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8,600 | In mathematics, especially in topology, a topological group G {\displaystyle G} is said to have no small subgroup if there exists a neighborhood U {\displaystyle U} of the identity that contains no nontrivial subgroup of G . {\displaystyle G.} An abbreviation '"NSS"' is sometimes used. | No small subgroup | 0.821208 |
8,601 | In theoretical physics, a short supermultiplet is a supermultiplet i.e. a representation of the supersymmetry algebra whose dimension is smaller than 2 N / 2 {\displaystyle 2^{N/2}} where N {\displaystyle N} is the number of real supercharges. The representations that saturate the bound are known as the long supermulti... | Short supermultiplet | 0.821205 |
8,602 | Quantum algorithms may also be stated in other models of quantum computation, such as the Hamiltonian oracle model.Quantum algorithms can be categorized by the main techniques used by the algorithm. Some commonly used techniques/ideas in quantum algorithms include phase kick-back, phase estimation, the quantum Fourier ... | Quantum algorithms | 0.821203 |
8,603 | In quantum computing, a quantum algorithm is an algorithm which runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation. A classical (or non-quantum) algorithm is a finite sequence of instructions, or a step-by-step procedure for solving a problem, w... | Quantum algorithms | 0.821203 |
8,604 | The quantum approximate optimization algorithm takes inspiration from quantum annealing, performing a discretized approximation of quantum annealing on a quantum circuit. It can be used to solve problems in graph theory. The algorithm makes use of classical optimization of quantum operations to maximize an objective fu... | Quantum algorithms | 0.821203 |
8,605 | Hybrid Quantum/Classical Algorithms combine quantum state preparation and measurement with classical optimization. These algorithms generally aim to determine the ground state eigenvector and eigenvalue of a Hermitian Operator. | Quantum algorithms | 0.821203 |
8,606 | After the successful conclusion of the First Survey, the spacecraft started its Second All Sky Survey on 14 February 2010. The last observations for the Second All Sky Survey were made on 28 May 2010.Some planned pointing list data from 2009 has been released publicly, along with a video visualization of the surveyed s... | Planck Satellite | 0.821198 |
8,607 | The HFI was sensitive between 100 and 857 GHz, using 52 bolometric detectors, manufactured by JPL/Caltech, optically coupled to the telescope through cold optics, manufactured by Cardiff University's School of Physics and Astronomy, consisting of a triple horn configuration and optical filters, a similar concept to tha... | Planck Satellite | 0.821198 |
8,608 | In theoretical physics, an invariant is an observable of a physical system which remains unchanged under some transformation. Invariance, as a broader term, also applies to the no change of form of physical laws under a transformation, and is closer in scope to the mathematical definition. Invariants of a system are de... | Invariant quantity | 0.821192 |
8,609 | This generally means that the form of the law (e.g. the type of differential equations used to describe the law) is unchanged in transformations so that no additional or different solutions are obtained. For example the rule describing Newton's force of gravity between two chunks of matter is the same whether they are ... | Invariant quantity | 0.821192 |
8,610 | In classical and quantum mechanics, invariance of space under translation results in momentum being an invariant and the conservation of momentum, whereas invariance of the origin of time, i.e. translation in time, results in energy being an invariant and the conservation of energy. In general, by Noether's theorem, an... | Invariant quantity | 0.821192 |
8,611 | Shared Variables are a feature of the programming language APL which allows APL programs running on one processor to share information with another processor. Although originally developed for mainframe computers, Shared Variables were also used in personal computer implementations of APL. Shared Variables could be use... | Shared Variables | 0.821181 |
