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15,300 | Two statements of the Pythagorean theorem (see figure above) A 2 + w 2 = b 2 {\displaystyle A^{2}+w^{2}=b^{2}} and B 2 + w 2 = a 2 {\displaystyle B^{2}+w^{2}=a^{2}} can be subtracted one from the other to eliminate w, and the result can be combined with 1 A + 1 B = 1 h {\displaystyle {\tfrac {1}{A}}+{\tfrac {1}{B}}={\t... | Crossed ladders problem | 0.814503 |
15,301 | Martin Gardner presents and discusses the problem in his book of mathematical puzzles published in 1979 and cites references to it as early as 1895. The crossed ladders problem may appear in various forms, with variations in name, using various lengths and heights, or requesting unusual solutions such as cases where al... | Crossed ladders problem | 0.814503 |
15,302 | The crossed ladders theorem was extended to crossed ladders within a triangle. In 2002, Harold Joseph Stengel (1947–2007), an American secondary school teacher of mathematics, proved the extended theorem.Let AC be the base of a triangle ABC. Let ladder (line) AD have its foot at A and intersect BC at D; likewise, let l... | Crossed ladders problem | 0.814503 |
15,303 | As the next example will show, present bases are a generalization of the bases of vector spaces. Then, the name "reference frame" can well replace "basis". Yet, contrary to the vector space case, a universal algebra might lack bases and, when it has them, their dimension sets might have different finite positive cardin... | Basis (universal algebra) | 0.8145 |
15,304 | This condition will define bases by the set L {\displaystyle L} of the I {\displaystyle I} -ary elementary functions of the algebra, which are certain functions ℓ {\displaystyle \ell } that take every m {\displaystyle m} as argument to get some algebra element as value ℓ ( m ) . {\displaystyle \ell (m).} In fact, they ... | Basis (universal algebra) | 0.8145 |
15,305 | A basis (or reference frame) of a (universal) algebra is a function b {\displaystyle b} that takes some algebra elements as values b ( i ) {\displaystyle b(i)} and satisfies either one of the following two equivalent conditions. Here, the set of all b ( i ) {\displaystyle b(i)} is called the basis set, whereas several ... | Basis (universal algebra) | 0.8145 |
15,306 | In universal algebra, a basis is a structure inside of some (universal) algebras, which are called free algebras. It generates all algebra elements from its own elements by the algebra operations in an independent manner. It also represents the endomorphisms of an algebra by certain indexings of algebra elements, which... | Basis (universal algebra) | 0.8145 |
15,307 | Still it is allowed, in case it serves some purpose. When the space is zero-dimensional, its ordered basis is empty. Then, being the empty function, it is a present basis. Yet, since this space only contains the null vector and its only endomorphism is the identity, any function b from any set I {\displaystyle I} (even... | Basis (universal algebra) | 0.8145 |
15,308 | In fact, though the functions ℓ {\displaystyle \ell } formally have an infinity of vectors in every argument, the linear combinations they evaluate never require infinitely many addenda c i m ( i ) {\displaystyle c_{i}m(i)} and each ℓ {\displaystyle \ell } determines a finite subset J of I {\displaystyle I} that contai... | Basis (universal algebra) | 0.8145 |
15,309 | (Also, several other generalizations of linear independence for universal algebras do not imply present independence.) The functions m for the inner condition correspond to the square arrays of field elements (namely, usual vector-space square matrices) that serve to build the endomorphisms of vector spaces (namely, li... | Basis (universal algebra) | 0.8145 |
15,310 | This other condition will define bases by the set E of the endomorphisms of the algebra, which are the homomorphisms from the algebra into itself, through its analytic representation ϱ {\displaystyle \varrho } by a basis. The latter is a function that takes every endomorphism e as argument to get a function m as value:... | Basis (universal algebra) | 0.8145 |
15,311 | (When an algebra operation has a single algebra element as argument, the value of such a composed function is the one that the operation takes from the value of a single previously computed I {\displaystyle I} -ary function as in composition. When it does not, such compositions require that many (or none for a nullary ... | Basis (universal algebra) | 0.8145 |
