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84,400 | Every irrational rotation algebra Aθ is simple, that is, it does not contain any proper closed two-sided ideals other than { 0 } {\displaystyle \{0\}} and itself. Every irrational rotation algebra has a unique tracial state. The irrational rotation algebras are nuclear. | Noncommutative torus | 0.783334 |
84,401 | In particular, the noncommutative tori with θ rational are Morita equivalent to the classical torus. On the other hand, the noncommutative tori with θ irrational are simple C*-algebras. == References == | Noncommutative torus | 0.783334 |
84,402 | The K-theory of Aθ is Z2 in both even dimension and odd dimension, and so does not distinguish the irrational rotation algebras. But as an ordered group, K0 ≃ Z + θZ. Therefore, two noncommutative tori Aθ and Aη are isomorphic if and only if either θ + η or θ − η is an integer.Two irrational rotation algebras Aθ and Aη... | Noncommutative torus | 0.783334 |
84,403 | In mathematics, and more specifically in the theory of C*-algebras, the noncommutative tori Aθ, also known as irrational rotation algebras for irrational values of θ, form a family of noncommutative C*-algebras which generalize the algebra of continuous functions on the 2-torus. Many topological and geometric propertie... | Noncommutative torus | 0.783334 |
84,404 | Agroecology also includes the idea of holistic management. This approach stems from the work of Allan Savory who claims that planned grazing can improve soil health and reverse the effects of desertification by increasing biomass. Desertification occurs when the soil carbon content is severely depleted, greatly reducin... | Soil regeneration and climate change | 0.783334 |
84,405 | Over time, these nutrients will be depleted from the soil. Agroecology is an overarching category of approaches to creating a more sustainable agricultural system and increasing the health of soil. These conservation agricultural practices utilize many techniques and resources to maintain healthy soil. | Soil regeneration and climate change | 0.783334 |
84,406 | The first-generation iPhone was released on June 29, 2007, and multiple new hardware iterations with new iOS releases have been released since.2007 Amazon Kindle In 2007, Amazon released the original Amazon Kindle e-book reader. While not the first of its kind, the Kindle led the charge to popularizing e-books and e-re... | Timeline of United States inventions (after 1991) | 0.783333 |
84,407 | The high surface area further allows for fast charging and discharging. The practicality of nanowire batteries is reasoned that a laptop computer that runs on a regular lithium-ion battery for two hours could potentially operate up to 20 hours using a nanowire battery without recharging, which would be a considerable a... | Timeline of United States inventions (after 1991) | 0.783333 |
84,408 | In addition, HVLS fans have been found to be more beneficial than traditional HVAC fans as they are more cost effective as well as having a stronger cooling effect while helping to maintain a constant temperature and humidity. Using 10 overhead airfoil blades in a barn, the HVLS fan was invented in 1998 by Walter Boyd ... | Timeline of United States inventions (after 1991) | 0.783333 |
84,409 | On January 28, 1994, Fossum filed U.S. patent #5,471,515, which was issued to him on November 28, 1995.1994 DNA computing DNA computing uses DNA, biochemistry and molecular biology, instead of the traditional silicon-based computer technologies. DNA computing, or, more generally, molecular computing, is a fast-developi... | Timeline of United States inventions (after 1991) | 0.783333 |
84,410 | Global coverage pertains to research and application, which is updated on a monthly basis. This database accesses periodicals, conference proceedings, technical reports, trade journals (including newsletter items), patents, books, and press releases. Non-serial publications are also covered. Research and application pe... | CSA Illumina | 0.783333 |
84,411 | CSA (formerly Cambridge Scientific Abstracts) was a division of Cambridge Information Group and provider of online databases, based in Bethesda, Maryland, before merging with ProQuest of Ann Arbor, Michigan, in 2007. CSA hosted databases of abstracts and developed taxonomic indexing of scholarly articles. These databas... | CSA Illumina | 0.783333 |
