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32,500 | In the geometry of hyperbolic 4-space, the order-5 5-cell honeycomb is one of five compact regular space-filling tessellations (or honeycombs). With Schläfli symbol {3,3,3,5}, it has five 5-cells around each face. Its dual is the 120-cell honeycomb, {5,3,3,3}. | Order-5 5-cell honeycomb | 0.803867 |
32,501 | In mathematics, MacMahon's master theorem (MMT) is a result in enumerative combinatorics and linear algebra. It was discovered by Percy MacMahon and proved in his monograph Combinatory analysis (1916). It is often used to derive binomial identities, most notably Dixon's identity. | MacMahon master theorem | 0.803866 |
32,502 | Consider a matrix A = ( 0 1 − 1 − 1 0 1 1 − 1 0 ) . {\displaystyle A={\begin{pmatrix}0&1&-1\\-1&0&1\\1&-1&0\end{pmatrix}}.} Compute the coefficients G(2n, 2n, 2n) directly from the definition: G ( 2 n , 2 n , 2 n ) = ( x 2 − x 3 ) 2 n ( x 3 − x 1 ) 2 n ( x 1 − x 2 ) 2 n = ∑ k = 0 2 n ( − 1 ) k ( 2 n k ) 3 , {\displays... | MacMahon master theorem | 0.803866 |
32,503 | Although various q-Dixon identities have been known for decades, except for a Krattenthaler–Schlosser extension (1999), the proper q-analog of MMT remained elusive. After Garoufalidis–Lê–Zeilberger's quantum extension (2006), a number of noncommutative extensions were developed by Foata–Han, Konvalinka–Pak, and Etingof... | MacMahon master theorem | 0.803866 |
32,504 | In 1962, Good found a short proof of Dixon's identity from MMT. In 1969, Cartier and Foata found a new proof of MMT by combining algebraic and bijective ideas (built on Foata's thesis) and further applications to combinatorics on words, introducing the concept of traces. Since then, MMT has become a standard tool in en... | MacMahon master theorem | 0.803866 |
32,505 | In the monograph, MacMahon found so many applications of his result, he called it "a master theorem in the Theory of Permutations." He explained the title as follows: "a Master Theorem from the masterly and rapid fashion in which it deals with various questions otherwise troublesome to solve." The result was re-derived... | MacMahon master theorem | 0.803866 |
32,506 | One can ask whether all contractible manifolds are homeomorphic to a ball. For dimensions 1 and 2, the answer is classical and it is "yes". In dimension 2, it follows, for example, from the Riemann mapping theorem. Dimension 3 presents the first counterexample: the Whitehead manifold. | Whitehead continuum | 0.803866 |
32,507 | In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to R 3 . {\displaystyle \mathbb {R} ^{3}.} J. H. C. Whitehead (1935) discovered this puzzling object while he was trying to prove the Poincaré conjecture, correcting an error in an earlier paper Whitehead (1934, theo... | Whitehead continuum | 0.803866 |
32,508 | The Whitehead manifold is defined as X = S 3 ∖ W , {\displaystyle X=S^{3}\setminus W,} which is a non-compact manifold without boundary. It follows from our previous observation, the Hurewicz theorem, and Whitehead's theorem on homotopy equivalence, that X is contractible. | Whitehead continuum | 0.803866 |
32,509 | Physical optics is also the name of an approximation commonly used in optics, electrical engineering and applied physics. In this context, it is an intermediate method between geometric optics, which ignores wave effects, and full wave electromagnetism, which is a precise theory. The word "physical" means that it is mo... | Physical optics | 0.803865 |
32,510 | One can therefore see that C ℓ 0 , 3 ( R ) ≅ H ⊗ D {\displaystyle C\ell _{0,3}(\mathbb {R} )\cong \mathbb {H} \otimes \mathbb {D} } where D = Cℓ1,0(R) is the algebra spanned by {1, ω}, the algebra of the split-complex numbers. Equivalently, C ℓ 0 , 3 ( R ) ≅ H ⊕ H . {\displaystyle C\ell _{0,3}(\mathbb {R} )\cong \mathb... | Split-biquaternion | 0.803865 |
32,511 | A split-biquaternion is ring isomorphic to the Clifford algebra Cℓ0,3(R). This is the geometric algebra generated by three orthogonal imaginary unit basis directions, {e1, e2, e3} under the combination rule e i e j = { − 1 i = j , − e j e i i ≠ j {\displaystyle e_{i}e_{j}={\begin{cases}-1&i=j,\\-e_{j}e_{i}&i\neq j\end{... | Split-biquaternion | 0.803865 |
