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94,800 | In the 1970s, researchers studied resilience in relation to child psychology and the exposure to certain risks. Resilience was used to describe people who have “the ability to recover from adversity.” One of the many researchers was Professor Sir Michael Rutter, who was concerned with a combination of risk experiences ... | Resilience (engineering and construction) | 0.778974 |
94,801 | The SHV (safe high voltage) connector is a type of RF connector used for terminating a coaxial cable. The connector uses a bayonet mount similar to those of the BNC and MHV connectors, but is easily distinguished due to its very thick and protruding insulator. This insulation geometry makes SHV connectors safer for han... | SHV connector | 0.778972 |
94,802 | Process–architecture–optimization is a development model for central processing units (CPUs) that Intel adopted in 2016. Under this three-phase (three-year) model, every microprocessor die shrink is followed by a microarchitecture change and then by one or more optimizations. It replaced the two-phase (two-year) tick–t... | Process–architecture–optimization model | 0.778971 |
94,803 | In topology, the Tychonoff plank is a topological space defined using ordinal spaces that is a counterexample to several plausible-sounding conjectures. It is defined as the topological product of the two ordinal spaces {\displaystyle } and {\displaystyle } , where ω {\displaystyle \omega } is the first infinite ordi... | Tychonoff plank | 0.778971 |
94,804 | For the MQ, its shape parameter largely determines its interpolation error. There exist a number of mathematical theories concerning the family of multiquadric radial basis functions and providing some suggestions on the choice of the shape parameter.The Kansa method are widely been applied in computational sciences. I... | Kansa method | 0.778969 |
94,805 | This method does not need weighted residuals (Galerkin), integration, or advanced mathematics. E. J. Kansa in very early 1990s made the first attempt to extend radial basis function (RBF), which was then quite popular in scattered data processing and function approximation, to the solution of partial differential equat... | Kansa method | 0.778969 |
94,806 | The Kansa method is a computer method used to solve partial differential equations. Its main advantage is it is very easy to understand and program on a computer. It is much less complicated than the finite element method. Another advantage is it works well on multi variable problems. | Kansa method | 0.778969 |
94,807 | Mathematicians prefer the latter for its rigorous and solid theoretical foundation, while engineering users often employ the former since it is easier and simpler and produces the sound results in the majority of cases. Substituting Eq. (4) or (5) into Eqs. (1–3) yields the resulting algebraic equation system: A α = b ... | Kansa method | 0.778969 |
94,808 | Leibniz expressed the law in the following terms in 1701: In any supposed continuous transition, ending in any terminus, it is permissible to institute a general reasoning, in which the final terminus may also be included (Cum Prodiisset).In a 1702 letter to French mathematician Pierre Varignon subtitled “Justification... | Law of continuity | 0.778968 |
94,809 | The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore,... | Harmonic oscillator (quantum) | 0.778967 |
94,810 | In the phase space formulation of quantum mechanics, eigenstates of the quantum harmonic oscillator in several different representations of the quasiprobability distribution can be written in closed form. The most widely used of these is for the Wigner quasiprobability distribution. The Wigner quasiprobability distribu... | Harmonic oscillator (quantum) | 0.778967 |
94,811 | The Sri Lanka Institute of Nanotechnology (Sinhala: ශ්රී ලංකා නිනිති තාක්ෂණ ආයතනය; Tamil: இலங்கை நனோ தொழில்நுட்ப நிறுவனம்) (abbreviated and commonly known as SLINTEC) is a Sri Lankan research institute specialising in the field of nanotechnology. It was incorporated in 2008 as a public-private partnership between the ... | Sri Lanka Institute of Nanotechnology | 0.778967 |
94,812 | SLINTEC was launched officially on 1 December 2008, with initial work taking place in a garage; formal research work commenced on 12 August 2009, operating out of MAS Holdings' Silueta complex within the Biyagama Export Processing Zone, with research staff drawn from local universities. In 2010, NANCO was merged into S... | Sri Lanka Institute of Nanotechnology | 0.778967 |
