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84,000 | Smoothness of the map Π {\displaystyle \Pi } can be regarded as a technicality, in that any continuous homomorphism will automatically be smooth.We can alternatively describe a representation of a Lie group G {\displaystyle G} as a linear action of G {\displaystyle G} on a vector space V {\displaystyle V} . Notationall... | Representation theory of Lie groups | 0.783539 |
84,001 | It is always possible to pass from a representation of a Lie group G to a representation of its Lie algebra g . {\displaystyle {\mathfrak {g}}.} If Π: G → GL(V) is a group representation for some vector space V, then its pushforward (differential) at the identity, or Lie map, π: g → End V {\displaystyle \pi :{\mathfrak... | Representation theory of Lie groups | 0.783539 |
84,002 | In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous ... | Representation theory of Lie groups | 0.783539 |
84,003 | Over a field, the free algebra on n indeterminates can be constructed as the tensor algebra on an n-dimensional vector space. For a more general coefficient ring, the same construction works if we take the free module on n generators. The construction of the free algebra on E is functorial in nature and satisfies an ap... | Noncommutative polynomial ring | 0.783539 |
84,004 | For example, X1X2 does not equal X2X1. More generally, one can construct the free algebra R⟨E⟩ on any set E of generators. Since rings may be regarded as Z-algebras, a free ring on E can be defined as the free algebra Z⟨E⟩. | Noncommutative polynomial ring | 0.783539 |
84,005 | . , i k {\displaystyle a_{i_{1},i_{2},...,i_{k}}} are said to be "coefficients" of these polynomials, and the R-algebra R⟨X1,...,Xn⟩ is called the "non-commutative polynomial algebra over R in n indeterminates". Note that unlike in an actual polynomial ring, the variables do not commute. | Noncommutative polynomial ring | 0.783539 |
84,006 | In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the polynomial ring may be regarded as a free commutative algebra. | Noncommutative polynomial ring | 0.783539 |
84,007 | Using this lemma, we can prove Hoeffding's inequality. As in the theorem statement, suppose X1, ..., Xn are n independent random variables such that X i ∈ {\displaystyle X_{i}\in } almost surely for all i, and let S n = X 1 + ⋯ + X n {\displaystyle S_{n}=X_{1}+\cdots +X_{n}} . | Hoeffding's inequality | 0.783537 |
84,008 | Then Hoeffding's theorem states that, for all t > 0, P ( S n − E ≥ t ) ≤ exp ( − 2 t 2 ∑ i = 1 n ( b i − a i ) 2 ) P ( | S n − E | ≥ t ) ≤ 2 exp ( − 2 t 2 ∑ i = 1 n ( b i − a i ) 2 ) {\displaystyle {\begin{aligned}\operatorname {P} \left(S_{n}-\mathrm {E} \left\geq t\right)&\leq \exp \left(-{\frac {2t^{2}}{\s... | Hoeffding's inequality | 0.783537 |
84,009 | In probability theory, Hoeffding's inequality provides an upper bound on the probability that the sum of bounded independent random variables deviates from its expected value by more than a certain amount. Hoeffding's inequality was proven by Wassily Hoeffding in 1963.Hoeffding's inequality is a special case of the Azu... | Hoeffding's inequality | 0.783537 |
84,010 | A remarkable formula for the asymptotic behavior of the 6-j symbol was first conjectured by Ponzano and Regge and later proven by Roberts. The asymptotic formula applies when all six quantum numbers j1, ..., j6 are taken to be large and associates to the 6-j symbol the geometry of a tetrahedron. If the 6-j symbol is de... | 6-j symbol | 0.783537 |
84,011 | In representation theory, 6-j symbols are matrix coefficients of the associator isomorphism in a tensor category. For example, if we are given three representations Vi, Vj, Vk of a group (or quantum group), one has a natural isomorphism ( V i ⊗ V j ) ⊗ V k → V i ⊗ ( V j ⊗ V k ) {\displaystyle (V_{i}\otimes V_{j})\otime... | 6-j symbol | 0.783537 |
84,012 | Test of a JavaScript engine using simple algorithms. It displays results of other visitors (best, average, poorest) with the same operating system and browser version. It has an additional statistics page with browser ranking. | SunSpider JavaScript Benchmark | 0.783536 |
84,013 | Kleiber's law relating animal metabolism to size, and allometric laws in general The two-thirds power law, relating speed to curvature in the human motor system. The Taylor's law relating mean population size and variance of populations sizes in ecology Neuronal avalanches The species richness (number of species) in cl... | Broken power law | 0.783533 |
