riemann invariants $R=u_t+c u_x$ and $S=u_t-c u_x$, the variational wave equation is equivalent to $R_t-c R_x=\tilde c (R^2-S^2)$ and $S_t+c S_x=-\tilde c (R^2-S^2)$ with $\tilde c=c'/(4c)$. An upwind scheme is defined for this system. We assume that the the speed $c$ is positive, increasing and both $c$ and its derivative are bounded away from zero and that $R|_{t=0}, S|_{t=0}\in L^1\cap L^3$ are nonpositive. The numerical scheme is illustrated on several examples.


The correct relatively stable category for idempotent modules

Abstract: We answer a question posed by Carlson, Peng, and Wheeler, and demonstrate that in general Rickard modules in relatively stable categories are not idempotent modules even if one localizes with respect to a tensor ideal subcategory. We also show that there is a modification one can make so as to recover the idempotent behaviour.


Special correspondences and Chow traces of Landweber-Novikov operations

Abstract: We prove that the function field of a variety which possesses a special correspondence in the sense of M. Rost preserves the rationality of cycles of small codimensions. This fact was proven by Vishik in the case of quadrics and played the crucial role in his construction of fields with $u$-invariant $2^r+1$. The main technical tools are algebraic cobordism of Levine-Morel, generalized Rost degree formula and divisibility of Chow traces of certain Landweber-Novikov operations. As a direct application of our methods we prove the Vishik's Theorem for all $F_4$-varieties.


Contribution in combinatorics in commutative algebra.(ph-d thesis)

Abstract: In the first chapter we present new results related on monomial ideals of Borel type. Also, we introduce a new class of monomial ideals, called $\de$-fixed ideals, which generalize the class of $p$-Borel ideals and we extend several results to this new class. In the second chapter, we compute the generic initial ideal, with repect to the reverse lexicographic order, of an ideal which define a complete intersection of embedding dimension three with strong Lefschetz property and we show that it is an almost reverse lexicographic ideal. This enable us to give a proof for Moreno's conjecture in the case $n=3$ and characteristic zero. Also, we prove that the $d$-component of the generic initial ideal, with respect to the reverse lexicographic order, of an ideal generated by a regular sequence of homogeneous polynomials of degree $d$ is revlex, in a particular, but important, case.


On derived equivalence classification of gentle two-cycle algebras

Abstract: We classify, up to derived (equivalently, tilting-cotilting) equivalence all nondegenerate gentle two-cycle algebras. We also give a partial classification and formulate a conjecture in the degenerate case.


Rate of Escape on Free Products

Abstract: Suppose we are given the free product $V$ of a finite family of finite or countable sets $(V_i)_{i\in\mathcal{I}}$ and probability measures on each $V_i$, which govern random walks on it. We consider a transient random walk on the free product arising naturally from the random walks on the $V_i$. We prove the existence of the rate of escape with respect to the block length, that is, the speed, at which the random walk escapes to infinity, and furthermore we compute formulas for it. For this purpose, we present three different techniques providing three different, equivalent formulas.


Rate of Escape on the Lamplighter Tree

Abstract: Suppose we are given a homogeneous tree $\mathcal{T}_q$ of degree $q\geq 3$, where at each vertex sits a lamp, which can be switched on or off. This structure can be described by the wreath product $(\mathbb{Z}/2)\wr \Gamma$, where $\Gamma=\ast_{i=1}^q \mathbb{Z}/2$ is the free product group of $q$ factors $\mathbb{Z}/2$. We consider a transient random walk on a Cayley graph of $(\mathbb{Z}/2)\wr \Gamma$, for which we want to compute lower and upper bounds for the rate of escape, that is, the speed at which the random walk flees to infinity.


Acceleration of Lamplighter Random Walks

Abstract: Suppose we are given an infinite, finitely generated group $G$ and a transient random walk on the wreath product $(\mathbb{Z}/ 2\mathbb{Z})\wr G$, such that its projection on $G$ is transient and has finite first moment. This random walk can be interpreted as a lamplighter random walk on $G$. Our aim is to show that the random walk on the wreath product escapes to infinity with respect to a suitable (pseudo-)metric faster than its projection onto $G$. We also address the case where the pseudo-metric is the length of a shortest ``travelling salesman tour''. In this context, and excluding some degenerate cases if $G=\mathbb{Z}$, the linear rate of escape is strictly bigger than the rate of escape of the lamplighter random walk's projection on $G$.


Fisher Lecture: Dimension Reduction in Regression

Abstract: Beginning with a discussion of R. A. Fisher's early written remarks that relate to dimension reduction, this article revisits principal components as a reductive method in regression, develops several model-based extensions and ends with descriptions of general approaches to model-based and model-free dimension reduction in regression. It is argued that the role for principal components and related methodology may be broader than previously seen and that the common practice of conditioning on observed values of the predictors may unnecessarily limit the choice of regression methodology.


