On Some Autotopisms Of Non-Steiner Central Loops

Abstract: An algebraic process for the construction of an autotopism for a non-Steiner C-loop is described and this is demonstrated with an example using a known finite C-loop. In every C-loop, two of its parastrophes are not equivalent(equal) it, if and only if both the first and second components of the constructed autotopism and its inverse autotopism are not equal to the identity map. Hence, none of the other three parastrophes is equivalent(equal) to the C-loop. It is proved that the set of autotopisms that prevent a C-loop from being a Steiner loop forms a Steiner triple system.


On central loops and the central square property

Abstract: The representation sets of a central square C-loop are investigated. Isotopes of central square C-loops of exponent 4 are shown to be both C-loops and A-loops.


Algebraic properties of some varieties of central loops

Abstract: Isotopes of C-loops with unique non-identity squares are shown to be both C-loops and A-loops. The relationship between C-loops and Steiner loops is further studied. Central loops with the weak and cross inverse properties are also investigated. C-loops are found to be Osborn loops if every element in them are squares.


Invariant measure for a three dimensional nonlinear wave equation

Abstract: We study the long time behavior of the subcritical (subcubic) defocussing nonlinear wave equation on the three dimensional ball, for random data of low regularity. We prove that for a large set of radial initial data in $\cap_{s<1/2} H^s(B(0,1))$ the equation is (globally in time) well posed and we construct an invariant measure.


Random data Cauchy theory for supercritical wave equations I: Local theory

Abstract: We study the local existence of strong solutions for the cubic nonlinear wave equation with data in $H^s(M)$, $s<1/2$, where $M$ is a three dimensional compact riemannian manifold. This problem is supercritical and can be shown to be strongly ill-posed (in the Hadamard sense). However, after a suitable randomization, we are able to construct local strong solution for a large set of initial data in $H^s(M)$, where $s\geq 1/4$ in the case of a boundary less manifold and $s\geq 8/21$ in the case of a manifold with boundary.


Random data Cauchy theory for supercritical wave equations II : A global existence result

Abstract: We prove that the subquartic wave equation on the three dimensional ball $\Theta$, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in $\cap_{s<1/2} H^s(\Theta)$. We obtain this result as a consequence of a general random data Cauchy theory for supercritical wave equations developed in our previous work \cite{BT2} and invariant measure considerations which allow us to obtain also precise large time dynamical informations on our solutions.


Abstract homotopical methods for theoretical computer science

Abstract: The purpose of this paper is to collect the homotopical methods used in the development of the theory of flows initialized by author's paper ``A model category for the homotopy theory of concurrency''. It is presented generalizations of the classical Whitehead theorem inverting weak homotopy equivalences between CW-complexes using weak factorization systems. It is also presented methods of calculation of homotopy limits and homotopy colimits using Quillen adjunctions and Reedy categories.


Explicit computations of all finite index bimodules for a family of II_1 factors

Abstract: We study II_1 factors M and N associated with good generalized Bernoulli actions of groups having an infinite almost normal subgroup with the relative property (T). We prove the following rigidity result: every finite index M-N-bimodule (in particular, every isomorphism between M and N) is described by a commensurability of the groups involved and a commensurability of their actions. The fusion algebra of finite index M-M-bimodules is identified with an extended Hecke fusion algebra, providing the first explicit computations of the fusion algebra of a II_1 factor. We obtain in particular explicit examples of II_1 factors with trivial fusion algebra, i.e. only having trivial finite index subfactors.


Standard Relations of Multiple Polylogarithm Values at Roots of Unity

Abstract: Let $N$ be a positive integer. In this paper we shall study the special values of multiple polylogarithms at $N$th roots of unity, called multiple polylogarithm values (MPVs) of level $N$. These objects are generalizations of multiple zeta values and alternating Euler sums, which was studied by Euler, and more recently, many mathematicians and theoretical physicists.. Our primary goal in this paper is to investigate the relations among the MPVs of the same weight and level by using the regularized double shuffle relations, regularized distribution relations, lifted versions of such relations from lower weights, and seeded relations which are produced by relations of weight one MPVs. We call relations from the above four families \emph{standard}. Let $d(w,N)$ be the $\Q$-dimension of $\Q$-span of all MPVs of weight $w$ and level $N$. Then we obtain upper bound for $d(w,N)$ by the standard relations which in general are no worse or no better than the one given by Deligne and Goncharov depending on whether $N$ is a prime-power or not, respectively, except for 2- and 3-powers, in which case standard relations seem to be often incomplete whereas Deligne shows that their bound should be sharp by a variant of Grothedieck's period conjecture. This suggests that in general there should be other linear relations among MPVs besides the standard relations, some of which are written down in this paper explicitly with good numerical verification. We also provide a few conjectures which are supported by our computational evidence.


Conditional large and moderate deviations for sums of discrete random variables. Combinatoric applications

Abstract: We prove large and moderate deviation principles for the distribution of an empirical mean conditioned by the value of the sum of discrete i.i.d. random variables. Some applications for combinatoric problems are discussed.


Coherent systems of genus 0, III: Computation of flips for k=1

Abstract: In this paper we continue the investigation of coherent systems of type $(n,d,k)$ on the projective line which are stable with respect to some value of a parameter $\alpha$. We consider the case $k=1$ and study the variation of the moduli spaces with $\alpha$. We determine inductively the first and last moduli spaces and the flip loci, and give an explicit description for ranks 2 and 3. We also determine the Hodge polynomials explicitly for ranks 2 and 3 and in certain cases for arbitrary rank.


Stable curves and screens on fatgraphs

Abstract: The mapping class group invariant ideal cell decomposition of the Teichmueller space of a punctured surface times an open simplex has been used in a number of computations. This paper answers a question about the asymptotics of this decomposition, namely, in a given cell of the decomposition, which curves can be short? Screens are a new combinatorial structure which provide an answer to this question. The heart of the calculation here involves Ptolemy transformations and the triangle inequalities on lambda lengths.


Coarse and synthetic Weil-Petersson geometry: quasi-flats, geodesics, and relative hyperbolicity

Abstract: We analyze the coarse geometry of the Weil-Petersson metric on Teichm\"uller space, focusing on applications to its synthetic geometry (in particular the behavior of geodesics). We settle the question of the strong relative hyperbolicity of the Weil-Petersson metric via consideration of its coarse quasi-isometric model, the "pants graph." We show that in dimension~3 the pants graph is strongly relatively hyperbolic with respect to naturally defined product regions and show any quasi-flat lies a bounded distance from a single product. For all higher dimensions there is no non-trivial collection of subsets with respect to which it strongly relatively hyperbolic; this extends a theorem of [BDM] in dimension 6 and higher into the intermediate range (it is hyperbolic if and only if the dimension is 1 or 2 [BF]). Stability and relative stability of quasi-geodesics in dimensions up through 3 provide for a strong understanding of the behavior of geodesics and a complete description of the CAT(0)-boundary of the Weil-Petersson metric via curve-hierarchies and their associated "boundary laminations."


On strict inclusions in hierarchies of convex bodies

Abstract: Let $\mathcal I_k$ be the class of convex $k$-intersection bodies in $\mathbb{R}^n$ (in the sense of Koldobsky) and $\mathcal I_k^m$ be the class of convex origin-symmetric bodies all of whose $m$-dimensional central sections are $k$-intersection bodies. We show that 1) $\mathcal I_k^m\not\subset \mathcal I_k^{m+1}$, $k+3\le m<n$, and 2) $\mathcal I_l \not\subset \mathcal I_k$, $1\le k<l < n-3$.


On l^p norms of weighted mean matrices

Abstract: We study $l^{p}$ operator norms of weighted mean matrices using the approaches of Kaluza-Szeg\"o and Redheffer. As an application, we prove a conjecture of Bennett.


Integrable operators and squares of Hankel Matrices

Abstract: In this note, we find sufficient conditions for an operator with kernel of the form $A(x)B(y)-A(x)B(y)/(x-y)$ (which we call a Tracy-Widom type operator) to be the square of a Hankel operator. We consider two contexts: infinite matrices on $\ell^2$, and integral operators on the Hardy space $H^2(\mathbb{T})$. The results can be applied to the discrete Bessel kernel, which is significant in random matrix theory.


Modified Shephard's problem on projections of convex bodies

Abstract: We disprove a conjecture of A. Koldobsky asking whether it is enough to compare $(n-2)$-derivatives of the projection functions of two symmetric convex bodies in the Shephard problem in order to get a positive answer in all dimensions.


Descent on Elliptic Curves and Hilbert's Tenth Problem

Abstract: Descent via an isogeny on an elliptic curve is used to construct two subrings of the field of rational numbers, which are complementary in a strong sense, and for which Hilbert's Tenth Problem is undecidable. This method further develops that of Poonen, who used elliptic divisibility sequences to obtain undecidability results for some large subrings of the rational numbers.


Heisenberg Uncertainty Principle for the q-Bessel Fourier transform

Abstract: In this paper we uses an I.I. Hirschman-W. Beckner entropy argument to give an uncertainty inequality for the $q$-Bessel Fourier transform: $$ \mathcal{F}_{q,v}f(x)=c_{q,v}\int_{0}^{\infty}f(t)j_{v}(xt,q^{2})t^{2v +1}d_{q}t, $$ where $j_v(x,q)$ is the normalized Hahn-Exton $q$-Bessel function.


Subsets of F_p^n without three term arithmetic progressions have several large Fourier coefficients

Abstract: Suppose that f : F_p^n -> [0,1] has expected value t in [p^(-n/9),1] (so, the density t can be quite low!). Furthermore, suppose that support(f) has no three-term arithmetic progressions. Then, we develop non-trivial lower bounds for f_j, which is the jth largest Fourier coefficient of f. This result is similar in spirit to that appearing in an earlier paper [1] by the author; however, in that paper the focus was on the ``small'' Fourier coefficients, whereas here the focus is on the ``large'' Fourier coefficients. Furthermore, the proof in the present paper requires much more sophisticated arguments than those of that other paper.


Quasi-isometries Between Tubular Groups

Abstract: We give a method of constructing maps between tubular groups inductively according to a set of strategies. This map will be a quasi-isometry exactly when the set of strategies is consistent. Conversely, if there exists a quasi-isometry between tubular groups, then there is a consistent set of strategies for them. There is an algorithm that will in finite time either produce a consistent set of strategies or decide that such a set does not exist. Consequently, this algorithm decides whether or not the groups are quasi-isometric.


