Volume preserving subgroups of A and K and singularities in unimodular geometry

Abstract: For a germ of a smooth map f and a subgroup G_V of any of the Mather groups G for which the source or target diffeomorphisms preserve some given volume form V in the source or in the target we study the G_V-moduli space of f that parameterizes the G_V-orbits inside the G-orbit of f. We find, for example, that this moduli space vanishes for A-equivalence with volume-preserving target diffeomorphisms and A-stable maps f and for K-equivalence with volume-preserving source diffeomorphisms and K-simple maps f. On the other hand, there are A-stable maps f with infinite-dimensional moduli space for A-equivalence with volume-preserving source diffeomorphisms.


On Pointed Hopf Algebras with Weyl Groups of exceptional type

Abstract: All -1-type pointed Hopf algebras and central quantum linear spaces with Weyl groups of exceptional type are found. It is proved that every non -1-type pointed Hopf algebra with real $G(H)$ is infinite dimensional and every central quantum linear space over finite group is finite dimensional. It is proved that except a few cases Nichols algebras of reducible Yetter-Drinfeld modules over Weyl groups of exceptional type are infinite dimensional.


Construction of Local Conservation Laws by Generalized Isometric Embeddings of Vector Bundles

Abstract: This article uses Cartan-K\"ahler theory to construct local conservation laws from covariantly closed vector valued differential forms, objects that can be given, for example, by harmonic maps between two Riemannian manifolds. We apply the article's main result to construct conservation laws for covariant divergence free energy-momentum tensors. We also generalize the local isometric embedding of surfaces in the analytic case by applying the main result to vector bundles of rank two over any surface.


Quasiflats in CAT(0) complexes

Abstract: We show that if X is a piecewise Euclidean 2-complex with a cocompact isometry group, then every 2-quasiflat in X is at finite Hausdorff distance from a subset which is locally flat outside a compact set, and asymptotically conical.


Low and high frequency approximations to eigenvibrations of string with double contrasts

Abstract: We study eigenvibrations for inhomogeneous string consisting of two parts with strongly contrasting stiffness and mass density. In this work we treat a critical case for the high frequency approximations, namely the case when the order of mass density inhomogeneity is the same as the order of stiffness inhomogeneity, with heavier part being softer. The limit problem for high frequency approximations depends nonlinearly on the spectral parameter. The quantization of the spectral semiaxies is applied in order to get a close approximations of eigenvalues as well as eigenfunctions for the prime problem under perturbation.


Effectively closed sets of measures and randomness

Abstract: We show that if a real $x$ is strongly Hausdorff $h$-random, where $h$ is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure $\mu$ such that the $\mu$-measure of the basic open cylinders shrinks according to $h$. The proof uses a new method to construct measures, based on effective (partial) continuous transformations and a basis theorem for $\Pi^0_1$-classes applied to closed sets of probability measures. We use the main result to give a new proof of Frostman's Lemma, to derive a collapse of randomness notions for Hausdorff measures, and to provide a characterization of effective Hausdorff dimension similar to Frostman's Theorem.


The Non-triviality of the Grope Filtrations of The Knot and Link Concordance Groups

Abstract: We consider the Grope filtration of the classical knot concordance group that was introduced in a paper of Cochran, Orr and Teichner. Our main result is that successive quotients at each stage in this filtration have infinite rank. We also establish the analogous result for the Grope filtration of the concordance group of string links consisting of more than one component.


A homotopy-theoretic universal property of Leinster's operad for weak omega-categories

Abstract: We explain how any cofibrantly generated weak factorisation system on a category may be equipped with a universally and canonically determined choice of cofibrant replacement. We then apply this to the theory of weak omega-categories, showing that the universal and canonical cofibrant replacement of the operad for strict omega-categories is precisely Leinster's operad for weak omega-categories.


Volume estimates for equiangular hyperbolic Coxeter polyhedra

Abstract: An equiangular hyperbolic Coxeter polyhedron is a hyperbolic polyhedron where all dihedral angles are equal to \pi/n for some fixed integer n at least 2. It is a consequence of Andreev's theorem that either n=3 and the polyhedron has all ideal vertices or that n=2. Volume estimates are given for all equiangular hyperbolic Coxeter polyhedra.


The core of ideals in arbitrary characteristic

Abstract: We prove formulas for the core of ideals that apply in arbitrary characteristic.


On the Loss of Compactness in the Vectorial Heteroclinic Connection Problem

Abstract: We give an alternative proof of the theorem of Alikakos-Fusco [AF] concerning existence of heteroclinic solutions to a Hamiltonian ODE system on the whole real line which arises in the theory of phase transitions. Our method is variational but differs from the original artificial constraint method of [AF] and establishes existence by analysing the loss of compactness in minimising sequences of the action in the appropriate functional space. Our assumptions are slightly different from those considered previously and also imply a priori estimates for the solution.


Projective background of the infinitesimal rigidity of frameworks

Abstract: We present proofs of two classical theorems. The first one, due to Darboux and Sauer, states that infinitesimal rigidity is a projective invariant; the other one establishes relations (infinitesimal Pogorelov maps) between the infinitesimal motions of a Euclidean framework and of its hyperbolic and spherical images. The arguments use the static formulation of infinitesimal rigidity. The duality between statics and kinematics is established through the principles of virtual work. A geometric approach to statics, due essentially to Grassmann, makes both theorems straightforward. Besides, it provides a simple derivation of the formulas both for the Darboux-Sauer correspondence and for the infinitesimal Pogorelov maps.


When is a Connection a Levi-Civita Connection?

Abstract: We consider the more general question as to when a connection is a metric connection. There are two aspects to this investigation: first, the determination of the integrability conditions that ensure the existence of a local parallel metric in the neighbourhood of a given point and second, the characterization of the topological obstruction to a globally defined parallel metric.


Four Properties of Reproducing Kernel Hilbert Spaces

Abstract: A reproducing kernel Hilbert space (RKHS) has four well-known easily derived properties. Since these properties are usually not emphasized as a simple means of gaining insight into RKHS structure, they are singled out and proved in this brief article.


Discrete surfaces of constant mean curvature

Abstract: We propose a unified definition for discrete analogues of constant mean curvature surfaces in spaces of constant curvature as a special case of discrete special isothermic nets. B\"acklund transformations and Lawson's correspondence are discussed. It is shown that the definition generalizes previous definitions and a construction for discrete cmc surfaces of revolution in space forms is provided.


Correlated Link Shadow Fading in Multi-hop Wireless Networks

Abstract: Accurate representation of the physical layer is required for analysis and simulation of multi-hop networking in sensor, ad hoc, and mesh networks. This paper investigates, models, and analyzes the correlations that exist in shadow fading between links in multi-hop networks. Radio links that are geographically proximate often experience similar environmental shadowing effects and thus have correlated fading. We describe a measurement procedure and campaign to measure a large number of multi-hop networks in an ensemble of environments. The measurements show statistically significant correlations among shadowing experienced on different links in the network, with correlation coefficients up to 0.33. We propose a statistical model for the shadowing correlation between link pairs which shows strong agreement with the measurements, and we compare the new model with an existing shadowing correlation model of Gudmundson (1991). Finally, we analyze multi-hop paths in three and four node networks using both correlated and independent shadowing models and show that independent shadowing models can underestimate the probability of route failure by a factor of two or greater.


Subgroup theorem for valuated groups and the CSA property

Abstract: A valuated group with normal forms is a group with an integer-valued length function satisfying some Lyndon's axioms and an additional axiom considered by Hurley. We prove a subgroup theorem for valuated groups with normal forms analogous to Grushko-Neumann's theorem. We study also the CSA property in such groups.


The Taylor expansion of Ruelle L-function at the origin and the Borel regulator

Abstract: We will prove that Ruelle L-function for a cuspidal local system on an odd dimensional hyperbolic manifold with finite volume satisfies a functional equation and an analog of the Riemann hypothesis. We will also compute its Laurent expansion at the origin and will prove that the second coefficient coincides with a rational multiple of the volume up to a certain contribution from cusps. Moreover if the dimension is three we will identify the leading coefficient. Both of them will be intepreted by the Borel regulator in algebraic K-theory. Also a relation with the L^2-torsion will be discussed.


Covering shadows with a smaller volume

Abstract: For each i = 1, ..., n constructions are given for convex bodies K and L in n-dimensional Euclidean space such that each rank i orthogonal projection of K can be translated inside the corresponding projection of L, even though K has strictly larger m-th intrinsic volumes (i.e. V_m(K) > V_m(L)) for all m > i. It is then shown that, for each i = 1, ..., n, there is a class of bodies C{n,i}, called i-cylinder bodies of R^n, such that, if the body L with i-dimensional covering shadows is an i-cylinder body, then K will have smaller n-volume than L. The families C{n,i} are shown to form a strictly increasing chain of subsets C{n,1} < C{n,2} < ... < C{n,n-1} < C{n,n}, where C{n,1} is precisely the collection of centrally symmetric compact convex sets in n-dimensional space, while C{n,n} is the collection of all compact convex sets in n-dimensional space. Members of each family C{n,i} are seen to play a fundamental role in relating covering conditions for projections to the theory of mixed volumes, and members of C{n,i} are shown to satisfy certain geometric inequalities. Related open questions are also posed.


An Exceptional Representation of Sp(4,F_q)

Abstract: We describe a folklore construction of an exceptional representation of Sp(4,F_q). This representation has the following remarkable combination of properties, namely it is cuspidal, degenerate and unipotent.


Structural stability of finite dispersion-relation preserving schemes

Abstract: The goal of this work is to determine classes of travelling solitary wave solutions for a differential approximation of a finite difference scheme by means of a hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurance of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such a behavior is referred here to has a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solution of the original continuous equations. This paper extends our previous work about classical schemes to dispersion-relation preserving schemes.


Convergence Rates of Nonparametric Posterior Distributions

Abstract: We study the asymptotic behavior of posterior distributions. We present general posterior convergence rate theorems, which extend several results on posterior convergence rates provided by Ghosal and Van der Vaart (2000), Shen and Wasserman (2001) and Walker, Lijor and Prunster (2007). Our main tools are the Hausdorff $\alpha$-entropy introduced by Xing and Ranneby (2008) and a new notion of prior concentration, which is a slight improvement of the usual prior concentration provided by Ghosal and Van der Vaart (2000). We apply our results to several statistical models.


