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https://en.wikipedia.org/wiki/Multiple%20comparisons%20problem	In statistics, the multiple comparisons, multiplicity or multiple testing problem occurs when one considers a set of statistical inferences simultaneously or infers a subset of parameters selected based on the observed values. The more inferences are made, the more likely erroneous inferences become. Several statistical techniques have been developed to address that problem, typically by requiring a stricter significance threshold for individual comparisons, so as to compensate for the number of inferences being made. History The problem of multiple comparisons received increased attention in the 1950s with the work of statisticians such as Tukey and Scheffé. Over the ensuing decades, many procedures were developed to address the problem. In 1996, the first international conference on multiple comparison procedures took place in Tel Aviv. Definition Multiple comparisons arise when a statistical analysis involves multiple simultaneous statistical tests, each of which has a potential to produce a "discovery". A stated confidence level generally applies only to each test considered individually, but often it is desirable to have a confidence level for the whole family of simultaneous tests. Failure to compensate for multiple comparisons can have important real-world consequences, as illustrated by the following examples: Suppose the treatment is a new way of teaching writing to students, and the control is the standard way of teaching writing. Students in the two groups can be compared in terms of grammar, spelling, organization, content, and so on. As more attributes are compared, it becomes increasingly likely that the treatment and control groups will appear to differ on at least one attribute due to random sampling error alone. Suppose we consider the efficacy of a drug in terms of the reduction of any one of a number of disease symptoms. As more symptoms are considered, it becomes increasingly likely that the drug will appear to be an improvement over existing drugs in terms of at least one symptom. In both examples, as the number of comparisons increases, it becomes more likely that the groups being compared will appear to differ in terms of at least one attribute. Our confidence that a result will generalize to independent data should generally be weaker if it is observed as part of an analysis that involves multiple comparisons, rather than an analysis that involves only a single comparison. For example, if one test is performed at the 5% level and the corresponding null hypothesis is true, there is only a 5% risk of incorrectly rejecting the null hypothesis. However, if 100 tests are each conducted at the 5% level and all corresponding null hypotheses are true, the expected number of incorrect rejections (also known as false positives or Type I errors) is 5. If the tests are statistically independent from each other (i.e. are performed on independent samples), the probability of at least one incorrect rejection is approximate
https://en.wikipedia.org/wiki/Abhyankar%27s%20lemma	In mathematics, Abhyankar's lemma (named after Shreeram Shankar Abhyankar) allows one to kill tame ramification by taking an extension of a base field. More precisely, Abhyankar's lemma states that if A, B, C are local fields such that A and B are finite extensions of C, with ramification indices a and b, and B is tamely ramified over C and b divides a, then the compositum AB is an unramified extension of A. See also Finite extensions of local fields References . Theorem 3, page 504. . , p. 279. . Theorems in algebraic geometry Lemmas in algebra Algebraic number theory Theorems in abstract algebra
https://en.wikipedia.org/wiki/Idan%20Shum	Idan Shum (; born 26 March 1976) is an Israeli former footballer. References External links Profile and biography of Idan Shum on Maccabi Haifa's official website Profile and statistics of Idan Shum on One.co.il 1976 births Living people Israeli Jews Israeli men's footballers Footballers from Kfar Saba Hapoel Kfar Saba F.C. players Hapoel Tzafririm Holon F.C. players Maccabi Haifa F.C. players Hapoel Petah Tikva F.C. players Hapoel Rishon LeZion F.C. players Maccabi Herzliya F.C. players Maccabi Netanya F.C. players Hapoel Haifa F.C. players FC Spartak Vladikavkaz players Liga Leumit players Israeli Premier League players Expatriate men's footballers in Russia Israeli expatriate sportspeople in Russia Israel men's under-21 international footballers Israeli people of Moldovan-Jewish descent Men's association football midfielders
https://en.wikipedia.org/wiki/Pushforward	The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" operations it defines Pushforward (homology), the map induced in homology by a continuous map between topological spaces Pushforward measure, measure induced on the target measure space by a measurable function Pushout (category theory), the categorical dual of pullback Direct image sheaf, the pushforward of a sheaf by a map Fiberwise integral, the direct image of a differential form or cohomology by a smooth map, defined by "integration on the fibres" Transfer operator, the pushforward on the space of measurable functions; its adjoint, the pull-back, is the composition or Koopman operator zh:推出
https://en.wikipedia.org/wiki/Inversion	Inversion or inversions may refer to: Arts Inversion (artwork), a 2005 temporary sculpture in Houston, Texas Inversion (music), a term with various meanings in music theory and musical set theory Inversions (novel) by Iain M. Banks Inversion (video game), a 2012 third person shooter for Xbox 360, PlayStation 3, and PC Inversions (EP), the 2014 extended play album by American rock music ensemble The Colourist Inversions (album), a 2019 album by Belinda O'Hooley Inversion (film), a 2016 Iranian film Linguistics and language Inversion (linguistics), grammatical constructions where two expressions switch their order of appearance Inversion (prosody), the reversal of the order of a foot's elements in poetry Anastrophe, a figure of speech also known as an inversion Mathematics and logic Additive inverse Involution (mathematics), a function that is its own inverse (when applied twice, the starting value is obtained) Inversion (discrete mathematics), any item that is out of order in a sequence Inverse element Inversive geometry#Circle inversion, a transformation of the Euclidean plane that maps generalized circles to generalized circles Inversion in a point, or point reflection, a kind of isometric (distance-preserving) transformation in a Euclidean space Inversion transformation, a conformal transformation (one which preserves angles of intersection) Method of inversion, the image of a harmonic function in a sphere (or plane); see Method of image charges Multiplicative inverse, the reciprocal of a number (or any other type of element for which a multiplication function is defined) Matrix inversion, an operation on a matrix that results in its multiplicative inverse Model inversion Set inversion Natural sciences Biology and medicine Inversion (evolutionary biology), a hypothesis about the evolution of the dorsoventral axis in animals Inversion (kinesiology), movement of the sole towards the median plane Chromosomal inversion, where a segment of a chromosome is reversed end-to-end Inversion therapy, the practice of hanging upside down (heart higher than head) for supposed health benefits Geology Inversion (geology), the relative uplift of a previously basinal area resulting from local shortening, in structural geology Relief inversion, when a previous depression becomes a landform that stands out from its surroundings Seismic inversion, transforming seismic reflection data into a quantitative rock-property description of a geological formation Physics and chemistry Island of inversion, a group of elements with abnormal nuclear shell structure Nitrogen inversion, a chemical process in which a trigonal nitrogen-containing structure turns inside-out Population inversion, in statistical mechanics, when a system exists in state with more members in an excited state than in lower-energy states Pyramidal inversion, a chemical process in which a trigonal structure turns inside-out Inverted sugar syrup, a chemical reaction
https://en.wikipedia.org/wiki/Algebraic%20modeling%20language	Algebraic modeling languages (AML) are high-level computer programming languages for describing and solving high complexity problems for large scale mathematical computation (i.e. large scale optimization type problems). One particular advantage of some algebraic modeling languages like AIMMS, AMPL, GAMS, Gekko, MathProg, Mosel, and OPL is the similarity of their syntax to the mathematical notation of optimization problems. This allows for a very concise and readable definition of problems in the domain of optimization, which is supported by certain language elements like sets, indices, algebraic expressions, powerful sparse index and data handling variables, constraints with arbitrary names. The algebraic formulation of a model does not contain any hints how to process it. An AML does not solve those problems directly; instead, it calls appropriate external algorithms to obtain a solution. These algorithms are called solvers and can handle certain kind of mathematical problems like: linear problems integer problems (mixed integer) quadratic problems mixed complementarity problems mathematical programs with equilibrium constraints constrained nonlinear systems general nonlinear problems non-linear programs with discontinuous derivatives nonlinear integer problems global optimization problems stochastic optimization problems Core elements The core elements of an AML are: a modeling language interpreter (the AML itself) solver links user interfaces (UI) data exchange facilities Design principles Most AML follow certain design principles: a balanced mix of declarative and procedural elements open architecture and interfaces to other systems different layers with separation of: model and data model and solution methods model and operating system model and interface Data driven model generation Most modeling languages exploit the similarities between structured models and relational databases by providing a database access layer, which enables the modelling system to directly access data from external data sources (e.g. these table handlers for AMPL). With the refinement of analytic technologies applied to business processes, optimization models are becoming an integral part of decision support systems; optimization models can be structured and layered to represent and support complex business processes. In such applications, the multi-dimensional data structure typical of OLAP systems can be directly mapped to the optimization models and typical MDDB operations can be translated into aggregation and disaggregation operations on the underlying model History Algebraic modelling languages find their roots in matrix-generator and report-writer programs (MGRW), developed in the late seventies. Some of these are MAGEN, MGRW (IBM), GAMMA.3, DATAFORM and MGG/RWG. These systems simplified the communication of problem instances to the solution algorithms and the generation of a readable report of the results. An early ma
https://en.wikipedia.org/wiki/Spatial%20statistics	Spatial statistics is a field of applied statistics dealing with spatial data. It involves stochastic processes (random fields, point processes), sampling, smoothing and interpolation, regional (areal unit) and lattice (gridded) data, point patterns, as well as image analysis and stereology. See also Geostatistics Modifiable areal unit problem Spatial analysis Spatial econometrics Statistical geography Spatial epidemiology Spatial network Statistical shape analysis References Applied statistics Statistics
https://en.wikipedia.org/wiki/Everyday%20Mathematics	Everyday Mathematics is a pre-K and elementary school mathematics curriculum, developed by the University of Chicago School Mathematics Project (not to be confused with the University of Chicago School of Mathematics). The program, now published by McGraw-Hill Education, has sparked debate. History Everyday Mathematics curriculum was developed by the University of Chicago School Math Project (or UCSMP ) which was founded in 1983. Work on it started in the summer of 1985. The 1st edition was released in 1998 and the 2nd in 2002. A third edition was released in 2007 and a fourth in 2014-2015. Curriculum structure Below is an outline of the components of EM as they are generally seen throughout the curriculum. Lessons A typical lesson outlined in one of the teacher’s manuals includes three parts Teaching the Lesson—Provides main instructional activities for the lesson. Ongoing Learning and Practice—Supports previously introduced concepts and skills; essential for maintaining skills. Differentiation Options—Includes options for supporting the needs of all students; usually an extension of Part 1, Teaching the Lesson. Daily Routines Every day, there are certain things that each EM lesson requires the student to do routinely. These components can be dispersed throughout the day or they can be part of the main math lesson. Math Messages—These are problems, displayed in a manner chosen by the teacher, that students complete before the lesson and then discuss as an opener to the main lesson. Mental Math and Reflexes—These are brief (no longer than 5 min) sessions “…designed to strengthen children's number sense and to review and advance essential basic skills…” (Program Components 2003). Math Boxes—These are pages intended to have students routinely practice problems independently. Home Links/Study Links—Everyday homework is sent home. Grades K-3 they are called Home Links and 4-6 they are Study Links. They are meant to reinforce instruction as well as connect home to the work at school. Supplemental Aspects Beyond the components already listed, there are supplemental resources to the program. The two most common are games and explorations. Games— “…Everyday Mathematics sees games as enjoyable ways to practice number skills, especially those that help children develop fact power…” (Program Components 2003). Therefore, authors of the series have interwoven games throughout daily lessons and activities. Scientific support for the curriculum What Works Clearinghouse ( or WWC ) reviewed the evidence in support of the Everyday Mathematics program. Of the 61 pieces of evidence submitted by the publisher, 57 did not meet the WWC minimum standards for scientific evidence, four met evidence standards with reservations, and one of those four showed a statistically significant positive effect. Based on the four studies considered, the WWC gave Everyday Math a rating of "Potentially Positive Effect" with the four studies showing a mean improv
https://en.wikipedia.org/wiki/Arif%20Zaman	Arif Zaman is a Pakistani mathematician, academic scientist, and a retired professor of Statistics and Mathematics from Syed Babar Ali School of Science and Engineering, Lahore University of Management Sciences (LUMS), Lahore, Pakistan. Before joining LUMS in 1994, he also served in the Statistics Department at Purdue University and at Florida State University. Zaman attended Harvey Mudd College, where he completed his B.S. in Mathematics in 1976. He received an M.A. in Applied Mathematics in 1977 at the Claremont Graduate School, and his PhD in Statistics at Stanford University in 1981. In his doctoral thesis, he studied de Finetti's theorem and its possible turn out in Markov chain. His dissertation was supervised by Persi Diaconis. Works Arif Zaman (1984), "An Approximation Theorem for Finite Markov Exchangeability", Annals of Applied Probability, volume 4, page 223–229. "Random Binary Matrices in Bio-ecological Ecology - Instituting a Good Neighbor Policy", Environmental and Ecological Statistics, 9, No. 4, 405–421, 2002, (with D. Simberloff). References External links 20th-century Pakistani mathematicians 21st-century Pakistani mathematicians Claremont Graduate University alumni Florida State University faculty Harvey Mudd College alumni Academic staff of Lahore University of Management Sciences Living people Pakistani statisticians Purdue University faculty Stanford University alumni Year of birth missing (living people)
https://en.wikipedia.org/wiki/Abhyankar%E2%80%93Moh%20theorem	In mathematics, the Abhyankar–Moh theorem states that if is a complex line in the complex affine plane , then every embedding of into extends to an automorphism of the plane. It is named after Shreeram Shankar Abhyankar and Tzuong-Tsieng Moh, who published it in 1975. More generally, the same theorem applies to lines and planes over any algebraically closed field of characteristic zero, and to certain well-behaved subsets of higher-dimensional complex affine spaces. References . Theorems in algebraic geometry
https://en.wikipedia.org/wiki/David%20Orrell	David John Orrell is a Canadian writer and mathematician. He received his doctorate in mathematics from the University of Oxford. His work in the prediction of complex systems such as the weather, genetics and the economy has been featured in New Scientist, the Financial Times, The Economist, Adbusters, BBC Radio, Russia-1, and CBC TV. He now conducts research and writes in the areas of systems biology and economics, and runs a mathematical consultancy Systems Forecasting. He is the son of theatre historian and English professor John Orrell. His books have been translated into over ten languages. Apollo's Arrow: The Science of Prediction and the Future of Everything was a national bestseller and finalist for the 2007 Canadian Science Writers' Award. Economyths: Ten Ways Economics Gets It Wrong was a finalist for the 2011 National Business Book Award. Criticism of use of mathematical models A consistent topic in Orrell’s work is the limitations of mathematical models, and the need to acknowledge these limitations if we are to understand the causes of forecast error. He argues for example that errors in weather prediction are caused primarily by model error, rather than the butterfly effect. Economic models are seen as particularly unrealistic. In Truth or Beauty: Science and the Quest for Order, he suggests that many such theories, along with areas of physics such as string theory, are motivated largely by the desire to conform with a traditional scientific aesthetic, that is currently being subverted by developments in complexity science. Quantum theory of money and value Orrell is considered a leading proponent of quantum finance and quantum economics. In The Evolution of Money (coauthored with journalist Roman Chlupatý) and a series of articles he proposed a quantum theory of money and value, which states that money has dualistic properties because it combines the properties of an owned and valued thing, with those of abstract number. The fact that these two sides of money are incompatible leads to its complex and often unpredictable behavior. In Quantum Economics: The New Science of Money he argued that these dualistic properties feed up to affect the economy as a whole. Books Revised and extended edition of 2010 book. Published in the U.S. as The Future of Everything: The Science of Prediction. See also Anticipatory science forecasts Complex systems Mathematical model Computer model Chaos theory Systems biology Quantum economics References External links David Orrell's homepage Systems Forecasting Video of talk on money given for Marshall McLuhan lecture, Berlin 2015 Video of talk on prediction given at TEDx Park Kultury, Moscow in 2012 ABC News - Good Morning America (excerpts from Apollo's Arrow) National Post's review of Apollo's Arrow Sunday Times review of Truth or Beauty David Orrell Interview with Rishabh Chaddha on Substack.com Canadian non-fiction writers Canadian mathematicians Systems biologists Compl
https://en.wikipedia.org/wiki/Connected%20Mathematics	Connected Mathematics is a comprehensive mathematics program intended for U.S. students in grades 6–8. The curriculum design, text materials for students, and supporting resources for teachers were created and have been progressively refined by the Connected Mathematics Project (CMP) at Michigan State University with advice and contributions from many mathematics teachers, curriculum developers, mathematicians, and mathematics education researchers. The current third edition of Connected Mathematics is a major revision of the program to reflect new expectations of the Common Core State Standards for Mathematics and what the authors have learned from over twenty years of field experience by thousands of teachers working with millions of middle grades students. This CMP3 program is now published in paper and electronic form by Pearson Education. Core principles The first edition of Connected Mathematics, developed with financial support from the National Science Foundation, was designed to provide instructional materials for middle grades mathematics based on the 1989 Curriculum and Evaluation Standards and the 1991 Professional Standards for Teaching Mathematics from the National Council of Teachers of Mathematics. These Standards implied four core features of the curriculum. Comprehensive coverage of mathematical concepts and skills in four content strands—number, algebra, geometry and measurement, and probability and statistics; connections between concepts and methods of the four major content strands and between the abstractions of mathematics and their embodiment in real-world problem contexts; Instructional materials that make classrooms lively places where students learn by solving problems and sharing their thinking with others and where teachers encourage and support students to wonder, to ask questions, and to enjoy learning and using mathematics; Developing students' understanding of mathematical concepts, principles, procedures, and habits of mind and the disposition to use mathematical reasoning in making sense of new situations and solving problems Those principles have been a consistent guide to the development and refinement of the Connected Mathematics program for over twenty years. The first edition was published in 1995; a major revision, also supported by National Science Foundation funding, was published in 2006; and the current third edition was published in 2014. In the third edition, the collection of units was expanded to cover Common Core Standards for both grade eight and Algebra I. Each CMP grade level course aims to advance student understanding, skills, and problem solving in every content strand with increasing sophistication and challenge over the middle school grades. The problem tasks for students are designed to make connections within mathematics, between mathematics and other subject areas, and/or to real-world settings that appeal to students. Curriculum units consist of 3–5 investigations, each fo
https://en.wikipedia.org/wiki/Completion%20of%20a%20ring	In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have a simpler structure than general ones, and Hensel's lemma applies to them. In algebraic geometry, a completion of a ring of functions R on a space X concentrates on a formal neighborhood of a point of X: heuristically, this is a neighborhood so small that all Taylor series centered at the point are convergent. An algebraic completion is constructed in a manner analogous to completion of a metric space with Cauchy sequences, and agrees with it in the case when R has a metric given by a non-Archimedean absolute value. General construction Suppose that E is an abelian group with a descending filtration of subgroups. One then defines the completion (with respect to the filtration) as the inverse limit: This is again an abelian group. Usually E is an additive abelian group. If E has additional algebraic structure compatible with the filtration, for instance E is a filtered ring, a filtered module, or a filtered vector space, then its completion is again an object with the same structure that is complete in the topology determined by the filtration. This construction may be applied both to commutative and noncommutative rings. As may be expected, when the intersection of the equals zero, this produces a complete topological ring. Krull topology In commutative algebra, the filtration on a commutative ring R by the powers of a proper ideal I determines the Krull (after Wolfgang Krull) or I-adic topology on R. The case of a maximal ideal is especially important, for example the distinguished maximal ideal of a valuation ring. The basis of open neighbourhoods of 0 in R is given by the powers In, which are nested and form a descending filtration on R: (Open neighborhoods of any r ∈ R are given by cosets r + In.) The completion is the inverse limit of the factor rings, pronounced "R I hat". The kernel of the canonical map from the ring to its completion is the intersection of the powers of I. Thus is injective if and only if this intersection reduces to the zero element of the ring; by the Krull intersection theorem, this is the case for any commutative Noetherian ring which is an integral domain or a local ring. There is a related topology on R-modules, also called Krull or I-adic topology. A basis of open neighborhoods of a module M is given by the sets of the form The completion of an R-module M is the inverse limit of the quotients This procedure converts any module over R into a complete topological module over . Examples The ring of p-adic integers is obtained by completing the ring of integers at the ideal (p). Let R = K[x1,...,xn] be the polynomial ring in n variables over a field K and be the maximal ideal generated by
https://en.wikipedia.org/wiki/Integral%20element	In commutative algebra, an element b of a commutative ring B is said to be integral over A, a subring of B, if there are n ≥ 1 and aj in A such that That is to say, b is a root of a monic polynomial over A. The set of elements of B that are integral over A is called the integral closure of A in B. It is a subring of B containing A. If every element of B is integral over A, then we say that B is integral over A, or equivalently B is an integral extension of A. If A, B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial). The case of greatest interest in number theory is that of complex numbers integral over Z (e.g., or ); in this context, the integral elements are usually called algebraic integers. The algebraic integers in a finite extension field k of the rationals Q form a subring of k, called the ring of integers of k, a central object of study in algebraic number theory. In this article, the term ring will be understood to mean commutative ring with a multiplicative identity. Examples Integral closure in algebraic number theory There are many examples of integral closure which can be found in algebraic number theory since it is fundamental for defining the ring of integers for an algebraic field extension (or ). Integral closure of integers in rationals Integers are the only elements of Q that are integral over Z. In other words, Z is the integral closure of Z in Q. Quadratic extensions The Gaussian integers are the complex numbers of the form , and are integral over Z. is then the integral closure of Z in . Typically this ring is denoted . The integral closure of Z in is the ring This example and the previous one are examples of quadratic integers. The integral closure of a quadratic extension can be found by constructing the minimal polynomial of an arbitrary element and finding number-theoretic criterion for the polynomial to have integral coefficients. This analysis can be found in the quadratic extensions article. Roots of unity Let ζ be a root of unity. Then the integral closure of Z in the cyclotomic field Q(ζ) is Z[ζ]. This can be found by using the minimal polynomial and using Eisenstein's criterion. Ring of algebraic integers The integral closure of Z in the field of complex numbers C, or the algebraic closure is called the ring of algebraic integers. Other The roots of unity, nilpotent elements and idempotent elements in any ring are integral over Z. Integral closure in geometry In geometry, integral closure is closely related with normalization and normal schemes. It is the first step in resolution of singularities since it gives a process for resolving singularities of codimension 1. For example, the integral closure of is the ring since geometrically, the first ring corresponds to the -plane unioned with the -plane. They have a co
