Michiel Hazewinkel is a Dutch mathematician, Emeritus Professor of Mathematics at the Centre for Mathematics and Computer and the University of Amsterdam known for his 1978 book Formal groups and applications and as editor of the Encyclopedia of Mathematics. Born in Amsterdam to Jan Hazewinkel and Geertrude Hendrika Werner, Hazewinkel studied at the University of Amsterdam, he received his BA in Mathematics and Physics in 1963, his MA in Mathematics with a minor in Philosophy in 1965 and his PhD in 1969 under supervision of Frans Oort and Albert Menalda for the thesis "Maximal Abelian Extensions of Local Fields". After graduation Hazewinkel started his academic career as Assistant Professor at the University of Amsterdam in 1969. In 1970 he became Associate Professor at the Erasmus University Rotterdam, where in 1972 he was appointed Professor of Mathematics at the Econometric Institute. Here he was thesis advisor of Roelof Stroeker, M. van de Vel, Jo Ritzen, Gerard van der Hoek. From 1973 to 1975 he was Professor at the Universitaire Instelling Antwerpen, where Marcel van de Vel was his PhD student.
From 1982 to 1985 he was appointed part-time Professor Extraordinarius in Mathematics at the Erasmus Universiteit Rotterdam, part-time Head of the Department of Pure Mathematics at the Centre for Mathematics and Computer in Amsterdam. In 1985 he was appointed Professor Extraordinarius in Mathematics at the University of Utrecht, where he supervised the promotion of Frank Kouwenhoven, Huib-Jan Imbens, J. Scholma and F. Wainschtein. At the Centre for Mathematics and Computer CWI in Amsterdam in 1988 he became Professor of Mathematics and head of the Department of Algebra and Geometry until his retirement in 2008. Hazewinkel has been managing editor for journals as Nieuw Archief voor Wiskunde since 1977, he was managing editor for the book series Mathematics and Its Applications for Kluwer Academic Publishers in 1977. Hazewinkel was member of 15 professional societies in the field of Mathematics, participated in numerous administrative tasks in institutes, Program Committee, Steering Committee, Consortiums and Boards.
In 1994 Hazewinkel was elected member of the International Academy of Computer Sciences and Systems. Hazewinkel has authored and edited several books, numerous articles. Books, selection: 1970. Géométrie algébrique-généralités-groupes commutatifs. With Michel Demazure and Pierre Gabriel. Masson & Cie. 1976. On invariants, canonical forms and moduli for linear, finite dimensional, dynamical systems. With Rudolf E. Kalman. Springer Berlin Heidelberg. 1978. Formal groups and applications. Vol. 78. Elsevier. 1993. Encyclopaedia of Mathematics. Ed. Vol. 9. Springer. Articles, a selection: Hazewinkel, Michiel. "Moduli and canonical forms for linear dynamical systems II: The topological case". Mathematical Systems Theory. 10: 363–385. Doi:10.1007/BF01683285. Archived from the original on 12 December 2013. Hazewinkel, Michiel. "On Lie algebras and finite dimensional filtering". Stochastics. 7: 29–62. Doi:10.1080/17442508208833212. Archived from the original on 12 December 2013. Hazewinkel, M.. J.. "Nonexistence of finite-dimensional filters for conditional statistics of the cubic sensor problem".
Systems & Control Letters. 3: 331–340. Doi:10.1016/0167-691190074-9. Hazewinkel, Michiel. "The algebra of quasi-symmetric functions is free over the integers". Advances in Mathematics. 164: 283–300. Doi:10.1006/aima.2001.2017. Homepage
YouTube is an American video-sharing website headquartered in San Bruno, California. Three former PayPal employees—Chad Hurley, Steve Chen, Jawed Karim—created the service in February 2005. Google bought the site in November 2006 for US$1.65 billion. YouTube allows users to upload, rate, add to playlists, comment on videos, subscribe to other users, it offers a wide variety of corporate media videos. Available content includes video clips, TV show clips, music videos and documentary films, audio recordings, movie trailers, live streams, other content such as video blogging, short original videos, educational videos. Most of the content on YouTube is uploaded by individuals, but media corporations including CBS, the BBC, Hulu offer some of their material via YouTube as part of the YouTube partnership program. Unregistered users can only watch videos on the site, while registered users are permitted to upload an unlimited number of videos and add comments to videos. Videos deemed inappropriate are available only to registered users affirming themselves to be at least 18 years old.
