We study some questions concerning the structure of the set of spreading models of a separable infinite-dimensional Banach space $X$. In particular we give an example of a reflexive $X$ so that all spreading models of $X$ contain ${{\ell }_{1}}$ but none of them is isomorphic to ${{\ell }_{1}}$. We also prove that for any countable set $C$ of spreading models generated by weakly null sequences there is a spreading model generated by a weakly null sequence which dominates each element of $C$. In certain cases this ensures that $X$ admits, for each $\alpha \,<\,{{\omega }_{1}}$, a spreading model ${{\left( \tilde{x}_{i}^{\left( \alpha \right)} \right)}_{i}}$ such that if $\alpha \,<\,\beta $ then ${{\left( \tilde{x}_{i}^{\left( \alpha \right)} \right)}_{i}}$ is dominated by (and not equivalent to) ${{\left( \tilde{x}_{i}^{\left( \beta \right)} \right)}_{i}}$. Some applications of these ideas are used to give sufficient conditions on a Banach space for the existence of a subspace and an operator defined on the subspace, which is not a compact perturbation of a multiple of the inclusion map.

We study adjacency of equisingularity types of planar complex curve singularities in terms of their Enriques diagrams. The goal is, given two equisingularity types, to determine whether one of themis adjacent to the other. For linear adjacency a complete answer is obtained, whereas for arbitrary (analytic) adjacency a necessary condition and a sufficient condition are proved. We also obtain new examples of exceptional deformations, i.e, singular curves of type ${\mathcal{D}}'$ that can be deformed to a curve of type $\mathcal{D}$ without ${\mathcal{D}}'$ being adjacent to $\mathcal{D}$.

We study Tychonoff spaces $X$ with the property that, for all topological embeddings $X\,\to \,Y$, the induced map $C(Y)\,\to \,C(X)$ is an epimorphism of rings. Such spaces are called absolute $\mathcal{C}\mathcal{R}$-epic. The simplest examples of absolute $\mathcal{C}\mathcal{R}$-epic spaces are $\sigma $-compact locally compact spaces and Lindelöf $P$-spaces. We show that absolute $\mathcal{C}\mathcal{R}$-epic first countable spaces must be locally compact.

However, a “bad” class of absolute $\mathcal{C}\mathcal{R}$-epic spaces is exhibited whose pathology settles, in the negative, a number of open questions. Spaces which are not absolute $\mathcal{C}\mathcal{R}$-epic abound, and some are presented.

Let $f\,=\,\sum{_{n=1}^{\infty }{{a}_{f}}\left( n \right){{q}^{n}}}$ be a cusp form with integer weight $k\,\ge \,2$ that is not a linear combination of forms with complex multiplication. For $n\,\ge \,1$, let

$${{i}_{f}}\left( n \right)\,=\,\left\{ _{0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{otherwise}\text{.}}^{\max \left\{ i\,:\,{{a}_{f}}\left( n+j \right)\,=\,0\,\text{for}\,\text{all}\,0\,\,\le \,j\,\le \,i \right\}\,\,\,\,\,\,\,\text{if}\,{{\text{a}}_{f}}\left( n \right)\,=\,0,} \right.\,$$

Concerning bounded values of ${{i}_{f}}\left( n \right)$ we prove that for $\in \,>\,0$ there exists $M\,=\,M\left( \in ,\,f \right)$ such that $\#\left\{ n\,\le \,x\,:\,{{i}_{f}}\left( n \right)\,\le \,M \right\}\,\ge \,\left( 1-\in \right)x.$ Using results of Wu, we show that if $f$ is a weight 2 cusp form for an elliptic curve without complex multiplication, then ${{i}_{f}}\left( n \right)\,{{\ll }_{f,\in }}\,{{n}^{\frac{51}{134}+\in }}$. Using a result of David and Pappalardi, we improve the exponent to $\frac{1}{3}$ for almost all newforms associated to elliptic curves without complex multiplication. Inspired by a classical paper of Selberg, we also investigate ${{i}_{f}}\left( n \right)$ on the average using well known bounds on the Riemann Zeta function.

