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16,000 | Plasmon coupling is a phenomenon that occurs when two or more plasmonic particles approach each other to a distance below approximately one diameter's length. Upon the occurrence of plasmon coupling, the resonance of individual particles start to hybridize, and their resonance spectrum peak wavelength will shift (eithe... | Plasmon Coupling | 0.814023 |
16,001 | Computational theorist Leslie Valiant uses the term ecorithms to describe how many less exact systems and techniques like fuzzy logic (and "less robust" logic) can be applied to learning algorithms. Valiant essentially redefines machine learning as evolutionary. In general use, ecorithms are algorithms that learn from ... | Fuzzy Logic | 0.814023 |
16,002 | Fuzzy logic with evaluated syntax (sometimes also called Pavelka's logic), denoted by EVŁ, is a further generalization of mathematical fuzzy logic. While the above kinds of fuzzy logic have traditional syntax and many-valued semantics, in EVŁ syntax is also evaluated. This means that each formula has an evaluation. Axi... | Fuzzy Logic | 0.814023 |
16,003 | It has the axioms of BL plus an axiom of idempotence of conjunction, and its models are called G-algebras. Product fuzzy logic is the extension of basic fuzzy logic BL where conjunction is the product t-norm. It has the axioms of BL plus another axiom for cancellativity of conjunction, and its models are called product... | Fuzzy Logic | 0.814023 |
16,004 | Łukasiewicz fuzzy logic is the extension of basic fuzzy logic BL where standard conjunction is the Łukasiewicz t-norm. It has the axioms of basic fuzzy logic plus an axiom of double negation, and its models correspond to MV-algebras. Gödel fuzzy logic is the extension of basic fuzzy logic BL where conjunction is the Gö... | Fuzzy Logic | 0.814023 |
16,005 | The most important propositional fuzzy logics are: Monoidal t-norm-based propositional fuzzy logic MTL is an axiomatization of logic where conjunction is defined by a left continuous t-norm and implication is defined as the residuum of the t-norm. Its models correspond to MTL-algebras that are pre-linear commutative bo... | Fuzzy Logic | 0.814023 |
16,006 | The term fuzzy logic was introduced with the 1965 proposal of fuzzy set theory by Iranian Azerbaijani mathematician Lotfi Zadeh. Fuzzy logic had, however, been studied since the 1920s, as infinite-valued logic—notably by Łukasiewicz and Tarski.Fuzzy logic is based on the observation that people make decisions based on ... | Fuzzy Logic | 0.814023 |
16,007 | Fuzzy logic and probability address different forms of uncertainty. While both fuzzy logic and probability theory can represent degrees of certain kinds of subjective belief, fuzzy set theory uses the concept of fuzzy set membership, i.e., how much an observation is within a vaguely defined set, and probability theory ... | Fuzzy Logic | 0.814023 |
16,008 | The main advantage of using TSK over Mamdani is that it is computationally efficient and works well within other algorithms, such as PID control and with optimization algorithms. It can also guarantee the continuity of the output surface. However, Mamdani is more intuitive and easier to work with by people. Hence, TSK ... | Fuzzy Logic | 0.814023 |
16,009 | From the seventeenth century, many of the most important advances in mathematics appeared motivated by the study of physics, and this continued in the following centuries (although in the nineteenth century mathematics started to become increasingly independent from physics). The creation and development of calculus we... | Relationship between mathematics and physics | 0.814023 |
16,010 | The relationship between mathematics and physics has been a subject of study of philosophers, mathematicians and physicists since Antiquity, and more recently also by historians and educators. Generally considered a relationship of great intimacy, mathematics has been described as "an essential tool for physics" and ph... | Relationship between mathematics and physics | 0.814023 |
16,011 | (from the Turing–Wittgenstein debate) Do Gödel's incompleteness theorems imply that physical theories will always be incomplete? (from Stephen Hawking) Is mathematics invented or discovered? (millennia-old question, raised among others by Mario Livio) | Relationship between mathematics and physics | 0.814023 |