32,512 | {\displaystyle (u+v)(w+z)\oplus (u-v)(w-z).} This element is also the image of pq under the mapping into H ⊕ H . {\displaystyle \mathbb {H} \oplus \mathbb {H} .} Thus the products agree, the mapping is a homomorphism; and since it is bijective, it is an isomorphism. Though split-biquaternions form an eight-dimensional ... | Split-biquaternion | 0.803865 |
32,513 | The isomorphism mapping from split-biquaternions to H ⊕ H {\displaystyle \mathbb {H} \oplus \mathbb {H} } is given by p ↦ ( u + v ) ⊕ ( u − v ) , q ↦ ( w + z ) ⊕ ( w − z ) . {\displaystyle p\mapsto (u+v)\oplus (u-v),\quad q\mapsto (w+z)\oplus (w-z).} In H ⊕ H {\displaystyle \mathbb {H} \oplus \mathbb {H} } , the produc... | Split-biquaternion | 0.803865 |
32,514 | The direct sum of the division ring of quaternions with itself is denoted H ⊕ H {\displaystyle \mathbb {H} \oplus \mathbb {H} } . The product of two elements ( a ⊕ b ) {\displaystyle (a\oplus b)} and ( c ⊕ d ) {\displaystyle (c\oplus d)} is a c ⊕ b d {\displaystyle ac\oplus bd} in this direct sum algebra. Proposition: ... | Split-biquaternion | 0.803865 |
32,515 | Since elements {1, i, j, k} of the quaternion group can be taken as a basis of the space of split-biquaternions, it may be compared to a vector space. But split-complex numbers form a ring, not a field, so vector space is not appropriate. Rather the space of split-biquaternions forms a free module. This standard term o... | Split-biquaternion | 0.803865 |
32,516 | This algebra was introduced by William Kingdon Clifford in an 1873 article for the London Mathematical Society. It has been repeatedly noted in mathematical literature since then, variously as a deviation in terminology, an illustration of the tensor product of algebras, and as an illustration of the direct sum of alge... | Split-biquaternion | 0.803865 |
32,517 | In mathematics, a split-biquaternion is a hypercomplex number of the form q = w + x i + y j + z k {\displaystyle q=w+xi+yj+zk} where w, x, y, and z are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each coefficient w, x, y, z spans two real dimensions, the split-biquaternion is an ele... | Split-biquaternion | 0.803865 |
32,518 | The split-biquaternions should not be confused with the (ordinary) biquaternions previously introduced by William Rowan Hamilton. Hamilton's biquaternions are elements of the algebra C ℓ 2 ( C ) = H ⊗ C . {\displaystyle C\ell _{2}(\mathbb {C} )=\mathbb {H} \otimes \mathbb {C} .} C ℓ 3 , 0 ( R ) = H ⊗ C . {\displaystyle... | Split-biquaternion | 0.803865 |
32,519 | The following terms and compounds refer to the split-biquaternion algebra: elliptic biquaternions – Clifford 1873, Rooney 2007 Clifford biquaternion – Joly 1902, van der Waerden 1985 dyquaternions – Rosenfeld 1997 D ⊗ H {\displaystyle \mathbb {D} \otimes \mathbb {H} } where D = split-complex numbers – Bourbaki 1994, Ro... | Split-biquaternion | 0.803865 |
32,520 | Each atom of helium-4 is a boson particle, by virtue of its integer spin. A helium-3 atom is a fermion particle; it can form bosons only by pairing with another particle like itself at much lower temperatures. The discovery of superfluidity in helium-3 was the basis for the award of the 1996 Nobel Prize in Physics. Thi... | Superfluidity | 0.803863 |
32,521 | It is also a property of various other exotic states of matter theorized to exist in astrophysics, high-energy physics, and theories of quantum gravity. The theory of superfluidity was developed by Soviet theoretical physicists Lev Landau and Isaak Khalatnikov. Superfluidity often co-occurs with Bose–Einstein condensat... | Superfluidity | 0.803863 |
32,522 | Superfluid vacuum theory (SVT) is an approach in theoretical physics and quantum mechanics where the physical vacuum is viewed as superfluid. The ultimate goal of the approach is to develop scientific models that unify quantum mechanics (describing three of the four known fundamental interactions) with gravity. This ma... | Superfluidity | 0.803863 |
32,523 | Then P(x) is reducible if and only if at least one of the following conditions holds: The polynomial P(x) has a rational root (this can be determined using the rational root theorem). The resolvent cubic R3(y) has a root of the form α2, for some non-null rational number α (again, this can be determined using the ration... | Resolvent cubic | 0.803862 |
