id int32 0 100k | text stringlengths 21 3.54k | source stringlengths 1 124 | similarity float32 0.78 0.88 |
|---|---|---|---|
22,900 | When F is R or C, SL(n, F) is a Lie subgroup of GL(n, F) of dimension n2 − 1. The Lie algebra s l ( n , F ) {\displaystyle {\mathfrak {sl}}(n,F)} of SL(n, F) consists of all n × n matrices over F with vanishing trace. The Lie bracket is given by the commutator. | Special linear group | 0.809137 |
22,901 | Since the latter matrices can be uniquely expressed as the exponential of symmetric traceless matrices, then this latter topology is that of (n + 2)(n − 1)/2-dimensional Euclidean space. Thus, the group SL(n, R) has the same fundamental group as SO(n), that is, Z for n = 2 and Z2 for n > 2. In particular this means tha... | Special linear group | 0.809137 |
22,902 | A hermitian matrix of unit determinant and having positive eigenvalues can be uniquely expressed as the exponential of a traceless hermitian matrix, and therefore the topology of this is that of (n2 − 1)-dimensional Euclidean space. Since SU(n) is simply connected, we conclude that SL(n, C) is also simply connected, fo... | Special linear group | 0.809137 |
22,903 | Any invertible matrix can be uniquely represented according to the polar decomposition as the product of a unitary matrix and a hermitian matrix with positive eigenvalues. The determinant of the unitary matrix is on the unit circle while that of the hermitian matrix is real and positive and since in the case of a matri... | Special linear group | 0.809137 |
22,904 | where F× is the multiplicative group of F (that is, F excluding 0). These elements are "special" in that they form an algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries). When F is a finite field of order q, the notation SL(n, q) is ... | Special linear group | 0.809137 |
22,905 | It includes the ability to search across multiple information sources including help sources on the Internet such as the Microsoft Knowledge Base. Users can share and install help content to and from other computers running Windows XP or Windows Server 2003 or switch to help for other supported operating system. Help a... | DSQuery | 0.809136 |
22,906 | Let Φ(V) denote Φ with these free variables all quantified, with the quantified variables restricted to the level V. Then ∃V is an axiom. Remark: This schema asserts the existence of a "partial" universe, namely the level V, in which all properties Φ holding when the quantified variables range over all levels, also hol... | Scott–Potter set theory | 0.809136 |
22,907 | Any theorem whose proof requires X is then formulated conditionally as "If X holds, then..." Potter defines several systems using existence premises, including the following two: ZfU =df ZU + Ordinals; ZFU =df Separation + Reflection.Ordinals: For each (infinite) ordinal α, there exists a corresponding level Vα. Remark... | Scott–Potter set theory | 0.809136 |
22,908 | "Dedekind algebra" is Potter's name for a set closed under a unary injective operation, successor, whose domain contains a unique element, zero, absent from its range. Because the theory of Dedekind algebras is categorical (all models are isomorphic), any such algebra can proxy for the natural numbers. Although Potter ... | Scott–Potter set theory | 0.809136 |
22,909 | Includes no equivalents of Choice or the axiom schema of Replacement.Hence ZU is closer to the Zermelo set theory of 1908, namely ZFC minus Choice, Replacement, and Foundation. It is stronger than this theory, however, since cardinals and ordinals can be defined, despite the absence of Choice, using Scott's trick and t... | Scott–Potter set theory | 0.809136 |
22,910 | Admits nonwellfounded collections. However Potter (2004) never invokes such collections, and all sets (collections which are contained in a level) are wellfounded. No theorem in Potter would be overturned if an axiom stating that all collections are sets were added to ZU. | Scott–Potter set theory | 0.809136 |
22,911 | In physics, a projectile launched with specific initial conditions will have a range. It may be more predictable assuming a flat Earth with a uniform gravity field, and no air resistance. The horizontal ranges of a projectile are equal for two complementary angles of projection with the same velocity. The following app... | Range of a projectile | 0.809135 |