84,014 | One way of stating Turán's theorem is the following: Any graph G = (V, E) contains an independent set of size at least |V|/(D+1), where D = 2|E|/|V| is the average degree of the graph. | Pessimistic estimator | 0.783531 |
84,015 | Agricultural and Forest Biotechnology Forest Entomology and Forest Pathology Agricultural Plant Pathology and Entomology Biological Control of Fungal Pathogens and Insect Pests Microbial Bioinformatics and Phylogenetics Mycology and Fungal Biodiversity | Forestry and Agricultural Biotechnology Institute | 0.783525 |
84,016 | The Forestry and Agricultural Biotechnology Institute (FABI) (Pretoria) was established in 1997 and is located on the University of Pretoria campus. The goal of the institute is to help the development of novel food and fibre crops, that will clearly contribute to global economic development and food security. FABI was... | Forestry and Agricultural Biotechnology Institute | 0.783525 |
84,017 | In physics, the distorted Schwarzschild metric is the metric of a standard/isolated Schwarzschild spacetime exposed in external fields. In numerical simulation, the Schwarzschild metric can be distorted by almost arbitrary kinds of external energy–momentum distribution. However, in exact analysis, the mature method to ... | Distorted Schwarzschild metric | 0.783525 |
84,018 | The Cahn–Ingold–Prelog priority rules (often abbreviated as the CIP system) were first published in 1966; allowing enantiomers to be more easily and accurately described. The same year saw first successful enantiomeric separation by gas chromatography an important development as the technology was in common use at the ... | Asymmetric synthesis | 0.783525 |
84,019 | The Chlorobi-RRM RNA motif is a conserved RNA structure identified by bioinformatics. It is found within bacteria in the phylum Chlorobiota (formerly Chlorobi), and is exclusively detected in the presumed 5' untranslated regions (5' UTRs) of genes that encode putative RNA-binding proteins. Since many RNA-binding protei... | Chlorobi-RRM RNA motif | 0.783524 |
84,020 | Future observations with JWST and the ELTs might improve the sample of Y-dwarfs with observed spectra. Y-dwarfs are dominated by deep spectral features of methane, water vapor and possibly absorption features of ammonia and water ice. Vertical mixing, clouds, metallicity, photochemistry, lightning, impact shocks and me... | T-type star | 0.783521 |
84,021 | An optical telescope is a telescope that gathers and focuses light mainly from the visible part of the electromagnetic spectrum, to create a magnified image for direct visual inspection, to make a photograph, or to collect data through electronic image sensors. There are three primary types of optical telescope: Refrac... | Optical telescopes | 0.783519 |
84,022 | In this generation of telescopes, the mirror is usually very thin, and is kept in an optimal shape by an array of actuators (see active optics). This technology has driven new designs for future telescopes with diameters of 30, 50 and even 100 meters. Relatively cheap, mass-produced ~2 meter telescopes have recently be... | Optical telescopes | 0.783519 |
84,023 | However, conceptually it is easier to derive by considering a reflecting telescope. If an extended object with angular size α {\displaystyle \alpha } is observed through a telescope, then due to the Laws of reflection and Trigonometry the size of the image projected onto the focal plane will be s = tan ( α ) f . {\di... | Optical telescopes | 0.783519 |
84,024 | A set E ⊂ R {\displaystyle E\subset \mathbb {R} } is compact if it is closed and bounded. This definition also holds for Euclidean space of any finite dimension, R n {\displaystyle \mathbb {R} ^{n}} , but it is not valid for metric spaces in general. The equivalence of the definition with the definition of compactness ... | Real Analysis | 0.783518 |
84,025 | Lebesgue integration is a mathematical construction that extends the integral to a larger class of functions; it also extends the domains on which these functions can be defined. The concept of a measure, an abstraction of length, area, or volume, is central to Lebesgue integral probability theory. | Real Analysis | 0.783518 |