Variation of Periods Modulo p in Arithmetic Dynamics

Abstract: Let F : V --> V be a self-morphism of a quasiprojective variety defined over a number field K and let P be a point in V(K) with infinite orbit under iteration of F. For each prime ideal p of good reduction, let m_p(F,P) be the size of the F-orbit of the reduction of P modulo p. Fix any e > 0. We show that for almost all primes p, in the sense of analytic density, the orbit size m_p(F,P) is larger than (log(N(p)))^(1-e), where N(p) is the norm of the ideal p.


The patch topology and the ultrafilter topology on the prime spectrum of a commutative ring

Abstract: Let R be a commutative ring and let Spec(R) denote the collection of prime ideals of R. We define a topology on Spec(R) by using ultrafilters and demonstrate that this topology is identical to the well known patch or constructible topology. The proof is accomplished by use of a von Neumann regular ring canonically associated with $R$.


Conjugacy classes and invariant subrings of R-automorphisms of R[x]

Abstract: We consider the group G of R-automorphisms of the polynomial ring R[x] especially in the case where R is the ring of integers modulo n. We describe conjugacy classes in G, and in the case where n = 4, we describe more explicitly the structure of G and determine all rings of invariants of R[x] with respect to subgroups of G.


Noncommutative geometry through monoidal categories I

Abstract: After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasi-coherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, Hopf-Galois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions in commutative geometry (e.g. faithfully flat descent theory, principal fibrations, equivariant and infinitesimal geometry) can be interpreted as noncommutative geometric constructions applied to commutative objects. For such generalized geometry we define global invariants constructing cyclic objects from which we derive Hochschild, cyclic and periodic cyclic homology (with coefficients) in the standard way.


Regulatory Dynamics on Random Networks: Asymptotic Periodicity and Modularity

Abstract: We study the dynamics of discrete-time regulatory networks on random digraphs. For this we define ensembles of deterministic orbits of random regulatory networks, and introduce some statistical indicators related to the long-term dynamics of the system. We prove that, in a random regulatory network, initial conditions converge almost surely to a periodic attractor. We study the subnetworks, which we call modules, where the periodic asymptotic oscillations are concentrated. We proof that those modules are dynamically equivalent to independent regulatory networks.


Polynomials with a common composite

Abstract: Let f and g be nonconstant polynomials over an arbitrary field K. In this paper we study the intersection of the polynomial rings K[f] and K[g], and in particular we ask whether this intersection is larger than K. We completely resolve this question when K has characteristic zero, and in positive characteristic we present various results, examples, and algorithms.


An accurate finite element method for elliptic interface problems

Abstract: A finite element method for elliptic problems with discontinuous coefficients is presented. The discontinuity is assumed to take place along a closed smooth curve. The proposed method allows to deal with meshes that are not adapted to the discontinuity line. The (nonconforming) finite element space is enriched with local basis functions. We prove an optimal convergence rate in the $H^1$--norm. Numerical tests confirm the theoretical results.


K theory of smooth complete toric varieties and related spaces

Abstract: The K-rings of non-singular complex pro jective varieties as well as quasi- toric manifolds were described in terms of generators and relations in an earlier work of the author with V. Uma. In this paper we obtain a similar description for complete non-singular toric varieties. Indeed, our approach enables us to obtain such a description for the more general class of torus manifolds with locally standard torus action and orbit space a homology polytope.


About the isotropy constant of random convex sets

Abstract: Let K be the symmetric convex hull of m independent random vectors uniformly distributed on the unit sphere of R^n. We prove that, for every $\delta>0$, the isotropy constant of K is bounded by a constant $c(\delta)$ with high probability, provided that $m\geq (1+\delta)n$.


k-Ordered Hamilton cycles in digraphs

Abstract: Given a digraph D, the minimum semi-degree of D is the minimum of its minimum indegree and its minimum outdegree. D is k-ordered Hamiltonian if for every ordered sequence of k distinct vertices there is a directed Hamilton cycle which encounters these vertices in this order. Our main result is that every digraph D of sufficiently large order n with minimum semi-degree at least (n+k)/2 -1 is k-ordered Hamiltonian. The bound on the minimum semi-degree is best possible. An undirected version of this result was proved earlier by Kierstead, S\'ark\"ozy and Selkow.


Involutions on 3-Manifolds and Self-dual, Binary Codes

Abstract: We study a correspondence between orientation reversing involutions on compact 3-manifolds with only isolated fixed points and binary, self-dual codes. We show in particular that every such code can be obtained from such an involution. We further relate doubly even codes to Pin^- -structures and Spin-manifolds.


Parameter Estimation in Manneville-Pomeau Processes

Abstract: In this work we study a class of stochastic processes $\{X_t\}_{t\in\N}$, where $X_t = (\phi \circ T_s^t)(X_0)$ is obtained from the iterations of the transformation T_s, invariant for an ergodic probability \mu_s on [0,1] and a continuous by part function $\phi:[0,1] \to \R$. We consider here $T_s:[0,1]\to [0,1]$ the Manneville-Pomeau transformation. The autocorrelation function of the resulting process decays hyperbolically (or polynomially) and we obtain efficient methods to estimate the parameter s from a finite time series. As a consequence we also estimate the rate of convergence of the autocorrelation decay of these processes. We compare different estimation methods based on the periodogram function, on the smoothed periodogram function, on the variance of the partial sum and on the wavelet theory.


Secant dimensions of low-dimensional homogeneous varieties

Abstract: We completely describe the higher secant dimensions of all connected homogeneous projective varieties of dimension at most 3, in all possible equivariant embeddings. In particular, we calculate these dimensions for all Segre-Veronese embeddings of P^1 * P^1, P^1 * P^1 * P^1, and P^2 * P^1, as well as for the variety F of incident point-line pairs in P^2. For P^2 * P^1 and F the results are new, while the proofs for the other two varieties are more compact than existing proofs. Our main tool is the second author's tropical approach to secant dimensions.


Exchangeable partitions derived from Markovian coalescents with simultaneous multiple collisions

Abstract: Kingman derived the Ewens sampling formula for random partitions from the genealogy model defined by a Poisson process of mutations along lines of descent governed by a simple coalescent process. M\"ohle described the recursion which determines the generalization of the Ewens sampling formula when the lines of descent are governed by a coalescent with multiple collisions. In a recent work by Dong, Gnedin and Pitman, authors exploit an analogy with the theory of regenerative composition and partition structures, and provide various characterizations of the associated exchangeable random partitions. This paper gives parallel results for the further generalized model with lines of descent following a coalescent with simultaneous multiple collisions.


Equivariant monads and equivariant lifts versus a 2-category of distributive laws

Abstract: Fix a monoidal category C. The 2-category of monads in the 2-category of C-actegories, colax C-equivarant functors, and C-equivariant natural transformations of colax functors, may be recast in terms of pairs consisting of a usual monad and a distributive law between the monad and the action of C, morphisms of monads respecting the distributive law, and transformations of monads satisfying some compatibility with the actions and distributive laws involved. The monads in this picture may be generalized to actions of monoidal categories, and actions of PRO-s in particular. If C is a PRO as well, then in special cases one gets various distributive laws of a given classical type, for example between a comonad and an endofunctor or between a monad and a comonad. The usual pentagons are in general replaced by multigons, and there are also ``mixed'' multigons involving two distinct distributive laws. Beck's bijection between the distributive laws and lifts of one monad to the Eilenberg-Moore category of another monad is here extended to an isomorphism of 2-categories. The lifts of maps of above mentioned pairs are colax C-equivariant. We finish with a short treatment of relative distributive laws between two pseudoalgebra structures which are relative with respect to the distributivity of two pseudomonads involved, what gives a hint toward the generalizations.


Analysis of the accuracy and convergence of equation-free projection to a slow manifold

Abstract: In [C.W. Gear, T.J. Kaper, I.G. Kevrekidis, and A. Zagaris, Projecting to a Slow Manifold: Singularly Perturbed Systems and Legacy Codes, SIAM J. Appl. Dyn. Syst. 4 (2005) 711-732], we developed a class of iterative algorithms within the context of equation-free methods to approximate low-dimensional, attracting, slow manifolds in systems of differential equations with multiple time scales. For user-specified values of a finite number of the observables, the m-th member of the class of algorithms (m = 0, 1, ...) finds iteratively an approximation of the appropriate zero of the (m+1)-st time derivative of the remaining variables and uses this root to approximate the location of the point on the slow manifold corresponding to these values of the observables. This article is the first of two articles in which the accuracy and convergence of the iterative algorithms are analyzed. Here, we work directly with explicit fast--slow systems, in which there is an explicit small parameter, epsilon, measuring the separation of time scales. We show that, for each m = 0, 1, ..., the fixed point of the iterative algorithm approximates the slow manifold up to and including terms of O(epsilon^m). Moreover, for each m, we identify explicitly the conditions under which the m-th iterative algorithm converges to this fixed point. Finally, we show that when the iteration is unstable (or converges slowly) it may be stabilized (or its convergence may be accelerated) by application of the Recursive Projection Method. Alternatively, the Newton-Krylov Generalized Minimal Residual Method may be used. In the subsequent article, we will consider the accuracy and convergence of the iterative algorithms for a broader class of systems-in which there need not be an explicit small parameter-to which the algorithms also apply.


Computation of the cover of Shimura curves $X_0(2) \to X(1)$ for the cyclic cubic field of discriminant 13^2

Abstract: We compute the canonical model of the cover of Shimura curves $X_0(2) \to X(1)$ for the cubic field of discriminant 13^2 described at the end of Elkies' paper "Shimura curves for level 3 subgroups of the (2,3,7) triangle group". Last, we list the coordinates of some rational CM points on X(1).


Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces

Abstract: This paper provides a fixed point theorem for asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces as well as new effective results on the Krasnoselski-Mann iterations of such mappings. The latter were found using methods from logic and the paper continues a case study in the general program of extracting effective data from prima-facie ineffective proofs in the fixed point theory of such mappings.


On the solutions of a boundary value problem arising in free convection with prescribed heat flux

Abstract: For given $a\in\R$, c<0, we are concerned with the solution $f^{}_b$ of the differential equation $f^{\prime\prime\prime}+ff^{\prime\prime}+\g(f^{\prime})=0$, satisfying the initial conditions $f(0)=a$, $f'(0)=b$, $f''(0)=c< 0$, where g is some nonnegative subquadratic locally Lipschitz function. It is proven that there exists $b_*>0$ such that $f^{}_b$ exists on $[0,+\infty)$ and is such that $f'_b(t)\to 0$ as $t\to+\infty$, if and only if $b\geq b_*$. This allows to answer questions about existence, uniqueness and boundedness of solutions to a boundary value problem arising in fluid mechanics, and especially in boundary layer theory.


