Stability of Tails and 4-Canonical Models

Abstract: We show that the GIT quotients of suitable loci in the Hilbert and Chow schemes of 4-canonically embedded curves of genus $g\ge 3$ are the moduli space $\bar{M}_g^{\text{ps}}$ of pseudo-stable curves constructed by Schubert in \cite{Schubert} using Chow varieties and 3-canonical models. The only new ingredient needed in the Hilbert scheme variant is a more careful analysis of the stability with respect to a certain 1-ps $\lambda$ of the $m^{\text{th}}$ Hilbert points of curves $X$ with elliptic tails. We compute the exact weight with which $\lambda$ acts, and not just the leading term in $m$ of this weight. A similar analysis of stability of curves with rational cuspidal tails allows us to determine the stable and semistable 4-canonical Chow loci. Although here the geometry of the quotient is more complicated because there are strictly semi-stable orbits, we are able to again identify it as $\bar{M}_g^{\text{ps}}$. Our computations yield, as byproducts, examples of both $m$-Hilbert unstable and $m$-Hilbert stable $X$ that are Chow strictly semi-stable.


Matricial formulae for partitions

Abstract: The exponential of the triangular matrix whose entries in the diagonal at distance $n$ from the principal diagonal are all equal to the sum of the inverse of the divisors of $n$ is the triangular matrix whose entries in the diagonal at distance $n$ from the principal diagonal are all equal to the number of partitions of $n$. A similar result is true for any pair of sequences satisfying a special recurrence.


Maximal plurisubharmonic models

Abstract: An analytic pair of dimension n and center V is a pair (V, M) where M is a complex manifold of (complex) dimension n and V is a closed totally real analytic submanifold of dimension n. To an analytic pair (V, M) we associate the class of the functions u from M to a positive bounded interval which are plurisubharmonic in M and such that u(p) = 0 for each p in V. If the class admits a maximal function u, the triple (V, M, u) is said to be a maximal plurisubharmonic model. After defining a pseudo-metric E(V,M) on the center V of an analytic pair (V, M) we prove (see Theorem 4.1, Theorem 5.1) that maximal plurisubharmonic models provide a natural generalization of the Monge-Ampere models introduced by Lempert and Szoke in [16].


Multifractal Formalism and Inequality involving Packing Dimension

Abstract: This article fits in many studies of multifractal analysis of measure. We took as a starting point the work of F. Ben Nasr in " Calculs de dimension de packing " to give a new inequality involving $Dim(\bar{X}^{\alpha})$ which would be, in certain cases, finer than the inequality established by L. Olsen in " A multifractal formalism " . Besides we elaborated an application of our result which gives a better inequality involving $Dim(\bar{X}^{\alpha})$.


Generalized Jacobi operators in Krein spaces

Abstract: A special class of generalized Jacobi operators which are self-adjoint in Krein spaces is presented. A description of the resolvent set of such operators in terms of solutions of the corresponding recurrence relations is given. In particular, special attention is paid to the periodic generalized Jacobi operators. Finally, the spectral properties of generalized Jacobi operators are applied to prove convergence results for Pad\'e approximants.


Comparison between shape optimization and volumic level set approximation for geometrical functionals

Abstract: We propose to differentiate a general curvature functional with two different approaches. In the first one we compute the derivative with the tools of shape optimization and in the second one we compute the derivative of a volumic approximation of the functional with respect to a level set function. We show that the two previous approaches give the same result.


Computing expected transition events in reducible Markov chains

Abstract: We present a closed-form, computable expression for the expected number of times any transition event occurs during the transient phase of a reducible Markov chain. Examples of events include time to absorption, number of visits to a state, traversals of a particular transition, loops from a state to itself, and arrivals to a state from a particular subset of states. We give an analogous expression for time-average events, which describe the steady-state behavior of reducible chains as well as the long-term behavior of irreducible chains.


Stabilizing Randomly Switched Systems

Abstract: This article is concerned with stability analysis and stabilization of randomly switched systems under a class of switching signals. The switching signal is modeled as a jump stochastic (not necessarily Markovian) process independent of the system state; it selects, at each instant of time, the active subsystem from a family of systems. Sufficient conditions for stochastic stability (almost sure, in the mean, and in probability) of the switched system are established when the subsystems do not possess control inputs, and not every subsystem is required to be stable. These conditions are employed to design stabilizing feedback controllers when the subsystems are affine in control. The analysis is carried out with the aid of multiple Lyapunov-like functions, and the analysis results together with universal formulae for feedback stabilization of nonlinear systems constitute our primary tools for control design


Sur la convergence des s\'eries trigonom\'etriques qui servent \`a repr\'esenter une fonction arbitraire entre des limites donn\'ees

Abstract: Dirichlet proves the general convergence of Fourier series, after pointing out errors in an earlier attempt by Cauchy. We transcribed from Crelle's Journal (1829) with numerous typographical corrections, and added a completed bibliography. Dirichlet prouve la convergence g\'en\'erale de la s\'eries de Fourier, apr\`es avoir montr\'e des erreurs dans un essai par Cauchy. Nous avons transcrit de Crelle's journal (1829) avec de nombreuses corrections typographiques, et avons ajout\'e une bibliographie compl\`ete.


Cores of Geometric Graphs

Abstract: Cameron and Kazanidis have recently shown that rank-3 graphs are either cores or have complete cores, and they asked whether this holds for all strongly regular graphs. We prove that this is true for the point graphs and line graphs of generalized quadrangles and that when the number of points is sufficiently large, it is also true for the block graphs of Steiner systems and orthogonal arrays.


Regular Maximal Monotone Multifunctions and Enlargements

Abstract: In this note we use recent results concerning the sum theorem for maximal monotone multifunctions in general Banach spaces to find new characterizations and properties of regular maximal monotone multifunctions and then use these to describe the domain of certain enlargements.


Parabolic principal Higgs bundles

Abstract: In ``Ramified G-bundles as parabolic bundles'' (J. Ramanujan Math. Soc. (2003) Vol. 18) with Balaji and Nagaraj we introduced the ramified principal bundles. The aim here is to define the Higgs bundles in the ramified context.


Insufficiency of the Brauer-Manin obstruction applied to etale covers

Abstract: Let k be any global field of characteristic not 2. We construct a k-variety X such that X(k) is empty, but for which the emptiness cannot be explained by the Brauer-Manin obstruction or even by the Brauer-Manin obstruction applied to finite etale covers.


Embeddings and chains of free groups

Abstract: We build two non-abelian CSA-groups in which maximal abelian subgroups are conjugate and divisible.


Localization theory for triangulated categories

Abstract: These notes provide an introduction to the theory of localization for triangulated categories. Localization is a machinery to formally invert morphisms in a category. We explain this formalism in some detail and we show how it is applied to triangulated categories.


Volume Growth and Curvature Decay of Complete Positively Curved K\"{a}hler Manifolds

Abstract: This paper constructs a class of complete K\"{a}hler metrics of positive holomorphic sectional curvature on ${\bf C}^n$ and finds that the constructed metrics satisfy the following properties: As the geodesic distance $\rho\to\infty,$ the volume of geodesic balls grows like $O(\rho^{\frac{2(\beta+1)n}{\beta+2}})$ and the Riemannian scalar curvature decays like $O(\rho^{-\frac{2(\beta+1)}{\beta+2}}),$ where $\beta\geq 0.$


Cesaro averages of Euler-like functions

Abstract: By Euler-like function we mean a function defined on the positive integers and associating to $n$ the product, over all primes $p$ dividing $n$, of 1 plus (or minus) the inverse of $p$ to the power $s$. We calculate the limit of the Cesaro mean of these functions.


On Discrete Subgroups of automorphism of $P^2_C$

Abstract: We study the geometry and dynamics of discrete subgroups $\Gamma$ of $\PSL(3,\mathbb{C})$ with an open invariant set $\Omega \subset \PC^2$ where the action is properly discontinuous and the quotient $\Omega/\Gamma$ contains a connected component whicis compact. We call such groups {\it quasi-cocompact}. In this case $\Omega/\Gamma$ is a compact complex projective orbifold and $\Omega$ is a {\it divisible set}. Our first theorem refines classical work by Kobayashi-Ochiai and others about complex surfaces with a projective structure: We prove that every such group is either virtually affine or complex hyperbolic. We then classify the divisible sets that appear in this way, the corresponding quasi-cocompact groups and the orbifolds $\Omega/\Gamma$. We also prove that excluding a few exceptional cases, the Kulkarni region of discontinuity coincides with the equicontinuity region and is the largest open invariant set where the action is properly discontinuous.


Asymptotics of Plancherel measures for $GL(n,q)$

Abstract: We introduce the Plancherel measure on the set of partition collections, which parameterize irreducible representations of order n general linear group over a finite field. We prove that as n goes to infinity, the random partitions from the random collections converge in finite dimencional distribution to independent random variables. We give explicit formulas for the corresponding limit distributions.


Classes of 3-regular graphs that are (7, 2)-edge-choosable

Abstract: A graph is (7, 2)-edge-choosable if, for every assignment of lists of size 7 to the edges, it is possible to choose two colors for each edge from its list so that no color is chosen for two incident edges. We show that every 3-edge-colorable graph is (7, 2)-edge-choosable and also that many non-3-edge-colorable 3-regular graphs are (7, 2)-edge-choosable.


Seshadri constants and surfaces of minimal degree

Abstract: In "Seshadri fibrations of algebraic surfaces" [arXiv:0709.2592v1] we showed that if the multiple point Seshadri constants of an ample line bundle on a smooth projective surface in very general points satisfy certain inequality then the surface is fibred by curves computing these constants. Here we characterize the border case of polarized surfaces whose Seshadri constants in general points fulfill the equality instead of inequality and which are not fibred by Seshadri curves. It turns out that these surfaces are the projective plane and surfaces of minimal degree.


Asymptotic Behavior of Solutions of a Free Boundary Problem Modelling the Growth of Tumors with Stokes Equations

Abstract: We study a free boundary problem modelling the growth of non-necrotic tumors with fluid-like tissues. The fluid velocity satisfies Stokes equations with a source determined by the proliferation rate of tumor cells which depends on the concentration of nutrients, subject to a boundary condition with stress tensor effected by surface tension. It is easy to prove that this problem has a unique radially symmetric stationary solution. By using a functional approach, we prove that there exists a threshold value $\gamma_*>0$ for the surface tension coefficient $\gamma$, such that in the case $\gamma>\gamma_*$ this radially symmetric stationary solution is asymptotically stable under small non-radial perturbations, whereas in the opposite case it is unstable.


