We consider a complete noncompact Riemannian manifold $M$ and give conditions on a compact submanifold $K\,\subset \,M$ so that the outward normal exponential map off the boundary of $K$ is a diffeomorphism onto $M\backslash K$. We use this to compactify $M$ and show that pinched negative sectional curvature outside $K$ implies $M$ has a compactification with a well-defined Hölder structure independent of $K$. The Hölder constant depends on the ratio of the curvature pinching. This extends and generalizes a 1985 result of Anderson and Schoen.

This paper deals with the extension of $\text{CR}$ functions from a manifold $M\,\subset \,{{\mathbb{C}}^{n}}$ into directions produced by higher order commutators of holomorphic and antiholomorphic vector fields. It uses the theory of complex “sectors” attached to real submanifolds introduced in recent joint work of the authors with D. Zaitsev. In addition, it develops a new technique of approximation of sectors by smooth discs.

We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further, we study conditions under which framed colored link cobordisms induce chain transformations between our formal brackets. We conjecture that for special choice of parameters, Khovanov and Lee homology theories of colored links are functorial (up to sign). Finally, we extend the Rasmussen invariant to links and give examples where this invariant is a stronger obstruction to sliceness than the multivariable Levine–Tristram signature.

In his seminal papers, Koblitz proposed curves for cryptographic use. For fast operations on these curves, these papers also initiated a study of the radix-$\tau $ expansion of integers in the number fields $\mathbb{Q}\left( \sqrt{-3} \right)$ and $\mathbb{Q}\left( \sqrt{-7} \right)$. The (window) nonadjacent form of $\tau $ -expansion of integers in $\mathbb{Q}\left( \sqrt{-7} \right)$ was first investigated by Solinas. For integers in $\mathbb{Q}\left( \sqrt{-3} \right)$, the nonadjacent form and the window nonadjacent form of the $\tau $ -expansion were studied. These are used for efficient point multiplications on Koblitz curves. In this paper, we complete the picture by producing the (window) nonadjacent radix-$\tau $ expansions for integers in all Euclidean imaginary quadratic number fields.

Littlewood–Paley analysis is generalized in this article. We show that the compactness of the Fourier support imposed on the analyzing function can be removed. We also prove that the Littlewood–Paley decomposition of tempered distributions converges under a topology stronger than the weak-star topology, namely, the inductive limit topology. Finally, we construct a multiparameter Littlewood–Paley analysis and obtain the corresponding “renormalization” for the convergence of this multiparameter Littlewood–Paley analysis.

This paper is the continuation of our previous work on the explicit determination of the structure of theta lifts for dual pairs $\left( {{\text{S}}_{{{\text{p}}_{2n}},}}\,O\left( V \right) \right)$ over a non-archimedean field $F$ of characteristic different than 2, where $n$ is the split rank of ${{\text{S}}_{{{\text{p}}_{2n}}}}$ and the dimension of the space $V$ (over $F$) is even. We determine the structure of theta lifts of tempered representations in terms of theta lifts of representations in discrete series.

We propose a new, constructive theory of moving frames for Lie pseudo-group actions on submanifolds. The moving frame provides an effective means for determining complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, and solving equivalence and symmetry problems arising in a broad range of applications.

The aim of this work is to provide a new approach for constructing $n$-dimensional Steinberg symbols on discrete valuation fields from $\left( n\,+\,1 \right)$-cocycles and to study reciprocity laws on curves related to these symbols.

Geometric intuition suggests that the Néron–Tate height of Heegner points on a rational elliptic curve $E$ should be asymptotically governed by the degree of its modular parametrisation. In this paper, we show that this geometric intuition asymptotically holds on average over a subset of discriminants. We also study the asymptotic behaviour of traces of Heegner points on average over a subset of discriminants and find a difference according to the rank of the elliptic curve. By the Gross–Zagier formulae, such heights are related to the special value at the critical point for either the derivative of the Rankin–Selberg convolution of $E$ with a certain weight one theta series attached to the principal ideal class of an imaginary quadratic field or the twisted $L$-function of $E$ by a quadratic Dirichlet character. Asymptotic formulae for the first moments associated with these $L$-series and $L$-functions are proved, and experimental results are discussed. The appendix contains some conjectural applications of our results to the problem of the discretisation of odd quadratic twists of elliptic curves.