32,524 | Then R3(y) = y3 + 2a2y2 + (a22 − 4a0)y − a12. There are two cases: If a1 ≠ 0 then R3(0) = −a12 < 0. Since R3(y) > 0 if y is large enough, then, by the intermediate value theorem, R3(y) has a root y0 with y0 > 0. | Resolvent cubic | 0.803862 |
32,525 | In algebra, a resolvent cubic is one of several distinct, although related, cubic polynomials defined from a monic polynomial of degree four: P ( x ) = x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 . {\displaystyle P(x)=x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}.} In each case: The coefficients of the resolvent cubic can be obta... | Resolvent cubic | 0.803862 |
32,526 | Suppose that the coefficients of P(x) belong to a field k whose characteristic is different from 2. In other words, we are working in a field in which 1 + 1 ≠ 0. Whenever roots of P(x) are mentioned, they belong to some extension K of k such that P(x) factors into linear factors in K. If k is the field Q of rational nu... | Resolvent cubic | 0.803862 |
32,527 | "It continues to outline that the earliest known case of AIDS in an adult IDU occurred in 1979 (mixed risk) and that known cases among IDUs increased rapidly from the 8 cases in 1980 (3 mixed risk), to 31 cases in 1981, to 160 cases in 1982, and to 340 cases in 1983. Statistics on the incidence of positive tests for HI... | Natural transfer | 0.803861 |
32,528 | "In an article published in AIDS: Cultural Analysis/Cultural Activism, author Douglas Crimp draws attention to anecdotal evidence about junkie pneumonia. Quoting: "Even these statistics are based on CDC epidemiology that continues to see the beginning of the epidemic as 1981 ... in spite of widespread anecdotal reporti... | Natural transfer | 0.803861 |
32,529 | Once it infects, viral replication takes place in virus factories in the cytoplasm. It was found that the genome of the family Marseilleviridae codes for about 28 different proteins. The capsid of the marseillevirus is about 250 nm wide with a geometry shape of an icosahedral. The replication of this virus usually occu... | Nucleocytoplasmic large DNA viruses | 0.803861 |
32,530 | Research in this area is conducted using molecular phylogenetics, comparing viral genomic sequences to determine relatedness. | Natural transfer | 0.803861 |
32,531 | In graph theory, a branch of mathematics, a half graph is a special type of bipartite graph. These graphs are called the half graphs because they have approximately half of the edges of a complete bipartite graph on the same vertices. The name was given to these graphs by Paul Erdős and András Hajnal. | Half graph | 0.80386 |
32,532 | Saharon Shelah's unstable formula theorem in model theory characterizes the stable theories (complete theories that have few types) by the nonexistence of countably infinite half graphs. Shelah defines a complete theory as having the order property if there exist a model M {\displaystyle M} of the theory, a formula ϕ (... | Half graph | 0.80386 |
32,533 | Faltings's theorem (formerly the Mordell conjecture) says that for any curve X of genus at least 2 over a number field k, the set X(k) is finite.Some of the great achievements of number theory amount to determining the rational points on particular curves. For example, Fermat's Last Theorem (proved by Richard Taylor an... | Rational point | 0.803859 |
32,534 | For a smooth hypersurface X of degree d in P n {\displaystyle \mathbb {P} ^{n}} over a field k of order q, Deligne's theorem gives the bound: | | X ( k ) | − ( q n − 1 + ⋯ + q + 1 ) | ≤ ( ( d − 1 ) n + 1 + ( − 1 ) n + 1 ( d − 1 ) d ) q ( n − 1 ) / 2 . {\displaystyle {\big |}|X(k)|-(q^{n-1}+\cdots +q+1){\big |}\leq {\bi... | Rational point | 0.803859 |
32,535 | On the other hand, in the terminology of algebraic geometry, the algebraic variety X over R {\displaystyle \mathbb {R} } is not empty, because the set of complex points X ( C ) {\displaystyle X(\mathbb {C} )} is not empty. More generally, for a scheme X over a commutative ring R and any commutative R-algebra S, the set... | Rational point | 0.803859 |
32,536 | Then a k-point of X means a section of this morphism, that is, a morphism a: Spec(k) → X such that the composition fa is the identity on Spec(k). This agrees with the previous definitions when X is an affine or projective variety (viewed as a scheme over k). When X is a variety over an algebraically closed field k, muc... | Rational point | 0.803859 |
