id int32 0 100k | text stringlengths 21 3.54k | source stringlengths 1 124 | similarity float32 0.78 0.88 |
|---|---|---|---|
80,200 | self-driving truck sales is expected to reach 60,000 by 2035 according to a report released by IHS Inc. in June 2016.As reported in June 1995 in Popular Science magazine, self-driving trucks were being developed for combat convoys, whereby only the lead truck would be driven by a human and the following trucks would re... | Autonomous vehicle | 0.785041 |
80,201 | Department of Energy (DOE) are nominating Cadmium Telluride (CdTe) for inclusion in the National Toxicology Program (NTP). This nomination is strongly supported by the National Renewable Energy Laboratory (NREL) and First Solar Inc. The material has the potential for widespread applications in photovoltaic energy gener... | Cadmium Telluride | 0.785038 |
80,202 | In quantum mechanics, given a projection valued measure of a measurable space X to the space of continuous endomorphisms upon a Hilbert space H, the projective space of the Hilbert space H is interpreted as the set of possible states Φ of a quantum system, the measurable space X is the value space for some quantum prop... | Projection-valued measure | 0.785037 |
80,203 | {\displaystyle \int _{X}^{\oplus }H_{x}\ d\mu (x).} The measure class of μ and the measure equivalence class of the multiplicity function x → dim Hx completely characterize the projection-valued measure up to unitary equivalence. A projection-valued measure π is homogeneous of multiplicity n if and only if the multipli... | Projection-valued measure | 0.785037 |
80,204 | Then π is a projection-valued measure on (X, M). Suppose π, ρ are projection-valued measures on (X, M) with values in the projections of H, K. π, ρ are unitarily equivalent if and only if there is a unitary operator U:H → K such that π ( E ) = U ∗ ρ ( E ) U {\displaystyle \pi (E)=U^{*}\rho (E)U\quad } for every E ∈ M. ... | Projection-valued measure | 0.785037 |
80,205 | A projection-valued measure π {\displaystyle \pi } on a measurable space ( X , M ) {\displaystyle (X,M)} , where M {\displaystyle M} is a σ-algebra of subsets of X {\displaystyle X} , is a mapping from M {\displaystyle M} to the set of self-adjoint projections on a Hilbert space H {\displaystyle H} (i.e. the orthogonal... | Projection-valued measure | 0.785037 |
80,206 | A measurement that can be performed by a projection-valued measure π is called a projective measurement. If X is the real number line, there exists, associated to π, a Hermitian operator A defined on H by A ( φ ) = ∫ R λ d π ( λ ) ( φ ) , {\displaystyle A(\varphi )=\int _{\mathbf {R} }\lambda \,d\pi (\lambda )(\varphi ... | Projection-valued measure | 0.785037 |
80,207 | {\displaystyle P_{\pi }(\psi ):E\mapsto \langle \psi \mid \pi (E)\psi \rangle } is a probability measure on X making the values of the observable into a random variable. | Projection-valued measure | 0.785037 |
80,208 | Surveillance tools are all means technological provided and used by the surveillance industry, police or military intelligence, and national security institutions that enable individual surveillance and mass surveillance. Steven Ashley in 2008 listed the following components used for surveillance: Primarily electronic ... | Surveillance tools | 0.785036 |
80,209 | In mathematics, particularly functional analysis, the Dunford–Schwartz theorem, named after Nelson Dunford and Jacob T. Schwartz, states that the averages of powers of certain norm-bounded operators on L1 converge in a suitable sense. | Dunford–Schwartz theorem | 0.785036 |
80,210 | Closed linear operators are a class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the spectrum and (with certain assumptions) functional calculus for such operators. Many impo... | Unbounded operator | 0.785035 |
80,211 | In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The term "unbounded operator" can be misleading, since "unbounded" should somet... | Unbounded operator | 0.785035 |
80,212 | There are four main chemical properties to account for when dealing with hydrogen that can come into contact with other materials even in normal atmospheric pressures and temperatures: The chemistry of hydrogen is very different from traditional chemicals. E.g., with oxidation in ambient environments. And neglecting th... | Hydrogen safety | 0.785035 |
