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12,900 | Set X = { 0 , 1 } {\displaystyle X=\{0,1\}} . The σ {\textstyle \sigma } -algebra on finite sets such as the one above is usually the power set, which is the set of all subsets (of a given set) and is denoted by ℘ ( ⋅ ) . {\textstyle \wp (\cdot ).} Sticking with this convention, we set In this simple case, the power se... | Measure space | 0.816671 |
12,901 | A measure space is a triple ( X , A , μ ) , {\displaystyle (X,{\mathcal {A}},\mu ),} where X {\displaystyle X} is a set A {\displaystyle {\mathcal {A}}} is a σ-algebra on the set X {\displaystyle X} μ {\displaystyle \mu } is a measure on ( X , A ) {\displaystyle (X,{\mathcal {A}})} In other words, a measure space consi... | Measure space | 0.816671 |
12,902 | A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the σ-algebra) and the method that is used for measuring (the measure). One important example of a measure spa... | Measure space | 0.816671 |
12,903 | A planar algebra is a representation of the planar operad; more precisely, it is a family of vector spaces ( P n , ± ) n ∈ N {\displaystyle ({\mathcal {P}}_{n,\pm })_{n\in \mathbb {N} }} , called n {\displaystyle n} -box spaces, on which acts the planar operad, i.e. for any tangle T {\displaystyle T} (with one output d... | Planar algebra | 0.816662 |
12,904 | The Temperley-Lieb planar algebra T L ( δ ) {\displaystyle {\mathcal {TL}}(\delta )} is generated by the planar tangles without input disk; its 3 {\displaystyle 3} -box space T L 3 , + ( δ ) {\displaystyle {\mathcal {TL}}_{3,+}(\delta )} is generated by Moreover, a closed string is replaced by a multiplication by δ {\d... | Planar algebra | 0.816662 |
12,905 | A (finite dimensional) Kac algebra "corresponds" (up to isomorphism) to an irreducible subfactor planar algebra of depth two.The subfactor planar algebra associated to an inclusion of finite groups, does not always remember the (core-free) inclusion.A Bisch-Jones subfactor planar algebra B J ( δ 1 , δ 2 ) {\displaystyl... | Planar algebra | 0.816662 |
12,906 | Note that e 1 = e N M {\displaystyle e_{1}=e_{N}^{M}} and i d = e M M {\displaystyle id=e_{M}^{M}} are biprojections; this can be seen as follows: A projection b {\displaystyle b} is a biprojection iff it is the Jones projection e K M {\displaystyle e_{K}^{M}} of an intermediate subfactor N ⊂ K ⊂ M {\displaystyle N\sub... | Planar algebra | 0.816662 |
12,907 | The map F 4 {\displaystyle {\mathcal {F}}^{4}} corresponds to four 1 {\displaystyle 1} -clicks of the outer star, so it's the identity map, and then a ¯ ¯ = a {\displaystyle {\overline {\overline {a}}}=a} . In the Kac algebra case, the contragredient is exactly the antipode, which, for a finite group, correspond to the... | Planar algebra | 0.816662 |
12,908 | Let N ⊂ M {\displaystyle N\subset M} be a finite index subfactor, and P {\displaystyle {\mathcal {P}}} the corresponding subfactor planar algebra. Assume that P {\displaystyle {\mathcal {P}}} is irreducible (i.e. P 1 , + = N ′ ∩ M 1 = C {\displaystyle {\mathcal {P}}_{1,+}=N'\cap M_{1}=\mathbb {C} } ). Let N ⊂ K ⊂ M {\d... | Planar algebra | 0.816662 |
12,909 | In mathematics, planar algebras first appeared in the work of Vaughan Jones on the standard invariant of a II1 subfactor. They also provide an appropriate algebraic framework for many knot invariants (in particular the Jones polynomial), and have been used in describing the properties of Khovanov homology with respect ... | Planar algebra | 0.816662 |
12,910 | The idea of the planar algebra is to be a diagrammatic axiomatization of the standard invariant. | Planar algebra | 0.816662 |
12,911 | The family of vector spaces ( T n , ± ) n ∈ N {\displaystyle ({\mathcal {T}}_{n,\pm })_{n\in \mathbb {N} }} generated by the planar tangles having 2 n {\displaystyle 2n} intervals on their output disk and a white (or black) ⋆ {\displaystyle \star } -marked interval, admits a planar algebra structure. | Planar algebra | 0.816662 |
12,912 | A semisimple and cosemisimple Hopf algebra over an algebraically closed field is encoded in a planar algebra defined by generators and relations, and "corresponds" (up to isomorphism) to a connected, irreducible, spherical, non degenerate planar algebra with non zero modulus δ {\displaystyle \delta } and of depth two.N... | Planar algebra | 0.816662 |