Abstract: Comment: Fisher Lecture: Dimension Reduction in Regression [arXiv:0708.3774]


Abstract: Rejoinder: Fisher Lecture: Dimension Reduction in Regression [arXiv:0708.3774]


Low upper bounds of ideals

Abstract: We show that there is a low T-upper bound for the class of K-trivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in the T-degrees below 0' for which there is a low T-upper bound.


Embedding Population Dynamics Models in Inference

Abstract: Increasing pressures on the environment are generating an ever-increasing need to manage animal and plant populations sustainably, and to protect and rebuild endangered populations. Effective management requires reliable mathematical models, so that the effects of management action can be predicted, and the uncertainty in these predictions quantified. These models must be able to predict the response of populations to anthropogenic change, while handling the major sources of uncertainty. We describe a simple ``building block'' approach to formulating discrete-time models. We show how to estimate the parameters of such models from time series of data, and how to quantify uncertainty in those estimates and in numbers of individuals of different types in populations, using computer-intensive Bayesian methods. We also discuss advantages and pitfalls of the approach, and give an example using the British grey seal population.


A General Framework for the Parametrization of Hierarchical Models

Abstract: In this paper, we describe centering and noncentering methodology as complementary techniques for use in parametrization of broad classes of hierarchical models, with a view to the construction of effective MCMC algorithms for exploring posterior distributions from these models. We give a clear qualitative understanding as to when centering and noncentering work well, and introduce theory concerning the convergence time complexity of Gibbs samplers using centered and noncentered parametrizations. We give general recipes for the construction of noncentered parametrizations, including an auxiliary variable technique called the state-space expansion technique. We also describe partially noncentered methods, and demonstrate their use in constructing robust Gibbs sampler algorithms whose convergence properties are not overly sensitive to the data.


Twisted Alexander Polynomials and Representation Shifts

Abstract: For any knot, the following are equivalent. (1) The infinite cyclic cover has uncountably many finite covers; (2) there exists a finite-image representation of the knot group for which the twisted Alexander polynomial vanishes; (3) the knot group admits a finite-image representation such that the image of the fundamental group of an incompressible Seifert surface is a proper subgroup of the image of the commutator subgroup of the knot group.


Derived classification of gentle algebras with two cycles

Abstract: We classify gentle algebras defined by quivers with two cycles under derived equivalence in a non degenerate case, by using some combinatorial invariants constructed from the quiver with relations defining these algebras. We also present a list of normal forms; any such algebra is derived equivalent to one of the algebras in the list.


Tropical Lines on Cubic Surfaces

Abstract: Given a tropical line $L$ and a smooth tropical surface $X$, we look at the position of $L$ on $X$. We introduce its primal and dual motif which are respectively a decorated graph and a subcomplex of the dual triangulation of $X$. They encode the combinatorial position of $L$ on $X$. We classify all possible motifs of tropical lines on general smooth tropical surfaces. This classification allows to give an upper bound for the number of tropical lines on a general smooth tropical surface with a given subdivision. We focus in particular on surfaces of degree three. As a concrete example, we look at tropical cubic surfaces dual to a fixed honeycomb triangulation, showing that a general surface contains exactly $27$ tropical lines.


Limits of zeros of polynomial sequences

Abstract: In the present paper we consider $F_k(x)=x^{k}-\sum_{t=0}^{k-1}x^t,$ the characteristic polynomial of the $k$-th order Fibonacci sequence, the latter denoted $G(k,l).$ We determine the limits of the real roots of certain odd and even degree polynomials related to the derivatives and integrals of $F_k(x),$ that form infinite sequences of polynomials, of increasing degree. In particular, as $k \to \infty,$ the limiting values of the zeros are determined, for both odd and even cases. It is also shown, in both cases, that the convergence is monotone for sufficiently large degree. We give an upper bound for the modulus of the complex zeros of the polynomials for each sequence. This gives a general solution related to problems considered by Dubeau 1989, 1993, Miles 1960, Flores 1967, Miller 1971 and later by the second author in the present paper, and Narayan 1997.


Calculating the image of the second Johnson-Morita representation

Abstract: Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus $g$ with one boundary component to $\wedge^3 H$, the third exterior product of the homology of the surface. Morita then extended Johnson's homomorphism to a homomorphism from the entire mapping class group to ${1/2} \wedge^3 H \semi \sp(H)$. This Johnson-Morita homomorphism is not surjective, but its image is finite index in ${1/2} \wedge^3 H \semi \sp(H)$. Here we give a description of the exact image of Morita's homomorphism. Further, we compute the image of the handlebody subgroup of the mapping class group under the same map.


Hilbert's Fifth Problem for Local Groups

Abstract: We solve Hilbert's fifth problem for local groups: every locally euclidean local group is locally isomorphic to a Lie group. Jacoby claimed a proof of this in 1957, but this proof is seriously flawed. We use methods from nonstandard analysis and model our solution after a treatment of Hilbert's fifth problem for global groups by Hirschfeld.