https://en.wikipedia.org/wiki/History%20of%20manifolds%20and%20varieties	The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology. Certain special classes of manifolds also have additional algebraic structure; they may behave like groups, for instance. In that case, they are called Lie Groups. Alternatively, they may be described by polynomial equations, in which case they are called algebraic varieties, and if they additionally carry a group structure, they are called algebraic groups. Nomenclature The term "manifold" comes from German Mannigfaltigkeit, by Bernhard Riemann. In English, "manifold" refers to spaces with a differentiable or topological structure, while "variety" refers to spaces with an algebraic structure, as in algebraic varieties. In Romance languages, manifold is translated as "variety" – such spaces with a differentiable structure are literally translated as "analytic varieties", while spaces with an algebraic structure are called "algebraic varieties". Thus for example, the French word "variété topologique" means topological manifold. In the same vein, the Japanese word "" (tayōtai) also encompasses both manifold and variety. ("" (tayō) means various.) Background Ancestral to the modern concept of a manifold were several important results of 18th and 19th century mathematics. The oldest of these was Non-Euclidean geometry, which considers spaces where Euclid's parallel postulate fails. Saccheri first studied this geometry in 1733. Lobachevsky, Bolyai, and Riemann developed the subject further 100 years later. Their research uncovered two types of spaces whose geometric structures differ from that of classical Euclidean space; these are called hyperbolic geometry and elliptic geometry. In the modern theory of manifolds, these notions correspond to manifolds with constant, negative and positive curvature, respectively. Carl Friedrich Gauss may have been the first to consider abstract spaces as mathematical objects in their own right. His theorema egregium gives a method for computing the curvature of a surface without considering the ambient space in which the surface lies. In modern terms, the theorem proved that the curvature of the surface is an intrinsic property. Manifold theory has come to focus exclusively on these intrinsic properties (or invariants), while largely ignoring the extrinsic properties of the ambient space. Another, more topological example of an intrinsic property of a manifold is the Euler characteristic. For a non-intersecting graph in the Euclidean plane, with V vertices (or corners), E edges and F faces (counting the exterior) Euler showed that V-E+F= 2. Thus 2 is called the Euler characteristic of the plane. By contrast, in 1813 Antoine-Jean Lhuilier showed that the Euler characteristic of the torus is 0, since the complete graph on seven points can be embedded into the torus. The Euler characteristic of other surfaces is a useful topological invari
https://en.wikipedia.org/wiki/National%20Gay%20Newspaper%20Guild	The National Gay Newspaper Guild is an organization of LGBT newspapers located in the United States. Through Rivendell Media, the guild gathers statistics on the readership of the member publications. Member publications Bay Area Reporter Bay Windows Between the Lines Dallas Voice Frontiers Philadelphia Gay News San Francisco Sentinel Washington Blade Windy City Times References National Gay Newspaper Guild
https://en.wikipedia.org/wiki/Zolt%C3%A1n%20Szab%C3%B3%20%28mathematician%29	Zoltán Szabó (born November 24, 1965) is a professor of mathematics at Princeton University known for his work on Heegaard Floer homology. Education and career Szabó received his B.A. from Eötvös Loránd University in Budapest, Hungary in 1990, and he received his Ph.D. from Rutgers University in 1994. Together with Peter Ozsváth, Szabó created Heegaard Floer homology, a homology theory for 3-manifolds. For this contribution to the field of topology, Ozsváth and Szabó were awarded the 2007 Oswald Veblen Prize in Geometry. In 2010, he was elected honorary member of the Hungarian Academy of Sciences. Selected publications . . Grid Homology for Knots and Links, American Mathematical Society, (2015) References External links Personal homepage 1965 births 20th-century Hungarian mathematicians 21st-century Hungarian mathematicians Members of the Hungarian Academy of Sciences Living people International Mathematical Olympiad participants Topologists Eötvös Loránd University alumni Rutgers University alumni Princeton University faculty
https://en.wikipedia.org/wiki/Mohsen%20Amiryoussefi	Mohsen Amiryousefi (, born 1972) is an Iranian director and screenwriter. A graduate in mathematics from Isfahan University, he completed his first short film in 1997 based on a story by Franz Kafka, after writing several screenplays for both screen and stage. Mohsen Amiryoussefi first came to prominence with his 2004 black comedy “Bitter Dream,” about a funeral director. He took home the Camera d’Or at that year's Cannes Film Festival 2004 as well as generous critical acclaim. Amiryoussefi belongs to the third generation of "Iranian New Wave". Awards He has received international awards for his critically acclaimed movies Khab- talkh and Atashkar. Filmography Khab-e talkh or Bitter Dream (2004) Atashkar or Fire Keeper (2009) Ashghal haye Doost Dashtani or Lovely Trashes (2012) See also Iranian New Wave Cinema of Iran Notes External links Iranian film directors 1972 births Living people
https://en.wikipedia.org/wiki/24-cell%20honeycomb	In four-dimensional Euclidean geometry, the 24-cell honeycomb, or icositetrachoric honeycomb is a regular space-filling tessellation (or honeycomb) of 4-dimensional Euclidean space by regular 24-cells. It can be represented by Schläfli symbol {3,4,3,3}. The dual tessellation by regular 16-cell honeycomb has Schläfli symbol {3,3,4,3}. Together with the tesseractic honeycomb (or 4-cubic honeycomb) these are the only regular tessellations of Euclidean 4-space. Coordinates The 24-cell honeycomb can be constructed as the Voronoi tessellation of the D4 or F4 root lattice. Each 24-cell is then centered at a D4 lattice point, i.e. one of These points can also be described as Hurwitz quaternions with even square norm. The vertices of the honeycomb lie at the deep holes of the D4 lattice. These are the Hurwitz quaternions with odd square norm. It can be constructed as a birectified tesseractic honeycomb, by taking a tesseractic honeycomb and placing vertices at the centers of all the square faces. The 24-cell facets exist between these vertices as rectified 16-cells. If the coordinates of the tesseractic honeycomb are integers (i,j,k,l), the birectified tesseractic honeycomb vertices can be placed at all permutations of half-unit shifts in two of the four dimensions, thus: (i+½,j+½,k,l), (i+½,j,k+½,l), (i+½,j,k,l+½), (i,j+½,k+½,l), (i,j+½,k,l+½), (i,j,k+½,l+½). Configuration Each 24-cell in the 24-cell honeycomb has 24 neighboring 24-cells. With each neighbor it shares exactly one octahedral cell. It has 24 more neighbors such that with each of these it shares a single vertex. It has no neighbors with which it shares only an edge or only a face. The vertex figure of the 24-cell honeycomb is a tesseract (4-dimensional cube). So there are 16 edges, 32 triangles, 24 octahedra, and 8 24-cells meeting at every vertex. The edge figure is a tetrahedron, so there are 4 triangles, 6 octahedra, and 4 24-cells surrounding every edge. Finally, the face figure is a triangle, so there are 3 octahedra and 3 24-cells meeting at every face. Cross-sections One way to visualize a 4-dimensional figure is to consider various 3-dimensional cross-sections. That is, the intersection of various hyperplanes with the figure in question. Applying this technique to the 24-cell honeycomb gives rise to various 3-dimensional honeycombs with varying degrees of regularity. A vertex-first cross-section uses some hyperplane orthogonal to a line joining opposite vertices of one of the 24-cells. For instance, one could take any of the coordinate hyperplanes in the coordinate system given above (i.e. the planes determined by xi = 0). The cross-section of {3,4,3,3} by one of these hyperplanes gives a rhombic dodecahedral honeycomb. Each of the rhombic dodecahedra corresponds to a maximal cross-section of one of the 24-cells intersecting the hyperplane (the center of each such (4-dimensional) 24-cell lies in the hyperplane). Accordingly, the rhombic dodecahedral honeycomb is the
https://en.wikipedia.org/wiki/Liouville%27s%20theorem%20%28differential%20algebra%29	In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in 1833 to 1841, places an important restriction on antiderivatives that can be expressed as elementary functions. The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. These are called nonelementary antiderivatives. A standard example of such a function is whose antiderivative is (with a multiplier of a constant) the error function, familiar from statistics. Other examples include the functions and Liouville's theorem states that elementary antiderivatives, if they exist, are in the same differential field as the function, plus possibly a finite number of applications of the logarithm function. Definitions For any differential field the of is the subfield Given two differential fields and is called a of if is a simple transcendental extension of (that is, for some transcendental ) such that This has the form of a logarithmic derivative. Intuitively, one may think of as the logarithm of some element of in which case, this condition is analogous to the ordinary chain rule. However, is not necessarily equipped with a unique logarithm; one might adjoin many "logarithm-like" extensions to Similarly, an is a simple transcendental extension that satisfies With the above caveat in mind, this element may be thought of as an exponential of an element of Finally, is called an of if there is a finite chain of subfields from to where each extension in the chain is either algebraic, logarithmic, or exponential. Basic theorem Suppose and are differential fields with and that is an elementary differential extension of Suppose and satisfy (in words, suppose that contains an antiderivative of ). Then there exist and such that In other words, the only functions that have "elementary antiderivatives" (that is, antiderivatives living in, at worst, an elementary differential extension of ) are those with this form. Thus, on an intuitive level, the theorem states that the only elementary antiderivatives are the "simple" functions plus a finite number of logarithms of "simple" functions. A proof of Liouville's theorem can be found in section 12.4 of Geddes, et al. See Lützen's scientific bibliography for a sketch of Liouville's original proof (Chapter IX. Integration in Finite Terms), its modern exposition and algebraic treatment (ibid. §61). Examples As an example, the field of rational functions in a single variable has a derivation given by the standard derivative with respect to that variable. The constants of this field are just the complex numbers that is, The function which exists in does not have an antiderivative in Its antiderivatives do, however, exist in the logarithmic extension Likewise, the function does not have an antiderivative in Its antiderivatives do not seem to satisfy the requirements of the theorem, since they are not (apparently) sums of ra
https://en.wikipedia.org/wiki/Luton/Dunstable%20urban%20area	The Luton/Dunstable Urban Area, according to the Office for National Statistics, is the conurbation (continuous built up area) including the settlements of Luton, Dunstable and Houghton Regis, in Bedfordshire, East of England. Despite straddling district boundaries the conurbation shares many facilities including an integrated bus service and the large Luton and Dunstable University Hospital. The conurbation is located in the southern part of the ceremonial county of Bedfordshire, England, and includes the unitary authority of Luton, and part of Central Bedfordshire. The current population (2021 census) is 286,803. This is an increase of 9% from the 2011 population of 258,018. Future growth The area is expected to grow due to development within and physical expansion of the three towns and large re-development of Luton, including redevelopment of the former Vauxhall Motors factory complex. The Luton & Dunstable Urban area is considered part of the Milton Keynes and South Midlands Sub Region, part of the East of England. the East of England Regional Spatial Strategy has outlined the identified the urban area for growth, as part of the Sustainable Communities Plan. It is also considered part of the London Commuter Belt. References External links Luton Population Geography of Bedfordshire Luton Urban areas of England
https://en.wikipedia.org/wiki/Cylindroid	Cylindroid may refer to: Elliptic cylinder, a cylinder with an ellipse as its cross-section An adjectival form of Cylinder (geometry), regardless of cross-section Plücker's conoid, a self-intersecting ruled surface whose cross-sections are pairs of crossing lines
https://en.wikipedia.org/wiki/Beatty%20sequence	In mathematics, a Beatty sequence (or homogeneous Beatty sequence) is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are named after Samuel Beatty, who wrote about them in 1926. Rayleigh's theorem, named after Lord Rayleigh, states that the complement of a Beatty sequence, consisting of the positive integers that are not in the sequence, is itself a Beatty sequence generated by a different irrational number. Beatty sequences can also be used to generate Sturmian words. Definition Any irrational number that is greater than one generates the Beatty sequence The two irrational numbers and naturally satisfy the equation . The two Beatty sequences and that they generate form a pair of complementary Beatty sequences. Here, "complementary" means that every positive integer belongs to exactly one of these two sequences. Examples When is the golden ratio , the complementary Beatty sequence is generated by . In this case, the sequence , known as the lower Wythoff sequence, is and the complementary sequence , the upper Wythoff sequence, is These sequences define the optimal strategy for Wythoff's game, and are used in the definition of the Wythoff array. As another example, for the square root of 2, , . In this case, the sequences are For and , the sequences are Any number in the first sequence is absent in the second, and vice versa. History Beatty sequences got their name from the problem posed in The American Mathematical Monthly by Samuel Beatty in 1926. It is probably one of the most often cited problems ever posed in the Monthly. However, even earlier, in 1894 such sequences were briefly mentioned by Lord Rayleigh in the second edition of his book The Theory of Sound. Rayleigh theorem Rayleigh's theorem (also known as Beatty's theorem) states that given an irrational number there exists so that the Beatty sequences and partition the set of positive integers: each positive integer belongs to exactly one of the two sequences. First proof Given let . We must show that every positive integer lies in one and only one of the two sequences and . We shall do so by considering the ordinal positions occupied by all the fractions and when they are jointly listed in nondecreasing order for positive integers j and k. To see that no two of the numbers can occupy the same position (as a single number), suppose to the contrary that for some j and k. Then = , a rational number, but also, not a rational number. Therefore, no two of the numbers occupy the same position. For any , there are positive integers such that and positive integers such that , so that the position of in the list is . The equation implies Likewise, the position of in the list is . Conclusion: every positive integer (that is, every position in the list) is of the form or of the form , but not both. The converse statement is also true: if p and q are two real numbers such t
https://en.wikipedia.org/wiki/Nagel%20point	In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concurrency of all three of the triangle's splitters. Construction Given a triangle , let be the extouch points in which the -excircle meets line , the -excircle meets line , and the -excircle meets line , respectively. The lines concur in the Nagel point of triangle . Another construction of the point is to start at and trace around triangle half its perimeter, and similarly for and . Because of this construction, the Nagel point is sometimes also called the bisected perimeter point, and the segments are called the triangle's splitters. There exists an easy construction of the Nagel point. Starting from each vertex of a triangle, it suffices to carry twice the length of the opposite edge. We obtain three lines which concur at the Nagel point. Relation to other triangle centers The Nagel point is the isotomic conjugate of the Gergonne point. The Nagel point, the centroid, and the incenter are collinear on a line called the Nagel line. The incenter is the Nagel point of the medial triangle; equivalently, the Nagel point is the incenter of the anticomplementary triangle. The isogonal conjugate of the Nagel point is the point of concurrency of the lines joining the mixtilinear touchpoint and the opposite vertex. Barycentric coordinates The un-normalized barycentric coordinates of the Nagel point are where is the semi-perimeter of the reference triangle . Trilinear coordinates The trilinear coordinates of the Nagel point are as or, equivalently, in terms of the side lengths History The Nagel point is named after Christian Heinrich von Nagel, a nineteenth-century German mathematician, who wrote about it in 1836. Early contributions to the study of this point were also made by August Leopold Crelle and Carl Gustav Jacob Jacobi. See also Mandart inellipse Trisected perimeter point References External links Nagel Point from Cut-the-knot Nagel Point, Clark Kimberling Spieker Conic and generalization of Nagel line at Dynamic Geometry Sketches Generalizes Spieker circle and associated Nagel line. Triangle centers fr:Cercles inscrit et exinscrits d'un triangle#Point de Nagel
https://en.wikipedia.org/wiki/Differential%20graded%20category	In mathematics, especially homological algebra, a differential graded category, often shortened to dg-category or DG category, is a category whose morphism sets are endowed with the additional structure of a differential graded -module. In detail, this means that , the morphisms from any object A to another object B of the category is a direct sum and there is a differential d on this graded group, i.e., for each n there is a linear map , which has to satisfy . This is equivalent to saying that is a cochain complex. Furthermore, the composition of morphisms is required to be a map of complexes, and for all objects A of the category, one requires . Examples Any additive category may be considered to be a DG-category by imposing the trivial grading (i.e. all vanish for ) and trivial differential (). A little bit more sophisticated is the category of complexes over an additive category . By definition, is the group of maps which do not need to respect the differentials of the complexes A and B, i.e., . The differential of such a morphism of degree n is defined to be , where are the differentials of A and B, respectively. This applies to the category of complexes of quasi-coherent sheaves on a scheme over a ring. A DG-category with one object is the same as a DG-ring. A DG-ring over a field is called DG-algebra, or differential graded algebra. Further properties The category of small dg-categories can be endowed with a model category structure such that weak equivalences are those functors that induce an equivalence of derived categories. Given a dg-category C over some ring R, there is a notion of smoothness and properness of C that reduces to the usual notions of smooth and proper morphisms in case C is the category of quasi-coherent sheaves on some scheme X over R. Relation to triangulated categories A DG category C is called pre-triangulated if it has a suspension functor and a class of distinguished triangles compatible with the suspension, such that its homotopy category Ho(C) is a triangulated category. A triangulated category T is said to have a dg enhancement C if C is a pretriangulated dg category whose homotopy category is equivalent to T. dg enhancements of an exact functor between triangulated categories are defined similarly. In general, there need not exist dg enhancements of triangulated categories or functors between them, for example stable homotopy category can be shown not to arise from a dg category in this way. However, various positive results do exist, for example the derived category D(A) of a Grothendieck abelian category A admits a unique dg enhancement. See also Differential algebra Graded (mathematics) Graded category Derivator References External links http://ncatlab.org/nlab/show/dg-category Homological algebra Categories in category theory
https://en.wikipedia.org/wiki/Nakagami%20distribution	The Nakagami distribution or the Nakagami-m distribution is a probability distribution related to the gamma distribution. The family of Nakagami distributions has two parameters: a shape parameter and a second parameter controlling spread . Characterization Its probability density function (pdf) is where Its cumulative distribution function is where P is the regularized (lower) incomplete gamma function. Parametrization The parameters and are and Parameter estimation An alternative way of fitting the distribution is to re-parametrize and m as σ = Ω/m and m. Given independent observations from the Nakagami distribution, the likelihood function is Its logarithm is Therefore These derivatives vanish only when and the value of m for which the derivative with respect to m vanishes is found by numerical methods including the Newton–Raphson method. It can be shown that at the critical point a global maximum is attained, so the critical point is the maximum-likelihood estimate of (m,σ). Because of the equivariance of maximum-likelihood estimation, one then obtains the MLE for Ω as well. Generation The Nakagami distribution is related to the gamma distribution. In particular, given a random variable , it is possible to obtain a random variable , by setting , , and taking the square root of : Alternatively, the Nakagami distribution can be generated from the chi distribution with parameter set to and then following it by a scaling transformation of random variables. That is, a Nakagami random variable is generated by a simple scaling transformation on a Chi-distributed random variable as below. For a Chi-distribution, the degrees of freedom must be an integer, but for Nakagami the can be any real number greater than 1/2. This is the critical difference and accordingly, Nakagami-m is viewed as a generalization of Chi-distribution, similar to a gamma distribution being considered as a generalization of Chi-squared distributions. History and applications The Nakagami distribution is relatively new, being first proposed in 1960. It has been used to model attenuation of wireless signals traversing multiple paths and to study the impact of fading channels on wireless communications. Related distributions Restricting m to the unit interval (q = m; 0 < q < 1) defines the Nakagami-q distribution, also known as distribution."The radius around the true mean in a bivariate normal random variable, re-written in polar coordinates (radius and angle), follows a Hoyt distribution. Equivalently, the modulus of a complex normal random variable does." With 2m = k, the Nakagami distribution gives a scaled chi distribution. With , the Nakagami distribution gives a scaled half-normal distribution. A Nakagami distribution is a particular form of generalized gamma distribution, with p = 2 and d = 2m See also Normal distribution Gamma distribution Modified half-normal distribution Normally distributed and uncorrelated doe
https://en.wikipedia.org/wiki/Isotomic%20conjugate	In geometry, the isotomic conjugate of a point with respect to a triangle is another point, defined in a specific way from and : If the base points of the lines on the sides opposite are reflected about the midpoints of their respective sides, the resulting lines intersect at the isotomic conjugate of . Construction We assume that is not collinear with any two vertices of . Let be the points in which the lines meet sidelines (extended if necessary). Reflecting in the midpoints of sides will give points respectively. The isotomic lines joining these new points to the vertices meet at a point (which can be proved using Ceva's theorem), the isotomic conjugate of . Coordinates If the trilinears for are , then the trilinears for the isotomic conjugate of are where are the side lengths opposite vertices respectively. Properties The isotomic conjugate of the centroid of triangle is the centroid itself. The isotomic conjugate of the symmedian point is the third Brocard point, and the isotomic conjugate of the Gergonne point is the Nagel point. Isotomic conjugates of lines are circumconics, and conversely, isotomic conjugates of circumconics are lines. (This property holds for isogonal conjugates as well.) See also Isogonal conjugate Triangle center References Robert Lachlan, An Elementary Treatise on Modern Pure Geometry, Macmillan and Co., 1893, page 57. Roger A. Johnson: Advanced Euclidean Geometry. Dover 2007, , pp. 157–159, 278 External links Pauk Yiu: Isotomic and isogonal conjugates Navneel Singhal: Isotomic and isogonal conjugates Triangle geometry zh:等角共轭
