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23,300 | Theaetetus was, like Plato, a disciple of Theodorus's; he worked on distinguishing different kinds of incommensurables, and was thus arguably a pioneer in the study of number systems. (Book X of Euclid's Elements is described by Pappus as being largely based on Theaetetus's work.) Euclid devoted part of his Elements to... | Number theorists | 0.808899 |
23,301 | Aristotle claimed that the philosophy of Plato closely followed the teachings of the Pythagoreans, and Cicero repeats this claim: Platonem ferunt didicisse Pythagorea omnia ("They say Plato learned all things Pythagorean").Plato had a keen interest in mathematics, and distinguished clearly between arithmetic and calcul... | Number theorists | 0.808899 |
23,302 | Aside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-mathematicians or through mathematical works from the early Hellenistic period. In the case of number theory, this means, by and large, Plato and Euclid, respectively. While Asian mathematics in... | Number theorists | 0.808899 |
23,303 | We know of no clearly arithmetical material in ancient Egyptian or Vedic sources, though there is some algebra in each. The Chinese remainder theorem appears as an exercise in Sunzi Suanjing (3rd, 4th or 5th century CE). (There is one important step glossed over in Sunzi's solution: it is the problem that was later sol... | Number theorists | 0.808899 |
23,304 | If some other method was used, the triples were first constructed and then reordered by c / a {\displaystyle c/a} , presumably for actual use as a "table", for example, with a view to applications. It is not known what these applications may have been, or whether there could have been any; Babylonian astronomy, for exa... | Number theorists | 0.808899 |
23,305 | Diophantine geometry should not be confused with the geometry of numbers, which is a collection of graphical methods for answering certain questions in algebraic number theory. Arithmetic geometry, however, is a contemporary term for much the same domain as that covered by the term Diophantine geometry. The term arithm... | Number theorists | 0.808899 |
23,306 | In 2023, Nature Biomedical Engineering wrote that "it is no longer possible to accurately distinguish" human-written text from text created by large language models, and that "It is all but certain that general-purpose large language models will rapidly proliferate... It is a rather safe bet that they will change many ... | Large language models | 0.808899 |
23,307 | For diagonal elements (when i = j), displacements perpendicular to the axis of rotation results in the above simplified version of the parallel axis theorem. The generalized version of the parallel axis theorem can be expressed in the form of coordinate-free notation as J = I + m , {\displaystyle \mathbf {J} =\mathbf ... | Parallel axis theorem | 0.808898 |
23,308 | The parallel axis theorem can be generalized to calculations involving the inertia tensor. Let Iij denote the inertia tensor of a body as calculated at the center of mass. Then the inertia tensor Jij as calculated relative to a new point is J i j = I i j + m ( | R | 2 δ i j − R i R j ) , {\displaystyle J_{ij}=I_{ij}+m\... | Parallel axis theorem | 0.808898 |
23,309 | The first term is the inertia matrix relative to the center of mass. The second and third terms are zero by definition of the center of mass R, ∑ i = 1 n m i ( r i − R ) = 0. {\displaystyle \sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )=0.} And the last term is the total mass of the system multiplied by the squar... | Parallel axis theorem | 0.808898 |
23,310 | The inertia matrix of a rigid system of particles depends on the choice of the reference point. There is a useful relationship between the inertia matrix relative to the center of mass R and the inertia matrix relative to another point S. This relationship is called the parallel axis theorem. Consider the inertia matri... | Parallel axis theorem | 0.808898 |
23,311 | In order to compare formulations of the parallel axis theorem using skew-symmetric matrices and the tensor formulation, the following identities are useful. Let be the skew symmetric matrix associated with the position vector R = (x, y, z), then the product in the inertia matrix becomes − = − 2 = . {\displaystyle ... | Parallel axis theorem | 0.808898 |
