Consensus optimization on manifolds

Abstract: The present paper considers distributed consensus algorithms that involve N agents evolving on a connected compact homogeneous manifold. The agents track no external reference and communicate their relative state according to a communication graph. The consensus problem is formulated in terms of the extrema of a cost function. This leads to efficient gradient algorithms to synchronize (i.e. maximizing the consensus) or balance (i.e. minimizing the consensus) the agents; a convenient adaptation of the gradient algorithms is used when the communication graph is directed and time-varying. The cost function is linked to a specific centroid definition on manifolds, introduced here as the induced arithmetic mean, that is easily computable in closed form and may be of independent interest for a number of manifolds. The special orthogonal group SO(n) and the Grassmann manifold Gr(p,n) are treated as original examples. A link is also drawn with the many existing results on the circle.


Embedding the bicyclic semigroup into countably compact topological semigroups

Abstract: We study algebraic and topological properties of topological semigroups containing a copy of the bicyclic semigroup C(p,q). We prove that each topological semigroup S with pseudocompact square contains no dense copy of C(p,q). On the other hand, we construct a (consistent) example of a pseudocompact (countably compact) Tychonov semigroup containing a copy of C(p,q).


Random walks in random Dirichlet environment are transient in dimension $d\ge 3$

Abstract: We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On $\Z^d$, RWDE are parameterized by a $2d$-uplet of positive reals. We prove that for all values of the parameters, RWDE are transient in dimension $d\ge 3$. We also prove that the Green function has some finite moments and we characterize the finite moments. Our result is more general and applies for example to finitely generated symmetric transient Cayley graphs. In terms of reinforced random walks it implies that directed edge reinforced random walks are transient for $d\ge 3$.


Forgetful maps between Deligne-Mostow ball quotients

Abstract: We study forgetful maps between Deligne-Mostow moduli spaces of weighted points on P^1, and classify the forgetful maps that extend to a map of orbifolds between the stable completions. The cases where this happens include the Livn\'e fibrations and the Mostow/Toledo maps between complex hyperbolic surfaces. They also include a retraction of a 3-dimensional ball quotient onto one of its 1-dimensional totally geodesic complex submanifolds.


Diophantine properties for q-analogues of Dirichlet's beta function at positive integers

Abstract: small In this paper, we define $q$-analogues of Dirichlet's beta function at positive integers, which can be written as $\beta_q(s)=\sum_{k\geq1}\sum_{d|k}\chi(k/d)d^{s-1}q^k$ for $s\in\N^*$, where $q$ is a complex number such that $|q|<1$ and $\chi$ is the non trivial Dirichlet character modulo 4. For odd $s$, these expressions are connected with the automorphic world, in particular with Eisenstein series of level 4. From this, we derive through Nesterenko's work the transcendance of the numbers $\beta_q(2s+1)$ for $q$ algebraic such that $0<|q|<1$. Our main result concerns the nature of the numbers $\beta_q(2s)$: we give a lower bound for the dimension of the vector space over $\Q$ spanned by $1,\beta_q(2),\beta_q(4),...,\beta_q(A)$, where $1/q\in\Z\setminus\{-1;1\}$ and $A$ is an even integer. As consequences, for $1/q\in\Z\setminus\{-1;1\}$, on the one hand there is an infinity of irrational numbers among $\beta_q(2),\beta_q(4),...$, and on the other hand at least one of the numbers $\beta_q(2),\beta_q(4),..., \beta_q(20)$ is irrational.


Uniform distribution of Galois conjugates and beta-conjugates of a Parry number near the unit circle and dichotomy of Perron numbers

Abstract: Concentration and equi-distribution, near the unit circle, in Solomyak's set, of the union of the Galois conjugates and the beta-conjugates of a Parry number $\beta$ are characterized by means of the Erd\H{o}s-Tur\'an approach, and its improvements by Mignotte and Amoroso, applied to the analytical function $f_{\beta}(z) = -1 + \sum_{i \geq 1} t_i z^i$ associated with the R\'enyi $\beta$-expansion $d_{\beta}(1)= 0.t_1 t_2 ...$ of unity. Mignotte's discrepancy function requires the knowledge of the factorization of the Parry polynomial of $\beta$. This one is investigated using theorems of Cassels, Dobrowolski, Pinner and Vaaler, Smyth, Schinzel in terms of cyclotomic, reciprocal non-cyclotomic and non-reciprocal factors. An upper bound of Mignotte's discrepancy function which arises from the beta-conjugates of $\beta$ which are roots of cyclotomic factors is linked to the Riemann hypothesis, following Amoroso. An equidistribution limit theorem, following Bilu's theorem, is formulated for the concentration phenomenon of conjugates of Parry numbers near the unit circle. Parry numbers are Perron numbers. Open problems on non-Parry Perron numbers are mentioned in the context of the existence of non-unique factorizations of elements of number fields into irreducible Perron numbers (Lind).


Extension of sections via adjoint ideals

Abstract: The paper contains theorems on extending sections of line bundles from divisors to the ambient space, inspired by various results of Siu, Kawamata, and especially Hacon-McKernan and Takayama. Applications are given to basepoint-freeness, to the invariance of analogues of plurigenera and of part of the boundary of the pseudo-effective cone in families, and to a simplified approach towards part of the results used for proving the existence of flips by Hacon-McKernan.


On Tits' Centre Conjecture for Fixed Point Subcomplexes

Abstract: We give a short and uniform proof of a special case of Tits' Centre Conjecture using a theorem of J-P. Serre and a result from our earlier work. We consider fixed point subcomplexes $X^H$ of the building $X = X(G)$ of a connected reductive algebraic group $G$, where $H$ is a subgroup of $G$.


The Eilenberg-Moore category and a Beck-type theorem for a Morita context

Abstract: The Eilenberg-Moore constructions and a Beck-type theorem for pairs of monads are described. More specifically, a notion of a {\em Morita context} comprising of two monads, two bialgebra functors and two connecting maps is introduced. It is shown that in many cases equivalences between categories of algebras are induced by such Morita contexts. The Eilenberg-Moore category of representations of a Morita context is constructed. This construction allows one to associate two pairs of adjoint functors with right adjoint functors having a common domain or a {\em double adjunction} to a Morita context. It is shown that, conversely, every Morita context arises from a double adjunction. The comparison functor between the domain of right adjoint functors in a double adjunction and the Eilenberg-Moore category of the associated Morita context is defined. The sufficient and necessary conditions for this comparison functor to be an equivalence (or for the {\em moritability} of a pair of functors with a common domain) are derived.


An Elementary Approach to a Model Problem of Lagerstrom

Abstract: The equation studied is u"+((n-1)/r)u'+epsilon u u'+ku'^{2}=0, with boundary conditions u(1)=0, u(infinity)=1. This model equation has been studied by many authors since it was introduced in the 1950s by P. A. Lagerstrom. We use an elementary approach to show that there is an infinite series solution which is uniformly convergent on 1<=r<infinity. The first few terms are easily derived, from which one quickly deduces the inner and outer asymptotic expansions, with no matching procedure or a priori assumptions about the nature of the expansion. We also give a short and elementary existence and uniqueness proof which covers all epsilon > 0, k >= 0, and n >= 1.


Ext-symmetry over quantum complete intersections

Abstract: We show that symmetry in the vanishing of cohomology holds for graded modules over quantum complete intersections. Moreover, symmetry holds for all modules if the algebra is symmetric.


Linear stochastic systems: a white noise approach

Abstract: Using the white noise setting, in particular the Wick product, the Hermite transform, and the Kondratiev space, we present a new approach to study linear stochastic systems, where randomness is also included in the transfer function. We prove BIBO type stability theorems for these systems, both in the discrete and continuous time cases. We also consider the case of dissipative systems for both discrete and continuous time systems. We further study $\ell_1$-$\ell_2$ stability in the discrete time case, and ${\mathbf L}_2$-${\mathbf L}_\infty$ stability in the continuous time case.


Stable pairs on elliptic K3 surfaces

Abstract: We study semistable pairs on elliptic K3 surfaces with a section: we construct a family of moduli spaces of pairs, related by wall crossing phenomena, which can be studied to describe the birational correspondence between moduli spaces of sheaves of rank 2 and Hilbert schemes on the surface. In the 4-dimensional case, this can be used to get the isomorphism between the moduli space and the Hilbert scheme described by Friedman.


Invariants of stationary AF-algebras and torsion subgroup of elliptic curves with complex multiplication

Abstract: Let G(A) be an AF-algebra given by periodic Bratteli diagram with the incidence matrix A in GL(n, Z). For a given polynomial p(x) in Z[x] we assign to G(A) a finite abelian group Z^n/p(A) Z^n. It is shown that if p(0)=1 or p(0)=-1 and Z[x]/(p(x)) is a principal ideal domain, then Z^n/p(A) Z^n is an invariant of the strong stable isomorphism class of G(A). For n=2 and p(x)=x-1 we conjecture a formula linking values of the invariant and torsion subgroup of elliptic curves with complex multiplication.


Enlargements of positive sets

Abstract: In this paper we introduce the notion of enlargement of a positive set in SSD spaces. To a maximally positive set $A$ we associate a family of enlargements $\E(A)$ and characterize the smallest and biggest element in this family with respect to the inclusion relation. We also emphasize the existence of a bijection between the subfamily of closed enlargements of $\E(A)$ and the family of so-called representative functions of $A$. We show that the extremal elements of the latter family are two functions recently introduced and studied by Stephen Simons. In this way we extend to SSD spaces some former results given for monotone and maximally monotone sets in Banach spaces.


A discrete Faa di Bruno's formula

Abstract: We derive some formulas that rule the behaviour of finite differences under composition of functions with vector values and arguments.


Principe d'Heisenberg et fonctions positives

Abstract: Starting from a problem in number theory, the article investigates the properties of the couples of Fourier transforms on the real line, f and g, real and even, f >= 0 out of an interval (-a, a) and f(0) < 0, g >= 0 out of an interval (-b, b) and g(0) < 0 . How small the product ab can be ? There is a strictly positive lower bound, the exact value is not known. The same problem is considered in several dimensions (where it is related to number theory, as the article points out).