https://en.wikipedia.org/wiki/Demographics%20of%20the%20Northwest%20Territories	The Northwest Territories is a territory of Canada. It has an area of 1,171,918 square kilometres and a population of 41,786 as of the 2016 Census. Population history Source: Statistics Canada,Canada's population . Statistics Canada. Last accessed September 28, 2006. with Social Science Federation of Canada for 1871–1901 Population geography Ten largest population centres Visible minorities and Indigenous peoples Languages French was made an official language in 1877 by the appointed government, after lengthy and bitter debate resulting from a speech from the throne in 1888 by Lt. Governor Joseph Royal. The members voted on more than one occasion to nullify and make English the only language used in the assembly. After some conflict with Ottawa and a decisive vote on January 19, 1892, the issue was put to rest as an English-only territory. In the early 1980s, the government of Northwest Territories was again under pressure by the federal government to reintroduce French as an official language. Some native members walked out of the assembly, protesting that they would not be permitted to speak their own language. The executive council appointed a special committee of MLAs to study the matter. They decided that if French was to be an official language, then so must the other languages in the territories. The Northwest Territories's Official Languages Act recognizes the following eleven official languages, which is more than any other political division in Canada: Chipewyan Cree English French Gwich’in Inuinnaqtun Inuktitut Inuvialuktun North Slavey South Slavey Tłįchǫ NWT residents have a right to use any of the above languages in a territorial court and in debates and proceedings of the legislature. However, laws are legally binding only in their French and English versions, and the government only publishes laws and other documents in the territory's other official languages when the legislature asks it to. Furthermore, access to services in any language is limited to institutions and circumstances where there is significant demand for that language or where it is reasonable to expect it given the nature of the services requested. In reality, this means that English language services are universally available and there is no guarantee that other languages, including French, will be used by any particular government service except for the courts. The 2006 Canadian census showed a population of 41,464.Of the 40,680 singular responses to the census question concerning 'mother tongue' the most commonly reported languages (official languages in bold) were: There were also about 40 single-language responses for Ukrainian; 35 for the Scandinavian languages, Slovak and Urdu; and 30 for Hungarian, the Iranian languages and Polish. In addition, there were also 320 responses of both English and a 'non-official language'; 15 of both French and a 'non-official language; 45 of both English and French, and about 400 people who either did n
https://en.wikipedia.org/wiki/Demographics%20of%20Yukon	Yukon is the westernmost of Canada's three northern territories. Its capital is Whitehorse. People from Yukon are known as Yukoners (). Unlike in other Canadian provinces and territories, Statistics Canada uses the entire territory as a single at-large census division. Population of Yukon: 40,232 (2021 Census) Percentage of Canadian population : 0.10% Population growth rate for 2007: +5.8% Population history Source: Statistics Canada Population geography Major communities Visible minorities and Indigenous peoples Languages The 2006 Canadian census showed a population of 30,372.Of the 29,940 singular responses to the census question concerning 'mother tongue' the most commonly reported languages were: There were also about 40 single-language responses for Ukrainian; 30 each for Czech and the Scandinavian languages; and about 25 single-language responses each for Italian and Japanese. In addition, there were also 130 responses of both English and a 'non-official language'; 10 of both French and a 'non-official language'; 110 of both English and French; and about 175 people who either did not respond to the question, or reported multiple non-official languages, or else gave some other unenumerated response. Yukon's official languages are English and French. (Figures shown are for the number of single language responses and the percentage of total single-language responses.) Religion The Majority of Christians in Yukon are Anglicans and Roman Catholics, with a small number of Presbyterians and members of the United Church of Canada. Migration Immigration The 2016 Canadian census counted a total of 4,410 immigrants living in Yukon.The most commonly reported countries of birth for these immigrants were: Internal migration A total of 7,400 people moved to Yukon from other parts of Canada between 1996 and 2006 while 10,505 people moved in the opposite direction. These movements resulted in a net influx of 230 from the Northwest Territories; and a net outmigration of 2,505 to Alberta, 915 to British Columbia and 115 to New Brunswick. There was a net influx of 120 francophones from Quebec during this period. All net inter-provincial and official minority movements of more than 100 persons are given. See also Demographics of Canada Notes References Yukon Yukon society
https://en.wikipedia.org/wiki/Algebraic%20differential%20equation	In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the concept of differential algebra used. The intention is to include equations formed by means of differential operators, in which the coefficients are rational functions of the variables (e.g. the hypergeometric equation). Algebraic differential equations are widely used in computer algebra and number theory. A simple concept is that of a polynomial vector field, in other words a vector field expressed with respect to a standard co-ordinate basis as the first partial derivatives with polynomial coefficients. This is a type of first-order algebraic differential operator. Formulations Derivations D can be used as algebraic analogues of the formal part of differential calculus, so that algebraic differential equations make sense in commutative rings. The theory of differential fields was set up to express differential Galois theory in algebraic terms. The Weyl algebra W of differential operators with polynomial coefficients can be considered; certain modules M can be used to express differential equations, according to the presentation of M. The concept of Koszul connection is something that transcribes easily into algebraic geometry, giving an algebraic analogue of the way systems of differential equations are geometrically represented by vector bundles with connections. The concept of jet can be described in purely algebraic terms, as was done in part of Grothendieck's EGA project. The theory of D-modules is a global theory of linear differential equations, and has been developed to include substantive results in the algebraic theory (including a Riemann-Hilbert correspondence for higher dimensions). Algebraic solutions It is usually not the case that the general solution of an algebraic differential equation is an algebraic function: solving equations typically produces novel transcendental functions. The case of algebraic solutions is however of considerable interest; the classical Schwarz list deals with the case of the hypergeometric equation. In differential Galois theory the case of algebraic solutions is that in which the differential Galois group G is finite (equivalently, of dimension 0, or of a finite monodromy group for the case of Riemann surfaces and linear equations). This case stands in relation with the whole theory roughly as invariant theory does to group representation theory. The group G is in general difficult to compute, the understanding of algebraic solutions is an indication of upper bounds for G. External links Differential equations Differential algebra
https://en.wikipedia.org/wiki/Spieker%20circle	In geometry, the incircle of the medial triangle of a triangle is the Spieker circle, named after 19th-century German geometer Theodor Spieker. Its center, the Spieker center, in addition to being the incenter of the medial triangle, is the center of mass of the uniform-density boundary of triangle. The Spieker center is also the point where all three cleavers of the triangle (perimeter bisectors with an endpoint at a side's midpoint) intersect each other. History The Spieker circle and Spieker center are named after Theodor Spieker, a mathematician and professor from Potsdam, Germany. In 1862, he published , dealing with planar geometry. Due to this publication, influential in the lives of many famous scientists and mathematicians including Albert Einstein, Spieker became the mathematician for whom the Spieker circle and center were named. Construction To find the Spieker circle of a triangle, the medial triangle must first be constructed from the midpoints of each side of the original triangle. The circle is then constructed in such a way that each side of the medial triangle is tangent to the circle within the medial triangle, creating the incircle. This circle center is named the Spieker center. Nagel points and lines Spieker circles also have relations to Nagel points. The incenter of the triangle and the Nagel point form a line within the Spieker circle. The middle of this line segment is the Spieker center. The Nagel line is formed by the incenter of the triangle, the Nagel point, and the centroid of the triangle. The Spieker center will always lie on this line. Nine-point circle and Euler line Spieker circles were first found to be very similar to nine-point circles by Julian Coolidge. At this time, it was not yet identified as the Spieker circle, but is referred to as the "P circle" throughout the book. The nine-point circle with the Euler line and the Spieker circle with the Nagel line are analogous to each other, but are not duals, only having dual-like similarities. One similarity between the nine-point circle and the Spieker circle deals with their construction. The nine-point circle is the circumscribed circle of the medial triangle, while the Spieker circle is the inscribed circle of the medial triangle. With relation to their associated lines, the incenter for the Nagel line relates to the circumcenter for the Euler line. Another analogous point is the Nagel point and the othocenter, with the Nagel point associated with the Spieker circle and the orthocenter associated with the nine-point circle. Each circle meets the sides of the medial triangle where the lines from the orthocenter, or the Nagel point, to the vertices of the original triangle meet the sides of the medial triangle. Spieker conic The nine-point circle with the Euler line was generalized into the nine-point conic. Through a similar process, due to the analogous properties of the two circles, the Spieker circle was also able to be generalized into the Spiek
https://en.wikipedia.org/wiki/Dima%20Grigoriev	Dima Grigoriev (Dmitry Grigoryev) (born 10 May 1954) is a mathematician. His research interests include algebraic geometry, symbolic computation and computational complexity theory in computer algebra, with over 130 published articles. Dima Grigoriev was born in Leningrad, Russia and graduated from the Leningrad State University, Dept. of Mathematics and Mechanics, in 1976 (Honours Diploma). During 1976–1992 he was with LOMI, Leningrad Department of the Steklov Mathematical Institute of the USSR Academy of Sciences. In 1979 he earned PhD (Candidate of Sciences) in Physics and Mathematics with thesis "Multiplicative Complexity of a Family of Bilinear Forms" (from LOMI, under the direction of Anatol Slissenko). In 1985 he earned Doctor of Science (higher doctorate) with thesis "Computational Complexity in Polynomial Algebra". Since 1988 until 1992 he was the head of Laboratory of algorithmic methods Leningrad Department of the Steklov Mathematical Institute. During 1992–1998 Grigoriev hold the position of full professor at Penn State University. Since 1998 he hold the position of Research Director at CNRS, University of Rennes 1, and since 2008 – Research Director at CNRS, Laboratory Paul Painleve University Lille 1 in France. He is member of editorial boards of the Journal Computational Complexity, Journal of Applicable Algebra in Engineering, Communications and Computations and Groups, Complexity, Cryptology. He is recipient of the Prize of Leningrad Mathematical Society (1984), Max Planck Research Award of the Max Planck Society, Germany (1994), and Humboldt Prize of Humboldt Foundation, Germany (2002), Invited Speaker of International Congress of Mathematicians, Berkeley, California, 1986. He has Erdős number 2 due to his collaborations with Andrew Odlyzko. References External links Russian mathematicians French mathematicians Living people 1954 births Research directors of the French National Centre for Scientific Research
https://en.wikipedia.org/wiki/Ideal%20sheaf	In algebraic geometry and other areas of mathematics, an ideal sheaf (or sheaf of ideals) is the global analogue of an ideal in a ring. The ideal sheaves on a geometric object are closely connected to its subspaces. Definition Let X be a topological space and A a sheaf of rings on X. (In other words, (X, A) is a ringed space.) An ideal sheaf J in A is a subobject of A in the category of sheaves of A-modules, i.e., a subsheaf of A viewed as a sheaf of abelian groups such that Γ(U, A) · Γ(U, J) ⊆ Γ(U, J) for all open subsets U of X. In other words, J is a sheaf of A-submodules of A. General properties If f: A → B is a homomorphism between two sheaves of rings on the same space X, the kernel of f is an ideal sheaf in A. Conversely, for any ideal sheaf J in a sheaf of rings A, there is a natural structure of a sheaf of rings on the quotient sheaf A/J. Note that the canonical map Γ(U, A)/Γ(U, J) → Γ(U, A/J) for open subsets U is injective, but not surjective in general. (See sheaf cohomology.) Algebraic geometry In the context of schemes, the importance of ideal sheaves lies mainly in the correspondence between closed subschemes and quasi-coherent ideal sheaves. Consider a scheme X and a quasi-coherent ideal sheaf J in OX. Then, the support Z of OX/J is a closed subspace of X, and (Z, OX/J) is a scheme (both assertions can be checked locally). It is called the closed subscheme of X defined by J. Conversely, let i: Z → X be a closed immersion, i.e., a morphism which is a homeomorphism onto a closed subspace such that the associated map i#: OX → i⋆OZ is surjective on the stalks. Then, the kernel J of i# is a quasi-coherent ideal sheaf, and i induces an isomorphism from Z onto the closed subscheme defined by J. A particular case of this correspondence is the unique reduced subscheme Xred of X having the same underlying space, which is defined by the nilradical of OX (defined stalk-wise, or on open affine charts). For a morphism f: X → Y and a closed subscheme ⊆ Y defined by an ideal sheaf J, the preimage ×Y X is defined by the ideal sheaf f⋆(J)OX = im(f⋆J → OX). The pull-back of an ideal sheaf J to the subscheme Z defined by J contains important information, it is called the conormal bundle of Z. For example, the sheaf of Kähler differentials may be defined as the pull-back of the ideal sheaf defining the diagonal X → X × X to X. (Assume for simplicity that X is separated so that the diagonal is a closed immersion.) Analytic geometry In the theory of complex-analytic spaces, the Oka-Cartan theorem states that a closed subset A of a complex space is analytic if and only if the ideal sheaf of functions vanishing on A is coherent. This ideal sheaf also gives A the structure of a reduced closed complex subspace. References Éléments de géométrie algébrique H. Grauert, R. Remmert: Coherent Analytic Sheaves. Springer-Verlag, Berlin 1984 Scheme theory Sheaf theory
https://en.wikipedia.org/wiki/List%20of%20AS%20Roma%20records%20and%20statistics	Records and statistics in relation to the Italian football club Associazione Sportiva Roma. Serie A records Updated 22 July 2020 Home victory: 9–0 v Cremonese, 13 October 1929 Away victory: 6–1 v Alessandria, 6 January 1935 & 6–1 v SPAL, 22 July 2020 Home draw with most goals: 4–4 v Catania, 31 May 1964 & 4–4 v Napoli, 20 October 2007 Away draw with most goals: 4–4 v Milan, 27 January 1935 & 4–4 v Chievo, 30 April 2006 Home defeat: 1–7 v Torino, 5 October 1947 Away defeat: 1–7 v Juventus, 6 March 1932 Most points in a season (3 pts per win): 87 (2016–17, 38 games) Most points in a season (2 pts per win): 43 (1982–83, 30 games) Most victories in a season: 28 (2016–17) Fewest victories in a season: 8 (1964–65, 34 games & 1992–93, 34 games) Fewest defeats in a season: 2 (1980–81, 34 games & 2001–02, 34 games) Most goals scored in a season (by team): 87 (1930–31, 34 games), 90 (2016–17, 38 games) Most goals scored in a season: 29 Rodolfo Volk (1930–31, 34 games) & Edin Džeko (2016–17, 37 games) Lowest goals against in a season (by team): 15 (1974–75, 30 games) Longest winning streak: 11 begun on 21 December 2005 (4–0 v Chievo), ended on 5 March 2006 (1–1 v Internazionale) Longest unbeaten run: 24 begun on 23 September 2001 (2–1 v Fiorentina), ended on 24 March 2002 (1–3 v Internazionale) & 24 begun on 1 November 2009 (2–1 v Bologna), ended on 25 April 2010 (1–2 v Sampdoria) Most appearances: 619, Francesco Totti Most goals scored: 250, Francesco Totti All competitions appearances AS Roma players with 300 or more appearances. Updated 23 October 2018 Players in bold are currently playing for Roma. Top all-time goalscorers Updated 28 July 2021 Players in bold are currently playing for Roma. Capocannoniere winners International honours won while playing at Roma FIFA World Cup The following players have won the FIFA World Cup while playing for A.S. Roma: Guido Masetti – 1934, 1938 Attilio Ferraris – 1934 Enrique Guaita – 1934 Aldo Donati – 1938 Eraldo Monzeglio – 1938 Pietro Serantoni – 1938 Bruno Conti – 1982 Thomas Berthold – 1990 Rudi Völler – 1990 Aldair – 1994 Vincent Candela – 1998 Cafu – 2002 Francesco Totti – 2006 Daniele De Rossi – 2006 Simone Perrotta – 2006 Paulo Dybala – 2022 FIFA Confederations Cup The following players have won the FIFA Confederations Cup while playing for A.S. Roma: Claudio Caniggia – 1992 Aldair – 1997 Cafu – 1997 Juan – 2009 Júlio Baptista – 2009 Antonio Rüdiger – 2017 UEFA European Championship The following players have won the UEFA European Championship while playing for A.S. Roma: Vincent Candela – 2000 Traianos Dellas – 2004 Bryan Cristante – 2020 Leonardo Spinazzola – 2020 Copa América The following players have won the Copa América while playing for A.S. Roma: Renato Gaúcho – 1989 Daniel Fonseca – 1995 Aldair – 1997 Antônio Carlos – 1999 Cafu – 1999 Mancini – 2004 Doni – 2007 Africa Cup of Nations The following players have won the Africa Cup of Na
https://en.wikipedia.org/wiki/Ratio%20distribution	A ratio distribution (also known as a quotient distribution) is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two (usually independent) random variables X and Y, the distribution of the random variable Z that is formed as the ratio Z = X/Y is a ratio distribution. An example is the Cauchy distribution (also called the normal ratio distribution), which comes about as the ratio of two normally distributed variables with zero mean. Two other distributions often used in test-statistics are also ratio distributions: the t-distribution arises from a Gaussian random variable divided by an independent chi-distributed random variable, while the F-distribution originates from the ratio of two independent chi-squared distributed random variables. More general ratio distributions have been considered in the literature. Often the ratio distributions are heavy-tailed, and it may be difficult to work with such distributions and develop an associated statistical test. A method based on the median has been suggested as a "work-around". Algebra of random variables The ratio is one type of algebra for random variables: Related to the ratio distribution are the product distribution, sum distribution and difference distribution. More generally, one may talk of combinations of sums, differences, products and ratios. Many of these distributions are described in Melvin D. Springer's book from 1979 The Algebra of Random Variables. The algebraic rules known with ordinary numbers do not apply for the algebra of random variables. For example, if a product is C = AB and a ratio is D=C/A it does not necessarily mean that the distributions of D and B are the same. Indeed, a peculiar effect is seen for the Cauchy distribution: The product and the ratio of two independent Cauchy distributions (with the same scale parameter and the location parameter set to zero) will give the same distribution. This becomes evident when regarding the Cauchy distribution as itself a ratio distribution of two Gaussian distributions of zero means: Consider two Cauchy random variables, and each constructed from two Gaussian distributions and then where . The first term is the ratio of two Cauchy distributions while the last term is the product of two such distributions. Derivation A way of deriving the ratio distribution of from the joint distribution of the two other random variables X , Y , with joint pdf , is by integration of the following form If the two variables are independent then and this becomes This may not be straightforward. By way of example take the classical problem of the ratio of two standard Gaussian samples. The joint pdf is Defining we have Using the known definite integral we get which is the Cauchy distribution, or Student's t distribution with n = 1 The Mellin transform has also been suggested for derivation of ratio distributions. In the case of p
https://en.wikipedia.org/wiki/Algebraic%20cycle	In mathematics, an algebraic cycle on an algebraic variety V is a formal linear combination of subvarieties of V. These are the part of the algebraic topology of V that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound insights into the structure of the variety. The most trivial case is codimension zero cycles, which are linear combinations of the irreducible components of the variety. The first non-trivial case is of codimension one subvarieties, called divisors. The earliest work on algebraic cycles focused on the case of divisors, particularly divisors on algebraic curves. Divisors on algebraic curves are formal linear combinations of points on the curve. Classical work on algebraic curves related these to intrinsic data, such as the regular differentials on a compact Riemann surface, and to extrinsic properties, such as embeddings of the curve into projective space. While divisors on higher-dimensional varieties continue to play an important role in determining the structure of the variety, on varieties of dimension two or more there are also higher codimension cycles to consider. The behavior of these cycles is strikingly different from that of divisors. For example, every curve has a constant N such that every divisor of degree zero is linearly equivalent to a difference of two effective divisors of degree at most N. David Mumford proved that, on a smooth complete complex algebraic surface S with positive geometric genus, the analogous statement for the group of rational equivalence classes of codimension two cycles in S is false. The hypothesis that the geometric genus is positive essentially means (by the Lefschetz theorem on (1,1)-classes) that the cohomology group contains transcendental information, and in effect Mumford's theorem implies that, despite having a purely algebraic definition, it shares transcendental information with . Mumford's theorem has since been greatly generalized. The behavior of algebraic cycles ranks among the most important open questions in modern mathematics. The Hodge conjecture, one of the Clay Mathematics Institute's Millennium Prize Problems, predicts that the topology of a complex algebraic variety forces the existence of certain algebraic cycles. The Tate conjecture makes a similar prediction for étale cohomology. Alexander Grothendieck's standard conjectures on algebraic cycles yield enough cycles to construct his category of motives and would imply that algebraic cycles play a vital role in any cohomology theory of algebraic varieties. Conversely, Alexander Beilinson proved that the existence of a category of motives implies the standard conjectures. Additionally, cycles are connected to algebraic K-theory by Bloch's formula, which expresses groups of cycles modulo rational equivalence as the cohomology of K-theory sheaves. Definition Let X be a scheme which is finite type over a field k. An algebraic r-cycle on X is
https://en.wikipedia.org/wiki/Shortest-path%20tree	In mathematics and computer science, a shortest-path tree rooted at a vertex v of a connected, undirected graph G is a spanning tree T of G, such that the path distance from root v to any other vertex u in T is the shortest path distance from v to u in G. In connected graphs where shortest paths are well-defined (i.e. where there are no negative-length cycles), we may construct a shortest-path tree using the following algorithm: Compute dist(u), the shortest-path distance from root v to vertex u in G using Dijkstra's algorithm or Bellman–Ford algorithm. For all non-root vertices u, we can assign to u a parent vertex pu such that pu is connected to u, and that dist(pu) + edge_dist(pu,u) = dist(u). In case multiple choices for pu exist, choose pu for which there exists a shortest path from v to pu with as few edges as possible; this tie-breaking rule is needed to prevent loops when there exist zero-length cycles. Construct the shortest-path tree using the edges between each node and its parent. The above algorithm guarantees the existence of shortest-path trees. Like minimum spanning trees, shortest-path trees in general are not unique. In graphs for which all edge weights are equal, shortest path trees coincide with breadth-first search trees. In graphs that have negative cycles, the set of shortest simple paths from v to all other vertices do not necessarily form a tree. For simple connected graphs, shortest-path trees can be used to suggest a non-linear relationship between two network centrality measures, closeness and degree. By assuming that the branches of the shortest-path trees are statistically similar for any root node in one network, one may show that the size of the branches depend only on the number of branches connected to the root vertex, i.e. to the degree of the root node. From this one deduces that the inverse of closeness, a length scale associated with each vertex, varies approximately linearly with the logarithm of degree. The relationship is not exact but it captures a correlation between closeness and degree in large number of networks constructed from real data and this success suggests that shortest-path trees can be a useful approximation in network analysis. See also Shortest path problem References References Spanning tree
