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Mathematics is the study ofrepresenting andreasoning about abstractobjects (such asnumbers,points,spaces,sets,structures, andgames). Mathematics is used throughout the world as an essential tool in many fields, includingnatural science,engineering,medicine, and thesocial sciences.Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such asstatistics andgame theory. Mathematicians also engage inpure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.(Full article...)

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Image 1

A statue depicting Zhang Heng at his tomb inNanyang, Henan province, China

Zhang Heng (Chinese:張衡; AD 78–139),courtesy namePingzi, formerlyromanizedChang Heng, was a Chinesepolymathic scientist and statesman who lived during theEastern Han dynasty. Educated in the capital cities ofLuoyang andChang'an, he achieved success as an astronomer, mathematician, seismologist, hydraulic engineer, inventor, geographer,cartographer,ethnographer, artist, poet, philosopher, politician, and literary scholar.

Zhang Heng began his career as a minor civil servant inNanyang. Eventually, he became Chief Astronomer, Prefect of the Majors for Official Carriages, and then Palace Attendant at the imperial court. His uncompromising stance on historical and calendrical issues led to his becoming a controversial figure, preventing him from rising to the status of Grand Historian. His political rivalry with the palaceeunuchs during the reign ofEmperor Shun (r. 125–144) led to his decision to retire from the central court to serve as an administrator ofHejian Kingdom in present-dayHebei. Zhang returned home to Nanyang for a short time, before being recalled to serve in the capital once more in 138. He died there a year later, in 139. (Full article...)
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Logic studies valid forms of inference likemodus ponens.

Logic is the study of correctreasoning. It includes bothformal andinformal logic. Formal logic is the study ofdeductively valid inferences orlogical truths. It examines how conclusions follow frompremises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated withinformal fallacies,critical thinking, andargumentation theory. Informal logic examines arguments expressed innatural language whereas formal logic usesformal language. When used as acountable noun, the term "a logic" refers to a specific logicalformal system that articulates aproof system. Logic plays a central role in many fields, such asphilosophy,mathematics,computer science, andlinguistics.

Logic studiesarguments, which consist of a set of premises that leads to aconclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to work." Premises and conclusions expresspropositions or claims that can be true or false. An important feature of propositions is their internal structure. For example, complex propositions are made up of simpler propositions linked bylogical vocabulary like $\land$ (and) or $\to$ (if...then). Simple propositions also have parts, like "Sunday" or "work" in the example. The truth of a proposition usually depends on the meanings of all of its parts. However, this is not the case for logically true propositions. They are true only because of their logical structure independent of the specific meanings of the individual parts. (Full article...)
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Elementary algebra studies which values solve equations formed using arithmetical operations.

Algebra is a branch ofmathematics that deals with abstract systems, known asalgebraic structures, and the manipulation ofexpressions within those systems. It is a generalization ofarithmetic that introducesvariables andalgebraic operations other than the standard arithmetic operations, such asaddition andmultiplication.

Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables.Linear algebra is a closely related field that investigateslinear equations and combinations of them calledsystems of linear equations. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. (Full article...)
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The manipulations of theRubik's Cube form theRubik's Cube group.

Inmathematics, agroup is aset with anoperation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation isassociative, it has anidentity element, and every element of the set has aninverse element. For example, theintegers with theaddition operation form a group.

The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers,geometric shapes andpolynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. (Full article...)
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Inclassical mechanics, theLaplace–Runge–Lenz vector (LRL vector) is avector used chiefly to describe the shape and orientation of theorbit of oneastronomical body around another, such as abinary star or a planet revolving around a star. Fortwo bodies interacting byNewtonian gravity, the LRL vector is aconstant of motion, meaning that it is the same no matter where it is calculated on the orbit; equivalently, the LRL vector is said to beconserved. More generally, the LRL vector is conserved in all problems in which two bodies interact by acentral force that varies as theinverse square of the distance between them; such problems are calledKepler problems.

