Multidimensional Gauss reduction theory for conjugacy classes of $\mathrm{SL}(n,\mathbb{Z})$

Oleg Karpenkov

Displaying similar documents to “Multidimensional Gauss reduction theory for conjugacy classes of $SL (n, ℤ)$ ”

Pell and Pell-Lucas numbers of the form $- 2^{a} - 3^{b} + 5^{c}$

Yunyun Qu, Jiwen Zeng (2020)

Czechoslovak Mathematical Journal

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In this paper, we find all Pell and Pell-Lucas numbers written in the form $- 2^{a} - 3^{b} + 5^{c}$ , in nonnegative integers $a$ , $b$ , $c$ , with $0 \leq max {a, b} \leq c$ .

The number of conjugacy classes of elements of the Cremona group of some given finite order

Jérémy Blanc (2007)

Bulletin de la Société Mathématique de France

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This note presents the study of the conjugacy classes of elements of some given finite order $n$ in the Cremona group of the plane. In particular, it is shown that the number of conjugacy classes is infinite if $n$ is even, $n = 3$ or $n = 5$ , and that it is equal to $3$ (respectively $9$ ) if $n = 9$ (respectively if $n = 15$ ) and to $1$ for all remaining odd orders. Some precise representative elements of the classes are given.

An approximation property of quadratic irrationals

Takao Komatsu (2002)

Bulletin de la Société Mathématique de France

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Let $α > 1$ be irrational. Several authors studied the numbers $ℓ^{m} (α) = inf {| y | : y \in Λ_{m}, y \neq 0},$ where $m$ is a positive integer and $Λ_{m}$ denotes the set of all real numbers of the form $y = ϵ_{0} α^{n} + ϵ_{1} α^{n - 1} + \dots + ϵ_{n - 1} α + ϵ_{n}$ with restricted integer coefficients $| ϵ_{i} | \leq m$ . The value of $ℓ^{1} (α)$ was determined for many particular Pisot numbers and $ℓ^{m} (α)$ for the golden number. In this paper the value of $ℓ^{m} (α)$ is determined for irrational numbers $α$ , satisfying $α^{2} = a α \pm 1$ with a positive integer $a$ .

On the unit group of a semisimple group algebra $𝔽_{q} S L (2, ℤ_{5})$

Rajendra K. Sharma, Gaurav Mittal (2022)

Mathematica Bohemica

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We give the characterization of the unit group of $𝔽_{q} S L (2, ℤ_{5})$ , where $𝔽_{q}$ is a finite field with $q = p^{k}$ elements for prime $p > 5,$ and $S L (2, ℤ_{5})$ denotes the special linear group of $2 \times 2$ matrices having determinant $1$ over the cyclic group $ℤ_{5}$ .

A note on multilinear Muckenhoupt classes for multiple weights

Songqing Chen, Huoxiong Wu, Qingying Xue (2014)

Studia Mathematica

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This paper is devoted to investigating the properties of multilinear $A_{P ⃗}$ conditions and $A_{(P ⃗, q)}$ conditions, which are suitable for the study of multilinear operators on Lebesgue spaces. Some monotonicity properties of $A_{P ⃗}$ and $A_{(P ⃗, q)}$ classes with respect to P⃗ and q are given, although these classes are not in general monotone with respect to the natural partial order. Equivalent characterizations of multilinear $A_{(P ⃗, q)}$ classes in terms of the linear $A_{p}$ classes are established. These results essentially improve...

On the length of rational continued fractions over $_{q} (X)$

S. Driss (2015)

Discussiones Mathematicae - General Algebra and Applications

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Let $_{q}$ be a finite field and $A (Y) \in_{q} (X, Y)$ . The aim of this paper is to prove that the length of the continued fraction expansion of $A (P); P \in_{q} [X]$ , is bounded.

$(0, 1)$ -matrices, discrepancy and preservers

LeRoy B. Beasley (2019)

Czechoslovak Mathematical Journal

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Let $m$ and $n$ be positive integers, and let $R = (r_{1}, ..., r_{m})$ and $S = (s_{1}, ..., s_{n})$ be nonnegative integral vectors. Let $A (R, S)$ be the set of all $m \times n$ $(0, 1)$ -matrices with row sum vector $R$ and column vector $S$ . Let $R$ and $S$ be nonincreasing, and let $F (R)$ be the $m \times n$ $(0, 1)$ -matrix, where for each $i$ , the $i$ th row of $F (R, S)$ consists of $r_{i}$ 1’s followed by $(n - r_{i})$ 0’s. Let $A \in A (R, S)$ . The discrepancy of A, $disc (A)$ , is the number of positions in which $F (R)$ has a 1 and $A$ has a 0. In this paper we investigate linear operators mapping $m \times n$ matrices over...

The sizes of the classes of $H^{(N)}$ -sets

Václav Vlasák (2014)

Fundamenta Mathematicae

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The class of $H^{(N)}$ -sets forms an important subclass of the class of sets of uniqueness for trigonometric series. We investigate the size of this class which is reflected by the family of measures (called polar) annihilating all sets from the class. The main aim of this paper is to answer in the negative a question stated by Lyons, whether the polars of the classes of $H^{(N)}$ -sets are the same for all N ∈ ℕ. To prove our result we also present a new description of $H^{(N)}$ -sets.