https://en.wikipedia.org/wiki/Kaplansky%20density%20theorem	In the theory of von Neumann algebras, the Kaplansky density theorem, due to Irving Kaplansky, is a fundamental approximation theorem. The importance and ubiquity of this technical tool led Gert Pedersen to comment in one of his books that, The density theorem is Kaplansky's great gift to mankind. It can be used every day, and twice on Sundays. Formal statement Let K− denote the strong-operator closure of a set K in B(H), the set of bounded operators on the Hilbert space H, and let (K)1 denote the intersection of K with the unit ball of B(H). Kaplansky density theorem. If is a self-adjoint algebra of operators in , then each element in the unit ball of the strong-operator closure of is in the strong-operator closure of the unit ball of . In other words, . If is a self-adjoint operator in , then is in the strong-operator closure of the set of self-adjoint operators in . The Kaplansky density theorem can be used to formulate some approximations with respect to the strong operator topology. 1) If h is a positive operator in (A−)1, then h is in the strong-operator closure of the set of self-adjoint operators in (A+)1, where A+ denotes the set of positive operators in A. 2) If A is a C-algebra acting on the Hilbert space H and u is a unitary operator in A−, then u is in the strong-operator closure of the set of unitary operators in A. In the density theorem and 1) above, the results also hold if one considers a ball of radius r > 0, instead of the unit ball. Proof The standard proof uses the fact that a bounded continuous real-valued function f is strong-operator continuous. In other words, for a net {aα} of self-adjoint operators in A, the continuous functional calculus a → f(a) satisfies, in the strong operator topology. This shows that self-adjoint part of the unit ball in A− can be approximated strongly by self-adjoint elements in A. A matrix computation in M2(A) considering the self-adjoint operator with entries 0 on the diagonal and a and a at the other positions, then removes the self-adjointness restriction and proves the theorem. See also Jacobson density theorem Notes References Kadison, Richard, Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, American Mathematical Society. . V.F.R.Jones von Neumann algebras; incomplete notes from a course. M. Takesaki Theory of Operator Algebras I Von Neumann algebras Theorems in functional analysis
https://en.wikipedia.org/wiki/Fakhrul%20Mulk	Nizamuddin Fakhrul Mulk () was an Islamic nobleman of the 11th and 12th century. His court included the algebra scholar Abu Bakr Karaji. Following the 1015 death of poet and ruler Sayyid Radi, Mulk was tasked with bringing his brother Sayyid Murtada back home. He was the recipient of at least five letters from the Sufi scholar al-Ghazali. References Muslim monarchs
https://en.wikipedia.org/wiki/Spectrum%20of%20theistic%20probability	Popularized by Richard Dawkins in The God Delusion, the spectrum of theistic probability is a way of categorizing one's belief regarding the probability of the existence of a deity. Atheism, theism, and agnosticism J. J. C. Smart argues that the distinction between atheism and agnosticism is unclear, and many people who have passionately described themselves as agnostics were in fact atheists. He writes that this mischaracterization is based on an unreasonable philosophical skepticism that would not allow us to make any claims to knowledge about the world. He proposes instead the following analysis: Let us consider the appropriateness or otherwise of someone (call him 'Philo') describing himself as a theist, atheist or agnostic. I would suggest that if Philo estimates the various plausibilities to be such that on the evidence before him the probability of theism comes out near to one he should describe himself as a theist and if it comes out near zero he should call himself an atheist, and if it comes out somewhere in the middle he should call himself an agnostic. There are no strict rules about this classification because the borderlines are vague. If need be, like a middle-aged man who is not sure whether to call himself bald or not bald, he should explain himself more fully. Dawkins' formulation In The God Delusion, Richard Dawkins posits that "the existence of God is a scientific hypothesis like any other." He goes on to propose a continuous "spectrum of probabilities" between two extremes of opposite certainty, which can be represented by seven "milestones". Dawkins suggests definitive statements to summarize one's place along the spectrum of theistic probability. These "milestones" are: Strong theist. 100% probability of God. In the words of Carl Jung: "I do not believe, I know." De facto theist. Very high probability but short of 100%. "I don't know for certain, but I strongly believe in God and live my life on the assumption that he is there." Leaning towards theism. Higher than 50% but not very high. "I am very uncertain, but I am inclined to believe in God." Completely impartial. Exactly 50%. "God's existence and non-existence are exactly equiprobable." Leaning towards atheism. Lower than 50% but not very low. "I do not know whether God exists but I'm inclined to be skeptical." De facto atheist. Very low probability, but short of zero. "I don't know for certain but I think God is very improbable, and I live my life on the assumption that he is not there." Strong atheist. "I know there is no God, with the same conviction as Jung knows there is one." Dawkins argues that while there appear to be plenty of individuals that would place themselves as "1" due to the strictness of religious doctrine against doubt, most atheists do not consider themselves "7" because atheism arises from a lack of evidence and evidence can always change a thinking person's mind. In print, Dawkins self-identified as a "6", though when interviewed by Bill
https://en.wikipedia.org/wiki/Cork%20county%20hurling%20team%20records%20and%20statistics	This is a list of Cork's record in the Munster and All-Ireland Senior Hurling Championship over the last few years. Overview 2000s 1990s 1980s 1970s 1960s 1950s Players Most championship appearances External links Cork GAA website Records and statistics County hurling team records and statistics
https://en.wikipedia.org/wiki/History%20of%20algebra	Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra (in fact, every proof must use the completeness of the real numbers, which is not an algebraic property). This article describes the history of the theory of equations, called here "algebra", from the origins to the emergence of algebra as a separate area of mathematics. Etymology The word "algebra" is derived from the Arabic word الجبر al-jabr, and this comes from the treatise written in the year 830 by the medieval Persian mathematician, Al-Khwārizmī, whose Arabic title, Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala, can be translated as The Compendious Book on Calculation by Completion and Balancing. The treatise provided for the systematic solution of linear and quadratic equations. According to one history, "[i]t is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the previous translation. The word 'al-jabr' presumably meant something like 'restoration' or 'completion' and seems to refer to the transposition of subtracted terms to the other side of an equation; the word 'muqabalah' is said to refer to 'reduction' or 'balancing'—that is, the cancellation of like terms on opposite sides of the equation. Arabic influence in Spain long after the time of al-Khwarizmi is found in Don Quixote, where the word 'algebrista' is used for a bone-setter, that is, a 'restorer'." The term is used by al-Khwarizmi to describe the operations that he introduced, "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. Stages of algebra Algebraic expression Algebra did not always make use of the symbolism that is now ubiquitous in mathematics; instead, it went through three distinct stages. The stages in the development of symbolic algebra are approximately as follows: Rhetorical algebra, in which equations are written in full sentences. For example, the rhetorical form of is "The thing plus one equals two" or possibly "The thing plus 1 equals 2". Rhetorical algebra was first developed by the ancient Babylonians and remained dominant up to the 16th century. Syncopated algebra, in which some symbolism is used, but which does not contain all of the characteristics of symbolic algebra. For instance, there may be a restriction that subtraction may be used only once within one side of an equation, which is not the case with symbolic algebra. Syncopated algebraic expression first appeared in Diophantus' Arithmetica (3rd century AD), followed by Brahmagupta's Brahma Sphuta Siddhanta (7th century).
https://en.wikipedia.org/wiki/Generator%20%28category%20theory%29	In mathematics, specifically category theory, a family of generators (or family of separators) of a category is a collection of objects in , such that for any two distinct morphisms in , that is with , there is some in and some morphism such that If the collection consists of a single object , we say it is a generator (or separator). Generators are central to the definition of Grothendieck categories. The dual concept is called a cogenerator or coseparator. Examples In the category of abelian groups, the group of integers is a generator: If f and g are different, then there is an element , such that . Hence the map suffices. Similarly, the one-point set is a generator for the category of sets. In fact, any nonempty set is a generator. In the category of sets, any set with at least two elements is a cogenerator. In the category of modules over a ring R, a generator in a finite direct sum with itself contains an isomorphic copy of R as a direct summand. Consequently, a generator module is faithful, i.e. has zero annihilator. References , p. 123, section V.7 External links Category theory
https://en.wikipedia.org/wiki/1%20%2B%202%20%2B%204%20%2B%208%20%2B%20%E2%8B%AF	In mathematics, is the infinite series whose terms are the successive powers of two. As a geometric series, it is characterized by its first term, 1, and its common ratio, 2. As a series of real numbers it diverges to infinity, so the sum of this series is infinity. However, it can be manipulated to yield a number of mathematically interesting results. For example, many summation methods are used in mathematics to assign numerical values even to a divergent series. For example, the Ramanujan summation of this series is −1, which is the limit of the series using the 2-adic metric. Summation The partial sums of are since these diverge to infinity, so does the series. It is written as : Therefore, any totally regular summation method gives a sum of infinity, including the Cesàro sum and Abel sum. On the other hand, there is at least one generally useful method that sums to the finite value of −1. The associated power series has a radius of convergence around 0 of only so it does not converge at Nonetheless, the so-defined function has a unique analytic continuation to the complex plane with the point deleted, and it is given by the same rule Since the original series is said to be summable (E) to −1, and −1 is the (E) sum of the series. (The notation is due to G. H. Hardy in reference to Leonhard Euler's approach to divergent series). An almost identical approach (the one taken by Euler himself) is to consider the power series whose coefficients are all 1, that is, and plugging in These two series are related by the substitution The fact that (E) summation assigns a finite value to shows that the general method is not totally regular. On the other hand, it possesses some other desirable qualities for a summation method, including stability and linearity. These latter two axioms actually force the sum to be −1, since they make the following manipulation valid: In a useful sense, is a root of the equation (For example, is one of the two fixed points of the Möbius transformation on the Riemann sphere). If some summation method is known to return an ordinary number for ; that is, not then it is easily determined. In this case may be subtracted from both sides of the equation, yielding so The above manipulation might be called on to produce −1 outside the context of a sufficiently powerful summation procedure. For the most well-known and straightforward sum concepts, including the fundamental convergent one, it is absurd that a series of positive terms could have a negative value. A similar phenomenon occurs with the divergent geometric series (Grandi's series), where a series of integers appears to have the non-integer sum These examples illustrate the potential danger in applying similar arguments to the series implied by such recurring decimals as and most notably . The arguments are ultimately justified for these convergent series, implying that and but the underlying proofs demand careful thinking about the in
https://en.wikipedia.org/wiki/Artin%E2%80%93Rees%20lemma	In mathematics, the Artin–Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis theorem. It was proved in the 1950s in independent works by the mathematicians Emil Artin and David Rees; a special case was known to Oscar Zariski prior to their work. An intuitive characterization of the lemma involves the notion that a submodule N of a module M over some ring A with specified ideal I holds a priori two topologies: one induced by the topology on M, and the other when considered with the I-adic topology over A. Then Artin-Rees dictates that these topologies actually coincide, at least when A is Noetherian and M finitely-generated. One consequence of the lemma is the Krull intersection theorem. The result is also used to prove the exactness property of completion. The lemma also plays a key role in the study of ℓ-adic sheaves. Statement Let I be an ideal in a Noetherian ring R; let M be a finitely generated R-module and let N a submodule of M. Then there exists an integer k ≥ 1 so that, for n ≥ k, Proof The lemma immediately follows from the fact that R is Noetherian once necessary notions and notations are set up. For any ring R and an ideal I in R, we set (B for blow-up.) We say a decreasing sequence of submodules is an I-filtration if ; moreover, it is stable if for sufficiently large n. If M is given an I-filtration, we set ; it is a graded module over . Now, let M be a R-module with the I-filtration by finitely generated R-modules. We make an observation is a finitely generated module over if and only if the filtration is I-stable. Indeed, if the filtration is I-stable, then is generated by the first terms and those terms are finitely generated; thus, is finitely generated. Conversely, if it is finitely generated, say, by some homogeneous elements in , then, for , each f in can be written as with the generators in . That is, . We can now prove the lemma, assuming R is Noetherian. Let . Then are an I-stable filtration. Thus, by the observation, is finitely generated over . But is a Noetherian ring since R is. (The ring is called the Rees algebra.) Thus, is a Noetherian module and any submodule is finitely generated over ; in particular, is finitely generated when N is given the induced filtration; i.e., . Then the induced filtration is I-stable again by the observation. Krull's intersection theorem Besides the use in completion of a ring, a typical application of the lemma is the proof of the Krull's intersection theorem, which says: for a proper ideal I in a commutative Noetherian ring that is either a local ring or an integral domain. By the lemma applied to the intersection , we find k such that for , Taking , this means or . Thus, if A is local, by Nakayama's lemma. If A is an integral domain, then one uses the determinant trick (that is a variant of the Cayley–Hamilton theorem and yields Nakayama's lemma): In the setup here, take u to be the identity
https://en.wikipedia.org/wiki/Maclaurin%27s%20inequality	In mathematics, Maclaurin's inequality, named after Colin Maclaurin, is a refinement of the inequality of arithmetic and geometric means. Let a1, a2, ..., an be positive real numbers, and for k = 1, 2, ..., n define the averages Sk as follows: The numerator of this fraction is the elementary symmetric polynomial of degree k in the n variables a1, a2, ..., an, that is, the sum of all products of k of the numbers a1, a2, ..., an with the indices in increasing order. The denominator is the number of terms in the numerator, the binomial coefficient Maclaurin's inequality is the following chain of inequalities: with equality if and only if all the ai are equal. For n = 2, this gives the usual inequality of arithmetic and geometric means of two numbers. Maclaurin's inequality is well illustrated by the case n = 4: Maclaurin's inequality can be proved using Newton's inequalities or generalised Bernoulli's inequality. See also Newton's inequalities Muirhead's inequality Generalized mean inequality Bernoulli's inequality References Real analysis Inequalities Symmetric functions
https://en.wikipedia.org/wiki/Dugald%20Macpherson	H. Dugald Macpherson is a mathematician and logician. He is Professor of Pure Mathematics at the University of Leeds. He obtained his DPhil from the University of Oxford in 1983 for his thesis entitled "Enumeration of Orbits of Infinite Permutation Groups" under the supervision of Peter Cameron. In 1997, he was awarded the Junior Berwick Prize by the London Mathematical Society. He continues to research into permutation groups and model theory. He is scientist in charge of the MODNET team at the University of Leeds. He co-authored the book Notes on Infinite Permutation Groups. References External links Prof. Macpherson's homepage Year of birth missing (living people) 20th-century British mathematicians 21st-century British mathematicians Living people Alumni of the University of Oxford Academics of the University of Leeds Model theorists Place of birth missing (living people)
https://en.wikipedia.org/wiki/Frobenius%20matrix	A Frobenius matrix is a special kind of square matrix from numerical mathematics. A matrix is a Frobenius matrix if it has the following three properties: all entries on the main diagonal are ones the entries below the main diagonal of at most one column are arbitrary every other entry is zero The following matrix is an example. Frobenius matrices are invertible. The inverse of a Frobenius matrix is again a Frobenius matrix, equal to the original matrix with changed signs outside the main diagonal. The inverse of the example above is therefore: Frobenius matrices are named after Ferdinand Georg Frobenius. The term Frobenius matrix may also be used for an alternative matrix form that differs from an Identity matrix only in the elements of a single row preceding the diagonal entry of that row (as opposed to the above definition which has the matrix differing from the identity matrix in a single column below the diagonal). The following matrix is an example of this alternative form showing a 4-by-4 matrix with its 3rd row differing from the identity matrix. An alternative name for this latter form of Frobenius matrices is Gauss transformation matrix, after Carl Friedrich Gauss. They are used in the process of Gaussian elimination to represent the Gaussian transformations. If a matrix is multiplied from the left (left multiplied) with a Gauss transformation matrix, a linear combination of the preceding rows is added to the given row of the matrix (in the example shown above, a linear combination of rows 1 and 2 will be added to row 3). Multiplication with the inverse matrix subtracts the corresponding linear combination from the given row. This corresponds to one of the elementary operations of Gaussian elimination (besides the operation of transposing the rows and multiplying a row with a scalar multiple). See also Elementary matrix, a special case of a Frobenius matrix with only one off-diagonal nonzero Notes References Gene H. Golub and Charles F. Van Loan (1996). Matrix Computations, third edition, Johns Hopkins University Press. (hardback), (paperback). Matrices
https://en.wikipedia.org/wiki/Base%20%28geometry%29	In geometry, a base is a side of a polygon or a face of a polyhedron, particularly one oriented perpendicular to the direction in which height is measured, or on what is considered to be the "bottom" of the figure. This term is commonly applied to triangles, parallelograms, trapezoids, cylinders, cones, pyramids, parallelepipeds and frustums. The side or point opposite the base is often called the apex or summit of the shape. Of a triangle In a triangle, any arbitrary side can be considered the base. The two endpoints of the base are called base vertices and the corresponding angles are called base angles. The third vertex opposite the base is called the apex. The extended base of a triangle (a particular case of an extended side) is the line that contains the base. When the triangle is obtuse and the base is chosen to be one of the sides adjacent to the obtuse angle, then the altitude dropped perpendicularly from the apex to the base intersects the extended base outside of the triangle. The area of a triangle is its half of the product of the base times the height (length of the altitude). For a triangle with opposite sides if the three altitudes of the triangle are called the area is: Given a fixed base side and a fixed area for a triangle, the locus of apex points is a straight line parallel to the base. Of a trapezoid or parallelogram Any of the sides of a parallelogram, or either (but typically the longer) of the parallel sides of a trapezoid can be considered its base. Sometimes the parallel opposite side is also called a base, or sometimes it is called a top, apex, or summit. The other two edges can be called the sides. Role in area and volume calculation Bases are commonly used (together with heights) to calculate the areas and volumes of figures. In speaking about these processes, the measure (length or area) of a figure's base is often referred to as its "base." By this usage, the area of a parallelogram or the volume of a prism or cylinder can be calculated by multiplying its "base" by its height; likewise, the areas of triangles and the volumes of cones and pyramids are fractions of the products of their bases and heights. Some figures have two parallel bases (such as trapezoids and frustums), both of which are used to calculate the extent of the figures. References Parts of a triangle Area Volume
https://en.wikipedia.org/wiki/Robert%20P.%20Dilworth	Robert Palmer Dilworth (December 2, 1914 – October 29, 1993) was an American mathematician. His primary research area was lattice theory; his biography at the MacTutor History of Mathematics archive states "it would not be an exaggeration to say that he was one of the main factors in the subject moving from being merely a tool of other disciplines to an important subject in its own right". He is best known for Dilworth's theorem relating chains and antichains in partial orders; he was also the first to study antimatroids . Dilworth was born in 1914 in Hemet, California, at that time a remote desert ranching town. He went to college at the California Institute of Technology, receiving his baccalaureate in 1936 and continuing there for his graduate studies. Dilworth's graduate advisor was Morgan Ward, a student of Eric Temple Bell, who was also on the Caltech faculty at the time. On receiving his Ph.D. in 1939, Dilworth took an instructorship at Yale University. While at Yale, he met and married his wife, Miriam White, with whom he eventually had two sons. He returned to Caltech as a faculty member in 1943, and spent the remainder of his academic career there. Dilworth advised 17 Ph.D. students and has 635 academic descendants listed at the Mathematics Genealogy Project, many through his student Juris Hartmanis, a noted complexity theorist. Other notable mathematicians advised by Dilworth include Curtis Greene and Alfred W. Hales. Selected bibliography . . . . . References and external links . . 20th-century American mathematicians Lattice theorists 1914 births 1993 deaths California Institute of Technology alumni California Institute of Technology faculty Yale University faculty People from Hemet, California Mathematicians from California
https://en.wikipedia.org/wiki/Bornology	In mathematics, especially functional analysis, a bornology on a set X is a collection of subsets of X satisfying axioms that generalize the notion of boundedness. One of the key motivations behind bornologies and bornological analysis is the fact that bornological spaces provide a convenient setting for homological algebra in functional analysis. This is becausepg 9 the category of bornological spaces is additive, complete, cocomplete, and has a tensor product adjoint to an internal hom, all necessary components for homological algebra. History Bornology originates from functional analysis. There are two natural ways of studying the problems of functional analysis: one way is to study notions related to topologies (vector topologies, continuous operators, open/compact subsets, etc.) and the other is to study notions related to boundedness (vector bornologies, bounded operators, bounded subsets, etc.). For normed spaces, from which functional analysis arose, topological and bornological notions are distinct but complementary and closely related. For example, the unit ball centered at the origin is both a neighborhood of the origin and a bounded subset. Furthermore, a subset of a normed space is a neighborhood of the origin (respectively, is a bounded set) exactly when it contains (respectively, it is contained in) a non-zero scalar multiple of this ball; so this is one instance where the topological and bornological notions are distinct but complementary (in the sense that their definitions differ only by which of and is used). Other times, the distinction between topological and bornological notions may even be unnecessary. For example, for linear maps between normed spaces, being continuous (a topological notion) is equivalent to being bounded (a bornological notion). Although the distinction between topology and bornology is often blurred or unnecessary for normed space, it becomes more important when studying generalizations of normed spaces. Nevertheless, bornology and topology can still be thought of as two necessary, distinct, and complementary aspects of one and the same reality. The general theory of topological vector spaces arose first from the theory of normed spaces and then bornology emerged from this general theory of topological vector spaces, although bornology has since become recognized as a fundamental notion in functional analysis. Born from the work of George Mackey (after whom Mackey spaces are named), the importance of bounded subsets first became apparent in duality theory, especially because of the Mackey–Arens theorem and the Mackey topology. Starting around the 1950s, it became apparent that topological vector spaces were inadequate for the study of certain major problems. For example, the multiplication operation of some important topological algebras was not continuous, although it was often bounded. Other major problems for which TVSs were found to be inadequate was in developing a more general theo
https://en.wikipedia.org/wiki/Box%20product	Box product may refer to: The scalar triple product of three vectors A cartesian product of topological spaces equipped with the box topology The cartesian product of graphs