Analysis of the Effect of Speed Limit Increases on Accident-Injury Severities

Abstract: The influence of speed limits on roadway safety has been a subject of continuous debate in the State of Indiana and nationwide. In Indiana, highway-related accidents result in about 900 fatalities and forty thousand injuries annually and place an incredible social and economic burden on the state. Still, speed limits posted on highways and other roads are routinely exceeded as individual drivers try to balance safety, mobility (speed), and the risks and penalties associated with law enforcement efforts. The speed-limit/safety issue has been a matter of considerable concern in Indiana since the state raised its speed limits on rural interstates and selected multilane highways on July 1, 2005. In this paper, the influence of the posted speed limit on the severity of vehicle accidents is studied using Indiana accident data from 2004 (the year before speed limits were raised) and 2006 (the year after speed limits were raised on rural interstates and some multi-lane non-interstate routes). Statistical models of the injury severity of different types of accidents on various roadway classes were estimated. The results of the model estimations showed that, for the speed limit ranges currently used, speed limits did not have a statistically significant effect on the severity of accidents on interstate highways. However, for some non-interstate highways, higher speed limits were found to be associated with higher accident severities - suggesting that future speed limit changes, on non-interstate highways in particular, need to be carefully assessed on a case-by-case basis.


Differentiable Categories, gerbes and G-structures

Abstract: The theories of strings and $D$-branes have motivated the development of non Abelian cohomology techniques in differential geometry, on the purpose to find a geometric interpretation of characteristic classes. The spaces studied here, like orbifolds are not often smooth. In classical differential geometry, non smooth spaces appear also naturally, for example in the theory of foliations, the space of leaves can be an orbifold with singularities. The scheme to study these structures is identical: classical tools used in differential geometry, like connections, curvature are adapted. The purpose of this paper is to present the notion of differential category which unifies all these points of view. This enables us to provide a geometric interpretation of 5-characteristic classes, and to interpret classical problems which appear in the theory of $G$-structures by using gerbes.


Asymptotic Stability of Stationary Solutions of a Free Boundary Problem Modeling the Growth of Tumors with Fluid Tissues

Abstract: This paper aims at proving asymptotic stability of the radial stationary solution of a free boundary problem modeling the growth of nonnecrotic tumors with fluid-like tissues. In a previous paper we considered the case where the nutrient concentration $\sigma$ satisfies the stationary diffusion equation $\Delta\sigma=f(\sigma)$, and proved that there exists a threshold value $\gamma_*>0$ for the surface tension coefficient $\gamma$, such that the radial stationary solution is asymptotically stable in case $\gamma>\gamma_*$, while unstable in case $\gamma<\gamma_*$. In this paper we extend this result to the case where $\sigma$ satisfies the non-stationary diffusion equation $\epsln\partial_t\sigma=\Delta\sigma-f(\sigma)$. We prove that for the same threshold value $\gamma_*$ as above, for every $\gamma>\gamma_*$ there is a corresponding constant $\epsln_0(\gamma)>0$ such that for any $0<\epsln<\epsln_0(\gamma)$ the radial stationary solution is asymptotically stable with respect to small enough non-radial perturbations, while for $0<\gamma<\gamma_*$ and $\epsln$ sufficiently small it is unstable under non-radial perturbations.


Isotriviality is equivalent to potential good reduction for endomorphisms of ${\mathbb P}^N$ over function fields

Abstract: Let $K=k(C)$ be the function field of a complete nonsingular curve $C$ over an arbitrary field $k$. The main result of this paper states that a morphism $\phi:{\mathbb P}^N_K\to{\mathbb P}^N_K$ is isotrivial if and only if it has potential good reduction at all places $v$ of $K$; this generalizes results of Benedetto for polynomial maps on ${\mathbb P}^1_K$ and Baker for arbitrary rational maps on ${\mathbb P}^1_K$. We offer two proofs: the first uses algebraic geometry and geometric invariant theory, and it is new even in the case N=1. The second proof uses non-archimedean analysis and dynamics, and it more directly generalizes the proofs of Benedetto and Baker. We will also give two applications. The first states that an endomorphism of ${\mathbb P}^N_K$ of degree at least two is isotrivial if and only if it has an isotrivial iterate. The second gives a dynamical criterion for whether (after base change) a locally free coherent sheaf ${\mathcal E}$ of rank $N+1$ on $C$ decomposes as a direct sum ${\mathcal L}\oplus...\oplus{\mathcal L}$ of $N+1$ copies of the same invertible sheaf ${\mathcal L}$.


Unranking permutations in transposition order and linear time

Abstract: An algorithm is presented for unranking permutations in transposition order: Given a seed s\in N, the algorithm produces a permutation P(s) that differs from the permutation P(s+1) by the transposition of two elements.


Global well-posedness for the $L^2$ critical Hartree equation on $\bbr^n$, $n\ge 3$

Abstract: We consider the initial value problem for the L^2-critical defocusing Hartree equation in R^n, n \ge 3. We show that the problem is globally well posed in H^s(R^n) when 1>s> \frac{2(n-2)}{3n-4}$. We use the "I-method" combined with a local in time Morawetz estimate for the smoothed out solution.


On the non-existence of some Steiner $t$-$(v,k)$ trades of certain volumes

Abstract: Mahmoodian and Soltankhah $\cite{MMS}$ conjectured that there does not exist any $t$-$(v,k)$ trade of volume $s_{i}< s <s_{i+1}$, where $s_{i}=2^{t+1}-2^{t-i}, i=0,1,..., t-1$. Also they showed that the conjecture is true for $i=0$. In this paper we prove the correctness of this conjecture for Steiner trades.


Constructing regular graphs with smallest defining number

Abstract: In a given graph $G$, a set $S$ of vertices with an assignment of colors is a {\sf defining set of the vertex coloring of $G$}, if there exists a unique extension of the colors of $S$ to a $\Cchi(G)$-coloring of the vertices of $G$. A defining set with minimum cardinality is called a {\sf smallest defining set} (of vertex coloring) and its cardinality, the {\sf defining number}, is denoted by $d(G, \Cchi)$. Let $ d(n, r, \Cchi = k)$ be the smallest defining number of all $r$-regular $k$-chromatic graphs with $n$ vertices. Mahmoodian et. al \cite{rkgraph} proved that, for a given $k$ and for all $n \geq 3k$, if $r \geq 2(k-1)$ then $d(n, r, \Cchi = k)=k-1$. In this paper we show that for a given $k$ and for all $n < 3k$ and $r\geq 2(k-1)$, $d(n, r, \Cchi=k)=k-1$.


Some Divisibility Properties in Ring of Polynomials over a Unique Factorization Domain

Abstract: Using polynomial evaluation, we give some useful criteria to answer questions about divisibility of polynomials. This allows us to develop interesting results concerning the prime elements in the domain of coefficients. In particular, it is possible to prove that under certain conditions, the domain of coefficients must have infinitely many prime elements. We give alternative characterizations for D-rings and present various examples.


A remark on the rational cohomology of $\bar{S}_{1,n}$

Abstract: We focus on the rational cohomology of Cornalba's moduli space of spin curves of genus 1 with $n$ marked points. In particular, we show that both its first and its third cohomology group vanish and the second cohomology group is generated by boundary classes.


New Upper Bound for the Edge Folkman Number Fe(3,5;13)

Abstract: In this paper we prove that the edge Folkman number Fe(3,5;13) is not greater than 21.


Equivalent Definitions for Uniform Exponential Trichotomy of Evolution Operators in Banach Spaces

Abstract: The aim of this paper is to give necessary and sufficient conditions for the uniform exponential trichotomy property of nonlinear evolution operators in Banach spaces. The obtained results are generalizations for infinite-dimensional case of some well-known results of Elaydi and Hajek on exponential trichotomy of differential systems.


Nonuniform Behaviors for Skew-Evolution Semiflows in Banach Spaces

Abstract: The paper emphasizes some asymptotic behaviors for skew-evolution semiflows in Banach spaces. These are defined by means of evolution semiflows and evolution cocycles. Some characterizations which generalize classical results are also provided. The approach is from nonuniform point of view.


The Relative Capacity

Abstract: The purpose of this article is to introduce the relative $p$-capacity $\Cap_{p,\Omega}$ with respect to an open set $\Omega$ in $\IR^N$. It is a Choquet capacity on the closure of $\Omega$ and extends the classical $p$-capacity $\Cap_p$ in the sense that $\Cap_{p,\Omega}=\Cap_p$ if $\Omega=\IR^N$. The importance of the relative $p$-capacity stems from the fact that a large class of Sobolev functions defined on a 'bad domain' admit a trace on the boundary $\partial\Omega$ which is then unique up to $\Cap_{p,\Omega}$-polar set. As an application we prove a characterization of $W^{1,p}_0(\Omega)$ for open sets $\Omega\subset\IR^N$.


Quasi-co-Frobenius corings as Galois comodules

Abstract: We compare several quasi-Frobenius-type properties for corings that appeared recently in literature and provide several new characterizations for each of these properties. By applying the theory of Galois comodules with a firm coinvariant ring, we can characterize a locally quasi-Frobenius (quasi-co-Frobenius) coring as a locally projective generator in its category of comodules.


Smooth Structures and Einstein Metrics on $CP^2#5,6,7\bar{CP^2}$

Abstract: We show that each of the topological 4-manifolds $CP^2#k\bar{CP^2}, for $k = 6, 7$ admits a smooth structure which has an Einstein metric of scalar curvature $s > 0$, a smooth structure which has an Einstein metric with $s < 0$ and infinitely many non-diffeomorphic smooth structures which do not admit Einstein metrics. We show that there are infinitely many manifolds homeomorphic non-diffeomorphic to $CP^2#5\bar{CP^2}$ which do not admit an Einstein metric. We also exhibit new higher dimensional examples of manifolds carrying Einstein metrics of both positive and negative scalar curvature. The main ingredients are recent constructions of exotic symplectic or complex manifolds with small topological numbers.


An extension problem for convex functions

Abstract: We give a statement on extension with estimates of convex functions defined on a linear subspace, inspired by similar extension results concerning metrics on positive line bundles


On invariants for Legendrian knots

Abstract: Suppose that L is a null--homologous Legendrian knot in a contact 3--manifold. We determine the connection between the sutured invariant of the complement of L and the Legendrian invariant defined by Lisca, Ozsvath, Stipsicz and Szabo. In particular, we derive a vanishing theorem for the Legendrian invariant in the presence of Giroux torsion in the complement of the knot, and reprove several known properties of the Legendrian invariant from this perspective.


On the extendability of elliptic surfaces of rank two and higher

Abstract: We study threefolds X in a projective space having as hyperplane section a smooth surface with an elliptic fibration. We first give a general theorem about the possible embeddings of such surfaces with Picard number two. More precise results are then proved for Weierstrass fibrations, both of rank two and higher. In particular we prove that a Weierstrass fibration of rank two that is not a K3 surface is not hyperplane section of a locally complete intersection threefold and we give some conditions, for many embeddings of Weierstrass fibrations of any rank, under which every such threefold must be a cone.