On the conjugate type vector and the structure of a normal subgroup

Ruifang Chen, Lujun Guo (2022)

Czechoslovak Mathematical Journal

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Let $N$ be a normal subgroup of a group $G$ . The structure of $N$ is given when the $G$ -conjugacy class sizes of $N$ is a set of a special kind. In fact, we give the structure of a normal subgroup $N$ under the assumption that the set of $G$ -conjugacy class sizes of $N$ is $(p_{1 n_{1}}^{a_{1 n_{1}}}, \dots, p_{11}^{a_{11}}, 1) \times \dots \times (p_{r n_{r}}^{a_{r n_{r}}}, \dots, p_{r 1}^{a_{r 1}}, 1)$ , where $r > 1$ , $n_{i} > 1$ and $p_{i j}$ are distinct primes for $i \in {1, 2, \dots, r}$ , $j \in {1, 2, \dots, n_{i}}$ .

-simplicity of interval max-min matrices

Ján Plavka, Štefan Berežný (2018)

Kybernetika

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A matrix $A$ is said to have $𝐗$ -simple image eigenspace if any eigenvector $x$ belonging to the interval $𝐗 = {x : \underset{̲}{x} \leq x \leq \bar{x}}$ containing a constant vector is the unique solution of the system $A \otimes y = x$ in $𝐗$ . The main result of this paper is an extension of $𝐗$ -simplicity to interval max-min matrix $𝐀 = {A : \underset{̲}{A} \leq A \leq \bar{A}}$ distinguishing two possibilities, that at least one matrix or all matrices from a given interval have $𝐗$ -simple image eigenspace. $𝐗$ -simplicity of interval matrices in max-min algebra are studied and equivalent conditions for...

A new characterization for the simple group $PSL (2, p^{2})$ by order and some character degrees

Behrooz Khosravi, Behnam Khosravi, Bahman Khosravi, Zahra Momen (2015)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group and $p$ a prime number. We prove that if $G$ is a finite group of order $| PSL (2, p^{2}) |$ such that $G$ has an irreducible character of degree $p^{2}$ and we know that $G$ has no irreducible character $θ$ such that $2 p ∣ θ (1)$ , then $G$ is isomorphic to $PSL (2, p^{2})$ . As a consequence of our result we prove that $PSL (2, p^{2})$ is uniquely determined by the structure of its complex group algebra.

$L_{p, q}$ spaces

Joseph Kupka

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CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the $L_{p, q}$ spaces................................. 114. Integral representation of bounded linear functionals on $L_{p, q} (B)$ ........ 235. Examples in $L_{p, q}$ theory...................................................................................

On row-sum majorization

Farzaneh Akbarzadeh, Ali Armandnejad (2019)

Czechoslovak Mathematical Journal

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Let $𝕄_{n, m}$ be the set of all $n \times m$ real or complex matrices. For $A, B \in 𝕄_{n, m}$ , we say that $A$ is row-sum majorized by $B$ (written as $A ≺^{rs} B$ ) if $R (A) ≺ R (B)$ , where $R (A)$ is the row sum vector of $A$ and $≺$ is the classical majorization on $ℝ^{n}$ . In the present paper, the structure of all linear operators $T : 𝕄_{n, m} \to 𝕄_{n, m}$ preserving or strongly preserving row-sum majorization is characterized. Also we consider the concepts of even and circulant majorization on $ℝ^{n}$ and then find the linear preservers of row-sum majorization of these relations on $𝕄_{n, m}$ . ...

Controllable and tolerable generalized eigenvectors of interval max-plus matrices

Matej Gazda, Ján Plavka (2021)

Kybernetika

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By max-plus algebra we mean the set of reals $ℝ$ equipped with the operations $a \oplus b = max {a, b}$ and $a \otimes b = a + b$ for $a, b \in ℝ .$ A vector $x$ is said to be a generalized eigenvector of max-plus matrices $A, B \in ℝ (m, n)$ if $A \otimes x = λ \otimes B \otimes x$ for some $λ \in ℝ$ . The investigation of properties of generalized eigenvectors is important for the applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. In this paper the properties of matrices and vectors with inexact (interval)...

On operators from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ or to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$

Christian Samuel (2010)

Colloquium Mathematicae

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We show that every operator from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ is compact when 1 ≤ p,q < s and that every operator from $ℓ_{s}$ to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$ is compact when 1/p + 1/q > 1 + 1/s.

On hereditary normality of $ω^{*}$ , Kunen points and character $ω_{1}$

Sergei Logunov (2021)

Commentationes Mathematicae Universitatis Carolinae

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We show that $ω^{*} ∖ {p}$ is not normal, if $p$ is a limit point of some countable subset of $ω^{*}$ , consisting of points of character $ω_{1}$ . Moreover, such a point $p$ is a Kunen point and a super Kunen point.

Displaying similar documents to “Multidimensional Gauss reduction theory for conjugacy classes of SL ( n , ℤ ) ”

Displaying similar documents to “Multidimensional Gauss reduction theory for conjugacy classes of $SL (n, ℤ)$ ”