https://en.wikipedia.org/wiki/Higher-order%20statistics	In statistics, the term higher-order statistics (HOS) refers to functions which use the third or higher power of a sample, as opposed to more conventional techniques of lower-order statistics, which use constant, linear, and quadratic terms (zeroth, first, and second powers). The third and higher moments, as used in the skewness and kurtosis, are examples of HOS, whereas the first and second moments, as used in the arithmetic mean (first), and variance (second) are examples of low-order statistics. HOS are particularly used in the estimation of shape parameters, such as skewness and kurtosis, as when measuring the deviation of a distribution from the normal distribution. In statistical theory, one long-established approach to higher-order statistics, for univariate and multivariate distributions is through the use of cumulants and joint cumulants. In time series analysis, the extension of these is to higher order spectra, for example the bispectrum and trispectrum. An alternative to the use of HOS and higher moments is to instead use L-moments, which are linear statistics (linear combinations of order statistics), and thus more robust than HOS. References External links http://www.maths.leeds.ac.uk/Applied/news.dir/issue2/hos_intro.html https://web.archive.org/web/20061125033107/http://lpce.cnrs-orleans.fr/~ddwit/lalonde/lalonde_presentations/horbury2.pdf http://www.ics.uci.edu/~welling/publications/papers/RobCum-aistats.pdf Summary statistics
https://en.wikipedia.org/wiki/European%20association%20football%20club%20records%20and%20statistics	This article details men's professional football club records and statistics (individual and collective) in Europe. The records and stats look across all European clubs competing in the highest divisions and levels of European professional football, allowing for cross-competition comparison. Therefore, the coverage only considers for domestic competitions the top-division of the national league and its cups (, league cup, super cup); for continental competitions, all UEFA club competitions including – although recognized but not organized by UEFA – the Fairs Cup as the predecessor to the UEFA Cup; and additionally, on an intercontinental scale, both the FIFA Club World Cup and its defunct predecessor, the Intercontinental Cup, which was endorsed by UEFA (Europe) and CONMEBOL (South America). All competitions for men's european football clubs Individual records Most goals in a season in all club competitions Only the period starting from the implementation of the modern offside rule in 1925 is considered for this list. Under the revised offside rule introduced in 1925, a player would be deemed offside unless there were two opposing players (including the goalkeeper) positioned ahead of them. The list refers to goals in all national club competitions , organized by UEFA (excluding UEFA qualifying rounds) as the predecessor of the and club competitions (excluding the International Champions Cup) Does not include goals scored in the , in invitational tournaments and in the national team Club records Most consecutive national league titles Source: 14 – Skonto Riga (1991–2004) 14 – Lincoln Red Imps (2003–2016) 13 – Rosenborg (1992–2004) 13 – BATE Borisov (2006–2018) 12 – Ludogorets Razgrad (2012–present) 11 – Dinamo Zagreb (2006–2016) 11 – Bayern Munich (2013–present) 10 – MTK Budapest (1914, 1917–1925) 10 – BFC Dynamo (1979–1988) 10 – Dinamo Tbilisi (1990–1999) 10 – Pyunik (2001–2010) 10 – Sheriff Tiraspol (2001–2010) 10 – Red Bull Salzburg (2014–present) Longest unbeaten run across all competitions Source: 62 – Celtic (1915–1917) 60 – Union SG (1933–1935) After the introduction of UEFA club competitions (1955–56) 48 – Benfica (1963–1965) 45 – Dinamo Zagreb (2014–2015) 45 – Rijeka (2016–2017) 44 – Rangers (1992–1993) 43 – Juventus (2011–2012) 42 – Milan (1991–1992) 42 – Ajax (1995–1996) 40 – Fiorentina (1955–1956) 40 – Nottingham Forest (1978) 40 – Real Madrid (2016–2017) 40 – Red Star Belgrade (2020–2021) Most consecutive wins across all competitions Italic denotes record that was not achieved in country's top tier (unofficial record for non-professional leagues). 36 – Jersey Bulls in 2019–20 and 2020–21 32 – South Shields in 2016–17 27 – Hereford in 2015–16 27 – East Kilbride in 2016–17 27 – The New Saints in 2016–17 26 – Dresdner SC in 1942–43 26 – Ajax in 1971–72 26 – Salisbury City reserves in 2007–08 23 – Red Star Belgrade in 1999–2000 and 2000–01 23 – Bayern Munich in 2019–20 and 2020–21
https://en.wikipedia.org/wiki/MME	MME may stand for: M or Mme, the French abbreviation for Madame MME, the IATA code for Teesside International Airport, United Kingdom MME, Maths Made Easy, an academic resource provider in the United Kingdom MME (psychedelic), 2,4-dimethoxy-5-ethoxyamphetamine, a psychedelic drug Magyar Madártani és Természetvédelmi Egyesület, Hungarian Ornithological and Nature Conservation Society Mass mortality event, the naturally occurring death of large numbers of a species in a short period of time Master of Mechanical Engineering, a postgraduate degree; see British degree abbreviations Membrane metallo-endopeptidase, an enzyme also named neprilysin Michigan Merit Exam, a minimum-competency test for students Mitsubishi Motors Europe, the European subsidiary of Mitsubishi Motors Mobility Management Entity, a standardized entity in a System Architecture Evolution network dedicated to mobility management Morphine milligram equivalents, a measure of opioid potency utilized in pain management Middle East Eye, a London-based online news outlet covering events in the Middle East Multi-model ensemble, a climate ensemble used in climate change research Multimedia Extensions, an extension to Windows 3.0 to support multimedia VEB Mikroelektronik "Karl Marx" Erfurt, a division of Kombinat Mikroelektronik Erfurt, in East Germany Middle Market Enterprises, 250-2500 FTE, 50M – 300M turnover
https://en.wikipedia.org/wiki/Fair%20coin	In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin. In theoretical studies, the assumption that a coin is fair is often made by referring to an ideal coin. John Edmund Kerrich performed experiments in coin flipping and found that a coin made from a wooden disk about the size of a crown and coated on one side with lead landed heads (wooden side up) 679 times out of 1000. In this experiment the coin was tossed by balancing it on the forefinger, flipping it using the thumb so that it spun through the air for about a foot before landing on a flat cloth spread over a table. Edwin Thompson Jaynes claimed that when a coin is caught in the hand, instead of being allowed to bounce, the physical bias in the coin is insignificant compared to the method of the toss, where with sufficient practice a coin can be made to land heads 100% of the time. Exploring the problem of checking whether a coin is fair is a well-established pedagogical tool in teaching statistics. Probability space definition In probability theory, a fair coin is defined as a probability space , which is in turn defined by the sample space, event space, and probability measure. Using for heads and for tails, the sample space of a coin is defined as: The event space for a coin includes all sets of outcomes from the sample space which can be assigned a probability, which is the full power set . Thus, the event space is defined as: is the event where neither outcome happens (which is impossible and can therefore be assigned 0 probability), and is the event where either outcome happens, (which is guaranteed and can be assigned 1 probability). Because the coin is fair, the possibility of any single outcome is 50-50. The probability measure is then defined by the function: So the full probability space which defines a fair coin is the triplet as defined above. Note that this is not a random variable because heads and tails don't have inherent numerical values like you might find on a fair two-valued die. A random variable adds the additional structure of assigning a numerical value to each outcome. Common choices are or . Role in statistical teaching and theory The probabilistic and statistical properties of coin-tossing games are often used as examples in both introductory and advanced text books and these are mainly based in assuming that a coin is fair or "ideal". For example, Feller uses this basis to introduce both the idea of random walks and to develop tests for homogeneity within a sequence of observations by looking at the properties of the runs of identical values within a sequence. The latter leads on to a runs test. A time-series consisting of the result from tossing a fair coin is called a Bernoulli process. Fair results from a biased coin If a cheat has altered a coin to prefer
https://en.wikipedia.org/wiki/Franco%20P.%20Preparata	Franco P. Preparata is a computer scientist, the An Wang Professor, Emeritus, of Computer Science at Brown University. He is best known for his 1985 book "Computational Geometry: An Introduction" into which he blended salient parts of M. I. Shamos' doctoral thesis (Shamos appears as a co-author of the book). This book, which represents a snapshot of the disciplines as of 1985, has been for many years the standard textbook in the field, and has been translated into four foreign Languages (Russian, Japanese, Chinese, and Polish). He has made several contributions to the computational geometry, the most recent being the notion of "algorithmic degree" as a key feature to control robust implementations of geometric algorithms. In addition, Preparata has worked in many other areas of, or closely related to, computer science. His initial work was in coding theory, where he (independently and simultaneously) contributed the Berlekamp-Preparata codes (optimal convolution codes for burst-error correction) and the Preparata codes, the first known systematic class of nonlinear binary codes, with higher information content than corresponding linear BCH codes of the same length. Thirty years later these codes have been found relevant to quantum coding theory. In 1967, he substantially contributed to a model of system-level fault diagnosis, known today as the PMC (Preparata-Metze-Chien) model, which is a main issue in the design of highly dependable processing systems. This model is still the object of intense research today (as attested by the literature). Over the years, he was also active in research in parallel computation and VLSI theory. His 1979 paper (with Jean Vuillemin), still highly cited, presented the cube-connected-cycles (CCC), a parallel architecture that optimally emulates the hypercube interconnection. This interconnection was closely reflected in the architecture of the CM2 of Thinking Machines Inc., the first massive-parallel system in the VLSI era. His 1991 paper with Zhou and Kang on interconnection delays in VLSI was awarded the 1993 "Darlington Best Paper Award" by the IEEE Circuits and Systems Society. In the late nineties, (in joint work with G. Bilardi) he confronted the problem of the physical limitations (space and speed) of parallel computation, and formulated the conclusion that mesh connections are ultimately the only scalable massively parallel architectures. More recently the focus of his research has been Computational Biology. Among other results, he contributed (with Eli Upfal) a novel approach to DNA Sequencing by Hybridization, achieving sequencing lengths that are the square of what was previously known, which has attracted media coverage. The unifying character of these results in diverse research areas is the methodological approach, based on the construction of precise mathematical models and the use of sophisticated mathematical techniques. Preparata was born in Italy in December, 1935. He received a doctor
https://en.wikipedia.org/wiki/Adrien%20Pouliot%20Award	The Adrien Pouliot Award is presented annually by the Canadian Mathematical Society. The award is presented to individuals or teams in recognition of significant contributions to mathematics education in Canada. The inaugural award was presented in 1995. Persons and teams that are nominated for the award will have their applications considered for a period of three years. The award is named in honor of Canadian mathematician Adrien Pouliot. It should be distinguished with a different but similarly-named award, the Adrien Pouliot Prize of the Mathematical Association of Québec. Recipients of the Adrien Pouliot Award Source: Canadian Mathematical Society See also List of mathematics awards References External links Canadian Mathematical Society Awards of the Canadian Mathematical Society Mathematics education awards Teacher awards Awards established in 1995
https://en.wikipedia.org/wiki/Atomic%20domain	In mathematics, more specifically ring theory, an atomic domain or factorization domain is an integral domain in which every non-zero non-unit can be written in at least one way as a finite product of irreducible elements. Atomic domains are different from unique factorization domains in that this decomposition of an element into irreducibles need not be unique; stated differently, an irreducible element is not necessarily a prime element. Important examples of atomic domains include the class of all unique factorization domains and all Noetherian domains. More generally, any integral domain satisfying the ascending chain condition on principal ideals (ACCP) is an atomic domain. Although the converse is claimed to hold in Cohn's paper, this is known to be false. The term "atomic" is due to P. M. Cohn, who called an irreducible element of an integral domain an "atom". Motivation In this section, a ring can be viewed as merely an abstract set in which one can perform the operations of addition and multiplication; analogous to the integers. The ring of integers (that is, the set of integers with the natural operations of addition and multiplication) satisfy many important properties. One such property is the fundamental theorem of arithmetic. Thus, when considering abstract rings, a natural question to ask is under what conditions such a theorem holds. Since a unique factorization domain is precisely a ring in which an analogue of the fundamental theorem of arithmetic holds, this question is readily answered. However, one notices that there are two aspects of the fundamental theorem of the arithmetic: first, that any integer is the finite product of prime numbers, and second, that this product is unique up to rearrangement (and multiplication by units). Therefore, it is also natural to ask under what conditions particular elements of a ring can be "decomposed" without requiring uniqueness. The concept of an atomic domain addresses this. Definition Let R be an integral domain. If every non-zero non-unit x of R can be written as a product of irreducible elements, R is referred to as an atomic domain. (The product is necessarily finite, since infinite products are not defined in ring theory. Such a product is allowed to involve the same irreducible element more than once as a factor.) Any such expression is called a factorization of x. Special cases In an atomic domain, it is possible that different factorizations of the same element x have different lengths. It is even possible that among the factorizations of x there is no bound on the number of irreducible factors. If on the contrary the number of factors is bounded for every non-zero non-unit x, then R is a bounded factorization domain (BFD); formally this means that for each such x there exists an integer N such that if with none of the xi invertible then n < N. If such a bound exists, no chain of proper divisors from x to 1 can exceed this bound in length (since the quotient at every ste
https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7%20rule	In statistics, the 68–95–99.7 rule, also known as the empirical rule, is a shorthand used to remember the percentage of values that lie within an interval estimate in a normal distribution: 68%, 95%, and 99.7% of the values lie within one, two, and three standard deviations of the mean, respectively. In mathematical notation, these facts can be expressed as follows, where is the probability function, is an observation from a normally distributed random variable, (mu) is the mean of the distribution, and (sigma) is its standard deviation: The usefulness of this heuristic especially depends on the question under consideration. In the empirical sciences, the so-called three-sigma rule of thumb (or 3 rule) expresses a conventional heuristic that nearly all values are taken to lie within three standard deviations of the mean, and thus it is empirically useful to treat 99.7% probability as near certainty. In the social sciences, a result may be considered "significant" if its confidence level is of the order of a two-sigma effect (95%), while in particle physics, there is a convention of a five-sigma effect (99.99994% confidence) being required to qualify as a discovery. A weaker three-sigma rule can be derived from Chebyshev's inequality, stating that even for non-normally distributed variables, at least 88.8% of cases should fall within properly calculated three-sigma intervals. For unimodal distributions, the probability of being within the interval is at least 95% by the Vysochanskij–Petunin inequality. There may be certain assumptions for a distribution that force this probability to be at least 98%. Proof We have that doing the change of variable , we have and this integral is independent of and . We only need to calculate each integral for the cases . Cumulative distribution function These numerical values "68%, 95%, 99.7%" come from the cumulative distribution function of the normal distribution. The prediction interval for any standard score z corresponds numerically to . For example, , or , corresponding to a prediction interval of . This is not a symmetrical interval – this is merely the probability that an observation is less than . To compute the probability that an observation is within two standard deviations of the mean (small differences due to rounding): This is related to confidence interval as used in statistics: is approximately a 95% confidence interval when is the average of a sample of size . Normality tests The "68–95–99.7 rule" is often used to quickly get a rough probability estimate of something, given its standard deviation, if the population is assumed to be normal. It is also used as a simple test for outliers if the population is assumed normal, and as a normality test if the population is potentially not normal. To pass from a sample to a number of standard deviations, one first computes the deviation, either the error or residual depending on whether one knows the population mean or only e
https://en.wikipedia.org/wiki/Handshaking%20lemma	In graph theory, a branch of mathematics, the handshaking lemma is the statement that, in every finite undirected graph, the number of vertices that touch an odd number of edges is even. For example, if there is a party of people who shake hands, the number of people who shake an odd number of other people's hands is even. The handshaking lemma is a consequence of the degree sum formula, also sometimes called the handshaking lemma, according to which the sum of the degrees (the numbers of times each vertex is touched) equals twice the number of edges in the graph. Both results were proven by in his famous paper on the Seven Bridges of Königsberg that began the study of graph theory. Beyond the Seven Bridges of Königsberg Problem, which subsequently formalized Eulerian Tours, other applications of the degree sum formula include proofs of certain combinatorial structures. For example, in the proofs of Sperner's lemma and the mountain climbing problem the geometric properties of the formula commonly arise. The complexity class PPA encapsulates the difficulty of finding a second odd vertex, given one such vertex in a large implicitly-defined graph. Definitions and statement An undirected graph consists of a system of vertices, and edges connecting unordered pairs of vertices. In any graph, the degree of a vertex is defined as the number of edges that have as an endpoint. For graphs that are allowed to contain loops connecting a vertex to itself, a loop should be counted as contributing two units to the degree of its endpoint for the purposes of the handshaking lemma. Then, the handshaking lemma states that, in every finite graph, there must be an even number of vertices for which is an odd number. The vertices of odd degree in a graph are sometimes called odd nodes (or odd vertices); in this terminology, the handshaking lemma can be rephrased as the statement that every graph has an even number of odd nodes. The degree sum formula states that where is the set of nodes (or vertices) in the graph and is the set of edges in the graph. That is, the sum of the vertex degrees equals twice the number of edges. In directed graphs, another form of the degree-sum formula states that the sum of in-degrees of all vertices, and the sum of out-degrees, both equal the number of edges. Here, the in-degree is the number of incoming edges, and the out-degree is the number of outgoing edges. A version of the degree sum formula also applies to finite families of sets or, equivalently, multigraphs: the sum of the degrees of the elements (where the degree equals the number of sets containing it) always equals the sum of the cardinalities of the sets. Both results also apply to any subgraph of the given graph and in particular to its connected components. A consequence is that, for any odd vertex, there must exist a path connecting it to another odd vertex. Applications Euler paths and tours Leonhard Euler first proved the handshaking lemma in his work on t
https://en.wikipedia.org/wiki/Michael%20Kent%20%28computer%20specialist%29	Michael Kent was one of two founders of the Computer Group which used a statistics based sports betting to predict the outcome of college football. The group reportedly made millions each season. According to figures compiled at the time by Michael Kent, the Computer Group in 1983-84 earned almost $5 million from wagers on college and, occasionally, NFL games. Yet Michael Kent suspects that his records are incomplete. They do not account for personal bets made by Dr. Mindlin, or Billy Walters and Glen Walker or by the dozens of other associates who had access to the Computer Group's information. By the time everyone had exhausted Kent's forecasts in the 1983-84 sports year, the group was estimated to have earned $10 to $15 million. Kent invented the statistical models. He was 34 when he had created the first successful program for handicapping basketball and football games: together with his brother, Michael collected statistical data about every team to put all that info to his computer and update the program. The story was first reported by a national publication in the March 1986 Sports Illustrated. References External links Keyboard Cappers: A sports-betting history lesson, with a nod to the computer and the trailblazers who saw the future Gambling � The Story of the Computer Group Living people Year of birth missing (living people)
https://en.wikipedia.org/wiki/Uniform%20integrability	In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales. Measure-theoretic definition Uniform integrability is an extension to the notion of a family of functions being dominated in which is central in dominated convergence. Several textbooks on real analysis and measure theory use the following definition: Definition A: Let be a positive measure space. A set is called uniformly integrable if , and to each there corresponds a such that whenever and Definition A is rather restrictive for infinite measure spaces. A more general definition of uniform integrability that works well in general measures spaces was introduced by G. A. Hunt. Definition H: Let be a positive measure space. A set is called uniformly integrable if and only if where . For finite measure spaces the following result follows from Definition H: Theorem 1: If is a (positive) finite measure space, then a set is uniformly integrable if and only if Many textbooks in probability present Theorem 1 as the definition of uniform integrability in Probability spaces. When the space is -finite, Definition H yields the following equivalency: Theorem 2: Let be a -finite measure space, and be such that almost surely. A set is uniformly integrable if and only if , and for any , there exits such that whenever . In particular, the equivalence of Definitions A and H for finite measures follows immediately from Theorem 2; for this case, the statement in Definition A is obtained by taking in Theorem 2. Probability definition In the theory of probability, Definition A or the statement of Theorem 1 are often presented as definitions of uniform integrability using the notation expectation of random variables., that is, 1. A class of random variables is called uniformly integrable if: There exists a finite such that, for every in , and For every there exists such that, for every measurable such that and every in , . or alternatively 2. A class of random variables is called uniformly integrable (UI) if for every there exists such that , where is the indicator function . Tightness and uniform integrability One consequence of uniformly integrability of a class of random variables is that family of laws or distributions is tight. That is, for each , there exists such that for all . This however, does not mean that the family of measures is tight. (In any case, tightness would require a topology on in order to be defined.) Uniform absolute continuity There is another notion of uniformity, slightly different than uniform integrability, which also has many applications in probability and measure theory, and which does not require random variables to have a finite integral Definition: Suppose is a probability space. A classed of random variables is uniformly absolutely continuous with respect to if for any , there is
https://en.wikipedia.org/wiki/UCL%20Department%20of%20Science%20and%20Technology%20Studies	The UCL Department of Science and Technology Studies (STS) is an academic department in University College London, London, England. It is part of UCL's Faculty of Mathematics and Physical Sciences. The department offers academic training at both undergraduate and graduate (MSc and MPhil/PhD) levels. The department received its current name in 1995. It had been the "Department of History and Philosophy of Science" from 1938 to 1995, and the "Department of History and Method of Science" from 1921 to 1938. University College London was the first UK university to offer single honours undergraduate degrees in this interdisciplinary subject, launching its BSc in history and philosophy of science in 1993. Two related BSc degrees followed shortly thereafter. At UCL, science and technology studies (abbreviated "STS") includes three specialist research clusters: "history of science," "philosophy of science," and "science, culture, and democracy". In 2022 STS accepted its first cohort for an MSc in Science Communication. The department offices are located on UCL's campus in Gordon Square, Bloomsbury, London. References External links UCL Department of Science and Technology Studies website Educational institutions established in 1994 Science and Technology Studies History of science and technology in England Science and technology studies Science and technology in London 1994 establishments in England