80,213 | Liquid hydrogen has a slightly different chemistry compared to other cryogenic chemicals, as trace accumulated air can easily contaminate liquid hydrogen and form an unstable mixture with detonative capabilities similar to TNT and other highly explosive materials. Because of this, liquid hydrogen requires complex stora... | Hydrogen safety | 0.785035 |
80,214 | Friedrich Wilhelm Karl Ernst Schröder (25 November 1841 in Mannheim, Baden, Germany – 16 June 1902 in Karlsruhe, Germany) was a German mathematician mainly known for his work on algebraic logic. He is a major figure in the history of mathematical logic, by virtue of summarizing and extending the work of George Boole, A... | Ernst Schröder (mathematician) | 0.785034 |
80,215 | 1 Vorlesungen über die Algebra der Logik (Exakte Logik), Volume 2, Abt. 2 Algebra und Logik der Relative, der Vorlesungen über die Algebra der Logik 3, Volume 3, Abt. 1 Schröder, E., 1898. | Ernst Schröder (mathematician) | 0.785034 |
80,216 | Teubner. Reprints: 1966, Chelsea; 2000, Thoemmes Press. Vorlesungen über die Algebra der Logik (Exakte Logik), Volume 1, Vorlesungen über die Algebra der Logik (Exakte Logik), Volume 2, Abt. | Ernst Schröder (mathematician) | 0.785034 |
80,217 | Schröder, E., 1890–1905. Vorlesungen über die Algebra der Logik, 3 vols. Leipzig: B.G. | Ernst Schröder (mathematician) | 0.785034 |
80,218 | Löwenheim stated and proved the Löwenheim theorem (later reproved and strengthened by Thoralf Skolem, whose name became attached to it together with Löwenheim's) in Peircian notation. In fact, there is no reference in Löwenheim's paper to any logic other than Peirce's. | Ernst Schröder (mathematician) | 0.785034 |
80,219 | Contrasting Frege with Schröder and C. S. Peirce, however, Hilary Putnam (1982) writes: When I started to trace the later development of logic, the first thing I did was to look at Schröder's Vorlesungen über die Algebra der Logik, ... third volume is on the logic of relations (Algebra und Logik der Relative, 1895). Th... | Ernst Schröder (mathematician) | 0.785034 |
80,220 | Schröder's influence on the early development of the predicate calculus, mainly by popularising C. S. Peirce's work on quantification, is at least as great as that of Frege or Peano. For an example of the influence of Schröder's work on English-speaking logicians of the early 20th century, see Clarence Irving Lewis (19... | Ernst Schröder (mathematician) | 0.785034 |
80,221 | He developed Boole's algebra into a calculus of relations, based on composition of relations as a multiplication. The Schröder rules relate alternative interpretations of a product of relations. The Vorlesungen is a prolix affair, only a small part of which has been translated into English. | Ernst Schröder (mathematician) | 0.785034 |
80,222 | Vol. 2 is in two parts, the second published posthumously, edited by Eugen Müller. The Vorlesungen was a comprehensive and scholarly survey of algebraic logic up to the end of the 19th century, one that had a considerable influence on the emergence of mathematical logic in the 20th century. | Ernst Schröder (mathematician) | 0.785034 |
80,223 | Unlike Boole, Schröder fully appreciated duality. John Venn and Christine Ladd-Franklin both warmly cited this short book of Schröder's, and Charles Sanders Peirce used it as a text while teaching at Johns Hopkins University. Schröder's masterwork, his Vorlesungen über die Algebra der Logik, was published in three volu... | Ernst Schröder (mathematician) | 0.785034 |
80,224 | dissertation. Schröder (1877) was a concise exposition of Boole's ideas on algebra and logic, which did much to introduce Boole's work to continental readers. The influence of the Grassmanns, especially Robert's little-known Formenlehre, is clear. | Ernst Schröder (mathematician) | 0.785034 |
80,225 | Schröder also made original contributions to algebra, set theory, lattice theory, ordered sets and ordinal numbers. Along with Georg Cantor, he codiscovered the Cantor–Bernstein–Schröder theorem, although Schröder's proof (1898) is flawed. Felix Bernstein (1878–1956) subsequently corrected the proof as part of his Ph.D... | Ernst Schröder (mathematician) | 0.785034 |
