Saturated subfields and invariants of finite groups

Abstract: Every subfield $\kk(\phi)$ of the field of rational functions $\kk(x_1,...,x_n)$ is contained in a unique maximal subfield of the form $\kk(\psi)$. The element $\psi$ is called generative for the element $\phi$. A subfield of $\kk(x_1,...,x_n)$ is called saturated if it contains a generative element of each its element. We study the saturation property for subfields of invariants $\kk(x_1,...,x_n)^G$, where $G$ is a finite group of automorphisms of the field $\kk(x_1,...,x_n)$.


Multiple solutions for the $p-$laplace operator with critical growth

Abstract: In this note we show the existence of at least three nontrivial solutions to the following quasilinear elliptic equation $-\Delta_p u = |u|^{p^*-2}u + \lambda f(x,u)$ in a smooth bounded domain $\Omega$ of $\R^N$ with homogeneous Dirichlet boundary conditions on $\partial\Omega$, where $p^*=Np/(N-p)$ is the critical Sobolev exponent and $\Delta_p u =div(|\nabla u|^{p-2}\nabla u)$ is the $p-$laplacian. The proof is based on variational arguments and the classical concentrated compactness method.


On slim double Lie groupoids

Abstract: We prove that every slim double Lie groupoid with proper core action is completely determined by a factorization of a certain canonically defined "diagonal" Lie groupoid.


On independent sets in purely atomic probability spaces with geometric distribution

Abstract: We are interested in constructing concrete independent events in purely atomic probability spaces with geometric distribution. Among other facts we prove that there are uncountable many sequences of independent events.


Algebraic characterization of the isometries of the hyperbolic 5-space

Abstract: Using the representation of the isometries as 2x2 invertible matrices over the division algebra $\H$ of quaternions, we give an algebraic characterization of the dynamical types of the orientation-preserving isometries of the hyperbolic 5-space. We also determine the conjugacy classes and the conjugacy classes of centralizers or the z-classes in $GL(2, \H)$.


The Annihilating-Ideal Graph of Commutative Rings I

Abstract: Let $R$ be a commutative ring with ${\Bbb{A}}(R)$ its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the {\it annihilating-ideal graph} of $R$, denoted by ${\Bbb{AG}}(R)$. It is the (undirected) graph with vertices ${\Bbb{A}}(R)^*:={\Bbb{A}}(R)\setminus\{(0)\}$, and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. First, we study some finiteness conditions of ${\Bbb{AG}}(R)$. For instance, it is shown that if $R$ is not a domain, then ${\Bbb{AG}}(R)$ has ACC (resp., DCC) on vertices if and only if $R$ is Noetherian (resp., Artinian). Moreover, the set of vertices of ${\Bbb{AG}}(R)$ and the set of nonzero proper ideals of $R$ have the same cardinality when $R$ is either an Artinian or a decomposable ring. This yields for a ring $R$, ${\Bbb{AG}}(R)$ has $n$ vertices $(n\geq 1)$ if and only if $R$ has only $n$ nonzero proper ideals. Next, we study the connectivity of ${\Bbb{AG}}(R)$. It is shown that ${\Bbb{AG}}(R)$ is a connected graph and $diam(\Bbb{AG})(R)\leq 3$ and if ${\Bbb{AG}}(R)$ contains a cycle, then $gr({\Bbb{AG}}(R))\leq 4$. Also, rings $R$ for which the graph ${\Bbb{AG}}(R)$ is complete or star, are characterized, as well as rings $R$ for which every vertex of ${\Bbb{AG}}(R)$ is a prime (or maximal) ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs.


The Annihilating-Ideal Graph of Commutative Rings II

Abstract: In this paper we continue our study of annihilating-ideal graph of commutative rings, that was introduced in Part I (see [5]). Let $R$ be a commutative ring with ${\Bbb{A}}(R)$ its set of ideals with nonzero annihilator and $Z(R)$ its set of zero divisors. The annihilating-ideal graph of $R$ is defined as the (undirected) graph ${\Bbb{AG}}(R)$ that its vertices are $\Bbb{A}(R)^* =\Bbb{A}(R)\hspace{-1mm}\setminus\{(0)\}$ in which for every distinct vertices $I$ and $J$, $I\hspace{-0.6mm}-\hspace{-1.7mm}-\hspace{-1.7mm}-\hspace{-0.5mm}J$ is an edge if and only if $IJ=(0)$. First, we study the diameter of ${\Bbb{AG}}(R)$. A complete characterization for the possible diameter is given exclusively in terms of the ideals of $R$ when either $R$ is a Noetherian ring or $Z(R)$ is not an ideal of $R$. Next, we study coloring of annihilating-ideal graphs. Among other results, we characterize when either $\chi({\Bbb{AG}}(R))\leq 2$ or $R$ is reduced and $\chi({\Bbb{AG}}(R))\leq \infty$. Also it is shown that for each reduced ring $R$, $\chi(\Bbb{AG}(R))= cl(\Bbb{AG}(R))$. Moreover, if $\chi(\Bbb{AG}(R))$ is finite, then $R$ has a finite number of minimal primes, and if $n$ is this number, then $\chi(\Bbb{AG}(R))= cl(\Bbb{AG}(R))= n$. Finally, we show that for a Noetherian ring $R$, $cl(\Bbb{AG}(R))$ is finite if and only if for every ideal $I$ of $R$ with $I^2=(0)$, $I$ has finite number of $R$-submodules.


Nonparametric goodness-of fit testing in quantum homodyne tomography with noisy data

Abstract: In the framework of quantum optics, we study the problem of goodness-of-fit testing in a severely ill-posed inverse problem. A novel testing procedure is introduced and its rates of convergence are investigated under various smoothness assumptions. The procedure is derived from a projection-type estimator, where the projection is done in $\mathbb{L}_2$ distance on some suitably chosen pattern functions. The proposed methodology is illustrated with simulated data sets.


Uniqueness of pairings in Hopf-cyclic cohomology

Abstract: We show that all pairings defined in the literature extending Connes-Moscovici characteristic map in Hopf-cyclic cohomology are isomorphic as natural transformations of derived double functors.


The Stadium Theorem

Abstract: A proof of a curious planar embedding theorem.


Maximizing orbits for higher dimensional convex billiards

Abstract: The main result of this paper is, that for convex billiards in higher dimensions, in contrast with 2D case, for every point on the boundary and for every $n$ there always exist billiard trajectories developing conjugate points at the $n$-th collision with the boundary. We shall explain that this is a consequence of the following variational property of the billiard orbits in higher dimension. If a segment of an orbit is locally maximizing, then it can not pass too close to the boundary. This fact follows from the second variation formula for the Length functional. It turns out that this formula behaves differently with respect to "longitudinal" and "transversal" variations.


Baric structures on triangulated categories and coherent sheaves

Abstract: We introduce the notion of a "baric structure" on a triangulated category, as an abstraction of S. Morel's weight truncation formalism for mixed l-adic sheaves. We study these structures on the derived category D_G(X) of G-equivariant coherent sheaves on a G-scheme X. Our main result shows how to endow this derived category with a family of nontrivial baric structures when G acts on X with finitely many orbits. We also describe a general construction for producing a new t-structure on a triangulated category equipped with given t- and baric structures, and we prove that the staggered t-structures on D_G(X) introduced by the first author arise in this way.


Purity and decomposition theorems for staggered sheaves

Abstract: Two major results in the theory of l-adic mixed constructible sheaves are the purity theorem (every simple perverse sheaf is pure) and the decomposition theorem (every pure object in the derived category is a direct sum of shifts of simple perverse sheaves). In this paper, we prove analogues of these results for coherent sheaves. Specificially, we work with staggered sheaves, which form the heart of a certain t-structure on the derived category of equivariant coherent sheaves. We prove, under some reasonable hypotheses, that every simple staggered sheaf is pure, and that every pure complex of coherent sheaves is a direct sum of shifts of simple staggered sheaves.


Balanced HKT metrics and strong HKT metrics on hypercomplex manifolds

Abstract: A manifold (M,I,J,K) is called hypercomplex if I,J,K are complex structures satisfying quaternionic relations. A quaternionic Hermitian metric is called HKT (hyperkaehler with torsion) if $Id\omega_I = Jd \omega_J=Kd\omega_K$, where $\omega_I,\omega_J, \omega_K$ are Hermitian forms associated with I, J, K. A Hermitian metric $\omega$ on a complex manifold is called balanced if $d^*\omega=0$. We show that balanced HKT metrics are precisely the quaternionic Calabi-Yau metrics defined in terms of the quaternionic Monge-Ampere equation. In particular, a balanced HKT-metric is unique in its cohomology class, and it always exists if the quaternionic Calabi-Yau theorem is true. We investigate the cohomological properties of strong HKT metrics (the quaternionic Hermitian metrics, satisfying, in addition to the HKT condition, the relation $dd^c \omega=0$), and show that the space of strong HKT metrics is finite-dimensional. Using Howe's duality for representations of Sp(n), we prove a hyperkaehler version of Hodge-Riemann bilinear relations. We use it to show that a manifold admitting a balanced HKT-metric never admits a strong HKT-metric, if $\dim_\R M \geq 12$.


Open book decompositions and stable Hamiltonian structures

Abstract: We show that every open book decomposition of a contact 3-manifold can be represented (up to isotopy) by a smooth R-invariant family of pseudoholomorphic curves on its symplectization with respect to a suitable stable Hamiltonian structure. In the planar case, this family survives small perturbations, and thus gives a concrete construction of a stable finite energy foliation that has been used in various applications to planar contact manifolds, including the Weinstein conjecture and equivalence of strong and Stein fillability.


Entire curves, Integral sets and Principal bundles

Abstract: We compare the behaviour of entire curves and integral sets, in particular in relation to locally trivial fiber bundles, algebraic groups and finite ramified covers over semi-abelian varieties.


Chen--Ruan cohomology of some moduli spaces

Abstract: Let X be a compact connected Riemann surface of genus at least two. We compute the Chen--Ruan cohomology ring of the moduli space of stable PSL(2, C)--bundles of nontrivial second Stiefel--Whitney class over X.


Concentration of 1-Lipschitz maps into an infinite dimensional $\ell^p$-ball with $\ell^q$-distance function

Abstract: In this paper, we study the L\'{e}vy-Milman concentration phenomenon of 1-Lipschitz maps into infinite dimensional metric spaces. Our main theorem asserts that the concentration to an infinite dimensional $\ell^p$-ball with the $\ell^q$-distance function for $1\leq p<q\leq +\infty$ is equivalent to the concentration to the real line.