Commuting elements in conjugacy classes: An application of Hall's Marriage Theorem

Abstract: Let G be a finite group. Define a relation ~ on the conjugacy classes of G by setting C ~ D if there are representatives c \in C and d \in D such that cd = dc. In the case where G has a normal subgroup H such that G/H is cyclic, two theorems are proved concerning the distribution, between cosets of H, of pairs of conjugacy classes of G related by ~. One of the proofs involves an interesting application of the famous Marriage Theorem of Philip Hall. The paper concludes by discussing some aspects of these theorems and of the relation ~ in the particular cases of symmetric and general linear groups, and by mentioning an open question related to Frobenius groups.


Generically finite morphisms and formal neighborhoods of arcs

Abstract: We show that given a dominant morphism between two smooth varieties of the same dimension, the induced morphism between the formal neighborhoods of two arcs on these varieties is a closed embedding, of codimension given by the order of vanishing along the ramification subscheme.


Applications of a finite-dimensional duality principle to some prediction problems

Abstract: Some of the most important results in prediction theory and time series analysis when finitely many values are removed from or added to its infinite past have been obtained using difficult and diverse techniques ranging from duality in Hilbert spaces of analytic functions (Nakazi, 1984) to linear regression in statistics (Box and Tiao, 1975). We unify these results via a finite-dimensional duality lemma and elementary ideas from the linear algebra. The approach reveals the inherent finite-dimensional character of many difficult prediction problems, the role of duality and biorthogonality for a finite set of random variables. The lemma is particularly useful when the number of missing values is small, like one or two, as in the case of Kolmogorov and Nakazi prediction problems. The stationarity of the underlying process is not a requirement. It opens up the possibility of extending such results to nonstationary processes.


Nearly Optimal Patchy Feedbacks for Minimization Problems with Free Terminal Time

Abstract: The paper is concerned with a general optimization problem for a nonlinear control system, in the presence of a running cost and a terminal cost, with free terminal time. We prove the existence of a patchy feedback whose trajectories are all nearly optimal solutions, with pre-assigned accuracy.


Coarse categories I: foundations

Abstract: Following Roe and others (see, e.g., [MR1451755]), we (re)develop coarse geometry from the foundations, taking a categorical point of view. In this paper, we concentrate on the discrete case in which topology plays no role. Our theory is particularly suited to the development of the_Roe (C*-)algebras_ C*(X) and their K-theory on the analytic side; we also hope that it will be of use in the strictly geometric/algebraic setting of controlled topology and algebra. We leave these topics to future papers. Crucial to our approach are nonunital coarse spaces, and what we call _locally proper_ maps (which are actually implicit in [MR1988817]). Our_coarse category_ Crs generalizes the usual one: its objects are nonunital coarse spaces and its morphisms (locally proper) coarse maps modulo_closeness_. Crs is much richer than the usual unital coarse category. As such, it has all nonzero limits and all colimits. We examine various other categorical issues. E.g., Crs does not have a terminal object, so we substitute a_termination functor_ which will be important in the development of exponential objects (i.e., "function spaces") and also leads to a notion of_quotient coarse spaces_. To connect our methods with the standard methods, we also examine the relationship between Crs and the usual coarse category of Roe. Finally we briefly discuss some basic examples and applications. Topics include_metric coarse spaces_,_continuous control_ [MR1277522], metric and continuously controlled_coarse simplices_,_sigma-coarse spaces_ [MR2225040], and the relation between quotient coarse spaces and the K-theory of Roe algebras (of particular interest for continuously controlled coarse spaces).


Modules with reducible complexity

Abstract: For a commutative Noetherian local ring we define and study the class of modules having reducible complexity, a class containing all modules of finite complete intersection dimension. Various properties of this class of modules are given, together with results on the vanishing of homology and cohomology.


Is critical 2D percolation universal?

Abstract: The aim of this paper is to explore possible ways of extending Smirnov's proof of Cardy's formula for critical site-percolation on the triangular lattice to other cases (such as bond-percolation on the square lattice); the main question we address is that of the choice of the lattice embedding into the plane which gives rise to conformal invariance in the scaling limit. Even though we were not able to produce a complete proof, we believe that the ideas presented here go in the right direction.


On the Hochschild (co)homology of quantum exterior algebras

Abstract: We compute the Hochschild cohomology and homology of a class of quantum exterior algebras, with coefficients in twisted bimodules. As a result we obtain several interesting examples of the homological behavior of these algebras.


On support varieties for modules over complete intersections

Abstract: We show that, over a local complete intersection, every possible variety is realized as the cohomological support variety of some module. Moreover, we show that the projective variety of a complete indecomposable maximal Cohen-Macaulay module is connected.