Visible Points on Curves over Finite Fields

For a prime $p$ and an absolutely irreducible modulo $p$ polynomial $f(U,V) \in \Z[U,V]$ we obtain an asymptotic formulas for the number of solutions to the congruence $f(x,y) \equiv a \pmod p$ in positive integers $x \le X$, $y \le Y$, with the additional condition $\gcd(x,y)=1$. Such solutions have a natural interpretation as solutions which are visible from the origin. These formulas are derived on average over $a$ for a fixed prime $p$, and also on average over $p$ for a fixed integer $a$

Anabelian geometry and descent obstructions on moduli spaces

We study the section conjecture of anabelian geometry and the sufficiency of the finite descent obstruction to the Hasse principle for the moduli spaces of principally polarized abelian varieties and of curves over number fields. For the former we show that the section conjecture fails and the finite descent obstruction holds for a general class of adelic points, assuming several well-known conjectures. This is done by relating the problem to a local-global principle for Galois representations. For the latter, we prove some partial results that indicate that the finite descent obstruction suffices. We also show how this sufficiency implies the same for all hyperbolic curves.Comment: exposition improve

Cubic Curves, Finite Geometry and Cryptography

Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational points are also surveyed. A possible strengthening of the security of elliptic curve cryptography is proposed using a `shared secret' related to the group law. Cubic curves are also used in a new way to construct sets of points having various combinatorial and geometric properties that are of particular interest in finite Desarguesian planes.Comment: This is a version of our article to appear in Acta Applicandae Mathematicae. In this version, we have corrected a sentence in the third paragraph. The final publication is available at springerlink.com at http://www.springerlink.com/content/xh85647871215644

Efficient optical energy harvesting in self-accelerating beams

We report the experimental observation of energetically confined self-accelerating optical beams propagating along various convex trajectories. We show that, under an appropriate transverse compression of their spatial spectra, these self-accelerating beams can exhibit a dramatic enhancement of their peak intensity and a significant decrease of their transverse expansion, yet retaining both the expected acceleration profile and the intrinsic self-healing properties. We found our experimental results to be in excellent agreement with the numerical simulations. We expect further applications in such contexts where power budget and optimal spatial confinement can be important limiting factors

On the asymptotic evolution of finite energy Airy wavefunctions

In general, there is an inverse relation between the degree of localization of a wavefunction of a certain class and its transform representation dictated by the scaling property of the Fourier transform. We report that in the case of finite energy Airy wavepackets a simultaneous increase in their localization in the direct and transform domains can be obtained as the apodization parameter is varied. One consequence of this is that the far field diffraction rate of a finite energy Airy beam decreases as the beam localization at the launch plane increases. We analyse the asymptotic properties of finite energy Airy wavefunctions using the stationary phase method. We obtain one dominant contribution to the long term evolution that admits a Gaussian-like approximation, which displays the expected reduction of its broadening rate as the input localization is increased

Cherenkov radiation control via self-accelerating wave-packets

Cherenkov radiation is a ubiquitous phenomenon in nature. It describes electromagnetic radiation from a charged particle moving in a medium with a uniform velocity larger than the phase velocity of light in the same medium. Such a picture is typically adopted in the investigation of traditional Cherenkov radiation as well as its counterparts in different branches of physics, including nonlinear optics, spintronics and plasmonics. In these cases, the radiation emitted spreads along a “cone”, making it impractical for most applications. Here, we employ a self-accelerating optical pump wave-packet to demonstrate controlled shaping of one type of generalized Cherenkov radiation - dispersive waves in optical fibers. We show that, by tuning the parameters of the wave-packet, the emitted waves can be judiciously compressed and focused at desired locations, paving the way to such control in any physical system

SARS-CoV-2 variants, spike mutations and immune escape.

Although most mutations in the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) genome are expected to be either deleterious and swiftly purged or relatively neutral, a small proportion will affect functional properties and may alter infectivity, disease severity or interactions with host immunity. The emergence of SARS-CoV-2 in late 2019 was followed by a period of relative evolutionary stasis lasting about 11 months. Since late 2020, however, SARS-CoV-2 evolution has been characterized by the emergence of sets of mutations, in the context of 'variants of concern', that impact virus characteristics, including transmissibility and antigenicity, probably in response to the changing immune profile of the human population. There is emerging evidence of reduced neutralization of some SARS-CoV-2 variants by postvaccination serum; however, a greater understanding of correlates of protection is required to evaluate how this may impact vaccine effectiveness. Nonetheless, manufacturers are preparing platforms for a possible update of vaccine sequences, and it is crucial that surveillance of genetic and antigenic changes in the global virus population is done alongside experiments to elucidate the phenotypic impacts of mutations. In this Review, we summarize the literature on mutations of the SARS-CoV-2 spike protein, the primary antigen, focusing on their impacts on antigenicity and contextualizing them in the protein structure, and discuss them in the context of observed mutation frequencies in global sequence datasets

Effective methods for bulk RNA-seq deconvolution using scnRNA-seq transcriptomes

Background: RNA profiling technologies at single-cell resolutions, including single-cell and single-nuclei RNA sequencing (scRNA-seq and snRNA-seq, scnRNA-seq for short), can help characterize the composition of tissues and reveal cells that influence key functions in both healthy and disease tissues. However, the use of these technologies is operationally challenging because of high costs and stringent sample-collection requirements. Computational deconvolution methods that infer the composition of bulk-profiled samples using scnRNA-seq-characterized cell types can broaden scnRNA-seq applications, but their effectiveness remains controversial. Results: We produced the first systematic evaluation of deconvolution methods on datasets with either known or scnRNA-seq-estimated compositions. Our analyses revealed biases that are common to scnRNA-seq 10X Genomics assays and illustrated the importance of accurate and properly controlled data preprocessing and method selection and optimization. Moreover, our results suggested that concurrent RNA-seq and scnRNA-seq profiles can help improve the accuracy of both scnRNA-seq preprocessing and the deconvolution methods that employ them. Indeed, our proposed method, Single-cell RNA Quantity Informed Deconvolution (SQUID), which combines RNA-seq transformation and dampened weighted least-squares deconvolution approaches, consistently outperformed other methods in predicting the composition of cell mixtures and tissue samples. Conclusions: We showed that analysis of concurrent RNA-seq and scnRNA-seq profiles with SQUID can produce accurate cell-type abundance estimates and that this accuracy improvement was necessary for identifying outcomes-predictive cancer cell subclones in pediatric acute myeloid leukemia and neuroblastoma datasets. These results suggest that deconvolution accuracy improvements are vital to enabling its applications in the life sciences