Twisted support varieties

Abstract: We define and study twisted support varieties for modules over an Artin algebra, where the twist is induced by an automorphism of the algebra. Under a certain finite generation hypothesis, we show that the twisted variety of a module satisfies Dade's Lemma and is one dimensional precisely when the module is periodic with respect to the twisting automorphism. As a special case we obtain results on DTr-periodic modules over Frobenius algebras.


Reversible skew Laurent polynomial rings and deformations of Poisson automorphisms

Abstract: A skew Laurent polynomial ring R[x^{\pm 1};\alpha] is reversible if it has a reversing automorphism, that is, an automorphism of period two that transposes x and x^{-1} and restricts to an automorphism of R. We study invariants for reversing automorphisms and apply our methods to determine the rings of invariants of reversing automorphisms of two simple skew Laurent polynomial rings, namely a localization of the enveloping algebra of the two-dimensional non-abelian solvable Lie algebra and the coordinate ring of the quantum torus. These two skew Laurent polynomial rings are deformations of Poisson algebras and we interpret their reversing automorphisms and their invariants as deformations of Poisson automorphisms and their invariants.


Best constants for Lipschitz embeddings of metric spaces into c_0

Abstract: We answer a question of Aharoni by showing that every separable metric space can be Lipschitz 2-embedded into $c_0$ and this result is sharp; this improves earlier estimates of Aharoni, Assouad and Pelant. We use our methods to examine the best constant for Lipschitz embeddings of the classical $\ell_p-$spaces into $c_0$ and give other applications. We prove that if a Banach space embeds almost isometrically into $c_0$, then it embeds linearly almost isometrically into $c_0$. We also study Lipschitz embeddings into $c_0^+$.


Finite-dimensional simple Poisson modules

Abstract: We prove a result that can be applied to determine the finite-dimensional simple Poisson modules over a Poisson algebra and apply it to numerous examples. In the discussion of the examples, the emphasis is on the correspondence with the finite-dimensional simple modules over deformations and on the behaviour of finite-dimensional simple Poisson modules on the passage from a Poisson algebra to the Poisson subalgebra of invariants for the action of a finite group of Poisson automorphisms.


The solution of the Minkowski problem for open surfaces in Riemannian space

Abstract: Author reduces the Minkowski problem to the problem of construction the G-deformations preserving the product of principal curvatures for every point of surface in Riemannian space. G-deformation transfers every normal vector of surface in parallel along the path of the translation for each point of surface. The continuous G-deformations preserving the product of principal curvatures of surface with boundary are considered in this article. The equations of deformations which are obtained in this paper reduce to the nonlinear boundary-value problem. The method of construction continuous G-deformations preserving the product of principal curvatures of surface with boundary and its qualitative analysis are presented in this article


Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples

Abstract: In Section 1, we present a number of classical results concerning the Generalized Gamma Convolution (:GGC) variables, their Wiener-Gamma representations, and relation with the Dirichlet processes.To a GGC variable, one may associate a unique Thorin measure. Let $G$ a positive r.v. and $\Gamma_t(G)$ (resp. $\Gamma_t(1/G))$ the Generalized Gamma Convolution with Thorin measure $t$-times the law of $G$ (resp. the law of $1/G$). In Section 2, we compare the laws of $\Gamma_t(G)$ and $\Gamma_t(1/G)$.In Section 3, we present some old and some new examples of GGC variables, among which the lengths of excursions of Bessel processes straddling an independent exponential time.


Sharp Spectral Asymptotics for Dirac Energy

Abstract: I derive sharp semiclassical asymptotics of $\int |e_h(x,y,0)|^2\omega(x,y) dx dy$ where $e_h(x,y,\tau)$ is the Schwartz kernel of the spectral projector and $\omega(x,y)$ is singular as $x=y$. I also consider asymptotics of more general expressions.


A convenient category for directed homotopy

Abstract: We propose a convenient category for directed homotopy consisting of preordered topological spaces generated by cubes. Its main advantage is that, like the category of topological spaces generated by simplices suggested by J. H. Smith, it is locally presentable.


On the proximinality of ridge functions

Abstract: Using two results of Garkavi, Medvedev and Khavinson, we give sufficient conditions for proximinality of sums of two ridge functions with bounded and continuous summands in the spaces of bounded and continuous multivariate functions respectively. In the first case, we give an example which shows that the corresponding sufficient condition cannot be made weaker for some subsets of $\mathbb{R}^{n}$. In the second case, we obtain also a necessary condition for proximinality. All the results are furnished with plenty of examples. The results, examples and following discussions naturally lead us to a conjecture on the proximinality of the considered class of ridge functions. The main purpose of the paper is to draw readers' attention to this conjecture.


Epidemics on random graphs with tunable clustering

Abstract: In this paper, a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. It is investigated how these quantities varies with the clustering in the graph and it turns out for instance that, as the clustering increases, the epidemic threshold decreases. The network is modelled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if they share at least one group.


On the modularity of supersingular elliptic curves over certain totally real number fields

Abstract: We study generalisations to totally real fields of methods originating with Wiles and Taylor-Wiles. In view of the results of Skinner-Wiles on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction. Combining these, we then obtain some partial results on the modularity problem for semistable elliptic curves, and end by giving some applications of our results, for example proving the modularity of all semistable elliptic curves over $\mathbb{Q}(\sqrt{2})$.


Optimality and uniqueness of the (4,10,1/6) spherical code

Abstract: Linear programming bounds provide an elegant method to prove optimality and uniqueness of an (n,N,t) spherical code. However, this method does not apply to the parameters (4,10,1/6). We use semidefinite programming bounds instead to show that the Petersen code, which consists of the midpoints of the edges of the regular simplex in dimension 4, is the unique (4,10,1/6) spherical code.


Categorification of Wedderburn's basis for \mathbb{C}[S_n]

Abstract: M. Neunh{\"o}ffer studies in \cite{Ne} a certain basis of $\mathbb{C}[S_n]$ with the origins in \cite{Lu} and shows that this basis is in fact Wedderburn's basis. In particular, in this basis the right regular representation of $S_n$ decomposes into a direct sum of irreducible representations (i.e. Specht or cell modules). In the present paper we rediscover essentially the same basis with a categorical origin coming from projective-injective modules in certain subcategories of the BGG-category $\mathcal{O}$. An important role in our arguments is played by the dominant projective module in each of these categories. As a biproduct of the study of this dominant projective module we show that {\it Kostant's problem} (\cite{Jo}) has a negative answer for some simple highest weight module over the Lie algebra $\mathfrak{sl}_4$, which disproves the general belief that Kostant's problem should have a positive answer for all simple highest weight modules in type $A$.


Liftable D_4-Covers

Abstract: Let k be an algebraically closed field of characteristic p and let G be a subgroup of Aut(k[[t]]) be a faithful action on a local power series ring over k. Let R be a discrete valuation ring of characteristic 0 with residue field k. One asks, whether it is possible to find a faithful action G inside Aut(R[[t]]) which reduces to the given action, i.e. a lift to characteristic 0. We show that liftable actions exists in the case that G = D_4 and p = 2. In fact we introduce a family, the supersimple D_4 -actions, which can always be lifted to characteristic 0.


The asymptotic behaviour of recurrence coefficients for orthogonal polynomials with varying exponential weights

Abstract: We consider orthogonal polynomials $\{p_{n,N}(x)\}_{n=0}^{\infty}$ on the real line with respect to a weight $w(x)=e^{-NV(x)}$ and in particular the asymptotic behaviour of the coefficients $a_{n,N}$ and $b_{n,N}$ in the three term recurrence $x \pi_{n,N}(x) = \pi_{n+1,N}(x) + b_{n,N} \pi_{n,N}(x) + a_{n,N} \pi_{n-1,N}(x)$. For one-cut regular $V$ we show, using the Deift-Zhou method of steepest descent for Riemann-Hilbert problems, that the diagonal recurrence coefficients $a_{n,n}$ and $b_{n,n}$ have asymptotic expansions as $n \to \infty$ in powers of $1/n^2$ and powers of $1/n$, respectively.


Chess, Chance and Conspiracy

Abstract: Chess and chance are seemingly strange bedfellows. Luck and/or randomness have no apparent role in move selection when the game is played at the highest levels. However, when competition is at the ultimate level, that of the World Chess Championship (WCC), chess and conspiracy are not strange bedfellows, there being a long and colorful history of accusations levied between participants. One such accusation, frequently repeated, was that all the games in the 1985 WCC (Karpov vs Kasparov) were fixed and prearranged move by move. That this claim was advanced by a former World Champion, Bobby Fischer, argues that it ought be investigated. That the only published, concrete basis for this claim consists of an observed run of particular moves, allows this investigation to be performed using probabilistic and statistical methods. In particular, we employ imbedded finite Markov chains to evaluate run statistic distributions. Further, we demonstrate how both chess computers and game data bases can be brought to bear on the problem.


Maty's Biography of Abraham De Moivre, Translated, Annotated and Augmented

Abstract: November 27, 2004, marked the 250th anniversary of the death of Abraham De Moivre, best known in statistical circles for his famous large-sample approximation to the binomial distribution, whose generalization is now referred to as the Central Limit Theorem. De Moivre was one of the great pioneers of classical probability theory. He also made seminal contributions in analytic geometry, complex analysis and the theory of annuities. The first biography of De Moivre, on which almost all subsequent ones have since relied, was written in French by Matthew Maty. It was published in 1755 in the Journal britannique. The authors provide here, for the first time, a complete translation into English of Maty's biography of De Moivre. New material, much of it taken from modern sources, is given in footnotes, along with numerous annotations designed to provide additional clarity to Maty's biography for contemporary readers.


PAC Fields over Finitely Generated Fields

Abstract: We prove the following theorem for a finitely generated field $K$: Let $M$ be a Galois extension of $K$ which is not separably closed. Then $M$ is not PAC over $K$.


A Conversation with Robert V. Hogg

Abstract: Robert Vincent Hogg was born on November 8, 1924 in Hannibal, Missouri. He earned a Ph.D. in statistics at the University of Iowa in 1950, where his advisor was Allen Craig. Following graduation, he joined the mathematics faculty at the University of Iowa. He was the founding Chair when the Department of Statistics was created at Iowa in 1965 and he served in that capacity for 19 years. At Iowa he also served as Chair of the Quality Management and Productivity Program and the Hanson Chair of Manufacturing Productivity. He became Professor Emeritus in 2001 after 51 years on the Iowa faculty. He is a Fellow of the Institute of Mathematical Statistics and the American Statistical Association plus an Elected Member of the International Statistical Institute. He was President of the American Statistical Association (1988) and chaired two of its winter conferences (1992, 1994). He received the ASA's Founder's Award (1991) and the Gottfried Noether Award (2001) for contributions to nonparametric statistics. His publications through 1996 are described in Communications in Statistics--Theory and Methods (1996), 2467--2481. This interview was conducted on April 14, 2004 at the Department of Statistics, University of Florida, Gainesville, Florida, and revised in the summer of 2006.