https://en.wikipedia.org/wiki/Circumconic%20and%20inconic	In Euclidean geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle. Suppose are distinct non-collinear points, and let denote the triangle whose vertices are . Following common practice, denotes not only the vertex but also the angle at vertex , and similarly for and as angles in . Let the sidelengths of . In trilinear coordinates, the general circumconic is the locus of a variable point satisfying an equation for some point . The isogonal conjugate of each point on the circumconic, other than , is a point on the line This line meets the circumcircle of in 0,1, or 2 points according as the circumconic is an ellipse, parabola, or hyperbola. The general inconic is tangent to the three sidelines of and is given by the equation Centers and tangent lines Circumconic The center of the general circumconic is the point The lines tangent to the general circumconic at the vertices are, respectively, Inconic The center of the general inconic is the point The lines tangent to the general inconic are the sidelines of , given by the equations , , . Other features Circumconic Each noncircular circumconic meets the circumcircle of in a point other than , often called the fourth point of intersection, given by trilinear coordinates If is a point on the general circumconic, then the line tangent to the conic at is given by The general circumconic reduces to a parabola if and only if and to a rectangular hyperbola if and only if Of all triangles inscribed in a given ellipse, the centroid of the one with greatest area coincides with the center of the ellipse. The given ellipse, going through this triangle's three vertices and centered at the triangle's centroid, is called the triangle's Steiner circumellipse. Inconic The general inconic reduces to a parabola if and only if in which case it is tangent externally to one of the sides of the triangle and is tangent to the extensions of the other two sides. Suppose that and are distinct points, and let As the parameter ranges through the real numbers, the locus of is a line. Define The locus of is the inconic, necessarily an ellipse, given by the equation where A point in the interior of a triangle is the center of an inellipse of the triangle if and only if the point lies in the interior of the triangle whose vertices lie at the midpoints of the original triangle's sides. For a given point inside that medial triangle, the inellipse with its center at that point is unique. The inellipse with the largest area is the Steiner inellipse, also called the midpoint inellipse, with its center at the triangle's centroid. In general, the ratio of the inellipse's area to the triangle's area, in terms of the unit-sum barycentric coordinates of the inellipse's center, is which is maximized by the centroid's barycentric c
https://en.wikipedia.org/wiki/Morse%E2%80%93Palais%20lemma	In mathematics, the Morse–Palais lemma is a result in the calculus of variations and theory of Hilbert spaces. Roughly speaking, it states that a smooth enough function near a critical point can be expressed as a quadratic form after a suitable change of coordinates. The Morse–Palais lemma was originally proved in the finite-dimensional case by the American mathematician Marston Morse, using the Gram–Schmidt orthogonalization process. This result plays a crucial role in Morse theory. The generalization to Hilbert spaces is due to Richard Palais and Stephen Smale. Statement of the lemma Let be a real Hilbert space, and let be an open neighbourhood of the origin in Let be a -times continuously differentiable function with that is, Assume that and that is a non-degenerate critical point of that is, the second derivative defines an isomorphism of with its continuous dual space by Then there exists a subneighbourhood of in a diffeomorphism that is with inverse, and an invertible symmetric operator such that Corollary Let be such that is a non-degenerate critical point. Then there exists a -with--inverse diffeomorphism and an orthogonal decomposition such that, if one writes then See also References Calculus of variations Hilbert spaces Lemmas in analysis
https://en.wikipedia.org/wiki/Misiurewicz%20point	In mathematics, a Misiurewicz point is a parameter value in the Mandelbrot set (the parameter space of complex quadratic maps) and also in real quadratic maps of the interval for which the critical point is strictly pre-periodic (i.e., it becomes periodic after finitely many iterations but is not periodic itself). By analogy, the term Misiurewicz point is also used for parameters in a multibrot set where the unique critical point is strictly pre-periodic (This term makes less sense for maps in greater generality that have more than one free critical point because some critical points might be periodic and others not). These points are named after the Polish-American mathematician Michał Misiurewicz, who was the first to study them. Mathematical notation A parameter is a Misiurewicz point if it satisfies the equations: and: so: where: is a critical point of , and are positive integers, denotes the -th iterate of . Name The term "Misiurewicz point" is used ambiguously: Misiurewicz originally investigated maps in which all critical points were non-recurrent; that is, in which there exists a neighborhood for every critical point that is not visited by the orbit of this critical point. This meaning is firmly established in the context of the dynamics of iterated interval maps. Only in very special cases does a quadratic polynomial have a strictly periodic and unique critical point. In this restricted sense, the term is used in complex dynamics; a more appropriate one would be Misiurewicz–Thurston points (after William Thurston, who investigated post-critically finite rational maps). Quadratic maps A complex quadratic polynomial has only one critical point. By a suitable conjugation any quadratic polynomial can be transformed into a map of the form which has a single critical point at . The Misiurewicz points of this family of maps are roots of the equations: Subject to the condition that the critical point is not periodic, where: k is the pre-period n is the period denotes the n-fold composition of with itself i.e. the nth iteration of . For example, the Misiurewicz points with k=2 and n=1, denoted by M2,1, are roots of: The root c=0 is not a Misiurewicz point because the critical point is a fixed point when c=0, and so is periodic rather than pre-periodic. This leaves a single Misiurewicz point M2,1 at c = −2. Properties of Misiurewicz points of complex quadratic mapping Misiurewicz points belong to, and are dense in, the boundary of the Mandelbrot set. If is a Misiurewicz point, then the associated filled Julia set is equal to the Julia set and means the filled Julia set has no interior. If is a Misiurewicz point, then in the corresponding Julia set all periodic cycles are repelling (in particular the cycle that the critical orbit falls onto). The Mandelbrot set and Julia set are locally asymptotically self-similar around Misiurewicz points. Types Misiurewicz points can be classified according to several criteria. One
https://en.wikipedia.org/wiki/Skew%20apeirohedron	In geometry, a skew apeirohedron is an infinite skew polyhedron consisting of nonplanar faces or nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface. Skew apeirohedra have also been called polyhedral sponges. Many are directly related to a convex uniform honeycomb, being the polygonal surface of a honeycomb with some of the cells removed. Characteristically, an infinite skew polyhedron divides 3-dimensional space into two halves. If one half is thought of as solid the figure is sometimes called a partial honeycomb. Regular skew apeirohedra According to Coxeter, in 1926 John Flinders Petrie generalized the concept of regular skew polygons (nonplanar polygons) to regular skew polyhedra (apeirohedra). Coxeter and Petrie found three of these that filled 3-space: There also exist chiral skew apeirohedra of types {4,6}, {6,4}, and {6,6}. These skew apeirohedra are vertex-transitive, edge-transitive, and face-transitive, but not mirror symmetric . Beyond Euclidean 3-space, in 1967 C. W. L. Garner published a set of 31 regular skew polyhedra in hyperbolic 3-space. Gott's regular pseudopolyhedrons J. Richard Gott in 1967 published a larger set of seven infinite skew polyhedra which he called regular pseudopolyhedrons, including the three from Coxeter as {4,6}, {6,4}, and {6,6} and four new ones: {5,5}, {4,5}, {3,8}, {3,10}. Gott relaxed the definition of regularity to allow his new figures. Where Coxeter and Petrie had required that the vertices be symmetrical, Gott required only that they be congruent. Thus, Gott's new examples are not regular by Coxeter and Petrie's definition. Gott called the full set of regular polyhedra, regular tilings, and regular pseudopolyhedra as regular generalized polyhedra, representable by a {p,q} Schläfli symbol, with by p-gonal faces, q around each vertex. However neither the term "pseudopolyhedron" nor Gott's definition of regularity have achieved wide usage. Crystallographer A.F. Wells in 1960's also published a list of skew apeirohedra. Melinda Green published many more in 1998. Prismatic forms There are two prismatic forms: {4,5}: 5 squares on a vertex (Two parallel square tilings connected by cubic holes.) {3,8}: 8 triangles on a vertex (Two parallel triangle tilings connected by octahedral holes.) Other forms {3,10} is also formed from parallel planes of triangular tilings, with alternating octahedral holes going both ways. {5,5} is composed of 3 coplanar pentagons around a vertex and two perpendicular pentagons filling the gap. Gott also acknowledged that there are other periodic forms of the regular planar tessellations. Both the square tiling {4,4} and triangular tiling {3,6} can be curved into approximating infinite cylinders in 3-space. Theorems He wrote some theorems: For every regular polyhedron {p,q}: (p-2)(q-2)<4. For Every regular tessellation: (p-2)(q-2)=4. For every regular pseudopolyhedron: (p-2)*(q-2)>4. The n
https://en.wikipedia.org/wiki/Separation%20property	Separation property may refer to: Separation property (finance), a concept used to simplify the process of building a portfolio of financial assets Prewellordering in mathematics, a component of set theory Separation axiom in mathematics, a concept in topology
https://en.wikipedia.org/wiki/Laurence%20Chisholm%20Young	Laurence Chisholm Young (14 July 1905 – 24 December 2000) was a British mathematician known for his contributions to measure theory, the calculus of variations, optimal control theory, and potential theory. He was the son of William Henry Young and Grace Chisholm Young, both prominent mathematicians. He moved to the US in 1949 but never sought American citizenship. The concept of Young measure is named after him: he also introduced the concept of the generalized curve and a concept of generalized surface which later evolved in the concept of varifold. The Young integral also is named after him and has now been generalised in the theory of rough paths. Life and academic career Laurence Chisholm Young was born in Göttingen, the fifth of the six children of William Henry Young and Grace Chisholm Young. He held positions of Professor at the University of Cape Town, South Africa, and at the University of Wisconsin-Madison. He was also a chess grandmaster. Selected publications Books , available from the Internet archive. . . Papers . , memoir presented by Stanisław Saks at the session of 16 December 1937 of the Warsaw Society of Sciences and Letters. The free PDF copy is made available by the RCIN –Digital Repository of the Scientifics Institutes. . . . . . . . . See also Bounded variation Caccioppoli set Measure theory Varifold Notes References Biographical and general references , including a reply by L. C. Young himself (pages 109–112). . Scientific references . One of the most complete monographs on the theory of Young measures, strongly oriented to applications in continuum mechanics of fluids. . A thorough scrutiny of Young measures and their various generalization is in Chapter 3 from the perspective of convex compactifications. . . An extended version of with a list of Almgren's publications. External links Obituary on University of Wisconsin web site 20th-century British mathematicians Alumni of Trinity College, Cambridge Mathematical analysts Scientists from Göttingen 1905 births 2000 deaths Variational analysts British historians of mathematics Instituto Nacional de Matemática Pura e Aplicada researchers
https://en.wikipedia.org/wiki/Hodges%E2%80%93Lehmann%20estimator	In statistics, the Hodges–Lehmann estimator is a robust and nonparametric estimator of a population's location parameter. For populations that are symmetric about one median, such as the Gaussian or normal distribution or the Student t-distribution, the Hodges–Lehmann estimator is a consistent and median-unbiased estimate of the population median. For non-symmetric populations, the Hodges–Lehmann estimator estimates the "pseudo–median", which is closely related to the population median. The Hodges–Lehmann estimator was proposed originally for estimating the location parameter of one-dimensional populations, but it has been used for many more purposes. It has been used to estimate the differences between the members of two populations. It has been generalized from univariate populations to multivariate populations, which produce samples of vectors. It is based on the Wilcoxon signed-rank statistic. In statistical theory, it was an early example of a rank-based estimator, an important class of estimators both in nonparametric statistics and in robust statistics. The Hodges–Lehmann estimator was proposed in 1963 independently by Pranab Kumar Sen and by Joseph Hodges and Erich Lehmann, and so it is also called the "Hodges–Lehmann–Sen estimator". Definition In the simplest case, the "Hodges–Lehmann" statistic estimates the location parameter for a univariate population. Its computation can be described quickly. For a dataset with n measurements, the set of all possible two-element subsets of it has n(n − 1)/2 elements. For each such subset, the mean is computed; finally, the median of these n(n − 1)/2 averages is defined to be the Hodges–Lehmann estimator of location. The Hodges–Lehmann statistic also estimates the difference between two populations. For two sets of data with m and n observations, the set of two-element sets made of them is their Cartesian product, which contains m × n pairs of points (one from each set); each such pair defines one difference of values. The Hodges–Lehmann statistic is the median of the m × n differences. Estimating the population median of a symmetric population For a population that is symmetric, the Hodges–Lehmann statistic estimates the population's median. It is a robust statistic that has a breakdown point of 0.29, which means that the statistic remains bounded even if nearly 30 percent of the data have been contaminated. This robustness is an important advantage over the sample mean, which has a zero breakdown point, being proportional to any single observation and so liable to being misled by even one outlier. The sample median is even more robust, having a breakdown point of 0.50. The Hodges–Lehmann estimator is much better than the sample mean when estimating mixtures of normal distributions, also. For symmetric distributions, the Hodges–Lehmann statistic has greater efficiency than does the sample median. For the normal distribution, the Hodges-Lehmann statistic is nearly as efficient as the samp
https://en.wikipedia.org/wiki/P%E2%80%93P%20plot	In statistics, a P–P plot (probability–probability plot or percent–percent plot or P value plot) is a probability plot for assessing how closely two data sets agree, or for assessing how closely a dataset fits a particular model. It works by plotting the two cumulative distribution functions against each other; if they are similar, the data will appear to be nearly a straight line. This behavior is similar to that of the more widely used Q–Q plot, with which it is often confused. Definition A P–P plot plots two cumulative distribution functions (cdfs) against each other: given two probability distributions, with cdfs "F" and "G", it plots as z ranges from to As a cdf has range [0,1], the domain of this parametric graph is and the range is the unit square Thus for input z the output is the pair of numbers giving what percentage of f and what percentage of g fall at or below z. The comparison line is the 45° line from (0,0) to (1,1), and the distributions are equal if and only if the plot falls on this line. The degree of deviation makes it easy to visually identify how different the distributions are, but because of sampling error, even samples drawn from identical distributions will not appear identical. Example As an example, if the two distributions do not overlap, say F is below G, then the P–P plot will move from left to right along the bottom of the square – as z moves through the support of F, the cdf of F goes from 0 to 1, while the cdf of G stays at 0 – and then moves up the right side of the square – the cdf of F is now 1, as all points of F lie below all points of G, and now the cdf of G moves from 0 to 1 as z moves through the support of G. (need a graph for this paragraph) Use As the above example illustrates, if two distributions are separated in space, the P–P plot will give very little data – it is only useful for comparing probability distributions that have nearby or equal location. Notably, it will pass through the point (1/2, 1/2) if and only if the two distributions have the same median. P–P plots are sometimes limited to comparisons between two samples, rather than comparison of a sample to a theoretical model distribution. However, they are of general use, particularly where observations are not all modelled with the same distribution. However, it has found some use in comparing a sample distribution from a known theoretical distribution: given n samples, plotting the continuous theoretical cdf against the empirical cdf would yield a stairstep (a step as z hits a sample), and would hit the top of the square when the last data point was hit. Instead one only plots points, plotting the observed kth observed points (in order: formally the observed kth order statistic) against the k/(n + 1) quantile of the theoretical distribution. This choice of "plotting position" (choice of quantile of the theoretical distribution) has occasioned less controversy than the choice for Q–Q plots. The resulting goodness of fit of t
https://en.wikipedia.org/wiki/Pangkor%20Airport	Pangkor Airport is an airport on Pangkor Island, Manjung District, Perak, Malaysia. Airlines and destinations Traffic and statistics See also List of airports in Malaysia References External links Short Take-Off and Landing Airports (STOL) at Malaysia Airports Holdings Berhad Aviation Photos: Pangkor Island (PKG / WMPA) at Airliners.net Airports in Perak Manjung District
https://en.wikipedia.org/wiki/Quadratics	Quadratics is a six-part Canadian instructional television series produced by TVOntario in 1993. The miniseries is part of the Concepts in Mathematics series. The program uses computer animation to demonstrate quadratic equations and their corresponding functions in the Cartesian coordinate system. Synopsis Each program involves two robots, Edie and Charon, who work on an assembly line in a high-tech factory. The robots discuss their desire to learn about quadratic equations, and they are subsequently provided with lessons that further their education. Episodes References 1993 Canadian television series debuts 1993 Canadian television series endings Canadian children's education television series TVO original programming Mathematics education television series 1990s Canadian children's television series Canadian television series with live action and animation
https://en.wikipedia.org/wiki/Deborah%20and%20Franklin%20Haimo%20Awards%20for%20Distinguished%20College%20or%20University%20Teaching%20of%20Mathematics	The Deborah and Franklin Tepper Haimo Awards for Distinguished College or University Teaching of Mathematics are awards given by the Mathematical Association of America to recognize college or university teachers "who have been widely recognized as extraordinarily successful and whose teaching effectiveness has been shown to have had influence beyond their own institutions." The Haimo awards are the highest teaching honor bestowed by the MAA. The awards were established in 1993 by Deborah Tepper Haimo and named after Haimo and her husband Franklin Haimo. After the first year of the award (when seven awards were given) up to three awards are given every year. Winners The winners of the award have been: 1993: Joseph Gallian, Robert V. Hogg, Anne Lester Hudson, Frank Morgan, V. Frederick Rickey, Doris Schattschneider, and Philip D. Straffin Jr. 1994: Paul Halmos, Justin Jesse Price, and Alan Tucker 1995: Robert L. Devaney, Lisa Mantini, and David S. Moore 1996: Thomas Banchoff, Edward M. Landesman, and Herbert Wilf 1997: Carl C. Cowen, Carl Pomerance, and T. Christine Stevens 1998: Colin Adams, Rhonda Hatcher, and Rhonda Hughes 1999: Joel Brawley, Robert W. Case, and Joan Hutchinson 2000: Arthur T. Benjamin, Donald S. Passman, and Gary W. Towsley 2001: Edward Burger, Evelyn Silvia, and Leonard F. Klosinki 2002: Dennis DeTurck, Paul Sally, and Edward Spitznagel Jr. 2003: Judith Grabiner, Ranjan Roy, and Paul Zeitz 2004: Thomas Garrity, Andy Liu, and Olympia Nicodemi 2005: Gerald L. Alexanderson, Aparna Higgins, and Deborah Hughes Hallett 2006: Jacqueline Dewar, Keith Stroyan, and Judy L. Walker 2007: Jennifer Quinn, Michael Starbird, and Gilbert Strang 2008: Annalisa Crannell, Kenneth I. Gross, and James A. Morrow 2009: Michael Bardzell, David Pengelley, and Vali Siadat 2010: Curtis Bennett, Michael Dorff, and Allan J. Rossman 2011: Erica Flapan, Karen Rhea, and Zvezdelina Stankova 2012: Matthew DeLong, Susan Loepp, and Cynthia Wyels 2013: Matthias Beck, Margaret M. Robinson, and Francis Su 2014: Carl Lee, Gavin LaRose, and Andrew Bennett 2015: Judith Covington, Brian Hopkins, and Shahriar Shahriari 2016: Satyan Devadoss, Tyler Jarvis, and Glen Van Brummelen 2017: Janet Barnett, Caren Diefenderfer, and Tevian Dray 2018: Gary Gordon, Hortensia Soto, and Ron Taylor 2019: Suzanne Dorée, Carl Lee, and Jennifer Switkes 2020: Federico Ardila, Mark Tomforde, and Suzanne Weekes 2021: Dave Kung, David Austin, and Elaine Kasimatis 2022: Pamela E. Harris, Darren Narayan, and Robin Wilson 2023: Carol S. Schumacher, Sarah C. Koch, and Adriana Salerno See also List of mathematics awards References Mathematics education awards American education awards Awards established in 1991 1991 establishments in the United States Awards of the Mathematical Association of America American science and technology awards
https://en.wikipedia.org/wiki/Exact%20couple	In mathematics, an exact couple, due to , is a general source of spectral sequences. It is common especially in algebraic topology; for example, Serre spectral sequence can be constructed by first constructing an exact couple. For the definition of an exact couple and the construction of a spectral sequence from it (which is immediate), see . For a basic example, see Bockstein spectral sequence. The present article covers additional materials. Exact couple of a filtered complex Let R be a ring, which is fixed throughout the discussion. Note if R is , then modules over R are the same thing as abelian groups. Each filtered chain complex of modules determines an exact couple, which in turn determines a spectral sequence, as follows. Let C be a chain complex graded by integers and suppose it is given an increasing filtration: for each integer p, there is an inclusion of complexes: From the filtration one can form the associated graded complex: which is doubly-graded and which is the zero-th page of the spectral sequence: To get the first page, for each fixed p, we look at the short exact sequence of complexes: from which we obtain a long exact sequence of homologies: (p is still fixed) With the notation , the above reads: which is precisely an exact couple and is a complex with the differential . The derived couple of this exact couple gives the second page and we iterate. In the end, one obtains the complexes with the differential d: The next lemma gives a more explicit formula for the spectral sequence; in particular, it shows the spectral sequence constructed above is the same one in more traditional direct construction, in which one uses the formula below as definition (cf. Spectral sequence#The spectral sequence of a filtered complex). Sketch of proof: Remembering , it is easy to see: where they are viewed as subcomplexes of . We will write the bar for . Now, if , then for some . On the other hand, remembering k is a connecting homomorphism, where x is a representative living in . Thus, we can write: for some . Hence, modulo , yielding . Next, we note that a class in is represented by a cycle x such that . Hence, since j is induced by , . We conclude: since , Proof: See the last section of May. Exact couple of a double complex A double complex determines two exact couples; whence, the two spectral sequences, as follows. (Some authors call the two spectral sequences horizontal and vertical.) Let be a double complex. With the notation , for each with fixed p, we have the exact sequence of cochain complexes: Taking cohomology of it gives rise to an exact couple: By symmetry, that is, by switching first and second indexes, one also obtains the other exact couple. Example: Serre spectral sequence The Serre spectral sequence arises from a fibration: For the sake of transparency, we only consider the case when the spaces are CW complexes, F is connected and B is simply connected; the general case involves more technicalit
https://en.wikipedia.org/wiki/Fermi%20coordinates	In the mathematical theory of Riemannian geometry, there are two uses of the term Fermi coordinates. In one use they are local coordinates that are adapted to a geodesic. In a second, more general one, they are local coordinates that are adapted to any world line, even not geodesical. Take a future-directed timelike curve , being the proper time along in the spacetime . Assume that is the initial point of . Fermi coordinates adapted to are constructed this way. Consider an orthonormal basis of with parallel to . Transport the basis along making use of Fermi-Walker's transport. The basis at each point is still orthonormal with parallel to and is non-rotated (in a precise sense related to the decomposition of Lorentz transformations into pure transformations and rotations) with respect to the initial basis, this is the physical meaning of Fermi-Walker's transport. Finally construct a coordinate system in an open tube , a neighbourhood of , emitting all spacelike geodesics through with initial tangent vector , for every . A point has coordinates where is the only vector whose associated geodesic reaches for the value of its parameter and is the only time along for that this geodesic reaching exists. If itself is a geodesic, then Fermi-Walker's transport becomes the standard parallel transport and Fermi's coordinates become standard Riemannian coordinates adapted to . In this case, using these coordinates in a neighbourhood of , we have , all Christoffel symbols vanish exactly on . This property is not valid for Fermi's coordinates however when is not a geodesic. Such coordinates are called Fermi coordinates and are named after the Italian physicist Enrico Fermi. The above properties are only valid on the geodesic. The Fermi-Coordinates adapted to a null geodesic is provided by Mattias Blau, Denis Frank, and Sebastian Weiss. Notice that, if all Christoffel symbols vanish near , then the manifold is flat near . See also Proper reference frame (flat spacetime)#Proper coordinates or Fermi coordinates Geodesic normal coordinates Fermi-Walker transport Christoffel symbols Isothermal coordinates References Riemannian geometry Coordinate systems in differential geometry