22,912 | "HistoneDB 2.0 - with variants", a database of histones and their variants maintained by National Center for Biotechnology Information, currently serves as the most comprehensive manually curated resource on histones and their variants that follows the new unified phylogeny-based nomenclature of histone variants. "Hist... | Histone variants | 0.809134 |
22,913 | On the other hand, if there exists an algorithm that can solve the optimization problem efficiently, then it can be utilized in solving the decision problem by comparing the input with the optimal value. Another theme in literature is to identify what are the "hard" problems. Researchers who study the 0-1 QKP often per... | 0-1 quadratic knapsack problem | 0.809134 |
22,914 | Researchers have studied 0-1 quadratic knapsack problems for decades. One focus is to find effective algorithms or effective heuristics, especially those with an outstanding performance solving real world problems. The relationship between the decision version and the optimization version of the 0-1 QKP should not be i... | 0-1 quadratic knapsack problem | 0.809134 |
22,915 | WJCC, February 1957. Newell and Simon were recognized with the ACM Turing Award in 1975 for having "made basic contributions to artificial intelligence, the psychology of human cognition, and list processing". The problem of machine translation for natural language processing led Victor Yngve at Massachusetts Institute... | Linked list | 0.809134 |
22,916 | Linked lists were developed in 1955–1956, by Allen Newell, Cliff Shaw and Herbert A. Simon at RAND Corporation and Carnegie Mellon University as the primary data structure for their Information Processing Language (IPL). IPL was used by the authors to develop several early artificial intelligence programs, including th... | Linked list | 0.809134 |
22,917 | The concept was introduced by mathematicians Edgar H. Brown and Samuel Gitler in a 1973 paper.In topology, Brown–Gitler spectrum is related to the concepts of the Segal conjecture (proven in 1984) and the Burnside ring. | Brown–Gitler spectrum | 0.809132 |
22,918 | In the mathematical discipline of topology, the Brown–Gitler spectrum is a spectrum whose cohomology is a certain cyclic module over the Steenrod algebra.Brown–Gitler spectra are defined by the isomorphism: Σ n A / { Sq i: 2 i > n } A ≅ G ( n ) . {\displaystyle \Sigma ^{n}A/\{\operatorname {Sq} ^{i}:2i>n\}A\cong G(n).} | Brown–Gitler spectrum | 0.809132 |
22,919 | The medal for the Nobel Prize in Chemistry is identical in design to the Nobel Prize in Physics medal. The reverse of the physics and chemistry medals depict the Goddess of Nature in the form of Isis as she emerges from clouds holding a cornucopia. The Genius of Science holds the veil which covers Nature's 'cold and au... | Nobel Prize in Chemistry | 0.809128 |
22,920 | Alfred Nobel stipulated in his last will and testament that his money be used to create a series of prizes for those who confer the "greatest benefit on mankind" in physics, chemistry, peace, physiology or medicine, and literature. Though Nobel wrote several wills during his lifetime, the last was written a little over... | Nobel Prize in Chemistry | 0.809128 |
22,921 | A Chemistry Nobel Prize laureate earns a gold medal, a diploma bearing a citation, and a sum of money. | Nobel Prize in Chemistry | 0.809128 |
22,922 | The Nobel Foundation then reached an agreement on guidelines for how the Nobel Prize should be awarded. In 1900, the Nobel Foundation's newly created statutes were promulgated by King Oscar II. According to Nobel's will, The Royal Swedish Academy of Sciences were to award the Prize in Chemistry. | Nobel Prize in Chemistry | 0.809128 |
22,923 | The first Nobel Prize in Chemistry was awarded in 1901 to Jacobus Henricus van 't Hoff, of the Netherlands, "for his discovery of the laws of chemical dynamics and osmotic pressure in solutions". From 1901 to 2022, the award has been bestowed on a total of 189 individuals. The 2022 Nobel Prize in Chemistry was awarded ... | Nobel Prize in Chemistry | 0.809128 |
22,924 | The Nobel Prize in Chemistry (Swedish: Nobelpriset i kemi) is awarded annually by the Royal Swedish Academy of Sciences to scientists in the various fields of chemistry. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895, awarded for outstanding contributions in chemistry, physics, liter... | Nobel Prize in Chemistry | 0.809128 |