Moduli of Continuity of Quasiregular Mappings

Abstract: This thesis consists of Chapters 1 and 2. The main results are contained in the two preprints and two published papers, listed below. Chapter 1 deals with conformal invariants in the euclidean space Rn; n >= 2; and their interrelation. In particular, conformally invariant metrics and balls of the respective metric spaces are studied. Another theme in Chapter 1 is the study of quasiconformal maps with identity boundary values in two diferent cases, the unit ball and the whole space minus two points. These results are based on the two preprints: R. Klen, V. Manojlovic and M. Vuorinen: Distortion of two point normalized quasiconformal mappings, arXiv:0808.1219[math.CV], 13 pp., V. Manojlovic and M. Vuorinen: On quasiconformal maps with identity boundary values, arXiv:0807.4418[math.CV], 16 pp. Chapter 2 deals with harmonic quasiregular maps. Topics studied are: Preservation of modulus of continuity, in particular Lipschitz continuity, from the boundary to the interior of domain in case of harmonic quasiregular maps and quasiisometry property of harmonic quasiconformal maps. Chapter 2 is based mainly on the two published papers: M. Arsenovic, V. Kojic and M. Mateljevic: On Lipschitz continuity of harmonic quasiregular maps on the unit ball in Rn., Ann. Acad. Sci. Fenn. Math. 33 (2008), no. 1, 315-318. V. Kojic and M. Pavlovic: Subharmonicity of jfjp for quasiregular harmonic functions, with applications, J. Math. Anal. Appl. 342 (2008) 742-746


Maximal Inequalities in Bilateral Grand Lebesque Spaces Over Unbounded Measure

Abstract: In this paper non-asymptotic exact rearrangement invariant norm estimates are derived for the maximum distribution of the family elements of some rearrangement invariant (r.i.) space over unbounded measure in the entropy terms and in the terms of generic chaining. We consider some applications in the martingale theory and in the theory of Fourier series.


Support of Borelian Measures in Separable Banach Spaces

Abstract: We prove in this article that every Borelian measure, for example, the distribution of a random variable, in separable Banach space has a support which is compact embedded Banach subspace; and prove that if the norm of the random variable belongs to some exponential Orlicz space, then the new subspace can be choose such that the norm of this variable in the new space also belongs to other exponential Orlicz space.


Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon Nikodym property

Abstract: We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincar\'e inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure spaces, in terms of directional derivatives in the direction of tangent vectors to suitable rectifiable curves.


Champs alg\'ebriques et foncteur de Picard

Abstract: This text is my thesis, defended in June 2007, in the status it was at this time. The most important results are contained in the article "Foncteur de Picard d'un champ alg\'ebrique" to appear in "Mathematische Annalen" (see the preprint arXiv:0711.4545). In the article, some results have been added, and some previous results have been strengthened. However, the proofs of the results contained in the appendix (concerning the smooth-\'etale cohomology on an algebraic stack) have been removed. The thesis is only put on the ArXiv to provide a more lasting reference than my webpage for these proofs.


Ratliff-Rush Filtration, regularity and depth of Higher Associated graded modules: Part II

Abstract: Let $(A,\m)$ be a Noetherian local ring, let $M$ be a finitely generated \CM $A$-module of dimension $r \geq 2$ and let $I$ be an ideal of definition for $M$. Set $L^I(M) = \bigoplus_{n\geq 0}M/I^{n+1}M$. In part one of this paper we showed that $L^I(M)$ is a module over $\R$, the Rees algebra of $I$ and we gave many applications of $L^I(M)$ to study the associated graded module, $G_I(M)$. In this paper we give many further applications of our technique; most notable is a reformulation of a classical result due to Narita in terms of the Ratliff-Rush filtration. This reformulation can be extended to all dimensions $\geq 2$.


Positivity of relative canonical bundles for families of canonically polarized manifolds

Abstract: Given an effectively parameterized family of canonically polarized manifolds the Kaehler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle. We use a global elliptic equation to show that this metric is strictly positive. For degenerating families we obtain a singular hermitian metric. Main application is an analytic proof of the quasi-projectivity of the moduli space of canonically polarized manifolds. Further applications in arXiv:1002.4858v2.


Coupled vortex equations and Moduli: Deformation theoretic Approach and Kaehler Geometry

Abstract: We investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact Kaehler manifold X. These solutions are known to be related to polystable triples via a Kobayashi-Hitchin type correspondence. Using a characterization of infinitesimal deformations in terms of the cohomology of a certain elliptic double complex, we construct a Hermitian structure on these moduli spaces. This Hermitian structure is proved to be Kaehler. The proof involves establishing a fiber integral formula for the Hermitian form. We compute the curvature tensor of this Kaehler form. When X is a Riemann surface, the holomorphic bisectional curvature turns out to be semi--positive. It is shown that in the case where X is a smooth complex projective variety, the Kaehler form is the Chern form of a Quillen metric on a certain determinant line bundle.


On behavior of solvable ideals of Lie algebras under outer derivations

Abstract: Let $L$ be a finite dimensional Lie algebra over a field $F$. It is well known that the solvable radical $S(L)$ of the algebra $L$ is a characteristic ideal of $L$ if $\char F=0$ and there are counterexamples to this statement in case $\char F=p>0$. We prove that the sum $S(L)$ of all solvable ideals of a Lie algebra $L$ (not necessarily finite dimensional) is a characteristic ideal of $L$ in the following cases: 1) $\char F=0;$ 2) $S(L)$ is solvable and its derived length is less than $\log_{2}p.$ Some estimations (in characteristic 0) for the derived length of ideals $I+D(I)+... +D^{k}(I)$ are obtained where $I$ is a solvable ideal of $L$ and $D\in Der(L).$


Proof of a dynamical Bogomolov conjecture for lines under polynomial actions

Abstract: We prove a dynamical version of the Bogomolov conjecture in the special case of lines in affine space A^m under the action of a map (f_1,...,f_m) where each f_i is a polynomial in Q-bar[X] of the same degree.


The dynamical Mordell-Lang problem for etale maps

Abstract: We prove a dynamical version of the Mordell-Lang conjecture for etale endomorphisms of quasiprojective varieties. We use p-adic methods inspired by the work of Skolem, Mahler, and Lech, combined with methods from algebraic geometry. As special cases of our result we obtain a new proof of the classical Mordell-Lang conjecture for cyclic subgroups of a semiabelian variety, and we also answer positively a question of Keeler/Rogalski/Stafford for critically dense sequences of closed points of a Noetherian integral scheme.


Homological interpretation of extensions and biextensions of 1-motives

Abstract: Let k be a separably closed field. Let K_i=[A_i \to B_i] (for i=1,2,3) be three 1-motives defined over k. We define the geometrical notions of extension of K_1 by K_3 and of biextension of (K_1,K_2) by K_3. We then compute the homological interpretation of these new geometrical notions: namely, the group Biext^0(K_1,K_2;K_3) of automorphisms of any biextension of (K_1,K_2) by K_3 is canonically isomorphic to the cohomology group Ext^0(K_1 \otimes K_2,K_3), and the group Biext^1(K_1,K_2;K_3) of isomorphism classes of biextensions of (K_1,K_2) by K_3 is canonically isomorphic to the cohomology group Ext^1(K_1 \otimes K_2,K_3).


Surfaces with p_g=q=1, K^2=8 and nonbirtional bicanonical map

Abstract: We prove that if the bicanonical map of a minimal surface of general type S with p_{g}=q=1 and K^2=8 is non birational, then it is a double cover onto a rational surface. An application of this theorem is the complete classification of minimal surfaces of general type with p_{g}=q=1, K^2=8 and nonbirational bicanonical map.


A note on $l^p$ norms of weighted mean matrices

Abstract: We present some results concerning the $l^p$ norms of weighted mean matrices. These results can be regarded as analogues to a result of Bennett concerning weighted Carleman's inequalities.


On homogenization of space-time dependent and degenerate random flows II

Abstract: We study the long time behavior (homogenization) of a diffusion in random medium with time and space dependent coefficients. The diffusion coefficient may degenerate. In Stochastic Process. Appl. (2007) (to appear), an invariance principle is proved for the critical rescaling of the diffusion. Here, we generalize this approach to diffusions whose space-time scaling differs from the critical one.


Siciak-Zahariuta extremal functions, analytic discs and polynomial hulls

Abstract: We prove two disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space. We use these formulas to characterize the polynomial hull of an arbitrary compact subset of complex affine space in terms of analytic discs. Similar results in previous work of ours required the subsets to be connected.


Ergodic actions of compact quantum groups from solutions of the conjugate equations

Abstract: We use a tensor C*-category with conjugates and two quasitensor functors into the category of Hilbert spaces to define a *-algebra depending functorially on this data. If one of them is tensorial, we can complete in the maximal C*-norm. A particular case of this construction allows us to begin with solutions of the conjugate equations and associate ergodic actions of quantum groups on the C*-algebra in question. The quantum groups involved are A_u(Q) and B_u(Q).


The Bi-Musquash Conjecture

Abstract: D.R. Woodall's definition of a 'musquash' is extended to that of a 'bi-musquash' and the existence/uniqueness of these is discussed.


Derived categories of cubic fourfolds

Abstract: We discuss the structure of the derived category of coherent sheaves on cubic fourfolds of three types: Pfaffian cubics, cubics containing a plane and singular cubics, and discuss its relation to the rationality of these cubics.


Seidel-Smith Cohomology for Tangles

Abstract: We generalize the "symplectic Khovanov cohomology" of Seidel and Smith to tangles using the notion of symplectic valued topological field theory introduced by Wehrheim and Woodward.


A bracket polynomial for graphs

Abstract: A knot diagram has an associated looped interlacement graph, obtained from the intersection graph of the Gauss diagram by attaching loops to the vertices that correspond to negative crossings. This construction suggests an extension of the Kauffman bracket to an invariant of looped graphs, and an extension of Reidemeister equivalence to an equivalence relation on looped graphs. The graph bracket polynomial can be defined recursively using the same pivot and local complementation operations used to define the interlace polynomial, and it gives rise to a graph Jones polynomial that is invariant under the graph Reidemeister moves.


On a doubly nonlinear diffusion model of chemotaxis with prevention of overcrowding

Abstract: This paper addresses the existence and regularity of weak solutions for a fully parabolic model of chemotaxis, with prevention of overcrowding, that degenerates in a two-sided fashion, including an extra nonlinearity represented by a $p$-Laplacian diffusion term. To prove the existence of weak solutions, a Schauder fixed-point argument is applied to a regularized problem and the compactness method is used to pass to the limit. The local H\"older regularity of weak solutions is established using the method of intrinsic scaling. The results are a contribution to showing, qualitatively, to what extent the properties of the classical Keller-Segel chemotaxis models are preserved in a more general setting. Some numerical examples illustrate the model.