16,012 | What is the geometry of physical space? What is the origin of the axioms of mathematics? How does the already existing mathematics influence in the creation and development of physical theories? | Relationship between mathematics and physics | 0.814023 |
16,013 | Some of the problems considered in the philosophy of mathematics are the following: Explain the effectiveness of mathematics in the study of the physical world: "At this point an enigma presents itself which in all ages has agitated inquiring minds. How can it be that mathematics, being after all a product of human tho... | Relationship between mathematics and physics | 0.814023 |
16,014 | In recent times the two disciplines have most often been taught separately, despite all the interrelations between physics and mathematics. This led some professional mathematicians who were also interested in mathematics education, such as Felix Klein, Richard Courant, Vladimir Arnold and Morris Kline, to strongly adv... | Relationship between mathematics and physics | 0.814023 |
16,015 | In these coordinates the matrix g(x) satisfies g(0) = I and the lines t ↦ tv are geodesics through 0. Euler's equations imply the matrix equation g(v)v = v,a key result, usually called the Gauss lemma. Geometrically it states that Taking polar coordinates (r,θ), it follows that the metric has the form ds2 = dr2 + G(r,θ... | Smooth surface | 0.814023 |
16,016 | Any other closed Riemannian 2-manifold M of constant Gaussian curvature, after scaling the metric by a constant factor if necessary, will have one of these three surfaces as its universal covering space. In the orientable case, the fundamental group Γ of M can be identified with a torsion-free uniform subgroup of G and... | Smooth surface | 0.814023 |
16,017 | The simply connected surfaces of constant curvature 0, +1 and –1 are the Euclidean plane, the unit sphere in E3, and the hyperbolic plane. Each of these has a transitive three-dimensional Lie group of orientation preserving isometries G, which can be used to study their geometry. Each of the two non-compact surfaces ca... | Smooth surface | 0.814023 |
16,018 | Gauss's Theorema Egregium, the "Remarkable Theorem", shows that the Gaussian curvature of a surface can be computed solely in terms of the metric and is thus an intrinsic invariant of the surface, independent of any isometric embedding in E3 and unchanged under coordinate transformations. In particular isometries of su... | Smooth surface | 0.814023 |
16,019 | The classical approach of Gauss to the differential geometry of surfaces was the standard elementary approach which predated the emergence of the concepts of Riemannian manifold initiated by Bernhard Riemann in the mid-nineteenth century and of connection developed by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in... | Smooth surface | 0.814023 |
16,020 | {\displaystyle H^{*}(X,A)=\operatorname {Ext} ^{*}(\mathbb {Z} _{X},A).} Here Ext is taken in the abelian category of sheaves of abelian groups on X, and Z X {\displaystyle \mathbb {Z} _{X}} is the sheaf of locally constant Z {\displaystyle \mathbb {Z} } -valued functions.For a commutative Noetherian local ring R with ... | Ext groups | 0.814016 |
16,021 | Group cohomology is defined by H ∗ ( G , M ) = Ext Z ∗ ( Z , M ) {\displaystyle H^{*}(G,M)=\operatorname {Ext} _{\mathbb {Z} }^{*}(\mathbb {Z} ,M)} , where G is a group, M is a representation of G over the integers, and Z {\displaystyle \mathbb {Z} } is the group ring of G.For an algebra A over a field k and an A-b... | Ext groups | 0.814016 |
16,022 | Moreover, for a fixed ring R, Ext is a functor in each variable (contravariant in A, covariant in B). For a commutative ring R and R-modules A and B, ExtiR(A, B) is an R-module (using that HomR(A, B) is an R-module in this case). For a non-commutative ring R, ExtiR(A, B) is only an abelian group, in general. If R is an... | Ext groups | 0.814016 |