8,612 | Without polymerase chain reaction development, there are many advancements in the field of bacterial study and protein expression study that would not have come to fruition. The development of the theory and process of polymerase chain reaction is essential but the invention of the thermal cycler is equally as importan... | History of biochemistry | 0.821176 |
8,613 | Biochemistry studies the chemical properties of important biological molecules, like proteins, and in particular the chemistry of enzyme-catalyzed reactions. The biochemistry of cell metabolism and the endocrine system has been extensively described. Other areas of biochemistry include the genetic code (DNA, RNA), prot... | History of biochemistry | 0.821176 |
8,614 | Nevertheless, several sources cite German chemist Carl Neuberg as having coined the term for the new discipline in 1903, and some credit it to Franz Hofmeister.The subject of study in biochemistry is the chemical processes in living organisms, and its history involves the discovery and understanding of the complex comp... | History of biochemistry | 0.821176 |
8,615 | The history of biochemistry can be said to have started with the ancient Greeks who were interested in the composition and processes of life, although biochemistry as a specific scientific discipline has its beginning around the early 19th century. Some argued that the beginning of biochemistry may have been the discov... | History of biochemistry | 0.821176 |
8,616 | As in the majority of early sciences, the Islamic world contributed significantly to early biological advancements as well as alchemical advancements; especially with the introduction of clinical trials and clinical pharmacology presented in Avicenna's The Canon of Medicine. On the side of chemistry, early advancements... | History of biochemistry | 0.821176 |
8,617 | In a sense, the study of biochemistry can be considered to have started in ancient times, for example when biology first began to interest society—as the ancient Chinese developed a system of medicine based on yin and yang, and also the five phases, which both resulted from alchemical and biological interests. Its begi... | History of biochemistry | 0.821176 |
8,618 | In 1960, the biochemist Robert K. Crane revealed his discovery of the sodium-glucose cotransport as the mechanism for intestinal glucose absorption. This was the very first proposal of a coupling between the fluxes of an ion and a substrate that has been seen as sparking a revolution in biology. This discovery, however... | History of biochemistry | 0.821176 |
8,619 | However, in 1926, James B. Sumner showed that the enzyme urease was a pure protein and crystallized it; Sumner did likewise for the enzyme catalase in 1937. The conclusion that pure proteins can be enzymes was definitively proved by Northrop and Stanley, who worked on the digestive enzymes pepsin (1930), trypsin, and c... | History of biochemistry | 0.821176 |
8,620 | He named the enzyme that brought about the fermentation of sucrose zymase. In 1907 he received the Nobel Prize in Chemistry "for his biochemical research and his discovery of cell-free fermentation". Following Buchner's example; enzymes are usually named according to the reaction they carry out. | History of biochemistry | 0.821176 |
8,621 | Since then, biochemistry has advanced, especially since the mid-20th century, with the development of new techniques such as chromatography, X-ray diffraction, NMR spectroscopy, radioisotopic labelling, electron microscopy and molecular dynamics simulations. These techniques allowed for the discovery and detailed analy... | History of biochemistry | 0.821176 |
8,622 | Polymerase chain reaction (PCR) is the primary gene amplification technique that has revolutionized modern biochemistry. Polymerase chain reaction was developed by Kary Mullis in 1983. There are four steps to a proper polymerase chain reaction: 1) denaturation 2) extension 3) insertion (of gene to be expressed) and fin... | History of biochemistry | 0.821176 |
8,623 | One of the most prolific of these modern biochemists was Hans Krebs who made huge contributions to the study of metabolism. Krebs was a student of extremely important Otto Warburg, and wrote a biography of Warburg by that title in which he presents Warburg as being educated to do for biological chemistry what Fischer d... | History of biochemistry | 0.821176 |
