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31,500 | The analog of the field F in such theories is the curvature form of the connection, which is represented in a gauge by a Lie algebra-valued one-form A. The Yang–Mills field F is then defined by F = d A + A ∧ A . {\displaystyle \mathbf {F} =d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} .} In the abelian case, such as ele... | Differential form | 0.80441 |
31,501 | Similar considerations describe the geometry of gauge theories in general. The 2-form ⋆ F {\displaystyle {\star }\mathbf {F} } , which is dual to the Faraday form, is also called Maxwell 2-form. Electromagnetism is an example of a U(1) gauge theory. | Differential form | 0.80441 |
31,502 | tensor components and the above-mentioned forms have different physical dimensions. Moreover, by decision of an international commission of the International Union of Pure and Applied Physics, the magnetic polarization vector has been called J → {\displaystyle {\vec {J}}} for several decades, and by some publishers J; ... | Differential form | 0.80441 |
31,503 | Differential forms are part of the field of differential geometry, influenced by linear algebra. Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to Élie Cartan with reference to his 1899 paper. Some aspects of the exterior al... | Differential form | 0.80441 |
31,504 | These conventions correspond to interpreting the integrand as a differential form, integrated over a chain. In measure theory, by contrast, one interprets the integrand as a function f with respect to a measure μ and integrates over a subset A, without any notion of orientation; one writes ∫ A f d μ = ∫ f d μ {\textst... | Differential form | 0.80441 |
31,505 | In terms of its chemical reactivity, however, singlet oxygen is far more reactive toward organic compounds. It is responsible for the photodegradation of many materials but can be put to constructive use in preparative organic chemistry and photodynamic therapy. | Dioxygen molecule | 0.804408 |
31,506 | In photosynthesis, singlet oxygen can be produced from the light-harvesting chlorophyll molecules. One of the roles of carotenoids in photosynthetic systems is to prevent damage caused by produced singlet oxygen by either removing excess light energy from chlorophyll molecules or quenching the singlet oxygen molecules ... | Dioxygen molecule | 0.804408 |
31,507 | Neurocomputing is a peer-reviewed scientific journal covering research on artificial intelligence, machine learning, and neural computation. It was established in 1989 and is published by Elsevier. The editor-in-chief is Zidong Wang (Brunel University London). Independent scientometric studies noted that despite being ... | Neurocomputing (journal) | 0.804403 |
31,508 | Intelligent design proponents have also occasionally appealed to broader teleological arguments outside of biology, most notably an argument based on the fine-tuning of universal constants that make matter and life possible and that are argued not to be solely attributable to chance. These include the values of fundame... | Evolutionary informatics | 0.804399 |
31,509 | The term was substituted into drafts of the book, directly replacing references to creation science and creationism, after the 1987 Supreme Court's Edwards v. Aguillard decision barred the teaching of creation science in public schools on constitutional grounds. From the mid-1990s, the intelligent design movement (IDM)... | Evolutionary informatics | 0.804399 |
31,510 | Intelligent design (ID) is a pseudoscientific argument for the existence of God, presented by its proponents as "an evidence-based scientific theory about life's origins". Proponents claim that "certain features of the universe and of living things are best explained by an intelligent cause, not an undirected process s... | Evolutionary informatics | 0.804399 |
31,511 | The first to be discovered was Strassen's algorithm, devised by Volker Strassen in 1969 and often referred to as "fast matrix multiplication". The optimal number of field operations needed to multiply two square n × n matrices up to constant factors is still unknown. This is a major open question in theoretical compute... | Coppersmith–Winograd algorithm | 0.804398 |