80,226 | Schröder's early work on formal algebra and logic was written in ignorance of the British logicians George Boole and Augustus De Morgan. Instead, his sources were texts by Ohm, Hankel, Hermann Grassmann, and Robert Grassmann (Peckhaus 1997: 233–296). In 1873, Schröder learned of Boole's and De Morgan's work on logic. T... | Ernst Schröder (mathematician) | 0.785034 |
80,227 | In fact, certain well-known chaotic systems, such as the Lorenz attractor and the Rössler map, are conventionally described as a system of three first-order differential equations that can combine into a single (although rather complicated) jerk equation. Another example of a jerk equation with nonlinearity in the magn... | Chaotic behavior | 0.785033 |
80,228 | , x ¨ , x ˙ , x ) = 0 {\displaystyle J\left({\overset {...}{x}},{\ddot {x}},{\dot {x}},x\right)=0} are sometimes called jerk equations. It has been shown that a jerk equation, which is equivalent to a system of three first order, ordinary, non-linear differential equations, is in a certain sense the minimal setting for... | Chaotic behavior | 0.785033 |
80,229 | In physics, jerk is the third derivative of position, with respect to time. As such, differential equations of the form J ( x . . . | Chaotic behavior | 0.785033 |
80,230 | Robotics is another area that has recently benefited from chaos theory. Instead of robots acting in a trial-and-error type of refinement to interact with their environment, chaos theory has been used to build a predictive model. Chaotic dynamics have been exhibited by passive walking biped robots. | Chaotic behavior | 0.785033 |
80,231 | Micro process engineering is the science of conducting chemical or physical processes (unit operations) inside small volumina, typically inside channels with diameters of less than 1 mm (microchannels) or other structures with sub-millimeter dimensions. These processes are usually carried out in continuous flow mode, a... | Microreaction technology | 0.785031 |
80,232 | A five-by-five is a performance in which a player accumulates a total of five in five statistical categories—points, rebounds, assists, steals, and blocks—in a single game. Statistics for steals and blocks were not kept in the NBA until the 1973–74 season, so all NBA five-by-fives are known only from that season onward... | Triple double | 0.785031 |
80,233 | In league history, the record for most career double-doubles is 968, held by Wilt Chamberlain. Special double-doubles are rare. One such achievement is sometimes called a 20–20, double double-double, or double-20, when a player accumulates 20 or more in two statistics in a game. | Triple double | 0.785031 |
80,234 | All facts based on data since the 1985–86 season: Greatest five-by-fives (most of each stat): Hakeem Olajuwon, on March 10, 1987, became the first in NBA history to record a six-by-five (at least 6 each of all five statistics: points, rebounds, assists, blocks, steals). It took nearly twenty years for the second offici... | Triple double | 0.785031 |
80,235 | Note that we have essentially proved Pr ( A ¯ ∣ ⋀ B ∈ S B ¯ ) ≥ 1 − x ( A ) {\displaystyle \Pr \left({\overline {A}}\mid \bigwedge _{B\in S}{\overline {B}}\right)\geq 1-x(A)} . To get the desired probability, we write it in terms of conditional probabilities applying Bayes' theorem repeatedly. Hence, Pr ( A 1 ¯ ∧ ⋯ ∧ A... | Lovász local lemma | 0.78503 |
80,236 | We have from Bayes' theorem Pr ( A ∣ ⋀ B ∈ S B ¯ ) = Pr ( A ∧ ⋀ B ∈ S 1 B ¯ ∣ ⋀ B ∈ S 2 B ¯ ) Pr ( ⋀ B ∈ S 1 B ¯ ∣ ⋀ B ∈ S 2 B ¯ ) . {\displaystyle \Pr \left(A\mid \bigwedge _{B\in S}{\overline {B}}\right)={\frac {\Pr \left(A\wedge \bigwedge _{B\in S_{1}}{\overline {B}}\mid \bigwedge _{B\in S_{2}}{\overline {B}}\right)... | Lovász local lemma | 0.78503 |
80,237 | Note that, as is often the case with probabilistic arguments, this theorem is nonconstructive and gives no method of determining an explicit element of the probability space in which no event occurs. However, algorithmic versions of the local lemma with stronger preconditions are also known (Beck 1991; Czumaj and Schei... | Lovász local lemma | 0.78503 |
