On the Essential Dimension of Some Semi-Direct Products

2002 ◽  
Vol 45 (3) ◽  
pp. 422-427 ◽  
Author(s):  
Arne Ledet
Keyword(s):  
Upper Bound ◽  
Prime Power ◽  
Rational Numbers ◽  
Direct Products ◽  
Essential Dimension

AbstractWe give an upper bound on the essential dimension of the group over the rational numbers, when q is a prime power.

scholarly journals The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

2012 ◽  
Vol 10 (3) ◽  
pp. 455-488 ◽  
Author(s):  
Indranil Biswas ◽  
Ajneet Dhillon ◽  
Nicole Lemire
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Vector Bundles ◽  
Upper Bounds ◽  
Essential Dimension ◽  
Parabolic Structure ◽  
Moduli Stack

AbstractWe find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our calculations also give lower bounds on the essential dimension of the semistable locus inside the moduli stack of vector bundles of rank r and degree d without parabolic structure.

scholarly journals On linear recurrence sequences with polynomial coefficients

1996 ◽  
Vol 38 (2) ◽  
pp. 147-155 ◽  
Author(s):  
A. J. van der Poorten ◽  
I. E. Shparlinski
Keyword(s):  
Recurrence Relation ◽  
Upper Bound ◽  
Rational Numbers ◽  
Negative Integer ◽  
Linear Recurrence ◽  
Initial Values ◽  
Polynomial Coefficients ◽  
Image Position ◽  
Recurrence Sequences

We consider sequences (Ah)defined over the field ℚ of rational numbers and satisfying a linear homogeneous recurrence relationwith polynomial coefficients sj;. We shall assume without loss of generality, as we may, that the sj, are defined over ℤ and the initial values A0A]…, An−1 are integer numbers. Also, without loss of generality we may assume that S0 and Sn have no non-negative integer zero. Indeed, any other case can be reduced to this one by making a shift h → h – l – 1 where l is an upper bound for zeros of the corresponding polynomials (and which can be effectively estimated in terms of their heights)

scholarly journals Congruence preserving expansions of nilpotent algebras

2019 ◽  
Vol 30 (01) ◽  
pp. 167-179
Author(s):  
Erhard Aichinger ◽  
Gábor Horváth
Keyword(s):  
Finite Type ◽  
Prime Power ◽  
Prime Power Order ◽  
Direct Products ◽  
Modular Varieties ◽  
Nilpotent Algebras

We characterize those algebras that have infinitely many polynomially inequivalent expansions among all finite nilpotent algebras of finite type in congruence modular varieties that are direct products of algebras of prime power order.

On the triple tensor product of prime-power groups

2020 ◽  
Vol 23 (5) ◽  
pp. 879-892
Author(s):  
S. Hadi Jafari ◽  
Halimeh Hadizadeh
Keyword(s):  
Tensor Product ◽  
Nilpotent Group ◽  
Upper Bound ◽  
Prime Power ◽  
Tensor Products ◽  
Lower Central Series ◽  
Central Series ◽  
Derived Subgroup

AbstractLet G be a finite p-group, and let {\otimes^{3}G} be its triple tensor product. In this paper, we obtain an upper bound for the order of {\otimes^{3}G}, which sharpens the bound given by G. Ellis and A. McDermott, [Tensor products of prime-power groups, J. Pure Appl. Algebra 132 1998, 2, 119–128]. In particular, when G has a derived subgroup of order at most p, we classify those groups G for which the bound is attained. Furthermore, by improvement of a result about the exponent of {\otimes^{3}G} determined by G. Ellis [On the relation between upper central quotients and lower central series of a group, Trans. Amer. Math. Soc. 353 2001, 10, 4219–4234], we show that, when G is a nilpotent group of class at most 4, {\exp(\otimes^{3}G)} divides {\exp(G)}.

scholarly journals On 'Best' Rational Approximations to $\pi$ and $\pi+e$

2020 ◽  
Author(s):  
Bertrand Teguia Tabuguia
Keyword(s):  
Power Series ◽  
Unit Circle ◽  
Infinite Series ◽  
Rational Number ◽  
Upper Bound ◽  
Continued Fraction ◽  
Series Representation ◽  
Rational Numbers ◽  
Similar Process ◽  
Rational Approximations

Through the half-unit circle area computation using the integration of the corresponding curve power series representation, we deduce a slow converging positive infinite series to $\pi$. However, by studying the remainder of that series we establish sufficiently close inequalities with equivalent lower and upper bound terms allowing us to estimate, more precisely, how the series approaches $\pi$. We use the obtained inequalities to compute up to four-digit denominator, what are in this sense, the best rational numbers that can replace $\pi$. It turns out that the well-known convergents of the continued fraction of $\pi$, $22/7$ and $355/113$ called, respectively, Yuel\"{u} and Mil\"{u} in China are the only ones found. Thus we apply a similar process to find rational estimations to $\pi+e$ where $e$ is taken as the power series of the exponential function evaluated at $1$. For rational numbers with denominators less than $2000$, the convergent $920/157$ of the continued fraction of $\pi+e$ turns out to be the only rational number of this type.

scholarly journals On the Chromatic Number of the Erdős-Rényi Orthogonal Polarity Graph

10.37236/4893 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Xing Peng ◽  
Michael Tait ◽  
Craig Timmons
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Prime Power ◽  
Constant Factor ◽  
Irreducible Polynomials ◽  
Orthogonal Polarity