22,925 | While posthumous nominations are not permitted, awards can occur if the individual died in the months between the nomination and the decision of the prize committee. The award in chemistry requires the significance of achievements being recognized is "tested by time". In practice it means that the lag between the disco... | Nobel Prize in Chemistry | 0.809128 |
22,926 | The Nobel Laureates in chemistry are selected by a committee that consists of five members elected by the Royal Swedish Academy of Sciences. In its first stage, several thousand people are asked to nominate candidates. These names are scrutinized and discussed by experts until only the laureates remain. This slow and t... | Nobel Prize in Chemistry | 0.809128 |
22,927 | Biology was in its infancy in Nobel's day and no award was established. The Economist argued there is no Nobel Prize for mathematics either, another major discipline, and added that Nobel's stipulation of no more than three winners is not readily applicable to modern physics, where progress is typically made through hu... | Nobel Prize in Chemistry | 0.809128 |
22,928 | In recent years, the Nobel Prize in Chemistry has drawn criticism from chemists who feel that the prize is more frequently awarded to non-chemists than to chemists. In the 30 years leading up to 2012, the Nobel Prize in Chemistry was awarded ten times for work classified as biochemistry or molecular biology, and once t... | Nobel Prize in Chemistry | 0.809128 |
22,929 | An advantage of small signal models is they can be solved directly, while large signal nonlinear models are generally solved iteratively, with possible convergence or stability issues. By simplification to a linear model, the whole apparatus for solving linear equations becomes available, for example, simultaneous equa... | Transistor models | 0.809127 |
22,930 | Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations, for which there are textbooks by Ernst Schröder, Clarence Lewis, and Gunther Schmidt. A deeper analysis of relations involves decomposing them into subsets called conce... | Left-unique relation | 0.809126 |
22,931 | {\displaystyle X\times Y.} A binary relation is called a homogeneous relation when X = Y. A binary relation is also called a heterogeneous relation when it is not necessary that X = Y. Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfy... | Left-unique relation | 0.809126 |
22,932 | These include, among others: the "is greater than", "is equal to", and "divides" relations in arithmetic; the "is congruent to" relation in geometry; the "is adjacent to" relation in graph theory; the "is orthogonal to" relation in linear algebra.A function may be defined as a special kind of binary relation. Binary re... | Left-unique relation | 0.809126 |
22,933 | In recreational mathematics and fractal geometry, inverse power-of-three lengths occur in the constructions leading to the Koch snowflake, Cantor set, Sierpinski carpet and Menger sponge, in the number of elements in the construction steps for a Sierpinski triangle, and in many formulas related to these sets. There are... | Power of three | 0.809123 |
22,934 | In graph theory, powers of three appear in the Moon–Moser bound 3n/3 on the number of maximal independent sets of an n-vertex graph, and in the time analysis of the Bron–Kerbosch algorithm for finding these sets. Several important strongly regular graphs also have a number of vertices that is a power of three, includin... | Power of three | 0.809123 |
22,935 | In number theory, all powers of three are perfect totient numbers. The sums of distinct powers of three form a Stanley sequence, the lexicographically smallest sequence that does not contain an arithmetic progression of three elements. A conjecture of Paul Erdős states that this sequence contains no powers of two other... | Power of three | 0.809123 |
22,936 | If z ∈ L then there is a proof π such that Vπ (z) accepts for every Ri. All clauses are satisfied if xi = π(i) and the auxiliary variables are added correctly. If input z ∉ L then For every assignment to x1,...,xl and yR's, the corresponding proof π(i) = xi causes the Verifier to reject for half of all R ∈ {0,1}r(|z|).... | MAX-3SAT | 0.80912 |