The Lie algebra perturbation lemma

Abstract: Let g be a differential graded Lie algebra and suppose given a contraction of chain complexes of g onto a general chain complex M. We show that the data determine an sh-Lie algebra structure on M, that is, a coalgebra perturbation of the coalgebra differential on the cofree coaugmented differential graded cocommutative coalgebra S' on the suspension of M, a Lie algebra twisting cochain from the perturbed coalgebra S" to the given Lie algebra g, and an extension of this Lie algebra twisting cochain to a contraction of chain complexes from the Cartan-Chevalley-Eilenberg coalgebra on g onto S" which is natural in the data. This extends a result established in a joint paper of the author with J. Stashef [Forum math. 14 (2002), 847-868, math.AG/9906036] where only the particular where M is the homology of g has been explored.


Gaps in the differential forms spectrum on cyclic coverings

Abstract: We are interested in the spectrum of the Hodge-de Rham operator on a cyclic covering $X$ over a compact manifold $M$ of dimension $n+1$. Let $\Sigma$ be a hypersurface in $M$ which does not disconnect $M$ and such that $M-\Sigma$ is a fundamental domain of the covering. If the cohomology group $H^{n/2 (\Sigma)$ is trivial, we can construct for each $N \in \N$ a metric $g=g_N$ on $M$, such that the Hodge-de Rham operator on the covering $(X,g)$ has at least $N$ gaps in its (essential) spectrum. If $H^{n/2}(\Sigma) \ne 0$, the same statement holds true for the Hodge-de Rham operators on $p$-forms provided $p \notin \{n/2,n/2+1\}$.


Extreme values of zeta and L-functions

Abstract: We introduce a "resonance" method to produce large values of $|\zeta(1/2+it)|$ and large and small central values of $L$-functions.


On ground fields of arithmetic hyperbolic reflection groups

Abstract: Using authors's methods of 1980, 1981, some explicit finite sets of number fields containing ground fields of arithmetic hyperbolic reflection groups are defined, and good bounds of their degrees (over Q) are obtained. For example, degree of the ground field of any arithmetic hyperbolic reflection group in dimension at least 6 is bounded by 56. These results could be important for further classification. We also formulate a mirror symmetric conjecture to finiteness of the number of arithmetic hyperbolic reflection groups which was established in full generality recently.


The group of automorphisms of a real rational surface is n-transitive

Abstract: Let X be a rational nonsingular compact connected real algebraic surface. Denote by Aut(X) the group of real algebraic automorphisms of X. We show that the group Aut(X) acts n-transitively on X, for all natural integers n. As an application we give a new and simpler proof of the fact that two rational nonsingular compact connected real algebraic surfaces are isomorphic if and only if they are homeomorphic as topological surfaces.


Critical points of inner functions, nonlinear partial differential equations, and an extension of Liouville's theorem

Abstract: We establish an extension of Liouville's classical representation theorem for solutions of the partial differential equation $\Delta u=4 e^{2u}$ and combine this result with methods from nonlinear elliptic PDE to construct holomorphic maps with prescribed critical points and specified boundary behaviour. For instance, we show that for every Blaschke sequence $\{z_j\}$ in the unit disk there is always a Blaschke product with $\{z_j\}$ as its set of critical points. Our work is closely related to the Berger-Nirenberg problem in differential geometry.


The Drinfel'd Double and Twisting in Stringy Orbifold Theory

Abstract: This paper exposes the fundamental role that the Drinfel'd double $\dkg$ of the group ring of a finite group $G$ and its twists $\dbkg$, $\beta \in Z^3(G,\uk)$ as defined by Dijkgraaf--Pasquier--Roche play in stringy orbifold theories and their twistings. The results pertain to three different aspects of the theory. First, we show that $G$--Frobenius algebras arising in global orbifold cohomology or K-theory are most naturally defined as elements in the braided category of $\dkg$--modules. Secondly, we obtain a geometric realization of the Drinfel'd double as the global orbifold $K$--theory of global quotient given by the inertia variety of a point with a $G$ action on the one hand and more stunningly a geometric realization of its representation ring in the braided category sense as the full $K$--theory of the stack $[pt/G]$. Finally, we show how one can use the co-cycles $\beta$ above to twist a) the global orbifold $K$--theory of the inertia of a global quotient and more importantly b) the stacky $K$--theory of a global quotient $[X/G]$. This corresponds to twistings with a special type of 2--gerbe.


A critical constant for the k nearest neighbour model

Abstract: Let P be a Poisson process of intensity one in a square S_n of area n. For a fixed integer k, join every point of P to its k nearest neighbours, creating an undirected random geometric graph G_{n,k}. We prove that there exists a critical constant c such that for c'<c, G_{n,c'log n} is disconnected with probability tending to 1 as n tends to infinity, and for c'>c G_{n,c'\log n} is connected with probability tending to 1 as n tends to infinity. This answers a question previously posed by the authors.


On the fifth order KdV equation: local well-posedness and lack of uniform continuity of the solution map

Abstract: In this paper we prove that the fifth order equation arising from the KdV hierarchy $ \partial_tu + \partial_x^5u + c_1\partial_x u\partial_x^2u + c_2u\partial_x^3u = 0 $ is locally well-posed in $ H^s(\mathbb{R}) $ for $ s> 5/2. Also, we prove the solution map of the equation is not uniformly continuous for $s>0$.


Comparing $GL_n$-Representations by Characteristic-Free Isomorphisms between Generalized Schur Algebras

Abstract: Isomorphisms are constructed between generalized Schur algebras in different degrees. The construction covers both the classical case (of general linear groups over infinite fields of arbitrary characteristic) and the quantized case (in type $A$, for any non-zero value of the quantum parameter $q$). The construction does not depend on the characteristic of the underlying field or the choice of $q \neq 0$. The proof combines a combinatorial construction with comodule structures and Ringel duality. Applications range from equivalences of categories to results on the structure and cohomology of Schur algebras to identities of decomposition numbers and also of $p$-Kostka numbers, in both cases reproving and generalizing row and column removal rules.


Ultrapowers of Banach algebras and modules

Abstract: The Arens products are the standard way of extending the product from a Banach algebra $\mc A$ to its bidual $\mc A''$. Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that if $\mc A$ is Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known for C$^*$-algebras. Our main tool is a Principle of Local Reflexivity result for modules and algebras.


BDDC and FETI-DP under Minimalist Assumptions

Abstract: The FETI-DP, BDDC and P-FETI-DP preconditioners are derived in a particulary simple abstract form. It is shown that their properties can be obtained from only on a very small set of algebraic assumptions. The presentation is purely algebraic and it does not use any particular definition of method components, such as substructures and coarse degrees of freedom. It is then shown that P-FETI-DP and BDDC are in fact the same. The FETI-DP and the BDDC preconditioned operators are of the same algebraic form, and the standard condition number bound carries over to arbitrary abstract operators of this form. The equality of eigenvalues of BDDC and FETI-DP also holds in the minimalist abstract setting. The abstract framework is explained on a standard substructuring example.


Tate motives and the fundamental group

Abstract: Let k be a number field, and let S be a finite set of k-rational points of P^1. We relate the Deligne-Goncharov contruction of the motivic fundamental group of X:=P^1_k- S to the Tannaka group scheme of the category of mixed Tate motives over X.


Whittaker unitary dual of affine graded Hecke algebras of type E

Abstract: This paper gives the classification of the Whittaker unitary dual for affine graded Hecke algebras of type E. By the Iwahori-Matsumoto involution, this is equivalent also to the classification of the spherical unitary dual for type E. This work completes the classification of the Whittaker Iwahori-spherical unitary dual, or equivalently, the spherical unitary dual, of split linear algebraic p-adic groups.


Combinatorial fiber bundles and fragmentation of a fiberwise PL-homeomorphism

Abstract: With a compact PL manifold X we associate a category T(X). The objects of T(X) are all combinatorial manifolds of type X, and morphisms are combinatorial assemblies. We prove that the homotopy equivalence BT (X) \approx BPL(X) holds, where PL(X) is the simplicial group of PL-homeomorphisms. Thus the space BT(X) is a canonical countable (as a CW-complex) model of BPL(X). As a result, we obtain functorial pure combinatorial models for PL fiber bundles with fiber X and a PL polyhedron B as the base. Such a model looks like a T(X)-coloring of some triangulation K of B. The vertices of K are colored by objects of T(X) and the arcs are colored by morphisms in such a way that the diagram arising from the 2-skeleton of K is commutative. Comparing with the classical results of geometric topology, we obtain combinatorial models of the real Grassmannian in small dimensions: BT(S^{n-1}) \approx BO(n) for n=1,2,3,4. The result is proved in a sequence of results on similar models of B\PL(X). Special attention is paid to the main noncompact case X=R^n and to the tangent bundle and Gauss functor of a combinatorial manifold. The trick that makes the proof possible is a collection of lemmas on "fragmentation of a fiberwise homeomorphism", a generalization of the folklore lemma on fragmentation of an isotopy.


Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces

Abstract: We prove effective equidistribution, with polynomial rate, for large closed orbits of semisimple groups on homogeneous spaces, under certain technical restrictions (notably, the acting group should have finite centralizer in the ambient group). The proofs make extensive use of spectral gaps, and also of a closing lemma for such actions.


Moments of the critical values of families of elliptic curves, with applications

Abstract: We make conjectures on the moments of the central values of the family of all elliptic curves and on the moments of the first derivative of the central values of a large family of positive rank curves. In both cases the order of magnitude is the same as that of the moments of the central values of an orthogonal family of L-functions. Notably, we predict that the critical values of all rank 1 elliptic curves is logarithmically larger than the rank 1 curves in the positive rank family. Furthermore, as arithmetical applications we make a conjecture on the distribution of a_p's amongst all rank 2 elliptic curves, and also show how the Riemann hypothesis can be deduced from sufficient knowledge of the first moment of the positive rank family (based on an idea of Iwaniec).