On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon

Abstract: The spectrum of a weighted Dirac comb on the Thue-Morse quasicrystal is investigated, and characterized up to a measure zero set, by means of the Bombieri-Taylor conjecture, for Bragg peaks, and of another conjecture that we call Aubry-Godr\`eche-Luck conjecture, for the singular continuous component. The decomposition of the Fourier transform of the weighted Dirac comb is obtained in terms of tempered distributions. We show that the asymptotic arithmetics of the $p$-rarefied sums of the Thue-Morse sequence (Dumont; Goldstein, Kelly and Speer; Grabner; Drmota and Skalba,...), namely the fractality of sum-of-digits functions, play a fundamental role in the description of the singular continous part of the spectrum, combined with some classical results on Riesz products of Peyri\`ere and M. Queff\'elec. The dominant scaling of the sequences of approximant measures on a part of the singular component is controlled by certain inequalities in which are involved the class number and the regulator of real quadratic fields.


A simple presentation of the handlebody group of genus 2

Abstract: Using a similar algorithm to Hatcher-Thurston's algorithm for finding a presentation of the mapping class group of a surface, Wajnryb succeeded to find a presentation for the handlebody group. This is long and complicated. In this note I simplify Wajnryb's presentation for the handlebody group of genus g = 2.


Extension of functions with bounded finite differences

Abstract: We prove that functions defined on a lattice in a finite dimensional torus with bounded finite differences can be smoothly extended to the whole torus, and relate the bounds on the extension's derivatives with bounds on the original function's finite differences.


Classifications of linear operators preserving elliptic, positive and non-negative polynomials

Abstract: We characterize all linear operators on finite or infinite-dimensional spaces of univariate real polynomials preserving the sets of elliptic, positive, and non-negative polynomials, respectively. This is done by means of Fischer-Fock dualities, Hankel forms, and convolutions with non-negative measures. We also establish higher-dimensional analogs of these results. In particular, our classification theorems solve the questions raised in [9] originating from entire function theory and the literature pertaining to Hilbert's 17th problem.


A non-solvable Galois extension of $\Q$ ramified at 2 only

Abstract: In this paper, we show the existence of a non-solvable Galois extension of $\Q$ which is unramified outside 2. The extension $K$ we construct has degree $2251731094732800=2^{19}(3\cdot 5\cdot 17\cdot 257)^2$ and has root discriminant $\delta_K <2^{{47/8}}=58.68...$, and is totally complex.


Words with intervening neighbours in infinite Coxeter groups are reduced

Abstract: Consider a graph with vertex set S. A word in the alphabet S has the intervening neighbours property if any two occurrences of the same letter are separated by all its graph neighbours. For a Coxeter graph, words represent group elements. Speyer recently proved that words with the intervening neighbours property are irreducible if the group is infinite and irreducible. We present a new and shorter proof using the root automaton for recognition of irreducible words.


$R$-polynomials of finite monoids of Lie type

Abstract: This paper concerns the combinatorics of the orbit Hecke algebra associated with the orbit of a two sided Weyl group action on the Renner monoid of a finite monoid of Lie type, $M$. It is shown by Putcha in \cite{Putcha97} that the Kazhdan-Lusztig involution (\cite{KL79}) can be extended to the orbit Hecke algebra which enables one to define the $R$-polynomials of the intervals contained in a given orbit. Using the $R$-polynomials, we calculate the M\"obius function of the Bruhat-Chevalley ordering on the orbits. Furthermore, we provide a necessary condition for an interval contained in a given orbit to be isomorphic to an interval in some Weyl group.


Exactly solved models of polyominoes and polygons

Abstract: This chapter deals with the exact enumeration of certain classes of self-avoiding polygons and polyominoes on the square lattice. We present three general approaches that apply to many classes of polyominoes. The common principle to all of them is a recursive description of the polyominoes which then translates into a functional equation satisfied by the generating function. The first approach applies to classes of polyominoes having a linear recursive structure and results in a rational generating function. The second approach applies to classes of polyominoes having an algebraic recursive structure and results in an algebraic generating function. The third approach, commonly called the Temperley method, is based on the action of adding a new column to the polyominoes. We conclude by discussing some open questions.


Notes on Algebraic Cycles and Homotopy theory

Abstract: We show that a conjecture by Lawson holds, that is, the inclusion from the Chow variety $C_{p,d}(P^n)$ of all effective algebraic p-cycles of degree d in n-dimensional projective space to the space of effective algebraic p-cycles is 2d-connected. As a result, the homotopy and homology groups of $C_{p,d}(P^n)$ are calculated up to 2d. We also show an analogous statement for Chow variety $C_{p,d}(\P^n)$ over algebraically closed fields of arbitrary characteristic and compute their etale homotopy groups up to 2d.


Translation-finite sets, and weakly compact derivations from $\lp{1}(\Z_+)$ to its dual

Abstract: We characterize those derivations from the convolution algebra $\ell^1({\mathbb Z}_+)$ to its dual which are weakly compact. In particular, we provide examples which are weakly compact but not compact. The characterization is combinatorial, in terms of "translation-finite" subsets of ${\mathbb Z}_+$, and we investigate how this notion relates to other notions of "smallness" for infinite subsets of ${\mathbb Z}_+$. In particular, we show that a set of strictly positive Banach density cannot be translation-finite; the proof has a Ramsey-theoretic flavour.


On the special values of certain Rankin-Selberg L-functions and applications to odd symmetric power L-functions of modular forms

Abstract: We prove an algebraicity result for the central critical value of certain Rankin-Selberg L-functions for GL(n) x GL(n-1). This is a generalization and refinement of some results of Harder, Kazhdan-Mazur-Schmidt, Mahnkopf, and Kasten-Schmidt. As an application of this result, we prove algebraicity results for certain critical values of the fifth and the seventh symmetric power L-functions attached to a holomorphic cusp form. Assuming Langlands functoriality one can prove similar algebraicity results for the special values of any odd symmetric power L-function. We also prove a conjecture of Blasius and Panchishkin on twisted L-values in some cases. We comment on the compatibility of our results with Deligne's conjecture on the critical values of motivic L-functions. These results, as in the above mentioned works, are, in general, based on a nonvanishing hypothesis on certain archimedean integrals.


Alternating Euler sums at the negative integers

Abstract: We study three special Dirichlet series, two of them alternating, related to the Riemann zeta function. These series are shown to have extensions to the entire complex plane and we find their values at the negative integers (or residues at poles). These values are given in terms of Bernoulli and Euler numbers.


Where to place a hole to achieve a maximal escape rate

Abstract: A natural question of how the survival probability depends upon a position of a hole was seemingly never addressed in the theory of open dynamical systems. We found that this dependency could be very essential. The main results are related to the holes with equal sizes (measure) in the phase space of strongly chaotic maps. Take in each hole a periodic point of minimal period. Then the faster escape occurs through the hole where this minimal period assumes its maximal value. The results are valid for all finite times (starting with the minimal period) which is unusual in dynamical systems theory where typically statements are asymptotic when time tends to infinity. It seems obvious that the bigger the hole is the bigger is the escape through that hole. Our results demonstrate that generally it is not true, and that specific features of the dynamics may play a role comparable to the size of the hole.


Continuous Wavelets on Compact Manifolds

Abstract: Let $\bf M$ be a smooth compact oriented Riemannian manifold, and let $\Delta_{\bf M}$ be the Laplace-Beltrami operator on ${\bf M}$. Say $0 \neq f \in \mathcal{S}(\RR^+)$, and that $f(0) = 0$. For $t > 0$, let $K_t(x,y)$ denote the kernel of $f(t^2 \Delta_{\bf M})$. We show that $K_t$ is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator $f(t^2\Delta)$ on $\RR^n$. We define continuous ${\cal S}$-wavelets on ${\bf M}$, in such a manner that $K_t(x,y)$ satisfies this definition, because of its localization near the diagonal. Continuous ${\cal S}$-wavelets on ${\bf M}$ are analogous to continuous wavelets on $\RR^n$ in $\mathcal{S}(\RR^n)$. In particular, we are able to characterize the H$\ddot{o}$lder continuous functions on ${\bf M}$ by the size of their continuous ${\mathcal{S}}-$wavelet transforms, for H$\ddot{o}$lder exponents strictly between 0 and 1. If $\bf M$ is the torus $\TT^2$ or the sphere $S^2$, and $f(s)=se^{-s}$ (the ``Mexican hat'' situation), we obtain two explicit approximate formulas for $K_t$, one to be used when $t$ is large, and one to be used when $t$ is small.


Log canonical thresholds on smooth varieties: the Ascending Chain Condition

Abstract: Building on results of Koll\'ar, we prove Shokurov's ACC Conjecture for log canonical thresholds on smooth varieties, and more generally, on varieties with quotient singularities.


Understanding Terrorist Organizations with a Dynamic Model

Abstract: Terrorist organizations change over time because of processes such as recruitment and training as well as counter-terrorism (CT) measures, but the effects of these processes are typically studied qualitatively and in separation from each other. Seeking a more quantitative and integrated understanding, we constructed a simple dynamic model where equations describe how these processes change an organization's membership. Analysis of the model yields a number of intuitive as well as novel findings. Most importantly it becomes possible to predict whether counter-terrorism measures would be sufficient to defeat the organization. Furthermore, we can prove in general that an organization would collapse if its strength and its pool of foot soldiers decline simultaneously. In contrast, a simultaneous decline in its strength and its pool of leaders is often insufficient and short-termed. These results and other like them demonstrate the great potential of dynamic models for informing terrorism scholarship and counter-terrorism policy making.


Towards Proving Legendre's Conjecture

Abstract: Legendre's conjecture states that there is a prime number between n^2 and (n+1)^2 for every positive integer n. We consider the following question : for all integer n>1 and a fixed integer k<=n does there exist a prime number such that kn < p < (k+1)n ? Bertrand-Chebyshev theorem answers this question affirmatively for k=1. A positive answer for k=n would prove Legendre's conjecture. In this paper, we show that one can determine explicitly a number N(k) such that for all n >= N(k), there is at least one prime between kn and (k+1)n. Our proof is based on Erdos's proof of Bertrand-Chebyshev theorem and uses elementary combinatorial techniques without appealing to the prime number theorem.


Laumon Spaces and the Calogero-Sutherland Integrable System

Abstract: This paper contains a proof of a conjecture of Braverman concerning Laumon quasiflag spaces. We consider the generating function Z(m), whose coefficients are the integrals of the equivariant Chern polynomial (with variable m) of the tangent bundles of the Laumon spaces. We prove Braverman's conjecture, which states that Z(m) coincides with the eigenfunction of the Calogero-Sutherland hamiltonian, up to a simple factor which we specify. This conjecture was inspired by the work of Nekrasov in the affine \hat{sl}_n setting, where a similar conjecture is still open.