84,412 | The Aerospace and High Technology Database is a bibliographic database covering the space sciences as well as aeronautics and astronautics. This database also provides access to complementary disciplines, such as electronics, chemistry, physics, geoscience, and communications. The print equivalent is International Aero... | CSA Illumina | 0.783333 |
84,413 | Nuclear technology is also part of this database covering: Fluid flow, Hydraulics, Pneumatics, and Vacuum Technology. Heat and Thermodynamics covers Industrial furnaces, process heating, space heating, air conditioning, refrigeration, and cryogenics. ISMEC, Mechanical Engineering Abstracts(ISSN 0896-7113) (OCLC 1727445... | CSA Illumina | 0.783333 |
84,414 | The target audience is specialists, researchers, and engineers.Further subject coverage includes all aspects of Mechanical engineering. Other subjects, which are covered in this database, are Aerospace engineering, Automotive engineering, Naval architecture and Marine engineering, Railroad engineering, and Materials ha... | CSA Illumina | 0.783333 |
84,415 | As of May 2010, this database contained more than 215,700 records.Engineering Information is a co-publisher of this database. Global in scope, broad subject coverage comprises theory and application in mechanical engineering, engineering management, and Production engineering. Format coverage consists of scientific jou... | CSA Illumina | 0.783333 |
84,416 | Another title is "Information service in Mechanical Engineering Bulletin".Temporal coverage for Mechanical Engineering Abstracts is from 1981 to 2002. Current information, beyond its temporal coverage, is not part of this database. Current information is located in Mechanical & Transportation Engineering Abstracts (see... | CSA Illumina | 0.783333 |
84,417 | Mechanical Engineering Abstracts is a continuation of the formerly named "ISMEC Bulletin" (v.1, no.1, July 10, 1973) (ISSN 0306-0039), which appears to have ceased under this title in December, 1987 (v. 20, no.6). ISMEC Bulletin was published by Cambridge Scientific Abstracts. Now, Mechanical Engineering Abstracts is a... | CSA Illumina | 0.783333 |
84,418 | The CSA High Technology Research Database is a major abstracting and indexing database and category in the CSA Illumina database structure. It is considered to be the online equivalent of International Aerospace Abstracts (IAA) and Scientific and Technical Aerospace Reports (STAR from 1962 to 1993). These are also list... | CSA Illumina | 0.783333 |
84,419 | If d {\displaystyle d} is even, then define e {\displaystyle e} through the relation d = 2 e {\displaystyle d=2e} ; if d {\displaystyle d} is odd, define e {\displaystyle e} through the relation d = 2 e − 1 {\displaystyle d=2e-1} . Let q ∗ = q − 1 {\displaystyle q^{*}=q-1} . Then, A q ( n , d , w ) ≤ ⌊ n q ∗ w ⌊ ( n − ... | Johnson bound | 0.783332 |
84,420 | If d = 2 t + 2 {\displaystyle d=2t+2} , A q ( n , d ) ≤ q n ∑ i = 0 t ( n i ) ( q − 1 ) i + ( n t + 1 ) ( q − 1 ) t + 1 A q ( n , d , t + 1 ) . {\displaystyle A_{q}(n,d)\leq {\frac {q^{n}}{\sum _{i=0}^{t}{n \choose i}(q-1)^{i}+{\frac {{n \choose t+1}(q-1)^{t+1}}{A_{q}(n,d,t+1)}}}}.} Theorem 2 (Johnson bound for A q ( n... | Johnson bound | 0.783332 |
84,421 | {\displaystyle A_{q}(n,d,w)=\max _{C\in C_{q}(n,d,w)}|C|.} Theorem 1 (Johnson bound for A q ( n , d ) {\displaystyle A_{q}(n,d)} ): If d = 2 t + 1 {\displaystyle d=2t+1} , A q ( n , d ) ≤ q n ∑ i = 0 t ( n i ) ( q − 1 ) i + ( n t + 1 ) ( q − 1 ) t + 1 − ( d t ) A q ( n , d , d ) A q ( n , d , t + 1 ) . {\displaystyle A... | Johnson bound | 0.783332 |
84,422 | Many modulation techniques have been specifically designed to have constant envelope modulation, i.e., the minimum possible crest factor of 1:1. In general, modulation techniques that have smaller crest factors usually transmit more bits per second than modulation techniques that have higher crest factors. This is beca... | Crest factor | 0.783331 |
84,423 | Electrical engineering — for describing the quality of an AC power waveform Vibration analysis — for estimating the amount of impact wear in a bearing Radio and audio electronics — for estimating the headroom required in a signal chainMusic has a widely varying crest factor. Typical values for a processed mix are aroun... | Crest factor | 0.783331 |