12,913 | The subfactor planar algebras are completely classified for index at most 5 {\displaystyle 5} and a bit beyond. This classification was initiated by Uffe Haagerup. It uses (among other things) a listing of possible principal graphs, together with the embedding theorem and the jellyfish algorithm.A subfactor planar alge... | Planar algebra | 0.816662 |
12,914 | A planar algebra ( P n , ± ) {\displaystyle ({\mathcal {P}}_{n,\pm })} is a subfactor planar algebra if and only if it is the standard invariant of an extremal subfactor N ⊆ M {\displaystyle N\subseteq M} of index = δ 2 {\displaystyle =\delta ^{2}} , with P n , + = N ′ ∩ M n − 1 {\displaystyle {\mathcal {P}}_{n,+}=N'\... | Planar algebra | 0.816662 |
12,915 | ∪ {\displaystyle \delta \in \{2\cos(\pi /n)|n=3,4,5,...\}\cup } For δ {\displaystyle \delta } as above, let I {\displaystyle {\mathcal {I}}} be the null ideal (generated by elements a {\displaystyle a} with ⟨ a | a ⟩ = 0 {\displaystyle \langle a\vert a\rangle =0} ). Then the quotient T L ( δ ) / I {\displaystyle {\mat... | Planar algebra | 0.816662 |
12,916 | No-ghost theorem: The planar algebra T L ( δ ) {\displaystyle {\mathcal {TL}}(\delta )} has no ghost (i.e. element a {\displaystyle a} with ⟨ a | a ⟩ < 0 {\displaystyle \langle a\vert a\rangle <0} ) if and only if δ ∈ { 2 cos ( π / n ) | n = 3 , 4 , 5 , . . . } | Planar algebra | 0.816662 |
12,917 | In statistics, an augmented Dickey–Fuller test (ADF) tests the null hypothesis that a unit root is present in a time series sample. The alternative hypothesis is different depending on which version of the test is used, but is usually stationarity or trend-stationarity. It is an augmented version of the Dickey–Fuller t... | Augmented Dickey–Fuller test | 0.816659 |
12,918 | Finite differences were introduced by Brook Taylor in 1715 and have also been studied as abstract self-standing mathematical objects in works by George Boole (1860), L. M. Milne-Thomson (1933), and Károly Jordan (1939). Finite differences trace their origins back to one of Jost Bürgi's algorithms (c. 1592) and work by ... | Difference calculus | 0.816657 |
12,919 | {\displaystyle \Delta (x)=f(x+1)-f(x).} A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives. There are many similarities between difference equations and differential equations, specially in the solving methods. | Difference calculus | 0.816657 |
12,920 | A finite difference is a mathematical expression of the form f (x + b) − f (x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, e... | Difference calculus | 0.816657 |
12,921 | Such generalizations are useful for constructing different modulus of continuity. The generalized difference can be seen as the polynomial rings R. It leads to difference algebras. Difference operator generalizes to Möbius inversion over a partially ordered set. As a convolution operator: Via the formalism of incidence... | Difference calculus | 0.816657 |
12,922 | Neuromechanics is an interdisciplinary field that combines biomechanics and neuroscience to understand how the nervous system interacts with the skeletal and muscular systems to enable animals to move. In a motor task, like reaching for an object, neural commands are sent to motor neurons to activate a set of muscles, ... | Neuromechanics | 0.816656 |
12,923 | Theoretical computer science – including data structures and algorithms, theory of computation, information theory and coding theory, programming language theory, and formal methods Computer systems – including computer architecture and computer engineering, computer performance analysis, concurrency, and distributed c... | Computer scientist | 0.816655 |
12,924 | Computer scientists are often hired by software publishing firms, scientific research and development organizations where they develop the theories that allow new technologies to be developed. Computer scientists are also employed by educational institutions such as universities. Computer scientists can follow more pra... | Computer scientist | 0.816655 |