Thus thehydrogen atom is a Kepler problem, since it comprises two charged particles interacting byCoulomb's law ofelectrostatics, another inverse-square central force. The LRL vector was essential in the firstquantum mechanical derivation of thespectrum of the hydrogen atom, before the development of theSchrödinger equation. However, this approach is rarely used today. (Full article...)
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Title page of the first edition of Wright'sCertaine Errors in Navigation (1599)

Edward Wright (baptised 8 October 1561; died November 1615) was an English mathematician andcartographer noted for his bookCertaine Errors in Navigation (1599; 2nd ed., 1610), which for the first time explained the mathematical basis of theMercator projection by building on the works ofPedro Nunes, and set out a reference table giving the linear scale multiplication factor as a function oflatitude, calculated for eachminute of arc up to a latitude of 75°. This was in fact a table of values of theintegral of the secant function, and was the essential step needed to make practical both the making and the navigational use of Mercator charts.

Wright was born atGarveston inNorfolk and educated atGonville and Caius College, Cambridge, where he became afellow from 1587 to 1596. In 1589 the college granted him leave afterElizabeth I requested that he carry outnavigational studies witha raiding expedition organised by theEarl of Cumberland to theAzores to capture Spanishgalleons. The expedition's route was the subject of the first map to be prepared according to Wright's projection, which was published inCertaine Errors in 1599. The same year, Wright created and published the first world map produced in England and the first to use the Mercator projection sinceGerardus Mercator's original 1569 map. (Full article...)
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Bust of Shen at theBeijing Ancient Observatory

Shen Kuo (Chinese:沈括; 1031–1095) orShen Gua,courtesy nameCunzhong (存中) andpseudonymMengqi (now usually given asMengxi)Weng (夢溪翁), was a Chinesepolymath, scientist, and statesman of theNorthern Song dynasty. Shen was a master in many fields of study includingmathematics,optics, andhorology. In his career as a civil servant, he became afinance minister, governmental state inspector, head official for theBureau of Astronomy in the Song court, Assistant Minister of Imperial Hospitality, and also served as anacademic chancellor. At court his political allegiance was to the Reformist faction known as theNew Policies Group, headed byChancellor Wang Anshi (1021–1085).

In hisDream Pool Essays orDream Torrent Essays (夢溪筆談;Mengxi Bitan) of 1088, Shen was the first to describe the magnetic needlecompass, which would be used for navigation (first described in Europe byAlexander Neckam in 1187). Shen discovered the concept oftrue north in terms ofmagnetic declination towards thenorth pole, with experimentation of suspended magnetic needles and "the improvedmeridian determined by Shen's [astronomical] measurement of the distance between thepole star and true north". This was the decisive step in human history to make compasses more useful for navigation, and may have been a concept unknown in Europefor another four hundred years (evidence of German sundials made circa 1450 show markings similar toChinese geomancers' compasses in regard to declination). (Full article...)
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Portrait,c. 1910, after 1620 original

Johannes Kepler (27 December 1571 – 15 November 1630) was a Germanastronomer,mathematician,astrologer,natural philosopher andmusic theorist. He is a key figure in the 17th-centuryScientific Revolution, best known for hislaws of planetary motion, and his booksAstronomia nova,Harmonice Mundi, andEpitome Astronomiae Copernicanae. The variety and impact of his work made Kepler one of the founders and fathers of modernastronomy, thescientific method,natural science, andmodern science. He has been described as the "father ofscience fiction" for his novelSomnium.

Kepler was a mathematics teacher at aseminary school inGraz, where he became an associate ofPrince Hans Ulrich von Eggenberg. Later he became an assistant to the astronomerTycho Brahe inPrague, and eventually the imperial mathematician toEmperor Rudolf II and his two successorsMatthias andFerdinand II. He also taught mathematics inLinz, and was an adviser toGeneral Wallenstein. (Full article...)
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One of Molyneux's celestial globes, which is displayed inMiddle Temple Library – from the frontispiece of theHakluyt Society's 1889 reprint ofA Learned Treatise of Globes, both Cœlestiall and Terrestriall, one of the English editions of Robert Hues'Latin workTractatus de Globis (1594)

Emery Molyneux (/ˈɛməriˈmɒlɪnoʊ/EM-ər-eeMOL-in-oh; died June 1598) was an EnglishElizabethan maker ofglobes,mathematical instruments andordnance. His terrestrial and celestial globes, first published in 1592, were the first to be made in England and the first to be made by an Englishman.