84,424 | To control the charged particle beam, appropriate electric and magnetic fields must be created. There are software packages to help in the design and understanding of the magnets, RF cavities, and other elements that create these fields. Codes include ACE3P COMSOL Multiphysics CST Studio Suite OPERA VSim | Methodical Accelerator Design | 0.78333 |
84,425 | A charged particle accelerator is a complex machine that takes elementary charged particles and accelerates them to very high energies. Accelerator physics is a field of physics encompassing all the aspects required to design and operate the equipment and to understand the resulting dynamics of the charged particles. T... | Methodical Accelerator Design | 0.78333 |
84,426 | Because the superregenerative detectors tend to receive the strongest signal and ignore other signals in the nearby spectrum, the superregen works best with bands that are relatively free of interfering signals. Due to Nyquist's theorem, its quenching frequency must be at least twice the signal bandwidth. But quenching... | Regenerative radio receiver | 0.78333 |
84,427 | Medical records (MR) and Health information technicians (HIT) are described as having the following duties according to the U.S. Bureau of Labor Statistics' Occupational Outlook Handbook: assemble patients' health information including medical history, symptoms, examination results, diagnostic tests, treatment methods,... | Health information management | 0.783329 |
84,428 | In mathematics, an Ockham algebra is a bounded distributive lattice with a dual endomorphism, that is, an operation ~ satisfying ~(x ∧ y) = ~x ∨ ~y, ~(x ∨ y) = ~x ∧ ~y, ~0 = 1, ~1 = 0. They were introduced by Berman (1977), and were named after William of Ockham by Urquhart (1979). Ockham algebras form a variety. Examp... | Ockham algebras | 0.783328 |
84,429 | In the field of molecular biology, the ETS (E26 transformation-specific or E-twenty-six. (Erythroblast Transformation Specific)) family is one of the largest families of transcription factors and is unique to animals. There are 29 genes in humans, 28 in the mouse, 10 in Caenorhabditis elegans and 9 in Drosophila. The f... | ETS transcription factor family | 0.783327 |
84,430 | SOC has become established as a strong candidate for explaining a number of natural phenomena, including: The magnitude of earthquakes (Gutenberg–Richter law) and frequency of aftershocks (Omori law) Fluctuations in economic systems such as financial markets (references to SOC are common in econophysics) The evolution ... | Self-organizing criticality | 0.783327 |
84,431 | The Bertrand–Diguet–Puiseux theorem asserts that K ( p ) = lim r → 0 + 3 2 π r − C ( r ) π r 3 = lim r → 0 + 12 π r 2 − A ( r ) π r 4 . {\displaystyle K(p)=\lim _{r\to 0^{+}}3{\frac {2\pi r-C(r)}{\pi r^{3}}}=\lim _{r\to 0^{+}}12{\frac {\pi r^{2}-A(r)}{\pi r^{4}}}.} The theorem is closely related to the Gauss–Bonnet the... | Bertrand–Diguet–Puiseux theorem | 0.783327 |
84,432 | In the mathematical study of the differential geometry of surfaces, the Bertrand–Diguet–Puiseux theorem expresses the Gaussian curvature of a surface in terms of the circumference of a geodesic circle, or the area of a geodesic disc. The theorem is named for Joseph Bertrand, Victor Puiseux, and Charles François Diguet.... | Bertrand–Diguet–Puiseux theorem | 0.783327 |
84,433 | Like the other Manufacturing USA institutes, ARM operates as a membership-based consortium with more than 200 national members spanning industry, academia, and government. ARM periodically releases separate technology and workforce development project calls. Members then form teams to bid for funding. The project calls... | Advanced Robotics for Manufacturing | 0.783326 |
84,434 | Advanced Robotics for Manufacturing (ARM), also known as ARM Institute, is a consortium created in 2017 through a Department of Defense grant won by Carnegie Mellon University. ARM is structured as a public-private partnership and the Manufacturing USA Institutes, a network of 16 institutes dedicated to advancing techn... | Advanced Robotics for Manufacturing | 0.783326 |
84,435 | This description allows a better understanding to the entire geology of that area. Can be used to decide whether there is potential for oil or natural gas that exists in these rocks. | Stratigraphic Column | 0.783326 |
84,436 | Principle of Uniformitarianism: defined in the authoritative Glossary of Geology as "the fundamental principle or doctrine that geologic processes and natural laws now operating to modify the Earth's crust have acted in the same regular manner and with essentially the same intensity throughout geologic time, and that p... | Stratigraphic Column | 0.783326 |