12,925 | A computer scientist is a scholar who specializes in the academic study of computer science.Computer scientists typically work on the theoretical side of computation, as opposed to the hardware side on which computer engineers mainly focus (although there is overlap). Although computer scientists can also focus their w... | Computer scientist | 0.816655 |
12,926 | Most computer scientists are required to possess a PhD, M.S. in computer science, or other similar fields like Information and Computer Science (CIS), or a closely related discipline such as mathematics or physics. | Computer scientist | 0.816655 |
12,927 | Various pieces of modern symbolic notation were introduced in this period, notably: 1556: Niccolò Tartaglia introduces parenthesis. 1557: Robert Recorde introduces the equal sign. 1591: François Viète's New algebra shows the modern notational algebraic manipulation. | Scientific breakthrough | 0.816651 |
12,928 | c. 700 BC: Pythagoras's theorem is discovered by Baudhayana in the Hindu Shulba Sutras in Upanishadic India. However, Indian mathematics, especially North Indian mathematics, generally did not have a tradition of communicating proofs, and it is not fully certain that Baudhayana or Apastamba knew of a proof. | Scientific breakthrough | 0.816651 |
12,929 | c. 600 BC: Thales of Miletus discovers Thales's theorem. | Scientific breakthrough | 0.816651 |
12,930 | 300 BC: Euclid discovers the Euclidean algorithm. 3rd century BC: Pingala in Mauryan India discovers the binomial coefficients in a combinatorial context and the additive formula for generating them ( n r ) = ( n − 1 r ) + ( n − 1 r − 1 ) {\displaystyle {\tbinom {n}{r}}={\tbinom {n-1}{r}}+{\tbinom {n-1}{r-1}}} , i.e. a... | Scientific breakthrough | 0.816651 |
12,931 | c. 300 BC: Euclid proves the Fundamental Theorem of Arithmetic. c. | Scientific breakthrough | 0.816651 |
12,932 | c. 500 BC: Hippasus, a Pythagorean, discovers irrational numbers. 4th century BC: Thaetetus shows that square roots are either integer or irrational. 4th century BC: Thaetetus enumerates the Platonic solids, an early work in graph theory. | Scientific breakthrough | 0.816651 |
12,933 | 3rd century BC: Archimedes develops the field of statics, introducing notions such as the center of gravity, mechanical equilibrium, the study of levers, and hydrostatics. 350-50 BC: Clay tablets from (possibly Hellenistic-era) Babylon describe the mean speed theorem. | Scientific breakthrough | 0.816651 |
12,934 | 2100 BC: The concept of area is first recognised in Babylonian clay tablets, and 3-dimensional volume is discussed in an Egyptian papyrus. This begins the study of geometry. Early 2nd millennium BC: Similar triangles and side-ratios are studied in Egypt (e.g. in the Rhind Mathematical Papyrus, a copy of an older Middle... | Scientific breakthrough | 0.816651 |
12,935 | 2012 – Higgs boson is discovered at CERN (confirmed to 99.999% certainty) 2012 – Photonic molecules are discovered at MIT 2014 – Exotic hadrons are discovered at the LHCb 2014 – Photonic metamaterials are discovered to make passive daytime radiative cooling possible by Raman et al. 2016 – The LIGO team detected gravita... | Scientific breakthrough | 0.816651 |
12,936 | 1856 – Robert Forester Mushet develops a process for the decarbonisation, and re-carbonisation of iron, thorough the addition of a calculated quantity of spiegeleisen, to produce cheap, consistently high quality steel. 1858 – Rudolf Virchow: cells can only arise from pre-existing cells 1859 – Charles Darwin and Alfred ... | Scientific breakthrough | 0.816651 |
12,937 | 1827 – Evariste Galois development of group theory 1827 – Georg Ohm: Ohm's law (Electricity) 1827 – Amedeo Avogadro: Avogadro's law (Gas law) 1828 – Friedrich Wöhler synthesized urea, refuting vitalism 1830 – Nikolai Lobachevsky created Non-Euclidean geometry 1831 – Michael Faraday discovers electromagnetic induction 1... | Scientific breakthrough | 0.816651 |
12,938 | 1802 – Jean-Baptiste Lamarck: teleological evolution 1805 – John Dalton: Atomic Theory in (chemistry) 1820 – Hans Christian Ørsted discovers that a current passed through a wire will deflect the needle of a compass, establishing the deep relationship between electricity and magnetism (electromagnetism). 1820 – Michael ... | Scientific breakthrough | 0.816651 |