Abstract: As the title ``Generalized regularity and solution concepts for differential equations'' suggests, the main topic of my thesis is the investigation of generalized solution concepts for differential equations, in particular first order hyperbolic partial differential equations with real-valued, non-smooth coefficients and their characteristic system of ordinary differential equations. In Colombeau theory there have been developed existence results that yield solutions for ordinary and partial differential equations beyond the scope of classical approaches. Nevertheless this comes at the price of sacrificing regularity (in general a Colombeau solution may even lack a distributional shadow). It is prevailing in the Colombeau setting that the question of mere existence of solutions is much easier to answer than to determine their regularity properties (i.e. if a distributional shadow exists and how regular it is). In order order to address these regularity question and encouraged by the fact that the solution of a (homogeneous) first order partial differential equation can be written as a pullback of the initial condition by the characteristic backward flow, a main topic of my thesis deals with the microlocal analysis of pullbacks of c-bounded Colombeau generalized functions. Another topic is the comparsion of Colombeau techniques for solving ordinary and partial differential equations to other generalized solution concepts, which has led to a joint article with G\"unther H\"ormann. A useful tool for this purpose is the concept of a generalized graph, which has been developed in the thesis.


A counter example on nontangential convergence for oscillatory integrals

Abstract: Consider the solution of the time-dependent Schr{\"o}dinger equation with initial data $f$. It is shown in \cite{artikel} that there exists $f$ in the Sobolev space $H^s(\RR), s=n/2$ such that tangential convergence can not be widened to convergence regions. In this paper we show that the corresponding result holds when $-\Delta_x$ is replaced by an operator $\phi(D)$, with special conditions on $\phi$.


Sharp bounds for symmetric and asymmetric Diophantine approximation

Abstract: In 2004, J.C. Tong found bounds for the approximation quality of a regular continued fraction convergent of a rational number, expressed in bounds for both the previous and next approximation. We sharpen his results with a geometric method and give both sharp upper and lower bounds. We also calculate the asymptotic frequency that these bounds occur.


Generalized Thomas hyperplane sections and relations between vanishing cycles

Abstract: As is remarked by B. Totaro, R. Thomas essentially proved that the Hodge conjecture is inductively equivalent to the existence of a hyperplane section, called a generalized Thomas hyperplane section, such that the restriction to it of a given primitive Hodge class does not vanish. We study the relations between the vanishing cycles in the cohomology of a general fiber, and show that each relation between the vanishing cycles of type (0,0) with unipotent monodromy around a singular hyperplane section defines a primitive Hodge class such that this singular hyperplane section is a generalized Thomas hyperplane section if and only if the pairing between a given primitive Hodge class and some of the constructed primitive Hodge classes does not vanish. This is a generalization of a construction by P. Griffiths.


Analysis of Metric Distances and Volumes of Hippocampi Indicates Different Morphometric Changes over Time in Dementia of Alzheimer Type and Nondemented Subjects

Abstract: In this article, we analyze the morphometry of hippocampus in subjects with very mild dementia of Alzheimer's type (DAT) and nondemented controls and how it changes over a two-year period. Morphometric differences with respect to a template hippocampus were measured by the metric distance obtained from the Large Deformation Diffeomorphic Metric Mapping (LDDMM) algorithm which was previously used to calculate dense one-to-one correspondence vector fields between the shapes. LDDMM assigns metric distances on the space of anatomical images thereby allowing for the direct comparison and quantization of morphometric changes. We use various statistical methods to compare the metric distances in a cross-sectional and longitudinal manner. At baseline, the metric distances for demented subjects are found not to be significantly different from those for nondemented subjects. At follow-up, the metric distances for demented subjects were significantly larger compared to nondemented subjects. The metric distances for demented subjects increased significantly from baseline to follow-up but not for nondemented subjects. We also use the metric distances in logistic regression for diagnostic discrimination of subjects. We compare metric distances with the volumes and obtain similar results. In classification, the model that uses volume, metric distance, and volume loss over time together performs better in detecting DAT. Thus, metric distances with respect to a template computed via LDDMM can be a powerful tool in detecting differences in shape in cross-sectional as well as longitudinal studies.


On the opposite of the category of rings

Abstract: We define a faithful contravariant functor NCSpec from the category of rings to the category of ringed spaces, and show that if R is a commutative ring then NCSpec(R) may be viewed as a completion of Spec(R) in an appropriate sense. We then explain how the spaces NCSpec(R) may be glued, and study quasicoherent sheaves on them. As an example, we compute the category of quasicoherent sheaves on a space constructed from a skew-polynomial ring R by an analogue of the Proj construction.


On the number of matrices and a random matrix with prescribed row and column sums and 0-1 entries

Abstract: We consider the set Sigma(R,C) of all mxn matrices having 0-1 entries and prescribed row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n). We prove an asymptotic estimate for the cardinality |Sigma(R, C)| via the solution to a convex optimization problem. We show that if Sigma(R, C) is sufficiently large, then a random matrix D in Sigma(R, C) sampled from the uniform probability measure in Sigma(R,C) with high probability is close to a particular matrix Z=Z(R,C) that maximizes the sum of entropies of entries among all matrices with row sums R, column sums C and entries between 0 and 1. Similar results are obtained for 0-1 matrices with prescribed row and column sums and assigned zeros in some positions.


Polytopal complexes: maps, chain complexes and... necklaces

Abstract: The notion of polytopal map between two polytopal complexes is defined. Surprisingly, this definition is quite simple and extends naturally those of simplicial and cubical maps. It is then possible to define an induced chain map between the associated chain complexes. Finally, we use this new tool to give the first combinatorial proof of the splitting necklace theorem of Alon. The paper ends with open questions, such as the existence of Sperner's lemma for a polytopal complex or the existence of a cubical approximation theorem.


Electromagnetism and geometry

Abstract: This work is an introduction to modern mathematical physics. We begin with Maxwell laws and vector calculus, pass next to consider the action and the Feynman integral in quantum mechanics, next relativity and differential geometry to formulate the electromagnetic laws in intrinsic form. Next we see gravitation to study the covariant derivative. We end with the electromagnetic bundle U(1). It contains the know-how + the know-why. The text is written in Spanish. 340 pps.


The M\"{o}bius Function of a Restricted Composition Poset

Abstract: We study a poset of compositions restricted by part size under a partial ordering introduced by Bj\"{o}rner and Stanley. We show that our composition poset $C_{d+1}$ is isomorphic to the poset of words $A_d^*$. This allows us to use techniques developed by Bj\"{o}rner to study the M\"{o}bius function of $C_{d+1}$. We use counting arguments and shellability as avenues for proving that the M\"{o}bius function is $\mu(u,w)=(-1)^{|u|+|w|}{w\choose u}_{dn}$, where ${w\choose u}_{dn}$ is the number of $d$-normal embeddings of $u$ in $w$. We then prove that the formal power series whose coefficients are given by the zeta and the M\"{o}bius functions are both rational. Following in the footsteps of Bj\"{o}rner and Reutenauer and Bj\"{o}rner and Sagan, we rely on definitions to prove rationality in one case, and in another case we use finite-state automata.


Local inverses of shift maps along orbits of flows

Abstract: Let $M$ be a smooth manifold and $F$ be a vector field on $M$. My article ["Smooth shifts along trajectories of flows", Topol. Appl. 130 (2003) 183-204, arXiv:math/0106199] concerning the homotopy types of the group of diffeomorphisms preserving orbits of $F$ contains two errors. They imply that the principal statement of that paper holds under additional assumptions on $F$. Unfortunately this result was essentially used in another paper of mine ["Homotopy types of stabilizers and orbits of Morse functions on surfaces" Ann. Glob. Anal. Geom., 29 no. 3, (2006) 241-285, arXiv:math/0310067]. The aim of this article is to expose the results of the first paper in a right way, extend them to a larger class of flows with degenerate singularities, and show that the results of the second paper remain true.


Homology representations arising from the half cube

Abstract: We construct a CW decomposition $C_n$ of the $n$-dimensional half cube in a manner compatible with its structure as a polytope. For each $3 \leq k \leq n$, the complex $C_n$ has a subcomplex $C_{n, k}$, which coincides with the clique complex of the half cube graph if $k = 4$. The homology of $C_{n, k}$ is concentrated in degree $k-1$ and furthermore, the $(k-1)$-st Betti number of $C_{n, k}$ is equal to the $(k-2)$-nd Betti number of the complement of the $k$-equal real hyperplane arrangement. These Betti numbers, which also appear in theoretical computer science, numerical analysis and engineering, are the coefficients of a certain Pascal-like triangle (Sloane's sequence A119258). The Coxeter groups of type $D_n$ act naturally on the complexes $C_{n, k}$, and thus on the associated homology groups.


Relative Chow-Kunneth decompositions for conic bundles and Prym varieties

Abstract: We construct a relative Chow-Kunneth decomposition for a conic bundle over a surface such that the middle projector gives the Prym variety of the associated double covering of the discriminant of the conic bundle. This gives a refinement (up to an isogeny) of Beauville's theorem on the relation between the intermediate Jacobian of the conic bundle and the Prym variety of the double covering.


On a Theorem on sums of the form 1+2^(2^n)+2^(2^n+1)+...+2^(2^n+m) and a result linking Fermat with Mersenne numbers

Abstract: In his book "250 Problems in Elementary Number Theory", W.Sierpinski shows that the numbers 1+2^(2^n)+2^(2^n+1) are divisible by 21; for n=1,2,.... In this paper, we prove a similar but more general result.Consider the natural numbers of the form I(n.m)= 1+2^(2^n)+2^(2^n+1)+...+2^(2^n+m).In Theorem 1 we prove that for every odd integer N greater than 1, there exist infinitely many natural numbers n and m such that the integers I(n.m) are divisible by N. We give an explicit construction of the numbers n and m, for a given N. As an example, when N=31, and with n=4k and m=94+124i, the numbers I(n,m) are divisible by 31. A similar example is offered for N=(31)(7)=217. In Theorem 2, we prove a result pertaining to Mersenne numbers.There are also three Corollaries in this work, one of which deals with Fermat numbers.


The Christoffel-Darboux Kernel

Abstract: A review of the uses of the CD kernel in the spectral theory of orthogonal polynomials, concentrating on recent results.


Highest weight categories arising from Khovanov's diagram algebra I: cellularity

Abstract: This is the first of four articles studying some slight generalisations H(n,m) of Khovanov's diagram algebra, as well as quasi-hereditary covers K(n,m) of these algebras in the sense of Rouquier, and certain infinite dimensional limiting versions. In this article we prove that H(n,m) is a cellular symmetric algebra and that K(n,m) is a cellular quasi-hereditary algebra. In subsequent articles, we relate these algebras to level two blocks of degenerate cyclotomic Hecke algebras, parabolic category O and the general linear supergroup, respectively.