https://en.wikipedia.org/wiki/Michael%20Waterman	Michael Spencer Waterman (born June 28, 1942) is a Professor of Biology, Mathematics and Computer Science at the University of Southern California (USC), where he holds an Endowed Associates Chair in Biological Sciences, Mathematics and Computer Science. He previously held positions at Los Alamos National Laboratory and Idaho State University. Education and early life Waterman grew up near Bandon, Oregon, and earned a bachelor's degree in Mathematics from Oregon State University, followed by a PhD in statistics and probability from Michigan State University in 1969. Research and career Waterman is one of the founders and current leaders in the area of computational biology. He focuses on applying mathematics, statistics, and computer science techniques to various problems in molecular biology. His work has contributed to some of the most widely used tools in the field. In particular, the Smith-Waterman algorithm (developed with Temple F. Smith) is the basis for many sequence alignment programs. In 1988, Waterman and Eric Lander published a landmark paper describing a mathematical model for fingerprint mapping. This work formed one of the theoretical cornerstones for many of the later DNA mapping and sequencing projects, especially the Human Genome Project. A 1995 paper by Idury and Waterman introduced Eulerian-De Bruijn sequence assembly which is widely used in next-generation sequencing projects. With Pavel A. Pevzner (a former postdoctoral researcher in his lab), he began the international conference Research in Computational Molecular Biology (RECOMB), and he is a founding editor of the Journal of Computational Biology. Waterman also authored one of the earliest textbooks in the field: Introduction to Computational Biology. Awards and honors With Cyrus Chothia and David Haussler, Waterman was awarded the 2015 Dan David Prize for his contributions to the field of bioinformatics. He was awarded an Honorary Doctorate from Tel Aviv University in 2011, and an Honorary Doctorate from the University of Southern Denmark in 2013. Waterman has been a member of the US American Academy of Arts and Sciences since 1995, a member of the US National Academy of Engineering since 2012, a member of the Chinese Academy of Sciences since 2013, and a member of the US National Academy of Sciences since 2001. He has been an academician of the French Academy of Sciences since 2005. Waterman was elected an ISCB Fellow in 2009 by the International Society for Computational Biology and was awarded their ISCB Senior Scientist Award in 2009. Personal life Waterman has written a memoir, Getting Outside, of a childhood spent on an isolated livestock ranch on the southern coast of Oregon in the mid-twentieth century. References Living people 1942 births People from Bandon, Oregon 20th-century American mathematicians 21st-century American mathematicians American bioinformaticians 21st-century American biologists University of Southern California faculty Idaho St
https://en.wikipedia.org/wiki/Gauss%E2%80%93Codazzi%20equations	In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi formulas) are fundamental formulas which link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold. The equations were originally discovered in the context of surfaces in three-dimensional Euclidean space. In this context, the first equation, often called the Gauss equation (after its discoverer Carl Friedrich Gauss), says that the Gauss curvature of the surface, at any given point, is dictated by the derivatives of the Gauss map at that point, as encoded by the second fundamental form. The second equation, called the Codazzi equation or Codazzi-Mainardi equation, states that the covariant derivative of the second fundamental form is fully symmetric. It is named for Gaspare Mainardi (1856) and Delfino Codazzi (1868–1869), who independently derived the result, although it was discovered earlier by Karl Mikhailovich Peterson. Formal statement Let be an n-dimensional embedded submanifold of a Riemannian manifold P of dimension . There is a natural inclusion of the tangent bundle of M into that of P by the pushforward, and the cokernel is the normal bundle of M: The metric splits this short exact sequence, and so Relative to this splitting, the Levi-Civita connection of P decomposes into tangential and normal components. For each and vector field Y on M, Let The Gauss formula now asserts that is the Levi-Civita connection for M, and is a symmetric vector-valued form with values in the normal bundle. It is often referred to as the second fundamental form. An immediate corollary is the Gauss equation for the curvature tensor. For , where is the Riemann curvature tensor of P and R is that of M. The Weingarten equation is an analog of the Gauss formula for a connection in the normal bundle. Let and a normal vector field. Then decompose the ambient covariant derivative of along X into tangential and normal components: Then Weingarten's equation: DX is a metric connection in the normal bundle. There are thus a pair of connections: ∇, defined on the tangent bundle of M; and D, defined on the normal bundle of M. These combine to form a connection on any tensor product of copies of TM and T⊥M. In particular, they defined the covariant derivative of : The Codazzi–Mainardi equation is Since every immersion is, in particular, a local embedding, the above formulas also hold for immersions. Gauss–Codazzi equations in classical differential geometry Statement of classical equations In classical differential geometry of surfaces, the Codazzi–Mainardi equations are expressed via the second fundamental form (L, M, N): The Gauss formula, depending on how one chooses to define the Gaussian curvature, may be a tautology. It can be stated as where (e, f, g) are the components of the f
https://en.wikipedia.org/wiki/Mioara%20Mugur-Sch%C3%A4chter	Mioara Mugur-Schächter is a French-Romanian physicist, specialized in fundamental quantum mechanics, probability theory and information theory. She is also an epistemologist (methodologist) of scientific knowledge generation. As a professor of theoretical physics at the University of Reims, she founded the Laboratory of Quantum Mechanics and Information Structures, which she directed until 1997. She is currently president of the Centre pour la Synthèse d'une Épistémologie Formalisée. Biography Born in Romania, she arrived in France in 1962 from Bucharest. Her PhD thesis, of which the whole content had been elaborated beforehand in Bucharest and sent to Louis de Broglie, was published in a volume prefaced by de Broglie and published in the collection "Les grands problèmes des sciences", Gauthiers Villars, Paris, 1964. She became a professor of theoretical physics at the University of Reims Champagne-Ardenne, where she founded the Laboratory for Quantum Mechanics and Information Structure, which she directed until 1997. Selected publications Étude du caractère complet de la théorie quantique (1964) The quantum mechanical one-system formalism, joint probabilities and locality, in Quantum Mechanics a half Century Later, J. L. Lopes and M. Paty, eds., Reidel, pp. 107-146, 1977 Study of Wigner’s Theorem on Joint Probabilities, Found. Phys., Vol. 9, pp. 389-404, 1979. Le concept nouveau de fonctionnelle d’opacité d’une statistique. Etude des relations entre la loi des grands nombres, l’entropie informationnelle et l’entropie statistique, Anns. de l’Inst. H. Poincaré, Section A, vol XXXII, no. 1, pp. 33-71, 1980 The Probabilistic-Informational Concept of an Opacity Functional, (en collab. Avec N. Hadjissavas), Kybernetes, pp.189-193, Vol. 11(3), 1982. Toward a Factually induced Spacetime Quantum Logic, Found. of Phys., Vol. 22, No. 7, pp. 963-994, 1992 Quantum Probabilities, Komogorov probabilities, and Informational Probabilities, Int. J. Theor. Phys., Vol. 33, No.1, pp. 53-90, 1994. See also Constructivist epistemology References External links Her web site Introduction of the collective book "QUANTUM MECHANICS, MATHEMATICS, COGNITION AND ACTION : Proposals for a formalized epistemology" Article "Quantum mechanics versus a method of relativized conceptualization" Year of birth missing (living people) Living people Romanian emigrants to France Academic staff of the University of Reims Champagne-Ardenne French physicists Quantum physicists
https://en.wikipedia.org/wiki/Monthly%20Labor%20Review	The Monthly Labor Review (MLR) is published by the U.S. Bureau of Labor Statistics (BLS). Issues often focus on a particular topic. Most articles are by BLS staff. Annually since 1969, the Lawrence R. Klein Award has been awarded to authors of articles appearing in the Monthly Labor Review, generally one to BLS authors and one to non-BLS authors. History In 1915, under commissioner Royal Meeker, BLS began publishing the Monthly Review, with a circulation of 8,000. The name became Monthly Labor Review in 1918, and circulation rose to 20,000 in June 1920. The journal has published its articles on the web for a decade. In 2008 the journal ceased to publish a bound paper edition, and now publishes only online. References External links The Monthly Labor Review web site Open access journals Monthly journals Bureau of Labor Statistics Academic journals published by the United States government 1915 establishments in the United States Publications established in 1915 United States Department of Labor publications
https://en.wikipedia.org/wiki/David%20Fowler%20%28mathematician%29	David Herbert Fowler (28 April 1937 – 13 April 2004) was a historian of Greek mathematics who published work on pre-Eudoxian ratio theory (using the process he called anthyphairesis). He disputed the standard story of Greek mathematical discovery, in which the discovery of the phenomenon of incommensurability came as a shock. Fowler was also the translator of René Thom's book Structural Stability and Morphogenesis from French (Stabilité strukturelle et morphogénèse) into English. References Obituary in The Guardian, 3 May 2004 by Christopher Zeeman. Obituary in The Independent, 24 May 2004. External links Bibliography Book Review by Fernando Q. Gouvêa of The Mathematics of Plato's Academy Memorial symposium organized in his honor at Warwick, 9 November 2004. British historians of mathematics 20th-century British mathematicians 21st-century British mathematicians 1937 births 2004 deaths Fowler, David Alumni of Gonville and Caius College, Cambridge People educated at Rossall School
https://en.wikipedia.org/wiki/S.%20R.%20Srinivasa%20Varadhan	Sathamangalam Ranga Iyengar Srinivasa Varadhan, (born 2 January 1940) is an Indian American mathematician. He is known for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviations. He is regarded as one of the fundamental contributors to the theory of diffusion processes with an orientation towards the refinement and further development of Itô’s stochastic calculus. In the year 2007, he became the first Asian to win the Abel Prize. Early life and education Srinivasa was born into a Hindu Tamil Brahmin Iyengar family in 1940 in Chennai (then Madras). In 1953, his family migrated to Kolkata. He grew up in Chennai and Kolkata. Varadhan received his undergraduate degree in 1959 and his postgraduate degree in 1960 from Presidency College, Chennai. He received his doctorate from ISI in 1963 under C R Rao, who arranged for Andrey Kolmogorov to be present at Varadhan's thesis defence. He was one of the "famous four" (the others being R Ranga Rao, K R Parthasarathy, and Veeravalli S Varadarajan) in ISI during 1956–1963. Career Since 1963, he has worked at the Courant Institute of Mathematical Sciences at New York University, where he was at first a postdoctoral fellow (1963–66), strongly recommended by Monroe D Donsker. Here he met Daniel Stroock, who became a close colleague and co-author. In an article in the Notices of the American Mathematical Society, Stroock recalls these early years: Varadhan is currently a professor at the Courant Institute. He is known for his work with Daniel W Stroock on diffusion processes, and for his work on large deviations with Monroe D Donsker. He has chaired the Mathematical Sciences jury for the Infosys Prize from 2009 and was the chief guest in 2020. Awards and honours Varadhan's awards and honours include the National Medal of Science (2010) from President Barack Obama, "the highest honour bestowed by the United States government on scientists, engineers and inventors". He also received the Birkhoff Prize (1994), the Margaret and Herman Sokol Award of the Faculty of Arts and Sciences, New York University (1995), and the Leroy P Steele Prize for Seminal Contribution to Research (1996) from the American Mathematical Society, awarded for his work with Daniel W Stroock on diffusion processes. He was awarded the Abel Prize in 2007 for his work on large deviations with Monroe D Donsker. In 2008, the Government of India awarded him the Padma Bhushan. and in 2023, he was awarded India's second highest civilian honor Padma Vibhushan. He also has two honorary degrees from Université Pierre et Marie Curie in Paris (2003) and from Indian Statistical Institute in Kolkata, India (2004). Varadhan is a member of the US National Academy of Sciences (1995), and the Norwegian Academy of Science and Letters (2009). He was elected to Fellow of the American Academy of Arts and Sciences (1988), the Third World Academy of Sciences (1988), the Institute of Mathematical St
https://en.wikipedia.org/wiki/Mazinho%20Oliveira	Waldemar Aureliano de Oliveira Filho, usually known as Mazinho Oliveira (born 26 December 1965), is a retired Brazilian footballer who played as a forward. Career statistics Club International References External links 1965 births Living people Brazilian men's footballers Brazilian expatriate men's footballers Brazil men's international footballers Santos FC players FC Bayern Munich footballers Sport Club Internacional players Club Athletico Paranaense players Clube Atlético Bragantino players CR Flamengo footballers Kashima Antlers players Kawasaki Frontale players Campeonato Brasileiro Série A players J1 League players Expatriate men's footballers in Japan Bundesliga players Expatriate men's footballers in Germany People from Guarujá Men's association football forwards Footballers from São Paulo (state)
https://en.wikipedia.org/wiki/William%20Fulton%20%28mathematician%29	William Edgar Fulton (born August 29, 1939) is an American mathematician, specializing in algebraic geometry. Education and career He received his undergraduate degree from Brown University in 1961 and his doctorate from Princeton University in 1966. His Ph.D. thesis, written under the supervision of Gerard Washnitzer, was on The fundamental group of an algebraic curve. Fulton worked at Princeton and Brandeis University from 1965 until 1970, when he began teaching at Brown. In 1987 he moved to the University of Chicago. He is, as of 2011, a professor at the University of Michigan. Fulton is known as the author or coauthor of a number of popular texts, including Algebraic Curves and Representation Theory. Awards and honors In 1996 he received the Steele Prize for mathematical exposition for his text Intersection Theory. Fulton is a member of the United States National Academy of Sciences since 1997; a fellow of the American Academy of Arts and Sciences from 1998, and was elected a foreign member of the Royal Swedish Academy of Sciences in 2000. In 2010, he was awarded the Steele Prize for Lifetime Achievement. In 2012 he became a fellow of the American Mathematical Society. Selected works Algebraic Curves: An Introduction To Algebraic Geometry, with Richard Weiss. New York: Benjamin, 1969. Reprint ed.: Redwood City, CA, USA: Addison-Wesley, Advanced Book Classics, 1989. . Full text online. See also Fulton–Hansen connectedness theorem References External links Fulton's home page at the University of Michigan 1939 births Living people 20th-century American mathematicians 21st-century American mathematicians Algebraic geometers Fellows of the American Mathematical Society Members of the United States National Academy of Sciences Members of the Royal Swedish Academy of Sciences Princeton University alumni University of Michigan faculty People from Naugatuck, Connecticut Mathematicians from Connecticut Brandeis University faculty Fellows of the American Academy of Arts and Sciences
https://en.wikipedia.org/wiki/Geometric%20programming	A geometric program (GP) is an optimization problem of the form where are posynomials and are monomials. In the context of geometric programming (unlike standard mathematics), a monomial is a function from to defined as where and . A posynomial is any sum of monomials. Geometric programming is closely related to convex optimization: any GP can be made convex by means of a change of variables. GPs have numerous applications, including component sizing in IC design, aircraft design, maximum likelihood estimation for logistic regression in statistics, and parameter tuning of positive linear systems in control theory. Convex form Geometric programs are not in general convex optimization problems, but they can be transformed to convex problems by a change of variables and a transformation of the objective and constraint functions. In particular, after performing the change of variables and taking the log of the objective and constraint functions, the functions , i.e., the posynomials, are transformed into log-sum-exp functions, which are convex, and the functions , i.e., the monomials, become affine. Hence, this transformation transforms every GP into an equivalent convex program. In fact, this log-log transformation can be used to convert a larger class of problems, known as log-log convex programming (LLCP), into an equivalent convex form. Software Several software packages exist to assist with formulating and solving geometric programs. MOSEK is a commercial solver capable of solving geometric programs as well as other non-linear optimization problems. CVXOPT is an open-source solver for convex optimization problems. GPkit is a Python package for cleanly defining and manipulating geometric programming models. There are a number of example GP models written with this package here. GGPLAB is a MATLAB toolbox for specifying and solving geometric programs (GPs) and generalized geometric programs (GGPs). CVXPY is a Python-embedded modeling language for specifying and solving convex optimization problems, including GPs, GGPs, and LLCPs. See also Signomial Clarence Zener References Convex optimization
https://en.wikipedia.org/wiki/Hilbert%20projection%20theorem	In mathematics, the Hilbert projection theorem is a famous result of convex analysis that says that for every vector in a Hilbert space and every nonempty closed convex there exists a unique vector for which is minimized over the vectors ; that is, such that for every Finite dimensional case Some intuition for the theorem can be obtained by considering the first order condition of the optimization problem. Consider a finite dimensional real Hilbert space with a subspace and a point If is a or of the function defined by (which is the same as the minimum point of ), then derivative must be zero at In matrix derivative notation Since is a vector in that represents an arbitrary tangent direction, it follows that must be orthogonal to every vector in Statement Detailed elementary proof Proof by reduction to a special case It suffices to prove the theorem in the case of because the general case follows from the statement below by replacing with Consequences : If then which implies : Let where is the underlying scalar field of and define which is continuous and linear because this is true of each of its coordinates The set is closed in because is closed in and is continuous. The kernel of any linear map is a vector subspace of its domain, which is why is a vector subspace of : Let The Hilbert projection theorem guarantees the existence of a unique such that (or equivalently, for all ). Let so that and it remains to show that The inequality above can be rewritten as: Because and is a vector space, and which implies that The previous inequality thus becomes or equivalently, But this last statement is true if and only if every Thus Properties Expression as a global minimum The statement and conclusion of the Hilbert projection theorem can be expressed in terms of global minimums of the followings functions. Their notation will also be used to simplify certain statements. Given a non-empty subset and some define a function A of if one exists, is any point in such that in which case is equal to the of the function which is: Effects of translations and scalings When this global minimum point exists and is unique then denote it by explicitly, the defining properties of (if it exists) are: The Hilbert projection theorem guarantees that this unique minimum point exists whenever is a non-empty closed and convex subset of a Hilbert space. However, such a minimum point can also exist in non-convex or non-closed subsets as well; for instance, just as long is is non-empty, if then If is a non-empty subset, is any scalar, and are any vectors then which implies: Examples The following counter-example demonstrates a continuous linear isomorphism for which Endow with the dot product, let and for every real let be the line of slope through the origin, where it is readily verified that Pick a real number and define by (so this map scal
https://en.wikipedia.org/wiki/Strip%20algebra	Strip Algebra is a set of elements and operators for the description of carbon nanotube structures, considered as a subgroup of polyhedra, and more precisely, of polyhedra with vertices formed by three edges. This restriction is imposed on the polyhedra because carbon nanotubes are formed of sp2 carbon atoms. Strip Algebra was developed initially for the determination of the structure connecting two arbitrary nanotubes, but has also been extended to the connection of three identical nanotubes Background Graphitic systems are molecules and crystals formed of carbon atoms in sp2 hybridization. Thus, the atoms are arranged on a hexagonal grid. Graphite, nanotubes, and fullerenes are examples of graphitic systems. All of them share the property that each atom is bonded to three others (3-valent). The relation between the number of vertices, edges and faces of any finite polyhedron is given by Euler's polyhedron formula: where e, f and v are the number of edges, faces and vertices, respectively, and g is the genus of the polyhedron, i.e., the number of "holes" in the surface. For example, a sphere is a surface of genus 0, while a torus is of genus 1. Nomenclature A substrip is identified by a pair of natural numbers measuring the position of the last ring in parentheses, together with the turns induced by the defect ring. The number of edges of the defect can be extracted from these. Elements A Strip is defined as a set of consecutive rings, that is able to be joined with others, by sharing a side of the first or last ring. Numerous complex structures can be formed with strips. As said before, strips have both at the beginning and at the end two connections. With strips only, can be formed two of them. Operators Given the definition of a strip, a set of operations may be defined. These are necessary to find out the combined result of a set of contiguous strips. Addition of two strips: (upcoming) Turn Operators: (upcoming) Inversion of a strip: (upcoming) Applications Strip Algebra has been applied to the construction of nanotube heterojunctions, and was first implemented in the CoNTub v1.0 software, which makes it possible to find the precise position of all the carbon rings needed to produce a heterojunction with arbitrary indices and chirality from two nanotubes. References Carbon nanotubes
https://en.wikipedia.org/wiki/Semimartingale	In probability theory, a real valued stochastic process X is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the largest class of processes with respect to which the Itô integral and the Stratonovich integral can be defined. The class of semimartingales is quite large (including, for example, all continuously differentiable processes, Brownian motion and Poisson processes). Submartingales and supermartingales together represent a subset of the semimartingales. Definition A real valued process X defined on the filtered probability space (Ω,F,(Ft)t ≥ 0,P) is called a semimartingale if it can be decomposed as where M is a local martingale and A is a càdlàg adapted process of locally bounded variation. An Rn-valued process X = (X1,…,Xn) is a semimartingale if each of its components Xi is a semimartingale. Alternative definition First, the simple predictable processes are defined to be linear combinations of processes of the form Ht = A1{t > T} for stopping times T and FT -measurable random variables A. The integral H · X for any such simple predictable process H and real valued process X is This is extended to all simple predictable processes by the linearity of H · X in H. A real valued process X is a semimartingale if it is càdlàg, adapted, and for every t ≥ 0, is bounded in probability. The Bichteler–Dellacherie Theorem states that these two definitions are equivalent . Examples Adapted and continuously differentiable processes are continuous finite variation processes, and hence semimartingales. Brownian motion is a semimartingale. All càdlàg martingales, submartingales and supermartingales are semimartingales. Itō processes, which satisfy a stochastic differential equation of the form dX = σdW + μdt are semimartingales. Here, W is a Brownian motion and σ, μ are adapted processes. Every Lévy process is a semimartingale. Although most continuous and adapted processes studied in the literature are semimartingales, this is not always the case. Fractional Brownian motion with Hurst parameter H ≠ 1/2 is not a semimartingale. Properties The semimartingales form the largest class of processes for which the Itō integral can be defined. Linear combinations of semimartingales are semimartingales. Products of semimartingales are semimartingales, which is a consequence of the integration by parts formula for the Itō integral. The quadratic variation exists for every semimartingale. The class of semimartingales is closed under optional stopping, localization, change of time and absolutely continuous change of probability measure (see Girsanov's Theorem). If X is an Rm valued semimartingale and f is a twice continuously differentiable function from Rm to Rn, then f(X) is a semimartingale. This is a consequence of Itō's lemma. The property of being a semimartingale is preserved under shrinking the filtration.
