On some identities involving spherical means

Gianfranco Cimmino

Displaying similar documents to “On some identities involving spherical means”

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.

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.

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...

On generalized square-full numbers in an arithmetic progression

Angkana Sripayap, Pattira Ruengsinsub, Teerapat Srichan (2022)

Czechoslovak Mathematical Journal

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Let $a$ and $b \in ℕ$ . Denote by $R_{a, b}$ the set of all integers $n > 1$ whose canonical prime representation $n = p_{1}^{α_{1}} p_{2}^{α_{2}} \dots p_{r}^{α_{r}}$ has all exponents $α_{i}$ $(1 \leq i \leq r)$ being a multiple of $a$ or belonging to the arithmetic progression $a t + b$ , $t \in ℕ_{0} : = ℕ \cup {0}$ . All integers in $R_{a, b}$ are called generalized square-full integers. Using the exponent pair method, an upper bound for character sums over generalized square-full integers is derived. An application on the distribution of generalized square-full integers in an arithmetic progression is given. ...

Structure fractals and para-quaternionic geometry

Julian Ławrynowicz, Massimo Vaccaro (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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It is well known that starting with real structure, the Cayley-Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions $p = 2, 4$ and $8$ , respectively, but the procedure fails for $p = 16$ in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are $n = 2^{7}$ . Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower...

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...

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 $K_{2 n - 2} (O_{F})$ .

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...

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 $a$ .

The Golomb space is topologically rigid

Taras O. Banakh, Dario Spirito, Sławomir Turek (2021)

Commentationes Mathematicae Universitatis Carolinae

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The Golomb space $ℕ_{τ}$ is the set $ℕ$ of positive integers endowed with the topology $τ$ generated by the base consisting of arithmetic progressions ${a + b n : n \geq 0}$ with coprime $a, b$ . We prove that the Golomb space $ℕ_{τ}$ is topologically rigid in the sense that its homeomorphism group is trivial. This resolves a problem posed by T. Banakh at Mathoverflow in 2017.

On the least almost-prime in arithmetic progressions

Liuying Wu (2024)

Czechoslovak Mathematical Journal

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Let $𝒫_{2}$ denote a positive integer with at most $2$ prime factors, counted according to multiplicity. For integers $a$ , $q$ such that $(a, q) = 1$ , let $𝒫_{2} (q, a)$ denote the least $𝒫_{2}$ in the arithmetic progression ${n q + a}_{n = 1}^{\infty}$ . It is proved that for sufficiently large $q$ , we have $𝒫_{2} (q, a) ≪ q^{1.825} .$ This result constitutes an improvement upon that of J. Li, M. Zhang and Y. Cai (2023), who obtained $𝒫_{2} (q, a) ≪ q^{1.8345} .$

Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions

G. Letac, J. Wesołowski (2011)

Bulletin de la Société Mathématique de France

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If the space $𝒬$ of quadratic forms in $ℝ^{n}$ is splitted in a direct sum $𝒬_{1} \oplus ... \oplus 𝒬_{k}$ and if $X$ and $Y$ are independent random variables of $ℝ^{n}$ , assume that there exist a real number $a$ such that $E (X | X + Y) = a (X + Y)$ and real distinct numbers $b_{1}, . . ., b_{k}$ such that $E (q (X) | X + Y) = b_{i} q (X + Y)$ for any $q$ in $𝒬_{i} .$ We prove that this happens only when $k = 2$ , when $ℝ^{n}$ can be structured in a Euclidean Jordan algebra and when $X$ and $Y$ have Wishart distributions corresponding to this structure.