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89,900 | Network Accelerated Motion Estimation and Reduction (NAMER) is a retrospective motion correction technique that utilizes convolutional neural networks (CNNs), a class of neural networks designed to process and learn from visual information such as images. This is a follow-up from the authors of the TAMER paper titled N... | MRI artifact | 0.7811 |
89,901 | In recent years, neural networks have generated a great deal of interest by outperforming traditional methods on longstanding problems across many fields. Machine learning, and by extension neural networks, have been used in many facets of MRI — for instance, speeding up image reconstruction, or improving reconstructio... | MRI artifact | 0.7811 |
89,902 | In the geometry of hyperbolic 3-space, the order-7 dodecahedral honeycomb a regular space-filling tessellation (or honeycomb). | Order-3-8 pentagonal honeycomb | 0.781097 |
89,903 | The (arithmetic, or field) norm of a Hurwitz quaternion a + bi + cj + dk, given by a2 + b2 + c2 + d2, is always an integer. By a theorem of Lagrange every nonnegative integer can be written as a sum of at most four squares. Thus, every nonnegative integer is the norm of some Lipschitz (or Hurwitz) quaternion. More prec... | Hurwitz integral quaternion | 0.781097 |
89,904 | A Hurwitz integer is called irreducible if it is not 0 or a unit and is not a product of non-units. A Hurwitz integer is irreducible if and only if its norm is a prime number. The irreducible quaternions are sometimes called prime quaternions, but this can be misleading as they are not primes in the usual sense of comm... | Hurwitz integral quaternion | 0.781097 |
89,905 | The modern concept of extraterrestrial life is based on assumptions that were not commonplace during the early days of astronomy. The first explanations for the celestial objects seen in the night sky were based on mythology. The Greek scholars from Ancient Greece were the first to consider that the universe is inheren... | Extraterrestrial being | 0.781096 |
89,906 | Scientists search for biosignatures within the Solar System by studying planetary surfaces and examining meteorites. Some claim to have identified evidence that microbial life has existed on Mars. In 1996, a controversial report stated that structures resembling nanobacteria were discovered in a meteorite, ALH84001, fo... | Extraterrestrial being | 0.781096 |
89,907 | The major problem with their incongruence is that, at Planck scale (a fundamental small unit of length) lengths, general relativity predicts a smooth, flowing surface, while quantum mechanics predicts a random, warped surface, which are nowhere near compatible. Superstring theory resolves this issue, replacing the clas... | Super-string theory | 0.781095 |
89,908 | General relativity typically deals with situations involving large mass objects in fairly large regions of spacetime whereas quantum mechanics is generally reserved for scenarios at the atomic scale (small spacetime regions). The two are very rarely used together, and the most common case that combines them is in the s... | Super-string theory | 0.781095 |
89,909 | Since strings can have an infinite number of modes, the symmetry used to describe string theory is based on infinite dimensional Lie algebras. Some Kac–Moody algebras that have been considered as symmetries for M-theory have been E10 and E11 and their supersymmetric extensions. | Super-string theory | 0.781095 |
89,910 | One of the deepest open problems in theoretical physics is formulating a theory of quantum gravity. Such a theory incorporates both the theory of general relativity, which describes gravitation and applies to large-scale structures, and quantum mechanics or more specifically quantum field theory, which describes the ot... | Super-string theory | 0.781095 |
89,911 | However, the known symmetries including supersymmetry currently restrict the spinors to 32-components—which limits the number of dimensions to 11 (or 12 if you include two time dimensions.) Some physicists (e.g., John Baez et al.) have speculated that the exceptional Lie groups E6, E7 and E8 having maximum orthogonal s... | Super-string theory | 0.781095 |
89,912 | Investigating how a string theory may include fermions in its spectrum led to the invention of supersymmetry (in the West) in 1971, a mathematical transformation between bosons and fermions. String theories that include fermionic vibrations are now known as "superstring theories". Since its beginnings in the seventies ... | Super-string theory | 0.781095 |