32,537 | Given a field k, and an algebraically closed extension K of k, an affine variety X over k is the set of common zeros in Kn of a collection of polynomials with coefficients in k: f 1 ( x 1 , … , x n ) = 0 , ⋮ f r ( x 1 , … , x n ) = 0. {\displaystyle {\begin{aligned}&f_{1}(x_{1},\ldots ,x_{n})=0,\\&\qquad \quad \vdots \... | Rational point | 0.803859 |
32,538 | The Mordell–Weil theorem says that for an elliptic curve (or, more generally, an abelian variety) X over a number field k, the abelian group X(k) is finitely generated. Computer algebra programs can determine the Mordell–Weil group X(k) in many examples, but it is not known whether there is an algorithm that always suc... | Rational point | 0.803859 |
32,539 | It is harder to determine whether a curve of genus 1 has a rational point. The Hasse principle fails in this case: for example, by Ernst Selmer, the cubic curve 3 x 3 + 4 y 3 + 5 z 3 = 0 {\displaystyle 3x^{3}+4y^{3}+5z^{3}=0} in P 2 {\displaystyle \mathbb {P} ^{2}} has a point over all completions of Q , {\displaystyle... | Rational point | 0.803859 |
32,540 | In higher dimensions, even more is true: every smooth cubic in P n {\displaystyle \mathbb {P} ^{n}} over Q {\displaystyle \mathbb {Q} } has a rational point when n ≥ 9, by Roger Heath-Brown. More generally, Birch's theorem says that for any odd positive integer d, there is an integer N such that for all n ≥ N, every hy... | Rational point | 0.803859 |
32,541 | In the case of a hypersurface X of degree d in P n {\displaystyle \mathbb {P} ^{n}} over a number field, there are good results when d is much smaller than n, often based on the Hardy–Littlewood circle method. For example, the Hasse–Minkowski theorem says that the Hasse principle holds for quadric hypersurfaces over a ... | Rational point | 0.803859 |
32,542 | In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point. ... | Rational point | 0.803859 |
32,543 | Deployed on 13 July 2020, TechEdSat-10 tested radio communication devices, precision deorbit technologies for National Oceanic and Atmospheric Administration (NOAA) and others, radiation tolerant electronics, and artificial intelligence hardware for future experiments. | TechEdSat | 0.803859 |
32,544 | TechEdSat-13 is a 3U CubeSat that is intended to test and validate three different technologies, including an Exo-Brake. Along with other cubesats (PAN-A and B, GEARRS-3, SteamSat-2, STORK-3, ADLER-1) it was launched on the third successful flight of Virgin Orbit's LauncherOne launch vehicle on 13 January 2022 as part ... | TechEdSat | 0.803859 |
32,545 | The above groups can be described algebraically as well as geometrically. Given the curve y2 = x3 + ax + b over the field K (whose characteristic we assume to be neither 2 nor 3), and points P = (xP, yP) and Q = (xQ, yQ) on the curve, assume first that xP ≠ xQ (case 1). Let y = sx + d be the equation of the line that i... | Weierstrass equation | 0.803859 |
32,546 | The mismatch negativity (MMN) or mismatch field (MMF) is a component of the event-related potential (ERP) to an odd stimulus in a sequence of stimuli. It arises from electrical activity in the brain and is studied within the field of cognitive neuroscience and psychology. It can occur in any sensory system, but has mos... | Mismatch negativity | 0.803859 |
32,547 | Most of the data obtained, unfortunately, do not allow the objective measurement of the accuracy of these stimulus representations. In addition, recent cognitive neuroscience seems to have succeeded in extracting such a measure, however. This is the mismatch negativity (MMN), a component of the event-related potential ... | Mismatch negativity | 0.803859 |
32,548 | Without these representations our ability to understand spoken language, for example, would be seriously impaired. Cognitive neuroscience has consequently emphasised the importance of understanding brain mechanisms of sensory information processing, that is, the sensory prerequisites of cognition. | Mismatch negativity | 0.803859 |
32,549 | Excessive dihedral can destabilize the Dutch roll, so configurations with highly swept wings require anhedral to offset the wing sweep contribution to L β {\displaystyle L_{\beta }} . The damping term is dominated by the product of the roll damping and the yaw damping derivatives, these are both negative, so their prod... | Aircraft stability | 0.803852 |