16,023 | Boolean-valued models can be used to give semantics to syntactic forcing; the price paid is that the semantics is not 2-valued ("true or false"), but assigns truth values from some complete Boolean algebra. Given a forcing poset P, there is a corresponding complete Boolean algebra B, often obtained as the collection of... | Boolean-valued model | 0.814015 |
16,024 | The truth values of the atomic formulas can be used to reconstruct the truth values of more complicated formulas, using the structure of the Boolean algebra. For propositional connectives, this is easy; one simply applies the corresponding Boolean operators to the truth values of the subformulae. For example, if φ(x) a... | Boolean-valued model | 0.814015 |
16,025 | Here a B-valued relation on VB is a function from VB × VB to B. To avoid confusion with the usual equality and membership, these are denoted by ‖x = y‖ and ‖x ∈ y‖ for x and y in VB. They are defined as follows: ‖x ∈ y‖ is defined to be Σt ∈ Dom(y) ‖x = t‖ ∧ y(t) ("x is in y if it is equal to something in y"). ‖x = y‖ ... | Boolean-valued model | 0.814015 |
16,026 | The Boolean-valued model MB is obtained by applying the above construction inside M. The restriction to transitive models is not serious, as the Mostowski collapsing theorem implies that every "reasonable" (well-founded, extensional) model is isomorphic to a transitive one. (If the model M is not transitive things get ... | Boolean-valued model | 0.814015 |
16,027 | Given a complete Boolean algebra B there is a Boolean-valued model denoted by VB, which is the Boolean-valued analogue of the von Neumann universe V. (Strictly speaking, VB is a proper class, so we need to reinterpret what it means to be a model appropriately.) Informally, the elements of VB are "Boolean-valued sets". | Boolean-valued model | 0.814015 |
16,028 | Fix a complete Boolean algebra B and a first-order language L; the signature of L will consist of a collection of constant symbols, function symbols, and relation symbols. A Boolean-valued model for the language L consists of a universe M, which is a set of elements (or names), together with interpretations for the sym... | Boolean-valued model | 0.814015 |
16,029 | So we get an ordinary model of ZF set theory by starting from M, a Boolean algebra B, and an ultrafilter U on B. (The model of ZF constructed like this is not transitive. In practice one applies the Mostowski collapsing theorem to turn this into a transitive model.) We have seen that forcing can be done using Boolean-v... | Boolean-valued model | 0.814015 |
16,030 | If g is a homomorphism from a Boolean algebra B to a Boolean algebra C and MB is any B-valued model of ZF (or of any other theory for that matter) we can turn MB into a C -valued model by applying the homomorphism g to the value of all formulas. In particular if C is {true, false} we get a {true, false}-valued model. T... | Boolean-valued model | 0.814015 |
16,031 | The ultrafilter U on B is defined to be the set of elements b of B that are greater than some element of (the image of) G. Given an ultrafilter U on a Boolean algebra, we get a homomorphism to {true, false} by mapping U to true and its complement to false. Conversely, given such a homomorphism, the inverse image of tru... | Boolean-valued model | 0.814015 |
16,032 | It can be constructed as the algebra of regular open sets in the topological space of P (with underlying set P, and a base given by the sets Up of elements q with q≤p). The map from the poset P to the complete Boolean algebra B is not injective in general. The map is injective if and only if P has the following propert... | Boolean-valued model | 0.814015 |
16,033 | Construct a complete Boolean algebra B as the complete Boolean algebra "generated by" the poset P. Construct an ultrafilter U on B (or equivalently a homomorphism from B to the Boolean algebra {true, false}) from the generic subset G of P. Use the homomorphism from B to {true, false} to turn the Boolean-valued model MB... | Boolean-valued model | 0.814015 |
16,034 | In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take values in some fixed complete Boolean algebra. Boolean-valued models were ... | Boolean-valued model | 0.814015 |
16,035 | In the fields of geometry and biochemistry, a triple helix (PL: triple helices) is a set of three congruent geometrical helices with the same axis, differing by a translation along the axis. This means that each of the helices keeps the same distance from the central axis. As with a single helix, a triple helix may be ... | Triple helix | 0.814011 |