15,312 | Let I = { a , b , c , … } {\displaystyle I=\{{\mathsf {a,b,c,}}\ldots \}} be an "alphabet", namely a (usually finite) set of objects called "letters". Let W denote the corresponding set of words or "strings", which will be denoted as in strings, namely either by writing their letters in sequence or by ϵ {\displaystyle ... | Basis (universal algebra) | 0.8145 |
15,313 | In the universal algebra corresponding to a vector space with finite dimension the bases essentially are the ordered bases of this vector space. Yet, this will come after several details. When the vector space is finite-dimensional, for instance I = { 0 , 1 , … , n − 1 } {\displaystyle I=\{0,1,\ldots ,n-1\}} with n > 0... | Basis (universal algebra) | 0.8145 |
15,314 | In the mathematical field of graph theory, the triangle graph is a planar undirected graph with 3 vertices and 3 edges, in the form of a triangle.The triangle graph is also known as the cycle graph C 3 {\displaystyle C_{3}} and the complete graph K 3 {\displaystyle K_{3}} . | Triangle graph | 0.814496 |
15,315 | The most simple EDAs assume that decision variables are independent, i.e. p ( X 1 , X 2 ) = p ( X 1 ) ⋅ p ( X 2 ) {\displaystyle p(X_{1},X_{2})=p(X_{1})\cdot p(X_{2})} . Therefore, univariate EDAs rely only on univariate statistics and multivariate distributions must be factorized as the product of N {\displaystyle N} ... | Estimation of Distribution Algorithm | 0.814496 |
15,316 | The general procedure of an EDA is outlined in the following: t := 0 initialize model M(0) to represent uniform distribution over admissible solutions while (termination criteria not met) do P := generate N>0 candidate solutions by sampling M(t) F := evaluate all candidate solutions in P M(t + 1) := adjust_model(P, F, ... | Estimation of Distribution Algorithm | 0.814496 |
15,317 | Estimation of distribution algorithms (EDAs), sometimes called probabilistic model-building genetic algorithms (PMBGAs), are stochastic optimization methods that guide the search for the optimum by building and sampling explicit probabilistic models of promising candidate solutions. Optimization is viewed as a series o... | Estimation of Distribution Algorithm | 0.814496 |
15,318 | The next stage of EDAs development was the use of multivariate factorizations. In this case, the joint probability distribution is usually factorized in a number of components of limited size | π i | ≤ K , ∀ i ∈ 1 , 2 , … , N {\displaystyle |\pi _{i}|\leq K,~\forall i\in 1,2,\dots ,N} . p ( X 1 , … , X N ) = ∏ i = 1 N ... | Estimation of Distribution Algorithm | 0.814496 |
15,319 | Probability collectives (PC) Hill climbing with learning (HCwL) Estimation of multivariate normal algorithm (EMNA) Estimation of Bayesian networks algorithm (EBNA) Stochastic hill climbing with learning by vectors of normal distributions (SHCLVND) Real-coded PBIL Selfish Gene Algorithm (SG) Compact Differential Evoluti... | Estimation of Distribution Algorithm | 0.814496 |
15,320 | Other uses of dipole magnets to deflect moving particles include isotope mass measurement in mass spectrometry, and particle momentum measurement in particle physics. Such magnets are also used in traditional televisions, which contain a cathode ray tube, which is essentially a small particle accelerator. Their magnets... | Bending magnet | 0.814494 |
15,321 | In particle physics, hypercolor is a hypothetical attractive force that binds prequarks together by the exchange of hypergluons, analogous to the exchange of gluons by the color force, which binds quarks together. | Hypercolor (physics) | 0.814493 |
15,322 | The Weierstrass preparation theorem can be used to show that the ring of germs of analytic functions in n variables is a Noetherian ring, which is also referred to as the Rückert basis theorem. | Distinguished polynomial | 0.814492 |
15,323 | For example, this applies to the ring of integers in a p-adic field. In this case the theorem says that a power series f(z) can always be uniquely factored as πn·u(z)·p(z), where u(z) is a unit in the ring of power series, p(z) is a distinguished polynomial (monic, with the coefficients of the non-leading terms each in... | Distinguished polynomial | 0.814492 |
