Study
Study or studies may refer to:
General
- In particular, a clinical trial
People
Other
See also
Endgame study
An endgame study, or just study, is a composed chess endgame position—that is, one that has been made up rather than one from an actual game—presented as a sort of puzzle, in which the aim of the solver is to find a way for one side (usually White) to win or draw, as stipulated, against any moves the other side plays.
Composed studies
Composed studies predate the modern form of chess. Shatranj studies exist in manuscripts from the 9th century, and the earliest treatises on modern chess by the likes of Luis Ramirez Lucena and Pedro Damiano (late 15th and early 16th century) also include studies. However, these studies often include superfluous pieces, added to make the position look more "game-like", but which take no part in the actual solution (something that is never done in the modern study). Various names were given to these positions (Damiano, for example, called them "subtleties"); the first book which called them "studies" appears to be Chess Studies, an 1851 publication by Josef Kling and Bernhard Horwitz, which is sometimes also regarded as the starting point for the modern endgame study. The form is considered to have been raised to an art in the late 19th century, with A. A. Troitsky and Henri Rinck particularly important in this respect.
Study (art)
In art, a study is a drawing, sketch or painting done in preparation for a finished piece, or as visual notes. Studies are often used to understand the problems involved in rendering subjects and to plan the elements to be used in finished works, such as light, color, form, perspective and composition. Studies can have more impact than more-elaborately planned work, due to the fresh insights the artist gains while exploring the subject. The excitement of discovery can give a study vitality. Even when layers of the work show changes the artist made as more was understood, the viewer shares more of the artist's sense of discovery. Written notes alongside visual images add to the import of the piece as they allow the viewer to share the artist's process of getting to know the subject.
Studies inspired some of the first 20th century conceptual art, where the creative process itself becomes the subject of the piece. Since the process is what is all-important in studies and conceptual art, the viewer may be left with no material object of art.
Guido Fubini
Guido Fubini (19 January 1879 – 6 June 1943) was an Italian mathematician, known for Fubini's theorem and the Fubini–Study metric.
Born in Venice, he was steered towards mathematics at an early age by his teachers and his father, who was himself a teacher of mathematics. In 1896 he entered the Scuola Normale Superiore di Pisa, where he studied under the notable mathematicians Ulisse Dini and Luigi Bianchi. He gained some early fame when his 1900 doctoral thesis, entitled Clifford's parallelism in elliptic spaces, was discussed in a widely read work on differential geometry published by Bianchi in 1902.
After earning his doctorate, he took up a series of professorships. In 1901 he began teaching at the University of Catania in Sicily; shortly afterwards he moved to the University of Genoa; and in 1908 he moved to the Politecnico in Turin and then the University of Turin, where he would stay for a few decades.
During this time his research focused primarily on topics in mathematical analysis, especially differential equations, functional analysis, and complex analysis; but he also studied the calculus of variations, group theory, non-Euclidean geometry, and projective geometry, among other topics. With the outbreak of World War I, he shifted his work towards more applied topics, studying the accuracy of artillery fire; after the war, he continued in an applied direction, applying results from this work to problems in electrical circuits and acoustics.
Fubini–Study metric
In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, complex projective space CPn endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study.
A Hermitian form in (the vector space) Cn+1 defines a unitary subgroup U(n+1) in GL(n+1,C). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U(n+1) action; thus it is homogeneous. Equipped with a Fubini–Study metric, CPn is a symmetric space. The particular normalization on the metric depends on the application. In Riemannian geometry, one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the (2n+1)-sphere. In algebraic geometry, one uses a normalization making CPn a Hodge manifold.
Construction
The Fubini–Study metric arises naturally in the quotient space construction of complex projective space.
Specifically, one may define CPn to be the space consisting of all complex lines in Cn+1, i.e., the quotient of Cn+1\{0} by the equivalence relation relating all complex multiples of each point together. This agrees with the quotient by the diagonal group action of the multiplicative group C* = C \ {0}:
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