Convergence to Weighted Fractional Brownian Sheets

Abstract: We define weighted fractional Brownian sheets, which are a class of Gaussian random fields with four parameters that include fractional Brownian sheets as special cases, and we give some of their properties. We show that for certain values of the parameters the weighted fractional Brownian sheets are obtained as limits in law of occupation time fluctuations of a stochastic particle model. In contrast with some known approximations of fractional Brownian sheets which use a kernel in a Volterra type integral representation of fractional Brownian motion with respect to ordinary Brownian motion, our approximation does not make use of a kernel.


Transgression and twisted anomaly cancellation formulas on odd dimensional manifolds

Abstract: We compute the transgressed forms of some modularly invariant characteristic forms,which are related to the twisted elliptic genera. We study the modularity properties of these secondary characteristic forms and relations among them. We also get some twisted anomaly cancellation formulas on some odd dimensional manifolds.


On the existence of embeddings into modules of finite homological dimensions

Abstract: Let R be a commutative Noetherian local ring. We show that R is Gorenstein if and only if every finitely generated R-module can be embedded in a finitely generated R-module of finite projective dimension. This extends a result of Auslander and Bridger to rings of higher Krull dimension, and it also improves a result due to Foxby where the ring is assumed to be Cohen-Macaulay.


A Local Relative Trace Formula for F*\SL(2,F)

Abstract: In this note, we derive explicitly the local relative trace formula for the symmetric space F*\SL(2,F) at the level of Lie algebras, where F is a p-adic field of residue characteristic greater than two and F* is the set of invertible elements in F. This is perhaps one of the simplest non-trivial analogs of the trace formula, and also a motivating example for the author's work (in preparation) on the relative trace formula.


Partial Correlation Estimation by Joint Sparse Regression Models

Abstract: In this paper, we propose a computationally efficient approach -- space(Sparse PArtial Correlation Estimation)-- for selecting non-zero partial correlations under the high-dimension-low-sample-size setting. This method assumes the overall sparsity of the partial correlation matrix and employs sparse regression techniques for model fitting. We illustrate the performance of space by extensive simulation studies. It is shown that space performs well in both non-zero partial correlation selection and the identification of hub variables, and also outperforms two existing methods. We then apply space to a microarray breast cancer data set and identify a set of hub genes which may provide important insights on genetic regulatory networks. Finally, we prove that, under a set of suitable assumptions, the proposed procedure is asymptotically consistent in terms of model selection and parameter estimation.


AV-Courant algebroids and generalized CR structures

Abstract: We construct a generalization of Courant algebroids which are classified by the third cohomology group $H^3(A,V)$, where $A$ is a Lie Algebroid, and $V$ is an $A$-module. We see that both Courant algebroids and $\mathcal{E}^1(M)$ structures are examples of them. Finally we introduce generalized CR structures on a manifold, which are a generalization of generalized complex structures, and show that every CR structure and contact structure is an example of a generalized CR structure.


Asymptotically hyperbolic manifolds with polyhomogeneous metric

Abstract: We analyze the resolvent and define the scattering matrix for asymptotically hyperbolic manifolds with metrics which have a polyhomogeneous expansion near the boundary, and also prove that there is always an essential singularity of the resolvent in this setting. We use this analysis to prove an inverse result for conformally compact odd dimensional Einstein manifolds.


Second order Poincar\'e inequalities and CLTs on Wiener space

Abstract: We prove infinite-dimensional second order Poincar\'e inequalities on Wiener space, thus closing a circle of ideas linking limit theorems for functionals of Gaussian fields, Stein's method and Malliavin calculus. We provide two applications: (i) to a new "second order" characterization of CLTs on a fixed Wiener chaos, and (ii) to linear functionals of Gaussian-subordinated fields.


Large Deviations estimates for some non-local equations I. Fast decaying kernels and explicit bounds

Abstract: We study large deviations for some non-local parabolic type equations. We show that, under some assumptions on the non-local term, problems defined in a bounded domain converge with an exponential rate to the solution of the problem defined in the whole space. We compute this rate in different examples, with different kernels defining the non-local term, and it turns out that the estimate of convergence depends strongly on the decay at infinity of that kernel.


A note on the convergence of parametrised non-resonant invariant manifolds

Abstract: Truncated Taylor series representations of invariant manifolds are abundant in numerical computations. We present an aposteriori method to compute the convergence radii and error estimates of analytic parametrisations of non-resonant local invariant manifolds of a saddle of an analytic vector field, from such a truncated series. This enables us to obtain local enclosures, as well as existence results, for the invariant manifolds.


Support varieties and the Hochschild cohomology ring modulo nilpotence

Abstract: This is a survey paper based on my talks at the 41st Symposium on Ring Theory and Representation Theory, held in Shizuoka University, Japan in September 2008, and will appear in the conference proceedings. The paper begins with a brief introduction to the use of Hochschild cohomology in developing the theory of support varieties for a module over an artin algebra, by Snashall and Solberg (Proc. London Math. Soc.(3) 88 (2004), 705-732). The paper then describes the current status of research concerning the structure of the Hochschild cohomology ring modulo nilpotence.


The 2D Ising model near criticality: a FK percolation analysis

Abstract: We study the 2d-Ising model defined on finite boxes at temperatures that are below but very close from the critical point. When the temperature approaches the critical point and the size of the box grows fast enough, we establish large deviations estimates on FK-percolation events that concern the phenomenon of phase coexistence.


Special L-values of t-motives: a conjecture

Abstract: We propose a conjecture on special values of $ L $-functions in a function field context with positive characteristic coefficients. For $ M $ a uniformizable $ t $-motive with everywhere good reduction we conjecture a relation between the value of the Goss $ L $-function $ L(M^\vee, s) $ at $ s = 0 $ and the uniformization of the abelian $ t $-module associated with $ M $. When $ M $ is a power of the Carlitz $ t $-motive the conjecture specializes to a theorem of Anderson and Thakur on Carlitz zeta values. Beyond this case we present numerical evidence.


Invariant elliptic curves as attractors in the projective plane

Abstract: Let f be a rational self-map of P^2 which leaves invariant an elliptic curve C with strictly negative transverse Lyapunov exponent. We show that C is an attractor, i.e. it possesses a dense orbit and its basin is of strictly positive measure.


Dynamics of postcritically bounded polynomial semigroups III: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles

Abstract: We investigate the dynamics of polynomial semigroups (semigroups generated by a family of polynomial maps on the Riemann sphere) and the random dynamics of polynomials on the Riemann sphere. Combining the dynamics of semigroups and the fiberwise (random) dynamics, we give a classification of polynomial semigroups $G$ such that $G$ is generated by a compact family $\Gamma $, the planar postcritical set of $G$ is bounded, and $G$ is (semi-) hyperbolic. In one of the classes, we have that for almost every sequence $\gamma \in \Gamma ^{\Bbb{N}}$, the Julia set $J_{\gamma}$ of $\gamma $ is a Jordan curve but not a quasicircle, the unbounded component of the Fatou set $F_{\gamma}$ of $\gamma$ is a John domain, and the bounded component of $F_{\gamma}$ is not a John domain. Note that this phenomenon does not hold in the usual iteration of a single polynomial. Moreover, we consider the dynamics of polynomial semigroups $G$ such that the planar postcritical set of $G$ is bounded and the Julia set is disconnected. Those phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are systematically investigated.


Birkhoff normal form and splitting methods for semi linear Hamiltonian PDEs. Part I: Finite dimensional discretization

Abstract: We consider {\em discretized} Hamiltonian PDEs associated with a Hamiltonian function that can be split into a linear unbounded operator and a regular nonlinear part. We consider splitting methods associated with this decomposition. Using a finite dimensional Birkhoff normal form result, we show the almost preservation of the {\em actions} of the numerical solution associated with the splitting method over arbitrary long time, provided the Sobolev norms of the initial data is small enough, and for asymptotically large level of space approximation. This result holds under {\em generic} non resonance conditions on the frequencies of the linear operator and on the step size. We apply this results to nonlinear Schr\"odinger equations as well as the nonlinear wave equation.}


Quantales of open groupoids

Abstract: It is well known that inverse semigroups are closely related to \'etale groupoids. In particular, it has recently been shown that there is a (non-functorial) equivalence between localic \'etale groupoids, on one hand, and complete and infinitely distributive inverse semigroups (abstract complete pseudogroups), on the other. This correspondence is mediated by a class of quantales, known as inverse quantal frames, that are obtained from the inverse semigroups by a simple join completion that yields an equivalence of categories. Hence, we can regard abstract complete pseudogroups as being essentially ``the same'' as inverse quantal frames, and in this paper we exploit this fact in order to find a suitable replacement for inverse semigroups in the context of open groupoids that are not necessarily \'etale. The interest of such a generalization lies in the importance and ubiquity of open groupoids in areas such as operator algebras, differential geometry and topos theory, and we achieve it by means of a class of quantales, called open quantal frames, which generalize inverse quantal frames and whose properties we study in detail. The resulting correspondence between quantales and open groupoids is not a straightforward generalization of the previous results concerning \'etale groupoids, and it depends heavily on the existence of inverse semigroups of local bisections of the quantales involved.


A note on Vasiu-Zink windows

Abstract: We propose a notion of frames and windows that allows an alternative proof of the Vasiu-Zink classification of p-divisible groups over ramified complete regular local rings by their Breuil windows.


Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras

Abstract: We define the i-restriction and i-induction functors on the category O of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.


Notes on Artin-Tate motives

Abstract: We first study the weight structure on the triangulated category of Artin-Tate motives over a perfect base field k, building on results of Bondarko's. We then study the t-structure on the triangulated category of Artin-Tate motives, when k is algebraic over the rationals, generalizing a result of Levine's. We finally study the interaction of the weight structure and the t-structure. When k is a number field, this will give a useful criterion identifying the weight structure via realizations.


A note on the subword complexes in Coxeter groups

Abstract: We prove that the Stanley--Reisner ideal of the Alexander dual of the subword complexes in Coxeter groups has linear quotients with respect to the lexicographical order of the minimal monomial generators. As a consequence, we obtain a shelling order on the facets of the subword complex. We relate some invariants of the subword complexes or of their dual with invariants of the word. For a particular class of subword complexes, we prove that the Stanley--Reisner ring is a complete intersection ring.


Averaging lemmas with a force term in the transport equation

Abstract: We obtain several averaging lemmas for transport operator with a force term. These lemmas improve the regularity yet known by not considering the force term as part of an arbitrary right-hand side. Two methods are used: local variable changes or stationary phase. These new results are subjected to two non degeneracy assumptions. We characterize the optimal conditions of these assumptions to compare the obtained regularities according to the space and velocity variables. Our results are mainly in $L^2$, and for constant force, in $L^p$ for $1<p \leq 2$.


