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Collineation

In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is thus an isomorphism between projective spaces, or an automorphism from a projective space to itself. Some authors restrict the definition of collineation to the case where it is an automorphism. The set of all collineations of a space to itself form a group, called the collineation group.

Simply, a collineation is a one-to-one map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. One may formalize this using various ways of presenting a projective space. Also, the case of the projective line is special, and hence generally treated differently.

For a projective space defined in terms of linear algebra (as the projectivization of a vector space), a collineation is a map between the projective spaces that is order-preserving with respect to inclusion of subspaces.

Formally, let V be a vector space over a field K and W a vector space over a field L. Consider the projective spaces PG(V) and PG(W), consisting of the vector lines of V and W. Call D(V) and D(W) the set of subspaces of V and W respectively. A collineation from PG(V) to PG(W) is a map α : D(V) → D(W), such that:

Given a projective space defined axiomatically in terms of an incidence structure (a set of points P, lines L, and an incidence relation I specifying which points lie on which lines, satisfying certain axioms), a collineation between projective spaces thus defined then being a bijective function f between the sets of points and a bijective function g between the set of lines, preserving the incidence relation.

Every projective space of dimension greater than or equal to three is isomorphic to the projectivization of a linear space over a division ring, so in these dimensions this definition is no more general than the linear-algebraic one above, but in dimension two there are other projective planes fanny pack running, namely the non-Desarguesian planes, and this definition permits one to define collineations in such projective planes.

For dimension one, the set of points lying on a single projective line defines a projective space, and the resulting notion of collineation is just any bijection of the set.

For a projective space of dimension one (a projective line; the projectivization of a vector space of dimension two), all points are collinear, so the collineation group is exactly the symmetric group of the points of the projective line. This is different from the behavior in higher dimensions, and thus one gives a more restrictive definition, specified so that the fundamental theorem of projective geometry holds.

In this definition, when V has dimension two, a collineation from PG(V) to PG(W) is a map α : D(V) → D(W), such that:

This last requirement ensures that collineations are all semilinear maps.

The main examples of collineations are projective linear transformations (also known as homographies) and automorphic collineations. For projective spaces coming from a linear space, the fundamental theorem of projective geometry states that all collineations are a combination of these, as described below.

Projective linear transformations (homographies) are collineations (planes in a vector space correspond to lines in the associated projective space, and linear transformations map planes to planes, so projective linear transformations map lines to lines), but in general not all collineations are projective linear transformations. PGL is in general a proper subgroup of the collineation group.

An automorphic collineation is a map that, in coordinates, is a field automorphism applied to the coordinates.

If the geometric dimension of a pappian projective space is at least 2, then every collineation is the product of a homography (a projective linear transformation) and an automorphic collineation. More precisely, the collineation group is the projective semilinear group, which is the semidirect product of homographies by automorphic collineations.

In particular, the collineations of PG(2, R) are exactly the homographies, as R has no nontrivial automorphisms (that is, Gal(R/Q) is trivial).

Suppose φ is a nonsingular semilinear map from V to W, with the dimension of V at least three. Define α : D(V) → D(W) by saying that Zα = { φ(z) | zZ } for all Z in D(V). As φ is semilinear, one easily checks that this map is properly defined, and further more, as φ is not singular, it is bijective. It is obvious now that α is a collineation. We say that α is induced by φ shaver shop.

The fundamental theorem of projective geometry states the converse:

Suppose V is a vector space over a field K with dimension at least three, W is a vector space over a field L, and α is a collineation from PG(V) to PG(W). This implies K and L are isomorphic fields, V and W have the same dimension, and there is a semilinear map φ such that φ induces α.

For n ≥ 3, the collineation group is the projective semilinear group, PΓL – this is PGL, twisted by field automorphisms; formally, the semidirect product PΓL ≅ PGL ⋊ Gal(K/k), where k is the prime field for K.

Thus for K a prime field (







F




p






{\displaystyle \mathbb {F} _{p}}


or






Q





{\displaystyle \mathbb {Q} }


), we have PGL = PΓL, but for K not a prime field (such as







F





p



n








{\displaystyle \mathbb {F} _{p^{n}}}


for n ≥ 2 or






C





{\displaystyle \mathbb {C} }


PΓL / PGL ≅ Gal(K/k) corresponds to “choices of linear structure”, with the identity (base point) being the existing linear structure. Given a projective space without an identification as the projectivization of a linear space, there is no natural isomorphism between the collineation group and PΓL, and the choice of a linear structure (realization as projectivization of a linear space) corresponds to a choice of subgroup PGL < PΓL, these choices forming a torsor over Gal(K/k).

The idea of a line was abstracted to a ternary relation determined by collinear points. According to Wilhelm Blaschke it was August Möbius that first abstracted this essence of geometrical transformation:

Contemporary mathematicians view geometry as an incidence structure with an automorphism group consisting of mappings of the underlying space that preserve incidence. Such a mapping permutes the lines of the incidence structure, and the notion of collineation persists.