80,238 | In probability theory, if a large number of events are all independent of one another and each has probability less than 1, then there is a positive (possibly small) probability that none of the events will occur. The Lovász local lemma allows one to relax the independence condition slightly: As long as the events are ... | Lovász local lemma | 0.78503 |
80,239 | γ a 2 = + 1 if 1 ≤ a ≤ p γ a 2 = − 1 if p + 1 ≤ a ≤ p + q γ a γ b = − γ b γ a if a ≠ b . {\displaystyle {\begin{matrix}\gamma _{a}^{2}&=&+1&{\mbox{if}}&1\leq a\leq p\\\gamma _{a}^{2}&=&-1&{\mbox{if}}&p+1\leq a\leq p+q\\\gamma _{a}\gamma _{b}&=&-\gamma _{b}\gamma _{a}&{\mbox{if}}&a\neq b.\ \\\end{matrix}}} Such a basis ... | Clifford module | 0.78503 |
80,240 | We will need to study anticommuting matrices (AB = −BA) because in Clifford algebras orthogonal vectors anticommute A ⋅ B = 1 2 ( A B + B A ) = 0. {\displaystyle A\cdot B={\frac {1}{2}}(AB+BA)=0.} For the real Clifford algebra R p , q {\displaystyle \mathbb {R} _{p,q}} , we need p + q mutually anticommuting matrices, o... | Clifford module | 0.78503 |
80,241 | Developed by Ettore Majorana, this Clifford module enables the construction of a Dirac-like equation without complex numbers, and its elements are called Majorana spinors. The four basis vectors are the three Pauli matrices and a fourth antihermitian matrix. The signature is (+++−). For the signatures (+−−−) and (−−−+)... | Clifford module | 0.78503 |
80,242 | In mathematics, a Clifford module is a representation of a Clifford algebra. In general a Clifford algebra C is a central simple algebra over some field extension L of the field K over which the quadratic form Q defining C is defined. The abstract theory of Clifford modules was founded by a paper of M. F. Atiyah, R. Bo... | Clifford module | 0.78503 |
80,243 | In organic chemistry, a carbyne is a general term for any compound whose structure consists of an electrically neutral carbon atom connected by a single covalent bond and has three non-bonded electrons. The carbon atom has either one or three unpaired electrons, depending on its excitation state; making it a radical. T... | Carbyne | 0.785029 |
80,244 | CP2K is a freely available (GPL) quantum chemistry and solid state physics program package, written in Fortran 2008, to perform atomistic simulations of solid state, liquid, molecular, periodic, material, crystal, and biological systems. It provides a general framework for different methods: density functional theory (... | CP2K | 0.785029 |
80,245 | Kühne, Thomas; Iannuzzi, Marcella; et al. (2020). "CP2K: An electronic structure and molecular dynamics software package - Quickstep: Efficient and accurate electronic structure calculations". Journal of Chemical Physics. | CP2K | 0.785029 |
80,246 | "All-electron ab-initio molecular dynamics". Physical Chemistry Chemical Physics. 2 (10): 2105–2112. | CP2K | 0.785029 |
80,247 | "The Gaussian and augmented-plane-wave density functional method for ab initio molecular dynamics simulations". Theoretical Chemistry Accounts: Theory, Computation, and Modeling. 103 (2): 124–140. | CP2K | 0.785029 |
80,248 | Lippert, Gerald; Hutter, Jürg; Parrinello, Michele (1997). "A hybrid Gaussian and plane wave density functional scheme". Molecular Physics. 92 (3): 477–487. | CP2K | 0.785029 |
80,249 | Every commutative unital ring R has a so-called real closure rcl(R) and this is unique up to a unique ring homomorphism over R. This means that rcl(R) is a real closed ring and there is a (not necessarily injective) ring homomorphism r: R → r c l ( R ) {\displaystyle r:R\to rcl(R)} such that for every ring homomorphism... | Real closed ring | 0.785029 |
80,250 | By definition, H(A) consists of all elements f in A with the property −N ≤ f ≤ N for some natural number N. Applied to the examples above, this means that the rings of bounded (semi-algebraic/definable) continuous functions are all real closed. The support map from the real spectrum of a real closed ring to its Zariski... | Real closed ring | 0.785029 |
80,251 | The category RCR of real closed rings which has real closed rings as objects and ring homomorphisms as morphisms has the following properties:Arbitrary products, direct limits and inverse limits (in the category of commutative unital rings) of real closed rings are again real closed. The fibre sum of two real closed ri... | Real closed ring | 0.785029 |