A partial $A_\infty$-structure on the cohomology of $C_n\times C_m$

Abstract: Suppose k is a field of characteristic 2, and $n,m\geq 4$ powers of 2. Then the $A_\infty$-structure of the group cohomology algebras $H^*(C_n,k)$ and $(H^*(C_m,k)$ are well known. We give results characterizing an $A_\infty$-structure on $H^*(C_n\times C_m,k)$ including limits on non-vanishing low-arity operations and an infinite family of non-vanishing higher operations.


Asymptotic regimes for the occupancy scheme of multiplicative cascades

Abstract: In the classical occupancy scheme, one considers a fixed discrete probability measure ${\bf p}=(p_i: {i\in{\cal I}})$ and throws balls independently at random in boxes labeled by ${\cal I}$, such that $p_i$ is the probability that a given ball falls into the box $i$. In this work, we are interested in asymptotic regimes of this scheme in the situation induced by a refining sequence $({\bf p}(k) : k\in\N)$ of random probability measures which arise from some multiplicative cascade. Our motivation comes from the study of the asymptotic behavior of certain fragmentation chains


Birational Calabi-Yau 3-folds and BPS state counting

Abstract: This paper contains some applications of Bridgeland-Douglas stability conditions on triangulated categories, and Joyce's work on counting invariants of semistable objects, to the study of birational geometry. We introduce the notion of motivic Gopakumar-Vafa invariants as counting invariants of D2-branes, and show that they are invariant under birational transformations between Calabi-Yau 3-folds. The result is similar to the fact that birational Calabi-Yau 3-folds have the same betti numbers or Hodge numbers.


Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic. Part I

Abstract: We prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic $(0,p)$. As a first application we provide a smooth solution (answer) to a conjecture (question) of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence in arbitrary unramified mixed characteristic $(0,p)$ of integral canonical models of projective Shimura varieties of Hodge type with respect to h--hyperspecial subgroups as pro-\'etale covers of N\'eron models; this forms progress towards the proof of conjectures of Milne and Reimann. Though the second application was known before in some cases, its proof is new and more of a principle.


On certain period relations for cusp forms on GL_n

Abstract: Let $\pi$ be a regular algebraic cuspidal automorphic representation of ${\rm GL}_n({\mathbb A}_F)$ for a number field $F$. We consider certain periods attached to $\pi$. These periods were originally defined by Harder when $n=2$, and later by Mahnkopf when $F = {\mathbb Q}$. In the first part of the paper we analyze the behaviour of these periods upon twisting $\pi$ by algebraic Hecke characters. In the latter part of the paper we consider Shimura's periods associated to a modular form. If $\phi_{\chi}$ is the cusp form associated to a character $\chi$ of a quadratic extension, then we relate the periods of $\phi_{\chi^n}$ to those of $\phi_{\chi}$, and as a consequence give another proof of Deligne's conjecture on the critical values of symmetric power $L$-functions associated to dihedral modular forms. Finally, we make some remarks on the symmetric fourth power $L$-functions.


Iterating the Pimsner construction

Abstract: For $A$ a $C^*$-algebra, $E_1, E_2$ two Hilbert bimodules over $A$, and a fixed isomorphism $\chi : E_1\otimes_AE_2\to E_2\otimes_AE_1$, we consider the problem of computing the $K$-theory of the Cuntz-Pimsner algebra ${\mathcal O}_{E_2\otimes_A{\mathcal O}_{E_1}}$ obtained by extending the scalars and by iterating the Pimsner construction. The motivating examples are a commutative diagram of Douglas and Howe for the Toeplitz operators on the quarter plane, and the Toeplitz extensions associated by Pimsner and Voiculescu to compute the $K$-theory of a crossed product. The applications are for Hilbert bimodules arising from rank two graphs and from commuting endomorphisms of abelian $C^*$-algebras.


On the explicit construction of higher deformations of partition statistics

Abstract: The modularity of the partition generating function has many important consequences, for example asymptotics and congruences for $p(n)$. In a series of papers the author and Ono \cite{BO1,BO2} connected the rank, a partition statistic introduced by Dyson, to weak Maass forms, a new class of functions which are related to modular forms and which were first considered in \cite{BF}. Here we do a further step towards understanding how weak Maass forms arise from interesting partition statistics by placing certain 2-marked Durfee symbols introduced by Andrews \cite{An1} into the framework of weak Maass forms. To do this we construct a new class of functions which we call quasiweak Maass forms because they have quasimodular forms as components. As an application we prove two conjectures of Andrews. It seems that this new class of functions will play an important role in better understanding weak Maass forms of higher weight themselves, and also their derivatives. As a side product we introduce a new method which enables us to prove transformation laws for generating functions over incomplete lattices.


On the subdivision of small categories

Abstract: We present an intrinsic and concrete development of the subdivision of small categories, give some simple examples and derive its fundamental properties. As an application, we deduce an alternative way to compare the homotopy categories of spaces and small categories, by using partially ordered sets. This yields a new conceptual proof to the well-known fact that these two homotopy categories are equivalent.


Cyclotomic $q$-Schur algebras associated to the Ariki-Koike algebra

Abstract: Let $S$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $H_{n,r}$ of rank $n$, introduced by Dipper-James-Mathas. For each $p = (r_1, ..., r_g)$ such that $r_1 + ... + r_g = r$, we define a subalgebra $S^p$ of $S$ and its quotient algebra $\bar S^p$. It is shown that $S^p$ is a standardly based algebra and $\bar S^p$ is a cellular algebra. By making use of these algebras, we show that certain decomposition numbers for $S$ can be expressed as a product of decomposition numbers for cyclotomic $q$-Schur algebras associated to smaller Ariki_koike algebras $H_{n_k,r_k}$.


Fields of invariants of finite linear groups

Abstract: The survey is devoted to the rationality question of finite linear groups. We concentrate on lower-dimensional cases, especially on the case of dimension four.


Relative log convergent cohomology and relative rigid cohomology I

Abstract: In this paper, we develop the theory of relative log convergent cohomology. We prove the coherence of relative log convergent cohomology in certain case by using the comparison theorem between relative log convergent cohomlogy and relative log crystalline cohomology, and we relates relative log convergent cohomology to relative rigid cohomology to show the validity of Berthelot's conjecture on the coherence and the overconvergence of relative rigid cohomology for proper smooth families when they admit nice proper log smooth compactification to which the coefficient extends logarithmically.


Relative log convergent cohomology and relative rigid cohomology II

Abstract: In this paper, we develop the theory of relative log convergent cohomology of radius $\lambda$ ($0 < \lambda \leq 1$), which is a generalization of the notion of relative log convergent cohomology in the previous paper. By comparing this cohomology with relative log crystalline cohomology, relative rigid cohomology and its variants and by using some technique of hypercovering, we prove a version of Berthelot's conjecture on the overconvergence of relative rigid cohomology for proper smooth families.


Statistical properties of a generalized threshold network model

Abstract: The threshold network model is a type of finite random graphs. In this paper, we introduce a generalized threshold network model. A pair of vertices with random weights is connected by an edge when real-valued functions of the pair of weights belong to given Borel sets. We extend several known limit theorems for the number of prescribed subgraphs to show that the strong law of large numbers can be uniform convergence. We also prove two limit theorems for the local and global clustering coefficients.


On the Graf's addition theorem for Hahn Exton q-Bessel function

Abstract: In this paper we study the positivity of the generalized $q$-translation associated with the $q$-Bessel Hahn Exton function which is deduced by a new formulation of the Graf's addition formula related to this function.


Random environment on coloured trees

Abstract: In this paper, we study a regular rooted coloured tree with random labels assigned to its edges, where the distribution of the label assigned to an edge depends on the colours of its endpoints. We obtain some new results relevant to this model and also show how our model generalizes many other probabilistic models, including random walk in random environment on trees, recursive distributional equations and multi-type branching random walk on $\mathbb{R}$.


Algebraic Connections vs. Algebraic {$\cD$}-modules: inverse and direct images

Abstract: In the dictionary between the language of (algebraic integrable) connections and that of (algebraic) $\cD$-modules, to compare the definitions of inverse images for connections and $\cD$-modules is easy. But the comparison between direct images for connections (the classical construction of the Gauss-Manin connection for smooth morphisms) and for $\cD$-modules, although known to specialists, has been explicitly proved only recently in a paper of Dimca, Maaref, Sabbah and Saito in 2000, where the authors' main technical tool was M. Saito's equivalence between the derived category of $\cD$-modules and a localized category of differential complexes. The aim of this short paper is to give a simplified summary of the [DMSS] argument, and to propose an alternative proof of this comparison which is simpler, in the sense that it does not use Saito equivalence. Moreover, our alternative strategy of comparison works in a context which is a precursor to the Gauss-Manin connection (at the level of $f^{-1}\cD_Y$-modules, for a morphism $f:X\to Y$), and may be of some intrinsic interest.


On decomposition numbers with Jantzen filtration of cyclotomic $q$-Schur algebras

Abstract: Let $\Sc(\vL)$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $\He_{n,r}$, introduced by Dipper-James-Mathas. In this paper, we consider $v$-decomposition numbers of $\Sc(\vL)$, namely decomposition numbers with respect to the Jantzen filtrations of Weyl modules. We prove, as a $v$-analogue of the result obtained by Shoji-Wada, a product formula for $v$-decomposition numbers of $\Sc(\vL)$, which asserts that certain $v$-decomposition numbers are expressed as a product of $v$-decomposition numbers for various cyclotomic $q$-Schur algebras associated to Ariki-koike algebras $\He_{n_i,r_i}$ of smaller rank. Moreover we prove a similar formula for $v$-decomposition numbers of $\He_{n,r}$ by using a Schur functor.


On the divisor function and the Riemann zeta-function in short intervals

Abstract: We obtain, for $T^\epsilon \le U=U(T)\le T^{1/2-\epsilon}$, asymptotic formulas for $$ \int_T^{2T}(E(t+U) - E(t))^2 dt,\quad \int_T^{2T}(\Delta(t+U) - \Delta(t))^2 dt, $$ where $\Delta(x)$ is the error term in the classical divisor problem, and $E(T)$ is the error term in the mean square formula for $|\zeta(1/2+it)|$. Upper bounds of the form $O_\epsilon(T^{1+\epsilon}U^2)$ for the above integrals with biquadrates instead of square are shown to hold for $T^{3/8} \le U =U(T) \ll T^{1/2}$. The connection between the moments of $E(t+U) - E(t)$ and $|\zeta(1/2+it)|$ is also given. Generalizations to some other number-theoretic error terms are discussed.