22,937 | The PCP theorem implies that there exists an ε > 0 such that (1-ε)-approximation of MAX-3SAT is NP-hard. Proof: Any NP-complete problem L ∈ P C P ( O ( log ( n ) ) , O ( 1 ) ) {\displaystyle L\in {\mathsf {PCP}}(O(\log(n)),O(1))} by the PCP theorem. For x ∈ L, a 3-CNF formula Ψx is constructed so that x ∈ L ⇒ Ψx is s... | MAX-3SAT | 0.80912 |
22,938 | MAX-3SAT(B) is the restricted special case of MAX-3SAT where every variable occurs in at most B clauses. Before the PCP theorem was proven, Papadimitriou and Yannakakis showed that for some fixed constant B, this problem is MAX SNP-hard. Consequently, with the PCP theorem, it is also APX-hard. This is useful because MA... | MAX-3SAT | 0.80912 |
22,939 | Håstad demonstrates a tighter result than Theorem 1 i.e. the best known value for ε. He constructs a PCP Verifier for 3-SAT that reads only 3 bits from the Proof. For every ε > 0, there is a PCP-verifier M for 3-SAT that reads a random string r of length O ( log ( n ) ) {\displaystyle O(\log(n))} and computes query p... | MAX-3SAT | 0.80912 |
22,940 | MAX-3SAT is a problem in the computational complexity subfield of computer science. It generalises the Boolean satisfiability problem (SAT) which is a decision problem considered in complexity theory. It is defined as: Given a 3-CNF formula Φ (i.e. with at most 3 variables per clause), find an assignment that satisfies... | MAX-3SAT | 0.80912 |
22,941 | When atoms interact to form a chemical bond, the atomic orbitals of each atom are said to combine in a process called orbital hybridisation. The two most common types of bonds are sigma bonds (usually formed by hybrid orbitals) and pi bonds (formed by unhybridized p orbitals for atoms of main group elements). The geome... | Bond angle | 0.809114 |
22,942 | (The angles θ11, θ22, θ33, and θ44 are always zero and that this relationship can be modified for a different number of peripheral atoms by expanding/contracting the square matrix.) 0 = | cos θ 11 cos θ 12 cos θ 13 cos θ 14 cos θ 21 cos θ 22 cos θ 23 cos θ 24 cos θ 31 cos θ 32 cos θ 33 cos θ 34 ... | Bond angle | 0.809114 |
22,943 | As it gives models to all consistent theories without distinction, it gives no reason to accept or reject any axiom as long as the theory remains consistent, but regards all consistent axiomatic theories as referring to equally existing worlds. It gives no indication on which axiomatic system should be preferred as a f... | Foundations of Mathematics | 0.809113 |
22,944 | Gödel's completeness theorem establishes an equivalence in first-order logic between the formal provability of a formula and its truth in all possible models. Precisely, for any consistent first-order theory it gives an "explicit construction" of a model described by the theory; this model will be countable if the lang... | Foundations of Mathematics | 0.809113 |
22,945 | Concepts of vector spaces emerged from the conception of barycentric coordinates by Möbius in 1827, to the modern definition of vector spaces and linear maps by Peano in 1888. Geometry was no more limited to three dimensions. These concepts did not generalize numbers but combined notions of functions and sets which wer... | Foundations of Mathematics | 0.809113 |
22,946 | Mathematicians had attempted to solve all of these problems in vain since the time of the ancient Greeks. Abel and Galois's works opened the way for the developments of group theory (which would later be used to study symmetry in physics and other fields), and abstract algebra. | Foundations of Mathematics | 0.809113 |
22,947 | For the first time, the limits of mathematics were explored. Niels Henrik Abel (1802–1829), a Norwegian, and Évariste Galois, (1811–1832) a Frenchman, investigated the solutions of various polynomial equations, and proved that there is no general algebraic solution to equations of degree greater than four (Abel–Ruffini... | Foundations of Mathematics | 0.809113 |
22,948 | Isaac Newton (1642–1727) in England and Leibniz (1646–1716) in Germany independently developed the infinitesimal calculus on a basis that required new foundations. In particular, Leibniz described infinitesimals as numbers that are infinitely close to zero, a concept that does not fit in previous foundational framework... | Foundations of Mathematics | 0.809113 |