Universal recursive formulae for Q-curvatures

Abstract: We formulate and discuss two conjectures concerning recursive formulae for Branson's $Q$-curvatures. The proposed formulae describe all $Q$-curvatures on manifolds of all even dimensions in terms of respective lower order $Q$-curvatures and lower order GJMS-operators. They are universal in the dimension of the underlying space. The recursive formulae are generated by an algorithm which rests on the theory of residue families. We attempt to resolve the algorithm by formulating a conjectural description of the coefficients in the recursive formulae in terms of interpolation polynomials associated to compositions of natural numbers. We prove that the conjectures cover $Q_4$ and $Q_6$ for general metrics, and $Q_8$ for conformally flat metrics. The result for $Q_8$ is proved here for the first time. Moreover, we display explicit (conjectural) formulae for $Q$-curvatures of order up to 16, and test high order cases for round spheres and Einstein metrics.


Boosting Algorithms: Regularization, Prediction and Model Fitting

Abstract: We present a statistical perspective on boosting. Special emphasis is given to estimating potentially complex parametric or nonparametric models, including generalized linear and additive models as well as regression models for survival analysis. Concepts of degrees of freedom and corresponding Akaike or Bayesian information criteria, particularly useful for regularization and variable selection in high-dimensional covariate spaces, are discussed as well. The practical aspects of boosting procedures for fitting statistical models are illustrated by means of the dedicated open-source software package mboost. This package implements functions which can be used for model fitting, prediction and variable selection. It is flexible, allowing for the implementation of new boosting algorithms optimizing user-specified loss functions.


Comment: Boosting Algorithms: Regularization, Prediction and Model Fitting

Abstract: The authors are doing the readers of Statistical Science a true service with a well-written and up-to-date overview of boosting that originated with the seminal algorithms of Freund and Schapire. Equally, we are grateful for high-level software that will permit a larger readership to experiment with, or simply apply, boosting-inspired model fitting. The authors show us a world of methodology that illustrates how a fundamental innovation can penetrate every nook and cranny of statistical thinking and practice. They introduce the reader to one particular interpretation of boosting and then give a display of its potential with extensions from classification (where it all started) to least squares, exponential family models, survival analysis, to base-learners other than trees such as smoothing splines, to degrees of freedom and regularization, and to fascinating recent work in model selection. The uninitiated reader will find that the authors did a nice job of presenting a certain coherent and useful interpretation of boosting. The other reader, though, who has watched the business of boosting for a while, may have quibbles with the authors over details of the historic record and, more importantly, over their optimism about the current state of theoretical knowledge. In fact, as much as ``the statistical view'' has proven fruitful, it has also resulted in some ideas about why boosting works that may be misconceived, and in some recommendations that may be misguided. [arXiv:0804.2752]


On Oliver's p-group conjecture

Abstract: Let S be a p-group for an odd prime p. Bob Oliver conjectures that a certain characteristic subgroup X(S) always contains the Thompson subgroup J(S). We obtain a reformulation of the conjecture as a statement about modular representations of p-groups. Using this we verify Oliver's conjecture for groups where S/X(S) has nilpotence class at most two.


A characterization of domains in $\mathbf C^2$ with noncompact automorphism group

Abstract: Let $D$ be a bounded domain in $\mathbf C^2$ with a non-compact group of holomorphic automorphisms. Model domains for $D$ are obtained under the hypothesis that at least one orbit accumulates at a boundary point near which the boundary is smooth, real analytic and of finite type.


Abstract: Comment on ``Boosting Algorithms: Regularization, Prediction and Model Fitting'' [arXiv:0804.2752]


A General Correspondence between Averages and Integrals

Abstract: Recent work has generalized the Furstenberg correspondence between sets of integers and dynamical systems to versions which involve sequences of finite graphs or sequences of $L^\infty$ functions. We give a unified version of the theorem subsuming all these generalizations.


Largest Laplacian Eigenvalue and Degree Sequences of Trees

Abstract: We investigate the structure of trees that have greatest maximum eigenvalue among all trees with a given degree sequence. We show that in such an extremal tree the degree sequence is non-increasing with respect to an ordering of the vertices that is obtained by breadth-first search. This structure is uniquely determined up to isomorphism. We also show that the maximum eigenvalue in such classes of trees is strictly monotone with respect to majorization.


Rejoinder: Boosting Algorithms: Regularization, Prediction and Model Fitting

Abstract: Rejoinder to ``Boosting Algorithms: Regularization, Prediction and Model Fitting'' [arXiv:0804.2752]


Reductions of locally conformal symplectic structures and de Rham cohomology tangent to a foliation

Abstract: We propose a produre of reduction a locally conformal symplectic structure. This procedure of reduction can be applied to wide class of submanifolds. There are no local obstructions for this procedure. But there are global obstructions. We find a necessary and sufficient condition when this reduction holds in terms of the special kind of de Rham cohomology class (tangent to the characteristic foliation) of the Lee form.


On certain nonlinear elliptic PDE and quasiconfomal mapps between Euclidean surfaces

Abstract: We mainly investigate some properties of quasiconformal mappings between smooth 2-dimensional surfaces with boundary in the Euclidean space, satisfying certain partial differential equations (inequalities) concerning Laplacian, and in particular satisfying Laplace equation and show that that these mappings are Lipschitz. Conformal parametrization of such surfaces and the method developed in our paper \cite{km} have important role in this paper.dan curves and is extended to the case of $C^{2,\alpha}$ surfaces with smooth and compact boundary.


Topological Hochschild and cyclic homology for Differential graded categories

Abstract: We define a topological Hochschild (THH) and cyclic (TC) homology theory for differential graded (dg) categories and construct several non-trivial natural transformations from algebraic K-theory to THH(-). In an intermediate step, we prove that the homotopy theory of dg categories is Quillen equivalent, through a four step zig-zag of Quillen equivalences, to the homotopy theory of Eilenberg-Mac Lane spectral categories. Finally, we show that over the rationals two dg categories are topological equivalent if and only if they are quasi-equivalent.


On three-parametric Lie groups as quasi-Kaehler manifolds with Killing Norden metric

Abstract: A 3-parametric family of 6-dimensional quasi-Kaehler manifolds with Norden metric is constructed on a Lie group. This family is characterized geometrically. The condition for such a 6-manifold to be isotropic Kaehler is given.


Existence and behavior of asymmetric traveling wave solutions to thin film equation

Abstract: We proved the existence and uniqueness of a traveling wave solution to the thin film equation with a Navier slip condition at the liquid-solid interface. We obtain explicit lower and upper bounds for the solution and an absolute error estimate of approximation of a solution to the thin films equation by the traveling-wave solution.


Quasi-Kaehler manifolds with a pair of Norden metrics

Abstract: The basic class of the non-integrable almost complex manifolds with a pair of Norden metrics are considered. The interconnections between corresponding quantities at the transformation between the two Levi-Civita connections are given. A 4-parametric family of 4-dimensional quasi-Kaehler manifolds with Norden metric is characterized with respect to the associated Levi-Civita connection.


A Lie group as a 4-dimensional quasi-Kaehler manifold with Norden metric

Abstract: A 4-parametric family of 4-dimensional quasi-Kaehler manifolds with Norden metric is constructed on a Lie group. This family is characterized geometrically. The condition for such a 4-manifold to be isotropic Kaehler is given.


On the geometry of quasi-Kaehler manifolds with Norden metric

Abstract: The basic class of the non-integrable almost complex manifolds with Norden metric is considered. Its curvature properties are studied. The isotropic Kaehler type of investigated manifolds is introduced and characterized geometrically.


Tangent Bundles with Sasaki Metric and Almost Hypercomplex Pseudo-Hermitian Structure

Abstract: The tangent bundle as a $4n$-manifold is equipped with an almost hypercomplex pseudo-Hermitian structure and it is characterized with respect to the relevant classifications. A number of 8-dimensional examples of the considered type of manifold are received from the known explicit examples in that manner.


Cyclic Cohomology of the Weyl Algebra

Abstract: We give an explicit formula for symplectically basic representatives of the cyclic cohomology of the Weyl algebra. This paper can be seen as cyclic addendum to the paper by Feigin, Felder and Shoikhet, where the analogous Hochschild case was treated. As an application, we prove a generalization of a Theorem of Nest and Tsygan concerning the relation of the Todd class and the cyclic cohomology of the differential operators on a complex manifold.


Some four-dimensional almost hypercomplex pseudo-Hermitian manifolds

Abstract: In this paper, a lot of examples of four-dimensional manifolds with an almost hypercomplex pseudo-Hermitian structure are constructed in several explicit ways. The received 4-manifolds are characterized by their linear invariants in the known aspects.


On the equivariant and the non-equivariant main conjecture for imaginary quadratic fields

Abstract: The Iwasawa main conjecture fields has been an important tool to study the arithmetic of special values of $L$-functions of Hecke characters of imaginary quadratic fields. To obtain the finest possible invariants it is important to know the main conjecture for all prime numbers $p$ and also to have an equivariant version at disposal. In this paper we first prove the main conjecture for imaginary quadratic fields for all prime numbers $p$, improving earlier results by Rubin. From this we deduce the equivariant main conjecture in the case that a certain $\mu$-invariant vanishes. For prime numbers $p\nmid 6$ which split in $K$, this is a theorem by a result of Gillard.


Uniform lamda-adjustment and mu-approximation in Banach spaces

Abstract: We introduce a new concept of perturbation of closed linear subspaces and operators in Banach spaces called uniform lambda-adjustment which is weaker than perturbations by small gap, operator norm, q-norm, and K2-approximation. In arbitrary Banach spaces some of the classical Fredholm stability theorems remain true under uniform lambda-adjustment, while other fail. However, uniformly lambda-adjusted subspaces and linear operators retain their (semi--)Fredholm properties in a Banach space which dual is Fr\'{e}chet-Urysohn in weak* topology. We also introduce another concept of perturbation called uniform mu-approximation which is weaker than perturbations by small gap, norm, and compact convergence, yet stronger than uniform lambda-adjustment. We present Fredholm stability theorems for uniform mu-approximation in arbitrary Banach spaces and a theorem on stability of Riesz kernels and ranges for commuting closed essentially Kato operators. Finally, we define the new concepts of a tuple of subspaces and of a complex of subspaces in Banach spaces, and present stability theorems for index and defect numbers of Fredholm tuples and complexes under uniform lambda-adjustment and uniform mu-approximation.