8,624 | The Call Level Interface (CLI) is an application programming interface (API) and software standard to embed Structured Query Language (SQL) code in a host program as defined in a joint standard by the International Organization for Standardization (ISO) and International Electrotechnical Commission (IEC): ISO/IEC 9075-... | Call Level Interface | 0.821173 |
8,625 | In software engineering, profiling ("program profiling", "software profiling") is a form of dynamic program analysis that measures, for example, the space (memory) or time complexity of a program, the usage of particular instructions, or the frequency and duration of function calls. Most commonly, profiling information... | Profiling (computer programming) | 0.821171 |
8,626 | In mathematics and its applications, particularly to phase transitions in matter, a Stefan problem is a particular kind of boundary value problem for a system of partial differential equations (PDE), in which the boundary between the phases can move with time. The classical Stefan problem aims to describe the evolution... | Stefan problem | 0.821159 |
8,627 | More recent phylogenetic tree/MSA methods use heuristics to isolate high-scoring, but not necessarily optimal, trees. The MALIGN method uses a maximum-parsimony technique to compute a multiple alignment by maximizing a cladogram score, and its companion POY uses an iterative method that couples the optimization of the ... | Phylogenetic inference | 0.821159 |
8,628 | It has recently been shown that, when topologies and ancestral sequence reconstruction are the desired output, choosing one criterion over another is not crucial. Instead, using the most complex nucleotide substitution model, GTR+I +G, leads to similar results for the inference of tree topology and ancestral sequences.... | Phylogenetic inference | 0.821159 |
8,629 | Many forms of molecular phylogenetics are closely related to and make extensive use of sequence alignment in constructing and refining phylogenetic trees, which are used to classify the evolutionary relationships between homologous genes represented in the genomes of divergent species. The phylogenetic trees constructe... | Phylogenetic inference | 0.821159 |
8,630 | Computational phylogenetics is the application of computational algorithms, methods, and programs to phylogenetic analyses. The goal is to assemble a phylogenetic tree representing a hypothesis about the evolutionary ancestry of a set of genes, species, or other taxa. For example, these techniques have been used to exp... | Phylogenetic inference | 0.821159 |
8,631 | Molecular phylogenetics methods rely on a defined substitution model that encodes a hypothesis about the relative rates of mutation at various sites along the gene or amino acid sequences being studied. At their simplest, substitution models aim to correct for differences in the rates of transitions and transversions i... | Phylogenetic inference | 0.821159 |
8,632 | The most naive way of identifying the most parsimonious tree is simple enumeration - considering each possible tree in succession and searching for the tree with the smallest score. However, this is only possible for a relatively small number of sequences or species because the problem of identifying the most parsimoni... | Phylogenetic inference | 0.821159 |
8,633 | Ad hoc methods exist for some classes of transcendental equations in one variable to transform them into algebraic equations which then might be solved. | Transcendental equation | 0.821154 |
8,634 | In applied mathematics, a transcendental equation is an equation over the real (or complex) numbers that is not algebraic, that is, if at least one of its sides describes a transcendental function. Examples include: x = e − x x = cos x 2 x = x 2 {\displaystyle {\begin{aligned}x&=e^{-x}\\x&=\cos x\\2^{x}&=x^{2}\end{al... | Transcendental equation | 0.821154 |
8,635 | Some transcendental systems of high-order equations can be solved by “separation” of the unknowns, reducing them to algebraic equations. The following can also be used when solving transcendental equations/inequalities: If x 0 {\displaystyle x_{0}} is a solution to the equation f ( x ) = g ( x ) {\displaystyle f(x)=g(x... | Transcendental equation | 0.821154 |
8,636 | If the unknown, say x, occurs only in exponents: applying the natural logarithm to both sides may yield an algebraic equation, e.g. 4 x = 3 x 2 − 1 ⋅ 2 5 x {\displaystyle 4^{x}=3^{x^{2}-1}\cdot 2^{5x}} transforms to x ln 4 = ( x 2 − 1 ) ln 3 + 5 x ln 2 {\displaystyle x\ln 4=(x^{2}-1)\ln 3+5x\ln 2} , which simplif... | Transcendental equation | 0.821154 |