80,252 | In mathematics, a real closed ring (RCR) is a commutative ring A that is a subring of a product of real closed fields, which is closed under continuous semi-algebraic functions defined over the integers. | Real closed ring | 0.785029 |
80,253 | There are many different characterizations of real closed fields. For example, in terms of maximality (with respect to algebraic extensions): a real closed field is a maximally orderable field; or, a real closed field (together with its unique ordering) is a maximally ordered field. Another characterization says that t... | Real closed ring | 0.785029 |
80,254 | A real closed ring is a reduced, commutative unital ring A which has the following properties: The set of squares of A is the set of nonnegative elements of a partial order ≤ on A and (A,≤) is an f-ring. Convexity condition: For all a, b in A, if 0 ≤ a ≤ b then b | a2. For every prime ideal p of A, the residue class ri... | Real closed ring | 0.785029 |
80,255 | the ring A of all continuous semi-algebraic functions on a semi-algebraic set of a real closed field (with values in that field). Also, the subring of all bounded (in any sense) functions in A is real closed. (generalizing the previous example) the ring of all (bounded) continuous definable functions on a definable set... | Real closed ring | 0.785029 |
80,256 | A chamfer or is a transitional edge between two faces of an object. Sometimes defined as a form of bevel, it is often created at a 45° angle between two adjoining right-angled faces. Chamfers are frequently used in machining, carpentry, furniture, concrete formwork, mirrors, and to facilitate assembly of many mechanica... | Chamfered edge | 0.785027 |
80,257 | Sankey diagrams are often used in fields of science, especially physics. They are used to represent energy inputs, useful output, and wasted output. | Sankey diagram | 0.785026 |
80,258 | In the resonator mode, the plasma density does not exceed the critical density. A standing electromagnetic wave, which is confined by a resonator cavity, penetrates the plasma and sustains it in the regions of highest field intensity. The geometry of this region determines the spatial distribution of the plasma. Plasma... | Resonator mode | 0.785026 |
80,259 | An event is something that happens at a point in space at an instant of time, or more formally a point in spacetime. The transformations connect the space and time coordinates of an event as measured by an observer in each frame.They supersede the Galilean transformation of Newtonian physics, which assumes an absolute ... | Lorentz transform | 0.785025 |
80,260 | In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations a... | Lorentz transform | 0.785025 |
80,261 | In the case of the Lorentz group, the exponential map is just the matrix exponential. Globally, the exponential map is not one-to-one, but in the case of the Lorentz group, it is surjective (onto). Hence any group element in the connected component of the identity can be expressed as an exponential of an element of the... | Lorentz transform | 0.785025 |
80,262 | A group parameter is a component of a coordinate vector representing an arbitrary element of the Lie algebra with respect to some basis. A basis, then, is a set of generators being a basis of the Lie algebra in the usual vector space sense. The exponential map from the Lie algebra to the Lie group, provides a one-to-on... | Lorentz transform | 0.785025 |
80,263 | In summary, a Lie algebra is defined as a vector space V over a field of numbers, and with a binary operation (called a Lie bracket in this context) on the elements of the vector space, satisfying the axioms of bilinearity, alternatization, and the Jacobi identity. Here the operation is the commutator which satisfies... | Lorentz transform | 0.785025 |
80,264 | Lorentz generators can be added together, or multiplied by real numbers, to obtain more Lorentz generators. In other words, the set of all Lorentz generators together with the operations of ordinary matrix addition and multiplication of a matrix by a number, forms a vector space over the real numbers. The generators Jx... | Lorentz transform | 0.785025 |
