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🔗 Mathematics

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any ${\displaystyle m\times n}$ matrix via an extension of the polar decomposition.

Specifically, the singular value decomposition of an ${\displaystyle m\times n}$ real or complex matrix ${\displaystyle \mathbf {M} }$ is a factorization of the form ${\displaystyle \mathbf {U\Sigma V^{*}} }$, where ${\displaystyle \mathbf {U} }$ is an ${\displaystyle m\times m}$ real or complex unitary matrix, ${\displaystyle \mathbf {\Sigma } }$ is an ${\displaystyle m\times n}$ rectangular diagonal matrix with non-negative real numbers on the diagonal, and ${\displaystyle \mathbf {V} }$ is an ${\displaystyle n\times n}$ real or complex unitary matrix. If ${\displaystyle \mathbf {M} }$ is real, ${\displaystyle \mathbf {U} }$ and ${\displaystyle \mathbf {V} =\mathbf {V^{*}} }$ are real orthonormal matrices.

The diagonal entries ${\displaystyle \sigma _{i}=\Sigma _{ii}}$ of ${\displaystyle \mathbf {\Sigma } }$ are known as the singular values of ${\displaystyle \mathbf {M} }$. The number of non-zero singular values is equal to the rank of ${\displaystyle \mathbf {M} }$. The columns of ${\displaystyle \mathbf {U} }$ and the columns of ${\displaystyle \mathbf {V} }$ are called the left-singular vectors and right-singular vectors of ${\displaystyle \mathbf {M} }$, respectively.

The SVD is not unique. It is always possible to choose the decomposition so that the singular values ${\displaystyle \Sigma _{ii}}$are in descending order. In this case, Σ (but not always U and V) is uniquely determined by M.

The term sometimes refers to the compact SVD, a similar decomposition ${\displaystyle \mathbf {M} =\mathbf {U\Sigma V^{*}} }$ in which Σ is square diagonal of size ${\displaystyle r\times r}$, where ${\displaystyle r\leq \min\{m,n\}}$ is the rank of M, and has only the non-zero singular values. In this variant, ${\displaystyle \mathbf {U} }$ is an ${\displaystyle m\times r}$ matrix and ${\displaystyle \mathbf {V} }$ is an ${\displaystyle n\times r}$ matrix, such that ${\displaystyle \mathbf {U^{*}U} =\mathbf {V^{*}V} =\mathbf {I} _{r\times r}}$.

Mathematical applications of the SVD include computing the pseudoinverse, matrix approximation, and determining the rank, range, and null space of a matrix. The SVD is also extremely useful in all areas of science, engineering, and statistics, such as signal processing, least squares fitting of data, and process control.

🔗 United States 🔗 Biography 🔗 Mathematics 🔗 United States/Vermont

Zerah Colburn (September 1, 1804 – March 2, 1840) was a child prodigy of the 19th century who gained fame as a mental calculator.

🔗 Mathematics

Pentagramma mirificum (Latin for miraculous pentagram) is a star polygon on a sphere, composed of five great circle arcs, all of whose internal angles are right angles. This shape was described by John Napier in his 1614 book Mirifici logarithmorum canonis descriptio (Description of the wonderful rule of logarithms) along with rules that link the values of trigonometric functions of five parts of a right spherical triangle (two angles and three sides). The properties of pentagramma mirificum were studied, among others, by Carl Friedrich Gauss.

🔗 Biography 🔗 Mathematics 🔗 Military history 🔗 Military history/Military biography 🔗 Cryptography 🔗 Cryptography/Computer science 🔗 Military history/European military history 🔗 Military history/British military history

William Thomas "Bill" Tutte OC FRS FRSC (; 14 May 1917 – 2 May 2002) was a British codebreaker and mathematician. During the Second World War, he made a brilliant and fundamental advance in cryptanalysis of the Lorenz cipher, a major Nazi German cipher system which was used for top-secret communications within the Wehrmacht High Command. The high-level, strategic nature of the intelligence obtained from Tutte's crucial breakthrough, in the bulk decrypting of Lorenz-enciphered messages specifically, contributed greatly, and perhaps even decisively, to the defeat of Nazi Germany. He also had a number of significant mathematical accomplishments, including foundation work in the fields of graph theory and matroid theory.

Tutte's research in the field of graph theory proved to be of remarkable importance. At a time when graph theory was still a primitive subject, Tutte commenced the study of matroids and developed them into a theory by expanding from the work that Hassler Whitney had first developed around the mid 1930s. Even though Tutte's contributions to graph theory have been influential to modern graph theory and many of his theorems have been used to keep making advances in the field, most of his terminology was not in agreement with their conventional usage and thus his terminology is not used by graph theorists today. "Tutte advanced graph theory from a subject with one text (D. Kőnig's) toward its present extremely active state."