Inequalities of Hardy-Sobolev type in Carnot-Carath\'eodory spaces

Abstract: We consider various types of Hardy-Sobolev inequalities on a Carnot-Carath\'eodory space $(\Om, d)$ associated to a system of smooth vector fields $X=\{X_1, X_2,...,X_m\}$ on $\RR^n$ satisfying the H\"ormander's finite rank condition $rank Lie[X_1,...,X_m] \equiv n$. One of our main concerns is the trace inequality \int_{\Om}|\phi(x)|^{p}V(x)dx\leq C\int_{\Om}|X\phi|^{p}dx,\qquad \phi\in C^{\infty}_{0}(\Om), where $V$ is a general weight, i.e., a nonnegative locally integrable function on $\Om$, and $1<p<+\infty$. Under sharp geometric assumptions on the domain $\Om\subset \Rn$ that can be measured equivalently in terms of subelliptic capacities or Hausdorff contents, we establish various forms of Hardy-Sobolev type inequalities.


Abstract: We present two new forms in which the Frechet differential of a power series in a Banach algebra can be expressed in terms of absolutely convergent series involving the commutant $C(T):A\mapsto [A,T]$.Then we apply the results to the analytic functional calculus in a complex Banach space.


Note on the Cantor-Bendixson rank of limit groups

Abstract: We show that the Cantor-Bendixson rank of a limit group is finite as well as that of a limit group of a linear group.


Monotonicity of Subelliptic Estimates on Rigid Pseudoconvex Domains

Abstract: This paper presents monotonicity of subelliptic estimates on rigid pseudoconvex domains. As an application of monotonicity, we will show that if a rigid monomial domain is of finite type in the D'Angelo's sense, then the sharp subelliptic estimate of this domain equals the reciprocal of the type.


Relative Riemann-Zariski spaces

Abstract: In this paper we study relative Riemann-Zariski spaces attached to a morphism of schemes and generalizing the classical Riemann-Zariski space of a field. We prove that similarly to the classical RZ spaces, the relative ones can be described either as projective limits of schemes in the category of locally ringed spaces or as certain spaces of valuations. We apply these spaces to prove the following two new results: a strong version of stable modification theorem for relative curves; a decomposition theorem which asserts that any separated morphism between quasi-compact and quasi-separated schemes factors as a composition of an affine morphism and a proper morphism. (In particular, we obtain a new proof of Nagata's compactification theorem.)


Information Preserving Component Analysis: Data Projections for Flow Cytometry Analysis

Abstract: Flow cytometry is often used to characterize the malignant cells in leukemia and lymphoma patients, traced to the level of the individual cell. Typically, flow cytometric data analysis is performed through a series of 2-dimensional projections onto the axes of the data set. Through the years, clinicians have determined combinations of different fluorescent markers which generate relatively known expression patterns for specific subtypes of leukemia and lymphoma -- cancers of the hematopoietic system. By only viewing a series of 2-dimensional projections, the high-dimensional nature of the data is rarely exploited. In this paper we present a means of determining a low-dimensional projection which maintains the high-dimensional relationships (i.e. information) between differing oncological data sets. By using machine learning techniques, we allow clinicians to visualize data in a low dimension defined by a linear combination of all of the available markers, rather than just 2 at a time. This provides an aid in diagnosing similar forms of cancer, as well as a means for variable selection in exploratory flow cytometric research. We refer to our method as Information Preserving Component Analysis (IPCA).


Movable algebraic singularities of second-order ordinary differential equations

Abstract: Any nonlinear equation of the form y''=\sum_{n=0}^N a_n(z)y^n has a (generally branched) solution with leading order behaviour proportional to (z-z_0)^{-2/(N-1)} about a point z_0, where the coefficients a_n are analytic at z_0 and a_N(z_0)\ne 0. We consider the subclass of equations for which each possible leading order term of this form corresponds to a one-parameter family of solutions represented near z_0 by a Laurent series in fractional powers of z-z_0. For this class of equations we show that the only movable singularities that can be reached by analytic continuation along finite-length curves are of the algebraic type just described. This work generalizes previous results of S. Shimomura. The only other possible kind of movable singularity that might occur is an accumulation point of algebraic singularities that can be reached by analytic continuation along infinitely long paths ending at a finite point in the complex plane. This behaviour cannot occur for constant coefficient equations in the class considered. However, an example of R. A. Smith shows that such singularities do occur in solutions of a simple autonomous second-order differential equation outside the class we consider here.


The sign of Galois representations attached to automorphic forms for unitary groups

Abstract: We determine the sign of the polarization of any polarized irreducible factor of a Galois representation attached to a polarized cohomological cuspidal automorphic form of Gl_n of a CM field: it is always +1, as was conjectured by Gross.


Spinning rough disk moving in a rarefied medium

Abstract: A spinning rough disk moves through a rarefied medium on the plane. The roughness is formed by small cavities on the disk boundary. The medium is so rare that mutual interaction of particles can be neglected. All collisions of particles with the disk are perfectly elastic; there may happen multiple collisions in the cavities. We calculate the force of resistance acting on the body and examine how it depends on the kind of roughness (shape of the cavities). We show that the nonzero transversal component of the force generally appears, resulting in deflection of the disk trajectory. In several simple cases the trajectory is determined. We compare our results with the ones known in the literature. It is known that there is a transversal force acting on a spinning body (most often a sphere or a cylinder) moving in a rarefied gas, due to nonelastic interaction of gas particles with the body. We propose another mechanism of creating the transversal force, resulting from multiple reflections of particles from the body.


Analytic Functions of a General Matrix Variable

Abstract: Recent innovations on the differential calculus for functions of non-commuting variables, begun for a quaternionic variable, are now extended to the case of a general matrix over the complex numbers. The expansion of F(X+Delta) is given to first order in Delta for general matrix variables X and Delta that do not commute with each other.


Characterization of Compact Subsets of $\mathcal{A}^p$ with Respect to Weak Topology

Abstract: In this brief article we characterize the relatively compact subsets of $\mathcal{A}^p$ for the topology $\sigma(\mathcal{A}^p,\mathcal{R}^q)$ (see below), by the weak compact subsets of $L^p$ . The spaces $\mathcal{R}^q$ endowed with the weak topology induced by $\mathcal{A}^p$, was recently employed to create the convex risk theory of random processes. The weak compact sets of $\mathcal{A}^p$ are important to characterize the so-called Lebesgue property of convex risk measures, to give a complete description of the Makcey topology on $\mathcal{R}^q$ and for their use in the optimization theory.


Forcing the Strong Lefschetz and the Maximal Rank Properties

Abstract: Three basic properties that standard graded artinian $k$-algebras may or may not enjoy are the Weak and Strong Lefschetz Properties and the Maximal Rank Property (respectively WLP, SLP, and MRP). In this paper we will assume that the base field $k$ has characteristic zero. It is known that SLP implies MRP, which in turn implies WLP, but that both implications are strict. However, it surprisingly turned out that the set of Hilbert functions admitting any algebras with WLP coincides with the corresponding set for SLP (and therefore with that for MRP). J. Migliore and the first author, using Green's theorem and a result of Wiebe, characterized the Hilbert functions forcing all algebras to enjoy WLP. The purpose of this note is to prove the corresponding characterizations for both SLP and MRP. Unsurprisingly (or surprisingly??), the two characterizations coincide, but they define a class of Hilbert functions strictly smaller than that determined for WLP. Our methods include the Herzog-Popescu theorem on quotients of $k$-algebras modulo a general form, a result of Wiebe, and gins and stable ideals. At the end, we will also discuss the importance of assuming that the characteristic be zero, and we will exhibit a class of codimension 2 monomial complete intersections for which SLP (but not MRP) fails in positive characteristic.


Uniform observability of hidden Markov models and filter stability for unstable signals

Abstract: A hidden Markov model is called observable if distinct initial laws give rise to distinct laws of the observation process. Observability implies stability of the nonlinear filter when the signal process is tight, but this need not be the case when the signal process is unstable. This paper introduces a stronger notion of uniform observability which guarantees stability of the nonlinear filter in the absence of stability assumptions on the signal. By developing certain uniform approximation properties of convolution operators, we subsequently demonstrate that the uniform observability condition is satisfied for various classes of filtering models with white-noise type observations. This includes the case of observable linear Gaussian filtering models, so that standard results on stability of the Kalman--Bucy filter are obtained as a special case.


Hitting Time Statistics and Extreme Value Theory

Abstract: We consider discrete time dynamical systems and show the link between Hitting Time Statistics (the distribution of the first time points land in asymptotically small sets) and Extreme Value Theory (distribution properties of the partial maximum of stochastic processes). This relation allows to study Hitting Time Statistics with tools from Extreme Value Theory, and vice versa. We apply these results to non-uniformly hyperbolic systems and prove that a multimodal map with an absolutely continuous invariant measure must satisfy the classical extreme value laws (with no extra condition on the speed of mixing, for example). We also give applications of our theory to higher dimensional examples, for which we also obtain classical extreme value laws and exponential hitting time statistics (for balls). We extend these ideas to the subsequent returns to asymptotically small sets, linking the Poisson statistics of both processes.


A Groebner basis for the secant ideal of the second hypersimplex

Abstract: We determine a Groebner basis for the secant ideal of the toric ideal associated to the second hypersimplex, with respect to any circular term order. The Groebner basis of the secant ideal requires polynomials of odd degree up to n. This shows that the circular term order is 2-delightful, resolving a conjecture of Drton, Sturmfels, and the author. The proof uses Groebner degenerations for secant ideals, combinatorial characterizations of the secant ideals of monomial ideals, and the relations between secant ideals and prolongations.


A Note about proving non-$\Gamma$ under a finite non-microstates free Fisher information Assumption

Abstract: We prove that if $X_{1},...,X_{n} (n >1)$ are selfadjoints in a $W^{*}$-probability space with finite non-microstates free Fisher information, then the von Neumann algebra $W^{*}(X_{1},...,X_{n})$ they generate doesn't have property $\Gamma$ (especially is not amenable). This is an analog of a well-known result of Voiculescu for microstates free entropy. We also prove factoriality under finite non-microstates entropy.