23,312 | Interferon alfa-2b is an antiviral or antineoplastic drug. It is a recombinant form of the protein Interferon alpha-2 that was originally sequenced and produced recombinantly in E. coli in the laboratory of Charles Weissmann at the University of Zurich, in 1980. It was developed at Biogen, and ultimately marketed by Sc... | Interferon alfa 2b | 0.808896 |
23,313 | Thereafter follows a alkyl shift and an electrocyclic ring opening. Proceeding through a shift, the walk rearrangement of norcaradienes is expected to proceed suprafacially with a retention of stereochemistry. Experimental observations, however, show that the 1,5-shifts of norcaradienes proceed antarafacially. Theore... | 1,5-hydride shift | 0.808893 |
23,314 | In a thermal hydride shift, a hydride moves three atoms. The Woodward–Hoffmann rules dictate that it would proceed in an antarafacial shift. Although such a shift is symmetry allowed, the Mobius topology required in the transition state prohibits such a shift because it is geometrically impossible, which accounts for ... | 1,5-hydride shift | 0.808893 |
23,315 | A sigmatropic reaction in organic chemistry is a pericyclic reaction wherein the net result is one σ-bond is changed to another σ-bond in an uncatalyzed intramolecular reaction. The name sigmatropic is the result of a compounding of the long-established sigma designation from single carbon–carbon bonds and the Greek wo... | 1,5-hydride shift | 0.808893 |
23,316 | A shift involves the shift of 1 substituent (hydride, alkyl, or aryl) down 5 atoms of a π system. Hydrogen has been shown to shift in both cyclic and open-chain compounds at temperatures at or above 200 ˚C. These reactions are predicted to proceed suprafacially, via a Hückel-topology transition state. Photoirradiation... | 1,5-hydride shift | 0.808893 |
23,317 | How Data Happened: A History from the Age of Reason to the Age of Algorithms is a 2023 non-fiction book written by Columbia University professors Chris Wiggins and Matthew L. Jones. The book explores the history of data and statistics from the end of the 18th century to the present day. | How Data Happened | 0.808891 |
23,318 | The book starts at the end of the 18th century, when European states began tabulating physical resources, and ends at the present day, when algorithms manipulate our personal information as a commodity. It looks at the rise of data and statistics, and how early statistical methods were used to justify eugenics, quantif... | How Data Happened | 0.808891 |
23,319 | The Church–Rosser property of the lambda calculus means that evaluation (β-reduction) can be carried out in any order, even in parallel. This means that various nondeterministic evaluation strategies are relevant. However, the lambda calculus does not offer any explicit constructs for parallelism. One can add construct... | Lambda-term bound variables | 0.808887 |
23,320 | Used to avoid name collisions. ( ( λ x . M ) E ) → ( M ) {\textstyle ((\lambda x.M)\ E)\rightarrow (M)}: β-reduction, replacing the bound variables with the argument expression in the body of the abstraction.If De Bruijn indexing is used, then α-conversion is no longer required as there will be no name collisions. If ... | Lambda-term bound variables | 0.808887 |
23,321 | In the simplest form of lambda calculus, terms are built using only the following rules: x {\displaystyle x}: Some lambda term, a character or string representing a parameter, or mathematical/logical value. ( λ x . M ) {\textstyle (\lambda x.M)}: A lambda abstraction is a function definition M {\textstyle M} , where th... | Lambda-term bound variables | 0.808887 |
23,322 | Lambda calculus (also written as λ-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathema... | Lambda-term bound variables | 0.808887 |
23,323 | In lambda calculus, a library would take the form of a collection of previously defined functions, which as lambda-terms are merely particular constants. The pure lambda calculus does not have a concept of named constants since all atomic lambda-terms are variables, but one can emulate having named constants by setting... | Lambda-term bound variables | 0.808887 |