84,437 | Stratigraphy is a branch of geology that concerns the order and relative position of geologic strata and their relationship to the geologic time scale. The relative time sequencing requires the analysis of the order and position of layers of archaeological remains and the structure of a particular set of strata. The co... | Stratigraphic Column | 0.783326 |
84,438 | A stratigraphic column is a representation used in geology and its subfield of stratigraphy to describe the vertical location of rock units in a particular area. A typical stratigraphic column shows a sequence of sedimentary rocks, with the oldest rocks on the bottom and the youngest on top. In areas that are more geol... | Stratigraphic Column | 0.783326 |
84,439 | ", and nylon velvet was too expensive and its stiffer fibers caused problems, leading to the choice and eventual standardization on cotton velveteen. While first demonstrated with bacteria, velveteen based replica plating has also become a standard technique in the microbiology of eukaryotes, such as yeast. == Referenc... | Replica plating | 0.783325 |
84,440 | The gauge algebra 24 decomposes as ( ( 8 , 1 ) 0 ( 1 , 3 ) 0 ( 1 , 1 ) 0 ( 3 , 2 ) − 5 6 ( 3 ¯ , 2 ) 5 6 ) . {\displaystyle {\begin{pmatrix}(8,1)_{0}\\(1,3)_{0}\\(1,1)_{0}\\(3,2)_{-{\frac {5}{6}}}\\({\bar {3}},2)_{\frac {5}{6}}\end{pmatrix}}~.} This 24 is a real representation, so the last two terms need explanation. B... | Georgi–Glashow model | 0.783325 |
84,441 | This yields a mechanism for proton decay, and the rate of proton decay can be predicted from the dynamics of the model. However, proton decay has not yet been observed experimentally, and the resulting lower limit on the lifetime of the proton contradicts the predictions of this model. Nevertheless, the elegance of the... | Georgi–Glashow model | 0.783325 |
84,442 | In particle physics, the Georgi–Glashow model is a particular Grand Unified Theory (GUT) proposed by Howard Georgi and Sheldon Glashow in 1974. In this model, the Standard Model gauge groups SU(3) × SU(2) × U(1) are combined into a single simple gauge group SU(5). The unified group SU(5) is then thought to be spontaneo... | Georgi–Glashow model | 0.783325 |
84,443 | The length of the shortest path between these two vertices is called the length of the spindle. The disproof of the Hirsch conjecture relies on the following theorem, referred to as the strong d-step theorem for spindles. Theorem (Santos) Let P {\displaystyle P} be a d-spindle. | Hirsch conjecture | 0.783324 |
84,444 | Theorem The following statements are equivalent: δ ( P ) ≤ n − d {\displaystyle \delta (P)\leq n-d} for all d-dimensional polytopes P {\displaystyle P} with n facets. δ ( P ) ≤ d {\displaystyle \delta (P)\leq d} for all d-dimensional polytopes P {\displaystyle P} with 2d facets.In other words, in order to prove or disp... | Hirsch conjecture | 0.783324 |
84,445 | The sources are writings of all sorts, from "literary novels and specialist journals to everyday newspapers and magazines and from Hansard to the language of blogs, emails, and social media". This may be contrasted with similar databases that sample only a specific kind of writing. The corpus is generally available onl... | Oxford English Corpus | 0.783323 |
84,446 | It was not until one year after his death in 1955 that his writings were published for the world to read. His works were also supported by the writings of a group of Catholic thinkers, which includes Pope Benedict XVI. His book The Phenomenon of Man has been dissected by astrophysicists and cosmologists, and is now vie... | Omega Point | 0.783319 |
84,447 | Energy exists in two basic modes: "Tangential Energy": energy that can be measured by physics. "Radial Energy": spiritual energy which accumulates into a higher state as time progresses.Teilhard defines Radial Energy as becoming more concentrated and available as it is a critical element in man's evolution. The theory ... | Omega Point | 0.783319 |
84,448 | Mathematical physicist Frank Tipler generalized Teilhard's term Omega Point to describe what he alleges is the ultimate fate of the universe as required by the laws of physics: roughly, Tipler argues that quantum mechanics is inconsistent unless the future of every point in spacetime contains an intelligent observer to... | Omega Point | 0.783319 |