Stable rank for inclusions of C*-algebras

Abstract: When a unital \ca $A$ has topological stable rank one (write $\tsr(A) = 1$), we know that $\tsr(pAp) \leq 1$ for a non-zero projection $p \in A$. When, however, $\tsr(A) \geq 2$, it is generally faluse. We prove that if a unital C*-algebra $A$ has a simple unital C*-subalgebra $D$ of $A$ with common unit such that $D$ has \PSP and $\sup_{p\in P(D)}\tsr(pAp) < \infty$, then $\tsr(A) \leq 2.$ As an application let $A$ be a simple unital \ca with $\tsr(A) = 1$ and \PSP, $\{G_k\}_{k=1}^n$ finite groups, $\af_k$ actions from $G_k$ to ${\rm Aut}((...((A\times_{\af_1}G_1)\times_{\af_2} G_2)...)\times_{\af_{k-1}}G_{k-1}).$ $(G_0 = \{1\})$ Then $$ \tsr((... ((A\times_{\af_1}G_1)\times_{\af_2} G_2)...)\times_{\af_n}G_n) \leq 2. $$


Asymptotic Blocking Probabilities in Loss Networks with Subexponential Demands

Abstract: The analysis of stochastic loss networks has long been of interest in computer and communications networks and is becoming important in the areas of service and information systems. In traditional settings, computing the well known Erlang formula for blocking probability in these systems becomes intractable for larger resource capacities. Using compound point processes to capture stochastic variability in the request process, we generalize existing models in this framework and derive simple asymptotic expressions for blocking probabilities. In addition, we extend our model to incorporate reserving resources in advance. Although asymptotic, our experiments show an excellent match between derived formulas and simulation results even for relatively small resource capacities and relatively large values of blocking probabilities.


Boundaries for algebras of holomorphic functions on Banach spaces

Abstract: We study the relations between boundaries for algebras of holomorphic functions on Banach spaces and complex convexity of their balls. In addition, we show that the Shilov boundary for algebras of holomorphic functions on an order continuous sequence space $X$ is the unit sphere $S_X$ if $X$ is locally c-convex. In particular, it is shown that the unit sphere of the Orlicz-Lorentz sequence space $\lambda_{\phi, w}$ is the Shilov boundary for algebras of holomorphic functions on $\lambda_{\phi, w}$ if $\phi$ satisfies the $\delta_2$-condition.


Bishop's Theorem and Differentiability of a subspace of $C_b(K)$

Abstract: Let $K$ be a Hausdorff space and $C_b(K)$ be the Banach algebra of all complex bounded continuous functions on $K$. We study the G\^{a}teaux and Fr\'echet differentiability of subspaces of $C_b(K)$. Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace $H$ of $C_b(K)$ is a dense $G_\delta$ subset of $H$, if $K$ is compact. This gives a generalized Bishop's theorem, which says that the closure of the set of strong peak point for $H$ is the smallest closed norming subset of $H$. The classical Bishop's theorem was proved for a separating subalgebra $H$ and a metrizable compact space $K$. In the case that $X$ is a complex Banach space with the Radon-Nikod\'ym property, we show that the set of all strong peak functions in $A_b(B_X)=\{f\in C_b(B_X) : f|_{B_X^\circ} {is holomorphic}\}$ is dense. As an application, we show that the smallest closed norming subset of $A_b(B_X)$ is the closure of the set of all strong peak points for $A_b(B_X)$. This implies that the norm of $A_b(B_X)$ is G\^{a}teaux differentiable on a dense subset of $A_b(B_X)$, even though the norm is nowhere Fr\'echet differentiable when $X$ is nontrivial. We also study the denseness of norm attaining holomorphic functions and polynomials. Finally we investigate the existence of numerical Shilov boundary.


Abstract: The Loewy length of the descent algebra of type D_{2m+1} is shown to be m+2, for m \geq 2, by providing an upper bound that agrees with the lower bound in \cite{BonnafePfeiffer2006}. The bound is obtained by showing that the length of the longest path in the quiver of the descent algebra of D_{2m+1} is at most m+1. To achieve this bound, the geometric approach to the descent algebra is used, in which the descent algebra of a finite Coxeter group is identified with an algebra associated to the reflection arrangement of the group.


Classification of uniformly outer actions of Z^2 on UHF algebras

Abstract: We give a complete classification up to cocycle conjugacy of uniformly outer actions of Z^2 on UHF algebras. In particular, it is shown that any two uniformly outer actions of Z^2 on a UHF algebra of infinite type are cocycle conjugate. We also classify them up to outer conjugacy.


Classification of outer actions of Z^N on O_2

Abstract: We will show that any two outer actions of Z^N on O_2 are cocycle conjugate.


Holder stability of diffeomorphisms

Abstract: We prove that a $C^2$ diffeomorphism $f$ of a compact manifold $M$ satisfies Axiom A and the strong transversality condition if and only if it is H\"{o}lder stable, that is, any $C^1$ diffeomorphism $g$ of $M$ sufficiently $C^1$ close to $f$ is conjugate to $f$ by a homeomorphism which is H\"{o}lder on the whole manifold.


A recursive online algorithm for the estimation of time-varying ARCH parameters

Abstract: In this paper we propose a recursive online algorithm for estimating the parameters of a time-varying ARCH process. The estimation is done by updating the estimator at time point $t-1$ with observations about the time point $t$ to yield an estimator of the parameter at time point $t$. The sampling properties of this estimator are studied in a non-stationary context -- in particular, asymptotic normality and an expression for the bias due to non-stationarity are established. By running two recursive online algorithms in parallel with different step sizes and taking a linear combination of the estimators, the rate of convergence can be improved for parameter curves from H\"{o}lder classes of order between 1 and 2.


Limiting distributions of curves under geodesic flow on hyperbolic manifolds

Abstract: We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic $n$-manifold under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is shown that under the geodesic flow, the normalized parameter measure on the curve gets asymptotically equidistributed with respect to the normalized natural Riemannian measure on the unit tangent bundle of a closed totally geodesically immersed submanifold. Moreover, if this immersed submanifold is a proper subset, then a lift of the curve to the universal covering space $T^1(H^n)$ is mapped into a proper subsphere of the ideal boundary sphere $\partial H^n$ under the visual map. This proper subsphere can be realized as the ideal boundary of an isometrically embedded hyperbolic subspace in $H^n$ covering the closed immersed submanifold. In particular, if the visual map does not send a lift of the curve into a proper subsphere of $\partial H^n$, then under the geodesic flow the curve gets asymptotically equidistributed on the unit tangent bundle of the manifold with respect to the normalized natural Riemannian measure. The proof uses dynamical properties of unipotent flows on finite volume homogeneous spaces of SO(n,1).


On dynamical smash product

Abstract: In the theory of dynamical Yang-Baxter equation, with any Hopf algebra $H$ and a certain $H$-module and $H$-comodule algebra $L$ (base algebra) one associates a monoidal category. Given an algebra $A$ in that category, one can construct an associative algebra $A\rtimes L$, which is a generalization of the ordinary smash product when $A$ is an ordinary $H$-algebra. We study this "dynamical smash product" and its modules induced from one-dimensional representation of the subalgebra $L$. In particular, we construct an analog of the Galois map $A\otimes_{A^H} A\to A\otimes H^*$.


Wavelet block thresholding for samples with random design: a minimax approach under the $L^p$ risk

Abstract: We consider the regression model with (known) random design. We investigate the minimax performances of an adaptive wavelet block thresholding estimator under the $\mathbb{L}^p$ risk with $p\ge 2$ over Besov balls. We prove that it is near optimal and that it achieves better rates of convergence than the conventional term-by-term estimators (hard, soft,...).


Polynomial identities and noncommutative versal torsors

Abstract: To any cleft Hopf Galois object, i.e., any algebra H[t] obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle t, we attach two "universal algebras" A(H,t) and U(H,t). The algebra A(H,t) is obtained by twisting the multiplication of H with the most general two-cocycle u formally cohomologous to t. The cocycle u takes values in the field of rational functions on H. By construction, A(H,t) is a cleft H-Galois extension of a "big" commutative algebra B(H,t). Any "form" of H[t] can be obtained from A(H,t) by a specialization of B(H,t) and vice versa. If the algebra H[t] is simple, then A(H,t) is an Azumaya algebra with center B(H,t). The algebra U(H,t) is constructed using a general theory of polynomial identities that we set up for arbitrary comodule algebras; it is the universal comodule algebra in which all comodule algebra identities of H[t] are satisfied. We construct an embedding of U(H,t) into A(H,t); this embedding maps the center Z(H,t) of U(H,t) into B(H,t) when the algebra H[t] is simple. In this case, under an additional assumption, A(H,t) is isomorphic to B(H,t) \otimes_{Z(H,t)} U(H,t), thus turning A(H,t) into a central localization of U(H,t). We work out these constructions in full detail for the four-dimensional Sweedler algebra.


Full Bayesian analysis for a class of jump-diffusion models

Abstract: A new Bayesian significance test is adjusted for jump detection in a diffusion process. This is an advantageous procedure for temporal data having extreme valued outliers, like financial data, pluvial or tectonic forces records and others.


Exploring spatial nonlinearity using additive approximation

Abstract: We propose to approximate the conditional expectation of a spatial random variable given its nearest-neighbour observations by an additive function. The setting is meaningful in practice and requires no unilateral ordering. It is capable of catching nonlinear features in spatial data and exploring local dependence structures. Our approach is different from both Markov field methods and disjunctive kriging. The asymptotic properties of the additive estimators have been established for $\alpha$-mixing spatial processes by extending the theory of the backfitting procedure to the spatial case. This facilitates the confidence intervals for the component functions, although the asymptotic biases have to be estimated via (wild) bootstrap. Simulation results are reported. Applications to real data illustrate that the improvement in describing the data over the auto-normal scheme is significant when nonlinearity or non-Gaussianity is pronounced.


Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions

Abstract: In this paper a new class of generalized backward doubly stochastic differential equations is investigated. This class involves an integral with respect to an adapted continuous increasing process. A probabilistic representation for viscosity solutions of semi-linear stochastic partial differential equations with a Neumann boundary condition is given.