15,324 | There is an analogous result, also referred to as the Weierstrass preparation theorem, for the ring of formal power series over complete local rings A: for any power series f = ∑ n = 0 ∞ a n t n ∈ A ] {\displaystyle f=\sum _{n=0}^{\infty }a_{n}t^{n}\in A]} such that not all a n {\displaystyle a_{n}} are in the maximal... | Distinguished polynomial | 0.814492 |
15,325 | A related result is the Weierstrass division theorem, which states that if f and g are analytic functions, and g is a Weierstrass polynomial of degree N, then there exists a unique pair h and j such that f = gh + j, where j is a polynomial of degree less than N. In fact, many authors prove the Weierstrass preparation a... | Distinguished polynomial | 0.814492 |
15,326 | Herbert G. Baker's list of 14 "ideal weed characteristics", published in the 1965 volume The Genetics of Colonizing Species, has been the basis for investigation into characteristics which could contribute to invasion success of plants. Since Baker first proposed this list, researchers have debated whether or not parti... | Invasion genetics | 0.814491 |
15,327 | The ecological success of many invaders despite these apparent genetic limitations suggests a "genetic paradox of invasion", for which a number of answers have been proposed.One of the possible resolutions for the genetic paradox of invasion is that most bottlenecks experienced by introduced species are typically not s... | Invasion genetics | 0.814491 |
15,328 | Since its publication in 1965, The Genetics of Colonizing Species helped to motivate research which would provide a theoretical and empirical foundation for invasion genetics. However, the term invasion genetics only first appeared in the literature in 1998, and the first published definition appeared in 2005. The succ... | Invasion genetics | 0.814491 |
15,329 | The evolutionary modern synthesis in the early 20th century brought together Charles Darwin's theory of evolution by natural selection and classical genetics through the development of population genetics, which provided the conceptual basis for studying how evolutionary processes shape variation in populations. This d... | Invasion genetics | 0.814491 |
15,330 | Hybridization and its contribution to evolution was a subject of interest for G. Ledyard Stebbins, who noted in a 1959 review that the introduction of European species of the genus Tragopogon to North America had led to hybrid speciation; this example was also discussed by Herbert G. Baker in The Genetics of Colonizing... | Invasion genetics | 0.814491 |
15,331 | Charles Elton formed the basis for examining biological invasions as a unified issue in his 1958 monograph, The Ecology of Invasions by Animals and Plants, drawing together case studies of species introductions. Other important events in the study of invasive species include a series of issues published by the Scientif... | Invasion genetics | 0.814491 |
15,332 | Invasion genetics, due to its focus on the biology of introduced species, is useful for identifying potential invasive species and developing practices for managing biological invasions. It is distinguished from the broader study of invasive species because it is less directly concerned with the impacts of biological i... | Invasion genetics | 0.814491 |
15,333 | Invasion genetics is the area of study within biology that examines evolutionary processes in the context of biological invasions. Invasion genetics considers how genetic and demographic factors affect the success of a species introduced outside of its native range, and how the mechanisms of evolution, such as natural ... | Invasion genetics | 0.814491 |
15,334 | Baker subsequently elaborated a series of "ideal weed characteristics" in an article in The Genetics of Colonizing Species, which included traits such as the ability to tolerate environmental variation, dispersal ability, and the ability to tolerate generalist herbivores and pathogens. While some of the traits, such as... | Invasion genetics | 0.814491 |
15,335 | Invasion genetics can be used to understand the processes involved at each stage of a biological invasion. Many of the foundational questions of invasion genetics focused on processes involved during establishment and spread. As early as 1955, Herbert G. Baker proposed that self-fertilization would be a favourable trai... | Invasion genetics | 0.814491 |