LRSA: A new computational method for analyzing time course microarray data

Abstract: Motivation: Time course data obtained from biological samples subject to specific treatments can be very useful for revealing complex and novel biological phenomena. Although an increasing number of time course microarray datasets becomes available, most of them contain few biological replicates and time points. So far there are few computational methods that can effectively reveal differentially expressed genes and their patterns in such data. Results: We have proposed a new two-step nonparametric statistical procedure, LRSA, to reveal differentially expressed genes and their expression trends in temporal microarray data. We have also employed external controls as a surrogate to estimate false discovery rates and thus to guide the discovery of differentially expressed genes. Our results showed that LRSA reveals substantially more differentially expressed genes and have much lower than two other methods, STEM and ANOVA, in both real data and the simulated data. Our computational results are confirmed using real-time PCRs. Contact: wuw2@upmc.edu


Call option prices based on Bessel processes

Abstract: As a complement to some recent work by Pal and Protter, "Strict local martingales, bubbles, and no early exercise", we show that the call option prices associated with the Bessel strict local martingales are integrable over time, and we discuss the probability densities obtained thus.


The SL(2)-type and Base Change

Abstract: The SL(2)-type of any smooth, irreducible and unitarizable representation of GL(n) over a p-adic field was defined by Venkatesh. We provide a natural way to extend the definition to all smooth and irreducible representations. For unitarizable representations we show that the SL(2)-type of a representation is preserved under base change with respect to any finite extension. The Klyachko model of a smooth, irreducible and unitarizable representation \pi of GL(n) depends only on the SL(2)-type of \pi. As a consequence we observe that the Klyachko model of \pi and of its base-change are of the same type.


Complete Constant Mean Curvature surfaces and Bernstein type Theorems in $\mathbb{M}^2\times \mathbb{R}$

Abstract: In this paper we study constant mean curvature surfaces $\Sigma$ in a product space, $\mathbb{M}^2\times \mathbb{R}$, where $\mathbb{M}^2$ is a complete Riemannian manifold. We assume the angle function $\nu = \meta{N}{\partial_t}$ does not change sign on $\Sigma$. We classify these surfaces according to the infimum $c(\Sigma)$ of the Gaussian curvature of the projection of $\Sigma$. When $H \neq 0$ and $c(\Sigma)\geq 0$, then $\Sigma $ is a cylinder over a complete curve with curvature 2H. If H=0 and $c(\Sigma) \geq 0$, then $\Sigma$ must be a vertical plane or $\Sigma$ is a slice $\mathbb{M}^2 \times {t}$, or $\mathbb{M}^2 \equiv \mathbb{R}^2$ with the flat metric and $\Sigma$ is a tilted plane (after possibly passing to a covering space). When $c(\Sigma)<0$ and $H>\sqrt{-c(\Sigma)} /2$, then $\Sigma$ is a vertical cylinder over a complete curve of $\mathbb{M}^2$ of constant geodesic curvature $2H$. This result is optimal. We also prove a non-existence result concerning complete multi-graphs in $\mathbb{M}^2\times \mathbb{R}$, when $c(\mathbb{M}^2)<0$.


On the Cayley semigroup of a finite aperiodic semigroup

Abstract: Let $S$ be a finite semigroup. In this paper we introduce the functions $\phi_s:S^* \to S^*$, first defined by Rhodes, given by $\phi_s([a_1,a_2 ,...,a_n]) = [sa_1,sa_1a_2,..., sa_1a_2 ... a_n]$. We show that if $S$ is a finite aperiodic semigroup, then the semigroup generated by the functions $\{\phi_s\}_{s \in S}$ is finite and aperiodic.


A New Invariant Metric and Applications

Abstract: We construct a new invariant metric for a compact subgroup of the automorphism group of a domain in complex space. Applications are provided.


The Range of a Class of Classifiable Separable Simple Amenable C*-Algebras

Abstract: We study the range of a classifiable class ${\cal A}$ of unital separable simple amenable $C^*$-algebras which satisfy the Universal Coefficient Theorem. The class ${\cal A}$ contains all unital simple AH-algebras. We show that all unital simple inductive limits of dimension drop circle $C^*$-algebras are also in the class. This unifies some of the previous known classification results for unital simple amenable $C^*$-algebras. We also show that there are many other $C^*$-algebras in the class. We prove that, for any partially ordered, simple weakly unperforated rationally Riesz group $G_0$ with order unit $u,$ any countable abelian group $G_1,$ any metrizable Choquet simplex $S,$ and any surjective affine continuous map $r: S\to S_u(G_0)$ (where $S_u(G_0)$ is the state space of $G_0$) which preserves extremal points, there exists one and only one (up to isomorphism) unital separable simple amenable $C^*$-algebra $A$ in the classifiable class ${\cal A}$ such that $$ ((K_0(A), K_0(A)_+, [1_A]), K_1(A), T(A), \lambda_A)=((G_0, (G_0)_+, u), G_1,S, r).


The base change fundamental lemma for central elements in parahoric Hecke algebras

Abstract: Clozel and Labesse proved the base change fundamental lemma for spherical Hecke algebras attached to an unramified group over a p-adic field. This paper proves an analogous fundamental lemma for centers of parahoric Hecke algebras attached to the same class of groups. This provides an ingredient needed for the author's program to study Shimura varieties with parahoric level structure at p.


Combinatorial bases for multilinear parts of free algebras with double compatible brackets

Abstract: Let X be an ordered alphabet. Lie_2(n) (and P_2(n) respectively) are the multilinear parts of the free Lie algebra (and the free Poisson algebra respectively) on X with a pair of compatible Lie brackets. In this paper, we prove the dimension formulas for these two algebras conjectured by B. Feigin by constructing bases for Lie_2(n) (and P_2(n)) from combinatorial objects. We also define a complementary space Eil_2(n) to Lie_2(n), give a pairing between Lie_2(n) and Eil_2(n), and show that the pairing is perfect.


Riesz basis for strongly continuous groups

Abstract: Given a Hilbert space and the generator of a strongly continuous group on this Hilbert space. If the eigenvalues of the generator have a uniform gap, and if the span of the corresponding eigenvectors is dense, then these eigenvectors form a Riesz basis (or unconditional basis) of the Hilbert space. Furthermore, we show that none of the conditions can be weakened. However, if the eigenvalues (counted with multiplicity) can be grouped into subsets of at most $K$ elements, and the distance between the groups is (uniformly) bounded away from zero, then the spectral projections associated to the groups form a Riesz family. This implies that if in every range of the spectral projection we construct an orthonormal basis, then the union of these bases is a Riesz basis in the Hilbert space.


Tropical enumerative invariants of F_0 and F_2

Abstract: There is an equation relating numbers of curves on F_0 satisfying incidence conditions and numbers of curves on F_2 satisfying incidence conditions. The purpose of this paper is to give a tropical proof of this equation in the case of rational curves. We use induction on the degree and two Kontsevich-type formulas for curves on F_0 and on F_2. The formula for F_2 was not known before and is proved using tropical geometry.


Scattering for small energy solutions of NLS with periodic potential in 1D

Abstract: We prove scattering for small solutions to of nonlinear Schroedinger equations in 1D with a space periodic potential


A central limit theorem for the rescaled L\'evy area of two-dimensional fractional Brownian motion with Hurst index $H<1/4$

Abstract: Let $B=(B^{(1)},B^{(2)})$ be a two-dimensional fractional Brownian motion with Hurst index $\alpha\in (0,1/4)$. Using an analytic approximation $B(\eta)$ of $B$ introduced in \cite{Unt08}, we prove that the rescaled L\'evy area process $(s,t)\to \eta^{\half(1-4\alpha)}\int_s^t dB_{t_1}^{(1)}(\eta) \int_s^{t_1} dB_{t_2}^{(2)}(\eta)$ converges in law to $W_t-W_s$ where $W$ is a Brownian motion independent from $B$. The method relies on a very general scheme of analysis of singularities of analytic functions, applied to the moments of finite-dimensional distributions of the L\'evy area.


On sequential Monte Carlo, partial rejection control and approximate Bayesian computation

Abstract: We present a sequential Monte Carlo sampler variant of the partial rejection control algorithm, and show that this variant can be considered as a sequential Monte Carlo sampler with a modified mutation kernel. We prove that the new sampler can reduce the variance of the incremental importance weights when compared with standard sequential Monte Carlo samplers. We provide a study of theoretical properties of the new algorithm, and make connections with some existing algorithms. Finally, the sampler is adapted for application under the challenging "likelihood free," approximate Bayesian computation modelling framework, where we demonstrate superior performance over existing likelihood-free samplers.


Generalized mean curvature flow in Carnot groups

Abstract: In this paper we study the generalized mean curvature flow of sets in the sub-Riemannian geometry of Carnot groups. We extend to our context the level sets method and the weak (viscosity) solutions introduced in the Euclidean setting by Evans-Spruck and Chen-Giga-Goto. We establish two special cases of the comparison principle, existence, uniqueness and basic geometric properties of the flow.


Discreteness Criteria and the Hyperbolic Geometry of Palindroms

Abstract: We consider non-elementary representations of two generator free groups in $PSL(2,\mathbb{C})$, not necessarily discrete or free, $G = < A, B >$. A word in $A$ and $B$, $W(A,B)$, is a palindrome if it reads the same forwards and backwards. A word in a free group is {\sl primitive} if it is part of a minimal generating set. Primitive elements of the free group on two generators can be identified with the positive rational numbers. We study the geometry of palindromes and the action of $G$ in $\HH^3$ whether or not $G$ is discrete. We show that there is a {\sl core geodesic} $\L$ in the convex hull of the limit set of $G$ and use it to prove three results: the first is that there are well defined maps from the non-negative rationals and from the primitive elements to $\L$; the second is that $G$ is geometrically finite if and only if the axis of every non-parabolic palindromic word in $G$ intersects $\L$ in a compact interval; the third is a description of the relation of the pleating locus of the convex hull boundary to the core geodesic and to palindromic elements.


Periodic orbits of linear endomorphisms on the 2-torus and its lattices

Abstract: Counting periodic orbits of endomorphisms on the 2-torus is considered, with special focus on the relation between global and local aspects and between the dynamical zeta function on the torus and its analogue on finite lattices. The situation on the lattices, up to local conjugacy, is completely determined by the determinant, the trace and a third invariant of the matrix defining the toral endomorphism.


K-processes, scaling limit and aging for the trap model in the complete graph

Abstract: We study K-processes, which are Markov processes in a denumerable state space, all of whose elements are stable, with the exception of a single state, starting from which the process enters finite sets of stable states with uniform distribution. We show how these processes arise, in a particular instance, as scaling limits of the trap model in the complete graph, and subsequently derive aging results for those models in this context.