For a prime power $q$, let $ER_q$ denote the Erdős-Rényi orthogonal polarity graph. We prove that if $q$ is an even power of an odd prime, then $\chi ( ER_{q}) \leq 2 \sqrt{q} + O ( \sqrt{q} / \log q)$. This upper bound is best possible up to a constant factor of at most 2. If $q$ is an odd power of an odd prime and satisfies some condition on irreducible polynomials, then we improve the best known upper bound for $\chi(ER_{q})$ substantially. We also show that for sufficiently large $q$, every $ER_q$ contains a subgraph that is not 3-chromatic and has at most 36 vertices.

scholarly journals On Rational Approximations to $\pi+e$: 920/157 Is the 'Best' Rational Approximation to $\pi+e$ with a Denominator of at Most Three Digits

2020 ◽  
Author(s):  
Bertrand Teguia Tabuguia
Keyword(s):  
Power Series ◽  
Unit Circle ◽  
Infinite Series ◽  
Rational Number ◽  
Upper Bound ◽  
Series Representation ◽  
Rational Numbers ◽  
Similar Process ◽  
Rational Approximations ◽  
Best Rational Approximation

Through the half-unit circle area computation using the integration of the corresponding curve power series representation, we deduce a slow converging positive infinite series to $\pi$. However, by studying the remainder of that series we establish sufficiently close inequalities with equivalent lower and upper bound terms allowing us to estimate, more precisely, how the series approaches $\pi$. We use the obtained inequalities to compute up to four-digit denominator, what are in this sense, the best rational numbers that can replace $\pi$. It turns out that the well-known $22/7$ and $355/113$ called, respectively, Yuel\"{u} and Mil\"{u} in China are the only ones found. This is not so surprising when one considers the empirical computations around these two rational approximations to $\pi$. Thus we apply a similar process to find rational estimations to $\pi+e$ where $e$ is taken as the power series of the exponential function evaluated at $1$. For rational numbers with denominators less than $2000$, $920/157$ turns out to be the only rational number of this type.

scholarly journals Set Systems with Restricted $t$-wise Intersections Modulo Prime Powers

10.37236/255 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Rudy X. J. Liu
Keyword(s):  
Upper Bound ◽  
Prime Power ◽  
Set Systems ◽  
Composite Moduli ◽  
Element Set

We give a polynomial upper bound on the size of set systems with restricted $t$-wise intersections modulo prime powers. Let $t\geq 2$. Let $p$ be a prime and $q=p^{\alpha}$ be a prime power. Let ${\cal L}=\{l_1,l_2,\ldots,l_s\}$ be a subset of $\{0, 1, 2, \ldots, q-1\}$. If ${\cal F}$ is a family of subsets of an $n$ element set $X$ such that $|F_{1}\cap \cdots \cap F_{t}| \pmod{q} \in {\cal L}$ for any collection of $t$ distinct sets from ${\cal F}$ and $|F| \pmod{q} \notin {\cal L}$ for every $F\in {\cal F}$, then $$ |{\cal F}|\leq {t(t-1)\over2}\sum_{i=0}^{2^{s-1}}{n\choose i}. $$ Our result extends a theorem of Babai, Frankl, Kutin, and Štefankovič, who studied the $2$-wise case for prime power moduli, and also complements a result of Grolmusz that no polynomial upper bound holds for non-prime-power composite moduli.

scholarly journals Upper Bounds for the Essential Dimension of E7

2013 ◽  
Vol 56 (4) ◽  
pp. 795-800 ◽  
Author(s):  
Mark L. MacDonald
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Essential Dimension ◽  
Connected Group ◽  
Simply Connected

Abstract.This paper gives a new upper bound for the essential dimension and the essential 2-dimension of the split simply connected group of type E7 over a field of characteristic not 2 or 3. In particular, ed(E7) ≤ 29, and ed(E7, 2) ≤ 27.

scholarly journals О размере множества произведения множеств рациональных чисел

2020 ◽  
Vol 21 (1) ◽  
pp. 357-363
Author(s):  
Юрий Николаевич Штейников
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Mathematical Expectation ◽  
Lower Estimate ◽  
Main Tool ◽  
Rational Numbers ◽  
Prime Numbers ◽  
Random Subset ◽  
First Time

For the first time  in the article [1] was established non-trivial lower bounds on the size of the set of products of rational numbers, the numerators and denominators of which are limited to a certain quantity $Q$. Roughly speaking, it was shown that the size of the product deviates from the maximum by no less than $$\exp \Bigl\{(9 + o(1)) \frac{\log Q}{\sqrt{\log{\log Q}}}\Bigl\}$$ times. In the article [7], the index of $ \log{\log Q} $ was improved from $ 1/2 $ to $ 1 $, and the proof of the main result on the set of fractions was fundamentally different. This proof, its argument was based on the search for a special large subset of the original set of rational numbers, the set of numerators and denominators of which were pairwise mutually prime numbers. The main tool was the consideration of random subsets. A lower estimate was obtained for the mathematical expectation of the size of this random subset. There, it was possible to obtain an upper bound for the multiplicative energy of the considered set. The lower bound for the number of products and the upper bound for the multiplicative energy of the set are close to optimal results. In this article, we propose the following scheme. In general, we follow the scheme of the proof of the article [1], while modifying some steps and introducing some additional optimizations, we also improve the index from $1/2$ to $1-\varepsilon$ for an arbitrary positive $\varepsilon>0$.

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