23,324 | Many features observed in geology are planes or lines, and their orientation is commonly referred to as their attitude. These attitudes are specified with two angles. For a line, these angles are called the trend and the plunge. The trend is the compass direction of the line, and the plunge is the downward angle it mak... | Orientation (geometry) | 0.808886 |
23,325 | The definition can be vastly generalized, for example to intersections along subvarieties instead of just at points, or to arbitrary complete varieties. In algebraic topology, the intersection number appears as the Poincaré dual of the cup product. Specifically, if two manifolds, X and Y, intersect transversely in a ma... | Intersection multiplicity | 0.808884 |
23,326 | In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for tangency. One needs a definition of intersection number in order to state r... | Intersection multiplicity | 0.808884 |
23,327 | This connection between a geometric notion of intersection and a homological notion of a derived tensor product has been influential and led in particular to several homological conjectures in commutative algebra. Serre's Tor formula states: let X be a regular variety, V and W two subvarieties of complementary dimensio... | Intersection multiplicity | 0.808884 |
23,328 | The second potential problem is that even if the intersection is zero-dimensional, it may be non-transverse, for example, if V is a plane curve and W is one of its tangent lines. The first problem requires the machinery of intersection theory, discussed above in detail, which replaces V and W by more convenient subvari... | Intersection multiplicity | 0.808884 |
23,329 | Structure-based combinatorial protein engineering (SCOPE) is a synthetic biology technique for creating gene libraries (lineages) of defined composition designed from structural and probabilistic constraints of the encoded proteins. The development of this technique was driven by fundamental questions about protein str... | Structure-based combinatorial protein engineering | 0.808884 |
23,330 | While the innovation is often popularly credited for using steam in baking ovens, leading to a different crust characteristic, it is notable for including procedures for high milling of grains (see Vienna grits), cracking them incrementally instead of mashing them with one pass; as well as better processes for growing ... | Sacchromyces cerevisiae | 0.808883 |
23,331 | The covering problem of Rado is an unsolved problem in geometry concerning covering planar sets by squares. It was formulated in 1928 by Tibor Radó and has been generalized to more general shapes and higher dimensions by Richard Rado. | Covering problem of Rado | 0.808878 |
23,332 | The HPA axis is involved in the neurobiology and pathophysiology of mood disorders and functional illnesses, including anxiety disorder, bipolar disorder, insomnia, posttraumatic stress disorder, borderline personality disorder, ADHD, major depressive disorder, burnout, chronic fatigue syndrome, fibromyalgia, irritable... | Hypothalamic–pituitary–adrenal axis | 0.808874 |
23,333 | In the Poincaré half-plane model of the hyperbolic plane (see Hyperbolic motions), one can establish the relation of Φ to a with Euclidean geometry. Let Q be the semicircle with diameter on the x-axis that passes through the points (1,0) and (0,y), where y > 1. Since Q is tangent to the unit semicircle centered at the ... | Angle of parallelism | 0.808873 |
23,334 | In hyperbolic geometry, angle of parallelism Π ( a ) {\displaystyle \Pi (a)} , is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length a between the right angle and the vertex of the angle of parallelism. Given a point not o... | Angle of parallelism | 0.808873 |
23,335 | The angle of parallelism was developed in 1840 in the German publication "Geometrische Untersuchungen zur Theory der Parallellinien" by Nikolai Lobachevsky. This publication became widely known in English after the Texas professor G. B. Halsted produced a translation in 1891. (Geometrical Researches on the Theory of Pa... | Angle of parallelism | 0.808873 |