89,913 | CEBAF Large Acceptance Spectrometer (CLAS) is a nuclear and particle physics detector located in the experimental Hall B at Jefferson Laboratory in Newport News, Virginia, United States. It is used to study the properties of the nuclear matter by the collaboration of over 200 physicists (CLAS Collaboration) from many c... | CLAS detector | 0.781094 |
89,914 | Arizona State University - Tempe, AZ California State University - Dominguez Hills, CA Carnegie Mellon University - Pittsburgh, PA Catholic University of America - Washington, DC CEA-Saclay - Gif-sur-Yvette, France Christopher Newport University, Newport News, VA Florida International University - Miami, FL Florida Sta... | CLAS detector | 0.781094 |
89,915 | A diagram of the CLAS detector is shown in the Figure, as well as a photograph of the detector when it was partially pulled open for maintenance. The physics target is at the center. Charged particles are detected in almost all directions, excluding the very forward (beam) and backward (beam) directions, and also exclu... | CLAS detector | 0.781094 |
89,916 | The CLAS detector was notable among devices in the area of hadronic particle physics in that it had a very large acceptance; in other words, it measured the momentum and angles of almost all of the particles produced in the electron-proton collisions. Roughly spherical, the detector measured 30 feet across. It surround... | CLAS detector | 0.781094 |
89,917 | This theorem is remarkable for it allows one to construct a measure by first defining it on a small algebra of sets, where its sigma additivity could be easy to verify, and then this theorem guarantees its extension to a sigma-algebra. The proof of this theorem is not trivial, since it requires extending μ 0 {\displays... | Hahn–Kolmogorov theorem | 0.781093 |
89,918 | There can be more than one extension of a pre-measure to the generated σ-algebra, if the pre-measure is not σ {\displaystyle \sigma } -finite, even if the extensions themselves are σ {\displaystyle \sigma } -finite (see example "Via rationals" below). | Hahn–Kolmogorov theorem | 0.781093 |
89,919 | First extend μ {\displaystyle \mu } to an outer measure μ ∗ {\displaystyle \mu ^{*}} on the power set 2 X {\displaystyle 2^{X}} of X {\displaystyle X} by and then restrict it to the set B {\displaystyle {\mathcal {B}}} of μ ∗ {\displaystyle \mu ^{*}} -measurable sets (that is, Carathéodory-measurable sets), which is th... | Hahn–Kolmogorov theorem | 0.781093 |
89,920 | {\displaystyle \sigma (R).} The Carathéodory's extension theorem states that it is also sufficient, that is, there exists a measure μ ′: σ ( R ) → {\displaystyle \mu ^{\prime }:\sigma (R)\to } such that μ ′ {\displaystyle \mu ^{\prime }} is an extension of μ ; {\displaystyle \mu ;} that is, μ ′ | R = μ . {\displaystyl... | Hahn–Kolmogorov theorem | 0.781093 |
89,921 | Let R {\displaystyle R} be a ring of sets on X {\displaystyle X} and let μ: R → {\displaystyle \mu :R\to } be a pre-measure on R , {\displaystyle R,} meaning that for all sets A ∈ R {\displaystyle A\in R} for which there exists a countable decomposition A = ⋃ i = 1 ∞ A i {\displaystyle A=\bigcup _{i=1}^{\infty }A_{i}}... | Hahn–Kolmogorov theorem | 0.781093 |
89,922 | This situation is ultimately unsatisfactory, as the weak solution u might not have enough derivatives for the expression Lu to be well-defined in the classical sense. The elliptic regularity theorem guarantees that, provided f is square-integrable, u will in fact have 2k square-integrable weak derivatives. In particula... | Elliptic differential operator | 0.781092 |
89,923 | The topological Tverberg theorem was proved for prime r by Barany, Shlosman and Szucs. Matousek: 162--163 presents a proof using deleted joins. The theorem was proved for r a prime-power by Ozaydin, and later by Volovikov and Sarkaria. | Tverberg theorem | 0.781091 |
89,924 | In discrete geometry, Tverberg's theorem, first stated by Helge Tverberg in 1966, is the result that sufficiently many points in d-dimensional Euclidean space can be partitioned into subsets with intersecting convex hulls. Specifically, for any positive integers d, r and any set of ( d + 1 ) ( r − 1 ) + 1 {\displaystyl... | Tverberg theorem | 0.781091 |
