Average Frobenius Distributions for Elliptic Curves: Extremal Primes and Koblitz\u27s Conjecture

Let E be a rational elliptic curve, and let p be a rational prime of good reduction. Let a_p denote the trace of the Frobenius endomorphism of $E$ at the prime $p$, and let N_p be the number of rational points on the reduced curve E modulo p. In this dissertation we investigate two different questions regarding the statistical distribution of these two quantities. We say p is a champion prime of E if when N_p is as large as possible in accordance with the Hasse bound. In a similar vein, we say p is a trailing prime of E when N_p is as small as possible in accordance with the Hasse bound. Together, we say that these primes constitute the extremal primes of E. The first result of this dissertation establishes an asymptotic on the number of elliptic curve champion primes that are less than a real number X in an average sense. As an immediate corollary, we also gain asymptotics on the average number of trailing primes less than and the average number of extremal primes less than X. In 1988, Koblitz made a conjecture on how often N_p is prime for any fixed rational elliptic curve. Balog, Cojocaru, and David have proved that Koblitz\u27s conjecture is true on average for rational elliptic curves. The second result of this dissertation generalizes their average result to elliptic curves over certain higher number fields

Average Frobenius Distributions for Elliptic Curves: Extremal Primes and Koblitz\u27s Conjecture

Let E be a rational elliptic curve, and let p be a rational prime of good reduction. Let a_p denote the trace of the Frobenius endomorphism of $E$ at the prime $p$, and let N_p be the number of rational points on the reduced curve E modulo p. In this dissertation we investigate two different questions regarding the statistical distribution of these two quantities. We say p is a champion prime of E if when N_p is as large as possible in accordance with the Hasse bound. In a similar vein, we say p is a trailing prime of E when N_p is as small as possible in accordance with the Hasse bound. Together, we say that these primes constitute the extremal primes of E. The first result of this dissertation establishes an asymptotic on the number of elliptic curve champion primes that are less than a real number X in an average sense. As an immediate corollary, we also gain asymptotics on the average number of trailing primes less than and the average number of extremal primes less than X. In 1988, Koblitz made a conjecture on how often N_p is prime for any fixed rational elliptic curve. Balog, Cojocaru, and David have proved that Koblitz\u27s conjecture is true on average for rational elliptic curves. The second result of this dissertation generalizes their average result to elliptic curves over certain higher number fields

The Mayfly Newsletter

The Mayfly Newsletter is the official newsletter of the Permanent Committee of the International Conferences on Ephemeroptera

The Mayfly Newsletter

The Mayfly Newsletter is the official newsletter of the Permanent Committee of the International Conferences on Ephemeroptera

Blockchain Technology Adoption Status and Strategies

Purpose: The purpose of this paper is to review the acceptance and future use of blockchain technology. Given the rapid technological changes, this paper focuses on a managerial overview and framework of how the blockchain, including its implementations such as Bitcoin have advanced and how blockchain can be utilized in large-scale, enterprise environments. The paper begins with a technological overview that covers the history of the technology, as well as describing the computational, cryptographic theory that serves as the basis for its notable security features. This paper also covers several key application areas such as finance, accounting, and marketplaces where blockchain technology is seeing major investments from some of the world’s largest organizations. Analysis Methods: Triangulation is utilized for this paper, which combines multiple methodologies, such as qualitative and quantitative methods, as complementary components for improving research study accuracy. The triangulation methods chosen for this paper include a secondary data environment analysis, a text analysis, and financial analysis in order to successfully manage and review the adoption diffusion of innovative technologies like blockchain. The blockchain stands to disrupt many areas of society with the proper application and thus it is important to examine its use with as many viewpoints as possible. Contributions and Conclusion: The contribution this paper describes the potential drivers and drawbacks of blockchain technology in real world applications and highlights the managerial implications of its use. This paper also expands the theoretical contributions for identifying blockchain technology progress on the diffusion of innovation curve. As it stands, the blockchain is within the innovation stage in terms of its application in multi-national enterprises, but with major firms making investments, the blockchain could see growing normalization and acceptance, and at an inflection point akin to the Internet of the 1990s

Deterioration modeling of steel moment resisting frames using finite-length plastic hinge force-based beam-column elements

