Gamma-invariant ideals in Iwasawa algebras

Abstract: Let kG be the completed group algebra of a uniform pro-p group G with coefficients in a field k of characteristic p. We study right ideals I in kG that are invariant under the action of another uniform pro-p group Gamma. We prove that if I is non-zero then an irreducible component of the characteristic support of kG/I must be contained in a certain finite union of rational linear subspaces of Spec gr kG. The minimal codimension of these subspaces gives a lower bound on the homological height of I in terms of the action of a certain Lie algebra on G/G^p. If we take Gamma to be G acting on itself by conjugation, then Gamma-invariant right ideals of kG are precisely the two-sided ideals of kG, and we obtain a non-trivial lower bound on the homological height of a possible non-zero two-sided ideal. For example, when G is open in SL_n(\Zp) this lower bound equals 2n - 2. This gives a significant improvement of the results of Ardakov, Wei and Zhang on reflexive ideals in Iwasawa algebras.


A new secant method for unconstrained optimization

Abstract: We present a gradient-based algorithm for unconstrained minimization derived from iterated linear change of basis. The new method is equivalent to linear conjugate gradient in the case of a quadratic objective function. In the case of exact line search it is a secant method. In practice, it performs comparably to BFGS and DFP and is sometimes more robust.


Fitting Martingales To Given Marginals

Abstract: We consider the problem of finding a real valued martingale fitting specified marginal distributions. For this to be possible, the marginals must be increasing in the convex order and have constant mean. We show that, under the extra condition that they are weakly continuous, the marginals can always be fitted in a unique way by a martingale which lies in a particular class of strong Markov processes. It is also shown that the map that this gives from the sets of marginal distributions to the martingale measures is continuous. Furthermore, we prove that it is the unique continuous method of fitting martingale measures to the marginal distributions.


The Penrose Transform for Complex Projective Space

Abstract: Various complexes of differential operators are constructed on complex projective space via the Penrose transform, which also computes their cohomology.


Distortion elements in $Diff^\infty(R/Z)$

Abstract: We consider the group of smooth diffeomorphisms of the circle. We show that any recurrent $f$ (in the sense that $\{f^n\}_{n \in Z}$ is not discrete) is in fact a distortion element (in the sense that its iterates can be written as short compositions involving finitely many smooth diffeomorphisms). Thus rotations are distortion elements.


Khovanov homology, open books, and tight contact structures

Abstract: We define the reduced Khovanov homology of an open book (S,h), and we identify a distinguished "contact element" in this group which may be used to establish the tightness or non-fillability of contact structures compatible with (S,h). Our construction generalizes the relationship between the reduced Khovanov homology of a link and the Heegaard Floer homology of its branched double cover. As an application, we give combinatorial proofs of tightness for several contact structures which are not Stein-fillable. Lastly, we investigate a comultiplication structure on the reduced Khovanov homology of an open book which parallels the comultiplication on Heegaard Floer homology defined previously by the first author.


Decomposable Principal Component Analysis

Abstract: We consider principal component analysis (PCA) in decomposable Gaussian graphical models. We exploit the prior information in these models in order to distribute its computation. For this purpose, we reformulate the problem in the sparse inverse covariance (concentration) domain and solve the global eigenvalue problem using a sequence of local eigenvalue problems in each of the cliques of the decomposable graph. We demonstrate the application of our methodology in the context of decentralized anomaly detection in the Abilene backbone network. Based on the topology of the network, we propose an approximate statistical graphical model and distribute the computation of PCA.


Le probl\`eme des diviseurs pour des formes binaires de degr\'e 4

Abstract: We study the average order of the divisor function, as it ranges over the values of binary quartic forms that are reducible over the rationals.


A Rejoinder to Mackintosh and some Remarks on the Concept of General Intelligence

Abstract: In 2000 Nicholas J. Mackintosh (2000) published an article in "Nature" referring to the concept of general intelligence ("g") claiming that there is clear empirical evidence for the existence of the g factor and psychologists are "united in their support of g". Surprisingly, his view remained yet unchallenged although this issue is by no means as clear-cut as Mackintosh argues. Let us therefore attempt to clarify some common but unfortunately major misconceptions about g, which Mackintosh, following Jensen's (1998) precedent, recounted in his "Nature" article. The bottom line is that Spearman's g does not exist, that this has been known and acknowledged by leading scholars (Guttman, 1992; Thurstone, 1947) of factor analysis for decades so that the task of objectively defining human intelligence remains unfinished.


A spline interpretation of Eulerian numbers

Abstract: In this paper, we explore the interrelationship between Eulerian numbers and B splines. Specifically, using B splines, we give the explicit formulas of the refined Eulerian numbers, and descents polynomials. Moreover, we prove that the coefficients of descent polynomials $D_d^n(t)$ are log-concave. This paper also provides a new approach to study Eulerian numbers and descent polynomials.


Cluster-tilted algebras and their intermediate coverings

Abstract: We construct the intermediate coverings of cluster-tilted algebras by defining the generalized cluster categories. These generalized cluster categories are Calabi-Yau triangulated categories with fraction CY-dimension and have also cluster tilting objects (subcategories). Furthermore we study the representations of these intermediate coverings of cluster-tilted algebras.


Complete minimal surfaces in R3 with a prescribed coordinate function

Abstract: In this paper we construct complete simply connected minimal surfaces with a prescribed coordinate function. Moreover, we prove that these surfaces are dense in the space of all minimal surfaces with this coordinate function (with the topology of the smooth convergence on compact sets).


Strong Spectral Gaps for Compact Quotients of Products of $\PSL(2,\bbR)$

Abstract: The existence of a strong spectral gap for quotients $\Gamma\bs G$ of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the Ramanujan-Selberg Conjectures. If $G$ has no compact factors then for general lattices a strong spectral gap can still be established, however, there is no uniformity and no effective bounds are known. This note is concerned with the strong spectral gap for an irreducible co-compact lattice $\Gamma$ in $G=\PSL(2,\bbR)^d$ for $d\geq 2$ which is the simplest and most basic case where the congruence subgroup property is not known. The method used here gives effective bounds for the spectral gap in this setting.


On finite-index extensions of subgroups of free groups

Abstract: We study the lattice of finite-index extensions of a given finitely generated subgroup $H$ of a free group $F$. This lattice is finite and we give a combinatorial characterization of its greatest element, which is the commensurator of $H$. This characterization leads to a fast algorithm to compute the commensurator, which is based on a standard algorithm from automata theory. We also give a sub-exponential and super-polynomial upper bound for the number of finite-index extensions of $H$, and we give a language-theoretic characterization of the lattice of finite-index subgroups of $H$. Finally, we give a polynomial time algorithm to compute the malnormal closure of $H$.


How to Draw Tropical Planes

Abstract: The tropical Grassmannian parameterizes tropicalizations of linear spaces, while the Dressian parameterizes all planes in $\TP^{n-1}$. We study these parameter spaces and we compute them explicitly for $n \leq 7$. Planes are identified with matroid subdivisions and with arrangements of trees. These representations are used to draw pictures.


Weighted Hardy and singular operators in Morrey spaces

Abstract: We study the weighted boundedness of the Cauchy singular integral operator $S_\Gm$ in Morrey spaces $L^{p,\lambda}(\Gm)$ on curves satisfying the arc-chord condition, for a class of "radial type" almost monotonic weights. The non-weighted boundedness is shown to hold on an arbitrary Carleson curve. We show that the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces $L^{p,\lambda}(0,\ell), \ell>0$. We find conditions for weighted Hardy operators to be bounded in Morrey spaces. To cover the case of curves we also extend the boundedness of the Hardy-Littlewood maximal operator in Morrey spaces, known in the Euclidean setting, to the case of Carleson curves. Key words and phrases: Morrey space, singular operator, Hardy operator, Hardy-Littlewood maximal operator, weighted estimate.


Fat-Tailed Distributions and Levy Processes

Abstract: The notion that natural disasters can be controlled is, of course, farcical; history is permeated with examples of countless failed attempts at this pointless task; it is synonymous with trying to build a perpetual motion machine. Nonetheless, there are ways to reduce their impact on human communities, particularly by looking away from the normal hypothesis.


Rankin's method and Jacobi forms of several variables

Abstract: Following Rankin's method, D. Zagier computed the $n$-th Rankin-Cohen bracket of a modular form $g$ of weight $k_1$ with the Eisenstein series of weight $k_2$ and then computed the inner product of this Rankin-Cohen bracket with a cusp form $f$ of weight $k = k_1+k_2+2n$ and showed that this inner product gives, upto a constant, the special value of the Rankin-Selberg convolution of $f$ and $g$. This result was generalized to Jacobi forms of degree 1 by Y. Choie and W. Kohnen. In this paper, we generalize this result to Jacobi forms defined over ${\mathcal H} \times {\mathbb C}^{(g, 1)}$.


Tropical R maps and Affine Geometric Crystals

Abstract: By modifying the method in [KNO], certain affine geometric crystals are realized in affinization of the fundamental representation $W(\varpi_1)_l$ and the tropical R maps for the affine geometric crystals are described explicitly. We also define prehomogeneous geometric crystals and show that for a positive geometric crystal, the connectedness of the corresponding ultra-discretized crystal is the sufficient condition for prehomogeneity of the positive geometric crystal. Moreover, the uniqueness of tropical R maps is discussed.


Embedding operators into strongly continuous semigroups

Abstract: We study linear operators $T$ on Banach spaces for which there exists a $C_0$-semigroup $(T(t))_{t\geq 0}$ such that $T=T(1)$. We present a necessary condition in terms of the spectral value 0 and give classes of examples where this can or cannot be achieved.


Graphs, Frobenius functionals, and the classical Yang-Baxter equation

Abstract: A Lie algebra is Frobenius if it admits a linear functional F such that the Kirillov form F([x,y]) is non-degenerate. If g is the m-th maximal parabolic subalgebra P(n,m) of sl(n) this occurs precisely when (n,m) = 1. We define a "cyclic" functional F on P(n,m) and prove it is non-degenerate using properties of certain graphs associated to F. These graphs also provide in some cases readily computable associated solutions of the classical Yang-Baxter equation. We also define a local ring associated to each connected loopless graph from which we show that the graph can be reconstructed. Finally, we examine the seaweed Lie algebras of Dergachev and Kirillov from our perspective.


Maximal regularity for parabolic partial differential equations on manifolds with cylindrical ends

Abstract: We give a short, simple proof of maximal regularity for linear parabolic evolution equations on manifolds with cylindrical ends by making use of pseudodifferential parametrices and the concept of R-boundedness for the resolvent.


Simple Cohen-Macaulay Codimension 2 Singularities

Abstract: In this article, we provide a complete list of simple Cohen-Macaulay codimension 2 singularities together with a list of adjacencies which is complete in the case of fat point and space curve singularities.