12,939 | 1600 – William Gilbert: Earth's magnetic field 1608 – Earliest record of an optical telescope 1609 – Johannes Kepler: first two laws of planetary motion 1610 – Galileo Galilei: Sidereus Nuncius: telescopic observations 1614 – John Napier: use of logarithms for calculation 1619 – Johannes Kepler: third law of planetary ... | Scientific breakthrough | 0.816651 |
12,940 | The Greeks make numerous advances in mathematics and astronomy through the Archaic, Classical and Hellenistic periods. | Scientific breakthrough | 0.816651 |
12,941 | General chemistry (sometimes referred to as "gen chem") is offered by colleges and universities as an introductory level chemistry course usually taken by students during their first year. The course is usually run with a concurrent lab section that gives students an opportunity to experience a laboratory environment a... | General chemistry | 0.816649 |
12,942 | The concepts taught in a typical general chemistry course are as follows: Stoichiometry Conservation of energy Conservation of mass Elementary atomic theory Periodic table and periodicity Law of constant composition Gas laws Nuclear chemistry Solubility Acid-base chemistry Chemical bonding Chemical kinetics Thermodynam... | General chemistry | 0.816649 |
12,943 | Measure-preserving dynamical system Ergodic theory Mixing (mathematics) Almost periodic function Symbolic dynamics Time scale calculus Arithmetic dynamics Sequential dynamical system Graph dynamical system Topological dynamical system | List of dynamical systems and differential equations topics | 0.816649 |
12,944 | Acids and bases Atomic structure Bonding and chemical interactions Chemical kinetics Electrochemistry Equilibrium Solutions Stoichiometry The gas phase Thermochemistry Redox reactions | General chemistry | 0.816649 |
12,945 | This makes a simply passing grade not acceptable for students with medical school aspirations. General chemistry professors have been known to make tests worth a large portion of the course, and make them more challenging than the material presents itself as. Grade deflation, purposely adjusting the grades of a course ... | General chemistry | 0.816649 |
12,946 | Students who enroll in AP chemistry in high school, a course that mirrors what is covered in college, could be perceived as having an advantage over students who do not come to college with a strong chemistry background. Students who wish to be competitive in applying to medical schools try to achieve success in genera... | General chemistry | 0.816649 |
12,947 | Students who are enrolled in general chemistry often desire to become doctors, researchers, and educators. Because of the demands of these fields, professors believe that the level of rigor that is associated with general chemistry should be elevated from that of a typical introductory course. This has led to this cour... | General chemistry | 0.816649 |
12,948 | Students in colleges and universities looking to follow the "pre-medical" track are required to pass general chemistry as the Association of American Medical Colleges requires at least one full year of chemistry. In order for students to apply to medical school, they must pass the medical college admission test, or MCA... | General chemistry | 0.816649 |
12,949 | Exponential growth Malthusian catastrophe Exponential response formula Simple harmonic motion Phasor (physics) RLC circuit Resonance Impedance Reactance Musical tuning Orbital resonance Tidal resonance Oscillator Harmonic oscillator Electronic oscillator Floquet theory Fundamental frequency Oscillation (Vibration) Fund... | List of dynamical systems and differential equations topics | 0.816649 |
12,950 | Examples of differential equations Autonomous system (mathematics) Picard–Lindelöf theorem Peano existence theorem Carathéodory existence theorem Numerical ordinary differential equations Bendixson–Dulac theorem Gradient conjecture Recurrence plot Limit cycle Initial value problem Clairaut's equation Singular solution ... | List of dynamical systems and differential equations topics | 0.816649 |