E_0-dilation of strongly commuting CP_0-semigroups

Abstract: We prove that every strongly commuting pair of CP_0-semigroups has a minimal E_0-dilation. This is achieved in two major steps, interesting in themselves: 1: we show that a strongly commuting pair of CP_0-semigroups can be represented via a two parameter product system representation; 2: we prove that every fully coisometric product system representation has a fully coisometric, isometric dilation. In particular, we obtain that every commuting pair of CP_0-semigroups on B(H), H finite dimensional, has an E_0-dilation.


On Excess of the Odious Primes

Abstract: We give a more strong heuristic justification of our conjecture on the excess of the odious primes.


On the discriminant of elliptic curves with non-trivial torsion

Abstract: For those elliptic curves defined over the rational with non--trivial torsion subgroup, we find a tight relationship between the torsion subgroup itself and a Galois group naturally arising from the curve.


Harmonic analysis on local fields and adelic spaces I

Abstract: We develop a harmonic analysis on objects of some category $C_2$ of infinite-dimensional filtered vector spaces over a finite field. It includes two-dimensional local fields and adelic spaces of algebraic surfaces defined over a finite field. The main result is the theory of the Fourier transform on these objects and two-dimensional Poisson formulas.


The spectral data for Hamiltonian stationary Lagrangian tori in R^4

Abstract: This article determines the spectral data, in the integrable systems sense, for all weakly conformally immersed Hamiltonian stationary Lagrangian in $\R^4$. This enables us to describe their moduli space and the locus of branch points of such an immersion. This is also an informative example in integrable systems geometry, since the group of ambient isometries acts non-trivially on the spectral data and the relevant energy functional (the area) need not be constant under deformations by higher flows.


On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions

Abstract: We consider a two-component competition-diffusion system with equal diffusion coefficients and inhomogeneous Dirichlet boundary conditions. When the interspecific competition parameter tends to infinity, the system solution converges to that of a freeboundary problem. If all stationary solutions of this limit problem are non-degenerate and if a certain linear combination of the boundary data does not identically vanish, then for sufficiently large interspecific competition, all non-negative solutions of the competition-diffusion system converge to stationary states as time tends to infinity. Such dynamics are much simpler than those found for the corresponding system with either homogeneous Neumann or homogeneous Dirichlet boundary conditions.


A note on the Hayman-Wu theorem

Abstract: The Hayman-Wu theorem states that the preimage of a line or circle L under a conformal mapping from the unit disc to a simply-connected domain U has total Euclidean length bounded by an absolute constant. The best possible constant is known to lie in the interval [pi^2, 4 pi), thanks to work of {\O}yma and Rohde. Earlier, Brown Flinn showed that the total length is at most pi^2 in the special case in which U contains L. Let r be the anti-M\"obius map that fixes L pointwise. In this note we extend the sharp bound pi^2 to the case where each connected component of the intersection of U with r(U) is bounded by one arc of U and its image under r. We also strengthen the bounds slightly by replacing Euclidean length with the strictly larger spherical length restricted to the unit disc.


The Poisson Kernel for Hardy Algebras

Abstract: This note contributes to a circle of ideas that we have been developing recently in which we view certain abstract operator algebras $H^{\infty}(E)$, which we call Hardy algebras, and which are noncommutative generalizations of classical $H^{\infty}$, as spaces of functions defined on their spaces of representations. We define a generalization of the Poisson kernel, which ``reproduces'' the values, on $\mathbb{D}((E^{\sigma})^*)$, of the ``functions'' coming from $H^{\infty}(E)$. We present results that are natural generalizations of the Poisson integral formuala. They also are easily seen to be generalizations of formulas that Popescu developed. We relate our Poisson kernel to the idea of a characteristic operator function and show how the Poisson kernel identifies the ``model space'' for the canonical model that can be attached to a point in the disc $\mathbb{D}((E^{\sigma})^*)$. We also connect our Poission kernel to various "point evaluations" and to the idea of curvature.


A new approach to the giant component problem

Abstract: We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n tend to infinity. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability all the components are small, and other conditions that imply that with high probability there is a giant component and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the results by Molloy and Reed on the size of the largest component in a random graph with a given degree sequence. We further obtain a new sharp result for the giant component just above the threshold, generalizing the case of G(n,p) with np=1+omega(n)n^{-1/3}, where omega(n) tends to infinity arbitrarily slowly. Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs.


Canonical connections on paracontact manifolds

Abstract: The canonical paracontact connection is defined and it is shown that its torsion is the obstruction the paracontact manifold to be paraSasakian. A $\mathcal{D}$-homothetic transformation is determined as a special gauge transformation. The $\eta$-Einstein manifold are defined, it is prove that their scalar curvature is a constant and it is shown that in the paraSasakian case these spaces can be obtained from Einstein paraSasakian manifolds with a $\mathcal{D}$-homothetic transformations. It is shown that an almost paracontact structure admits a connection with totally skew-symmetric torsion if and only if the Nijenhuis tensor of the paracontact structure is skew-symmetric and the defining vector field is Killing.


A note on the radial solutions for the supercritical Henon equation

Abstract: We prove the existence of a positive radial solution for the H\'enon equation with arbitrary growth. The solution is found by means of a shooting method and turns out to be an increasing function of the radial variable. Some numerical experiments suggest the existence of many positive oscillating solutions.


Equations for Chow and Hilbert Quotients

Abstract: We give explicit equations for the Chow and Hilbert quotients of a projective scheme X by the action of an algebraic torus T in an auxiliary toric variety. As a consequence we provide GIT descriptions of these canonical quotients, and obtain other GIT quotients of X by variation of GIT quotient. We apply these results to find equations for the moduli space \bar{M}_{0,n} of stable genus zero n-pointed curves as a subvariety of a smooth toric variety defined via tropical methods.


Universal L^s -rate-optimality of L^r-optimal quantizers by dilatation and contraction

Abstract: Let $ r, s>0 $. For a given probability measure $P$ on $\mathbb{R}^d$, let $(\alpha_n)_{n \geq 1}$ be a sequence of (asymptotically) $L^r(P)$- optimal quantizers. For all $\mu \in \mathbb{R}^d $ and for every $\theta >0$, one defines the sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$ by : $\forall n \geq 1, \alpha_n^{\theta, \mu} = \mu + \theta(\alpha_n - \mu) = \{\mu + \theta(a- \mu), a \in \alpha_n \} $. In this paper, we are interested in the asymptotics of the $L^s$-quantization error induced by the sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$. We show that for a wide family of distributions, the sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$ is $L^s$-rate-optimal. For the Gaussian and the exponential distributions, one shows how to choose the parameter $\theta$ such that $(\alpha_n^{\theta, \mu})_{n \geq 1}$ satisfies the empirical measure theorem and probably be asymptotically $L^s$-optimal.


The Overconvergent Site II. Cohomology

Abstract: We prove that rigid cohomology can be computed as the cohomology of a site analogous to the crystalline site. Berthelot designed rigid cohomology as a common generalization of crystalline and Monsky-Washnitzer cohomology. Unfortunately, unlike the former, the functoriality of the theory is not built-in. We defined somewhere else the "overconvergent site" which is functorially attached to an algebraic variety and proved that the category of modules of finite presentation on this ringed site is equivalent to the category of over- convergent isocrystals on the variety. We show here that their cohomology also coincides.


Large continuum, oracles

Abstract: Our main theorem is about iterated forcing for making the continuum larger than aleph_2. We present a generalization of math.LO/0303294 which is dealing with oracles for random, etc., replacing aleph_1, aleph_2 by lambda,lambda^+ (starting with lambda=lambda^{<lambda}>aleph_1). Well, instead of properness we demand absolute c.c.c. So we get, e.g. the continuum is lambda^+ but we can get cov(meagre)=lambda. We give some applications. As in math.LO/0303294, it is a "partial" countable support iteration but it is c.c.c.


A solid angle polynomial with negative coefficients

Abstract: This article has been replaced by arXiv:0906.4031


An extended version of additive K-theory

Abstract: There are two infinitesimal (i.e., additive) versions of the $K$-theory of a field $F$: one was introduced by Cathelineau, which is an $F$-module, and another one introduced by Bloch-Esnault, which is an $F^*$-module. Both versions are equipped with a regulator map, when $F$ is the field of complex numbers. In our short paper we will introduce an extended version of Cathelineau's group, and a complex-valued regulator map given by the entropy. We will also give a comparison map between our extended version and Cathelineau's group. Our results were motivated by two unrelated sources: Neumann's work on the extended Bloch group (which is isomorphic to indecomposable $K_3$ of the complex numbers), and the study of singularities of generating series of hypergeometric multisums. Final version.


On the geometry of the moduli space of spin curves

Abstract: We determine the smooth locus and the locus of canonical singularities in the Cornalba compactification \bar S_g of the moduli space S_g of spin curves, i.e., smooth curves of genus g with a theta characteristic. Moreover, the following lifting result for pluricanonical forms is proved: Every pluricanonical form on the smooth locus of \bar S_g extends holomorphically to a desingularisation of \bar S_g.


On the girth of random Cayley graphs

Abstract: We prove that random d-regular Cayley graphs of the symmetric group asymptotically almost surely have girth at least (log_{d-1}|G|)^{1/2}/2 and that random d-regular Cayley graphs of simple algebraic groups over F_q asymptotically almost surely have girth at least log_{d-1}|G|/dim(G). For the symmetric p-groups the girth is between log log |G| and (log|G|)^alpha with alpha<1. Several conjectures and open questions are presented.


Polynomials with PSL(2) monodromy

Abstract: Let K be a field of characteristic p>0, and let q be a power of p. We determine all polynomials f in K[t]\K[t^p] of degree q(q-1)/2 such that the Galois group of f(t)-u over K(u) has a transitive normal subgroup isomorphic to PSL_2(q), subject to a certain ramification hypothesis. As a consequence, we describe all polynomials f in K[t], of degree not a power of p, such that f is functionally indecomposable over K but f decomposes over an extension of K. Moreover, except for one ramification setup (which is treated in the companion paper arxiv:0707.1837), we describe all indecomposable polynomials f in K[t] of non-p-power degree which are exceptional, in the sense that x-y is the only absolutely irreducible factor of f(x)-f(y) which lies in K[x,y]. It is known that, when K is finite, a polynomial f is exceptional if and only if it induces a bijection on infinitely many finite extensions of K.


