Split (1)

train · 40.2k rows

source stringlengths 31 168	text stringlengths 51 3k
https://en.wikipedia.org/wiki/Schanuel%27s%20lemma	In mathematics, especially in the area of algebra known as module theory, Schanuel's lemma, named after Stephen Schanuel, allows one to compare how far modules depart from being projective. It is useful in defining the Heller operator in the stable category, and in giving elementary descriptions of dimension shifting. Statement Schanuel's lemma is the following statement: If 0 → K → P → M → 0 and 0 → K′ → P′ → M → 0 are short exact sequences of R-modules and P and P′ are projective, then K ⊕ P′ is isomorphic to K′ ⊕ P. Proof Define the following submodule of P ⊕ P′, where φ : P → M and φ′ : P′ → M: The map π : X → P, where π is defined as the projection of the first coordinate of X into P, is surjective. Since φ′ is surjective, for any p in P, one may find a q in P′ such that φ(p) = φ′(q). This gives (p,q) X with π(p,q) = p. Now examine the kernel of the map π: We may conclude that there is a short exact sequence Since P is projective this sequence splits, so X ≅ K′ ⊕ P. Similarly, we can write another map π : X → P′, and the same argument as above shows that there is another short exact sequence and so X ≅ P′ ⊕ K. Combining the two equivalences for X gives the desired result. Long exact sequences The above argument may also be generalized to long exact sequences. Origins Stephen Schanuel discovered the argument in Irving Kaplansky's homological algebra course at the University of Chicago in Autumn of 1958. Kaplansky writes: Early in the course I formed a one-step projective resolution of a module, and remarked that if the kernel was projective in one resolution it was projective in all. I added that, although the statement was so simple and straightforward, it would be a while before we proved it. Steve Schanuel spoke up and told me and the class that it was quite easy, and thereupon sketched what has come to be known as "Schanuel's lemma." Notes Homological algebra Module theory
https://en.wikipedia.org/wiki/Pregeometry	In mathematics and physics, pregeometry has several meanings: Pregeometry (model theory), another name for a matroid Pregeometry (physics), a structure from which geometry arises
https://en.wikipedia.org/wiki/Pregeometry%20%28physics%29	In physics, a pregeometry is a hypothetical structure from which the geometry of the universe develops. Some cosmological models feature a pregeometric universe before the Big Bang. The term was championed by John Archibald Wheeler in the 1960s and 1970s as a possible route to a theory of quantum gravity. Since quantum mechanics allowed a metric to fluctuate, it was argued that the merging of gravity with quantum mechanics required a set of more fundamental rules regarding connectivity that were independent of topology and dimensionality. Where geometry could describe the properties of a known surface, the physics of a hypothetical region with predefined properties, "pregeometry" might allow one to work with deeper underlying rules of physics that were not so strongly dependent on simplified classical assumptions about the properties of space. No single proposal for pregeometry has gained wide consensus support in the physics community. Some notions related to pregeometry predate Wheeler, other notions depart considerably from his outline of pregeometry but are still associated with it. A 2006 paper provided a survey and critique of pregeometry or near-pregeometry proposals up to that time. A summary of these is given below: Discrete spacetime by Hill A proposal anticipating Wheeler's pregeometry, though assuming some geometric notions embedded in quantum mechanics and special relativity. A subgroup of Lorentz transformations with only rational coefficients is deployed. Energy and momentum variables are restricted to a certain set of rational numbers. Quantum wave functions work out to be a special case semi-periodical functions though the nature of wave functions is ambiguous since the energy-momentum space cannot be uniquely interpreted. Discrete-space structure by Dadic and Pisk Spacetime as an unlabeled graph whose topological structure entirely characterizes the graph. Spatial points are related to vertices. Operators define the creation or annihilation of lines which develop into a Fock space framework. This discrete-space structure assumes the metric of spacetime and assumes composite geometric objects so it is not a pregeometric scheme in line with Wheeler's original conception of pregeometry. Pregeometric graph by Wilson Spacetime is described by a generalized graph consisting of a very large or infinite set of vertices paired with a very large or infinite set of edges. From that graph emerge various constructions such as vertices with multiple edges, loops, and directed edges. These in turn support formulations of the metrical foundation of space-time. Number theory pregeometry by Volovich Spacetime as a non-Archimedean geometry over a field of rational numbers and a finite Galois field where rational numbers themselves undergo quantum fluctuations. Causal sets by Bombelli, Lee, Meyer and Sorkin All of spacetime at very small scales is a causal set consisting of locally finite set of elements with a partial order linked to the not
https://en.wikipedia.org/wiki/Gauss%27s%20lemma%20%28Riemannian%20geometry%29	In Riemannian geometry, Gauss's lemma asserts that any sufficiently small sphere centered at a point in a Riemannian manifold is perpendicular to every geodesic through the point. More formally, let M be a Riemannian manifold, equipped with its Levi-Civita connection, and p a point of M. The exponential map is a mapping from the tangent space at p to M: which is a diffeomorphism in a neighborhood of zero. Gauss' lemma asserts that the image of a sphere of sufficiently small radius in TpM under the exponential map is perpendicular to all geodesics originating at p. The lemma allows the exponential map to be understood as a radial isometry, and is of fundamental importance in the study of geodesic convexity and normal coordinates. Introduction We define the exponential map at by where is the unique geodesic with and tangent and is chosen small enough so that for every the geodesic is defined. So, if is complete, then, by the Hopf–Rinow theorem, is defined on the whole tangent space. Let be a curve differentiable in such that and . Since , it is clear that we can choose . In this case, by the definition of the differential of the exponential in applied over , we obtain: So (with the right identification ) the differential of is the identity. By the implicit function theorem, is a diffeomorphism on a neighborhood of . The Gauss Lemma now tells that is also a radial isometry. The exponential map is a radial isometry Let . In what follows, we make the identification . Gauss's Lemma states: Let and . Then, For , this lemma means that is a radial isometry in the following sense: let , i.e. such that is well defined. And let . Then the exponential remains an isometry in , and, more generally, all along the geodesic (in so far as is well defined)! Then, radially, in all the directions permitted by the domain of definition of , it remains an isometry. Proof Recall that We proceed in three steps: : let us construct a curve such that and . Since , we can put . Therefore, where is the parallel transport operator and . The last equality is true because is a geodesic, therefore is parallel. Now let us calculate the scalar product . We separate into a component parallel to and a component normal to . In particular, we put , . The preceding step implies directly: We must therefore show that the second term is null, because, according to Gauss's Lemma, we must have: : Let us define the curve Note that Let us put: and we calculate: and Hence We can now verify that this scalar product is actually independent of the variable , and therefore that, for example: because, according to what has been given above: being given that the differential is a linear map. This will therefore prove the lemma. We verify that : this is a direct calculation. Since the maps are geodesics, Since the maps are geodesics, the function is constant. Thus, See also Riemannian geometry Metric tensor Refer
https://en.wikipedia.org/wiki/Robertson%20Centre%20for%20Biostatistics	The Robertson Centre for Biostatistics is a specialised biostatistical research centre in Glasgow, Scotland. It is part of the College of Medical, Veterinary and Life Sciences and the Institute of Health and Wellbeing at the University of Glasgow. All scales of research are carried out at the centre from multi-site clinical trials to small scale research projects. The centre also has interests in the development of novel informatics solutions for clinical research, statistical issues in epidemiology and health economic evaluation. History The centre led the WOSCOP study (New England Journal of Medicine 1995; 333:1301-7) which found that treatment with Pravastatin significantly reduced the risk of myocardial infarction and the risk of death from cardiovascular causes without adversely affecting the risk of death from noncardiovascular causes in men with moderate hypercholesterolaemia and no history of myocardial infarction. The Robertson Centre joined with the Glasgow Clinical Research Facility and Greater Glasgow and Clyde NHS R&D division in November 2007 to create a UKCRN registered Clinical Trials Unit - the Glasgow Clinical Trials Unit. References External links Robertson Centre for Biostatistics website University of Glasgow website The Glasgow Clinical Research Facility website The WOSCOP study Statistical service organizations University of Glasgow Biostatistics Hillhead
https://en.wikipedia.org/wiki/Glasgow%20Clinical%20Trials%20Unit	The Glasgow Clinical Trials Unit (CTU) is a collaborative research establishment in Glasgow, Scotland. It comprises the Glasgow Clinical Research Facility, the Robertson Centre for Biostatistics and Greater Glasgow and Clyde NHS R&D division. History In November 2007 the UKCRN gave the Glasgow Clinical Trials Unit registered Clinical Trials Unit status. References External links Glasgow Clinical Trials Unit NHS Greater Glasgow and Clyde The Glasgow Clinical Research Facility Robertson Centre for Biostatistics University of Glasgow Research institutes in Scotland Organisations based in Glasgow Science and technology in Glasgow 2007 establishments in Scotland
https://en.wikipedia.org/wiki/Organization%20for%20Security%20and%20Co-operation%20in%20Europe%20statistics	1 These countries are currently not participating in the EU's single market (EEA), but the EU has common external Customs Union agreements with Turkey (EU-Turkey Customs Union in force since 1995), Andorra (since 1991) and San Marino (since 2002). Monaco participates in the EU customs union through its relationship with France; its ports are administered by the French. Vatican City has a customs union in effect with Italy. 2 Monaco, San Marino and Vatican City are not members of Schengen, but act as such via their open borders with France and Italy, respectively. 3 Switzerland is not an official member of EEA but has bilateral agreements largely with same content, making it virtual member. Euler diagram Notes statistics
https://en.wikipedia.org/wiki/Hayley%20Yelling	Hayley Higham (born 3 January 1974, in Dorchester) is a British runner. She is the sister-in-law of fellow British runner Liz Yelling. She works as a Maths teacher at Sir William Borlase's Grammar School in Marlow, Buckinghamshire and runs for the Windsor, Slough, Eton and Hounslow Athletic Club. She has competed for England in the Commonwealth Games and for Great Britain in a number of competitions. In December 2004 she won the European Cross Country Championship in Heringsdorf. In December 2009 she won the European Cross Country Championship in Dublin after coming out of retirement from competitive running. She followed this up a month later by coming fourth in the 2010 International Edinburgh Cross Country, fourteen seconds after winner Tirunesh Dibaba over the freezing 5.8 kilometre course. Career highlights British National Championships 2002 - 1st, 5,000 m 2003 - 1st, 10,000 m 2003 - 1st, 5,000 m 2006 - 1st, 5,000 m Other competitions 2004 - 1st, European Cross Country Championships 2007 - 1st, Cross Internacional de San Sebastián 2008 - 1st, Belfast International Cross Country 2009 - 1st, European Cross Country Championships Personal bests References External links Living people 1974 births English female long-distance runners British female long-distance runners English female middle-distance runners World Athletics Championships athletes for Great Britain Sportspeople from Dorchester, Dorset European Cross Country Championships winners AAA Championships winners
https://en.wikipedia.org/wiki/Chaotic%20bubble	Chaotic bubbles within physics and mathematics, occur in cases when there are any dynamic processes that generate bubbles that are nonlinear. Many exhibit mathematically chaotic patterns consistent with chaos theory. In most systems, they arise out of a forcing pressure that encounters some kind of resistance or shear factor, but the details vary depending on the particular context. The most widely known example is bubbles in various forms of liquid. Although there may have been an earlier use of the term, it was used in 1987 specifically in connection with a model of the motion of a single bubble in a fluid subject to periodically driven pressure oscillations (Smereka, Birnir, and Banerjee, 1987). For an overview of models of single-bubble dynamics see Feng and Leal (1997). There is extensive literature on nonlinear analysis of the dynamics of bubbles in liquids, with important contributions from Werner Lauterborn (1976). Lauterborn and Cramer (1981) also applied chaos theory to acoustics, in which bubble dynamics play a crucial part. This includes analysis of chaotic dynamics in an acoustic cavitation bubble field in a liquid (Lauterborn, Holzfuss, and Bilio, 1994). The study of the role of shear stresses in non-Newtonian fluids has been done by Li, Mouline, Choplin, and Midoux (1997). A somewhat related field, the study of controlling such chaotic bubble dynamics (control of chaos), converts them to periodic oscillations, and has an important application to gas–solids in fluidized bed reactors, also applicable to the ammoxidation of propylene to acrylonitrile (Kaart, Schouten, and van den Bleek, 1999). Sarnobat et al.) study the behavior of electrostatic fields on chaotic bubbling in attempt to control the chaos into a lower order periodicity. References Further reading Chaos theory
https://en.wikipedia.org/wiki/Renan%20Lavigne	Renan Lavigne (born 1 November 1974 in Longjumeau) is a professional squash player from France. Career statistics Listed below. PSA Titles (9) All Results for Renan Lavigne in PSA World's Tour tournament PSA Tour Finals (Runner-Up) (6) External links PSA player profile French male squash players 1974 births Living people Competitors at the 2009 World Games
https://en.wikipedia.org/wiki/Indonesians%20in%20South%20Korea	Indonesians in South Korea numbered 34,514 individuals , down from 41,599 in 2009 according to South Korean government statistics. More than 90% of those are estimated to be migrant workers employed on short-term contracts. In 2007 the South Korean government extended the validity of Indonesians' working permits from three to five years, and has modified the recruitment process in order to improve working conditions. Indonesian workers in South Korea are paid an average of US$1,000 per month. The Indonesian government signed its first memorandum of understanding with the South Korean government about the provision of labourers to South Korea in 2004, after having signed similar agreements with Jordan, Kuwait, and Malaysia. Indonesia's official news agency ANTARA claimed there were 600,000 illegal Indonesian workers in South Korea as of 2006, making up almost 87% of the estimated 692,000 illegal Indonesian workers worldwide. See also Koreans in Indonesia References Demographics of South Korea Korea, South Ethnic groups in South Korea
https://en.wikipedia.org/wiki/Maurice%20Audin	Maurice Audin (14 February 1932 – c. 21 June 1957) was a renowned French mathematics assistant at the University of Algiers, a member of the Algerian Communist Party and an activist in the anticolonialist cause, who died under torture by the French state during the Battle of Algiers. In the centre of Algiers, beside the university, the intersection of streets bearing the names of several other heroes of the Algerian Revolution is called the Place Maurice-Audin. He is also memorialized by the Maurice Audin Prize, sponsored by the Société de Mathématiques Appliquées et Industrielles, the Société Mathématique de France, and others, and granted biennially to an Algerian mathematician working in Algeria and a French mathematician working in France. Family and childhood He is the son of Louis Audin (1900-1977) and Alphonsine Fort (1902-1989), who married in 1923 in Koléa (Algeria); they both came from modest families, he from Lyon workers, she from peasants from the Mitidja. When Maurice was born, his father was commander of the gendarmerie brigade in Béja, in the French protectorate of Tunisia. Later, Louis Audin was assigned to metropolitan France, then he passed a competitive examination and became a postman in Algiers. Academic studies and career Son of a soldier, Maurice Audin became an and, in 1942, entered the sixth grade at the preparatory military school of Hammam Righa. Then, in 1946, he was admitted to the Autun School.In 1948, giving up a career as an officer, he returned to the elementary mathematics class in Algiers (at the Gautier school). He studied mathematics at the University of Algiers, obtaining his degree in June 1953, then a diploma of higher education in July. In February 1953, he was recruited as assistant to Professor René de Possel, a post he saved in 1954. He also worked on a thesis on “linear equations in a vector space” as part of a state doctorate from mathematics. In January 1953, he married (1931-2019); they had three children: Michèle (1954), Louis (1955-2006) and Pierre (April 1957). Algerian War Maurice and Josette Audin were part of the anti-colonialist minority of the French in Algeria, whose desire is the independence of Algeria, which is also the position of the Algerian Communist Party. The latter was banned on 13 September 1955 and became an underground organization, negotiating with the FLN. The Audin family took part in certain illegal operations: in September 1956, Maurice together with his sister (Charlye, born in 1925) and his brother-in-law (Christian Buono), organized the clandestine exfiltration abroad of Larbi Bouhali, first secretary of PCA. In January 1957, following the numerous attacks perpetrated against the population by the FLN, the so-called “Battle of Algiers” operation was launched, for which General Massu's 10th parachute division held police powers in the area of Algiers. This unit engaged in massive torture and summary executions. Paul Teitgen notes 3,024 disappearances in one y
https://en.wikipedia.org/wiki/Isabelle%20Stoehr	Isabelle Stoehr (born June 9, 1979, in Tours) is a professional squash player from France. Career statistics Listed below Professional Tour Titles (9) All Results for Isabelle Stoehr in WISPA World's Tour tournament Note: (ret) = retired, min = minutes, h = hours WISPA Tour Finals (runner-up) (6) References External links French female squash players 1979 births Living people Sportspeople from Tours, France Competitors at the 2005 World Games
https://en.wikipedia.org/wiki/Exceptional%20inverse%20image%20functor	In mathematics, more specifically sheaf theory, a branch of topology and algebraic geometry, the exceptional inverse image functor is the fourth and most sophisticated in a series of image functors for sheaves. It is needed to express Verdier duality in its most general form. Definition Let f: X → Y be a continuous map of topological spaces or a morphism of schemes. Then the exceptional inverse image is a functor Rf!: D(Y) → D(X) where D(–) denotes the derived category of sheaves of abelian groups or modules over a fixed ring. It is defined to be the right adjoint of the total derived functor Rf! of the direct image with compact support. Its existence follows from certain properties of Rf! and general theorems about existence of adjoint functors, as does the unicity. The notation Rf! is an abuse of notation insofar as there is in general no functor f! whose derived functor would be Rf!. Examples and properties If f: X → Y is an immersion of a locally closed subspace, then it is possible to define f!(F) := f∗ G, where G is the subsheaf of F of which the sections on some open subset U of Y are the sections s ∈ F(U) whose support is contained in X. The functor f! is left exact, and the above Rf!, whose existence is guaranteed by abstract nonsense, is indeed the derived functor of this f!. Moreover f! is right adjoint to f!, too. Slightly more generally, a similar statement holds for any quasi-finite morphism such as an étale morphism. If f is an open immersion, the exceptional inverse image equals the usual inverse image. Duality of the exceptional inverse image functor Let be a smooth manifold of dimension and let be the unique map which maps everything to one point. For a ring , one finds that is the shifted -orientation sheaf. On the other hand, let be a smooth -variety of dimension . If denotes the structure morphism then is the shifted canonical sheaf on . Moreover, let be a smooth -variety of dimension and a prime invertible in . Then where denotes the Tate twist. Recalling the definition of the compactly supported cohomology as lower-shriek pushforward and noting that below the last means the constant sheaf on and the rest mean that on , , and the above computation furnishes the -adic Poincaré duality from the repeated application of the adjunction condition. References treats the topological setting treats the case of étale sheaves on schemes. See Exposé XVIII, section 3. gives the duality statements. Sheaf theory
https://en.wikipedia.org/wiki/Superb%20Internet	Superb Internet Technologies is a web hosting and Internet domain registrar company in business since 1996. According to Netcraft's January 2008 statistics, Superb Internet hosts over 340,000 unique sites, making it one of the world's 60 largest web hosts. Services offered by Superb Internet include basic web hosting, dedicated hosting, VPS hosting, as well as reseller hosting. Superb Internet is led by CEO Jeremy Gulban, and is currently a private company. Superb Internet's corporate offices are located in Honolulu, Hawaii, and with data centers located coast-to-coast in the United States in McLean, Virginia and South Seattle, Washington. Purchase of HopOne Internet and Superb Internet was completed by CherryRoad Technologies in May 2019. References External links Dedicated Web Hosting Company Web hosting Internet mirror services
https://en.wikipedia.org/wiki/Parker%20vector	In mathematics, especially the field of group theory, the Parker vector is an integer vector that describes a permutation group in terms of the cycle structure of its elements. Definition The Parker vector P of a permutation group G acting on a set of size n, is the vector whose kth component for k = 1, ..., n is given by: where ck(g) is the number of k-cycles in the cycle decomposition of g. Applications The Parker vector can assist in the recognition of Galois groups. References Permutation groups
https://en.wikipedia.org/wiki/Al-Qarara	al-Qarara () is a Palestinian town located north of Khan Yunis, in the Khan Yunis Governorate of the southern Gaza Strip. According to the Palestinian Central Bureau of Statistics, al-Qarara had a population of 29,004 inhabitants in 2017, References Towns in the Gaza Strip Khan Yunis Governorate Municipalities of the State of Palestine
