On the angles between certain arithmetically defined subspaces of ${\bf C}^n$

Robert Brooks

Displaying similar documents to “On the angles between certain arithmetically defined subspaces of $𝐂^{n}$ ”

Generalized symmetry classes of tensors

Gholamreza Rafatneshan, Yousef Zamani (2020)

Czechoslovak Mathematical Journal

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Let $V$ be a unitary space. For an arbitrary subgroup $G$ of the full symmetric group $S_{m}$ and an arbitrary irreducible unitary representation $Λ$ of $G$ , we study the generalized symmetry class of tensors over $V$ associated with $G$ and $Λ$ . Some important properties of this vector space are investigated.

$L_{p, q}$ 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...................................................................................

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.

On the higher power moments of cusp form coefficients over sums of two squares

Guodong Hua (2022)

Czechoslovak Mathematical Journal

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Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $Γ = SL (2, ℤ)$ . Denote by $λ_{f} (n)$ the $n$ th normalized Fourier coefficient of $f$ . We are interested in the average behaviour of the sum $\sum_{a^{2} + b^{2} \leq x} λ_{f}^{j} (a^{2} + b^{2})$ for $x \geq 1$ , where $a, b \in ℤ$ and $j \geq 9$ is any fixed positive integer. In a similar manner, we also establish analogous results for the normalized coefficients of Dirichlet expansions of associated symmetric power $L$ -functions and Rankin-Selberg $L$ -functions.

Lower bounds for the largest eigenvalue of the gcd matrix on ${1, 2, \dots, n}$

Jorma K. Merikoski (2016)

Czechoslovak Mathematical Journal

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Consider the $n \times n$ matrix with $(i, j)$ ’th entry $gcd (i, j)$ . Its largest eigenvalue $λ_{n}$ and sum of entries $s_{n}$ satisfy $λ_{n} > s_{n} / n$ . Because $s_{n}$ cannot be expressed algebraically as a function of $n$ , we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S. Hong, R. Loewy (2004). We also conjecture that $λ_{n} > 6 π^{- 2} n log n$ for all $n$ . If $n$ is large enough, this follows from F. Balatoni (1969).

Asymptotic values of modular multiplicities for ${GL}_{2}$

Sandra Rozensztajn (2014)

Journal de Théorie des Nombres de Bordeaux

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We study the irreducible constituents of the reduction modulo $p$ of irreducible algebraic representations $V$ of the group ${Res}_{K / ℚ_{p}} {GL}_{2}$ for $K$ a finite extension of $ℚ_{p}$ . We show that asymptotically, the multiplicity of each constituent depends only on the dimension of $V$ and the central character of its reduction modulo $p$ . As an application, we compute the asymptotic value of multiplicities that are the object of the Breuil-Mézard conjecture.

Groups satisfying the two-prime hypothesis with a composition factor isomorphic to PSL $_{2} (q)$ for $q \geq 7$

Mark L. Lewis, Yanjun Liu, Hung P. Tong-Viet (2018)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group and write $cd (G)$ for the degree set of the complex irreducible characters of $G$ . The group $G$ is said to satisfy the two-prime hypothesis if for any distinct degrees $a, b \in cd (G)$ , the total number of (not necessarily different) primes of the greatest common divisor $gcd (a, b)$ is at most $2$ . We prove an upper bound on the number of irreducible character degrees of a nonsolvable group that has a composition factor isomorphic to PSL $_{2} (q)$ for $q \geq 7$ .

On higher moments of Hecke eigenvalues attached to cusp forms

Guodong Hua (2022)

Czechoslovak Mathematical Journal

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Let $f$ , $g$ and $h$ be three distinct primitive holomorphic cusp forms of even integral weights $k_{1}$ , $k_{2}$ and $k_{3}$ for the full modular group $Γ = SL (2, ℤ)$ , respectively, and let $λ_{f} (n)$ , $λ_{g} (n)$ and $λ_{h} (n)$ denote the $n$ th normalized Fourier coefficients of $f$ , $g$ and $h$ , respectively. We consider the cancellations of sums related to arithmetic functions $λ_{g} (n)$ , $λ_{h} (n)$ twisted by $λ_{f} (n)$ and establish the following results: $\sum_{n \leq x} λ_{f} (n) λ_{g} {(n)}^{i} λ_{h} {(n)}^{j} ≪_{f, g, h, ε} x^{1 - 1 / 2^{i + j} + ε}$ for any $ε > 0$ , where $1 \leq i \leq 2$ , $j \geq 5$ are any fixed positive integers.

On sharp characters of type ${- 1, 0, 2}$

Alireza Abdollahi, Javad Bagherian, Mahdi Ebrahimi, Maryam Khatami, Zahra Shahbazi, Reza Sobhani (2022)

Czechoslovak Mathematical Journal

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For a complex character $χ$ of a finite group $G$ , it is known that the product $sh (χ) = \prod_{l \in L (χ)} (χ (1) - l)$ is a multiple of $| G |$ , where $L (χ)$ is the image of $χ$ on $G - {1}$ . The character $χ$ is said to be a sharp character of type $L$ if $L = L (χ)$ and $sh (χ) = | G |$ . If the principal character of $G$ is not an irreducible constituent of $χ$ , then the character $χ$ is called normalized. It is proposed as a problem by P. J. Cameron and M. Kiyota, to find finite groups $G$ with normalized sharp characters of type ${- 1, 0, 2}$ . Here we prove that such a group with nontrivial...

On Kakeya–Nikodym averages, $L^{p}$ -norms and lower bounds for nodal sets of eigenfunctions in higher dimensions

Matthew D. Blair, Christopher D. Sogge (2015)

Journal of the European Mathematical Society

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We extend a result of the second author [27, Theorem 1.1] to dimensions $d \geq 3$ which relates the size of $L^{p}$ -norms of eigenfunctions for $2 < p < 2 (d + 1) / d - 1$ to the amount of $L^{2}$ -mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee [22] and a variable coefficient variant of an " $ϵ$ removal lemma" of Tao and Vargas [35]. We also use Hörmander’s [20] $L^{2}$ oscillatory integral theorem and the Cartan–Hadamard theorem to show that, under the assumption of nonpositive...

Displaying similar documents to “On the angles between certain arithmetically defined subspaces of 𝐂 n ”

Displaying similar documents to “On the angles between certain arithmetically defined subspaces of $𝐂^{n}$ ”