On Finitely Generated Models of Theories with at Most Countably Many Nonisomorphic Finitely Generated Models

Abstract: We study finitely generated models of countable theories, having at most countably many nonisomorphic finitely generated models. We intro- duce a notion of rank of finitely generated models and we prove, when T has at most countably many nonisomorphic finitely generated models, that every finitely generated model has an ordinal rank. This rank is used to give a prop- erty of finitely generated models analogue to the Hopf property of groups and also to give a necessary and sufficient condition for a finitely generated model to be prime of its complete theory. We investigate some properties of limit groups of equationally noetherian groups, in respect to their ranks.


Minimal volume $k$-point lattice $d$-simplices

Abstract: We extend the results of Bey, Hen, and Wills (http://arxiv.org/abs/math/0606089). In this paper, we show that, up to equivalence under unimodular transformations, there is exactly one class of $d$-simplices having $k \ge 1$ interior lattice points and minimal volume $\frac{1}{d!}(dk+1)$.


Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions

Abstract: Using the bijection between partitions and vacillating tableaux, we establish a correspondence between pairs of noncrossing free Dyck paths of length $2n$ and noncrossing partitions of $[2n+1]$ with $n+1$ blocks. In terms of the number of up steps at odd positions, we find a characterization of Dyck paths constructed from pairs of noncrossing free Dyck paths by using the Labelle merging algorithm.


The Conjecture of Nowicki on Weitzenboeck derivations of polynomial algebras

Abstract: The Weitzenboeck theorem states that the algebra of constants of a linear locally nilpotent derivation of the polynomial algebra K[Z]=K[z_1,...,z_m] in m variables over a field K of characteristic 0 is finitely generated. If m=2n and the Jordan normal form of the derivation consists of Jordan cells of size 2 only, we may assume that K[Z]=K[X,Y] and the derivation sends y_i to x_i and x_i to 0, i=1,...,n. Nowicki conjectured that the algebra of constants of this derivation is generated by x_1,...,x_n and x_iy_j-x_jy_i, i<j. Recently this conjecture was confirmed in the Ph.D. thesis of Khoury, and also by Derksen. In this paper we give an elementary proof of the conjecture of Nowicki. Then we find a very simple system of defining relations of the algebra of constants which corresponds to the reduced Groebner basis of the related ideal with respect to a suitable admissible order, and present an explicit basis of the algebra of constants as a vector space.


The Aleph-zero or zero dichotomy

Abstract: This paper proves the existence of a dichotomy which being formally derived from the topological successiveness of w-order leads to the same absurdity of Zeno's Dichotomy II. It also derives a contradictory result from the first Zeno's Dichotomy.


Recursion and w-order

Abstract: This paper has been withdrawn by the author due to an error.


Margin-adaptive model selection in statistical learning

Abstract: A classical condition for fast learning rates is the margin condition, first introduced by Mammen and Tsybakov. We tackle in this paper the problem of adaptivity to this condition in the context of model selection, in a general learning framework. Actually, we consider a weaker version of this condition that allows one to take into account that learning within a small model can be much easier than within a large one. Requiring this "strong margin adaptivity" makes the model selection problem more challenging. We first prove, in a general framework, that some penalization procedures (including local Rademacher complexities) exhibit this adaptivity when the models are nested. Contrary to previous results, this holds with penalties that only depend on the data. Our second main result is that strong margin adaptivity is not always possible when the models are not nested: for every model selection procedure (even a randomized one), there is a problem for which it does not demonstrate strong margin adaptivity.


Superswapping

Abstract: Supertask theory is used here to prove a contradictory result which involves the consistency of w-order and the Axiom of Infinity.


A disturbing supertask

Abstract: This paper examines the consistency of w-order by means of a supertask that functions as a supertrap for the assumed existence of w-ordered collections, which are simultaneously complete (as is required by the Actual infinity) and uncompletable (because no last element completes them).


On the $S_n$-equivariant Euler characteristic of moduli spaces of hyperelliptic curves

Abstract: The generating function for $S_n$-equivariant Euler characteristics of moduli spaces of pointed hyperelliptic curves for any genus g>1 is calculated. This answer generalizes the known ones for genera 2 and 3 and answers obtained by J. Bergstrom for any genus and n<8 points.


A construction of spherical designs from finite graphs with the theory of crystal lattice

Abstract: We want to introduce a construction of spherical designs from finite graphs with the theory of crystal lattice. We start from a finite graph, and we consider standard realization of the crystal lattices as the maximal Abelian covering of the graph. Then, we take the set of vectors which form the crystal lattice. If every vector has the same norm, then we can consider a finite set on Euclidean sphere, and then we get a spherical design. In this paper, we observe the results by numerical calculations. We tried constructing vectors from various finite graphs, strongly regular graphs, distance regular graphs, and so on. We also introduce some facts and conjectures.


Demystifying Double Robustness: A Comparison of Alternative Strategies for Estimating a Population Mean from Incomplete Data

Abstract: When outcomes are missing for reasons beyond an investigator's control, there are two different ways to adjust a parameter estimate for covariates that may be related both to the outcome and to missingness. One approach is to model the relationships between the covariates and the outcome and use those relationships to predict the missing values. Another is to model the probabilities of missingness given the covariates and incorporate them into a weighted or stratified estimate. Doubly robust (DR) procedures apply both types of model simultaneously and produce a consistent estimate of the parameter if either of the two models has been correctly specified. In this article, we show that DR estimates can be constructed in many ways. We compare the performance of various DR and non-DR estimates of a population mean in a simulated example where both models are incorrect but neither is grossly misspecified. Methods that use inverse-probabilities as weights, whether they are DR or not, are sensitive to misspecification of the propensity model when some estimated propensities are small. Many DR methods perform better than simple inverse-probability weighting. None of the DR methods we tried, however, improved upon the performance of simple regression-based prediction of the missing values. This study does not represent every missing-data problem that will arise in practice. But it does demonstrate that, in at least some settings, two wrong models are not better than one.


Comment: Demystifying Double Robustness: A Comparison of Alternative Strategies for Estimating a Population Mean from Incomplete Data

Abstract: Comment on ``Demystifying Double Robustness: A Comparison of Alternative Strategies for Estimating a Population Mean from Incomplete Data'' [arXiv:0804.2958]


Comment: Performance of Double-Robust Estimators When ``Inverse Probability'' Weights Are Highly Variable

Abstract: Comment on ``Performance of Double-Robust Estimators When ``Inverse Probability'' Weights Are Highly Variable'' [arXiv:0804.2958]


Comment: Understanding OR, PS and DR

Abstract: Comment on ``Understanding OR, PS and DR'' [arXiv:0804.2958]


Rejoinder: Demystifying Double Robustness: A Comparison of Alternative Strategies for Estimating a Population Mean from Incomplete Data

Abstract: Rejoinder to ``Demystifying Double Robustness: A Comparison of Alternative Strategies for Estimating a Population Mean from Incomplete Data'' [arXiv:0804.2958]


On some Moduli spaces of stable vector bundles on cubic and quartic threefolds

Abstract: We study certain moduli spaces of stable vector bundles of rank two on cubic and quartic threefolds. In many cases under consideration, it turns out that the moduli space is complete and irreducible and a general member has vanishing intermediate cohomology. In one case, all except one component of the moduli space has such vector bundles.


Measuring Traffic

Abstract: A traffic performance measurement system, PeMS, currently functions as a statewide repository for traffic data gathered by thousands of automatic sensors. It has integrated data collection, processing and communications infrastructure with data storage and analytical tools. In this paper, we discuss statistical issues that have emerged as we attempt to process a data stream of 2 GB per day of wildly varying quality. In particular, we focus on detecting sensor malfunction, imputation of missing or bad data, estimation of velocity and forecasting of travel times on freeway networks.


Splitting type, global sections and Chern classes for torsion free sheaves on P^N

Abstract: In this paper we compare a torsion free sheaf $\FF$ on $\PP^N$ and the free vector bundle $\oplus_{i=1}^n\OPN(b_i)$ having same rank and splitting type. We show that the first one has always "less" global sections, while it has a higher second Chern class. In both cases bounds for the difference are found in terms of the maximal free subsheaves of $\FF$. As a consequence we obtain a direct, easy and more general proof of the "Horrocks' splitting criterion", also holding for torsion free sheaves, and lower bounds for the Chern classes $c_i(\FF(t))$ of twists of $\FF$, only depending on some numerical invariants of $\FF$. Especially, we prove for rank $n$ torsion free sheaves on $\PP^N$, whose splitting type has no gap (i.e. $b_i\geq b_{i+1}\geq b_i-1$ for every $i=1, ...,n-1$), the following formula for the discriminant: \[ \Delta(\FF):=2nc_2-(n-1)c_1^2\geq -{1/12}n^2(n^2-1)\] Finally in the case of rank $n$ reflexive sheaves we obtain polynomial upper bounds for the absolute value of the higher Chern classes $c_3(\FF(t)), ..., c_n(\FF(t))$, for the dimension of the cohomology modules $H^i\FF(t)$ and for the Castelnuovo-Mumford regularity of $\FF$; these polynomial bounds only depend only on $c_1(\FF)$, $c_2(\FF)$, the splitting type of $\FF$ and $t$.


Higher order invariants of Levi degenerate hypersurfaces

Abstract: The first part of this paper considers higher order CR invariants of three dimensional hypersurfaces of finite type. Using a full normal form we give a complete characterization of hypersurfaces with trivial local automorphism group, and analogous results for finite groups. The second part considers hypersurfaces of finite Catlin multitype and the Kohn-Nirenberg phenomenon in higher dimensions. We give a necessary condition for local convexifiability of a class of pseudoconvex hypersurfaces in $\mathbb C^{n+1}$.


The Convenient Setting for non-Quasianalytic Denjoy--Carleman Differentiable Mappings

Abstract: For Denjoy--Carleman differential function classes $C^M$ where the weight sequence $M=(M_k)$ is logarithmically convex, stable under derivations, and non-quasianalytic of moderate growth, we prove the following: A mapping is $C^M$ if it maps $C^M$-curves to $C^M$-curves. The category of $C^M$-mappings is cartesian closed in the sense that $C^M(E,C^M(F,G))\cong C^M(E\x F, G)$ for convenient vector spaces. Applications to manifolds of mappings are given: The group of $C^M$-diffeomorphisms is a $C^M$-Lie group but not better.


