Vol.35 (1) 2013 — 3 articles found.

The Cuntz Semigroup of Some Spaces of Dimension at Most Two

C. R. Math. Rep. Acad. Sci. Canada Vol. 35 (1) 2013, pp. 22–32
Vol.35 (1) 2013
Leonel Robert Details
(Received: 2012-09-12 , Revised: 2013-03-26 )
(Received: 2012-09-12 , Revised: 2013-03-26 )

Leonel Robert, Department of Mathematics, University of Louisiana at Lafayette, Lafayette, USA; e-mail: lrobert@louisiana.edu

Abstract/Résumé:

It is shown that the Cuntz semigroup of a space with dimension at most two, and with second cohomology of its compact subsets equal to zero, is isomorphic to the ordered semigroup of lower semicontinuous functions on the space with values in the natural numbers with the infinity adjoined. This computation is then used to obtain the Cuntz semigroup of all compact surfaces. A converse to the first computation is also proven: if the Cuntz semigroup of a separable C*-algebra is isomorphic

Il est montré que le semi-groupe de Cuntz d’un espace de dimension au plus deux, et avec cohomologie deuxième de ses sous-ensembles compacts égales à zéro, est isomorphe au semi-groupe ordonné de fonctions semi-continue inférieurement sur l’espace de baisse avec des valeurs au entiers naturels augmentée à l’infini. Ce calcul est ensuite utilisé pour obtenir le semi-groupe de Cuntz de toutes les surfaces compacts. Un inverse du premier calcul est également prouvé: si le semi-groupe de Cuntz d’un C*-algèbre séparable est isomorphe aux fonctions semi-continue inférieurement de une space topoligique à valeurs dans les entiers naturels augmentée, alors la C*-algébre est commutative à stabilisation près, et son spectrum satisfait aux conditions dimensionnelles et cohomologique mentionné ci-dessus.

Keywords: Cuntz Semigroup

AMS Subject Classification: $C^*$-modules 46L08

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Construction of the Heptadecagon and Quadratic Reciprocity

C. R. Math. Rep. Acad. Sci. Canada Vol. 35 (1) 2013, pp. 16–21
Vol.35 (1) 2013
Yuri Burda; Liudmyla Kadets Details
(Received: 2012-03-26 )
(Received: 2012-03-26 )

Yuri Burda, University of Toronto; e-mail: yburda@math.toronto.edu

Liudmyla Kadets, University of Toronto; e-mail: lucy.kadets@math.toronto.edu

Abstract/Résumé:

In this note we present a construction of a regular 17-gon using ruler and compass. We relate steps in this construction to quadratic reciprocity and some trigonometric identities.

Dans cette note, nous présentons une construction à la règle et au compas de l’heptadécagone. Nous éstablisson des liens les étapes de cette construction de réciprocité quadratique et des identitiés trigonométriques.

Keywords: Heptadecagon, Quadratic Reciprocity

AMS Subject Classification: 11A20

PDF(click to download): Construction of the Heptadecagon and Quadratic Reciprocity

On the Korselt Set of a Squarefree Composite Number

C. R. Math. Rep. Acad. Sci. Canada Vol. 35 (1) 2013, pp. 1–15
Vol.35 (1) 2013
Ibrahim Al-Rasasi; Othman Echi; Nejib Ghanmi Details
(Received: 2012-09-01 , Revised: 2012-11-18 )
(Received: 2012-09-01 , Revised: 2012-11-18 )

Ibrahim Al-Rasasi, King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics PO Box 5046, Dhahran 31261, Saudi Arabia e-mail: irasasi@kfupm.edu.sa

Othman Echi, King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics PO Box 5046, Dhahran 31261, Saudi Arabia e-mail: echi@kfupm.edu.sa; othechi@yahoo.com

Nejib Ghanmi, Umm Al-Qura University, University College in Makkah, Department of Mathematics, Azizia PO.Box 2064, Makkah, Kingdom of Saudi Arabia e-mail: naghanmi@uqu.edu.sa; neghanmi@yahoo.fr

Abstract/Résumé:

Let \(\alpha\in \mathbb{Z}\setminus \{0\}\). A positive composite squarefree integer \(N\) is said to be an \(\alpha\)-Korselt number (\(K_{\alpha}\)-number, for short) if \(N\neq \alpha\) and \(p-\alpha\) divides \(N-\alpha\) for each prime divisor \(p\) of \(N\). By the Korselt set of \(N\), we mean the set of all \(\alpha\in \mathbb{Z}\setminus \{0\}\) such that \(N\) is a \(K_{\alpha}\)-number. This set will be denoted by \(\mathcal{KS}(N)\).

In a recent paper , Bouallegue–Echi–Pinch have asked whether there are infinitely many squarefree composite numbers with empty Korselt set. This paper aims to solve this question by showing that for each prime number \(q\geq 19\), \(6q\) has an empty Korselt set.

We also show that for each integer \(l\geq 3\), there are infinitely many squarefree composite numbers with \(l\) prime divisors whose Korselt sets are empty.

Soit \(N\) un nombre composé sans facteur carré et \(\alpha\in \mathbb{Z} \setminus \{0\}\). On dit que \(N\) est \(\alpha\)-Korselt si \(N\neq \alpha\) et \(p-\alpha\) divise \(N-\alpha\) pour tout facteur premier \(p\) de \(N\).

L’ensemble constitué de tous les \(\alpha\) tels que \(N\) est \(\alpha\)-Korselt, noté \(\mathcal{KS}(N)\), est appelé l’ensemble de Korselt de \(N\).

Bouallegue–Echi–Pinch se sont posés la question d’existence d’une infinité de nombres composés sans facteur carré possédant des ensembles de Korselt vides.

Dans ce papier on donne une réponse positive à cette question en démontrant que pour tout premier \(q\geq 19\), \(6q\) a un ensemble de Korselt vide.

On prouve aussi que pour tout entier \(l\geq 3\), il existe une infinité de nombres composés sans facteur carré ayant \(l\) facteurs premiers et d’ensembles de Korselt vides.

Keywords: Carmichael number, Korselt number, Korselt set, prime number, squarefree composite number

AMS Subject Classification: Algorithms; complexity 11Y16

PDF(click to download): On the Korselt Set of a Squarefree Composite Number

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