22,949 | For over 2,000 years, Euclid's Elements stood as a perfectly solid foundation for mathematics, as its methodology of rational exploration guided mathematicians, philosophers, and scientists well into the 19th century. The Middle Ages saw a dispute over the ontological status of the universals (platonic Ideas): Realism ... | Foundations of Mathematics | 0.809113 |
22,950 | It gives a rigorous foundation of infinitesimal calculus based on the set of real numbers, arguably resolving the Zeno paradoxes and Berkeley's arguments. Mathematicians such as Karl Weierstrass (1815–1897) discovered pathological functions such as continuous, nowhere-differentiable functions. Previous conceptions of a... | Foundations of Mathematics | 0.809113 |
22,951 | Cauchy (1789–1857) started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors. In his 1821 work Cours d'Analyse he defines infinitely small quantities in terms of decreasing seque... | Foundations of Mathematics | 0.809113 |
22,952 | For the time being we probably cannot answer this question ... In some cases these questions may be sufficiently answered through the study of formal theories, in disciplines such as reverse mathematics and computational complexity theory. As noted by Weyl, formal logical systems also run the risk of inconsistency; in ... | Foundations of Mathematics | 0.809113 |
22,953 | Virtually all mathematical theorems today can be formulated as theorems of set theory. The truth of a mathematical statement, in this view, is represented by the fact that the statement can be derived from the axioms of set theory using the rules of formal logic. Merely the use of formalism alone does not explain sever... | Foundations of Mathematics | 0.809113 |
22,954 | After many failed attempts to derive the parallel postulate from other axioms, the study of the still hypothetical hyperbolic geometry by Johann Heinrich Lambert (1728–1777) led him to introduce the hyperbolic functions and compute the area of a hyperbolic triangle (where the sum of angles is less than 180°). Then the ... | Foundations of Mathematics | 0.809113 |
22,955 | The norms p {\displaystyle p} and q {\displaystyle q} are equivalent if and only if they induce the same topology on X . {\displaystyle X.} Any two norms on a finite-dimensional space are equivalent but this does not extend to infinite-dimensional spaces. | L2 norm | 0.809112 |
22,956 | Mohawk Data Sciences subsequently produced an improved magnetic tape encoder in 1965, which was somewhat successfully marketed as a keypunch replacement. The rise of microprocessors and inexpensive computer terminals led to the development of additional key-to-tape and key-to-disk systems from smaller companies such as... | IBM 059 | 0.809111 |
22,957 | In the 1950s, Remington Rand introduced the UNITYPER, which enabled data entry directly to magnetic tape for UNIVAC systems. Mohawk Data Sciences subsequently produced an improved magnetic tape encoder in 1965, which was somewhat successfully marketed as a keypunch replacement. In the mid-1970s, the rise of microproces... | IBM 059 | 0.809111 |
22,958 | A phased approach to investment decisions for development arose in large-scale projects for mechanical and chemical engineering, particularly since the 1940s. One source described eight phases. In 1958, the American Association of Cost Engineers created four standard cost estimate type classifications to match these de... | Phase-gate process | 0.809107 |
22,959 | In 1661, Boyle defined elements as "those primitive and simple Bodies of which the mixt ones are said to be composed, and into which they are ultimately resolved. "In 1718, Étienne François Geoffroy's Affinity Table made use of several aspects — (1) tabular grouping and (2) correlation with chemical affinity — that wou... | History of the periodic table | 0.809107 |
22,960 | Pauli demonstrated that this was not the case. Instead, the Pauli exclusion principle was developed, not upon a mathematical basis, but upon the previous developments in alignment with chemistry. This rule states that no electrons can coexist in the same quantum state, and showed, in conjunction with empirical observat... | History of the periodic table | 0.809107 |
22,961 | Bohr proposed that elements in the same group behaved similarly because they have similar electron configurations, and that noble gases had filled valence shells; this forms the basis of the modern octet rule. Bohr's study of spectroscopy and chemistry was not usual among theoretical atomic physicists. | History of the periodic table | 0.809107 |