Representation theorems are proved for Banach ideal spaces with the Fatou property which are built by the Calderón–Lozanovskiĭ construction. Factorization theorems for operators in spaces more general than the Lebesgue ${{L}^{p}}$ spaces are investigated. It is natural to extend the Gagliardo theorem on the Schur test and the Rubio de Francia theorem on factorization of the Muckenhoupt ${{A}_{p}}$ weights to reflexive Orlicz spaces. However, it turns out that for the scales far from ${{L}^{p}}$-spaces this is impossible. For the concrete integral operators it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces are not valid. Representation theorems for the Calderón–Lozanovskiĭ construction are involved in the proofs.

We study the gap (= “projection norm” = “graph distance”) topology of the space of all (not necessarily bounded) self-adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show the surprising result that this space is connected in contrast to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary.

We consider an asymptotically flat Lorentzian manifold of dimension (1, 3). An inequality is derived which bounds the Riemannian curvature tensor in terms of the $\text{ADM}$ energy in the general case with second fundamental form. The inequality quantifies in which sense the Lorentzian manifold becomes flat in the limit when the $\text{ADM}$ energy tends to zero.

The present paper is concerned with the study of the ${{L}^{2}}$ cohomology spaces of negatively curved manifolds. The first half presents a finiteness and vanishing result obtained under some curvature assumptions, while the second half identifies a class of metrics having non-trivial ${{L}^{2}}$ cohomology for degree equal to the half dimension of the space. For the second part we rely on the existence and regularity properties of the solution for the heat equation for forms.

We introduce some non-linear transformations from the set of Hausdorff moment sequences into itself; among them is the one defined by the formula: $T\left( {{\left( {{a}_{n}} \right)}_{n}} \right)\,=\,1/\left( {{a}_{0}}\,+\cdots +\,{{a}_{n}} \right).$ We give some examples of Hausdorff moment sequences arising from the transformations and provide the corresponding measures: one of these sequences is the reciprocal of the harmonic numbers ${{\left( 1+1/2\,+\cdots +\,1/\left( n+1 \right) \right)}^{-1}}.$

We introduce and study several notions of amenability for unitary corepresentations and $*$-representations of algebraic quantum groups, which may be used to characterize amenability and co-amenability for such quantum groups. As a background for this study, we investigate the associated tensor ${{C}^{*}}$-categories.

In the paper we formulate a Covering Property Axiom, $\text{CP}{{\text{A}}_{\text{prism}}}$, which holds in the iterated perfect set model, and show that it implies the following facts, of which (a) and (b) are the generalizations of results of J. Steprāns.

(a) There exists a family $\mathcal{F}$ of less than continuum many ${{C}^{1}}$ functions from $\mathbb{R}$ to $\mathbb{R}$ such that ${{\mathbb{R}}^{2}}$ is covered by functions from $\mathcal{F}$, in the sense that for every $\left\langle x,\,y \right\rangle \,\in \,{{\mathbb{R}}^{2}}$${{\mathbb{R}}^{2}}$ there exists an $f\,\in \,\mathcal{F}$ such that either $f\left( x \right)\,=\,y$ or $f\left( y \right)\,=\,x$.

(b) For every Borel function $f:\,\mathbb{R}\,\to \,\mathbb{R}$ there exists a family $\mathcal{F}$ of less than continuum many “${{C}^{1}}$” functions (i.e., differentiable functions with continuous derivatives, where derivative can be infinite) whose graphs cover the graph of $f$.

(c) For every $n\,>\,0$ and a ${{D}^{n}}$ function $f:\,\mathbb{R}\,\to \,\mathbb{R}$ there exists a family $\mathcal{F}$ of less than continuum many ${{C}^{n}}$ functions whose graphs cover the graph of $f$.

We also provide the examples showing that in the above properties the smoothness conditions are the best possible. Parts (b), (c), and the examples are closely related to work of A. Olevskiî.

We prove that it is relatively consistent with $ZFC$ that in any perfect Polish space, for every nonmeager set $A$ there exists a nowhere dense Cantor set $C$ such that $A\,\cap \,C$ is nonmeager in $C$. We also examine variants of this result and establish a measure theoretic analog.

