Let $g$ be a totally positive function of finite type, that is, $ĝ\left(\xi \right)={\prod }_{\nu =1}^M(1+2\pi i{\delta }_{\nu }\xi {)}^{-1}$ for ${\delta }_{\nu }\in \mathbb{R}$, ${\delta }_{\nu }\ne 0$, and $M\ge 2$, and let $\alpha ,\beta >0$. Then the set $\left\{e^{2\pi i\beta lt}g\right(t-\alpha k):k,l\in \mathbb{Z}\}$ is a frame for $L^2\left(\mathbb{R}\right)$ if and only if $\alpha \beta <1$. This result is a first positive contribution to a conjecture of Daubechies from 1990. Until now, the complete characterization of lattice parameters $\alpha$, $\beta$ that generate a frame has been known for only six window functions $g$. Our main result now yields an uncountable class of window functions. As a by-product of the proof method, we also derive new sampling theorems in shift-invariant spaces and obtain the correct Nyquist rate.

This article studies a distinguished collection of so-called generalized Heegner cycles in the product of a Kuga–Sato variety with a power of a CM elliptic curve. Its main result is a $p$-adic analogue of the Gross–Zagier formula which relates the images of generalized Heegner cycles under the $p$-adic Abel–Jacobi map to the special values of certain $p$-adic Rankin $L$-series at critical points that lie outside their range of classical interpolation.

In this article, we classify $n$-dimensional ($n\ge 4$) complete Bach-flat gradient shrinking Ricci solitons. More precisely, we prove that any $4$-dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat and hence a finite quotient of the Gaussian shrinking soliton ${\mathbb{R}}^4$ or the round cylinder ${\mathbb{S}}^3\times \mathbb{R}$. More generally, for $n\ge 5$, a Bach-flat gradient shrinking Ricci soliton is either Einstein, or a finite quotient of the Gaussian shrinking soliton ${\mathbb{R}}^n$ or the product $N^{n-1}\times \mathbb{R}$, where $N^{n-1}$ is Einstein.

We study degenerations of complex surfaces with no vanishing cycles first described by J. Wahl. Given such a degeneration, we construct an exceptional vector bundle on the general fiber $Y$ in the case $H^{2,0}\left(Y\right)=H^1\left(Y\right)=0$. For $Y={\mathbb{P}}^2$, we show that our construction establishes a bijective correspondence between the possible singular surfaces and the set of exceptional bundles on $Y$ modulo a natural equivalence relation.