Unipotent flows on products of $SL(2,K)/\Gamma$'s

Abstract: We give a simplified and a direct proof of a special case of Ratner's theorem on closures and uniform distribution of individual orbits of unipotent flows; namely, the case of orbits of the diagonally embedded unipotent subgroup acting on $SL(2,K)/\Gamma_1\times ...\times SL(2,K)/\Gamma_n$, where $K$ is a locally compact field of characteristic 0 and each $\Gamma_i$ is a cocompact discrete subgroup of $SL(2,K)$. This special case of Ratner's theorem plays a crucial role in the proofs of uniform distribution of Heegner points by Vatsal, and Mazur conjecture on Heegner points by C. Cornut; and their generalizations in their joint work on CM-points and quaternion algebras. A purpose of the article is to make the ergodic theoretic results accessible to a wide audience.


Estimation in hidden Markov models via efficient importance sampling

Abstract: Given a sequence of observations from a discrete-time, finite-state hidden Markov model, we would like to estimate the sampling distribution of a statistic. The bootstrap method is employed to approximate the confidence regions of a multi-dimensional parameter. We propose an importance sampling formula for efficient simulation in this context. Our approach consists of constructing a locally asymptotically normal (LAN) family of probability distributions around the default resampling rule and then minimizing the asymptotic variance within the LAN family. The solution of this minimization problem characterizes the asymptotically optimal resampling scheme, which is given by a tilting formula. The implementation of the tilting formula is facilitated by solving a Poisson equation. A few numerical examples are given to demonstrate the efficiency of the proposed importance sampling scheme.


A limit result for a system of particles in random environment

Abstract: We consider an infinite system of particles in one dimension, each particle performs independant Sinai's random walk in random environment. Considering an instant $t$, large enough, we prove a result in probability showing that the particles are trapped in the neighborhood of well defined points of the lattice depending on the random environment the time $t$ and the starting point of the particles.


On the scaling limit of a singular integral operator

Abstract: The scaling limit and Schauder bounds are derived for a singular integral operator arising from a difference equation approach to monodromy problems.


Penalized nonparametric mean square estimation of the coefficients of diffusion processes

Abstract: We consider a one-dimensional diffusion process $(X_t)$ which is observed at $n+1$ discrete times with regular sampling interval $\Delta$. Assuming that $(X_t)$ is strictly stationary, we propose nonparametric estimators of the drift and diffusion coefficients obtained by a penalized least squares approach. Our estimators belong to a finite-dimensional function space whose dimension is selected by a data-driven method. We provide non-asymptotic risk bounds for the estimators. When the sampling interval tends to zero while the number of observations and the length of the observation time interval tend to infinity, we show that our estimators reach the minimax optimal rates of convergence. Numerical results based on exact simulations of diffusion processes are given for several examples of models and illustrate the qualities of our estimation algorithms.


Monogenic Functions in Conformal Geometry

Abstract: Monogenic functions are basic to Clifford analysis. On Euclidean space they are defined as smooth functions with values in the corresponding Clifford algebra satisfying a certain system of first order differential equations, usually referred to as the Dirac equation. There are two equally natural extensions of these equations to a Riemannian spin manifold only one of which is conformally invariant. We present a straightforward exposition.


Reflecting recollements

Abstract: A recollement describes one triangulated category T as "glued together" from two others, S and U. The definition is not symmetrical in S and U, but this note shows how S and U can be interchanged when T has a Serre functor.


Characterization of count data distributions involving additivity and binomial subsampling

Abstract: In this paper we characterize all the $r$-parameter families of count distributions (satisfying mild conditions) that are closed under addition and under binomial subsampling. Surprisingly, few families satisfy both properties and the resulting models consist of the $r$th-order univariate Hermite distributions. Among these, we find the Poisson ($r=1$) and the ordinary Hermite distributions ($r=2$).


On martingale approximations

Abstract: Consider additive functionals of a Markov chain $W_k$, with stationary (marginal) distribution and transition function denoted by $\pi$ and $Q$, say $S_n=g(W_1)+...+g(W_n)$, where $g$ is square integrable and has mean 0 with respect to $\pi$. If $S_n$ has the form $S_n=M_n+R_n$, where $M_n$ is a square integrable martingale with stationary increments and $E(R_n^2)=o(n)$, then $g$ is said to admit a martingale approximation. Necessary and sufficient conditions for such an approximation are developed. Two obvious necessary conditions are $E[E(S_n|W_1)^2]=o(n)$ and $\lim_{n\to \infty}E(S_n^2)/n<\infty$. Assuming the first of these, let $\Vert g\Vert^2_+=\limsup_{n\to \infty}E(S_n^2)/n$; then $\Vert\cdot\Vert_+$ defines a pseudo norm on the subspace of $L^2(\pi)$ where it is finite. In one main result, a simple necessary and sufficient condition for a martingale approximation is developed in terms of $\Vert\cdot\Vert_+$. Let $Q^*$ denote the adjoint operator to $Q$, regarded as a linear operator from $L^2(\pi)$ into itself, and consider co-isometries ($QQ^*=I$), an important special case that includes shift processes. In another main result a convenient orthonormal basis for $L_0^2(\pi)$ is identified along with a simple necessary and sufficient condition for the existence of a martingale approximation in terms of the coefficients of the expansion of $g$ with respect to this basis.


The Laguerre process and generalized Hartman--Watson law

Abstract: In this paper, we study complex Wishart processes or the so-called Laguerre processes $(X_t)_{t\geq0}$. We are interested in the behaviour of the eigenvalue process; we derive some useful stochastic differential equations and compute both the infinitesimal generator and the semi-group. We also give absolute-continuity relations between different indices. Finally, we compute the density function of the so-called generalized Hartman--Watson law as well as the law of $T_0:=\inf\{t,\det(X_t)=0\}$ when the size of the matrix is 2.


Constructive decomposition of a function of two variables as a sum of functions of one variable

Abstract: Given a compact set $K$ in the plane, which does not contain any triple of points forming a vertical and a horizontal segment, and a map $f\in C(K)$, we give a construction of functions $g,h\in C(\mathbb R)$ such that $f(x,y)=g(x)+h(y)$ for all $(x,y)\in K$. This provides a constructive proof of a part of Sternfeld's theorem on basic embeddings in the plane. In our proof the set $K$ is approximated by a finite set of points.


Rational maps between moduli spaces of curves and Gieseker-Petri divisors

Abstract: We perform an intersection theoretic study of the rational map between two different moduli spaces of stable curves which associates to a curve its corresponding Brill-Noether locus (in the case this locus has virtual dimension 1). We then use these results to describe the cone of moving divisors on M_g. Several other applications to moduli spaces of Prym varieties are presented. In a different direction, we prove that the locus in M_g of curves failing to satisfy the Gieseker-Petri theorem is supported in codimension 1 for every possible type of linear series.


Some algorithms for semi-invariants of quivers

Abstract: We present some theorems and algorithms for calculating perpendicular categories and locally semi-simple decompositions. We implemented a computer program {\sc TETIVA} based on these algorithms and we offer this program for everybody's use.


External edge condition and group cohomologies associated with the quantum Clebsch-Gordan condition

Abstract: In this article we determine the structure of a twisted first cohomology group of the first homology of a trivalent graph with a coefficient associated with the quantum Clebsch-Gordan condition. As an application we give a characterization of a combinatorial property, the external edge condition, which is defined by the author in the study of the Heisenberg representation on the TQFT-module.


Partial magmatic bialgebras

Abstract: A partial magmatic bialgebra, (T;S)-magmatic bialgebra where T \subset S are subsets of the set of positive integers, is a vector space endowed with an n-ary operation for each n in S and an m-ary co-operation for each m in T satisfying some compatibility and unitary relations. We prove an analogue of the Poincar\'e-Birkhoff-Witt theorem for these partial magmatic bialgebras.


Amenability of ultraproducts of Banach algebras

Abstract: We study when certain properties of Banach algebras are stable under ultrapower constructions. In particular, we consider when every ultrapower of $\mc A$ is Arens regular, and give some evidence that this is if and only if $\mc A$ is isomorphic to a closed subalgebra of operators on a super-reflexive Banach space. We show that such ideas are closely related to whether one can sensibly define an ultrapower of a dual Banach algebra. We study how tensor products of ultrapowers behave, and apply this to study the question of when every ultrapower of $\mc A$ is amenable. We provide an abstract characterisation in terms of something like an approximate diagonal, and consider when every ultrapower of a C$^*$-algebra, or a group $L^1$-convolution algebra, is amenable.


Braided-Lie bialgebras associated to Kac-Moody algebras

Abstract: Braided-Lie bialgebras have been introduced by Majid, as the Lie versions of Hopf algebras in braided categories. In this paper we extend previous work of Majid and of ours to show that there is a braided-Lie bialgebra associated to each inclusion of Kac-Moody bialgebras. Doing so, we obtain many new examples of infinite-dimensional braided-Lie bialgebras. We analyze further the case of untwisted affine Kac-Moody bialgebras associated to finite-dimensional simple Lie algebras. The inclusion we study is that of the finite-type algebra in the affine algebra. This braided-Lie bialgebra is isomorphic to the current algebra over the simple Lie algebra, now equipped with a braided cobracket. We give explicit expressions for this braided cobracket for the simple Lie algebra $sl_3$.


Global properties of indefinite metrics with parallel Weyl tensor

Abstract: This is an exposition of some recent results on ECS manifolds, by which we mean pseudo-Riemannian manifolds of dimensions greater than 3 that are neither conformally flat nor locally symmetric, and have parallel Weyl tensor. All ECS metrics are indefinite. We state two classification theorems, describing the local structure of ECS manifolds, and outline an argument showing that compact ECS manifolds exist in infinitely many dimensions greater than 4. We also discuss some properties of compact manifolds that admit ECS metrics, and provide a list of open questions about compact ECS manifolds.


On the quiver of the descent algebra

Abstract: We study the quiver of the descent algebra of a finite Coxeter group W. The results include a derivation of the quiver of the descent algebra of types A and B. Our approach is to study the descent algebra as an algebra constructed from the reflection arrangement associated to W.


Embedding $FD(\omega)$ into $\mathcal{P}_s$ densely

Abstract: Let $\mathcal{P}_s$ be the lattice of degrees of non-empty $\Pi_1^0$ subsets of $2^\omega$ under Medvedev reducibility. Binns and Simpson proved that $FD(\omega)$, the free distributive lattice on countably many generators, is lattice-embeddable below any non-zero element in $\mathcal{P}_s$. Cenzer and Hinman proved that $\mathcal{P}_s$ is dense, by adapting the Sacks Preservation and Sacks Coding Strategies used in the proof of the density of the c.e.\ Turing degrees. With a construction that is a modification of the one by Cenzer and Hinman, we improve on the result of Binns and Simpson by showing that for any $\mathcal{U} <_s \mathcal{V}$, we can lattice embed $FD(\omega)$ into $\mathcal{P}_s$ strictly between $deg_s(\mathcal{U})$ and $deg_s(\mathcal{V)}$. We also note that, in contrast to the infinite injury in the proof of the Sacks Density Theorem, in our proof all injury is finite, and that this is also true for the proof of Cenzer and Hinman, if a straightforward simplification is made.