Rigidity and uniruling for Lagrangian submanifolds

Abstract: This paper explores the topology of monotone Lagrangian submanifolds $L$ inside a symplectic manifold $M$ by exploiting the relationships between the quantum homology of $M$ and various quantum structures associated to the Lagrangian $L$.


Strong generic vanishing and a higher dimensional Castelnuovo-de Franchis inequality

Abstract: We extend to manifolds of arbitrary dimension the Castelnuovo-de Franchis inequality for surfaces. The proof is based on the theory of Generic Vanishing and higher regularity, and on the Evans-Griffith Syzygy Theorem in commutative algebra. Along the way we give a positive answer, in the setting of K\"ahler manifolds, to a question of Green-Lazarsfeld on the vanishing of higher direct images of Poincar\'e bundles. We indicate generalizations to arbitrary Fourier-Mukai transforms.


Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation

Abstract: We give new proofs of two basic results in number theory: The law of quadratic reciprocity and the sign of the Gauss sum. We show that these results are encoded in the relation between the discrete Fourier transform and the action of the Weyl element in the Weil representation modulo p,q and pq.


Multivariable operator-valued Nevanlinna-Pick interpolation: a survey

Abstract: The theory of Nevanlinna-Pick and Carath\'eodory-Fej\'er interpolation for matrix- and operator-valued Schur class functions on the unit disk is now well established. Recent work has produced extensions of the theory to a variety of multivariable settings, including the ball and the polydisk (both commutative and noncommutative versions), as well as a time-varying analogue. Largely independent of this is the recent Nevanlinna-Pick interpolation theorem by P.S. Muhly and B. Solel for an abstract Hardy algebra set in the context of a Fock space built from a $W^*$-correspondence E over a $W^{*}$-algebra $\cA$ and a *-representation $\sigma$ of $\cA$. In this review we provide an exposition of the Muhly-Solel interpolation theory accessible to operator theorists, and explain more fully the connections with the already existing interpolation literature. The abstract point evaluation first introduced by Muhly-Solel leads to a tensor-product type functional calculus in the main examples. A second kind of point-evaluation for the $W^*$-correspondence Hardy algebra, also introduced by Muhly and Solel, is here further investigated, and a Nevanlinna-Pick theorem in this setting is proved. It turns out that, when specified for examples, this alternative point-evaluation leads to an operator-argument functional calculus and corresponding Nevanlinna-Pick interpolation. We also discuss briefly several Nevanlinna-Pick interpolation results for Schur classes that do not fit into the Muhly-Solel $W^*$-correspondence formalism.


Operator-Valued $L_{q}\to L_{p}$ Fourier Multipliers

Abstract: This paper is withdrawn due to some gaps in the proof


On Exceptional Collections on Grassmannians

Abstract: The author was informed that the result in the original version had been obtained earlier by K. Ueda (arXiv:math/0503355 [math.AG]). The paper is retracted.


SO(3) quantum invariants are dense

Abstract: We show that when $r \geq 5$ is prime, the SO(3) Witten-Reshetikhin-Turaev quantum invariants for three-manifolds at the level $r$ form a dense set in the complex plane. This confirms a conjecture of Larsen and Wang.


Algebraic points of small height missing a union of varieties

Abstract: Let $K$ be a number field, $\overline{\mathbb Q}$, or the field of rational functions on a smooth projective curve over a perfect field, and let $V$ be a subspace of $K^N$, $N \geq 2$. Let $Z_K$ be a union of varieties defined over $K$ such that $V \nsubseteq Z_K$. We prove the existence of a point of small height in $V \setminus Z_K$, providing an explicit upper bound on the height of such a point in terms of the height of $V$ and the degree of a hypersurface containing $Z_K$, where dependence on both is optimal. This generalizes and improves upon the previous results of the author. As a part of our argument, we provide a basic extension of the function field version of Siegel's lemma of J. Thunder to an inequality with inhomogeneous heights. As a corollary of the method, we derive an explicit lower bound for the number of algebraic integers of bounded height in a fixed number field.


Relaxed commutant lifting: existence of a unique solution

Abstract: In this paper we present necessary and sufficient conditions for the existence of a unique solution to the relaxed commutant lifting problem. The obtained conditions are more complicated than those for the classical commutant lifting setting, and earlier obtained sufficient conditions turn out not to be necessary conditions. It is also shown that these conditions simplify in certain special cases.


Inequalities for analytic functions with the derivative in H1

Abstract: It is proved an inequality - integrated analogue of the Hardy inequality and as application simplified proof of the theorem of S. A. Vinogradov for the bounded Toeplitz operators on the space of functions analytic and bounded in the unit disc is given.


Commuting-Square Subfactors and Central Sequences

Abstract: Let $M_0 \subset M_1$ be a finite-index infinite-depth hyperfinite $II_1$ subfactor and $\omega$ a free ultrafilter of the natural numbers. We show that if this subfactor is constructed from a commuting square then the central sequence inclusion $M_0^{\omega} \cap M_1' \subset (M_0)_{\omega}$ has infinite Pimnser-Popa index. We will also demonstrate this result for certain infinite-depth hyperfinite subfactors coming from groups.


Improved lower bound on the size of Kakeya sets over finite fields

Abstract: In a recent breakthrough, Dvir showed that every Kakeya set in $\F^n$ must be of cardinality at least $c_n |\F|^n$ where $c_n \approx 1/n!$. We improve this lower bound to $\beta^n |\F|^n$ for a constant $\beta > 0$. This pins down the growth of the leading constant to the right form as a function of $n$.


The Conjugacy Problem in the Grigorchuk Group is polynomial time decidable

Abstract: In this paper we prove that the Conjugacy Problem in the Grigorchuk group $\Gamma$ has polynomial time complexity.


An observation about submatrices

Abstract: Let M be an arbitrary Hermitian matrix of order n, and k be a positive integer less than or equal to n. We show that if k is large, the distribution of eigenvalues on the real line is almost the same for almost all principal submatrices of M of order k. The proof uses results about random walks on symmetric groups and concentration of measure. In a similar way, we also show that almost all k x n submatrices of M have almost the same distribution of singular values.


Unification theorems in algebraic geometry

Abstract: In this paper, for a given finitely generated algebra (an algebraic structure with arbitrary operations and no predicates) A we study finitely generated limit algebras of A, approaching them via model theory and algebraic geometry. Along the way we lay down foundations of algebraic geometry over arbitrary algebraic structures.


A Geometrical Approach to Hilbert-Schmidt Operators

Abstract: We give a Riemannian structure to the set $\Sigma$ of positive invertible unitized Hilbert-Schmidt operators, by means of the trace inner product. This metric makes of $\Sigma$ a nonpositively curved, simply connected and metrically complete Hilbert manifold. The manifold $\Sigma$ is a universal model for symmetric spaces of the noncompact type: any such space can be isometrically embedded into $\Sigma$. We give an intrinsic algebraic characterization of convex closed submanifolds $M$. We study the group of isometries of such submanifolds: we prove that $G_M$, the Banach-Lie group generated by $M$, acts isometrically and transitively on $M$. Moreover, $G_M$ admits a polar decomposition relative to $M$, namely $G_M\simeq M\times K$ as Hilbert manifolds (here $K$ is the isotropy of $p=1$ for the action $I_g: p\mapsto gpg^*$), and also $G_M/K\simeq M$ so $M$ is an homogeneous space. We obtain several decomposition theorems by means of geodesically convex submanifolds $M$. These decompositions are obtained \textit{via} a nonlinear but analytic orthogonal projection $\Pi_M:\Sigma\to M$, a map which is a contraction for the geodesic distance. As a byproduct, we prove the isomorphism $NM\simeq\Sigma$ (here $NM$ stands for the normal bundle of a convex closed submanifold $M$). Writing down the factorizations for fixed ${\rm e}^a$, we obtain ${\rm e}^a={\rm e}^x{\rm e}^v{\rm e}^x$ with ${\rm e}^x\in M$ and $v$ orthogonal to $M$ at $p=1$. As a corollary we obtain decompositions for the full group of invertible elements $G\simeq M\times \exp(T_1M^{\perp})\times K$.


Hopf-Rinow Theorem in the Sato Grassmannian

Abstract: Let $U_2({\cal H})$ be the Banach-Lie group of unitary operators in the Hilbert space ${\cal H}$ which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit $$\{upu^*: u\in U_2({\cal H})\},$$ of an infinite projection $p$ in ${\cal H}$. This orbit coincides with the connected component of $p$ in the Hilbert-Schmidt restricted Grassmannian $Gr_{res}(p)$ (also known in the literature as the Sato Grassmannian) corresponding to the polarization ${\cal H}=p({\cal H})\oplus p({\cal H})^\perp$. It is known that the components of $Gr_{res}(p)$ are differentiable manifolds. Here we give a simple proof of the fact that $Gr_{res}^0(p)$ is a smooth submanifold of the affine Hilbert space $p+{\cal B}_2({\cal H})$, where ${\cal B}_2({\cal H})$ denotes the space of Hilbert-Schmidt operators of ${\cal H}$. We prove that the geodesics of the natural connection, which are of the form $\gamma(t)=e^{tz}pe^{-tz}$, for $z$ a $p$-codiagonal anti-hermitic element of ${\cal B}_2({\cal H})$, have minimal length provided that $\|z\|\le \pi/2$. Note that the condition is given in terms of the usual operator norm, a fact which implies that there exist minimal geodesics of arbitrary length. Also we show that any two points $p_1,p_2\in Gr_{res}^0(p)$ are joined by a minimal geodesic. If moreover $\|p_1-p_2\|<1$, the minimal geodesic is unique. Finally, we replace the 2-norm by the $k$-Schatten norm ($k>2$), and prove that the geodesics are also minimal for these norms, up to a critical value of $t$, which is estimated also in terms of the usual operator norm. In the process, minimality results in the $k$-norms are also obtained for the group $U_2({\cal H})$.


Weak Riemannian manifolds from finite index subfactors

Abstract: Let $N\subset M$ be a finite Jones' index inclusion of II$_1$ factors, and denote by $U_N\subset U_M$ their unitary groups. In this paper we study the homogeneous space $U_M/U_N$, which is a (infinite dimensional) differentiable manifold, diffeomorphic to the orbit $$ {\cal O}(p) =\{u p u^*: u\in U_M\} $$ of the Jones projection $p$ of the inclusion. We endow ${\cal O}(p) $ with a Riemannian metric, by means of the trace on each tangent space. These are pre-Hilbert spaces (the tangent spaces are not complete), therefore ${\cal O}(p)$ is a weak Riemannian manifold. We show that ${\cal O}(p)$ enjoys certain properties similar to classic Hilbert-Riemann manifolds. Among them, metric completeness of the geodesic distance, uniqueness of geodesics of the Levi-Civita connection as minimal curves, and partial results on the existence of minimal geodesics. For instance, around each point $p_1$ of ${\cal O}(p)$, there is a ball $\{q\in {\cal O}(p):\|q-p_1\|<r\}$ (of uniform radius $r$) of the usual norm of $M$, such that any point $p_2$ in the ball is joined to $p_1$ by a unique geodesic, which is shorter than any other piecewise smooth curve lying inside this ball. We also give an intrinsic (algebraic) characterization of the directions of degeneracy of the submanifold inclusion ${\cal O}(p)\subset {\cal P}(M_1)$, where the last set denotes the Grassmann manifold of the von Neumann algebra generated by $M$ and $p$.