Tail asymptotics for a random sign Lindley recursion

Abstract: We investigate the tail behaviour of the steady state distribution of a stochastic recursion that generalises Lindley's recursion. This recursion arises in queuing systems with dependent interarrival and service times, and includes alternating service systems and carousel storage systems as special cases. We obtain precise tail asymptotics in three qualitatively different cases, and compare these with existing results for Lindley's recursion and for alternating service systems.


Differential systems with Fuchsian linear part: correction and linearization, normal forms and multiple orthogonal polynomials

Abstract: Differential systems with a Fuchsian linear part are studied in regions including all the singularities in the complex plane of these equations. Such systems are not necessarily analytically equivalent to their linear part (they are not linearizable) and obstructions are found as a unique nonlinear correction after which the system becomes formally linearizable. More generally, normal forms are found. The corrections and the normal forms are found constructively. Expansions in multiple orthogonal polynomials and their generalization to matrix-valued polynomials are instrumental to these constructions.


A Reconstruction Algorithm for Photoacoustic Imaging based on the Nonuniform FFT

Abstract: Fourier reconstruction algorithms significantly outperform conventional back-projection algorithms in terms of computation time. In photoacoustic imaging, these methods require interpolation in the Fourier space domain, which creates artifacts in reconstructed images. We propose a novel reconstruction algorithm that applies the one-dimensional nonuniform fast Fourier transform to photoacoustic imaging. It is shown theoretically and numerically that our algorithm avoids artifacts while preserving the computational effectiveness of Fourier reconstruction.


A Chevalley's theorem in class C^r

Abstract: Let W be a finite reflection group acting orthogonally on R^n, P be the Chevalley polynomial mapping determined by an integrity basis of the algebra of W-invariant polynomials, and h be the highest degree of the coordinate polynomials in P.There exists a linear mapping from (C^r(R^n))^W to C^[r/h](R^n), f\mapsto F such that f=F \circ P, continuous for the natural Fr\'echet topologies. A general counterexample shows that this result is the best possible. The proof uses techniques of division by linear forms and a study of compensation phenomenons. An extension to P^{-1}(R^n) of invariant formally holomorphic regular fields is needed.


Edge percolation on a random regular graph of low degree

Abstract: Consider a uniformly random regular graph of a fixed degree $d\ge3$, with $n$ vertices. Suppose that each edge is open (closed), with probability $p(q=1-p)$, respectively. In 2004 Alon, Benjamini and Stacey proved that $p^*=(d-1)^{-1}$ is the threshold probability for emergence of a giant component in the subgraph formed by the open edges. In this paper we show that the transition window around $p^*$ has width roughly of order $n^{-1/3}$. More precisely, suppose that $p=p(n)$ is such that $\omega:=n^{1/3}|p-p^*|\to\infty$. If $p<p^*$, then with high probability (whp) the largest component has $O((p-p^*)^{-2}\log n)$ vertices. If $p>p^*$, and $\log\omega\gg\log\log n$, then whp the largest component has about $n(1-(p\pi+q)^d)\asymp n(p-p^*)$ vertices, and the second largest component is of size $(p-p^*)^{-2}(\log n)^{1+o(1)}$, at most, where $\pi=(p\pi+q)^{d-1},\pi\in(0,1)$. If $\omega$ is merely polylogarithmic in $n$, then whp the largest component contains $n^{2/3+o(1)}$ vertices.


Topologie sur l'ensemble des parties positives d'un r\'eseau

Abstract: We define a notion of {\it positive part} of a lattice $\Lambda$ and we endow the set of such positive parts with a topology. We then study some properties of this topology, by comparing it with the one of $V^*/\RM_{> 0}$, where $V^*$ is the dual vector space of $\RM \otimes_\ZM \Lambda$.


The universal Cannon--Thurston maps and the boundary of the curve complex

Abstract: In genus two and higher, the fundamental group of a closed surface acts naturally on the curve complex of the surface with one puncture. Combining ideas from previous work of Kent--Leininger--Schleimer and Mitra, we construct a universal Cannon--Thurston map from a subset of the circle at infinity for the closed surface group onto the boundary of the curve complex of the once-punctured surface. Using the techniques we have developed, we also show that the boundary of this curve complex is locally path-connected.


Semicontinuity properties of Kazhdan-Lusztig cells

Abstract: Computations in small Coxeter groups or dihedral groups suggest that the partition into Kazhdan-Lusztig cells with unequal parameters should obey to some semicontinuity phenomenon (as the parameters vary). The aim of this paper is to provide a rigorous theoretical background for supporting this intuition that will allow to state several precise conjectures.


The universal Kummer congruences

Abstract: Let $p$ be a prime. In this paper, we present a detailed $p$-adic analysis to factorials and double factorials and their congruences. We give good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number ${{\hat B_n}\over n}$ when $n$ is divisible by $p-1$. Using these we then establish the universal Kummer congruences modulo powers of a prime $p$ for the divided universal Bernoulli numbers ${{\hat B_n}\over n}$ when $n$ is divisible by $p-1$.


Applications of computational invariant theory to Kobayashi hyperbolicity and to Green-Griffiths algebraic degeneracy

Abstract: A major unsolved problem (according to Demailly 1997) towards the Kobayashi hyperbolicity conjecture in optimal degree is to understand jet differentials of germs of holomorphic discs that are invariant under any reparametrization of the source. The underlying group action is not reductive, but we provide a complete algorithm to generate all invariants, in arbitrary dimension n and for jets of arbitrary order k. Two main new situations are studied in great details. For jets of order 4 in dimension 4, we establish that the algebra of Demailly-Semple invariants is generated by 2835 polynomials, while the algebra of bi-invariants is generated by 16 mutually independent polynomials sharing 41 groebnerized syzygies. Nonconstant entire holomorphic curves valued in an algebraic 3-fold (resp. 4-fold) X^3 in P^4 (C) (resp. X^4 in P^5(C)) of degree d satisfy global differential equations as soon as d > 71 (resp. d > 258). A useful asymptotic formula for the Euler-Poincare characteristic of Schur bundles in terms of Giambelli's determinants is derived. For jets of order 5 in dimension 2, we establish that the algebra of Demailly-Semple invariants is generated by 56 polynomials, while the algebra of bi-invariants is generated by 17 mutually independent polynomials sharing 105 groebnerized syzygies.


Some 6-dimensional Hamiltonian S^1 manifolds

Abstract: In an earlier paper we explained how to convert the problem of symplectically embedding one 4-dimensional ellipsoid into another into the problem of embedding a certain set of disjoint balls into \CP^2 by using a new way to desingularize orbifold blow ups Z of the weighted projective space \CP^2_{1,m,n}. We now use a related method to construct symplectomorphisms of these spaces Z. This allows us to construct some well known Fano 3-folds (including the Mukai--Umemura 3-fold) in purely symplectic terms, using a classification by Tolman of a particular class of Hamiltonian S^1-manifolds. We also show that these manifolds are uniquely determined by their fixed point data up to equivariant symplectomorphism. As part of this argument we show that the symplectomorphism group of a certain weighted blow up of a weighted projective plane is connected.


Infinite divisibility of Smith matrices

Abstract: Given an arithmetical function $f$, by $f(a, b)$ and $f[a, b]$ we denote the function $f$ evaluated at the greatest common divisor $(a, b)$ of positive integers $a$ and $b$ and evaluated at the least common multiple $[a, b]$ respectively. A positive semi-definite matrix $A=(a_{ij})$ with $a_{ij}\ge 0$ for all $i$ and $j$ is called infinitely divisible if the fractional Hadamard power $A^{\circ r}=(a_{ij}^r)$ is positive semi-definite for every nonnegative real number $r$. Let $S=\{x_1, ..., x_n\}$ be a set of $n$ distinct positive integers. In this paper, we show that if $f$ is a multiplicative function such that $(f*\mu)(d)\ge 0$ whenever $d|x$ for any $x\in S$, then the $n\times n$ matrices $(f(x_i, x_j))$, $(\frac{1}{f[x_i, x_j]})$ and $(\frac{f(x_i, x_j)}{f[x_i, x_j]})$ are infinitely divisible. Finally we extend these results to the Dirichlet convolution case which produces infinitely many examples of infinitely divisible matrices. Our results extend the results obtained previously by Bourque, Ligh, Bhatia, Hong, Lee, Lindqvist and Seip.


Alg\`ebres enveloppantes \`a homotopie pr\`es, homologie et cohomologie

Abstract: We present an unified construction for algebras and modules homologies and cohomologies, in the case of associative, commuttaive, Lie and Gerstenhaber algebras. We make a distinction between the linear part of the construction of algebras and cogebras, characterized by the symmetries of the defining relations and the structure itself which appears as a differential on these algebras and cogebras.


Formal and Informal Model Selection with Incomplete Data

Abstract: Model selection and assessment with incomplete data pose challenges in addition to the ones encountered with complete data. There are two main reasons for this. First, many models describe characteristics of the complete data, in spite of the fact that only an incomplete subset is observed. Direct comparison between model and data is then less than straightforward. Second, many commonly used models are more sensitive to assumptions than in the complete-data situation and some of their properties vanish when they are fitted to incomplete, unbalanced data. These and other issues are brought forward using two key examples, one of a continuous and one of a categorical nature. We argue that model assessment ought to consist of two parts: (i) assessment of a model's fit to the observed data and (ii) assessment of the sensitivity of inferences to unverifiable assumptions, that is, to how a model described the unobserved data given the observed ones.


Discrete Affine Surfaces based on Quadrangular Meshes

Abstract: In this paper we are interested in defining affine structures on discrete quadrangular surfaces of the affine three-space. We introduce, in a constructive way, two classes of such surfaces, called respectively indefinite and definite surfaces. The underlying meshes for indefinite surfaces are asymptotic nets satisfying a non-degeneracy condition, while the underlying meshes for definite surfaces are non-degenerate conjugate nets satisfying a certain natural condition. In both cases we associate to any of these nets several discrete affine invariant quantities: a metric, a normal and a co-normal vector fields, and a mean curvature. Moreover, we derive structural and compatibility equations which are shown to be necessary and sufficient conditions for the existence of a discrete quadrangular surface with a given affine structure.