32,550 | One month later, the Phillips group independently reported the wild-type GFP structure in Nature Biotechnology. These crystal structures provided vital background on chromophore formation and neighboring residue interactions. Researchers have modified these residues by directed and random mutagenesis to produce the wid... | Green Fluorescent Protein | 0.803849 |
32,551 | The spell checker included the relatively rare, for the time, option to check on-the-fly. It also added document statistics display, including the number of characters, pages, etc.The two headlining features of the Plus was the addition of on-screen text styles, not just underlining and italic, that continued to update... | 1st Word | 0.803849 |
32,552 | In statistics, non-sampling error is a catch-all term for the deviations of estimates from their true values that are not a function of the sample chosen, including various systematic errors and random errors that are not due to sampling. Non-sampling errors are much harder to quantify than sampling errors.Non-sampling... | Non-sampling error | 0.803849 |
32,553 | Discrete Optimization, 6:79–91, 2009. 2-dim partial order: A 2-dimensional partial order is a k-dimensional partial order for k = 2. k-dim partial order: A poset is a k-dimensional partial order iff it can be embedded into the k-dimensional Euclidean space in such a way that each node is represented by a k-dimensional ... | Notation for theoretic scheduling problems | 0.803848 |
32,554 | level order: Given precedence relation where each vertex of a given level ℓ (i.e. the length of the longest directed path starting from this vertex is ℓ) is a predecessor of all the vertices of level ℓ − 1. interval order: Given precedence relation for which one can associate to each vertex an interval in the real line... | Notation for theoretic scheduling problems | 0.803848 |
32,555 | In mathematical analysis and computer science, functions which are Z-order, Lebesgue curve, Morton space-filling curve, Morton order or Morton code map multidimensional data to one dimension while preserving locality of the data points. It is named in France after Henri Lebesgue, who studied it in 1904, and named in th... | Z-order curve | 0.803847 |
32,556 | Similar techniques can of course be used where multiple instructions are involved, as long as the combined instruction length is adjusted accordingly. For example, in this same example, if it is required to clear the rest of each array entry to nulls immediately after the 100 byte field copied, an additional clear inst... | Loop unwinding | 0.803846 |
32,557 | Since the benefits of loop unrolling are frequently dependent on the size of an array—which may often not be known until run time—JIT compilers (for example) can determine whether to invoke a "standard" loop sequence or instead generate a (relatively short) sequence of individual instructions for each element. This fle... | Loop unwinding | 0.803846 |
32,558 | Thus, there can be a trade-off between the two optimizations. Possible increased register usage in a single iteration to store temporary variables, which may reduce performance, though much will depend on possible optimizations. Apart from very small and simple code, unrolled loops that contain branches are even slower... | Loop unwinding | 0.803846 |
32,559 | However, it appeared at the end of the 19th century that Euclid's postulates were not sufficient for characterizing geometry. The first complete axiom system for geometry was given only at the end of the 19th century by David Hilbert. At the same time, it appeared that both synthetic methods and analytic methods can be... | Computational synthetic geometry | 0.803842 |
32,560 | So the term "synthetic geometry" was introduced to refer to the older methods that were, before Descartes, the only known ones. According to Felix Klein Synthetic geometry is that which studies figures as such, without recourse to formulae, whereas analytic geometry consistently makes use of such formulae as can be wri... | Computational synthetic geometry | 0.803842 |
32,561 | Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic method for proving all results from a few basic properties initially called postulate, and at present called axioms. The term "synthetic geometry" has been coined... | Computational synthetic geometry | 0.803842 |