https://en.wikipedia.org/wiki/Kline%20sphere%20characterization	In mathematics, a Kline sphere characterization, named after John Robert Kline, is a topological characterization of a two-dimensional sphere in terms of what sort of subset separates it. Its proof was one of the first notable accomplishments of R. H. Bing; Bing gave an alternate proof using brick partitioning in his paper Complementary domains of continuous curves A simple closed curve in a two-dimensional sphere (for instance, its equator) separates the sphere into two pieces upon removal. If one removes a pair of points from a sphere, however, the remainder is connected. Kline's sphere characterization states that the converse is true: If a nondegenerate locally connected metric continuum is separated by any simple closed curve but by no pair of points, then it is a two-dimensional sphere. References Bing, R. H., The Kline sphere characterization problem, Bulletin of the American Mathematical Society 52 (1946), 644–653. Theorems in topology Topology
https://en.wikipedia.org/wiki/Baqa%20ash-Sharqiyya	Baqa ash-Sharqiyya () is a Palestinian town in the northern West Bank, located northeast of Tulkarm in the Tulkarm Governorate. According to the Palestinian Central Bureau of Statistics (PCBS), the town had a population of 4,892 inhabitants in 2017. Refugees made up 20.4% of the Baqa ash-Sharqiyya's population in 1997. Approximately to the west, on the other side of the Green Line, lies Baqa al-Gharbiyye, ("the western bouquet of flowers") which is under Israeli jurisdiction. Both towns were originally one town, known as Baqa, until the aftermath of the 1948 Arab-Israeli War. Prior to the Second Intifada, Baqa ash-Sharqiyya consisted of 4,000 dunams; Israel confiscated about 2,000 dunams of land in order to build the Israeli West Bank barrier. History Ceramic from the Hellenistic, early and late Roman, Byzantine and the Middle Ages have been found here. In 1265, Baqa ash-Sharqiyya was among the estates Sultan Baibars handed to his followers, after he had defeated the Crusaders; the whole of Baqa ash-Sharqiyya was given to Emir 'Ala' al-Din Aidakin al-Bunduqdar al-Salihi. Ottoman era During early Ottoman rule, in 1596, Baqa ash-Sharqiyya was located in the nahiya of Qaqun in the Sanjak of Nablus. It had a population of 35 Muslim households, and paid taxes on wheat, barley, summer crops, olives, goats and/or bees, and a press for olives or grapes; a total of 14,000 akçe. In 1838 it was noted as a village, Bakah, in the western Esh-Sha'rawiyeh administrative region, north of Nablus. In 1870/1871 (1288 AH), an Ottoman census listed the village in the nahiya (sub-district) of al-Sha'rawiyya al-Sharqiyya. In 1882 the PEF's Survey of Western Palestine described Baqa as "a very small hamlet on high ground, with olives. It has a well to the south and a little Mukam (Muslim tomb) to the north; scattered olives surround it, and there are two or three palms close by. A few houses stand separate, on the south east, near a second Mukam, called Abu Nar ("Father of Fire")." A stone with Arabic inscriptions located over the entrance of the old village mosque could be the beginning of an endowment (waqf) text. A population list from about 1887 showed that Baqa ash-Sharqiyya had about 180 inhabitants, all Muslim. British Mandate era In the 1922 census of Palestine conducted by the British Mandate authorities, Baqa Sharkiyeh had a population of 269, all Muslims, increasing by the 1931 census to 330, still all Muslim, in 67 houses. In the 1945 statistics the population of Baqa ash-Sharqiyya consisted of 480, all Muslim, with a land area of 3,986 dunams, according to an official land and population survey. Of this, 173 dunams were designated for plantations and irrigable land, 2,870 for cereals, while 14 dunams were built-up areas. Jordanian era In the wake of the 1948 Arab–Israeli War, and after the 1949 Armistice Agreements, Baqa ash-Sharqiyya came under Jordanian rule. As part of the 1949 armistice agreements following the 1948 Arab-Israeli
https://en.wikipedia.org/wiki/Singular%20spectrum%20analysis	In time series analysis, singular spectrum analysis (SSA) is a nonparametric spectral estimation method. It combines elements of classical time series analysis, multivariate statistics, multivariate geometry, dynamical systems and signal processing. Its roots lie in the classical Karhunen (1946)–Loève (1945, 1978) spectral decomposition of time series and random fields and in the Mañé (1981)–Takens (1981) embedding theorem. SSA can be an aid in the decomposition of time series into a sum of components, each having a meaningful interpretation. The name "singular spectrum analysis" relates to the spectrum of eigenvalues in a singular value decomposition of a covariance matrix, and not directly to a frequency domain decomposition. Brief history The origins of SSA and, more generally, of subspace-based methods for signal processing, go back to the eighteenth century (Prony's method). A key development was the formulation of the spectral decomposition of the covariance operator of stochastic processes by Kari Karhunen and Michel Loève in the late 1940s (Loève, 1945; Karhunen, 1947). Broomhead and King (1986a, b) and Fraedrich (1986) proposed to use SSA and multichannel SSA (M-SSA) in the context of nonlinear dynamics for the purpose of reconstructing the attractor of a system from measured time series. These authors provided an extension and a more robust application of the idea of reconstructing dynamics from a single time series based on the embedding theorem. Several other authors had already applied simple versions of M-SSA to meteorological and ecological data sets (Colebrook, 1978; Barnett and Hasselmann, 1979; Weare and Nasstrom, 1982). Ghil, Vautard and their colleagues (Vautard and Ghil, 1989; Ghil and Vautard, 1991; Vautard et al., 1992; Ghil et al., 2002) noticed the analogy between the trajectory matrix of Broomhead and King, on the one hand, and the Karhunen–Loeve decomposition (Principal component analysis in the time domain), on the other. Thus, SSA can be used as a time-and-frequency domain method for time series analysis — independently from attractor reconstruction and including cases in which the latter may fail. The survey paper of Ghil et al. (2002) is the basis of the section of this article. A crucial result of the work of these authors is that SSA can robustly recover the "skeleton" of an attractor, including in the presence of noise. This skeleton is formed by the least unstable periodic orbits, which can be identified in the eigenvalue spectra of SSA and M-SSA. The identification and detailed description of these orbits can provide highly useful pointers to the underlying nonlinear dynamics. The so-called ‘Caterpillar’ methodology is a version of SSA that was developed in the former Soviet Union, independently of the mainstream SSA work in the West. This methodology became known in the rest of the world more recently (Danilov and Zhigljavsky, Eds., 1997; Golyandina et al., 2001; Zhigljavsky, Ed., 2010; Golyandina and Z
https://en.wikipedia.org/wiki/Orlando%20Magic%20all-time%20roster	The following is a list of players, both past and current, who appeared at least in one game for the Orlando Magic NBA franchise. Players Note: Statistics are correct through the end of the season. A \|- \|align="left"\| \|\| align="center"\|F \|\| align="left"\|San Jose State \|\| align="center"\|1 \|\| align="center"\| \|\| 46 \|\| 1,205 \|\| 239 \|\| 72 \|\| 563 \|\| 26.2 \|\| 5.2 \|\| 1.6 \|\| 12.2 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F/C \|\| align="left"\|Oral Roberts \|\| align="center"\|3 \|\| align="center"\|– \|\| 216 \|\| 3,930 \|\| 1,042 \|\| 114 \|\| 855 \|\| 18.2 \|\| 4.8 \|\| 0.5 \|\| 4.0 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G/F \|\| align="left"\|UCLA \|\| align="center"\|3 \|\| align="center"\|– \|\| 190 \|\| 5,541 \|\| 567 \|\| 484 \|\| 2,566 \|\| 29.2 \|\| 3.0 \|\| 2.5 \|\| 13.5 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|Virginia \|\| align="center"\|1 \|\| align="center"\| \|\| 26 \|\| 227 \|\| 25 \|\| 36 \|\| 52 \|\| 8.7 \|\| 1.0 \|\| 1.4 \|\| 2.0 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F/C \|\| align="left"\|Villanova \|\| align="center"\|1 \|\| align="center"\| \|\| 18 \|\| 178 \|\| 32 \|\| 4 \|\| 23 \|\| 9.9 \|\| 1.8 \|\| 0.2 \|\| 1.3 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|Fresno State \|\| align="center"\|1 \|\| align="center"\| \|\| 29 \|\| 856 \|\| 83 \|\| 148 \|\| 348 \|\| 29.5 \|\| 2.9 \|\| 5.1 \|\| 12.0 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F/C \|\| align="left"\|Penn State \|\| align="center"\|2 \|\| align="center"\|– \|\| 162 \|\| 3,394 \|\| 534 \|\| 169 \|\| 1,486 \|\| 21.0 \|\| 3.3 \|\| 1.0 \|\| 9.2 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F \|\| align="left"\|Wake Forest \|\| align="center"\|2 \|\| align="center"\|– \|\| 35 \|\| 747 \|\| 178 \|\| 50 \|\| 171 \|\| 21.3 \|\| 5.1 \|\| 1.4 \|\| 4.9 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G/F \|\| align="left"\|Illinois \|\| align="center" bgcolor="#CFECEC"\|10 \|\| align="center"\|– \|\| bgcolor="#CFECEC"\|692 \|\| 22,440 \|\| 3,667 \|\| 1,937 \|\| 10,650 \|\| 32.4 \|\| 5.3 \|\| 2.8 \|\| 15.4 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F \|\| align="left"\|California \|\| align="center"\|3 \|\| align="center"\|– \|\| 188 \|\| 4,299 \|\| 1,028 \|\| 143 \|\| 2,148 \|\| 22.9 \|\| 5.5 \|\| 0.8 \|\| 11.4 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F \|\| align="left"\|Alabama \|\| align="center"\|2 \|\| align="center"\|– \|\| 139 \|\| 2,098 \|\| 615 \|\| 65 \|\| 1,005 \|\| 15.1 \|\| 4.4 \|\| 0.5 \|\| 7.2 \|\| align=center\| \|- \|align="left" bgcolor="#CCFFCC"\|x \|\| align="center"\|G \|\| align="left"\|North Carolina \|\| align="center"\|3 \|\| align="center"\|– \|\| 172 \|\| 4,884 \|\| 857 \|\| 796 \|\| 2,448 \|\| 28.4 \|\| 5.0 \|\| 4.6 \|\| 14.2 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|Michigan State \|\| align="center"\|1 \|\| align="center"\| \|\| 5 \|\| 27 \|\| 1 \|\| 1 \|\| 6 \|\| 5.4 \|\| 0.2 \|\| 0.2 \|\| 1.2 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F \|\| align="left"\|Illinois \|\| align="center"\|1 \|\| align="center"\| \|\| 1 \|\| 4 \|\| 1 \|\| 0 \|\| 2 \|\| 4.0 \|\| 1.0 \|\| 0.0 \|\| 2.0 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|Arizona \|\| align="center"\|1 \|\| align="center"\| \|\| 49 \|\| 1,070 \|\| 119 \|
https://en.wikipedia.org/wiki/Protestantism%20in%20the%20Dominican%20Republic	Protestants in the Dominican Republic represent a sizeable minority of the population. There are several figures for the number of Protestants in the country. In 2019, statistics estimated that 26% of the country was Protestant. The 2020 Latinobarometer survey noted that 20% of the population were evangelical Protestant, with a sizeable number of non-evangelical Protestants. In the same year, the World Religion Database noted that Protestants comprised 8.43% of the population. Morgan Foley was the leader of the Protestantism for women in the 19th century. During the 1820s, Protestants migrated to the Dominican Republic from the United States. West Indian Protestants arrived on the island late 19th and the early 20th centuries, and by the 1920s, several Protestant organizations were established all throughout the country, which added diversity to the religious representation in the Dominican Republic. Many of the Protestant groups in DR had connections with organizations in the United States including Evangelical groups like the Assemblies of God, the Evangelical Church of the Dominican Republic (a united Methodist-Presbyterian church), and the Seventh-day Adventist Church. These groups dominated the Protestant movement in the earlier part of the 20th century, but in the 1960s and 1970s Pentecostal churches saw the most growth. Protestant denominations active in the Dominican Republic now include: Assembly of God Church of God Baptist Pentecostal Seventh-day Adventist Church Missionaries, Episcopalians, Jehovah's Witnesses, Mormons, and Mennonites, also travel to the island. Jehovah's Witnesses, specifically, have been known to be migrating (more so during the last decade) to the Dominican Republic where they feel there is a "great need" for evangelizing their faith. However they are not seen as Protestant denomination by mainstream Christianity. See also Religion in the Dominican Republic Evangelical Church of the Dominican Republic Catholic Church in the Dominican Republic Afro-American religion Religion in Latin America References Christianity in the Dominican Republic Dominican Republic Protestantism in the Caribbean
https://en.wikipedia.org/wiki/Hilbert%20number	In number theory, a branch of mathematics, a Hilbert number is a positive integer of the form (). The Hilbert numbers were named after David Hilbert. The sequence of Hilbert numbers begins 1, 5, 9, 13, 17, ... ) Properties The Hilbert number sequence is the arithmetic sequence with , meaning the Hilbert numbers follow the recurrence relation . The sum of a Hilbert number amount of Hilbert numbers (1 number, 5 numbers, 9 numbers, etc.) is also a Hilbert number. Hilbert primes A Hilbert prime is a Hilbert number that is not divisible by a smaller Hilbert number (other than 1). The sequence of Hilbert primes begins 5, 9, 13, 17, 21, 29, 33, 37, 41, 49, ... . A Hilbert prime is not necessarily a prime number; for example, 21 is a composite number since . However, 21 a Hilbert prime since neither 3 nor 7 (the only factors of 21 other than 1 and itself) are Hilbert numbers. It follows from multiplication modulo 4 that a Hilbert prime is either a prime number of the form (called a Pythagorean prime), or a semiprime of the form . References External links Eponymous numbers in mathematics Integer sequences
https://en.wikipedia.org/wiki/Warren%20Goldfarb	Warren David Goldfarb (born 1949) is Walter Beverly Pearson Professor of Modern Mathematics and Mathematical Logic at Harvard University. He specializes in the history of analytic philosophy and in logic, most notably the classical decision problem. Education and career He received his A.B. and philosophy Ph.D. from Harvard University under the supervision of Burton Dreben, and has been a member of the Harvard faculty since 1975. He received tenure in 1982, the only philosopher to be promoted to tenure at Harvard between 1962 and 1999. Prof. Goldfarb is also one of the founders of the Harvard Gay & Lesbian Caucus and was one of the first openly gay Harvard faculty members. Philosophical work Goldfarb was an editor of volumes III–V of Kurt Gödel's Collected Works. He has also published articles on important analytic philosophers, including Frege, Russell, Wittgenstein's early and later work, Carnap and Quine. Selected publications Books The Decision Problem: Solvable Classes of Quantificational Formulas, Addison-Wesley, 1979. (with Burton Dreben). Deductive Logic, Hackett, 2003. Articles "Logic in the Twenties: the Nature of the Quantifier," The Journal of Symbolic Logic (1979) "I want you to bring me a slab: Remarks on the opening sections of the Philosophical Investigations," Synthese (1983) "Kripke on Wittgenstein on Rules," The Journal of Philosophy (1985) "Poincare Against the Logicists," in History and Philosophy of Modern Mathematics (University of Minnesota Press, 1987) "Wittgenstein on Understanding," Midwest Studies in Philosophy (1992) "Frege's Conception of Logic," in Future Pasts: The Analytic Tradition in the 20th Century, Juliet Floyd and Sanford Shieh, editors (Oxford University Press, 2001) “Rule-Following Revisited," in Wittgenstein and the Philosophy of Mind, Jonathan Ellis and Daniel Guevara, editors (Oxford University Press, 2012). References External links Goldfarb's web page at Harvard University Works by Warren Goldfarb at PhilPapers Living people American logicians Mathematical logicians American historians of mathematics Harvard College alumni Harvard University faculty Harvard University Department of Philosophy faculty American gay men 1949 births Harvard Graduate School of Arts and Sciences alumni 20th-century American LGBT people 21st-century American LGBT people
https://en.wikipedia.org/wiki/Ramified%20forcing	In the mathematical discipline of set theory, ramified forcing is the original form of forcing introduced by to prove the independence of the continuum hypothesis from Zermelo–Fraenkel set theory. Ramified forcing starts with a model of set theory in which the axiom of constructibility, , holds, and then builds up a larger model of Zermelo–Fraenkel set theory by adding a generic subset of a partially ordered set to , imitating Kurt Gödel's constructible hierarchy. Dana Scott and Robert Solovay realized that the use of constructible sets was an unnecessary complication, and could be replaced by a simpler construction similar to John von Neumann's construction of the universe as a union of sets for ordinals . Their simplification was originally called "unramified forcing" , but is now usually just called "forcing". As a result, ramified forcing is only rarely used. References . . . Forcing (mathematics)
https://en.wikipedia.org/wiki/High%20probability%20instruction	The high probability instruction (HPI) treatment is a behaviorist psychological treatment based on the idea of positive reinforcement. It consists of the idea of reinforcing an instruction with a low probability of compliance by using the reinforcement of an instruction with a high probability. Sources Luce Doze (2005), under the direction of Ph.D Esteve Freixa i Bacquet (University of Picardie, France) and Mrs. Rivière (University of Lille-III, France). Treatment by HPI of an autistic child - (An example of complementarity between fundamental research and clinical practice from an autistic child case). Results shown at the 2005 seminar of the Association Picardie de Pratiques Cognitives et Comportementales (Picard Association of Cognitive and Behaviorist Practices). Ardoin, S. P., Martens, B. K., & Wolfe, L. A. (1999). Using high-probability instruction sequences with fading to increase student compliance during transitions. Journal of Applied Behavior Analysis, 32, 339-351. Psychological theories
https://en.wikipedia.org/wiki/Dickson%20polynomial	In mathematics, the Dickson polynomials, denoted , form a polynomial sequence introduced by . They were rediscovered by in his study of Brewer sums and have at times, although rarely, been referred to as Brewer polynomials. Over the complex numbers, Dickson polynomials are essentially equivalent to Chebyshev polynomials with a change of variable, and, in fact, Dickson polynomials are sometimes called Chebyshev polynomials. Dickson polynomials are generally studied over finite fields, where they sometimes may not be equivalent to Chebyshev polynomials. One of the main reasons for interest in them is that for fixed , they give many examples of permutation polynomials; polynomials acting as permutations of finite fields. Definition First kind For integer and in a commutative ring with identity (often chosen to be the finite field ) the Dickson polynomials (of the first kind) over are given by The first few Dickson polynomials are They may also be generated by the recurrence relation for , with the initial conditions and . The coefficients are given at several places in the OEIS with minute differences for the first two terms. Second kind The Dickson polynomials of the second kind, , are defined by They have not been studied much, and have properties similar to those of Dickson polynomials of the first kind. The first few Dickson polynomials of the second kind are They may also be generated by the recurrence relation for , with the initial conditions and . The coefficients are also given in the OEIS. Properties The are the unique monic polynomials satisfying the functional equation where and . They also satisfy a composition rule, The also satisfy a functional equation for , , with and . The Dickson polynomial is a solution of the ordinary differential equation and the Dickson polynomial is a solution of the differential equation Their ordinary generating functions are Links to other polynomials By the recurrence relation above, Dickson polynomials are Lucas sequences. Specifically, for , the Dickson polynomials of the first kind are Fibonacci polynomials, and Dickson polynomials of the second kind are Lucas polynomials. By the composition rule above, when α is idempotent, composition of Dickson polynomials of the first kind is commutative. The Dickson polynomials with parameter give monomials. The Dickson polynomials with parameter are related to Chebyshev polynomials of the first kind by Since the Dickson polynomial can be defined over rings with additional idempotents, is often not related to a Chebyshev polynomial. Permutation polynomials and Dickson polynomials A permutation polynomial (for a given finite field) is one that acts as a permutation of the elements of the finite field. The Dickson polynomial (considered as a function of with α fixed) is a permutation polynomial for the field with elements if and only if is coprime to . proved that any integral polynomial that is a permutat
https://en.wikipedia.org/wiki/Permutation%20polynomial	In mathematics, a permutation polynomial (for a given ring) is a polynomial that acts as a permutation of the elements of the ring, i.e. the map is a bijection. In case the ring is a finite field, the Dickson polynomials, which are closely related to the Chebyshev polynomials, provide examples. Over a finite field, every function, so in particular every permutation of the elements of that field, can be written as a polynomial function. In the case of finite rings Z/nZ, such polynomials have also been studied and applied in the interleaver component of error detection and correction algorithms. Single variable permutation polynomials over finite fields Let be the finite field of characteristic , that is, the field having elements where for some prime . A polynomial with coefficients in (symbolically written as ) is a permutation polynomial of if the function from to itself defined by is a permutation of . Due to the finiteness of , this definition can be expressed in several equivalent ways: the function is onto (surjective); the function is one-to-one (injective); has a solution in for each in ; has a unique solution in for each in . A characterization of which polynomials are permutation polynomials is given by (Hermite's Criterion) is a permutation polynomial of if and only if the following two conditions hold: has exactly one root in ; for each integer with and , the reduction of has degree . If is a permutation polynomial defined over the finite field , then so is for all and in . The permutation polynomial is in normalized form if and are chosen so that is monic, and (provided the characteristic does not divide the degree of the polynomial) the coefficient of is 0. There are many open questions concerning permutation polynomials defined over finite fields. Small degree Hermite's criterion is computationally intensive and can be difficult to use in making theoretical conclusions. However, Dickson was able to use it to find all permutation polynomials of degree at most five over all finite fields. These results are: A list of all monic permutation polynomials of degree six in normalized form can be found in . Some classes of permutation polynomials Beyond the above examples, the following list, while not exhaustive, contains almost all of the known major classes of permutation polynomials over finite fields. permutes if and only if and are coprime (notationally, ). If is in and then the Dickson polynomial (of the first kind) is defined by These can also be obtained from the recursion with the initial conditions and . The first few Dickson polynomials are: If and then permutes GF(q) if and only if . If then and the previous result holds. If is an extension of of degree , then the linearized polynomial with in , is a linear operator on over . A linearized polynomial permutes if and only if 0 is the only root of in . This condition can be expressed algebraically as