84,026 | The study of issues of convergence for sequences of functions eventually gave rise to Fourier analysis as a subdiscipline of mathematical analysis. Investigation of the consequences of generalizing differentiability from functions of a real variable to ones of a complex variable gave rise to the concept of holomorphic ... | Real Analysis | 0.783518 |
84,027 | Various ideas from real analysis can be generalized from the real line to broader or more abstract contexts. These generalizations link real analysis to other disciplines and subdisciplines. For instance, generalization of ideas like continuous functions and compactness from real analysis to metric spaces and topologic... | Real Analysis | 0.783518 |
84,028 | The distinction between pointwise and uniform convergence is important when exchanging the order of two limiting operations (e.g., taking a limit, a derivative, or integral) is desired: in order for the exchange to be well-behaved, many theorems of real analysis call for uniform convergence. For example, a sequence of ... | Real Analysis | 0.783518 |
84,029 | Absolutely continuous functions are continuous: consider the case n = 1 in this definition. The collection of all absolutely continuous functions on I is denoted AC(I). Absolute continuity is a fundamental concept in the Lebesgue theory of integration, allowing the formulation of a generalized version of the fundamenta... | Real Analysis | 0.783518 |
84,030 | For instance, in contrast to the behavior of finite sums, rearranging the terms of an infinite series may result in convergence to a different number (see the article on the Riemann rearrangement theorem for further discussion). An example of a convergent series is a geometric series which forms the basis of one of Zen... | Real Analysis | 0.783518 |
84,031 | The concept of limit was informally introduced for functions by Newton and Leibniz, at the end of the 17th century, for building infinitesimal calculus. For sequences, the concept was introduced by Cauchy, and made rigorous, at the end of the 19th century by Bolzano and Weierstrass, who gave the modern ε-δ definition, ... | Real Analysis | 0.783518 |
84,032 | Roughly speaking, a limit is the value that a function or a sequence "approaches" as the input or index approaches some value. (This value can include the symbols ± ∞ {\displaystyle \pm \infty } when addressing the behavior of a function or sequence as the variable increases or decreases without bound.) The idea of a l... | Real Analysis | 0.783518 |
84,033 | The topology induced by metric d {\displaystyle d} turns out to be identical to the standard topology induced by order < {\displaystyle <} . Theorems like the intermediate value theorem that are essentially topological in nature can often be proved in the more general setting of metric or topological spaces rather than... | Real Analysis | 0.783518 |
84,034 | Many of the theorems of real analysis are consequences of the topological properties of the real number line. The order properties of the real numbers described above are closely related to these topological properties. As a topological space, the real numbers has a standard topology, which is the order topology induce... | Real Analysis | 0.783518 |
84,035 | The theorems of real analysis rely on the properties of the real number system, which must be established. The real number system consists of an uncountable set ( R {\displaystyle \mathbb {R} } ), together with two binary operations denoted + and ⋅, and an order denoted <. The operations make the real numbers a field, ... | Real Analysis | 0.783518 |
84,036 | The real numbers have various lattice-theoretic properties that are absent in the complex numbers. Also, the real numbers form an ordered field, in which sums and products of positive numbers are also positive. Moreover, the ordering of the real numbers is total, and the real numbers have the least upper bound property... | Real Analysis | 0.783518 |
84,037 | Important results include the Bolzano–Weierstrass and Heine–Borel theorems, the intermediate value theorem and mean value theorem, Taylor's theorem, the fundamental theorem of calculus, the Arzelà-Ascoli theorem, the Stone-Weierstrass theorem, Fatou's lemma, and the monotone convergence and dominated convergence theore... | Real Analysis | 0.783518 |