Molyneux was known as amathematician and maker ofmathematical instruments such ascompasses andhourglasses. He became acquainted with many prominent men of the day, including the writerRichard Hakluyt and the mathematiciansRobert Hues andEdward Wright. He also knew the explorersThomas Cavendish,Francis Drake,Walter Raleigh andJohn Davis. Davis probably introduced Molyneux to his own patron, the London merchant William Sanderson, who largely financed the construction of the globes. When completed, the globes were presented toElizabeth I. Larger globes were acquired by royalty, noblemen and academic institutions, while smaller ones were purchased as practical navigation aids for sailors and students. The globes were the first to be made in such a way that they were unaffected by the humidity at sea, and they came into general use on ships. (Full article...)
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Cantor,c. 1910

Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔːr/KAN-tor;German:[ˈɡeːɔʁkˈfɛʁdinantˈluːtvɪçˈfiːlɪpˈkantoːɐ̯]; 3 March [O.S. 19 February] 1845 – 6 January 1918) was a mathematician who played a pivotal role in the creation ofset theory, which has become afundamental theory in mathematics. Cantor established the importance ofone-to-one correspondence between the members of two sets, definedinfinite andwell-ordered sets, and proved that thereal numbers are more numerous than thenatural numbers. Cantor's method of proof of this theorem implies the existence of aninfinity of infinities. He defined thecardinal andordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.

Originally, Cantor's theory oftransfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such asLeopold Kronecker andHenri Poincaré and later fromHermann Weyl andL. E. J. Brouwer, whileLudwig Wittgenstein raisedphilosophical objections; seeControversy over Cantor's theory. Cantor, a devoutLutheran Christian, believed the theory had been communicated to him by God. SomeChristian theologians (particularlyneo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God – on one occasion equating the theory of transfinite numbers withpantheism – a proposition that Cantor vigorously rejected. Not all theologians were against Cantor's theory; prominent neo-scholastic philosopherKonstantin Gutberlet [de;it] was in favor of it and CardinalJohann Baptist Franzelin accepted it as a valid theory (after Cantor made some important clarifications). (Full article...)
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Damage fromHurricane Katrina in 2005. Actuaries need to estimate long-term levels of such damage in order to accurately price property insurance, set appropriatereserves, and design appropriatereinsurance and capital management strategies.

Anactuary is a professional with advanced mathematical skills who deals with the measurement and management ofrisk and uncertainty. These risks can affect both sides of thebalance sheet and requireasset management,liability management, and valuation skills. Actuaries provide assessments of financial security systems, with a focus on their complexity, their mathematics, and their mechanisms. The name of the corresponding academic discipline isactuarial science.

While the concept of insurance dates to antiquity, the concepts needed to scientifically measure and mitigate risks have their origins in 17th-century studies of probability and annuities. Actuaries in the 21st century require analytical skills, business knowledge, and an understanding of human behavior and information systems; actuaries use this knowledge to design programs that manage risk, by determining if the implementation of strategies proposed for mitigating potential risks does not exceed the expected cost of those risks actualized. Thesteps needed to become an actuary, including education and licensing, are specific to a given country, with various additional requirements applied by regional administrative units; however, almost all processes impart universal principles of risk assessment, statistical analysis, and risk mitigation, involving rigorously structured training and examination schedules, taking many years to complete. (Full article...)
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Rejewski,c. 1932

Marian Adam Rejewski (Polish:[ˈmarjanrɛˈjɛfskʲi]^ⓘ; 16 August 1905 – 13 February 1980) was a Polishmathematician andcryptologist who in late 1932 reconstructed the sight-unseen German militaryEnigma cipher machine, aided by limited documents obtained byFrench military intelligence.

Over the next nearly seven years, Rejewski and fellow mathematician-cryptologistsJerzy Różycki andHenryk Zygalski, working at thePolish General Staff'sCipher Bureau, developed techniques and equipment for decrypting the Enigma ciphers, even as the Germans introduced modifications to their Enigma machines and encryption procedures. Rejewski's contributions included thecryptologic card catalog and thecryptologic bomb. (Full article...)
Image 13

Kaczynski after his arrest in 1996

Theodore John Kaczynski (/kəˈzɪnski/ ^ⓘkə-ZIN-skee; May 22, 1942 – June 10, 2023), also known as theUnabomber (/ˈjuːnəbɒmər/ ^ⓘYOO-nə-bom-ər), was an American mathematician anddomestic terrorist. A mathematicsprodigy, he abandoned his academic career in 1969 to pursue areclusive primitive lifestyle andlone wolf terrorism campaign.