Courant algebroids and Poisson Geometry

Abstract: Given a manifold M with an action of a quadratic Lie algebra d, such that all stabilizer algebras are co-isotropic in d, we show that the product M\times d becomes a Courant algebroid over M. If the bilinear form on d is split, the choice of transverse Lagrangian subspaces g_1, g_2 of d defines a bivector field on M, which is Poisson if (d,g_1,g_2) is a Manin triple. In this way, we recover the Poisson structures of Lu-Yakimov, and in particular the Evens-Lu Poisson structures on the variety of Lagrangian Grassmannians and on the de Concini-Procesi compactifications. Various Poisson maps between such examples are interpreted in terms of the behaviour of Lagrangian splittings under Courant morphisms.


The Zassenhaus variety of a reductive Lie algebra in positive characteristic

Abstract: Let g be the Lie algebra of a connected reductive group G over an algebraically closed field k of characteristic p>0. Let $Z$ be the centre of the universal enveloping algebra U=U(g) of g. Its maximal spectrum is called the Zassenhaus variety of g. We show that, under certain mild assumptions on G, the field of fractions Frac(Z) of Z is G-equivariantly isomorphic to the function field of the dual space g* with twisted G-action. In particular Frac(Z) is rational. This confirms a conjecture J. Alev. Furthermore we show that Z is a unique factorisation domain, confirming a conjecture of A. Braun and C. Hajarnavis. Recently, A. Premet used the above result about Frac(Z), a result of Colliot-Thelene, Kunyavskii, Popov and Reichstein and reduction mod p arguments to show that the Gelfand-Kirillov conjecture cannot hold for simple complex Lie algebras that are not of type A, C or G_2.


Analytic equivalence of normal crossing functions on a real analytic manifold

Abstract: By Hironaka Desingularization Theorem, any real analytic function has only normal crossing singularities after a suitable modification. We focus on the analytic equivalence of such functions with only normal crossing singularities. We prove that for such functions $C^{\infty}$ right equivalence implies analytic equivalence. We prove moreover that the cardinality of the set of equivalence classes is zero or countable.


Trace Forms of Symbol Algebras

Abstract: Let S be a symbol algebra. The trace form of S is computed and it is shown how this form can be used to determine whether S is a division algebra or not. In addition, the exterior powers of the trace form of S are computed.


Concentration of the integral norm of idempotents

Abstract: This is a companion paper of a recent one, entitled {\sl Integral concentration of idempotent trigonometric polynomials with gaps}. New results of the present work concern $L^1$ concentration, while the above mentioned paper deals with $L^p$-concentration. Our aim here is two-fold. At the first place we try to explain methods and results, and give further straightforward corollaries. On the other hand, we push forward the methods to obtain a better constant for the possible concentration (in $L^1$ norm) of an idempotent on an arbitrary symmetric measurable set of positive measure. We prove a rather high level $\gamma_1>0.96$, which contradicts strongly the conjecture of Anderson et al. that there is no positive concentration in $L^1$ norm. The same problem is considered on the group $\mathbb{Z}/q\mathbb{Z}$, with $q$ say a prime number. There, the property of absolute integral concentration of idempotent polynomials fails, which is in a way a positive answer to the conjecture mentioned above. Our proof uses recent results of B. Green and S. Konyagin on the Littlewood Problem.


Coagulation, diffusion and the continuous Smoluchowski equation

Abstract: The Smoluchowski equation is a system of partial differential equations modelling the diffusion and binary coagulation of a large collection of tiny particles. The mass parameter may be indexed either by positive integers, or by positive reals, these corresponding to the discrete or the continuous form of the equations. In dimension at least 3, we derive the continuous Smoluchowski PDE as a kinetic limit of a microscopic model of Brownian particles liable to coalesce, using a similar method to that used to derive the discrete form of the equations in Hammond and Rezakhanlou [4]. The principal innovation is a correlation-type bound on particle locations that permits the derivation in the continuous context while simplifying the arguments of [4]. We also comment on the scaling satisfied by the continuous Smoluchowski PDE, and its potential implications for blow-up of solutions of the equations.


Quadratic perturbations of quadratic codimension-four centers

Abstract: We study the stratum in the set of all quadratic differential systems $\dot{x}=P_2(x,y), \dot{y}=Q_2(x,y)$ with a center, known as the codimension-four case $Q_4$. It has a center and a node and a rational first integral. The limit cycles under small quadratic perturbations in the system are determined by the zeros of the first Poincar\'e-Pontryagin-Melnikov integral $I$. We show that the orbits of the unperturbed system are elliptic curves, and $I$ is a complete elliptic integral. Then using Picard-Fuchs equations and the Petrov's method (based on the argument principle), we set an upper bound of eight for the number of limit cycles produced from the period annulus around the center.


On the Kontsevich integral for knotted trivalent graphs

Abstract: We construct an extension of the Kontsevich integral of knots to knotted trivalent graphs, which commutes with orientation switches, edge deletions, edge unzips, and connected sums. In 1997 Murakami and Ohtsuki [MO] first constructed such an extension, building on Drinfel'd's theory of associators. We construct a step by step definition, using elementary Kontsevich integral methods, to get a one-parameter family of corrections that all yield invariants well behaved under the graph operations above.


On the graded center of the stable category of a finite $p$-group

Abstract: We show that for any finite $p$-group $P$ of rank at least 2 and any algebraically closed field $k$ of characteristic $p$ the graded center $Z^*(\modbar(kP))$ of the stable module category of finite-dimensional $kP$-modules has infinite dimension in each odd degree, and if $p=2$ also in each even degree. In particular, this provides examples of symmetric algebras $A$ for which $Z^0(\modbar(A))$ is not finite-dimensional, answering a question raised in [10]


On (n,d)-perfect rings

Abstract: In this paper, we introduce the notion of "(n,d)-perfect rings" which is in some way a generalization of the notion of "S-rings". After we give some basic results of this rings and we survey the relationship between "A(n) property" and "(n,d)-perfect property". Finally, we investigate the "(n,d)-perfect property" in pullback rings.


A candidate for a solution to Wall's D(2) problem

Abstract: We show that Wall's D(2) problem, the Realization problem and the Relation Gap problem could all be solved if it could be shown that the deficiency of a certain group is, as intuition would suggest, less than -1. Note the paper has been withdrawn. A presentation of *_p (C_p x C_p)with deficiency -1 is given on p35 of: Cynthia Hog-Angeloni, Beitrage zum (einfachen) homotopietyp zweidimensionaler komplexe zu freein produkten und anderen gruppentheoretischen konstruktionen : PhD thesis, Frankfurt/Main 1988


The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes

Abstract: We prove localization and Zariski-Mayer-Vietoris for higher Grothendieck-Witt groups, alias hermitian $K$-groups, of schemes admitting an ample family of line-bundles. No assumption on the characteristic is needed, and our schemes can be singular. Along the way, we prove additivity, fibration and approximation theorems for the hermitian $K$-theory of exact categories with weak equivalences and duality.


The theory of disconjugacy for a second order linear differential equation

Abstract: This is an introduction to the theory of disconjugacy for a second order linear differential equation. We give new proofs of some of basic results and obtain new sufficient conditions for disconjugacy (in particular, on the whole real axis).


Transformations of hypergeometric elliptic integrals

Abstract: The paper classifies algebraic transformations of Gauss hypergeometric functions with the local exponent differences $(1/2,1/4,1/4)$, $(1/2,1/3,1/6)$ and $(1/3,1/3,1/3)$. These form a special class of algebraic transformations of Gauss hypergeometric functions, of arbitrary high degree. The Gauss hypergeometric functions can be identified as elliptic integrals on the genus 1 curves $y=x^3-x$ or $y=x^3-1$. Especially interesting are algebraic transformations of the hypergeometric functions into themselves; these transformations come from isogenies of the respective elliptic curves.


The Ascending Chain Condition for log canonical thresholds on l.c.i. varieties

Abstract: Shokurov's ACC Conjecture says that the set of all log canonical thresholds on varieties of bounded dimension satisfies the Ascending Chain Condition. This conjecture was proved for log canonical thresholds on smooth varieties in [EM1]. Here we use this result and inversion of adjunction to establish the conjecture for locally complete intersection varieties.


Liouville type of theorems with weights for the Navier-Stokes equations and the Euler equations

Abstract: We study Liouville type of theorems for the Navier-Stokes and the Euler equations on $\Bbb R^N$, $N\geq 2$. Specifically, we prove that if a weak solution $(v,p)$ satisfies $|v|^2 +|p| \in L^1 (0,T; L^1(\Bbb R^N, w_1(x)dx))$ and $\int_{\Bbb R^N} p(x,t)w_2 (x)dx \geq0$ for some weight functions $w_1(x)$ and $w_2 (x)$, then the solution is trivial, namely $v=0$ almost everywhere on $\Bbb R^N \times (0, T)$. Similar results hold for the MHD Equations on $\Bbb R^N$, $N\geq3$.


A note on Fibonacci-type polynomials

Abstract: We opt to study the convergence of maximal real roots of certain Fibonacci-type polynomials given by $G_n=x^kG_{n-1}+G_{n-2}$. The special cases $k=1$ and $k=2$ are found in [4] and [7], respectively.


Gotzmann lexsegment ideals

Abstract: In this paper we characterize the componentwise lexsegment ideals which are componentwise linear and the lexsegment ideals generated in one degree which are Gotzmann.


Convergence rates of posterior distributions for observations without the iid structure

Abstract: The classical condition on the existence of uniformly exponentially consistent tests for testing the true density against the complement of its arbitrary neighborhood has been widely adopted in study of asymptotics of Bayesian nonparametric procedures. Because we follow a Bayesian approach, it seems to be more natural to explore alternative and appropriate conditions which incorporate the prior distribution. In this paper we supply a new prior-dependent integration condition to establish general posterior convergence rate theorems for observations which may not be independent and identically distributed. The posterior convergence rates for such observations have recently studied by Ghosal and van der Vaart \cite{ghv1}. We moreover adopt the Hausdorff $\alpha$-entropy given by Xing and Ranneby \cite{xir1}\cite{xi1}, which is also prior-dependent and smaller than the widely used metric entropies. These lead to extensions of several existing theorems. In particular, we establish a posterior convergence rate theorem for general Markov processes and as its application we improve on the currently known posterior rate of convergence for a nonlinear autoregressive model.