94,813 | Due to a lack of funds, however, the project stagnated until the efforts of Ravi Silva, a Sri Lankan professor and nanotechnology specialist at the University of Surrey, who persuaded five private companies to consider investing in the venture. SLINTEC was incorporated as a private company in April 2008. In September o... | Sri Lanka Institute of Nanotechnology | 0.778967 |
94,814 | The National Nanotechnology Initiative was formed informally in the early 2000s as a collaborative effort between a group of expatriate Sri Lankan scientists and officials from the NSF and the MoST, seeking to promote nanotechnology research in Sri Lanka. The Initiative cited several other related objectives: developin... | Sri Lanka Institute of Nanotechnology | 0.778967 |
94,815 | The Academy of the Sri Lanka Institute of Nanotechnology (also known as SLINTEC Academy) is a private non-profit graduate school, founded as SLINTEC's knowledge dissemination arm. The school offers MPhil and PhD degrees in Nano- and Advanced Sciences, and was formally inaugurated on 22 September 2017 at the Lakshman Ka... | Sri Lanka Institute of Nanotechnology | 0.778967 |
94,816 | As part of the agreement, NFCL invested a further US$0.8 million as a seed fund for SLINTEC to develop second- and third-generation nanotechnology-based fertilizer products for NFCL. Another with LAUGFS Holdings, wherein the two companies would establish a pilot plant to produce titanium dioxide from the country's larg... | Sri Lanka Institute of Nanotechnology | 0.778967 |
94,817 | {\displaystyle {\frac {S_{n}}{n}}\ {\xrightarrow {p}}\ 0,\qquad {\frac {S_{n}}{n}}\ {\xrightarrow {a.s. }}0,\qquad {\text{as}}\ \ n\to \infty .} On the other hand, the central limit theorem states that the sums Sn scaled by the factor n−½ converge in distribution to a standard normal distribution. | Law of the iterated logarithm | 0.778965 |
94,818 | The law of iterated logarithms operates “in between” the law of large numbers and the central limit theorem. There are two versions of the law of large numbers — the weak and the strong — and they both state that the sums Sn, scaled by n−1, converge to zero, respectively in probability and almost surely: S n n → p 0 , ... | Law of the iterated logarithm | 0.778965 |
94,819 | In probability theory, the law of the iterated logarithm describes the magnitude of the fluctuations of a random walk. The original statement of the law of the iterated logarithm is due to A. Ya. Khinchin (1924). Another statement was given by A. N. Kolmogorov in 1929. | Law of the iterated logarithm | 0.778965 |
94,820 | Hartman–Wintner (1940) generalized LIL to random walks with increments with zero mean and finite variance. De Acosta (1983) gave a simple proof of the Hartman–Wintner version of the LIL.Chung (1948) proved another version of the law of the iterated logarithm for the absolute value of a brownian motion.Strassen (1964) s... | Law of the iterated logarithm | 0.778965 |
94,821 | Database description messages contain descriptions of the topology of the autonomous system or area. They convey the contents of the link-state database (LSDB) for the area from one router to another. Communicating a large LSDB may require several messages to be sent by having the sending device designated as a master ... | Open Shortest Path First | 0.778963 |
94,822 | On networks (same subnet) with networks type of: Broadcast Non-Broadcast Multi-Access (NBMA)A system of designated router (DR) and backup designated router (BDR), is used to reducing network traffic by providing a source for routing updates. This is done using multicast addresses: 224.0.0.5, all routers in the topology... | Open Shortest Path First | 0.778963 |
94,823 | This means that it can also be used on non-IP networks, such as optical networks. OSPF-TE is used in GMPLS networks as a means to describe the topology over which GMPLS paths can be established. GMPLS uses its own path setup and forwarding protocols, once it has the full network map. In the Resource Reservation Protoco... | Open Shortest Path First | 0.778963 |
94,824 | OSPF-TE is an extension to OSPF extending the expressivity to allow for traffic engineering and use on non-IP networks. Using OSPF-TE, more information about the topology can be exchanged using opaque LSA carrying type–length–value elements. These extensions allow OSPF-TE to run completely out of band of the data plane... | Open Shortest Path First | 0.778963 |