On Integral Operators with Operator Valued Kernels

Abstract: Here Lq-Lp boundedness of integral operator with operator-valued kernels is studied and the main result is applied to convolution operators. Using these results Besov space regularity for Fourier multiplier operator is established.


Ekedahl-Oort strata and Kottwitz-Rapoport strata

Abstract: We study the moduli space A_g of g-dimensional principally polarized abelian varieties in positive characteristic, and its variant A_I with Iwahori level structure. Both supersingular Ekedahl-Oort strata and supersingular Kottwitz-Rapoport strata are isomorphic to disjoint unions of Deligne-Lusztig varieties (see [Hoeve 2008] and [Goertz, Yu 2008], resp.). Here we compare these isomorphisms. We also give an explicit description of Kottwitz-Rapoport strata contained in the supersingular locus in the general parahoric case. Finally, we show that every Ekedahl-Oort stratum is isomorphic to a parahoric Kottwitz-Rapoport stratum.


Vector partition functions and index of transversally elliptic operators

Abstract: Let G be a torus acting linearly on a complex vector space M, and let X be the list of weights of G in M. We determine the equivariant K-theory of the open subset of M consisting of points with finite stabilizers. We identify it to the space DM(X) of functions on the lattice of weights of G, satisfying the cocircuit difference equations associated to X, introduced by Dahmen--Micchelli in the context of the theory of splines in order to study vector partition functions. This allows us to determine the range of the index map from G-transversally elliptic operators on M to generalized functions on G and to prove that the index map is an isomorphism on the image. This is a setting studied by Atiyah-Singer which is in a sense universal for index computations.


A bicategorical version of Masuoka's theorem. Applications to bimodules over functor categories and to firm bimodules

Abstract: We give a bicategorical version of the main result of A. Masuoka ({Corings and invertible bimodules,} {\em Tsukuba J. Math.} \textbf{13} (1989), 353--362) which proposes a non-commutative version of the fact that for a faithfully flat extension of commutative rings $R \subseteq S$, the relative Picard group $Pic(S/R)$ is isomorphic to the Amitsur 1--cohomology group $H^1(S/R,U)$ with coefficients in the units functor $U$.


Galois sections for abelianized fundamental groups

Abstract: Given a smooth projective curve $X$ of genus at least 2 over a number field $k$, Grothendieck's Section Conjecture predicts that the canonical projection from the \'etale fundamental group of $X$ onto the absolute Galois group of $k$ has a section if and only if the curve has a rational point. We show that there exist curves where the above map has a section over each completion of $k$ but not over $k$. In the appendix Victor Flynn gives explicit examples in genus 2. Our result is a consequence of a more general investigation of the existence of sections for the projection of the \'etale fundamental group `with abelianized geometric part' onto the Galois group. We give a criterion for the existence of sections in arbitrary dimension and over arbitrary perfect fields, and then study the case of curves over local and global fields more closely. We also point out the relation to the elementary obstruction of Colliot-Th\'el\`ene and Sansuc.


Equivalence of robust stabilization and robust performance via feedback

Abstract: One approach to robust control for linear plants with structured uncertainty as well as for linear parameter-varying (LPV) plants (where the controller has on-line access to the varying plant parameters) is through linear-fractional-transformation (LFT) models. Control issues to be addressed by controller design in this formalism include robust stability and robust performance. Here robust performance is defined as the achievement of a uniform specified $L^{2}$-gain tolerance for a disturbance-to-error map combined with robust stability. By setting the disturbance and error channels equal to zero, it is clear that any criterion for robust performance also produces a criterion for robust stability. Counter-intuitively, as a consequence of the so-called Main Loop Theorem, application of a result on robust stability to a feedback configuration with an artificial full-block uncertainty operator added in feedback connection between the error and disturbance signals produces a result on robust performance. The main result here is that this performance-to-stabilization reduction principle must be handled with care for the case of dynamic feedback compensation: casual application of this principle leads to the solution of a physically uninteresting problem, where the controller is assumed to have access to the states in the artificially-added feedback loop. Application of the principle using a known more refined dynamic-control robust stability criterion, where the user is allowed to specify controller partial-state dimensions, leads to correct robust-performance results. These latter results involve rank conditions in addition to Linear Matrix Inequality (LMI) conditions.


Stickelberger elements and Kolyvagin systems

Abstract: In this paper, we construct (many) Kolyvagin systems out of Stickelberger elements, utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. We show how to apply this construction to prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and deduce the main conjecture of Iwasawa theory for totally real fields (for totally odd characters). Although the main results of this paper have already been established by Wiles, our approach provides another example (which slightly differs from the case of Stark elements) on how to study Kolyvagin systems of core rank r > 1 (in the sense of Mazur and Rubin). Also, by making use of the 'rigidity' of the collection of Kolyvagin systems, we establish a link between the Stickelberger elements and the Rubin-Stark elements.


Degree formula for connective K-theory

Abstract: We apply the degree formula for connective $K$-theory to study rational contractions of algebraic varieties. Examples include rationally connected varieties and complete intersections.


Universal Malliavin Calculus in Fock and L\'{e}vy-It\^{o} Spaces

Abstract: We review and extend Lindsay's work on abstract gradient and divergence operators in Fock space over a general complex Hilbert space. Precise expressions for the domains are given, the $L^2$-equivalence of norms is proved and an abstract version of the It\^{o}-Skorohod isometry is established. We then outline a new proof of It\^{o}'s chaos expansion of complex L\'{e}vy-It\^{o} space in terms of multiple Wiener-L\'{e}vy integrals based on Brownian motion and a compensated Poisson random measure. The duality transform now identifies L\'{e}vy-It\^{o} space as a Fock space. We can then easily obtain key properties of the gradient and divergence of a general L\'{e}vy process. In particular we establish maximal domains of these operators and obtain the It\^{o}-Skorohod isometry on its maximal domain.


Quantum groups and quantization of Weyl group symmetries of Painlev\'e systems

Abstract: We shall construct the quantized q-analogues of the birational Weyl group actions arising from nilpotent Poisson algebras, which are conceptual generalizations, proposed by Noumi and Yamada, of the B\"acklund transformations for Painlev\'e equations. Consider a quotient Ore domain of the lower nilpotent part of a quantized universal enveloping algebra of arbitrary symmetrizable Kac-Moody type. Then non-integral powers of the image of the Chevalley generators generate the quantized q-analogue of the birational Weyl group action. Using the same method, we shall reconstruct the quantized B\"acklund transformations of q-Painlev\'e equations constructed by Hasegawa. We shall also prove that any subquotient integral domain of a quantized universal enveloping algebra of finite or affine type is an Ore domain.


On Bogovski\u{\i} and regularized Poincar\'e integral operators for de Rham complexes on Lipschitz domains

Abstract: We study integral operators related to a regularized version of the classical Poincar\'e path integral and the adjoint class generalizing Bogovski\u{\i}'s integral operator, acting on differential forms in $R^n$. We prove that these operators are pseudodifferential operators of order -1. The Poincar\'e-type operators map polynomials to polynomials and can have applications in finite element analysis. For a domain starlike with respect to a ball, the special support properties of the operators imply regularity for the de Rham complex without boundary conditions (using Poincar\'e-type operators) and with full Dirichlet boundary conditions (using Bogovski\u{\i}-type operators). For bounded Lipschitz domains, the same regularity results hold, and in addition we show that the cohomology spaces can always be represented by $C^\infty$ functions.


Explicit double shuffle relations and a generalization of Euler's decomposition formula

Abstract: We give an explicit formula for the shuffle relation in a general double shuffle framework that specializes to double shuffle relations of multiple zeta values and multiple polylogarithms. As an application, we generalize the well-known decomposition formula of Euler that expresses the product of two Riemann zeta values as a sum of double zeta values to a formula that expresses the product of two multiple polylogarithm values as a sum of other multiple polylogarithm values.


Characterizations of discrete Sugeno integrals as polynomial functions over distributive lattices

Abstract: We give several characterizations of discrete Sugeno integrals over bounded distributive lattices, as particular cases of lattice polynomial functions, that is, functions which can be represented in the language of bounded lattices using variables and constants. We also consider the subclass of term functions as well as the classes of symmetric polynomial functions and weighted minimum and maximum functions, and present their characterizations, accordingly. Moreover, we discuss normal form representations of these functions.


Liberation of orthogonal Lie groups

Abstract: We show that under suitable assumptions, we have a one-to-one correspondence between classical groups and free quantum groups, in the compact orthogonal case. We classify the groups under correspondence, with the result that there are exactly 6 of them: $O_n,S_n,H_n,B_n,S_n',B_n'$. We investigate the representation theory aspects of the correspondence, with the result that for $O_n,S_n,H_n,B_n$, this is compatible with the Bercovici-Pata bijection. Finally, we discuss some more general classification problems in the compact orthogonal case, notably with the construction of a new quantum group.


A connection between the stochastic heat equation and fractional Brownian motion, and a simple proof of a result of Talagrand

Abstract: We give a new representation of fractional Brownian motion with Hurst parameter H<=1/2 using stochastic partial differential equations. This representation allows us to use the Markov property and time reversal, tools which are not usually available for fractional Brownian motion. We then give simple proofs that fractional Brownian motion does not hit points in the critical dimension, and that it does not have double points in the critical dimension. These facts were already known, but our proofs are quite simple and use some ideas of Levy.


The Laguerre polynomials preserve real-rootedness

Abstract: The linear transformation that sends $x^n$ to the n'th Laguerre polynomial preserves real-rootedness.


Operator-valued Fourier multipliers in Besov spaces and its applications

Abstract: The present paper, is devoted to investigation of operator--valued Fourier multiplier theorems from $B_{q_{1},r}^{s}$ to $B_{q_{2},r}^{s}$, optimal embedding of Besov spaces, the separability and positivity of differential operators. Here, we show that these differential operators generate analytic semigroup.