8,637 | If the unknown x occurs only in linear expressions inside arguments of hyperbolic functions, unfolding them by their defining exponential expressions and substituting y = e x p ( x ) {\displaystyle y=exp(x)} yields an algebraic equation, e.g. 3 cosh x = 4 + sinh ( 2 x − 6 ) {\displaystyle 3\cosh x=4+\sinh(2x-6)} un... | Transcendental equation | 0.821154 |
8,638 | If only real numbers are considered, x = 0 {\displaystyle x=0} is not a solution, as it leads to a non-real subexpression log 5 ( − 1 ) {\displaystyle \log _{5}(-1)} in the given equation. This requires the original equation to consist of integer-coefficient linear combinations of logarithms w.r.t. a unique base, and... | Transcendental equation | 0.821154 |
8,639 | If the unknown x occurs only in arguments of a logarithm function: applying exponentiation to both sides may yield an algebraic equation, e.g. 2 log 5 ( 3 x − 1 ) − log 5 ( 12 x + 1 ) = 0 {\displaystyle 2\log _{5}(3x-1)-\log _{5}(12x+1)=0} transforms, using exponentiation to base 5. {\displaystyle 5.} to ( 3 x − 1 ... | Transcendental equation | 0.821154 |
8,640 | {\displaystyle n,m,l,...} might all be transformed to arguments of the form, say, sin x {\displaystyle \sin x} . After that, substituting y = sin ( x ) {\displaystyle y=\sin(x)} yields an algebraic equation, e.g. sin ( x + a ) = ( cos 2 x ) − 1 {\displaystyle \sin(x+a)=(\cos ^{2}x)-1} transforms to ( sin x ) ... | Transcendental equation | 0.821154 |
8,641 | High-fidelity data (HiFi) includes data that was produced by a person or Stochastic Process that closely matches the operational context of interest. For example, in wing design optimization, high-fidelity data uses physical models in simulation that produce results that closely match the wing in a similar real-world s... | Multifidelity simulation | 0.821152 |
8,642 | Multifidelity (or multi-fidelity) methods leverage both low- and high-fidelity data in order to maximize the accuracy of model estimates, while minimizing the cost associated with parametrization. They have been successfully used in impedance cardiography, wing-design optimization, robotic learning, computational biome... | Multifidelity simulation | 0.821152 |
8,643 | Low-fidelity data (LoFi) includes any data that was produced by a person or Stochastic Process that deviates from the real-world system of interest. For example, LoFi data can be produced by models of a physical system that use approximations to simulate the system, rather than modeling the system in an exhaustive mann... | Multifidelity simulation | 0.821152 |
8,644 | A soft-switching three-level inverter (S3L inverter) is a high-efficiency power electronic inverter intended, in particular, for use with three-phase drives, as a grid-tie inverter for photovoltaic installations or wind turbines and in power supplies. The topology was developed in 2009 at HTWG Konstanz (Constance Unive... | Soft-switching three-level inverter | 0.82115 |
8,645 | To improve their efficiency, high-power inverters (from about 10 kW) frequently make use of a technique referred to as a three-level design (three-level inverter).Forming the basis of the S3L inverter is a hard-switching three-level inverter of this kind with a T-type topology. This base design is supplemented by a snu... | Soft-switching three-level inverter | 0.82115 |
8,646 | Any group that is virtually abelian. Any nilpotent group. Any semidirect product N ⋊ H {\displaystyle N\rtimes H} where N is nilpotent and H is finite. Any semidirect product N ⋊ H {\displaystyle N\rtimes H} where N is finite and H is nilpotent.Gromov's theorem says that a finitely generated group is virtually nilpoten... | Virtually nilpotent | 0.821145 |
8,647 | Any semidirect product N ⋊ H {\displaystyle N\rtimes H} where N is finite and H is free. Any free product H ∗ K {\displaystyle H*K} , where H and K are both finite. (For example, the modular group PSL ( 2 , Z ) {\displaystyle \operatorname {PSL} (2,\mathbb {Z} )} . )It follows from Stalling's theorem that any torsion... | Virtually nilpotent | 0.821145 |
8,648 | In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group G is said to be virtually P if there is a finite index subgroup H ≤ G {\displaystyle H\leq G}... | Virtually nilpotent | 0.821145 |