94,825 | OSPF is an interior gateway protocol (IGP) for routing Internet Protocol (IP) packets within a single routing domain, such as an autonomous system. It gathers link state information from available routers and constructs a topology map of the network. The topology is presented as a routing table to the Internet Layer wh... | Open Shortest Path First | 0.778963 |
94,826 | A network is divided into OSPF areas that are logical groupings of hosts and networks. An area includes its connecting router having an interface for each connected network link. Each router maintains a separate link-state database for the area whose information may be summarized towards the rest of the network by the ... | Open Shortest Path First | 0.778963 |
94,827 | The topology is presented as a routing table to the Internet Layer for routing packets by their destination IP address. OSPF supports Internet Protocol Version 4 (IPv4) and Internet Protocol Version 6 (IPv6) networks and supports the Classless Inter-Domain Routing (CIDR) addressing model. OSPF is widely used in large e... | Open Shortest Path First | 0.778963 |
94,828 | Open Shortest Path First (OSPF) is a routing protocol for Internet Protocol (IP) networks. It uses a link state routing (LSR) algorithm and falls into the group of interior gateway protocols (IGPs), operating within a single autonomous system (AS). OSPF gathers link state information from available routers and construc... | Open Shortest Path First | 0.778963 |
94,829 | Unlike the simpler 1,3-diketone–containing compound acetylacetone (which forms acetylacetonate complexes with metals), the entire skeleton of curcumin is in resonance with the 1,3-dicarbonyl section, making the backbone an extended conjugated system. Investigations of the structure have shown that the positive charge i... | Rosocyanine | 0.778963 |
94,830 | The use of azides is very important in a variety of different applications in organic and biological chemistry. Azides are used in the research of drug applications, in materials science, and also all throughout biology. A majority of the research that is conducted on azides pertains to the catalyst that is needed to c... | 4-Chlorophenyl azide | 0.778962 |
94,831 | A simple way to make the iminium ion is to react an amine with formaldehyde so water leaves the reaction and favors the iminium creation. An example of this reaction is seen below: Aryl azides such as 4-chlorophenyl azide are important in click chemistry. 4-chlorophenyl azide is versatile in joining different molecules... | 4-Chlorophenyl azide | 0.778962 |
94,832 | When connected in an aromatic ring, the extra charge on the outside is an important aspect of the molecules reactivity. This extra outside charge is able to stabilize the positive charge on the middle nitrogen and allow for the release of nitrogen gas, an important step in reactions that include 4-chlorophenyl azide. T... | 4-Chlorophenyl azide | 0.778962 |
94,833 | 4-Chlorophenyl azide is an organic aryl azide compound with the chemical formula C6H4ClN3. The geometry between the nitrogen atoms in the azide functional group is approximately linear while the geometry between the nitrogen and the carbon of the benzene is trigonal planar. | 4-Chlorophenyl azide | 0.778962 |
94,834 | In differential geometry, a Sasakian manifold (named after Shigeo Sasaki) is a contact manifold ( M , θ ) {\displaystyle (M,\theta )} equipped with a special kind of Riemannian metric g {\displaystyle g} , called a Sasakian metric. | Sasakian manifold | 0.778962 |
94,835 | Sasakian manifolds were introduced in 1960 by the Japanese geometer Shigeo Sasaki. There was not much activity in this field after the mid-1970s, until the advent of String theory. Since then Sasakian manifolds have gained prominence in physics and algebraic geometry, mostly due to a string of papers by Charles P. Boye... | Sasakian manifold | 0.778962 |
94,836 | A deep theorem due to Conley asserts continuation invariance: Conley index is invariant under certain deformations of the dynamical system. Computation of the index can, therefore, be reduced to the case of the diffeomorphism or a vector field whose invariant sets are well understood. If the index is nontrivial then th... | Conley index theory | 0.778959 |