Type-II Matrices and Combinatorial Structures

Abstract: Type-II matrices are a class of matrices used by Jones in his work on spin models. In this paper we show that type-II matrices arise naturally in connection with some interesting combinatorial and geometric structures.


A new family of exceptional polynomials in characteristic two

Abstract: We produce a new family of polynomials f(x) over fields K of characteristic 2 which are exceptional, in the sense that f(x)-f(y) has no absolutely irreducible factors in K[x,y] besides the scalar multiples of x-y; when K is finite, this condition is equivalent to saying there are infinitely many finite extensions L/K for which the map c --> f(c) is bijective on L. Our polynomials have degree (2^e-1)*2^(e-1), where e is odd. Combined with our previous paper arxiv:0707.1835, this completes the classification of indecomposable exceptional polynomials of degree not a power of the characteristic. The strategy of our proof is to identify the curves that can arise as the Galois closure of the branched cover P^1 --> P^1 induced by an exceptional polynomial f. In this case, the curves turn out to be x^(q+1)+y^(q+1)=a+T(xy), where T(z)=z^(q/2)+z^(q/4)+...+z. Our proofs rely on new properties of ramification in Galois covers of curves, as well as the computation of the automorphism groups of all curves in a certain 2-parameter family.


Computing the complete CS decomposition

Abstract: An algorithm is developed to compute the complete CS decomposition (CSD) of a partitioned unitary matrix. Although the existence of the CSD has been recognized since 1977, prior algorithms compute only a reduced version (the 2-by-1 CSD) that is equivalent to two simultaneous singular value decompositions. The algorithm presented here computes the complete 2-by-2 CSD, which requires the simultaneous diagonalization of all four blocks of a unitary matrix partitioned into a 2-by-2 block structure. The algorithm appears to be the only fully specified algorithm available. The computation occurs in two phases. In the first phase, the unitary matrix is reduced to bidiagonal block form, as described by Sutton and Edelman. In the second phase, the blocks are simultaneously diagonalized using techniques from bidiagonal SVD algorithms of Golub, Kahan, and Demmel. The algorithm has a number of desirable numerical features.


Foundations of the calculus of variations in generalized function algebras

Abstract: We propose the use of algebras of generalized functions for the analysis of certain highly singular problems in the calculus of variations. After a general study of extremal problems on open subsets of Euclidean space in this setting we introduce the first and second variation of a variational problem. We then derive necessary (Euler-Lagrange equations) and sufficient conditions for extremals. The concept of association is used to obtain connections to a distributional description of singular variational problems. We study variational symmetries and derive an appropriate version of N\"other's theorem. Finally, a number of applications to geometry, mechanics, elastostatics and elastodynamics are presented.


On a theorem in multi-parameter potential theory

Abstract: We prove a theorem on additive Levy processes and give applications


On a general theorem for additive Levy processes

Abstract: We prove a new theorem on additive Levy processes and show that this theorem implies several proved theorems and a hard conjectured theorem.


Jones Pairs

Abstract: Motivated by Jones' braid group representations constructed from spin models, we define {\sl a Jones pair} to be a pair of $\nbyn$ matrices $(A,B)$ such that the endomorphisms $X_A$ and $\D_B$ form a representation of a braid group. When $A$ and $B$ are type-II matrices, we call $(A,B)$ {\sl an invertible Jones pair}. We develop the theory of Jones pairs in this thesis. Our aim is to study the connections among association schemes, spin models and four-weight spin models using the viewpoint of Jones pairs. We use Nomura's method to construct a pair of algebras from the matrices $(A,B)$, which we call the Nomura algebras of $(A,B)$. These algebras become the central tool in this thesis. We explore their properties in Chapters \ref{Nomura} and \ref{IINom}. In Chapter \ref{JP}, we introduce Jones pairs. We prove the equivalence of four-weight spin models and invertible Jones pairs. We extend some existing concepts for four-weight spin models to Jones pairs. In Chapter \ref{SpinModels}, we provide new proofs for some well-known results on the Bose-Mesner algebras associated with spin models. We document the main results of the thesis in Chapter \ref{InvJP}. We prove that every four-weight spin model comes from a symmetric spin model (up to odd-gauge equivalence). We present four Bose-Mesner algebras associated to each four-weight spin model. We study the relations among these algebras. In particular, we provide a strategy to search for four-weight spin models. This strategy is analogous to the method given by Bannai, Bannai and Jaeger for finding spin models.


Hausdorrf dimension for level sets and k-multiple times

Abstract: We compute the Hausdorff dimension of the zero set of an additive Levy process.


The topology of terminal quartic 3-folds

Abstract: In this thesis, I determine a bound on the defect of terminal Gorenstein quartic 3-folds. More generally, I study the defect of terminal Gorenstein Fano 3-folds of Picard rank 1 and genus at least 3. I state a geometric "motivation" of non Q-factoriality in the case of quartics.


The Gauss-Bonnet-Grotemeyer Theorem in spaces of constant curvature

Abstract: In 1963, K.P.Grotemeyer proved an interesting variant of the Gauss-Bonnet Theorem. Let M be an oriented closed surface in the Euclidean space R^3 with Euler characteristic \chi(M), Gauss curvature G and unit normal vector field n. Grotemeyer's identity replaces the Gauss-Bonnet integrand G by the normal moment <a,n>^2G, where $a$ is a fixed unit vector. Grotemeyer showed that the total integral of this integrand is (2/3)pi times chi(M). We generalize Grotemeyer's result to oriented closed even-dimesional hypersurfaces of dimension n in an (n+1) ndimensional space form N^{n+1}(k).


Topologies on the space of holomorphic functions

Abstract: We show that a fairly arbitrary Frechet space topology on the space of holomorphic functions on a domain controls the topology of uniform convergence on compact sets. In fact it turns out that the result we present can be proved more simply using the closed graph theorem. However, we believe that the techniques presented here may be used to prove a more interesting result. Details to appear later.


Dynamical systems associated with crossed products

Abstract: In this paper, we consider both algebraic crossed products of commutative complex algebras A with the integers under an automorphism of A, and Banach algebra crossed products of commutative C^*-algebras A with the integers under an automorphism of A. We investigate, in particular, connections between algebraic properties of these crossed products and topological properties of naturally associated dynamical systems. For example, we draw conclusions about the ideal structure of the crossed product by investigating the dynamics of such a system. To begin with, we recall results in this direction in the context of an algebraic crossed product and give simplified proofs of generalizations of some of these results. We also investigate new questions, for example about ideal intersection properties of algebras properly between the coefficient algebra A and its commutant A'. Furthermore, we introduce a Banach algebra crossed product and study the relation between the structure of this algebra and the topological dynamics of a naturally associated system.


Maltsiniotis's first conjecture for K_1

Abstract: We show that K_1 of an exact category agrees with K_1 of the associated triangulated derivator. More generally we show that K_1 of a Waldhausen category with cylinders and a saturated class of weak equivalences coincides with K_1 of the associated right pointed derivator.


The $\ell^2$-homology of even Coxeter groups

Abstract: Given a Coxeter system (W,S), there is an associated CW-complex, Sigma, on which W acts properly and cocompactly. We prove that when the nerve L of (W,S) is a flag triangulation of the 3-sphere, then the reduced $\ell^2$-homology of Sigma vanishes in all but the middle dimension.


Application of Extended Kalman Filter to Tactical Ballistic Missile Re-entry Problem

Abstract: The objective is to investigate the advantages and performance of Extended Kalman Filter for the estimation of non-linear system where linearization takes place about a trajectory that was continually updated with the state estimates resulting from the measurement. Here tactile ballistic missile Re-entry problem is taken as a nonlinear system model and Extended Kalman Filter technique is used to estimate the positions and velocities at the X and Y direction at different values of ballistic coefficients. The result shows that the method gives better estimation with the increase of ballistic coefficient.


The Elements of an Analysis of the Functions of a Set

Abstract: The variant of calculation of functions of set and their application is offered. In particular: the new measure of system of sets generalizing classical concept of a measure is entered; the variation of set that has allowed to construct a derivative of function of set on a measure is entered, and to receive necessary conditions of optimization in terms of function of set.


On plate decompositions of cone multipliers

Abstract: An important inequality due to Wolff on plate decompositions of cone multipliers is known to have consequences for a variety of problems in harmonic analysis. We observe that the range in Wolff's inequality, for the conic and the spherical versions, can be improved by using bilinear restriction results. We also use this inequality to give some improved estimates on square functions associated to decompositions of cone multipliers in low dimensions. This gives a new L^4 bound for the cone multiplier operator in three dimensions.


A reinterpretation of Emerton's $p$-adic Banach spaces

Abstract: It is shown that the $p$-adic Banach spaces introduced by Emerton are isomorphic to the cohomology groups of the sheaf of continuous $\Q_{p}$-valued functions on a certain space. Some applications of this result are discussed.


The uniqueness of the helicoid in the Lorentz-Minkowski space L3

Abstract: In this paper we deal with the uniqueness of the Lorentzian helicoid and Enneper's surface among properly embedded maximal surfaces with lightlike boundary of mirror symmetry in the Lorentz-Minkowski space L3.


Existence of positive solutions for nonlinear systems

Abstract: This paper deals with the existence of positive solutions for the nonlinear system q(t)\phi(p(t)u'_{i}(t)))'+f^{i}(t,\textbf{u})=0,\quad 0<t<1,\quad i=1,2,...,n. This system often arises in the study of positive radial solutions of nonlinear elliptic system. Here $\textbf{u}=(u_{1},...,u_{n})$ and $f^{i}, i=1,2,...,n$ are continuous and nonnegative functions, $p(t), q(t)\hbox{\rm :} [0,1]\to (0,\oo)$ are continuous functions. Moreover, we characterize the eigenvalue intervals for (q(t)\phi(p(t)u'_{i}(t)))'+\lambda h_{i}(t)g^{i} (\textbf{u})=0, \quad 0<t<1,\quad i=1,2,...,n. The proof is based on a well-known fixed point theorem in cones.


Strong convergence of modified Ishikawa iterations for nonlinear mappings

Abstract: In this paper, we prove a strong convergence theorem of modified Ishikawa iterations for relatively asymptotically nonexpansive mappings in Banach space. Our results extend and improve the recent results by Nakajo, Takahashi, Kim, Xu, Matsushita and some others.