https://en.wikipedia.org/wiki/Jubb%20al-Shami	Jubb al-Shami () is a hamlet east of Homs. According to the Central Bureau of Statistics (CBS), its population was 35 in 2004. The inhabitants lived in five households. References External links Wikimapia Panoramio Populated places in Homs District
https://en.wikipedia.org/wiki/Chernoff%27s%20distribution	In probability theory, Chernoff's distribution, named after Herman Chernoff, is the probability distribution of the random variable where W is a "two-sided" Wiener process (or two-sided "Brownian motion") satisfying W(0) = 0. If then V(0, c) has density where gc has Fourier transform given by and where Ai is the Airy function. Thus fc is symmetric about 0 and the density ƒZ = ƒ1. Groeneboom (1989) shows that where is the largest zero of the Airy function Ai and where . In the same paper, Groeneboom also gives an analysis of the process . The connection with the statistical problem of estimating a monotone density is discussed in Groeneboom (1985). Chernoff's distribution is now known to appear in a wide range of monotone problems including isotonic regression. The Chernoff distribution should not be confused with the Chernoff geometric distribution (called the Chernoff point in information geometry) induced by the Chernoff information. History Groeneboom, Lalley and Temme state that the first investigation of this distribution was probably by Chernoff in 1964, who studied the behavior of a certain estimator of a mode. In his paper, Chernoff characterized the distribution through an analytic representation through the heat equation with suitable boundary conditions. Initial attempts at approximating Chernoff's distribution via solving the heat equation, however, did not achieve satisfactory precision due to the nature of the boundary conditions. The computation of the distribution is addressed, for example, in Groeneboom and Wellner (2001). The connection of Chernoff's distribution with Airy functions was also found independently by Daniels and Skyrme and Temme, as cited in Groeneboom, Lalley and Temme. These two papers, along with Groeneboom (1989), were all written in 1984. References Continuous distributions Stochastic processes
https://en.wikipedia.org/wiki/Propositiones%20ad%20Acuendos%20Juvenes	The medieval Latin manuscript Propositiones ad Acuendos Juvenes () is one of the earliest known collections of recreational mathematics problems. The oldest known copy of the manuscript dates from the late 9th century. The text is attributed to Alcuin of York (died 804.) Some editions of the text contain 53 problems, others 56. It has been translated into English by John Hadley, with annotations by John Hadley and David Singmaster. The manuscript contains the first known occurrences of several types of problem, including three river-crossing problems: Problem 17: The jealous husbands problem. In Alcuin's version of this problem, three men, each with a sister, must cross a boat which can carry only two people, so that a woman whose brother is not present is never left in the company of another man,, p. 111. Problem 18: The problem of the wolf, goat, and cabbage, p. 112., and Problem 19: Propositio de viro et muliere ponderantibus plaustrum. In this problem, a man and a woman of equal weight, together with two children, each of half their weight, wish to cross a river using a boat which can only carry the weight of one adult;, p. 112. a so-called "barrel-sharing" problem: Problem 12: A certain father died and left as an inheritance to his three sons 30 glass flasks, of which 10 were full of oil, another 10 were half full, while another 10 were empty. Divide the oil and flasks so that an equal share of the commodities should equally come down to the three sons, both of oil and glass;, p. 109. The number of solutions to this problem for n of each type of flask are terms of Alcuin's sequence. a variant of the jeep problem: Problem 52: A certain head of household ordered that 90 modia of grain be taken from one of his houses to another 30 leagues away. Given that this load of grain can be carried by a camel in three trips and that the camel eats one modius per league, how many modia were left over at the end of the journey?, pp. 124–125. and three packing problems: Problem 27: Proposition concerning a quadrangular city. There is a quadrangular city which has one side of 1100 feet, another side of 1000 feet, a front of 600 feet, and a final side of 600 feet. I want to put some houses there so that each house is 40 feet long and 30 feet wide. Let him say, he who wishes, How many houses ought the city to contain? Problem 28: Proposition concerning a triangular city. There is a triangular city which has one side of 100 feet, another side of 100 feet, and a third of 90 feet. Inside of this, I want to build a structure of houses, however, in such a way that each house is 20 feet in length, 10 feet in width. Let him say, he who can, How many houses should be contained? Problem 29: Proposition concerning a round city. There is a city which is 8000 feet in circumference. Let him say, he who is able, How many houses should the city contain, such that each [house] is 30 feet long, and 20 feet wide? Some further problems are: Problem 5: A merchant
https://en.wikipedia.org/wiki/Polyhedral%20symbol	The polyhedral symbol is sometimes used in coordination chemistry to indicate the approximate geometry of the coordinating atoms around the central atom. One or more italicised letters indicate the geometry, e.g. TP-3 which is followed by a number that gives the coordination number of the central atom. The polyhedral symbol can be used in naming of compounds, in which case it is followed by the configuration index. Polyhedral symbols Configuration index The first step in determining the configuration index is to assign a priority number to each coordinating ligand according to the Cahn-Ingold-Prelog priority rules, (CIP rules). The preferred ligand takes the lowest priority number. For example, of the ligands acetonitrile, chloride ion and pyridine thepriority number assigned are chloride, 1; acetonitrile,2; pyridene,3. Each coordination type has a different procedure for specifying the configuration index and these are outlined in below. T shaped (TS-3) The configuration index is a single digit which is defined as the priority number of the ligand on the stem of the "T". Seesaw (SS- 4) The configuration index has two digits which are the priority numbers of the ligands separated by the largest angle. The lowest priority number of the pair is quoted first. Square planar (SP-4) The configuration index is a single digit which is the priority number of the ligand trans to the highest priority ligand. (If there are two possibilities the principle of trans difference is applied). As an example, (acetonitrile)dichlorido(pyridine)platinum(II) complex where the Cl ligands may be trans or cis to one another. The ligand priority numbers are, applying the CIP rules: two chlorides of priority number 1 acetonitrile priority 2 pyridine priority 3 In the trans case the configuration index is 1 giving the name(SP-4-1)-(acetonitrile)dichlorido(pyridine)platinum(II). In the cis case both of the organic ligands are trans to a chloride so to choose the trans difference is considered and the greater is between 1 and three therefore the name is (SP-4-3)-(acetonitrile)dichlorido(pyridine)platinum(II). Octahedral (OC-6) The configuration index has two digits. The first digit is the priority number of the ligand trans to the highest priority ligand. This pair is then used to define the reference axis of the octahedron. The second digit is the priority number of the ligand trans to the highest priority ligand in the plane perpendicular to the reference axis. Square pyramidal (SPY-4) The configuration index is a single digit which is the priority number of the ligand trans to the ligand of lowest priority in the plane perpendicular to the 4 fold axis. (If there is more than one choice then the highest numerical value second digit is taken.) NB this procedure gives the same result as SP-4, however in this case the polyhedral symbol specifies that the complex is non-planar. Square pyramidal (SPY-5) There are two digits. The first digit is the priority numb
https://en.wikipedia.org/wiki/Eduardo%20Ledesma	Eduardo Fabián Ledesma Trinidad (born 7 August 1985) is a Paraguayan football midfielder. Honours Lanús Argentine Primera División (1): 2007 Apertura External links Argentine Primera statistics at Fútbol XXI 1985 births Living people Paraguayan men's footballers Men's association football midfielders Club Olimpia footballers Club Atlético Lanús footballers Rosario Central footballers Argentine Primera División players Paraguayan expatriate men's footballers Expatriate men's footballers in Argentina Expatriate men's footballers in Ecuador L.D.U. Quito footballers
https://en.wikipedia.org/wiki/Carlos%20Arce%20%28footballer%2C%20born%201985%29	Carlos Arce (born 4 February 1985, in Lanús) is an Argentine football defender. He currently plays for Club Atlético Lanús in Argentina. External links Argentine Primera statistics 1985 births Living people Footballers from Lanús Argentine people of Basque descent Argentine men's footballers Men's association football defenders Argentine Primera División players Club Atlético Lanús footballers
https://en.wikipedia.org/wiki/Marcos%20Ram%C3%ADrez%20%28footballer%29	Marcos Ramírez (born 25 April 1983 in Avellaneda) is an Argentine football defender. He currently plays for Defensa y Justicia. External links Argentine Primera statistics 1983 births Footballers from Avellaneda Argentine men's footballers Men's association football defenders Defensa y Justicia footballers Club Atlético Independiente footballers Club de Gimnasia y Esgrima La Plata footballers Godoy Cruz Antonio Tomba footballers Chacarita Juniors footballers Argentine Primera División players Living people
https://en.wikipedia.org/wiki/1951%20CCCF%20Championship	The 1951 CCCF Championship was played in Panama City, Panama, from 25 February to 4 March. Most member countries did not participate because of a polio epidemic. Final standings Statistics Goalscorers External links CCCF Championship 1951 on RSSSF Archive Match Details CCCF Championship Cccf Championship, 1951 International association football competitions hosted by Panama Cccf Championship, 1951 CCCF CCCF February 1951 sports events in North America March 1951 sports events in North America Sports competitions in Panama City 20th century in Panama City
https://en.wikipedia.org/wiki/Log-logistic%20distribution	In probability and statistics, the log-logistic distribution (known as the Fisk distribution in economics) is a continuous probability distribution for a non-negative random variable. It is used in survival analysis as a parametric model for events whose rate increases initially and decreases later, as, for example, mortality rate from cancer following diagnosis or treatment. It has also been used in hydrology to model stream flow and precipitation, in economics as a simple model of the distribution of wealth or income, and in networking to model the transmission times of data considering both the network and the software. The log-logistic distribution is the probability distribution of a random variable whose logarithm has a logistic distribution. It is similar in shape to the log-normal distribution but has heavier tails. Unlike the log-normal, its cumulative distribution function can be written in closed form. Characterization There are several different parameterizations of the distribution in use. The one shown here gives reasonably interpretable parameters and a simple form for the cumulative distribution function. The parameter is a scale parameter and is also the median of the distribution. The parameter is a shape parameter. The distribution is unimodal when and its dispersion decreases as increases. The cumulative distribution function is where , , The probability density function is Alternative parameterization An alternative parametrization is given by the pair in analogy with the logistic distribution: Properties Moments The th raw moment exists only when when it is given by where B is the beta function. Expressions for the mean, variance, skewness and kurtosis can be derived from this. Writing for convenience, the mean is and the variance is Explicit expressions for the skewness and kurtosis are lengthy. As tends to infinity the mean tends to , the variance and skewness tend to zero and the excess kurtosis tends to 6/5 (see also related distributions below). Quantiles The quantile function (inverse cumulative distribution function) is : It follows that the median is , the lower quartile is and the upper quartile is . Applications Survival analysis The log-logistic distribution provides one parametric model for survival analysis. Unlike the more commonly used Weibull distribution, it can have a non-monotonic hazard function: when the hazard function is unimodal (when ≤ 1, the hazard decreases monotonically). The fact that the cumulative distribution function can be written in closed form is particularly useful for analysis of survival data with censoring. The log-logistic distribution can be used as the basis of an accelerated failure time model by allowing to differ between groups, or more generally by introducing covariates that affect but not by modelling as a linear function of the covariates. The survival function is and so the hazard function is The log-logistic distribution with shape param
https://en.wikipedia.org/wiki/Laplace%20operators%20in%20differential%20geometry	In differential geometry there are a number of second-order, linear, elliptic differential operators bearing the name Laplacian. This article provides an overview of some of them. Connection Laplacian The connection Laplacian, also known as the rough Laplacian, is a differential operator acting on the various tensor bundles of a manifold, defined in terms of a Riemannian- or pseudo-Riemannian metric. When applied to functions (i.e. tensors of rank 0), the connection Laplacian is often called the Laplace–Beltrami operator. It is defined as the trace of the second covariant derivative: where T is any tensor, is the Levi-Civita connection associated to the metric, and the trace is taken with respect to the metric. Recall that the second covariant derivative of T is defined as Note that with this definition, the connection Laplacian has negative spectrum. On functions, it agrees with the operator given as the divergence of the gradient. If the connection of interest is the Levi-Civita connection one can find a convenient formula for the Laplacian of a scalar function in terms of partial derivatives with respect to a coordinate system: where is a scalar function, is absolute value of the determinant of the metric (absolute value is necessary in the pseudo-Riemannian case, e.g. in General Relativity) and denotes the inverse of the metric tensor. Hodge Laplacian The Hodge Laplacian, also known as the Laplace–de Rham operator, is a differential operator acting on differential forms. (Abstractly, it is a second order operator on each exterior power of the cotangent bundle.) This operator is defined on any manifold equipped with a Riemannian- or pseudo-Riemannian metric. where d is the exterior derivative or differential and δ is the codifferential. The Hodge Laplacian on a compact manifold has nonnegative spectrum. The connection Laplacian may also be taken to act on differential forms by restricting it to act on skew-symmetric tensors. The connection Laplacian differs from the Hodge Laplacian by means of a Weitzenböck identity. Bochner Laplacian The Bochner Laplacian is defined differently from the connection Laplacian, but the two will turn out to differ only by a sign, whenever the former is defined. Let M be a compact, oriented manifold equipped with a metric. Let E be a vector bundle over M equipped with a fiber metric and a compatible connection, . This connection gives rise to a differential operator where denotes smooth sections of E, and T*M is the cotangent bundle of M. It is possible to take the -adjoint of , giving a differential operator The Bochner Laplacian is given by which is a second order operator acting on sections of the vector bundle E. Note that the connection Laplacian and Bochner Laplacian differ only by a sign: Lichnerowicz Laplacian The Lichnerowicz Laplacian is defined on symmetric tensors by taking to be the symmetrized covariant derivative. The Lichnerowicz Laplacian is then defined by ,
https://en.wikipedia.org/wiki/Nonlinear%20eigenproblem	In mathematics, a nonlinear eigenproblem, sometimes nonlinear eigenvalue problem, is a generalization of the (ordinary) eigenvalue problem to equations that depend nonlinearly on the eigenvalue. Specifically, it refers to equations of the form where is a vector, and is a matrix-valued function of the number . The number is known as the (nonlinear) eigenvalue, the vector as the (nonlinear) eigenvector, and as the eigenpair. The matrix is singular at an eigenvalue . Definition In the discipline of numerical linear algebra the following definition is typically used. Let , and let be a function that maps scalars to matrices. A scalar is called an eigenvalue, and a nonzero vector is called a right eigevector if . Moreover, a nonzero vector is called a left eigevector if , where the superscript denotes the Hermitian transpose. The definition of the eigenvalue is equivalent to , where denotes the determinant. The function is usually required to be a holomorphic function of (in some domain ). In general, could be a linear map, but most commonly it is a finite-dimensional, usually square, matrix. Definition: The problem is said to be regular if there exists a such that . Otherwise it is said to be singular. Definition: An eigenvalue is said to have algebraic multiplicity if is the smallest integer such that the th derivative of with respect to , in is nonzero. In formulas that but for . Definition: The geometric multiplicity of an eigenvalue is the dimension of the nullspace of . Special cases The following examples are special cases of the nonlinear eigenproblem. The (ordinary) eigenvalue problem: The generalized eigenvalue problem: The quadratic eigenvalue problem: The polynomial eigenvalue problem: The rational eigenvalue problem: where are rational functions. The delay eigenvalue problem: where are given scalars, known as delays. Jordan chains Definition: Let be an eigenpair. A tuple of vectors is called a Jordan chain iffor , where denotes the th derivative of with respect to and evaluated in . The vectors are called generalized eigenvectors, is called the length of the Jordan chain, and the maximal length a Jordan chain starting with is called the rank of . Theorem: A tuple of vectors is a Jordan chain if and only if the function has a root in and the root is of multiplicity at least for , where the vector valued function is defined as Mathematical software The eigenvalue solver package SLEPc contains C-implementations of many numerical methods for nonlinear eigenvalue problems. The NLEVP collection of nonlinear eigenvalue problems is a MATLAB package containing many nonlinear eigenvalue problems with various properties. The FEAST eigenvalue solver is a software package for standard eigenvalue problems as well as nonlinear eigenvalue problems, designed from density-matrix representation in quantum mechanics combined with contour integration techniques. The MATLAB toolb
https://en.wikipedia.org/wiki/Matrix%20polynomial	In mathematics, a matrix polynomial is a polynomial with square matrices as variables. Given an ordinary, scalar-valued polynomial this polynomial evaluated at a matrix A is where I is the identity matrix. A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. A matrix polynomial identity is a matrix polynomial equation which holds for all matrices A in a specified matrix ring Mn(R). Characteristic and minimal polynomial The characteristic polynomial of a matrix A is a scalar-valued polynomial, defined by . The Cayley–Hamilton theorem states that if this polynomial is viewed as a matrix polynomial and evaluated at the matrix A itself, the result is the zero matrix: . The characteristic polynomial is thus a polynomial which annihilates A. There is a unique monic polynomial of minimal degree which annihilates A; this polynomial is the minimal polynomial. Any polynomial which annihilates A (such as the characteristic polynomial) is a multiple of the minimal polynomial. It follows that given two polynomials P and Q, we have if and only if where denotes the jth derivative of P and are the eigenvalues of A with corresponding indices (the index of an eigenvalue is the size of its largest Jordan block). Matrix geometrical series Matrix polynomials can be used to sum a matrix geometrical series as one would an ordinary geometric series, If I − A is nonsingular one can evaluate the expression for the sum S. See also Latimer–MacDuffee theorem Matrix exponential Matrix function Notes References . . Matrix theory Polynomials
https://en.wikipedia.org/wiki/Minkowski%20content	The Minkowski content (named after Hermann Minkowski), or the boundary measure, of a set is a basic concept that uses concepts from geometry and measure theory to generalize the notions of length of a smooth curve in the plane, and area of a smooth surface in space, to arbitrary measurable sets. It is typically applied to fractal boundaries of domains in the Euclidean space, but it can also be used in the context of general metric measure spaces. It is related to, although different from, the Hausdorff measure. Definition For , and each integer m with , the m-dimensional upper Minkowski content is and the m-dimensional lower Minkowski content is defined as where is the volume of the (n−m)-ball of radius r and is an -dimensional Lebesgue measure. If the upper and lower m-dimensional Minkowski content of A are equal, then their common value is called the Minkowski content Mm(A). Properties The Minkowski content is (generally) not a measure. In particular, the m-dimensional Minkowski content in Rn is not a measure unless m = 0, in which case it is the counting measure. Indeed, clearly the Minkowski content assigns the same value to the set A as well as its closure. If A is a closed m-rectifiable set in Rn, given as the image of a bounded set from Rm under a Lipschitz function, then the m-dimensional Minkowski content of A exists, and is equal to the m-dimensional Hausdorff measure of A. See also Gaussian isoperimetric inequality Geometric measure theory Isoperimetric inequality in higher dimensions Minkowski–Bouligand dimension Footnotes References . . Measure theory Geometry Analytic geometry Dimension theory Dimension Measures (measure theory) Fractals Hermann Minkowski
https://en.wikipedia.org/wiki/Diane%20Lambert	Diane Marie Lambert is an American statistician known for her work on zero-inflated models, a method for extending Poisson regression to applications such as the statistics of manufacturing defects in which one can expect to observe a large number of zeros. A former Bell Labs Fellow, she is a research scientist for Google, where she lists her current research areas as "algorithms and theory, data mining and modeling, and economics and electronic commerce". Education and career Lambert earned her Ph.D. in 1978 from the University of Rochester. Her dissertation, supervised by W. Jackson Hall, was P-Values: Asymptotics and Robustness. In the early part of her career, she worked as a faculty member at Carnegie Mellon University. As an assistant professor there, she did pioneering work on the confidentiality of statistical information. She earned tenure at Carnegie Mellon, but moved to Bell Labs in 1986. At Bell Labs, she became head of statistics, and a Bell Labs Fellow. She moved again to Google in 2005. Recognition Lambert became a Fellow of the American Statistical Association in 1991. She is also a Fellow of the Institute of Mathematical Statistics, was executive secretary of the institute from 1990 to 1993, and was one of the institute's Medallion Lecturers in 1995. References Year of birth missing (living people) Living people American statisticians Women statisticians University of Rochester alumni Carnegie Mellon University faculty Fellows of the American Statistical Association Fellows of the Institute of Mathematical Statistics Google people