16,036 | In the Standard Model of electroweak interactions of particle physics, the weak hypercharge is a quantum number relating the electric charge and the third component of weak isospin. It is frequently denoted Y W {\displaystyle Y_{\mathsf {W}}} and corresponds to the gauge symmetry U(1).It is conserved (only terms that a... | Weak hypercharge | 0.814003 |
16,037 | One advantage is that NIR can typically penetrate much further into a sample than mid infrared radiation. Near-infrared spectroscopy is, therefore, not a particularly sensitive technique, but it can be very useful in probing bulk material with little or no sample preparation. The molecular overtone and combination band... | Near-infrared spectroscopy | 0.813999 |
16,038 | Near-infrared spectroscopy is based on molecular overtone and combination vibrations. Such transitions are forbidden by the selection rules of quantum mechanics. As a result, the molar absorptivity in the near-IR region is typically quite small. (NIR absorption bands are typically 10–100 times weaker than the correspon... | Near-infrared spectroscopy | 0.813999 |
16,039 | The learning algorithm for an n-node random neural network that includes feedback loops (it is also a recurrent neural network) is of computational complexity O(n^3) (the number of computations is proportional to the cube of n, the number of neurons). The random neural network can also be used with other learning algor... | Random neural network | 0.813999 |
16,040 | The RNN is a recurrent model, i.e. a neural network that is allowed to have complex feedback loops. A highly energy-efficient implementation of random neural networks was demonstrated by Krishna Palem et al. using the Probabilistic CMOS or PCMOS technology and was shown to be c. 226–300 times more efficient in terms of... | Random neural network | 0.813999 |
16,041 | Cells whose internal excitatory state has a positive value are allowed to send out spikes of either kind to other cells in the network according to specific cell-dependent spiking rates. The model has a mathematical solution in steady-state which provides the joint probability distribution of the network in terms of th... | Random neural network | 0.813999 |
16,042 | The random neural network (RNN) is a mathematical representation of an interconnected network of neurons or cells which exchange spiking signals. It was invented by Erol Gelenbe and is linked to the G-network model of queueing networks as well as to Gene Regulatory Network models. Each cell state is represented by an i... | Random neural network | 0.813999 |
16,043 | S. Basterrech, S. Mohamed, G. Rubino, M. Soliman. "Levenberg-Marquardt Training Algorithms for Random Neural Networks," Computer Journal, 54 (1), 125–135, 2011. Michael Georgiopoulos, Cong Li and Taskin Kocak. "Learning in the feed-forward random neural network: A critical review", Performance Evaluation, 68 (4): 361–3... | Random neural network | 0.813999 |
16,044 | Pedro Casas, Sandrine Vaton. "On the use of random neural networks for traffic matrix estimation in large-scale IP networks", IWCMC 2010: 326–330, 2010. S. Basterrech, G. Rubino, "Random Neural Network as Supervised Learning Tool," Neural Network World, 25(5), 457-499, doi:10.14311/NNW.2015.25.024, 2015. | Random neural network | 0.813999 |
16,045 | Stelios Timotheou. "The Random Neural Network: A Survey", Comput. | Random neural network | 0.813999 |
16,046 | A novel weight initialization method for the random neural network. In Fifth International Symposium on Neural Networks (ISNN), Beijing, China, 2008. | Random neural network | 0.813999 |
16,047 | Nonnegative least squares learning for the random neural network. In Proceedings of the 18th International Conference on Artificial Neural Networks, Prague, Czech Republic, pages 195–204, 2008. S. Timotheou. | Random neural network | 0.813999 |
16,048 | A denial of service detector based on maximum likelihood detection and the random neural network. Computer Journal, 50(6):717–727, November 2007. S. Timotheou. | Random neural network | 0.813999 |