Clifford indices for vector bundles on curves

Abstract: For smooth projective curves the Clifford index is an important invariant which provides a bound for the dimension of the space of sections of a line bundle. This is the first step in distinguishing curves of the same genus. In this paper we generalise this to introduce Clifford indices for semistable vector bundles on curves. We study these invariants, giving some basic properties and carrying out some computations for small ranks and for general and some special curves. For curves whose classical Clifford index is two, we compute all values of our new Clifford indices.


Unconditional bases and strictly convex dual renormings

Abstract: We present equivalent conditions for a space $X$ with an unconditional basis to admit an equivalent norm with a strictly convex dual norm.


Bouvier's conjecture

Abstract: This paper deals with Bouvier's conjecture which sustains that finite-dimensional non-Noetherian Krull domains need not be Jaffard


t-Class semigroups of Noetherian domains

Abstract: The t-class semigroup of an integral domain is the semigroup of fractional t-ideals modulo its subsemigroup of nonzero principal ideals with the operation induced by ideal t-multiplication. This paper investigates ring-theoretic properties of a Noetherian domain that reflect reciprocally in the Clifford or Boolean property of its t-class semigroup.


Gromov-Witten invariants of toric Calabi-Yau threefolds

Abstract: Based on the large N duality relating topological string theory on Calabi-Yau 3-folds and Chern-Simons theory on 3-manifolds, M. Aganagic, A. Klemm, M. Marino and C. Vafa proposed the topological vertex (hep-th/0305132), an algorithm on computing Gromov-Witten invariants in all genera of any non-singular toric Calabi-Yau 3-fold. In this expository article, we describe the mathematical theory of the topological vertex developed by J. Li, K. Liu, J. Zhou, and the author (math/0408426).


Divided power structures and chain complexes

Abstract: We interpret divided power structures on the homotopy groups of simplicial commutative rings as having a counterpart in divided power structures on chain complexes coming from a non-standard symmetric monoidal structure.


Fast rotating Bose-Einstein condensates in an asymmetric trap

Abstract: We investigate the effect of the anisotropy of a harmonic trap on the behaviour of a fast rotating Bose-Einstein condensate. This is done in the framework of the 2D Gross-Pitaevskii equation and requires a symplectic reduction of the quadratic form defining the energy. This reduction allows us to simplify the energy on a Bargmann space and study the asymptotics of large rotational velocity. We characterize two regimes of velocity and anisotropy; in the first one where the behaviour is similar to the isotropic case, we construct an upper bound: a hexagonal Abrikosov lattice of vortices, with an inverted parabola profile. The second regime deals with very large velocities, a case in which we prove that the ground state does not display vortices in the bulk, with a 1D limiting problem. In that case, we show that the coarse grained atomic density behaves like an inverted parabola with large radius in the deconfined direction but keeps a fixed profile given by a Gaussian in the other direction. The features of this second regime appear as new phenomena.


A Construction of Biorthogonal Wavelets With a Compact Operator

Abstract: We present a construction of biorthogonal wavelets using a compact operator which allows to preserve or increase some properties: regularity/vanishing moments, parity, compact supported. We build then a simple algorithm which computes new filters.


Generalized power method for sparse principal component analysis

Abstract: In this paper we develop a new approach to sparse principal component analysis (sparse PCA). We propose two single-unit and two block optimization formulations of the sparse PCA problem, aimed at extracting a single sparse dominant principal component of a data matrix, or more components at once, respectively. While the initial formulations involve nonconvex functions, and are therefore computationally intractable, we rewrite them into the form of an optimization program involving maximization of a convex function on a compact set. The dimension of the search space is decreased enormously if the data matrix has many more columns (variables) than rows. We then propose and analyze a simple gradient method suited for the task. It appears that our algorithm has best convergence properties in the case when either the objective function or the feasible set are strongly convex, which is the case with our single-unit formulations and can be enforced in the block case. Finally, we demonstrate numerically on a set of random and gene expression test problems that our approach outperforms existing algorithms both in quality of the obtained solution and in computational speed.


Counting decomposable multivariate polynomials

Abstract: A polynomial f (multivariate over a field) is decomposable if f = g(h) with g univariate of degree at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and an approximation to their number over a finite field. The relative error in our approximations is exponentially decaying in the input size.


A Note on Toric Varieties Associated to Moduli Spaces

Abstract: In this note we give a brief review of the construction of a toric variety $\mathcal{V}$ coming from a genus $g \geq 2$ Riemann surface $\Sigma^g$ equipped with a trinion, or pair of pants, decomposition. This was outlined by J. Hurtubise and L. C. Jeffrey in \cite{JH1}. In \cite{T1} A. Tyurin used this construction on a certain collection of trinion decomposed surfaces to produce a variety $DM_g$ -- the so-called Delzant model of moduli space -- for each genus $g.$ We conclude this note with some basic facts about the moment polytopes of the varieties $\mathcal{V}.$ In particular, we show that the varieties $DM_g$ constructed by Tyurin, and claimed to be smooth, are in fact singular for $g \geq 3.$


The structure of maximal zero-sum free Sequences

Abstract: Let n be an integer, and consider finite sequences of elements of the group Z/nZ x Z/nZ. Such a sequence is called zero-sum free, if no subsequence has sum zero. It is known that the maximal length of such a zero-sum free sequence is 2n-2, and Gao and Geroldinger conjectured that every zero-sum free sequence of this length contains an element with multiplicity at least n-2. By recent results of Gao, Geroldinger and Grynkiewicz, it essentially suffices to verify the conjecture for n prime. Now fix a sequence (a_i) of length 2n-2 with maximal multiplicity of elements at most n-3. There are different approeaches to show that (a_i) contains a zero-sum; some work well when (a_i) does contain elements with high multiplicity, others work well when all multiplicities are small. The aim of this article is to initiate a systematic approach to property B via the highest occurring multiplicities. Our main results are the following: denote by m_1 >= m_2 the two maximal multiplicities of (a_i), and suppose that n is sufficiently big and prime. Then (a_i) contains a zero-sum in any of the following cases: when m_2 >= 2/3n, when m_1 > (1-c)n, and when m_2 < cn, for some constant c > 0 not depending on anything.


Twenty Digits of Some Integrals of the Prime Zeta Function

Abstract: The double sum sum_(s >= 1) sum_p 1/(p^s log p^s) = 2.00666645... over the inverse of the product of prime powers p^s and their logarithms, is computed to 24 decimal digits. The sum covers all primes p and all integer exponents s>=1. The calculational strategy is adopted from Cohen's work which basically looks at the fraction as the underivative of the Prime Zeta Function, and then evaluates the integral by numerical methods.


On the stability of the overconvergence under the direct image by a proper smooth morphism

Abstract: Up to a translation in the language of arithmetic $\D$-modules, we prove a conjecture of Berthelot on the preservation of the overconvergence under the direct image by a smooth proper morphism of varieties over a perfect field of characteristic $p>0$.


The falling appart of the tagged fragment and the asymptotic disintegration of the Brownian height fragmentation

Abstract: We present a further analysis of the fragmentation at heights of the normalized Brownian excursion. Specifically we study a representation for the mass of a tagged fragment in terms of a Doob transformation of the 1/2-stable subordinator and use it to study its jumps; this accounts for a description of how a typical fragment falls apart. These results carry over to the height fragmentation of the stable tree. Additionally, the sizes of the fragments in the Brownian fragmentation when it is about to reduce to dust are described in a limit theorem.


The Depth of a Hypersubstitution

Abstract: For given depth of a we derive a formula for the depth of the image of that term under a given hypersubstitution.


Multi-Hypersubstitutions and Colored Solid Varieties

Abstract: Hypersubstitutions are mappings which map operation symbols to terms. Terms can be visualized by trees. Hypersubstitutions can be extended to mappings defined on sets of trees. The nodes of the trees, describing terms, are labelled by operation symbols and by colors, i.e. certain positive integers. We are interested in mappings which map differently colored operation symbols to different terms. In this paper we extend the theory of hypersubstitutions and solid varieties to multi-hypersubstitutions and colored solid varieties. We develop the interconnections between such colored terms and multi-hypersubstitutions and the equational theory of Universal Algebra. The collection of all varieties of a given type forms a complete lattice which is very complex and difficult to study; multi-hypersubstitutions and colored solid varieties offer a new method to study complete sublattices of this lattice.


On some cohomological properties of almost complex manifolds

Abstract: We study a special type of almost complex structures, called pure and full and introduced by T.J. Li and W. Zhang, in relation to symplectic structures and Hard Lefschetz condition. We provide sufficient conditions to the existence of the above type of almost complex structures on compact quotients of Lie groups by discrete subgroups. We obtain families of pure and full almost complex structures on compact nilmanifolds and solvmanifolds. Some of these families are parametrized by real 2-forms which are anti-invariant with respect to the almost complex structures.


Further Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions

Abstract: For relatively prime positive integers u_0 and r, we consider the arithmetic progression {u_k := u_0+k*r} (0 <= k <= n). Define L_n := lcm{u_0,u_1,...,u_n} and let a >= 2 be any integer. In this paper, we show that, for integers alpha,r >= a and n >= 2*alpha*r, we have L_n >= u_0*r^{alpha+a-2}*(r+1)^n. In particular, letting a = 2 yields an improvement to the best previous lower bound on L_n (obtained by Hong and Yang) for all but three choices of alpha,r >= 2.


Some equivariant constructions in noncommutative algebraic geometry

Abstract: We here present rudiments of an approach to geometric actions in noncommutative algebraic geometry, based on geometrically admissible actions of monoidal categories. This generalizes the usual (co)module algebras over Hopf algebras which provide affine examples. We introduce a compatibility of monoidal actions and localizations which is a distributive law. There are satisfactory notions of equivariant objects, noncommutative fiber bundles and quotients in this setup.


A note on the abelianizations of finite-index subgroups of the mapping class group

Abstract: For some $g \geq 3$, let $\Gamma$ be a finite index subgroup of the mapping class group of a genus $g$ surface (possibly with boundary components and punctures). An old conjecture of Ivanov says that the abelianization of $\Gamma$ should be finite. In this note, we prove two theorems supporting this conjecture. For the first, let $T_x$ denote the Dehn twist about a simple closed curve $x$. For some $n \geq 1$, we have $T_x^n \in \Gamma$. We prove that $T_x^n$ is torsion in the abelianization of $\Gamma$. Our second result shows that the abelianization of $\Gamma$ is finite if $\Gamma$ contains a "large chunk" (in a certain technical sense) of the Johnson kernel, that is, the subgroup of the mapping class group generated by twists about separating curves. This generalizes work of Hain and Boggi.