https://en.wikipedia.org/wiki/Exponential%20dispersion%20model	In probability and statistics, the class of exponential dispersion models (EDM) is a set of probability distributions that represents a generalisation of the natural exponential family. Exponential dispersion models play an important role in statistical theory, in particular in generalized linear models because they have a special structure which enables deductions to be made about appropriate statistical inference. Definition Univariate case There are two versions to formulate an exponential dispersion model. Additive exponential dispersion model In the univariate case, a real-valued random variable belongs to the additive exponential dispersion model with canonical parameter and index parameter , , if its probability density function can be written as Reproductive exponential dispersion model The distribution of the transformed random variable is called reproductive exponential dispersion model, , and is given by with and , implying . The terminology dispersion model stems from interpreting as dispersion parameter. For fixed parameter , the is a natural exponential family. Multivariate case In the multivariate case, the n-dimensional random variable has a probability density function of the following form where the parameter has the same dimension as . Properties Cumulant-generating function The cumulant-generating function of is given by with Mean and variance Mean and variance of are given by with unit variance function . Reproductive If are i.i.d. with , i.e. same mean and different weights , the weighted mean is again an with with . Therefore are called reproductive. Unit deviance The probability density function of an can also be expressed in terms of the unit deviance as where the unit deviance takes the special form or in terms of the unit variance function as . Examples A lot of very common probability distributions belong to the class of EDMs, among them are: normal distribution, Binomial distribution, Poisson distribution, Negative binomial distribution, Gamma distribution, Inverse Gaussian distribution, and Tweedie distribution. References Statistical models
https://en.wikipedia.org/wiki/Grubbs%27s%20test	In statistics, Grubbs's test or the Grubbs test (named after Frank E. Grubbs, who published the test in 1950), also known as the maximum normalized residual test or extreme studentized deviate test, is a test used to detect outliers in a univariate data set assumed to come from a normally distributed population. Definition Grubbs's test is based on the assumption of normality. That is, one should first verify that the data can be reasonably approximated by a normal distribution before applying the Grubbs test. Grubbs's test detects one outlier at a time. This outlier is expunged from the dataset and the test is iterated until no outliers are detected. However, multiple iterations change the probabilities of detection, and the test should not be used for sample sizes of six or fewer since it frequently tags most of the points as outliers. Grubbs's test is defined for the following hypotheses: H0: There are no outliers in the data set Ha: There is exactly one outlier in the data set The Grubbs test statistic is defined as with and denoting the sample mean and standard deviation, respectively. The Grubbs test statistic is the largest absolute deviation from the sample mean in units of the sample standard deviation. This is the two-sided test, for which the hypothesis of no outliers is rejected at significance level α if with tα/(2N),N−2 denoting the upper critical value of the t-distribution with N − 2 degrees of freedom and a significance level of α/(2N). One-sided case Grubbs's test can also be defined as a one-sided test, replacing α/(2N) with α/N. To test whether the minimum value is an outlier, the test statistic is with Ymin denoting the minimum value. To test whether the maximum value is an outlier, the test statistic is with Ymax denoting the maximum value. Related techniques Several graphical techniques can be used to detect outliers. A simple run sequence plot, a box plot, or a histogram should show any obviously outlying points. A normal probability plot may also be useful. See also Chauvenet's criterion Peirce's criterion Q test Studentized residual Tau distribution References Further reading Statistical tests Statistical outliers
https://en.wikipedia.org/wiki/Yates%20analysis	In statistics, a Yates analysis is an approach to analyzing data obtained from a designed experiment, where a factorial design has been used. Full- and fractional-factorial designs are common in designed experiments for engineering and scientific applications. In these designs, each factor is assigned two levels, typically called the low and high levels, and referred to as "-" and "+". For computational purposes, the factors are scaled so that the low level is assigned a value of -1 and the high level is assigned a value of +1. A full factorial design contains all possible combinations of low/high levels for all the factors. A fractional factorial design contains a carefully chosen subset of these combinations. The criterion for choosing the subsets is discussed in detail in the fractional factorial designs article. Formalized by Frank Yates, a Yates analysis exploits the special structure of these designs to generate least squares estimates for factor effects for all factors and all relevant interactions. The Yates analysis can be used to answer the following questions: What is the ranked list of factors? What is the goodness-of-fit (as measured by the residual standard deviation) for the various models? The mathematical details of the Yates analysis are given in chapter 10 of Box, Hunter, and Hunter (1978). The Yates analysis is typically complemented by a number of graphical techniques such as the dex mean plot and the dex contour plot ("dex" stands for "design of experiments"). Yates Order Before performing a Yates analysis, the data should be arranged in "Yates order". That is, given k factors, the kth column consists of 2(k - 1) minus signs (i.e., the low level of the factor) followed by 2(k - 1) plus signs (i.e., the high level of the factor). For example, for a full factorial design with three factors, the design matrix is Determining the Yates order for fractional factorial designs requires knowledge of the confounding structure of the fractional factorial design. Output A Yates analysis generates the following output. A factor identifier (from Yates order). The specific identifier will vary depending on the program used to generate the Yates analysis. Dataplot, for example, uses the following for a 3-factor model. 1 = factor 1 2 = factor 2 3 = factor 3 12 = interaction of factor 1 and factor 2 13 = interaction of factor 1 and factor 3 23 = interaction of factor 2 and factor 3 123 = interaction of factors 1, 2, and 3 A ranked list of important factors. That is, least squares estimated factor effects ordered from largest in magnitude (most significant) to smallest in magnitude (least significant). A t-value for the individual factor effect estimates. The t-value is computed as where e is the estimated factor effect and se is the standard deviation of the estimated factor effect. The residual standard deviation that results from the model with the single term only. That is, the residual standard deviation from the model
https://en.wikipedia.org/wiki/George%20Mostow	George Daniel Mostow (July 4, 1923 – April 4, 2017) was an American mathematician, renowned for his contributions to Lie theory. He was the Henry Ford II (emeritus) Professor of Mathematics at Yale University, a member of the National Academy of Sciences, the 49th president of the American Mathematical Society (1987–1988), and a trustee of the Institute for Advanced Study from 1982 to 1992. The rigidity phenomenon for lattices in Lie groups he discovered and explored is known as Mostow rigidity. His work on rigidity played an essential role in the work of three Fields medalists, namely Grigori Margulis, William Thurston, and Grigori Perelman. In 1993 he was awarded the American Mathematical Society's Leroy P. Steele Prize for Seminal Contribution to Research. In 2013, he was awarded the Wolf Prize in Mathematics "for his fundamental and pioneering contribution to geometry and Lie group theory." Biography George (Dan) Mostow was born in 1923 in Boston, Massachusetts. His parents were Jews from Ukraine who immigrated to the United States in the early 20th century. He received his Ph.D. from Harvard University in 1948, with a thesis written under the supervision of Garrett Birkhoff. His academic appointments had been at Syracuse University from 1949 to 1952, at Johns Hopkins University from 1952 to 1961, and at Yale University from 1961 until his retirement in 1999. Mostow was elected to the National Academy of Sciences in 1974, served as the President of the American Mathematical Society in 1987 and 1988, and was a Trustee of the Institute for Advanced Study in Princeton, New Jersey from 1982 to 1992. He was awarded the AMS Leroy P. Steele Prize for Seminal Contribution to Research in 1993 for his book Strong rigidity of locally symmetric spaces (1973). He died on April 4, 2017. See also Strong rigidity Superrigidity Hochschild–Mostow group References Science 20 October 1978: Vol. 202. no. 4365, pp. 297–298. Pierre Deligne and Daniel Mostow, Commensurabilities among lattices in PU(1,n). Annals of Mathematics Studies, 132. Princeton University Press, 1993 Roger Howe, editor, Discrete groups in geometry and analysis. Papers in Honor of G. D. Mostow on His Sixtieth Birthday (Conference held at Yale University, New Haven, CT, USA, March 23–25, 1986), Progress in Mathematics, Vol. 67. Birkhäuser, Boston–Basel–Stuttgart George Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, no. 78, Princeton University Press, Princeton, 1973 Alexander Lubotzky, Tannaka duality for discrete groups. American Journal of Mathematics Vol. 102, pp. 663 – 689, 1980 External links AMS Presidents: A Timeline at the American Mathematical Society website. 1923 births 2017 deaths 20th-century American mathematicians 21st-century American mathematicians American people of Ukrainian-Jewish descent Members of the United States National Academy of Sciences Presidents of the American Mathematical Society Wolf Prize in Mathem
https://en.wikipedia.org/wiki/Lis%20%28linear%20algebra%20library%29	Lis (Library of Iterative Solvers for linear systems, pronounced [lis]) is a scalable parallel software library for solving discretized linear equations and eigenvalue problems that mainly arise in the numerical solution of partial differential equations by using iterative methods. Although it is designed for parallel computers, the library can be used without being conscious of parallel processing. Features Lis provides facilities for: Automatic program configuration NUMA aware hybrid implementation with MPI and OpenMP Exchangeable dense and sparse matrix storage formats Basic linear algebra operations for dense and sparse matrices Parallel iterative methods for linear equations and eigenvalue problems Parallel preconditioners for iterative methods Quadruple precision floating point operations Performance analysis Command-line interface to solvers and benchmarks Example A C program to solve the linear equation is written as follows: #include <stdio.h> #include "lis_config.h" #include "lis.h" LIS_INT main(LIS_INT argc, char* argv[]) { LIS_MATRIX A; LIS_VECTOR b, x; LIS_SOLVER solver; LIS_INT iter; double time; lis_initialize(&argc, &argv); lis_matrix_create(LIS_COMM_WORLD, &A); lis_vector_create(LIS_COMM_WORLD, &b); lis_vector_create(LIS_COMM_WORLD, &x); lis_input_matrix(A, argv[1]); lis_input_vector(b, argv[2]); lis_vector_duplicate(A, &x); lis_solver_create(&solver); lis_solver_set_optionC(solver); lis_solve(A, b, x, solver); lis_solver_get_iter(solver, &iter); lis_solver_get_time(solver, &time); printf("number of iterations = %d\n", iter); printf("elapsed time = %e\n", time); lis_output_vector(x, LIS_FMT_MM, argv[3]); lis_solver_destroy(solver); lis_matrix_destroy(A); lis_vector_destroy(b); lis_vector_destroy(x); lis_finalize(); return 0; } System requirements The installation of Lis requires a C compiler. The Fortran interface requires a Fortran compiler, and the algebraic multigrid preconditioner requires a Fortran 90 compiler. For parallel computing environments, an OpenMP or MPI library is required. Both the Matrix Market and Harwell-Boeing formats are supported to import and export user data. Packages that use Lis Gerris OpenModelica OpenGeoSys SICOPOLIS STOMP Diablo Kiva Notus Solis GeMA openCFS numgeo freeCappuccino Andromeda Yelmo See also List of numerical libraries Conjugate gradient method Biconjugate gradient stabilized method (BiCGSTAB) Generalized minimal residual method (GMRES) Eigenvalue algorithm Lanczos algorithm Arnoldi iteration Krylov subspace Multigrid method References External links Development repository on GitHub Prof. Jack Dongarra's freely available linear algebra software page Netlib repository (Courtesy of Netlib Project) Fedora packages (Courtesy of Fedora Project) Gentoo packages (Courtesy of Gentoo Linux Project) AUR packages (Courtesy of Arch Linux Community) FreeBSD packages (Co
https://en.wikipedia.org/wiki/Homological%20dimension	Homological dimension may refer to the global dimension of a ring. It may also refer to any other concept of dimension that is defined in terms of homological algebra, which includes: Projective dimension of a module, based on projective resolutions Injective dimension of a module, based on injective resolutions Weak dimension of a module, or flat dimension, based on flat resolutions Weak global dimension of a ring, based on the weak dimension of its modules Cohomological dimension of a group Homological algebra
https://en.wikipedia.org/wiki/Homothetic%20center	In geometry, a homothetic center (also called a center of similarity or a center of similitude) is a point from which at least two geometrically similar figures can be seen as a dilation or contraction of one another. If the center is external, the two figures are directly similar to one another; their angles have the same rotational sense. If the center is internal, the two figures are scaled mirror images of one another; their angles have the opposite sense. General polygons If two geometric figures possess a homothetic center, they are similar to one another; in other words they must have the same angles at corresponding points and differ only in their relative scaling. The homothetic center and the two figures need not lie in the same plane; they can be related by a projection from the homothetic center. Homothetic centers may be external or internal. If the center is internal, the two geometric figures are scaled mirror images of one another; in technical language, they have opposite chirality. A clockwise angle in one figure would correspond to a counterclockwise angle in the other. Conversely, if the center is external, the two figures are directly similar to one another; their angles have the same sense. Circles Circles are geometrically similar to one another and mirror symmetric. Hence, a pair of circles has both types of homothetic centers, internal and external, unless the centers are equal or the radii are equal; these exceptional cases are treated after general position. These two homothetic centers lie on the line joining the centers of the two given circles, which is called the line of centers (Figure 3). Circles with radius zero can also be included (see exceptional cases), and negative radius can also be used, switching external and internal. Computing homothetic centers For a given pair of circles, the internal and external homothetic centers may be found in various ways. In analytic geometry, the internal homothetic center is the weighted average of the centers of the circles, weighted by the opposite circle's radius – distance from center of circle to inner center is proportional to that radius, so weighting is proportional to the opposite radius. Denoting the centers of the circles by and their radii by and denoting the center by , this is: The external center can be computed by the same equation, but considering one of the radii as negative; either one yields the same equation, which is: More generally, taking both radii with the same sign (both positive or both negative) yields the inner center, while taking the radii with opposite signs (one positive and the other negative) yields the outer center. Note that the equation for the inner center is valid for any values (unless both radii zero or one is the negative of the other), but the equation for the external center requires that the radii be different, otherwise it involves division by zero. In synthetic geometry, two parallel diameters are drawn, one
https://en.wikipedia.org/wiki/Demographics%20of%20the%20Swiss%20Federal%20Council	The tables below show information and statistics about the members of the Swiss Federal Council (in German: Bundesrat, in French: conseil fédéral, in Italian: consiglio federale), or Federal Councilors (in German: Bundesräte, in French: conseillers fédéraux, in Italian: consiglieri federali). The Swiss Federal Council (, , , ) is the seven-member executive council which constitutes the government as well as the head of state of Switzerland. Each of the seven Federal Councillors heads a department of the Swiss federal government. The members of the Federal Council are elected for a term of four years by both chambers of the federal parliament sitting together as the Federal Assembly. Each Councillor is elected individually by secret ballot by an absolute majority of votes. Since 1848, the seven Councillors have never been replaced simultaneously, thus guaranteeing a continuity of the government. Once elected for a four-year-term, Federal Councillors can neither be voted out of office by a motion of no confidence nor can they be impeached. Reelection is possible for an indefinite number of terms, and it has historically been extremely rare for Parliament not to reelect a sitting Councillor and this has only happened four times. In practice, therefore, Councillors serve until they decide to resign and retire to private life, usually after three to five terms of office. Parties Time in office The following tables do not include councilors currently in office. Age (oldest and youngest) Lifespan References The Swiss Confederation: A brief guide 2006, edited by the Swiss Federal Chancellery. , compiled by the services of the Swiss Parliament. Clive H. Church (2004). The Politics and Government of Switzerland. Palgrave Macmillan. . External links Chronological index of Federal Councillors, on the official website of the Swiss Federal Council. List Swiss Federal Council Swiss Federal Council
https://en.wikipedia.org/wiki/Gr%C3%A9gory%20Leca	Grégory Leca (born 22 August 1980) is a French former professional footballer who played for FC Metz and Stade Malherbe Caen. He featured as a midfielder and as a defender. Career statistics Club References External links 1980 births Living people Men's association football midfielders Footballers from Metz French men's footballers FC Metz players Stade Malherbe Caen players Ligue 1 players Ligue 2 players Championnat National players Corsica men's international footballers French people of Corsican descent
https://en.wikipedia.org/wiki/Count%20data	In statistics, count data is a statistical data type describing countable quantities, data which can take only the counting numbers, non-negative integer values {0, 1, 2, 3, ...}, and where these integers arise from counting rather than ranking. The statistical treatment of count data is distinct from that of binary data, in which the observations can take only two values, usually represented by 0 and 1, and from ordinal data, which may also consist of integers but where the individual values fall on an arbitrary scale and only the relative ranking is important. Count variables An individual piece of count data is often termed a count variable. When such a variable is treated as a random variable, the Poisson, binomial and negative binomial distributions are commonly used to represent its distribution. Graphical examination Graphical examination of count data may be aided by the use of data transformations chosen to have the property of stabilising the sample variance. In particular, the square root transformation might be used when data can be approximated by a Poisson distribution (although other transformation have modestly improved properties), while an inverse sine transformation is available when a binomial distribution is preferred. Relating count data to other variables Here the count variable would be treated as a dependent variable. Statistical methods such as least squares and analysis of variance are designed to deal with continuous dependent variables. These can be adapted to deal with count data by using data transformations such as the square root transformation, but such methods have several drawbacks; they are approximate at best and estimate parameters that are often hard to interpret. The Poisson distribution can form the basis for some analyses of count data and in this case Poisson regression may be used. This is a special case of the class of generalized linear models which also contains specific forms of model capable of using the binomial distribution (binomial regression, logistic regression) or the negative binomial distribution where the assumptions of the Poisson model are violated, in particular when the range of count values is limited or when overdispersion is present. See also Index of dispersion Empirical distribution function Frequency distribution Further reading Statistical data types Countable quantities Units of amount
