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17,100 | In mathematics, specifically group theory, the identity component of a group G refers to several closely related notions of the largest connected subgroup of G containing the identity element. In point set topology, the identity component of a topological group G is the connected component G0 of G that contains the ide... | Component group | 0.813182 |
17,101 | The quotient group G/G0 is called the group of components or component group of G. Its elements are just the connected components of G. The component group G/G0 is a discrete group if and only if G0 is open. If G is an algebraic group of finite type, such as an affine algebraic group, then G/G0 is actually a finite gro... | Component group | 0.813182 |
17,102 | The meaning of these for a closed ribbon is explained and also that of the writhing number of a closed curve. Some simple examples are given, some of which may be relevant to the structure of chromatin. Analysis of DNA topology uses three values: L = linking number - the number of times one DNA strand wraps around the ... | Nucleic acid double helix | 0.813179 |
17,103 | tensor The tensor product algebra of associative algebras is the tensor product of the algebras as the modules with component multiplication The tensor algebra of a vector space or a module V is the direct sum of all tensor powers V ⊗ n {\displaystyle V^{\otimes n}} with the multiplication given by tensor product. triv... | Glossary of ring theory | 0.813172 |
17,104 | For ring-theoretic concepts in the language of modules, see also Glossary of module theory. For specific types of algebras, see also: Glossary of field theory and Glossary of Lie groups and Lie algebras. Since, currently, there is no glossary on not-necessarily-associative algebra-structures in general, this glossary i... | Glossary of ring theory | 0.813172 |
17,105 | Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject. For the items in commutative algebra (the theory of commutative rings), see glossary of commutative algebra. | Glossary of ring theory | 0.813172 |
17,106 | opposite Given a ring R, its opposite ring Rop has the same underlying set as R, the addition operation is defined as in R, but the product of s and r in Rop is rs, while the product is sr in R. order An order of an algebra is (roughly) a subalgebra that is also a full lattice. Ore A left Ore domain is a (non-commutati... | Glossary of ring theory | 0.813172 |
17,107 | differential A differential algebra is an algebra together with a derivation. direct A direct product of a family of rings is a ring given by taking the cartesian product of the given rings and defining the algebraic operations component-wise. | Glossary of ring theory | 0.813172 |
17,108 | derivation 1. A derivation of a possibly-non-associative algebra A over a commutative ring R is an R-linear endomorphism that satisfies the Leibniz rule. 2. The derivation algebra of an algebra A is the subalgebra of the endomorphism algebra of A that consists of derivations. | Glossary of ring theory | 0.813172 |
17,109 | A biomarker can be any kind of molecule indicating the existence, past or present, of living organisms. In the fields of geology and astrobiology, biomarkers, versus geomarkers, are also known as biosignatures. The term biomarker is also used to describe biological involvement in the generation of petroleum. | Biological marker | 0.813167 |
17,110 | More recently developed 'binary optimizers' for various platforms, some claiming novelty but, nevertheless, essentially using the same (or similar) techniques described above, include: IBM Automatic Binary Optimizer for z/OS (ABO) (2015) IBM Automatic Binary Optimizer for z/OS (ABO) Trial Cloud Service (2020) The Sun S... | Binary optimizer | 0.813164 |
17,111 | Suppose that the surface X {\displaystyle X} is a closed Riemann surface of genus g {\displaystyle g} and that ω {\displaystyle \omega } is a nonzero holomorphic 1-form on X {\displaystyle X} , with zeroes of order d 1 , … , d m {\displaystyle d_{1},\ldots ,d_{m}} . Then the Riemann–Roch theorem implies that ∑ j = 1 m ... | Translation surface | 0.813159 |