12,951 | Deterministic system (mathematics) Linear system Partial differential equation Dynamical systems and chaos theory Chaos theory Chaos argument Butterfly effect 0-1 test for chaos Bifurcation diagram Feigenbaum constant Sharkovskii's theorem Attractor Strange nonchaotic attractor Stability theory Mechanical equilibrium A... | List of dynamical systems and differential equations topics | 0.816649 |
12,952 | Pendulum Inverted pendulum Double pendulum Foucault pendulum Spherical pendulum Kinematics Equation of motion Dynamics (mechanics) Classical mechanics Isolated physical system Lagrangian mechanics Hamiltonian mechanics Routhian mechanics Hamilton-Jacobi theory Appell's equation of motion Udwadia–Kalaba equation Celesti... | List of dynamical systems and differential equations topics | 0.816649 |
12,953 | Particle swarm optimization Self-propelled particles Swarm intelligence | List of dynamical systems and differential equations topics | 0.816649 |
12,954 | Political administrative boundaries (local, state and federal), suburbs, postal, and cadastral are approximated by mesh block allocations. This is so that statistics can be easily made available for the many types of uses needed, such as non-standard areas like water catchments etc. Mesh blocks are about four or five t... | Mesh blocks | 0.816648 |
12,955 | In Australia mesh blocks were developed by the Australian Bureau of Statistics from 2006, and released in February 2008, although they were not fully implemented until 2011. The 2016 Australian Statistical Geography Standard contains 358,122 mesh blocks in Australia. They are intended to be the basic unit which compris... | Mesh blocks | 0.816648 |
12,956 | New Zealand's countrywide meshblock framework was first set up in 1976, although the term dates back to at least the 1916 census. The meshblock pattern is updated each year. It comprised 41,376 meshblocks at the 2006 census, increasing to 46,637 in 2013, 53,589 in 2018 and 57,539 in 2023.Meshblocks are defined by Stati... | Mesh blocks | 0.816648 |
12,957 | The Genetics and Molecular Research Journal is a peer-reviewed open-access scientific journal in the fields of biology and medicine, edited and published monthly in the fields of genetics, molecular biology, proteomics, genomics and evolution. The journal was listed on Beall's list before it was taken down in 2017. A h... | Genetics and Molecular Research | 0.816647 |
12,958 | For example, 99.999... = 100. The real numbers also have an important but highly technical property called the least upper bound property. It can be shown that any ordered field, which is also complete, is isomorphic to the real numbers. The real numbers are not, however, an algebraically closed field, because they do ... | Number set | 0.816644 |
12,959 | The study of these questions belongs to number theory. Goldbach's conjecture is an example of a still unanswered question: "Is every even number the sum of two primes?" One answered question, as to whether every integer greater than one is a product of primes in only one way, except for a rearrangement of the primes, w... | Number set | 0.816644 |
12,960 | But most further development of the theory of primes in Europe dates to the Renaissance and later eras.In 1796, Adrien-Marie Legendre conjectured the prime number theorem, describing the asymptotic distribution of primes. Other results concerning the distribution of the primes include Euler's proof that the sum of the ... | Number set | 0.816644 |
12,961 | Prime numbers have been studied throughout recorded history. They are positive integers that are only divisible by 1 and themselves. Euclid devoted one book of the Elements to the theory of primes; in it he proved the infinitude of the primes and the fundamental theorem of arithmetic, and presented the Euclidean algori... | Number set | 0.816644 |
12,962 | One standard definition (there are slight variants) defines solutions of differential equations of the form F ( x , y , y ′ , ⋯ , y ( n ) ) = 0 {\displaystyle F\left(x,y,y',\cdots ,y^{(n)}\right)=0} ,where F {\displaystyle F} is a polynomial with constant coefficients, as algebraically transcendental or differentially ... | Transcendentally transcendental function | 0.816643 |