80,265 | Lactotransferrin has been used in the synthesis of fluorescent gold quantum clusters, which has potential applications in nanotechnology. | Lactoferrin | 0.785023 |
80,266 | Let g ≅ TeG be the tangent space of a Lie group G at the identity (its Lie algebra). G acts on itself by left translation L: G × G → G {\displaystyle L:G\times G\to G} such that for a given g ∈ G we have L g: G → G where L g ( h ) = g h , {\displaystyle L_{g}:G\to G\quad {\mbox{where}}\quad L_{g}(h)=gh,} and this induc... | Maurer–Cartan equation | 0.785022 |
80,267 | The Baltimore Underground Science Space (BUGSS) is a non-profit synthetic biology and biotechnology makerspace laboratory for science enthusiasts, hobbyists, and professionals to practice, share and learn about the biological sciences. BUGSS is closely aligned with do-it-yourself biology and the Maryland science commun... | Baltimore Underground Science Space | 0.785022 |
80,268 | This reflects the fact that in many applications only such lattices are considered. One then usually is looking for the smallest set that has the property of being a fixed point of the function f. Abstract interpretation makes ample use of the Knaster–Tarski theorem and the formulas giving the least and greatest fixpoi... | Knaster-Tarski theorem | 0.785022 |
80,269 | More generally, if f is monotonic, then the least fixpoint of f is the stationary limit of f α(0), taking α over the ordinals, where f α is defined by transfinite induction: f α+1 = f (f α) and f γ for a limit ordinal γ is the least upper bound of the f β for all β ordinals less than γ. The dual theorem holds for the g... | Knaster-Tarski theorem | 0.785022 |
80,270 | Since complete lattices cannot be empty (they must contain a supremum and infimum of the empty set), the theorem in particular guarantees the existence of at least one fixed point of f, and even the existence of a least fixed point (or greatest fixed point). In many practical cases, this is the most important implicati... | Knaster-Tarski theorem | 0.785022 |
80,271 | In mathematics, a path in a topological space X {\displaystyle X} is a continuous function from the closed unit interval {\displaystyle } into X . {\displaystyle X.} Paths play an important role in the fields of topology and mathematical analysis. For example, a topological space for which there exists a path connecti... | Arc (topology) | 0.785022 |
80,272 | This fact may be used to prove minimization results for continuous convex functionals, in the same way that the Bolzano–Weierstrass theorem is used for continuous functions on Rd. Among several variants, one simple statement is as follows: If f: H → R is a convex continuous function such that f(x) tends to +∞ when ‖x‖ ... | Complex Hilbert space | 0.78502 |
80,273 | In a Hilbert space H, a sequence {xn} is weakly convergent to a vector x ∈ H when for every v ∈ H. For example, any orthonormal sequence {fn} converges weakly to 0, as a consequence of Bessel's inequality. Every weakly convergent sequence {xn} is bounded, by the uniform boundedness principle. Conversely, every bounded ... | Complex Hilbert space | 0.78502 |
80,274 | In the 1940s, Israel Gelfand, Mark Naimark and Irving Segal gave a definition of a kind of operator algebras called C*-algebras that on the one hand made no reference to an underlying Hilbert space, and on the other extrapolated many of the useful features of the operator algebras that had previously been studied. The ... | Complex Hilbert space | 0.78502 |
80,275 | On the other hand, in the early 1930s it became clear that classical mechanics can be described in terms of Hilbert space (Koopman–von Neumann classical mechanics) and that certain properties of classical dynamical systems can be analyzed using Hilbert space techniques in the framework of ergodic theory.The algebra of ... | Complex Hilbert space | 0.78502 |
80,276 | The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book on quantum mechanics and the theory of groups.The significance of the concept of a Hilbert space was underlined with the realization that it offers one of the best mathematical formulations of quantum mechanics. In short, the s... | Complex Hilbert space | 0.78502 |
