Displaying similar documents to “Counting invertible matrices and uniform distribution”

New bounds on the length of finite pierce and Engel series

P. Erdös, J. O. Shallit (1991)

Journal de théorie des nombres de Bordeaux

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Every real number x , 0 < x ≤ 1 , has an essentially unique expansion as a Pierce series : x = 1 x 1 - 1 x 1 x 2 + 1 x 1 x 2 x 3 - ⋯ where the x i form a strictly increasing sequence of positive integers. The expansion terminates if and only if x is rational. Similarly, every positive real number y has a unique expansion as an Engel series : y = 1 y 1 - 1 y 1 y 2 + 1 y 1 y 2 y 3 + ⋯ where the y i form a (not necessarily strictly) increasing sequence of positive integers. If the expansion is infinite, we require that the sequence yi...

On units of some fields of the form ℚ ( 2 , p , q , - l )

Mohamed Mahmoud Chems-Eddin (2023)

Mathematica Bohemica

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Let p ≡ 1 ( mod 8 ) and q ≡ 3 ( mod 8 ) be two prime integers and let ℓ ∉ { - 1 , p , q } be a positive odd square-free integer. Assuming that the fundamental unit of ℚ ( 2 p ) has a negative norm, we investigate the unit group of the fields ℚ ( 2 , p , q , - ℓ ) .

Polydisc slicing in ℂ n

Krzysztof Oleszkiewicz, Aleksander Pełczyński (2000)

Studia Mathematica

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Let D be the unit disc in the complex plane ℂ. Then for every complex linear subspace H in ℂ n of codimension 1, v o l 2 n - 2 ( D n - 1 ) ≤ v o l 2 n - 2 ( H ∩ D n ) ≤ 2 v o l 2 n - 2 ( D n - 1 ) . The lower bound is attained if and only if H is orthogonal to the versor e j of the jth coordinate axis for some j = 1,...,n; the upper bound is attained if and only if H is orthogonal to a vector e j + σ e k for some 1 ≤ j < k ≤ n and some σ ∈ ℂ with |σ| = 1. We identify ℂ n with ℝ 2 n ; by v o l k ( · ) we denote the usual k-dimensional volume in ℝ 2 n . The result is a complex counterpart of Ball’s [B1]...

On the binary expansions of algebraic numbers

David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, Carl Pomerance (2004)

Journal de Théorie des Nombres de Bordeaux

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Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1’s in the binary expansions of real algebraic numbers. A central result is that if a real y has algebraic degree D &gt; 1 , then the number # ( | y | , N ) of 1-bits in the expansion of | y | through bit position N satisfies # ( | y | , N ) &gt; C N 1 / D for a positive number C (depending on y ) and sufficiently large N . This in itself establishes the transcendency...

Extension of CR functions to «wedge type» domains

Andrea D&amp;amp;#039;Agnolo, Piero D&amp;#039;Ancona, Giuseppe Zampieri (1991)

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

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Let X be a complex manifold, S a generic submanifold of X R , the real underlying manifold to X . Let Ω be an open subset of S with ∂ ⁡ Ω analytic, Y a complexification of S . We first recall the notion of Ω -tuboid of X and of Y and then give a relation between; we then give the corresponding result in terms of microfunctions at the boundary. We relate the regularity at the boundary for ∂ ¯ b to the extendability of C ⁢ R functions on Ω to Ω -tuboids of X . Next, if X has complex dimension 2, we give results...

Note on the Hilbert 2-class field tower

Abdelmalek Azizi, Mohamed Mahmoud Chems-Eddin, Abdelkader Zekhnini (2022)

Mathematica Bohemica

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Let k be a number field with a 2-class group isomorphic to the Klein four-group. The aim of this paper is to give a characterization of capitulation types using group properties. Furthermore, as applications, we determine the structure of the second 2-class groups of some special Dirichlet fields 𝕜 = ℚ ( d , - 1 ) , which leads to a correction of some parts in the main results of A. Azizi and A. Zekhini (2020).

The unit group of some fields of the form ℚ ( 2 , p , q , - l )

Moha Ben Taleb El Hamam (2024)

Mathematica Bohemica

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Let p and q be two different prime integers such that p ≡ q ≡ 3 ( mod 8 ) with ( p / q ) = 1 , and l a positive odd square-free integer relatively prime to p and q . In this paper we investigate the unit groups of number fields 𝕃 = ℚ ( 2 , p , q , - l ) .