The initial version of the Birch and Swinnerton-Dyer conjecture concerned asymptotics for partial Euler products for an elliptic curve $L$-function at $s\,=\,1$. Goldfeld later proved that these asymptotics imply the Riemann hypothesis for the $L$-function and that the constant in the asymptotics has an unexpected factor of $\sqrt{2}$. We extend Goldfeld's theorem to an analysis of partial Euler products for a typical $L$-function along its critical line. The general $\sqrt{2}$ phenomenon is related to second moments, while the asymptotic behavior (over number fields) is proved to be equivalent to a condition that in a precise sense seems much deeper than the Riemann hypothesis. Over function fields, the Euler product asymptotics can sometimes be proved unconditionally.

Decompositions of spectral type are obtained for closed Hilbert space operators with empty resolvent set, but whose square has closure which is spectral. Krein space situations are also discussed.

With applications in mind we establish a summation formula for the coefficients of a general Dirichlet series satisfying a suitable functional equation. Among a number of consequences we derive a generalization of an elegant divisor sum bound due to F. V. Atkinson.

In this article we prove some new results on projective normality, normal presentation and higher syzygies for surfaces of general type, not necessarily smooth, embedded by adjoint linear series. Some of the corollaries of more general results include: results on property ${{N}_{p}}$ associated to ${{K}_{S}}\,\otimes \,{{B}^{\otimes n}}$ where $B$ is base-point free and ample divisor with $B\,\otimes \,{{K}^{*}}\,nef,$ results for pluricanonical linear systems and results giving effective bounds for adjoint linear series associated to ample bundles. Examples in the last section show that the results are optimal.

We provide a porosity-based approach to the differentiability and continuity of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone $K$ with non-empty interior. We also show that the set of nowhere $K$-monotone functions has a $\sigma $-porous complement in the space of continuous functions endowed with the uniform metric.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and without complex multiplication. Let $K$ be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes $p\,\le \,x$ for which $\mathbb{Q}\left( {{\pi }_{p}} \right)\,=\,K$, where ${{\pi }_{p}}$ denotes the Frobenius endomorphism of $E$ at $p$. More precisely, under a generalized Riemann hypothesis we show that this number is ${{O}_{E}}\left( {{x}^{17/18}}\,\log x \right)$, and unconditionally we show that this number is ${{O}_{E,K}}\left( \frac{x{{\left( \log \,\log x \right)}^{13/12}}}{{{\left( \log x \right)}^{25/24}}} \right)$ We also prove that the number of imaginary quadratic fields $K$, with − disc $K\,\le \,x$ and of the form $K\,=\,\mathbb{Q}({{\pi }_{p}})$, is ${{\gg }_{E}}\,\log \,\log \,\log \,x$ for $x\,\ge \,{{x}_{0}}\left( E \right)$. These results represent progress towards a 1976 Lang–Trotter conjecture.

Let $k$ be a field of characteristic $0,\,R\,=\,k\left[ {{x}_{1}},\,\ldots ,\,{{x}_{d}} \right]$ be a polynomial ring, and $m$ its maximal homogeneous ideal. Let $I\,\subset \,R$ be a homogeneous ideal in $R$. Let $\lambda (M)$ denote the length of an $R$-module $M$. In this paper, we show that

$$\underset{x\to \infty }{\mathop{\lim }}\,\,\frac{\lambda \left( H_{m}^{0}\left( R/{{I}^{n}} \right) \right)}{{{n}^{d}}}\,=\,\underset{x\to \infty }{\mathop{\lim }}\,\,\frac{\lambda \left( \text{Ext}_{R}^{d}\left( R/{{I}^{n}},\,R\left( -d \right) \right) \right)}{{{n}^{d}}}$$

always exists. This limit has been shown to be $e(I)/d!$ for $m$-primary ideals $I$ in a local Cohen–Macaulay ring, where $e(I)$ denotes the multiplicity of $I$. But we find that this limit may not be rational in general. We give an example for which the limit is an irrational number thereby showing that the lengths of these extension modules may not have polynomial growth.

We are interested in Poisson structures transverse to nilpotent adjoint orbits in a complex semi-simple Lie algebra, and we study their polynomial nature. Furthermore, in the case of $s{{l}_{n}}$, we construct some families of nilpotent orbits with quadratic transverse structures.