Solutions for a class of iterated singular equations

Abstract: Some fundamental solutions of radial type for a class of iterated elliptic singular equations including the iterated Euler equation are given.


Ergodic theory of amenable semigroup actions

Abstract: In this paper, among other things, we state and prove the mean ergodic theorem for amenable semigroup algebras.


Some further remarks on good sets

Abstract: We show that in n-fold cartesian product, n >= 4, a related component need not be a full component. We also prove that when n >= 4, uniform boundedness of lengths of geodesics is not a necessary condition for boundedness of solutions of (1) for bounded function f.


Weighted composition operators on weighted Bergman spaces of bounded symmetric domains

Abstract: In this paper, we study the weighted compositon operators on weighted Bergman spaces of bounded symmetric domains. The necessary and sufficient conditions for a weighted composition operator $W_{\phi,\psi}$ to be bounded and compact are studied by using the Carleson measure techniques. In the last section, we study the Schatten p-class weighted composition operators.


Infinite dimensional differential games with hybrid controls

Abstract: A two-person zero-sum infinite dimensional differential game of infinite duration with discounted payoff involving hybrid controls is studied. The minimizing player is allowed to take continuous, switching and impulse controls whereas the maximizing player is allowed to take continuous and switching controls. By taking strategies in the sense of Elliott--Kalton, we prove the existence of value and characterize it as the unique viscosity solution of the associated system of quasi-variational inequalities.


Borel hierarchies in infinite products of Polish spaces

Abstract: Let H be a product of countably infinite number of copies of an uncountable Polish space X. Let $\Sigma_\xi$ $(\bar {\Sigma}_\xi)$ be the class of Borel sets of additive class \xi for the product of copies of the discrete topology on X (the Polish topology on X), and let ${\cal B} = \cup_{\xi < \omega_1} \bar{\Sigma}_\xi$. We prove in the L\'{e}vy--Solovay model that \bar{\Sigma}_\xi =\Sigma_{\xi}\cap {\cal B} for $1 \leq \xi < \omega_1$.


Law of iterated logarithm for NA sequences with non-identical distributions

Abstract: Based on a law of the iterated logarithm for independent random variables sequences, an iterated logarithm theorem for NA sequences with non-identical distributions is obtained. The proof is based on a Kolmogrov-type exponential inequality.


Fractional extensions of some boundary value problems in oil strata

Abstract: In the present paper, we solve three boundary value problems related to the temperature field in oil strata -- the fractional extensions of the incomplete lumped formulation and lumped formulation in the linear case and the fractional generalization of the incomplete lumped formulation in the radial case. By using the Caputo differintegral operator and the Laplace transform, the solutions are obtained in integral forms where the integrand is expressed in terms of the convolution of some auxiliary functions of Wright function type. A generalization of the Laplace transform convolution theorem, known as Efros' theorem is widely used.


A variant of Davenport's constant

Abstract: Let p be a prime number. Let G be a finite abelian p-group of exponent n (written additively) and A be a non-empty subset of $]n[:= \{1,2,..., n\}$ such that elements of A are incongruent modulo p and non-zero modulo p. Let $k \geq D(G)/|A|$ be any integer where D(G) denotes the well-known Davenport's constant. In this article, we prove that for any sequence g_1, g_2, ..., g_k (not necessarily distinct) in G, one can always extract a subsequence g_{i_1}, g_{i_2}, ..., g_{i_\ell} with $1\leq \ell \leq k$ such that \begin{equation*} \sum_{j=1}^\ell a_{j}g_{i_j} = 0 {in} G, \end{equation*} where a_j \in A for all j. We provide examples where this bound cannot be improved. Furthermore, for the cyclic groups, we prove some sharp results in this direction. In the last section, we explore the relation between this problem and a similar problem with prescribed length. The proof of Theorem~1 uses group-algebra techniques, while for the other theorems, we use elementary number theory techniques.


Weak quantization of Poisson structures

Abstract: In this paper we prove that any Poisson structure on a sheaf of Lie algebroids admits a weak deformation quantization, and give a sufficient condition for such a Poisson structure to admit an actual deformation quantization. We also answer the corresponding classification problems. In the complex symplectic case, we recover in particular some results of Nest-Tsygan and Polesello-Schapira. We begin the paper with a recollection of known facts about deformation theory of cosimplicial differential graded Lie algebras.


Hidden Life of Riemann's Zeta Function 1. Arrow, Bow, and Targets

Abstract: The Riemann Hypothesis is reformulated as statements about eigenvalues of some matrices entries of which are defined via Taylor coefficient of the zeta function. These eigenvalues demonstrate interesting visual patterns allowing one to state a number of conjectures.


Ratio Asymptotic of Hermite-Pad\'e Orthogonal Polynomials for Nikishin Systems. II

Abstract: We prove ratio asymptotic for sequences of multiple orthogonal polynomials with respect to a Nikishin system of measures ${\mathcal{N}}(\sigma_1,...,\sigma_m)$ such that for each $k$, the support of $\sigma_k$ consists of an interval $\widetilde{\Delta}_k$, on which $\sigma_k^{\prime} > 0$ almost everywhere, and a set without accumulation points in $\mathbb{R} \setminus \widetilde{\Delta}_k$.


Coarse topology, enlargeability, and essentialness

Abstract: Using methods from coarse topology we show that fundamental classes of closed enlargeable manifolds map non-trivially both to the rational homology of their fundamental groups and to the K-theory of the corresponding reduced C*-algebras. Our proofs do not depend on the Baum--Connes conjecture and provide independent confirmation for specific predictions derived from this conjecture.


A geometric model for Hochschild homology of Soergel bimodules

Abstract: An important step in the calculation of the triply graded link homology theory of Khovanov and Rozansky is the determination of the Hochschild homology of Soergel bimodules for SL(n). We present a geometric model for this Hochschild homology for any simple group G, as equivariant intersection homology of B x B-orbit closures in G. We show that, in type A these orbit closures are equivariantly formal for the conjugation T-action. We use this fact to show that in the case where the corresponding orbit closure is smooth, this Hochschild homology is an exterior algebra over a polynomial ring on generators whose degree is explicitly determined by the geometry of the orbit closure, and describe its Hilbert series, proving a conjecture of Jacob Rasmussen.


Functions of q-positive type

Abstract: In this paper we characterize the subspace of $\mathcal{L}_{q,1,v}$ of function which are the q-Bessel Fourier transform of positive functions in $\mathcal{L}_{q,1,v}$. As application we give a q-version of the Bochner's theorem.


Optimal Non-Linear Models for Sparsity and Sampling

Abstract: Given a set of vectors (the data) in a Hilbert space H, we prove the existence of an optimal collection of subspaces minimizing the sum of the square of the distances between each vector and its closest subspace in the collection. This collection of subspaces gives the best sparse representation for the given data, in a sense defined in the paper, and provides an optimal model for sampling in union of subspaces. The results are proved in a general setting and then applied to the case of low dimensional subspaces of R^N and to infinite dimensional shift-invariant spaces in L^2(R^d). We also present an iterative search algorithm for finding the solution subspaces. These results are tightly connected to the new emergent theories of compressed sensing and dictionary design, signal models for signals with finite rate of innovation, and the subspace segmentation problem.


Exit problems associated with affine reflection groups

Abstract: We obtain a formula for the distribution of the first exit time of Brownian motion from the alcove of an affine Weyl group. In most cases the formula is expressed compactly, in terms of Pfaffians. Expected exit times are derived in the type \~A case. The results extend to other Markov processes. We also give formulas for the real eigenfunctions of the Dirichlet and Neumann Laplacians on alcoves, observing that the `Hot Spots' conjecture of J. Rauch is true for alcoves.


Generalized motion of level sets by functions of their curvatures on Riemannian manifolds

Abstract: We consider the generalized evolution of compact level sets by functions of their normal vectors and second fundamental forms on a Riemannian manifold M. The level sets of a function $u:M\to\mathbb{R}$ evolve in such a way whenever u solves an equation $u_{t}+F(Du, D^{2}u)=0$, for some real function F satisfying a geometric condition. We show existence and uniqueness of viscosity solutions to this equation under the assumptions that M has nonnegative curvature, F is continuous off Du=0, (degenerate) elliptic, and locally invariant by parallel translation. We then prove that this approach is geometrically consistent, hence it allows to define a generalized evolution of level sets by very general, singular functions of their curvatures. For instance, these assumptions on F are satisfied when F is given by the evolutions of level sets by their mean curvature (even in arbitrary codimension) or by their positive Gaussian curvature. We also prove that the generalized evolution is consistent with the classical motion by the corresponding function of the curvature, whenever the latter exists. When M is not of nonnegative curvature, the same results hold if one additionally requires that F is uniformly continuous with respect to D^2 u. Finally we give some counterexamples showing that several well known properties of the evolutions in $\mathbb{R}^{n}$ are no longer true when M has negative sectional curvature.


Toeplitz and Hankel operators and Dixmier traces on the unit ball of \mathbb C^n

Abstract: We compute the Dixmier trace of pseudo-Toeplitz operators on the Fock space. As an application we find a formula for the Dixmier trace of the product of commutators of Toeplitz operators on the Hardy and weighted Bergman spaces on the unit ball of \mathbb C^d. This generalizes an earlier work of Helton-Howe for the usual trace of the anti-symmetrization of Toeplitz operators.


Cotorsion pairs generated by modules of bounded projective dimension

Abstract: We apply the theory of cotorsion pairs to study closure properties of classes of modules with finite projective dimension with respect to direct limit operations and to filtrations. We also prove that if the ring is an order in an $\aleph_0$-noetherian ring Q of small finitistic dimension 0, then the cotorsion pair generated by the modules of projective dimension at most one is of finite type if and only if Q has big finitistic dimension 0. This applies, for example, to semiprime Goldie rings and Cohen Macaulay noetherian commutative rings.


Proper actions of lamplighter groups associated with free groups

Abstract: Given a finite group $H$ and a free group $F_n$, we prove that the wreath product $H\wr F_n$ admits a metrically proper, isometric action on a Hilbert space.


A GIT Construction of Moduli Spaces of Stable Maps in Positive Characteristic

Abstract: In a previous paper, the author and David Swinarski constructed the moduli spaces of stable maps, \bar M_g,n(P^r,d), via geometric invariant theory (GIT). That paper required the base field to be the complex numbers, a restriction which this paper removes: here the coarse moduli spaces of stable maps are constructed via GIT over a more general base.