22,962 | He introduced the word transition to describe the elements now known as transition metals or transition elements.In the 1910s and 1920s, pioneering research into quantum mechanics led to new developments in atomic theory and small changes to the periodic table. In the 19th century, Mendeleev had already asserted that t... | History of the periodic table | 0.809107 |
22,963 | He made an exception for the first shell to only contain two electrons. These postulates were introduced on the basis of Rydberg's rule which Niels Bohr had used not in chemistry, but in physics, to apply to the orbits of electrons around the nucleus. In the Langmuir paper, he introduced the rule as 2N2 where N was a p... | History of the periodic table | 0.809107 |
22,964 | However, these studies could not unambiguously determine the first element with 5f electrons and therefore the first element in the actinide series; it was thus also referred to as the "thoride" or "uranide" series until it was later found that the series began with actinium.In light of these observations and an appare... | History of the periodic table | 0.809107 |
22,965 | The elements from actinium to uranium were instead believed to form part of a fourth series of transition metals because of their high oxidation states; accordingly, they were placed in groups 3 through 6.In 1940, neptunium and plutonium were the first transuranic elements to be discovered; they were placed in sequence... | History of the periodic table | 0.809107 |
22,966 | He determined that the rare-earth elements (lanthanides), 13 of which were known, lay within that gap. Although Mendeleev knew of lanthanum, cerium, and erbium, they were previously unaccounted for in the table because their total number and exact order were not known; Mendeleev still could not fit them in his table by... | History of the periodic table | 0.809107 |
22,967 | None of the proposals was accepted immediately, and many contemporary chemists found it too abstract to have any meaningful value. Of those chemists that proposed their categorizations, Mendeleev strove to back his work and promote his vision of periodicity, Meyer did not promote his work very actively, and Newlands di... | History of the periodic table | 0.809107 |
22,968 | The chemists' approach during the period of the Old Quantum Theory (1913 to 1925) was incorporated into the understanding of the electron shells and orbitals under current quantum mechanics. A real pioneer who gave us the foundation for our current model of electrons is Irving Langmuir. In his 1919 paper, he postulated... | History of the periodic table | 0.809107 |
22,969 | Despite the completion of the seventh period, experimental chemistry of some transactinides has been shown to be inconsistent with the periodic law. In the 1990s, Ken Czerwinski at University of California, Berkeley observed similarities between rutherfordium and plutonium and between dubnium and protactinium, rather t... | History of the periodic table | 0.809107 |
22,970 | Systems-based design is a proven process of unit load component cost optimization based on an understanding of how the pallet, packaging and material handling equipment interact during product distribution and storage to design the unit load component parts. A systems-based approach to unit load design uses software to... | Unit load | 0.809106 |
22,971 | Methylated DNA immunoprecipitation (MeDIP or mDIP) is a large-scale (chromosome- or genome-wide) purification technique in molecular biology that is used to enrich for methylated DNA sequences. It consists of isolating methylated DNA fragments via an antibody raised against 5-methylcytosine (5mC). This technique was fi... | Methylated DNA | 0.809105 |
22,972 | A functor F: C → D between preadditive categories is additive if it is an abelian group homomorphism on each hom-set in C. If the categories are additive, then a functor is additive if and only if it preserves all biproduct diagrams. That is, if B is a biproduct of A1, ... , An in C with projection morphisms pk and inj... | Additive category | 0.809105 |
22,973 | We give an alternative definition. Define a semiadditive category to be a category (note: not a preadditive category) which admits a zero object and all binary biproducts. It is then a remarkable theorem that the Hom sets naturally admit an abelian monoid structure. A proof of this fact is given below. An additive cate... | Additive category | 0.809105 |