12,963 | The exponential function, logarithm, and the trigonometric and hyperbolic functions. The generalized hypergeometric functions, including special cases such as Bessel functions (except some special cases which are algebraic). | Transcendentally transcendental function | 0.816643 |
12,964 | A hypertranscendental function or transcendentally transcendental function is a transcendental analytic function which is not the solution of an algebraic differential equation with coefficients in Z {\displaystyle \mathbb {Z} } (the integers) and with algebraic initial conditions. | Transcendentally transcendental function | 0.816643 |
12,965 | The problem of unit commitment involves finding the least-cost dispatch of available generation resources to meet the electrical load. Generating resources can include a wide range of types: Nuclear Thermal (using coal, gas, other fossil fuels, or biomass) Renewables (including hydro, wind, wave-power, and solar)The ke... | Optimal power flow | 0.816642 |
12,966 | These programs typically make use of mathematical optimization techniques such linear programming, quadratic programming, and mixed integer programming. Multiple elements of a power system can be modelled. A power-flow study calculates the loading on transmission lines and the power necessary to be generated at generat... | Optimal power flow | 0.816642 |
12,967 | Power system long-term optimization focuses on optimizing the multi-year expansion and retirement plan for generation, transmission, and distribution facilities. The optimization problem will typically consider the long term investment cash flow and a simplified version of OPF / UC (Unit commitment), to make sure the p... | Optimal power flow | 0.816642 |
12,968 | When the red homozygous flower is paired with the white homozygous flower, the result yields a pink snapdragon flower. The pink snapdragon is the result of incomplete dominance. A similar type of incomplete dominance is found in the four o'clock plant wherein pink color is produced when true-bred parents of white and r... | Homozygous dominant | 0.816637 |
12,969 | Incomplete dominance (also called partial dominance, semi-dominance, intermediate inheritance, or occasionally incorrectly co-dominance in reptile genetics) occurs when the phenotype of the heterozygous genotype is distinct from and often intermediate to the phenotypes of the homozygous genotypes. The phenotypic result... | Homozygous dominant | 0.816637 |
12,970 | We will replace each random step by a deterministic step that keeps the conditional expectation of Q at or above |V|/(D+1). This will ensure a successful outcome, that is, one in which the independent set S has size at least |V|/(D+1), realizing the bound in Turán's theorem. Given that the first t steps have been taken... | Method of conditional probabilities | 0.816635 |
12,971 | In mathematics and computer science, the probabilistic method is used to prove the existence of mathematical objects with desired combinatorial properties. The proofs are probabilistic — they work by showing that a random object, chosen from some probability distribution, has the desired properties with positive probab... | Method of conditional probabilities | 0.816635 |
12,972 | The consistency strength of KPω is given by the Bachmann–Howard ordinal. KP fails to prove some common theorems in set theory, such as the Mostowski collapse lemma. | Kripke-Platek set theory | 0.816632 |
12,973 | The second core section includes long surveys of 26 branches of research mathematics such as algebraic geometry and combinatorial group theory. The third describes 38 important mathematical problems and theorems such as the four color theorem, the Birch and Swinnerton-Dyer conjecture, and the Halting problem. A collect... | The Princeton Companion to Mathematics | 0.816628 |
12,974 | The book concentrates primarily on modern pure mathematics rather than applied mathematics, although it does also cover both applications of mathematics and the mathematics that relates to those applications; it provides a broad overview of the significant ideas and developments in research mathematics. It is organized... | The Princeton Companion to Mathematics | 0.816628 |