17,112 | This gives S ∖ Σ {\displaystyle S\setminus \Sigma } the structure of a Riemann surface, which extends to the entire surface S {\displaystyle S} by Riemann's theorem on removable singularities. In addition, the differential d z {\displaystyle dz} where z: U → C {\displaystyle z:U\to \mathbb {C} } is any chart defined ab... | Translation surface | 0.813159 |
17,113 | L-Photo-leucine and L-photo-methionine are analogs of the naturally occurring L-leucine and L-methionine amino acids that are endogenously incorporated into the primary sequence of proteins during synthesis using the normal translation machinery. They are then ultraviolet light (UV)-activated to covalently crosslink pr... | Photoreactive amino acid | 0.813151 |
17,114 | However, compounds of adjugates can be expressed using adjugates of compounds, and vice versa. From the identities C r ( C s ( A ) ) C r ( adj s ( A ) ) = ( det A ) r I , {\displaystyle C_{r}(C_{s}(A))C_{r}(\operatorname {adj} _{s}(A))=(\det A)^{r}I,} C r ( C s ( A ) ) adj r ( C s ( A ) ) = ( det C s ( A ) ) I , {\... | Compound matrix | 0.81315 |
17,115 | {\displaystyle J={\begin{pmatrix}&&1\\&\cdots &\\1&&\end{pmatrix}}.} Then Jacobi's theorem states that the r th higher adjugate matrix is: adj r ( A ) = J C n − r ( S A S ) T J . {\displaystyle \operatorname {adj} _{r}(A)=JC_{n-r}(SAS)^{T}J.} | Compound matrix | 0.81315 |
17,116 | He was a lifelong regular contributor to the Educational Times. His correspondents included the logicians William Stanley Jevons and Charles Sanders Peirce. He also corresponded, and argued in print, with the young Bertrand Russell, and reviewed Alfred North Whitehead's 1898 Universal Algebra for Mind magazine. | Calculus of equivalent statements | 0.81315 |
17,117 | MacColl is known for three main accomplishments in formal and philosophical logic: During 1877–1879, while working out a problem involving integration, he published a four-part article establishing the first known variant of the propositional calculus, terming it the "calculus of equivalent statements", preceding Gottl... | Calculus of equivalent statements | 0.81315 |
17,118 | In linear algebra, a branch of mathematics, a (multiplicative) compound matrix is a matrix whose entries are all minors, of a given size, of another matrix. Compound matrices are closely related to exterior algebras, and their computation appears in a wide array of problems, such as in the analysis of nonlinear time-va... | Compound matrix | 0.81315 |
17,119 | If 1 ≤ r ≤ min(m, n), then Cr (AT) = Cr (A)T. If 1 ≤ r ≤ min(m, n), then Cr (A*) = Cr (A)*. Cr (AB) = Cr (A) Cr (B), which is closely related to Cauchy–Binet formula.Assume in addition that A is a square matrix of size n. Then: Cn (A) = det A. If A has one of the following properties, then so does Cr (A): Upper triangu... | Compound matrix | 0.81315 |
17,120 | It follows immediately from Jacobi's theorem that C r ( A ) J ( C n − r ( S A S ) ) T J = ( det A ) I ( n r ) . {\displaystyle C_{r}(A)\,J(C_{n-r}(SAS))^{T}J=(\det A)I_{\binom {n}{r}}.} Taking adjugates and compounds does not commute. | Compound matrix | 0.81315 |
17,121 | In physics and quantum chemistry, specifically density functional theory, the Kohn–Sham equation is the non-interacting Schrödinger equation (more clearly, Schrödinger-like equation) of a fictitious system (the "Kohn–Sham system") of non-interacting particles (typically electrons) that generate the same density as any ... | Kohn–Sham equations | 0.813149 |
17,122 | Incremental heuristic search has been extensively used in robotics, where a larger number of path planning systems are based on either D* (typically earlier systems) or D* Lite (current systems), two different incremental heuristic search algorithms. | Incremental heuristic search | 0.813147 |
17,123 | Heuristic search algorithms, often based on A*, use heuristic knowledge in the form of approximations of the goal distances to focus the search and solve search problems potentially much faster than uninformed search algorithms. The resulting search problems, sometimes called dynamic path planning problems, are graph s... | Incremental heuristic search | 0.813147 |
17,124 | One of these students was Edwin Catmull. Catmull had just come from The Boeing Company and had been working on his degree in physics. Growing up on Disney, Catmull loved animation yet quickly discovered that he did not have the talent for drawing. | Triangle primitives | 0.813146 |