31,512 | The conceptual idea of these algorithms is similar to Strassen's algorithm: a way is devised for multiplying two k × k-matrices with fewer than k3 multiplications, and this technique is applied recursively. The laser method has limitations to its power, and cannot be used to show that ω < 2.3725. Duan, Wu and Zhou iden... | Coppersmith–Winograd algorithm | 0.804398 |
31,513 | The matrix multiplication exponent, usually denoted ω, is the smallest real number for which any two n × n {\displaystyle n\times n} matrices over a field can be multiplied together using n ω + o ( 1 ) {\displaystyle n^{\omega +o(1)}} field operations. This notation is commonly used in algorithms research, so that algo... | Coppersmith–Winograd algorithm | 0.804398 |
31,514 | In theoretical computer science, the computational complexity of matrix multiplication dictates how quickly the operation of matrix multiplication can be performed. Matrix multiplication algorithms are a central subroutine in theoretical and numerical algorithms for numerical linear algebra and optimization, so finding... | Coppersmith–Winograd algorithm | 0.804398 |
31,515 | There is current interest in the search of quantum spin ices, materials in which the laws of quantum mechanics now become needed to describe the behavior of the magnetic moments. Magnetic ions other than dysprosium (Dy) and holmium (Ho) are required to generate a quantum spin ice, with praseodymium (Pr), terbium (Tb) a... | Spin ice | 0.804396 |
31,516 | Most importantly, it is the long-range magnetostatic dipole–dipole interaction, and not the nearest-neighbor exchange, that causes the frustration and the consequential two-in, two-out rule that leads to the spin ice phenomenology.For a tetrahedron in a two-in, two-out state, the magnetization field is divergent-free; ... | Spin ice | 0.804396 |
31,517 | Interferometry inherently depends on the wave nature of the object. As pointed out by de Broglie in his PhD thesis, particles, including electrons can behave like waves (the so-called wave–particle duality, now explained in the general framework of quantum mechanics). One of the first interferometry experiments with el... | Electron interferometer | 0.804396 |
31,518 | In number theory, an equidigital number is a natural number in a given number base that has the same number of digits as the number of digits in its prime factorization in the given number base, including exponents but excluding exponents equal to 1. For example, in base 10, 1, 2, 3, 5, 7, and 10 (2 × 5) are equidigita... | Equidigital number | 0.804395 |
31,519 | Phosphoenolpyruvate (2-phosphoenolpyruvate, PEP) is the ester derived from the enol of pyruvate and phosphate. It exists as an anion. PEP is an important intermediate in biochemistry. It has the highest-energy phosphate bond found (−61.9 kJ/mol) in organisms, and is involved in glycolysis and gluconeogenesis. In plants... | Phosphoenolpyruvic acid | 0.804394 |
31,520 | More advanced methods use the optimality criterion of maximum likelihood, often within a Bayesian framework, and apply an explicit model of evolution to phylogenetic tree estimation. Identifying the optimal tree using many of these techniques is NP-hard, so heuristic search and optimization methods are used in combinat... | Evolutionary tree | 0.804393 |
31,521 | Phylogenetic trees composed with a nontrivial number of input sequences are constructed using computational phylogenetics methods. Distance-matrix methods such as neighbor-joining or UPGMA, which calculate genetic distance from multiple sequence alignments, are simplest to implement, but do not invoke an evolutionary m... | Evolutionary tree | 0.804393 |
31,522 | This view is supported by research in molecular genetics. Thus, James (2009) concluded that "it is now widely accepted that... radiality and dorsiventrality are but extremes of a continuous spectrum. In fact, it is simply the timing of the KNOX gene expression". | Evolutionary history of plants | 0.804393 |
31,523 | Examples include the descent rate of raindrops, convective clouds, simplifications of the atmospheric radiative transfer on the basis of atmospheric radiative transfer codes, and cloud microphysics. Radiative parameterizations are important to both atmospheric and oceanic modeling alike. Atmospheric emissions from diff... | Parametrization (climate) | 0.804392 |
31,524 | locally finite if closures of finite sets are finite.Triviality, modularity and local modularity pass to the associated geometry and are preserved under localization. If S {\displaystyle S} is a locally modular homogeneous pregeometry and a ∈ S ∖ cl ( ∅ ) {\displaystyle a\in S\setminus {\text{cl}}(\varnothing )} then t... | Pregeometry (model theory) | 0.804391 |