🔗 Mathematics 🔗 Philosophy 🔗 Socialism

The mathematical manuscripts of Karl Marx are a manuscript collection of Karl Marx's mathematical notes where he attempted to derive the foundations of infinitesimal calculus from first principles.

The notes that Marx took have been collected into four independent treatises: On the Concept of the Derived Function, On the Differential, On the History of Differential Calculus, and Taylor's Theorem, MacLaurin's Theorem, and Lagrange's Theory of Derived Functions, along with several notes, additional drafts, and supplements to these four treatises. These treatises attempt to construct a rigorous foundation for calculus and use historical materialism to analyze the history of mathematics.

Marx's contributions to mathematics did not have any impact on the historical development of calculus, and he was unaware of many more recent developments in the field at the time, such as the work of Cauchy. However, his work in some ways anticipated, but did not influence, some later developments in 20th century mathematics. These manuscripts, which are from around 1873–1883, were not published in any language until 1968 when they were published in the Soviet Union alongside a Russian translation. Since their publication, Marx's independent contributions to mathematics have been analyzed in terms of both his own historical and economic theories, and in light of their potential applications of nonstandard analysis.

🔗 Mathematics 🔗 Biology 🔗 Physics 🔗 Plants

Tendril perversion, often referred to in context as simply perversion, is a geometric phenomenon found in helical structures such as plant tendrils, in which a helical structure forms that is divided into two sections of opposite chirality, with a transition between the two in the middle. A similar phenomenon can often be observed in kinked helical cables such as telephone handset cords.

The phenomenon was known to Charles Darwin, who wrote in 1865,

A tendril ... invariably becomes twisted in one part in one direction, and in another part in the opposite direction... This curious and symmetrical structure has been noticed by several botanists, but has not been sufficiently explained.

The term "tendril perversion" was coined by Goriely and Tabor in 1998 based on the word perversion found in the 19th Century science literature. "Perversion" is a transition from one chirality to another and was known to James Clerk Maxwell, who attributed it to the topologist J. B. Listing.

Tendril perversion can be viewed as an example of spontaneous symmetry breaking, in which the strained structure of the tendril adopts a configuration of minimum energy while preserving zero overall twist.

Tendril perversion has been studied both experimentally and theoretically. Gerbode et al. have made experimental studies of the coiling of cucumber tendrils. A detailed study of a simple model of the physics of tendril perversion was made by MacMillen and Goriely in the early 2000s. Liu et al. showed in 2014 that "the transition from a helical to a hemihelical shape, as well as the number of perversions, depends on the height to width ratio of the strip's cross-section."

Generalized tendril perversions were put forward by Silva et al., to include perversions that can be intrinsically produced in elastic filaments, leading to a multiplicity of geometries and dynamical properties.

🔗 Computer science 🔗 Mathematics

In computational complexity theory, the unique games conjecture (often referred to as UGC) is a conjecture made by Subhash Khot in 2002. The conjecture postulates that the problem of determining the approximate value of a certain type of game, known as a unique game, has NP-hard computational complexity. It has broad applications in the theory of hardness of approximation. If the unique games conjecture is true and P ≠ NP, then for many important problems it is not only impossible to get an exact solution in polynomial time (as postulated by the P versus NP problem), but also impossible to get a good polynomial-time approximation. The problems for which such an inapproximability result would hold include constraint satisfaction problems, which crop up in a wide variety of disciplines.

The conjecture is unusual in that the academic world seems about evenly divided on whether it is true or not.

🔗 Biography 🔗 Mathematics 🔗 Biography/science and academia 🔗 History of Science 🔗 Switzerland 🔗 Genealogy

The Bernoulli family (German pronunciation: [bɛʁˈnʊli]) of Basel was a patrician family, notable for having produced eight mathematically gifted academics who, among them, contributed substantially to the development of mathematics and physics during the early modern period.

🔗 Mathematics

Von Neumann's elephant is a problem in recreational mathematics, consisting of constructing a planar curve in the shape of an elephant from only four fixed parameters. It originated from a discussion between physicists John von Neumann and Enrico Fermi.

🔗 Mathematics 🔗 Economics 🔗 Politics 🔗 Game theory

In game theory, the Nash equilibrium, named after the mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.

In terms of game theory, if each player has chosen a strategy, and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and their corresponding payoffs constitutes a Nash equilibrium.

Stated simply, Alice and Bob are in Nash equilibrium if Alice is making the best decision she can, taking into account Bob's decision while his decision remains unchanged, and Bob is making the best decision he can, taking into account Alice's decision while her decision remains unchanged. Likewise, a group of players are in Nash equilibrium if each one is making the best decision possible, taking into account the decisions of the others in the game as long as the other parties' decisions remain unchanged.

Nash showed that there is a Nash equilibrium for every finite game: see further the article on strategy.