8,649 | Let ν {\displaystyle \nu } be a measure on ( Y , B ) {\displaystyle (Y,{\mathcal {B}})} , and k: Y × X → {\displaystyle k:Y\times X\to } a measurable function with respect to the product σ {\displaystyle \sigma } -algebra A ⊗ B {\displaystyle {\mathcal {A}}\otimes {\mathcal {B}}} such that ∫ Y k ( y , x ) ν ( d y ) = ... | Probability kernel | 0.82114 |
8,650 | This modified rotation matrix can be rewritten as an exponential function. P a b = − G 2 R = I − P a b + P a b = e G β {\displaystyle {\begin{aligned}P_{ab}&=-G^{2}\\R&=I-P_{ab}+\leftP_{ab}=e^{G\beta }\\\end{aligned}}} Analysis is often easier in terms of these generators, rather than the full rotation matrix. Analysi... | Infinitesimal rotation | 0.82114 |
8,651 | Approximating Δθ as θ/N, where N is a large number, a rotation of θ about the axis may be represented as: R = ( I + A θ N ) N ≈ e A θ . {\displaystyle R=\left(I+{\frac {A\theta }{N}}\right)^{N}\approx e^{A\theta }.} It can be seen that Euler's theorem essentially states that all rotations may be represented in this for... | Infinitesimal rotation | 0.82114 |
8,652 | The International Journal of Computational Geometry and Applications (IJCGA) is a bimonthly journal published since 1991, by World Scientific. It covers the application of computational geometry in design and analysis of algorithms, focusing on problems arising in various fields of science and engineering such as compu... | International Journal of Computational Geometry and Applications | 0.821133 |
8,653 | If instead of a uniform downwards gravitational force we consider two bodies orbiting with the mutual gravitation between them, we obtain Kepler's laws of planetary motion. The derivation of these was one of the major works of Isaac Newton and provided much of the motivation for the development of differential calculus... | Trajectory | 0.821121 |
8,654 | It employs the mathematics of differential calculus (which was also initiated by Newton in his youth). Over the centuries, countless scientists have contributed to the development of these two disciplines. | Trajectory | 0.821121 |
8,655 | In the gravitational field of a point mass or a spherically-symmetrical extended mass (such as the Sun), the trajectory of a moving object is a conic section, usually an ellipse or a hyperbola. This agrees with the observed orbits of planets, comets, and artificial spacecraft to a reasonably good approximation, althoug... | Trajectory | 0.821121 |
8,656 | The most important cases are sequences of real or complex numbers, and these spaces, together with linear subspaces, are known as sequence spaces. Operators on these spaces are known as sequence transformations. Bounded linear operators over Banach space form a Banach algebra in respect to the standard operator norm. T... | Mathematical operator | 0.821121 |
8,657 | From the point of view of functional analysis, calculus is the study of two linear operators: the differential operator d d t {\displaystyle {\frac {d}{dt}}} , and the Volterra operator ∫ 0 t {\displaystyle \int _{0}^{t}} . | Mathematical operator | 0.821121 |
8,658 | The Laplace transform is another integral operator and is involved in simplifying the process of solving differential equations. Given f = f(s), it is defined by: | Mathematical operator | 0.821121 |
8,659 | Operators are also involved in probability theory, such as expectation, variance, and covariance. Indeed, every covariance is basically a dot product; every variance is a dot product of a vector with itself, and thus is a quadratic norm; every standard deviation is a norm (square root of the quadratic norm); the corres... | Mathematical operator | 0.821121 |
8,660 | In mathematics, an operator is generally a mapping or function that acts on elements of a space to produce elements of another space (possibly and sometimes required to be the same space). There is no general definition of an operator, but the term is often used in place of function when the domain is a set of function... | Mathematical operator | 0.821121 |
8,661 | Three operators are key to vector calculus: Grad (gradient), (with operator symbol ∇ {\displaystyle \nabla } ) assigns a vector at every point in a scalar field that points in the direction of greatest rate of change of that field and whose norm measures the absolute value of that greatest rate of change. Div (divergen... | Mathematical operator | 0.821121 |