16,049 | April 2004. Gerardo Rubino, Pierre Tirilly, Martín Varela "Evaluating Users' Satisfaction in Packet Networks Using Random Neural Networks", ICANN (1) 2006: 303–312, 2006. Gülay Öke and Georgios Loukas. | Random neural network | 0.813999 |
16,050 | E. Gelenbe, Z.-H. Mao and Y-D. Li "Function approximation by random neural networks with a bounded number of layers", 'Differential Equations and Dynamical Systems', 12 (1&2), 143–170, Jan. | Random neural network | 0.813999 |
16,051 | Samir Mohamed, Gerardo Rubino, Martín Varela "Performance evaluation of real-time speech through a packet network: a random neural networks-based approach", Perform. Eval. 57 (2): 141–161, 2004. | Random neural network | 0.813999 |
16,052 | Fourneau '"Random neural networks with multiple classes of signals", Neural Computation, 11, 721–731, 1999. Ugur Halici "Reinforcement learning with internal expectation for the random neural network", European Journal of Operational Research 126 (2): 288–307, 2000. Aristidis Likas, Andreas Stafylopatis "Training the r... | Random neural network | 0.813999 |
16,053 | E. Gelenbe, Z. H. Mao, and Y. D. Li, "Function approximation with the random neural network", IEEE Trans. Neural Networks, 10, (1), January 1999. E. Gelenbe, J.M. | Random neural network | 0.813999 |
16,054 | 1529–1543, October 1996. E. Gelenbe, T. Feng, K.R.R. Krishnan Neural network methods for volumetric magnetic resonance imaging of the human brain, Proceedings of the IEEE, Vol. | Random neural network | 0.813999 |
16,055 | E. Gelenbe, C. Cramer, M. Sungur, P. Gelenbe "Traffic and video quality in adaptive neural compression", Multimedia Systems, 4, 357–369, 1996. C. Cramer, E. Gelenbe, H. Bakircioglu Low bit rate video compression with neural networks and temporal sub-sampling, Proceedings of the IEEE, Vol. 84, No. 10, pp. | Random neural network | 0.813999 |
16,056 | 5, no. 1, pp. 154–164, 1993. E. Gelenbe, V. Koubi, F. Pekergin, Dynamical random neural network approach to the traveling salesman problem, Proc. | Random neural network | 0.813999 |
16,057 | ), New York, Pergamon, pp. 139–147, 1992. E. Gelenbe, Learning in the recurrent random neural network, Neural Computation, vol. | Random neural network | 0.813999 |
16,058 | Artificial Neural Networks, Helsinki, pp. 307–312, 1991. E. Gelenbe, F. Batty, Minimum cost graph covering with the random neural network, Computer Science and Operations Research, O. Balci (ed. | Random neural network | 0.813999 |
16,059 | E. Gelenbe, Stability of the random neural network model, Neural Computation, vol. 2, no. 2, pp. 239–247, 1990. | Random neural network | 0.813999 |
16,060 | References SourcesE. Gelenbe, Random neural networks with negative and positive signals and product form solution, Neural Computation, vol. 1, no. 4, pp. 502–511, 1989. | Random neural network | 0.813999 |
16,061 | The method has been applied to two molecules of interest in organic photochemistry - ethylene and cyclobutene. The photodynamics of ethylene involves both covalent and ionic electronic excited states and the return to the ground state proceeds through a pyramidalized geometry. For the photoinduced ring opening of cyclo... | Ab initio multiple spawning | 0.813996 |
16,062 | The ab initio multiple spawning, or AIMS, method is a time-dependent formulation of quantum chemistry. In AIMS, nuclear dynamics and electronic structure problems are solved simultaneously. Quantum mechanical effects in the nuclear dynamics are included, especially the nonadiabatic effects which are crucial in modeling... | Ab initio multiple spawning | 0.813996 |
16,063 | Physics that incorporates elements of either QM or ER (or both) is said to be modern physics. It is in this latter sense that the term is generally used.Modern physics is often encountered when dealing with extreme conditions. Quantum mechanical effects tend to appear when dealing with "lows" (low temperatures, small d... | Modern Physics | 0.813993 |
16,064 | In a literal sense, the term modern physics means up-to-date physics. In this sense, a significant portion of so-called classical physics is modern. However, since roughly 1890, new discoveries have caused significant paradigm shifts: especially the advent of quantum mechanics (QM) and relativity (ER). | Modern Physics | 0.813993 |