89,925 | The case d = 1 states that any 2r-1 points on the real line can be partitioned into r subsets with intersecting convex hulls. Indeed, if the points are x1 < x2 < ... < x2r < x2r-1, then the partition into Ai = {xi, x2r-i} for i in 1,...,r satisfies this condition (and it is unique). For r = 2, Tverberg's theorem states... | Tverberg theorem | 0.781091 |
89,926 | The disorder may be diagnosed by either the analysis of the molecular genetics of the individual, or by detection of abnormal metabolite levels in blood and/or plasma. Individuals with this deficiency have an elevated amount of 3-hydroxyglutarate excreted through the urine; a heightened level of C4-OH acylcarnitine in ... | Hydroxyacyl-Coenzyme A dehydrogenase | 0.781091 |
89,927 | Space.com The Huffington Post Yahoo! News Universe Today Toronto Star Astronomy (magazine) == References == | Miniverse | 0.781091 |
89,928 | But since black holes lack a surface, acceleration due to gravity increases exponentially, but never reaches a final value as there is no considered surface in a singularity. In quantum mechanics, the black hole emits Hawking radiation and so it can come to thermal equilibrium with a gas of radiation (not compulsory). ... | White Holes | 0.78109 |
89,929 | In mathematics, a stacky curve is an object in algebraic geometry that is roughly an algebraic curve with potentially "fractional points" called stacky points. A stacky curve is a type of stack used in studying Gromov–Witten theory, enumerative geometry, and rings of modular forms. Stacky curves are deeply related to 1... | Stacky curve | 0.781088 |
89,930 | The generalization of GAGA for stacky curves is used in the derivation of algebraic structure theory of rings of modular forms.The study of stacky curves is used extensively in equivariant Gromov–Witten theory and enumerative geometry. == References == | Stacky curve | 0.781088 |
89,931 | A stacky curve is uniquely determined (up to isomorphism) by its coarse space X (a smooth quasi-projective curve over k), a finite set of points xi (its stacky points) and integers ni (its ramification orders) greater than 1. The canonical divisor of X {\displaystyle {\mathfrak {X}}} is linearly equivalent to the sum o... | Stacky curve | 0.781088 |
89,932 | In mathematics, the Kervaire semi-characteristic, introduced by Michel Kervaire (1956), is an invariant of closed manifolds M of dimension 4 n + 1 {\displaystyle 4n+1} taking values in Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } , given by k ( M ) = ∑ i = 0 2 n dim H 2 i ( M , R ) mod 2 {\displaystyle k(M)=\su... | Kervaire semi-characteristic | 0.781087 |
89,933 | The Causes of Evolution is a 1932 book on evolution by J.B.S. Haldane (1990 edition ISBN 0-691-02442-1), based on a series of January 1931 lectures entitled "A Re-examination of Darwinism". It was influential in the founding of population genetics and the modern synthesis. | The Causes of Evolution | 0.781084 |
89,934 | This leads to the same Lie algebra, because the inverse map on G can be used to identify left invariant vector fields with right invariant vector fields, and acts as −1 on the tangent space Te. The Lie algebra structure on Te can also be described as follows: the commutator operation (x, y) → xyx−1y−1on G × G sends (e,... | P-adic Lie group | 0.781081 |
89,935 | Simple Lie groups are sometimes defined to be those that are simple as abstract groups, and sometimes defined to be connected Lie groups with a simple Lie algebra. For example, SL(2, R) is simple according to the second definition but not according to the first. They have all been classified (for either definition). | P-adic Lie group | 0.781081 |
89,936 | Problems about Lie groups are often solved by first solving the corresponding problem for the Lie algebras, and the result for groups then usually follows easily. For example, simple Lie groups are usually classified by first classifying the corresponding Lie algebras. We could also define a Lie algebra structure on Te... | P-adic Lie group | 0.781081 |
89,937 | {\displaystyle {\mathfrak {g}}.} Thus the Lie bracket on g {\displaystyle {\mathfrak {g}}} is given explicitly by = e.This Lie algebra g {\displaystyle {\mathfrak {g}}} is finite-dimensional and it has the same dimension as the manifold G. The Lie algebra of G determines G up to "local isomorphism", where two Lie grou... | P-adic Lie group | 0.781081 |