Hessenberg Pairs of Linear Transformations

Abstract: Let $\fld$ denote a field and $V$ denote a nonzero finite-dimensional vector space over $\fld$. We consider an ordered pair of linear transformations $A: V \to V$ and $A^*: V \to V$ that satisfy (i)--(iii) below. Each of $A, A^*$ is diagonalizable on $V$. There exists an ordering $\lbrace V_i \rbrace_{i=0}^d$ of the eigenspaces of $A$ such that A^* V_i \subseteq V_0 + V_1 + ... + V_{i+1} \qquad \qquad (0 \leq i \leq d), where $V_{-1} = 0$, $V_{d+1}= 0$. There exists an ordering $\lbrace V^*_i \rbrace_{i=0}^{\delta}$ of the eigenspaces of $A^*$ such that A V^*_i \subseteq V^*_0 + V^*_1 + ... +V^*_{i+1} \qquad \qquad (0 \leq i \leq \delta), where $V^*_{-1} = 0$, $V^*_{\delta+1}= 0$. We call such a pair a {\it Hessenberg pair} on $V$. In this paper we obtain some characterizations of Hessenberg pairs. We also explain how Hessenberg pairs are related to tridiagonal pairs.


Trace operators of modulation, alpha modulation and Besov spaces

Abstract: In this paper, we consider the trace theorem for modulation spaces, alpha modulation spaces and Besov spaces. For the modulation space, we obtain the sharp results.


Some Examples of Dynamics for Gelfand Tsetlin Patterns

Abstract: We give examples of stochastic processes in the Gelfand Tsetlin cone in which each component evolves independently apart from a blocking and pushing interaction. The processes give couplings to certain conditioned Markov processes, last passage times and asymetric exclusion processes. An example of a cone valued process whose components cannot escape past a wall at the origin is also considered.


A Wreath Product Approach to Classical Subgroup Theorems

Abstract: We provide elementary proofs of the Nielsen-Schreier Theorem and the Kurosh Subgroup Theorem via wreath products. Our proofs are diagrammatic in nature and work simultaneously in the abstract and profinite categories. A new proof that open subgroups of quasifree profinite groups are quasifree is also given.


Homogeneous interpolation on ten points

Abstract: In this paper we prove that for all pairs $(d,m)$ with $d/m \geq 174/55$, the linear system of plane curves of degree $d$ with ten general base points of multiplicity $m$ has the expected dimension.


The central value of the Rankin-Selberg $L$-functions

Abstract: Let $f$ be a Maass form for $SL(3, \mathbb{Z})$ which is fixed and $u_j$ be an orthonormal basis of even Maass forms for $SL(2, \mathbb{Z}),$ we prove an asymptotic formula for the average of the product of the Rankin-Selberg $L$-function of $f$ and $u_j$ and the $L$-function of $u_j$ at the central value 1/2. This implies simultaneous nonvanishing results of these $L$-functions at $1/2.$


Bounds for $GL(3)\times GL(2)$ $L$-functions and GL(3) $L$-functions

Abstract: In this paper, we will give the subconvexity bounds for self dual GL(3) $L-$functions in the $t$ aspect as well as subconvexity bounds for self dual $GL(3)\times GL(2)$ $L-$functions in the GL(2) spectral aspect.


New Presentations of Thompson's Groups and Applications

Abstract: We find new presentations for the Thompson's groups $F$, the derived group $F^{'}$ and the intermediate group $D$. These presentations have a common ground in that their relators are the same and only the generating sets differ. As an application of these presentations we extract the following consequences: the cost of the group $F^{'}$ is 1 hence the cost cannot decide the (non)amenability question of $F$; the $II_1$ factor $L(F^{'})$ is inner asymptotically abelian and the reduced $C^*$-algebra of $F$ is not residually finite dimensional.


On the average indices of closed geodesics on positively curved Finsler spheres

Abstract: In this paper, we prove that on every Finsler $n$-sphere $(S^n, F)$ for $n\ge 6$ with reversibility $\lambda$ and flag curvature $K$ satisfying $(\frac{\lambda}{\lambda+1})^2<K\le 1$, either there exist infinitely many prime closed geodesics or there exist $[\frac{n}{2}]-2$ closed geodesics possessing irrational average indices. If in addition the metric is bumpy, then there exist $n-3$ closed geodesics possessing irrational average indices provided the number of closed geodesics is finite.


Stability of closed characteristics on symmetric compact convex hypersurfaces in $\R^{2n}$

Abstract: In this article, let $\Sigma\subset\R^{2n}$ be a compact convex hypersurface which is symmetric with respect to the origin. We prove that if $\Sg$ carries finitely many geometrically distinct closed characteristics, then at least $n-1$ of them must be non-hyperbolic; if $\Sg$ carries exactly $n$ geometrically distinct closed characteristics, then at least two of them must be elliptic.


On the Burns-Epstein invariants of spherical CR 3-manifolds

Abstract: In this paper we develop a method to compute the Burns-Epstein invariant of a spherical CR homology sphere, up to an integer, from its holonomy representation. As application, we give a formula for the Burns-Epstein invariant, modulo an integer, of a spherical CR structure on a Seifert fibered homology sphere in terms of its holonomy representation.


On a generalization of P(3,n)

Abstract: By using techniques of poset representation theory, we present a formula for the number of partitions of a positive integer into three polygonal numbers.


Stability of closed characteristics on compact hypersurfaces in $\R^{2n}$ under pinching condition

Abstract: In this article, let $\Sigma\subset\R^{2n}$ be a compact convex hypersurface which is $(r, R)$-pinched with $\frac{R}{r}<\sqrt{{3/2}}$. Then $\Sg$ carries at least two strictly elliptic closed characteristics; moreover, $\Sg$ carries at least $2[\frac{n+2}{4}]$ non-hyperbolic closed characteristics.


Indirect Cross-validation for Density Estimation

Abstract: A new method of bandwidth selection for kernel density estimators is proposed. The method, termed indirect cross-validation, or ICV, makes use of so-called selection kernels. Least squares cross-validation (LSCV) is used to select the bandwidth of a selection-kernel estimator, and this bandwidth is appropriately rescaled for use in a Gaussian kernel estimator. The proposed selection kernels are linear combinations of two Gaussian kernels, and need not be unimodal or positive. Theory is developed showing that the relative error of ICV bandwidths can converge to 0 at a rate of $n^{-1/4}$, which is substantially better than the $n^{-1/10}$ rate of LSCV. Interestingly, the selection kernels that are best for purposes of bandwidth selection are very poor if used to actually estimate the density function. This property appears to be part of the larger and well-documented paradox to the effect that "the harder the estimation problem, the better cross-validation performs." The ICV method uniformly outperforms LSCV in a simulation study, a real data example, and a simulated example in which bandwidths are chosen locally.


Empirical study of indirect cross-validation

Abstract: In this paper we provide insight into the empirical properties of indirect cross-validation (ICV), a new method of bandwidth selection for kernel density estimators. First, we describe the method and report on the theoretical results used to develop a practical-purpose model for certain ICV parameters. Next, we provide a detailed description of a numerical study which shows that the ICV method usually outperforms least squares cross-validation (LSCV) in finite samples. One of the major advantages of ICV is its increased stability compared to LSCV. Two real data examples show the benefit of using both ICV and a local version of ICV.


On the total mean curvature of non-rigid surfaces

Abstract: Using Green's theorem we reduce the variation of the total mean curvature of a smooth surface in the Euclidean 3-space to a line integral of a special vector field and obtain the following well-known theorem as an immediate consequence: the total mean curvature of a closed smooth surface in the Euclidean 3-space is stationary under an infinitesimal flex.


Equivalence of Control Systems with Linear Systems on Lie Groups and Homogeneous Spaces

Abstract: The aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphism to a linear system on a Lie group or a homogeneous space if and only the vector fields of the system are complete and generate a finite dimensional Lie algebra. A vector field on a connected Lie group is linear if its flow is a one parameter group of automorphisms. An affine vector field is obtained by adding a left invariant one. Its projection on a homogeneous space, whenever it exists, is still called affine. Affine vector fields on homogeneous spaces can be characterized by their Lie brackets with the projections of right invariant vector fields. A linear system on a homogeneous space is a system whose drift part is affine and whose controlled part is invariant. The main result is based on a general theorem on finite dimensional algebras generated by complete vector fields, closely related to a theorem of Palais, and which have its own interest. The present proof makes use of geometric control theory arguments.


Multiplicities of the discrete series

Abstract: The purpose of this paper is to show that the multiplicities of a discrete series representation relatively to a compact subgroup can be "computed" geometrically, in the way predicted by the "qantization commutes with reduction" principle of Guillemin-Sternberg.


Cutoff phenomena for random walks on random regular graphs

Abstract: The cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite Markov chains to equilibrium. Many natural families of chains are believed to exhibit cutoff, and yet establishing this fact is often extremely challenging. An important such family of chains is the random walk on $\G(n,d)$, a random $d$-regular graph on $n$ vertices. It is well known that almost every such graph for $d\geq 3$ is an expander, and even essentially Ramanujan, implying a mixing-time of $O(\log n)$. According to a conjecture of Peres, the simple random walk on $\G(n,d)$ for such $d$ should then exhibit cutoff with high probability. As a special case of this, Durrett conjectured that the mixing time of the lazy random walk on a random 3-regular graph is w.h.p. $(6+o(1))\log_2 n$. In this work we confirm the above conjectures, and establish cutoff in total-variation, its location and its optimal window, both for simple and for non-backtracking random walks on $\G(n,d)$. Namely, for any fixed $d\geq3$, the simple random walk on $\G(n,d)$ w.h.p. has cutoff at $\frac{d}{d-2}\log_{d-1} n$ with window order $\sqrt{\log n}$. Surprisingly, the non-backtracking random walk on $\G(n,d)$ w.h.p. has cutoff already at $\log_{d-1} n$ with constant window order. We further extend these results to $\G(n,d)$ for any $d=n^{o(1)}$ that grows with $n$ (beyond which the mixing time is O(1)), where we establish concentration of the mixing time on one of two consecutive integers.


Configurations of infinitely near points

Abstract: We present a survey of some aspects and new results on configurations, i.e. disjoint unions of constellations of infinitely near points, local and global theory, with some applications and results on generalized Enriques diagrams, singular foliations, and linear systems defined by clusters.


Potential functions via toric degenerations

Abstract: This is a short companion paper to arXiv:0810.3470. We construct an integrable system on an open subset of a Fano manifold equipped with a toric degeneration, and compute the potential function for its Lagrangian torus fibers if the central fiber is a toric Fano variety admitting a small resolution.