The use of empirically calibrated moment-rotation models that account for strength and stiffness deterioration of steel frame members is paramount in evaluating the performance of steel structures prone to collapse under seismic loading. These deterioration models are typically used as zero-length springs in a concentrated plasticity formulation; however, a calibration procedure is required when they are used to represent the moment-curvature (M−χ) behavior in distributed plasticity formulations because the resulting moment-rotation (M−θ) response depends on the element integration method. A plastic hinge integration method for using deterioration models in force-based elements is developed and validated using flexural stiffness modifications parameters to recover the exact solution for linear problems while ensuring objective softening response. To guarantee accurate results in both the linear and nonlinear range of response, the flexural stiffness modification parameters are computed at the beginning of the analysis as a function of the user-specified plastic hinge length. With this approach, moment-rotation models that account for strength and stiffness deterioration can be applied in conjunction with force-based plastic hinge beam-column elements to support collapse prediction without increased modeling complexity

Application of reliability-based robustness assessment of steel moment resisting frame structures under post-mainshock cascading events

This paper proposes a reliability-based framework for quantifying structural robustness considering the occurrence of a major earthquake (mainshock) and subsequent cascading hazard events, such as aftershocks that are triggered by the mainshock. These events can significantly increase the probability of failure of buildings, especially for structures that are damaged during the mainshock. The application of the proposed framework is exemplified through three numerical case studies. The case studies correspond to three SAC steel moment frame buildings of three, nine, and 20 stories, which were designed to pre-Northridge codes and standards. Two-dimensional nonlinear finite-element models of the buildings are developed with the Open System for Earthquake Engineering Simulation framework (OpenSees), using a finite length plastic hinge beam model and a bilinear constitutive law with deterioration, and are subjected to multiple mainshock-aftershock seismic sequences. For the three buildings analyzed herein, it is shown that the structural reliability under a single seismic event can be significantly different from that under a sequence of seismic events. The reliability based robustness indicator shows that the structural robustness is influenced by the extent to which a structure can distribute damage

Dynamic Time-Warping Correction for Shifts in Ultrahigh Resolving Power Ion Mobility Spectrometry and Structures for Lossless Ion Manipulations

Detection of arrival time shifts between ion mobility spectrometry (IMS) separations can limit achievable resolving power (Rp), particularly when multiple separations are summed or averaged, as commonly practiced in IMS. Such variations can be apparent in higher Rp measurements and are particularly evident in long path length traveling wave structures for lossless ion manipulations (SLIM) IMS due to their typically much longer separation times. Here, we explore data processing approaches employing single value alignment (SVA) and nonlinear dynamic time warping (DTW) to correct for variations between IMS separations, such as due to pressure fluctuations, to enable more effective spectrum summation for improving Rp and detection of low-intensity species. For multipass SLIM IMS separations, where narrow mobility range measurements have arrival times that can extend to several seconds, the SVA approach effectively corrected for such variations and significantly improved Rp for summed separations. However, SVA was much less effective for broad mobility range separations, such as obtained with multilevel SLIM IMS. Changes in ions’ arrival times were observed to be correlated with small pressure changes, with approximately 0.6% relative arrival time shifts being common, sufficient to result in a loss of Rp for summed separations. Comparison of the approaches showed that DTW alignment performed similarly to SVA when used over a narrow mobility range but was significantly better (providing narrower peaks and higher signal intensities) for wide mobility range data. We found that the DTW approach increased Rp by as much as 115% for measurements in which 50 IMS separations over 2 s were summed. We conclude that DTW is superior to SVA for ultra-high-resolution broad mobility range SLIM IMS separations and leads to a large improvement in effective Rp, correcting for ion arrival time shifts regardless of the cause, as well as improving the detectability of low-abundance species. Our tool is publicly available for use with universal ion mobility format (.UIMF) and text (.txt) files

The Relative Influence of Habitat Amount and Configuration on Genetic Structure Across Multiple Spatial Scales

Despite strong interest in understanding how habitat spatial structure shapes the genetics of populations, the relative importance of habitat amount and configuration for patterns of genetic differentiation remains largely unexplored in empirical systems. In this study, we evaluate the relative influence of, and interactions among, the amount of habitat and aspects of its spatial configuration on genetic differentiation in the pitcher plant midge, Metriocnemus knabi. Larvae of this species are found exclusively within the water-filled leaves of pitcher plants (Sarracenia purpurea) in a system that is naturally patchy at multiple spatial scales (i.e., leaf, plant, cluster, peatland). Using generalized linear mixed models and multimodel inference, we estimated effects of the amount of habitat, patch size, interpatch distance, and patch isolation, measured at different spatial scales, on genetic differentiation (F ST) among larval samples from leaves within plants, plants within clusters, and clusters within peatlands. Among leaves and plants, genetic differentiation appears to be driven by female oviposition behaviors and is influenced by habitat isolation at a broad (peatland) scale. Among clusters, gene flow is spatially restricted and aspects of both the amount of habitat and configuration at the focal scale are important, as is their interaction. Our results suggest that both habitat amount and configuration can be important determinants of genetic structure and that their relative influence is scale dependent