94,837 | 2,2,6,6-Tetramethylpiperidine, abbreviated TMP, HTMP, or TMPH, is an organic compound of the amine class. In appearance, it is a colorless liquid and has a "fishy", amine-like odor. This amine is used in chemistry as a hindered base (hindered amine). Although TMP finds limited use per se, its derivatives are a mainstay... | 2,2,6,6-Tetramethylpiperidine | 0.778958 |
94,838 | The generalized series at ξ ∈ C {\displaystyle \xi \in \mathbb {C} } is defined by ( z − ξ ) α ∑ k = 0 ∞ c k ( z − ξ ) k , with α , c k ∈ C and c 0 ≠ 0 , {\displaystyle (z-\xi )^{\alpha }\sum _{k=0}^{\infty }c_{k}(z-\xi )^{k},{\text{ with }}\alpha ,c_{k}\in \mathbb {C} {\text{ and }}c_{0}\neq 0,} which is known as Frob... | Fuchsian Theory | 0.778957 |
94,839 | The Fuchsian theory of linear differential equations, which is named after Lazarus Immanuel Fuchs, provides a characterization of various types of singularities and the relations among them. At any ordinary point of a homogeneous linear differential equation of order n {\displaystyle n} there exists a fundamental syste... | Fuchsian Theory | 0.778957 |
94,840 | This solution is considered to be the first theorem of graph theory.Euler also discovered the formula V − E + F = 2 {\displaystyle V-E+F=2} relating the number of vertices, edges, and faces of a convex polyhedron, and hence of a planar graph. The constant in this formula is now known as the Euler characteristic for the... | Leonhard Euler | 0.778956 |
94,841 | He proved that the relationship shown between even perfect numbers and Mersenne primes (which he had earlier proved) was one-to-one, a result otherwise known as the Euclid–Euler theorem. Euler also conjectured the law of quadratic reciprocity. The concept is regarded as a fundamental theorem within number theory, and h... | Leonhard Euler | 0.778956 |
94,842 | He proved that the sum of the reciprocals of the primes diverges. In doing so, he discovered the connection between the Riemann zeta function and prime numbers; this is known as the Euler product formula for the Riemann zeta function.Euler invented the totient function φ(n), the number of positive integers less than or... | Leonhard Euler | 0.778956 |
94,843 | Euler's interest in number theory can be traced to the influence of Christian Goldbach, his friend in the St. Petersburg Academy. Much of Euler's early work on number theory was based on the work of Pierre de Fermat. Euler developed some of Fermat's ideas and disproved some of his conjectures, such as his conjecture th... | Leonhard Euler | 0.778956 |
94,844 | Euler worked in almost all areas of mathematics, including geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory, and other areas of physics. He is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental int... | Leonhard Euler | 0.778956 |
94,845 | The 31 volumes of Series II, amounting to 10,660 pages, contain the works on mechanics, astronomy, and engineering. Series III contains 12 volumes on physics. Series IV, which contains the massive amount of Euler's correspondences, unpublished manuscripts, and notes only began compilation in 1967. The series is project... | Leonhard Euler | 0.778956 |
94,846 | Fortunately, the publication of Euler's Opera Omnia has made steady progress, with over 70 volumes (averaging 426 pages each) published by 2006 and 80 volumes published by 2022. These volumes are organized into four series. The first series compiles the works on analysis, algebra, and number theory; it consists of 29 v... | Leonhard Euler | 0.778956 |
94,847 | Euler has an extensive bibliography. His books include: Mechanica (1736) Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti (1744) (A method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimet... | Leonhard Euler | 0.778956 |
94,848 | Vanity of geometry! Throughout his stay in Berlin, Euler maintained a strong connection to the academy in St. Petersburg and also published 109 papers in Russia. | Leonhard Euler | 0.778956 |
94,849 | In 1753 he bought a house in Charlottenburg, in which he lived with his family and widowed mother.Euler became the tutor for Friederike Charlotte of Brandenburg-Schwedt, the Princess of Anhalt-Dessau and Frederick's niece. He wrote over 200 letters to her in the early 1760s, which were later compiled into a volume enti... | Leonhard Euler | 0.778956 |