The Epic Story of Maximum Likelihood

Abstract: At a superficial level, the idea of maximum likelihood must be prehistoric: early hunters and gatherers may not have used the words ``method of maximum likelihood'' to describe their choice of where and how to hunt and gather, but it is hard to believe they would have been surprised if their method had been described in those terms. It seems a simple, even unassailable idea: Who would rise to argue in favor of a method of minimum likelihood, or even mediocre likelihood? And yet the mathematical history of the topic shows this ``simple idea'' is really anything but simple. Joseph Louis Lagrange, Daniel Bernoulli, Leonard Euler, Pierre Simon Laplace and Carl Friedrich Gauss are only some of those who explored the topic, not always in ways we would sanction today. In this article, that history is reviewed from back well before Fisher to the time of Lucien Le Cam's dissertation. In the process Fisher's unpublished 1930 characterization of conditions for the consistency and efficiency of maximum likelihood estimates is presented, and the mathematical basis of his three proofs discussed. In particular, Fisher's derivation of the information inequality is seen to be derived from his work on the analysis of variance, and his later approach via estimating functions was derived from Euler's Relation for homogeneous functions. The reaction to Fisher's work is reviewed, and some lessons drawn.


On the Carleson measure criterion in linear systems theory

Abstract: In Ho, Russell, and Weiss, a Carleson measure criterion for admissibility of one-dimensional input elements with respect to diagonal semigroups is given. We extend their results from the Hilbert space situation $X=\ell_2$ and $L^2$--admissibility to the more general situation of $L^p$--admissibility on $\ell_q$--spaces. In case of analytic diagonal semigroups we present a new result that does not rely on Laplace transform methods. A comparison of both criteria leads to result of $L^p$--admissibility for reciprocal systems in the sense of Curtain.


Rank four vector bundles without theta divisor over a curve of genus two

Abstract: We show that the locus of stable rank four vector bundles without theta divisor over a smooth projective curve of genus two is in canonical bijection with the set of theta-characteristics. We give several descriptions of these bundles and compute the degree of the rational theta map.


Generalized SURE for Exponential Families: Applications to Regularization

Abstract: Stein's unbiased risk estimate (SURE) was proposed by Stein for the independent, identically distributed (iid) Gaussian model in order to derive estimates that dominate least-squares (LS). In recent years, the SURE criterion has been employed in a variety of denoising problems for choosing regularization parameters that minimize an estimate of the mean-squared error (MSE). However, its use has been limited to the iid case which precludes many important applications. In this paper we begin by deriving a SURE counterpart for general, not necessarily iid distributions from the exponential family. This enables extending the SURE design technique to a much broader class of problems. Based on this generalization we suggest a new method for choosing regularization parameters in penalized LS estimators. We then demonstrate its superior performance over the conventional generalized cross validation approach and the discrepancy method in the context of image deblurring and deconvolution. The SURE technique can also be used to design estimates without predefining their structure. However, allowing for too many free parameters impairs the performance of the resulting estimates. To address this inherent tradeoff we propose a regularized SURE objective. Based on this design criterion, we derive a wavelet denoising strategy that is similar in sprit to the standard soft-threshold approach but can lead to improved MSE performance.


Geodesics in large planar maps and in the Brownian map

Abstract: We study geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe geodesics starting from the distinguished point called the root, and we characterize the set S of all points that are connected to the root by more than one geodesic. The set S is dense in the Brownian map and homeomorphic to a non-compact real tree. Furthermore, for every x in S, the number of distinct geodesics from x to the root is equal to the number of connected components of the complement of {x} in S. In particular, points of the Brownian map can be connected to the root by at most three distinct geodesics. Our results have applications to the behavior of geodesics in large planar maps.


Real Paley-Wiener theorems and local spectral radius formulas

Abstract: We systematically develop real Paley-Wiener theory for the Fourier transform on R^d for Schwartz functions, L^p-functions and distributions, in an elementary treatment based on the inversion theorem. As an application, we show how versions of classical Paley-Wiener theorems can be derived from the real ones via an approach which does not involve domain shifting and which may be put to good use for other transforms of Fourier type as well. An explanation is also given why the easily applied classical Paley-Wiener theorems are unlikely to be able to yield information about the support of a function or distribution which is more precise than giving its convex hull, whereas real Paley-Wiener theorems can be used to reconstruct the support precisely, albeit at the cost of combinatorial complexity. We indicate a possible application of real Paley-Wiener theory to partial differential equations in this vein and furthermore we give evidence that a number of real Paley-Wiener results can be expected to have an interpretation as local spectral radius formulas. A comprehensive overview of the literature on real Paley-Wiener theory is included.


A Proposal of Multigrid Methods for Hermitian Positive Definite Linear Systems enjoying an order relation

Abstract: Given a multigrid procedure for linear systems with coefficient matrices $A_n$, we discuss the optimality of a related multigrid procedure with the same smoother and the same projector, when applied to properly related algebraic problems with coefficient matrices $B_n$: we assume that both $A_n$ and $B_n$ are positive definite with $A_n\le \vartheta B_n$, for some positive $\vartheta$ independent of $n$. In this context we prove the Two-Grid method optimality. We apply this elementary strategy for designing a multigrid solution for modifications of multilevel structured (Toeplitz, circulants, Hartley, sine ($\tau$ class) and cosine algebras) linear systems, in which the coefficient matrix is banded in a multilevel sense and Hermitian positive definite. In such a way, several linear systems arising from the approximation of integro-differential equations with various boundary conditions can be efficiently solved in linear time (with respect to the size of the algebraic problem). Some numerical experiments are presented and discussed, both with respect to Two-Grid and multigrid procedures.


Three Dimensional Corners: A Box Norm Proof

Abstract: In an additive group (G,+), a three-dimensional corner is the four points g, g+d(1,0,0), g+d(0,1,0), g+d(0,0,1), where g is in G^3, and d is a non-zero element of G. The Ramsey number of interest is R_3(G) the maximal cardinality of a subset of G^3 that does not contain a three-dimensional corner. Furstenberg and Katznelson have shown R_3(Z_N) is little-o of N^3, and in fact the corresponding result holds in all dimensions, a result that is a far reaching extension of the Szemeredi Theorem. We give a new proof of the finite field version of this fact, a proof that is a common generalization of the Gowers proof of Szemeredi's Theorem for four term progressions, and the result of Shkredov on two-dimensional corners. The principal tool are the Gowers Box Norms.


Torsion Invariants for Families

Abstract: We give an overview over the higher torsion invariants of Bismut-Lott, Igusa-Klein and Dwyer-Weiss-Williams, including some more or less recent developments.


On the size of identifying codes in binary hypercubes

Abstract: We consider identifying codes in binary Hamming spaces F^n, i.e., in binary hypercubes. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. Currently, the subject forms a topic of its own with several possible applications, for example, to sensor networks. Let C be a subset of F^n. For any subset X of F^n, denote by I_r(X)=I_r(C;X) the set of elements of C within distance r from at least one x in X. Now C is called an (r,<= l)-identifying code if the sets I_r(X) are distinct for all subsets X of size at most l. We estimate the smallest size of such codes with fixed l and r/n converging to some number rho in (0,1). We further show the existence of such a code of size O(n^{3/2}) for every fixed l and r slightly less than n/2, and give for l=2 an explicit construction of small such codes for r the integer part of n/2-1 (the largest possible value).


Torsion dans un produit de courbes elliptiques

Abstract: Let $A$ be an abelian variety defined over a number field $K$, the number of torsion points rational over a finite extension $L$ is bounded polynomially in terms of the degree $[L:K]$. We formulate a question suggesting the optimal exponent for this bound in terms of the dimension of the Mumford-Tate groups of the abelian subvarieties of $A$; we study the behaviour under product and then give a positive answer to our question when $A$ is the product of elliptic curves.


The Clustering Coefficient of a Scale-Free Random Graph

Abstract: We consider a random graph process in which, at each time step, a new vertex is added with m out-neighbours, chosen with probabilities proportional to their degree plus a strictly positive constant. We show that the expectation of the clustering coefficient of the graph process is asymptotically proportional to log n/n. Bollob\'as and Riordan have previously shown that when the constant is zero, the same expectation is asymptotically proportional to ((log n)^2)/n.


A Simple Sample Size Formula for Estimating Means of Poisson Random Variables

Abstract: In this paper, we derive an explicit sample size formula based a mixed criterion of absolute and relative errors for estimating means of Poisson random variables.


Distance graphs in vector spaces over finite fields, coloring and pseudo-randomness

Abstract: In this paper we systematically study various properties of the distance graph in ${\Bbb F}_q^d$, the $d$-dimensional vector space over the finite field ${\Bbb F}_q$ with $q$ elements. In the process we compute the diameter of distance graphs and show that sufficiently large subsets of $d$-dimensional vector spaces over finite fields contain every possible finite configurations.


Absolute continuity for some one-dimensional processes

Abstract: We introduce an elementary method for proving the absolute continuity of the time marginals of one-dimensional processes. It is based on a comparison between the Fourier transform of such time marginals with those of the one-step Euler approximation of the underlying process. We obtain some absolute continuity results for stochastic differential equations with H\"{o}lder continuous coefficients. Furthermore, we allow such coefficients to be random and to depend on the whole path of the solution. We also show how it can be extended to some stochastic partial differential equations and to some L\'{e}vy-driven stochastic differential equations. In the cases under study, the Malliavin calculus cannot be used, because the solution in generally not Malliavin differentiable.


Galois extensions for coquasi-Hopf algebras

Abstract: The notions of Galois and cleft extensions are generalized for coquasi-Hopf algebras. It is shown that such an extension over a coquasi-Hopf algebra is cleft if and only if it is Galois and has the normal basis property. A Schneider type theorem is proven for coquasi-Hopf algebras with bijective antipode. As an application, we generalize Schauenburg's bialgebroid construction for coquasi-Hopf algebras.


Grid Diagrams and Legendrian Lens Space Links

Abstract: Grid diagrams encode useful geometric information about knots in S^3. In particular, they can be used to combinatorially define the knot Floer homology of a knot K in S^3, and they have a straightforward connection to Legendrian representatives of K in (S^3, \xi_\st), where \xi_\st is the standard, tight contact structure. The definition of a grid diagram was extended to include a description for links in all lens spaces, resulting in a combinatorial description of the knot Floer homology of a knot K in L(p, q) for all p > 0. In the present article, we explore the connection between lens space grid diagrams and the contact topology of a lens space. Our hope is that an understanding of grid diagrams from this point of view will lead to new approaches to the Berge conjecture, which claims to classify all knots in S^3 upon which surgery yields a lens space.