32,562 | Simply discarding it gives absolute geometry, while negating it yields hyperbolic geometry. Other consistent axiom sets can yield other geometries, such as projective, elliptic, spherical or affine geometry. Axioms of continuity and "betweenness" are also optional, for example, discrete geometries may be created by dis... | Computational synthetic geometry | 0.803842 |
32,563 | There is no fixed axiom set for geometry, as more than one consistent set can be chosen. Each such set may lead to a different geometry, while there are also examples of different sets giving the same geometry. With this plethora of possibilities, it is no longer appropriate to speak of "geometry" in the singular. Hist... | Computational synthetic geometry | 0.803842 |
32,564 | The commutator of two operators acting on a Hilbert space is a central concept in quantum mechanics, since it quantifies how well the two observables described by these operators can be measured simultaneously. The uncertainty principle is ultimately a theorem about such commutators, by virtue of the Robertson–Schrödin... | Graded commutator | 0.803839 |
32,565 | Rings often do not support division. Thus, the commutator of two elements a and b of a ring (or any associative algebra) is defined differently by = a b − b a . {\displaystyle =ab-ba.} The commutator is zero if and only if a and b commute. | Graded commutator | 0.803839 |
32,566 | {\displaystyle \{a,b\}=ab+ba.} Sometimes + {\displaystyle _{+}} is used to denote anticommutator, while − {\displaystyle _{-}} is then used for commutator. The anticommutator is used less often, but can be used to define Clifford algebras and Jordan algebras and in the derivation of the Dirac equation in particle phy... | Graded commutator | 0.803838 |
32,567 | In linear algebra, if two endomorphisms of a space are represented by commuting matrices in terms of one basis, then they are so represented in terms of every basis. By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra. The anticommutator of two elements a and b of a ring... | Graded commutator | 0.803838 |
32,568 | Programs written in Java have a reputation for being slower and requiring more memory than those written in C++. However, Java programs' execution speed improved significantly with the introduction of just-in-time compilation in 1997/1998 for Java 1.1, the addition of language features supporting better code analysis (... | Java programming language | 0.803838 |
32,569 | The degree deg L {\displaystyle \operatorname {deg} L} of the line bundle L {\displaystyle L} over the surface Σ {\displaystyle \Sigma } is deg L = c 1 ( L ) = 1 2 π ∫ Σ i F {\displaystyle \operatorname {deg} L=c_{1}(L)={\frac {1}{2\pi }}\int _{\Sigma }iF} where c 1 ( L ) = c 1 ( L ) ∈ H 2 ( Σ ) {\displaystyle c_{... | Ginzburg–Landau theory | 0.803837 |
32,570 | In physics, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after Vitaly Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining thei... | Ginzburg–Landau theory | 0.803837 |
32,571 | In particle physics, any quantum field theory with a unique classical vacuum state and a potential energy with a degenerate critical point is called a Landau–Ginzburg theory. The generalization to N = (2,2) supersymmetric theories in 2 spacetime dimensions was proposed by Cumrun Vafa and Nicholas Warner in November 198... | Ginzburg–Landau theory | 0.803837 |
32,572 | The two-body Dirac equations of constraint dynamics rectify this flaw. The forms of these equations can not only be derived from quantum field theory they can also be derived purely in the context of Dirac's constraint dynamics and relativistic mechanics and quantum mechanics. Their structures, unlike the more familiar... | Two-body Dirac equations | 0.803836 |
32,573 | Keller, C.; Kanellakopulos, B. (April 1965). "Ternäre oxide des drei-bis siebenwertigen technetiums mit alkalien" . Journal of Inorganic and Nuclear Chemistry. | Sodium metatechnetate | 0.803835 |
32,574 | Stereoisomerism about double bonds arises because rotation about the double bond is restricted, keeping the substituents fixed relative to each other. If the two substituents on at least one end of a double bond are the same, then there is no stereoisomer and the double bond is not a stereocenter, e.g. propene, CH3CH=C... | Stereoisomerism | 0.803832 |