22,974 | The original example of an additive category is the category of abelian groups Ab. The zero object is the trivial group, the addition of morphisms is given pointwise, and biproducts are given by direct sums. More generally, every module category over a ring R is additive, and so in particular, the category of vector sp... | Additive category | 0.809105 |
22,975 | Given objects A1, ... , An and B1, ... , Bm in an additive category, we can represent morphisms f: A1 ⊕ ⋅⋅⋅ ⊕ An → B1 ⊕ ⋅⋅⋅ ⊕ Bm as m-by-n matrices ( f 11 f 12 ⋯ f 1 n f 21 f 22 ⋯ f 2 n ⋮ ⋮ ⋯ ⋮ f m 1 f m 2 ⋯ f m n ) {\displaystyle {\begin{pmatrix}f_{11}&f_{12}&\cdots &f_{1n}\\f_{21}&f_{22}&\cdots &f_{2n}\\\vdots &\vdot... | Additive category | 0.809105 |
22,976 | Curves, Jacobians, and Cryptography | Jacobian of a curve | 0.809102 |
22,977 | As a group, the Jacobian variety of a curve is isomorphic to the quotient of the group of divisors of degree zero by the subgroup of principal divisors, i.e., divisors of rational functions. This holds for fields that are not algebraically closed, provided one considers divisors and functions defined over that field. | Jacobian of a curve | 0.809102 |
22,978 | Over the complex numbers, the Jacobian variety can be realized as the quotient space V/L, where V is the dual of the vector space of all global holomorphic differentials on C and L is the lattice of all elements of V of the form : ω ↦ ∫ γ ω {\displaystyle :\ \omega \mapsto \int _{\gamma }\omega } where γ is a closed pa... | Jacobian of a curve | 0.809102 |
22,979 | The Jacobian variety is named after Carl Gustav Jacobi, who proved the complete version of the Abel–Jacobi theorem, making the injectivity statement of Niels Abel into an isomorphism. It is a principally polarized abelian variety, of dimension g, and hence, over the complex numbers, it is a complex torus. If p is a poi... | Jacobian of a curve | 0.809102 |
22,980 | In mathematics, the Jacobian variety J(C) of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of C, hence an abelian variety. | Jacobian of a curve | 0.809102 |
22,981 | A carbanion is one of several reactive intermediates in organic chemistry. In organic synthesis, organolithium reagents and Grignard reagents are commonly treated and referred to as "carbanions." This is a convenient approximation, although these species are generally clusters or complexes containing highly polar, but ... | CH acidity | 0.809097 |
22,982 | In organic chemistry, a carbanion is an anion in which carbon is negatively charged.Formally, a carbanion is the conjugate base of a carbon acid: R3CH + B− → R3C− + HBwhere B stands for the base. The carbanions formed from deprotonation of alkanes (at an sp3 carbon), alkenes (at an sp2 carbon), arenes (at an sp2 carbon... | CH acidity | 0.809097 |
22,983 | This is so because the remainder of the line is still infinitely long and therefore equivalent to the original line. If the finite segment is very short, then in the equivalent circuit it will be modelled by an L-network consisting of one element of d Z {\displaystyle \scriptstyle dZ} and one of d Y {\displaystyle \scr... | Primary line coefficients | 0.809097 |
22,984 | In real calculus, the derivative of 1/x = x−1 is given by the power rule with the power −1: d d x x − 1 = ( − 1 ) x ( − 1 ) − 1 = − x − 2 = − 1 x 2 . {\displaystyle {\frac {d}{dx}}x^{-1}=(-1)x^{(-1)-1}=-x^{-2}=-{\frac {1}{x^{2}}}.} The power rule for integrals (Cavalieri's quadrature formula) cannot be used to compute ... | Reciprocal function | 0.809094 |
22,985 | Positive computing is a technological design perspective that embraces psychological well-being and ethical practice, aiming at building a digital environment to support happier and healthier users. Positive computing develops approaches that integrate insights from psychology, education, neuroscience, and HCI with tec... | Positive computing | 0.809093 |
22,986 | Intelligence explosion is a possible outcome of future computer science and computer engineering research, and I. J. Good described the first superintelligent machine as "the last invention that man need ever make," because of the vast influence it would have on our species. Indeed, Nick Bostrom, in his book Superintel... | Positive computing | 0.809093 |