https://en.wikipedia.org/wiki/Stable%20roommates%20problem	In mathematics, economics and computer science, particularly in the fields of combinatorics, game theory and algorithms, the stable-roommate problem (SRP) is the problem of finding a stable matching for an even-sized set. A matching is a separation of the set into disjoint pairs ("roommates"). The matching is stable if there are no two elements which are not roommates and which both prefer each other to their roommate under the matching. This is distinct from the stable-marriage problem in that the stable-roommates problem allows matches between any two elements, not just between classes of "men" and "women". It is commonly stated as: In a given instance of the stable-roommates problem (SRP), each of 2n participants ranks the others in strict order of preference. A matching is a set of n disjoint pairs of participants. A matching M in an instance of SRP is stable if there are no two participants x and y, each of whom prefers the other to their partner in M. Such a pair is said to block M, or to be a blocking pair with respect to M. Solution Unlike the stable marriage problem, a stable matching may fail to exist for certain sets of participants and their preferences. For a minimal example of a stable pairing not existing, consider 4 people , , , and , whose rankings are: A:(B,C,D), B:(C,A,D), C:(A,B,D), D:(A,B,C) In this ranking, each of A, B, and C is the most preferable person for someone. In any solution, one of A, B, or C must be paired with D and the other two with each other (for example AD and BC), yet for anyone who is partnered with D, another member will have rated them highest, and D's partner will in turn prefer this other member over D. In this example, AC is a more favorable pairing than AD, but the necessary remaining pairing of BD then raises the same issue, illustrating the absence of a stable matching for these participants and their preferences. Algorithm An efficient algorithm was given in . The algorithm will determine, for any instance of the problem, whether a stable matching exists, and if so, will find such a matching. Irving's algorithm has O(n2) complexity, provided suitable data structures are used to implement the necessary manipulation of the preference lists and identification of rotations. The algorithm consists of two phases. In Phase 1, participants propose to each other, in a manner similar to that of the Gale-Shapley algorithm for the stable marriage problem. Each participant orders the other members by preference, resulting in a preference list—an ordered set of the other participants. Participants then propose to each person on their list, in order, continuing to the next person if and when their current proposal is rejected. A participant will reject a proposal if they already hold a proposal from someone they prefer. A participant will also reject a previously-accepted proposal if they later receive a proposal that they prefer. In this case, the rejected participant will then propose to the next per
https://en.wikipedia.org/wiki/Evolutionary%20acquisition%20of%20neural%20topologies	Evolutionary acquisition of neural topologies (EANT/EANT2) is an evolutionary reinforcement learning method that evolves both the topology and weights of artificial neural networks. It is closely related to the works of Angeline et al. and Stanley and Miikkulainen. Like the work of Angeline et al., the method uses a type of parametric mutation that comes from evolution strategies and evolutionary programming (now using the most advanced form of the evolution strategies CMA-ES in EANT2), in which adaptive step sizes are used for optimizing the weights of the neural networks. Similar to the work of Stanley (NEAT), the method starts with minimal structures which gain complexity along the evolution path. Contribution of EANT to neuroevolution Despite sharing these two properties, the method has the following important features which distinguish it from previous works in neuroevolution. It introduces a genetic encoding called common genetic encoding (CGE) that handles both direct and indirect encoding of neural networks within the same theoretical framework. The encoding has important properties that makes it suitable for evolving neural networks: It is complete in that it is able to represent all types of valid phenotype networks. It is closed, i.e. every valid genotype represents a valid phenotype. (Similarly, the encoding is closed under genetic operators such as structural mutation and crossover.) These properties have been formally proven. For evolving the structure and weights of neural networks, an evolutionary process is used, where the exploration of structures is executed at a larger timescale (structural exploration), and the exploitation of existing structures is done at a smaller timescale (structural exploitation). In the structural exploration phase, new neural structures are developed by gradually adding new structures to an initially minimal network that is used as a starting point. In the structural exploitation phase, the weights of the currently available structures are optimized using an evolution strategy. Performance EANT has been tested on some benchmark problems such as the double-pole balancing problem, and the RoboCup keepaway benchmark. In all the tests, EANT was found to perform very well. Moreover, a newer version of EANT, called EANT2, was tested on a visual servoing task and found to outperform NEAT and the traditional iterative Gauss–Newton method. Further experiments include results on a classification problem References External links BEACON Blog: What is neuroevolution? Artificial neural networks Evolutionary algorithms Evolutionary computation
https://en.wikipedia.org/wiki/List%20of%20Birmingham%20City%20F.C.%20records%20and%20statistics	Birmingham City Football Club is a professional association football club based in the city of Birmingham, England. Founded in 1875 as Small Heath Alliance, the club turned professional in 1885 and three years later, under the name of Small Heath F.C. Ltd, was the first football club to become a limited company with a board of directors. They were later known as Birmingham before adopting their current name in 1943. Elected to the newly formed Second Division of the Football League in 1892, they have never dropped below the third tier of English football. They were also pioneers of European football competition, taking part in the inaugural season of the Inter-Cities Fairs Cup. The list encompasses the major honours won by Birmingham City, records set by the club, their managers and their players, and details of their performance in European competition. The player records section itemises the club's leading goalscorers and those who have made most appearances in first-team competitions. It also records notable achievements by Birmingham players on the international stage, and the highest transfer fees paid and received by the club. Attendance records at St Andrew's, the club's home ground since 1906, are also included. All figures are correct as of 25 July 2020. Honours Birmingham's first ever silverware was the Walsall Cup which they won in 1883. Their first honour in national competitive football was the inaugural championship of the Football League Second Division in 1892–93. The majority of their success came in the period from the mid-1950s to the early 1960s. Promoted to the First Division in 1955, in the following season they achieved their highest league finish of sixth place and their second FA Cup final appearance. They went on to reach two successive finals of the Inter-Cities Fairs Cup, and won their only major trophy, the League Cup, for the first time in 1963, a success not repeated until 2011. In the 1994–95 season they completed the "lower-division double", of the Division Two (level 3) title and the Football League Trophy, a cup competition open to teams from the third and fourth tiers of English football; this was the first time the golden goal was used to decide the winner of a senior English cup final. Birmingham City's honours and achievements include the following: European competition Inter-Cities Fairs Cup Finalists (2): 1960, 1961 The Football League Second Division / The Championship (level 2) Champions (4): 1892–93, 1920–21, 1947–48, 1954–55 Runners up (7): 1893–94, 1900–01, 1902–03, 1971–72, 1984–85, 2006–07, 2008–09 Promotion (2): 1979–80, 2001–02 Third Division / Division Two (level 3) Champions (1): 1994–95 Runners up (1): 1991–92 Domestic cup competition FA Cup Finalists (2): 1930–31, 1955–56 League Cup Winners (2): 1962–63, 2010–11 Finalists (1): 2000–01 Football League Trophy and predecessors Winners (2): 1990–91, 1994–95 Wartime competition Football League South Champions (1): 1945–
https://en.wikipedia.org/wiki/Facundo%20Mart%C3%ADnez	Facundo Martín Martínez Montagnoli (born 2 April 1985 in Buenos Aires) is an Argentine-Uruguayan footballer playing for Universidad Católica. External links Player profile Player statistics 1985 births Living people Argentine men's footballers Uruguayan men's footballers Men's association football midfielders Rampla Juniors players Club Atlético River Plate footballers Montevideo Wanderers F.C. players C.D. Universidad Católica del Ecuador footballers Argentine Primera División players Ecuadorian Serie A players Expatriate men's footballers in Ecuador Expatriate men's footballers in Uruguay Footballers from Buenos Aires
https://en.wikipedia.org/wiki/Postnikov%20system	In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree agrees with the truncated homotopy type of the original space . Postnikov systems were introduced by, and are named after, Mikhail Postnikov. Definition A Postnikov system of a path-connected space is an inverse system of spaces with a sequence of maps compatible with the inverse system such that The map induces an isomorphism for every . for . Each map is a fibration, and so the fiber is an Eilenberg–MacLane space, . The first two conditions imply that is also a -space. More generally, if is -connected, then is a -space and all for are contractible. Note the third condition is only included optionally by some authors. Existence Postnikov systems exist on connected CW complexes, and there is a weak homotopy-equivalence between and its inverse limit, so , showing that is a CW approximation of its inverse limit. They can be constructed on a CW complex by iteratively killing off homotopy groups. If we have a map representing a homotopy class , we can take the pushout along the boundary map , killing off the homotopy class. For this process can be repeated for all , giving a space which has vanishing homotopy groups . Using the fact that can be constructed from by killing off all homotopy maps , we obtain a map . Main property One of the main properties of the Postnikov tower, which makes it so powerful to study while computing cohomology, is the fact the spaces are homotopic to a CW complex which differs from only by cells of dimension . Homotopy classification of fibrations The sequence of fibrations have homotopically defined invariants, meaning the homotopy classes of maps , give a well defined homotopy type . The homotopy class of comes from looking at the homotopy class of the classifying map for the fiber . The associated classifying map is , hence the homotopy class is classified by a homotopy class called the n-th Postnikov invariant of , since the homotopy classes of maps to Eilenberg-Maclane spaces gives cohomology with coefficients in the associated abelian group. Fiber sequence for spaces with two nontrivial homotopy groups One of the special cases of the homotopy classification is the homotopy class of spaces such that there exists a fibration giving a homotopy type with two non-trivial homotopy groups, , and . Then, from the previous discussion, the fibration map gives a cohomology class in , which can also be interpreted as a group cohomology class. This space can be considered a higher local system. Examples of Postnikov towers Postnikov tower of a K(G,n) One of the conceptually simplest cases of a Postnikov tower is that of the Eilenberg–Maclane space . This gives a tower with Postnikov tower of S2 The Postnikov tower for the sphere is a special case whose
https://en.wikipedia.org/wiki/Inverse%20mean%20curvature%20flow	In the mathematical fields of differential geometry and geometric analysis, inverse mean curvature flow (IMCF) is a geometric flow of submanifolds of a Riemannian or pseudo-Riemannian manifold. It has been used to prove a certain case of the Riemannian Penrose inequality, which is of interest in general relativity. Formally, given a pseudo-Riemannian manifold and a smooth manifold , an inverse mean curvature flow consists of an open interval and a smooth map from into such that where is the mean curvature vector of the immersion . If is Riemannian, if is closed with , and if a given smooth immersion of into has mean curvature which is nowhere zero, then there exists a unique inverse mean curvature flow whose "initial data" is . Gerhardt's convergence theorem A simple example of inverse mean curvature flow is given by a family of concentric round hyperspheres in Euclidean space. If the dimension of such a sphere is and its radius is , then its mean curvature is . As such, such a family of concentric spheres forms an inverse mean curvature flow if and only if So a family of concentric round hyperspheres forms an inverse mean curvature flow when the radii grow exponentially. In 1990, Claus Gerhardt showed that this situation is characteristic of the more general case of mean-convex star-shaped smooth hypersurfaces of Euclidean space. In particular, for any such initial data, the inverse mean curvature flow exists for all positive time and consists only of mean-convex and star-shaped smooth hypersurfaces. Moreover the surface area grows exponentially, and after a rescaling that fixes the surface area, the surfaces converge smoothly to a round sphere. The geometric estimates in Gerhardt's work follow from the maximum principle; the asymptotic roundness then becomes a consequence of the Krylov-Safonov theorem. In addition, Gerhardt's methods apply simultaneously to more general curvature-based hypersurface flows. As is typical of geometric flows, IMCF solutions in more general situations often have finite-time singularities, meaning that often cannot be taken to be of the form . Huisken and Ilmanen's weak solutions Following the seminal works of Yun Gang Chen, Yoshikazu Giga, and Shun'ichi Goto, and of Lawrence Evans and Joel Spruck on the mean curvature flow, Gerhard Huisken and Tom Ilmanen replaced the IMCF equation, for hypersurfaces in a Riemannian manifold , by the elliptic partial differential equation for a real-valued function on . Weak solutions of this equation can be specified by a variational principle. Huisken and Ilmanen proved that for any complete and connected smooth Riemannian manifold which is asymptotically flat or asymptotically conic, and for any precompact and open subset of whose boundary is a smooth embedded submanifold, there is a proper and locally Lipschitz function on which is a positive weak solution on the complement of and which is nonpositive on ; moreover such a function is uniquely determi
https://en.wikipedia.org/wiki/Quasi-derivative	In mathematics, the quasi-derivative is one of several generalizations of the derivative of a function between two Banach spaces. The quasi-derivative is a slightly stronger version of the Gateaux derivative, though weaker than the Fréchet derivative. Let f : A → F be a continuous function from an open set A in a Banach space E to another Banach space F. Then the quasi-derivative of f at x0 ∈ A is a linear transformation u : E → F with the following property: for every continuous function g : [0,1] → A with g(0)=x0 such that g′(0) ∈ E exists, If such a linear map u exists, then f is said to be quasi-differentiable at x0. Continuity of u need not be assumed, but it follows instead from the definition of the quasi-derivative. If f is Fréchet differentiable at x0, then by the chain rule, f is also quasi-differentiable and its quasi-derivative is equal to its Fréchet derivative at x0. The converse is true provided E is finite-dimensional. Finally, if f is quasi-differentiable, then it is Gateaux differentiable and its Gateaux derivative is equal to its quasi-derivative. References Banach spaces Generalizations of the derivative
https://en.wikipedia.org/wiki/List%20of%20International%20Mathematical%20Olympiad%20participants	The International Mathematical Olympiad (IMO) is an annual international high school mathematics competition focused primarily on pre-collegiate mathematics, and is the oldest of the international science olympiads. The awards for exceptional performance include medals for roughly the top half participants, and honorable mentions for participants who solve at least one problem perfectly. This is a list of participants who have achieved notability. This includes participants that went on to become notable mathematicians, participants who won medals at an exceptionally young age, or participants who scored highly. Exceptionally young medalists High-scoring participants The following table lists all IMO Winners who have won at least three gold medals, with corresponding years and non-gold medals received noted (P denotes a perfect score.) Notable participants A number of IMO participants have gone on to become notable mathematicians. The following IMO participants have either received a Fields Medal, an Abel Prize, a Wolf Prize or a Clay Research Award, awards which recognise groundbreaking research in mathematics; a European Mathematical Society Prize, an award which recognizes young researchers; or one of the American Mathematical Society's awards (a Blumenthal Award in Pure Mathematics, Bôcher Memorial Prize in Analysis, Cole Prize in Algebra, Cole Prize in Number Theory, Fulkerson Prize in Discrete Mathematics, Steele Prize in Mathematics, or Veblen Prize in Geometry and Topology) recognizing research in specific mathematical fields. Grigori Perelman proved the Poincaré conjecture (one of the seven Millennium Prize Problems), and Yuri Matiyasevich gave a negative solution of Hilbert's tenth problem. G denotes an IMO gold medal, S denotes a silver medal, B denotes a bronze medal, and P denotes a perfect score. IMO medalists have also gone on to become notable computer scientists. The following IMO medalists have received a Nevanlinna Prize, a Knuth Prize, or a Gödel Prize; these awards recognise research in theoretical computer science. G denotes an IMO gold medal, S denotes a silver medal, B denotes a bronze medal, and P denotes a perfect score. See also Provincial Mathematical Olympiad List of mathematics competitions List of International Mathematical Olympiads Notes References External links General information on the IMO International Mathematical Olympiad International Mathematical Olympiad participants
https://en.wikipedia.org/wiki/Varimax%20rotation	In statistics, a varimax rotation is used to simplify the expression of a particular sub-space in terms of just a few major items each. The actual coordinate system is unchanged, it is the orthogonal basis that is being rotated to align with those coordinates. The sub-space found with principal component analysis or factor analysis is expressed as a dense basis with many non-zero weights which makes it hard to interpret. Varimax is so called because it maximizes the sum of the variances of the squared loadings (squared correlations between variables and factors). Preserving orthogonality requires that it is a rotation that leaves the sub-space invariant. Intuitively, this is achieved if, (a) any given variable has a high loading on a single factor but near-zero loadings on the remaining factors and if (b) any given factor is constituted by only a few variables with very high loadings on this factor while the remaining variables have near-zero loadings on this factor. If these conditions hold, the factor loading matrix is said to have "simple structure," and varimax rotation brings the loading matrix closer to such simple structure (as much as the data allow). From the perspective of individuals measured on the variables, varimax seeks a basis that most economically represents each individual—that is, each individual can be well described by a linear combination of only a few basis functions. One way of expressing the varimax criterion formally is this: Suggested by Henry Felix Kaiser in 1958, it is a popular scheme for orthogonal rotation (where all factors remain uncorrelated with one another). Rotation in factor analysis A summary of the use of varimax rotation and of other types of factor rotation is presented in this article on factor analysis. Implementations In the R programming language the varimax method is implemented in several packages including stats (function varimax( )), or in contributed packages including GPArotation or psych. In SAS varimax rotation is available in PROC FACTOR using ROTATE = VARIMAX. See also Factor analysis Empirical orthogonal functions Q methodology Rotation matrix Notes External links Factor rotations in Factor Analyses by Herve Abdi About Varimax Properties of Principal Components http://www2.imm.dtu.dk/pubdb/views/edoc_download.php/4041/pdf/imm4041.pdf Factor analysis
https://en.wikipedia.org/wiki/Cauchy-continuous%20function	In mathematics, a Cauchy-continuous, or Cauchy-regular, function is a special kind of continuous function between metric spaces (or more general spaces). Cauchy-continuous functions have the useful property that they can always be (uniquely) extended to the Cauchy completion of their domain. Definition Let and be metric spaces, and let be a function from to Then is Cauchy-continuous if and only if, given any Cauchy sequence in the sequence is a Cauchy sequence in Properties Every uniformly continuous function is also Cauchy-continuous. Conversely, if the domain is totally bounded, then every Cauchy-continuous function is uniformly continuous. More generally, even if is not totally bounded, a function on is Cauchy-continuous if and only if it is uniformly continuous on every totally bounded subset of Every Cauchy-continuous function is continuous. Conversely, if the domain is complete, then every continuous function is Cauchy-continuous. More generally, even if is not complete, as long as is complete, then any Cauchy-continuous function from to can be extended to a continuous (and hence Cauchy-continuous) function defined on the Cauchy completion of this extension is necessarily unique. Combining these facts, if is compact, then continuous maps, Cauchy-continuous maps, and uniformly continuous maps on are all the same. Examples and non-examples Since the real line is complete, continuous functions on are Cauchy-continuous. On the subspace of rational numbers, however, matters are different. For example, define a two-valued function so that is when is less than but when is greater than (Note that is never equal to for any rational number ) This function is continuous on but not Cauchy-continuous, since it cannot be extended continuously to On the other hand, any uniformly continuous function on must be Cauchy-continuous. For a non-uniform example on let be ; this is not uniformly continuous (on all of ), but it is Cauchy-continuous. (This example works equally well on ) A Cauchy sequence in can be identified with a Cauchy-continuous function from to defined by If is complete, then this can be extended to will be the limit of the Cauchy sequence. Generalizations Cauchy continuity makes sense in situations more general than metric spaces, but then one must move from sequences to nets (or equivalently filters). The definition above applies, as long as the Cauchy sequence is replaced with an arbitrary Cauchy net. Equivalently, a function is Cauchy-continuous if and only if, given any Cauchy filter on then is a Cauchy filter base on This definition agrees with the above on metric spaces, but it also works for uniform spaces and, most generally, for Cauchy spaces. Any directed set may be made into a Cauchy space. Then given any space the Cauchy nets in indexed by are the same as the Cauchy-continuous functions from to If is complete, then the extension of the function to will giv
https://en.wikipedia.org/wiki/Chris%20Holmes%20%28mathematician%29	Christopher C. Holmes is a British statistician. He has held the position of Professor of Biostatistics in Genomics in the Nuffield Department of Clinical Medicine and the Department of Statistics at the University of Oxford since September 2014, a post that carries with it a Fellowship of St Anne's College, Oxford. Previously he was titular Professor of Biostatistics and a Fellow of Lincoln College. After working in industry he completed his doctorate in Bayesian statistics at Imperial College, London, supervised by Adrian Smith. Holmes's research interests are in spatial statistics, Bayesian non-parametrics and statistical problems in genetics. He is one of the co-founders of the Oxford-Man Institute. Holmes was awarded the 2003 Research Prize and the 2009 Guy Medal in Bronze by the Royal Statistical Society. References External links Chris Holmes' webpage, Department of Statistics, Oxford University Christopher C. Holmes, Mathematics Genealogy Project English mathematicians Living people Year of birth missing (living people) English statisticians Bayesian statisticians Statutory Professors of the University of Oxford Fellows of Lincoln College, Oxford Fellows of St Anne's College, Oxford