https://en.wikipedia.org/wiki/Fred%20Swan	Fred Swan is an American painter who resides in Barre, Vermont. He graduated from the United States Naval Academy, and then taught mathematics at Spaulding High School. A self-taught artist, Swan is best known for his comforting, warm landscapes which take up to 500 hours to complete. Typical of these is Blue Moon which, as with many of Swan's paintings, features houses and is highly detailed but could be criticised for an idealised, 'chocolate box' style. Swan's paintings are highly commercial and have been adapted for calendars and jigsaw puzzles and are sold as prints. Swan won the 1979 Saturday Evening Post Cover Contest, and his art is featured in several famous collections, including those of Johnson and Johnson, Malcolm Forbes, and the Vermont Council on the Arts. His paintings have also been featured in Yankee Magazine and Vermont Life Magazine. References External links Champlain Collection – Swan's publisher 20th-century American painters American male painters 21st-century American painters 21st-century American male artists People from Barre, Vermont Year of birth missing (living people) Living people Artists from Vermont Place of birth missing (living people) Educators from Vermont United States Naval Academy alumni 20th-century American male artists
https://en.wikipedia.org/wiki/International%20rankings%20of%20South%20Korea	The following are international rankings of South Korea. Economy Education Environment Health & Safety Note: In the case of statistics with potentially conflicting meanings, the rankings have been converted to reflect the same direction - Positive statistics rank higher, while negative statistics rank lower. Industry Innovation Politics, Law and Military Science & Technology Society & Quality of Life Tourism Transportation See also International rankings of North Korea International rankings of China International rankings of Japan References South Korea
https://en.wikipedia.org/wiki/Coarse%20topology	In mathematics, coarse topology is a term in comparison of topologies which specifies the partial order relation of a topological structure to other one(s). Specifically, the coarsest topology may refer to: Initial topology, the most coarse topology in a certain category of topologies Trivial topology, the most coarse topology possible on a given set See also Weak topology, an example of topology coarser than the standard one Fine topology (disambiguation)
https://en.wikipedia.org/wiki/Tanaka%27s%20formula	In the stochastic calculus, Tanaka's formula for the Brownian motion states that where Bt is the standard Brownian motion, sgn denotes the sign function and Lt is its local time at 0 (the local time spent by B at 0 before time t) given by the L2-limit One can also extend the formula to semimartingales. Properties Tanaka's formula is the explicit Doob–Meyer decomposition of the submartingale \|Bt\| into the martingale part (the integral on the right-hand side, which is a Brownian motion), and a continuous increasing process (local time). It can also be seen as the analogue of Itō's lemma for the (nonsmooth) absolute value function , with and ; see local time for a formal explanation of the Itō term. Outline of proof The function \|x\| is not C2 in x at x = 0, so we cannot apply Itō's formula directly. But if we approximate it near zero (i.e. in [−ε, ε]) by parabolas and use Itō's formula, we can then take the limit as ε → 0, leading to Tanaka's formula. References (Example 5.3.2) Equations Martingale theory Probability theorems Stochastic calculus
https://en.wikipedia.org/wiki/Minimal%20polynomial%20%28field%20theory%29	In field theory, a branch of mathematics, the minimal polynomial of an element of a field extension is, roughly speaking, the polynomial of lowest degree having coefficients in the field, such that is a root of the polynomial. If the minimal polynomial of exists, it is unique. The coefficient of the highest-degree term in the polynomial is required to be 1. More formally, a minimal polynomial is defined relative to a field extension and an element of the extension field . The minimal polynomial of an element, if it exists, is a member of , the ring of polynomials in the variable with coefficients in . Given an element of , let be the set of all polynomials in such that . The element is called a root or zero of each polynomial in More specifically, Jα is the kernel of the ring homomorphism from F[x] to E which sends polynomials g to their value g(α) at the element α. Because it is the kernel of a ring homomorphism, Jα is an ideal of the polynomial ring F[x]: it is closed under polynomial addition and subtraction (hence containing the zero polynomial), as well as under multiplication by elements of F (which is scalar multiplication if F[x] is regarded as a vector space over F). The zero polynomial, all of whose coefficients are 0, is in every since for all and . This makes the zero polynomial useless for classifying different values of into types, so it is excepted. If there are any non-zero polynomials in , i.e. if the latter is not the zero ideal, then is called an algebraic element over , and there exists a monic polynomial of least degree in . This is the minimal polynomial of with respect to . It is unique and irreducible over . If the zero polynomial is the only member of , then is called a transcendental element over and has no minimal polynomial with respect to . Minimal polynomials are useful for constructing and analyzing field extensions. When is algebraic with minimal polynomial , the smallest field that contains both and is isomorphic to the quotient ring , where is the ideal of generated by . Minimal polynomials are also used to define conjugate elements. Definition Let E/F be a field extension, α an element of E, and F[x] the ring of polynomials in x over F. The element α has a minimal polynomial when α is algebraic over F, that is, when f(α) = 0 for some non-zero polynomial f(x) in F[x]. Then the minimal polynomial of α is defined as the monic polynomial of least degree among all polynomials in F[x] having α as a root. Properties Throughout this section, let E/F be a field extension over F as above, let α ∈ E be an algebraic element over F and let Jα be the ideal of polynomials vanishing on α. Uniqueness The minimal polynomial f of α is unique. To prove this, suppose that f and g are monic polynomials in Jα of minimal degree n > 0. We have that r := f−g ∈ Jα (because the latter is closed under addition/subtraction) and that m := deg(r) < n (because the polynomials are monic of the same degree)
https://en.wikipedia.org/wiki/Minimal%20polynomial%20%28linear%20algebra%29	In linear algebra, the minimal polynomial of an matrix over a field is the monic polynomial over of least degree such that . Any other polynomial with is a (polynomial) multiple of . The following three statements are equivalent: is a root of , is a root of the characteristic polynomial of , is an eigenvalue of matrix . The multiplicity of a root of is the largest power such that strictly contains . In other words, increasing the exponent up to will give ever larger kernels, but further increasing the exponent beyond will just give the same kernel. Formally, is the nilpotent index of . If the field is not algebraically closed, then the minimal and characteristic polynomials need not factor according to their roots (in ) alone, in other words they may have irreducible polynomial factors of degree greater than . For irreducible polynomials one has similar equivalences: divides , divides , the kernel of has dimension at least . the kernel of has dimension at least . Like the characteristic polynomial, the minimal polynomial does not depend on the base field. In other words, considering the matrix as one with coefficients in a larger field does not change the minimal polynomial. The reason for this differs from the case with the characteristic polynomial (where it is immediate from the definition of determinants), namely by the fact that the minimal polynomial is determined by the relations of linear dependence between the powers of : extending the base field will not introduce any new such relations (nor of course will it remove existing ones). The minimal polynomial is often the same as the characteristic polynomial, but not always. For example, if is a multiple of the identity matrix, then its minimal polynomial is since the kernel of is already the entire space; on the other hand its characteristic polynomial is (the only eigenvalue is , and the degree of the characteristic polynomial is always equal to the dimension of the space). The minimal polynomial always divides the characteristic polynomial, which is one way of formulating the Cayley–Hamilton theorem (for the case of matrices over a field). Formal definition Given an endomorphism on a finite-dimensional vector space over a field , let be the set defined as where is the space of all polynomials over the field . is a proper ideal of . Since is a field, is a principal ideal domain, thus any ideal is generated by a single polynomial, which is unique up to a unit in . A particular choice among the generators can be made, since precisely one of the generators is monic. The minimal polynomial is thus defined to be the monic polynomial that generates . It is the monic polynomial of least degree in . Applications An endomorphism of a finite-dimensional vector space over a field is diagonalizable if and only if its minimal polynomial factors completely over into distinct linear factors. The fact that there is only one factor for every eige
https://en.wikipedia.org/wiki/Religion%20in%20Transnistria	Pridnestrovian Moldavian Republic (Transnistria) official statistics show that 91 percent of the Transnistrian population adhere to Eastern Orthodox Christianity, with 4 percent adhering to the Catholic Church. Roman Catholics are mainly located in Northern Transnistria, where a notable Polish minority is living. Transnistria's government has supported the restoration and construction of new Orthodox churches. It affirms that the republic has freedom of religion and states that 114 religious beliefs and congregations are officially registered. However, as recently as 2009, registration hurdles were met with by some religious groups, notably the Jehovah's Witnesses. References Demographics of Transnistria History of Eastern Europe History of Transnistria Transnistrian culture
https://en.wikipedia.org/wiki/Ada%20Dietz	Ada K. Dietz (October 7, 1888 – January 12, 1981) was an American weaver best known for her 1949 monograph Algebraic Expressions in Handwoven Textiles, which defines a novel method for generating weaving patterns based on algebraic patterns. Her method employs the expansion of multivariate polynomials to devise a weaving scheme. Dietz' work is still well-regarded today, by both weavers and mathematicians. Along with the references listed below, Griswold (2001) cites several additional articles on her work. Algebraic weaving Ada Dietz developed her algebraic method in 1946 while living in Long Beach, California. An avid weaver, Dietz drew upon her experience as a former math teacher to devise a threading pattern based on a cubic binomial expansion. She describes her idea as follows: "Taking the cube of a binomial [ (x + y)3], I approached [the pattern] in the way applied algebraic problems are approached - by letting x equal one unknown and y equal the other unknown. "In this case, x equaled the first and second harnesses, and y equaled the third and fourth harnesses. Then it was simply a matter of expanding the cube of the binomial and substituting the values of x and y to write the threading draft." (Dietz, 1949) A piece based on the formula (a + b + c + d + e + f)2, submitted to the Little Loomhouse Country Fair in Louisville, Kentucky received such a positive response, which prompted a collaboration between Dietz and Little Loomhouse's founder, Lou Tate. The fruits of the collaboration included the booklet Algebraic Expressions in Handwoven Textiles and a traveling exhibit which continued throughout the 1950s. History and development Dietz was a high school biology and math teacher when she met Ruth E. Foster, a professional weaver with the Hewson Studios in Los Angeles. Foster's work inspired Dietz to begin studying weaving at Wayne University in Detroit under Nellie Sargent Johnson. Her experiments in writing weaving drafts began in Johnson's classes. It was later when Dietz and Foster were driving north to study at the Banff School of Fine Arts in Canada that she began using mathematical equations. She wanted "a reason for writing a draft in a definite way", and went to the mathematical equations she had worked with for so long. See also Mathematics and fiber arts References Sources Redfield, Gail (1959). "Variations on an Algebraic Equation". Handweaver & Craftsman (Summer): 46-49 1880s births 1950 deaths American weavers American textile designers Mathematical artists Women textile artists
https://en.wikipedia.org/wiki/Pseudomedian	In statistics, the pseudomedian is a measure of centrality for data-sets and populations. It agrees with the median for symmetric data-sets or populations. In mathematical statistics, the pseudomedian is also a location parameter for probability distributions. Description The pseudomedian of a distribution is defined to be a median of the distribution of , where and are independent, each with the same distribution . When is a symmetric distribution, the pseudomedian coincides with the median; otherwise this is not generally the case. The Hodges–Lehmann statistic, defined as the median of all of the midpoints of pairs of observations, is a consistent estimator of the pseudomedian. Like the set of medians, the pseudomedian is well defined for all probability distributions, even for the many distributions that lack modes or means. Pseudomedian filter in signal processing In signal processing there is another definition of pseudomedian filter for discrete signals. For a time series of length 2N + 1, the pseudomedian is defined as follows. Construct N + 1 sliding windows each of length N + 1. For each window, compute the minimum and maximum. Across all N + 1 windows, find the maximum minimum and the minimum maximum. The pseudomedian is the average of these two quantities. See also Hodges–Lehmann estimator Median filter Lulu smoothing References Means Summary statistics
https://en.wikipedia.org/wiki/Benjamin%20Kagan	Veniamin Fyodorovich Kagan (; 10 March 1869 – 8 May 1953) was a Russian and Soviet mathematician and expert in geometry. He is the maternal grandfather of mathematicians Yakov Sinai and Grigory Barenblatt. Biography Kagan was born in Shavli, in the Kovno Governorate of the Russian Empire (now Šiauliai, Lithuania) in 1869, to a poor Lithuanian Jewish family. In 1871 his family moved to Yekaterinoslav (now Dnipro), where he grew up. Kagan entered the Imperial Novorossiya University in Odesa in 1887, but was expelled for revolutionary activities in 1889. He was put on probation and sent back to Yekaterinoslav. He studied mathematics on his own and in 1892 passed the state exam at Kyiv University. In 1894 Kagan moved to Saint Petersburg where he continued his studies with Andrey Markov and Konstantin Posse. They tried to help him to obtain an academic position, but Kagan's Jewish background was an obstacle. Only in 1897 was he allowed to become a dozent at the Imperial Novorossiya University, where he continued to work until 1923. His students in the theory of relativity class he taught in 1921-22 included Nikolai Papaleksi, Alexander Frumkin and Igor Tamm. Kagan worked at Moscow State University where he held the Geometry Chair from 1923 till 1952. In 1924 he joined Otto Schmidt in drawing up plans for the Great Soviet Encyclopedia. Mathematical work He published over 100 mathematical papers in different parts of geometry, particularly on hyperbolic geometry and on Riemannian geometry. He received the Stalin Prize in 1943. He founded the science publisher Mathesis in Odesa. He was a director of the mathematics and natural sciences department of the Great Soviet Encyclopedia. He wrote a definitive biography of Nikolai Lobachevsky and edited his collected works (5 volumes, 1946–1951). Kagan's doctoral students include Viktor Wagner and Isaak Yaglom. Trivia He's a minor character in The Fourth Prose (1930) by Osip Mandelstam. External links Biography – in the "Kstati" newspaper (in Russian) 1869 births 1953 deaths Mathematicians from the Russian Empire People from Šiauliai People from Kovno Governorate Academic staff of Moscow State University Odesa University alumni Taras Shevchenko National University of Kyiv alumni Recipients of the Stalin Prize Recipients of the Order of the Red Banner of Labour Lithuanian Jews Soviet Jews Soviet mathematicians Geometers Differential geometers
https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Alabama	This is a list of law enforcement agencies in the U.S. state of Alabama. According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 417 law enforcement agencies employing 11,631 sworn police officers, about 251 for each 100,000 residents. State Agencies Alabama Law Enforcement Agency Alabama Department of Conservation and Natural Resources Alabama Department of Conservation and Natural Resources#Wildlife and Freshwater Fisheries Alabama Board of Pardons and Parole Alabama State Parks Ranger Service Alabama State Lands Security Alabama Department of Corrections Bureau of Special Investigations Alabama Department of Mental Health Police Alabama Securities Commission Alabama State Port Authority Police Alabama Department of Insurance State Fire Marshal's Office Marshals of the Alabama Appellate Courts Alabama National Guard Military Police/Security Forces (under state gubernatorial control unless federalized under Title 10 of the United States Code) County sheriff agencies Autauga County Sheriff's Office (Alabama) Baldwin County Sheriff's Office (Alabama) Barbour County Sheriff's Office (Alabama) Bibb County Sheriff's Department(Alabama) Blount County Sheriff's Office Bullock County Sheriff's Office Butler County Sheriff's Office Calhoun County Sheriff's Office Chambers County Sheriff's Office Cherokee County Sheriff's Department Chilton County Sheriff's Department Choctaw County Sheriff's Department Clarke County Sheriff's Office Clay County Sheriff's Office Cleburne County Sheriff's Office Coffee County Sheriff's Office Colbert County Sheriff's Department Conecuh County Sheriff's Office Coosa County Sheriff's Office Covington County Sheriff's Office Crenshaw County Sheriff's Office Cullman County Sheriff's Office Dale County Sheriff's Office Dallas County Sheriff's Office DeKalb County Sheriff's Office Elmore County Sheriff's Office Escambia County Sheriff's Department Etowah County Sheriff's Office Fayette County Sheriff's Office Franklin County Sheriff's Office Geneva County Sheriff's Office Greene County Sheriff's Office Hale County Sheriff's Office Henry County Sheriff's Office Houston County Sheriff's Department Jackson County Sheriff's Office Jefferson County Sheriff's Department Lamar County Sheriff's Office Lauderdale County Sheriff's Office Lawrence County Sheriff's Office Lee County Sheriff's Department Limestone County Sheriff's Office Lowndes County Sheriff's Office Macon County Sheriff's Office Madison County Sheriff's Department Marengo County Sheriff's Office Marion County Sheriff's Office Marshall County Sheriff's Office Mobile County Sheriff's Department Monroe County Sheriff's Office Montgomery County Sheriff's Office Morgan County Sheriff's Office Perry County Sheriff's Office Pickens County Sheriff's Office Pike County Sheriff's Office Randolph County Sheriff's Office Russell County Sheriff's Office Saint Clair County Sheriff's Office Shelby County Sher

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