31,525 | The pregeometry ( S , cl ) {\displaystyle (S,{\text{cl}})} is said to be: trivial (or degenerate) if cl ( X ) = ⋃ { cl ( a ) ∣ a ∈ X } {\displaystyle {\text{cl}}(X)=\bigcup \{{\text{cl}}(a)\mid a\in X\}} for all non-empty X ⊆ S {\displaystyle X\subseteq S} . modular if any two closed finite dimensional sets X , Y ⊆ S {... | Pregeometry (model theory) | 0.804391 |
31,526 | The linear algebra concepts of independent set, generating set, basis and dimension can all be expressed using the cl {\displaystyle {\text{cl}}} -operator alone. A pregeometry is an abstraction of this situation: we start with an arbitrary set S {\displaystyle S} and an arbitrary operator cl {\displaystyle {\text{cl}}... | Pregeometry (model theory) | 0.804391 |
31,527 | If S {\displaystyle S} is any set we may define cl ( A ) = A {\displaystyle {\text{cl}}(A)=A} for all A ⊆ S {\displaystyle A\subseteq S} . This pregeometry is a trivial, homogeneous, locally finite geometry. | Pregeometry (model theory) | 0.804391 |
31,528 | Since a pregeometry satisfies the Steinitz exchange property all bases are of the same cardinality, hence we may define the dimension of A {\displaystyle A} over D {\displaystyle D} , written as dim D A {\displaystyle {\text{dim}}_{D}A} , as the cardinality of any basis of A {\displaystyle A} over D {\displaystyle D} .... | Pregeometry (model theory) | 0.804391 |
31,529 | Given a pregeometry ( S , cl ) {\displaystyle (S,{\text{cl}})} its associated geometry (sometimes referred in the literature as the canonical geometry) is the geometry ( S ′ , cl ′ ) {\displaystyle (S',{\text{cl}}')} where S ′ = { cl ( a ) ∣ a ∈ S ∖ cl ( ∅ ) } {\displaystyle S'=\{{\text{cl}}(a)\mid a\in S\setminus {\te... | Pregeometry (model theory) | 0.804391 |
31,530 | Given a countable first-order language L and an L-structure M, any definable subset D of M that is strongly minimal gives rise to a pregeometry on the set D. The closure operator here is given by the algebraic closure in the model-theoretic sense. A model of a strongly minimal theory is determined up to isomorphism by ... | Pregeometry (model theory) | 0.804391 |
31,531 | An automorphism of a pregeometry ( S , cl ) {\displaystyle (S,{\text{cl}})} is a bijection σ: S → S {\displaystyle \sigma :S\to S} such that σ ( cl ( X ) ) = cl ( σ ( X ) ) {\displaystyle \sigma ({\text{cl}}(X))={\text{cl}}(\sigma (X))} for any X ⊆ S {\displaystyle X\subseteq S} . A pregeometry S {\displaystyle S} is s... | Pregeometry (model theory) | 0.804391 |
31,532 | Let V {\displaystyle V} be a κ {\displaystyle \kappa } -dimensional affine space over a field F {\displaystyle F} . Given a set define its closure to be its affine hull (i.e. the smallest affine subspace containing it). This forms a homogeneous ( κ + 1 ) {\displaystyle (\kappa +1)} -dimensional geometry. An affine spac... | Pregeometry (model theory) | 0.804391 |
31,533 | The dimension of A {\displaystyle A} coincides with the transcendence degree trdeg ( K ( A ) / K ) {\displaystyle {\text{trdeg}}(K(A)/K)} . In model theory, the case of L {\displaystyle L} being algebraically closed and K {\displaystyle K} its prime field is especially important. While vector spaces are modular and aff... | Pregeometry (model theory) | 0.804391 |
31,534 | Let L / K {\displaystyle L/K} be a field extension. The set L {\displaystyle L} becomes a pregeometry if we define cl ( A ) = { x ∈ L: x is algebraic over K ( A ) } {\displaystyle {\text{cl}}(A)=\{x\in L:x{\text{ is algebraic over }}K(A)\}} for A ⊆ L {\displaystyle A\subseteq L} . The set A {\displaystyle A} is indepen... | Pregeometry (model theory) | 0.804391 |
31,535 | Vector spaces are considered to be the prototypical example of modularity. V {\displaystyle V} is locally finite if and only if F {\displaystyle F} is finite. V {\displaystyle V} is not a geometry, as the closure of any nontrivial vector is a subspace of size at least 2 {\displaystyle 2} . The associated geometry of a ... | Pregeometry (model theory) | 0.804391 |
31,536 | Let F {\displaystyle F} be a field (a division ring actually suffices) and let V {\displaystyle V} be a vector space over F {\displaystyle F} . Then V {\displaystyle V} is a pregeometry where closures of sets are defined to be their span. The closed sets are the linear subspaces of V {\displaystyle V} and the notion of... | Pregeometry (model theory) | 0.804391 |