16,065 | In general, quantum and relativistic effects are believed to exist across all scales, although these effects may be very small at human scale. While quantum mechanics is compatible with special relativity (See: Relativistic quantum mechanics), one of the unsolved problems in physics is the unification of quantum mechan... | Modern Physics | 0.813993 |
16,066 | Modern physics is a branch of physics that developed in the early 20th century and onward or branches greatly influenced by early 20th century physics. Notable branches of modern physics include quantum mechanics, special relativity and general relativity. Classical physics is typically concerned with everyday conditio... | Modern Physics | 0.813993 |
16,067 | These are generally considered to be the topics regarded as the "core" of the foundation of modern physics: | Modern Physics | 0.813992 |
16,068 | Feedstock is one of the main sources of environmental impacts in the biofuel production, the source of this impacts are related to the field operation to grow, handle and transport the biomass to the biorefinery gate. Agricultural residues are the feedstock with the lowest environmental impact followed by lignocellulos... | C16 Biosciences | 0.813991 |
16,069 | Underlying every C*-algebra is a *-semigroup. An important instance is the algebra Mn(C) of n-by-n matrices over C, with the conjugate transpose as involution. If X is a set, the set of all binary relations on X is a *-semigroup with the * given by the converse relation, and the multiplication given by the usual compos... | Monoid with involution | 0.813988 |
16,070 | In terms of examples for these notions, in the *-semigroup of binary relations on a set, the partial isometries are the relations that are difunctional. The projections in this *-semigroup are the partial equivalence relations.The partial isometries in a C*-algebra are exactly those defined in this section. In the case... | Monoid with involution | 0.813988 |
16,071 | As of March 2017, OpenBiome had provided over 20,000 treatments to 50 states and 7 countries.In 2015, OpenBiome announced the launch of PersonalBiome, a stool banking program through which individuals could store their stool for future use in fecal transplantation after microbial dysbiosis.OpenBiome was founded in 2012... | OpenBiome | 0.813985 |
16,072 | Embryogenesis, Ontogeny and Developmental biology - a discipline that has given rise to many scientific field theories. Bioenergetics - the study of energy exchange on the molecular level of living systems. Biological psychiatry, Neurology, Psychoneuroimmunology | Radiobiology | 0.81398 |
16,073 | The interactions between organisms and electromagnetic fields (EMF) and ionizing radiation can be studied in a number of ways: Radiation physics Radiation chemistry Molecular and cell biology Molecular genetics Cell death and apoptosis High and low-level electromagnetic radiation and health Specific absorption rates of... | Radiobiology | 0.81398 |
16,074 | Radiobiology experiments typically make use of a radiation source which could be: An isotopic source, typically 137Cs or 60Co. A particle accelerator generating high energy protons, electrons or charged ions. Biological samples can be irradiated using either a broad, uniform beam, or using a microbeam, focused down to ... | Radiobiology | 0.81398 |
16,075 | The bridge is synthetically incorporated at the 2', 4'-position of the ribose to afford a 2', 4'-BNA monomer. The monomers can be incorporated into oligonucleotide polymeric structures using standard phosphoramidite chemistry. BNAs are structurally rigid oligo-nucleotides with increased binding affinities and stability... | Bridged Nucleic Acids | 0.813975 |
16,076 | The Kobon triangle problem is an unsolved problem in combinatorial geometry first stated by Kobon Fujimura (1903-1983). The problem asks for the largest number N(k) of nonoverlapping triangles whose sides lie on an arrangement of k lines. Variations of the problem consider the projective plane rather than the Euclidean... | Kobon triangle problem | 0.813974 |