23,336 | sin B E C = sinh B C sinh C E = sinh B C sinh D A = sinh B C sin A C D sinh C A = sinh B C cos A C B sinh C A = sinh B C tanh C A tanh C B sinh C A = cosh B C cosh C A = cosh B C cosh C B cosh A B = 1 cosh A B . {\displaystyle \sin BEC={\frac {\sinh {BC}}{\sinh {CE}}}={\frac {\... | Angle of parallelism | 0.808873 |
23,337 | The semicircle Q has its center at (x, 0), x < 0, so its radius is 1 − x. Thus, the radius squared of Q is x 2 + y 2 = ( 1 − x ) 2 , {\displaystyle x^{2}+y^{2}=(1-x)^{2},} hence x = 1 2 ( 1 − y 2 ) . {\displaystyle x={\tfrac {1}{2}}(1-y^{2}).} The metric of the Poincaré half-plane model of hyperbolic geometry parametri... | Angle of parallelism | 0.808873 |
23,338 | Fixed-point computation refers to the process of computing an exact or approximate fixed point of a given function. In its most common form, we are given a function f that satisfies the condition to the Brouwer fixed-point theorem, that is: f is continuous and maps the unit d-cube to itself. The Brouwer fixed-point the... | Fixed-point computation | 0.808873 |
23,339 | The unit interval is denoted by E := {\displaystyle E:=} , and the unit d-dimensional cube is denoted by Ed. A continuous function f is defined on Ed (from Ed to itself). Often, it is assumed that f is not only continuous but also Lipschitz continuous, that is, for some constant L, | f ( x ) − f ( y ) | ≤ L ⋅ | x − y ... | Fixed-point computation | 0.808873 |
23,340 | However, there are classes of functions for which finding an approximate root is equivalent to finding an approximate fixed point. One example is the class of functions g such that g ( x ) + x {\displaystyle g(x)+x} maps Ed to itself (that is: g ( x ) + x {\displaystyle g(x)+x} is in Ed for all x in Ed). This is becaus... | Fixed-point computation | 0.808873 |
23,341 | A discrete function is a function defined on a subset of Z d {\displaystyle \mathbb {Z} ^{d}} (the d-dimensional integer grid). There are several discrete fixed-point theorems, stating conditions under which a discrete function has a fixed point. For example, the Iimura-Murota-Tamura theorem states that (in particular)... | Fixed-point computation | 0.808873 |
23,342 | Banach's fixed-point theorem implies that, when fixed-point iteration is applied to a contraction mapping, the error after t iterations is in O ( L t ) {\displaystyle O(L^{t})} . Therefore, the number of evaluations required for a δ-relative fixed-point is approximately log L ( δ ) = log ( δ ) / log ( L ) = log ... | Fixed-point computation | 0.808873 |
23,343 | In theoretical physics, the 't Hooft–Polyakov monopole is a topological soliton similar to the Dirac monopole but without the Dirac string. It arises in the case of a Yang–Mills theory with a gauge group G, coupled to a Higgs field which spontaneously breaks it down to a smaller group H via the Higgs mechanism. It was ... | 't Hooft–Polyakov monopole | 0.808869 |
23,344 | "Nuclear Energy: Principles", Physics Department, Faculty of Science, Mansoura University, Mansoura, Egypt; apparently excerpted from notes from the University of Washington Department of Mechanical Engineering; themselves apparently summarized from Bodansky, D. (1996), Nuclear Energy: Principles, Practices, and Prospe... | Prompt criticality | 0.808862 |
23,345 | {\displaystyle x.} ) Evidently Lagarde was aware that Arab mathematicians, in the "rhetorical" stage of algebra's development, often used that word to represent the unknown quantity. He surmised that "nothing could be more natural" ("Nichts war also natürlicher...") than for the initial of the Arabic word—romanized as ... | Greek geometric algebra | 0.80886 |
23,346 | It is the claim that algebraic x {\displaystyle x} is the abbreviation of a supposed loanword from Arabic in Old Spanish. The theory originated in 1884 with the German orientalist Paul de Lagarde, shortly after he published his edition of a 1505 Spanish/Arabic bilingual glossary in which Spanish cosa ("thing") was pair... | Greek geometric algebra | 0.80886 |
23,347 | It has been suggested that he later settled on x {\displaystyle x} (in place of z {\displaystyle z} ) for the first unknown because of its relatively greater abundance in the French and Latin typographical fonts of the time.Three alternative theories of the origin of algebraic x {\displaystyle x} were suggested in the ... | Greek geometric algebra | 0.80886 |
