Monotonicity of the period function for some planar differential systems. Part I: Conservative and quadratic systems

A. Raouf Chouikha

Displaying similar documents to “Monotonicity of the period function for some planar differential systems. Part I: Conservative and quadratic systems”

Positivity of quadratic base change $L$ -functions

Hervé Jacquet, Chen Nan (2001)

Bulletin de la Société Mathématique de France

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We show that certain quadratic base change $L$ -functions for $Gl (2)$ are non-negative at their center of symmetry.

Minimal $𝒮$ -universality criteria may vary in size

Noam D. Elkies, Daniel M. Kane, Scott Duke Kominers (2013)

Journal de Théorie des Nombres de Bordeaux

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In this note, we give simple examples of sets $𝒮$ of quadratic forms that have minimal $𝒮$ -universality criteria of multiple cardinalities. This answers a question of Kim, Kim, and Oh [KKO05] in the negative.

Weighted Erdős-Kac type theorem over quadratic field in short intervals

Xiaoli Liu, Zhishan Yang (2022)

Czechoslovak Mathematical Journal

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Let $𝕂$ be a quadratic field over the rational field and $a_{𝕂} (n)$ be the number of nonzero integral ideals with norm $n$ . We establish Erdős-Kac type theorems weighted by $a_{𝕂} {(n)}^{l}$ and $a_{𝕂} {(n^{2})}^{l}$ of quadratic field in short intervals with $l \in ℤ^{+}$ . We also get asymptotic formulae for the average behavior of $a_{𝕂} {(n)}^{l}$ and $a_{𝕂} {(n^{2})}^{l}$ in short intervals.

A twisted class number formula and Gross's special units over an imaginary quadratic field

Saad El Boukhari (2023)

Czechoslovak Mathematical Journal

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Let $F / k$ be a finite abelian extension of number fields with $k$ imaginary quadratic. Let $O_{F}$ be the ring of integers of $F$ and $n \geq 2$ a rational integer. We construct a submodule in the higher odd-degree algebraic $K$ -groups of $O_{F}$ using corresponding Gross’s special elements. We show that this submodule is of finite index and prove that this index can be computed using the higher “twisted” class number of $F$ , which is the cardinal of the finite algebraic $K$ -group $K_{2 n - 2} (O_{F})$ .

The number of minimum points of a positive quadratic form

G. L. Watson

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CONTENTSIntroduction.......................................................................................61. Definition of certain special forms...........................................62. Statement of results...................................................................83. Proof of Theorem 2.....................................................................94. Preliminaries for Theorem 1.....................................................105. Further preliminaries for Theorem...

Optimality of the Width- $w$ Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases

Clemens Heuberger, Daniel Krenn (2013)

Journal de Théorie des Nombres de Bordeaux

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We consider digit expansions $\sum_{j = 0}^{ℓ - 1} Φ^{j} (d_{j})$ with an endomorphism $Φ$ of an Abelian group. In such a numeral system, the $w$ -NAF condition (each block of $w$ consecutive digits contains at most one nonzero) is shown to minimise the Hamming weight over all expansions with the same digit set if and only if it fulfills the subadditivity condition (the sum of every two expansions of weight $1$ admits an optimal $w$ -NAF). This result is then applied to imaginary quadratic bases, which are used for scalar...

On some identities involving spherical means

Gianfranco Cimmino (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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For every positive definite quadratic form in $n$ variables the reciprocal of the square root of the discriminant is equal to the arithmetic mean of the values assumed by the form on the $n-1$ sphere centered at $0$ and with radius $1$ raised to the ( $-\frac{n}{2}$ )-th. power. Various consequences are deduced from this, in particular a simplification of some calculations from which one obtains the possibility of solving linear systems using spherical means rather than determinants.

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 $a$ .

Sumsets in quadratic residues

I. D. Shkredov (2014)

Acta Arithmetica

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We describe all sets $A \subseteq_{p}$ which represent the quadratic residues $R \subseteq_{p}$ in the sense that R = A + A or R = A ⨣ A. Also, we consider the case of an approximate equality R ≈ A + A and R ≈ A ⨣ A and prove that A is then close to a perfect difference set.

Lower bound for class numbers of certain real quadratic fields

Mohit Mishra (2023)

Czechoslovak Mathematical Journal

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Let $d$ be a square-free positive integer and $h (d)$ be the class number of the real quadratic field $ℚ (\sqrt{d}) .$ We give an explicit lower bound for $h (n^{2} + r)$ , where $r = 1, 4$ . Ankeny and Chowla proved that if $g > 1$ is a natural number and $d = n^{2 g} + 1$ is a square-free integer, then $g ∣ h (d)$ whenever $n > 4$ . Applying our lower bounds, we show that there does not exist any natural number $n > 1$ such that $h (n^{2 g} + 1) = g$ . We also obtain a similar result for the family $ℚ (\sqrt{n^{2 g} + 4})$ . As another application, we deduce some criteria for a class group of prime power order to be...

Invariance of the parity conjecture for $p$ -Selmer groups of elliptic curves in a $D_{2 p^{n}}$ -extension

Thomas de La Rochefoucauld (2011)

Bulletin de la Société Mathématique de France

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We show a $p$ -parity result in a $D_{2 p^{n}}$ -extension of number fields $L / K$ ( $p \geq 5$ ) for the twist $1 \oplus η \oplus τ$ : $W (E / K, 1 \oplus η \oplus τ) = {(- 1)}^{〈1 \oplus η \oplus τ, X_{p} (E / L)〉}$ , where $E$ is an elliptic curve over $K$ , $η$ and $τ$ are respectively the quadratic character and an irreductible representation of degree $2$ of $Gal (L / K) = D_{2 p^{n}}$ , and $X_{p} (E / L)$ is the $p$ -Selmer group. The main novelty is that we use a congruence result between $ε_{0}$ -factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the $p$ -parity conjecture...