12,975 | {\displaystyle x.} The theorem also guarantees that ‖ f φ ‖ = ‖ φ ‖ X ′ {\displaystyle \left\|f_{\varphi }\right\|=\|\varphi \|_{X^{\prime }}} and ‖ f φ R ‖ = ‖ φ R ‖ X R ′ . {\displaystyle \left\|f_{\varphi _{\mathbb {R} }}\right\|=\left\|\varphi _{\mathbb {R} }\right\|_{X_{\mathbb {R} }^{\prime }}.} | Dual vector | 0.816627 |
12,976 | Explicitly, this real inner product on X R {\displaystyle X_{\mathbb {R} }} is defined by ⟨ x | y ⟩ R := Re ⟨ x | y ⟩ {\displaystyle \langle x|y\rangle _{\mathbb {R} }:=\operatorname {Re} \langle x|y\rangle } for all x , y ∈ X {\displaystyle x,y\in X} and it induces the same norm on X {\displaystyle X} as ⟨ ⋅ | ⋅ ⟩ {... | Dual vector | 0.816627 |
12,977 | Let X be a topological vector space (TVS) with continuous dual space X ′ . {\displaystyle X'.} For any subset H of X ′ , {\displaystyle X',} the following are equivalent: H is equicontinuous; H is contained in the polar of some neighborhood of 0 {\displaystyle 0} in X; the (pre)polar of H is a neighborhood of 0 {\displ... | Dual vector | 0.816627 |
12,978 | In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If V is a vector space over a field k, the set of all linear functionals from V to k is itself a vector space over k ... | Dual vector | 0.816627 |
12,979 | There are numerous problems arising especially from applications that have to do with various notions of flows in networks, for example: Max flow min cut theorem | Algorithmic graph theory | 0.816625 |
12,980 | Within computer science, causal and non-causal linked structures are graphs that are used to represent networks of communication, data organization, computational devices, the flow of computation, etc. For instance, the link structure of a website can be represented by a directed graph, in which the vertices represent ... | Algorithmic graph theory | 0.816625 |
12,981 | Matrix structures include the incidence matrix, a matrix of 0's and 1's whose rows represent vertices and whose columns represent edges, and the adjacency matrix, in which both the rows and columns are indexed by vertices. In both cases a 1 indicates two adjacent objects and a 0 indicates two non-adjacent objects. The ... | Algorithmic graph theory | 0.816625 |
12,982 | A simpler proof considering only 633 configurations was given twenty years later by Robertson, Seymour, Sanders and Thomas.The autonomous development of topology from 1860 and 1930 fertilized graph theory back through the works of Jordan, Kuratowski and Whitney. Another important factor of common development of graph t... | Algorithmic graph theory | 0.816625 |
12,983 | Tait's reformulation generated a new class of problems, the factorization problems, particularly studied by Petersen and Kőnig. The works of Ramsey on colorations and more specially the results obtained by Turán in 1941 was at the origin of another branch of graph theory, extremal graph theory. The four color problem r... | Algorithmic graph theory | 0.816625 |
12,984 | (italics as in the original).The first textbook on graph theory was written by Dénes Kőnig, and published in 1936. Another book by Frank Harary, published in 1969, was "considered the world over to be the definitive textbook on the subject", and enabled mathematicians, chemists, electrical engineers and social scientis... | Algorithmic graph theory | 0.816625 |
12,985 | The fusion of ideas from mathematics with those from chemistry began what has become part of the standard terminology of graph theory. In particular, the term "graph" was introduced by Sylvester in a paper published in 1878 in Nature, where he draws an analogy between "quantic invariants" and "co-variants" of algebra a... | Algorithmic graph theory | 0.816625 |
12,986 | Enumerative graph theory then arose from the results of Cayley and the fundamental results published by Pólya between 1935 and 1937. These were generalized by De Bruijn in 1959. Cayley linked his results on trees with contemporary studies of chemical composition. | Algorithmic graph theory | 0.816625 |
12,987 | This study had many implications for theoretical chemistry. The techniques he used mainly concern the enumeration of graphs with particular properties. | Algorithmic graph theory | 0.816625 |