23,348 | By tradition, the first unknown variable in an algebraic problem is nowadays represented by the symbol x {\displaystyle {\mathit {x}}} and if there is a second or a third unknown, then these are labeled y {\displaystyle {\mathit {y}}} and z {\displaystyle {\mathit {z}}} respectively. Algebraic x {\displaystyle x} is co... | Greek geometric algebra | 0.80886 |
23,349 | Ancient Egyptian algebra dealt mainly with linear equations while the Babylonians found these equations too elementary, and developed mathematics to a higher level than the Egyptians.The Rhind Papyrus, also known as the Ahmes Papyrus, is an ancient Egyptian papyrus written c. 1650 BC by Ahmes, who transcribed it from a... | Greek geometric algebra | 0.80886 |
23,350 | Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Gottfried Leibniz was the first, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa, ordinate, tangent, chord, and the p... | Greek geometric algebra | 0.80886 |
23,351 | Al-Hassār, a mathematician from Morocco specializing in Islamic inheritance jurisprudence during the 12th century, developed the modern symbolic mathematical notation for fractions, where the numerator and denominator are separated by a horizontal bar. This same fractional notation appeared soon after in the work of Fi... | Greek geometric algebra | 0.80886 |
23,352 | This is the operation which Al-Khwarizmi originally described as al-jabr. The name "algebra" comes from the "al-jabr" in the title of his book. R. Rashed and Angela Armstrong write: "Al-Khwarizmi's text can be seen to be distinct not only from the Babylonian tablets, but also from Diophantus' Arithmetica. | Greek geometric algebra | 0.80886 |
23,353 | Ssy-yüan yü-chien《四元玉鑒》, or Precious Mirror of the Four Elements, was written by Chu Shih-chieh in 1303 and it marks the peak in the development of Chinese algebra. The four elements, called heaven, earth, man and matter, represented the four unknown quantities in his algebraic equations. The Ssy-yüan yü-chien deals wi... | Greek geometric algebra | 0.80886 |
23,354 | He also developed the concepts of the maxima and minima of curves in order to solve cubic equations which may not have positive solutions. He understood the importance of the discriminant of the cubic equation and used an early version of Cardano's formula to find algebraic solutions to certain types of cubic equations... | Greek geometric algebra | 0.80886 |
23,355 | He only considered positive roots and he did not go past the third degree. He also saw a strong relationship between geometry and algebra.In the 12th century, Sharaf al-Dīn al-Tūsī (1135–1213) wrote the Al-Mu'adalat (Treatise on Equations), which dealt with eight types of cubic equations with positive solutions and fiv... | Greek geometric algebra | 0.80886 |
23,356 | Omar Khayyám (c. 1050 – 1123) wrote a book on Algebra that went beyond Al-Jabr to include equations of the third degree. Omar Khayyám provided both arithmetic and geometric solutions for quadratic equations, but he only gave geometric solutions for general cubic equations since he mistakenly believed that arithmetic so... | Greek geometric algebra | 0.80886 |
23,357 | "The application of areas" is only a part of geometric algebra and it is thoroughly covered in Euclid's Elements. An example of geometric algebra would be solving the linear equation a x = b c . {\displaystyle ax=bc.} | Greek geometric algebra | 0.80886 |
23,358 | It is sometimes alleged that the Greeks had no algebra, but this is inaccurate. By the time of Plato, Greek mathematics had undergone a drastic change. The Greeks created a geometric algebra where terms were represented by sides of geometric objects, usually lines, that had letters associated with them, and with this n... | Greek geometric algebra | 0.80886 |
23,359 | Data is a work written by Euclid for use at the schools of Alexandria and it was meant to be used as a companion volume to the first six books of the Elements. The book contains some fifteen definitions and ninety-five statements, of which there are about two dozen statements that serve as algebraic rules or formulas. ... | Greek geometric algebra | 0.80886 |