80,277 | John von Neumann coined the term abstract Hilbert space in his work on unbounded Hermitian operators. Although other mathematicians such as Hermann Weyl and Norbert Wiener had already studied particular Hilbert spaces in great detail, often from a physically motivated point of view, von Neumann gave the first complete ... | Complex Hilbert space | 0.78502 |
80,278 | In 1907, Frigyes Riesz and Ernst Sigismund Fischer independently proved that the space L2 of square Lebesgue-integrable functions is a complete metric space. As a consequence of the interplay between geometry and completeness, the 19th century results of Joseph Fourier, Friedrich Bessel and Marc-Antoine Parseval on tri... | Complex Hilbert space | 0.78502 |
80,279 | Prior to the development of Hilbert spaces, other generalizations of Euclidean spaces were known to mathematicians and physicists. In particular, the idea of an abstract linear space (vector space) had gained some traction towards the end of the 19th century: this is a space whose elements can be added together and mul... | Complex Hilbert space | 0.78502 |
80,280 | If the x k {\displaystyle x_{k}} are random variables, this reproduces the usual definition of a (discrete) martingale: the expectation of x n + 1 {\displaystyle x_{n+1}} , conditioned on x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} , is equal to x n {\displaystyle x_{n}} . Hilbert spaces are also used throughout t... | Complex Hilbert space | 0.78502 |
80,281 | As Stapleton (1995) puts it, "the analysis of variance may be viewed as the decomposition of the squared length of a vector into the sum of the squared lengths of several vectors, using the Pythagorean Theorem." The theory of martingales can be formulated in Hilbert spaces. A martingale in a Hilbert space is a sequence... | Complex Hilbert space | 0.78502 |
80,282 | In particular, if two random variables X {\displaystyle X} and Y {\displaystyle Y} (in L 2 ( Ω , P ) {\displaystyle L^{2}(\Omega ,P)} ) are independent, then the centered random variables X − E ( X ) {\displaystyle X-E(X)} and Y − E ( Y ) {\displaystyle Y-E(Y)} are orthogonal. (This means that the two variables have ze... | Complex Hilbert space | 0.78502 |
80,283 | If F ≤ B {\displaystyle {\mathcal {F}}\leq {\mathcal {B}}} is a sigma subalgebra of B {\displaystyle {\mathcal {B}}} , then the conditional expectation E {\displaystyle E} is the orthogonal projection of X {\displaystyle X} onto the subspace of L 2 ( Ω , P ) {\displaystyle L^{2}(\Omega ,P)} consisting of the F {\displ... | Complex Hilbert space | 0.78502 |
80,284 | In the Hilbert space view, this is the orthogonal projection of X {\displaystyle X} onto the kernel of the expectation operator, which a continuous linear functional on the Hilbert space (in fact, the inner product with the constant random variable 1), and so this kernel is a closed subspace. The conditional expectatio... | Complex Hilbert space | 0.78502 |
80,285 | In probability theory, Hilbert spaces also have diverse applications. Here a fundamental Hilbert space is the space of random variables on a given probability space, having class L 2 {\displaystyle L^{2}} (finite first and second moments). A common operation in statistics is that of centering a random variable by subtr... | Complex Hilbert space | 0.78502 |
80,286 | The net effect of the upward and downward movements creates a decrease in bottom hole pressure. This decrease in pressure is referred to as Swab pressure. Both surge and swab pressures are affected by: Velocity of the pipe, or tripping speed Fluid density Fluid viscosity Fluid gel strength Well bore geometry (annular c... | Well control | 0.78502 |
80,287 | Tripping pipe into and out of a well is another common operation during completions and workovers. Unfortunately, statistics indicate that most kicks occur during trips. Therefore, understanding the basic concepts of tripping is a major concern in completion/workover operations. | Well control | 0.78502 |