16,077 | In quantum field theory, and especially asymptotically free quantum field theories, an observable is infrared safe if it does not depend on the low energy/long distance physics of the theory. Such observables can therefore be calculated reliably using perturbative methods and then compared to experiment. An example of ... | Infrared safety (particle physics) | 0.813973 |
16,078 | In abstract algebraic logic, a branch of mathematical logic, the Leibniz operator is a tool used to classify deductive systems, which have a precise technical definition and capture a large number of logics. The Leibniz operator was introduced by Wim Blok and Don Pigozzi, two of the founders of the field, as a means to... | Abstract algebraic hierarchy | 0.813969 |
16,079 | For instance, protoalgebraic logics, forming the widest class in the hierarchy – i.e., the one that lies in the bottom of the hierarchy and contains all other classes – are characterized by the monotonicity of the Leibniz operator on their theories. Other notable classes are formed by the equivalential logics, the weak... | Abstract algebraic hierarchy | 0.813969 |
16,080 | The Leibniz operator and the study of various of its properties that may or may not be satisfied for particular sentential logics have given rise to what is now known as the abstract algebraic hierarchy or Leibniz hierarchy of sentential logics. Logics are classified in various levels of this hierarchy depending on how... | Abstract algebraic hierarchy | 0.813969 |
16,081 | In the mathematical field of group theory, Lagrange's theorem is a theorem that states that for any finite group G, the order (number of elements) of every subgroup of G divides the order of G. The theorem is named after Joseph-Louis Lagrange. The following variant states that for a subgroup H {\displaystyle H} of a fi... | Lagrange's theorem (group theory) | 0.813963 |
16,082 | Any prime divisor q {\displaystyle q} of the Mersenne number 2 p − 1 {\displaystyle 2^{p}-1} satisfies 2 p ≡ 1 ( mod q ) {\displaystyle 2^{p}\equiv 1{\pmod {q}}} (see modular arithmetic), meaning that the order of 2 {\displaystyle 2} in the multiplicative group ( Z / q Z ) ∗ {\displaystyle (\mathbb {Z} /q\mathbb {Z} )^... | Lagrange's theorem (group theory) | 0.813963 |
16,083 | These special cases were known long before the general theorem was proved. The theorem also shows that any group of prime order is cyclic and simple, since the subgroup generated by any non-identity element must be the whole group itself. Lagrange's theorem can also be used to show that there are infinitely many primes... | Lagrange's theorem (group theory) | 0.813963 |
16,084 | A consequence of the theorem is that the order of any element a of a finite group (i.e. the smallest positive integer number k with ak = e, where e is the identity element of the group) divides the order of that group, since the order of a is equal to the order of the cyclic subgroup generated by a. If the group has n ... | Lagrange's theorem (group theory) | 0.813963 |
16,085 | Because V contains all disjoint transpositions in A4, gvg−1 ∈ V. Hence, gvg−1 ∈ H ⋂ V = K. Since gvg−1 ≠ v, we have demonstrated that there is a third element in K. But earlier we assumed that |K| = 2, so we have a contradiction. Therefore, our original assumption that there is a subgroup of order 6 is not true and con... | Lagrange's theorem (group theory) | 0.813963 |
16,086 | V = {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}.Let K = H ⋂ V. Since both H and V are subgroups of A4, K is also a subgroup of A4. From Lagrange's theorem, the order of K must divide both 6 and 4, the orders of H and V respectively. The only two positive integers that divide both 6 and 4 are 1 and 2. | Lagrange's theorem (group theory) | 0.813963 |
16,087 | The converse of Lagrange's theorem states that if d is a divisor of the order of a group G, then there exists a subgroup H where |H| = d. We will examine the alternating group A4, the set of even permutations as the subgroup of the Symmetric group S4. A4 = {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3), (1 2 3), (1 3 2), (1 2 ... | Lagrange's theorem (group theory) | 0.813963 |