94,850 | Concerned about the continuing turmoil in Russia, Euler left St. Petersburg in June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia. He lived for 25 years in Berlin, where he wrote several hundred articles. In 1748 his text on functions called the Introductio in... | Leonhard Euler | 0.778956 |
94,851 | The most notable of these approximations are Euler's method and the Euler–Maclaurin formula.Euler helped develop the Euler–Bernoulli beam equation, which became a cornerstone of engineering. Besides successfully applying his analytic tools to problems in classical mechanics, Euler applied these techniques to celestial ... | Leonhard Euler | 0.778956 |
94,852 | Some of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of the Bernoulli numbers, Fourier series, Euler numbers, the constants e and π, continued fractions, and integrals. He integrated Leibniz's differential calculus with Newton's Method of Fluxions,... | Leonhard Euler | 0.778956 |
94,853 | One of the foremost examples was Euler's solution to the Basel problem, a then 150-year-old unsolved in Analysis. The problem consisted on finding the result of the sum of the infinite series of the naturals' squared integers reciprocals; Euler eventually found that this series equaled exactly π2/6. Euler has also cred... | Leonhard Euler | 0.778956 |
94,854 | The nobility, suspicious of the academy's foreign scientists, cut funding for Euler and his colleagues and prevented the entrance of foreign and non-aristocratic students into the Gymnasium and universities.Conditions improved slightly after the death of Peter II in 1730 and the German-influenced Anna of Russia assumed... | Leonhard Euler | 0.778956 |
94,855 | Johann Bernoulli's two sons, Daniel and Nicolaus, entered into service at the Imperial Russian Academy of Sciences in Saint Petersburg in 1725, leaving Euler with the assurance they would recommend him to a post when one was available. On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Ru... | Leonhard Euler | 0.778956 |
94,856 | In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory. In breaking ground for this new field, Euler created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions, and the analytic theory of continued fractions. For example, h... | Leonhard Euler | 0.778956 |
94,857 | He found a way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis. He invented the calculus of variations and formulated the Euler–Lagrange equation for reducing optimization problems in this area to the solution of differential equations. Euler pioneered the use of ana... | Leonhard Euler | 0.778956 |
94,858 | The development of infinitesimal calculus was at the forefront of 18th-century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs... | Leonhard Euler | 0.778956 |
94,859 | Euler is credited for popularizing the Greek letter π {\displaystyle \pi } (lowercase pi) to denote the ratio of a circle's circumference to its diameter, as well as first using the notation f ( x ) {\displaystyle f(x)} for the value of a function, the letter i {\displaystyle i} to express the imaginary unit − 1 {\disp... | Leonhard Euler | 0.778956 |
94,860 | Leonhard Euler ( OY-lər, German: ; 15 April 1707 – 18 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician, and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number t... | Leonhard Euler | 0.778956 |
94,861 | Given the information on footprint and capabilities, the uCDN may proceed to the initial selection of a dCDN—first on the basis of footprint and then on the basis of capabilities. However, such procedures may lead to suboptimal or incorrect decisions; for example when the dCDN is selected on the basis of footprint, it ... | Content delivery network interconnection | 0.778955 |
94,862 | The Internet Engineering Task Force (IETF) (see Standardisation status section) defines five interfaces required to interconnect a pair of CDNs from a technical perspective, as depicted in Figure 2. The interfaces are control plane interfaces operating at the application layer that aim to reuse or leverage existing pro... | Content delivery network interconnection | 0.778955 |
94,863 | acib is developing more environmentally friendly and economic processes for the biotechnological, pharmaceutical and chemical industries. All these processes are modelled on methods and tools from nature. Research areas include biocatalysis and chemical analytics, enzyme technologies and protein engineering, microbial ... | Austrian Centre of Industrial Biotechnology | 0.778952 |