8,662 | In many cases this requires comparing a given sequence with previously studied ones. The comparison between the strings becomes complicated when insertions, deletions and mutations occur in a string. A survey and taxonomy of the key algorithms for sequence comparison for bioinformatics is presented by Abouelhoda & Ghan... | Sequence mining | 0.821117 |
8,663 | String mining typically deals with a limited alphabet for items that appear in a sequence, but the sequence itself may be typically very long. Examples of an alphabet can be those in the ASCII character set used in natural language text, nucleotide bases 'A', 'G', 'C' and 'T' in DNA sequences, or amino acids for protei... | Sequence mining | 0.821117 |
8,664 | The Physics and Astronomy Classification Scheme (PACS) is a scheme developed in 1970 by the American Institute of Physics (AIP) for classifying scientific literature using a hierarchical set of codes. PACS has been used by over 160 international journals, including the Physical Review series since 1975. Since 2016, Ame... | Physics and Astronomy Classification Scheme | 0.821112 |
8,665 | This enables marine prokaryotes to thrive as extremophiles in harsh environments as cold as the ice surface of Antarctica, studied in cryobiology, as hot as undersea hydrothermal vents, or in high saline conditions as (halophiles). Some marine prokaryotes live symbiotically in or on the bodies of other marine organisms... | Marine prokaryotes | 0.821098 |
8,666 | Newton's notation for differentiation, also called the dot notation, places a dot over the function name to represent a time derivative. If y = f ( t ) {\displaystyle y=f(t)} , then y ˙ {\displaystyle {\dot {y}}} and y ¨ {\displaystyle {\ddot {y}}} denote, respectively, the first and second derivatives of y {\displayst... | Instantaneous velocities | 0.821095 |
8,667 | The N-terminal domain contains a three-layer alpha/beta/alpha sandwich that is homologous with the Rossmann fold (CATH class 3.40.50.170) of which the major feature is a long beta sheet that is composed of nine mostly parallel beta strands. SurEstructural domain has a similar topology to the N-terminal protein domain o... | SurE, survival protein E | 0.821095 |
8,668 | In molecular biology, the protein domain surE refers to survival protein E. It was originally found that cells that did not contain this protein, could not survive in the stationary phase, at above normal temperatures, and in high-salt media. Hence the name, survival protein E. It is a metal ion-dependent phosphatase t... | SurE, survival protein E | 0.821095 |
8,669 | Let us establish the evident fact that the group Z3 = Z/3Z is indeed cyclic. Via Cayley's theorem, Z3 is a subgroup of the symmetric group S3. Now, Z 3 = { e , ( 1 2 3 ) , ( 1 3 2 ) } {\displaystyle \mathbb {Z} _{3}=\{e,(1\ 2\ 3),(1\ 3\ 2)\}} S 3 = { e , ( 1 2 ) , ( 1 3 ) , ( 2 3 ) , ( 1 2 3 ) , ( 1 3 2 ) } {\displayst... | Schreier's subgroup lemma | 0.821095 |
8,670 | Aquatic science is the study of the various bodies of water that make up our planet including oceanic and freshwater environments. Aquatic scientists study the movement of water, the chemistry of water, aquatic organisms, aquatic ecosystems, the movement of materials in and out of aquatic ecosystems, and the use of wat... | Aquatic sciences | 0.821092 |
8,671 | Geological oceanographers and marine geologists use sampling to examine the history of sea-floor spreading, plate tectonics, thermohaline circulation, and climates. In addition, they study undersea volcanos as well as mantle (geology) and hydrothermal circulation. Their research helps us to better understand the events... | Aquatic sciences | 0.821092 |
8,672 | In addition, a chemical oceanographer might use chemistry to better understand how ocean currents move seawater and how the ocean affects the climate. They might also search for ocean resources that could be beneficial, such as products that have medicinal properties. The field of oceanography also consists of geologic... | Aquatic sciences | 0.821092 |
8,673 | Oceanography refers to the study of the physical, chemical, and biological characteristics of oceanic environments. Oceanographers study the history, current condition, and future of the planet's oceans. They also study marine life and ecosystems, ocean circulation, plate tectonics, the geology of the seafloor, and the... | Aquatic sciences | 0.821092 |
