Prime Optimization Shipilevsky, Yuly
Statistics, optimization & information computing,
06/2021, Volume:
9, Issue:
2
Journal Article
Open access
This is a pioneering work, introducing a novel class of optimization of objective functions over subsets of primeonly integer points. We show a rich variety of Prime Optimization and mixed problems.
Let E/\mathbb {Q} be an elliptic curve and p an odd prime where E has good supersingular reduction. Let F_1 denote the characteristic power series of the Pontryagin dual of the fine Selmer group of E ...over the cyclotomic \mathbb {Z}_p-extension of \mathbb {Q} and let F_2 denote the greatest common divisor of Pollack’s plus and minus p-adic L-functions or Sprung’s sharp and flat p-adic L-functions attached to E, depending on whether a_p(E)=0 or a_p(E)\ne 0. We study a link between the divisors of F_1 and F_2 in the Iwasawa algebra. This gives new insights into problems posed by Greenberg and Pollack–Kurihara on these elements.
Eye cues have been shown to stimulate rapid, reflexive, unconscious processing and in many experimental settings to cue increased prosocial and decreased antisocial behaviour. Eye cues are being ...widely applied in public policy to reduce crime and antisocial behaviour. Recently, failed replication attempts and two meta-analyses examining the eye cue effect on generosity have raised doubts regarding earlier findings. Much of the wider evidence on eye cues has still not been systematically reviewed, notably that which is most relevant to its practical application: the effect of eye cues on antisocial behaviour. Given the evidence of humans' heightened sensitivity to threat and negative information, we hypothesized that the watching eyes effect would be more consistent in studies examining antisocial behaviour. In our meta-analysis of 15 experiments from 13 research papers we report a reduction in the risk of antisocial behaviour of 35% when eye cues are present. By contrast, systematic reviews have suggested CCTV cameras reduce crime by only 16%. We conclude that there is sufficient evidence of a watching eyes effect on antisocial behaviour to justify their use in the very low-cost and potentially high-impact real-world interventions that are proliferating in public policy, particularly in the UK.
Our meta-analysis of 15 experiments involving 2035 participants shows that photographs and/or stylized images of eyes reduced antisocial behaviour by 35%. Our findings support public policy initiatives employing pictures of ‘watching eyes’ to reduce crime. Furthermore, in an age when we are watched more than at any time in modern history – both online and on the street – our findings highlight an urgent need to fully understand the effect that perceived surveillance, feeling watched, has on our decisions and actions.
In this article, we present some rare counterexamples, which are related to (strong) persistence property and (nearly) normally torsion-freeness of monomial ideals. They may be useful for researchers ...in this field to construct the other counterexamples refuting some conjectures.
There are two different notions for symbolic powers of ideals existing in the literature, one defined in terms of associated primes, the other in terms of minimal primes. Elaborating on an idea known ...to Eisenbud, Herzog, Hibi, Hoa, and Trung, we interpret both notions of symbolic powers as suitable saturations of the ordinary powers. We prove a binomial expansion formula for saturated powers of sums of ideals. This gives a uniform treatment to an array of existing and new results on both notions of symbolic powers of such sums: binomial expansion formulas, computations of depth and regularity, and criteria for the equality of ordinary and symbolic powers.
ON THE SET OF KRONECKER NUMBERS GOSWAMI, SAYAN; HUANG, WEN; WU, XIAOSHENG
Bulletin of the Australian Mathematical Society,
03/2024
Journal Article
Peer reviewed
Abstract A positive even number is said to be a Maillet number if it can be written as the difference between two primes, and a Kronecker number if it can be written in infinitely many ways as the ...difference between two primes. It is believed that all even numbers are Kronecker numbers. We study the division and multiplication of Kronecker numbers and show that these numbers are rather abundant. We prove that there is a computable constant k and a set D consisting of at most 720 computable Maillet numbers such that, for any integer n , $kn$ can be expressed as a product of a Kronecker number and a Maillet number in D . We also prove that every positive rational number can be written as a ratio of two Kronecker numbers.
We use some elementary arguments to obtain a new bound on bilinear sums with weighted Kloosterman sums which complements those recently obtained by E. Kowalski, P. Michel and W. Sawin (2020).
We show that a positive proportion of the primes have the property that if any one of its digits in base 10, including its infinitely many leading 0 digits, is replaced by a different digit, then the ...resulting number is composite.
Abstract
Let denote the maximal number of different primes that may occur in the order of a finite solvable group
G
, all elements of which have orders divisible by at most
n
distinct primes. We show ...that for all . As an application, we improve on a recent bound by Hung and Yang for arbitrary finite groups.
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