94,864 | The Austrian Centre of Industrial Biotechnology (ACIB) is an international research institution for industrial biotechnology. Research facilities are located in Graz, Linz, Innsbruck, Tulln and Vienna, with additional research sites in Heidelberg, Bielefeld, Pavia, Rzeszów, Barcelona, Canterbury and Taiwan. The adminis... | Austrian Centre of Industrial Biotechnology | 0.778952 |
94,865 | Scientists also already knew that nitrogen formed the dominant portion of the atmosphere, but manmade chemistry had yet to establish a means to fix it. Then, in 1909, German chemist Fritz Haber successfully fixed atmospheric nitrogen in a laboratory. This success had extremely attractive military, industrial and agricu... | History of the Haber process | 0.778952 |
94,866 | A few years later, Haber used results published by Nernst on the chemical equilibrium of ammonia and his own familiarity with high pressure chemistry and the liquefaction of air, to develop a new nitrogen fixation process. He had no precise information on the parameters to impose on the system, but at the conclusion of... | History of the Haber process | 0.778952 |
94,867 | In 1905, German chemist Fritz Haber published Thermodynamik technischer Gasreaktionen (The Thermodynamics of Technical Gas Reactions), a book more concerned about the industrial application of chemistry than to its theoretical study. In it, Haber inserted the results of his study of the equilibrium equation of ammonia:... | History of the Haber process | 0.778952 |
94,868 | When one of his assistants was killed following the accidental explosion of a device, Le Chatelier decided to end his research.Americans Bradley and Lovejoy, specialists in electrochemistry, developed a method of producing nitric acid using electric arcs. The industrial manufacture of nitric acid using this method bega... | History of the Haber process | 0.778952 |
94,869 | A major consumer of Chilean saltpeter, Germany saltpeter imports totaled 350,000 tonnes in 1900. Twelve years later, it imported 900,000 tonnes. The United States was in much better position due to the Guano Islands Act.In the years between 1890 and 1900, chemistry advanced on several fronts, and more scientists attemp... | History of the Haber process | 0.778952 |
94,870 | This element also has two meta states, 45mCr, the more stable one, and 59mCr, the least stable isotope or isomer. 53Cr is the radiogenic decay product of 53Mn. Chromium isotopic contents are typically combined with manganese isotopic contents and have found application in isotope geology. | Chromium-43 | 0.778952 |
94,871 | In this case L(a,b)c = abc + cba with inner product (a,b) = −Tr ab. After removing a factor of √(-1), this is Koecher's construction applied to complex conjugation on E = Hn(C). F is the direct sum of two copies of the Cayley numbers, regarded as 1 by 2 matrices. This triple system is obtained by Koecher's construction... | Jordan frame (Jordan algebra) | 0.778952 |
94,872 | Then F = E−1 is a simple Euclidean Jordan triple system, obtained by restricting the triple system on E to F. Koecher exhibits explicit involutions of simple Euclidean Jordan algebras directly (see below). These Jordan triple systems correspond to irreducible Hermitian symmetric spaces given by Siegel domains of the se... | Jordan frame (Jordan algebra) | 0.778952 |
94,873 | The remaining cases are constructed uniformly by Koecher using involutions of simple Euclidean Jordan algebras. Let E be a simple Euclidean Jordan algebra and τ a Jordan algebra automorphism of E of period 2. Thus E = E+1 ⊕ E−1 has an eigenspace decomposition for τ with E+1 a Jordan subalgebra and E−1 a module. | Jordan frame (Jordan algebra) | 0.778952 |
94,874 | Requiring the existence of an involution σ leads directly to a Jordan triple structure on V as above. As for all Jordan triple structures, fixing c in E, the operators Lc(a) = L(a,c) give E a Jordan algebra structure, determined by e. The operators L(a,b) themselves come from a Jordan algebra structure as above if and ... | Jordan frame (Jordan algebra) | 0.778952 |
94,875 | These formulas, originally derived for Jordan algebras, work equally well for Jordan triple systems. The account in Koecher (1969) develops the theory of bounded symmetric domains starting from the standpoint of 3-graded Lie algebras. For a given finite-dimensional vector space E, Koecher considers finite-dimensional L... | Jordan frame (Jordan algebra) | 0.778952 |