A note on the Markoff condition and central words

Abstract: We define Markoff words as certain factors appearing in bi-infinite words satisfying the Markoff condition. We prove that these words coincide with central words, yielding a new characterization of Christoffel words.


Quantum Structures for Lagrangian Submanifolds

Abstract: We discuss various algebraic quantum structures associated to monotone Lagrangian submanifolds and we present a number of applications, computations and examples.


Algebraic G-functions associated to matrices over a group-ring

Abstract: Given a square matrix with elements in the group-ring of a group, one can consider the sequence formed by the trace (in the sense of the group-ring) of its powers. We prove that the corresponding generating series is an algebraic $G$-function (in the sense of Siegel) when the group is free of finite rank. Consequently, it follows that the norm of such elements is an exactly computable algebraic number, and their Green function is algebraic. Our proof uses the notion of rational and algebraic power series in non-commuting variables and is an easy application of a theorem of Haiman. Haiman's theorem uses results of linguistics regarding regular and context-free language. On the other hand, when the group is free abelian of finite rank, then the corresponding generating series is a $G$-function. We ask whether the latter holds for general hyperbolic groups. This version has an expanded introduction following suggestions from Lehner, Voiculescu and others.


Minimal Distortion Bending and Morphing of Compact Manifolds

Abstract: Let $M$ and $N$ be compact smooth oriented Riemannian $n$-manifolds without boundary embedded in $\mathbb{R}^{n+1}$. Several problems about minimal distortion bending and morphing of $M$ to $N$ are posed. Cost functionals that measure distortion due to stretching or bending produced by a diffeomorphism $h:M \to N$ are defined, and new results on the existence of minima of these cost functionals are presented. In addition, the definition of a morph between two manifolds $M$ and $N$ is given, and the theory of minimal distortion morphing of compact manifolds is reviewed.


On the torsion of Drinfeld modules of rank two

Abstract: We study rational points and torsion points on Drinfeld modular curves defined over rational function fields. As a consequence we derive a conjecture of Schweizer describing completely the torsion of Drinfeld modules of rank two over $\Bbb F_2(T)$ implying Poonen's uniform boundedness conjecture in this particular case.


Hurwitz-Hodge Integrals, the E6 and D4 root systems, and the Crepant Resolution Conjecture

Abstract: Let G be the group A_4 or Z_2xZ_2. We compute the integral of \lambda_g on the Hurwitz locus H_G\subset M_g of curves admitting a degree 4 cover of P^1 having monodromy group G. We compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the E_6 and D_4 root systems respectively. As an application, we prove the Crepant Resolution Conjecture for the orbifolds [C^3/A_4] and [C^3/(Z_2xZ_2)].


Conjugacy and Dynamics in Thompson's Groups

Abstract: We give a unified solution to the conjugacy problem for Thompson's groups F, T, and V. The solution uses strand diagrams, which are similar in spirit to braids and generalize tree-pair diagrams for elements of Thompson's groups. Strand diagrams are closely related to piecewise-linear functions for elements of Thompson's groups, and we use this correspondence to investigate the dynamics of elements of F. Though many of the results in this paper are known, our approach is new, and it yields elegant proofs of several old results.


Virtual Yang-Baxter cocycle invariants

Abstract: We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter cocycle invariants for classical knots but provide extra information about virtual knots and links. In particular, they provide a method for detecting non-classicality of virtual knots and links.


On hitting times and fastest strong stationary times for skip-free and more general chains

Abstract: An (upward) skip-free Markov chain with the set of nonnegative integers as state space is a chain for which upward jumps may be only of unit size; there is no restriction on downward jumps. In a 1987 paper, Brown and Shao determined, for an irreducible continuous-time skip-free chain and any d, the passage time distribution from state 0 to state d. When the nonzero eigenvalues nu_j of the generator are all real, their result states that the passage time is distributed as the sum of d independent exponential random variables with rates nu_j. We give another proof of their theorem. In the case of birth-and-death chains, our proof leads to an explicit representation of the passage time as a sum of independent exponential random variables. Diaconis and Miclo recently obtained the first such representation, but our construction is much simpler. We obtain similar (and new) results for a fastest strong stationary time T of an ergodic continuous-time skip-free chain with stochastically monotone time-reversal started in state 0, and we also obtain discrete-time analogs of all our results. In the paper's final section we present extensions of our results to more general chains.


The spectral sequence of an equivariant chain complex and homology with local coefficients

Abstract: We study the spectral sequence associated to the filtration by powers of the augmentation ideal on the (twisted) equivariant chain complex of the universal cover of a connected CW-complex X. In the process, we identify the d^1 differential in terms of the coalgebra structure of H_*(X,\k), and the \k\pi_1(X)-module structure on the twisting coefficients. In particular, this recovers in dual form a result of Reznikov, on the mod p cohomology of cyclic p-covers of aspherical complexes. This approach provides information on the homology of all Galois covers of X. It also yields computable upper bounds on the ranks of the cohomology groups of X, with coefficients in a prime-power order, rank one local system. When X admits a minimal cell decomposition, we relate the linearization of the equivariant cochain complex of the universal abelian cover to the Aomoto complex, arising from the cup-product structure of H^*(X,\k), thereby generalizing a result of Cohen and Orlik.


Canonical 2-forms on the moduli space of Riemann surfaces

Abstract: As was shown by Harer the second homology of ${\mathbb M}_g$, the moduli space of compact Riemann surfaces of genus $g$, is of rank 1, provided $g \geq 3$. This means a nontrivial second de Rham cohomology class on ${\mathbb M}_g$ is unique up to constant factor. But several canonical 2-forms on the moduli space have been constructed in various geometric contexts, and differ from each other. In this article we review some of constructions in order to provide materials for future research on "secondary geometry" of the moduli space ${\mathbb M}_g$.


Normal approximation for nonlinear statistics using a concentration inequality approach

Abstract: Let $T$ be a general sampling statistic that can be written as a linear statistic plus an error term. Uniform and non-uniform Berry--Esseen type bounds for $T$ are obtained. The bounds are the best possible for many known statistics. Applications to U-statistics, multisample U-statistics, L-statistics, random sums and functions of nonlinear statistics are discussed.


Continuation of solutions of coupled dynamical systems

Abstract: Recently, the synchronization of coupled dynamical systems has been widely studied. Synchronization is referred to as a process wherein two (or many) dynamical systems are adjusted to a common behavior as time goes to infinity, due to coupling or forcing. Therefore, before discussing synchronization, a basic problem on continuation of the solution must be solved: For given initial conditions, can the solution of coupled dynamical systems be extended to the infinite interval $[0,+\infty)$? In this paper, we propose a general model of coupled dynamical systems, which includes previously studied systems as special cases, and prove that under the assumption of QUAD, the solution of the general model exists on $[0,+\infty)$.


Lefschetz numbers for C*-algebras

Abstract: Using Poincare duality, we formulate a formula of Lefschetz type which computes the Lefschetz number of an endomorphism of a separable, nuclear C*-algebra satisfying Poincare duality and the Kunneth theorem. (The Lefschetz number of an endomorphism is the graded trace of the induced map on K-theory tensored with the complex numbers, as in the classical case.) We then consider endomorphisms of Cuntz-Krieger algebras O_A. An endomorphism has an invariant, which is a permutation of an infinite set, and the contracting and expanding behavior of this permutation describes the Lefschetz number of the endomorphism. Using this description we derive a closed polynomial formula for the Lefschetz number depending on the matrix A and the presentation of the endomorphism.


A Lefschetz fixed-point formula for certain orbifold C*-algebras

Abstract: Using Poincar\'e duality in K-theory, we state and prove a Lefschetz fixed point formula for endomorphisms of cross product C*-algebras $C_0(X)\cross G$ coming from covariant pairs. Here $G$ is assumed countable, $X$ a manifold, and $X\cross G$ cocompact and proper. The formula in question expresses the graded trace of the map on rationalized K-theory of $C_0(X)\cross G$ induced by the endomorphism, \emph{i.e.} the Lefschetz number, in terms of fixed orbits and representation-theoretic data connected with certain isotropy subgroups of the isotropy group at that point. The technique is to use noncommutative Poinca\'e duality and the formal Lefschetz lemma of the second author.


Correction to: The scaling limit behaviour of periodic stable-like processes

Abstract: Correction to Bernoulli (2006), 12, 551--570 http://projecteuclid.org/euclid.bj/1151525136


Dynamical sensitivity of the infinite cluster in critical percolation

Abstract: In dynamical percolation, the status of every bond is refreshed according to an independent Poisson clock. For graphs which do not percolate at criticality, the dynamical sensitivity of this property was analyzed extensively in the last decade. Here we focus on graphs which percolate at criticality, and investigate the dynamical sensitivity of the infinite cluster. We first give two examples of bounded degree graphs, one which percolates for all times at criticality and one which has exceptional times of nonpercolation. We then make a nearly complete analysis of this question for spherically symmetric trees with spherically symmetric edge probabilities bounded away from 0 and 1. One interesting regime occurs when the expected number of vertices at the nth level that connect to the root at a fixed time is of order n(\log n)^\alpha. R. Lyons (1990) showed that at a fixed time, there is an infinite cluster a.s. if and only if \alpha >1. We prove that the probability that there is an infinite cluster at all times is 1 if \alpha > 2, while this probability is 0 if 1<\alpha \le 2. Within the regime where a.s. there is an infinite cluster at all times, there is yet another type of ``phase transition'' in the behavior of the process: if the expected number of vertices at the nth level connecting to the root at a fixed time is of order n^\theta with \theta > 2, then the number of connected components of the set of times in [0,1] at which the root does not percolate is finite a.s., while if 1<\theta < 2, then the number of such components is infinite with positive probability.