17,125 | The Association for Computing Machinery (ACM) is the world's largest educational and scientific computing society. It has its own Code of Ethics and another set of ethical principles that were also approved by the IEEE as the standard for teaching and practicing software engineering. These codes are Code of Ethics and ... | Programming ethics | 0.813142 |
17,126 | Ohio University also trains their computer science majors on the field of programming ethics. They expect every student to understand and follow ACM's principles and responsibilities. | Programming ethics | 0.813142 |
17,127 | This guideline prohibits programmer involvement in such unlawful actions. Improve their ability to create safe, reliable, and useful quality software. Since technology advances faster year by year, and so does virtual criminality, the need of well-structured and designed programs is increasing. Computer systems get old... | Programming ethics | 0.813142 |
17,128 | Software developers should write programs that can protect users’ private information and that can avoid other undesired people to have unauthorized access to it (Code of Ethics and Professional Conduct). Honor confidentiality. Unless required by law or any other ethical guideline, a programmer must keep secret any add... | Programming ethics | 0.813142 |
17,129 | He worked with several professional-ethics advocates to write ethical guidelines for creating codes of Ethics. He formed part of the creation of Code of Ethics and Professional Conduct, adopted by the Association for Computing Machinery (ACM) in 1992, and Software Engineering Code of Ethics and Professional Practice, a... | Programming ethics | 0.813142 |
17,130 | Those working in the generativist tradition often regard structuralist approaches as outdated and superseded. For example, Mitchell Marcus writes that structural linguistics was "fundamentally inadequate to process the full range of natural language". Holland writes that Chomsky had "decisively refuted Saussure". Simil... | Structural linguistics | 0.813141 |
17,131 | What is more, Saussure abandoned evolutionary linguistics altogether and, instead, defined synchronic analysis as the study of the language system; and diachronic analysis as the study of language change. With such precaution, structural explanation of language is analogous to structuralism in biology which explains st... | Structural linguistics | 0.813141 |
17,132 | In physics, the meter water equivalent (often m.w.e. or mwe) is a standard measure of cosmic ray attenuation in underground laboratories. A laboratory at a depth of 1000 m.w.e is shielded from cosmic rays equivalently to a lab 1,000 m (3,300 ft) below the surface of a body of water. Because laboratories at the same dep... | Metre water equivalent | 0.813141 |
17,133 | G acts on F by restriction of action of S. If the fixed field of this action is M, then, by the fundamental theorem of Galois theory, the Galois group of F/M is G. On the other hand, it is an open problem whether every finite group is the Galois group of a field extension of the field Q of the rational numbers. Igor Sh... | Solvable by radicals | 0.813138 |
17,134 | Choose a field K and a finite group G. Cayley's theorem says that G is (up to isomorphism) a subgroup of the symmetric group S on the elements of G. Choose indeterminates {xα}, one for each element α of G, and adjoin them to K to get the field F = K({xα}). Contained within F is the field L of symmetric rational functio... | Solvable by radicals | 0.813138 |
17,135 | Originally, the theory had been developed for algebraic equations whose coefficients are rational numbers. It extends naturally to equations with coefficients in any field, but this will not be considered in the simple examples below. These permutations together form a permutation group, also called the Galois group of... | Solvable by radicals | 0.813138 |
17,136 | 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 permutations (or rearrangements) of the roots such that any algebraic equation satisfied ... | Solvable by radicals | 0.813138 |
17,137 | His solution contained a gap, which Cauchy considered minor, though this was not patched until the work of the Norwegian mathematician Niels Henrik Abel, who published a proof in 1824, thus establishing the Abel–Ruffini theorem. While Ruffini and Abel established that the general quintic could not be solved, some parti... | Solvable by radicals | 0.813138 |