8,674 | Limnologists who specialize in the biology field only study the living aquatic organisms that are present in a certain freshwater environment. They aim to understand various aspects of the organisms, such as their history, their life cycles, and their populations. These scientists study living organisms in order to sup... | Aquatic sciences | 0.821092 |
8,675 | Another aspect that they may examine are the effects of a nonnative species on native populations of aquatic life. Most ecological limnologists conduct their studies in laboratory settings, where their hypotheses can be tested, verified, and controlled. Another area of study under limnology is biology. | Aquatic sciences | 0.821092 |
8,676 | One of those areas of study is ecology, particularly the ecology of water systems. The ecology of water systems focuses on the organisms that live in freshwater environments and how they are affected by changes in their habitat. For example, a limnologist specializing in ecology could study how chemical or temperature ... | Aquatic sciences | 0.821092 |
8,677 | Physical oceanographers also study topics such as the transmission of light and sound through water and the effects that the ocean has on weather and climate. All of these fields are intertwined. In order for an oceanographer to succeed in their field, they need to have an adequate understanding of other related scienc... | Aquatic sciences | 0.821092 |
8,678 | However, they were reluctant to adopt this axiom: Fraenkel stated "that Replacement was too strong an axiom for 'general set theory'", while "Skolem only wrote that 'we could introduce' Replacement".Von Neumann worked on the problems of Zermelo set theory and provided solutions for some of them: A theory of ordinals Pr... | Von Neumann–Bernays–Gödel set theory | 0.821091 |
8,679 | A recursive computer program succinctly captures the construction of a class from a given formula. The definition of this program does not depend on the proof of the class existence theorem. However, the proof is needed to prove that the class constructed by the program satisfies the given formula and is built using th... | Von Neumann–Bernays–Gödel set theory | 0.821091 |
8,680 | Also, its proof—instead of invoking finitely many NBG axioms—inductively describes how to use NBG axioms to construct a class satisfying a given formula. For every formula, this description can be turned into a constructive existence proof that is in NBG. Therefore, this metatheorem can generate the NBG proofs that rep... | Von Neumann–Bernays–Gödel set theory | 0.821091 |
8,681 | The proof starts by applying the transformation rules to the given formula to produce a transformed formula. Since this formula is equivalent to the given formula, the proof is completed by proving the class existence theorem for transformed formulas. Gödel pointed out that the class existence theorem "is a metatheorem... | Von Neumann–Bernays–Gödel set theory | 0.821091 |
8,682 | If a formula contains no free set variables other than x 1 , … , x n , {\displaystyle x_{1},\dots ,x_{n},} then bound variables that are nested within q {\displaystyle q} quantifiers are replaced with x n + q {\displaystyle x_{n+q}} . These variables have (quantifier) nesting depth q {\displaystyle q} . Proof of the cl... | Von Neumann–Bernays–Gödel set theory | 0.821091 |
8,683 | {\displaystyle (x_{1},t)\in F\land (t,x_{2})\in G.} However, since the class existence theorem is stated for subscripted variables, this formula does not have the form expected by the induction hypothesis. This problem is solved by replacing the variable t {\displaystyle t} with x 3 . | Von Neumann–Bernays–Gödel set theory | 0.821091 |
8,684 | The theorem's proof will be done in two steps: Transformation rules are used to transform the given formula ϕ {\displaystyle \phi } into an equivalent formula that simplifies the inductive part of the proof. For example, the only logical symbols in the transformed formula are ¬ {\displaystyle \neg } , ∧ {\displaystyle ... | Von Neumann–Bernays–Gödel set theory | 0.821091 |
8,685 | It allows us to write: The following axioms and definitions are needed for the proof of the class existence theorem. Axiom of extensionality. If two classes have the same elements, then they are identical. | Von Neumann–Bernays–Gödel set theory | 0.821091 |