23,360 | 'Abd al-Hamīd ibn Turk authored a manuscript entitled Logical Necessities in Mixed Equations, which is very similar to al-Khwarzimi's Al-Jabr and was published at around the same time as, or even possibly earlier than, Al-Jabr. The manuscript gives exactly the same geometric demonstration as is found in Al-Jabr, and in... | Greek geometric algebra | 0.80886 |
23,361 | Aleph Objects, Inc. was a small manufacturing company based in Loveland, Colorado. Their business model focused around the development of Open-source hardware for 3D printing with full support for Free and open-source software. | Aleph Objects | 0.808858 |
23,362 | In 3DForged.com's review of the TAZ 6, Brent Hale called the TAZ 6 "the best overall 3D printer I have ever used." However it was the less expensive model, the LulzBot Mini 2 that was named Best Intermediate Printer of the Year for 2019 by Tom's Guide. The Mediahq agreed, naming the Lulzbot Mini 2 the best 3D printer f... | Aleph Objects | 0.808858 |
23,363 | In its 2014 Ultimate Guide to 3D Printing special issue, Make Magazine awarded the Lulzbot TAZ "best documentation" of all the hobbyist-grade 3D printers that were tested.The TAZ 4 was released in early 2014 with several key improvements to the TAZ platform to make it more robust and capable. This included a newly engi... | Aleph Objects | 0.808858 |
23,364 | HA + B ↽ − − ⇀ A − + BH + {\displaystyle {\ce {HA + B <=> A- + BH+}}} In this representation both the base, B, and the conjugate base, A−, are shown carrying a lone pair of electrons and the proton, which is a Lewis acid, is transferred between them. Lewis later wrote "To restrict the group of acids to those substances... | Proton donor | 0.808855 |
23,365 | A case can be made that Percy MacMahon (1915) was the first to prove a result in the field with his study of "lattice permutations". In particular MacMahon shows that the permutations which can be divided into two decreasing subsequences (i.e., the 123-avoiding permutations) are counted by the Catalan numbers.Another e... | Permutation patterns | 0.808849 |
23,366 | {\displaystyle (q+Qe)(r+Re)=(qr+\gamma R^{*}Q)+(Rq+Qr^{*})e.} Zorn’s German language description of this Cayley–Dickson construction contributed to the persistent use of this eponym describing the construction of composition algebras. Cohl Furey has proposed that octonion algebras can be utilized in an attempt to recon... | Cayley algebra | 0.808849 |
23,367 | Another octonion may be written r + Re. Then with * denoting the conjugation in the quaternion algebra, their product is ( q + Q e ) ( r + R e ) = ( q r + γ R ∗ Q ) + ( R q + Q r ∗ ) e . | Cayley algebra | 0.808849 |
23,368 | It follows that the invertible elements in any octonion algebra form a Moufang loop, as do the elements of unit norm. The construction of general octonion algebras over an arbitrary field k was described by Leonard Dickson in his book Algebren und ihre Zahlentheorie (1927) (Seite 264) and repeated by Max Zorn. The prod... | Cayley algebra | 0.808849 |
23,369 | Up to F-algebra isomorphism, there is a unique split octonion algebra over any field F. When F is algebraically closed or a finite field, these are the only octonion algebras over F. Octonion algebras are always non-associative. They are, however, alternative algebras, alternativity being a weaker form of associativity... | Cayley algebra | 0.808849 |
23,370 | In mathematics, an octonion algebra or Cayley algebra over a field F is a composition algebra over F that has dimension 8 over F. In other words, it is a unital non-associative algebra A over F with a non-degenerate quadratic form N (called the norm form) such that N ( x y ) = N ( x ) N ( y ) {\displaystyle N(xy)=N(x)N... | Cayley algebra | 0.808849 |
23,371 | SAS is perhaps best known for its technical conference with the Federation of Analytical Chemistry and Spectroscopy Societies and short courses on various aspects of spectroscopy and data analysis. The society publishes the scientific journal Applied Spectroscopy. SAS is affiliated with American Institute of Physics (A... | Society for Applied Spectroscopy | 0.808848 |
23,372 | The Society for Applied Spectroscopy (SAS) is an organization promoting research and education in the fields of spectroscopy, optics, and analytical chemistry. Founded in 1958, it is currently headquartered in Frederick, MD. In 2006 it had about 2,000 members worldwide. | Society for Applied Spectroscopy | 0.808848 |