89,938 | This shows that the space of left invariant vector fields (vector fields satisfying Lg*Xh = Xgh for every h in G, where Lg* denotes the differential of Lg) on a Lie group is a Lie algebra under the Lie bracket of vector fields. Any tangent vector at the identity of a Lie group can be extended to a left invariant vector... | P-adic Lie group | 0.781081 |
89,939 | To get around these problems we give the general definition of the Lie algebra of a Lie group (in 4 steps): Vector fields on any smooth manifold M can be thought of as derivations X of the ring of smooth functions on the manifold, and therefore form a Lie algebra under the Lie bracket = XY − YX, because the Lie bracke... | P-adic Lie group | 0.781081 |
89,940 | The preceding description can be made more rigorous as follows. The Lie algebra of a closed subgroup G of GL(n, C), may be computed as Lie ( G ) = { X ∈ M ( n ; C ) | exp ( t X ) ∈ G for all t in R } , {\displaystyle \operatorname {Lie} (G)=\{X\in M(n;\mathbb {C} )|\operatorname {exp} (tX)\in G{\text{ for all }}t{\... | P-adic Lie group | 0.781081 |
89,941 | The Lie algebra of the general linear group GL(n, C) of invertible matrices is the vector space M(n, C) of square matrices with the Lie bracket given by = AB − BA. If G is a closed subgroup of GL(n, C) then the Lie algebra of G can be thought of informally as the matrices m of M(n, C) such that 1 + εm is in G, where ε... | P-adic Lie group | 0.781081 |
89,942 | To every Lie group we can associate a Lie algebra whose underlying vector space is the tangent space of the Lie group at the identity element and which completely captures the local structure of the group. Informally we can think of elements of the Lie algebra as elements of the group that are "infinitesimally close" t... | P-adic Lie group | 0.781081 |
89,943 | Some examples of groups that are not Lie groups (except in the trivial sense that any group having at most countably many elements can be viewed as a 0-dimensional Lie group, with the discrete topology), are: Infinite-dimensional groups, such as the additive group of an infinite-dimensional real vector space, or the sp... | P-adic Lie group | 0.781081 |
89,944 | Semisimple Lie groups are Lie groups whose Lie algebra is a product of simple Lie algebras. They are central extensions of products of simple Lie groups.The identity component of any Lie group is an open normal subgroup, and the quotient group is a discrete group. The universal cover of any connected Lie group is a sim... | P-adic Lie group | 0.781081 |
89,945 | But there are also just five "exceptional Lie algebras" that do not fall into any of these families. E8 is the largest of these. Lie groups are classified according to their algebraic properties (simple, semisimple, solvable, nilpotent, abelian), their connectedness (connected or simply connected) and their compactness... | P-adic Lie group | 0.781081 |
89,946 | The structure of an abelian Lie algebra is mathematically uninteresting (since the Lie bracket is identically zero); the interest is in the simple summands. Hence the question arises: what are the simple Lie algebras of compact groups? It turns out that they mostly fall into four infinite families, the "classical Lie a... | P-adic Lie group | 0.781081 |
89,947 | The mathematical object capturing this structure is called a Lie algebra (Lie himself called them "infinitesimal groups"). It can be defined because Lie groups are smooth manifolds, so have tangent spaces at each point. The Lie algebra of any compact Lie group (very roughly: one for which the symmetries form a bounded ... | P-adic Lie group | 0.781081 |
89,948 | The Computer Science Network (CSNET) was a computer network funded by the NSF that began operation in 1981. Its purpose was to extend networking benefits for computer science departments at academic and research institutions that could not be directly connected to ARPANET due to funding or authorization limitations. It... | Packet switched | 0.781081 |
89,949 | Molecular cloning is one of the most commonly used procedures in molecular biology. A gene of interest may be inserted into a plasmid vector via ligation, and the plasmid is then transformed into Escherichia coli cells. However, not all the plasmids transformed into cells may contain the desired gene insert, and checki... | Blue-white screen | 0.781081 |