15,336 | In radiometry, radiant intensity is the radiant flux emitted, reflected, transmitted or received, per unit solid angle, and spectral intensity is the radiant intensity per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. These are directional quantiti... | Spectral intensity | 0.81449 |
15,337 | The strongest possible evidence for de novo emergence is the inference of the specific "enabling" mutation(s) that created coding potential, typically through the analysis of smaller sequence regions, termed microsyntenic regions, of closely related species. One challenge in applying synteny-based methods is that synte... | De novo gene birth | 0.814489 |
15,338 | With the development of technologies such as RNA-seq and Ribo-seq, eukaryotic genomes are now known to be pervasively transcribed and translated. Many ORFs that are either unannotated, or annotated as long non-coding RNAs (lncRNAs), are translated at some level, either in a condition or tissue-specific manner. Though i... | De novo gene birth | 0.814489 |
15,339 | In addition to its significance for the field of evolutionary biology, de novo gene birth has implications for human health. It has been speculated that novel genes, including de novo genes, may play an outsized role in species-specific traits; however, many species-specific genes lack functional annotation. Neverthele... | De novo gene birth | 0.814489 |
15,340 | The category of pointed sets and pointed maps has both products and coproducts, but it is not a distributive category. It is also an example of a category where 0 × A {\displaystyle 0\times A} is not isomorphic to 0 {\displaystyle 0} .Many algebraic structures are pointed sets in a rather trivial way. For example, grou... | Pointed set | 0.814488 |
15,341 | One textbook notes that "This formal completion of sets and partial maps by adding 'improper', 'infinite' elements was reinvented many times, in particular, in topology (one-point compactification) and in theoretical computer science. "The category of pointed sets and pointed maps is isomorphic to the coslice category ... | Pointed set | 0.814488 |
15,342 | Pointed maps are the homomorphisms of these algebraic structures. The class of all pointed sets together with the class of all based maps form a category. In this category the pointed singleton sets ( { a } , a ) {\displaystyle (\{a\},a)} are initial objects and terminal objects, i.e. they are zero objects. | Pointed set | 0.814488 |
15,343 | In mathematics, a pointed set (also based set or rooted set) is an ordered pair ( X , x 0 ) {\displaystyle (X,x_{0})} where X {\displaystyle X} is a set and x 0 {\displaystyle x_{0}} is an element of X {\displaystyle X} called the base point, also spelled basepoint. : 10–11 Maps between pointed sets ( X , x 0 ) {\displ... | Pointed set | 0.814488 |
15,344 | In mathematical physics, a Pöschl–Teller potential, named after the physicists Herta Pöschl (credited as G. Pöschl) and Edward Teller, is a special class of potentials for which the one-dimensional Schrödinger equation can be solved in terms of special functions. | Pöschl–Teller potential | 0.814484 |
15,345 | The Molecular Modeling Database (MMDB) is a database of experimentally determined three-dimensional biomolecular structures and hosted by the National Center for Biotechnology Information. | Molecular Modeling Database | 0.814483 |
15,346 | A Master of Physics honours (or MPhys (Hons)) degree is a specific master's degree for courses in the field of physics. | Master of Physics | 0.814472 |
15,347 | It is usual for there to be some variation in the MPhys schemes, to allow for students to study the area of physics which most interests them. For example, Lancaster University's physics department offer the following schemes: MPhys Physics MPhys Physics, Astrophysics and Cosmology MPhys Physics with Particle Physics a... | Master of Physics | 0.814472 |
15,348 | Properties characterize their bearers; they express what their bearer is like.Substance is a key concept in ontology, the latter in turn part of metaphysics, which may be classified into monist, dualist, or pluralist varieties according to how many substances or individuals are said to populate, furnish, or exist in th... | Substance theory | 0.814469 |
