Every positive integer is the order of an ordinary abelian variety over $${{\mathbb {F}}}_2$$
Abstract

We show that for every integer$$m > 0$$$m>0$, there is an ordinary abelian variety over $${{\mathbb {F}}}_2$$${F}_{2}$that has exactlymrational points.

Authors:
;
Award ID(s):
Publication Date:
NSF-PAR ID:
10305476
Journal Name:
Research in Number Theory
Volume:
7
Issue:
4
ISSN:
2522-0160
Publisher:
Hemiwicking is the phenomena where a liquid wets a textured surface beyond its intrinsic wetting length due to capillary action and imbibition. In this work, we derive a simple analytical model for hemiwicking in micropillar arrays. The model is based on the combined effects of capillary action dictated by interfacial and intermolecular pressures gradients within the curved liquid meniscus and fluid drag from the pillars at ultra-low Reynolds numbers$${\boldsymbol{(}}{{\bf{10}}}^{{\boldsymbol{-}}{\bf{7}}}{\boldsymbol{\lesssim }}{\bf{Re}}{\boldsymbol{\lesssim }}{{\bf{10}}}^{{\boldsymbol{-}}{\bf{3}}}{\boldsymbol{)}}$$$\left({10}^{-7}\lesssim \mathrm{Re}\lesssim {10}^{-3}\right)$. Fluid drag is conceptualized via a critical Reynolds number:$${\bf{Re}}{\boldsymbol{=}}\frac{{{\bf{v}}}_{{\bf{0}}}{{\bf{x}}}_{{\bf{0}}}}{{\boldsymbol{\nu }}}$$$\mathrm{Re}=\frac{{v}_{0}{x}_{0}}{\nu }$, wherev0corresponds to the maximum wetting speed on a flat, dry surface andx0is the extension length of the liquidmore »
We present a proof of concept for a spectrally selective thermal mid-IR source based on nanopatterned graphene (NPG) with a typical mobility of CVD-grown graphene (up to 3000$$\hbox {cm}^2\,\hbox {V}^{-1}\,\hbox {s}^{-1}$$${\text{cm}}^{2}\phantom{\rule{0ex}{0ex}}{\text{V}}^{-1}\phantom{\rule{0ex}{0ex}}{\text{s}}^{-1}$), ensuring scalability to large areas. For that, we solve the electrostatic problem of a conducting hyperboloid with an elliptical wormhole in the presence of anin-planeelectric field. The localized surface plasmons (LSPs) on the NPG sheet, partially hybridized with graphene phonons and surface phonons of the neighboring materials, allow for the control and tuning of the thermal emission spectrum in the wavelength regime from$$\lambda =3$$$\lambda =3$to 12$$\upmu$$$\mu$m by adjusting themore »
The proximity of many strongly correlated superconductors to density-wave or nematic order has led to an extensive search for fingerprints of pairing mediated by dynamical quantum-critical (QC) fluctuations of the corresponding order parameter. Here we study anisotropics-wave superconductivity induced by anisotropic QC dynamical nematic fluctuations. We solve the non-linear gap equation for the pairing gap$$\Delta (\theta ,{\omega }_{m})$$$\Delta \left(\theta ,{\omega }_{m}\right)$and show that its angular dependence strongly varies below$${T}_{{\rm{c}}}$$${T}_{c}$. We show that this variation is a signature of QC pairing and comes about because there are multiples-wave pairing instabilities with closely spaced transition temperatures$${T}_{{\rm{c}},n}$$${T}_{c,n}$. Taken alone, each instability would produce a gap$$\Deltamore »$\Delta \left(\theta ,{\omega }_{m}\right)$that changes sign$$8n$$$8n$times along the Fermi surface. We show that the equilibrium gap$$\Delta (\theta ,{\omega }_{m})$$$\Delta \left(\theta ,{\omega }_{m}\right)$is a superposition of multiple components that are nonlinearly induced below the actual$${T}_{{\rm{c}}}={T}_{{\rm{c}},0}$$${T}_{c}={T}_{c,0}$, and get resonantly enhanced at$$T={T}_{{\rm{c}},n}\ <\ {T}_{{\rm{c}}}$$$T={T}_{c,n}\phantom{\rule{0ex}{0ex}}<\phantom{\rule{0ex}{0ex}}{T}_{c}$. This gives rise to strong temperature variation of the angular dependence of$$\Delta (\theta ,{\omega }_{m})$$$\Delta \left(\theta ,{\omega }_{m}\right)$. This variation progressively disappears away from a QC point. 4. Abstract We evaluate the$$a_1(1260) \rightarrow \pi \sigma (f_0(500))$$${a}_{1}\left(1260\right)\to \pi \sigma \left({f}_{0}\left(500\right)\right)$decay width from the perspective that the$$a_1(1260)$$${a}_{1}\left(1260\right)$resonance is dynamically generated from the pseudoscalar–vector interaction and the$$\sigma $$$\sigma$arises from the pseudoscalar–pseudoscalar interaction. A triangle mechanism with$$a_1(1260) \rightarrow \rho \pi $$${a}_{1}\left(1260\right)\to \rho \pi$followed by$$\rho \rightarrow \pi \pi $$$\rho \to \pi \pi$and a fusion of two pions within the loop to produce the$$\sigma $$$\sigma$provides the mechanism for this decay under these assumptions for the nature of the two resonances. We obtain widths of the order of 13–22 MeV. Present experimental results differ substantially from each other, suggesting that extra efforts should be devoted to the precise extraction of this important partial decaymore » 5. Abstract We study the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the$$\ell _0$$${\ell }_{0}$-norm of the vector. Our main results are new improved bounds on the minimal$$\ell _0$$${\ell }_{0}$-norm of solutions to systems$$A\varvec{x}=\varvec{b}$$$Ax=b$, where$$A\in \mathbb {Z}^{m\times n}$$$A\in {Z}^{m×n}$,$${\varvec{b}}\in \mathbb {Z}^m$$$b\in {Z}^{m}$and$$\varvec{x}$$$x$is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In certain cases, we give polynomial time algorithms for computing solutions with$$\ell _0${\ell }_{0}$-norm satisfying the obtained bounds. We show that our bounds aremore »