Bifurcation and Secondary Bifurcation of Heavy Periodic Hydroelastic Travelling Waves

Abstract: The paper deals with a problem of interaction between hydrodynamics and mechanics of nonlinear elastic bodies. The existence question for two-dimensional symmetric steady waves travelling on the surface of a deep ocean beneath a heavy elastic membrane is analyzed as a problem in bifurcation theory. The behaviour of the two-dimensional cross-section of the membrane is modelled as a thin (unshearable), heavy, hyperelastic Cosserat rod, following Antman's elasticity theory, and the fluid beneath is supposed to be in steady 2D irrotational motion under gravity. Assuming that gravity and the density of the undeformed membrane are prescribed, the free parameters of the problem are the speed of the wave and drift velocity of the membrane. The analysis relies upon a conformal formulation of the hydro-elastic problem developed in previous papers; the basic tool for the study of the bifurcation picture is the implicit function theorem, under some non-resonance assumptions. The most interesting part of the final result is the existence of a symmetry-breaking 'third sheet' of solutions, which bifurcates from primary sheets, and is a hydro-elastic analogue of the phenomenon known as 'Wilton ripples' in the surface tension case.


Brownian Brownian Motion-1

Abstract: A classical model of Brownian motion consists of a heavy molecule submerged into a gas of light atoms in a closed container. In this work we study a 2D version of this model, where the molecule is a heavy disk of mass M and the gas is represented by just one point particle of mass m = 1, which interacts with the disk and the walls of the container via elastic collisions. Chaotic behavior of the particles is ensured by convex (scattering) walls of the container. We prove that the position and velocity of the disk, in an appropriate time scale, converge, as M tends to infinity, to a Brownian motion (possibly, inhomogeneous); the scaling regime and the structure of the limit process depend on the initial conditions. Our proofs are based on strong hyperbolicity of the underlying dynamics, fast decay of correlations in systems with elastic collisions (billiards), and methods of averaging theory.


Nearly-optimal estimates for the stability problem in Hardy spaces

Abstract: We continue the work of \cite{TLNT}. Let $E$ be a non-Blaschke subset of the unit disc $\mathbb{D}$ of the complex plane $\mathbb{C}$. Fixed $1\leq p\leq \infty$, let $H^p(\mathbb{D})$ be the Hardy space of holomorphic functions in the disk whose boundary value function is in $L^p(\partial \mathbb{D})$. Fixed $0<R<1$. For $\epsilon >0$ define C_p(\varepsilon, R) = \sup \{\sup_{|z| \leq R}|g(z)|: g\in H^p, \|g\|_p\leq 1, |g(\zeta)| \leq \varepsilon \forall \zeta\in E\}. In this paper we find upper and lower bounds for $C_p(\epsilon, R)$ when $\epsilon$ is small for any non-Blaschke set $E$. The bounds are nearly-optimal for many such sets $E$, including sets contained in a compact subset of $\mathbb{D}$ and sets contained in a finite union of Stolz angles.


Note on potential theory for functions in Hardy classes

Abstract: The purpose of this note is to show that the set functions defined in \cite{trong-tuyen} can be suitably extended to all subsets $E$ of the unit disk $\mathbb{D}$. In particular we obtain uniform nearly-optimal estimates for the following quantity D_p(E,\epsilon, R) = \sup \{\sup_{|z| \leq R}|g(z)|: g\in H^p, ||g||_{H^p}\leq 1, (1-|\zeta |)|g(\zeta)| \leq \epsilon \forall \zeta\in E\}.


The Powers Sum of spatial CPD-semigroups and CP-semigroups

Abstract: We define spatial CPD-semigroup and construct their Powers sum. We construct the Powers sum for general spatial CP-semigroups. In both cases, we show that the product system of that Powers sum is the product of the spatial product systems of its factors. We show that on the domain of intersection, pointwise bounded CPD-semigroups on the one side and Schur CP-semigroups on the other, the constructions coincide. This summarizes all known results about Powers sums and generalizes them considerably.


$\omega$-Lie algebras

Abstract: We study a certain generalization of Lie algebras where the Jacobian of three elements does not vanish but is equal to an expression depending on a skew-symmetric bilinear form.


Sprouts game on compact surfaces

Abstract: Sprouts is a two-player topological game, invented in 1967 by Michael Paterson and John Conway. The game starts with p spots drawn on a sheet of paper, and lasts at most 3p-1 moves: the player who makes the last move wins. Sprouts is a very intricate game and the best known manual analysis only achieved to find a winning strategy up to p=7 spots. Recent computer analysis reached up to p=32. The standard game is played on a plane, or equivalently on a sphere. In this article, we generalize and study the game on any compact surface. First, we describe the possible moves on a compact surface, and the way to implement them in a program. Then, we show that we only need to consider a finite number of surfaces to analyze the game with p spots on any compact surface: if we take a surface with a genus greater than some limit genus, then the game on this surface is equivalent to the game on some smaller surface. Finally, with computer calculation, we observe that the winning player on orientable surfaces seems to be always the same one as on a plane, whereas there are significant differences on non-orientable surfaces.


Extending the Set of Quadratic Exponential Vectors

Abstract: We extend the square of white noise algebra over the step functions on R to the test function space of bounded square-integrable functions on R^d, and we show that in the Fock representation the exponential vectors exist for all test functions bounded by 1/2.


A rank inequality for the Tate Conjecture over global function fields

Abstract: Following D. Ramakrishnan, we explain how L. Lafforgue's modularity theorem and an analytic theorem of H. Jacquet and J. Shalika can be applied to prove the following result related to the Tate Conjecture: for a smooth, projective, geometrically-connected variety defined over a global function field, the algebraic rank is less than or equal to the analytic rank. Also discussed is the analogous (open) question for number fields and an easy extension of Lafforgue's theorem to remove the "finite-order character" assumption. All results are likely "known to the experts", but don't appear to be written down.


C*-envelopes of tensor algebras for multivariable dynamics

Abstract: We give a new very concrete description of the C*-envelope of the tensor algebra associated to multivariable dynamical system. In the surjective case, this C*-envelope is described as a crossed product by an endomorphism, and as a groupoid C*-algebra. In the non-surjective case, it is a full corner of a such an algebra. We also show that when the space is compact, then the C*-envelope is simple if and only if the system is minimal.


Fixation Probability for Competing Selective Sweeps

Abstract: We consider a biological population in which a beneficial mutation is undergoing a selective sweep when a second beneficial mutation arises at a linked locus and we investigate the probability that both mutations will eventually fix in the population. Previous work has dealt with the case where the second mutation to arise confers a smaller benefit than the first. In that case population size plays almost no role. Here we consider the opposite case and observe that, by contrast, the probability of both mutations fixing can be heavily dependent on population size. Indeed the key parameter is $\rho N$, the product of the population size and the recombination rate between the two selected loci. If $\rho N$ is small, the probability that both mutations fix can be reduced through interference to almost zero while for large $\rho N$ the mutations barely influence one another. The main rigorous result is a method for calculating the fixation probability of a double mutant in the large population limit.


On Families of (Phi,Gamma)-modules

Abstract: Berger and Colmez introduced a theory of families of overconvergent \'etale (Phi,Gamma)-modules associated to families of p-adic Galois representations over p-adic Banach algebras. However, in contrast with the classical theory of (Phi,Gamma)-modules, the functor they obtain is not an equivalence of categories. In this paper, we prove that when the base is an affinoid space, every family of (overconvergent) \'etale (Phi,Gamma)-modules can locally be converted into a family of p-adic representations in a unique manner, providing the "local" equivalence. There is a global mod p obstruction related to the moduli of residual representations.


Bounds for the annealed return probability on large finite percolation clusters

Abstract: Bounds for the expected return probability of the delayed random walk on finite clusters of an invariant percolation on transitive unimodular graphs are derived. They are particularly suited for the case of critical Bernoulli percolation and the associated heavy-tailed cluster size distributions. The upper bound relies on the fact that cartesian products of finite graphs with cycles of a certain minimal size are Hamiltonian. For critical Bernoulli bond percolation on the homogeneous tree this bound is sharp. The asymptotic type of the expected return probability for large times t in this case is of order of the 3/4'th power of 1/t.


On Rank Problems for Planar Webs and Projective Structures

Abstract: We present old and recent results on rank problems and linearizability of geodesic planar webs.


A simple proof of a theorem of Fukaya and Oh

Abstract: We study the moduli space of pseudo pointed holomorphic disks with boundaries mapped in the zero section of the cotangent bundle of a manifold. We define perturbations of the equation for which it is possible to describe explicitly all the solutions of the problem in terms of Morse graphs on the manifold. In particular, this proves that the $A_\infty$ structure of the zero section of the cotangent bundle is equivalent to the Morse $A_\infty$ structure of the base manifold.


Homotopy Equivalences induced by Balanced Pairs

Abstract: We introduce the notion of balanced pair of additive subcategories in an abelian category. We give sufficient conditions under which the balanced pair of subcategories gives rise to equivalent homotopy categories of complexes. As an application, we prove that for a left-Gorenstein ring, there exists a triangle-equivalence between the homotopy category of its Gorenstein projective modules and the homotopy category of its Gorenstein injective modules, which restricts to a triangle-equivalence between the homotopy category of projective modules and the homotopy category of injective modules. In the case of commutative Gorenstein rings we prove that up to a natural isomorphism our equivalence extends Iyengar-Krause's equivalence.


Permutations sortable by n-4 passes through a stack

Abstract: We characterise and enumerate permutations that are sortable by n-4 passes through a stack. We conjecture the number of permutations sortable by n-5 passes, and also the form of a formula for the general case n-k, which involves a polynomial expression.


Complete convergence of message passing algorithms for some satisfiability problems

Abstract: In this paper we analyze the performance of Warning Propagation, a popular message passing algorithm. We show that for 3CNF formulas drawn from a certain distribution over random satisfiable 3CNF formulas, commonly referred to as the planted-assignment distribution, running Warning Propagation in the standard way (run message passing until convergence, simplify the formula according to the resulting assignment, and satisfy the remaining subformula, if necessary, using a simple "off the shelf" heuristic) results in a satisfying assignment when the clause-variable ratio is a sufficiently large constant.


Enrichments over symmetric Picard categories

Abstract: Categorical rings were introduced by Jibladze and Pirashvili in their paper "Third Mac Lane cohomology via categorical rings", Journal of Homotopy and related structures, 2, 2007, 187-216. We call those "2-rings". In these notes we present basic definitions and results regarding 2-modules. This is work in progress.


