An Easy Proof of Hurwitz's Theorem

2002 ◽  
Vol 109 (10) ◽  
pp. 916 ◽  
Author(s):  
Manuel Benito ◽  
J. Javier Escribano
Keyword(s):  
Easy Proof ◽  
Hurwitz’S Theorem

An Easy Proof of Hurwitz's Theorem

2002 ◽  
Vol 109 (10) ◽  
pp. 916-918
Author(s):  
Manuel Benito ◽  
J. Javier Escribano
Keyword(s):  
Easy Proof ◽  
Hurwitz’S Theorem

An Easy Proof for Some Classical Theorems in Plane Geometry

1992 ◽  
Vol 35 (4) ◽  
pp. 560-568 ◽  
Author(s):  
C. Thas
Keyword(s):  
Projective Plane ◽  
Projective Geometry ◽  
The Other ◽  
Plane Geometry ◽  
The Real ◽  
Easy Proof ◽  
Special Cases ◽  
Euclidean Planes ◽  
Straight Lines ◽  
Short Method

AbstractThe main result of this paper is a theorem about three conies in the complex or the real complexified projective plane. Is this theorem new? We have never seen it anywhere before. But since the golden age of projective geometry so much has been published about conies that it is unlikely that no one noticed this result. On the other hand, why does it not appear in the literature? Anyway, it seems interesting to "repeat" this property, because several theorems in connection with straight lines and (or) conies in projective, affine or euclidean planes are in fact special cases of this theorem. We give a few classical examples: the theorems of Pappus-Pascal, Desargues, Pascal (or its converse), the Brocard points, the point of Miquel. Finally, we have never seen in the literature a proof of these theorems using the same short method see the proof of the main theorem).

An Easy Proof of the Irrationality of

2019 ◽  
Vol 92 (1) ◽  
pp. 71-71
Author(s):  
Poo-Sung Park
Keyword(s):  
Easy Proof

scholarly journals TENSOR PRODUCTS OF STEINBERG ALGEBRAS

2019 ◽  
pp. 1-16 ◽  
Author(s):  
SIMON W. RIGBY
Keyword(s):  
Tensor Products ◽  
Isomorphism Problem ◽  
Universal Property ◽  
Easy Proof

We prove that $A_{R}(G)\otimes _{R}A_{R}(H)\cong A_{R}(G\times H)$ if $G$ and $H$ are Hausdorff ample groupoids. As part of the proof, we give a new universal property of Steinberg algebras. We then consider the isomorphism problem for tensor products of Leavitt algebras, and show that no diagonal-preserving isomorphism exists between $L_{2,R}\otimes L_{3,R}$ and $L_{2,R}\otimes L_{2,R}$ . In fact, there are no unexpected diagonal-preserving isomorphisms between tensor products of finitely many Leavitt algebras. We give an easy proof that every $\ast$ -isomorphism of Steinberg algebras over the integers preserves the diagonal, and it follows that $L_{2,\mathbb{Z}}\otimes L_{3,\mathbb{Z}}\not \cong L_{2,\mathbb{Z}}\otimes L_{2,\mathbb{Z}}$ (as $\ast$ -rings).

scholarly journals An easy proof of the Rogers-Ramanujan identities

1983 ◽  
Vol 16 (2) ◽  
pp. 235-241 ◽  
Author(s):  
D.M. Bressoud
Keyword(s):  
Easy Proof

scholarly journals Section problems for configuration spaces of surfaces

2019 ◽  
pp. 1-29
Author(s):  
Lei Chen
Keyword(s):  
Finite Type ◽  
Braid Groups ◽  
Configuration Spaces ◽  
Easy Proof ◽  
Genus Formula

In this paper, we give a close-to-sharp answer to the basic questions: When is there a continuous way to add a point to a configuration of [Formula: see text] ordered points on a surface [Formula: see text] of finite type so that all the points are still distinct? When this is possible, what are all the ways to do it? More precisely, let PConf[Formula: see text] be the space of ordered [Formula: see text]-tuple of distinct points in [Formula: see text]. Let [Formula: see text] be the map given by [Formula: see text]. We classify all continuous sections of [Formula: see text] up to homotopy by proving the following: (1) If [Formula: see text] and [Formula: see text], any section of [Formula: see text] is either “adding a point at infinity” or “adding a point near [Formula: see text]”. (We define these two terms in Sec. 2.1; whether we can define “adding a point near [Formula: see text]” or “adding a point at infinity” depends in a delicate way on properties of [Formula: see text].) (2) If [Formula: see text] a [Formula: see text]-sphere and [Formula: see text], any section of [Formula: see text] is “adding a point near [Formula: see text]”; if [Formula: see text] and [Formula: see text], the bundle [Formula: see text] does not have a section. (We define this term in Sec. 3.2). (3) If [Formula: see text] a surface of genus [Formula: see text] and for [Formula: see text], we give an easy proof of [D. L. Gonçalves and J. Guaschi, On the structure of surface pure braid groups, J. Pure Appl. Algebra 182 (2003) 33–64, Theorem 2] that the bundle [Formula: see text] does not have a section.

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