23,373 | Perturbation theory has been used in a large number of different settings in physics and applied mathematics. Examples of the "collection of equations" D {\displaystyle D} include algebraic equations,differential equations (e.g., the equations of motion and commonly wave equations), thermodynamic free energy in statist... | Small perturbations | 0.808846 |
23,374 | Perturbation theory in quantum mechanics is fairly accessible, as the quantum notation allows expressions to be written in fairly compact form, thus making them easier to comprehend. This resulted in an explosion of applications, ranging from the Zeeman effect to the hyperfine splitting in the hydrogen atom. Despite th... | Small perturbations | 0.808845 |
23,375 | These well-developed perturbation methods were adopted and adapted to solve new problems arising during the development of quantum mechanics in 20th century atomic and subatomic physics. Paul Dirac developed quantum perturbation theory in 1927 to evaluate when a particle would be emitted in radioactive elements. This w... | Small perturbations | 0.808845 |
23,376 | Quite often, these are differential equations, thus, the letter "D". The process is generally mechanical, if laborious. One begins by writing the equations D {\displaystyle D} so that they split into two parts: some collection of equations D 0 {\displaystyle D_{0}} which can be solved exactly, and some additional remai... | Small perturbations | 0.808845 |
23,377 | From representing the six special cases, we see a clear advantage of the moment matrix representation, i.e., it allows a unified representation for seemingly diverse types of knowledge, including linear equations, joint and conditional distributions, and ignorance. The unification is significant not only for knowledge ... | Linear belief function | 0.808844 |
23,378 | Second, according to statistics, μ 2 − μ 1 ( Σ 11 ) − 1 Σ 12 {\displaystyle \mu _{2}-\mu _{1}(\Sigma _{11})^{-1}\Sigma _{12}} is the conditional mean of Y given X = 0; Σ 22 − Σ 21 ( Σ 11 ) − 1 Σ 12 {\displaystyle \Sigma _{22}-\Sigma _{21}(\Sigma _{11})^{-1}\Sigma _{12}} is the conditional covariance matrix of Y given X... | Linear belief function | 0.808844 |
23,379 | Thus, a direct observation can be represented as: M ( X ) = ( μ 0 ) {\displaystyle M(X)=\left({\begin{array}{*{20}c}\mu \\0\end{array}}\right)} Third, suppose one is completely ignorant about X. This is a very thorny case in Bayesian statistics since the density function does not exist. By using the fully swept moment ... | Linear belief function | 0.808844 |
23,380 | Linear belief functions are an extension of the Dempster–Shafer theory of belief functions to the case when variables of interest are continuous. Examples of such variables include financial asset prices, portfolio performance, and other antecedent and consequent variables. The theory was originally proposed by Arthur ... | Linear belief function | 0.808844 |
23,381 | Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic (including in plants) and help inform public health and plant health interventions. Models use basic assumptions or collected statistics along with mathematics to find parameters for various infectious diseases an... | Epidemiological models | 0.808844 |
23,382 | In physics, particularly in quantum field theory, configurations of a physical system that satisfy classical equations of motion are called "on the mass shell" or simply more often on shell; while those that do not are called "off the mass shell", or off shell. In quantum field theory, virtual particles are termed off ... | On-shell solution | 0.808842 |
23,383 | The ideas in Lagrangian mechanics have numerous applications in other areas of physics, and can adopt generalized results from the calculus of variations. | Cyclic coordinate | 0.808842 |
23,384 | In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his 1788 work, Mécanique analytique.Lagrangian mechanics descri... | Cyclic coordinate | 0.808842 |