Large sample asymptotics for the two-parameter Poisson--Dirichlet process

Abstract: This paper explores large sample properties of the two-parameter $(\alpha,\theta)$ Poisson--Dirichlet Process in two contexts. In a Bayesian context of estimating an unknown probability measure, viewing this process as a natural extension of the Dirichlet process, we explore the consistency and weak convergence of the the two-parameter Poisson--Dirichlet posterior process. We also establish the weak convergence of properly centered two-parameter Poisson--Dirichlet processes for large $\theta+n\alpha.$ This latter result complements large $\theta$ results for the Dirichlet process and Poisson--Dirichlet sequences, and complements a recent result on large deviation principles for the two-parameter Poisson--Dirichlet process. A crucial component of our results is the use of distributional identities that may be useful in other contexts.


Dilatation structures in sub-riemannian geometry

Abstract: Based on the notion of dilatation structure arXiv:math/0608536, we give an intrinsic treatment to sub-riemannian geometry, started in the paper arXiv:0706.3644 . Here we prove that regular sub-riemannian manifolds admit dilatation structures. From the existence of normal frames proved by Bellaiche we deduce the rest of the properties of regular sub-riemannian manifolds by using the formalism of dilatation structures.


Vector Optimization by Two Objective Functions

Abstract: Recently wide application in engineering-economic problems was received with problems of vector optimization. Development of methods of the decision of these problems it is executed in works A. Messac and others. Complexity of the offered methods consists in construction of an aggregate objective function. In the given work an algorithm of the solution of a vector optimization problem is suggested carry out by use analytical representation of Pareto cone.


Z_2 actions on complexes with three non-trivial cells

Abstract: In this paper, we study $Z_2$ actions on a cell complex X having the cohomology ring isomorphic to that of the wedge sum $P^2 (n) V S^{3n}$ or $S^n V S^{2n} V S^{3n}$. We determine the possible fixed point sets depending on whether or not X is totally non-homologous to zero in $X_{Z_2}$ and give examples realizing the possible cases.


Coupling hidden Markov models for the discovery of Cis-regulatory modules in multiple species

Abstract: Cis-regulatory modules (CRMs) composed of multiple transcription factor binding sites (TFBSs) control gene expression in eukaryotic genomes. Comparative genomic studies have shown that these regulatory elements are more conserved across species due to evolutionary constraints. We propose a statistical method to combine module structure and cross-species orthology in de novo motif discovery. We use a hidden Markov model (HMM) to capture the module structure in each species and couple these HMMs through multiple-species alignment. Evolutionary models are incorporated to consider correlated structures among aligned sequence positions across different species. Based on our model, we develop a Markov chain Monte Carlo approach, MultiModule, to discover CRMs and their component motifs simultaneously in groups of orthologous sequences from multiple species. Our method is tested on both simulated and biological data sets in mammals and Drosophila, where significant improvement over other motif and module discovery methods is observed.


Group cocycles and the ring of affiliated operators

Abstract: In this article we study cocycles of discrete countable groups with values in l^2(G) and the ring of affiliated operators UG. We clarify properties of the first cohomology of a group G with coefficients in l^2(G) and answer several questions from [CTV]. Moreover, we obtain strong results about the existence of free subgroups and the subgroup structure, provided the group has a positive first l^2-Betti number. We give numerous applications and examples of groups which satisfy our assumptions.


Structure and finiteness properties of subdirect products of groups

Abstract: We investigate the structure of subdirect products of groups, particularly their finiteness properties. We pay special attention to the subdirect products of free groups, surface groups and HNN extensions. We prove that a finitely presented subdirect product of free and surface groups virtually contains a term of the lower central series of the direct product or else fails to intersect one of the direct summands. This leads to a characterization of the finitely presented subgroups of the direct product of 3 free or surface groups, and to a solution to the conjugacy problem for arbitrary finitely presented subgroups of direct products of surface groups. We obtain a formula for the first homology of a subdirect product of two free groups and use it to show there is no algorithm to determine the first homology of a finitely generated subgroup.


A statistical approach to simultaneous mapping and localization for mobile robots

Abstract: Mobile robots require basic information to navigate through an environment: they need to know where they are (localization) and they need to know where they are going. For the latter, robots need a map of the environment. Using sensors of a variety of forms, robots gather information as they move through an environment in order to build a map. In this paper we present a novel sampling algorithm to solving the simultaneous mapping and localization (SLAM) problem in indoor environments. We approach the problem from a Bayesian statistics perspective. The data correspond to a set of range finder and odometer measurements, obtained at discrete time instants. We focus on the estimation of the posterior distribution over the space of possible maps given the data. By exploiting different factorizations of this distribution, we derive three sampling algorithms based on importance sampling. We illustrate the results of our approach by testing the algorithms with two real data sets obtained through robot navigation inside office buildings at Carnegie Mellon University and the Pontificia Universidad Catolica de Chile.


Uniqueness of the 2-universality Criterion

Abstract: Kim, Kim, and Oh gave a minimal criterion for the 2-universality of positive-definite integer-matrix quadratic forms. We show that this 2-universality criterion is unique in the sense of the uniqueness of the Conway-Schneeberger Fifteen Theorem.


Continuation unique a partir de l'infini conforme pour les metriques d'Einstein

Abstract: We prove the unique continuation property at the conformal infinity for asymptotically hyperbolic Einstein metrics.


Random-set methods identify distinct aspects of the enrichment signal in gene-set analysis

Abstract: A prespecified set of genes may be enriched, to varying degrees, for genes that have altered expression levels relative to two or more states of a cell. Knowing the enrichment of gene sets defined by functional categories, such as gene ontology (GO) annotations, is valuable for analyzing the biological signals in microarray expression data. A common approach to measuring enrichment is by cross-classifying genes according to membership in a functional category and membership on a selected list of significantly altered genes. A small Fisher's exact test $p$-value, for example, in this $2\times2$ table is indicative of enrichment. Other category analysis methods retain the quantitative gene-level scores and measure significance by referring a category-level statistic to a permutation distribution associated with the original differential expression problem. We describe a class of random-set scoring methods that measure distinct components of the enrichment signal. The class includes Fisher's test based on selected genes and also tests that average gene-level evidence across the category. Averaging and selection methods are compared empirically using Affymetrix data on expression in nasopharyngeal cancer tissue, and theoretically using a location model of differential expression. We find that each method has a domain of superiority in the state space of enrichment problems, and that both methods have benefits in practice. Our analysis also addresses two problems related to multiple-category inference, namely, that equally enriched categories are not detected with equal probability if they are of different sizes, and also that there is dependence among category statistics owing to shared genes. Random-set enrichment calculations do not require Monte Carlo for implementation. They are made available in the R package allez.


Jordan algebras over algebraic varieties

Abstract: We construct Jordan algebras over a locally ringed space using generalizations of the Tits process and the first Tits construction by Achhammer. Some general results on the structure of these algebras are obtained. Examples of Albert algebras over a Brauer-Severi variety with associated central simple algebra of degree 3 are given.


G-functions and multisum versus holonomic sequences

Abstract: The purpose of the paper is three-fold: (a) we prove that every sequence which is a multidimensional sum of a balanced hypergeometric term has an asymptotic expansion of Gevrey type-1 with rational exponents, (b) we construct a class of $G$-functions that come from enumerative combinatorics, and (c) we give a counterexample to a question of Zeilberger that asks whether holonomic sequences can be written as multisums of balanced hypergeometric terms. The proofs utilize the notion of a $G$-function, introduced by Siegel, and its analytic/arithmetic properties shown recently by Andr\'e.


Elevated soil lead: Statistical modeling and apportionment of contributions from lead-based paint and leaded gasoline

Abstract: While it is widely accepted that lead-based paint and leaded gasoline are primary sources of elevated concentrations of lead in residential soils, conclusions regarding their relative contributions are mixed and generally study specific. We develop a novel nonlinear regression for soil lead concentrations over time. It is argued that this methodology provides useful insights into the partitioning of the average soil lead concentration by source and time over large residential areas. The methodology is used to investigate soil lead concentrations from the 1987 Minnesota Lead Study and the 1990 National Lead Survey. Potential litigation issues are discussed briefly.


$G$-stable pieces and partial flag varieties

Abstract: We will use the combinatorics of the $G$-stable pieces to describe the closure relation of the partition of partial flag varieties in \cite[section 4]{L3}.


The size of a pond in 2D invasion percolation

Abstract: We consider invasion percolation on the square lattice. It has been proved by van den Berg, Peres, Sidoravicius and Vares, that the probability that the radius of a so-called pond is larger than n, differs at most a factor of order log n from the probability that in critical Bernoulli percolation the radius of an open cluster is larger than n. We show that these two probabilities are, in fact, of the same order. Moreover, we prove an analogous result for the volume of a pond.


Fibonacci-like sequences and shift spaces in symbolic dynamics

Abstract: We afford the problem of counting the blocks of a given length made with symbols drawn from an alphabet and relate this number to Fibonacci-like recurrent relations. The recurrence polynomia allows to calculate the limit ratio of two adjacent terms and this information is used to determine the topological entropy of a discrete numbers of associate shift spaces. We then describe a scheme to build a shift space with a pre-selected entropy.


Bernoulli free-boundary problems in strip-like domains and a property of permanent waves in water of finite depth

Abstract: We study weak solutions for a class of free boundary problems which includes as a special case the classical problem of traveling waves on water of finite depth. We show that such problems are equivalent to problems in fixed domains and study the regularity of their solutions. We also prove that in very general situations the free boundary is necessarily the graph of a function.


Harmonic contact metric structures, and submersions

Abstract: We study harmonic almost contact structures in the context of contact metric manifolds, and an analysis is carried out when such a manifold fibres over an almost Hermitian manifold, as exemplified by the Boothby-Wang fibration. Two types of almost contact metric warped products are also studied, relating their harmonicity to that of the almost Hermitian structure on the base or fibre.


A rate-independent model for the isothermal quasi-static evolution of shape-memory materials

Abstract: This note addresses a three-dimensional model for isothermal stress-induced transformation in shape-memory polycrystalline materials. We treat the problem within the framework of the energetic formulation of rate-independent processes and investigate existence and continuous dependence issues at both the constitutive relation and quasi-static evolution level. Moreover, we focus on time and space approximation as well as on regularization and parameter asymptotics.


Automorphisms of Galois Coverings of Generic $m$-Canonical Projections

Abstract: The automorphism group of the Galois covering induced by a pluri-canonical generic covering of a projective space is investigated. It is shown that by means of such coverings one obtains, in dimensions one and two, serieses of specific actions of the symmetric groups $S_d$ on curves and surfaces not deformable to an action of $S_d$ which is not the full automorphism group. As an application, new DIF $\ne$ DEF examples for $G$-varieties in complex and real geometry are given.