https://en.wikipedia.org/wiki/New%20Orleans%20Pelicans%20all-time%20roster	The following is a list of players, both past and current, who appeared at least in one game for the New Orleans Hornets/New Orleans Pelicans NBA franchise. Players Note: Statistics are correct through the end of the season. A to B \|- \|align="left"\| \|\| align="center"\|C \|\| align="left"\|Pittsburgh \|\| align="center"\|1 \|\| align="center"\| \|\| 58 \|\| 1,605 \|\| 514 \|\| 111 \|\| 438 \|\| 27.7 \|\| 8.9 \|\| 1.9 \|\| 7.6 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F/C \|\| align="left"\| Hyères-Toulon \|\| align="center"\|4 \|\| align="center"\|– \|\| 222 \|\| 3,353 \|\| 1,038 \|\| 130 \|\| 1,330 \|\| 15.1 \|\| 4.7 \|\| 0.6 \|\| 6.0 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|Fresno State \|\| align="center"\|1 \|\| align="center"\| \|\| 66 \|\| 1,360 \|\| 118 \|\| 79 \|\| 523 \|\| 20.6 \|\| 1.8 \|\| 1.2 \|\| 7.9 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|Virginia Tech \|\| align="center"\|3 \|\| align="center"\|– \|\| 143 \|\| 2,915 \|\| 392 \|\| 330 \|\| 1,414 \|\| 20.4 \|\| 2.7 \|\| 2.3 \|\| 9.9 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G/F \|\| align="left"\|Oklahoma State \|\| align="center"\|1 \|\| align="center"\| \|\| 22 \|\| 273 \|\| 47 \|\| 9 \|\| 103 \|\| 12.4 \|\| 2.1 \|\| 0.4 \|\| 4.7 \|\| align=center\| \|- \|align="left" bgcolor="#CCFFCC"\|x \|\| align="center"\|G \|\| align="left"\|Georgia Tech \|\| align="center"\|2 \|\| align="center"\|– \|\| 115 \|\| 2,144 \|\| 241 \|\| 338 \|\| 880 \|\| 18.6 \|\| 2.1 \|\| 2.9 \|\| 7.7 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F \|\| align="left"\|Wake Forest \|\| align="center"\|3 \|\| align="center"\|– \|\| 222 \|\| 5,588 \|\| 1,389 \|\| 283 \|\| 1,526 \|\| 25.2 \|\| 6.3 \|\| 1.3 \|\| 6.9 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F/C \|\| align="left"\|UNLV \|\| align="center"\|2 \|\| align="center"\|– \|\| 36 \|\| 393 \|\| 113 \|\| 14 \|\| 82 \|\| 10.9 \|\| 3.1 \|\| 0.4 \|\| 2.3 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F/C \|\| align="left"\|Blinn \|\| align="center"\|3 \|\| align="center"\|– \|\| 104 \|\| 2,034 \|\| 574 \|\| 77 \|\| 680 \|\| 19.6 \|\| 5.5 \|\| 0.7 \|\| 6.5 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|C \|\| align="left"\| FC Barcelona \|\| align="center"\|1 \|\| align="center"\| \|\| 29 \|\| 223 \|\| 50 \|\| 6 \|\| 78 \|\| 7.7 \|\| 1.7 \|\| 0.2 \|\| 2.7 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|Georgia Tech \|\| align="center"\|1 \|\| align="center"\| \|\| 23 \|\| 446 \|\| 45 \|\| 77 \|\| 139 \|\| 19.4 \|\| 2.0 \|\| 3.3 \|\| 6.0 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F \|\| align="left"\|California \|\| align="center"\|4 \|\| align="center"\|– \|\| 230 \|\| 6,981 \|\| 1,352 \|\| 242 \|\| 3,702 \|\| 30.4 \|\| 5.9 \|\| 1.1 \|\| 16.1 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F \|\| align="left"\|UCLA \|\| align="center"\|2 \|\| align="center"\|– \|\| 116 \|\| 3,950 \|\| 623 \|\| 297 \|\| 1,270 \|\| 34.1 \|\| 5.4 \|\| 2.6 \|\| 10.9 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|Fayetteville State \|\| align="center"\|2 \|\| align="center"\|– \|\| 93 \|\| 2,659 \|\| 274 \|\| 376 \|\| 982 \|\| 28.6 \|\| 2.9 \|\| 4.0 \|\| 10.6 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F/C \|\| align="left"\|UConn \|\| align="center"\|4 \|\| align=
https://en.wikipedia.org/wiki/Los%20Angeles%20Lakers%20accomplishments%20and%20records	This page details the all-time statistics, records, and other achievements pertaining to the Los Angeles Lakers. The Los Angeles Lakers are an American professional basketball team currently playing in the National Basketball Association. Laker individual accomplishments NBA MVP Kareem Abdul-Jabbar – 1976, 1977, 1980 Magic Johnson – 1987, 1989, 1990 Shaquille O'Neal – 2000 Kobe Bryant – 2008 NBA Finals MVP Jerry West – 1969 Wilt Chamberlain – 1972 Magic Johnson – 1980, 1982, 1987 Kareem Abdul-Jabbar – 1985 James Worthy – 1988 Shaquille O'Neal – 2000, 2001, 2002 Kobe Bryant – 2009, 2010 LeBron James – 2020 NBA Defensive Player of the Year Michael Cooper – 1987 NBA Coach of the Year Bill Sharman – 1972 Pat Riley – 1990 Del Harris – 1995 NBA Sixth Man of the Year Lamar Odom – 2011 NBA Executive of the Year Jerry West – 1995 Rookie of the Year Elgin Baylor – 1959 NBA All-Rookie First Team Bill Hewitt – 1969 Dick Garrett – 1970 Jim Price – 1973 Brian Winters – 1975 Norm Nixon – 1978 Magic Johnson – 1980 James Worthy – 1983 Byron Scott – 1984 Vlade Divac – 1990 Eddie Jones – 1995 Jordan Clarkson – 2015 Kyle Kuzma – 2018 NBA All-Rookie Second Team Nick Van Exel – 1994 Kobe Bryant – 1997 Travis Knight – 1997 D'Angelo Russell – 2016 Brandon Ingram – 2017 Lonzo Ball – 2018 J. Walter Kennedy Citizenship Award Michael Cooper – 1986 Magic Johnson – 1992 Ron Artest – 2011 Pau Gasol – 2012 NBA Community Assist Award Pau Gasol – 2012 NBA scoring champion George Mikan – 1949–1951 Jerry West – 1970 Shaquille O'Neal – 2000 Kobe Bryant – 2006, 2007 NBA assist leaders Jerry West – 1972 Magic Johnson – 1983, 1984, 1986, 1987 LeBron James – 2020 All-NBA First Team George Mikan – 1949–1954 Jim Pollard – 1949, 1950 Elgin Baylor – 1959–1965, 1967–1969 Jerry West – 1962–1967, 1970–1973 Gail Goodrich – 1974 Kareem Abdul-Jabbar – 1976, 1977, 1980, 1981, 1984, 1986 Magic Johnson – 1983–1991 Shaquille O'Neal – 1998, 2000–2004 Kobe Bryant – 2002–2004, 2006–2013 Anthony Davis – 2020 LeBron James – 2020 All-NBA Second Team Vern Mikkelsen – 1951–1953, 1955 Jim Pollard – 1952, 1954 Slater Martin – 1955, 1956 Clyde Lovellette – 1956 Dick Garmaker – 1957 Jerry West – 1968, 1969 Wilt Chamberlain – 1972 Kareem Abdul-Jabbar – 1978, 1979, 1983, 1985 Magic Johnson – 1982 Shaquille O'Neal – 1999 Kobe Bryant – 2000, 2001 Pau Gasol – 2011 Andrew Bynum – 2012 LeBron James – 2021 All-NBA Third Team James Worthy – 1990, 1991 Shaquille O'Neal – 1997 Kobe Bryant – 1999, 2005 Pau Gasol – 2009, 2010 Dwight Howard – 2013 LeBron James – 2019, 2022, 2023 NBA All-Defensive First Team Jerry West – 1970–1973 Wilt Chamberlain – 1972, 1973 Kareem Abdul-Jabbar – 1979–1981 Michael Cooper – 1982, 1984, 1985, 1987, 1988 Kobe Bryant – 2000, 2003, 2004, 2006–2011 Anthony Davis – 2020 NBA All-Defensive Second Team Jerry West – 1969 Kareem Abdul-Jabbar – 1976–1978, 1984 Michael Cooper – 1981, 1983, 1986 A. C. Green – 1989 Eddie Jones – 1998 Shaquille O'Neal – 2000, 2001, 2003 Kobe Bryant – 2001,
https://en.wikipedia.org/wiki/L%C3%AA%20V%C4%83n%20Thi%C3%AAm	Lê Văn Thiêm (29 March 1918 – 3 July 1991) was a Vietnamese scientist. Together with Hoàng Tụy, he is considered the father of Vietnam Mathematics society. He was the first director of the Vietnam Institute of Mathematics, and the first Headmaster of Hanoi National University of Education and Hanoi University of Science. Biography Lê Văn Thiêm was born in 1918 at Trung Lễ Commune, Đức Thọ District, Hà Tĩnh Province, to an intellectual family. He was the youngest of 13 brothers and sisters. After the death of his parents in 1930, he moved to live with his older brother in Quy Nhơn and attended the Collège de Quy Nhơn, where Thiêm stood out in science and mathematics. Within four years, he had completed the 9-year education (equivalent to K-12 system of the US) and went to University of Indochina for his higher education. Because of the humble scale of the university at that moment, no Math course was offered. Therefore, he enrolled in the PCB (Physics-Chemistry-Biology) class. In 1939, after passing the final term examination excellently, Thiêm was offered a scholarship to study at École Normale Supérieure. His education was interrupted by the outbreak of the Second World War, and only continues in 1941. He graduated with bachelor's degree of Mathematics within a year, rather than the conventional 3-year time. Under the direction of Professor Georges Valiron, he defended his Ph.D. dissertation successfully in Germany at 1945 and then moved to the University of Zurich where he met and worked with Rolf Herman Nevanlinna for some years.His contributions in Paris and Zurich placed him among the best researchers in mathematics in the 1940s. In 1949, following the call of Hồ Chí Minh, Thiêm returned to Vietnam to support the war of decolonizing Vietnam. Moving through many positions in North Vietnam's science institutions and government, in the mid-1950s he was appointed headmaster of the National University of Advanced Education and School of Basic Science (now the Hanoi National University of Education and the Hanoi University of Science, Vietnam National University, Hanoi). In the 1960s, he was one of the scientists who suggested opening two national high schools (Hanoi National University of Education High school and Hanoi University of Science) to foster Vietnamese mathematics talents. In 1970 he became the first director of the Vietnam Institute of Mathematics. Later, he founded and was first editor-in-chief of two of Vietnam's Mathematical Journals: Acta Mathematica Vietnamica (in Latin) and Vietnam Journal of Mathematics (in English). He was the host of Neal Koblitz in his lectures in Vietnam. Lê Văn Thiêm died in 1991 in Hồ Chí Minh City, aged 73. Public recognition Awards: The 3rd degree Nation Liberation Decoration The 2nd degree Labor Decoration The 1st degree Independence Decoration Hồ Chí Minh Award A scholarship for Young Vietnamese Mathematics Talents was named after him. He was the first Modern Mathematician of Vietnam to h
https://en.wikipedia.org/wiki/Lynn%20Schneemeyer	Lynn F. Schneemeyer (born c. 1952) is a former professor of Chemistry and Biochemistry, and former Associate Dean for Academic Affairs, College of Science and Mathematics at Montclair State University. Prior to that, Dr. Schneemeyer served as Vice Provost for Research and Graduate Education at Rutgers-Newark, and as National Science Foundation Program Officer for the Chemistry Division from 2002 to 2005. Dr. Schneemeyer's publications have appeared in numerous academic journals, including Solid State Sciences, Journal of Solid State Chemistry, Journal of Materials Research, and Nature. Awards include being named the 2003-2004 Sylvia M. Stoesser Lecturer in Chemistry. Research Lynn Schneemeyer's research interests cover a broad range of materials including electronic, optical, superconducting, chemical, and magnetic materials. The focus of Dr. Schneemeyer's research has been on the design, synthesis and characterization of new materials with unique characteristics and applications potential. Education College of Notre Dame of Maryland, B.A., 1973 Cornell University, M.S., 1976 Cornell University, Ph.D., 1978 Massachusetts Institute of Technology, 1978–1980 References External links Webpage at Montclair State University Cornell University alumni Fellows of the American Physical Society Notre Dame of Maryland University alumni Rutgers University faculty Living people Year of birth missing (living people)
https://en.wikipedia.org/wiki/Negacyclic%20convolution	In mathematics, negacyclic convolution is a convolution between two vectors a and b. It is also called skew circular convolution or wrapped convolution. It results from multiplication of a skew circulant matrix, generated by vector a, with vector b. See also Circular convolution theorem Bilinear maps Functional analysis Image processing
https://en.wikipedia.org/wiki/Simmie%2C%20Saskatchewan	Simmie is a hamlet in Bone Creek Rural Municipality No. 108, Saskatchewan, Canada. Listed as a designated place by Statistics Canada, the hamlet had a population of 41 in the Canada 2016 Census. The hamlet is located on Highway 343 and Highway 631, about 50 km southwest of Swift Current. Demographics In the 2021 Census of Population conducted by Statistics Canada, Simmie had a population of 25 living in 12 of its 18 total private dwellings, a change of from its 2016 population of 41. With a land area of , it had a population density of in 2021. See also List of communities in Saskatchewan Hamlets of Saskatchewan References Bone Creek No. 108, Saskatchewan Designated places in Saskatchewan Hamlets in Saskatchewan Division No. 4, Saskatchewan
https://en.wikipedia.org/wiki/Ren%C3%A9%20Borjas	René Borjas (23 December 1897 – 16 December 1931) was a Uruguayan footballer who played as a forward. He was member of Uruguay national team which won gold medal at 1928 Olympics. Career statistics International References External links René Borjas at databaseOlympics.com René Borjas at mwfc.com.uy 1897 births 1931 deaths Uruguayan men's footballers Footballers at the 1928 Summer Olympics Olympic footballers for Uruguay Olympic gold medalists for Uruguay Uruguay men's international footballers Uruguayan Primera División players Montevideo Wanderers F.C. players Club Nacional de Football players Olympic medalists in football Copa América-winning players Medalists at the 1928 Summer Olympics Men's association football forwards People from Minas, Uruguay
https://en.wikipedia.org/wiki/Church%27s%20thesis%20%28constructive%20mathematics%29	In constructive mathematics, Church's thesis is an axiom stating that all total functions are computable functions. The similarly named Church–Turing thesis states that every effectively calculable function is a computable function, thus collapsing the former notion into the latter. is stronger in the sense that with it every function is computable. The constructivist principle is fully formalizable, using formalizations of "function" and "computable" that depend on the theory considered. A common context is recursion theory as established since the 1930's. Adopting as a principle, then for a predicate of the form of a family of existence claims (e.g. below) that is proven not to be validated for all in a computable manner, the contrapositive of the axiom implies that this is then not validated by any total function (i.e. no mapping corresponding to ). It thus collapses the possible scope of the notion of function compared to the underlying theory, restricting it to the defined notion of computable function. The axiom in turn affects ones proof calculus, negating some common classical propositions. The axiom is incompatible with systems that validate total functional value associations and evaluations that are also proven not to be computable. For example, Peano arithmetic is such a system. Concretely, the constructive Heyting arithmetic with the thesis in its first-order formulation, as an additional axiom concerning associations between natural numbers is able to disprove some universally closed variants of instances of the principle of the excluded middle. It is in this way that the axiom is shown incompatible with . However, is equiconsistent with both as well as with the theory given by plus . That is, adding either the law of the excluded middle or Church's thesis does not make Heyting arithmetic inconsistent, but adding both does. Formal statement This principle has formalizations in various mathematical frameworks. Let denote Kleene's T predicate, so that e.g. validity of the predicate expresses that is the index of a total computable function. Note that there are also variations on and the value extracting , as functions with return values. Here they are expressed as primitive recursive predicates. Write to abbreviate , as the values plays a role in the principle's formulations. So the computable function with index terminates on with value iff . This -predicate of on triples may be expressed by , at the cost of introducing abbreviating notation involving the sign already used for arithmetic equality. In first-order theories such as , which cannot quantify over relations and function directly, may be stated as an axiom schema saying that for any definable total relation, which comprises a family of valid existence claims , the latter are computable in the sense of . For each formula of two variables, the schema includes the axiom In words: If for every there is a satisfying , then there is in fact an that
https://en.wikipedia.org/wiki/Z-group	In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct types of groups: in the study of finite groups, a Z-group is a finite group whose Sylow subgroups are all cyclic. in the study of infinite groups, a Z-group is a group which possesses a very general form of central series. in the study of ordered groups, a Z-group or -group is a discretely ordered abelian group whose quotient over its minimal convex subgroup is divisible. Such groups are elementarily equivalent to the integers . Z-groups are an alternative presentation of Presburger arithmetic. occasionally, (Z)-group is used to mean a Zassenhaus group, a special type of permutation group. Groups whose Sylow subgroups are cyclic Usage: , , , , In the study of finite groups, a Z-group is a finite group whose Sylow subgroups are all cyclic. The Z originates both from the German Zyklische and from their classification in . In many standard textbooks these groups have no special name, other than metacyclic groups, but that term is often used more generally today. See metacyclic group for more on the general, modern definition which includes non-cyclic p-groups; see for the stricter, classical definition more closely related to Z-groups. Every group whose Sylow subgroups are cyclic is itself metacyclic, so supersolvable. In fact, such a group has a cyclic derived subgroup with cyclic maximal abelian quotient. Such a group has the presentation : , where mn is the order of G(m,n,r), the greatest common divisor, gcd((r-1)n, m) = 1, and rn ≡ 1 (mod m). The character theory of Z-groups is well understood , as they are monomial groups. The derived length of a Z-group is at most 2, so Z-groups may be insufficient for some uses. A generalization due to Hall are the A-groups, those groups with abelian Sylow subgroups. These groups behave similarly to Z-groups, but can have arbitrarily large derived length . Another generalization due to allows the Sylow 2-subgroup more flexibility, including dihedral and generalized quaternion groups. Group with a generalized central series Usage: , The definition of central series used for Z-group is somewhat technical. A series of G is a collection S of subgroups of G, linearly ordered by inclusion, such that for every g in G, the subgroups Ag = ∩ { N in S : g in N } and Bg = ∪ { N in S : g not in N } are both in S. A (generalized) central series of G is a series such that every N in S is normal in G and such that for every g in G, the quotient Ag/Bg is contained in the center of G/Bg. A Z-group is a group with such a (generalized) central series. Examples include the hypercentral groups whose transfinite upper central series form such a central series, as well as the hypocentral groups whose transfinite lower central series form such a central series . Special 2-transitive groups Usage: A (Z)-group is a group faithfully represented as a doubly transitive permutation group in
https://en.wikipedia.org/wiki/Giorgos%20Economides	Giorgos Economides (; born 10 April 1990) is a Cypriot footballer who plays as a central midfielder for Olympiakos Nicosia and the Cyprus national team. Career statistics Club International Honours APOEL Cypriot Cup (1): 2007–08 External links Giorgos Economides at apoel.fc 1990 births Living people Cypriot men's footballers Cyprus men's under-21 international footballers Cyprus men's international footballers Cypriot First Division players Digenis Akritas Morphou FC players APOEL FC players Doxa Katokopias FC players AC Omonia players Men's association football midfielders Greek Cypriot people Olympiakos Nicosia players
https://en.wikipedia.org/wiki/Central%20Agency%20for%20Public%20Mobilization%20and%20Statistics	Central Agency for Public Mobilization and Statistics (CAPMAS; ) is the official statistical agency of Egypt that collects, processes, analyzes, and disseminates statistical data and conducts the census. CAPMAS was established by a Presidential Decree 2915 in 1964 and is the official provider of data, statistics, and reports. CAPMAS functions support state planning, decision-making and policy assessment but it has been criticized for acting as an information regulator and for failing to provide access to researchers. Researchers must obtain a permit from CAPMAS prior to doing research in the country, History A Freedom of Information Act was being considered in 2013 by the Parliament of Egypt, to help ensure high level of transparency and disclosure in line with international best practices but some doubted it would pass and as of 2016, there is no such law for Egypt. The agency participated in World Statistics Day (in October 2015), with activities honoring senior statisticians, holding workshops, and with the launching of their new website. Population growth One of the more important findings, by the agency, has been the type of population growth occurring in Egypt. Since 2014, Major General Abu Bakr al-Gendy, the head of CAPMAS, said that Egypt's population has been growing, for decades, at an unsustainable rate. He said the population grows at an alarming 1 million Egyptians every six months and called this type of growth "a virus" that must be addressed. See also Census in Egypt List of national and international statistical services References External links 1964 establishments in Egypt Government agencies established in 1964 Government agencies of Egypt Egypt
https://en.wikipedia.org/wiki/Ergebnisse%20der%20Mathematik%20und%20ihrer%20Grenzgebiete	Ergebnisse der Mathematik und ihrer Grenzgebiete/A Series of Modern Surveys in Mathematics is a series of scholarly monographs published by Springer Science+Business Media. The title literally means "Results in mathematics and related areas". Most of the books were published in German or English, but there were a few in French and Italian. There have been several sequences, or Folge: the original series, neue Folge, and 3.Folge. Some of the most significant mathematical monographs of 20th century appeared in this series. Original series The series started in 1932 with publication of Knotentheorie by Kurt Reidemeister as "Band 1" (English: volume 1). There seems to have been double numeration in this sequence. Neue Folge This sequence started in 1950 with the publication of Transfinite Zahlen by Heinz Bachmann. The volumes are consecutively numbered, designated as either "Band" or "Heft". A total of 100 volumes was published, often in multiple editions, but preserving the original numbering within the series. The ISSN for this sequence is 0071-1136. As of February 2008, P. R. Halmos, P. J. Hilton, R. Remmert, and B. Szökefalvi-Nagy are listed on the series' website as the editors of the defunct 2. Folge. 3. Folge This sequence started in 1983 with the publication of Galois module structure of algebraic integers by Albrecht Fröhlich. As of February 2008, the editor-in-chief is R. Remmert. External links Series of mathematics books
https://en.wikipedia.org/wiki/Hidden%20variable	Hidden variables may refer to: Confounding, in statistics, an extraneous variable in a statistical model that correlates (directly or inversely) with both the dependent variable and the independent variable Hidden transformation, in computer science, a way to transform a generic constraint satisfaction problem into a binary one by introducing new hidden variables Hidden-variable theories, in physics, the proposition that statistical models of physical systems (such as Quantum mechanics) are inherently incomplete, and that the apparent randomness of a system depends not on collapsing wave functions, but rather due to unseen or unmeasurable (and thus "hidden") variables Local hidden-variable theory, in quantum mechanics, a hidden-variable theory in which distant events are assumed to have no instantaneous (or at least faster-than-light) effect on local events Latent variables, in statistics, variables that are inferred from other observed variables See also Hidden dependency Hidden side effect Infrequent variables