23,385 | Kummer's theorem states that the number of carries involved in adding two numbers in base p {\displaystyle p} is equal to the exponent of the highest power of p {\displaystyle p} dividing a certain binomial coefficient. When several random numbers of many digits are added, the statistics of the carry digits bears an un... | Carry (arithmetic) | 0.808841 |
23,386 | {\displaystyle S={\tfrac {1}{2}}\int {\bigl }\,.} To have arbitrarily small charges requires that the U(1) is not the circle of unit complex numbers under multiplication, but the real numbers R under addition, which is only different in the global topology. | Higgs potential | 0.808838 |
23,387 | The Higgs mechanism was incorporated into modern particle physics by Steven Weinberg and Abdus Salam, and is an essential part of the Standard Model. In the Standard Model, at temperatures high enough that electroweak symmetry is unbroken, all elementary particles are massless. At a critical temperature, the Higgs fiel... | Higgs potential | 0.808838 |
23,388 | Generally, the upper electrode projects from the upper fixture surface, while the lower electrode is a flat conductive surface. In some cases, the bottom electrode can project above the bottom fixture, to conform to geometry or to better localize melting through the reduction of stray electric field. | Dielectric welding | 0.808838 |
23,389 | Several groups have applied atomic manipulation techniques for artistic purposes to demonstrate control over the adatom positions. These include various institutional logos and a movie called “A Boy and His Atom” composed of individual STM scans by IBM researchers. Several notable condensed matter physics experiments h... | Atomic manipulation | 0.808836 |
23,390 | These represent the extension of random processes to multiple dimensions, and are common in physics, where they are used in statistical mechanics to describe properties such as force or electric field that can vary continuously over three dimensions (or four dimensions, when time is included).These concepts originate i... | Statistical data type | 0.808828 |
23,391 | But the mapping of computer science data types to statistical data types depends on which categorization of the latter is being implemented. Other categorizations have been proposed. For example, Mosteller and Tukey (1977) distinguished grades, ranks, counted fractions, counts, amounts, and balances. | Statistical data type | 0.808828 |
23,392 | Because variables conforming only to nominal or ordinal measurements cannot be reasonably measured numerically, sometimes they are grouped together as categorical variables, whereas ratio and interval measurements are grouped together as quantitative variables, which can be either discrete or continuous, due to their n... | Statistical data type | 0.808828 |
23,393 | In statistics, groups of individual data points may be classified as belonging to any of various statistical data types, e.g. categorical ("red", "blue", "green"), real number (1.68, -5, 1.7e+6), odd number (1,3,5) etc. The data type is a fundamental component of the semantic content of the variable, and controls which... | Statistical data type | 0.808828 |
23,394 | More generally, this technique leads to deformation quantization, where the ★-product is taken to be a deformation of the algebra of functions on a symplectic manifold or Poisson manifold. However, as a natural quantization scheme (a functor), Weyl's map is not satisfactory. For example, the Weyl map of the classical a... | Geometric quantisation | 0.808826 |
23,395 | In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theor... | Geometric quantisation | 0.808826 |
23,396 | He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are... | Mathematical discipline | 0.808826 |
23,397 | Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis broadly includes the study of approximation and discretization wi... | Mathematical discipline | 0.808826 |
23,398 | Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz. It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the c... | Mathematical discipline | 0.808826 |
23,399 | In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support). For example, when asked how he came about his theorems, Gauss once replied "durch planmässiges Tattoniere... | Mathematical discipline | 0.808826 |
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