15,349 | Therefore, only God is a substance as Ens perfectissimus (most perfect being). Heidegger showed the inextricable relationship between the concept of substance and of subject, which explains why, instead of talking about "man" or "humankind", he speaks about the Dasein, which is not a simple subject, nor a substance.Alf... | Substance theory | 0.814469 |
15,350 | The school of Madhyamaka, namely Nāgārjuna, introduced the idea of the ontological void (śūnyatā). The Buddhist metaphysics Abhidharma presumes particular forces which determine the origin, persistence, aging and decay of everything in the world. Vasubandhu added a special force making a human, called "aprāpti" or "pṛt... | Substance theory | 0.814469 |
15,351 | In chapter 6 of book I the Physics Aristotle argues that any change must be analysed in reference to the property of an invariant subject: as it was before the change and thereafter. Thus, in his hylomorphic account of change, matter serves as a relative substratum of transformation, i.e., of changing (substantial) for... | Substance theory | 0.814469 |
15,352 | The following outline is provided as an overview of, and topical guide to, applied physics: Applied physics – physics intended for a particular technological or practical use. It is usually considered as a bridge or a connection between "pure" physics and engineering.Applied Physics – is the proper name of a journal fo... | Outline of applied physics | 0.814466 |
15,353 | Applied physics can be described as all of the following: Branch of science Branch of physics Branch of applied science Branch of engineering | Outline of applied physics | 0.814466 |
15,354 | American Institute of Physics Journal of Applied Physics Applied Physics Letters Japan Society of Applied Physics Japanese Journal of Applied Physics Applied Physics Express IOP Publishing Journal of Physics D: Applied Physics Springer Berlin Heidelberg New York Applied Physics Applied Physics A Applied Physics B Topic... | Outline of applied physics | 0.814466 |
15,355 | International Union of Pure and Applied Physics Harvard School of Engineering and Applied Sciences Applied Physics Laboratory, Johns Hopkins University National Institute of Physics, University of the Philippines Diliman Institute of Mathematical Science and Physics, University of the Philippines Los Baños School of Pu... | Outline of applied physics | 0.814466 |
15,356 | Bishop discover vitamin E as essential for rat pregnancy, originally calling it "food factor X" until 1925. In 1925 Hart discovered that iron absorption requires trace amounts of copper. In 1927 Adolf Otto Reinhold Windaus synthesized vitamin D, for which he won the Nobel Prize in Chemistry in 1928. | Human diet | 0.814466 |
15,357 | In the early 20th century, Carl von Voit and Max Rubner independently measured caloric energy expenditure in different species of animals, applying principles of physics in nutrition. In 1906, Edith G. Willcock and Frederick Hopkins showed that the amino acid tryptophan aids the well-being of mice but it did not assure... | Human diet | 0.814466 |
15,358 | Robert Boyle advanced chemistry. Sanctorius measured body weight. Physician Herman Boerhaave modeled the digestive process. Physiologist Albrecht von Haller worked out the difference between nerves and muscles. | Human diet | 0.814466 |
15,359 | Memetic algorithms have been successfully applied to a multitude of real-world problems. Although many people employ techniques closely related to memetic algorithms, alternative names such as hybrid genetic algorithms are also employed. Researchers have used memetic algorithms to tackle many classical NP problems. To ... | Memetic algorithm | 0.814464 |
15,360 | The learning method/meme used has a significant impact on the improvement results, so care must be taken in deciding which meme or memes to use for a particular optimization problem. The frequency and intensity of individual learning directly define the degree of evolution (exploration) against individual learning (exp... | Memetic algorithm | 0.814464 |
15,361 | The Prouhet–Thue–Morse constant was shown to be transcendental by Kurt Mahler in 1929.He also showed that the number ∑ i = 0 ∞ t n α n {\displaystyle \sum _{i=0}^{\infty }t_{n}\,\alpha ^{n}} is also transcendental for any algebraic number α, where 0 < |α| < 1. Yann Bugaeud proved that the Prouhet–Thue–Morse constant ha... | Prouhet–Thue–Morse constant | 0.814464 |