Kaczynski murdered 3 people and injured 23 others between 1978 and 1995 in a nationwidemail bombing campaign against people he believed to be advancingmodern technology and the destruction of thenatural environment. He authored a roughly 35,000-wordmanifesto andsocial critique,Industrial Society and Its Future (1995), which opposes all forms of technology, rejectsleftism andfascism, advocates cultural primitivism, and ultimately suggests violentrevolution. (Full article...)
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General relativity, also known as thegeneral theory of relativity, and asEinstein's theory of gravity, is thegeometric theory ofgravitation published byAlbert Einstein in May 1916 and is the accepted description of the gravitation of macroscopic objects inmodern physics. Generalrelativity generalizesspecial relativity and refinesIsaac Newton'slaw of universal gravitation, providing a unified description of gravity as a geometric property ofspace andtime, or four-dimensionalspacetime. In particular, thecurvature of spacetime is directly related to theenergy,momentum andstress of whatever is present, includingmatter andradiation. The relation is specified by theEinstein field equations, a system of second-orderpartial differential equations.John Archibald Wheeler summarized it: "Space-time tells matter how to move; matter tells space-time how to curve."

Newton's law of universal gravitation, which describes gravity in classical mechanics, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gravitation inclassical physics. These predictions concern the passage of time, thegeometry of space, the motion of bodies infree fall, and the propagation of light, and includegravitational time dilation,gravitational lensing, thegravitational redshift of light, theShapiro time delay andsingularities/black holes. So far, alltests of general relativity have been in agreement with the theory. The time-dependent solutions of general relativity enable us to extrapolate the history of the universe into the past and future, and have provided the modern framework forcosmology, thus leading to the discovery of theBig Bang andcosmic microwave background radiation. Despite the introduction of a number ofalternative theories, general relativity continues to be the simplest theory consistent withexperimental data. (Full article...)
Image 15
The regular triangular tiling of the plane, whose symmetries are described by the affine symmetric groupS̃₃

Theaffine symmetric groups are a family ofmathematical structures that describe the symmetries of thenumber line and the regulartriangular tiling of the plane, as well as related higher-dimensional objects. In addition to this geometric description, the affine symmetric groups may be defined in other ways: as collections ofpermutations (rearrangements) of the integers (..., −2, −1, 0, 1, 2, ...) that are periodic in a certain sense, or in purely algebraic terms as agroup with certaingenerators and relations. They are studied incombinatorics andrepresentation theory.

A finitesymmetric group consists of all permutations of afinite set. Each affine symmetric group is an infiniteextension of a finite symmetric group. Many important combinatorial properties of the finite symmetric groups can be extended to the corresponding affine symmetric groups.Permutation statistics such asdescents andinversions can be defined in the affine case. As in the finite case, the natural combinatorial definitions for these statistics also have a geometric interpretation. (Full article...)

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three lines connecting corresponding vertices of a larger triangle on the left and a smaller one on the right converge at a point further to the right called the "center of perspectivity"

Desargues's theorem

Credit: User:Jujutacular, based on an original byUser:DynaBlast

Inprojective geometry,Desargues' theorem states that two triangles are inperspective axiallyif and only if they are in perspective centrally. Lines through the triangle sides meet in pairs atcollinear points along the axis of perspectivity. Lines through corresponding pairs ofvertices on the triangles meet at a point called the center of perspectivity.

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Image 1
Two greedy colorings of the samecrown graph using different vertex orders. The right example generalises to 2-colorable graphs withn vertices, where the greedy algorithm expendsn/2 colors.

In the study ofgraph coloring problems inmathematics andcomputer science, agreedy coloring orsequential coloring is a coloring of thevertices of agraph formed by agreedy algorithm that considers the vertices of the graph in sequence and assigns eachvertex itsfirst available color. Greedy colorings can be found in linear time, but they do not, in general, use the minimum number of colors possible.