Conductor inequalities and criteria for Sobolev-Lorentz two-weight inequalities

Abstract: In this paper we present integral conductor inequalities connecting the Lorentz p,q-(quasi)norm of a gradient of a function to a one-dimensional integral of the p,q-capacitance of the conductor between two level surfaces of the same function. These inequalities generalize an inequality obtained by the second author in the case of the Sobolev norm. Such conductor inequalities lead to necessary and sufficient conditions for Sobolev-Lorentz type inequalities involving two arbitrary measures.


Small parts in the Bernoulli sieve

Abstract: Sampling from a random discrete distribution induced by a `stick-breaking' process is considered. Under a moment condition, it is shown that the asymptotics of the sequence of occupancy numbers, and of the small-parts counts (singletons, doubletons, etc) can be read off from a limiting model involving a unit Poisson point process and a self-similar renewal process on the halfline.


Beyond Chaos

Abstract: The first part of this paper defines recursive interactions by means of logistic functions and derives a general result on the way interacting systems evolve in attractors. It also defines the notion of coevolution trajectory and presents a new family of attractors: orbital attractors (including single, irregular, folded, complex and discontinuous orbits). The second part summarizes the results of a first experimental analysis of recursive interactions in both binary and multiple interactions. Among other results, this analysis reveals that interacting systems may easily evolve from chaos to order.


Degeneracy loci of families of Dirac operators

Abstract: Generalizing some results from R. Leung's thesis, we compute, in rational cohomology, the Poincare dual of the degeneracy locus of the family of Dirac operators parameterized by the moduli space of projectively anti-self-dual $\SO(3)$ connections. This is the first step in a program to derive a relation between the Donaldson and spin invariants.


Integral equalities for functions of unbounded spectral operators in Banach spaces

Abstract: We investigate a limiting procedure for extending local integral operator equalities to the global ones and to applying it to obtaining generalizations of the Newton-Leibnitz formula for operator-valued maps for a wide class of unbounded operators.


Determinant computations for some classes of Toeplitz-Hankel matrices

Abstract: The purpose of this paper is to compute the asymptotics of determinants of finite sections of operators that are trace class perturbations of Toeplitz operators. For example, we consider the asymptotics in the case where the matrices are of the form $ (a_{i-j} \pm a_{i+j+1-k})_{i,j=0... N-1} $ with $k$ is fixed. We will show that this example as well as some general classes of operators have expansions that are similar to those that appear in the Strong Szeg\"{o} Limit Theorem. We also obtain exact identitities for some of the determinants that are analogous to the one derived independently by Geronimo and Case and by Borodin and Okounkov for finite Toeplitz matrices. These problems were motivated by considering certain statistical quantities that appear in random matrix theory.


(q,t)-analogues and GL_n(F_q)

Abstract: We start with a (q,t)-generalization of a binomial coefficient. It can be viewed as a polynomial in t that depends upon an integer q, with combinatorial interpretations when q is a positive integer, and algebraic interpretations when q is the order of a finite field. These (q,t)-binomial coefficients and their interpretations generalize further in two directions, one relating to column-strict tableaux and Macdonald's ``seventh variation'' of Schur functions, the other relating to permutation statistics and Hilbert series from the invariant theory of GL_n(F_q).


Refined convergence for the Boolean model

Abstract: In a previous work, two of the authors proposed a new proof of a well known convergence result for the scaled elementary connected vacant component in the high intensity Boolean model towards the Crofton cell of the Poisson hyperplane process. In this paper, we consider the particular case of the two-dimensional Boolean model where the grains are discs with random radii. We investigate the second-order term in this convergence when the Boolean model and the Poisson line process are coupled on the same probability space. A precise coupling between the Boolean model and the Poisson line process is first established, a result of directional convergence in distribution for the difference of the two sets involved is derived as well.


A characterization of dimension free concentration in terms of transportation inequalities

Abstract: The aim of this paper is to show that a probability measure concentrates independently of the dimension like a gaussian measure if and only if it verifies Talagrand's $\T_2$ transportation-cost inequality. This theorem permits us to give a new and very short proof of a result of Otto and Villani. Generalizations to other types of concentration are also considered. In particular, one shows that the Poincar\'e inequality is equivalent to a certain form of dimension free exponential concentration. The proofs of these results rely on simple Large Deviations techniques.


Unfolding a Codimension-Two, Discontinuous, Andronov-Hopf Bifurcation

Abstract: We present an unfolding of the codimension-two scenario of the simultaneous occurrence of a discontinuous bifurcation and an Andronov-Hopf bifurcation in a piecewise-smooth, continuous system of autonomous ordinary differential equations in the plane. We find the Hopf cycle undergoes a grazing bifurcation that may be very shortly followed by a saddle-node bifurcation of the orbit. We derive scaling laws for the bifurcation curves that emanate from the codimension-two bifurcation.


Teichm\"uller Structures and Dual Geometric Gibbs Type Measure Theory for Continuous Potentials

Abstract: The Gibbs measure theory for smooth potentials is an old and beautiful subject and has many important applications in modern dynamical systems. For continuous potentials, it is impossible to have such a theory in general. However, we develop a dual geometric Gibbs type measure theory for certain continuous potentials in this paper following some ideas and techniques from Teichm\"uller theory for Riemann surfaces. Furthermore, we prove that the space of those continuous potentials has a Teichm\"uller structure. Moreover, this Teichm\"uller structure is a complete structure and is the completion of the space of smooth potentials under this Teichm\"uller structure. Thus our dual geometric Gibbs type theory is the completion of the Gibbs measure theory for smooth potentials from the dual geometric point of view.


Subellipticity of the $\bar\partial$-Neumann problem on a weakly $q$-pseudoconvex/concave domain

Abstract: For a domain $D$ of $\mathbb{C}^n$ which is weakly $q$-pseudoconvex or $q$-pseudoconcave we give a sufficient condition for subelliptic estimates for the $\bar{\partial}$-Neumann problem. The paper extends to domains which are not necessarily pseudoconvex, the results and the techniques of Catlin. MSC: 32D10, 32U05, 32V25


Feedback Control and the Arrow of Time

Abstract: The purpose of this paper is to highlight the central role that the time asymmetry of stability plays in feedback control. We show that this provides a new perspective on the use of doubly-infinite or semi-infinite time axes for signal spaces in control theory. We then focus on the implication of this time asymmetry in modeling uncertainty, regulation and robust control. We point out that modeling uncertainty and the ease of control depend critically on the direction of time. We also discuss the relationship of this control-based time-arrow with the well known arrows of time in physics.


A unified construction yielding precisely Hilbert and James sequences spaces

Abstract: Following James' approach, we shall define the Banach space $J(e)$ for each vector $e=(e_1,e_2,...,e_d) \in \Bbb{R}^d$ with $ e_1 \ne 0$. The construction immediately implies that J(1) coincides with the Hilbert space $i_2$ and that $J(1;-1)$ coincides with the celebrated quasireflexive James space $J$. The results of this paper show that, up to an isomorphism, there are only the following two possibilities: (i) either $J(e)$ is isomorphic to $l_2$, if $e_1+e_2+...+e_d\ne 0$ (ii) or $J(e)$ is isomorphic to $J$. Such a dichotomy also holds for every separable Orlicz sequence space $l_M$.


Siegel modular forms mod p

Abstract: We prove a vanishing theorem for one forms on the moduli stack of principally polarized abelian varieties of genus g>1 with level structure N over fields of characteristic p different from two. This is used to compute the Picard groups of these stacks. As an application we obtain results one congruences between integral Siegel modular forms.


Some applications of the Beta function

Abstract: This short note deals with some applications of the Beta function


Interlaced processes on the circle

Abstract: When two Markov operators commute, it suggests that we can couple two copies of one of the corresponding processes. We explicitly construct a number of couplings of this type for a commuting family of Markov processes on the set of conjugacy classes of the unitary group, using a dynamical rule inspired by the RSK algorithm. Our motivation for doing this is to develop a parallel programme, on the circle, to some recently discovered connections in random matrix theory between reflected and conditioned systems of particles on the line. One of the Markov chains we consider gives rise to a family of Gibbs measures on `bead configurations' on the infinite cylinder. We show that these measures have determinantal structure and compute the corresponding space-time correlation kernel.


Ruan's Conjecture on Singular symplectic flops

Abstract: We prove that the orbifold quantum ring is preserved under singular symplectic flops. Hence we verify Ruan's conjecture for this case.


Singular symplectic flops and Ruan cohomology

Abstract: In this paper, we study the symplectic geometry of singular conifolds of the finite group quotient $$ W_r=\{(x,y,z,t)|xy-z^{2r}+t^2=0 \}/\mu_r(a,-a,1,0), r\geq 1, $$ which we call orbi-conifolds. The related orbifold symplectic conifold transition and orbifold symplectic flops are constructed. Let $X$ and $Y$ be two symplectic orbifolds connected by such a flop. We study orbifold Gromov-Witten invariants of exceptional classes on $X$ and $Y$ and show that they have isomorphic Ruan cohomologies. Hence, we verify a conjecture of Ruan.


Derived Categories and Birational Geometry

Abstract: A survey article for AMS Summer Institute at Seattle in 2005.


Finite generation of a canonical ring

Abstract: The purpose of this note is to review an algebraic proof of the finite generation theorem due to Birkar-Cascini-Hacon-McKernan whose method is based on the Minimal Model Program. A survey article for Current Development in Mathematics 2007.


Bayesian computation for statistical models with intractable normalizing constants

Abstract: This paper deals with some computational aspects in the Bayesian analysis of statistical models with intractable normalizing constants. In the presence of intractable normalizing constants in the likelihood function, traditional MCMC methods cannot be applied. We propose an approach to sample from such posterior distributions. The method can be thought as a Bayesian version of the MCMC-MLE approach of Geyer and Thompson (1992). To the best of our knowledge, this is the first general and asymptotically consistent Monte Carlo method for such problems. We illustrate the method with examples from image segmentation and social network modeling. We study as well the asymptotic behavior of the algorithm and obtain a strong law of large numbers for empirical averages.