84,038 | Compactness is a concept from general topology that plays an important role in many of the theorems of real analysis. The property of compactness is a generalization of the notion of a set being closed and bounded. (In the context of real analysis, these notions are equivalent: a set in Euclidean space is compact if an... | Real Analysis | 0.783518 |
84,039 | Although this definition gives the Darboux integral the appearance of being a special case of the Riemann integral, they are, in fact, equivalent, in the sense that a function is Darboux integrable if and only if it is Riemann integrable, and the values of the integrals are equal. In fact, calculus and real analysis te... | Real Analysis | 0.783518 |
84,040 | In complex analysis, it is natural to define differentiation via holomorphic functions, which have a number of useful properties, such as repeated differentiability, expressibility as power series, and satisfying the Cauchy integral formula. In real analysis, it is usually more natural to consider differentiable, smoot... | Real Analysis | 0.783518 |
84,041 | This feature may cause packet analyzers that are unaware or uncertain about the use of checksum offload to report invalid checksums in outbound packets that have not yet reached the network adapter. This will only occur for packets that are intercepted before being transmitted by the network adapter; all packets transm... | TCP acceleration | 0.783515 |
84,042 | In molecular biology, the interferon-inducible GTPase (IIGP) family of proteins is thought to play a role in intracellular defence. IIGP is predominantly associated with the Golgi apparatus and also localises to the endoplasmic reticulum and exerts a distinct role in IFN-induced intracellular membrane trafficking or pr... | Interferon-inducible GTPase | 0.783514 |
84,043 | The Quarterly Review of Biology is a peer-reviewed scientific journal covering all aspects of biology. It was established in 1926 by Raymond Pearl. In the 1960s it was purchased by the Stony Brook Foundation when the editor H. Bentley Glass became academic vice president of Stony Brook University. The editor-in-chief i... | The Quarterly Review of Biology | 0.783514 |
84,044 | Stronger results are known for restricted classes of polynomials or algebraic numbers. If P(x) is not reciprocal then M ( P ) ≥ M ( x 3 − x − 1 ) ≈ 1.3247 {\displaystyle M(P)\geq M(x^{3}-x-1)\approx 1.3247} and this is clearly best possible. If further all the coefficients of P are odd then M ( P ) ≥ M ( x 2 − x − 1 ) ... | Lehmer conjecture | 0.783514 |
84,045 | The Royal Belgian Institute for Space Aeronomy (BIRA-IASB) (Dutch: Koninklijk Belgisch Instituut voor Ruimte-Aeronomie - BIRA, French: Institut royal d'aéronomie spatiale de Belgique - IASB) is a Belgian federal scientific research institute. Created in 1964, its main tasks are research and public service in space aero... | Belgian Institute for Space Aeronomy | 0.783513 |
84,046 | BIRA-IASB studies the Earth's stratosphere by looking at ozone depletion and related key parameters like volcanic aerosols, polar stratospheric clouds and halogenated compounds. Long-term changes are evaluated and forecasting capabilities have been developed. The research supports the Montreal Protocol on the protectio... | Belgian Institute for Space Aeronomy | 0.783513 |
84,047 | BIRA-IASB focuses on Chemistry & Physics of Atmospheres and Space Plasma Physics designs and builds instruments to monitor atmospheres and the space environment operates Belgian experiments on board the International Space Station and other satellites participates in international measurement programmes compares observ... | Belgian Institute for Space Aeronomy | 0.783513 |
84,048 | In mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetic function, by means of an inverse Mellin transform. | Perron formula | 0.783512 |
84,049 | One stronger form of the circle packing theorem, on representing planar graphs by systems of tangent circles, states that every polyhedral graph can be represented by the vertices and edges of a polyhedron with a midsphere. Equivalently, any convex polyhedron can be transformed into a combinatorially equivalent form, w... | Midsphere | 0.783509 |
84,050 | Iron–sulfur clusters are molecular ensembles of iron and sulfide. They are most often discussed in the context of the biological role for iron–sulfur proteins, which are pervasive. Many Fe–S clusters are known in the area of organometallic chemistry and as precursors to synthetic analogues of the biological clusters (s... | Iron-sulfur cluster | 0.783509 |