89,950 | The Discovery Investigations were a series of scientific cruises and shore-based investigations into the biology of whales in the Southern Ocean. They were funded by the British Colonial Office and organised by the Discovery Committee in London, which was formed in 1918. They were intended to provide the scientific bac... | Discovery Investigations | 0.78108 |
89,951 | For example, there is a one-to-one correspondence between vector bundles and locally free sheaves of O X {\displaystyle {\mathcal {O}}_{X}} -modules. This paradigm applies to real vector bundles, complex vector bundles, or vector bundles in algebraic geometry (where O {\displaystyle {\mathcal {O}}} consists of smooth f... | Global sections functor | 0.781078 |
89,952 | Similarly, schemes, the foundational notion of spaces in algebraic geometry, are locally ringed spaces that are locally isomorphic to the spectrum of a ring. Given a ringed space, a sheaf of modules is a sheaf M {\displaystyle {\mathcal {M}}} such that on every open set U {\displaystyle U} of X {\displaystyle X} , M ( ... | Global sections functor | 0.781078 |
89,953 | In several geometrical disciplines, including algebraic geometry and differential geometry, the spaces come along with a natural sheaf of rings, often called the structure sheaf and denoted by O X {\displaystyle {\mathcal {O}}_{X}} . Such a pair ( X , O X ) {\displaystyle (X,{\mathcal {O}}_{X})} is called a ringed spac... | Global sections functor | 0.781078 |
89,954 | Three of the most fundamental operations which can be performed on square matrices are matrix addition, multiplication by a scalar, and matrix multiplication. These are exactly those operations necessary for defining a polynomial function of an n × n matrix A {\displaystyle A} . If we recall from basic calculus that ma... | Generalized eigenvector | 0.781078 |
89,955 | {\displaystyle A={\begin{pmatrix}1&1\\0&1\end{pmatrix}}.} Then there is only one eigenvalue, λ = 1 {\displaystyle \lambda =1} , and its algebraic multiplicity is m = 2 {\displaystyle m=2} . Notice that this matrix is in Jordan normal form but is not diagonal. | Generalized eigenvector | 0.781078 |
89,956 | The matrix A = ( 5 1 − 2 4 0 5 2 2 0 0 5 3 0 0 0 4 ) {\displaystyle A={\begin{pmatrix}5&1&-2&4\\0&5&2&2\\0&0&5&3\\0&0&0&4\end{pmatrix}}} has an eigenvalue λ 1 = 5 {\displaystyle \lambda _{1}=5} of algebraic multiplicity μ 1 = 3 {\displaystyle \mu _{1}=3} and an eigenvalue λ 2 = 4 {\displaystyle \lambda _{2}=4} of algeb... | Generalized eigenvector | 0.781078 |
89,957 | Consider the problem of solving the system of linear ordinary differential equations where x = ( x 1 ( t ) x 2 ( t ) ⋮ x n ( t ) ) , x ′ = ( x 1 ′ ( t ) x 2 ′ ( t ) ⋮ x n ′ ( t ) ) , {\displaystyle \mathbf {x} ={\begin{pmatrix}x_{1}(t)\\x_{2}(t)\\\vdots \\x_{n}(t)\end{pmatrix}},\quad \mathbf {x} '={\begin{pmatrix}x_{1}... | Generalized eigenvector | 0.781078 |
89,958 | In the preceding sections we have seen techniques for obtaining the n {\displaystyle n} linearly independent generalized eigenvectors of a canonical basis for the vector space V {\displaystyle V} associated with an n × n {\displaystyle n\times n} matrix A {\displaystyle A} . These techniques can be combined into a proc... | Generalized eigenvector | 0.781078 |
89,959 | The theorem may be generalized in a variety of ways. | Frobenius theorem (differential topology) | 0.781078 |
89,960 | The statement does not generalize to higher degree forms, although there is a number of partial results such as Darboux's theorem and the Cartan-Kähler theorem. | Frobenius theorem (differential topology) | 0.781078 |
89,961 | In classical mechanics, the integrability of a system's constraint equations determines whether the system is holonomic or nonholonomic. In microeconomic theory, Frobenius' theorem can be used to prove the existence of a solution to the problem of integrability of demand functions. | Frobenius theorem (differential topology) | 0.781078 |
89,962 | Despite being named for Ferdinand Georg Frobenius, the theorem was first proven by Alfred Clebsch and Feodor Deahna. Deahna was the first to establish the sufficient conditions for the theorem, and Clebsch developed the necessary conditions. Frobenius is responsible for applying the theorem to Pfaffian systems, thus pa... | Frobenius theorem (differential topology) | 0.781078 |