15,362 | The 4-neighbor square grid allows a no-winner situation, and the 8-neighbor square grid allows a two-winner situation. Consequently, connectedness properties in R 2 {\displaystyle \mathbb {R} ^{2}} , such as the Jordan curve theorem, do not generalize to Z 2 {\displaystyle \mathbb {Z} ^{2}} under either graph structure... | Simple closed curve | 0.814463 |
15,363 | In image processing, a binary picture is a discrete square grid of 0 and 1, or equivalently, a compact subset of Z 2 {\displaystyle \mathbb {Z} ^{2}} . Topological invariants on R 2 {\displaystyle \mathbb {R} ^{2}} , such as number of components, might fail to be well-defined for Z 2 {\displaystyle \mathbb {Z} ^{2}} if... | Simple closed curve | 0.814463 |
15,364 | The Jordan curve theorem was independently generalized to higher dimensions by H. Lebesgue and L. E. J. Brouwer in 1911, resulting in the Jordan–Brouwer separation theorem. The proof uses homology theory. It is first established that, more generally, if X is homeomorphic to the k-sphere, then the reduced integral homol... | Simple closed curve | 0.814463 |
15,365 | The Jordan curve theorem can be proved from the Brouwer fixed point theorem (in 2 dimensions), and the Brouwer fixed point theorem can be proved from the Hex theorem: "every game of Hex has at least one winner", from which we obtain a logical implication: Hex theorem implies Brouwer fixed point theorem, which implies J... | Simple closed curve | 0.814463 |
15,366 | Alternatively, a Jordan curve is the image of a continuous map φ: → R2 such that φ(0) = φ(1) and the restriction of φ to [0,1) is injective. The first two conditions say that C is a continuous loop, whereas the last condition stipulates that C has no self-intersection points. With these definitions, the Jordan curve t... | Simple closed curve | 0.814463 |
15,367 | A Jordan curve or a simple closed curve in the plane R2 is the image C of an injective continuous map of a circle into the plane, φ: S1 → R2. A Jordan arc in the plane is the image of an injective continuous map of a closed and bounded interval into the plane. It is a plane curve that is not necessarily smooth nor alg... | Simple closed curve | 0.814463 |
15,368 | If X is an n-dimensional compact connected submanifold of Rn+1 (or Sn+1) without boundary, its complement has 2 connected components. There is a strengthening of the Jordan curve theorem, called the Jordan–Schönflies theorem, which states that the interior and the exterior planar regions determined by a Jordan curve in... | Simple closed curve | 0.814463 |
15,369 | A few molecules have a tetrahedral geometry with no central atom. An inorganic example is tetraphosphorus (P4) which has four phosphorus atoms at the vertices of a tetrahedron and each bonded to the other three. An organic example is tetrahedrane (C4H4) with four carbon atoms each bonded to one hydrogen and the other t... | Inverted tetrahedral geometry | 0.814463 |
15,370 | Inversion of tetrahedra occurs widely in organic and main group chemistry. The Walden inversion illustrates the stereochemical consequences of inversion at carbon. Nitrogen inversion in ammonia also entails transient formation of planar NH3. | Inverted tetrahedral geometry | 0.814463 |
15,371 | Even when an allele is selectively neutral, selection acting on nearby genes may also change its allele frequency through hitchhiking or background selection. While heterozygosity at a given locus decreases over time as alleles become fixed or lost in the population, variation is maintained in the population through ne... | Allele frequencies | 0.814462 |
15,372 | Population genetics describes the genetic composition of a population, including allele frequencies, and how allele frequencies are expected to change over time. The Hardy–Weinberg law describes the expected equilibrium genotype frequencies in a diploid population after random mating. Random mating alone does not chang... | Allele frequencies | 0.814462 |
15,373 | Algorithmic topology, or computational topology, is a subfield of topology with an overlap with areas of computer science, in particular, computational geometry and computational complexity theory. A primary concern of algorithmic topology, as its name suggests, is to develop efficient algorithms for solving problems t... | Algorithmic topology | 0.81446 |