84,449 | While BTs have become popular for modeling the artificial intelligence in computer games such as Halo and Spore, these types of trees are very different from the ones described on this page, and are closer to a combination of hierarchical finite state machines or decision trees. Soccer-player modeling has also been a s... | Behavior tree | 0.783314 |
84,450 | The formal semantics of behavior trees is given via a process algebra and its operational semantics. The semantics has been used as the basis for developing simulation, model checking and failure modes and effects analysis. | Behavior tree | 0.783314 |
84,451 | Behavior trees are a formal, graphical modelling language used primarily in systems and software engineering. Behavior trees employ a well-defined notation to unambiguously represent the hundreds or even thousands of natural language requirements that are typically used to express the stakeholder needs for a large-scal... | Behavior tree | 0.783314 |
84,452 | As a result, the term behavior engineering was introduced to describe the processes that exploit behavior trees to construct systems. The term "behavior engineering" has previously been used in a specialized area of Artificial Intelligence – robotics research. The present use embraces a much broader rigorous formalizat... | Behavior tree | 0.783314 |
84,453 | Behavior trees and the concepts for their application in systems and software engineering were originally developed by Dromey with first publication of some of the key ideas in 2001. Early publications on this work used the terms "genetic software engineering" and "genetic design" to describe the application of behavio... | Behavior tree | 0.783314 |
84,454 | Single and composite or integrated behavior tree forms are both important in the application of behavior trees in systems and software engineering. Requirement behavior trees: Initially, individual requirement behavior trees (RBTs) are used to capture all the fragments of behavior in each individual natural language re... | Behavior tree | 0.783314 |
84,455 | Let ( X , Y , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle (X,Y,\langle \cdot ,\cdot \rangle )} be a dual pair of vector spaces over the field F {\displaystyle \mathbb {F} } of real numbers R {\displaystyle \mathbb {R} } or complex numbers C . {\displaystyle \mathbb {C} .} For any B ⊆ X {\displaystyle B\subseteq X} and any y ∈ Y , {\dis... | Strong dual space | 0.783314 |
84,456 | In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X {\displaystyle X} is the continuous dual space X ′ {\displaystyle X^{\prime }} of X {\displaystyle X} equipped with the strong (dual) topology or the topology of uniform convergence on bounded subsets of... | Strong dual space | 0.783314 |
84,457 | {\displaystyle \left\langle x,x^{\prime }\right\rangle :=x^{\prime }(x).} In the special case when X {\displaystyle X} is a locally convex space, the strong topology on the (continuous) dual space X ′ {\displaystyle X^{\prime }} (that is, on the space of all continuous linear functionals f: X → F {\displaystyle f:X\to ... | Strong dual space | 0.783314 |
84,458 | So a subset B ⊆ X {\displaystyle B\subseteq X} is called bounded if and only if This is equivalent to the usual notion of bounded subsets when X {\displaystyle X} is given the weak topology induced by Y , {\displaystyle Y,} which is a Hausdorff locally convex topology. Let B {\displaystyle {\mathcal {B}}} denote the fa... | Strong dual space | 0.783314 |
84,459 | The review claims that neuroscience is irrelevant to the study of psychoanalysis.Kirkus Reviews stated the book is "somewhat scattershot, but Doidge's personal stories, enthusiasm for his subject and admiration for its researchers keep the reader engaged. "In a review of the book for the Society for Psychoanalysis and ... | The Brain that Changes Itself | 0.783311 |
84,460 | However, in spite of significant efforts, perfect reliability as envisaged in the initial ARPANET specification turned out to be impossible to provide – a reality that became increasingly obvious once the ARPANET grew well beyond its initial four-node topology. The ARPANET thus provided a strong case for the inherent l... | End to end principle | 0.783311 |