31,537 | It turns out that many fundamental concepts of linear algebra – closure, independence, subspace, basis, dimension – are available in the general framework of pregeometries. In the branch of mathematical logic called model theory, infinite finitary matroids, there called "pregeometries" (and "geometries" if they are sim... | Pregeometry (model theory) | 0.804391 |
31,538 | Pregeometry, and in full combinatorial pregeometry, are essentially synonyms for "matroid". They were introduced by Gian-Carlo Rota with the intention of providing a less "ineffably cacophonous" alternative term. Also, the term combinatorial geometry, sometimes abbreviated to geometry, was intended to replace "simple m... | Pregeometry (model theory) | 0.804391 |
31,539 | {\displaystyle \scriptstyle -\left(2+{\frac {3}{4}}\right).} Any such sum of a whole plus a part can be converted to an improper fraction by applying the rules of adding unlike quantities. This tradition is, formally, in conflict with the notation in algebra where adjacent symbols, without an explicit infix operator, d... | Mixed numeral | 0.804391 |
31,540 | A number is a rational number precisely when it can be written in that form (i.e., as a common fraction). However, the word fraction can also be used to describe mathematical expressions that are not rational numbers. Examples of these usages include algebraic fractions (quotients of algebraic expressions), and express... | Mixed numeral | 0.804391 |
31,541 | A recursively enumerable language is a formal language for which there exists a Turing machine (or other computable function) that will halt and accept when presented with any string in the language as input but may either halt and reject or loop forever when presented with a string not in the language. Contrast this t... | Recognizable language | 0.804391 |
31,542 | In mathematics, logic and computer science, a formal language is called recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i.e., if there exi... | Recognizable language | 0.804391 |
31,543 | The theorem refines Chebyshev's inequality by including the factor of 4/9, made possible by the condition that the distribution be unimodal. It is common, in the construction of control charts and other statistical heuristics, to set λ = 3, corresponding to an upper probability bound of 4/81= 0.04938..., and to constru... | Vysochanskij–Petunin inequality | 0.804391 |
31,544 | In probability theory, the Vysochanskij–Petunin inequality gives a lower bound for the probability that a random variable with finite variance lies within a certain number of standard deviations of the variable's mean, or equivalently an upper bound for the probability that it lies further away. The sole restrictions o... | Vysochanskij–Petunin inequality | 0.804391 |
31,545 | Of the fifty-seven stars included in the new almanac, these two had no traditional names. The RAF insisted that all of the stars must have names, so new names were invented for them. These names have been approved by the IAU WGSN.The book Star Names: Their Lore and Meaning by R. H. Allen (1899) has had effects on star ... | Stellar designation | 0.804391 |
31,546 | Organoxenon chemistry is the study of the properties of organoxenon compounds, which contain carbon to xenon chemical bonds. The first organoxenon compounds were divalent, such as (C6F5)2Xe. The first tetravalent organoxenon compound, , was synthesized in 2004. | Organoxenon compound | 0.804388 |
31,547 | In statistics, the uncertainty coefficient, also called proficiency, entropy coefficient or Theil's U, is a measure of nominal association. It was first introduced by Henri Theil and is based on the concept of information entropy. | Uncertainty coefficient | 0.804387 |
31,548 | As the disturbance propagates towards the edge of the pond, however, the waves will smooth out and will appear to be nearly circular. The van Cittert–Zernike theorem has important implications for radio astronomy. With the exception of pulsars and masers, all astronomical sources are spatially incoherent. Nevertheless,... | Van Cittert–Zernike theorem | 0.804386 |