22,987 | When endowed with this topology, the real line R {\displaystyle \mathbb {R} } is a T5 space. Given two subsets say A {\displaystyle A} and B {\displaystyle B} of R {\displaystyle \mathbb {R} } with A ¯ ∩ B = A ∩ B ¯ = ∅ , {\displaystyle {\overline {A}}\cap B=A\cap {\overline {B}}=\varnothing ,} where A ¯ {\displaystyle... | Euclidean topology | 0.809093 |
22,988 | The Euclidean norm on R n {\displaystyle \mathbb {R} ^{n}} is the non-negative function ‖ ⋅ ‖: R n → R {\displaystyle \|\cdot \|:\mathbb {R} ^{n}\to \mathbb {R} } defined by Like all norms, it induces a canonical metric defined by d ( p , q ) = ‖ p − q ‖ . {\displaystyle d(p,q)=\|p-q\|.} The metric d: R n × R n → R {\d... | Euclidean topology | 0.809093 |
22,989 | In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} by the Euclidean metric. | Euclidean topology | 0.809093 |
22,990 | Idaho National Laboratories describes Digital Twin as "the merging of integrated and connected data, sensors and instrumentation, artificial intelligence, and online monitoring into a single cohesive unit. "It is a critical capability of model-based systems engineering (MBSE) and the foundation for a Digital twin, whic... | Digital thread | 0.809091 |
22,991 | Big data analytics and artificial intelligence used in conjunction with Digital Thread are increasingly more required in smart manufacturing applications. Big data analytics is a "prerequisite for managing highly variable" data of smart manufacturing processes, gathered through digital thread. Artificial Intelligence c... | Digital thread | 0.809091 |
22,992 | A key aspect of digital thread is the Internet of things, whose "cyber-physical systems, sensors, and so-called smart devices" are an important source of the data required by digital thread. "The ability to gather massive amounts of data through the aspired omnipresence of sensors furthermore fuels the emergence of oth... | Digital thread | 0.809091 |
22,993 | "DeepLynx is an ontological data warehouse with timeseries data support". It was primarily authored by John Darrington and Cristopher Ritter to tackle Model-Based Systems Engineering (MBSE) tool integrations and warehousing, and has evolved to enable support for digital twin. | Digital thread | 0.809091 |
22,994 | A vector space, the m-dimensional superspace, then appears as the m-fold Cartesian product of these one-dimensional Λ . {\displaystyle \Lambda .} It can be shown that this is essentially equivalent to an algebra with m generators, but this requires work. | Grassman variable | 0.809091 |
22,995 | Note that the c-numbers form a subalgebra of Λ {\displaystyle \Lambda } , but the a-numbers do not (they are a subspace, not a subalgebra). The definition of Grassmann numbers allows mathematical analysis to be performed, in analogy to analysis on complex numbers. That is, one may define superholomorphic functions, def... | Grassman variable | 0.809091 |
22,996 | Products of an odd number of Grassmann variables anti-commute with each other; such a product is often called an a-number. Products of an even number of Grassmann variables commute (with all Grassman numbers); they are often called c-numbers. By abuse of terminology, an a-number is sometimes called an anticommuting c-n... | Grassman variable | 0.809091 |
22,997 | The spinor space is defined as the Grassmann or exterior algebra ⋀ W {\displaystyle \textstyle {\bigwedge }W} of the space of Weyl spinors W {\displaystyle W} (and anti-spinors W ¯ {\displaystyle {\overline {W}}} ), such that the wave functions of n fermions belong in ⋀ n W {\displaystyle \textstyle {\bigwedge }^{n}W} ... | Grassman variable | 0.809091 |
22,998 | In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra over the complex numbers. The special case of a 1-dimensional algebra is known as a dual number. Grassmann numbers saw an early use in physics to express... | Grassman variable | 0.809091 |
22,999 | A beam axle is typically simple in design, rugged, and inexpensive to manufacture. Only one universal joint or constant-velocity joint is needed at each steered and driven wheel and none are needed at non-steered wheels; this reduces maintenance requirements and manufacturing costs compared to independent suspensions, ... | Solid axle | 0.809085 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.