89,963 | The notation is defined in the article on one-forms. During his development of axiomatic thermodynamics, Carathéodory proved that if ω {\displaystyle \omega } is an integrable one-form on an open subset of R n {\displaystyle \mathbb {R} ^{n}} , then ω = f d g {\displaystyle \omega =fdg} for some scalar functions f , g ... | Frobenius theorem (differential topology) | 0.781078 |
89,964 | An example is shown in the diagram on the right. If the one-form is integrable, then loops exactly close upon themselves, and each surface would be 2-dimensional. Frobenius' theorem states that this happens precisely when ω ∧ d ω = 0 {\displaystyle \omega \wedge d\omega =0} over all of the domain, where ω := a d x + b ... | Frobenius theorem (differential topology) | 0.781078 |
89,965 | There are thus two forms of the theorem: one which operates with distributions, that is smooth subbundles D of the tangent bundle TM; and the other which operates with subbundles of the graded ring Ω(M) of all forms on M. These two forms are related by duality. If D is a smooth tangent distribution on M, then the annih... | Frobenius theorem (differential topology) | 0.781078 |
89,966 | The infinite-dimensional version of the Frobenius theorem also holds on Banach manifolds. The statement is essentially the same as the finite-dimensional version. Let M be a Banach manifold of class at least C2. Let E be a subbundle of the tangent bundle of M. The bundle E is involutive if, for each point p ∈ M and pai... | Frobenius theorem (differential topology) | 0.781078 |
89,967 | Specifically, they must satisfy relations of the form L i L j u ( x ) − L j L i u ( x ) = ∑ k c i j k ( x ) L k u ( x ) {\displaystyle L_{i}L_{j}u(x)-L_{j}L_{i}u(x)=\sum _{k}c_{ij}^{k}(x)L_{k}u(x)} for 1 ≤ i, j ≤ r, and all C2 functions u, and for some coefficients ckij(x) that are allowed to depend on x. In other word... | Frobenius theorem (differential topology) | 0.781078 |
89,968 | Let U be an open set in a manifold M, Ω1(U) be the space of smooth, differentiable 1-forms on U, and F be a submodule of Ω1(U) of rank r, the rank being constant in value over U. The Frobenius theorem states that F is integrable if and only if for every p in U the stalk Fp is generated by r exact differential forms. Ge... | Frobenius theorem (differential topology) | 0.781078 |
89,969 | A projective space is a topological space, as endowed with the quotient topology of the topology of a finite dimensional real vector space. Let S be the unit sphere in a normed vector space V, and consider the function that maps a point of S to the vector line passing through it. This function is continuous and surject... | Projective space over a division algebra | 0.781075 |
89,970 | Given a vector space V over a field K, the projective space P(V) is the set of equivalence classes of V \ {0} under the equivalence relation ~ defined by x ~ y if there is a nonzero element λ of K such that x = λy. If V is a topological vector space, the quotient space P(V) is a topological space, endowed with the quot... | Projective space over a division algebra | 0.781075 |
89,971 | So a projective variety is the set of points in a projective space, whose homogeneous coordinates are common zeros of a set of homogeneous polynomials.Any affine variety can be completed, in a unique way, into a projective variety by adding its points at infinity, which consists of homogenizing the defining polynomials... | Projective space over a division algebra | 0.781075 |
89,972 | In the multivariate case, the consideration of complex zeros is also needed, but not sufficient: one must also consider zeros at infinity. For example, Bézout's theorem asserts that the intersection of two plane algebraic curves of respective degrees d and e consists of exactly de points if one consider complex points ... | Projective space over a division algebra | 0.781075 |
89,973 | Originally, algebraic geometry was the study of common zeros of sets of multivariate polynomials. These common zeros, called algebraic varieties belong to an affine space. It appeared soon, that in the case of real coefficients, one must consider all the complex zeros for having accurate results. For example, the funda... | Projective space over a division algebra | 0.781075 |
89,974 | The Spherics (Greek: Σφαιρικά Sphairiká) is a three-volume treatise on spherical geometry written by the Greek mathematician Theodosius of Bithynia in the 2nd century BC. Book I and the first half of Book II establish basic geometric constructions needed for spherical geometry using the tools of Euclidean solid geometr... | Theodosius' Spherics | 0.781074 |