Different choices of the sequence of vertices will typically produce different colorings of the given graph, so much of the study of greedy colorings has concerned how to find a good ordering. There always exists an ordering that produces an optimal coloring, but although such orderings can be found for many special classes of graphs, they are hard to find in general. Commonly used strategies for vertex ordering involve placing higher-degree vertices earlier than lower-degree vertices, or choosing vertices with fewer available colors in preference to vertices that are less constrained. (Full article...)
Image 2
A Garden of Eden inConway's Game of Life, discovered by R. Banks in 1971. The cells outside the image are all dead (white).

In acellular automaton, aGarden of Eden is a configuration that has no predecessor. It can be theinitial configuration of the automaton but cannot arise in any other way.
John Tukey named these configurations after theGarden of Eden inAbrahamic religions, which was created out of nowhere.

A Garden of Eden is determined by the state of every cell in the automaton (usually a one- or two-dimensional infinitesquare lattice of cells). However, for any Garden of Eden there is a finite pattern (a subset of cells and their states, called anorphan) with the same property of having no predecessor, no matter how the remaining cells are filled in.
A configuration of the whole automaton is a Garden of Eden if and only if it contains an orphan.
For one-dimensional cellular automata, orphans and Gardens of Eden can be found by an efficient algorithm, but for higherdimensions this is anundecidable problem. Nevertheless, computer searches have succeeded in finding these patterns inConway's Game of Life. (Full article...)
Image 3
Viète's formula, as printed in Viète'sVariorum de rebus mathematicis responsorum, liber VIII (1593)

Inmathematics,Viète's formula is the followinginfinite product ofnested radicals representing twice thereciprocal of the mathematical constantπ:
${\begin{aligned}{\frac {2}{\pi }}&={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots \\[5mu]<br>&={\sqrt {\frac {1}{2}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {\frac {1}{2}}}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {\frac {1}{2}}}}}}}\cdots <br>\end{aligned}}$

It can also be represented as
${\frac {2}{\pi }}=\prod _{n=1}^{\infty }\cos {\frac {\pi }{2^{n+1}}}.$ (Full article...)
Image 4

Official portrait, 2014

José Félix Mendieta Villarroel (born 15 November 1958) is a Bolivian politician and trade unionist who served as a member of theChamber of Deputies fromCochabamba, representing circumscription 28 from 2010 to 2015.

Though educated inpedagogy, Mendieta spent most of his career in commercial driving, climbing the ranks of the sector's trade unions to eventually become general secretary of the Sacaba Mixed Motor Transport Union. Though traditionally conservative, under the leadership of figures like Mendieta, many of the country's drivers' unions were reoriented towards the left. (Full article...)
Image 5
Ronald Paul "Ron" Fedkiw (born February 27, 1968) is afull professor in theStanford University department ofcomputer science and a leading researcher in the field ofcomputer graphics, focusing on topics relating to physically based simulation of natural phenomena and machine learning. His techniques have been employed in manymotion pictures. He has earned recognition at the80th Academy Awards and the87th Academy Awards as well as from theNational Academy of Sciences.

His first Academy Award was awarded for developing techniques that enabled many technically sophisticated adaptations including the visual effects in 21st century movies in theStar Wars,Harry Potter,Terminator, andPirates of the Caribbean franchises. Fedkiw has designed aplatform that has been used to create many of the movie world's most advanced special effects since it was first used on theT-X character inTerminator 3: Rise of the Machines. His second Academy Award was awarded for computer graphics techniques for special effects for large scale destruction. Although he has won an Oscar for his work, he does not design thevisual effects that use his technique. Instead, he has developed a system that other award-winning technicians and engineers have used to create visual effects for some of the world's most expensive and highest-grossing movies. (Full article...)
Image 6

Acube is athree-dimensional solid object ingeometry. A cube has eight vertices and twelve straight edges of the same length, so that these edges form sixsquare faces of the same size. It is an example of apolyhedron.