Some remarks on the Stanley's depth for multigraded modules

Abstract: We show that the Stanley's conjecture holds for any multigraded $S$-module $M$ with $\sdepth(M)=0$, where $S=K[x_1,...,x_n]$. Also, we give some bounds for the Stanley depth of the powers of the maximal irrelevant ideal in $S$.


Invariant conformal metrics on S^n

Abstract: In this paper we use the relationship between conformal metrics on the sphere and horospherically convex hypersurfaces in the hyperbolic space for giving sufficient conditions on a conformal metric to be radial under some constrain on the eigenvalues of its Schouten tensor. Also, we study conformal metrics on the sphere which are invariant by a $k-$parameter subgroup of conformal diffeomorphisms of the sphere, giving a bound on its maximum dimension. Moreover, we classify conformal metrics on the sphere whose eigenvalues of the Shouten tensor are all constant (we call them \emph{isoparametric conformal metrics}), and we use a classification result for radial conformal metrics which are solution of some $\sigma _k -$Yamabe type problem for obtaining existence of rotational spheres and Delaunay-type hypersurfaces for some classes of Weingarten hypersurfaces in $\h ^{n+1}$.


Communication-optimal parallel and sequential QR and LU factorizations

Abstract: We present parallel and sequential dense QR factorization algorithms that are both optimal (up to polylogarithmic factors) in the amount of communication they perform, and just as stable as Householder QR. We prove optimality by extending known lower bounds on communication bandwidth for sequential and parallel matrix multiplication to provide latency lower bounds, and show these bounds apply to the LU and QR decompositions. We not only show that our QR algorithms attain these lower bounds (up to polylogarithmic factors), but that existing LAPACK and ScaLAPACK algorithms perform asymptotically more communication. We also point out recent LU algorithms in the literature that attain at least some of these lower bounds.


Conformal dimension: Cantor sets and moduli

Abstract: In this paper we give several conditions for a space to be minimal for conformal dimension. We show that there are sets of zero length and conformal dimension 1 thus answering a question of Bishop and Tyson. Another sufficient condition for minimality is given in terms of a modulus of a system of measures in the sense of Fuglede \cite{Fug}. It implies in particular that there are many sets $E\subset\mathbb{R}$ of zero length such that $X\times Y$ is minimal for conformal dimension for every compact $Y$.


Integer Points in Backward Orbits

Abstract: A theorem of J. Silverman states that a forward orbit of a rational map $\phi(z)$ on $\mathbb P^1(K)$ contains finitely many $S$-integers in the number field $K$ when $(\phi\circ\phi)(z)$ is not a polynomial. We state an analogous conjecture for the backward orbits using a general $S$-integrality notion based on the Galois conjugates of points. This conjecture is proven for the map $\phi(z)=z^d$, and consequently Chebyshev polynomials, by uniformly bounding the number of Galois orbits for $z^n-\beta$ when $\beta\not =0$ is a non-root of unity. In general, our conjecture is true provided that the number of Galois orbits for $\phi^n(z)-\beta$ is bounded independently of $n$.


Discretely ordered groups

Abstract: We consider group orders and right-orders which are discrete, meaning there is a least element which is greater than the identity. We note that free groups cannot be given discrete orders, although they do have right-orders which are discrete. More generally, we give necessary and sufficient conditions that a given orderable group can be endowed with a discrete order. In particular, every orderable group G embeds in a discretely orderable group. We also consider conditions on right-orderable groups to be discretely right-orderable. Finally, we discuss a number of illustrative examples involving discrete orderability, including the Artin braid groups and Bergman's non-locally-indicable right orderable groups.


A simple estimation of the maximal rank of tensors with two slices by row and column operations, symmetrization and induction

Abstract: The determination of the maximal ranks of a set of a given type of tensors is a basic problem both in theory and application. In statistical applications, the maximal rank is related to the number of necessary parameters to be built in a tensor model. Based on this classical theorem by Bosch we will show the tight bound for 2 x n x n tensors by simple row and column operations, symmetrization and mathematical induction, which has been given by several authors based on eigenvalue theories.


Complex manifolds with generating tangent bundles

Abstract: Let $M$ be a close complex manifold and $TM$ its holomorphic tangent bundle. We prove that if the global holomorphic sections of tangent bundle generate each fibre, then $M$ is a complex homogeneous manifold. Our proof depends on the complex version of Chow-Rashevskii theorem in Carnot-Caratheodory spaces.


A dozen integrals: Russell-style

Abstract: On June 15, 1876, the Proceedings of the Royal Society of London published a paper by Mr. W. H. L. Russell entitled "On certain integrals". The paper starts with "The following are certain integrals which will, I hope, be found interesting". The rest of the paper is simply a list of 12 integrals. We provide a list of 12 integrals that we hope would be interesting.


A construction of Frobenius manifolds with logarithmic poles and applications

Abstract: A construction theorem for Frobenius manifolds with logarithmic poles is established. This is a generalization of a theorem of Hertling and Manin. As an application we prove a generalization of the reconstruction theorem of Kontsevich and Manin for projective smooth varieties with convergent Gromov-Witten potential. A second application is a construction of Frobenius manifolds out of a variation of polarized Hodge structures which degenerates along a normal crossing divisor when certain generation conditions are fulfilled.


Posterior Convergence and Model Estimation in Bayesian Change-point Problems

Abstract: We study the posterior distribution of the Bayesian multiple change-point regression problem when the number and the locations of the change-points are unknown. While it is relatively easy to apply the general theory to obtain the $O(1/\sqrt{n})$ rate up to some logarithmic factor, showing the exact parametric rate of convergence of the posterior distribution requires additional work and assumptions. Additionally, we demonstrate the asymptotic normality of the segment levels under these assumptions. For inferences on the number of change-points, we show that the Bayesian approach can produce a consistent posterior estimate. Finally, we argue that the point-wise posterior convergence property as demonstrated might have bad finite sample performance in that consistent posterior for model selection necessarily implies the maximal squared risk will be asymptotically larger than the optimal $O(1/\sqrt{n})$ rate. This is the Bayesian version of the same phenomenon that has been noted and studied by other authors.


Formal topology and constructive mathematics: the Gelfand and Stone-Yosida representation theorems

Abstract: We present a constructive proof of the Stone-Yosida representation theorem for Riesz spaces motivated by considerations from formal topology. This theorem is used to derive a representation theorem for f-algebras. In turn, this theorem implies the Gelfand representation theorem for C*-algebras of operators on Hilbert spaces as formulated by Bishop and Bridges. Our proof is shorter, clearer, and we avoid the use of approximate eigenvalues.


Invariant manifolds for analytic dynamical systems over ultrametric fields

Abstract: We give an exposition of the theory of invariant manifolds around a fixed point, in the case of time-discrete, analytic dynamical systems over a complete ultrametric field K. Typically, we consider an analytic manifold M modelled on an ultrametric Banach space over K, an analytic self-map f of M, and a fixed point p of f. Under suitable conditions on the tangent map of f at p, we construct a centre-stable manifold, a centre manifold, respectively, an r-stable manifold around p, for a given positive real number r not exceeding 1. The invariant manifolds are useful in the theory of Lie groups over local fields, where they allow results to be extended to the case of positive characteristic which previously were only available in characteristic zero (i.e., for p-adic Lie groups).


Recognizing trace graphs of closed braids

Abstract: To a closed braid in a solid torus we associate a trace graph in a thickened torus in such a way that closed braids are isotopic if and only if their trace graphs can be related by trihedral and tetraherdal moves. For closed braids with a fixed number of strands, we recognize trace graphs up to isotopy and trihedral moves in polynomial time with respect to the braid length.


Cusp areas of Farey manifolds and applications to knot theory

Abstract: This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey tesselation of the hyperbolic plane. The bounds on cusp area lead to explicit bounds on the volume of Dehn fillings of these manifolds, for example sharp bounds on volumes of hyperbolic closed 3-braids in terms of the Schreier normal form of the associated braid word. Finally, these results are applied to derive relations between the Jones polynomial and the volume of hyperbolic knots, and to disprove a related conjecture.


Smoothing noisy spectroscopic data with many-knot spline method

Abstract: In this paper, we present the development of a many-knot spline method derived to remove the statistical noise in the spectroscopic data. This method is an expansion of the B-spline method. Compared to the B-spline method, the many-knot spline method is significantly faster.


Perturbation method for determining the group of invariance of hierarchical models

Abstract: We propose a perturbation method for determining the (largest) group of invariance of a toric ideal defined in Aoki and Takemura [2008a]. In the perturbation method, we investigate how a generic element in the row space of the configuration defining a toric ideal is mapped by a permutation of the indeterminates. Compared to the proof in Aoki and Takemura [2008a] which was based on stabilizers of a subset of indeterminates, the perturbation method gives a much simpler proof of the group of invariance. In particular, we determine the group of invariance for a general hierarchical model of contingency tables in statistics, under the assumption that the numbers of the levels of the factors are generic. We prove that it is a wreath product indexed by a poset related to the intersection poset of the maximal interaction effects of the model.


Multiplicity one theorem for (GL(n+1,R),GL(n,R))

Abstract: Let F be either R or C. Consider the standard embedding GL(n,F)<GL(n+1,F) and the action of GL(n,F) on GL(n+1,F) by conjugation. In this paper we show that any GL(n,F)-invariant distribution on GL(n+1,F) is invariant with respect to transposition. We show that this implies that for any irreducible admissible smooth Frechet representations $\pi$ of GL(n+1,F) and $\tau$ of GL(n,F), $$dim Hom_{GL(n,F)}(\pi,\tau) \leq 1.$$ For p-adic fields those results were proven in [AGRS].


Efficient rare-event simulation for the maximum of heavy-tailed random walks

Abstract: Let $(X_n:n\geq 0)$ be a sequence of i.i.d. r.v.'s with negative mean. Set $S_0=0$ and define $S_n=X_1+... +X_n$. We propose an importance sampling algorithm to estimate the tail of $M=\max \{S_n:n\geq 0\}$ that is strongly efficient for both light and heavy-tailed increment distributions. Moreover, in the case of heavy-tailed increments and under additional technical assumptions, our estimator can be shown to have asymptotically vanishing relative variance in the sense that its coefficient of variation vanishes as the tail parameter increases. A key feature of our algorithm is that it is state-dependent. In the presence of light tails, our procedure leads to Siegmund's (1979) algorithm. The rigorous analysis of efficiency requires new Lyapunov-type inequalities that can be useful in the study of more general importance sampling algorithms.


Two orbits: When is one in the closure of the other?

Abstract: Let $G$ be a connected linear algebraic group, let $V$ be a finite dimensional algebraic $G$-module, and let $\mathcal O_1$, $\mathcal O_2$ be two $G$-orbits in $V$. We describe a constructive way to find out whether $\mathcal O_1$ lies in the closure of $\mathcal O_2$ or not.