Manin's conjecture on a nonsingular quartic del Pezzo surface

Abstract: Given a nonsingular quartic del Pezzo surface, a conjecture of Manin predicts the density of rational points on the open subset of the surface formed by deleting the lines. We prove that this prediction is of the correct order of magnitude for a particular surface.


A basis of bideterminants for the coordinate ring of the orthogonal group

Abstract: We give a basis of bideterminants for the coordinate ring K[O(n)] of the orthogonal group O(n,K), where K is an infinite field of characteristic not 2. The bideterminants are indexed by pairs of Young tableaux which are O(n)-standard in the sense of King-Welsh. We also give an explicit filtration of K[O(n)] as an O(n,K)-bimodule, whose factors are isormorphic to the tensor product of orthogonal analogues of left and right Schur modules.


Model category extensions of the Pirashvili-S{\l}omi\'{n}ska theorems

Abstract: We describe the class of semi-stable model categories, which generalize the equivalence of finite products and coproducts in abelian and stable model categories, and use this to establish Morita equivalences among categories of functors. We provide a construction of pairs of small categories--known as conjugate pairs--whose associated categories of diagrams are Quillen equivalent in the semi-stable setting. We frame our development in the context of Morita theory, following Slominska's work on similar questions for categories of functors enriched over and taking values in R-modules.


It\^o's formula for the $L_{p}$-norm of stochastic $W^{1}_{p}$-valued processes

Abstract: We prove It\^o's formula for the $L_{p}$-norm of a stochastic $W^{1}_{p}$-valued processes appearing in the theory of SPDEs in divergence form.


Schur Positivity and the $q$-Log-convexity of the Narayana Polynomials

Abstract: Using Schur positivity and the principal specialization of Schur functions, we provide a proof of a recent conjecture of Liu and Wang on the $q$-log-convexity of the Narayana polynomials, and a proof of the second conjecture that the Narayana transformation preserves the log-convexity. Based on a formula of Br\"and$\mathrm{\acute{e}}$n which expresses the $q$-Narayana numbers as the specializations of Schur functions, we derive several symmetric function identities using the Littlewood-Richardson rule for the product of Schur functions, and obtain the strong $q$-log-convexity of the Narayana polynomials and the strong $q$-log-concavity of the $q$-Narayana numbers.


Transcendence of Power Series for Some Number Theoretic Functions

Abstract: We give a new proof of Fatou's theorem: {\em if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function.} This result is applied to show that for any non--trivial completely multiplicative function from $\mathbb{N}$ to $\{-1,1\}$, the series $\sum_{n=1}^\infty f(n)z^n$ is transcendental over $\mathbb{Z}[z]$; in particular, $\sum_{n=1}^\infty \lambda(n)z^n$ is transcendental, where $\lambda$ is Liouville's function. The transcendence of $\sum_{n=1}^\infty \mu(n)z^n$ is also proved.


On the Structure of the Fusion Ideal

Abstract: We prove that there is a finite level-independent bound on the number of relations defining the fusion ring of positive energy representations of the loop group of a simple, simply connected Lie group. As an illustration, we compute the fusion ring of $G_2$ at all levels.


On the comparison of positive elements of a C*-algebra by lower semicontinuous traces

Abstract: It is shown in this paper that two positive elements of a C*-algebra agree on all lower semicontinuous traces if and only if they are equivalent in the sense of Cuntz and Pedersen. A similar result is also obtained in the more general case where the two elements are comparable by their values on the lower semicontinuous traces. This result is used to give a characterization of the functions on the cone of lower semicontinuous traces of a stable C*-algebra that arise from positive elements of the algebra.


On a certain asymptotic relationship involving $\vartheta(t) - \lfloor t \rfloor$ and $t^{1/2}$

Abstract: Let $\lfloor t \rfloor$ denote the greatest positive integer less than or equal to a given positive real number $t$ and $\vartheta(t)$ the Chebyshev $\vartheta$-function. In this paper, we prove a certain asymptotic relationship involving $\vartheta(t) - \lfloor t \rfloor $ and $t^{1/2}$.


Comparing 2-handle additions to a genus 2 boundary component

Abstract: We prove that knots obtained by attaching a band to a split link satisfy the cabling conjecture. We also give new proofs that unknotting number one knots are prime and that genus is superadditive under band sum. Additionally, we prove a collection of results comparing two 2-handle additions to a genus two boundary component of a compact, orientable 3-manifold. These results give a near complete solution to a conjecture of Scharlemann and provide evidence for a conjecture of Scharlemann and Wu. The proofs make use of a new theorem concerning the effects of attaching a 2-handle to a suture in the boundary of a sutured manifold.


SiZer for Censored Density and Hazard Estimation

Abstract: The SiZer method is extended to nonparametric hazard estimation and also to censored density and hazard estimation. The new method allows quick, visual statistical inference about the important issue of statistically significant increases and decreases in the smooth curve estimate. This extension has required the opening of a new avenue of research on the interface between statistical inference and scale space.


On a six-parameter generalized Burr XII distribution

Abstract: In this paper, we derive a probability density function that generalizes the Burr XII distribution. The cumulative distribution function and the $n^{th}$ moment of the generalized distribution are obtained while the distribution of some order statistics of the distribution are established. A theorem that relate the new distribution to another statistical distribution is established.


On type III generalized half logistic distribution

Abstract: It is well known that generalized models is attracting the attention of researchers in recent times because of their flexibilities. Particularly, the logistic model has been generalized and applied by many authors while the half logistic distribution has not recieve much attention in term of its generalization. In this paper, we considered a generalized form of half logistic model called type III generalized half logistic distribution. We obtained its probability density function, the cumulative distribution function, the $n^{th}$ moment, the median, the mode, the 100$p$-percentage points and the order statistics of the generalized distribution are established.


Distinguished principal series representations for GLn over a p-adic field

Abstract: In the following article, we give a description of the distingushed irreducible principal series representations of the general linear group over a p-adic field in terms of inducing datum. This provides a counter-example to a conjecture of Jacquet about distinction (Conjecture 1 in U.K Anandavardhanan, "Distinguished non-Archimedean representations ", Proc. Hyderabad Conference on Algebra and Number Theory, 2005, 183-192).


A positive solution to a conjecture of A. Katok for diffeomorphism case

Abstract: In this paper, using Pesin theory and Liao theory, we give a positive solution to a conjecture of A. Katok.


A Tamed 3D Navier-Stokes Equation in Domains

Abstract: In this paper, we analyze a tamed 3D Navier-Stokes equation in uniform $C^2$-domains (not necessarily bounded), which obeys the scaling invariance principle, and prove the existence and uniqueness of strong solutions to this tamed equation. In particular, if there exists a bounded solution to the classical 3D Navier-Stokes equation, then this solution satisfies our tamed equation. Moreover, the existence of a global attractor for the tamed equation in bounded domains is also proved. As simple applications, some well known results for the classical Navier-Stokes equations in unbounded domains are covered.


The Hochschild cohomology ring of the standard Podles quantum sphere

Abstract: The cup and cap product in twisted Hochschild (co)homology is computed for the standard quantum 2-sphere and used to construct a cyclic 2-cocycle that represents the fundamental Hochschild class.


Some Critical Issues for the "Equation-Free" Approach to Multiscale Modeling

Abstract: The "equation-free'' approach has been proposed in recent years as a general framework for developing multiscale methods to efficiently capture the macroscale behavior of a system using only the microscale models. In this paper, we take a close look at some of the algorithms proposed under the "equation-free'' umbrella, the projective integrators and the patch dynamics. We discuss some very simple examples in the context of the "equation-free'' approach. These examples seem to indicate that while its general philosophy is quite attractive and indeed similar to many other approaches in concurrent multiscale modeling, there are severe limitations to the specific implementation proposed by the equation-free approach.


Bayesian outlier detection in Capital Asset Pricing Model

Abstract: We propose a novel Bayesian optimisation procedure for outlier detection in the Capital Asset Pricing Model. We use a parametric product partition model to robustly estimate the systematic risk of an asset. We assume that the returns follow independent normal distributions and we impose a partition structure on the parameters of interest. The partition structure imposed on the parameters induces a corresponding clustering of the returns. We identify via an optimisation procedure the partition that best separates standard observations from the atypical ones. The methodology is illustrated with reference to a real data set, for which we also provide a microeconomic interpretation of the detected outliers.


Geodesically Complete Lorentzian Metrics on Some Homogeneous 3 Manifolds

Abstract: In this work it is shown that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient for 3-dimensional non unimodular Lie groups. As a consequence it is possible to identify, amongst the compact locally homogeneous Lorentzian 3-manifolds with non compact (local) isotropy group, those that are geodesically complete.


Counterpropagating Two-Soliton Solutions in the FPU Lattice

Abstract: We study the interaction of small amplitude, long wavelength solitary waves in the Fermi-Pasta-Ulam model with general nearest-neighbor interaction potential. We establish global-in-time existence and stability of counter-propagating solitary wave solutions. These solutions are close to the linear superposition of two solitary waves for large positive and negative values of time; for intemediate values of time these solutions describe the interaction of two counterpropagating pulses. These solutions are stable with respect to perturbations in $\ell^2$ and asymptotically stable with respect to perturbations which decay exponentially at spatial $\pm \infty$.}


H\"older Regularity of Two-Dimensional Almost-Minimal Sets in $\R^n$

Abstract: We give a different and probably more elementary proof of a good part of Jean Taylor's regularity theorem for Almgren almost-minimal sets of dimension 2 in $\R^3$. We use this opportunity to settle some details about almost-minimal sets, extend a part of Taylor's result to almost-minimal sets of dimension 2 in $\R^n$, and give the expected characterization of the closed sets $E$ of dimension 2 in $\R^3$ that are minimal, in the sense that $H^2(E\setminus F) \leq H^2(F\setminus E)$ for every closed set $F$ such that there is a bounded set $B$ so that $F=E$ out of $B$ and $F$ separates points of $\R^3 \setminus B$ that $E$ separates.


Haar Shifts, Commutators, and Hankel Operators

Abstract: Hankel operators lie at the junction of analytic and real-variables. We will explore this junction, from the point of view of Haar shifts and commutators. An decomposition of the commutator [H,b] into paraproducts is presented.


Confidence Sets Based on Penalized Maximum Likelihood Estimators in Gaussian Regression

Abstract: Confidence intervals based on penalized maximum likelihood estimators such as the LASSO, adaptive LASSO, and hard-thresholding are analyzed. In the known-variance case, the finite-sample coverage properties of such intervals are determined and it is shown that symmetric intervals are the shortest. The length of the shortest intervals based on the hard-thresholding estimator is larger than the length of the shortest interval based on the adaptive LASSO, which is larger than the length of the shortest interval based on the LASSO, which in turn is larger than the standard interval based on the maximum likelihood estimator. In the case where the penalized estimators are tuned to possess the `sparsity property', the intervals based on these estimators are larger than the standard interval by an order of magnitude. Furthermore, a simple asymptotic confidence interval construction in the `sparse' case, that also applies to the smoothly clipped absolute deviation estimator, is discussed. The results for the known-variance case are shown to carry over to the unknown-variance case in an appropriate asymptotic sense.


