Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system ℝ by adding two elements: + ∞ and − ∞, where the infinities are treated as actual numbers. It is useful in describing the algebra on infinities and the various limiting behaviors in calculus and mathematical analysis in the theory of measure and integration; the affinely extended real number system is denoted R ¯ or or ℝ ∪. When the meaning is clear from context, the symbol +∞ is written as ∞, it is useful to describe the behavior of a function f, as either the argument x or the function value f gets "infinitely large" in some sense. For example, consider the function f = x − 2; the graph of this function has a horizontal asymptote at y = 0. Geometrically, when moving farther to the right along the x -axis, the value of 1 x 2 approaches 0; this limiting behavior is similar to the limit of a function at a real number, except that there is no real number to which x approaches. By adjoining the elements + ∞ and − ∞ to R, it enables a formulation of a "limit at infinity", with topological properties similar to those for R.

To make things formal, the Cauchy sequences definition of R allows defining + ∞ as the set of all sequences of rational numbers, such that every M ∈ R is associated with a corresponding N ∈ N for which a n > M for all n > N. The definition of − ∞ can be constructed similarly. In measure theory, it is useful to allow sets that have infinite measure and integrals whose value may be infinite; such measures arise out of calculus. For example, in assigning a measure to R that agrees with the usual length of intervals, this measure must be larger than any finite real number; when considering improper integrals, such as ∫ 1 ∞ d x x the value "infinity" arises. It is useful to consider the limit of a sequence of functions, such as f n = { 2 n, if 0 ≤ x ≤ 1 n 0, if 1 n < x ≤ 1 Without allowing functions to take on infinite values, such essential results as the monotone convergence theorem and the dominated convergence theorem would not make sense. The affinely extended real number system can be turned into a ordered set, by defining − ∞ ≤ a ≤ + ∞ for all a.

With this order topology, R ¯ has the desirable property of compactness: every subset of R ¯ has a supremum and an infimum. Moreover, with this topology, R ¯ is homeomorphic to the unit interval, thus the topology is metrizable, corresponding to the ordinary metric on this interval. There is no metric, an extension of the ordinary metric on R. In this topology, a set U is a neighborhood of + ∞, if and only if it contains a set for some real number a; the notion of the neighborhood of − ∞ can be defined similarly. Using this characterizat

Echo of the Mountain

Echo of the Mountain is a 2014 Mexican documentary film about Santos de la Torre directed by Nicolás Echevarría. It was one of fourteen films shortlisted by Mexico to be their submission for the Academy Award for Best Foreign Language Film at the 88th Academy Awards, but it lost out to 600 Miles; this documentary focuses on the work and life of Huichol artist Santos de la Torre, whose paneled mural has been displayed, since its inauguration by former Mexican president Ernesto Zedillo in 1997, at the Palais Royal station. However, the story that the audience learns from Santos is that he wasn't invited to its unveiling and how it was wrongly assembled; this is only how the film begins reflecting the oblivion and marginalization in which Santos and his people live in his own country. Echevarría's camera follows the process of Santos de la Torre and his family helping him in the making of a new mural that illustrates the history and religious practices of the Huichol people and their pilgrimage to Wirikuta, the sacred place for them where he goes to ask their gods for permission to create his new work.

Echo of the Mountain had screenings from 2014 to 2015 at festivals such as Cinéma du Réel, Chicago International Film Festival and the Berlin International Film Festival. At the Chicago Film Festival where it was awarded for Best Documentary, a review from The Focus Pull read "Echevarría’s film is powerful in its revealing of the stories and thoughts that inform a single piece of art that will undoubtedly be transposed into new cultures. It’s both a successful document of the creation of an affecting work of art, an affecting work of art itself." Jay Weissberg of Variety celebrated the cinematography of the documentary writing "clouds reflected in puddles, awe-inspiring landscapes, the indescribable warmth of wrinkled faces: All these are lovingly captured via a masterful use of framing and a sophisticated use of focal ranges". Clarence Tsui from The Hollywood Reporter wrote "Echevarria’s documentary captures his life and Huichol culture vividly and poetically with some fluid camerawork. With the whole piece revolving around the de la Torre’s process of making a new mural, Echo of the Mountain resembles a ritual in itself."

Official website Echo of the Mountain on IMDb


ATP synthase subunit s, mitochondrial is an enzyme that in humans is encoded by the ATP5S gene. This gene encodes a subunit of mitochondrial ATP synthase. Mitochondrial ATP synthase catalyzes ATP synthesis, utilizing an electrochemical gradient of protons across the inner membrane during oxidative phosphorylation. ATP synthase is composed of two linked multi-subunit complexes: the soluble catalytic core, F1, the membrane-spanning component, Fo, comprising the proton channel; this gene encodes the subunit s known as factor B, of the proton channel. This subunit is necessary for the energy transduction activity of the ATP synthase complexes. Alternatively spliced transcript variants encoding different isoforms have been identified. Human ATP5S genome location and ATP5S gene details page in the UCSC Genome Browser. ATP5S: ATP synthase, H+ transporting, mitochondrial Fo complex subunit s