89,975 | This approach differs from other quantitative methods of Greek astronomy such as the analemma (orthographic projection), stereographic projection, or trigonometry (a fledgling subject introduced by Theodosius' contemporary Hipparchus). It also differs from the approach taken in Menelaus' Spherics, a treatise of the sam... | Theodosius' Spherics | 0.781074 |
89,976 | Spherical distances and radii are treated as Euclidean distances in the surrounding 3-dimensional space. The relationship between planes is described in terms of dihedral angle. As in the Elements, there is no concept of angle measure or trigonometry per se. | Theodosius' Spherics | 0.781074 |
89,977 | Several of the definitions and theorems in the Spherics were used without mention in Euclid's Phenomena and two extant works by Autolycus concerning motions of the celestial sphere, all written about two centuries before Theodosius. It has been speculated that the this tradition of Greek "Spherics" – founded in the axi... | Theodosius' Spherics | 0.781074 |
89,978 | A great deal of evidence for neutrino oscillation has been collected from many sources, over a wide range of neutrino energies and with many different detector technologies. The 2015 Nobel Prize in Physics was shared by Takaaki Kajita and Arthur B. McDonald for their early pioneering observations of these oscillations.... | Atmospheric neutrino oscillation | 0.781073 |
89,979 | The function f(z) = exp(–1/z) has the asymptotic series 0 + 0z + ... with an error bound of the form above in the region |arg(z)| < θ for any θ < π/2, but is not given by the Borel sum of its asymptotic series. This shows that the number π/2 in Watson's theorem cannot be replaced by any smaller number (unless the bound... | Borel summation | 0.781071 |
89,980 | This integral converges for all z ≥ 0, so the original divergent series is Borel summable for all such z. This function has an asymptotic expansion as z tends to 0 that is given by the original divergent series. This is a typical example of the fact that Borel summation will sometimes "correctly" sum divergent asymptot... | Borel summation | 0.781071 |
89,981 | Carleman's theorem shows that a function is uniquely determined by an asymptotic series in a sector provided the errors in the finite order approximations do not grow too fast. More precisely it states that if f is analytic in the interior of the sector |z| < C, Re(z) > 0 and |f(z)| < |bnz|n in this region for all n, t... | Borel summation | 0.781071 |
89,982 | A Tauberian theorem provides conditions under which convergence of one summation method implies convergence under another method. The principal Tauberian theorem for Borel summation provides conditions under which the weak Borel method implies convergence of the series. Theorem (Hardy 1992, 9.13). If A is (wB) summable... | Borel summation | 0.781071 |
89,983 | There are always many different functions with any given asymptotic expansion. However, there is sometimes a best possible function, in the sense that the errors in the finite-dimensional approximations are as small as possible in some region. Watson's theorem and Carleman's theorem show that Borel summation produces s... | Borel summation | 0.781071 |
89,984 | }}\\&=e^{t}\sin(e^{t}).\end{aligned}}} At z = 2 the Borel sum is given by ∫ 0 ∞ e t sin ( e 2 t ) d t = ∫ 1 ∞ sin ( u 2 ) d u = π 8 − S ( 1 ) < ∞ , {\displaystyle \int _{0}^{\infty }e^{t}\sin(e^{2t})\,dt=\int _{1}^{\infty }\sin(u^{2})\,du={\sqrt {\frac {\pi }{8}}}-S(1)<\infty ,} where S(x) is the Fresnel integral. ... | Borel summation | 0.781071 |
89,985 | Considering instead the weak Borel transform, the partial sums are given by AN(z) = (1 − zN+1)/(1 − z), and so the weak Borel sum is lim t → ∞ e − t ∑ n = 0 ∞ 1 − z n + 1 1 − z t n n ! = lim t → ∞ e − t 1 − z ( e t − z e t z ) = 1 1 − z , {\displaystyle \lim _{t\rightarrow \infty }e^{-t}\sum _{n=0}^{\infty }{\frac {1-z... | Borel summation | 0.781071 |
89,986 | In topology, a branch of mathematics, an abstract stratified space, or a Thom–Mather stratified space is a topological space X that has been decomposed into pieces called strata; these strata are manifolds and are required to fit together in a certain way. Thom–Mather stratified spaces provide a purely topological sett... | Topologically stratified space | 0.78107 |