Explicit solutions of division problems for matrices of polynomials

Abstract: We present integral representations of solutions to division problems involving matrices of polynomials in several complex variables. We also find estimates of the polynomial degree of the solutions by means of careful degree estimates of the so-called Hefer forms which are components of the representations.


Mean value one of prime-pair constants

Abstract: For k greater than 1 and r different from 0, let pi^k_{2r}(x) denote the number of prime pairs (p,p^k+2r) with p not exceeding (large) x. By the Bateman-Horn conjecture, the function pi^k_{2r}(x) should be asymptotic to (2/k)C^k_{2r}li_2(x), with certain specific constants C^k_{2r}. Heuristic arguments lead to the conjecture that these constants have mean value one, just like the Hardy-Littlewood constants C_{2r} for prime pairs (p,p+2r). The conjecture is supported by extensive numerical work.


Group actions and Helly's theorem

Abstract: We describe a connection between the combinatorics of generators for certain groups and the combinatorics of Helly's 1913 theorem on convex sets. We use this connection to prove fixed point theorems for actions of these groups on nonpositively curved metric spaces. These results are encode d in a property that we introduce called ``property $\FA_r$'', which reduces to Serre's property $\FA$ when $r=1$. The method applies to $S$-arithmetic groups in higher $\Q$-rank, to simplex reflection groups (including some non-arithmetic ones), and to higher rank Chevalley groups over polynomial and other rings (for example $\SL_n(\Z[x_1,..., x_d]), n>2$).


Transcendence of the Gaussian Liouville number and relatives

Abstract: {\em The Liouville number}, denoted $l$, is defined by $$l:=0.100101011101101111100...,$$ where the $n$th bit is given by ${1/2}(1+\gl(n))$; here $\gl$ is the Liouville function for the parity of prime divisors of $n$. Presumably the Liouville number is transcendental, though at present, a proof is unattainable. Similarly, define {\em the Gaussian Liouville number} by $$\gamma:=0.110110011100100111011...$$ where the $n$th bit reflects the parity of the number of rational Gaussian primes dividing $n$, 1 for even and 0 for odd. In this paper, we prove that the Gaussian Liouville number and its relatives are transcendental. One such relative is the number $$\sum_{k=0}^\infty\frac{2^{3^k}}{2^{3^k2}+2^{3^k}+1}=0.101100101101100100101...,$$ where the $n$th bit is determined by the parity of the number of prime divisors that are equivalent to 2 modulo 3. We use methods similar to that of Dekking's proof of the transcendence of the Thue--Morse number \cite{Dek1} as well as a theorem of Mahler's \cite{Mahl1}. (For completeness we provide proofs of all needed results.) This method involves proving the transcendence of formal power series arising as generating functions of completely multiplicative functions.


Gauss-Manin connection and t-adic geometry

Abstract: We show that the de Rham cohomology of any separated and smooth rigid variety over a field of Laurent series of characteristic zero carries a natural formal meromorphic connection, which we call the Gauss-Manin connection. We compare it with the Gauss-Manin connection of a proper and smooth variety over a curve, and with the Gauss-Manin connection of the Milnor fibration at an isolated complex hypersurface singularity.


Multiplicative Properties of the Slice Filtration

Abstract: We show that the slice filtration introduced by Voevodsky is compatible in a suitable sense with the symmetric monoidal structure in the category of motivic symmetric T-spectra constructed by Jardine. It follows from this compatibility that the zero slice of the sphere spectrum s_{0}(1) is a ring spectrum and that for every symmetric T-spectrum X, and every integer n; the n-slice s_{n}(X) is a module over s_{0}(1). In particular, if the base scheme is a field of characteristic zero, we have that all the slices s_{n}(X) are big motives in the sense of Voevodsky. We also get as a corollary that the smash product of symmetric T-spectra induces pairings in the motivic Atiyah-Hirzebruch spectral sequence.


Schottky groups cannot act on $\mathbb{P}^{2n}_{\mathbb{C}}$ as subgroups of $PSL(2n+1,\Bbb{C})$

Abstract: In this paper we look at a special type of discrete subgroups of $PSL_{n+1}(\Bbb{C})$ called Schottky groups. We develop some basic properties of these groups and their limit set when $n > 1$, and we prove that Schottky groups only occur in odd dimensions, {\it i.e.}, they cannot be realized as subgroups of $PSL_{2n+1}(\Bbb{C})$.


Automorphisms of $P_8$ singularities and the complex crystallographic groups

Abstract: The paper completes the study of symmetries of parabolic function singularities with relation to complex crystallographic groups that was started in \cite{GM,X9}. We classify smoothable automorphisms of $P_8$ singularities which split the kernel of the intersection form on the second homology. For such automorphisms, the monodromy groups acting on the duals to the eigenspaces with degenerate intersection form are then identified as some of complex affine reflection groups tabled in \cite{P}.


Linearizing a certain family of nonlinear differential equations

Abstract: We show how to reduce the problem of solving members of a certain family of nonlinear differential equations to that of solving some corresponding linear differential equations.


Linear Fractional Stable Sheets: wavelet expansion and sample path properties

Abstract: In this paper we give a detailed description of the random wavelet series representation of real-valued linear fractional stable sheet introduced in Ayache, Roueff and Xiao (2007). By using this representation, in the case where the sample paths are continuous, an anisotropic uniform and quasi-optimal modulus of continuity of these paths is obtained as well as an upper bound for their behavior at infinity and around the coordinate axes. The Hausdorff dimensions of the range and graph of these stable random fields are then derived.


Noncommutative geometry as a functor

Abstract: In this note the noncommutative geometry is interpreted as a functor, whose range is a family of the operator algebras. Some examples are given and a program is sketched.


On the transverse invariant for bindings of open books

Abstract: Consider a transverse knot which is the binding of an open book for the ambient contact manifold. In this paper, we show that the transverse invariants defined by Lisca, Ozsvath, Stipsicz, and Szabo (LOSS) are nonvanishing for such transverse knots. This is true regardless of whether or not the ambient contact structure is tight. We also prove a vanishing theorem for LOSS's Legendrian and transverse invariants. As a corollary, we show that if (T,\pi) is an open book with connected binding, then the complement of T has no Giroux torsion.


Eigenvalue multiplicity and volume growth

Abstract: Let $G$ be a finite group with symmetric generating set $S$, and let $c = \max_{R > 0} |B(2R)|/|B(R)|$ be the doubling constant of the corresponding Cayley graph, where $B(R)$ denotes an $R$-ball in the word-metric with respect to $S$. We show that the multiplicity of the $k$th eigenvalue of the Laplacian on the Cayley graph of $G$ is bounded by a function of only $c$ and $k$. More specifically, the multiplicity is at most $\exp((\log c)(\log c + \log k))$. Similarly, if $X$ is a compact, $n$-dimensional Riemannian manifold with non-negative Ricci curvature, then the multiplicity of the $k$th eigenvalue of the Laplace-Beltrami operator on $X$ is at most $\exp(n^2 + n log k)$. The first result (for $k=2$) yields the following group-theoretic application. There exists a normal subgroup $N$ of $G$, with $[G : N] \leq \alpha(c)$, and such that $N$ admits a homomorphism onto the cyclic group $Z_M$, where $M \geq |G|^{\delta(c)}$ and $\alpha(c), \delta(c) > 0$ are explicit functions depending only on $c$. This is the finitary analog of a theorem of Gromov which states that every infinite group of polynomial growth has a subgroup of finite index which admits a homomorphism onto the integers. This addresses a question of Trevisan, and is proved by scaling down Kleiner's proof of Gromov's theorem. In particular, we replace the space of harmonic functions of fixed polynomial growth by the second eigenspace of the Laplacian on the Cayley graph of $G$.


On coproducts in varieties, quasivarieties and prevarieties

Abstract: If the free algebra F on one generator in a variety V of algebras (in the sense of universal algebra) has a subalgebra free on two generators, must it also have a subalgebra free on three generators? In general, no; but yes if F generates the variety V. Generalizing the argument, it is shown that if we are given an algebra and subalgebras, A_0\supseteq ... \supseteq A_n, in a prevariety (SP-closed class of algebras) P such that A_n generates P, and also subalgebras B_i\subseteq A_{i-1} (0<i\leq n) such that for each i>0 the subalgebra of A_{i-1} generated by A_i and B_i is their coproduct in P, then the subalgebra of A generated by B_1, ..., B_n is the coproduct in P of these algebras. Some further results on coproducts are noted: If P satisfies the amalgamation property, one has the stronger "transitivity" statement: if A has a finite family of subalgebras (B_i)_{i\in I} such that the subalgebra of A generated by the B_i is their coproduct, and each B_i has a finite family of subalgebras (C_{ij})_{j\in J_i} with the same property, then the subalgebra of A generated by all the C_{ij} is their coproduct. For P a residually small prevariety or an arbitrary quasivariety, relationships are proved between the least number of algebras needed to generate P as a prevariety or quasivariety, and behavior of the coproduct operation in P. It is shown by example that for B a subgroup of the group S = Sym(\Omega) of all permutations of an infinite set \Omega, the group S need not have a subgroup isomorphic over B to the coproduct with amalgamation S \cP_B S. But under weak additional hypotheses on B, the question remains open.


Threshold solutions for the focusing 3d cubic Schroedinger equation

Abstract: We study the focusing 3d cubic NLS equation with H^1 data at the mass-energy threshold, namely, when M[u_0]E[u_0] = M[Q]E[Q]. In earlier works of Holmer-Roudenko and Duyckaerts-Holmer-Roudenko, the behavior of solutions (i.e., scattering and blow up in finite time) is classified when M[u_0]E[u_0] < M[Q]E[Q]. In this paper, we first exhibit 3 special solutions: e^{it}Q and Q^+, Q^-; here Q is the ground state, and Q^+, Q^- exponentially approach the ground state solution in the positive time direction, meanwhile Q^+ having finite time blow up and Q^- scattering in the negative time direction. Secondly, we classify solutions at this threshold and obtain that up to \dot{H}^{1/2} symmetries, they behave exactly as the above three special solutions, or scatter and blow up in both time directions as the solutions below the mass-energy threshold. These results are obtained by studying the spectral properties of the linearized Schroedinger operator in this mass-supercritical case, establishing relevant modulational stability and careful analysis of the exponentially decaying solutions to the linearized equation.