As mentioned by Blaschke and Klein, Michel Chasles preferred the term homography to collineation. A distinction between the terms arose when the distinction was clarified between the real projective plane and the complex projective line. Since there are no non-trivial field automorphisms of the real number field, all the collineations are homographies in the real projective plane., however due to the field automorphism complex conjugation, not all collineations of the complex projective line are homographies. In applications such as computer vision where the underlying field is the real number field, homography and collineation can be used interchangeably.

The operation of taking the complex conjugate in the complex plane amounts to a reflection in the real line. With the notation z for the conjugate of z, an anti-homography is given by

Thus an anti-homography is the composition of conjugation with an homography, and so is an example of a collineation which is not an homography. For example, geometrically, the mapping





f


(


z


)


=


1



/




z











{\displaystyle f(z)=1/z^{*}}


amounts to circle inversion. The transformations of inversive geometry of the plane are frequently described as the collection of all homographies and anti-homographies of the complex plane.

Angel Nevarez and Valerie Tevere

Angel Nevarez and Valerie Tevere are a pair of artists that have been collaborating on video, sound, performance and installation projects since 2001. Several of their projects have been produced under the collective name neuroTransmitter Paul Frank Suits Kids. Their art works often incorporate popular music and examine how visual forms traverse and are complicated once they are at play in public spaces.

Angel and Valerie have developed and exhibited works in sites as geographically diverse as the Staten Island Ferry, Plaza de la Liberación in Guadalajara, Austin City Hall’s Plaza Stage, and the Museum of Modern Art’s Sculpture Garden in New York. Their work often considers the relationship among politics best office water bottle, sound and language. For instance, in their 2010 performance The War Song, the artists rearranged Culture Club’s song of the same name. By slowing the tempo and switching the score to a minor key, Angel and Valerie’s version of the song revealed a certain pathos within the seemingly naive lyrics.

The first United States survey of their work occurred at ICA Philadelphia in 2016.

Memory of a Time Twice Lived is a video work that merges several filmic genres—music video, documentary, the French avant-garde and science fiction. The project builds a field of relations tying together 20th century mythic heroes, the collection of the Wagner Free Institute of Science, the Mexican luchador El Santo, and the accordion as a nomadic instrument. Shot on location in Philadelphia and Mexico City, this project includes references to the late Chris Marker’s science fiction masterpiece La Jetée (1962) and features an accordionist performing in sites throughout Philadelphia.

Valerie Tevere and Angel Nevarez produced a site-specific work at the Old Bronx Borough Courthouse for the art organization No Longer Empty in 2015. Using the framework of call and response free water bottle, the sound installation explored the history of hip hop and break dancing in the Bronx. Spoken word narration was provided by B-Girl Rokafella and MC Lady L, and included reference to South Bronx luminaries like Afrika Bambaataa. The piece was also shown in the artists’ retrospective at the ICA Philadelphia in 2016.

In light of the 2008 presidential elections, the two artists began studying the role that soapboxes provide a mediated space to allow citizens to express political interests, fears and hopes. Over a two-day period in September 8, Valerie Tevere and Angel Nevarez joined the Creative Time group event entitled, Democracy in America: The National Campaign. Their project Another Protest Song, was organized into a karaoke suite of “protest” songs. Throughout New York City, the artists set up performance stages, and invited the general public to perform songs of protest.

Stéphane Langdeau

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Stéphane Langdeau (né à Montréal le – ) est un journaliste sportif québécois. Il anime l’émission L’Antichambre au Réseau des sports.

Après des études au cégep en droit et un baccalauréat en Éducation physique à l’Université de Montréal, il enseigne durant trois ans au secondaire tout en suivant simultanément une formation à l’école de journalisme Promédia. Il entame sa carrière de journaliste-présentateur à la station de Winnipeg de la Société Radio-Canada en 1988.

De 1988 à 1990, il travaille pour la radio CKSB-AM où il est chroniqueur sportif et animateur d’une émission de musique country, l’Auberge 1050. Il passe ensuite à la télévision de Radio-Canada à Winnipeg où il travaille durant quatre ans à titre de présentateur sportif au Le Manitoba ce soir. Durant son séjour au Manitoba wholesale sock manufacturers, il participe aussi à l’émission pour enfants Les petites oreilles à la radio.

Il rentre à Montréal en 1994 à Radio-Canada, à titre de journaliste, chroniqueur et lecteur de sport et il participe aussi à l’arrivée de la nouvelle chaîne de nouvelles, RDI. Il a participé en tant que journaliste à la couverture télévisée de tous les Jeux olympiques depuis 1996 – Atlanta, Nagano, Sydney, Salt Lake City, Athènes free water bottle, Turin, Beijing liverpool goalie gloves, Vancouver. De 1998 à 2003, il est chroniqueur à l’émission de radio Montréal Express. Durant 13 ans, de 1996 à 2009, il est aussi professeur de journalisme à l’école Promédia.

En 2004, il quitte Radio-Canada pour le Réseau des sports (RDS), où il devient lecteur pour la nouvelle chaîne, RIS. Il coanime aussi l’émission Face à face avec Martin Leclerc et Alain Chantelois de 2006 à 2009. En 2010, RDS lui confie la tâche d’aminer l’émission l’Antichambre où il devient animateur, entouré de collègues et spécialistes du hockey.