15,374 | He uses instanton gauge theory, the geometrization theorem of 3-manifolds, and subsequent work of Greg Kuperberg on the complexity of knottedness detection. The connect-sum decomposition of 3-manifolds is also implemented in Regina, has exponential run-time and is based on a similar algorithm to the 3-sphere recognitio... | Algorithmic topology | 0.81446 |
15,375 | In statistics and social sciences, an antecedent variable is a variable that can help to explain the apparent relationship (or part of the relationship) between other variables that are nominally in a cause and effect relationship. In a regression analysis, an antecedent variable would be one that influences both the i... | Antecedent variable | 0.81446 |
15,376 | One measure of cable manufacturing and installation quality is how closely the characteristic impedance adheres to the nominal impedance along its length. Impedance changes can be caused by variations in geometry along the cable length. In turn, these can be caused by a faulty manufacturing process or by faulty install... | 50 Ω | 0.814459 |
15,377 | The connection with Green's theorem can be understood in terms of integration in polar coordinates: in polar coordinates, area is computed by the integral ∫ θ 1 2 ( r ( θ ) ) 2 d θ , {\textstyle \int _{\theta }{\tfrac {1}{2}}(r(\theta ))^{2}\,d\theta ,} where the form being integrated is quadratic in r, meaning that th... | Planimeter | 0.814459 |
15,378 | In 1993 the American Association of Physics Teachers Metric Committee specified that the radian should explicitly appear in quantities only when different numerical values would be obtained when other angle measures were used, such as in the quantities of angle measure (rad), angular speed (rad/s), angular acceleration... | Unit angle | 0.814459 |
15,379 | Thus the existence of the measure μ {\displaystyle \mu } is equivalent to (1). Using a representation theorem for positive polynomials on , one can reformulate (1) as a condition on Hankel matrices. See Shohat & Tamarkin 1943 and Krein & Nudelman 1977 for more details. | Problem of moments | 0.814459 |
15,380 | One way to prove these results is to consider the linear functional φ {\displaystyle \varphi } that sends a polynomial P ( x ) = ∑ k a k x k {\displaystyle P(x)=\sum _{k}a_{k}x^{k}} to ∑ k a k m k . {\displaystyle \sum _{k}a_{k}m_{k}.} If mkn are the moments of some measure μ supported on , then evidently Vice versa, i... | Problem of moments | 0.814459 |
15,381 | So the linear functional Λ {\displaystyle \Lambda } is positive for all the non-negative polynomials in the univariate case. By Haviland's theorem, the linear functional has a measure form, that is Λ ( x n ) = ∫ − ∞ ∞ x n d μ {\displaystyle \Lambda (x^{n})=\int _{-\infty }^{\infty }x^{n}d\mu } . A condition of similar ... | Problem of moments | 0.814459 |
15,382 | In the classical setting, μ is a measure on the real line, and M is the sequence { xn: n = 0, 1, 2, ... }. In this form the question appears in probability theory, asking whether there is a probability measure having specified mean, variance and so on, and whether it is unique. There are three named classical moment pr... | Problem of moments | 0.814459 |
15,383 | The polar planimeter is restricted by design to measuring areas within limits determined by its size and geometry. However, the linear type has no restriction in one dimension, because it can roll. Its wheels must not slip, because the movement must be constrained to a straight line. | Planimeter | 0.814459 |
15,384 | They achieved acceptance as an international standard in parallel to the SI system of units and are commonly used outside of the physics community in both engineering and industry. Addition of the constant c would be needed to define units for all dimensions used in physics, as in the SI. The SI system made the transit... | Conventional electrical unit | 0.814458 |
15,385 | A conventional electrical unit (or conventional unit where there is no risk of ambiguity) is a unit of measurement in the field of electricity which is based on the so-called "conventional values" of the Josephson constant, the von Klitzing constant agreed by the International Committee for Weights and Measures (CIPM) ... | Conventional electrical unit | 0.814458 |