32,575 | In the computer science fields of knowledge engineering and ontology, the Sigma knowledge engineering environment is an open source computer program for the development of formal ontologies. It is designed for use with the Suggested Upper Merged Ontology. It originally included only the Vampire theorem prover as its co... | Sigma knowledge engineering environment | 0.803831 |
32,576 | The trigonometry of a tetrahedron explains the relationships between the lengths and various types of angles of a general tetrahedron. | Trigonometry of a tetrahedron | 0.803829 |
32,577 | Take the tetrahedron X {\displaystyle X} , and consider the point P i {\displaystyle P_{i}} as an apex. The Alternating sines theorem is given by the following identity:One may view the two sides of this identity as corresponding to clockwise and counterclockwise orientations of the surface. | Trigonometry of a tetrahedron | 0.803829 |
32,578 | By the spherical law of cosines:Combining the two equations gives the following result: Making cos λ {\displaystyle \cos \lambda } the subject:Thus, using the cosine law and some basic trigonometry:Thus:So: R i k {\displaystyle R_{ik}} and R i l {\displaystyle R_{il}} are obtained by permutation of the edge lengths. ... | Trigonometry of a tetrahedron | 0.803829 |
32,579 | Take the face F i {\displaystyle F_{i}} ; the edges will have lengths d j k , d j l , d k l {\displaystyle d_{jk},d_{jl},d_{kl}} and the respective opposite angles are given by α l , i , α k , i , α j , i {\displaystyle \alpha _{l,i},\alpha _{k,i},\alpha _{j,i}} . The usual laws for planar trigonometry of a triangle ho... | Trigonometry of a tetrahedron | 0.803829 |
32,580 | Consider the projective (spherical) triangle at the point P i {\displaystyle P_{i}} ; the vertices of this projective triangle are the three lines that join P i {\displaystyle P_{i}} with the other three vertices of the tetrahedron. The edges will have spherical lengths α i , j , α i , k , α i , l {\displaystyle \alpha... | Trigonometry of a tetrahedron | 0.803829 |
32,581 | For any neighborhood U {\displaystyle U} of p ∈ M {\displaystyle p\in M} there exists a neighborhood V ⊂ U , p ∈ V {\displaystyle V\subset U,p\in V} such that there exists no timelike curve that passes through V {\displaystyle V} more than once. For any neighborhood U {\displaystyle U} of p ∈ M {\displaystyle p\in M} t... | Causality conditions | 0.803828 |
32,582 | In the study of Lorentzian manifold spacetimes there exists a hierarchy of causality conditions which are important in proving mathematical theorems about the global structure of such manifolds. These conditions were collected during the late 1970s.The weaker the causality condition on a spacetime, the more unphysical ... | Causality conditions | 0.803828 |
32,583 | In physics, the Painlevé conjecture is a theorem about singularities among the solutions to the n-body problem: there are noncollision singularities for n ≥ 4.The theorem was proven for n ≥ 5 in 1988 by Jeff Xia and for n=4 in 2014 by Jinxin Xue. | Painlevé conjecture | 0.803828 |
32,584 | Although no P450 has yet been shown to be essential in a microbe, the CYP105 family is highly conserved with a representative in every streptomycete genome sequenced so far. Due to the solubility of bacterial P450 enzymes, they are generally regarded as easier to work with than the predominantly membrane bound eukaryot... | Cytochrome P450 | 0.803827 |
32,585 | Whereas statistics and data analysis procedures generally yield their output in numeric or tabular form, graphical techniques allow such results to be displayed in some sort of pictorial form. They include plots such as scatter plots, histograms, probability plots, spaghetti plots, residual plots, box plots, block plot... | Statistical graphics | 0.803827 |
32,586 | Statistical graphics, also known as statistical graphical techniques, are graphics used in the field of statistics for data visualization. | Statistical graphics | 0.803827 |
32,587 | Because of their backgrounds, quantitative analysts draw from various forms of mathematics: statistics and probability, calculus centered around partial differential equations, linear algebra, discrete mathematics, and econometrics. Some on the buy side may use machine learning. The majority of quantitative analysts ha... | Quantitative analyst | 0.803826 |