A Lost Theorem: Definite Integrals in Asymptotic Setting

Abstract: We present a simple yet rigorous theory of integration that is based on two axioms rather than on a construction involving Riemann sums. With several examples we demonstrate how to set up integrals in applications of calculus without using Riemann sums. In our axiomatic approach even the proof of the existence of the definite integral (which does use Riemann sums) becomes slightly more elegant than the conventional one. We also discuss an interesting connection between our approach and the history of calculus. The article is written for readers who teach calculus and its applications. It might be accessible to students under a teacher's supervision and suitable for senior projects on calculus, real analysis, or history of mathematics.


Hartman-Mycielski functor of non-metrizable compacta

Abstract: We investigate some topological properties of a normal functor $H$ introduced earlier by Radul which is a certain functorial compactification of the Hartman-Mycielski construction $HM$. We show that $H$ is open and find the condition when $HX$ is an absolute retract homeomorphic to the Tychonov cube.


The Arf-Kervaire invariant of framed manifolds as an obstruction to embeddability

Abstract: We prove that no $14$-connected (resp. $30$-connected) stably parallelizable manifold $N^{30}$ (resp. $N^{62}$) of dimension $30$ (resp. $62$) with the Arf-Kervaire invariant 1 can be smoothly embedded into $\mathbb{R}^{36}$ (resp. $\mathbb{R}^{83}$).


Preprojective algebras and cluster algebras

Abstract: We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups.


Cram\'{e}r asymptotics for finite time first passage probabilities of general L\'{e}vy processes

Abstract: We derive the exact asymptotics of $P(\sup_{u\leq t}X(u) > x)$ if $x$ and $t$ tend to infinity with $x/t$ constant, for a L\'{e}vy process $X$ that admits exponential moments. The proof is based on a renewal argument and a two-dimensional renewal theorem of H\"{o}glund (1990).


Plane Jacobian Conjecture for rational polynomials

Abstract: A non-zero constant Jacobian polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ is invertible if $P$ and $Q$ are rational polynomials.


Harold Jeffreys's Theory of Probability Revisited

Abstract: Published exactly seventy years ago, Jeffreys's Theory of Probability (1939) has had a unique impact on the Bayesian community and is now considered to be one of the main classics in Bayesian Statistics as well as the initiator of the objective Bayes school. In particular, its advances on the derivation of noninformative priors as well as on the scaling of Bayes factors have had a lasting impact on the field. However, the book reflects the characteristics of the time, especially in terms of mathematical rigor. In this paper we point out the fundamental aspects of this reference work, especially the thorough coverage of testing problems and the construction of both estimation and testing noninformative priors based on functional divergences. Our major aim here is to help modern readers in navigating in this difficult text and in concentrating on passages that are still relevant today.


Higher Green's functions for modular forms

Abstract: Higher Green functions are real-valued functions of two variables on the upper half plane which are bi-invariant under the action of a congruence subgroup, have logarithmic singularity along the diagonal, but instead of the usual equation $\Delta f=0$ we have equation $\Delta f = k(1-k) f$. Here $k$ is a positive integer. Properties of these functions are related to the space of modular forms of weight $2k$. In the case when there are no cusp forms of weight $2k$ it was conjectured that the values of the Green function at points of complex multiplication are algebraic multiples of logarithms of algebraic numbers. We show that this conjecture can be proved in any particular case if one constructs a family of elements of certain higher Chow groups on the power of a family of elliptic curves. These families have to satisfy certain properties. A different family of elements of Higher Chow groups is needed for a different point of complex multiplication. We give an example of such families, thereby proving the conjecture for the case when the group is $PSL_2(\mathbf Z)$, $k=2$ and one of the arguments is $\sqrt{-1}$.


A summation by Gencev

Abstract: Gencev has recently reported a closed form summation for an infinite series involving the harmonic numbers and the central binomial numbers. This note indicates a possible approach to the proof involving the dilogarithm function.


Representations of unramified U(2,2) over a p-adic field I: representations of non-integral level

Abstract: Let F_0 be a non-archimedean local field of odd residual characteristic and let G be the unramified unitary group U(2,2) defined over F_0. In this paper, we give a classification of the irreducible smooth representations of G of non-integral level using the Hecke algebraic method developed by Allen Moy for GSp(4).


Computable counter-examples to the Brouwer fixed-point theorem

Abstract: This paper is an overview of results that show the Brouwer fixed-point theorem (BFPT) to be essentially non-constructive and non-computable. The main results, the counter-examples of Orevkov and Baigger, imply that there is no procedure for finding the fixed point in general by giving an example of a computable function which does not fix any computable point. Research in reverse mathematics has shown the BFPT to be equivalent to the weak K\"onig lemma in RCA$_0$ (the system of recursive comprehension) and this result is illustrated by relating the weak K\"onig lemma directly to the Baigger example.


Stability of Universal Equivalence of Groups under Free Constructions

Abstract: In 1971 J. Stallings introduced a generalisation of amalgamated products of groups -- called a pregroup, which is a particular kind of a partial group. He defined the universal group U(P) of a pregroup P to be a universal object (in the sense of category theory) extending the partial operations on P to group operations on U(P). This turns out to be a versatile and convenient generalisation of classical group constructions: HNN-extensions and amalgamated products. In this respect the following general question arises. Which properties of pregroups, or relations between pregroups, carry over to the respective universal groups? In this paper it is proved that universal equivalence of pregroups extends to universal equivalence of universal groups. Applications to free products with amalgamation and HNN-extensions are then described.


A Blaschke-type condition for analytic and subharmonic functions and application to contraction operators

Abstract: Let $E$ be a closed set on the unit circle. We find a Blaschke-type condition, optimal in a sense of the order, on the Riesz measure of a subharmonic function $v$ in the unit disk with a certain growth at the direction of $E$. In particular case when $E$ is a finite set, and $v=\log|f|$ with an analytic function $f$, our result agrees with the recent one by A. Borichev, L. Golinskii and S. Kupin. An application to contractions close to unitary operators in the Hilbert space is given.


A structural approach to subset-sum problems

Abstract: We discuss a structural approach to subset-sum problems in additive combinatorics. The core of this approach are Freiman-type structural theorems, many of which will be presented through the paper. These results have applications in various areas, such as number theory, combinatorics and mathematical physics.


Poisson automorphisms and quiver moduli

Abstract: A factorization formula for certain automorphisms of a Poisson algebra associated to a quiver is proved, which involves framed versions of moduli spaces of quiver representations. This factorization formula is related to wall-crossing formulas for Donaldson-Thomas type invariants of M. Kontsevich and Y. Soibelman.


A tessellation for algebraic surfaces in CP3

Abstract: We present an explicit algorithm for tessellating the algebraic surfaces (real 4-manifolds) F(n) embedded in CP3 defined by the equation z0^n + z1^n + z2^n + z3^n = 0 in the standard homogeneous coordinates [z0, z1, z2, z3], where n is any positive integer. Note that F(4) in particular is a K3 surface. Our tessellation contains a minimal number of vertices, namely the n-th roots of unity in the six standard projective lines CP1 in CP3, which form an obvious framework for constructing a natural tessellation of F(n). Our tessellation is invariant under the action of the obvious isomorphism group of F(n) induced by permutations and phase multiplications of the coordinates, and the action is transitive on the set of 4-cells. The tessellation is built upon a similar triangulation for the corresponding algebraic curves in CP2.


Stability Functions

Abstract: In this article we discuss the role of stability functions in geometric invariant theory and apply stability function techniques to problems in toric geometry. In particular we show how one can use these techniques to recover results of Burns-Guillemin-Uribe and Shiffman-Tate-Zelditch on asymptotic properties of sections of holomorphic line bundles over toric varieties.


On generators of bounded ratios of minors for totally positive matrices

Abstract: We provide a method for factoring all bounded ratios of the form $$\det A(I_1|I_1')\det A(I_2|I_2')/\det A(J_1|J_1')\det A(J_2|J_2')$$ where $A$ is a totally positive matrix, into a product of more elementary ratios each of which is bounded by 1, thus giving a new proof of Skandera's result. The approach we use generalizes the one employed by Fallat et al. in their work on principal minors. We also obtain a new necessary condition for a ratio to be bounded for the case of non-principal minors.


A generalization of Ostrowski inequality on time scales for k points

Abstract: In this paper we first generalize the Ostrowski inequality on time scales for k points and then unify corresponding continuous and discrete versions. We also point out some particular Ostrowski type inequalities on time scales as special cases.


An Ostrowski-Gruss type inequality on time scales

Abstract: In this paper we derive a new inequality of Ostrowski-Gruss type on time scales and thus unify corresponding continuous and discrete versions. We also apply our result to the quantum calculus case.


Fundamental classes of negatively curved manifolds cannot be represented by products of manifolds

Abstract: Not every singular homology class is the push-forward of the fundamental class of some manifold. In the same spirit, one can study the following problem: Which singular homology classes are the push-forward of the fundamental class of a given type of manifolds? In the present article, we show that the fundamental classes of negatively curved manifolds cannot be represented by a non-trivial product of manifolds. This observation sheds some light on the functorial semi-norm on singular homology given by products of compact surfaces.


A Conversation with Seymour Geisser

Abstract: Seymour Geisser received his bachelor's degree in Mathematics from the City College of New York in 1950, and his M.A. and Ph.D. degrees in Mathematical Statistics at the University of North Carolina in 1952 and 1955, respectively. He then held positions at the National Bureau of Standards and the National Institute of Mental Health until 1961. From 1961 until 1965, he was Chief of the Biometry Section at the National Institute of Arthritis and Metabolic Diseases, and also held the position of Professorial Lecturer at the George Washington University from 1960 to 1965. From 1965 to 1970, he was the founding Chair of the Department of Statistics at the State University of New York, Buffalo, and in 1971, he became the founding Director of the School of Statistics at the University of Minnesota, remaining in that position until 2001. He held visiting professorships at Iowa State University, 1960; University of Wisconsin, 1964; University of Tel-Aviv (Israel), 1971; University of Waterloo (Canada), 1972; Stanford University, 1976, 1977, 1988; Carnegie Mellon University, 1976; University of the Orange Free State (South Africa), 1978, 1993; Harvard University, 1981; University of Chicago, 1985; University of Warwick (England), 1986; University of Modena (Italy), 1996; and National Chiao Tung University (Taiwan), 1998. He was the Lady Davis Visiting Professor, Hebrew University of Jerusalem, 1991, 1994, 1999, and the Schor Scholar, Merck Research Laboratories, 2002-2003. He was a Fellow of the Institute of Mathematical Statistics and the American Statistical Association.