A Kind of Compact Quantum Semigroups

Abstract: We show that the quantum family of all maps from a finite space to a finite dimensional compact quantum semigroup has a canonical quantum semigroup structure.


Unifying Practical Uncertainty Representations: I. Generalized P-Boxes

Abstract: There exist several simple representations of uncertainty that are easier to handle than more general ones. Among them are random sets, possibility distributions, probability intervals, and more recently Ferson's p-boxes and Neumaier's clouds. Both for theoretical and practical considerations, it is very useful to know whether one representation is equivalent to or can be approximated by other ones. In this paper, we define a generalized form of usual p-boxes. These generalized p-boxes have interesting connections with other previously known representations. In particular, we show that they are equivalent to pairs of possibility distributions, and that they are special kinds of random sets. They are also the missing link between p-boxes and clouds, which are the topic of the second part of this study.


An iterative method for numerical integration of rational functions

Abstract: The rational Landen transformations are used to produce a highly efficient numerical method for the integration of rational functions.


An invitation to toric degenerations

Abstract: This is an expository paper which explores the ideas of the authors' paper "From Affine Geometry to Complex Geometry", arXiv:0709.2290. We explain the basic ideas of the latter paper by going through a large number of concrete, increasingly complicated examples.


A class of logarithmic integrals

Abstract: We present a systematic study of integrals over [0,1] where the integrand is of the form Q(x) log log 1/x. Here Q is a rational function.


The integrals in Gradshteyn and Ryzhik. Part 11: The incomplete beta function

Abstract: The table of Gradshteyn and Ryzhik contains some integrals that can be expressed in terms of the incomplete beta function. We describe some elementary properties of this function and use them to check some of the formulas in the mentioned table.


Exterior algebras and two conjectures on finite abelian groups

Abstract: Let G be a finite abelian group with |G|>1. Let a_1,...,a_k be k distinct elements of G and let b_1,...,b_k be (not necessarily distinct) elements of G, where k is a positive integer smaller than the least prime divisor of |G|. We show that there is a permutation $\pi$ on {1,...,k} such that a_1b_{\pi(1)},...,a_kb_{\pi(k)} are distinct, provided that any other prime divisor of |G| (if there is any) is greater than k!. This in particular confirms the Dasgupta-Karolyi-Serra-Szegedy conjecture for abelian p-groups. We also pose a new conjecture involving determinants and characters, and show that its validity implies Snevily's conjecture for abelian groups of odd order. Our methods involve exterior algebras and characters.


Exponential sums nondegenerate relative to a lattice

Abstract: Our previous theorems on exponential sums often did not apply or did not give sharp results when certain powers of a variable appearing in the polynomial were divisible by p. We remedy that defect in this paper by systematically applying "p-power reduction," making it possible to strengthen and extend our earlier results.


On almost universal mixed sums of squares and triangular numbers

Abstract: In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than 2719 can be represented by the famous Ramanujan form $x^2+y^2+10z^2$, equivalently the form $2x^2+5y^2+4T_z$ represents all integers greater than 1359, where $T_z$ denotes the triangular number $z(z+1)/2$. Given positive integers $a,b,c$ we employ modular forms and the theory of quadratic forms to determine completely when the general form $ax^2+by^2+cT_z$ represents sufficiently large integers and establish similar results for the forms $ax^2+bT_y+cT_z$ and $aT_x+bT_y+cT_z$. Here are some consequences of our main theorems: (i) All sufficiently large odd numbers have the form $2ax^2+y^2+z^2$ if and only if all prime divisors of $a$ are congruent to 1 modulo 4. (ii) The form $ax^2+y^2+T_z$ is almost universal (i.e., it represents sufficiently large integers) if and only if each odd prime divisor of $a$ is congruent to 1 or 3 modulo 8. (iii) $ax^2+T_y+T_z$ is almost universal if and only if all odd prime divisors of $a$ are congruent to 1 modulo 4. (iv) When $v_2(a)\not=3$, the form $aT_x+T_y+T_z$ is almost universal if and only if all odd prime divisors of $a$ are congruent to 1 modulo 4 and $v_2(a)\not=5,7,...$, where $v_2(a)$ is the 2-adic order of $a$.


On the (ir)rationality of Kontsevich weights

Abstract: We compute the weight of a Kontsevich graph in deformation quantization. Up to rationals, the result is Zeta(3)^2/Pi^6.


Algebraic Values of Transcendental Functions at Algebraic Points

Abstract: In this paper, the authors will prove that any subset of $\overline{\QQ}$ can be the exceptional set of some transcendental entire function. Furthermore, we could generalize this theorem to a much more general version and present a unified proof.


Unifying Practical Uncertainty Representations: II. Clouds

Abstract: There exist many simple tools for jointly capturing variability and incomplete information by means of uncertainty representations. Among them are random sets, possibility distributions, probability intervals, and the more recent Ferson's p-boxes and Neumaier's clouds, both defined by pairs of possibility distributions. In the companion paper, we have extensively studied a generalized form of p-box and situated it with respect to other models . This paper focuses on the links between clouds and other representations. Generalized p-boxes are shown to be clouds with comonotonic distributions. In general, clouds cannot always be represented by random sets, in fact not even by 2-monotone (convex) capacities.


Positivity and Kleiman transversality in equivariant K-theory of homogeneous spaces

Abstract: We prove the conjectures of Graham-Kumar and Griffeth-Ram concerning the alternation of signs in the structure constants for torus-equivariant K-theory of generalized flag varieties G/P. These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with finitely many orbits. The computation of the coefficients in the expansion of the equivariant K-class of a subvariety in terms of Schubert classes is reduced to an Euler characteristic using the homological transversality theorem for non-transitive group actions due to S. Sierra. A vanishing theorem, when the subvariety has rational singularities, shows that the Euler characteristic is a sum of at most one term--the top one--with a well-defined sign. The vanishing is proved by suitably modifying a geometric argument due to M. Brion in ordinary K-theory that brings Kawamata-Viehweg vanishing to bear.


Strongly t-logarithmic t-generating sets: Geometric properties of some soluble groups

Abstract: We introduce the concept of a strongly t-logarithmic t-generating set for a Z[t,t^{-1}]-module, which enables us to prove that a large class of soluble groups are not almost convex. We also prove some results about dead-end depth.


Breuil's classification of $p$-divisible groups over regular local rings of arbitrary dimension

Abstract: Let $k$ be a perfect field of characteristic $p \geq 3$. We classify $p$-divisible groups over regular local rings of the form $W(k)[[t_1,...,t_r,u]]/(u^e+pb_{e-1}u^{e-1}+...+pb_1u+pb_0)$, where $b_0,...,b_{e-1}\in W(k)[[t_1,...,t_r]]$ and $b_0$ is an invertible element. This classification was in the case $r = 0$ conjectured by Breuil and proved by Kisin.


Locally Connected HL Compacta

Abstract: It is consistent with MA plus not CH that there is a locally connected hereditarily Lindelof compact space which is not metrizable.


A remark on Khovanov homology and two-fold branched covers

Abstract: Examples of knots and links distinguished by the total rank of their Khovanov homology but sharing the same two-fold branched cover are given. As a result, Khovanov homology does not yield an invariant of two-fold branched covers.


On satellites in semi-abelian categories: Homology without projectives

Abstract: Working in a semi-abelian context, we use Janelidze's theory of generalised satellites to study universal properties of the Everaert long exact homology sequence. This results in a new definition of homology which does not depend on the existence of projective objects. We explore the relations with other notions of homology, and thus prove a version of the higher Hopf formulae. We also work out some examples.


Curvature structure of self-dual 4-manifolds

Abstract: We show the existence of a modified Cliff(1,1) structure compatible with an Osserman 0-model of signature (2,2). We then apply this algebraic result to certain classes of pseudo-Riemannian manifolds of signature (2,2). We obtain a new characterization of the Weyl curvature tensor of an (anti-)self-dual manifold and we prove some new results regarding (Jordan) Osserman manifolds.


SPM Bulletin 25

Abstract: Contents: 1. Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces, I; 2. Frechet-Urysohn fans in free topological groups; 3. Packing index of subsets in Polish groups; 4. Symmetric monochromatic subsets in colorings of the Lobachevsky plane; 5. Structural Ramsey theory of metric spaces and topological dynamics of isometry groups; 6. Distinguishing Number of Countable Homogeneous Relational Structures; 7. Indestructible colourings and rainbow Ramsey theorems; 8. Products of Borel subgroups; 9. Selection theorems and treeability; 10. Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces, IV; 11. A property of Cp[0, 1]; 12. A Dedekind Finite Borel Set; 13. Aronszajn Compacta; 14. A strong antidiamond principle compatible with CH; 15. On the strength of Hausdorff's gap condition; 16. Nonhomogeneous analytic families of trees; 17. Reasonable non-Radon-Nikodym ideals; 18. Continuity and related forcing; 19. An exact Ramsey principle for block sequences; 20. Baire reflection; 21. Tukey classes of ultrafilters on; 22. Countably determined compact abelian groups; 23. A topological reflection principle equivalent to Shelah's Strong Hypothesis; 24. Superfilters, Ramsey theory, and van der Waerden's Theorem.


Notes on the Heegaard-Floer Link Surgery Spectral Sequence

Abstract: This largely technical paper is divided into two parts: part I: An account of P. Ozsvath and Z. Szabo's construction of the link surgery spectral sequence. There are no new results here, but this part slightly modifies and expands their proof and is included as an aid to part II. part II: Some modifications of the spectral sequence suitable for knot Floer homology; an exposition of the invariance of the spectral sequence, under a suitable equivalence, from alterations of the underlying bouquet; and an account of the morphisms induced on the spectral sequences by suitable four dimensional cobordisms and how they depend upon the four manifold. This is the author's effort to investigate a question at the end of the introduction to P. Ozsvath's and Z. Szabo's paper "On the Heegaard-Floer homology of double branched covers."


More bijective Catalan combinatorics on permutations and on signed permutations

Abstract: In this paper, we construct bijections between Dyck paths, noncrossing partitions, and 231-avoiding permutations, which send the area statistic on Dyck paths to the inversion number on noncrossing partitions and on 231-avoiding permutations. This bijection has the additional property that it simultaneously sends the major index on Dyck paths to the sum of the major index and the inverse major index on noncrossing partitions and on 231-avoiding permutations, respectively. Moreover, we provide generalizations of these constructions to the group of signed permutations.


Quandle-like Structures From Groups

Abstract: We define a class of quandle-like structures called pseudoquandles and analyze some of their algebraic properties.


