skip to main content

Title: Large prime gaps and progressions with few primes
We show that the existence of arithmetic progressions with few primes, with a quantitative bound on ''few'', implies the existence of larger gaps between primes less than x than is currently known unconditionally. In particular, we derive this conclusion if there are certain types of exceptional zeros of Dirichlet L-functions.
Alessandro Zaccagnini
Award ID(s):
Publication Date:
Journal Name:
Rivista di matematica della Università di Parma
Page Range or eLocation-ID:
Sponsoring Org:
National Science Foundation
More Like this
  1. Elkins, Christopher A. (Ed.)
    ABSTRACT Monitoring the prevalence of SARS-CoV-2 variants is necessary to make informed public health decisions during the COVID-19 pandemic. PCR assays have received global attention, facilitating a rapid understanding of variant dynamics because they are more accessible and scalable than genome sequencing. However, as PCR assays target only a few mutations, their accuracy could be reduced when these mutations are not exclusive to the target variants. Here we introduce PRIMES, an algorithm that evaluates the sensitivity and specificity of SARS-CoV-2 variant-specific PCR assays across different geographical regions by incorporating sequences deposited in the GISAID database. Using PRIMES, we determined that the accuracy of several PCR assays decreased when applied beyond the geographic scope of the study in which the assays were developed. Subsequently, we used this tool to design Alpha and Delta variant-specific PCR assays for samples from Illinois, USA. In silico analysis using PRIMES determined the sensitivity/specificity to be 0.99/0.99 for the Alpha variant-specific PCR assay and 0.98/1.00 for the Delta variant-specific PCR assay in Illinois, respectively. We applied these two variant-specific PCR assays to six local sewage samples and determined the dominant SARS-CoV-2 variant of either the wild type, the Alpha variant, or the Delta variant. Using next-generationmore »sequencing (NGS) of the spike (S) gene amplicons of the Delta variant-dominant samples, we found six mutations exclusive to the Delta variant (S:T19R, S:Δ156/157, S:L452R, S:T478K, S:P681R, and S:D950N). The consistency between the variant-specific PCR assays and the NGS results supports the applicability of PRIMES. IMPORTANCE Monitoring the introduction and prevalence of variants of concern (VOCs) and variants of interest (VOIs) in a community can help the local authorities make informed public health decisions. PCR assays can be designed to keep track of SARS-CoV-2 variants by measuring unique mutation markers that are exclusive to the target variants. However, the mutation markers may not be exclusive to the target variants because of regional and temporal differences in variant dynamics. We introduce PRIMES, an algorithm that enables the design of reliable PCR assays for variant detection. Because PCR is more accessible, scalable, and robust for sewage samples than sequencing technology, our findings will contribute to improving global SARS-CoV-2 variant surveillance.« less
  2. We propose that emotional priming may be an effective approach to scaffold the creation of rich stories. There are relatively few emotion-based approaches to support users to create, instead of consume, rich stories. Emotional priming is the technique of using emotion- related stimuli to affect human’s executive control and affective processing. It has been researched mostly in terms of human’s behaviors and decision making. We conducted a within-subjects study with 12 participants to investigate the effects of emotional priming induced through an interactive application on storytelling quality. Two conditions of priming were compared to a baseline condition of no priming. In the first condition, the application primes participants by having asking them to perceive and recognize varying emotional stimuli (perception-based priming). In the second condition, the application primes participants by having them produce varying emotional facial expressions (production- based priming). Analyses show that emotional priming resulted in richer storytelling than no emotional priming, and that the production-based emotional priming condition resulted in statistically richer stories being told by participants. We discuss the possibility of integrating interactive emotional priming into storytelling applications.
  3. Abstract Starting with the work of Serre, Katz, and Swinnerton-Dyer, theta operators have played a key role in the study of $p$-adic and $\textrm{mod}\; p$ modular forms and Galois representations. This paper achieves two main results for theta operators on automorphic forms on PEL-type Shimura varieties: (1) the analytic continuation at unramified primes $p$ to the whole Shimura variety of the $\textrm{mod}\; p$ reduction of $p$-adic Maass–Shimura operators a priori defined only over the $\mu $-ordinary locus, and (2) the construction of new $\textrm{mod}\; p$ theta operators that do not arise as the $\textrm{mod}\; p$ reduction of Maass–Shimura operators. While the main accomplishments of this paper concern the geometry of Shimura varieties and consequences for differential operators, we conclude with applications to Galois representations. Our approach involves a careful analysis of the behavior of Shimura varieties and enables us to obtain more general results than allowed by prior techniques, including for arbitrary signature, vector weights, and unramified primes in CM fields of arbitrary degree.
  4. The cerebellum is emerging as a powerful regulator of cognitive and affective processing and memory in both humans and animals and has been implicated in affective disorders. How the cerebellum supports affective function remains poorly understood. The short-latency (just a few milliseconds) functional connections that were identified between the cerebellum and amygdala—a structure crucial for the processing of emotion and valence—more than four decades ago raise the exciting, yet untested, possibility that a cerebellum-amygdala pathway communicates information important for emotion. The major hurdle in rigorously testing this possibility is the lack of knowledge about the anatomy and functional connectivity of this pathway. Our initial anatomical tracing studies in mice excluded the existence of a direct monosynaptic connection between the cerebellum and amygdala. Using transneuronal tracing techniques, we have identified a novel disynaptic circuit between the cerebellar output nuclei and the basolateral amygdala. This circuit recruits the understudied intralaminar thalamus as a node. Using ex vivo optophysiology and super-resolution microscopy, we provide the first evidence for the functionality of the pathway, thus offering a missing mechanistic link between the cerebellum and amygdala. This discovery provides a connectivity blueprint between the cerebellum and a key structure of the limbic system. As such,more »it is the requisite first step toward obtaining new knowledge about cerebellar function in emotion, thus fundamentally advancing understanding of the neurobiology of emotion, which is perturbed in mental and autism spectrum disorders.« less
  5. Abstract The control of tetrahedral carbon stereocentres remains a focus of modern synthetic chemistry and is enabled by their configurational stability. By contrast, trisubstituted nitrogen 1 , phosphorus 2 and sulfur compounds 3 undergo pyramidal inversion, a fundamental and well-recognized stereochemical phenomenon that is widely exploited 4 . However, the stereochemistry of oxonium ions—compounds bearing three substituents on a positively charged oxygen atom—is poorly developed and there are few applications of oxonium ions in synthesis beyond their existence as reactive intermediates 5,6 . There are no examples of configurationally stable oxonium ions in which the oxygen atom is the sole stereogenic centre, probably owing to the low barrier to oxygen pyramidal inversion 7 and the perception that all oxonium ions are highly reactive. Here we describe the design, synthesis and characterization of a helically chiral triaryloxonium ion in which inversion of the oxygen lone pair is prevented through geometric restriction to enable it to function as a determinant of configuration. A combined synthesis and quantum calculation approach delineates design principles that enable configurationally stable and room-temperature isolable salts to be generated. We show that the barrier to inversion is greater than 110 kJ mol −1 and outline processes for resolution. This constitutes,more »to our knowledge, the only example of a chiral non-racemic and configurationally stable molecule in which the oxygen atom is the sole stereogenic centre.« less