31,549 | The van Cittert–Zernike theorem, named after physicists Pieter Hendrik van Cittert and Frits Zernike, is a formula in coherence theory that states that under certain conditions the Fourier transform of the intensity distribution function of a distant, incoherent source is equal to its complex visibility. This implies t... | Van Cittert–Zernike theorem | 0.804386 |
31,550 | Such a process would require a sufficient number of samples to satisfy the Nyquist sampling theorem; this is many more observations than are needed to approximately reconstruct the brightness distribution of the source. Astronomers therefore take advantage of physical constraints on the brightness distribution of astro... | Van Cittert–Zernike theorem | 0.804386 |
31,551 | The van Cittert–Zernike theorem is crucial to the measurement of the brightness distribution of a source. With two telescopes, a radio astronomer (or an infrared or submillimeter astronomer) can measure the correlation between the electric field at the two dishes due to some point from the source. By measuring this cor... | Van Cittert–Zernike theorem | 0.804386 |
31,552 | The van Cittert–Zernike theorem can be used to calculate the partial spatial coherence of radiation from a free-electron laser. | Van Cittert–Zernike theorem | 0.804386 |
31,553 | Suppose we have a situation identical to that considered when the van Cittert–Zernike theorem was derived, except that the medium is now heterogeneous. We therefore introduce the transmission function of the medium, K ( l , m , P , ν ) {\displaystyle K(l,m,P,\nu )} . Following a similar derivation as before, we find th... | Van Cittert–Zernike theorem | 0.804386 |
31,554 | The van Cittert–Zernike theorem assumes that the source is quasi-monochromatic. That is, if the source emits light over a range of frequencies, Δ ν {\displaystyle \Delta \nu } , with mean frequency ν {\displaystyle \nu } , then it should satisfy Δ ν ν ≲ 1 {\displaystyle {\frac {\Delta \nu }{\nu }}\lesssim 1} Moreover, ... | Van Cittert–Zernike theorem | 0.804386 |
31,555 | Consider two very distant parallel planes, both perpendicular to the line of sight, and let's call them source plane and observation plane; If Γ 12 ( u , v , 0 ) {\displaystyle \Gamma _{12}(u,v,0)} is the mutual coherence function between two points in the observation plane, then Γ 12 ( u , v , 0 ) = ∬ I ( l , m ) e − ... | Van Cittert–Zernike theorem | 0.804386 |
31,556 | The first peak in g(r) (the RDF) of the Eigen structure is similar to the equilibrium, standard RDF, only slightly more ordered, while the first peak of the Zundel structure is actually split into two peaks. The actual proton transfer (PT) event was then traced (after synchronizing all PT events so that t=0 is the actu... | Grotthus mechanism | 0.804385 |
31,557 | Infer.NET is a free and open source .NET software library for machine learning. It supports running Bayesian inference in graphical models and can also be used for probabilistic programming. | Infer.NET | 0.804381 |
31,558 | Infer.NET follows a model-based approach and is used to solve different kinds of machine learning problems including standard problems like classification, recommendation or clustering, customized solutions and domain-specific problems. The framework is used in various different domains such as bioinformatics, epidemio... | Infer.NET | 0.804381 |
31,559 | But numbers register the departure from theory with an authority and finesse that no qualitative technique can duplicate, and that departure is often enough to start a search (Kuhn, 1961, p. 180).In classical physics, the theory and definitions which underpin measurement are generally deterministic in nature. In contra... | Quantitative research | 0.804379 |
31,560 | Quantitative research is generally closely affiliated with ideas from 'the scientific method', which can include: The generation of models, theories and hypotheses The development of instruments and methods for measurement Experimental control and manipulation of variables Collection of empirical data Modeling and anal... | Quantitative research | 0.804379 |
31,561 | Introgression is an important source of genetic variation in natural populations and may contribute to adaptation and even adaptive radiation. It can occur across hybrid zones due to chance, selection or hybrid zone movement. There is evidence that introgression is a ubiquitous phenomenon in plants and animals, includi... | Introgressive hybridization | 0.804378 |