Real AlphaBeta-Geometries

Abstract: By a real alphabeta-geometry we mean a four-dimensional manifold M equipped with a neutral metric h such that (M,h) admits both an integrable distribution of alpha-planes and an integrable distribution of beta-planes. We obtain a local characterization of the metric when at least one of the distributions is parallel (i.e., is a Walker geometry) and the three-dimensional distribution spanned by the alpha- and beta-distributions is integrable. The case when both distributions are parallel, which has been called two-sided Walker geometry, is obtained as a special case. We also consider real \alpha\beta-geometries for which the corresponding spinors are both multiple Weyl principal spinors.


Symmetric Functions and Caps

Abstract: Given a finite subset S in F_p^d, let a(S) be the number of distinct r-tuples (x_1,...,x_r) in S such that x_1+...+x_r = 0. We consider the "moments" F(m,n) = sum_|S|=n a(S)^m. Specifically, we present an explicit formula for F(m,n) as a product of two matrices, ultimately yielding a polynomial in q=p^d. The first matrix is independent of n while the second makes no mention of finite fields. However, the complexity of calculating each grows with m. The main tools here are the Schur-Weyl duality theorem, and some elementary properties of symmetric functions. This problem is closely to the study of maximal caps.


The Haar system in the preduals of hyperfinite factors

Abstract: We shall present examples of Schauder bases in the preduals to the hyperfinite factors of types $\hbox{II}_1$, $\hbox{II}_\infty$, $\hbox{III}_\lambda$, $0 < \lambda \leq 1$. In the semifinite (respectively, purely infinite) setting, these systems form Schauder bases in any associated separable symmetric space of measurable operators (respectively, in any non-commutative $L^p$-space).


Knots yielding homeomorphic lens spaces by Dehn surgery

Abstract: We show that there exist infinitely many pairs of distinct knots in the 3-sphere such that each pair can yield homeomorphic lens spaces by the same Dehn surgery. Moreover, each knot of the pair can be chosen to be a torus knot, a satellite knot or a hyperbolic knot, except that both cannot be satellite knots simultaneously. This exception is shown to be unavoidable by the classical theory of binary quadratic forms.


Unbounded Fredholm modules and double operator integrals

Abstract: In noncommutative geometry one is interested in invariants such as the Fredholm index or spectral flow and their calculation using cyclic cocycles. A variety of formulae have been established under side conditions called summability constraints. These can be formulated in two ways, either for spectral triples or for bounded Fredholm modules. We study the relationship between these by proving various properties of the map on unbounded self adjoint operators $D$ given by $f(D)=D(1+D^2)^{-1/2}$. In particular we prove commutator estimates which are needed for the bounded case. In fact our methods work in the setting of semifinite noncommutative geometry where one has $D$ as an unbounded self adjoint linear operator affiliated with a semi-finite von Neumann algebra $\aM$. More precisely we show that for a pair $D,D_0$ of such operators with $D-D_0$ a bounded self-adjoint linear operator from $\aM$ and $ ({\bf 1}+D_0^2)^{-1/2}\in \sE$, where $\sE$ is a noncommutative symmetric space associated with $\aM$, then $$ \Vert f(D) - f (D_0) \Vert_{\sE} \leq C\cdot \Vert D-D_0\Vert_{\aM}. $$ This result is further used to show continuous differentiability of the mapping between an odd $\sE$-summable spectral triple and its bounded counterpart.


Non- (quantum) differentiable $C^1$-functions in the spaces with trivial Boyd indices

Abstract: If E is a separable symmetric sequence space with trivial Boyd indices and $\cC^E$ is the corresponding ideal of compact operators, then there exists a $C^1$-function $f_E$, a self-adjoint element $W\in \cC^E$ and a densely defined closed symmetric derivation $\delta$ on $\cC^E$ such that $W \in Dom \delta$, but $f_E(W) \notin Dom \delta$.


Simple homotopy invariance of higher signatures

Abstract: We prove that the higher signature for any close oriented manifold is a simple-homotopy invariant.


Bayesian nonparametric estimators derived from conditional Gibbs structures

Abstract: We consider discrete nonparametric priors which induce Gibbs-type exchangeable random partitions and investigate their posterior behavior in detail. In particular, we deduce conditional distributions and the corresponding Bayesian nonparametric estimators, which can be readily exploited for predicting various features of additional samples. The results provide useful tools for genomic applications where prediction of future outcomes is required.


Central limit theorem for a many-server queue with random service rates

Abstract: Given a random variable $N$ with values in ${\mathbb{N}}$, and $N$ i.i.d. positive random variables $\{\mu_k\}$, we consider a queue with renewal arrivals and $N$ exponential servers, where server $k$ serves at rate $\mu_k$, under two work conserving routing schemes. In the first, the service rates $\{\mu_k\}$ need not be known to the router, and each customer to arrive at a time when some servers are idle is routed to the server that has been idle for the longest time (or otherwise it is queued). In the second, the service rates are known to the router, and a customer that arrives to find idle servers is routed to the one whose service rate is greatest. In the many-server heavy traffic regime of Halfin and Whitt, the process that represents the number of customers in the system is shown to converge to a one-dimensional diffusion with a random drift coefficient, where the law of the drift depends on the routing scheme. A related result is also provided for nonrandom environments.


Stein's method for discrete Gibbs measures

Abstract: Stein's method provides a way of bounding the distance of a probability distribution to a target distribution $\mu$. Here we develop Stein's method for the class of discrete Gibbs measures with a density $e^V$, where $V$ is the energy function. Using size bias couplings, we treat an example of Gibbs convergence for strongly correlated random variables due to Chayes and Klein [Helv. Phys. Acta 67 (1994) 30--42]. We obtain estimates of the approximation to a grand-canonical Gibbs ensemble. As side results, we slightly improve on the Barbour, Holst and Janson [Poisson Approximation (1992)] bounds for Poisson approximation to the sum of independent indicators, and in the case of the geometric distribution we derive better nonuniform Stein bounds than Brown and Xia [Ann. Probab. 29 (2001) 1373--1403].


Stability of the slow manifold in the primitive equations

Abstract: We show that, under reasonably mild hypotheses, the solution of the forced--dissipative rotating primitive equations of the ocean loses most of its fast, inertia--gravity, component in the small Rossby number limit as $t\to\infty$. At leading order, the solution approaches what is known as "geostrophic balance" even under ageostrophic, slowly time-dependent forcing. Higher-order results can be obtained if one further assumes that the forcing is time-independent and sufficiently smooth. If the forcing lies in some Gevrey space, the solution will be exponentially close to a finite-dimensional "slow manifold" after some time.


A concrete co-existential map that is not confluent

Abstract: We give a concrete example of a co-existential map between continua that is not confluent.


Ordinary differential system in dinension six with affine Weyl group symmetry of type $D_4^{(2)}$

Abstract: We find a three-parameter family of ordinary differential systems in dimension six with affine Weyl group symmetry of type $D_4^{(2)}$. This is the second example which gave higher order Painlev\'e type systems of type $D_{4}^{(2)}$. We show that we give its symmetry and holomorphy conditions. These symmetries, holomorphy conditions and invariant divisors are new.


Hyperbolic conservation laws on manifolds with limited regularity

Abstract: We introduce a formulation of the initial and boundary value problem for nonlinear hyperbolic conservation laws posed on a differential manifold endowed with a volume form, possibly with a boundary; in particular, this includes the important case of Lorentzian manifolds. Only limited regularity is assumed on the geometry of the manifold. For this problem, we establish the existence and uniqueness of an L1 semi-group of weak solutions satisfying suitable entropy and boundary conditions.


A Short History of Markov Chain Monte Carlo: Subjective Recollections from Incomplete Data

Abstract: We attempt to trace the history and development of Markov chain Monte Carlo (MCMC) from its early inception in the late 1940s through its use today. We see how the earlier stages of Monte Carlo (MC, not MCMC) research have led to the algorithms currently in use. More importantly, we see how the development of this methodology has not only changed our solutions to problems, but has changed the way we think about problems.


On the largest component of a random graph with a subpower-law degree sequence in a subcritical phase

Abstract: A uniformly random graph on $n$ vertices with a fixed degree sequence, obeying a $\gamma$ subpower law, is studied. It is shown that, for $\gamma>3$, in a subcritical phase with high probability the largest component size does not exceed $n^{1/\gamma+\varepsilon_n}$, $\varepsilon_n=O(\ln\ln n/\ln n)$, $1/\gamma$ being the best power for this random graph. This is similar to the best possible $n^{1/(\gamma-1)}$ bound for a different model of the random graph, one with independent vertex degrees, conjectured by Durrett, and proved recently by Janson.


The density of integral points on hypersurfaces of degree at least four

Abstract: Let $f$ be a polynomial of degree at least four with integer-valued coefficients. We establish new bounds for the density of integer solutions to the equation $f=0$, using an iterated version of Heath-Browns $q$-analogue of van der Corput's method of exponential sums.


Krull dimension of solvable groups

Abstract: In this paper we prove that free solvable groups have finite Krull dimension. In fact, this is true for much wider class of solvable groups, termed rigid groups. Along the way we study the algebraic structure of the limit solvable groups (fully residually free solvable groups).


Morita Equivalence of Brandt Semigroup Algebras

Abstract: We prove that for every group $G$ and any two sets $I,J$, the Brandt semigroup algebras $\ell(B(I,G))$ and $\ell(B(J,G))$ are Morita equivalent with respect to the Morita theory of self-induced Banach algebras introduced by Gronbaek. As applications, we show that if $G$ is an amenable group, then for a wide class of Banach $\ell(B(I,G))$-bimodules $E$, and every $n>0$, the bounded Hochschild cohomology groups $H^n(\ell(B(I,G)),E^*)$ are trivial, and also, the notion of approximate amenability is not Morita invariant.


A binary tree representation for the 2-adic valuation of a sequence arising from a rational integral

Abstract: We present a binary tree that describes the 2-adic valuation of a sequence of coefficients arising from the evaluation of a rational integral.


Parseval frames for ICC groups

Abstract: We analyze Parseval frames generated by the action of an ICC group on a Hilbert space. We parametrize the set of all such Parseval frames by operators in the commutant of the corresponding representation. We characterize when two such frames are strongly disjoint. We prove an undersampling result showing that if the representation has a Parseval frame of norm $\frac{1}{\sqrt{N}}$, the Hilbert space is spanned by an orthonormal basis generated by a subgroup. As applications we obtain some sufficient conditions under which a unitary representation admits a Parseval frame which is spanned by an Riesz sequences generated by a subgroup. In particular, every subrepresentation of the left regular representation of a free group has this property.