8,686 | The power class of A {\displaystyle A} is P ( A ) = { x: x ⊆ A } . {\displaystyle {\mathcal {P}}(A)=\{x:x\subseteq A\}.} The extended version of the class existence theorem implies the existence of these classes. | Von Neumann–Bernays–Gödel set theory | 0.821091 |
8,687 | The axioms of pairing and regularity, which were needed for the proof of the class existence theorem, have been given above. NBG contains four other set axioms. Three of these axioms deal with class operations being applied to sets. Definition. | Von Neumann–Bernays–Gödel set theory | 0.821091 |
8,688 | ZFC's replacement schema has been replaced by a single axiom, but the axiom schema of class comprehension has been introduced. To produce a theory with finitely many axioms, the axiom schema of class comprehension is first replaced with finitely many class existence axioms. Then these axioms are used to prove the class... | Von Neumann–Bernays–Gödel set theory | 0.821091 |
8,689 | NBG is finitely axiomatizable, while ZFC and MK are not. A key theorem of NBG is the class existence theorem, which states that for every formula whose quantifiers range only over sets, there is a class consisting of the sets satisfying the formula. This class is built by mirroring the step-by-step construction of the ... | Von Neumann–Bernays–Gödel set theory | 0.821091 |
8,690 | A phenotypic trait, simply trait, or character state is a distinct variant of a phenotypic characteristic of an organism; it may be either inherited or determined environmentally, but typically occurs as a combination of the two. For example, having eye color is a character of an organism, while blue, brown and hazel v... | Biological trait | 0.821081 |
8,691 | E7 has an SU(8) subalgebra, as is evident by noting that in the 8-dimensional description of the root system, the first group of roots are identical to the roots of SU(8) (with the same Cartan subalgebra as in the E7). In addition to the 133-dimensional adjoint representation, there is a 56-dimensional "vector" represe... | E7 (mathematics) | 0.821079 |
8,692 | The compact real form of E7 is the isometry group of the 64-dimensional exceptional compact Riemannian symmetric space EVI (in Cartan's classification). It is known informally as the "quateroctonionic projective plane" because it can be built using an algebra that is the tensor product of the quaternions and the octoni... | E7 (mathematics) | 0.821079 |
8,693 | The split form, EV (or E7(7)), which has maximal compact subgroup SU(8)/{±1}, fundamental group cyclic of order 4 and outer automorphism group of order 2. EVI (or E7(-5)), which has maximal compact subgroup SU(2)·SO(12)/(center), fundamental group non-cyclic of order 4 and trivial outer automorphism group. EVII (or E7(... | E7 (mathematics) | 0.821079 |
8,694 | There is a unique complex Lie algebra of type E7, corresponding to a complex group of complex dimension 133. The complex adjoint Lie group E7 of complex dimension 133 can be considered as a simple real Lie group of real dimension 266. This has fundamental group Z/2Z, has maximal compact subgroup the compact form (see b... | E7 (mathematics) | 0.821079 |
8,695 | By means of a Chevalley basis for the Lie algebra, one can define E7 as a linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split (sometimes also known as “untwisted”) adjoint form of E7. Over an algebraically closed field,... | E7 (mathematics) | 0.821079 |
8,696 | In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7. The designation E7 comes from the Cartan–Killing classification of the complex simpl... | E7 (mathematics) | 0.821079 |
8,697 | Now substitute this expression for x into the bottom equation: 4 ( 3 − 3 2 y ) + 9 y = 15. {\displaystyle 4\left(3-{\frac {3}{2}}y\right)+9y=15.} This results in a single equation involving only the variable y {\displaystyle y} . Solving gives y = 1 {\displaystyle y=1} , and substituting this back into the equation for... | Homogeneous equation | 0.821077 |
8,698 | Very often, and in this article, the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers, but the theory and the algorithms apply for coefficients and solutions in any field. For solutions in an integral domain like the ring of the integers, or in other al... | Homogeneous equation | 0.821077 |
8,699 | In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject used in most modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer scie... | Homogeneous equation | 0.821077 |
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