Painl\'eve III and a singular linear statistics in Hermitian random matrix ensembles I

Abstract: In this paper, we study a certain linear statistics of the unitary Laguerre ensembles, motivated in part by an integrable quantum field theory at finite temperature. It transpires that this is equivalent to the characterization of a sequence of polynomials orthogonal with respect to the weight $$ w(x)=w(x,s):=x^{\al}\rme^{-x}\rme^{-s/x}, \quad 0\leq x<\infty, \al>0, s>0, $$ namely, the determination of the associated Hankel determinant and recurrence coefficients. Here $w(x,s)$ is the Laguerre weight $x^{\al}\:\rme^{-x}$ 'perturbed' by a multiplicative factor $\rme^{-s/x},$ which induces an infinitely strong zero at the origin. For polynomials orthogonal on the unit circle, a particular example where there are explicit formulas, the weight of which has infinitely strong zeros, was investigated by Pollazcek and Szeg\"o many years ago. Such weights are said to be 'singular' or irregular due to the violation of the Szeg\"o condition. In our problem, the linear statistics is a sum of the reciprocal of positive random variables $\{x_j:j=1,..,,n\};$ $\sum_{j=1}^{n}1/x_j.$ We show that the moment generating function, or the Laplace transform of the probability density function of this linear statistics is expressed as the ratio of Hankel determinants and as an integral of the combination of a particular third Painlev\'e function.


Vector space bases associated to vanishing ideals of points

Abstract: We discuss four different constructions of vector space bases associated to vanishing ideals of points. We show how to compute normal forms with respect to these bases and give new complexity bounds. As an application, we improve the computational algebra approach to the reverse engineering of gene regulatory networks.


Exceptional Times for the Dynamical Discrete Web

Abstract: The dynamical discrete web (DyDW),introduced in recent work of Howitt and Warren, is a system of coalescing simple symmetric one-dimensional random walks which evolve in an extra continuous dynamical time parameter \tau. The evolution is by independent updating of the underlying Bernoulli variables indexed by discrete space-time that define the discrete web at any fixed \tau. In this paper, we study the existence of exceptional (random) values of \tau where the paths of the web do not behave like usual random walks and the Hausdorff dimension of the set of exceptional such \tau. Our results are motivated by those about exceptional times for dynamical percolation in high dimension by H\"{a}ggstrom, Peres and Steif, and in dimension two by Schramm and Steif. The exceptional behavior of the walks in the DyDW is rather different from the situation for the dynamical random walks of Benjamini, H\"{a}ggstrom, Peres and Steif. For example, we prove that the walk from the origin S^\tau_0 violates the law of the iterated logarithm (LIL) on a set of \tau of Hausdorff dimension one. We also discuss how these and other results extend to the dynamical Brownian web, the natural scaling limit of the DyDW.


On the dimension of the Hilbert scheme of curves

Abstract: Consider a component of the Hilbert scheme whose general point corresponds to a degree d genus g smooth irreducible and nondegenerate curve in a projective variety X. We give lower bounds for the dimension of such a component when X is P^3, P^4 or a smooth quadric threefold in P^4 respectively. Those bounds make sense from the asymptotic viewpoint if we fix d and let g vary. Some examples are constructed using determinantal varieties to show the sharpness of the bounds for d and g in a certain range. The results can also be applied to study rigid curves.


On Universal Cycles of Labeled Graphs

Abstract: A universal cycle is a compact listing of a class of combinatorial objects. In this paper, we prove the existence of universal cycles of classes of labeled graphs, including simple graphs, trees, graphs with m edges, graphs with loops, graphs with multiple edges (with up to m duplications of each edge), directed graphs, hypergraphs, and k-uniform hypergraphs.


Circular Digraph Walks, k-Balanced Strings, Lattice Paths and Chebychev Polynomials

Abstract: We count the number of walks of length n on a k-node circular digraph that cover all k nodes in two ways. The first way illustrates the transfer-matrix method. The second involves counting various classes of height-restricted lattice paths. We observe that the results also count so-called k-balanced strings of length n, generalizing a 1996 Putnam problem.


The action of Hecke operators on hypergeometric functions

Abstract: We study the action of Hecke operators on the set of hypergeometric functions. We show that the spectrum of these operators is the set of powers n^a and that polylogarithms play a dominant role in the study of the corresponding eigenfunctions. As a corollary, we obtain a characterization of completely multiplicative hypergeometric coefficients.


Butterflies I: morphisms of 2-group stacks

Abstract: Weak morphisms of non-abelian complexes of length 2, or crossed modules, are morphisms of the associated 2-group stacks, or gr-stacks. We present a full description of the weak morphisms in terms of diagrams we call butterflies. We give a complete description of the resulting bicategory of crossed modules, which we show is fibered and biequivalent to the 2-stack of 2-group stacks. As a consequence we obtain a complete characterization of the non-abelian derived category of complexes of length 2. Deligne's analogous theorem in the case of Picard stacks and abelian sheaves becomes an immediate corollary. Commutativity laws on 2-group stacks are also analyzed in terms of butterflies, yielding new characterizations of braided, symmetric, and Picard 2-group stacks. Furthermore, the description of a weak morphism in terms of the corresponding butterfly diagram allows us to obtain a long exact sequence in non-abelian cohomology, removing a preexisting fibration condition on the coefficients short exact sequence.


Large deviations for infinite dimensional stochastic dynamical systems

Abstract: The large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation. The construction and justification of the approximations can be onerous, especially in the case where the process state is infinite dimensional. In this paper we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of Brownian noise. The approach is based on a variational representation for functionals of Brownian motion. Proofs of large deviations properties are reduced to demonstrating basic qualitative properties (existence, uniqueness and tightness) of certain perturbations of the original process.


Floer homology for negative line bundles and Reeb chords in pre-quantization spaces

Abstract: In this article we prove existence of Reeb orbits for Bohr-Sommerfeld Legendrians in certain pre-quantization spaces. We give a quantitative estimate from below. These estimates are obtained by studying Floer homology for fibre-wise quadratic Hamiltonian functions on negative line bundles.


Finite type invariants of words and Arnold's invariants

Abstract: We define a new finite type invariant for stably homeomorphic class of curves on compact oriented surfaces without boundaries and extend to a regular homotopy invariant for spherical curves.


Extension theorems for differential forms and Bogomolov-Sommese vanishing on log canonical varieties

Abstract: Given a p-form defined on the smooth locus of a normal variety, and a resolution of singularities, we study the problem of extending the pull-back of the p-form over the exceptional set of the desingularization. For log canonical pairs and for certain values of p, we show that an extension always exists, possibly with logarithmic poles along the exceptional set. As a corollary, it is shown that sheaves of reflexive differentials enjoy good pull-back properties. A natural generalization of the well-known Bogomolov-Sommese vanishing theorem to log canonical threefold pairs follows.


Reversibility of chordal SLE

Abstract: We prove that the chordal SLE$_{\kappa}$ trace is reversible for $\kappa\in(0,4]$.


Some Applications of the Isoperimetric Inequality for Integral Varifolds

Abstract: In this work the Isoperimetric Inequality for integral varifolds is used to obtain sharp estimates for the size of the set where the density quotient is small and to generalise Calder\'on's and Zygmund's theory of first order differentiability for functions in Lebesgue spaces from Lebesgue measure to integral varifolds.


Associative algebra deformations of the Connes-Moscovici's Hopf algebra $\mathcal{H}_1$

Abstract: We compute the second Hochschild cohomology space $HH^2(\mathcal{H}_1)$ of Connes-Moscovici's Hopf algebra $\mathcal{H}_1$, giving the infinitesimal deformations (up to equivalence) of the associative structure. $HH^2(\mathcal{H}_1)$ is shown to be one dimensional, and thus Connes-Moscovici's formal deformation of $\mathcal{H}_1$ using Rankin-Cohen brackets is unique up to equivalence.


Martingale approach to stochastic differential games of control and stopping

Abstract: We develop a martingale approach for studying continuous-time stochastic differential games of control and stopping, in a non-Markovian framework and with the control affecting only the drift term of the state-process. Under appropriate conditions, we show that the game has a value and construct a saddle pair of optimal control and stopping strategies. Crucial in this construction is a characterization of saddle pairs in terms of pathwise and martingale properties of suitable quantities.


A Sobolev Poincar\'e type inequality for integral varifolds

Abstract: In this work a local inequality is provided which bounds the distance of an integral varifold from a multivalued plane (height) by its tilt and mean curvature. The bounds obtained for the exponents of the Lebesgue spaces involved are shown to be sharp.


Second order rectifiability of integral varifolds of locally bounded first variation

Abstract: In this work it is shown that every integral varifold in an open subset of Euclidian space of locally bounded first variation can be covered by a countable collection of submanifolds of class C^2. Moreover, the mean curvature of each member of the collection agrees with the mean curvature of the varifold almost everywhere with respect to the varifold.


Pseudocyclic association schemes and strongly regular graphs

Abstract: Let X be a pseudocyclic association scheme in which all the nontrivial relations are strongly regular graphs with the same eigenvalues. We prove that the principal part of the first eigenmatrix of X is a linear combination of an incidence matrix of a symmetric design and the all-ones matrix. Amorphous pseudocyclic association schemes are examples of such association schemes whose associated symmetric design is trivial. We present several non-amorphous examples, which are either cyclotomic association schemes, or their fusion schemes. Special properties of symmetric designs guarantee the existence of further fusions, and the two known non-amorphous association schemes of class 4 discovered by van Dam and by the authors, are recovered in this way. We also give another pseudocyclic non-amorphous association scheme of class 7 on GF(2^{21}), and a new pseudocyclic amorphous association scheme of class 5 on GF(2^{12}).


Triangulated structures induced by simplicial descent categories

Abstract: The present paper is devoted to study the homotopy category associated with a simplicial descent category (D,s,E) (arXiv:0808.3684v2). We prove that the class E of equivalences has a calculus of left fractions over a quotient category of D modulo homotopy. We study the fiber/cofiber sequences induced by a (co)simplicial descent structure. Examples of such fiber/cofiber sequences are deduced for (commutative) differential graded algebras, simplicial sets or topological spaces. We prove that the homotopy category of a stable simplicial descent category is triangulated. In addition, these triangulated structures may be extended to the homotopy categories of diagram categories of D. As a corollary, we obtain the triangulated structures on: (filtered) derived categories of abelian categories, the derived category of DG-modules over a DG-category, the stable derived category of fibrant spectra and the localized category of mixed Hodge complexes.


(Co)Simplicial Descent Categories

Abstract: In this paper we study the question of how to transfer homotopic structure from the category sD of simplicial objects in a fixed category D to D. To this end we use a sort of homotopy colimit s : sD --> D, which we call simple functor. For instance, the Bousfield-Kan homotopy colimit in a Quillen simplicial model category is an example of simple functor. As a remarkable example outside the setting of Quillen models we include Deligne simple of mixed Hodge complexes. We prove here that the simple functor induces an equivalence on the corresponding localized categories. We also describe a natural structure of Brown category of cofibrant objects on sD. We use these facts to produce cofiber sequences on the localized category of D by E, which give rise to a natural Verdier triangulated structure in the stable case.