84,051 | The research involve researchers from the fields of physics, chemistry and biology. The aim is to understand how functional materials work at the atomic and molecular level. The Zernike Institute for Advanced Materials is involved in the whole string of research starting from design, through synthesis, device building,... | Zernike Institute for Advanced Materials | 0.783508 |
84,052 | The Zernike Institute for Advanced Materials is the department of nanoscience and materials science of the University of Groningen in the Netherlands. The institute is named after the Dutch Nobel prize winner Frits Zernike, famous for his development of phase contrast microscopy. The research of the Zernike Institute i... | Zernike Institute for Advanced Materials | 0.783508 |
84,053 | PowerPC (with the backronym Performance Optimization With Enhanced RISC – Performance Computing, sometimes abbreviated as PPC) is a reduced instruction set computer (RISC) instruction set architecture (ISA) created by the 1991 Apple–IBM–Motorola alliance, known as AIM. PowerPC, as an evolving instruction set, has been ... | Power PC | 0.783507 |
84,054 | The format of the competition was created by the Dutch programme creator Karen van Dijk (VPRO) and was inspired by the Eurovision Song Contest. Participating teams are tasked with the composition of a song using artificial intelligence. Each submission is then evaluated by a jury, which assesses the use of AI in the so... | AI Song Contest | 0.783506 |
84,055 | The AI Song Contest (Dutch: AI Songfestival) is an international music competition for songs that have been composed using artificial intelligence (AI). The inaugural edition took place on 12 May 2020 and was organised by the Dutch public broadcaster VPRO, in collaboration with NPO 3FM and NPO Innovation. Since 2021, t... | AI Song Contest | 0.783506 |
84,056 | The dual contouring algorithm was first published in the 2002 SIGGRAPH proceedings by Ju and Losasso, developed as an extension to both surface nets and marching cubes. It retains a dual vertex within the voxel but no longer at the center. Dual contouring leverages the position and normal of where the surface crosses t... | Isosurface | 0.783506 |
84,057 | In solid-state physics, the t-J model is a model first derived in 1977 from the Hubbard model by Józef Spałek to explain antiferromagnetic properties of the Mott insulators and taking into account experimental results about the strength of electron-electron repulsion in this materials. The model consider the materials ... | T-J model | 0.783504 |
84,058 | In quantum physics system's models are usually based on the Hamiltonian operator H ^ {\displaystyle {\hat {H}}} , corresponding to the total energy of that system, including both kinetic energy and potential energy. The t-J Hamiltonian can be derived from the H ^ {\displaystyle {\hat {H}}} of the Hubbard model using th... | T-J model | 0.783504 |
84,059 | The inherent geometry of film capacitor structure results in very low ohmic losses and a very low parasitic inductance, which makes them especially suitable for applications with very high surge currents (snubbers) and for AC power applications, or for applications at higher frequencies. Another feature of film capacit... | Film capacitor | 0.783501 |
84,060 | The Two Curious Puppies, one big and one little, are being pursued by a dog catcher until they hide in an abandoned house. They soon discover the house is owned by Sham-Fu, a magician who is unseen over the course of the short. As a result, each puppy (both of whom are separated from each other shortly after entering t... | Prest-O Change-O | 0.7835 |
84,061 | {\displaystyle P=\{~\{1,2\},\{3,4\},\{5,6\},\ldots \}.} The deleted integer topology is defined by letting X = ⋃ n ∈ N ( n − 1 , n ) ⊆ R {\displaystyle X={\begin{matrix}\bigcup _{n\in \mathbb {N} }(n-1,n)\subseteq \mathbb {R} \end{matrix}}} and P = { ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 3 ) , … } . {\displaystyle P={\left\{(0... | Partition topology | 0.7835 |
84,062 | In mathematics, the partition topology is a topology that can be induced on any set X {\displaystyle X} by partitioning X {\displaystyle X} into disjoint subsets P ; {\displaystyle P;} these subsets form the basis for the topology. There are two important examples which have their own names: The odd–even topology is th... | Partition topology | 0.7835 |