12,988 | The paper written by Leonhard Euler on the Seven Bridges of Königsberg and published in 1736 is regarded as the first paper in the history of graph theory. This paper, as well as the one written by Vandermonde on the knight problem, carried on with the analysis situs initiated by Leibniz. Euler's formula relating the n... | Algorithmic graph theory | 0.816625 |
12,989 | Graph theory plays an important role in electrical modeling of electrical networks, here, weights are associated with resistance of the wire segments to obtain electrical properties of network structures. Graphs are also used to represent the micro-scale channels of porous media, in which the vertices represent the por... | Algorithmic graph theory | 0.816625 |
12,990 | This approach is especially used in computer processing of molecular structures, ranging from chemical editors to database searching. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems. Similarly, in computa... | Algorithmic graph theory | 0.816625 |
12,991 | Graph theory is also used to study molecules in chemistry and physics. In condensed matter physics, the three-dimensional structure of complicated simulated atomic structures can be studied quantitatively by gathering statistics on graph-theoretic properties related to the topology of the atoms. Also, "the Feynman grap... | Algorithmic graph theory | 0.816625 |
12,992 | A directed graph or digraph is a graph in which edges have orientations. In one restricted but very common sense of the term, a directed graph is an ordered pair G = ( V , E ) {\displaystyle G=(V,E)} comprising: V {\displaystyle V} , a set of vertices (also called nodes or points); E ⊆ { ( x , y ) ∣ ( x , y ) ∈ V 2 and... | Algorithmic graph theory | 0.816625 |
12,993 | Subdivision containment is related to graph properties such as planarity. For example, Kuratowski's Theorem states: A graph is planar if it contains as a subdivision neither the complete bipartite graph K3,3 nor the complete graph K5.Another problem in subdivision containment is the Kelmans–Seymour conjecture: Every 5-... | Algorithmic graph theory | 0.816625 |
12,994 | Many graph properties are hereditary for minors, which means that a graph has a property if and only if all minors have it too. For example, Wagner's Theorem states: A graph is planar if it contains as a minor neither the complete bipartite graph K3,3 (see the Three-cottage problem) nor the complete graph K5.A similar ... | Algorithmic graph theory | 0.816625 |
12,995 | Many problems and theorems in graph theory have to do with various ways of coloring graphs. Typically, one is interested in coloring a graph so that no two adjacent vertices have the same color, or with other similar restrictions. One may also consider coloring edges (possibly so that no two coincident edges are the sa... | Algorithmic graph theory | 0.816625 |
12,996 | In mathematics, graphs are useful in geometry and certain parts of topology such as knot theory. Algebraic graph theory has close links with group theory. Algebraic graph theory has been applied to many areas including dynamic systems and complexity. | Algorithmic graph theory | 0.816625 |
12,997 | In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines). A distinction is made between undirected graphs... | Algorithmic graph theory | 0.816625 |
12,998 | Local feature size refers to several related concepts in computer graphics and computational geometry for measuring the size of a geometric object near a particular point. Given a smooth manifold M {\displaystyle M} , the local feature size at any point x ∈ M {\displaystyle x\in M} is the distance between x {\displayst... | Local feature size | 0.816622 |
12,999 | In the context of group theory, a sequence G 0 → f 1 G 1 → f 2 G 2 → f 3 ⋯ → f n G n {\displaystyle G_{0}\;{\xrightarrow {f_{1}}}\;G_{1}\;{\xrightarrow {f_{2}}}\;G_{2}\;{\xrightarrow {f_{3}}}\;\cdots \;{\xrightarrow {f_{n}}}\;G_{n}} of groups and group homomorphisms is called exact, if the image (or range) of each homo... | Numerical order | 0.816619 |
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