Equidistribution of polynomial sequences in function fields, with applications

We prove a function field analog of Weyl's classical theorem on equidistribution of polynomial sequences. Our result covers the case when the degree of the polynomial is greater than or equal to the characteristic of the field, which is a natural barrier when applying the Weyl differencing process to function fields. We also discuss applications to van der Corput and intersective sets in function fields.Comment: 24 page

Are Smell-Based Metrics Actually Useful in Effort-Aware Structural Change-Proneness Prediction? An Empirical Study

Bad code smells (also named as code smells) are symptoms of poor design choices in implementation. Existing studies empirically confirmed that the presence of code smells increases the likelihood of subsequent changes (i.e., change-proness). However, to the best of our knowledge, no prior studies have leveraged smell-based metrics to predict particular change type (i.e., structural changes). Moreover, when evaluating the effectiveness of smell-based metrics in structural change-proneness prediction, none of existing studies take into account of the effort inspecting those change-prone source code. In this paper, we consider five smell-based metrics for effort-aware structural change-proneness prediction and compare these metrics with a baseline of well-known CK metrics in predicting particular categories of change types. Specifically, we first employ univariate logistic regression to analyze the correlation between each smellbased metric and structural change-proneness. Then, we build multivariate prediction models to examine the effectiveness of smell-based metrics in effort-aware structural change-proneness prediction when used alone and used together with the baseline metrics, respectively. Our experiments are conducted on six Java open-source projects with up to 60 versions and results indicate that: (1) all smell-based metrics are significantly related to structural change-proneness, except metric ANS in hive and SCM in camel after removing confounding effect of file size; (2) in most cases, smell-based metrics outperform the baseline metrics in predicting structural change-proneness; and (3) when used together with the baseline metrics, the smell-based metrics are more effective to predict change-prone files with being aware of inspection effort

A prime analogue of the Erdös-Pomerance conjecture for elliptic curves

Let E/Q be an elliptic curve of rank ≥ 1 and b ∈ E(Q) a rational point of infinite order. For a prime p of good reduction, let gb(p) be the order of the cyclic group generated by the reduction b of b modulo p. We denote by ω(gb(p)) the number of distinct prime divisors of gb(p). Assuming the GRH, we show that the normal order of ω(gb(p)) is log log p. We also prove conditionally that there exists a normal distribution for the quantity ω(gb(p)) − log log p √log log p . The latter result can be viewed as an elliptic analogue of a conjecture of Erdös and Pomerance about the distribution of ω(fa(n)), where a is a natural number > 1 and fa(n) the order of a modulo n.Research partially supported by an NSERC Discovery Grant

A Generalization of the Turán Theorem and Its Applications

This article has been published in a revised form in the Canadian Mathematical Bulletin https://doi.org/10.4153/CMB-2004-056-7. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © 2004 Canadian Mathematical Bulletin.We axiomatize the main properties of the classical Turan Theorem in order to apply it to a general context. We provide applications in the cases of number fields, function fields, and geometrically irreducible varieties over a finite field.Research partially supported by NSERC PGSB scholarship and NSERC discovery grant

Prime analogues of the Erdős–Kac theorem for elliptic curves

This article is made available through Elsevier's open archive. This article is available here: https://doi.org/10.1016/j.jnt.2005.10.014 © 2006 Elsevier Inc. All rights reserved.Let E/Q be an elliptic curve. For a prime p of good reduction, let E(Fp) be the set of rational points defined over the finite field Fp. We denote by ω(#E(Fp)), the number of distinct prime divisors of #E(Fp). We prove that the quantity (assuming the GRH if E is non-CM) ω(#E(Fp)) − log logp √log logp distributes normally. This result can be viewed as a “prime analogue” of the Erdos–Kac theorem. We also study the normal distribution of the number of distinct prime factors of the exponent of E(Fp).Research partially supported by an NSERC Discovery Grant

The Erdős Theorem and the Halberstam Theorem in function fields

This is the Accepted Version of the paper published in the journal Acta Arithmetica in 2004. The final Version of Record is available here https://doi.org/10.4064/aa114-4-3Introduction. For n ∈ N, define ω(n) to be the number of distinct prime divisors of n. The Tur´an Theorem [9] concerns the second moment of ω(n) and it implies a result of Hardy and Ramanujan [4] that the normal order of ω(n) is log log n. Further development of probabilistic ideas led Erd˝os and Kac [2] to prove a remarkable refinement of the Hardy Ramanujan Theorem, namely, the existence of a normal distribution for ω(n).Research partially supported by an NSERC discovery grant

A Generalization of the Erdös-Kac Theorem and its Applications

This article has been published in a revised form in the Canadian Mathematical Bulletin https://doi.org/10.4153/CMB-2004-057-4. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © 2004 Canadian Mathematical Bulletin.We axiomatize the main properties of the classical Erdös-Kac Theorem in order to apply it to a general context. We provide applications in the cases of number fields, function fields, and geometrically irreducible varieties over a finite field.Research partially supported by an NSERC discovery grant

Prime Divisors of the Number of Rational Points on Elliptic Curves with Complex Multiplication

This is the peer reviewed version of the following article: Liu, Y.-R. (2005). Prime divisors of the number of rational points on elliptic curves with complex multiplication. Bulletin of the London Mathematical Society, 37(5), 658–664, which has been published in final form at https://doi.org/10.1112/s0024609305004558. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.Let E/Q be an elliptic curve. For a prime p of good reduction, let E(Fp) be the set of rational points defined over the finite field Fp. Denote by ω(#E(Fp)) the number of distinct prime divisors of #E(Fp). For an elliptic curve with complex multiplication, the normal order of ω(#E(Fp)) is shown to be log log p. The normal order of the number of distinct prime factors of the exponent of E(Fp) is also studied. 2000 Mathematics Subject Classification 11N37, 11G20.Research partially supported by an NSERC Discovery Grant