17,138 | After the discovery of del Ferro's work, he felt that Tartaglia's method was no longer secret, and thus he published his solution in his 1545 Ars Magna. His student Lodovico Ferrari solved the quartic polynomial; his solution was also included in Ars Magna. In this book, however, Cardano did not provide a "general form... | Solvable by radicals | 0.813138 |
17,139 | {\displaystyle {\begin{aligned}A&={\sqrt {2}}+{\sqrt {3}},\\B&={\sqrt {2}}-{\sqrt {3}},\\C&=-{\sqrt {2}}+{\sqrt {3}},\\D&=-{\sqrt {2}}-{\sqrt {3}}.\end{aligned}}} There are 24 possible ways to permute these four roots, but not all of these permutations are members of the Galois group. The members of the Galois group mu... | Solvable by radicals | 0.813138 |
17,140 | This widely generalizes the Abel–Ruffini theorem, which asserts that a general polynomial of degree at least five cannot be solved by radicals. Galois theory has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated (doubling the cube and trisecting th... | Solvable by radicals | 0.813138 |
17,141 | In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand. Galois int... | Solvable by radicals | 0.813138 |
17,142 | In mathematics, error analysis is the study of kind and quantity of error, or uncertainty, that may be present in the solution to a problem. This issue is particularly prominent in applied areas such as numerical analysis and statistics. | Error analysis (mathematics) | 0.813133 |
17,143 | The discrete dipole approximation is a flexible technique for computing scattering and absorption by targets of arbitrary geometry. The formulation is based on integral form of Maxwell equations. The DDA is an approximation of the continuum target by a finite array of polarizable points. | Electromagnetic modeling | 0.813131 |
17,144 | {\displaystyle (x,y)=(\cos t,\sin t).} Parametric representations are generally nonunique (see the "Examples in two dimensions" section below), so the same quantities may be expressed by a number of different parameterizations.In addition to curves and surfaces, parametric equations can describe manifolds and algebraic... | Parametric formula | 0.813131 |
17,145 | The underlying idea is that the expensive simulations with a detailed low-level system model are not always needed. Rather, for obtaining sufficient number of design points, approximate but faster evaluation methods are acceptable. Thus, the multilevel system modeling enables quick analysis of many design points using ... | MULTICUBE | 0.81313 |
17,146 | With the aim to reduce the design time of future embedded systems, the MULTICUBE project faces the problems related to multiobjective DSE of MPSoC platforms. The MULTICUBE project defines an automatic framework for DSE providing advanced methodologies for heuristic optimization and techniques for analyzing the effects ... | MULTICUBE | 0.81313 |
17,147 | Designing complex systems on chip many platform parameters has to be tuned. This is done in order to maximize platform performances while minimizing non functional costs such as the power consumption. This tuning phase is called Design Space Exploration (DSE). This process can be formalized as a multiobjective optimiza... | MULTICUBE | 0.81313 |
17,148 | MULTICUBE ("Multi-objective Design Space Exploration of MultiProcessor-SoC Architectures for Embedded Multimedia Applications") is a Seventh Framework Programme (FP7) project aimed to define innovative methods for the design optimization of computer architectures for the embedded system domain. | MULTICUBE | 0.81313 |
17,149 | In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from Ancient Greek: ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, hə-MO-tə-pee; , HOH-moh-t... | Homotopy classes | 0.813128 |
17,150 | A monoid whose operation is commutative is called a commutative monoid (or, less commonly, an abelian monoid). Commutative monoids are often written additively. Any commutative monoid is endowed with its algebraic preordering ≤, defined by x ≤ y if there exists z such that x + z = y. An order-unit of a commutative mono... | Complete monoid | 0.813128 |