Probability and Fourier duality for affine iterated function systems

Abstract: Let $d$ be a positive integer, and let $\mu$ be a finite measure on $\br^d$. In this paper we ask when it is possible to find a subset $\Lambda$ in $\br^d$ such that the corresponding complex exponential functions $e_\lambda$ indexed by $\Lambda$ are orthogonal and total in $L^2(\mu)$. If this happens, we say that $(\mu, \Lambda)$ is a spectral pair. This is a Fourier duality, and the $x$-variable for the $L^2(\mu)$-functions is one side in the duality, while the points in $\Lambda$ is the other. Stated this way, the framework is too wide, and we shall restrict attention to measures $\mu$ which come with an intrinsic scaling symmetry built in and specified by a finite and prescribed system of contractive affine mappings in $\br^d$; an affine iterated function system (IFS). This setting allows us to generate candidates for spectral pairs in such a way that the sets on both sides of the Fourier duality are generated by suitably chosen affine IFSs. For a given affine setup, we spell out the appropriate duality conditions that the two dual IFS-systems must have. Our condition is stated in terms of certain complex Hadamard matrices. Our main results give two ways of building higher dimensional spectral pairs from combinatorial algebra and spectral theory applied to lower dimensional systems.


Polynomial Bounds for Oscillation of Solutions of Fuchsian Systems

Abstract: We study the problem of placing effective upper bounds for the number of zeros of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of the class of Fuchsian systems of given dimension n having m singular points. As a function of n,m, this bound turns out to be double exponential in the precise sense explained in the paper. As a corollary, we obtain a solution of the so called restricted infinitesimal Hilbert 16th problem, an explicit upper bound for the number of isolated zeros of Abelian integrals which is polynomially growing as the Hamiltonian tends to the degeneracy locus. This improves the exponential bounds recently established by A. Glutsyuk and Yu. Ilyashenko.


Hecke-Clifford algebras and spin Hecke algebras III: the trigonometric type

Abstract: The notion of trigonometric spin double affine Hecke algebras (tsDaHa) and trigonometric double affine Hecke-Clifford algebras (tDaHCa) associated to classical Weyl groups are introduced. The PBW basis property is established. An algebra isomorphism relating tDaHCa to tsDaHa is obtained.


On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem

Abstract: We prove that the number of limit cycles generated by a small non-conservative perturbation of a Hamiltonian polynomial vector field on the plane, is bounded by a double exponential of the degree of the fields. This solves the long-standing tangential Hilbert 16th problem. The proof uses only the fact that Abelian integrals of a given degree are horizontal sections of a regular flat meromorphic connection (Gauss-Manin connection) with a quasiunipotent monodromy group.


Models of expansions of N with no end extensions

Abstract: We deal with models of Peano arithmetic (specifically with a question of Ali Enayat). The methods are from creature forcing. We find an expansion of N such that its theory has models with no (elementary) end extensions. In fact there is a Borel uncountable set of subsets of N such that expanding N by any uncountably many of them suffice. Also we find arithmetically closed A with no definably closed ultrafilter on it.


Continuous Bounded Cohomology of Topological Semigroups

Abstract: In this short note, we give some new results on continuous bounded cohomology groups of topological semigroups with values in complex field. We show that the second continuous bounded cohomology group of a compact metrizable semigroup, is a Banach space. Also, we study cohomology groups of amenable topological semigroups, and we show that cohomology groups of rank greater than one of a compact left or right amenable semigroup, are trivial. Also, we give some examples and applications about topological lattices.


On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues

Abstract: We prove that the isoperimetric inequality due to Hersch-Payne-Schiffer for the n-th nonzero Steklov eigenvalue of a bounded simply-connected planar domain is sharp for all n=1,2,... The equality is attained in the limit by a sequence of simply-connected domains degenerating to the disjoint union of n identical disks. We give a new proof of this inequality for n=2 and show that it is strict in this case. Related results are also obtained for the product of two consecutive Steklov eigenvalues.


Beilinson-Hodge cycles on semiabelian varieties

Abstract: Beilinson conjectured that all rational cycles of type (q,q) on the qth cohomology of a smooth complex algebraic variety should come from motivic cohomology. The purpose of this note is to prove this when the variety is a semiabelian variety or a product of curves. The proof is based on the study of invariants under the Mumford-Tate group.


Nilpotent orbits in classical Lie algebras over finite fields of characteristic 2 and the Springer correspondence

Abstract: Let $G$ be an adjoint algebraic group of type $B$, $C$, or $D$ over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the Lie algebra of $G$. In particular, for orthogonal Lie algebras in characteristic 2, the structure of component groups of nilpotent centralizers is determined and the number of nilpotent orbits over finite fields is obtained.


Representations of $asl_2$

Abstract: We study representations of the simple Lie antialgebra $asl_2$ introduced by Ovsienko. We show that representations of $asl_2$ are closely related to the famous ghost Casimir element of the universal enveloping algebra $osp(1|2)$. We prove that $asl_2$ has no non-trivial finite-dimensional representations; we define and classify some particular infinite-dimensional representations that we call weighted representations.


Nilpotent orbits in the dual of classical Lie algebras in characteristic 2 and the Springer correspondence

Abstract: Let $G$ be a simply connected algebraic group of type $B,C$ or $D$ over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the dual vector space of the Lie algebra of $G$. In particular, we classify the nilpotent orbits in the duals of symplectic and orthogonal Lie algebras over algebraically closed or finite fields of characteristic 2.


Up-and-Down and the Percentile-Finding Problem

Abstract: Up-and-Down (U&D) is a popular sequential design for estimating threshold percentiles in binary experiments. However, U&D application practices have stagnated, and significant gaps in understanding its properties persist. The first part of my work aims to fill gaps in U&D theory. New results concerning stationary distribution properties are proven. A second focus of this study is nonparametric U&D estimation. An improvement to isotonic regression called "centered isotonic regression" (CIR), and a new averaging estimator called "auto-detect" are introduced and their properties studied. Bayesian percentile-finding designs, most notably the continual reassessment method (CRM) developed for Phase I clinical trials, are also studied. In general, CRM convergence depends upon random run-time conditions -- meaning that convergence is not always assured. Small-sample behavior is studied as well. It is shown that CRM is quite sensitive to outlier sub-sequences of thresholds, resulting in highly variable small-sample behavior between runs under identical conditions. Nonparametric CRM variants exhibit a similar sensitivity. Ideas to combine the advantages of U&D and Bayesian designs are examined. A new approach is developed, using a hybrid framework, that evaluates the evidence for overriding the U&D allocation with a Bayesian one.


A Colding-Minicozzi Stability inequality and its applications

Abstract: We consider operators $L$ acting on functions on a Riemannian surface, $\Sigma$, of the form $L = \Delta + V +a K.$ Here $\Delta$ is the Laplacian of $\Sigma$, $V$ a non-negative potential on $\Sigma$, K the Gaussian curvature and $a$ is a non-negative constant. Such operators $L$ arise as the stability operator of $\Sigma$ immersed in a Riemannian 3-manifold with constant mean curvature (for particular choices of $V$ and $a$). We assume L is nonpositive acting on functions compactly supported on $\Sigma$ and we obtain results in the spirit of some theorems of Ficher-Colbrie-Schoen, Colding-Minicozzi, and Castillon. We extend these theorems to $a \leq 1/4$. We obtain results on the conformal type of $\Sigma$ and a distance (to the boundary) lemma.


Non-structure in lambda^{++} using instances of WGCH

Abstract: We try to redo, improve and continue the non-structure parts in some works on a.e.c., which uses weak diamond, in lambda^+ and lambda^{++} getting better and more results and do what is necessary for the book on a.e.c. Comparing with math.LO/9805146 we make the context closer to the examples, hence hopefully improve transparency, though losing some generality. Toward this we work also on the positive theory, i.e. structure side of "low frameworks" like almost good lambda-frames.


On the concentration and the convergence rate with a moment condition in first passage percolation

Abstract: We consider the first passage percolation model on the ${\bf Z}^d$ lattice. In this model, we assign independently to each edge $e$ a non-negative passage time $t(e)$ with a common distribution $F$. Let $a_{0,n}$ be the passage time from the origin to $(n,0,..., 0)$. Under the exponential tail assumption, Kesten (1993) and Talagrand (1995) investigated the concentration of $a_{0,n}$ from its mean using different methods. With this concentration and the exponential tail assumption, Alexander gave an estimate for the convergence rate for ${\bf E} a_{0,n}$. In this paper, focusing on a moment condition, we reinvestigate the concentration and the convergence rate for $a_{0,n}$ using a special martingale structure.


Categoricity and solvability of A.E.C., quite highly

Abstract: We investigate in ZFC what can be the family of large enough cardinals mu in which an a.e.c. K is categorical or even just solvable. We show that for not few cardinals lambda<mu there is a superlimit model in K_lambda. Moreover, our main result is that we can find a good lambda-frame s categorical in lambda such that K_s subseteq K_lambda. We then show how to use [Sh:705] to get categoricity in every large enough cardinality if K has cases of mu-amalgamation for enough mu and 2^mu<2^{mu^{+1}} <... < 2^{mu^{+n}}... for enough mu.


Twisted cyclic theory, equivariant KK theory and KMS States

Abstract: Recently, examples of an index theory for KMS states of circle actions were discovered, \cite{CPR2,CRT}. We show that these examples are not isolated. Rather there is a general framework in which we use KMS states for circle actions on a C*-algebra A to construct Kasparov modules and semifinite spectral triples. By using a residue construction analogous to that used in the semifinite local index formula we associate to these triples a twisted cyclic cocycle on a dense subalgebra of A. This cocycle pairs with the equivariant KK-theory of the mapping cone algebra for the inclusion of the fixed point algebra of the circle action in A. The pairing is expressed in terms of spectral flow between a pair of unbounded self adjoint operators that are Fredholm in the semifinite sense. A novel aspect of our work is the discovery of an eta cocycle that forms a part of our twisted residue cocycle. To illustrate our theorems we observe firstly that they incorporate the results in \cite{CPR2,CRT} as special cases. Next we use the Araki-Woods III_\lambda representations of the Fermion algebra to show that there are examples which are not Cuntz-Krieger systems.


Skew-product representations of multidimensional Dunkl Markov processes

Abstract: In this paper we obtain skew-product representations of the multidimensional Dunkl processes which generalize the skew-product decomposition in dimension 1 obtained in L. Gallardo and M. Yor. Some remarkable properties of the Dunkl martingales. S\'{e}minaire de Probabilit\'{e}s XXXIX, 2006. We also study the radial part of the Dunkl process, i.e. the projection of the Dunkl process on a Weyl chamber.


Exponential Lower Bounds for Quasimodes of Semiclassical Schr\"{o}dinger Operators

Abstract: We prove quantitative unique continuation results for the semiclassical Schrodinger operator on smooth, compact domains. These take the form of exponentially decreasing (in h) local L^{2} lower bounds for exponentially precise quasimodes. We also show that these lower bounds are sharp in h, and that, moreover, the hypothesized quasimode accuracy is also sharp.