84,461 | A new design method that couples the above bottom-up optimization with a preliminary system level top-down design has been formulated. The two step design process has been motivated by considering that constructal and modular design does not refer to any objective to be reached in the design process. A theoretical form... | Modular design | 0.783309 |
84,462 | Some authors observe that modular design has generated in the vehicle industry a constant increase of weight over time. Trancossi advanced the hypothesis that modular design can be coupled by some optimization criteria derived from the constructal law. In fact, the constructal law is modular for his nature and can appl... | Modular design | 0.783309 |
84,463 | Also known as the (Moran-)Gamma Process, the gamma process is a random process studied in mathematics, statistics, probability theory, and stochastics. The gamma process is a stochastic or random process consisting of independently distributed gamma distributions where N ( t ) {\displaystyle N(t)} represents the number... | Gamma process | 0.783308 |
84,464 | Lévy Processes and Stochastic Calculus by David Applebaum, CUP 2004, ISBN 0-521-83263-2. == References == | Gamma process | 0.783308 |
84,465 | Some Sun-like stars of spectral class F8 to G8 have been found to undergo enormous magnetic outbursts to produce so-called superflares (coronal mass ejections) that release between 100 and 10 million times more energy than the largest flares ever observed on the sun, making them brighten briefly by up to 20 times.Magne... | Kappa1 Ceti | 0.783308 |
84,466 | The comet was lost after its 1930 apparition as the 1935 apparition had poor viewing geometry, but was recovered in 1979. During perihelion in 1985, the comet was unobserved as it was on the far side of the Sun 1.9 AU from Earth. In 1990 the comet reached apparent magnitude 9 and was the best appearance since 1930. | 73P/Schwassmann–Wachmann | 0.783308 |
84,467 | In the 17th century Guillaume Amontons discovered a regular relationship between the pressure and temperature of a gas at constant volume. Some introductory physics textbooks still define the pressure-temperature relationship as Gay-Lussac's law. Gay-Lussac primarily investigated the relationship between volume and tem... | Gay-Lussac's law | 0.783307 |
84,468 | In the field of semiconductor physics, it is still universally called group IV. The group is also known as the tetrels (from the Greek word tetra, which means four), stemming from the Roman numeral IV in the group names, or (not coincidentally) from the fact that these elements have four valence electrons (see below). ... | Carbon group | 0.783307 |
84,469 | Donald Nelson, chair of the CODASYL COBOL committee, said in 1984 that "academics ... hate COBOL" and that computer science graduates "had 'hate COBOL' drilled into them".By the mid-1980s, there was also significant condescension towards COBOL in the business community from users of other languages, for example FORTRAN... | COBOL programming language | 0.783307 |
84,470 | The COBOL community has always been isolated from the computer science community. No academic computer scientists participated in the design of COBOL: all of those on the committee came from commerce or government. Computer scientists at the time were more interested in fields like numerical analysis, physics and syste... | COBOL programming language | 0.783307 |
84,471 | A Klein geometry (G, H) is said to be reductive and G/H a reductive homogeneous space if the Lie algebra h {\displaystyle {\mathfrak {h}}} of H has an H-invariant complement in g {\displaystyle {\mathfrak {g}}} . | Reductive homogeneous space | 0.783306 |
84,472 | A Klein geometry (G, H) is geometrically oriented if G is connected. (This does not imply that G/H is an oriented manifold). If H is connected it follows that G is also connected (this is because G/H is assumed to be connected, and G → G/H is a fibration). Given any Klein geometry (G, H), there is a geometrically orien... | Reductive homogeneous space | 0.783306 |
84,473 | When the distribution of X {\displaystyle X} (and therefore, of Z {\displaystyle Z} ) is not available, the DUDE replaces the unknown vector P Z i , z n ∖ i {\displaystyle \mathbf {P} _{Z_{i},z^{n\backslash i}}} with an empirical estimate obtained along the noisy sequence z n {\displaystyle z^{n}} itself, namely with μ... | Discrete Universal Denoiser | 0.783304 |