The Mukai pairing, I: a categorical approach

Abstract: We study the Hochschild homology of smooth spaces, emphasizing the importance of a pairing which generalizes Mukai's pairing on the cohomology of K3 surfaces. We show that integral transforms between derived categories of spaces induce, functorially, linear maps on homology. Adjoint functors induce adjoint linear maps with respect to the Mukai pairing. We define a Chern character with values in Hochschild homology, and we discuss analogues of the Hirzebruch-Riemann-Roch theorem and the Cardy Condition from physics. This is done in the context of a 2-category which has spaces as its objects and integral kernels as its 1-morphisms.


The structure of surfaces mapping to the moduli stack of canonically polarized varieties

Abstract: Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective surface that maps to the moduli stack, we employ extension properties of logarithmic pluri-forms to establish a strong relationship between the moduli map and the minimal model program of the surface. As a result, we can describe the fibration induced by the moduli map quite explicitly. A refined affirmative answer to Viehweg's conjecture for families over surfaces follows as a corollary.


Integral formulas for a class of curvature PDE's and applications to isoperimetric inequalities and to symmetry problems

Abstract: We prove integral formulas for closed hypersurfaces in C^{n+1}, which furnish a relation between elementary symmetric functions in the eigenvalues of the complex Hessian matrix of the defining function and the Levi curvatures of the hypersurface. Then we follow the Reilly approach to prove an isoperimetric inequality. As an application, we obtain the ``Soap Bubble Theorem'' for star-shaped domains with positive and constant Levi curvatures bounding the classical mean curvature from above.


Constant Mean Curvature Hypersurfaces in the (n+1)-Sphere by Gluing Spherical Building Blocks

Abstract: The techniques developed by Butscher in arXiv:math/0703469 for constructing constant mean curvature (CMC) hypersurfaces in the (n+1)-sphere by gluing together spherical building blocks are generalized to handle less symmetric initial configurations. The outcome is that the approximately CMC hypersurface obtained by gluing the initial configuration together can be perturbed into an exactly CMC hypersurface only when certain global geometric conditions are met. These `balancing conditions' are analogous to those that must be satisfied in the `classical' context of gluing constructions of CMC hypersurfaces in Euclidean space, although they are more restrictive in the (n+1)-sphere case. An example of an initial configuration is given which demonstrates this fact; and another example of an initial configuration is given which possesses no symmetries at all.


Approximate zero-one laws and sharpness of the percolation transition in a class of models including 2D Ising percolation

Abstract: One of the most well-known classical results for site percolation on the square lattice is the equation p_c + p_c^* = 1. In words, this equation means that for all values different from p_c of the parameter p the following holds: Either a.s. there is an infinite open cluster or a.s. there is an infinite closed `star' cluster. This result is closely related to the percolation transition being sharp: Below p_c the size of the open cluster of a given vertex is not only (a.s.) finite, but has a distrubtion with an exponential tail. The analog of this result has been proved by Higuchi in 1993 for two-dimensional Ising percolation, with fixed inverse temparature beta <beta_c, and as parameter the external field h. Using sharp-threshold results (approximate zero-one laws) and a modification of an RSW-like result by Bollobas and Riordan, we show that these results hold for a large class of percolation models where the vertex values can be `nicely' represented (in a sense which will be defined precisely) by i.i.d. random variables. We point out that the ordinary percolation model belongs obviously to this class, and we show that also the above mentionedIsing model belongs to it. We hope that our results improve insight in the Ising percolation model, and will help to show that many other (not yet analyzed) weakly dependent percolation models also belong to the abovementioned class.


Nearly optimal embeddings of trees

Abstract: In this paper we show how to find nearly optimal embeddings of large trees in several natural classes of graphs. The size of the tree T can be as large as a constant fraction of the size of the graph G, and the maximum degree of T can be close to the minimum degree of G. For example, we prove that any graph of minimum degree d without 4-cycles contains every tree of size \epsilon d^2 and maximum degree at most (1-2\epsilon)d - 2. As there exist d-regular graphs without 4-cycles of size O(d^2), this result is optimal up to constant factors. We prove similar nearly tight results for graphs of given girth, graphs with no complete bipartite subgraph K_{s,t}, random and certain pseudorandom graphs. These results are obtained using a simple and very natural randomized embedding algorithm, which can be viewed as a "self-avoiding tree-indexed random walk".


Lagrangian embeddings of the Klein bottle and combinatorial properties of mapping class groups

Abstract: A proof of non-existence of Lagrangian embeddings of the Klein bottle K in \CP^2 is given. We exploit the existence of a special embedding of K in a symplectic Lefschetz pencil on \CP^2 and study its monodromy. As the main technical tool, we develop the theory of mapping class groups, considered as quotients of special Artin braid groups, and obtain some new results about combinatorial structure of such groups.


Ellipses of minimal area and of minimal eccentricity circumscribed about a convex quadrilateral

Abstract: First, we fill in key gaps in Steiner's nice characterization of the most nearly circular ellipse which passes through the vertices of a convex quadrilateral, D. Steiner proved that there is only one pair of conjugate directions, M1 and M2, that belong to all ellipses of circumscription. Then he proves that if there is an ellipse, E, whose equal conjugate diameters possess the directional constants M1 and M2, then E must be an ellipse of circumscription which has minimal eccentricity. However, Steiner does not show the existence or uniqueness of such an ellipse. We prove that there is a unique ellipse of minimal eccentricity which passes through the vertices of D. We also show that there exists an ellipse which passes through the vertices of D and whose equal conjugate diameters possess the directional constants M1 and M2. We also show that there exists a unique ellipse of minimal area which passes through the vertices of D. Finally, we call a convex quadrilateral, D, bielliptic if the unique inscribed and circumscribed ellipses of minimal eccentricity have the same eccentricity. This generalizes the notion of bicentric quadrilaterals. In particular we show the existence of a bielliptic convex quadrilateral which is not bicentric.


Conjectures on Partitions of Integers as Summations of Primes

Abstract: In this short note we present a class of conjectures on partitions of integers as summations of primes, which are extensions of Goldbach conjecture.


Monodromy of the p-rank strata of the moduli space of curves

Abstract: We compute the Z/\ell and \ell-adic monodromy of every irreducible component of the moduli space M_g^f of curves of genus and and p-rank f. In particular, we prove that the Z/\ell-monodromy of every component of M_g^f is the symplectic group Sp_{2g}(Z/\ell) if g>=3 and \ell is a prime distinct from p. We give applications to the generic behavior of automorphism groups, Jacobians, class groups, and zeta functions of curves of given genus and p-rank.


Exact Computation of Minimum Sample Size for Estimation of Binomial Parameters

Abstract: It is a common contention that it is an ``impossible mission'' to exactly determine the minimum sample size for the estimation of a binomial parameter with prescribed margin of error and confidence level. In this paper, we investigate such a very old but also extremely important problem and demonstrate that the difficulty for obtaining the exact solution is not insurmountable. Unlike the classical approximate sample size method based on the central limit theorem, we develop a new approach for computing the minimum sample size that does not require any approximation. Moreover, our approach overcomes the conservatism of existing rigorous sample size methods derived from Bernoulli's theorem or Chernoff bounds. Our computational machinery consists of two essential ingredients. First, we prove that the minimum of coverage probability with respect to a binomial parameter bounded in an interval is attained at a discrete set of finite many values of the binomial parameter. This allows for reducing infinite many evaluations of coverage probability to finite many evaluations. Second, a recursive bounding technique is developed to further improve the efficiency of computation.


Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras

Abstract: We will prove that one-sided topological Markov shifts $(X_A,\sigma_A)$ and $(X_B,\sigma_B)$ for matrices $A$ and $B$ with entries in $\{0,1\}$ are topologically orbit equivalent if and only if there exists an isomorphism between the Cuntz-Krieger algebras ${\Cal O}_A$ and ${\Cal O}_B$ keeping their commutative $C^*$-subalgerbas $C(X_A)$ and $C(X_B)$. It is also equivalent to the condition that there exists a homeomorphism from $X_A$ to $X_B$ intertwining their topological full groups. We will also study structure of the automorphisms of ${\Cal O}_A$ keeping the commutative $C^*$-algebra $C(X_A)$.


Exact Computation of Minimum Sample Size for Estimating Proportion of Finite Population

Abstract: In this paper, we develop an exact method for the determination of the minimum sample size for estimating the proportion of a finite population with prescribed margin of error and confidence level. By characterizing the behavior of the coverage probability with respect to the proportion, we show that the computational complexity can be significantly reduced and bounded regardless population size.


Exact Computation of Minimum Sample size for Estimation of Poisson Parameters

Abstract: In this paper, we develop an approach for the exact determination of the minimum sample size for the estimation of a Poisson parameter with prescribed margin of error and confidence level. The exact computation is made possible by reducing infinite many evaluations of coverage probability to finite many evaluations. Such reduction is based on our discovery that the minimum of coverage probability with respect to a Poisson parameter bounded in an interval is attained at a discrete set of finite many values.


Cycle lengths in sparse graphs

Abstract: Let C(G) denote the set of lengths of cycles in a graph G. In the first part of this paper, we study the minimum possible value of |C(G)| over all graphs G of average degree d and girth g. Erdos conjectured that |C(G)| =\Omega(d^{\lfloor (g-1)/2\rfloor}) for all such graphs, and we prove this conjecture. In particular, the longest cycle in a graph of average degree d and girth g has length \Omega(d^{\lfloor (g-1)/2\rfloor}). The study of this problem was initiated by Ore in 1967 and our result improves all previously known lower bounds on the length of the longest cycle. Moreover, our bound cannot be improved in general, since known constructions of d-regular Moore Graphs of girth g have roughly that many vertices. We also show that \Omega(d^{\lfloor (g-1)/2\rfloor}) is a lower bound for the number of odd cycle lengths in a graph of chromatic number d and girth g. Further results are obtained for the number of cycle lengths in H-free graphs of average degree d. In the second part of the paper, motivated by the conjecture of Erdos and Gyarfas that every graph of minimum degree at least three contains a cycle of length a power of two, we prove a general theorem which gives an upper bound on the average degree of an n-vertex graph with no cycle of even length in a prescribed infinite sequence of integers. For many sequences, including the powers of two, our theorem gives the upper bound e^{O(\log^* n)} on the average degree of graph of order n with no cycle of length in the sequence, where \log^* n is the number of times the binary logarithm must be applied to n to get a number which is at most


A formula for a quartic integral: a survey of old proofs and some new ones

Abstract: We discuss several existing proofs of the value of a quartic integral and present a new proof that evolved from rational Landen transformations.