Asymptotic behaviour of thermoviscoelastic Berger plate

Abstract: System of partial differential equations with a convolution terms and non-local nonlinearity describing oscillations of plate due to Berger approach and with accounting for thermal regime in terms of Coleman-Gurtin and Gurtin-Pipkin law and fading memory of material is considered. The equation is transformed into a dynamical system in a suitable Hilbert space which asymptotic behaviour is analysed. Existence of the compact global attractor in this dynamical system and some of its properties are proved in this article. Main tool in analysis of asymptotic behaviour is stabilizability inequality.


Asymptotics of randomly stopped sums in the presence of heavy tails

Abstract: We study conditions under which $P(S_\tau>x)\sim P(M_\tau>x)\sim E\tau P(\xi_1>x)$ as $x\to\infty$, where $S_\tau$ is a sum $\xi_1+...+\xi_\tau$ of random size $\tau$ and $M_\tau$ is a maximum of partial sums $M_\tau=\max_{n\le\tau}S_n$. Here $\xi_n$, $n=1$, 2, ..., are independent identically distributed random variables whose common distribution is assumed to be subexponential. We consider mostly the case where $\tau$ is independent of the summands; also, in a particular situation, we deal with a stopping time. Also we consider the case where $E\xi>0$ and where the tail of $\tau$ is comparable with or heavier than that of $\xi$, and obtain the asymptotics $P(S_\tau>x) \sim E\tau P(\xi_1>x)+P(\tau>x/E\xi)$ as $x\to\infty$. This case is of a primary interest in the branching processes. In addition, we obtain new uniform (in all $x$ and $n$) upper bounds for the ratio $P(S_n>x)/P(\xi_1>x)$ which substantially improve Kesten's bound in the subclass ${\mathcal S}^*$ of subexponential distributions.


A hyperbolic Out(F_n)-complex

Abstract: For any finite collection $f_i$ of fully irreducible automorphisms of the free group $F_n$ we construct a connected $\delta$-hyperbolic $Out(F_n)$-complex in which each $f_i$ has positive translation length.


Survival of contact processes on the hierarchical group

Abstract: We consider contact processes on the hierarchical group, where sites infect other sites at a rate depending on their hierarchical distance, and sites become healthy with a constant recovery rate. If the infection rates decay too fast as a function of the hierarchical distance, then we show that the critical recovery rate is zero. On the other hand, we derive sufficient conditions on the speed of decay of the infection rates for the process to exhibit a nontrivial phase transition between extinction and survival. For our sufficient conditions, we use a coupling argument that compares contact processes on the hierarchical group with freedom two with contact processes on a renormalized lattice. An interesting novelty in this renormalization argument is the use of a result due to Rogers and Pitman on Markov functionals.


The abstract Titchmarsh-Weyl M-function for adjoint operator pairs and its relation to the spectrum

Abstract: In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M-function see the same singularities as the resolvent of a certain restriction $A_B$ of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces $\Sc$ and $\tilde{\Sc}$ such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M-function is analytic. We present three examples -- one involving a Hain-L\"{u}st type operator, one involving a perturbed Friedrichs operator and one involving a simple ordinary differential operators on a half line -- which together indicate that the abstract results are probably best possible.


Geometric Approach to the Weil-Petersson Symplectic Form

Abstract: In a family of compact, canonically polarized, complex manifolds the first variation of the lengths of closed geodesics is computed. As an application, we show the coincidence of the Fenchel-Nielsen and Weil-Petersson symplectic forms on the Teichmueller spaces of compact Riemann surfaces in a purely geometric way. The method can also be applied to situations like moduli spaces of weighted punctured Riemann surfaces, where the methods of Kleinian groups are not available.


Some questions about $\mathcal G$-bundles on curves

Abstract: We define the notion of a parahoric group scheme $\mathcal G$ over a smooth projective curve, and formulate four conjectures on the structure of the stack of $\mathcal G$-bundles, which generalize to this case well-known results on $G$-bundles with $G$ a constant reductive group. The conjectures concern the set of connected components, the uniformization by affine flag varieties of twisted loop groups, the Picard groups, and the space of global sections of a dominant line bundle. Since a first version of this paper was circulated, Heinloth [arXiv:0711.4450] has proved a good part of these conjectures.


Necessary and sufficient conditions for the existence of the q-optimal measure

Abstract: This paper presents the general form and essential properties of the q-optimal measure following the approach of Delbaen and Schachermayer (1996) and proves its existence under mild conditions. Most importantly, it states a necessary and sufficient condition for a candidate measure to be the q-optimal measure in the case even of signed measures. Finally, an updated characterization of the q-optimal measure for continuous asset price processes is presented in the light of the counterexample appearing in Cerny and Kallsen (2006) concerning Hobson's (2004) approach.


Hilbert-Chow morphism for non commutative Hilbert schemes and moduli spaces of linear representations

Abstract: Let $k$ be a commutative ring and let $R$ be a commutative $k-$algebra. The aim of this paper is to define and discuss some connection morphisms between schemes associated to the representation theory of a (non necessarily commutative) $R-$algebra $A. $ We focus on the scheme $\ran//\GL_n$ of the $n-$dimensional representations of $A, $ on the Hilbert scheme $\Hilb_A^n$ parameterizing the left ideals of codimension $n$ of $A$ and on the affine scheme Spec $\Gamma_R^n(A)^{ab} $ of the abelianization of the divided powers of order $n$ over $A. $ We give a generalization of the Grothendieck-Deligne norm map from $\Hilb_A^n$ to Spec $\Gamma_R^n(A)^{ab} $ which specializes to the Hilbert Chow morphism on the geometric points when $A$ is commutative and $k$ is an algebraically closed field. Describing the Hilbert scheme as the base of a principal bundle we shall factor this map through the moduli space $\ran//\GL_n$ giving a nice description of this Hilbert-Chow morphism, and consequently proving that it is projective.


Occupation times of subcritical branching immigration system with Markov motions

Abstract: We consider a branching system consisting of particles moving according to a Markov family in $\Rd$ and undergoing subcritical branching with a constant rate $V>0$. New particles immigrate to the system according to homogeneous space-time Poisson random field. The process of the fluctuations of the rescaled occupation time is studied with very mild assumptions on the Markov family. In this general setting a functional central limit theorem is proved. The subcriticality of the branching law is crucial for the limit behaviour and in a sense overwhelms the properties of the particles' motion. It is for this reason that the limit is the same for all dimensions and can be obtained for a wide class of Markov processes. Another consequence is the form of the limit - $\SP$-valued Wiener process with a simple temporal structure and a complicated spatial one. This behaviour contrasts sharply with the case of critical branching systems.


Hypergraph Ramsey numbers

Abstract: The Ramsey number r_k(s,n) is the minimum N such that every red-blue coloring of the k-tuples of an N-element set contains either a red set of size s or a blue set of size n, where a set is called red (blue) if all k-tuples from this set are red (blue). In this paper we obtain new estimates for several basic hypergraph Ramsey problems. We give a new upper bound for r_k(s,n) for k \geq 3 and s fixed. In particular, we show that r_3(s,n) \leq 2^{n^{s-2}\log n}, which improves by a factor of n^{s-2}/ polylog n the exponent of the previous upper bound of Erdos and Rado from 1952. We also obtain a new lower bound for these numbers, showing that there are constants c_1,c_2>0 such that r_3(s,n) \geq 2^{c_1 sn \log (n/s)} for all 4 \leq s \leq c_2n. When s is a constant, it gives the first superexponential lower bound for r_3(s,n), answering an open question posed by Erdos and Hajnal in 1972. Next, we consider the 3-color Ramsey number r_3(n,n,n), which is the minimum N such that every 3-coloring of the triples of an N-element set contains a monochromatic set of size n. Improving another old result of Erdos and Hajnal, we show that r_3(n,n,n) \geq 2^{n^{c \log n}}. Finally, we make some progress on related hypergraph Ramsey-type problems.


The isocohomological property, higher Dehn functions, and relatively hyperbolic groups

Abstract: The property that the polynomial cohomology with coefficients of a finitely generated discrete group is canonically isomorphic to the group cohomology is called the (weak) isocohomological property for the group. In the case when a group is of type $HF^\infty$, i.e. that has a classifying space with the homotopy type of a cellular complex with finitely many cells in each dimension, we show that the isocohomological property is equivalent to the universal cover of the classifying space satisfying polynomially bounded higher Dehn functions. If a group is hyperbolic relative to a collection of subgroups, each of which is polynomially combable (respectively $HF^\infty$ and isocohomological), then we show that the group itself has these respective properties too. Combining with the results of Connes-Moscovici and Dru{\c{t}}u-Sapir we conclude that a group satisfies the Novikov conjecture if it is relatively hyperbolic to subgroups that are of property RD, of type $HF^\infty$ and isocohomological.


The commutant of L(H) in its ultrapower may or may not be trivial

Abstract: Kirchberg asked in 2004 whether the commutant of L(H)$ in its (norm) ultrapower is trivial. Assuming the Continnuum Hypothesis, we prove that the answer depends on the choice of the ultrafilter.


KBSM of the product of a disk with two holes and S^{1}

Abstract: We introduce diagrams and Reidemeister moves for links in FxS^{1}, where F is an orientable surface. Using these diagrams we compute (in a new way) the Kauffman Bracket Skein Modules (KBSM) for D^{2}xS^{1} and AxS^{1}, where D^{2} is a disk and A is an annulus. Moreover, we also find the KBSM for the F_{0,3}xS^{1}, where F_{0,3} denotes a disk with two holes, and thus show that the module is free.


Equilibrium policies when preferences are time inconsistent

Abstract: This paper characterizes differentiable and subgame Markov perfect equilibria in a continuous time intertemporal decision problem with non-constant discounting. Capturing the idea of non commitment by letting the commitment period being infinitesimally small, we characterize the equilibrium strategies by a value function, which must satisfy a certain equation. The equilibrium equation is reminiscent of the classical Hamilton-Jacobi-Bellman equation of optimal control, but with a non-local term leading to differences in qualitative behavior. As an application, we formulate an overlapping generations Ramsey model where the government maximizes a utilitarian welfare function defined as the discounted sum of successive generations' lifetime utilities. When the social discount rate is different from the private discount rate, the optimal command allocation is time inconsistent and we retain subgame perfection as a principle of intergenerational equity. Existence of multiple subgame perfect equilibria is established. The multiplicity is due to the successive governments' inability to coordinate their beliefs and we single out one of them as (locally) renegotiation-proof. Decentralization can be achieved with both age and time dependent lump sum transfers and, long term distorting capital interest income taxes/subsidy.