https://en.wikipedia.org/wiki/Image%20functors%20for%20sheaves	In mathematics, especially in sheaf theory—a domain applied in areas such as topology, logic and algebraic geometry—there are four image functors for sheaves that belong together in various senses. Given a continuous mapping f: X → Y of topological spaces, and the category Sh(–) of sheaves of abelian groups on a topological space. The functors in question are direct image f∗ : Sh(X) → Sh(Y) inverse image f∗ : Sh(Y) → Sh(X) direct image with compact support f! : Sh(X) → Sh(Y) exceptional inverse image Rf! : D(Sh(Y)) → D(Sh(X)). The exclamation mark is often pronounced "shriek" (slang for exclamation mark), and the maps called "f shriek" or "f lower shriek" and "f upper shriek"—see also shriek map. The exceptional inverse image is in general defined on the level of derived categories only. Similar considerations apply to étale sheaves on schemes. Adjointness The functors are adjoint to each other as depicted at the right, where, as usual, means that F is left adjoint to G (equivalently G right adjoint to F), i.e. Hom(F(A), B) ≅ Hom(A, G(B)) for any two objects A, B in the two categories being adjoint by F and G. For example, f∗ is the left adjoint of f*. By the standard reasoning with adjointness relations, there are natural unit and counit morphisms and for on Y and on X, respectively. However, these are almost never isomorphisms—see the localization example below. Verdier duality Verdier duality gives another link between them: morally speaking, it exchanges "∗" and "!", i.e. in the synopsis above it exchanges functors along the diagonals. For example the direct image is dual to the direct image with compact support. This phenomenon is studied and used in the theory of perverse sheaves. Base Change Another useful property of the image functors is base change. Given continuous maps and , which induce morphisms and , there exists a canonical isomorphism . Localization In the particular situation of a closed subspace i: Z ⊂ X and the complementary open subset j: U ⊂ X, the situation simplifies insofar that for j∗=j! and i!=i∗ and for any sheaf F on X, one gets exact sequences 0 → j!j∗ F → F → i∗i∗ F → 0 Its Verdier dual reads i∗Ri! F → F → Rj∗j∗ F → i∗Ri! F[1], a distinguished triangle in the derived category of sheaves on X. The adjointness relations read in this case and . See also Six operations References treats the topological setting treats the case of étale sheaves on schemes. See Exposé XVIII, section 3. is another reference for the étale case. Sheaf theory
https://en.wikipedia.org/wiki/M.%20K.%20Fort%20Jr.	Marion Kirkland "Kirk" Fort Jr. (1921–1964) was an American mathematician, specializing in general topology. The topological spaces called Fort space and Arens–Fort space are named after him. Fort was born in 1921 in Spartanburg, South Carolina, where he graduated with an A.B. from Wofford College in 1941. He received an M.A. in 1944 from the University of Virginia, where he also received his Ph.D. in 1948, advised by Gordon Thomas Whyburn. He was at the University of Illinois from then until 1953, when he came to the University of Georgia (UGA). He later served as head of the UGA mathematics department, 1959–1963. In 1963 Fort became the first holder of the university's David C. Barrow Chair of Mathematics. He died in 1964 during a leave of absence at the Institute for Defense Analyses in Princeton, New Jersey. References 1921 births 1964 deaths People from Spartanburg, South Carolina Topologists Wofford College alumni University of Virginia alumni University of Georgia faculty University of Illinois Urbana-Champaign faculty 20th-century American mathematicians Mathematicians from South Carolina
https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders%20distribution	The Birnbaum–Saunders distribution, also known as the fatigue life distribution, is a probability distribution used extensively in reliability applications to model failure times. There are several alternative formulations of this distribution in the literature. It is named after Z. W. Birnbaum and S. C. Saunders. Theory This distribution was developed to model failures due to cracks. A material is placed under repeated cycles of stress. The jth cycle leads to an increase in the crack by Xj amount. The sum of the Xj is assumed to be normally distributed with mean nμ and variance nσ2. The probability that the crack does not exceed a critical length ω is where Φ() is the cdf of normal distribution. If T is the number of cycles to failure then the cumulative distribution function (cdf) of T is The more usual form of this distribution is: Here α is the shape parameter and β is the scale parameter. Properties The Birnbaum–Saunders distribution is unimodal with a median of β. The mean (μ), variance (σ2), skewness (γ) and kurtosis (κ) are as follows: Given a data set that is thought to be Birnbaum–Saunders distributed the parameters' values are best estimated by maximum likelihood. If T is Birnbaum–Saunders distributed with parameters α and β then T−1 is also Birnbaum-Saunders distributed with parameters α and β−1. Transformation Let T be a Birnbaum-Saunders distributed variate with parameters α and β. A useful transformation of T is . Equivalently . X is then distributed normally with a mean of zero and a variance of α2 / 4. Probability density function The general formula for the probability density function (pdf) is where γ is the shape parameter, μ is the location parameter, β is the scale parameter, and is the probability density function of the standard normal distribution. Standard fatigue life distribution The case where μ = 0 and β = 1 is called the standard fatigue life distribution. The pdf for the standard fatigue life distribution reduces to Since the general form of probability functions can be expressed in terms of the standard distribution, all of the subsequent formulas are given for the standard form of the function. Cumulative distribution function The formula for the cumulative distribution function is where Φ is the cumulative distribution function of the standard normal distribution. Quantile function The formula for the quantile function is where Φ −1 is the quantile function of the standard normal distribution. References External links Fatigue life distribution Continuous distributions
https://en.wikipedia.org/wiki/Tukey%20lambda%20distribution	Formalized by John Tukey, the Tukey lambda distribution is a continuous, symmetric probability distribution defined in terms of its quantile function. It is typically used to identify an appropriate distribution (see the comments below) and not used in statistical models directly. The Tukey lambda distribution has a single shape parameter, λ, and as with other probability distributions, it can be transformed with a location parameter, μ, and a scale parameter, σ. Since the general form of probability distribution can be expressed in terms of the standard distribution, the subsequent formulas are given for the standard form of the function. Quantile function For the standard form of the Tukey lambda distribution, the quantile function, (i.e. the inverse function to the cumulative distribution function) and the quantile density function ( are For most values of the shape parameter, , the probability density function (PDF) and cumulative distribution function (CDF) must be computed numerically. The Tukey lambda distribution has a simple, closed form for the CDF and / or PDF only for a few exceptional values of the shape parameter, for example: { 2, 1, , 0 } (see uniform distribution [case = 1] and the logistic distribution [case = 0]). However, for any value of both the CDF and PDF can be tabulated for any number of cumulative probabilities, , using the quantile function to calculate the value , for each cumulative probability , with the probability density given by , the reciprocal of the quantile density function. As is the usual case with statistical distributions, the Tukey lambda distribution can readily be used by looking up values in a prepared table. Moments The Tukey lambda distribution is symmetric around zero, therefore the expected value of this distribution, if it exists, is equal to zero. The variance exists for and is given by the formula (except when λ = 0) More generally, the n-th order moment is finite when and is expressed in terms of the beta function Β(x,y) (except when λ = 0) : Note that due to symmetry of the density function, all moments of odd orders, if they exist, are equal to zero. L-moments Differently from the central moments, L-moments can be expressed in a closed form. The L-moment of order r>1 is given by The first six L-moments can be presented as follows: Comments The Tukey lambda distribution is actually a family of distributions that can approximate a number of common distributions. For example, {\| class="wikitable" \|- \| = −1 \| approx. Cauchy \|- \| = 0 \| exactly logistic \|- \| = 0.14 \| approx. normal \|- \| = 0.5 \| strictly concave (-shaped) \|- \| = 1 \| exactly uniform \|- \| = 2 \| exactly uniform \|} The most common use of this distribution is to generate a Tukey lambda PPCC plot of a data set. Based on the PPCC plot, an appropriate model for the data is suggested. For example, if the best-fit of the curve to the data occurs for a value of at or near 0.14, then the data could be well-m
https://en.wikipedia.org/wiki/Lattice%20%28discrete%20subgroup%29	In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of Rn, this amounts to the usual geometric notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all lattices are relatively well understood. The theory is particularly rich for lattices in semisimple Lie groups or more generally in semisimple algebraic groups over local fields. In particular there is a wealth of rigidity results in this setting, and a celebrated theorem of Grigory Margulis states that in most cases all lattices are obtained as arithmetic groups. Lattices are also well-studied in some other classes of groups, in particular groups associated to Kac–Moody algebras and automorphisms groups of regular trees (the latter are known as tree lattices). Lattices are of interest in many areas of mathematics: geometric group theory (as particularly nice examples of discrete groups), in differential geometry (through the construction of locally homogeneous manifolds), in number theory (through arithmetic groups), in ergodic theory (through the study of homogeneous flows on the quotient spaces) and in combinatorics (through the construction of expanding Cayley graphs and other combinatorial objects). Generalities on lattices Informal discussion Lattices are best thought of as discrete approximations of continuous groups (such as Lie groups). For example, it is intuitively clear that the subgroup of integer vectors "looks like" the real vector space in some sense, while both groups are essentially different: one is finitely generated and countable, while the other is not finitely generated and has the cardinality of the continuum. Rigorously defining the meaning of "approximation of a continuous group by a discrete subgroup" in the previous paragraph in order to get a notion generalising the example is a matter of what it is designed to achieve. Maybe the most obvious idea is to say that a subgroup "approximates" a larger group is that the larger group can be covered by the translates of a "small" subset by all elements in the subgroups. In a locally compact topological group there are two immediately available notions of "small": topological (a compact, or relatively compact subset) or measure-theoretical (a subset of finite Haar measure). Note that since the Haar measure is a Radon measure, so it gives finite mass to compact subsets, the second definition is more general. The definition of a lattice used in mathematics relies upon the second meaning (in particular to include such examples as ) but the first also has its own interest (such lattices are called uniform). Other notions are coarse equivalence and the stronger quasi-isometry. Uniform lattices are quasi-isometric to their ambient groups, but non-uniform ones are not even coarsely equiva
https://en.wikipedia.org/wiki/Statistics%20Austria	Statistics Austria, known locally as Statistik Austria, is the official name of Austria's Federal Statistical Office (), the country's agency for collecting and publishing official statistics related to Austria. In 2000 a bill (federal law for statistics) transformed the Österreichisches Statistisches Zentralamt (Austrian Statistical Central Office) into the Statistik Austria. Statistik Austria is an independent, not profit-seeking institution with public rights, which has the duty to fulfill services of the Bundesstatistik (=Federal Statistics), the GDI (=Gender-related Development Index) for example is calculated by the Statistik Austria. The current director generals are Gabriela Petrovic, who handles administrative matters, and Tobias Thomas. Although Statistik Austria was validated as Austria's institution for statistics research, the organization itself was already founded in 1829 as with the name 'Statistical Bureau'. In 1840 it was renamed the Direktion der Administrativen Statistik, in 1863 again the K&K Statistische Zentralkommission; in the First Austrian Republic () from 1921 to 1938 (German Invasion) it was named Bundesamt für Statistik and after the Second World War, from 1945 to 1999 it bore the name Österreichisches Statistisches Zentralamt. References External links Official website Government of Austria Austria
https://en.wikipedia.org/wiki/Superrigidity	In mathematics, in the theory of discrete groups, superrigidity is a concept designed to show how a linear representation ρ of a discrete group Γ inside an algebraic group G can, under some circumstances, be as good as a representation of G itself. That this phenomenon happens for certain broadly defined classes of lattices inside semisimple groups was the discovery of Grigory Margulis, who proved some fundamental results in this direction. There is more than one result that goes by the name of Margulis superrigidity. One simplified statement is this: take G to be a simply connected semisimple real algebraic group in GLn, such that the Lie group of its real points has real rank at least 2 and no compact factors. Suppose Γ is an irreducible lattice in G. For a local field F and ρ a linear representation of the lattice Γ of the Lie group, into GLn (F), assume the image ρ(Γ) is not relatively compact (in the topology arising from F) and such that its closure in the Zariski topology is connected. Then F is the real numbers or the complex numbers, and there is a rational representation of G giving rise to ρ by restriction. See also Mostow rigidity theorem Local rigidity Notes References Gromov, M.; Pansu, P. Rigidity of lattices: an introduction. Geometric topology: recent developments (Montecatini Terme, 1990), 39–137, Lecture Notes in Math., 1504, Springer, Berlin, 1991. doi:10.1007/BFb0094289 Gromov, Mikhail; Schoen, Richard. Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Inst. Hautes Études Sci. Publ. Math. No. 76 (1992), 165–246. Ji, Lizhen. A summary of the work of Gregory Margulis. Pure Appl. Math. Q. 4 (2008), no. 1, Special Issue: In honor of Grigory Margulis. Part 2, 1–69. [Pages 17-19] Jost, Jürgen; Yau, Shing-Tung. Applications of quasilinear PDE to algebraic geometry and arithmetic lattices. Algebraic geometry and related topics (Inchon, 1992), 169–193, Conf. Proc. Lecture Notes Algebraic Geom., I, Int. Press, Cambridge, MA, 1993. Tits, Jacques. Travaux de Margulis sur les sous-groupes discrets de groupes de Lie. Séminaire Bourbaki, 28ème année (1975/76), Exp. No. 482, pp. 174–190. Lecture Notes in Math., Vol. 567, Springer, Berlin, 1977. Discrete groups
https://en.wikipedia.org/wiki/Shifted%20log-logistic%20distribution	The shifted log-logistic distribution is a probability distribution also known as the generalized log-logistic or the three-parameter log-logistic distribution. It has also been called the generalized logistic distribution, but this conflicts with other uses of the term: see generalized logistic distribution. Definition The shifted log-logistic distribution can be obtained from the log-logistic distribution by addition of a shift parameter . Thus if has a log-logistic distribution then has a shifted log-logistic distribution. So has a shifted log-logistic distribution if has a logistic distribution. The shift parameter adds a location parameter to the scale and shape parameters of the (unshifted) log-logistic. The properties of this distribution are straightforward to derive from those of the log-logistic distribution. However, an alternative parameterisation, similar to that used for the generalized Pareto distribution and the generalized extreme value distribution, gives more interpretable parameters and also aids their estimation. In this parameterisation, the cumulative distribution function (CDF) of the shifted log-logistic distribution is for , where is the location parameter, the scale parameter and the shape parameter. Note that some references use to parameterise the shape. The probability density function (PDF) is again, for The shape parameter is often restricted to lie in [-1,1], when the probability density function is bounded. When , it has an asymptote at . Reversing the sign of reflects the pdf and the cdf about . Related distributions When the shifted log-logistic reduces to the log-logistic distribution. When → 0, the shifted log-logistic reduces to the logistic distribution. The shifted log-logistic with shape parameter is the same as the generalized Pareto distribution with shape parameter Applications The three-parameter log-logistic distribution is used in hydrology for modelling flood frequency. Alternate parameterization An alternate parameterization with simpler expressions for the PDF and CDF is as follows. For the shape parameter , scale parameter and location parameter , the PDF is given by The CDF is given by The mean is and the variance is , where . References Continuous distributions
https://en.wikipedia.org/wiki/Probability%20plot%20correlation%20coefficient%20plot	The probability plot correlation coefficient (PPCC) plot is a graphical technique for identifying the shape parameter for a distributional family that best describes the data set. This technique is appropriate for families, such as the Weibull, that are defined by a single shape parameter and location and scale parameters, and it is not appropriate or even possible for distributions, such as the normal, that are defined only by location and scale parameters. Many statistical analyses are based on distributional assumptions about the population from which the data have been obtained. However, distributional families can have radically different shapes depending on the value of the shape parameter. Therefore, finding a reasonable choice for the shape parameter is a necessary step in the analysis. In many analyses, finding a good distributional model for the data is the primary focus of the analysis. The technique is simply "plot the probability plot correlation coefficients for different values of the shape parameter, and choose whichever value yields the best fit". Definition The PPCC plot is formed by: Vertical axis: Probability plot correlation coefficient; Horizontal axis: Value of shape parameter. That is, for a series of values of the shape parameter, the correlation coefficient is computed for the probability plot associated with a given value of the shape parameter. These correlation coefficients are plotted against their corresponding shape parameters. The maximum correlation coefficient corresponds to the optimal value of the shape parameter. For better precision, two iterations of the PPCC plot can be generated; the first is for finding the right neighborhood and the second is for fine tuning the estimate. The PPCC plot is used first to find a good value of the shape parameter. The probability plot is then generated to find estimates of the location and scale parameters and in addition to provide a graphical assessment of the adequacy of the distributional fit. The PPCC plot answers the following questions: What is the best-fit member within a distributional family? Does the best-fit member provide a good fit (in terms of generating a probability plot with a high correlation coefficient)? Does this distributional family provide a good fit compared to other distributions? How sensitive is the choice of the shape parameter? Comparing distributions In addition to finding a good choice for estimating the shape parameter of a given distribution, the PPCC plot can be useful in deciding which distributional family is most appropriate. For example, given a set of reliability data, one might generate PPCC plots for a Weibull, lognormal, gamma, and inverse Gaussian distributions, and possibly others, on a single page. This one page would show the best value for the shape parameter for several distributions and would additionally indicate which of these distributional families provides the best fit (as measured by the maximum probability plo
https://en.wikipedia.org/wiki/Tangential%20angle	In geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent line to the curve at the given point and the -axis. (Some authors define the angle as the deviation from the direction of the curve at some fixed starting point. This is equivalent to the definition given here by the addition of a constant to the angle or by rotating the curve.) Equations If a curve is given parametrically by , then the tangential angle at is defined (up to a multiple of ) by Here, the prime symbol denotes the derivative with respect to . Thus, the tangential angle specifies the direction of the velocity vector , while the speed specifies its magnitude. The vector is called the unit tangent vector, so an equivalent definition is that the tangential angle at is the angle such that is the unit tangent vector at . If the curve is parametrized by arc length , so , then the definition simplifies to In this case, the curvature is given by , where is taken to be positive if the curve bends to the left and negative if the curve bends to the right. Conversely, the tangent angle at a given point equals the definite integral of curvature up to that point: If the curve is given by the graph of a function , then we may take as the parametrization, and we may assume is between and . This produces the explicit expression Polar tangential angle In polar coordinates, the polar tangential angle is defined as the angle between the tangent line to the curve at the given point and ray from the origin to the point. If denotes the polar tangential angle, then , where is as above and is, as usual, the polar angle. If the curve is defined in polar coordinates by , then the polar tangential angle at is defined (up to a multiple of ) by . If the curve is parametrized by arc length as , , so , then the definition becomes . The logarithmic spiral can be defined a curve whose polar tangential angle is constant. See also Differential geometry of curves Whewell equation Subtangent References Further reading Analytic geometry Differential geometry
https://en.wikipedia.org/wiki/Valencin	Valencin () is a commune in the Isère department in southeastern France. Population See also Communes of the Isère department References External links INSEE statistics Communes of Isère Isère communes articles needing translation from French Wikipedia
https://en.wikipedia.org/wiki/Analysis%20situs	Analysis situs may refer to: Topology, originally called analysis situs, but the term is now obsolete "Analysis Situs" (paper), an 1895 article on topology by Henri Poincaré Analysis Situs (book), a 1922 book on topology by Oswald Veblen