The 2-adic valuation of a sequence arising from a rational integral

Abstract: We analyze properties of the 2-adic valuations of an integer sequence that originates from an explicit evaluation of a quartic integral. We also give a combinatorial interpretation of the valuations of this sequence. Connections with the orbits arising from the Collatz (3x+1) problem are discussed.


The integrals in Gradshteyn and Ryzhik. Part 6: the beta function

Abstract: We present the evaluation of definite integrals in the classical table by I. S. Gradshteyn and I. M. Ryzhik that can be reduced to the beta function.


The integrals in Gradshteyn and Ryzhik. Part 7: Elementary examples

Abstract: The classical table of integrals by I. S. Gradshteyn and I. M. Ryzhik contains some elementary integrals. We discuss their evaluations.


The integrals in Gradshteyn and Ryzhik. Part 8: Combinations of powers, exponentials and logarithms

Abstract: We present the evaluation of some definite integrals in the classical table by I. S. Gradshteyn and I. M. Ryzhik where the integrand is a combination of powers, exponentials and logarithms.


The integrals in Gradshteyn and Ryzhik. Part9: Combinations of logarithms, rational and trigonometric functions

Abstract: The classical table of integrals by I. S. Gradshteyn and I. M. Ryzhik contains many definite integrals where the integrand is the product of a rational function times the logarithm of another rational function. We begin the systematic evaluation of these integrals.


The bisymplectomorphism group of a bounded symmetric domain

Abstract: An Hermitian bounded symmetric domain in a complex vector space, given in its circled realization, is endowed with two natural symplectic forms: the flat form and the hyperbolic form. In a similar way, the ambient vector space is also endowed with two natural symplectic forms: the Fubini-Study form and the flat form. It has been shown in arXiv:math.DG/0603141 that there exists a diffeomorphism from the domain to the ambient vector space which puts in correspondence the above pair of forms. This phenomenon is called symplectic duality for Hermitian non compact symmetric spaces. In this article, we first give a different and simpler proof of this fact. Then, in order to measure the non uniqueness of this symplectic duality map, we determine the group of bisymplectomorphisms of a bounded symmetric domain, that is, the group of diffeomorphisms which preserve simultaneously the hyperbolic and the flat symplectic form. This group is the direct product of the compact Lie group of linear automorphisms with an infinite-dimensional Abelian group. This result appears as a kind of Schwarz lemma.


Gagliardo-Nirenberg inequalities on manifolds

Abstract: We prove Gagliardo-Nirenberg inequalities on some classes of manifolds, Lie groups and graphs.


Semigroups on Frechet spaces and equations with infinite delays

Abstract: In this paper, we show existence and uniqueness of a solution to a functional differential equation with infinite delay. We choose an appropriate Frechet space so as to cover a large class of functions to be used as initial functions to obtain existence and uniqueness of solutions.


Nonconforming h-p spectral element methods for elliptic problems

Abstract: In this paper we show that we can use a modified version of the h-p spectral element method proposed in \cite{duttora1,duttom,duttora2,tomarth} to solve elliptic problems with general boundary conditions to exponential accuracy on polygonal domains using nonconforming spectral element functions. A geometrical mesh is used in a neighbourhood of the corners. With this mesh we seek a solution which minimizes the sum of a weighted squared norm of the residuals in the partial differential equation and the squared norm of the residuals in the boundary conditions in fractional Sobolev spaces and enforce continuity by adding a term which measures the jump in the function and its derivatives at inter-element boundaries, in fractional Sobolev norms, to the functional being minimized. In the neighbourhood of the corners, modified polar coordinates are used and a global coordinate system elsewhere. A stability estimate is derived for the functional which is minimized based on the regularity estimate in \cite{babguo1}. We examine how to parallelize the method and show that the set of common boundary values consists of the values of the function at the corners of the polygonal domain. The method is faster than that proposed in \cite{duttora1,duttom,tomarth} and the h-p finite element method and stronger error estimates are obtained.


Reducing system of parameters and the Cohen--Macaulay property

Abstract: Let $R$ be a local ring and let ($x_1\biss x_r$) be part of a system of parameters of a finitely generated $R$-module $M,$ where $r < \dim_R M$. We will show that if ($y_1\biss y_r$) is part of a reducing system of parameters of $M$ with $(y_1\biss y_r)M=(x_1\biss x_r)M$ then $(x_1\biss x_r)$ is already reducing. Moreover, there is such a part of a reducing system of parameters of $M$ iff for all primes $P\in \supp M \cap V_R(x_1\biss x_r)$ with $\dim_R R/P = \dim_R M -r$ the localization $M_P$ of $M$ at $P$ is an $r$-dimensional \cm\ module over $R_P$. Furthermore, we will show that $M$ is a \cm module iff $y_d$ is a non zero divisor on $M/(y_1\biss y_{d-1})M$, where $(y_1\biss y_d)$ is a reducing system of parameters of $M$ ($d := \dim_R M$).


Corners of normal matrices

Abstract: We study various conditions on matrices $B$ and $C$ under which they can be the off-diagonal blocks of a partitioned normal matrix.


On a Result of Hardy and Ramanujan

Abstract: In this paper, we introduce some explicit approximations for the summation $\sum_{k\leq n}\Omega(k)$, where $\Omega(k)$ is the total number of prime factors of $k$.


Nice surjections on spaces of operators

Abstract: A bounded linear operator is said to be nice if its adjoint preserves extreme points of the dual unit ball. Motivated by a description due to Labuschagne and Mascioni \cite{LM} of such maps for the space of compact operators on a Hilbert space, in this article we consider a description of nice surjections on ${\mathcal K}(X,Y)$ for Banach spaces $X,Y$. We give necessary and sufficient conditions when nice surjections are given by composition operators. Our results imply automatic continuity of these maps with respect to other topologies on spaces of operators. We also formulate the corresponding result for ${\mathcal L}(X,Y)$ thereby proving an analogue of the result from \cite{LM} for $L^p$ ($1 <p \neq 2 <\infty$) spaces. We also formulate results when nice operators are not of the canonical form, extending and correcting the results from \cite{KS}.


Gromov--Witten invariants and quantum cohomology

Abstract: This article is an elaboration of a talk given at an international conference on Operator Theory, Quantum Probability, and Noncommutative Geometry held during December~20--23, 2004, at the Indian Statistical Institute, Kolkata. The lecture was meant for a general audience, and also prospective research students, the idea of the quantum cohomology based on the Gromov--Witten invariants. Of course there are many important aspects that are not discussed here.


Representations of homogeneous quantum L\'evy fields

Abstract: We study homogeneous quantum L\'{e}vy processes and fields with independent additive increments over a noncommutative *-monoid. These are described by infinitely divisible generating state functionals, invariant with respect to an endomorphic injective action of a symmetry semigroup. A strongly covariant GNS representation for the conditionally positive logarithmic functionals of these states is constructed in the complex Minkowski space in terms of canonical quadruples and isometric representations on the underlying pre-Hilbert field space. This is of much use in constructing quantum stochastic representations of homogeneous quantum L\'{e}vy fields on It\^{o} monoids, which is a natural algebraic way of defining dimension free, covariant quantum stochastic integration over a space-time indexing set.


Malliavin calculus of Bismut type without probability

Abstract: We translate in semigroup theory Bismut's way of the Malliavin calculus.


Stochastic integral representations of quantum martingales on multiple Fock space

Abstract: In this paper a quantum stochastic integral representation theorem is obtained for unbounded regular martingales with respect to multidimensional quantum noise. This simultaneously extends results of Parthasarathy and Sinha to unbounded martingales and those of the author to multidimensions.


On equivariant Dirac operators for $SU_q(2)$

Abstract: We explain the notion of minimality for an equivariant spectral triple and show that the triple for the quantum SU(2) group constructed by Chakraborty and Pal in \cite{c-p1} is minimal. We also give a decomposition of the spectral triple constructed by Dabrowski {\it et al} \cite{dlssv} in terms of the minimal triple constructed in \cite{c-p1}.


Reducible family of height three level algebras

Abstract: Let $R=k[x_1,..., x_r]$ be the polynomial ring in $r$ variables over an infinite field $k$, and let $M$ be the maximal ideal of $R$. Here a \emph{level algebra} will be a graded Artinian quotient $A$ of $R$ having socle $Soc(A)=0:M$ in a single degree $j$. The Hilbert function $H(A)=(h_0,h_1,... ,h_j)$ gives the dimension $h_i=\dim_k A_i$ of each degree-$i$ graded piece of $A$ for $0\le i\le j$. The embedding dimension of $A$ is $h_1$, and the \emph{type} of $A$ is $\dim_k \Soc (A)$, here $h_j$. The family $\Levalg (H)$ of level algebra quotients of $R$ having Hilbert function $H$ forms an open subscheme of the family of graded algebras or, via Macaulay duality, of a Grassmannian. We show that for each of the Hilbert functions $H=H_1=(1,3,4,4)$ and $H=H_2=(1,3,6,8,9,3)$ the family $LevAlg (H)$ parametrizing level Artinian algebras of Hilbert function $H$ has several irreducible components. We show also that these examples each lift to points. However, in the first example, an irreducible Betti stratum for Artinian algebras becomes reducible when lifted to points. These were the first examples we obtained of multiple components for $\Levalg(H)$ in embedding dimension three. We also show that the second example is the first in an infinite sequence of examples of type three Hilbert functions $H(c)$ in which also the number of components of LevAlg(H) gets arbitrarily large. The first case where the phenomenon of multiple components can occur (i.e. the lowest embedding dimension and then the lowest type) is that of dimension three and type two. Examples of this first case have been obtained by the authors and also by J.-O. Kleppe.


Local representations of the quantum Teichmuller space

Abstract: We introduce a certain type of representations for the quantum Teichmuller space of a punctured surface, which we call local representations. We show that, up to finitely many choices, these purely algebraic representations are classified by classical geometric data. We also investigate the family of intertwining operators associated to such a representations. In particular, we use these intertwiners to construct a natural fiber bundle over the Teichmuller space and its quotient under the action of the mapping class group. This construction also offers a convenient framework to exhibit invariants of surface diffeomorphisms.


On Hilbert's construction of positive polynomials

Abstract: In 1888, Hilbert described how to find real polynomials in more than one variable which take only non-negative values but are not a sum of squares of polynomials. His construction was so restrictive that no explicit examples appeared until the late 1960s. We revisit and generalize Hilbert's construction and present many such polynomials.