16,088 | However, there exist solvable groups that are not CLT (for example, A4) and CLT groups that are not supersolvable (for example, S4, the symmetric group of degree 4). There are partial converses to Lagrange's theorem. For general groups, Cauchy's theorem guarantees the existence of an element, and hence of a cyclic subg... | Lagrange's theorem (group theory) | 0.813963 |
16,089 | Lagrange's theorem raises the converse question as to whether every divisor of the order of a group is the order of some subgroup. This does not hold in general: given a finite group G and a divisor d of |G|, there does not necessarily exist a subgroup of G with order d. The smallest example is A4 (the alternating grou... | Lagrange's theorem (group theory) | 0.813963 |
16,090 | Lagrange's theorem can be extended to the equation of indexes between three subgroups of G. If we take K = {e} (e is the identity element of G), then = |G| and = |H|. Therefore, we can recover the original equation |G| = |H|. | Lagrange's theorem (group theory) | 0.813963 |
16,091 | With the later development of abstract groups, this result of Lagrange on polynomials was recognized to extend to the general theorem about finite groups which now bears his name. In his Disquisitiones Arithmeticae in 1801, Carl Friedrich Gauss proved Lagrange's theorem for the special case of ( Z / p Z ) ∗ {\displayst... | Lagrange's theorem (group theory) | 0.813963 |
16,092 | Lagrange himself did not prove the theorem in its general form. He stated, in his article Réflexions sur la résolution algébrique des équations, that if a polynomial in n variables has its variables permuted in all n! ways, the number of different polynomials that are obtained is always a factor of n!. (For example, if... | Lagrange's theorem (group theory) | 0.813963 |
16,093 | Furthermore, since the tangent bundle is dual to the cotangent bundle, multivector fields of degree k are dual to k-forms, and both are subsumed in the general concept of a tensor field, which is a section of some tensor bundle, often consisting of exterior powers of the tangent and cotangent bundles. A (k,0)-tensor fi... | Polyvector field | 0.813962 |
16,094 | A Polyvector field within Mathematics topology is concerned with the properties of a geometric object. A multivector field, polyvector field of degree k , or k-vector field, on a manifold M {\displaystyle M} , is a generalization of the notion of a vector field on a manifold. Whereas a vector field X ∈ Γ ( T M ) {\disp... | Polyvector field | 0.813962 |
16,095 | In quantum field theory, a branch of theoretical physics, crossing is the property of scattering amplitudes that allows antiparticles to be interpreted as particles going backwards in time. Crossing states that the same formula that determines the S-matrix elements and scattering amplitudes for particle A {\displaystyl... | Crossing (physics) | 0.813962 |
16,096 | The data were also useful as a monitor of the trapped radiation fluxes. The instrument was provided by Dr. C. Bostrom of the Applied Physics Laboratory of Johns Hopkins University. The instrument was turned off on 4 October 1991 because it was giving erroneous information. | International Ultraviolet Explorer | 0.813958 |
16,097 | Ultraviolet astronomy was impossible before the Space Age, and some of the first space telescopes were UV telescopes designed to observe this previously inaccessible region of the electromagnetic spectrum. One particular success was the second Orbiting Astronomical Observatory (OAO-2), which had a number of 20 cm (7.9 ... | International Ultraviolet Explorer | 0.813958 |
16,098 | In 1987, a star in the Large Magellanic Cloud exploded as a supernova. Designated SN 1987A, this event was of enormous importance to astronomy, as it was the closest known supernova to Earth, and the first visible to the naked eye, since Kepler's star in 1604 – before the invention of the telescope. The opportunity to ... | International Ultraviolet Explorer | 0.813958 |
16,099 | The IUE mission, by virtue of its very long duration and the fact that for most of its lifetime it provided astronomers' only access to UV light, had a major impact on astronomy. By the end of its mission it was considered by far the most successful and productive space observatory mission. For many years after the end... | International Ultraviolet Explorer | 0.813958 |
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