17,151 | In particular, any bounded lattice can be endowed with both a meet- and a join- monoid structure. The identity elements are the lattice's top and its bottom, respectively. Being lattices, Heyting algebras and Boolean algebras are endowed with these monoid structures. | Complete monoid | 0.813127 |
17,152 | In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups. In two-dimensional geometry, the infinite dihedral group represents the frieze group symmetry, p1m1, seen as an infinite set of parallel reflections along an axis. | Infinite dihedral group | 0.813125 |
17,153 | The optimization of the particle size distribution facilitates the pumping, mixing and transportation of foodstuff. Particle size analysis is usually done with any milled food, such as coffee, flour, cocoa powder. It is especially helpful with chocolate quality to ensure there is a consistent taste and feeling when eat... | Particle size measurement | 0.813122 |
17,154 | Particle size analyzers are used also in biology to measure protein aggregation. DLS is a particularly appreciated technique for the characterization of nanoparticles designed for drug delivery, such as vaccines. DLS instruments are for instance part of the quality control process for mRNA vaccines formulated in lipid ... | Particle size measurement | 0.813122 |
17,155 | To overcome this issue, LD PSAs require appropriate sampling and dilution systems, which increase capital investments and operational costs. Another approach is to apply multiple scattering correction models together with the optical models to compute the PSD. A large number of algorithms for multiple scattering correc... | Particle size measurement | 0.813122 |
17,156 | Minds and Machines is a peer-reviewed academic journal covering artificial intelligence, philosophy, and cognitive science.The journal was established in 1991 with James Henry Fetzer as founding editor-in-chief. It was published by Kluwer Academic Publishers but was taken over by Springer in 2021 (Springer Science+Busi... | Minds and Machines | 0.813122 |
17,157 | Dr. Mohammad Karamudini had been International Biology Olympiad coordinator for Iran since 1998. IrBO is running inside the rules and regulations of International Biology Olympiad, and is a regular participant and a member of it. Iran hosted the International Biology Olympiad in 2018. | Iranian Biology Olympiad | 0.81312 |
17,158 | Iranian Biology Olympiad (IrBO) is an annual multistage competition for Iranian high school students of the age of 17-18 in the field of biology. The first Iranian Biology Olympiad was launched under the auspices of the Ministry of Education, Iran in 1999. Since then, the four winners of the national competitions parti... | Iranian Biology Olympiad | 0.81312 |
17,159 | IrBO was found in 1998, when Dr. Mohammad Karamudini, a biology educationist participated in the 9th IBO held in Kiel Germany as an observer from Iran. The first Iranian team participated in the 10th IBO held in Uppsala, Sweden in 1999, in which Iranian students won 2 silver and 1 bronze medals. Here are the results fo... | Iranian Biology Olympiad | 0.81312 |
17,160 | IrBO tried to challenge and stimulate gifted students to expand their talents and to promote their career as biologists. IrBO also is focusing on biology as a valuable subject. In offering a wider syllabus than Iranian National Biology Curriculum, it allows gifted students to demonstrate their theoretical knowledge and... | Iranian Biology Olympiad | 0.81312 |
17,161 | IrBO is fully supported by The Ministry of Education, Iran. IrBO National committee is responsible for all theoretical and practical exams and the results. Each student will pass 4 round of exams during the national competitions prior to become a team member. The IrBO national committee comprises university professors,... | Iranian Biology Olympiad | 0.81312 |
17,162 | The LRG concept was developed by the GEN2PHEN project in conjunction with the European Bioinformatics Institute (EBI) and the National Center for Biotechnology Information (NCBI). The LRG homepage provides access to existing LRG sequences and allows the submission of requests for the creation of new LRGs. This page als... | Locus Reference Genomic | 0.813119 |