A volume form on the Khovanov invariant

Abstract: The Reidemeister torsion construction can be applied to the chain complex used to compute the Khovanov homology of a knot or a link. This defines a volume form on Khovanov homology. The volume form transforms correctly under Reidemeister moves to give an invariant volume on the Khovanov homology. In this paper, its construction and invariance under these moves is demonstrated. Also, some examples of the invariant are presented for particular choices for the bases of homology groups to obtain a numerical invariant of knots and links. In these examples, the algebraic torsion seen in the Khovanov chain complex when homology is computed over $\mathbb{Z}$ is recovered.


Homotopy theory of C*-algebras

Abstract: In this work we construct from ground up a homotopy theory of C*-algebras. This is achieved in parallel with the development of classical homotopy theory by first introducing an unstable model structure and second a stable model structure. The theory makes use of a full fledged import of homotopy theoretic techniques into the subject of C*-algebras. The spaces in C*-homotopy theory are certain hybrids of functors represented by C*-algebras and spaces studied in classical homotopy theory. In particular, we employ both the topological circle and the C*-algebra circle of complex-valued continuous functions on the real numbers which vanish at infinity. By using the inner workings of the theory, we may stabilize the spaces by forming spectra and bispectra with respect to either one of these circles or their tensor product. These stabilized spaces or spectra are the objects of study in stable C*-homotopy theory. The stable homotopy category of C*-algebras gives rise to invariants such as stable homotopy groups and bigraded cohomology and homology theories. We work out examples related to the emerging subject of noncommutative motives and zeta functions of C*-algebras. In addition, we employ homotopy theory to define a new type of K-theory of C*-algebras.


Pattern Recognition on Oriented Matroids: Three-Tope Committees

Abstract: A three-tope committee K* for a simple oriented matroid M is a 3-subset of its maximal covectors such that every positive halfspace of M contains at least two topes from K*. We consider three-tope committees as the vertex sets of triangles in graphs associated with the topes and enumerate them making use of the properties of the poset of convex subsets of the ground set of M.


Optimal sequential procedures with Bayes decision rules

Abstract: In this article, a general problem of sequential statistical inference for general discrete-time stochastic processes is considered. The problem is to minimize an average sample number given that Bayesian risk due to incorrect decision does not exceed some given bound. We characterize the form of optimal sequential stopping rules in this problem. In particular, we have a characterization of the form of optimal sequential decision procedures when the Bayesian risk includes both the loss due to incorrect decision and the cost of observations.


Homogeneous Lagrangian submanifolds of positive Euler characteristic

Abstract: We fully classify all Lagrangian submanifolds of a complex Grassmannian which are an orbit of a compact group of isometries and have positive Euler characteristic.


Factor complexity of infinite words associated with non-simple Parry numbers

Abstract: The factor complexity of the infinite word $\ubeta$ canonically associated to a non-simple Parry number $\beta$ is studied. Our approach is based on the notion of special factors introduced by Berstel and Cassaigne. At first, we give a handy method for determining infinite left special branches; this method is applicable to a broad class of infinite words which are fixed points of a primitive substitution. In the second part of the article, we focus on infinite words $\ubeta$ only. To complete the description of its special factors, we define and study $(a,b)$-maximal left special factors. This enables us to characterize non-simple Parry numbers $\beta$ for which the word $\ubeta$ has affine complexity.


The Inverse of a Nearly Banded Matrix

Abstract: A quantitative form of the Nullity Theorem is presented, which establishes a linear relation between the singular values of the two submatrices involved in the theorem up to the first order. The theorem is then extended to function spaces and a corresponding form in infinite dimension is discussed.


Traces on Hecke algebras and families of p-adic modular forms

Abstract: In this preprint we prove that any finite slope modular form fits into a p-adic family of modular forms which is indexed by the weight. Here, the term p-adic family means that p-adic congruences between weights entail certain p-adic congruences between the corresponding modular forms. We also show that the dimension of the slope subspace of the space of modular forms does not depend on the weight. Both statements are predicted by the Mazur-Gouvea Conjecture, which has been proven by Coleman using methods from rigid analytic geometry. In contrast our proof is based on a comparison of (topological) trace formulas.


Minimal Faithful Permutation Degrees for Irreducible Coxeter Groups and Binary Polyhedral Groups

Abstract: In this article we calculate the minimal faithful permutation degree for all of the irreducible Coxeter groups. We also exhibit new examples of finite groups that possess a quotient whose minimal degree is strictly greater than that of the group.


Generalized Bunce-Deddens algebras

Abstract: We define a broad class of crossed product C*-algebras of the form C(G)xG, where G is a discrete countable amenable residually finite group, and G is a profinite completion of G. We show that they are unital separable simple nuclear quasidiagonal C*-algebras, or real rank zero, stable rank one, with comparability of projections and with a unique trace.


Reduced free products of unital AH algebras and Blackadar and Kirchberg's MF algebras

Abstract: In the paper, we prove that reduced free products of unital AH algebras with respect to given faithful tracial states, in the sense of Voiculescu, are Blackadar and Kirhcberg's MF algebras. We also show that the reduced free products of unital AH algebras with respect to given faithful tracial states, under mild conditions, are not quasidiagonal. Therefore we conclude, for a large class of AH algebras, the Brown-Douglas-Fillmore extension semigroups of the reduced free products of these AH algebras with respect to given faithful tracial states are not groups. Our result is based on Haagerup and Thorbj{\o}rsen's work on the reduced C$^*$-algebras of free groups.


Toric ideals generated by circuits

Abstract: Let I be the toric ideal of a homogeneous normal configuration. We prove that I is generated by circuits if and only if each unbalanced circuit of I has a "connector" which is a linear combination of circuits with a square-free term. In particular if each circuit of I with non-square-free terms is balanced, then I is generated by circuits. As a consequence we prove that the toric ideal of a normal edge subring of a multigraph is generated by circuits with a square-free term.


Some characterizations of affinely full-dimensional factorial designs

Abstract: A new class of two-level non-regular fractional factorial designs is defined. We call this class an {\it affinely full-dimensional factorial design}, meaning that design points in the design of this class are not contained in any affine hyperplane in the vector space over $\mathbb{F}_2$. The property of the indicator function for this class is also clarified. A fractional factorial design in this class has a desirable property that parameters of the main effect model are simultaneously identifiable. We investigate the property of this class from the viewpoint of $D$-optimality. In particular, for the saturated designs, the $D$-optimal design is chosen from this class for the run sizes $r \equiv 5,6,7$ (mod 8).


A general type of twisted anomaly cancellation formulas

Abstract: For even dimensional manifolds, we prove some twisted anomaly cancellation formulas which generalize some well-known cancellation formulas. For odd dimensional manifolds, we obtain some modularly invariant characteristic forms by the Chern-Simons transgression and we also get some twisted anomaly cancellation formulas.


Hodge decomposition in the homology of long knots

Abstract: The paper describes a natural splitting in the rational homology and homotopy of the spaces of long knots. This decomposition presumably arises from the cabling maps in the same way as a natural decomposition in the homology of loop spaces arises from power maps. The generating function for the Euler characteristics of the terms of this splitting is presented. Based on this generating function we show that both the homology and homotopy ranks of the spaces in question grow at least exponentially. Using natural graph-complexes we show that this splitting on the level of the bialgebra of chord diagrams is exactly the splitting defined earlier by Dr. Bar-Natan. In the Appendix we present tables of computer calculations of the Euler characteristics. These computations give a certain optimism that the Vassiliev invariants of order > 20 can distinguish knots from their inverses.


An Analytic Approach to Stability

Abstract: The stability method is very useful for obtaining exact solutions of many extremal graph problems. Its key step is to establish the stability property which, roughly speaking, states that any two almost optimal graphs of the same order $n$ can be made isomorphic by changing o(n^2) edges. Here we show how the recently developed theory of graph limits can be used to give an analytic approach to stability. As an application, we present a new proof of the Erdos-Simonovits Stability Theorem. Also, we investigate various properties of the edit distance. In particular, we show that the combinatorial and fractional versions are within a constant factor from each other, thus answering a question of Goldreich, Krivelevich, Newman, and Rozenberg.


Face vectors of two-dimensional Buchsbaum complexes

Abstract: In this paper, we characterize all possible h-vectors of 2-dimensional Buchsbaum simplicial complexes.


Singular Hermitian-Einstein monopoles on the product of a circle and a Riemann surface

Abstract: In this paper, the moduli space of singular unitary Hermitian--Einstein monopoles on the product of a circle and a Riemann surface is shown to correspond to a moduli space of stable pairs on the Riemann surface. These pairs consist of a holomorphic vector bundle on the surface and a meromorphic automorphism of the bundle. The singularities of this automorphism correspond to the singularities of the singular monopole. We then consider the complex geometry of the moduli space; in particular, we compute dimensions, both from the complex geometric and the gauge theoretic point of view.


Parity condition for irreducibility of Heegaard splittings

Abstract: Casson and Gordon gave the rectangle condition for strong irreducibility of Heegaard splittings [1]. We give a parity condition for irreducibility of Heegaard splittings of irreducible manifolds. As an application, we give examples of non-stabilized Heegaard splittings by doing a single Dehn twist.


Monotonicity theorems for Laplace Beltrami operator on Riemannian manifolds

Abstract: For free boundary problems on Euclidean spaces, the monotonicity formulas of Alt-Caffarelli-Friedman and Caffarelli-Jerison-Kenig are cornerstones for the regularity theory as well as the existence theory. In this article we establish the analogs of these results for the Laplace-Beltrami operator on Riemannian manifolds. As an application we show that our monotonicity theorems can be employed to prove the Lipschitz continuity for the solutions of a general class of two-phase free boundary problems on Riemannian manifolds.


On the Adams-Riemann-Roch theorem in positive characteristic

Abstract: We give a new proof of the Adams-Riemann-Roch theorem for a smooth projective morphism $X\to Y$, in the situation where $Y$ is a regular scheme, which is quasi-projective over $\mF_p$. We also partially answer a question of B. K\"ock.


Representation dimension of extensions of hereditary algebras

Abstract: We show that if H is a hereditary finite dimensional algebra, M is a finitely generated H-module and B is a semisimple subalgebra of the endomorphism algebra of M, then the representation dimension of the corresponding triangular matrix algebra is less or equal to 3 whenever one of the following conditions hold: i) H is of finite representation type; ii) H is tame and M is a direct sum of regular and preprojective modules; iii) M has no self-extensions