Classification of compact ancient solutions to the curve shortening flow

Abstract: We consider an embedded convex ancient solution $\Gamma_t$ to the curve shortening flow in $\mathbb{R}^2$. We prove that there are only two possibilities: the family $\Gamma_t$ is either the family of contracting circles, which is a type I ancient solution, or the family of evolving Angenent ovals, which correspond to a type II ancient solution to the curve shortening flow. We also give a necessary and sufficient curvature condition for an embedded, closed ancient solution to the curve shortening flow to be convex.


The harmonic mean curvature flow of nonconvex surfaces in $\mathbb{R}^3$

Abstract: We consider a compact, star-shaped, mean convex hypersurface $\Sigma^2\subset \mathbb{R}^3$. We prove that in some cases the flow exists until it shrinks to a point in a spherical manner, which is very typical for convex surfaces as well (see \cite{An1}). We also prove that in the case we have a surface of revolution which is star-shaped and mean convex, a smooth solution always exists up to some finite time $T < \infty$ at which the flow shrinks to a point asymptotically spherically.


Gregory Trees, The Continuum, And Martin's Axiom

Abstract: We continue the investigation of Gregory trees and the Cantor Tree Property carried out by Hart and Kunen. We produce models of MA with the Continuum arbitrarily large in which there are Gregory trees, and in which there are no Gregory trees.


Permutation equivalent maximal irreducible Goppa codes

Abstract: We consider the problem of finding the number of permutation non-equivalent classical irreducible maximal Goppa codes having fixed parameters q, n and r from a group theory point of view.


Formes automorphes et theoremes de Riemann-Roch arithmetiques

Abstract: Nous construisons trois familles de formes automorphes au moyen du theoreme de Riemann-Roch arithmetique et de la formule de Lefschetz arithmetique. Deux de ces familles ont deja ete construites par Yoshikawa et notre construction met en lumiere leur origine arithmetique. ----- We construct three families of automorphic forms following the arithmetic Riemann-Roch theorem and the arithmetic Lefschetz formula. Two of these families were already constructed by Yoshikawa and our construction illuminates their arithmetic origin.


Correction to "Simplicial monoids and Segal categories"

Abstract: In this note we make a minor correction to the paper ``Simplicial monoids and Segal categories."


On 2-partitionable clutters and the MFMC property

Abstract: We introduce 2-partitionable clutters as the simplest case of the class of $k$-partitionable clutters and study some of their combinatorial properties. In particular, we study properties of the rank of the incidence matrix of these clutters and properties of their minors. A well known conjecture of Conforti and Cornu\'ejols \cite{ConfortiCornuejols,cornu-book} states: That all the clutters with the packing property have the max-flow min-cut property, i.e. are mengerian. Among the general classes of clutters known to verify the conjecture are: balanced clutters (Fulkerson, Hoffman and Oppenheim \cite{FulkersonHoffmanOppenheim}), binary clutters (Seymour \cite{Seymour}) and dyadic clutters (Cornu\'ejols, Guenin and Margot \cite{CornuejolsGueninMargot}). We find a new infinite family of 2-partitionable clutters, that verifies the conjecture. On the other hand we are interested in studying the normality of the Rees algebra associated to a clutter and possible relations with the Conforti and Cornu\'ejols conjecture. In fact this conjecture is equivalent to an algebraic statement about the normality of the Rees algebra \cite{rocky}.


Some reducible Specht modules for Iwahori--Hecke algebras of type $A$ with $q=-1$

Abstract: The reducibility of the Specht modules for the Iwahori--Hecke algebras in type $A$ is still open in the case where the defining parameter $q$ equals -1. We prove the reducibility of a large class of Specht modules for these algebras.


A non-crossing standard monomial theory

Abstract: The second author has introduced non-crossing tableaux, objects whose non-nesting analogues are semi-standard Young tableaux. We relate non-crossing tableaux to Gelfand-Tsetlin patterns and develop the non-crossing analogue of standard monomial theory. Leclerc and Zelevinsky's weakly separated sets are special cases of non-crossing tableaux, and we suggest that non-crossing tableaux may help illuminate the theory of weakly separated sets.


Analytic families of holomorphic iterated function systems

Abstract: This paper deals with analytic families of holomorphic iterated function systems. Using real analyticity of the pressure function (which we prove), we establish a classification theorem for analytic families of holomorphic iterated function systems which depend continuously on a parameter when the space of holomorphic iterated function systems is endowed with the "$\lambda$-topology". This classification theorem allows us to generalize some geometric results from the paper ("Lambda-Topology vs. Pointwise Topology", to appear in Ergod. Th. & Dynam. Sys.) of the authors, and gives us a better and clearer understanding of the global structure of the space of conformal IFSs.


Well-posedness for compressible Euler equations with physical vacuum singularity

Abstract: An important problem in the theory of compressible gas flows is to understand the singular behavior of vacuum states. The main difficulty lies in the fact that the system becomes degenerate at the vacuum boundary, where the characteristics coincide and have unbounded derivative. In this paper, we overcome this difficulty by presenting a new formulation and new energy spaces. We establish the local in time well-posedness of one-dimensional compressible Euler equations for isentropic flows with the physical vacuum singularity in some spaces adapted to the singularity.


Energy-efficient motion camouflage in three dimensions

Abstract: Recent observations suggest that one insect may camouflage its own motion whilst tracking another. Here we present a strategy by which one agent can efficiently track and intercept another agent, whilst camouflaging its own motion and minimizing its energy consumption


Planar Algebra of the Subgroup-Subfactor

Abstract: We give an identification between the planar algebra of the subgroup-subfactor $R \rtimes H \subset R \rtimes G$ and the $G$-invariant planar subalgebra of the planar algebra of the bipartite graph $\star_n$, where $n = [G : H]$. The crucial step in this identification is an exhibition of a model for the basic construction tower, and thereafter of the standard invariant, of $R \rtimes H \subset R \rtimes G$ in terms of operator matrices. We also obtain an identification between the planar algebra of the fixed algebra subfactor $R^G \subset R^H$ and the $G$-invariant planar subalgebra of the planar algebra of the `flip' of $\star_n $.


On isomorphism classes and invariants of low dimensional complex filiform Leibniz algebras (part 2)

Abstract: The paper is an implementation in low dimensional cases of the classification method presented before by Rakhimov and Bekbaev. We give a complete classification of a subclass of complex filiform Leibniz algebras obtained from the naturally graded non-Lie filiform Leibniz algebras. A hypothetic formula for the adapted number of isomorphism classes is given.


Coefficients of squares of Newman polynomials

Abstract: We show that there are polynomials $p_N$ of arbitrarily large degree $N$, with coefficients equal to 0 or 1 (Newman polynomials), such that $$ \liminf_{N \to \infty} N \Linf{p_N^2} \bigl / p_N^2(1) < 1, $$ where $\Linf{q}$ denotes the maximum coefficient of the polynomial $q$ and which, at the same time, are sparse: $p_N(1)/N \to 0$. This disproves a conjecture of Yu \cite{yu}. We build on some previous results of Berenhaut and Saidak \cite{berenhaut-saidak} and Dubickas \cite{dubickas} whose examples lacked the sparsity. This sparsity we create from these examples by randomization.


Cohomology algebra of the orbit space of free circle group actions on lens spaces

Abstract: Suppose that G=S^1 acts freely on a finitistic space X whose mod p cohomology ring isomorphic to that of a lens space L^{2m-1}(p;q_1,...,q_m). In this paper, we determine the mod p cohomology ring of the orbit space X/G. If the characteristic class \alpha\belongs H^2(X/G;Z_p) of the S^1-bundle S--> X--> X/G is nonzero, then mod p ndex of the action is deined to be the largest integer n such that \alpha^n is nonzero. We also show that the mod p index of a free action of S^1 on a lens space L^(2m-1)(p;q_1,...,q_m) is p-1, provided that \alpha is nonzero.


Quotients of Banach spaces with the Daugavet property

Abstract: We consider a general concept of Daugavet property with respect to a norming subspace. This concept covers both the usual Daugavet property and its weak$^*$ analogue. We introduce and study analogues for narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of $L_1[0,1]$ over an $\ell_1$-subspace can fail the Daugavet property. The latter answers a question posed to us by A. Pelczynski in the negative.


An algorithm for weighted fractional matroid matching

Abstract: Let M be a matroid on ground set E. A subset l of E is called a `line' when its rank equals 1 or 2. Given a set L of lines, a `fractional matching' in (M,L) is a nonnegative vector x indexed by the lines in L, that satisfies a system of linear constraints, one for each flat of M. Fractional matchings were introduced by Vande Vate, who showed that the set of fractional matchings is a half-integer relaxation of the matroid matching polytope. It was shown by Chang et al. that a maximum size fractional matching can be found in polynomial time. In this paper we give a polynomial time algorithm to find for any given weights on the lines in L, a maximum weight fractional matching.


Distal actions and shifted convolution property

Abstract: A locally compact group $G$ is said to have shifted convolution property (abbr. as SCP) if for every regular Borel probability measure $\mu$ on $G$, either $\sup_{x\in G} \mu ^n (Cx) \ra 0$ for all compact subsets $C$ of $G$, or there exist $x\in G$ and a compact subgroup $K$ normalised by $x$ such that $\mu^nx^{-n} \ra \omega_K$, the Haar measure on $K$. We first consider distality of factor actions of distal actions. It is shown that this holds in particular for factors under compact groups invariant under the action and for factors under the connected component of identity. We then characterize groups having SCP in terms of a readily verifiable condition on the conjugation action (point-wise distality). This has some interesting corollaries to distality of certain actions and Choquet Deny measures which actually motivated SCP and point-wise distal groups. We also relate distality of actions on groups to that of the extensions on the space of probability measures.


Alexandrov curvature of Kaehler curves

Abstract: We study the intrinsic geometry of a one-dimensional complex space provided with a Kaehler metric in the sense of Grauert. We show that if K is an upper bound for the Gaussian curvature on the regular locus, then the intrinsic metric has curvature at most K in the sense of Alexandrov.


Groupoid representations and modules over the convolution algebras

Abstract: The classical Serre-Swan's theorem defines a bijective correspondence between vector bundles and finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an etale Lie groupoid and the category of modules over its convolution algebra that are of finite type and of constant rank. Both of these constructions are functorially defined on the Morita bicategory of etale Lie groupoids and the given correspondence represents a natural equivalence between them.


Index and nullity of the Gauss map of the Costa-Hoffman-Meeks surfaces

Abstract: The aim of this work is to extend the results of S. Nayatani about the index and the nullity of the Gauss map of the Costa-Hoffman-Meeks surfaces for values of the genus bigger than 37. That allows us to state that these minimal surfaces are non degenerate for all the values of the genus in the sense of the definition of J. Perez and A. Ros.


