Hecke operators are used to investigate part of the ${{E}_{2}}$-term of the Adams spectral sequence based on elliptic homology. The main result is a derivation of $\text{Ex}{{\text{t}}^{1}}$ which combines use of classical Hecke operators and $p$-adic Hecke operators due to Serre.

Every weakly compact composition operator between weighted Banach spaces $H_{v}^{\infty }$ of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space $H_{v}^{\infty }$ are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces.

Let $M$ be a compact, connected, orientable 3-manifold whose boundary is a torus and whose interior admits a complete hyperbolic metric of finite volume. In this paper we show that if theminimal Culler-Shalen norm of a non-zero class in ${{H}_{1}}(\partial M)$ is larger than 8, then the finite surgery conjecture holds for $M$. This means that there are at most 5 Dehn fillings of $M$ which can yieldmanifolds having cyclic or finite fundamental groups and the distance between any slopes yielding such manifolds is at most 3.

This paper contains two essentially independent results in the invariant theory of finite groups. First we prove that, for any faithful representation of a non-trivial $p$-group over a field of characteristic $p$, the ring of vector invariants of $m$ copies of that representation is not Cohen-Macaulay for $m\,\ge \,3$. In the second section of the paper we use Poincaré series methods to produce upper bounds for the degrees of the generators for the ring of invariants as long as that ring is Gorenstein. We prove that, for a finite non-trivial group $G$ and a faithful representation of dimension $n$ with $n\,>\,1$, if the ring of invariants is Gorenstein then the ring is generated in degrees less than or equal to $n(\left| G \right|\,-\,1)$. If the ring of invariants is a hypersurface, the upper bound can be improved to $\left| G \right|$.

We investigate pointwise domination property in operator spaces generated by Lorentz sequence spaces.

We obtain an explicit formula for heat kernels of Lorentz cones, a family of classical symmetric cones. By this formula, the heat kernel of a Lorentz cone is expressed by a function of time $t$ and two eigenvalues of an element in the cone. We obtain also upper and lower bounds for the heat kernels of Lorentz cones.

The class of rings studied in this paper properly contains the class of right distributive rings which have at least one completely prime ideal in the Jacobson radical. Amongst other results we study prime and semiprime ideals, right noetherian rings with comparability and prove a structure theorem for rings with comparability. Several examples are also given.

In this paper we prove the convergence of arithmetic means of sequences generated by a periodic function $\varphi (X)$, moreover if $\varphi (X)$ satisfies a suitable symmetry condition, we prove that their limit is $\varphi (0)$. Applications of previous results are given to study other means of sequences and the behaviour of a class of recursive series.

Given a $\text{TQFT}$ in dimension $d\,+\,1$, and an infinite cyclic covering of a closed ($d\,+\,1$)-dimensional manifold $M$, we define an invariant taking values in a strong shift equivalence class of matrices. The notion of strong shift equivalence originated in R. Williams’ work in symbolic dynamics. The Turaev-Viro module associated to a $\text{TQFT}$ and an infinite cyclic covering is then given by the Jordan form of this matrix away from zero. This invariant is also defined if the boundary of $M$ has an ${{S}^{1}}$ factor and the infinite cyclic cover of the boundary is standard. We define a variant of a $\text{TQFT}$ associated to a finite group $G$ which has been studied by Quinn. In this way, we recover a link invariant due to D. Silver and S. Williams. We also obtain a variation on the Silver-Williams invariant, by using the $\text{TQFT}$ associated to $G$ in its unmodified form.

In this paper we study the boundedness of the commutators $[b,\,{{S}_{n}}]$ where $b$ is a $\text{BMO}$ function and ${{S}_{n}}$ denotes the $n$-th partial sum of the Fourier-Bessel series on $(0,\,\infty )$. Perturbing the measure by $\text{exp(}2\text{b)}$ we obtain that certain operators related to ${{S}_{n}}$ depend analytically on the functional parameter $b$.

We investigate the pairs $(X,\,\varepsilon )$ consisting of a smooth complex projective variety $X$ of dimension $n$ and an ample vector bundle $\varepsilon $ of rank $n\,-\,1$ on $X$ such that $\varepsilon $ has a section whose zero locus is a smooth elliptic curve.

Bessa [Be] proved that for given $n$ and ${{i}_{0}}$, there exists an $\varepsilon (\text{n,}\,{{i}_{0}})\,>\,0$ depending on $n$, ${{i}_{0}}$ such that if $M$ admits a metric $g$ satisfying $\text{Ri}{{\text{c}}_{(M,g)}}\ge n-1,\text{in}{{\text{j}}_{(M,g)}}\ge {{i}_{0}}\,>\,0$ and $\text{dia}{{\text{m}}_{(M,g)}}\,\ge \,\pi \,-\,\varepsilon $, then $M$ is diffeomorphic to the standard sphere. In this note, we improve this result by replacing a lower bound on the injectivity radius with a lower bound of the conjugate radius.

We study boundedness properties of the $q$-mean-square operator {{S}^{(q)}} on $E$-valued analytic martingales, where $E$ is a complex quasi-Banach space and $2\,\le \,q\,<\,\infty $. We establish that a.s. finiteness of ${{S}^{(q)}}$ for every bounded $E$-valued analytic martingale implies strong $(p,\,p)$-type estimates for ${{S}^{(q)}}$ and all $p\,\in \,(0,\,\infty )$. Our results yield new characterizations (in terms of analytic and stochastic properties of the function ${{S}^{(q)}}$) of the complex spaces $E$ that admit an equivalent $q$-uniformly $\text{PL}$-convex quasi-norm. We also obtain a vector-valued extension (and a characterization) of part of an observation due to Bourgain and Davis concerning the ${{L}^{p}}$-boundedness of the usual square-function on scalar-valued analytic martingales.

Let $R$ be a one-dimensional locally analytically irreducible Noetherian domain with finite residue fields. In this note it is shown that if $I$ is a finitely generated ideal of the ring $\text{Int(}R)$ of integer-valued polynomials such that for each $\text{x}\,\in \,R$ the ideal $I\text{(}x\text{)}=\{f(x)|f\in I\}$ is strongly $\text{n}$-generated, $n\,\ge \,2$, then $I$ is $\text{n}$-generated, and some variations of this result.

Let $(X,\,\left\| \,.\, \right\|)$ be a Minkowski space (finite dimensional Banach space) with unit ball $B$. Various definitions of surface area are possible in $X$. Here we explore the one given by Benson [1], [2]. In particular, we show that this definition is convex and give details about the nature of the solution to the isoperimetric problem.

In this paper we show that the Lie-group $\text{Pi}{{\text{n}}_{4}}$ is isomorphic to the semidirect product $\text{(S}{{\text{U}}_{2}}\times \text{S}{{\text{U}}_{2}})\text{Z/2}$ where $Z/2$ operates by flipping the factors. Using this structure theorem we prove a classification theorem for $\text{Pi}{{\text{n}}_{4}}$-bundles over a finite 4-complex $X$.