YouTube and its creators earn advertising revenue from Google AdSense, a program which targets ads according to site content and audience. The vast majority of its videos are free to view, but there are exceptions, including subscription-based premium channels, film rentals, as well as YouTube Music and YouTube Premium, subscription services offering premium and ad-free music streaming, ad-free access to all content, including exclusive content commissioned from notable personalities; as of February 2017, there were more than 400 hours of content uploaded to YouTube each minute, one billion hours of content being watched on YouTube every day. As of August 2018, the website is ranked as the second-most popular site in the world, according to Alexa Internet. YouTube has faced criticism over aspects of its operations, including its handling of copyrighted content contained within uploaded videos, its recommendation algorithms perpetuating videos that promote conspiracy theories and falsehoods, hosting videos ostensibly targeting children but containing violent and/or sexually suggestive content involving popular characters, videos of minors attracting pedophilic activities in their comment sections, fluctuating policies on the types of content, eligible to be monetized with advertising.
YouTube was founded by Chad Hurley, Steve Chen, Jawed Karim, who were all early employees of PayPal. Hurley had studied design at Indiana University of Pennsylvania, Chen and Karim studied computer science together at the University of Illinois at Urbana–Champaign. According to a story, repeated in the media and Chen developed the idea for YouTube during the early months of 2005, after they had experienced difficulty sharing videos, shot at a dinner party at Chen's apartment in San Francisco. Karim did not attend the party and denied that it had occurred, but Chen commented that the idea that YouTube was founded after a dinner party "was very strengthened by marketing ideas around creating a story, digestible". Karim said the inspiration for YouTube first came from Janet Jackson's role in the 2004 Super Bowl incident, when her breast was exposed during her performance, from the 2004 Indian Ocean tsunami. Karim could not find video clips of either event online, which led to the idea of a video sharing site.
Hurley and Chen said that the original idea for YouTube was a video version of an online dating service, had been influenced by the website Hot or Not. Difficulty in finding enough dating videos led to a change of plans, with the site's founders deciding to accept uploads of any type of video. YouTube began as a venture capital-funded technology startup from an $11.5 million investment by Sequoia Capital and an $8 million investment from Artis Capital Management between November 2005 and April 2006. YouTube's early headquarters were situated above a pizzeria and Japanese restaurant in San Mateo, California; the domain name www.youtube.com was activated on February 14, 2005, the website was developed over the subsequent months. The first YouTube video, titled Me at the zoo, shows co-founder Jawed Karim at the San Diego Zoo; the video was uploaded on April 23, 2005, can still be viewed on the site. YouTube offered the public a beta test of the site in May 2005; the first video to reach one million views was a Nike advertisement featuring Ronaldinho in November 2005.
Following a $3.5 million investment from Sequoia Capital in November, the site launched on December 15, 2005, by which time the site was receiving 8 million views a day. The site grew and, in July 2006, the company announced that more than 65,000 new videos were being uploaded every day, that the site was receiving 100 million video views per day. According to data published by market research company comScore, YouTube is the dominant provider of online video in the United States, with a market share of around 43% and more than 14 billion views of videos in May 2010. In May 2011, 48 hours of new videos were uploaded to the site every minute, which increased to 60 hours every minute in January 2012, 100 hours every minute in May 2013, 300 hours every minute in November 2014, 400 hours every minute in February 2017; as of January 2012, the site had 800 million unique users a month. It is estimated that in 2007 YouTube consumed as much bandwidth as the entire Internet in 2000. According to third-party web analytics providers and SimilarWeb, YouTube is the second-most visited website in the world, as of December 2016.
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though most classify up to homotopy equivalence. Although algebraic topology uses algebra to study topological problems, using topology to solve algebraic problems is sometimes possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Below are some of the main areas studied in algebraic topology: In mathematics, homotopy groups are used in algebraic topology to classify topological spaces; the first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. In algebraic topology and abstract algebra, homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group.
In homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign'quantities' to the chains of homology theory. A manifold is a topological space. Examples include the plane, the sphere, the torus, which can all be realized in three dimensions, but the Klein bottle and real projective plane which cannot be realized in three dimensions, but can be realized in four dimensions. Results in algebraic topology focus on global, non-differentiable aspects of manifolds. Knot theory is the study of mathematical knots. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone.
In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R 3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R 3 upon itself. A simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments and their n-dimensional counterparts. Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory; the purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. A CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory; this class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation. An older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones.
In the 1920s and 1930s, there was growing emphasis on investigating topological spaces by finding correspondences from them to algebraic groups, which led to the change of name to algebraic topology. The combinatorial topology name is still sometimes used to emphasize an algorithmic approach based on decomposition of spaces. In the algebraic approach, one finds a correspondence between spaces and groups that respects the relation of homeomorphism of spaces; this allows one to recast statements about topological spaces into statements about groups, which have a great deal of manageable structure making these statement easier to prove. Two major ways in which this can be done are through fundamental groups, or more homotopy theory, through homology and cohomology groups; the fundamental groups give us basic information about the structure of a topological space, but they are nonabelian and can be difficult to work with. The fundamental group of a simplicial complex does have a finite presentation.
Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated. Finitely generated abelian groups are classified and are easy to work with. In general, all constructions of algebraic topology are functorial. Fundamental groups and homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated morphisms correspond — a continuous mapping of spaces induces a group homomorphism on the associated groups, these homomorphisms can be used to show non-existence of mappings. One of the first mathematicians to work with different types of cohomology was Georges de Rham. One can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf co
In mathematics, the antipodal point of a point on the surface of a sphere is the point, diametrically opposite to it — so situated that a line drawn from the one to the other passes through the center of the sphere and forms a true diameter. This term applies to opposite points on any n-sphere. An antipodal point is sometimes called an antipode, a back-formation from the Greek loan word antipodes, which meant "opposite the feet". In mathematics, the concept of antipodal points is generalized to spheres of any dimension: two points on the sphere are antipodal if they are opposite through the centre. On a circle, such points are called diametrically opposite. In other words, each line through the centre intersects the sphere in two points, one for each ray out from the centre, these two points are antipodal; the Borsuk–Ulam theorem is a result from algebraic topology dealing with such pairs of points. It says that any continuous function from Sn to Rn maps some pair of antipodal points in Sn to the same point in Rn.
Here, Sn denotes the n-dimensional sphere in -dimensional space. The antipodal map A: Sn → Sn, defined by A = −x, sends every point on the sphere to its antipodal point, it is homotopic to the identity map if n is odd, its degree is n+1. If one wants to consider antipodal points as identified, one passes to projective space. An antipodal pair of a convex polygon is a pair of 2 points admitting 2 infinite parallel lines being tangent to both points included in the antipodal without crossing any other line of the convex polygon. Hazewinkel, Michiel, ed. "Antipodes", Encyclopedia of Mathematics, Springer Science+Business Media B. V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 "antipodal". PlanetMath
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, from a topological viewpoint they are the same; the word homeomorphism comes from the Greek words ὅμοιος = similar or same and μορφή = shape, introduced to mathematics by Henri Poincaré in 1895. Speaking, a topological space is a geometric object, the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other. However, this description can be misleading; some continuous deformations are not homeomorphisms, such as the deformation of a line into a point. Some homeomorphisms are not continuous deformations, such as the homeomorphism between a trefoil knot and a circle.
An often-repeated mathematical joke is that topologists can't tell the difference between a coffee cup and a donut, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in a cup's handle. A function f: X → Y between two topological spaces is a homeomorphism if it has the following properties: f is a bijection, f is continuous, the inverse function f − 1 is continuous. A homeomorphism is sometimes called a bicontinuous function. If such a function exists, X and Y are homeomorphic. A self-homeomorphism is a homeomorphism from a topological space onto itself. "Being homeomorphic" is an equivalence relation on topological spaces. Its equivalence classes are called homeomorphism classes; the open interval is homeomorphic to the real numbers R for any a < b.. The unit 2-disc D 2 and the unit square in R2 are homeomorphic. An example of a bicontinuous mapping from the square to the disc is, in polar coordinates, ↦.
The graph of a differentiable function is homeomorphic to the domain of the function. A differentiable parametrization of a curve is a homeomorphism between the domain of the parametrization and the curve. A chart of a manifold is an homeomorphism between an open subset of the manifold and an open subset of a Euclidean space; the stereographic projection is a homeomorphism between the unit sphere in R3 with a single point removed and the set of all points in R2. If G is a topological group, its inversion map. For any x ∈ G, the left translation y ↦ x y, the right translation y ↦ y x, the inner automorphism y ↦ x y x − 1 are homeomorphisms. Rm and Rn are not homeomorphic for m ≠ n; the Euclidean real line is not homeomorphic to the unit circle as a subspace of R2, since the unit circle is compact as a subspace of Euclidean R2 but the real line is not compact. The one-dimensional intervals and are not homeomorphic because no continuous bijection could be made; the third requirement, that f − 1 be continuous, is essential.