Stochastic equations with delay: optimal control via BSDEs and regular solutions of Hamilton-Jacobi-Bellman equations

Abstract: We consider an Ito stochastic differential equation with delay, driven by brownian motion, whose solution, by an appropriate reformulation, defines a Markov process $X$ with values in a space of continuous functions $\mathbf C$, with generator $\mathcal L$. We then consider a backward stochastic differential equation depending on $X$, with unknown processes $(Y,Z)$, and we study properties of the resulting system, in particular we identify the process $Z$ as a deterministic functional of $X$. We next prove that the forward-backward system provides a suitable solution to a class of parabolic partial differential equations on the space $\mathbf C$ driven by $\mathcal L$, and we apply this result to prove a characterization of the fair price and the hedging strategy for a financial market with memory effects. We also include applications to optimal stochastic control of differential equation with delay: in particular we characterize optimal controls as feedback laws in terms the process $X$.


Darboux transforms and spectral curves of Hamiltonian stationary Lagrangian tori

Abstract: The multiplier spectral curve of a conformal torus in the 4-sphere is essentially, see arXiv:0712.2311, given by all Darboux transforms of the conformal torus. In the particular case when the conformal immersion is a Hamiltonian stationary torus in Euclidean 4-space, the left normal of the immersion is harmonic, hence we can associate a second Riemann surface: the eigenline spectral curve of the left normal, as defined by Hitchin. We show that the multiplier spectral curve of a Hamiltonian stationary torus and the eigenline spectral curve of its left normal are biholomorphic Riemann surfaces of genus zero. Moreover, we prove that all Darboux transforms, which arise from generic points on the spectral curve, are Hamiltonian stationary whereas we also provide examples of Darboux transforms which are not even Lagrangian.


Base-point-free pencils on triple covers of smooth curves

Abstract: Let $X$ be a smooth algebraic curve. Suppose that there exists a triple covering $f : X \to Y$ where $Y$ is a smooth algebraic curve. In this paper, we investigate the existence of morphisms from $X$ to the projective line $\mathbf{P}^1$ which do not factor through the covering $f$. For this purpose, we generalize the classical results of Maroni concerning base-point-free pencils on trigonal curves to the case of triple covers of arbitrary smooth irrational curves.


Spiraling spectra of geodesic lines in negatively curved manifolds

Abstract: Given a negatively curved geodesic metric space M, we study the asymptotic penetration behaviour of geodesic lines of M in small neighbourhoods of closed geodesics and of other compact convex subsets of M. We define a spiraling spectrum which gives precise information on the asymptotic spiraling lengths of geodesic lines around these objects. We prove analogs of the theorems of Dirichlet, Hall and Cusick in this context. As a consequence, we obtain Diophantine approximation results of real numbers, complex numbers, or elements of the Heisenberg group by irrational quadratic ones.


On pro-p fundamental groups of marked arithmetic curves

Abstract: Let k be a global field, p an odd prime number different from char(k) and S, T disjoint, finite sets of primes of k. Let G_S^T(k)(p)=Gal(k_S^T(p)|k) be the Galois group of the maximal p-extension of k which is unramified outside S and completely split at T. We prove the existence of a finite set of primes S_0, which can be chosen disjoint from any given set M of Dirichlet density zero, such that the cohomology of G_{S\cup S_0}^T(k)(p) coincides with the etale cohomology of the associated marked arithmetic curve. In particular, cd G_{S\cup S_0}^T(k)(p)=2. Furthermore, we can choose S_0 in such a way that k_{S\cup S_0}^T(p) realizes the maximal p-extension k_\p(p) of the local field k_\p for all \p\in S\cup S_0, the cup-product H^1(G_{S\cup S_0}^T(k)(p),\F_p) \otimes H^1(G_{S\cup S_0}^T(k)(p),\F_p) --> H^2(G_{S\cup S_0}^T(k)(p),\F_p) is surjective and the decomposition groups of the primes in S establish a free product inside G_{S\cup S_0}^T(k)(p). This generalizes previous work of the author where similar results were shown in the case T=\emptyset under the restrictive assumption p\nmid Cl(k) and \zeta_p\notin k.


Making research on symmetric functions using MuPAD-Combinat

Abstract: We present an overview of the implementation of symmetric functions in MuPAD-Combinat. We also explain how to interface C++ programs with MuPAD in order to make efficient research during concrete work sessions.


Mock Jacobi forms in basic hypergeometric series

Abstract: We show that some $q$-series such as universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are restricted to torsion points and multiplied by suitable powers of $q$. And we prove that certain linear sums of $q$-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or completion by mock Jacobi forms. As an application, we obtain a relation between the rank and crank of a partition.


Equality of multiplicity free skew characters

Abstract: In this paper we show that two skew diagrams lambda/mu and alpha/beta can represent the same multiplicity free skew character [lambda/mu]=[alpha/beta] only in the the trivial cases when lambda/mu and alpha/beta are the same up to translation or rotation or if lambda=alpha is a staircase partition lambda=(l,l-1,...,2,1) and lambda/mu and alpha/beta are conjugate of each other.


A family of transversely nonsimple knots

Abstract: We apply knot Floer homology to exhibit an infinite family of transversely nonsimple prime knots starting with $10_{132}$. We also discuss the combinatorial relationship between grid diagrams, braids, and Legendrian and transverse knots in standard contact $\mathbb{R}^3$.


Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations

Abstract: In this article, we provide existence results for a general class of nonlocal and nonlinear second-order parabolic equations. The main motivation comes from front propagation theory in the cases when the normal velocity depends on the moving front in a nonlocal way. Among applications, we present level-set equations appearing in dislocations' theory and in the study of Fitzhugh-Nagumo systems.


Extrapolation of Threshold-Limited Null Measurement Frequencies

Abstract: The total measurable level of a pathogen is due to many sources, which produce a variety of pulses, overlapping in time, that rise suddenly and then decay. What is measured is the level of the total contribution of the sources at a given time. But since we are only capable of measuring the total level above some threshold $x_0$, we would like to predict the distribution below this level. Our principal model assumption is that of the asymptotic exponential decay of all pulses. We show that this implies a power law distribution for the frequencies of low amplitude observations. As a consequence, there is a simple extrapolation procedure for carrying the data to the region below $x_0$.


The Stochastic Heat Equation Driven by a Gaussian Noise: germ Markov Property

Abstract: Let $u=\{u(t,x);t \in [0,T], x \in {\mathbb{R}}^{d}\}$ be the process solution of the stochastic heat equation $u_{t}=\Delta u+ \dot F, u(0,\cdot)=0$ driven by a Gaussian noise $\dot F$, which is white in time and has spatial covariance induced by the kernel $f$. In this paper we prove that the process $u$ is locally germ Markov, if $f$ is the Bessel kernel of order $\alpha=2k,k \in \bN_{+}$, or $f$ is the Riesz kernel of order $\alpha=4k,k \in \bN_{+}$.


Minimal surfaces in circle bundles over Riemann surfaces

Abstract: For a compact 3-manifold $M$ which is a circle bundle over a compact Riemann surface $\Sigma$ with even Euler number $e(M)$, and with a Riemannian metric compatible with the bundle projection, there exists a compact minimal surface $S$ in $M$. $S$ is embedded and is a section of the restriction of the bundle to the complement of a finite number of points in $\Sigma$.


The mixing time evolution of Glauber dynamics for the mean-field Ising model

Abstract: We consider Glauber dynamics for the Ising model on the complete graph on $n$ vertices, known as the Curie-Weiss model. It is well-known that the mixing-time in the high temperature regime ($\beta < 1$) has order $n\log n$, whereas the mixing-time in the case $\beta > 1$ is exponential in $n$. Recently, Levin, Luczak and Peres proved that for any fixed $\beta < 1$ there is cutoff at time $[2(1-\beta)]^{-1} n\log n$ with a window of order $n$, whereas the mixing-time at the critical temperature $\beta=1$ is $\Theta(n^{3/2})$. It is natural to ask how the mixing-time transitions from $\Theta(n\log n)$ to $\Theta(n^{3/2})$ and finally to $\exp(\Theta(n))$. That is, how does the mixing-time behave when $\beta=\beta(n)$ is allowed to tend to 1 as $n\to\infty$. In this work, we obtain a complete characterization of the mixing-time of the dynamics as a function of the temperature, as it approaches its critical point $\beta_c=1$. In particular, we find a scaling window of order $1/\sqrt{n}$ around the critical temperature. In the high temperature regime, $\beta = 1 - \delta$ for some $0 < \delta < 1$ so that $\delta^2 n \to\infty$ with $n$, the mixing-time has order $(n/\delta)\log(\delta^2 n)$, and exhibits cutoff with constant 1/2 and window size $n/\delta$. In the critical window, $\beta = 1\pm \delta$ where $\delta^2 n$ is O(1), there is no cutoff, and the mixing-time has order $n^{3/2}$. At low temperature, $\beta = 1 + \delta$ for $\delta > 0$ with $\delta^2 n \to\infty$ and $\delta=o(1)$, there is no cutoff, and the mixing time has order $(n/\delta)\exp(({3/4}+o(1))\delta^2 n)$.


Bitwist 3-manifolds

Abstract: Our earlier twisted-face-pairing construction showed how to modify an arbitrary orientation-reversing face-pairing on a faceted 3-ball in a mechanical way so that the quotient is automatically a closed, orientable 3-manifold. The modifications were, in fact, parametrized by a finite set of positive integers, arbitrarily chosen, one integer for each edge class of the original face-pairing. This allowed us to find very simple face-pairing descriptions of many, though presumably not all, 3-manifolds. Here we show how to modify the construction to allow negative parameters, as well as positive parameters, in the twisted-face-pairing construction. We call the modified construction the bitwist construction. We prove that all closed connected orientable 3-manifolds are bitwist manifolds. As with the twist construction, we analyze and describe the Heegaard splitting naturally associated with a bitwist description of a manifold.


The Kohn Algorithm on Denjoy-Carleman Classes

Abstract: The equivalence of the Kohn finite ideal type and the D'Angelo finite type with the subellipticity of the $\bar\partial$-Neumann problem is extended to pseudoconvex domains in $C^n$ whose defining function is in a Denjoy-Carleman quasianalytic class closed under differentiation. The proof involves algebraic geometry over a ring of germs of Denjoy-Carleman quasianalytic functions that is not known to be Noetherian and that is intermediate between the ring of germs of real-analytic functions and the ring of germs of smooth functions. It is also shown that this type of ring of germs of Denjoy-Carleman functions satisfies the $\sqrt{acc}$ property, one of the strongest properties a non-Noetherian ring could possess.


Analogue of Sylvester-Cayley formula for invariants of ternary form

Abstract: The number $\nu_d(n)$ of linearly independed homogeneous invariants of degree $n$ for the ternary form of degree $d$ is calculated.