84,063 | Unless P {\displaystyle P} is trivial, at least one set in P {\displaystyle P} contains more than one point, and the elements of this set are topologically indistinguishable: the topology does not separate points. Hence X {\displaystyle X} is not a Kolmogorov space, nor a T1 space, a Hausdorff space or an Urysohn space... | Partition topology | 0.7835 |
84,064 | The trivial partitions yield the discrete topology (each point of X {\displaystyle X} is a set in P , {\displaystyle P,} so P = { { x }: x ∈ X } {\displaystyle P=\{~\{x\}~:~x\in X~\}} ) or indiscrete topology (the entire set X {\displaystyle X} is in P , {\displaystyle P,} so P = { X } {\displaystyle P=\{X\}} ). Any se... | Partition topology | 0.7835 |
84,065 | For air, n0 ≈ 1, n2 ≈ 4×10−23 m2/W for λ = 800 nm, and the critical power is Pcr ≈ 2.4 GW, corresponding to an energy of about 0.3 mJ for a pulse duration of 100 fs. For silica, n0 ≈ 1.453, n2 ≈ 2.4×10−20 m2/W, and the critical power is Pcr ≈ 2.8 MW. Kerr-induced self-focusing is crucial for many applications in laser ... | Self-focusing | 0.783498 |
84,066 | Nair proved a convergence analysis. For twice continuously differentiable functions, the LJ heuristic generates a sequence of iterates having a convergent subsequence. For this class of problems, Newton's method is the usual optimization method, and it has quadratic convergence (regardless of the dimension of the space... | Local unimodal sampling | 0.783495 |
84,067 | In practice, the LJ heuristic has been recommended for functions that need be neither convex nor differentiable nor locally Lipschitz: The LJ heuristic does not use a gradient or subgradient when one be available, which allows its application to non-differentiable and non-convex problems. Proposed by Luus and Jaakola, ... | Local unimodal sampling | 0.783495 |
84,068 | In computational engineering, Luus–Jaakola (LJ) denotes a heuristic for global optimization of a real-valued function. In engineering use, LJ is not an algorithm that terminates with an optimal solution; nor is it an iterative method that generates a sequence of points that converges to an optimal solution (when one ex... | Local unimodal sampling | 0.783495 |
84,069 | A code unit is the minimum bit combination that can represent a character in a character encoding (in computer science terms, it is the word size of the character encoding). For example, common code units include 7-bit, 8-bit, 16-bit, and 32-bit. In some encodings, some characters are encoded using multiple code units;... | Character code | 0.783495 |
84,070 | This is only an upper bound because not every matrix is invertible and thus usable as a key. The number of invertible matrices can be computed via the Chinese Remainder Theorem. I.e., a matrix is invertible modulo 26 if and only if it is invertible both modulo 2 and modulo 13. | Hill cipher | 0.783493 |
84,071 | In classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra. Invented by Lester S. Hill in 1929, it was the first polygraphic cipher in which it was practical (though barely) to operate on more than three symbols at once. The following discussion assumes an elementary knowled... | Hill cipher | 0.783493 |
84,072 | The basic Hill cipher is vulnerable to a known-plaintext attack because it is completely linear. An opponent who intercepts n 2 {\displaystyle n^{2}} plaintext/ciphertext character pairs can set up a linear system which can (usually) be easily solved; if it happens that this system is indeterminate, it is only necessar... | Hill cipher | 0.783493 |
84,073 | However, many extensions to the type system of ML, such as polymorphic recursion, can make the inference of the principal type undecidable. Other extensions, such as Haskell's generalized algebraic data types, destroy the principal type property of the language, requiring the use of type annotations or the compiler to ... | Principal type | 0.783492 |
84,074 | A stronger form of the twin prime conjecture, the Hardy–Littlewood conjecture (see below), postulates a distribution law for twin primes akin to the prime number theorem. On 17 April 2013, Yitang Zhang announced a proof that for some integer N that is less than 70 million, there are infinitely many pairs of primes that... | Twin Prime Conjecture | 0.783491 |
84,075 | The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture, which states that there are infinitely many primes p such that p + 2 is also prime. In 1849, de Polignac made the more general conje... | Twin Prime Conjecture | 0.783491 |