31,562 | More precisely, an orthonormal basis is a Hamel basis if and only if the Hilbert space is a finite-dimensional vector space. Completeness of an orthonormal system of vectors of a Hilbert space can be equivalently restated as: for every v ∈ H, if ⟨v, ek⟩ = 0 for all k ∈ B, then v = 0.This is related to the fact that the... | Hilbert Space | 0.804378 |
31,563 | The notion of an orthonormal basis from linear algebra generalizes over to the case of Hilbert spaces. In a Hilbert space H, an orthonormal basis is a family {ek}k ∈ B of elements of H satisfying the conditions: Orthogonality: Every two different elements of B are orthogonal: ⟨ek, ej⟩ = 0 for all k, j ∈ B with k ≠ j. N... | Hilbert Space | 0.804378 |
31,564 | A converse is also true in the sense that, for a non-negative operator A, there exists a unique non-negative square root B such that In a sense made precise by the spectral theorem, self-adjoint operators can usefully be thought of as operators that are "real". An element A of B(H) is called normal if A*A = AA*. Normal... | Hilbert Space | 0.804378 |
31,565 | This defines another bounded linear operator A*: H2 → H1, the adjoint of A. The adjoint satisfies A** = A. When the Riesz representation theorem is used to identify each Hilbert space with its continuous dual space, the adjoint of A can be shown to be identical to the transpose tA: H2* → H1* of A, which by definition s... | Hilbert Space | 0.804378 |
31,566 | The continuous linear operators A: H1 → H2 from a Hilbert space H1 to a second Hilbert space H2 are bounded in the sense that they map bounded sets to bounded sets. Conversely, if an operator is bounded, then it is continuous. The space of such bounded linear operators has a norm, the operator norm given by The sum and... | Hilbert Space | 0.804378 |
31,567 | Most spaces used in physics are separable, and since these are all isomorphic to each other, one often refers to any infinite-dimensional separable Hilbert space as "the Hilbert space" or just "Hilbert space". Even in quantum field theory, most of the Hilbert spaces are in fact separable, as stipulated by the Wightman ... | Hilbert Space | 0.804378 |
31,568 | In the mathematically rigorous formulation of quantum mechanics, developed by John von Neumann, the possible states (more precisely, the pure states) of a quantum mechanical system are represented by unit vectors (called state vectors) residing in a complex separable Hilbert space, known as the state space, well define... | Hilbert Space | 0.804378 |
31,569 | In this case, H is called the internal direct sum of the Vi. A direct sum (internal or external) is also equipped with a family of orthogonal projections Ei onto the ith direct summand Hi. These projections are bounded, self-adjoint, idempotent operators that satisfy the orthogonality condition The spectral theorem for... | Hilbert Space | 0.804378 |
31,570 | This subsection employs the Hilbert projection theorem. If C is a non-empty closed convex subset of a Hilbert space H and x a point in H, there exists a unique point y ∈ C that minimizes the distance between x and points in C, This is equivalent to saying that there is a point with minimal norm in the translated convex... | Hilbert Space | 0.804378 |
31,571 | More explicitly, suppose that the energy E is fixed, and let ΩE be the subset of the phase space consisting of all states of energy E (an energy surface), and let Tt denote the evolution operator on the phase space. The dynamical system is ergodic if every invariant measurable functions on ΩE is constant . An invariant... | Hilbert Space | 0.804378 |
31,572 | This can also be expressed by requiring that U be onto and ⟨Ux, Uy⟩ = ⟨x, y⟩ for all x, y ∈ H. The unitary operators form a group under composition, which is the isometry group of H. An element of B(H) is compact if it sends bounded sets to relatively compact sets. Equivalently, a bounded operator T is compact if, for ... | Hilbert Space | 0.804378 |
31,573 | The integrand is known as the Bergman kernel of D. This integral kernel satisfies a reproducing property A Bergman space is an example of a reproducing kernel Hilbert space, which is a Hilbert space of functions along with a kernel K(ζ, z) that verifies a reproducing property analogous to this one. The Hardy space H2(D... | Hilbert Space | 0.804378 |