A characterisation of the Hoffman-Wohlgemuth surfaces in terms of their symmetries

Abstract: For an embedded singly periodic minimal surface M with genus bigger than or equal to 4 and annular ends, some weak symmetry hypotheses imply its congruence with one of the Hoffman-Wohlgemuth examples. We give a very geometrical proof of this fact, along which they come out many valuable clues for the understanding of these surfaces.


On divergence form SPDEs with VMO coefficients

Abstract: We present several results on solvability in Sobolev spaces $W^{1}_{p}$ of SPDEs in divergence form in the whole space.


On Picture (2+1)-TQFTs

Abstract: The goal of the paper is an exposition of the simplest $(2+1)$-TQFTs in a sense following a pictorial approach. In the end, we fell short on details in the later sections where new results are stated and proofs are outlined. Comments are welcome and should be sent to the 4th author.


A conjecture on the forms of the roots of equations

Abstract: E30 in the Enestrom index. Translated from the Latin original "De formis radicum aequationum cuiusque ordinis coniectatio" (1733). For an equation of degree n, Euler wants to define a "resolvent equation" of degree n-1 whose roots are related to the roots of the original equation. Thus by solving the resolvent we can solve the original equation. In sections 2 to 7 he works this out for quadratic, cubic and biquadratic equations. Apparently he gives a new method for solving the quartic in section 5. Then in section 8 Euler says that he wants to try the same approach for solving the quintic equation and general nth degree equations. In the rest of the paper Euler tries to figure out in what cases resolvents will work. Two references I found useful were Chapter 14, p.p. 106-113 of C. Edward Sandifer, "The Early Mathematics of Leonhard Euler", published 2007 by The Mathematical Association of America and Olaf Neumann, "Cyclotomy: from Euler through Vandermonde to Gauss", p.p. 323-362 in the collection "Leonhard Euler: Life, Work and Legacy" edited by Bradley and Sandifer, 2007. Stacy Langton has given a lot of details about Euler's work on the theory of equations, and also some advice on the translation; of course any mistakes are my own. If Langton ends up writing anything about Euler' and the theory of equations I would highly recommend reading it.


Fibers of Generic Projections

Abstract: Let X be a smooth projective variety of dimension n in P^r. We study the fibers of a general linear projection pi: X --> P^{n+c}, with c > 0. When n is small it is classical that the degree of any fiber is bounded by n/c+1, but this fails for n >> 0. We describe a new invariant of the fiber that agrees with the degree in many cases and is always bounded by n/c+1. This implies, for example, that if we write a fiber as the disjoint union of schemes Y' and Y'' such that Y' is the union of the locally complete intersection components of Y, then deg Y'+deg Y''_red <= n/c+1 and this formula can be strengthened a little further. Our method also gives a sharp bound on the subvariety of P^r swept out by the l-secant lines of X for any positive integer l, and we discuss a corresponding bound for highly secant linear spaces of higher dimension. These results extend Ziv Ran's "Dimension+2 Secant Lemma".


Algebraic density property of homogeneous spaces

Abstract: Let $X$ be an affine algebraic variety with a transitive action of the algebraic automorphism group. Suppose that $X$ is equipped with several non-degenerate fixed point free $SL_2$-actions satisfying some mild additional assumption. Then we show that the Lie algebra generated by completely integrable algebraic vector fields on $X$ coincides with the set of all algebraic vector fields. In particular, we show that apart from a few exceptions this fact is true for any homogeneous space of form $G/R$ where $G$ is a linear algebraic group and $R$ is its proper reductive subgroup.


Semitoric integrable systems on symplectic 4-manifolds

Abstract: Let M be a symplectic 4-manifold. A semitoric integrable system on M is a pair of real-valued smooth functions J, H on M for which J generates a Hamiltonian S^1-action and the Poisson brackets {J,H} vanish. We shall introduce new global symplectic invariants for these systems; some of these invariants encode topological or geometric aspects, while others encode analytical information about the singularities and how they stand with respect to the system. Our goal is to prove that a semitoric system is completely determined by the invariants we introduce.


A Dedekind Finite Borel Set

Abstract: In this paper we prove three theorems about the theory of Borel sets in models of ZF without any form of the axiom of choice. We prove that if B is a G-delta-sigma set, then either B is countable or B contains a perfect subset. Second, we prove that if the real line is the countable union of countable sets, then there exists an F-sigma-delta set which is uncountable but contains no perfect subset. Finally, we construct a model of ZF in which we have an infinite Dedekind finite set of reals which is F-sigma-delta.


On divergence form SPDEs with VMO coefficients in a half space

Abstract: We extend several known results on solvability in the Sobolev spaces $W^{1}_{p}$, $p\in[2,\infty)$, of SPDEs in divergence form in $\bR^{d}_{+}$ to equations having coefficients which are discontinuous in the space variable.


Curvature flows and CMC hypersurfaces

Abstract: We give an overview of the existence and regularity results for curvature flows and how these flows can be used to solve some problems in geometry and physics.


On the question of ergodicity for minimal group actions on the circle

Abstract: This work is devoted to the study of minimal, smooth actions of finitely generated groups on the circle. We provide a sufficient condition for such an action to be ergodic (with respect to the Lebesgue measure), and we illustrate this condition by studying two relevant examples. Under an analogous hypothesis, we also deal with the problem of the zero Lebesgue measure for exceptional minimal sets. This hypothesis leads to many other interesting conclusions, mainly concerning the stationary and conformal measures. Moreover, several questions are left open. The methods work as well for codimension-one foliations, though the results for this case are not explicitly stated.


The moduli space of flat SU(2)-bundles over a nonorientable surface

Abstract: We study the topology of the moduli space of flat SU(2)-bundles over a nonorientable surface X. This moduli space may be identified with the space of homomorphisms Hom(\pi_1(X),SU(2)) modulo conjugation by SU(2). In particular, we compute the (rational) equivariant cohomology ring of Hom(\pi_1(X),SU(2)) and use this to compute the ordinary cohomology groups of the quotient Hom(\pi_1(X),SU(2))/SU(2). A key property is that the conjugation action is equivariantly formal.


Projections of Singular Vectors of Verma Modules over Rank 2 Kac-Moody Lie Algebras

Abstract: We prove an explicit formula for a projection of singular vectors in the Verma module over a rank 2 Kac-Moody Lie algebra onto the universal enveloping algebra of the Heisenberg Lie algebra and of $sl_{2}$ (Theorem 3). The formula is derived from a more general but less explicit formula due to Feigin, Fuchs and Malikov [Funct. Anal. Appl. 20 (1986), no. 2, 103-113]. In the simpler case of $A_{1}^{1}$ the formula was obtained in [Fuchs D., Funct. Anal. Appl. 23 (1989), no. 2, 154-156].


Simple SL(n)-Modules with Normal Closures of Maximal Torus Orbits

Abstract: Let $T$ be the subgroup of diagonal matrices in the group SL(n). The aim of this paper is to find all finite-dimensional simple rational SL(n)-modules $V$ with the following property: for each point $v\in V$ the closure $\bar{Tv}$ of its $T$-orbit is a normal affine variety. Moreover, for any SL(n)-module without this property a $T$-orbit with non-normal closure is constructed. The proof is purely combinatorial: it deals with the set of weights of simple SL(n)-modules. The saturation property is checked for each subset in the set of weights.


Words Maps and Spectra of Random Graph Lifts

Abstract: We begin with a new analysis of formal words. Let w be a formal word in letters g_1,...,g_k. The word map associated with w maps the permutations s_1,...,s_k in S_n to the permutation obtained by replacing for each i, every occurrence of g_i in w by s_i. We investigate the random variable X_w^n that counts the fixed points in this permutation when the s_i are selected uniformly at random. A major ingredient of our work is a new categorization of words which considerably extends the dichotomy of primitive vs. imprimitive words. We establish some results and make a few conjectures about the relation between the expectation E(X_w^n) and this new categorization. This analysis contributes deeply to our study of the spectra of random lifts of graphs. Let G be a connected graph, and let the infinite tree T be its universal cover space. If L and R are the spectral radii of G and T respectively, then, as shown by J. Friedman, for almost every n-lift H of G, all "new" eigenvalues of H are < O(L^(1/2)R^(1/2)). We improve this upper bound to O(L^(1/3)R^(2/3)), and our aforementioned conjectures suggest a possible approach to proving an upper bound of O(R). This is a generalization of the problem of bounding the second eigenvalue in a random 2d-regular graph. As an aside, we obtain a new conceptual and relatively simple proof of a theorem of A. Nica, which determines, for every fixed w, the limit distribution (as n \to \infty) of X_w^n. A surprising aspect of this theorem is that the answer depends only on the largest integer d so that w=u^d for some word u.


Examples of limits of Frobenius (type) structures: the singularity case

Abstract: We give examples of families of Frobenius type structures on the punctured plane and we study their limits at the boundary. We then discuss the existence of a limit Frobenius manifold. We also give an example of a logarithmic Frobenius manifold.


Zeta functions, heat kernels and spectral asymptotics on degenerating families of discrete tori

Abstract: By a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with generating set given by choosing a generator for each cyclic factor. In this article we study the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First we show that the sequence of heat kernels corresponding to the degenerating family converges, after re-scaling, to the heat kernel on an associated real torus. We then establish an asymptotic expansion, in the degeneration parameter, of the determinant of the combinatorial Laplacian. The zeta-regularized determinant of the Laplacian of the limiting real torus appears as the constant term in this expansion. On the other hand, using a classical theorem by Kirchhoff the determinant of the combinatorial Laplacian of a finite graph divided by the number of vertices equals the number of spanning trees, called the complexity, of the graph. As a result, we establish a precise connection between the complexity of the Cayley graphs of finite abelian groups and heights of real tori. It is also known that spectral determinants on discrete tori can be expressed using trigonometric functions and that spectral determinants on real tori can be expressed using modular forms on general linear groups. Another interpretation of our analysis is thus to establish a link between limiting values of certain products of trigonometric functions and modular forms. The heat kernel analysis which we employ uses a careful study of I-Bessel functions. Our methods extend to prove the asymptotic behavior of other spectral invariants through degeneration, such as special values of spectral zeta functions and Epstein-Hurwitz type zeta functions.


Shadows and intersections: stability and new proofs

Abstract: We give a short new proof of a version of the Kruskal-Katona theorem due to Lov\'asz. Our method can be extended to a stability result, describing the approximate structure of configurations that are close to being extremal, which answers a question of Mubayi. This in turn leads to another combinatorial proof of a stability theorem for intersecting families, which was originally obtained by Friedgut using spectral techniques and then sharpened by Keevash and Mubayi by means of a purely combinatorial result of Frankl. We also give an algebraic perspective on these problems, giving yet another proof of intersection stability that relies on expansion of a certain Cayley graph of the symmetric group, and an algebraic generalisation of Lov\'asz's theorem that answers a question of Frankl and Tokushige.