84,474 | The Discrete Universal Denoiser (DUDE) is a denoising scheme that estimates an unknown signal x n = ( x 1 … x n ) {\displaystyle x^{n}=\left(x_{1}\ldots x_{n}\right)} over a finite alphabet from a noisy version z n = ( z 1 … z n ) {\displaystyle z^{n}=\left(z_{1}\ldots z_{n}\right)} . While most denoising schemes in th... | Discrete Universal Denoiser | 0.783304 |
84,475 | Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1715, although an earlier version of the result was already mentioned in 1671 by James Gregory.Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analys... | Taylor's Theorem | 0.783303 |
84,476 | In calculus, Taylor's theorem gives an approximation of a k {\textstyle k} -times differentiable function around a given point by a polynomial of degree k {\textstyle k} , called the k {\textstyle k} -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order k {\textstyle k... | Taylor's Theorem | 0.783303 |
84,477 | The precise statement of the most basic version of Taylor's theorem is as follows: The polynomial appearing in Taylor's theorem is the k {\textstyle {\boldsymbol {k}}} -th order Taylor polynomial of the function f at the point a. The Taylor polynomial is the unique "asymptotic best fit" polynomial in the sense that if ... | Taylor's Theorem | 0.783303 |
84,478 | Let f ( x ) {\displaystyle f(x)} be any real-valued continuous function to be approximated by the Taylor polynomial. Step 1: Let F {\textstyle F} and G {\textstyle G} be functions. Set F {\textstyle F} and G {\textstyle G} to be Step 2: Properties of F {\textstyle F} and G {\textstyle G}: Similarly, Step 3: Use Cauchy ... | Taylor's Theorem | 0.783303 |
84,479 | The Annals of Mathematical Statistics was a peer-reviewed statistics journal published by the Institute of Mathematical Statistics from 1930 to 1972. It was superseded by the Annals of Statistics and the Annals of Probability. In 1938, Samuel Wilks became editor-in-chief of the Annals and recruited a remarkable editori... | Annals of Mathematical Statistics | 0.783303 |
84,480 | Although presented at an undergraduate level, The Geometry of the Octonions is not a textbook: its material is likely too specialized for an undergraduate course, and it lacks exercises or similar material that would be needed to use it as a textbook. Readers should be familiar with linear algebra, and some experience ... | The Geometry of the Octonions | 0.783302 |
84,481 | Reviewer Danail Brezov notes with disappointment that Clifford algebras, although very relevant to this material, are not covered.The second part of the book uses the octonions and the other division algebras associated with it to provide concrete descriptions of the Lie groups of geometric symmetries. These include ro... | The Geometry of the Octonions | 0.783302 |
84,482 | The book is subdivided into three parts, with the second part being the most significant. Its contents combine both a survey of past work in this area, and much of its authors' own researches.The first part explains the Cayley–Dickson construction, which constructs the complex numbers from the real numbers, the quatern... | The Geometry of the Octonions | 0.783302 |
84,483 | The Geometry of the Octonions is a mathematics book on the octonions, a system of numbers generalizing the complex numbers and quaternions, presenting its material at a level suitable for undergraduate mathematics students. It was written by Tevian Dray and Corinne Manogue, and published in 2015 by World Scientific. Th... | The Geometry of the Octonions | 0.783302 |
84,484 | (2000), Octonions, Jordan Algebras and Exceptional Groups, Springer Monographs in Mathematics, Berlin: Springer-Verlag, doi:10.1007/978-3-662-12622-6, ISBN 3-540-66337-1, MR 1763974 Ward, J. P. (1997), Quaternions and Cayley Numbers: Algebra and Applications, Mathematics and its Applications, vol. 403, Dordrecht: Kluwe... | The Geometry of the Octonions | 0.783302 |
84,485 | (2003), On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, Natick, Massachusetts: A K Peters, ISBN 1-56881-134-9, MR 1957212 Kantor, I. L.; Solodovnikov, A. S. (1989), Hypercomplex Numbers: An Elementary Introduction to Algebras, New York: Springer-Verlag, doi:10.1007/978-1-4612-3650-4, ISBN 0-387-... | The Geometry of the Octonions | 0.783302 |
84,486 | A Higgs prime, named after Denis Higgs, is a prime number with a totient (one less than the prime) that evenly divides the square of the product of the smaller Higgs primes. (This can be generalized to cubes, fourth powers, etc.) To put it algebraically, given an exponent a, a Higgs prime Hpn satisfies ϕ ( H p n ) | ∏ ... | Higgs prime | 0.783301 |