31,574 | Another consequence of this inequality is that the linear functional that evaluates a function f at a point of D is actually continuous on L2, h(D). The Riesz representation theorem implies that the evaluation functional can be represented as an element of L2, h(D). Thus, for every z ∈ D, there is a function ηz ∈ L2, h... | Hilbert Space | 0.804378 |
31,575 | Spectral theory also underlies certain aspects of the Fourier transform of a function. Whereas Fourier analysis decomposes a function defined on a compact set into the discrete spectrum of the Laplacian (which corresponds to the vibrations of a violin string or drum), the Fourier transform of a function is the decompos... | Hilbert Space | 0.804378 |
31,576 | Many of the applications of Hilbert spaces exploit the fact that Hilbert spaces support generalizations of simple geometric concepts like projection and change of basis from their usual finite dimensional setting. In particular, the spectral theory of continuous self-adjoint linear operators on a Hilbert space generali... | Hilbert Space | 0.804378 |
31,577 | Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a linear subspace or a subspace (the analog of "dropping the altitude" of a triangle) plays a significant role in optimization problems and other aspects of the theory. An element of ... | Hilbert Space | 0.804378 |
31,578 | They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term Hilbert space for t... | Hilbert Space | 0.804378 |
31,579 | In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spac... | Hilbert Space | 0.804378 |
31,580 | It focuses in particular on microbiology and genetics; molecular and cell biology; metabolism and physiology; signal transduction and development; biotechnology; phytopathology; and environmental microbiology ,and ecology. The journal was originally published by Gustav Fischer Verlag. | Microbiological Research | 0.804376 |
31,581 | Microbiological Research is an academic journal in microbiology, published by Elsevier. It has an impact factor of 6.7 as of 2022. The journal was established in 1896. The journal publishes research on prokaryotic and eukaryotic microorganisms such as yeasts, fungi, bacteria, archaea, and protozoa, as well as on intera... | Microbiological Research | 0.804376 |
31,582 | Let F: R n → R n {\displaystyle F\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}} be a C1 vector field with a critical point p, i.e., F(p) = 0, and let J denote the Jacobian matrix of F at p. If the matrix J has no eigenvalues with zero real parts then p is called hyperbolic. Hyperbolic fixed points may also be called hype... | Hyperbolic equilibrium point | 0.804375 |
31,583 | As perpendicularity is the relation of conjugate diameters of a circle, so hyperbolic orthogonality is the relation of conjugate diameters of rectangular hyperbolas. The placement of tie rods reinforcing a square assembly of girders is guided by the relation of conjugate diameters in a book on analytic geometry.Conjuga... | Conjugate diameter | 0.804371 |
31,584 | The American Society for Pharmacology and Experimental Therapeutics (ASPET) is a scientific society founded in late 1908 by John Jacob Abel of Johns Hopkins University (also the founder of the American Society for Biochemistry and Molecular Biology), with the aim of promoting the growth of pharmacological research. Man... | Molecular Interventions | 0.804371 |
31,585 | Round-trip engineering is closely related to traditional software engineering disciplines: forward engineering (creating software from specifications), reverse engineering (creating specifications from existing software), and reengineering (understanding existing software and modifying it). Round-trip engineering is of... | Round-trip engineering | 0.80437 |
31,586 | Michie Smith was selected to be superintendent. Starting in 1895 there was a rapid transfer of work and equipment from the Madras Observatory to Kodaikanal and the observatory was founded on 1 April 1899. The first observations were commenced at Kodaikanal in 1901.Partial List of Assistant Directors John Evershed 1906–... | Kodaikanal Solar Observatory | 0.804365 |
31,587 | As early as 1881, Mr. Blanford, then Meteorological Reporter to the Government of India, recommended "the improvement of the work of solar observations in order to obtain accurate measures of the sun’s heating power at the earth’s surface and its periodic variations". In May 1882, the government astronomer at Madras, N... | Kodaikanal Solar Observatory | 0.804365 |