A sufficient criterion for homotopy cartesianess

Abstract: Suppose given a commutative quadrangle in a Verdier triangulated category such that there exists an induced isomorphism on the horizontally taken cones. Suppose that the endomorphism ring of the initial or the terminal corner object of this quadrangle satisfies a finiteness condition. Then this quadrangle is homotopy cartesian.


On the Gr\"obner complexity of matrices

Abstract: In this paper we show that if for an integer matrix A the universal Gr\"obner basis of the associated toric ideal \Ideal_A coincides with the Graver basis of A, then the Gr\"obner complexity u(A) and the Graver complexity g(A) of its higher Lawrence liftings agree, too. We conclude that for the matrices A_{3\times 3} and A_{3\times 4}, defining the 3\times 3 and 3\times 4 transportation problems, we have u(A_{3\times 3})=g(A_{3\times 3})=9 and u(A_{3\times 4})=g(A_{3\times 4})\geq 27. Moreover, we prove u(A_{a,b})=g(A_{a,b})=2(a+b)/\gcd(a,b) for positive integers a,b and A_{a,b}=(\begin{smallmatrix} 1 & 1 & 1 & 1 0 & a & b & a+b \end{smallmatrix}).


The computation of the classes of some tori in the Grothendieck ring of varieties

Abstract: We establish a formula for the classes of certain tori in the Grothendieck ring of varieties, in terms of its lambda-structure. More explicitly, we will see that if L* is the torus of invertible elements in the n-dimensional separable k-algebra L, then the class of L* can be expressed as an alternating sum of the images of the spectrum of L under the lambda-operations, multiplied by powers of the Lefschetz class. This formula is suggested from the cohomology of the torus, illustrating a heuristic method that can be used in other situations. To prove the formula will require some rather explicit calculations in the Grothendieck ring. To be able to make these we introduce a homomorphism from the Burnside ring of the absolute Galois group of k, to the Grothendieck ring of varieties over k. In the process we obtain some information about the structure of the subring generated by zero-dimensional varieties.


Cyclic odd degree base change lifting for unitary groups in three variables

Abstract: Let E/F be a quadratic number (resp. p-adic) field extension, and F' an odd degree cyclic field extension of F. We establish a base-change functorial lifting of automorphic (resp. admissible) representations from the unitary group U(3,E/F) associated with E/F to the unitary group U(3,F'E/F'). As a consequence, we classify the invariant packets of U(3,F'E/F'), namely those which contain (irreducible) automorphic (resp. admissible) representations which are invariant under the action of the Galois group Gal(F'E/E). To do this we use the trace formula technique, and well-known results on the base-change lifting from U(3,E/F) to GL(3,E) and on the base-change lifting for the general linear groups. We also determine the invariance of individual representations, using Howe correspondence. This work is the first study of base change into an algebraic group whose packets are not all singletons, and which does not satisfy the "strong multiplicity one theorem." Novel phenomena are encountered: e.g. there are invariant packets where not every irreducible automorphic (resp. admissible) member is Galois-invariant. We also obtain local twisted character identities with respect to this base-change lifting.


2-gerbes and 2-Tate spaces

Abstract: We construct a central extension of the group of automorphisms of a 2-Tate vector space viewed as a discrete 2-group. This is done using an action of this 2-group on a 2-gerbe of gerbel theories. This central extension is used to define central extensions of double loop groups.


Mirror symmetry and tropical geometry

Abstract: Using tropical geometry we propose a mirror construction for monomial degenerations of Calabi-Yau varieties in toric Fano varieties. The construction reproduces the mirror constructions by Batyrev for Calabi-Yau hypersurfaces and by Batyrev and Borisov for Calabi-Yau complete intersections. We apply the construction to Pfaffian examples and recover the mirror given by Rodland for the degree 14 Calabi-Yau threefold in PP^6 defined by the Pfaffians of a general linear 7x7 skew-symmetric matrix. We provide the necessary background knowledge entering into the tropical mirror construction such as toric geometry, Groebner bases, tropical geometry, Hilbert schemes and deformations. The tropical approach yields an algorithm which we illustrate in a series of explicit examples.


The largest component in a subcritical random graph with a power law degree distribution

Abstract: It is shown that in a subcritical random graph with given vertex degrees satisfying a power law degree distribution with exponent $\gamma>3$, the largest component is of order $n^{1/(\gamma-1)}$. More precisely, the order of the largest component is approximatively given by a simple constant times the largest vertex degree. These results are extended to several other random graph models with power law degree distributions. This proves a conjecture by Durrett.


Moments and distribution of the local times of a transient random walk on $\Z^d$

Abstract: Consider an arbitrary transient random walk on $\Z^d$ with $d\in\N$. Pick $\alpha\in[0,\infty)$ and let $L_n(\alpha)$ be the spatial sum of the $\alpha$-th power of the $n$-step local times of the walk. Hence, $L_n(0)$ is the range, $L_n(1)=n+1$, and for integers $\alpha$, $L_n(\alpha)$ is the number of the $\alpha$-fold self-intersections of the walk. We prove a strong law of large numbers for $L_n(\alpha)$ as $n\to\infty$. Furthermore, we identify the asymptotic law of the local time in a random site uniformly distributed over the range. These results complement and contrast analogous results for recurrent walks in two dimensions recently derived by \v{C}ern\'y \cite{Ce07}. Although these assertions are certainly known to experts, we could find no proof in the literature in this generality.


Freeness with amalgamation, limit theorems and S-transform in non-commutative probability spaces of type B

Abstract: The present material addresses several problems left open in the Trans. AMS paper " Non-crossing cumulants of type B" of P. Biane, F. Goodman and A. Nica. The main result is that a type B non-commutative probability space can be studied in the framework of freeness with amalgamation. This view allows easy ways of constructing a version of the S-transform as well as proving analogue results to Central Limit Theorem and Poisson Limit Theorem.


On Selfadjoint Subspace of One-Speed Boltzmann Operator

Abstract: The aim of the paper is to obtain a description of the selfadjoint subspace of the one-speed Boltzmann operator. It is proved that this subspace is nontrivial if the collision integral is polynomial and the multiplication coefficient has a lattice of gaps. A similar result is shown to hold for the 3-dimensional transport operator. Keywords: completely nonselfadjoint, Boltzmann operator, uncertainly principle.


A note on the Verlinde bundles on elliptic curves

Abstract: We study the splitting properties of the Verlinde bundles over elliptic curves. Our methods rely on the explicit description of the moduli space of semistable vector bundles on elliptic curves, and on the analysis of the symmetric powers of the Schrodinger representation of the Theta group.


Sofic groups and profinite topology on free groups

Abstract: We give a definition of weakly sofic groups (w-sofic groups). Our definition is rather natural extension of the definition of sofic groups where instead of Hamming metric on symmetric groups we use general bi-invariant metrics on finite groups. The existence of non w-sofic groups is equivalent to some conjecture about profinite topology on free groups.


Hidden Life of Riemann's Zeta Function 2. Electrons and Trains

Abstract: The Riemann Hypothesis can be reformulated as statements about the eigenvalues of certain matrices whose entries are defined in terms of the Taylor coefficients of the zeta function. These eigenvalues exhibit interesting visual patterns allowing one to state a number of conjectures. The Hankel matrices introduced here are obtained, by rearranging of columns, from Toeplitz matrices whose eigenvalues were considered in arXiv:0707.1983 . The present paper is a continuation of that publication.


Alexander Duality and Serre's Property $(S_i)$ for Square-free Monomial Ideals

Abstract: In this note, we study Serre's property $(S_i)$, and its relation to Alexander duality for monomial ideals in a polynomial ring over a field. We describe ideals that define the non-Cohen-Macaulay- and the non-$(S_i)$-loci of finitely generated modules over regular rings, and show that minimal prime ideals in these loci are homogeneous, in the graded case. We show that a square-free monomial ideal has property $(S_i)$ if and only if its Alexander dual has a linear resolution up to homological degree $i-1$. We prove that for square-free monomial ideals, having property $(S_2)$ is equivalent to being locally connected in codimension 1.


Chern classes of Deligne-Mumford stacks and their coarse moduli spaces

Abstract: Let $X$ be a complex projective algebraic variety with Gorenstein quotient singularities and $\X$ a smooth Deligne-Mumford stack having $X$ as its coarse moduli space. We show that the CSM class $c^{SM}(X)$ coincides with the pushforward to $X$ of the total Chern class $c(T_{I\X})$ of the inertia stack $I\X$. We also show that the stringy Chern class $c_{str}(X)$ of $X$, whenever is defined, coincides with the pushforward to $X$ of the total Chern class $c(T_{II\X})$ of the double inertia stack $II\X$. Some consequences concerning stringy/orbifold Hodge numbers are deduced.


Circular law for non-central random matrices

Abstract: Let $(X_{jk})_{j,k\geq 1}$ be an infinite array of i.i.d. complex random variables, with mean 0 and variance 1. Let $\la_{n,1},...,\la_{n,n}$ be the eigenvalues of $(\frac{1}{\sqrt{n}}X_{jk})_{1\leq j,k\leq n}$. The strong circular law theorem states that with probability one, the empirical spectral distribution $\frac{1}{n}(\de_{\la_{n,1}}+...+\de_{\la_{n,n}})$ converges weakly as $n\to\infty$ to the uniform law over the unit disc $\{z\in\dC;|z|\leq1\}$. In this short note, we provide an elementary argument that allows to add a deterministic matrix $M$ to $(X_{jk})_{1\leq j,k\leq n}$ provided that $\mathrm{Tr}(MM^*)=O(n^2)$ and $\mathrm{rank}(M)=O(n^\al)$ with $\al<1$. Conveniently, the argument is similar to the one used for the non-central version of Wigner's and Marchenko-Pastur theorems.


Steiner-Minkowski Polynomials of Convex Sets in High Dimension, and Limit Entire Functions

Abstract: For a convex set (K) of the (n)-dimensional Euclidean space, the Steiner-Minkowski polynomial (M_K(t)) is defined as the (n)-dimensional Euclidean volume of the neighborhood of the radius (t). Being defined for positive (t), the Steiner-Minkowski polynomials are considered for all complex (t). The renormalization procedure for Steiner polynomial is proposed. The renormalized Steiner-Minkowski polynomials corresponding to all possible solid convex sets in all dimensions form a normal family in the whole complex plane. For each of the four families of convex sets: the Euclidean balls, the cubes, the regular cross-polytopes and the regular symplexes of dimensions (n), the limiting entire functions, as (n) tends to infinity, are calculated explicitly.