The cube is found in many popular cultures, including toys and games, the arts, optical illusions, and architectural buildings. Cubes can be found in crystal structures, science, and technological devices. It is also found in ancient texts, such asPlato's workTimaeus, which described a set of solids now calledPlatonic solids, associating a cube with theclassical element ofearth. Acube with unit length is the canonical unit ofvolume in three-dimensional space, relative to which other solid objects are measured. (Full article...)
Image 7
Translation of an English sentence to first-order logic

Logic translation is the process of representing a text in theformal language of alogical system. If the original text is formulated inordinary language then the termnatural language formalization is often used. An example is the translation of the English sentence "some men are bald" intofirst-order logic as $\exists x(M(x)\land B(x))$ . The purpose is to reveal thelogical structure ofarguments. This makes it possible to use the precise rules of formal logic to assess whether these arguments are correct. It can also guidereasoning by arriving at newconclusions.

Many of the difficulties of the process are caused byvague orambiguous expressions in natural language. For example, the English word "is" can mean that somethingexists, that it isidentical to something else, or that it has a certainproperty. This contrasts with the precise nature of formal logic, which avoids such ambiguities. Natural language formalization is relevant to various fields in thesciences andhumanities. It may play a key role forlogic in general since it is needed to establish a link between many forms of reasoning and abstract logical systems. The use ofinformal logic is an alternative to formalization since it analyzes the cogency of ordinary language arguments in their original form. Natural language formalization is distinguished from logic translations that convert formulas from one logical system into another, for example, frommodal logic to first-order logic. This form of logic translation is specifically relevant forlogic programming andmetalogic. (Full article...)
Image 8
The red graph is the dual graph of the blue graph, andvice versa.

In themathematical discipline ofgraph theory, thedual graph of aplanar graphG is a graph that has avertex for eachface ofG. The dual graph has anedge for each pair of faces inG that are separated from each other by an edge, and aself-loop when the same face appears on both sides of an edge. Thus, each edgee ofG has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of e. The definition of the dual depends on the choice of embedding of the graphG, so it is a property of plane graphs (graphs that are already embedded in the plane) rather thanplanar graphs (graphs that may be embedded but for which the embedding is not yet known). For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph.

Historically, the first form of graphduality to be recognized was the association of thePlatonic solids into pairs ofdual polyhedra. Graph duality is atopological generalization of the geometric concepts of dual polyhedra anddual tessellations, and is in turn generalized combinatorially by the concept of adual matroid. Variations of planar graph duality include a version of duality fordirected graphs, and duality for graphs embedded onto non-planar two-dimensional surfaces. (Full article...)
Image 9
A homomorphism from theflower snarkJ₅ into the cycle graphC₅.
It is also a retraction onto the subgraph on the central five vertices. ThusJ₅ is in fact homomorphically equivalent to thecoreC₅.

In themathematical field ofgraph theory, agraph homomorphism is a mapping between twographs that respects their structure. More concretely, it is a function between the vertex sets of two graphs that maps adjacentvertices to adjacent vertices.

Homomorphisms generalize various notions ofgraph colorings and allow the expression of an important class ofconstraint satisfaction problems, such as certainscheduling orfrequency assignment problems.
The fact that homomorphisms can be composed leads to rich algebraic structures: apreorder on graphs, adistributive lattice, and acategory (one for undirected graphs and one for directed graphs).
Thecomputational complexity of finding a homomorphism between given graphs is prohibitive in general, but a lot is known about special cases that are solvable inpolynomial time. Boundaries between tractable and intractable cases have been an active area of research. (Full article...)
Image 10

Berlin, 1959

Andrew Mattei Gleason (November 4, 1921 – October 17, 2008) was an Americanmathematician who made fundamental contributions to widely varied areas of mathematics, including the solution ofHilbert's fifth problem, and was a leader in reform and innovation in mathematics teaching at all levels.Gleason's theorem inquantum logic and theGreenwood–Gleason graph, an important example inRamsey theory, are named for him.

As a young World War II naval officer, Gleason broke German and Japanese military codes. After the war he spent his entire academic career atHarvard University, from which he retired in 1992. His numerous academic and scholarly leadership posts included chairmanship of the Harvard Mathematics Department and theHarvard Society of Fellows, and presidency of theAmerican Mathematical Society. He continued to advise the United States government oncryptographic security, and the Commonwealth of Massachusetts on mathematics education for children, almost until the end of his life. (Full article...)
Image 11
Incontrol system theory, and various branches of engineering, atransfer function matrix, or justtransfer matrix is a generalisation of thetransfer functions ofsingle-input single-output (SISO) systems tomultiple-input and multiple-output (MIMO) systems. Thematrix relates the outputs of the system to its inputs. It is a particularly useful construction forlinear time-invariant (LTI) systems because it can be expressed in terms of thes-plane.