Limit laws for the energy of a charged polymer

Abstract: In this paper we obtain the central limit theorems, moderate deviations and the laws of the iterated logarithm for the energy \[H_n=\sum_{1\le j<k\le n}\omega_j\omega_k1_{\{S_j=S_k\}}\] of the polymer $\{S_1,...,S_n\}$ equipped with random electrical charges $\{\omega_1,...,\omega_n\}$. Our approach is based on comparison of the moments between $H_n$ and the self-intersection local time \[Q_n=\sum_{1\le j<k\le n}1_{\{S_j=S_k\}}\] run by the $d$-dimensional random walk $\{S_k\}$. As partially needed for our main objective and partially motivated by their independent interest, the central limit theorems and exponential integrability for $Q_n$ are also investigated in the case $d\ge3$.


Explicit and Almost Explicit Spectral Calculations for Diffusion Operators

Abstract: The diffusion operator $$ H_D=-\frac12\frac d{dx}a\frac d{dx}-b\frac d{dx}=-\frac12\exp(-2B)\frac d{dx}a\exp(2B)\frac d{dx}, $$ where $B(x)=\int_0^x\frac ba(y)dy$, defined either on $R^+=(0,\infty)$ with the Dirichlet boundary condition at $x=0$, or on $R$, can be realized as a self-adjoint operator with respect to the density $\exp(2Q(x))dx$. The operator is unitarily equivalent to the Schr\"odinger-type operator $H_S=-\frac12\frac d{dx}a\frac d{dx}+V_{b,a}$, where $V_{b,a}=\frac12(\frac{b^2}a+b')$. We obtain an explicit criterion for the existence of a compact resolvent and explicit formulas up to the multiplicative constant 4 for the infimum of the spectrum and for the infimum of the essential spectrum for these operators. We give some applications which show in particular how $\inf\sigma(H_D)$ scales when $a=\nu a_0$ and $b=\gamma b_0$, where $\nu$ and $\gamma$ are parameters, and $a_0$ and $b_0$ are chosen from certain classes of functions. We also give applications to self-adjoint, multi-dimensional diffusion operators.


On mean central limit theorems for stationary sequences

Abstract: In this paper, we give estimates of the minimal ${\mathbb{L}}^1$ distance between the distribution of the normalized partial sum and the limiting Gaussian distribution for stationary sequences satisfying projective criteria in the style of Gordin or weak dependence conditions.


Global well-posedness of incompressible flow in porous media with critical diffusion in Besov spaces

Abstract: In this paper we study the model of heat transfer in a porous medium with a critical diffusion. We obtain global existence and uniqueness of solutions to the equations of heat transfer of incompressible fluid in Besov spaces $\dot B^{3/p}_{p,1}(\mathbb{R}^3)$ with $1\le p\le\infty$ by the method of modulus of continuity and Fourier localization technique.


On the topological entropy of families of braids

Abstract: A method for computing the topological entropy of each braid in an infinite family, making use of Dynnikov's coordinates on the boundary of Teichm\"uller space, is described. The method is illustrated on two two-parameter families of braids.


Joint continuity of the local times of fractional Brownian sheets

Abstract: Let $B^H=\{B^H(t),t\in{{\mathbb{R}}_+^N}\}$ be an $(N,d)$-fractional Brownian sheet with index $H=(H_1,...,H_N)\in(0,1)^N$ defined by $B^H(t)=(B^H_1(t),...,B^H_d(t)) (t\in {\mathbb{R}}_+^N),$ where $B^H_1,...,B^H_d$ are independent copies of a real-valued fractional Brownian sheet $B_0^H$. We prove that if $d<\sum_{\ell=1}^NH_{\ell}^{-1}$, then the local times of $B^H$ are jointly continuous. This verifies a conjecture of Xiao and Zhang (Probab. Theory Related Fields 124 (2002)). We also establish sharp local and global H\"{o}lder conditions for the local times of $B^H$. These results are applied to study analytic and geometric properties of the sample paths of $B^H$.


Minimal average degree aberration and the state polytope for experimental designs

Abstract: For a particular experimental design, there is interest in finding which polynomial models can be identified in the usual regression set up. The algebraic methods based on Groebner bases provide a systematic way of doing this. The algebraic method does not in general produce all estimable models but it can be shown that it yields models which have minimal average degree in a well-defined sense and in both a weighted and unweighted version. This provides an alternative measure to that based on "aberration" and moreover is applicable to any experimental design. A simple algorithm is given and bounds are derived for the criteria, which may be used to give asymptotic Nyquist-like estimability rates as model and sample sizes increase.


A quantized Tits-Kantor-Koecher algebra

Abstract: We propose a quantum analogue of a Tits-Kantor-Koecher algebra with a Jordan torus as an coordinated algebra by looking at the vertex operator construction over a Fock space.


Evaluations of the twisted Alexander polynomials of 2-bridge knots at $\pm 1$

Abstract: Let $H(p)$ be the set of 2-bridge knots $K$ whose group $G$ is mapped onto a non-trivial free product, $Z/2 * Z/p$, $p$ being odd. Then there is an algebraic integer $s_0$ such that for any $K$ in $H(p)$, $G$ has a parabolic representation $\rho$ into $SL(2, Z[s_0]) \subset SL(2,C)$. Let $\Delta(t)$ be the twisted Alexander polynomial associated to $\rho$. Then we prove that for any $K$ in $H(p)$, $\Delta(1)=-2s_0^{-1}$ and $\Delta(-1)=-2s_0^{-1}\mu^2$, where $s_0^{-1}, \mu \in Z[s_0]$. The number $\mu$ can be recursively evaluated.


Uniform deterministic equivalent of additive functionals and non-parametric drift estimation for one-dimensional recurrent diffusions

Abstract: Usually the problem of drift estimation for a diffusion process is considered under the hypothesis of ergodicity. It is less often considered under the hypothesis of null-recurrence, simply because there are fewer limit theorems and existing ones do not apply to the whole null-recurrent class. The aim of this paper is to provide some limit theorems for additive functionals and martingales of a general (ergodic or null) recurrent diffusion which would allow us to have a somewhat unified approach to the problem of non-parametric kernel drift estimation in the one-dimensional recurrent case. As a particular example we obtain the rate of convergence of the Nadaraya--Watson estimator in the case of a locally H\"{o}lder-continuous drift.


A theory of hierarchical consequence and conditionals

Abstract: We introduce A-ranked preferential structures and combine them with an accessibility relation. This framework allows us to formalize contrary to duty obligations. Representation results are proved.


Roadmap for preferential logics

Abstract: We give an overview of logical and semantical rules for nonmonotonic and related logics.


Reactive preferential structures and nonmonotonic consequence

Abstract: We introduce information bearing systems (IBRS) as an abstraction of many logical systems. We define a general semantics for IBRS, and show that IBRS generalize in a natural way preferential semantics and solve open representation problems.


Cumulativity without closure of the domain under finite unions

Abstract: For nonmonotonic logics, Cumulativity is an important rule. We show here that Cumulativity fans out into an infinity of different conditions, if the domain is not closed under finite unions.


Decoration invariants for horseshoe braids

Abstract: The Decoration Conjecture describes the structure of the set of braid types of Smale's horseshoe map ordered by forcing, providing information about the order in which periodic orbits can appear when a horseshoe is created. A proof of this conjecture is given for the class of so-called lone decorations, and it is explained how to calculate associated braid conjugacy invariants which provide additional information about forcing for horseshoe braids.


Isomonodromic deformatiion with an irregular singularity and hyperelliptic curve

Abstract: In this paper, we extend the result of Kitaev and Korotkin to the case where a monodromy-preserving deformation has an irregular singularity. For the monodromy-preserving deformation, we obtain the $\tau$-function whose deformation parameters are the positions of regular singularities and the parameter $t$ of an irregular singularity. Furthermore, the $\tau$-function is expressed by the hyperelliptic $\Theta$ function moving the argument $\z$ and the period $\B,$ where $t$ and the positions of regular singularities move $z$ and $\B,$ respectively.


Global Smooth Effects and Well-Posedness for the Derivative Nonlinear Schr\"odinger Equation with Small Rough Data

Abstract: \rm We obtain the global smooth effects for the solutions of the linear Schr\"odinger equation in anisotropic Lebesgue spaces. Applying these estimates, we study the Cauchy problem for the generalized elliptical and non-elliptical derivative nonlinear Schr\"odinger equations (DNLS) and get the global well posedness of solutions with small data in modulation spaces $M^{3/2}_{2,1}(\mathbb{R}^n)$. Noticing that $H^{\tilde{s}} \subset M^s_{2,1}$ $(\tilde{s}-s>n/2)$ is an optimal inclusion, we have shown the global well posedness of DNLS with a class of very rough data.


Identities for hyperelliptic P-functions of genus one, two and three in covariant form

Abstract: We give a covariant treatment of the quadratic differential identities satisfied by the P-functions on the Jacobian of smooth hyperelliptic curves of genera 1, 2 and 3.


Some properties of multivariate measures of concordance

Abstract: We explore the consequences of a set of axioms which extend Scarsini's axioms for bivariate measures of concordance to the multivariate case and exhibit the following results: (1) A method of extending measures of concordance from the bivariate case to arbitrarily high dimensions. (2) A formula expressing the measure of concordance of the random vectors $(\pm X_1,...,\pm X_n)$ in terms of the measures of concordance of the "marginal" random vectors $(X_{i_1},...,X_{i_k})$. (3) A method of expressing the measure of concordance of an odd-dimensional copula in terms of the measures of concordance of its even-dimensional marginals. (4) A family of relations which exist between the measures of concordance of the marginals of a given copula.


Top homology of hypergraph matching complexes, $p$-cycle complexes and Quillen complexes of symmetric groups

Abstract: We investigate the representation of a symmetric group $S_n$ on the homology of its Quillen complex at a prime $p$. For homology groups in small codimension, we derive an explicit formula for this representation in terms of the representations of symmetric groups on homology groups of $p$-uniform hypergraph matching complexes. We conjecture an explicit formula for the representation of $S_n$ on the top homology group of the corresponding hypergraph matching complex when $n \equiv 1 \bmod p$. Our conjecture follows from work of Bouc when $p=2$, and we prove the conjecture when $p=3$.


Some relative stable categories are compactly generated

Abstract: Let G be a finite group. The stable module category of G has been applied extensively in group representation theory. In particular, it has been used to great effect that it is a triangulated category which is compactly generated. Let H be a subgroup of G. It is possible to define a stable module category of G relative to H. It too is a triangulated category, but no non-trivial examples have been known where this relative stable category was compactly generated. We show here that the relative stable category is compactly generated if the group algebra of H has finite representation type. In characteristic p, this is equivalent to the Sylow p-subgroups of H being cyclic.