84,076 | An isolated prime (also known as single prime or non-twin prime) is a prime number p such that neither p − 2 nor p + 2 is prime. In other words, p is not part of a twin prime pair. For example, 23 is an isolated prime, since 21 and 25 are both composite. The first few isolated primes are 2, 23, 37, 47, 53, 67, 79, 83, ... | Twin Prime Conjecture | 0.783491 |
84,077 | The Hardy–Littlewood conjecture (named after G. H. Hardy and John Littlewood) is a generalization of the twin prime conjecture. It is concerned with the distribution of prime constellations, including twin primes, in analogy to the prime number theorem. Let π 2 ( x ) {\displaystyle \pi _{2}(x)} denote the number of pri... | Twin Prime Conjecture | 0.783491 |
84,078 | This assumption, which is suggested by the prime number theorem, implies the twin prime conjecture, as shown in the formula for π 2 ( x ) {\displaystyle \pi _{2}(x)} above. The fully general first Hardy–Littlewood conjecture on prime k-tuples (not given here) implies that the second Hardy–Littlewood conjecture is false... | Twin Prime Conjecture | 0.783491 |
84,079 | a ⋅ γ ν = a ν a ⋅ γ ν = a ν . {\displaystyle {\begin{aligned}a\cdot \gamma ^{\nu }&=a^{\nu }\\a\cdot \gamma _{\nu }&=a_{\nu }.\end{aligned}}} Like in tensor calculus, a change of index position can be achieved using the metric and the use of index gymnastics: γ μ = η μ ν γ ν γ μ = η μ ν γ ν . {\displaystyle {\begin{ali... | Spacetime algebra | 0.783491 |
84,080 | The spacetime algebra is not a division algebra, because it contains idempotent elements 1 2 ( 1 ± γ 0 γ i ) {\displaystyle {\tfrac {1}{2}}(1\pm \gamma _{0}\gamma _{i})} and nonzero zero divisors: ( 1 + γ 0 γ i ) ( 1 − γ 0 γ i ) = 0 {\displaystyle (1+\gamma _{0}\gamma _{i})(1-\gamma _{0}\gamma _{i})=0} . These can be i... | Spacetime algebra | 0.783491 |
84,081 | In the standard vector calculus formulation, two potential functions are used: the electric scalar potential, and the magnetic vector potential. Using the tools of STA, these two objects are combined into a single vector field A {\displaystyle A} , analogous to the electromagnetic four-potential in tensor calculus. In ... | Spacetime algebra | 0.783491 |
84,082 | In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4). According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of special relativity and relativistic spacetime. It is a vector space that a... | Spacetime algebra | 0.783491 |
84,083 | POST runs formal fellowship schemes with scientific societies and research councils, whereby PhD students can spend three months working at POST through an extension of their maintenance grants. These include: Arts and Humanities Research Council Biotechnology and Biological Sciences Research Council British Ecological... | Parliamentary Office of Science and Technology | 0.783489 |
84,084 | POST has eight science advisers, covering the fields of biology and health; physical sciences and ICT; environment and energy; and social sciences. Science advisers generally have a postgraduate qualification and science policy experience. | Parliamentary Office of Science and Technology | 0.783489 |
84,085 | The Large Electron–Positron Collider had four detectors, built around the four collision points within underground halls. Each was the size of a small house and was capable of registering the particles by their energy, momentum and charge, thus allowing physicists to infer the particle reaction that had happened and th... | Large Electron Positron Collider | 0.783488 |
84,086 | Surface nuclear magnetic resonance (SNMR), also known as magnetic resonance Sounding (MRS), is a geophysical technique specially designed for hydrogeology. It is based on the principle of nuclear magnetic resonance (NMR) and measurements can be used to indirectly estimate the water content of saturated and unsaturated ... | Surface nuclear magnetic resonance | 0.783487 |
84,087 | Clearly this is unstable, but at least in classical physics one can imagine that they are so carefully balanced they will remain pointing straight up indefinitely so long as they are not perturbed. Wiggling one of the upside-down pendulums will have a very different effect from before. The speed of propagation of the e... | Tachyonic field | 0.783486 |