31,588 | Studies on the structure and dynamics of the equatorial ionosphere and its response to the solar and interplanetary variability are being carried out. Studies of the equatorial electrojet and of the structure and dynamics of the equatorial ionosphere and its response to solar and interplanetary variability are being ma... | Kodaikanal Solar Observatory | 0.804365 |
31,589 | The displays are mainly pictorial, with a few models, a live solar image and the Fraunhofer spectrum also presented. The library is one of the observatory's proud possessions. It has a collection of astronomical literature, which is of archival value. The library maintains a skeletal collection of current literature in... | Kodaikanal Solar Observatory | 0.804365 |
31,590 | A high frequency Doppler radar was built indigenously and made operational to study F-region Skywave dynamics. A lacour magnetometer and a Watson magnetometer were installed and have been used regularly at the observatory since the early 1900s. They also have a broadband seismograph, GPS receiver and magnetic variomete... | Kodaikanal Solar Observatory | 0.804365 |
31,591 | The Kodaikanal Solar Observatory is a solar observatory owned and operated by the Indian Institute of Astrophysics. It is on the southern tip of the Palani Hills 4 kilometres (2.5 mi) from Kodaikanal. The Evershed effect was first detected at this observatory in January 1909. | Kodaikanal Solar Observatory | 0.804365 |
31,592 | In the simply typed lambda calculus, types T are either atomic type constants or function types of form T → T {\displaystyle T\rightarrow T} . Such types are monomorphic. Typical examples are the types used in arithmetic values: 3: Number add 3 4: Number add: Number -> Number -> Number Contrary to this, the untyped lam... | Let-polymorphism | 0.804362 |
31,593 | Combinatorial maps are used as efficient data structures in image representation and processing, in geometrical modeling. This model is related to simplicial complexes and to combinatorial topology. A combinatorial map is a boundary representation model; it represents object by its boundaries. | Combinatorial maps | 0.804361 |
31,594 | Both Peirce and Wittgenstein introduced the analogy of colour theory in order to illustrate the shades of meanings of words. Primary categories, like primary colours, are analytical representing the furthest we can go in terms of analysis and abstraction and include Quantity, Motion and Quality. Secondary categories, l... | Theory of Categories | 0.804361 |
31,595 | In the twentieth century the primacy of the division between the subjective and the objective, or between mind and matter, was disputed by, among others, Bertrand Russell and Gilbert Ryle. Philosophy began to move away from the metaphysics of categorisation towards the linguistic problem of trying to differentiate betw... | Theory of Categories | 0.804361 |
31,596 | Mutations to NR5A1 gene encoding steroidogenic factor 1 (SF-1) have been found in a small subset of men with non-obstructive male factor infertility where the cause is unknown. Results of one study investigating a cohort of 315 men revealed changes within the hinge region of SF-1 and no rare allelic variants in fertile... | Secondary infertility | 0.80436 |
31,597 | , x n {\displaystyle x_{1},x_{2},...,x_{n}} f ( x ) {\displaystyle f(x)} is the objective function h i ( x ) {\displaystyle h_{i}(x)} are m 1 {\displaystyle m_{1}} equality constraints g j ( x ) {\displaystyle g_{j}(x)} are m 2 {\displaystyle m_{2}} inequality constraints X {\displaystyle X} is a set constraint that in... | Design Optimization | 0.804359 |
31,598 | The formal mathematical (standard form) statement of the design optimization problem is minimize f ( x ) s u b j e c t t o h i ( x ) = 0 , i = 1 , … , m 1 g j ( x ) ≤ 0 , j = 1 , … , m 2 and x ∈ X ⊆ R n {\displaystyle {\begin{aligned}&{\operatorname {minimize} }&&f(x)\\&\operatorname {subject\;to} &&h_{i}(x)=0,\quad i=... | Design Optimization | 0.804359 |
31,599 | Practical design optimization problems are typically solved numerically and many optimization software exist in academic and commercial forms. There are several domain-specific applications of design optimization posing their own specific challenges in formulating and solving the resulting problems; these include, shap... | Design Optimization | 0.804359 |
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