89,987 | MAGNUM was one of the telescopes that observed a cosmic explosion billions of years away in 2005. The telescope was used for a long-term study of the size of the universe; for example it studied the Seyfert 1 galaxies NGC 5548, NGC 4051, NGC 3227, and NGC 7469 The telescope was located at the Haleakala Observatory and ... | Multicolor Active Galactic Nuclei Monitoring | 0.78107 |
89,988 | In mathematics and computer science, Recamán's sequence is a well known sequence defined by a recurrence relation. Because its elements are related to the previous elements in a straightforward way, they are often defined using recursion. It takes its name after its inventor Bernardo Recamán Santos, a Colombian mathema... | Recamán's sequence | 0.781069 |
89,989 | Nanotechnology in textiles is a branch of nano-science in which molecular systems at the nano-scale of size (1–100 Nanometre) are applied in the field of textiles to improve performance or add functions to textiles. Nanotechnology unites a variety of scientific fields, such as material science, physics, chemistry, biol... | Textile performance | 0.781069 |
89,990 | The Lambert W function was employed in the field of chemical engineering for modelling the porous electrode film thickness in a glassy carbon based supercapacitor for electrochemical energy storage. The Lambert W function turned out to be the exact solution for a gas phase thermal activation process where growth of car... | Lambert's W function | 0.781066 |
89,991 | In 1993, it was reported that the Lambert W function provides an exact solution to the quantum-mechanical double-well Dirac delta function model for equal charges—a fundamental problem in physics. Prompted by this, Rob Corless and developers of the Maple computer algebra system realized that "the Lambert W function has... | Lambert's W function | 0.781066 |
89,992 | Plastic waste is either industrial scrap (sometimes referred to as post industrial resin) or consumer waste. Scrap is generated during production and is usually handled differently. It can include flashings, trimmings, sprues and rejects. As it is collected at the point of manufacture it is clean, and of a known type a... | Recycled plastic | 0.781065 |
89,993 | In the case of several meteorite fields, such as Dar al Gani, Dhofar, and others, favorable light-colored geology consisting of basic rocks (clays, dolomites, and limestones) makes meteorites particularly easy to identify.Although meteorites had been sold commercially and collected by hobbyists for many decades, up to ... | Meteorite | 0.781064 |
89,994 | The diameter of the largest impactor to hit Earth on any given day is likely to be about 40 centimeters (16 inches), in a given year about four metres (13 ft), and in a given century about 20 m (66 ft). These statistics are obtained by the following: Over at least the range from five centimeters (2.0 inches) to roughly... | Meteorite | 0.781064 |
89,995 | To be clear, this property also holds for finite sequences A 1 , … , A n {\displaystyle A_{1},\ldots ,A_{n}} of pairwise disjoint sets (by letting A i := ∅ {\displaystyle A_{i}:=\varnothing } for all i > n {\displaystyle i>n} ).Conversely, it is easy to check that a family of sets that satisfy conditions 4-6 is a Dynki... | Dynkin system | 0.781063 |
89,996 | In geometry, a hexagon (from Greek ἕξ, hex, meaning "six", and γωνία, gonía, meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. | Hexagon | 0.781062 |
89,997 | Let ABCDEF be a hexagon formed by six tangent lines of a conic section. Then Brianchon's theorem states that the three main diagonals AD, BE, and CF intersect at a single point. In a hexagon that is tangential to a circle and that has consecutive sides a, b, c, d, e, and f, a + c + e = b + d + f . {\displaystyle a+c+e=... | Hexagon | 0.781062 |
89,998 | Psychoacoustics has long enjoyed a symbiotic relationship with computer science. Internet pioneers J. C. R. Licklider and Bob Taylor both completed graduate-level work in psychoacoustics, while BBN Technologies originally specialized in consulting on acoustics issues before it began building the first packet-switched n... | Psychoacoustics | 0.781061 |
89,999 | Fractional quantum mechanics Fractional Schrödinger equation Lévy process Fractional statistical mechanics | Functional integral | 0.781059 |
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