Consider for instance the function f: [ 0, 2 π ) → S 1 defined by f =. This function is bijective and continuous, but not a homeomorphism ( S
Ham sandwich theorem
In mathematical measure theory, for every positive integer n the ham sandwich theorem states that given n measurable "objects" in n-dimensional Euclidean space, it is possible to divide all of them in half with a single -dimensional hyperplane. It was proposed by Hugo Steinhaus and proved by Stefan Banach, years called the Stone–Tukey theorem after Arthur H. Stone and John Tukey; the ham sandwich theorem takes its name from the case when n = 3 and the three objects of any shape are a chunk of ham and two chunks of bread—notionally, a sandwich—which can all be bisected with a single cut. In two dimensions, the theorem is known as the pancake theorem because of having to cut two infinitesimally thin pancakes on a plate each in half with a single cut. According to Beyer & Zardecki, the earliest known paper about the ham sandwich theorem the n = 3 case of bisecting three solids with a plane, is by Steinhaus. Beyer and Zardecki's paper includes a translation of the 1938 paper, it attributes the posing of the problem to Hugo Steinhaus, credits Stefan Banach as the first to solve the problem, by a reduction to the Borsuk–Ulam theorem.
The paper poses the problem in two ways: first, formally, as "Is it always possible to bisect three solids, arbitrarily located, with the aid of an appropriate plane?" and second, informally, as "Can we place a piece of ham under a meat cutter so that meat and fat are cut in halves?" The paper offers a proof of the theorem. A more modern reference is Stone & Tukey, the basis of the name "Stone–Tukey theorem"; this paper proves the n-dimensional version of the theorem in a more general setting involving measures. The paper attributes the n = 3 case to Stanislaw Ulam, based on information from a referee; the two-dimensional variant of the theorem can be proved by an argument which appears in the fair cake-cutting literature. For each angle α ∈, we can bisect pancake #1 using a straight line in angle α; this means that we can take a straight knife, rotate it at every angle α ∈ and translate it appropriately for that particular angle, such that pancake #1 is bisected at each angle and corresponding translation.
When the knife is at angle 0, it cuts pancake #2, but the pieces are unequal. Define the'positive' side of the knife as the side in which the fraction of pancake #2 is larger. Define p as the fraction of pancake #2 at the positive side of the knife. P ≥ 1 / 2; when the knife is at angle 180, the knife is upside-down, so p ≤ 1 / 2. By the intermediate value theorem, there must be an angle in which p = 1 / 2. Cutting at that angle bisects both pancakes simultaneously; the ham sandwich theorem can be proved. This proof follows the one described by Steinhaus and others, attributed there to Stefan Banach, for the n = 3 case. In the field of Equivariant topology, this proof would fall under the configuration-space/tests-map paradigm. Let A1, A2, …, An denote the n objects that we wish to bisect. Let S be the unit -sphere embedded in n-dimensional Euclidean space R n, centered at the origin. For each point p on the surface of the sphere S, we can define a continuum of oriented affine hyperplanes perpendicular to the vector from the origin to p, with the "positive side" of each hyperplane defined as the side pointed to by that vector.
By the intermediate value theorem, every family of such hyperplanes contains at least one hyperplane that bisects the bounded object An: at one extreme translation, no volume of An is on the positive side, at the other extreme translation, all of An's volume is on the positive side, so in between there must be a translation that has half of An's volume on the positive side. If there is more than one such hyperplane in the family, we can pick one canonically by choosing the midpoint of the interval of translations for which An is bisected, thus we obtain, for each point p on the sphere S, a hyperplane π, perpendicular to the vector from the origin to p and that bisects An. Now we define a function f from the (n −
Lazar Aronovich Lyusternik was a Soviet mathematician. He is famous for his work in topology and differential geometry, to which he applied the variational principle. Using the theory he introduced, together with Lev Schnirelmann, he proved the theorem of the three geodesics, a conjecture by Henri Poincaré that every convex body in 3-dimensions has at least three simple closed geodesics; the ellipsoid with distinct but nearly equal axis is the critical case with three closed geodesics. The Lusternik–Schnirelmann theory, as it is called now, is based on the previous work by Poincaré, David Birkhoff, Marston Morse, it has led to numerous advances in differential topology. For this work Lyusternik received the Stalin Prize in 1946. In addition to serving as a professor of mathematics at Moscow State University, Lyusternik worked at the Steklov Mathematical Institute from 1934 to 1948 and the Lebedev Institute of Precise Mechanics and Computer Engineering from 1948 to 1955, he was a student of Nikolai Luzin.
In 1930 he became one of the initiators of the Egorov affair and one of the participants in the notorious political persecution of his teacher Nikolai Luzin known as the Luzin case or Luzin affair. Lusternik–Schnirelmann category Lyusternik's generalization of the Brunn–Minkowski theorem Pavel Aleksandrov et al. LAZAR' ARONOVICH LYUSTERNIK, Russ. Math. Surv. 15, 153-168. Pavel Aleksandrov, In memory of Lazar Aronovich Lyusternik, Russ. Math. Surv. 37, 145-147 Lazar Lyusternik at the Mathematics Genealogy Project