84,088 | The phenomenon of spontaneous symmetry breaking, which is closely related to tachyon condensation, plays a central part in many aspects of theoretical physics, including the Ginzburg–Landau and BCS theories of superconductivity. Other examples include the inflaton field in certain models of cosmic inflation (such as ne... | Tachyonic field | 0.783486 |
84,089 | Therefore, information never moves faster than light regardless of the presence or absence of tachyonic fields. Examples for tachyonic fields are all cases of spontaneous symmetry breaking. In condensed matter physics a notable example is ferromagnetism; in particle physics the best known example is the Higgs mechanism... | Tachyonic field | 0.783486 |
84,090 | In quantum field theory, a tachyon is a quantum of a field—usually a scalar field—whose squared mass is negative, and is used to describe spontaneous symmetry breaking: The existence of such a field implies the instability of the field vacuum; the field is at a local maximum rather than a local minimum of its potential... | Tachyonic field | 0.783486 |
84,091 | In string theory, tachyons have the same interpretation as in quantum field theory. However, string theory can, at least, in principle, not only describe the physics of tachyonic fields, but also predict whether such fields appear. Tachyonic fields indeed arise in many versions of string theory. In general, string theo... | Tachyonic field | 0.783486 |
84,092 | This process is known as tachyon condensation. A famous example is the condensation of the Higgs boson in the Standard Model of particle physics. In modern physics, all fundamental particles are regarded as localized excitations of fields. Tachyons are unusual because the instability prevents any such localized excitat... | Tachyonic field | 0.783486 |
84,093 | In physics, a tachyonic field, or simply tachyon, is a quantum field with an imaginary mass. Although tachyonic particles (particles that move faster than light) are a purely hypothetical concept that violate a number of essential physical principles, at least one field with imaginary mass, the Higgs field, is believed... | Tachyonic field | 0.783486 |
84,094 | In organic chemistry, ethenolysis is a chemical process in which internal olefins are degraded using ethylene (H2C=CH2) as the reagent. The reaction is an example of cross metathesis. The utility of the reaction is driven by the low cost of ethylene as a reagent and its selectivity. It produces compounds with terminal ... | Ethenolysis | 0.783485 |
84,095 | Lickorish's proof rested on the Lickorish twist theorem, which states that any orientable automorphism of a closed orientable surface is generated by Dehn twists along 3g − 1 specific simple closed curves in the surface, where g denotes the genus of the surface. Wallace's proof was more general and involved adding hand... | Lickorish–Wallace theorem | 0.783485 |
84,096 | In mathematics, the Lickorish–Wallace theorem in the theory of 3-manifolds states that any closed, orientable, connected 3-manifold may be obtained by performing Dehn surgery on a framed link in the 3-sphere with ±1 surgery coefficients. Furthermore, each component of the link can be assumed to be unknotted. The theore... | Lickorish–Wallace theorem | 0.783485 |
84,097 | Because Direct3D IHV drivers are kernel-mode, and the user-mode code is out of the IHV's hand, there is no chance for such optimizations to occur. Because the Direct3D runtime, the user-mode part that implements the API, cannot have explicit knowledge of the driver's inner workings, it cannot effectively support marsha... | Comparison of OpenGL and Direct3D | 0.783484 |
84,098 | Soc. Japan, 85, 369–432, https://doi.org/10.2151/jmsj.85.369. Kaspar, F., Niermann, D., Borsche, M., Fiedler, S., Keller, J., Potthast, R., Rösch, T., Spangehl, T., and Tinz, B., 2020: Regional atmospheric reanalysis activities at Deutscher Wetterdienst: review of evaluation results and application examples with a focu... | Meteorological reanalysis | 0.783483 |
84,099 | CommView puts the network adapter into promiscuous mode and captures network traffic. It also supports capturing packets from dial-up and virtual adapters (e.g. ADSL or 3G modems), as well as capturing loopback traffic. Captured traffic is then analyzed and the application displays network statistics and individual pac... | CommView | 0.783482 |
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