The Ramsey number for a triple of large cycles

Abstract: We find the asymptotic value of the Ramsey number for a triple of long cycles, where the lengths of the cycles are large but may have different parity.


On the distribution of conjugacy classes between the cosets of a finite group in a cyclic extension

Abstract: Let G be a finite group and H a normal subgroup such that G/H is cyclic. Given a conjugacy class g^G of G we define its centralizing subgroup to be HC_G(g). Let K be such that H\le K\le G. We show that the G-conjugacy classes contained in K whose centralizing subgroup is K, are equally distributed between the cosets of H in K. The proof of this result is entirely elementary. As an application we find expressions for the number of conjugacy classes of K under its own action, in terms of quantities relating only to the action of G.


Theta Series Associated with the Weil-Schroedinger Representation

Abstract: The Weil representation discovered by Andre Weil plays an important role in the study of the tranformation properties of theta series. In this paper, we define the Weil-Schroedinger representation of the Jacobi group and prove that the theta series associated with the Weil-Schroedinger representation is a Jacobi form with respect to a suitable arithmetic subgroup of the Jacobi modular group.


Quasi-Anosov diffeomorphisms of 3-manifolds

Abstract: In 1969, Hirsch posed the following problem: given a diffeomorphism, and a compact invariant hyperbolic set, describe its topology and restricted dynamics. We solve the problem where the hyperbolic invariant set is a closed 3-manifold: if the manifold is orientable, then it is a connected sum of tori and handles; otherwise it is a connected sum of tori and handles quotiented by involutions. The dynamics of the diffeomorphisms restricted to these manifolds, called quasi-Anosov diffeomorphisms, is also classified: it is the connected sum of DA-diffeomorphisms, quotiented by commuting involutions.


A Fixed Point Conjecture

Abstract: Inverse limits, unlike direct limits, can in general be void, [1]. The existence of fixed points for arbitrary mappings $T : X \longrightarrow X$ is conjectured to be equivalent with the fact that related direct limits of all finite partitions of X are not void.


A wildland fire model with data assimilation

Abstract: A wildfire model is formulated based on balance equations for energy and fuel, where the fuel loss due to combustion corresponds to the fuel reaction rate. The resulting coupled partial differential equations have coefficients that can be approximated from prior measurements of wildfires. An ensemble Kalman filter technique with regularization is then used to assimilate temperatures measured at selected points into running wildfire simulations. The assimilation technique is able to modify the simulations to track the measurements correctly even if the simulations were started with an erroneous ignition location that is quite far away from the correct one.


New q-Euler numbers and polynomials associated with p-adic q-integrals

Abstract: In this paper we study q-Euler numbers and polynomials by using p-adic q-fermionic integrals on Z_p. The methods to study q-Euler numbers and polynomials in this paper are new.


Theorie ergodique des fractions rationnelles sur un corps ultrametrique

Abstract: We make the first steps towards an understanding of the ergodic properties of a rational map defined over a complete algebraically closed non-archimedean field. For such a rational map R, we construct a natural invariant probability measure m_R which reprensents the asymptotic distribution of preimages of non-exceptional point. We show that this measure is exponentially mixing, and satisfies the central limit theorem. We prove some general bounds on the metric entropy of m_R, and on the topological entropy of R. We finally prove that rational maps with vanishing topological entropy have potential good reduction.


The magic functions and automorphisms of a domain

Abstract: We introduce the notion of magic functions of a general domain in d-dimensional complex space and show that the set of magic functions of a given domain is an intrinsic complex-geometric object. We determine the set of magic functions of the symmetrised bidisc G, and thereby find all automorphisms of G and a formula for the Caratheodory distance on G.


Heegaard genus and Property 'tau' for hyperbolic 3-manifolds

Abstract: We show that any finitely generated non-elementary Kleinian group has a co-final family of finite index normal subgroups with respect to which it has Property $\tau$. As a consequence, any closed hyperbolic 3-manifold has a co-final family of finite index normal subgroups for which the infimal Heegaard gradient is positive.


Symmetry classes connected with the magnetic Heisenberg ring

Abstract: We define symmetry classes and commutation symmetries in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigate them by means of tools from the representation theory of symmetric groups S_N such as decompositions of ideals of the group ring C[S_N], idempotents of C[S_N], discrete Fourier transforms of S_N, Littlewood-Richardson products. In particular, we determine smallest symmetry classes and stability subgroups of both single eigenvectors v and subspaces U of eigenvectors of the Hamiltonian of the magnet. The determination of the smallest symmetry class for U bases on an algorithm which calculates explicitely a generating idempotent for a non-direct sum of right ideals of C[S_N]. Let U be a subspace of eigenvectors of a a fixed eigenvalue \mu of the Hamiltonian with weight (r_1,r_2). If one determines the smallest symmetry class for every v in U then one can observe jumps of the symmetry behaviour. For ''generic'' v all smallest symmetry classes have the same maximal dimension d and structure. But U can contain linear subspaces on which the dimension of the smallest symmetry class of v jumps to a value smaller than d. Then the stability subgroup of v can increase. We can calculate such jumps explicitely. In our investigations we use computer calculations by means of the Mathematica packages PERMS and HRing.


Local convergence for alternating and averaged nonconvex projections

Abstract: The idea of a finite collection of closed sets having "strongly regular intersection" at a given point is crucial in variational analysis. We show that this central theoretical tool also has striking algorithmic consequences. Specifically, we consider the case of two sets, one of which we assume to be suitably "regular" (special cases being convex sets, smooth manifolds, or feasible regions satisfying the Mangasarian-Fromovitz constraint qualification). We then prove that von Neumann's method of "alternating projections" converges locally to a point in the intersection, at a linear rate associated with a modulus of regularity. As a consequence, in the case of several arbitrary closed sets having strongly regular intersection at some point, the method of "averaged projections" converges locally at a linear rate to a point in the intersection. Inexact versions of both algorithms also converge linearly.


A new method for the estimation of variance matrix with prescribed zeros in nonlinear mixed effects models

Abstract: We propose a new method for the Maximum Likelihood Estimator (MLE) of nonlinear mixed effects models when the variance matrix of Gaussian random effects has a prescribed pattern of zeros (PPZ). The method consists in coupling the recently developed Iterative Conditional Fitting (ICF) algorithm with the Expectation Maximization (EM) algorithm. It provides positive definite estimates for any sample size, and does not rely on any structural assumption on the PPZ. It can be easily adapted to many versions of EM.


On the precision of the spectral profile

Abstract: We examine the spectral profile bound of Goel, Montenegro and Tetali for the uniform mixing time of continuous-time random walk in reversible settings. We find that it is precise up to a log log factor, and that this log log factor cannot be improved.


A remark on utility streams

Abstract: We consider arbitrarily long, but finite utility streams, and some appropriate axioms.


On Zariski's multiplicity conjecture

Abstract: We discuss some features of the so-called Zariski's multiplicity problem especially the application of the work of A'Campo on the zeta function of a monodromy of an isolated singularity of a complex hypersurface to the problem.


Pointed and copointed Hopf algebras as cocycle deformations

Abstract: We show that all finite dimensional pointed Hopf algebras with the same diagram in the classification scheme of Andruskiewitsch and Schneider are cocycle deformations of each other. This is done by giving first a suitable characterization of such Hopf algebras, which allows for the application of a result of Masuoka about Morita-Takeuchi equivalence and of Schauenburg about Hopf Galois extensions. The "infinitesimal" part of the deforming cocycle and of the deformation determine the deformed multiplication and can be described explicitly in terms of Hochschild cohomology. Applications to, and results for copointed Hopf algebras are also considered.


Introduction to shape stability for a storage model

Abstract: We consider a new idea for a storage model on n nodes, namely stability of shape. These nodes support K neighborhoods S_i \subset {1, ..., n} and items arrive at the S_i as independent Poisson streams with rates lambda_i, i=1, ...,K. Upon arrival at S_i an item is stored at node j \in S_i where j is determined by some policy. Under natural conditions on the lambda_i we exhibit simple local policies such that the multidimensional process describing the evolution of the number of items at each node is positive recurrent (stable) in shape.


Numerical analysis of a nonlocal parabolic problem resulting from thermistor problem

Abstract: We analyze the spatially semidiscrete piecewise linear finite element method for a nonlocal parabolic equation resulting from thermistor problem. Our approach is based on the properties of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite element method. We assume minimal regularity of the exact solution that yields optimal order error estimate. The full discrete backward Euler method and the Crank-Nicolson-Galerkin scheme are also considered. Finally, a simple algorithm for solving the fully discrete problem is proposed.


On the index of the Heegner subgroup of elliptic curves

Abstract: Let E be an elliptic curve of conductor N and rank one over Q. So there is a non-constant morphism X+0(N) --> E defined over Q, where X+0(N) = X0(N)/wN and wN is the Fricke involution of the modular curve X+0(N). Under this morphism the traces of the Heegner points of X+0(N) map to rational points on E. In this paper we study the index I of the subgroup generated by all these traces on E(Q). We propose and also discuss a conjecture that says that if N is prime and I > 1, then either the number of connected components of the real locus X+0(N)(R) is greater than 1 or (less likely) the order S of the Tate-Safarevich group is non-trivial. This conjecture is backed by computations performed on each E that satisfies the above hypothesis in the range N < 129999. This paper was prepared for the proceedings of the Conference on Algorithmic Number Theory, Turku, May 8-11, 2007. We tried to make the paper as self contained as possible.


Harder-Narasimhan Filtrations and K-Groups of an Elliptic Curve

Abstract: Let $X$ be an elliptic curve over an algebraically closed field. We prove that some exact sub-categories of the category of vector bundles over $X$, defined using Harder-Narasimhan filtrations, have the same K-groups as the whole category.


On the Vertices of Indecomposable Modules Over Dihedral 2-Groups

Abstract: Let $k$ be an algebraically closed field of characteristic 2. We compute the vertices of all indecomposable $kD_8$-modules for the dihedral group $D_8$ of order 8. We also give a conjectural formula of the induced module of a string module from $kT_0$ to $kG$ where $G$ is a dihedral group of order $\geq 8$ and where $T_0$ is a dihedral subgroup of index 2 of $G$. Some cases where we verified this formula are given.