Symplectic fillings of links of quotient surface singularities

Abstract: We study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.


A New Hypoelliptic Operator on Almost CR Manifolds

Abstract: The aim of this paper is to present the construction, out of the Kohn-Rossi complex, of a new hypoelliptic operator $Q_{L}$ on almost CR manifolds equipped with a real structure. The operator acts on all (p,q)-forms, but when restricted to (p,0)-forms and (p,n)-forms it is a sum of squares up to sign factor and lower order terms. Therefore, only a finite type condition condition is needed to have hypoellipticity on those forms. However, outside these forms $Q_{L}$ may fail to be hypoelliptic, as it is shown in the example of the Heisenberg group. We also look at the Fredholm properties of $Q_{L}$ and show that the corresponding Fredholm index is zero.


Homotopy types of topological stacks

Abstract: We define the notion of {\em classifying space} of a topological stack and show that every topological stack \X has a classifying space X which is a topological space well-defined up to weak homotopy equivalence. Under a certain paracompactness condition on \X, we show that X is actually well-defined up to homotopy equivalence. These results are formulated in terms of functors from the category of topological stacks to the (weak) homotopy category of topological spaces. We prove similar results for (small) diagrams of topological stacks.


Meromorphic functions with linearly distributed values and Julia sets of rational functions

Abstract: If the preimage of a four-point set under a meromorphic function belongs to the real line, then the image of the real line is contained in a circle in the Riemann sphere. We include an application of this result to holomorphic dynamics: if the Julia set of a rational function is contained in a smooth curve then it is contained in a circle.


Some Characterizations of Domination

Abstract: We show that a cocycle has a dominated splitting if and only if there is a uniform exponential gap between singular values of its iterates. Then we consider sets $\Sigma$ in $GL(d,\mathbb{R})$ with the property that any cocycle with values in $\Sigma$ has a dominated splitting. We characterize these sets in terms of existence of invariant multicones, thus extending a 2-dimensional result by Avila, Bochi, and Yoccoz. We give an example showing how these multicones can fail to have convexity properties.


On the Koszul Betti numbers in positive characteristic

Abstract: We have observed a gap in one of the arguments in the main theorem. We choose to withdraw the paper until we rectify the gap.


Algorithmic enumeration of ideal classes for quaternion orders

Abstract: We provide algorithms to count and enumerate representatives of the (right) ideal classes of an Eichler order in a quaternion algebra defined over a number field. We analyze the run time of these algorithms and consider several related problems, including the computation of two-sided ideal classes, isomorphism classes of orders, connecting ideals for orders, and ideal principalization. We conclude by giving the complete list of definite Eichler orders with class number at most 2.


Minimal hypersurfaces in $\HH^n \times \R$, total curvature and index

Abstract: In this paper, we consider minimal hypersurfaces in the product space $\mathbb{H}^n \times \mathbb{R}$. We begin by studying examples of rotation hypersurfaces and hypersurfaces invariant under hyperbolic translations. We then consider minimal hypersurfaces with finite total curvature. This assumption implies that the corresponding curvature goes to zero uniformly at infinity. We show that surfaces with finite total intrinsic curvature have finite index. The converse statement is not true as shown by our examples which also serve as useful barriers.


On the Pfister Number of Quadratic Forms

Abstract: The generic quadratic form of even dimension n with trivial discriminant over an arbitrary field of characteristic different from 2 containing a square root of -1 can be written in the Witt ring as a sum of 2-fold Pfister forms using n-2 terms and not less. The number of 2-fold Pfister forms needed to express a quadratic form of dimension 6 with trivial discriminant is determined in various cases.


Tetrahedra on deformed spheres and integral group cohomology

Abstract: We show that for every injective continuous map f: S^2 --> R^3 there are four distinct points in the image of f such that the convex hull is a tetrahedron with the property that two opposite edges have the same length and the other four edges are also of equal length. This result represents a partial result for the topological Borsuk problem for R^3. Our proof of the geometrical claim, via Fadell-Husseini index theory, provides an instance where arguments based on group cohomology with integer coefficients yield results that cannot be accessed using only field coefficients.


Directed polymer in random environment and last passage percolation

Abstract: The sequence of random probability measures $\nu_n$ that gives a path of length $n$, $\unsur{n}$ times the sum of the random weights collected along the paths, is shown to satisfy a large deviations principle with good rate function the Legendre transform of the free energy of the associated directed polymer in a random environment. Consequences on the asymptotics of the typical number of paths whose collected weight is above a fixed proportion are then drawn.


Gibbs Sampling, Exponential Families and Orthogonal Polynomials

Abstract: We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions.


Comment: Gibbs Sampling, Exponential Families and Orthogonal Polynomials

Abstract: Comment on ``Gibbs Sampling, Exponential Families and Orthogonal Polynomials'' [arXiv:0808.3852]


Comment: Gibbs Sampling, Exponential Families, and Orthogonal Polynomials

Abstract: Comment on ``Gibbs Sampling, Exponential Families, and Orthogonal Polynomials'' [arXiv:0808.3852]


Covariate Balance in Simple, Stratified and Clustered Comparative Studies

Abstract: In randomized experiments, treatment and control groups should be roughly the same--balanced--in their distributions of pretreatment variables. But how nearly so? Can descriptive comparisons meaningfully be paired with significance tests? If so, should there be several such tests, one for each pretreatment variable, or should there be a single, omnibus test? Could such a test be engineered to give easily computed $p$-values that are reliable in samples of moderate size, or would simulation be needed for reliable calibration? What new concerns are introduced by random assignment of clusters? Which tests of balance would be optimal? To address these questions, Fisher's randomization inference is applied to the question of balance. Its application suggests the reversal of published conclusions about two studies, one clinical and the other a field experiment in political participation.


Comment: Lancaster Probabilities and Gibbs Sampling

Abstract: Comment on ``Lancaster Probabilities and Gibbs Sampling'' [arXiv:0808.3852]


Comment: On Random Scan Gibbs Samplers

Abstract: Comment on ``On Random Scan Gibbs Samplers'' [arXiv:0808.3852]


Parallel in Time Simulation of Multiscale Stochastic Chemical Kinetics

Abstract: A version of the time-parallel algorithm parareal is analyzed and applied to stochastic models in chemical kinetics. A fast predictor at the macroscopic scale (evaluated in serial) is available in the form of the usual reaction rate equations. A stochastic simulation algorithm is used to obtain an exact realization of the process at the mesoscopic scale (in parallel). The underlying stochastic description is a jump process driven by the Poisson measure. A convergence result in this arguably difficult setting is established suggesting that a homogenization of the solution is advantageous. We devise a simple but highly general such technique. Three numerical experiments on models representative to the field of computational systems biology illustrate the method. For non-stiff problems, it is shown that the method is able to quickly converge even when stochastic effects are present. For stiff problems we are instead able to obtain fast convergence to a homogenized solution. Overall, the method builds an attractive bridge between on the one hand, macroscopic deterministic scales and, on the other hand, mesoscopic stochastic ones. This construction is clearly possible to apply also to stochastic models within other fields.


Rejoinder: Gibbs Sampling, Exponential Families and Orthogonal Polynomials

Abstract: We are thankful to the discussants for their hard, interesting work. The main purpose of our paper was to give reasonably sharp rates of convergence for some simple examples of the Gibbs sampler. We chose examples from expository accounts where direct use of available techniques gave practically useless answers. Careful treatment of these simple examples grew into bivariate modeling and Lancaster families. Since bounding rates of convergence is our primary focus, let us begin there. [arXiv:0808.3852]


O-minimal homotopy and generalized (co)homology

Abstract: This article explains and extends semialgebraic homotopy theory (developed by H. Delfs and M. Knebusch) to o-minimal homotopy theory (over a field). The homotopy category of definable CW-complexes is equivalent to the homotopy category of topological CW-complexes (with continuous mappings). If the theory of the o-minimal expansion of a field is bounded, then these categories are equivalent to the homotopy category of weakly definable spaces. Similar facts hold for decreasing systems of spaces. As a result, generalized homology and cohomology theories on pointed weak polytopes uniquely correspond (up to an isomorphism) to the known topological generalized homology and cohomology theories on pointed CW-complexes.


Embeddings of C*-surfaces into weighted projective spaces

Abstract: Let V be a normal affine surface which admits a C*- and a C+-action. In this note we show that in many cases V can be embedded as a principal Zariski open subset into a hypersurface of a weighted projective space. In particular, we recover a result of D. Daigle and P. Russell.


Higher Rank Wavelets

Abstract: A theory of higher rank multiresolution analysis is given in the setting of abelian multiscalings. This theory enables the construction, from a higher rank MRA, of finite wavelet sets whose multidilations have translates forming an orthonormal basis in $L^2(\bR^d)$. While tensor products of uniscaled MRAs provide simple examples we construct many nonseparable higher rank wavelets. In particular we construct Latin square wavelets as rank 2 variants of Haar wavelets. Also we construct nonseparable scaling functions for rank 2 variants of Meyer wavelet scaling functions, and we construct the associated nonseparable wavelets with compactly supported Fourier transforms. On the other hand we show that compactly supported scaling functions for biscaled MRAs are necessarily separable.


A chain morphism for Adams operations on rational algebraic K-theory

Abstract: For any regular noetherian scheme X and every k>0, we define a chain morphism between two chain complexes whose homology with rational coefficients is isomorphic to the algebraic K-groups of X tensored by the field of rational numbers. It is shown that these morphisms induce in homology the Adams operations defined by Gillet and Soule or the ones defined by Grayson.


Quasi-analytische Zerlegungen

Abstract: The leaves in singular holomorphic foliation theory are examples of quasi-analytic layers. In the first part of our publication we are concerned with a theory of these subjects. A quasi-analytic decomposition of a complex manifold is a decomposition into pairwise disjoint connected quasi-analytic layers. These are holomorphic foliations in the sense of P. Stefan and K. Spallek. A very different but more usual conception of holomorphic foliations is develloped by P. Baum and R. Bott. It is based on holomorphic sheaf theory. In the second part we study the relation between quasi-analytic decompositions and singular holomorphic foliations in the sense of Baum and Bott.


Randomization Does Not Justify Logistic Regression

Abstract: The logit model is often used to analyze experimental data. However, randomization does not justify the model, so the usual estimators can be inconsistent. A consistent estimator is proposed. Neyman's non-parametric setup is used as a benchmark. In this setup, each subject has two potential responses, one if treated and the other if untreated; only one of the two responses can be observed. Beside the mathematics, there are simulation results, a brief review of the literature, and some recommendations for practice.