94,876 | In this case L ( a , b ) = a d {\displaystyle \displaystyle {L(a,b)=\mathrm {ad} \,}} defines a Jordan triple system on g − 1 {\displaystyle {\mathfrak {g}}_{-1}} . In the case of Euclidean Jordan algebras or triple systems the Kantor–Koecher–Tits construction can be identified with the Lie algebra of the Lie group of... | Jordan frame (Jordan algebra) | 0.778952 |
94,877 | {\displaystyle \displaystyle {=L(L(a,b)c,d)-L(c,L(b,a)d).}} Any such bilinear system is called a Euclidean Jordan triple system. By definition the operators L(a,b) form a Lie subalgebra of End E. The Kantor–Koecher–Tits construction gives a one-one correspondence between Jordan triple systems and 3-graded Lie algebras ... | Jordan frame (Jordan algebra) | 0.778952 |
94,878 | In fact for a Euclidean Jordan algebra E let L ( a , b ) = 2 ( + L ( a b ) ) . {\displaystyle \displaystyle {L(a,b)=2(+L(ab)).}} Then L(a,b) gives a bilinear map into End E such that L ( a , b ) ∗ = L ( b , a ) , L ( a , b ) c = L ( c , b ) a {\displaystyle \displaystyle {L(a,b)^{*}=L(b,a)},\,\,\,L(a,b)c=L(c,b)a} and ... | Jordan frame (Jordan algebra) | 0.778952 |
94,879 | Euclidean Jordan algebras can be used to construct Hermitian symmetric spaces of tube type. The remaining Hermitian symmetric spaces are Siegel domains of the second kind. They can be constructed using Euclidean Jordan triple systems, a generalization of Euclidean Jordan algebras. | Jordan frame (Jordan algebra) | 0.778952 |
94,880 | Aut EC is the complexification of the compact Lie group Aut E in GL(EC). This follows because the Lie algebras of Aut EC and Aut E consist of derivations of the complex and real Jordan algebras EC and E. Under the isomorphism identifying End EC with the complexification of End E, the complex derivations is identified w... | Jordan frame (Jordan algebra) | 0.778952 |
94,881 | The complexification of a simple Euclidean Jordan algebra is a simple complex Jordan algebra which is also separable, i.e. its trace form is non-degenerate. Conversely, using the existence of a real form of the Lie algebra of the structure group, it can be shown that every complex separable simple Jordan algebra is the... | Jordan frame (Jordan algebra) | 0.778952 |
94,882 | The inequality therefore follows in any special Euclidean Jordan algebra and hence in general.On the other hand, by the Krein–Milman theorem, the closed unit ball is the (closed) convex span of S. It follows that ||L(u)|| = 1, in the operator norm corresponding to either the inner product norm or spectral norm. Hence |... | Jordan frame (Jordan algebra) | 0.778952 |
94,883 | The inequality is easy to establish in non-exceptional simple Euclidean Jordan algebras, since each such Jordan algebra and its complexification can be realized as a subalgebra of some Hn(R) and its complexification Hn(C) ⊂ Mn(C). The spectral norm in Hn(C) is the usual operator norm. In that case, for unitary matrices... | Jordan frame (Jordan algebra) | 0.778952 |
94,884 | Each u in S has norm one. Moreover, if u = eia and v = eib, then ||uv|| ≤ 1. Indeed, by the Cohn–Shirshov theorem the unital Jordan subalgebra of E generated by a and b is special. | Jordan frame (Jordan algebra) | 0.778952 |
94,885 | The fundamental identity for the quadratic representation implies that ||Q(a)b|| ≤ ||a||2||b||. The spectral norm of an element a is defined in terms of C so depends only on a and not the particular Euclidean Jordan algebra in which it is calculated.The compact set S is the set of extreme points of the closed unit ball... | Jordan frame (Jordan algebra) | 0.778952 |
94,886 | Let E be a Euclidean Jordan algebra with the inner product given by the trace form. Let (ei) be a fixed Jordan frame in E. For given a in EC choose u in Γu(EC) such that ua = Σ αi ei with αi ≥ 0. Then the spectral norm ||a|| = max αi is independent of all choices. It is a norm on EC with ‖ a ∗ ‖ = ‖ a ‖ , ‖ { a , a ∗ ,... | Jordan frame (Jordan algebra) | 0.778952 |
94,887 | Let E be a Euclidean Jordan algebra with complexification A = EC = E + iE. The unit ball or disk D in A is just the convex bounded open set of elements a such the ||a|| < 1, i.e. the unit ball for the spectral norm. The tube domain T in A is the unbounded convex open set T = E + iC, where C is the open positive cone in... | Jordan frame (Jordan algebra) | 0.778952 |