https://en.wikipedia.org/wiki/Grey%20box	Grey or gray box may refer to: Science and technology Gray box testing, software testing Grey box model, in mathematics, statistics, and computational modelling Grey identification method, used in system identification Botany Jojoba (Simmondsia chinensis), also known as gray box bush Eucalyptus Grey box, many trees in the genus Eucalyptus, native to Australia, including: Eucalyptus argillacea (Kimberley grey box or northern grey box) Eucalyptus bosistoana (coast grey box or Gippsland grey box) Eucalyptus brownii (grey box) Eucalyptus hemiphloia (grey box); See Eucalyptus albens Eucalyptus largeana (grey box) Eucalyptus microcarpa (grey box, inland grey box or western grey box) Eucalyptus moluccana (grey box or coastal grey box) Eucalyptus normantonensis (grey box) Eucalyptus pilligaensis (narrow-leaved grey box, pilliga grey box) Eucalyptus quadrangulata (grey box) Eucalyptus rummeryi (grey box) Eucalyptus tectifica (grey box)
https://en.wikipedia.org/wiki/Box%E2%80%93Cox%20distribution	In statistics, the Box–Cox distribution (also known as the power-normal distribution) is the distribution of a random variable X for which the Box–Cox transformation on X follows a truncated normal distribution. It is a continuous probability distribution having probability density function (pdf) given by for y > 0, where m is the location parameter of the distribution, s is the dispersion, ƒ is the family parameter, I is the indicator function, Φ is the cumulative distribution function of the standard normal distribution, and sgn is the sign function. Special cases ƒ = 1 gives a truncated normal distribution. References Continuous distributions
https://en.wikipedia.org/wiki/Tami%20Sagher	Tami Sagher is an American comedy writer, producer, and actress. Biography A native of Chicago, Sagher studied mathematics at the University of Chicago before joining Boom Chicago and then Second City. Career TV Sagher has written for the TV shows 30 Rock, Psych, MADtv and Inside Amy Schumer. She was a staff writer on the CBS sitcom How I Met Your Mother, leaving before the show's final season. Sagher then spent two seasons as writer-producer on the Netflix series Orange is the New Black. From 2020 to 2021, Sagher was a writer-executive producer on the Hulu series Shrill. National Public Radio In addition, Sagher has contributed to This American Life. Performance Her performing includes playing an improv performer in Don't Think Twice, starring in the short film The Shabbos Goy, as well as various appearances on TV sitcoms and sketch shows. In particular, she appeared in Season 5 of Curb Your Enthusiasm. Honors Sagher has been nominated for 4 Writers Guild of America Awards: Three for MADtv; One (2008 in the category of Best Comedy Series) for the third season of 30 Rock. References External links American film actresses American television actresses American television producers American women television producers American television writers Living people American women television writers Writers Guild of America Award winners Place of birth missing (living people) Year of birth missing (living people) Upright Citizens Brigade Theater performers 21st-century American women
https://en.wikipedia.org/wiki/Computational%20statistics	Computational statistics, or statistical computing, is the bond between statistics and computer science, and refers to the statistical methods that are enabled by using computational methods. It is the area of computational science (or scientific computing) specific to the mathematical science of statistics. This area is also developing rapidly, leading to calls that a broader concept of computing should be taught as part of general statistical education. As in traditional statistics the goal is to transform raw data into knowledge, but the focus lies on computer intensive statistical methods, such as cases with very large sample size and non-homogeneous data sets. The terms 'computational statistics' and 'statistical computing' are often used interchangeably, although Carlo Lauro (a former president of the International Association for Statistical Computing) proposed making a distinction, defining 'statistical computing' as "the application of computer science to statistics", and 'computational statistics' as "aiming at the design of algorithm for implementing statistical methods on computers, including the ones unthinkable before the computer age (e.g. bootstrap, simulation), as well as to cope with analytically intractable problems" [sic]. The term 'Computational statistics' may also be used to refer to computationally intensive statistical methods including resampling methods, Markov chain Monte Carlo methods, local regression, kernel density estimation, artificial neural networks and generalized additive models. History Though computational statistics is widely used today, it actually has a relatively short history of acceptance in the statistics community. For the most part, the founders of the field of statistics relied on mathematics and asymptotic approximations in the development of computational statistical methodology. In statistical field, the first use of the term “computer” comes in an article in the Journal of the American Statistical Association archives by Robert P. Porter in 1891. The article discusses about the use of Hermann Hollerith’s machine in the 11th Census of the United States. Hermann Hollerith’s machine, also called tabulating machine, was an electromechanical machine designed to assist in summarizing information stored on punched cards. It was invented by Herman Hollerith (February 29, 1860 – November 17, 1929), an American businessman, inventor, and statistician. His invention of the punched card tabulating machine was patented in 1884, and later was used in the 1890 Census of the United States. The advantages of the technology were immediately apparent. the 1880 Census, with about 50 million people, and it took over 7 years to tabulate. While in the 1890 Census, with over 62 million people, it took less than a year. This marks the beginning of the era of mechanized computational statistics and semiautomatic data processing systems. In 1908, William Sealy Gosset performed his now well-known Monte Carlo metho
https://en.wikipedia.org/wiki/Generalized%20logistic%20distribution	The term generalized logistic distribution is used as the name for several different families of probability distributions. For example, Johnson et al. list four forms, which are listed below. Type I has also been called the skew-logistic distribution. Type IV subsumes the other types and is obtained when applying the logit transform to beta random variates. Following the same convention as for the log-normal distribution, type IV may be referred to as the logistic-beta distribution, with reference to the standard logistic function, which is the inverse of the logit transform. For other families of distributions that have also been called generalized logistic distributions, see the shifted log-logistic distribution, which is a generalization of the log-logistic distribution; and the metalog ("meta-logistic") distribution, which is highly shape-and-bounds flexible and can be fit to data with linear least squares. Definitions The following definitions are for standardized versions of the families, which can be expanded to the full form as a location-scale family. Each is defined using either the cumulative distribution function (F) or the probability density function (ƒ), and is defined on (-∞,∞). Type I The corresponding probability density function is: This type has also been called the "skew-logistic" distribution. Type II The corresponding probability density function is: Type III Here B is the beta function. The moment generating function for this type is The corresponding cumulative distribution function is: Type IV Where, B is the beta function and is the standard logistic function. The moment generating function for this type is This type is also called the "exponential generalized beta of the second type". The corresponding cumulative distribution function is: Relationship between types Type IV is the most general form of the distribution. The Type III distribution can be obtained from Type IV by fixing . The Type II distribution can be obtained from Type IV by fixing (and renaming to ). The Type I distribution can be obtained from Type IV by fixing . Fixing gives the standard logistic distribution. Type IV (logistic-beta) properties The Type IV generalized logistic, or logistic-beta distribution, with support and shape parameters , has (as shown above) the probability density function (pdf): where is the standard logistic function. The probability density functions for three different sets of shape parameters are shown in the plot, where the distributions have been scaled and shifted to give zero means and unity variances, in order to facilitate comparison of the shapes. In what follows, the notation is used to denote the Type IV distribution. Relationship with Gamma Distribution This distribution can be obtained in terms of the gamma distribution as follows. Let and independently, and let . Then . Symmetry If , then . Mean and variance By using the logarithmic expectations of the gamma distribut
https://en.wikipedia.org/wiki/G%C3%B6khan%20K%C3%B6k	Gökhan Kök (born 3 January 1981, in Bulanık) is a Turkish retired football defender. External links Profile & Statistics at Guardian's Stats Centre Profile at TFF.org.tr 1981 births Living people Turkish men's footballers Men's association football defenders Göztepe S.K. footballers Sakaryaspor footballers Kayseri Erciyesspor footballers Çaykur Rizespor footballers İnegölspor footballers People from Bulanık
https://en.wikipedia.org/wiki/Amy%20Cohen-Corwin	Amy Cohen-Corwin (formerly known as Amy C. Murray) is a professor emerita of mathematics at Rutgers University, and former Dean of University College at Rutgers University. In 2006, she was named Fellow of the American Association for the Advancement of Science. Cohen-Corwin is especially interested in the Korteweg–de Vries equation, cubic Schrödinger equation on the line, and improving undergraduate education, especially for future teachers. She worked on Project SEED whilst at the University of California, Berkeley in 1970 which fueled her interest in Mathematics education. Cohen-Corwin has held numerous organizational positions, including Co-organizer for the AIM (American Institute of Mathematics) and NSF (National Science Foundation)-sponsored workshop "Finding and Keeping Graduate Students in the Mathematical Sciences." Awards Louise Hay Award for Contributions to Mathematics Education, Joint Mathematics Meeting, Association for Women in Mathematics - 2013 Fellow, American Association for the Advancement of Science, elected 2006 Fellow, Association for Women in Mathematics, 2019 Education Ph.D., 1970, Mathematics, University of California at Berkeley, under the supervision of Murray H. Protter M.S., 1966, Mathematics, University of California at Berkeley A.B., 1964, Mathematics, Harvard University (Radcliffe College) References External links Rutgers Office for the Promotion of Women in Science, Technology, and Engineering Living people Rutgers University faculty Radcliffe College alumni UC Berkeley College of Letters and Science alumni 20th-century American mathematicians 21st-century American mathematicians American women mathematicians Fellows of the American Association for the Advancement of Science Fellows of the Association for Women in Mathematics 20th-century women mathematicians 21st-century women mathematicians Year of birth missing (living people) 20th-century American women 21st-century American women
https://en.wikipedia.org/wiki/Conway%E2%80%93Maxwell%E2%80%93Poisson%20distribution	In probability theory and statistics, the Conway–Maxwell–Poisson (CMP or COM–Poisson) distribution is a discrete probability distribution named after Richard W. Conway, William L. Maxwell, and Siméon Denis Poisson that generalizes the Poisson distribution by adding a parameter to model overdispersion and underdispersion. It is a member of the exponential family, has the Poisson distribution and geometric distribution as special cases and the Bernoulli distribution as a limiting case. Background The CMP distribution was originally proposed by Conway and Maxwell in 1962 as a solution to handling queueing systems with state-dependent service rates. The CMP distribution was introduced into the statistics literature by Boatwright et al. 2003 and Shmueli et al. (2005). The first detailed investigation into the probabilistic and statistical properties of the distribution was published by Shmueli et al. (2005). Some theoretical probability results of COM-Poisson distribution is studied and reviewed by Li et al. (2019), especially the characterizations of COM-Poisson distribution. Probability mass function and basic properties The CMP distribution is defined to be the distribution with probability mass function where : The function serves as a normalization constant so the probability mass function sums to one. Note that does not have a closed form. The domain of admissible parameters is , and , . The additional parameter which does not appear in the Poisson distribution allows for adjustment of the rate of decay. This rate of decay is a non-linear decrease in ratios of successive probabilities, specifically When , the CMP distribution becomes the standard Poisson distribution and as , the distribution approaches a Bernoulli distribution with parameter . When the CMP distribution reduces to a geometric distribution with probability of success provided . For the CMP distribution, moments can be found through the recursive formula Cumulative distribution function For general , there does not exist a closed form formula for the cumulative distribution function of . If is an integer, we can, however, obtain the following formula in terms of the generalized hypergeometric function: The normalizing constant Many important summary statistics, such as moments and cumulants, of the CMP distribution can be expressed in terms of the normalizing constant . Indeed, The probability generating function is , and the mean and variance are given by The cumulant generating function is and the cumulants are given by Whilst the normalizing constant does not in general have a closed form, there are some noteworthy special cases: , where is a modified Bessel function of the first kind. For integer , the normalizing constant can expressed as a generalized hypergeometric function: . Because the normalizing constant does not in general have a closed form, the following asymptotic expansion is of interest. Fix . Then, as , where the a
https://en.wikipedia.org/wiki/A%E2%88%9E-operad	{{DISPLAYTITLE:A∞-operad}} In the theory of operads in algebra and algebraic topology, an A∞-operad is a parameter space for a multiplication map that is homotopy coherently associative. (An operad that describes a multiplication that is both homotopy coherently associative and homotopy coherently commutative is called an E∞-operad.) Definition In the (usual) setting of operads with an action of the symmetric group on topological spaces, an operad A is said to be an A∞-operad if all of its spaces A(n) are Σn-equivariantly homotopy equivalent to the discrete spaces Σn (the symmetric group) with its multiplication action (where n ∈ N). In the setting of non-Σ operads (also termed nonsymmetric operads, operads without permutation), an operad A is A∞if all of its spaces A(n) are contractible. In other categories than topological spaces, the notions of homotopy and contractibility have to be replaced by suitable analogs, such as homology equivalences in the category of chain complexes. An-operads The letter A in the terminology stands for "associative", and the infinity symbols says that associativity is required up to "all" higher homotopies. More generally, there is a weaker notion of An-operad (n ∈ N), parametrizing multiplications that are associative only up to a certain level of homotopies. In particular, A1-spaces are pointed spaces; A2-spaces are H-spaces with no associativity conditions; and A3-spaces are homotopy associative H-spaces. A∞-operads and single loop spaces A space X is the loop space of some other space, denoted by BX, if and only if X is an algebra over an -operad and the monoid π0(X) of its connected components is a group. An algebra over an -operad is referred to as an -space. There are three consequences of this characterization of loop spaces. First, a loop space is an -space. Second, a connected -space X is a loop space. Third, the group completion of a possibly disconnected -space is a loop space. The importance of -operads in homotopy theory stems from this relationship between algebras over -operads and loop spaces. A∞-algebras An algebra over the -operad is called an -algebra. Examples feature the Fukaya category of a symplectic manifold, when it can be defined (see also pseudoholomorphic curve). Examples The most obvious, if not particularly useful, example of an -operad is the associative operad a given by . This operad describes strictly associative multiplications. By definition, any other -operad has a map to a which is a homotopy equivalence. A geometric example of an A∞-operad is given by the Stasheff polytopes or associahedra. A less combinatorial example is the operad of little intervals: The space consists of all embeddings of n disjoint intervals into the unit interval. See also Homotopy associative algebra operad E-infinity operad loop space References Abstract algebra Algebraic topology
https://en.wikipedia.org/wiki/E%E2%88%9E-operad	{{DISPLAYTITLE:E∞-operad}} In the theory of operads in algebra and algebraic topology, an E∞-operad is a parameter space for a multiplication map that is associative and commutative "up to all higher homotopies". (An operad that describes a multiplication that is associative but not necessarily commutative "up to homotopy" is called an A∞-operad.) Definition For the definition, it is necessary to work in the category of operads with an action of the symmetric group. An operad A is said to be an E∞-operad if all of its spaces E(n) are contractible; some authors also require the action of the symmetric group Sn on E(n) to be free. In other categories than topological spaces, the notion of contractibility has to be replaced by suitable analogs, such as acyclicity in the category of chain complexes. En-operads and n-fold loop spaces The letter E in the terminology stands for "everything" (meaning associative and commutative), and the infinity symbols says that commutativity is required up to "all" higher homotopies. More generally, there is a weaker notion of En-operad (n ∈ N), parametrizing multiplications that are commutative only up to a certain level of homotopies. In particular, E1-spaces are A∞-spaces; E2-spaces are homotopy commutative A∞-spaces. The importance of En- and E∞-operads in topology stems from the fact that iterated loop spaces, that is, spaces of continuous maps from an n-dimensional sphere to another space X starting and ending at a fixed base point, constitute algebras over an En-operad. (One says they are En-spaces.) Conversely, any connected En-space X is an n-fold loop space on some other space (called BnX, the n-fold classifying space of X). Examples The most obvious, if not particularly useful, example of an E∞-operad is the commutative operad c given by c(n) = *, a point, for all n. Note that according to some authors, this is not really an E∞-operad because the Sn-action is not free. This operad describes strictly associative and commutative multiplications. By definition, any other E∞-operad has a map to c which is a homotopy equivalence. The operad of little n-cubes or little n-disks is an example of an En-operad that acts naturally on n-fold loop spaces. See also operad A-infinity operad loop space References Abstract algebra Algebraic topology
https://en.wikipedia.org/wiki/Adams%20filtration	In mathematics, especially in the area of algebraic topology known as stable homotopy theory, the Adams filtration and the Adams–Novikov filtration allow a stable homotopy group to be understood as built from layers, the nth layer containing just those maps which require at most n auxiliary spaces in order to be a composition of homologically trivial maps. These filtrations, named after Frank Adams and Sergei Novikov, are of particular interest because the Adams (–Novikov) spectral sequence converges to them. Definition The group of stable homotopy classes between two spectra X and Y can be given a filtration by saying that a map has filtration n if it can be written as a composite of maps such that each individual map induces the zero map in some fixed homology theory E. If E is ordinary mod-p homology, this filtration is called the Adams filtration, otherwise the Adams–Novikov filtration. Homotopy theory
https://en.wikipedia.org/wiki/List%20of%20Rugby%20World%20Cup%20try%20scorers	This article lists charts each team's try scorers from the first Rugby World Cup to date. The list does not include penalty tries. Statistics correct after New Zealand vs South Africa, 28 October 2023. 7 tries Juan José Imhoff 6 tries Julián Montoya 4 tries Pablo Bouza Felipe Contepomi Ignacio Corleto Martín Gaitán Nicolás Sánchez Joaquín Tuculet 3 tries Lucas Borges Mateo Carreras Santiago Carreras Manuel Contepomi Santiago Cordero Tomás Cubelli Juan Martín Hernández Juan Manuel Leguizamón Matías Moroni Martín Terán 2 tries Horacio Agulla Diego Albanese Federico Martín Aramburú Lisandro Arbizu Emiliano Boffelli Juan Cruz Mallia Nicolás Fernández Miranda Lucas González Amorosino Juan Martín González Juan Lanza Guido Petti Agustín Pichot 1 try Patricio Albacete Matías Alemanno Alejandro Allub Gonzalo Bertranou Martín Bogado Rodrigo Bruni Santiago Chocobares José Cilley Matías Corral Agustín Creevy Rodrigo Crexell Jeronimo de la Fuente Julio Farías Cabello Santiago Fernández Juan Fernández Miranda Genaro Fessia Juan Figallo Hernán García Simón Fabio Gómez Agustín Gosio Omar Hasan Facundo Isa Rodrigo Isgro Martín Landajo Tomás Lavanini Gonzalo Longo Rolando Martín Federico Méndez Lucas Noguera Paz Patricio Noriega Juan Pablo Orlandi Rodrigo Roncero Ignacio Ruiz Joel Sclavi Leonardo Senatore Gonzalo Tiesi Federico Todeschini 14 tries Drew Mitchell 12 tries Adam Ashley-Cooper 11 tries Chris Latham 10 tries David Campese 8 tries Matt Giteau 7 tries Joe Roff 6 tries Matt C. Burke Tim Horan Lote Tuqiri 5 tries Berrick Barnes Matt P. Burke Tevita Kuridrani Stirling Mortlock David Pocock Mat Rogers 4 tries Rocky Elsom Marika Koroibete Michael Lynagh Ben McCalman George Smith 3 tries George Gregan Dane Haylett-Petty Michael Hooper Toutai Kefu Stephen Larkham Andrew Slack Ben Tune 2 tries Ben Donaldson David Codey Anthony Fainga'a Bernard Foley Michael Foley Adam Freier Will Genia Peter Grigg Phil Kearns Tolu Latu Andrew Leeds David Lyons Sean McMahon Mark Nawaqanitawase Jordan Petaia Marty Roebuck Brian Smith Damian Smith Morgan Turinui 1 try Ben Alexander Richie Arnold Al Baxter Kurtley Beale Angus Bell Tony Daly Jack Dempsey Owen Finegan Elton Flatley Nathan Grey Mark Hartill Reece Hodge Stephen Hoiles Rob Horne James Horwill Digby Ioane Rod Kafer Sekope Kepu Samu Kerevi Jason Little Salesi Ma'afu Pat McCabe Andrew McIntyre Fraser McReight Stephen Moore Dean Mumm James O'Connor Brett Papworth Jeremy Paul Simon Poidevin Dave Porecki John Roe Wendell Sailor Radike Samo Nathan Sharpe Rob Simmons Peter Slattery James Slipper Henry Speight Tiaan Straaus Ilivasi Tabua Joe Tomane Matt To'omua Steve Tuynman Suli Vunivalu Chris Whitaker Nic White David Wilson 5 tries D. T. H. van der Merwe 4 tries Al Charron Morgan Williams 3 tries Rod Snow 2 tries Aaron Carpenter Phil Mackenzie Kyle Nichols Pat Palmer Ryan Smith Winston Stanley Conor Trainor Paul Vaesen 1 try Aaron Abrams Mark Cardinal Andrew Coe Jamie Cudmore Craig Culpan Glen Ennis Matt Evans Sean Fauth R

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.