80,288 | As mentioned earlier, there are several computer and calculator programs that can estimate surge and swab pressures. Swabbing is detected by closely monitoring hole fill-up volumes during trips. For example, if three barrels of steel (tubing) are removed from the well and it takes only two barrels of fluid to fill the ... | Well control | 0.78502 |
80,289 | Fracture pressure can be expressed as a gradient (psi/ft), a fluid density equivalent (ppg), or by calculated total pressure at the formation (psi). Fracture gradients normally increase with depth due to increasing overburden pressure. Deep, highly compacted formations can require high fracture pressures to overcome th... | Well control | 0.78502 |
80,290 | Abnormal pore pressure is defined as any pore pressure that is greater than the hydrostatic pressure of the formation fluid occupying the pore space. It is sometimes called overpressure or geopressure. An abnormally pressured formation can often be predicted using well history, surface geology, downhole logs or geophys... | Well control | 0.78502 |
80,291 | Let G denote the subcollection of F consisting of all balls from the cN disjoint families A1,...,AcN. The less precise following statement is clearly true: every point x ∈ RN belongs to at most cN different balls from the subcollection G, and G remains a cover for E (every point y ∈ E belongs to at least one ball from ... | Besicovitch covering theorem | 0.785019 |
80,292 | In mathematical analysis, a Besicovitch cover, named after Abram Samoilovitch Besicovitch, is an open cover of a subset E of the Euclidean space RN by balls such that each point of E is the center of some ball in the cover. The Besicovitch covering theorem asserts that there exists a constant cN depending only on the d... | Besicovitch covering theorem | 0.785019 |
80,293 | k-mer frequency and spectrum variation is heavily used in metagenomics for both analysis and binning. In binning, the challenge is to separate sequencing reads into "bins" of reads for each organism (or operational taxonomic unit), which will then be assembled. TETRA is a notable tool that takes metagenomic samples and... | K-mer | 0.785018 |
80,294 | Because GC bonds are more thermally stable, sequences with higher GC content exhibit a higher Tm. In 1987, the Ad Hoc Committee on Reconciliation of Approaches to Bacterial Systematics proposed the use of ΔTm as factor in determining species boundaries as part of the phylogenetic species concept, though this proposal d... | K-mer | 0.785018 |
80,295 | In bioinformatics, k-mers are substrings of length k {\displaystyle k} contained within a biological sequence. Primarily used within the context of computational genomics and sequence analysis, in which k-mers are composed of nucleotides (i.e. A, T, G, and C), k-mers are capitalized upon to assemble DNA sequences, impr... | K-mer | 0.785018 |
80,296 | Two later articles help explain the actual mechanism underlying codon-pair deoptimization: codon-pair bias is the result of dinucleotide bias. By studying viruses and their hosts, both sets of authors were able to conclude that the molecular mechanism that results in the attenuation of viruses is an increase in dinucle... | K-mer | 0.785018 |
80,297 | This approach has also been used effectively to create an influenza vaccine as well a vaccine for Marek's disease herpesvirus (MDV). Notably, the codon-pair bias manipulation employed to attenuate MDV did not effectively reduce the oncogenicity of the virus, highlighting a potential weakness in the biotechnology applic... | K-mer | 0.785018 |
80,298 | Similarly, codon pair optimization, a combination of dinucelotide and codon optimization, has also been successfully used to increase expression.The most studied application of k-mers for decreasing translational efficiency is codon-pair manipulation for attenuating viruses in order to create vaccines. Researchers were... | K-mer | 0.785018 |
80,299 | The history of string theory spans several decades of intense research including two superstring revolutions. Through the combined efforts of many researchers, string theory has developed into a broad and varied subject with connections to quantum gravity, particle and condensed matter physics, cosmology, and pure math... | Second Superstring Revolution | 0.785018 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.