17,163 | Chemistry: I. Physical Chemistry: General Studies of Substance ( Moles, Molarity, Redox Reactions, etc.), Atomic structure (with concerned topics of Quantum Mechanics), States of matter, Chemical Thermodynamics and Chemical Kinetics, Equilibrium chemistry (Both Chemical equilibrium and Ionic equilibrium), Electrochemis... | Joint Entrance Examination – Advanced | 0.813118 |
17,164 | Mathematics: Higher Algebra (including certain topics from Linear Algebra), Combinatorics, Probability (including topics like Conditional probability, Law of total probability, Bayes theorem), Geometry, Coordinate system (Point and Lines, Circles, Parabolas, Ellipse and Hyperbola), Trigonometry (including the Inverse t... | Joint Entrance Examination – Advanced | 0.813118 |
17,165 | Since the starting of the examination in 1961, the syllabus majorly consists of topics that are taught in Indian High schools, from the curriculum of Class XI and Class XII. These include topics from Mathematics, Physics and Chemistry (Organic Chemistry, Inorganic Chemistry and Physical Chemistry). A recent change in t... | Joint Entrance Examination – Advanced | 0.813118 |
17,166 | JEE(Advanced) is conducted in two shifts of three hours each - Paper-1 and Paper-2, both the papers consist questions from three major subjects- Physics, Chemistry and Mathematics. Unlike most of the other exams, the type and the number of questions being asked in the paper varies from year to year, with an average of ... | Joint Entrance Examination – Advanced | 0.813118 |
17,167 | IIT Tirupati and IIT Palakkad were started in 2015 and four more institutes (IIT Bhilai, IIT Dharwad, IIT Goa, and IIT Jammu) opened in 2016. In 2018, to ensure minimum female enrollment of 14%, the IITs introduced "female-only" and "gender-neutral" seats based on 2017 enrollment statistics; and "super-numerary" seats ... | Joint Entrance Examination – Advanced | 0.813118 |
17,168 | In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts,... | Homology (mathematics) | 0.813115 |
17,169 | A map f: A → X {\displaystyle f:A\to X} is called a cofibration if given (1) a map h 0: X → Z {\displaystyle h_{0}:X\to Z} and (2) a homotopy g t: A → Z {\displaystyle g_{t}:A\to Z} , there exists a homotopy h t: X → Z {\displaystyle h_{t}:X\to Z} that extends h 0 {\displaystyle h_{0}} and such that h t ∘ f = g t {\dis... | Homotopy theory | 0.813114 |
17,170 | In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated, or Hausdorff, or a CW complex. In the same vein as above, a "map" is ... | Homotopy theory | 0.813114 |
17,171 | This can be proven by comparing the restrictions on (undirected) Dynkin diagrams with the restrictions on Coxeter diagrams of finite groups: formally, the Coxeter graph can be obtained from the Dynkin diagram by discarding the direction of the edges, and replacing every double edge with an edge labelled 4 and every tri... | H4 (mathematics) | 0.813109 |
17,172 | For the intersecting secants theorem and chord theorem the power of a point plays the role of an invariant: Intersecting secants theorem: For a point P {\displaystyle P} outside a circle c {\displaystyle c} and the intersection points S 1 , S 2 {\displaystyle S_{1},S_{2}} of a secant line g {\displaystyle g} with c {\d... | Power of a point | 0.813103 |
17,173 | Let c 1 , c 2 {\displaystyle c_{1},c_{2}} be two circles, E {\displaystyle E} their outer similarity point and g {\displaystyle g} a line through E {\displaystyle E} , which meets the two circles at four points G 1 , H 1 , G 2 , H 2 {\displaystyle G_{1},H_{1},G_{2},H_{2}} . From the defining property of point E {\displ... | Power of a point | 0.813103 |
17,174 | Solving the Problem of Apollonius Construction of the Malfatti circles: For a given triangle determine three circles, which touch each other and two sides of the triangle each. Spherical version of Malfatti's problem: The triangle is a spherical one.Essential tools for investigations on circles are the radical axis of ... | Power of a point | 0.813103 |
17,175 | Due to the Pythagorean theorem the number Π ( P ) {\displaystyle \Pi (P)} has the simple geometric meanings shown in the diagram: For a point P {\displaystyle P} outside the circle Π ( P ) {\displaystyle \Pi (P)} is the squared tangential distance | P T | {\displaystyle |PT|} of point P {\displaystyle P} to the circle ... | Power of a point | 0.813103 |