A Conversation with Monroe Sirken

Abstract: Born January 11, 1921 in New York City, Monroe Sirken grew up in a suburb of Pasadena, California. He earned B.A. and M.A. degrees in sociology at UCLA in 1946 and 1947, and a Ph.D. in 1950 in sociology with a minor in mathematics at the University of Washington in 1950 where Professor Z. W. Birnbaum was his mentor and thesis advisor. As a Post-Doctoral Fellow of the Social Science Research Council, Monroe spent 1950--1951 at the Statistics Laboratory, University of California at Berkeley and the Office of the Assistant Director for Research, U.S. Bureau of the Census in Suitland, Maryland. Monroe visited the Census Bureau at a time of great change in the use of sampling and survey methods, and decided to remain. He began his government career there in 1951 as a mathematical statistician, and moved to the National Office of Vital Statistics (NOVS) in 1953 where he was an actuarial mathematician and a mathematical statistician. He has held a variety of research and administrative positions at the National Center for Health Statistics (NCHS) and he was the Associate Director, Research and Methodology and the Director, Office of Research and Methodology until 1996 when he became a senior research scientist, the title he currently holds. Aside from administrative responsibilities, Monroe's major professional interests have been conducting and fostering survey and statistical research responsive to the needs of federal statistics. His interest in the design of rare and sensitive population surveys led to the development of network sampling which improves precision by linking multiple selection units to the same observation units. His interest in fostering research on the cognitive aspects of survey methods led to the establishment of permanent questionnaire design research laboratories, first at NCHS and later at other federal statistical agencies here and abroad.


Poisson type generators for L^1(R)

Abstract: We characterize the discrete sets L of the real line such that {f(t-l), l in L} span L^1(R), f being an L^1(R)-function whose Fourier transform behaves like the one of the Poisson function.


Model spaces results for the Gabor and Wavelet transforms

Abstract: We prove that the unique Gabor atom with analytical model space is the Gaussian function. We give an analogous result for the wavelet transform. For the general case we give a new approach to study the irregular Gabor and wavelet frames. We improve some results for Gabor atoms in the Feichtinger algebra, and for a special class of wavelets.


Torus fibrations and localization of index I

Abstract: We define a local Riemann-Roch number for an open symplectic manifold when a complete integrable system without Bohr-Sommerfeld fiber is provided on its end. In particular when a structure of a singular Lagrangian fibration is given on a closed symplectic manifold, its Riemann-Roch number is described as the sum of the number of nonsingular Bohr-Sommerfeld fibers and a contribution of the singular fibers. A key step of the proof is formally explained as a version of Witten's deformation applied to a Hilbert bundle.


Milnor $K$-group attached to a torus and Birch-Tate conjecture

Abstract: We formulate (and prove under a certain assumption) a conjecture relating the order of Somekawa's Milnor $K$-group attached to a torus $T$ and the value of the Artin $L$-function attached to the cocharacter group of $T$ (regarded as an Artin representation) at $s=-1$. The case $T=\G_m$ reduces to the classical Birch-Tate conjecture.


Lifts of matroid representations over partial fields

Abstract: There exist several theorems which state that when a matroid is representable over distinct fields F_1,...,F_k, it is also representable over other fields. We prove a theorem, the Lift Theorem, that implies many of these results. First, parts of Whittle's characterization of representations of ternary matroids follow from our theorem. Second, we prove the following theorem by Vertigan: if a matroid is representable over both GF(4) and GF(5), then it is representable over the real numbers by a matrix such that the absolute value of the determinant of every nonsingular square submatrix is a power of the golden ratio. Third, we give a characterization of the 3-connected matroids having at least two inequivalent representations over GF(5). We show that these are representable over the complex numbers. Additionally we provide an algebraic construction that, for any set of fields F_1,...,F_k, gives the best possible result that can be proven using the Lift Theorem.


Optimal stopping for L\'evy processes and affine functions

Abstract: This paper studies an optimal stopping problem for L\'evy processes. We give a justification of the form of the Snell envelope using standard results of optimal stopping. We also justify the convexity of the value function, and without a priori restriction to a particular class of stopping times, we deduce that the smallest optimal stopping time is necessarily a hitting time. We propose a method which allows to obtain the optimal threshold. Moreover this method allows to avoid long calculations of the integro-differential operatorused in the usual proofs.


A characterization of double covers of curves in terms of the ample cone of second symmetric product

Abstract: We investigate the nef cone spanned by the diagonal and the fibre classes of second symmetric product of a curve of genus $g$. This 2-dimensional nef cone gives a characterization of double covers of curves of genus $\le \frac{g-1}{8}$. This is a generalization of a result by Debarre.


Simulation of stochastic reaction-diffusion processes on unstructured meshes

Abstract: Stochastic chemical systems with diffusion are modeled with a reaction-diffusion master equation. On a macroscopic level, the governing equation is a reaction-diffusion equation for the averages of the chemical species. On a mesoscopic level, the master equation for a well stirred chemical system is combined with Brownian motion in space to obtain the reaction-diffusion master equation. The space is covered by an unstructured mesh and the diffusion coefficients on the mesoscale are obtained from a finite element discretization of the Laplace operator on the macroscale. The resulting method is a flexible hybrid algorithm in that the diffusion can be handled either on the meso- or on the macroscale level. The accuracy and the efficiency of the method are illustrated in three numerical examples inspired by molecular biology.


Principal basis in Cartan subalgebra

Abstract: Let $\Lie{g}$ be a simple complex Lie algebra and $\Lie{h}$ a Cartan subalgebra. In this article we explain how to obtain the principal basis of $\Lie{h}$ starting form a set of generators $\{p_1,...,p_r\}$,$r=\rank(\Lie{g})$, of the invariants polynomials $\Sgg$. For each invariant polynomial $p$, we define a $G$-equivariant map $Dp$ form $\Lie{g}$ to $\Lie{g}$. We show that the Gram-Schmidt orthogonalization of the elements $\{Dp_1(\rho^\vee), ... Dp_r(\rho^\vee) \}$ gives the principal basis of $\Lie{h}$. Similarly the orthogonalization of the elements $\{Dp_1(\rho), >... Dp_r(\rho) \}$ produces the principal basis of the Cartan subalgebra of $\lLie{g}$, the Langlands dual of $\Lie{g}$.


On the H\"ormander multiplier theorem and modulation spaces

Abstract: It is known that the Sobolev space $L^2_s$ with $s>n/2$ appeared in the H\"ormander multiplier theorem can be replaced by the Besov space $B^{2,1}_{n/2}$. On the other hand, the Besov space $B_{n/2}^{2,1}$ is continuously embedded in the modulation space $M^{2,1}_0$. In this paper, we consider the problem whether we can replace $B_{n/2}^{2,1}$ by $M^{2,1}_0$.


Geometry of Carnot--Carath\'{e}odory Spaces, Differentiability and Coarea Formula

Abstract: We give a simple proof of Gromov's Theorem on nilpotentization of vector fields, and exhibit a new method for obtaining quantitative estimates of comparing geometries of two different local Carnot groups in Carnot--Carath\'{e}odory spaces with $C^{1,\alpha}$-smooth basis vector fields, $\alpha\in[0,1]$. From here we obtain the similar estimates for comparing geometries of a Carnot--Carath\'{e}odory space and a local Carnot group. These two theorems imply basic results of the theory: Gromov type Local Approximation Theorems, and for $\alpha>0$ Rashevski\v{\i}-Chow Theorem and Ball--Box Theorem, etc. We apply the obtained results for proving $hc$-differentiability of mappings of Carnot--Carath\'{e}odory spaces with continuous horizontal derivatives. The latter is used in proving the coarea formula for some classes of contact mappings of Carnot--Carath\'{e}odory spaces.


The Anti-Symmetric GUE Minor Process

Abstract: Our study is initiated by a multi-component particle system underlying the tiling of a half hexagon by three species of rhombi. In this particle system species $j$ consists of $\lfloor j/2 \rfloor$ particles which are interlaced with neigbouring species. The joint probability density function (PDF) for this particle system is obtained, and is shown in a suitable scaling limit to coincide with the joint eigenvalue PDF for the process formed by the successive minors of anti-symmetric GUE matrices, which in turn we compute from first principles. The correlations for this process are determinantal and we give an explicit formula for the corresponding correlation kernel in terms of Hermite polynomials. Scaling limits of the latter are computed, giving rise to the Airy kernel, extended Airy kernel and bead kernel at the soft edge and in the bulk, as well as a new kernel at the hard edge.


Entire functions of exponential type, almost periodic in Besicovitch's sense on the real hyperplane

Abstract: Suppose that an almost periodic in Besicovitch's sense function $f(x)$ of several variables is the restriction to the real hyperplane of an entire function of exponential type $b$. Then its spectrum is contained in the ball of radius $b$ with the center in the origin.


Cluster algebras of finite type via Coxeter elements and principal minors

Abstract: We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in the simply connected semisimple algebraic group of the same Cartan-Killing type. In this realization, the cluster variables appear as certain (generalized) principal minors.


A proof of the Dalang-Morton-Willinger theorem

Abstract: We give a new proof of the Dalang-Morton-Willinger theorem, relating the no-arbitrage condition in stochastic securities market models to the existence of an equivalent martingale measure with bounded density for a $d$-dimensional stochastic sequence $(S_n)_{n=0}^N$ of stock prices. Roughly speaking, the proof is reduced to the assertion that under the no-arbitrage condition for N=1 and $S\in L^1$ there exists a strictly positive linear fucntional on $L^1$, which is bounded from above on a special subset of the subspace $K\subset L^1$ of investor's gains.


R-boundedness of smooth operator-valued functions

Abstract: In this paper we study $R$-boundedness of operator families $\mathcal{T}\subset \calL(X,Y)$, where $X$ and $Y$ are Banach spaces. Under cotype and type assumptions on $X$ and $Y$ we give sufficient conditions for $R$-boundedness. In the first part we show that certain integral operator are $R$-bounded. This will be used to obtain $R$-boundedness in the case that $\mathcal{T}$ is the range of an operator-valued function $T:\R^d\to \calL(X,Y)$ which is in a certain Besov space $B^{d/r}_{r,1}(\R^d;\calL(X,Y))$. The results will be applied to obtain $R$-boundedness of semigroups and evolution families, and to obtain sufficient conditions for existence of solutions for stochastic Cauchy problems.