In some systems, especially ones consisting entirely ofpassive components, it can be ambiguous which variables are inputs and which are outputs. In electrical engineering, a common scheme is to gather all the voltage variables on one side and all the current variables on the other regardless of which are inputs or outputs. This results in all the elements of the transfer matrix being in units ofimpedance. The concept of impedance (and hence impedance matrices) has been borrowed into other energy domains by analogy, especially mechanics and acoustics. (Full article...)
Image 12
Quadratrix (red); snapshot of E and F having completed 60% of their motions

Thequadratrix ortrisectrix of Hippias (also called thequadratrix of Dinostratus) is acurve which is created by auniform motion. It is traced out by the crossing point of twolines, one moving bytranslation at a uniform speed, and the other moving byrotation around one of its points at a uniform speed. An alternative definition as aparametric curve leads to an equivalence between the quadratrix, the image of theLambert W function, and the graph of the function $y=x\cot x$ .

The discovery of this curve is attributed to the Greek sophistHippias of Elis, around 420 BC. Historians of mathematics have suggested that Hippias used it to solve theangle trisection problem, hence its name as atrisectrix. Later around 350 BCDinostratus used it to solve the problem ofsquaring the circle, hence its name as aquadratrix.Dinostratus's theorem, used by Dinostratus to square the circle, relates an endpoint of the curve to the value ofπ. Both angle trisection and squaring the circle can be solved using a compass, a straightedge, and a given copy of this curve; however, they cannot be solved withcompass and straightedge alone. Although adense set of points on the curve can be constructed by compass and straightedge, allowing these problems to be approximated, the whole curve cannot be constructed in this way. (Full article...)

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... that the BritishNational Hospital Service Reserve trained volunteers to carry out first aid in the aftermath of a nuclear or chemical attack?
... that Latvian-Soviet artistKarlis Johansons exhibited a skeletaltensegrity form of theSchönhardt polyhedron seven years beforeErich Schönhardt's 1928 paper on its mathematics?
... thatGreen Day's "Wake Me Up When September Ends" became closely associated with the aftermath ofHurricane Katrina?
... thatHannah Fry used mathematics to compare Elizabeth II's Christmas messages with the lyrics ofSnoop Dogg?
... thatCarmel Naughton, having been told that girls were "stupid and couldn't do maths", sponsored aSTEM scholarship fund?

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...that the mathematicianGrigori Perelman was offered aFields Medal in 2006, in part for his proof of thePoincaré conjecture, which he declined?
...that a regularheptagon is theregular polygon with the fewest sides which is notconstructible with acompass and straightedge?
...that the regulartrigonometric functions and thehyperbolic trigonometric functions can be related without usingcomplex numbers through theGudermannian function?
...that theCatalan numbers solve a number of problems incombinatorics such as the number of ways to completely parenthesize an algebraic expression withn+1 factors?
...that aball can be cut up and reassembled into two balls, each the same size as the original (Banach-Tarski paradox)?
...that it is impossible to devise a single formula involving only polynomials and radicals for solving an arbitraryquintic equation?
...thatEuler found 59 moreamicable numbers while for 2000 years, only 3 pairs had been found before him?

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Georg Ferdinand Ludwig Philipp Cantor (December 3, 1845,St. Petersburg, Russia – January 6, 1918,Halle, Germany) was a German mathematician who is best known as the creator ofset theory. Cantor established the importance ofone-to-one correspondence between sets, definedinfinite andwell-ordered sets, and proved that thereal numbers are "more numerous" than thenatural numbers. In fact,Cantor's theorem implies the existence of an "infinity of infinities." He defined thecardinal andordinal numbers, and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.

Cantor's work encounteredresistance from mathematical contemporaries such asLeopold Kronecker andHenri Poincaré, and later fromHermann Weyl andL.E.J. Brouwer.Ludwig Wittgenstein raisedphilosophical objections. Nowadays, the vast majority of mathematicians who are neitherconstructivists norfinitists accept Cantor's work on transfinite sets and arithmetic, recognizing it as a majorparadigm shift.(Full article...)