17,176 | In elementary plane geometry, the power of a point is a real number that reflects the relative distance of a given point from a given circle. It was introduced by Jakob Steiner in 1826.Specifically, the power Π ( P ) {\displaystyle \Pi (P)} of a point P {\displaystyle P} with respect to a circle c {\displaystyle c} wit... | Power of a point | 0.813103 |
17,177 | For a second secant through E {\displaystyle E}: | E H 1 | ⋅ | E G 2 | = | E H 1 ′ | ⋅ | E G 2 ′ | {\displaystyle |EH_{1}|\cdot |EG_{2}|=|EH'_{1}|\cdot |EG'_{2}|} From the secant theorem one gets: The four points H 1 , G 2 , H 1 ′ , G 2 ′ {\displaystyle H_{1},G_{2},H'_{1},G'_{2}} lie on a circle.And analogously: The fo... | Power of a point | 0.813103 |
17,178 | Laguerre defined the power of a point P with respect to an algebraic curve of degree n to be the product of the distances from the point to the intersections of a circle through the point with the curve, divided by the nth power of the diameter d. Laguerre showed that this number is independent of the diameter (Laguerr... | Power of a point | 0.813103 |
17,179 | ISBN 978-0-387-95097-6. Montgomery, Hugh; Robert C. Vaughan (2005). Multiplicative Number Theory I. Classical Theory. Cambridge: Cambridge University Press. ISBN 978-0-521-84903-6. | Multiplicative number theory | 0.813102 |
17,180 | A large part of analytic number theory deals with multiplicative problems, and so most of its texts contain sections on multiplicative number theory. These are some well-known texts that deal specifically with multiplicative problems: Davenport, Harold (2000). Multiplicative Number Theory (3rd ed.). Berlin: Springer. | Multiplicative number theory | 0.813102 |
17,181 | There is also much interest in the size of the smallest prime in an arithmetic progression; Linnik's theorem gives an estimate. The twin prime conjecture, namely that there are an infinity of primes p such that p+2 is also prime, is the subject of active research. Chen's theorem shows that there are an infinity of prim... | Multiplicative number theory | 0.813102 |
17,182 | The distribution of primes numbers among residue classes modulo an integer is an area of active research. Dirichlet's theorem on primes in arithmetic progressions shows that there are an infinity of primes in each co-prime residue class, and the prime number theorem for arithmetic progressions shows that the primes are... | Multiplicative number theory | 0.813102 |
17,183 | Multiplicative number theory deals primarily in asymptotic estimates for arithmetic functions. Historically the subject has been dominated by the prime number theorem, first by attempts to prove it and then by improvements in the error term. The Dirichlet divisor problem that estimates the average order of the divisor ... | Multiplicative number theory | 0.813102 |
17,184 | The methods belong primarily to analytic number theory, but elementary methods, especially sieve methods, are also very important. The large sieve and exponential sums are usually considered part of multiplicative number theory. The distribution of prime numbers is closely tied to the behavior of the Riemann zeta funct... | Multiplicative number theory | 0.813102 |
17,185 | Multiplicative number theory is a subfield of analytic number theory that deals with prime numbers and with factorization and divisors. The focus is usually on developing approximate formulas for counting these objects in various contexts. The prime number theorem is a key result in this subject. The Mathematics Subjec... | Multiplicative number theory | 0.813102 |
17,186 | Query optimization is a feature of many relational database management systems and other databases such as NoSQL and graph databases. The query optimizer attempts to determine the most efficient way to execute a given query by considering the possible query plans.Generally, the query optimizer cannot be accessed direct... | Query optimization | 0.813101 |
17,187 | Classical query optimization assumes that query plans are compared according to one single cost metric, usually execution time, and that the cost of each query plan can be calculated without uncertainty. Both assumptions are sometimes violated in practice and multiple extensions of classical query optimization have bee... | Query optimization | 0.813101 |