15,386 | A multisection of the series of an analytic function f ( z ) = ∑ n = 0 ∞ a n ⋅ z n {\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}\cdot z^{n}} has a closed-form expression in terms of the function f ( x ) {\displaystyle f(x)}: ∑ m = 0 ∞ a q m + p ⋅ z q m + p = 1 q ⋅ ∑ k = 0 q − 1 ω − k p ⋅ f ( ω k ⋅ z ) , {\displaystyle... | Series multisection | 0.814457 |
15,387 | The direct algebra approach to the above cases integral multiplier lead to the following formula: X = D 10 m − 1 n 10 k − 1 , {\displaystyle X=D{\frac {10^{m}-1}{n10^{k}-1}},} where m is the number of digits of X, and D, the k-digit number shifted from the low end of X to the high end of n X, satisfies D < 10k. If the ... | Cyclic permutation of integer | 0.814452 |
15,388 | In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability p {\displaystyle p} and the value 0 with probability q = 1 − p {\displaystyle q=1-p} . Less formally, ... | Bernoulli random variable | 0.814452 |
15,389 | In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them. Usually, this involves determining a function that relates the size of an algorithm's input to the number of steps it takes (its ti... | Computationally expensive | 0.814449 |
15,390 | Run-time analysis is a theoretical classification that estimates and anticipates the increase in running time (or run-time or execution time) of an algorithm as its input size (usually denoted as n) increases. Run-time efficiency is a topic of great interest in computer science: A program can take seconds, hours, or ev... | Computationally expensive | 0.814449 |
15,391 | Prior to World War II, the center of organic chemistry research and industrial production was Germany. Students interested in pursuing a career in organic chemistry needed to learn German to read articles and textbooks, and often went to graduate school in Germany. When the war broke out, an effort to jump‑start a nati... | Organic Reactions | 0.814448 |
15,392 | The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appe... | Symmetry in mathematics | 0.814446 |
15,393 | So if the entries are written as A = (aij), then aij = aji, for all indices i and j. For example, the following 3×3 matrix is symmetric: {\displaystyle {\begin{bmatrix}1&7&3\\7&4&-5\\3&-5&6\end{bmatrix}}} Every square diagonal matrix is symmetric, since all off-diagonal entries are zero. Similarly, each diagonal eleme... | Symmetry in mathematics | 0.814446 |
15,394 | In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose (i.e., it is invariant under matrix transposition). Formally, matrix A is symmetric if A = A T . {\displaystyle A=A^{T}.} | Symmetry in mathematics | 0.814446 |
15,395 | In abstract algebra, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loo... | Symmetry in mathematics | 0.814446 |
15,396 | Given a polynomial, it may be that some of the roots are connected by various algebraic equations. For example, it may be that for two of the roots, say A and B, that A2 + 5B3 = 7. The central idea of Galois theory is to consider those permutations (or rearrangements) of the roots having the property that any algebraic... | Symmetry in mathematics | 0.814446 |
15,397 | An isometry is a distance-preserving map between metric spaces. Given a metric space, or a set and scheme for assigning distances between elements of the set, an isometry is a transformation which maps elements to another metric space such that the distance between the elements in the new metric space is equal to the d... | Symmetry in mathematics | 0.814446 |
15,398 | Knowledge of a Line symmetry can be used to simplify an ordinary differential equation through reduction of order.For ordinary differential equations, knowledge of an appropriate set of Lie symmetries allows one to explicitly calculate a set of first integrals, yielding a complete solution without integration. Symmetri... | Symmetry in mathematics | 0.814446 |
15,399 | A symmetry of a differential equation is a transformation that leaves the differential equation invariant. Knowledge of such symmetries may help solve the differential equation. A Line symmetry of a system of differential equations is a continuous symmetry of the system of differential equations. | Symmetry in mathematics | 0.814446 |
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