The modular branching rule for affine Hecke algebras of type A

Abstract: For the affine Hecke algebra of type A at roots of unity, we make explicit the correspondence between geometrically constructed simple modules and combinatorially constructed simple modules and prove the modular branching rule. The latter generalizes work by Vazirani.


A sectional characterization of the Dade group

Abstract: Let $k$ be a field of characteristic $p$, let $P$ be a finite $p$- group, where $p$ is an odd prime, and let $D(P)$ be the Dade group of endo-permutation $kP$-modules. It is known that $D(P)$ is detected via deflation--restriction by the family of all sections of $P$ which are elementary abelian of rank $\leq2$. In this paper, we improve this result by characterizing $D(P)$ as the limit (with respect to deflation--restriction maps and conjugation maps) of all groups $D(T/S)$ where $T/S$ runs through all sections of $P$ which are either elementary abelian of rank $\leq3$ or extraspecial of order $p^3$ and exponent $p$.


On a surprising relation between the Marchenko-Pastur law, rectangular and square free convolutions

Abstract: n this paper, we prove a result linking the square and the rectangular R-transforms, the consequence of which is a surprising relation between the square and rectangular versions the free additive convolutions, involving the Marchenko-Pastur law. Consequences on random matrices, on infinite divisibility and on the arithmetics of the square versions of the free additive and multiplicative convolutions are given.


Generators and defining relations for ring of differential operators on smooth affine algebraic variety in prime characteristic

Abstract: For ring of differential operators on smooth affine algebraic variety over perfect field of prime characteristic a set of algebra generators and a set of defining relations are found explicitly.


Stability of roots of polynomials under linear combinations of derivatives

Abstract: Let $T=\alpha_0 I + \alpha_1 D + ...+\alpha_n D^n$, where $D$ is the differentiation operator and $\alpha_0\not= 0$, and let $f$ be a square-free polynomial with large minimum root separation. We prove that the roots of $Tf$ are close to the roots of $f$ translated by $-\alpha_1/\alpha_0$.


Anti-Perfect Morse Stratification

Abstract: For an equivariant Morse stratification which contains a unique open stratum, we introduce the notion of equivariant antiperfection, which means the difference of the equivariant Morse series and the equivariant Poincare series achieves the maximal possible value (instead of the minimal possible value 0 in the equivariantly perfect case). We also introduce a weaker condition of local equivariant antiperfection. We prove that the Morse stratification of the Yang-Mills functional on the space of connections on a principal U(n)-bundle over a connected, closed, nonorientable surface is locally equivariantly Q-antiperfect when the rank n=2,3; we propose that it is actually equivariantly Q-antiperfect when n=2,3. Our proposal yields formulas of G-equivariant Poincare series of the representation variety of flat G-connections for the nonorientable surface where G=U(2), SU(2), U(3), SU(3). Our rank 2 formulas agree with formulas proved by T. Baird in arXiv:0806.1975. Baird verified our conjectural rank 3 formulas when the nonorientable surface is the real projective plane or the Klein bottle (arXiv:0901.1604); he proved our conjectural U(3) formula for any closed nonorientable surfaces by establishing equivariant Q-antiperfection in this case (arXiv:0902.4581).


Least Squares Methods for Equidistant Tree Reconstruction

Abstract: UPGMA is a heuristic method identifying the least squares equidistant phylogenetic tree given empirical distance data among $n$ taxa. We study this classic algorithm using the geometry of the space of all equidistant trees with $n$ leaves, also known as the Bergman complex of the graphical matroid for the complete graph $K_n$. We show that UPGMA performs an orthogonal projection of the data onto a maximal cell of the Bergman complex. We also show that the equidistant tree with the least (Euclidean) distance from the data is obtained from such an orthogonal projection, but not necessarily given by UPGMA. Using this geometric information we give an extension of the UPGMA algorithm. We also present a branch and bound method for finding the best equidistant tree. Finally, we prove that there are distance data among $n$ taxa which project to at least $(n-1)!$ equidistant trees.


Lagrangian Quantum Homology

Abstract: The present paper is mainly a survey of our work arXiv:0708.4221 and arXiv:0808.2440 but it also contains the announcement of some new results. Its main purpose is to present an accessible introduction to a technique allowing efficient calculations in Lagrangian Floer theory.


Viability for stochastic differential equations driven by fractional Brownian motion

Abstract: In this paper we prove a viability result for multidimensional, time dependent, stochastic differential equations driven by fractional Brownian motion with Hurst parameter1/2 < H < 1, using pathwise approach. The sufficient condition is also an alternative global existence result for the fractional differential equations with restrictions on the state.


The Scalar Curvature Equation on $S^3$

Abstract: We give existence results for solutions of the prescribed scalar curvature equation on $S^3$, when the curvature function is a positive Morse function and satisfies an index-count condition.


Fourier transform of function on locally compact Abelian groups taking value in Banach spaces

Abstract: We consider Fourier transform of vector-valued functions on a locally compact group $G$, which take value in a Banach space $X$, and are square-integrable in Bochner sense. If $G$ is a finite group then Fourier transform is a bounded operator. If $G$ is an infinite group then Fourier transform $F: L_2(G,X)\to L_2(\widehat G,X)$ is a bounded operator if and only if Banach space $X$ is isomorphic to a Hilbert one.


A priori Holder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps

Abstract: In this paper, we consider the following type of non-local (pseudo-differential) operators $\LL $ on $\R^d$: $$ \LL u(x) =\frac12 \sum_{i, j=1}^d \frac{\partial}{\partial x_i} (a_{ij}(x) \frac{\partial}{\partial x_j}) + \lim_{\eps \downarrow 0} \int_{\{y\in \R^d: |y-x|>\eps\}} (u(y)-u(x)) J(x, y) dy, $$ where $A(x)=(a_{ij}(x))_{1\leq i, j\leq d}$ is a measurable $d\times d$ matrix-valued function on $\R^d$ that is uniform elliptic and bounded and $J$ is a symmetric measurable non-trivial non-negative kernel on $\R^d\times \R^d$ satisfying certain conditions. Corresponding to $\LL$ is a symmetric strong Markov process $X$ on $\R^d$ that has both the diffusion component and pure jump component. We establish a priori H\"older estimate for bounded parabolic functions of $\LL$ and parabolic Harnack principle for positive parabolic functions of $\LL$. Moreover, two-sided sharp heat kernel estimates are derived for such operator $\LL$ and jump-diffusion $X$. In particular, our results apply to the mixture of symmetric diffusion of uniformly elliptic divergence form operator and mixed stable-like processes on $\R^d$. To establish these results, we employ methods from both probability theory and analysis.


The H\"older continuity of a class of 3-dimension ultraparabolic equations

Abstract: We obtained the $C^\alpha$ continuity for weak solutions of a class of ultraparabolic equations with measurable coefficients of the form $${\partial_t u} = \partial_x(a(x,y,t)\partial_x u) +b_0(x,y,t)\partial_x u+b(x,y,t)\partial_y u,$$ which generalized our recent results on KFP equations.


Fundamental Dominations in Graphs

Abstract: Nine variations of the concept of domination in a simple graph are identified as fundamental domination concepts, and a unified approach is introduced for studying them. For each variation, the minimum cardinality of a subset of dominating elements is the corresponding fundamental domination number. It is observed that, for each nontrivial connected graph, at most five of these nine numbers can be different, and inequalities between these five numbers are given. Finally, these fundamental dominations are interpreted in terms of the total graph of the given graph, a concept introduced by the second author in 1965. It is argued that the very first domination concept, defined by O. Ore in 1962 and under a different name by C. Berge in 1958, deserves to be called the most fundamental of graph dominations.


The Center of Mass for Spatial Branching Processes and an Application for Self-Interaction

Abstract: In this paper we prove that the center of mass of a supercritical branching-Brownian motion, or that of a supercritical super-Brownian motion tends to a limiting position almost surely, which, in a sense complements a result of Tribe on the final behavior of a critical super-Brownian motion. This is shown to be true also for a model where branching Brownian motion is modified by attraction/repulsion between particles. We then put this observation together with the description of the interacting system as viewed from its center of mass, and get the following asymptotic behavior: the system asymptotically becomes a branching Ornstein Uhlenbeck process (inward for attraction and outward for repulsion), but the origin is shifted to a random point which has normal distribution, and the Ornstein Uhlenbeck particles are not independent but constitute a system with a degree of freedom which is less by their number by precisely one.


Regular projections of graphs with at most three double points

Abstract: A generic immersion of a planar graph into the 2-space is said to be knotted if there does not exist a trivial embedding of the graph into the 3-space obtained by lifting the immersion with respect to the natural projection from the 3-space to the 2-space. In this paper we show that if a generic immersion of a planar graph is knotted then the number of double points of the immersion is more than or equal to three. To prove this, we also show that an embedding of a graph obtained from a generic immersion of the graph (does not need to be planar) with at most three double points is totally free if it contains neither a Hopf link nor a trefoil knot.


On a Construction of L. Hua for Positive Reproducing Kernels

Abstract: We study a positive reproducing kernel for holomorphic functions on a domain in a complex space. The technique is based on an idea of L. Hua. Applications are provided. These ideas were developed in another context (quantization of K\"{a}hler manifolds) by Berezin.


Nakajima's remark on Henn's proof

Abstract: We fill up a gap in Henn's proof concerning large automorphism groups of function fields of degree 1 over an algebraically closed field of positive characteristic.


Hybrid data regression modelling in measurement

Abstract: Measurement involves the determination of quantitative estimates of physical quantities from experiment, along with estimates of their associated uncertainties. Herewith an experimental system model is the key to extracting information from the experimental data. The measurement information obtained depends directly on the quality of the model. With this concern novel regression modelling techniques have been fashioned by data integration from computer-simulation and physical designed experiments. These techniques have allowed attaining the advanced level of model completeness, parsimony, and precision via approximation of the exact unknown model by mathematical product of available theoretical and appropriate empirical functions. The purpose of this approximation is to represent adequately the true model on the considered region of factor space with all advantages of theoretical modelling. This allows a further focus on the measurement science of issue. Pneumatic gauge hybrid data candidate model building, solving and validation reviled that such adequate models permit to attain minimum discrepancy from empirical evidence.


Karl Pearson's Theoretical Errors and the Advances They Inspired

Abstract: Karl Pearson played an enormous role in determining the content and organization of statistical research in his day, through his research, his teaching, his establishment of laboratories, and his initiation of a vast publishing program. His technical contributions had initially and continue today to have a profound impact upon the work of both applied and theoretical statisticians, partly through their inadequately acknowledged influence upon Ronald A. Fisher. Particular attention is drawn to two of Pearson's major errors that nonetheless have left a positive and lasting impression upon the statistical world.