17,188 | The goal of optimization is usually to generate all query plans that could be optimal for any of the possible parameter value combinations. This yields a set of relevant query plans. At run time, the best plan is selected out of that set once the true parameter values become known. The advantage of parametric query opt... | Query optimization | 0.813101 |
17,189 | Classical query optimization associates each query plan with one scalar cost value. Parametric query optimization assumes that query plan cost depends on parameters whose values are unknown at optimization time. Such parameters can for instance represent the selectivity of query predicates that are not fully specified ... | Query optimization | 0.813101 |
17,190 | Each different way typically requires different processing time. Processing times of the same query may have large variance, from a fraction of a second to hours, depending on the chosen method. The purpose of query optimization, which is an automated process, is to find the way to process a given query in minimum time... | Query optimization | 0.813101 |
17,191 | Query predicates are often highly correlated (for example, model='Accord' implies make='Honda'), and it is very hard to estimate the selectivity of the conjunct in general. Poor cardinality estimates and uncaught correlation are one of the main reasons why query optimizers pick poor query plans. This is one reason why ... | Query optimization | 0.813101 |
17,192 | One of the hardest problems in query optimization is to accurately estimate the costs of alternative query plans. Optimizers cost query plans using a mathematical model of query execution costs that relies heavily on estimates of the cardinality, or number of tuples, flowing through each edge in a query plan. Cardinali... | Query optimization | 0.813101 |
17,193 | Fixing these issues found in these plans can shave tens of percent execution time, and in some cases can cut two-dimensional searches to linear ones. One of the primary and simplest optimization checklists is to use operations that most RDMS are designed to execute efficiently. See Sargable. | Query optimization | 0.813101 |
17,194 | Multi-objective parametric query optimization generalizes parametric and multi-objective query optimization. Plans are compared according to multiple cost metrics and plan costs may depend on parameters whose values are unknown at optimization time. The cost of a query plan is therefore modeled as a function from a mul... | Query optimization | 0.813101 |
17,195 | Genetic engineering methods are weakly represented in film; Michael Clark, writing for the Wellcome Trust, calls the portrayal of genetic engineering and biotechnology "seriously distorted" in films such as The 6th Day. In Clark's view, the biotechnology is typically "given fantastic but visually arresting forms" while... | Genetically engineered | 0.813099 |
17,196 | These techniques are used to produce medicines such as insulin, human growth hormone, and vaccines, supplements such as tryptophan, aid in the production of food (chymosin in cheese making) and fuels. Other applications with genetically engineered bacteria could involve making them perform tasks outside their natural c... | Genetically engineered | 0.813099 |
17,197 | The three-state Potts CFT, also known as the Z 3 {\displaystyle \mathbb {Z} _{3}} parafermion CFT, is a conformal field theory in two dimensions. It is a minimal model with central charge c = 4 / 5 {\displaystyle c=4/5} . It is considered to be the simplest minimal model with a non-diagonal partition function in Viraso... | Critical three-state Potts model | 0.813093 |
17,198 | The local antiholomorphic W primaries similarly are given by 1 , ϵ ¯ , σ ¯ 1 , σ ¯ 2 , ψ ¯ 1 , ψ ¯ 2 {\displaystyle 1,{\bar {\epsilon }},{\bar {\sigma }}_{1},{\bar {\sigma }}_{2},{\bar {\psi }}_{1},{\bar {\psi }}_{2}} with the same scaling dimensions. Each field in the theory is either a combination of a holomorphic an... | Critical three-state Potts model | 0.813093 |
17,199 | The partition function is diagonal when expressed in terms of W-algebra characters (where traces are taken over irreducible representations of the W algebra, instead of over irreducible representations of the Virasoro algebra). Since χ 1 = χ 1 , 1 + χ 4 , 1 {\displaystyle \chi _{1}=\chi _{1,1}+\chi _{4,1}} and χ ϵ = χ ... | Critical three-state Potts model | 0.813093 |
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