32,588 | In 1969 Robert Merton promoted continuous stochastic calculus and continuous-time processes. Merton was motivated by the desire to understand how prices are set in financial markets, which is the classical economics question of "equilibrium", and in later papers he used the machinery of stochastic calculus to begin inv... | Quantitative analyst | 0.803826 |
32,589 | stock market. The field has grown to incorporate numerous approaches and techniques; see Outline of finance § Quantitative investing, Post-modern portfolio theory, Financial economics § Portfolio theory. In 1965 Paul Samuelson introduced stochastic calculus into the study of finance. | Quantitative analyst | 0.803826 |
32,590 | He was able to create a system, known broadly as card counting, which used probability theory and statistical analysis to successfully win blackjack games. His research was subsequently used during the 1980s and 1990s by investment management firms seeking to generate systematic and consistent returns in the U.S. | Quantitative analyst | 0.803826 |
32,591 | Although the language of finance now involves Itô calculus, management of risk in a quantifiable manner underlies much of the modern theory. Modern quantitative investment management was first introduced from the research of Edward Thorp, a mathematics professor at New Mexico State University (1961–1965) and University... | Quantitative analyst | 0.803826 |
32,592 | Quantitative developers, sometimes called quantitative software engineers, or quantitative engineers, are computer specialists that assist, implement and maintain the quantitative models. They tend to be highly specialised language technicians that bridge the gap between software engineers and quantitative analysts. Th... | Quantitative analyst | 0.803826 |
32,593 | FOQs typically are significantly better paid than those in back office, risk, and model validation. Although highly skilled analysts, FOQs frequently lack software engineering experience or formal training, and bound by time constraints and business pressures, tactical solutions are often adopted. See also structurer. | Quantitative analyst | 0.803826 |
32,594 | In quantum physics, a density matrix for an n-level quantum system is an n × n complex matrix ρ that is positive semidefinite with trace 1. If ρ can be expressed as ρ = ∑ i p i v i v i ∗ {\displaystyle \rho =\sum _{i}p_{i}v_{i}v_{i}^{*}} where p i > 0 {\displaystyle p_{i}>0} and Σ pi = 1, the set { p i , v i } {\displa... | Matrix square root | 0.803826 |
32,595 | In linear algebra and operator theory, given a bounded positive semidefinite operator (a non-negative operator) T on a complex Hilbert space, B is a square root of T if T = B* B, where B* denotes the Hermitian adjoint of B. According to the spectral theorem, the continuous functional calculus can be applied to obtain a... | Matrix square root | 0.803826 |
32,596 | Finally, if the distribution of income changes in favor of consumers who prefer the good in question, the demand will shift out. It is important to remember that factors that affect individual demand can also affect aggregate demand. However, net effects must be considered. The most important problem for micro- and mac... | Aggregation method | 0.803825 |
32,597 | A typical example is the aggregate production function. Another famous problem is Sonnenschein-Mantel-Debreu theorem. | Aggregation method | 0.803825 |
32,598 | In economics, an aggregate is a summary measure. It replaces a vector that is composed of many real numbers by a single real number, or a scalar. Consequently, there occur various problems that are inherent in the formulations that use aggregated variables.The aggregation problem is the difficult problem of finding a v... | Aggregation method | 0.803825 |
32,599 | One easy example of local homology is calculating the local homology of the cone (topology) of a space at the origin of the cone. Recall that the cone is defined as the quotient space C X = ( X × I ) / ( X × { 0 } ) , {\displaystyle CX=(X\times I)/(X\times \{0\}),} where X × { 0 } {\displaystyle X\times \{0\}} has the ... | Local homology | 0.803825 |
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