94,888 | The classification of simple Euclidean Jordan algebras was accomplished by Jordan, von Neumann & Wigner (1934), with details of the one exceptional algebra provided in the article immediately following theirs by Albert (1934). Using the Peirce decomposition, they reduced the problem to an algebraic problem involving mu... | Jordan frame (Jordan algebra) | 0.778952 |
94,889 | Because A is associative when A = R, C or H, it is immediate that Hr(A) is a Jordan algebra for r ≥ 3. A separate argument, given originally by Albert (1934), is required to show that H3(O) with Jordan product a∘b = 1/2(ab + ba) satisfies the Jordan identity = 0. There is a later more direct proof using the Freudentha... | Jordan frame (Jordan algebra) | 0.778952 |
94,890 | )Such an algebra A is called a Euclidean Hurwitz algebra. In A if λ(a)b = ab and ρ(a)b = ba, then: the involution is an antiautomorphism, i.e. (a b)*=b* a* a a* = ‖ a ‖2 1 = a* a λ(a*) = λ(a)*, ρ(a*) = ρ(a)*, so that the involution on the algebra corresponds to taking adjoints Re(a b) = Re(b a) if Re x = (x + x*)/2 = (... | Jordan frame (Jordan algebra) | 0.778952 |
94,891 | Let E be a simple Euclidean Jordan algebra. From the properties of the Peirce decomposition it follows that: If E has rank 2, then it has the form V ⊕ R for some inner product space V with Jordan product as described above. If E has rank r > 2, then there is a non-associative unital algebra A, associative if r > 3, equ... | Jordan frame (Jordan algebra) | 0.778952 |
94,892 | From the Jordan condition it follows that the Jordan algebra is power associative, i.e. the Jordan subalgebra generated by any single element a in E is actually an associative commutative algebra. Thus, defining an inductively by an = a (an−1), the following associativity relation holds: a m a n = a m + n , {\displayst... | Jordan frame (Jordan algebra) | 0.778952 |
94,893 | In mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries, i.e. invertible operators that take the cone onto itself. By the Koecher–Vinberg theorem these correspond to the cone of squares in finite-dimensional ... | Jordan frame (Jordan algebra) | 0.778952 |
94,894 | It was also under investigation for the treatment of sleep apnea, but no recent development has been reported for this indication. Seltorexant is under development by Minerva Neurosciences and Johnson & Johnson's Janssen Pharmaceuticals.Seltorexant is being explored at doses of 5 to 80 mg. In the early clinical trials ... | Seltorexant | 0.77895 |
94,895 | The orbifold construction. In physical terms, this describes a bosonic string propagating on a quotient orbifold. The construction of Frenkel–Lepowsky–Meurman was the first time orbifolds appeared in conformal field theory. Attached to the –1 involution of the Leech lattice, there is an involution h of VL, and an irred... | Conway-Norton conjecture | 0.77895 |
94,896 | The Frenkel–Lepowsky–Meurman construction starts with two main tools: The construction of a lattice vertex operator algebra VL for an even lattice L of rank n. In physical terms, this is the chiral algebra for a bosonic string compactified on a torus Rn/L. It can be described roughly as the tensor product of the group ... | Conway-Norton conjecture | 0.77895 |
94,897 | (Roberts 2009, p. 361) More recent work has simplified and clarified the last steps of the proof. Jurisich (Jurisich (1998), Jurisich, Lepowsky & Wilson (1995)) found that the homology computation could be substantially shortened by replacing the usual triangular decomposition of the Monster Lie algebra with a decompos... | Conway-Norton conjecture | 0.77895 |
94,898 | Using the Goddard–Thorn "no-ghost" theorem from string theory, the root multiplicities are found to be coefficients of J. One uses the Koike–Norton–Zagier infinite product identity to construct a generalized Kac–Moody Lie algebra by generators and relations. The identity is proved using the fact that Hecke operators ap... | Conway-Norton conjecture | 0.77895 |
94,899 | Richard Borcherds' proof of the conjecture of Conway and Norton can be broken into the following major steps: One begins with a vertex operator algebra V with an invariant bilinear form, an action of M by automorphisms, and with known decomposition of the homogeneous spaces of seven lowest degrees into irreducible M-re... | Conway-Norton conjecture | 0.77895 |
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