scholarly journals An improved bound on the least common multiple of polynomial sequences

2021 ◽  
Vol 32 (3) ◽  
pp. 891-899
Author(s):  
Ashwin Sah
Keyword(s):  
Common Multiple ◽  
Least Common Multiple ◽  
Polynomial Sequences

scholarly journals On the least common multiple of polynomial sequences at prime arguments

2021 ◽  
pp. 1-11
Author(s):  
Ayan Nath ◽  
Abhishek Jha
Keyword(s):  
Lower Bounds ◽  
Prime Divisor ◽  
Irreducible Polynomial ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Polynomial Sequences

Cilleruelo conjectured that if [Formula: see text] is an irreducible polynomial of degree [Formula: see text] then, [Formula: see text] In this paper, we investigate the analog of prime arguments, namely, [Formula: see text] where [Formula: see text] denotes a prime and obtain nontrivial lower bounds on it. Further, we also show some results regarding the greatest prime divisor of [Formula: see text]

scholarly journals On the least common multiple of random q-integers

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Carlo Sanna
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Probabilistic Model ◽  
Expected Value ◽  
Random Set ◽  
Common Multiple ◽  
Least Common Multiple

AbstractFor every positive integer n and for every $$\alpha \in [0, 1]$$ α ∈ [ 0 , 1 ] , let $${\mathcal {B}}(n, \alpha )$$ B ( n , α ) denote the probabilistic model in which a random set $${\mathcal {A}} \subseteq \{1, \ldots , n\}$$ A ⊆ { 1 , … , n } is constructed by picking independently each element of $$\{1, \ldots , n\}$$ { 1 , … , n } with probability $$\alpha $$ α . Cilleruelo, Rué, Šarka, and Zumalacárregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of $${\mathcal {A}}$$ A .Let q be an indeterminate and let $$[k]_q := 1 + q + q^2 + \cdots + q^{k-1} \in {\mathbb {Z}}[q]$$ [ k ] q : = 1 + q + q 2 + ⋯ + q k - 1 ∈ Z [ q ] be the q-analog of the positive integer k. We determine the expected value and the variance of $$X := \deg {\text {lcm}}\!\big ([{\mathcal {A}}]_q\big )$$ X : = deg lcm ( [ A ] q ) , where $$[{\mathcal {A}}]_q := \big \{[k]_q : k \in {\mathcal {A}}\big \}$$ [ A ] q : = { [ k ] q : k ∈ A } . Then we prove an almost sure asymptotic formula for X, which is a q-analog of the result of Cilleruelo et al.

scholarly journals Metaplectic flavor symmetries from magnetized tori

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Yahya Almumin ◽  
Mu-Chun Chen ◽  
Víctor Knapp-Pérez ◽  
Saúl Ramos-Sánchez ◽  
Michael Ratz ◽  
...  
Keyword(s):  
Zero Mode ◽  
Metaplectic Groups ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Yukawa Couplings ◽  
Flavor Symmetries ◽  
Euler’S Theorem ◽  
Modular Transformations ◽  
Analytic Expressions ◽  
Geometric Picture

Abstract We revisit the flavor symmetries arising from compactifications on tori with magnetic background fluxes. Using Euler’s Theorem, we derive closed form analytic expressions for the Yukawa couplings that are valid for arbitrary flux parameters. We discuss the modular transformations for even and odd units of magnetic flux, M, and show that they give rise to finite metaplectic groups the order of which is determined by the least common multiple of the number of zero-mode flavors involved. Unlike in models in which modular flavor symmetries are postulated, in this approach they derive from an underlying torus. This allows us to retain control over parameters, such as those governing the kinetic terms, that are free in the bottom-up approach, thus leading to an increased predictivity. In addition, the geometric picture allows us to understand the relative suppression of Yukawa couplings from their localization properties in the compact space. We also comment on the role supersymmetry plays in these constructions, and outline a path towards non-supersymmetric models with modular flavor symmetries.

Improved Least Common Multiple Effective Wavelength Method Used in Dual-Wavelength Interferometry

2021 ◽  
Vol 58 (7) ◽  
pp. 0712002
Author(s):  
郭小庭 Guo Xiaoting ◽  
刘晓军 Liu Xiaojun ◽  
雷自力 Lei Zili ◽  
杨文军 Yang Wenjun ◽  
徐龙 Xu Long
Keyword(s):  
Dual Wavelength ◽  
Effective Wavelength ◽  
Common Multiple ◽  
Least Common Multiple

On the least common multiple of Lucas subsequences

2013 ◽  
Vol 161 (4) ◽  
pp. 327-349 ◽  
Author(s):  
Shigeki Akiyama ◽  
Florian Luca
Keyword(s):  
Common Multiple ◽  
Least Common Multiple

On a Theorem Of Baer and Higman

1956 ◽  
Vol 8 ◽  
pp. 263-270 ◽  
Author(s):  
Seân Tobin
Keyword(s):  
Finite Number ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Derived Subgroup

1.1 Baer has shown (1) that if the fact that the exponent of a group is m (that is, m is the least common multiple of the periods of the elements) implies a limitation on the class of the group, then m must be a prime. Graham Higman has extended this result by proving (3) that for any given integer M there are at most a finite number of prime powers q other than primes, such that the fact that a group has exponent q implies a limitation on the class of the Mth derived subgroup.

scholarly journals Estimate of a certain least common multiple.

1960 ◽  
Vol 7 (1) ◽  
pp. 75-78
Author(s):  
D. J. Newman
Keyword(s):  
Common Multiple ◽  
Least Common Multiple
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