84,487 | The atomic number of calcium The third magic number in physics The International Astronomical Union shower number for Coma Berenicids | 20 (number) | 0.783301 |
84,488 | Bring's curve is a Riemann surface of genus four, whose fundamental polygon is a regular hyperbolic 20-sided icosagon, with an area equal to 12 π {\displaystyle 12\pi } by the Gauss-Bonnet theorem.The Happy Family of sporadic groups is made up of twenty finite simple groups that are all subquotients of the friendly gia... | 20 (number) | 0.783301 |
84,489 | When comparing modularity optimization methods, the two measures of importance are the speed and the resulting modularity value. A higher speed is better as it shows a method is more efficient than others and a higher modularity value is desirable as it points to having better-defined communities. The compared methods ... | Louvain modularity | 0.783294 |
84,490 | The Louvain method for community detection is a method to extract non-overlapping communities from large networks created by Blondel et al. from the University of Louvain (the source of this method's name). The method is a greedy optimization method that appears to run in time O ( n ⋅ log n ) {\displaystyle O(n\cdot ... | Louvain modularity | 0.783294 |
84,491 | The system of congruences solved by the Chinese remainder theorem may be rewritten as a system of linear Diophantine equations: x = a 1 + x 1 n 1 ⋮ x = a k + x k n k , {\displaystyle {\begin{aligned}x&=a_{1}+x_{1}n_{1}\\&\vdots \\x&=a_{k}+x_{k}n_{k},\end{aligned}}} where the unknown integers are x {\displaystyle x} and... | Linear congruence theorem | 0.783293 |
84,492 | The range ambiguity resolution techniques used with medium pulse repetition frequency radar can be seen as a special case of the Chinese remainder theorem. | Linear congruence theorem | 0.783293 |
84,493 | The result was later generalized with a complete solution called Da-yan-shu (大衍術) in Qin Jiushao's 1247 Mathematical Treatise in Nine Sections which was translated into English in early 19th century by British missionary Alexander Wylie. The notion of congruences was first introduced and used by Carl Friedrich Gauss in... | Linear congruence theorem | 0.783293 |
84,494 | Sunzi's work contains neither a proof nor a full algorithm. What amounts to an algorithm for solving this problem was described by Aryabhata (6th century). Special cases of the Chinese remainder theorem were also known to Brahmagupta (7th century), and appear in Fibonacci's Liber Abaci (1202). | Linear congruence theorem | 0.783293 |
84,495 | The earliest known statement of the theorem, as a problem with specific numbers, appears in the 3rd-century book Sunzi Suanjing by the Chinese mathematician Sunzi: There are certain things whose number is unknown. If we count them by threes, we have two left over; by fives, we have three left over; and by sevens, two a... | Linear congruence theorem | 0.783293 |
84,496 | {\displaystyle K.} Both Lagrange interpolation and Chinese remainder theorem assert the existence of a unique polynomial P ( X ) , {\displaystyle P(X),} of degree less than k {\displaystyle k} such that P ( x i ) = A i , {\displaystyle P(x_{i})=A_{i},} for every i . {\displaystyle i.} | Linear congruence theorem | 0.783293 |
84,497 | They are pairwise coprime if the x i {\displaystyle x_{i}} are all different. The remainder of the division by P i ( X ) {\displaystyle P_{i}(X)} of a polynomial P ( X ) {\displaystyle P(X)} is P ( x i ) {\displaystyle P(x_{i})} , by the polynomial remainder theorem. Now, let A 1 , … , A k {\displaystyle A_{1},\ldots ,... | Linear congruence theorem | 0.783293 |
84,498 | A special case of Chinese remainder theorem for polynomials is Lagrange interpolation. For this, consider k monic polynomials of degree one: P i ( X ) = X − x i . {\displaystyle P_{i}(X)=X-x_{i}.} | Linear congruence theorem | 0.783293 |
84,499 | The Chinese remainder theorem can be generalized to non-coprime moduli. Let m , n , a , b {\displaystyle m,n,a,b} be any integers, let g = gcd ( m , n ) {\displaystyle g=\gcd(m,n)} ; M = lcm ( m , n ) {\displaystyle M=\operatorname {lcm} (m,n)} , and consider the system of congruences: x ≡ a ( mod m ) x ≡ b ( mod n )... | Linear congruence theorem | 0.783293 |
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