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16,200 | The diagram is a schematic representation of the graph with vertices V = { 1 , 2 , 3 , 4 , 5 , 6 } {\displaystyle V=\{1,2,3,4,5,6\}} and edges E = { { 1 , 2 } , { 1 , 5 } , { 2 , 3 } , { 2 , 5 } , { 3 , 4 } , { 4 , 5 } , { 4 , 6 } } . {\displaystyle E=\{\{1,2\},\{1,5\},\{2,3\},\{2,5\},\{3,4\},\{4,5\},\{4,6\}\}.} In com... | Finite graph | 0.813899 |
16,201 | The Standard addition method, often used in analytical chemistry, quantifies the analyte present in an unknown. This method is useful for analyzing complex samples where a matrix effect interferes with the analyte signal. In comparison to the calibration curve method, the standard addition method has the advantage of t... | Standard addition method | 0.813897 |
16,202 | Abstract class Accessibility Abstract method Abstraction (computer science) Access control Access modifiers Accessor method Adapter pattern Aspect-oriented | Index of object-oriented programming articles | 0.813895 |
16,203 | Immutable object (also called immutable value) Information hiding Inheritance Initialize Inline function Inner class Instance (computer science) Instance method Instance variable (also called data member) Interaction diagram Interface Inversion of control (IoC) Iterator | Index of object-oriented programming articles | 0.813895 |
16,204 | Object (computer science) Object type OOPSLA – annual conference on Object-Oriented Programming, Systems, Languages, and Applications Open/closed principle Orthogonality Overload | Index of object-oriented programming articles | 0.813895 |
16,205 | Classical Electrodynamics is a textbook written by theoretical particle and nuclear physicist John David Jackson. The book originated as lecture notes that Jackson prepared for teaching graduate-level electromagnetism first at McGill University and then at the University of Illinois at Urbana-Champaign. Intended for gr... | Classical Electrodynamics (book) | 0.813894 |
16,206 | Jackson wrote that this is in acknowledgement of the fact virtually all undergraduate textbooks on electrodynamics employ SI units and admitted he had "betrayed" an agreement he had with Edward Purcell that they would support each other in the use of Gaussian units. In the third edition, some materials, such as those o... | Classical Electrodynamics (book) | 0.813894 |
16,207 | Advanced topics treated in the first edition include magnetohydrodynamics, plasma physics, the vector form of Kirchhoff's diffraction theory, special relativity, and radiation emitted by moving and colliding charges. Jackson's choice of these topics is aimed at students interested in theoretical physics in general and ... | Classical Electrodynamics (book) | 0.813894 |
16,208 | According to a 2015 review of Andrew Zangwill's Modern Electrodynamics in the American Journal of Physics, "he classic electrodynamics text for the past four decades has been the monumental work by J. D. Jackson, the book from which most current-generation physicists took their first course." | Classical Electrodynamics (book) | 0.813894 |
16,209 | Reviewer Royce Zia from the Virginia Polytechnic Institute wrote that according to many students and professors, a major problem with the first edition of the book was how mathematically heavy the book was, which distracted students from the essential physics. In the second edition, many issues were addressed, more ins... | Classical Electrodynamics (book) | 0.813894 |
16,210 | He wrote that Modern Electrodynamics by Andrew Zangwill offers a "stimulating fresh look" on this subject.James Russ, an experimental high-energy physicist at the Carnegie Mellon University, was of the opinion that examples are challenging, and the fine points of physics are often left as exercises. He added that Moder... | Classical Electrodynamics (book) | 0.813894 |
16,211 | Fox took an advanced course on electrodynamics in 1965 using the first edition of Jackson and taught graduate electrodynamics for the first time in 1978 using the second edition.Jagdish Mehra, a physicist and historian of science, wrote that Jackson's text is not as good as the book of the same name by Julian Schwinger... | Classical Electrodynamics (book) | 0.813894 |
16,212 | However, the problems in Jackson do not pertain to other branches of physics, such as condensed-matter physics and biophysics. For optimal results, one must fill in the steps between equations and solve a lot of practice problems. Suggested readings and references are valuable. | Classical Electrodynamics (book) | 0.813894 |
16,213 | Physicist Wayne Saslow from Texas A&M University observed that some important new applications were added to the text, such as fiber optics and dielectric waveguides, which are crucial in modern communications technology, and synchrotron light sources, responsible for advances in condensed-matter physics, and that frag... | Classical Electrodynamics (book) | 0.813894 |
16,214 | (hyperbolic cosine addition formula). The commutative and associative laws are functional equations. In its familiar form, the associative law is expressed by writing the binary operation in infix notation, but if we write f(a, b) instead of a ○ b then the associative law looks more like a conventional functional equat... | Functional equation | 0.813892 |
16,215 | In 1993, the IUPAC proposed the shorter name "dalton" (with symbol "Da") for the unified atomic mass unit. As with other unit names such as watt and newton, "dalton" is not capitalized in English, but its symbol, "Da", is capitalized. The name was endorsed by the International Union of Pure and Applied Physics (IUPAP) ... | Atomic mass units | 0.813891 |
16,216 | The atomic mass constant can also be expressed as its energy-equivalent, muc2. The 2018 CODATA recommended values are: The megaelectronvolt mass-equivalent (MeV/c2) is commonly used as a unit of mass in particle physics, and these values are also important for the practical determination of relative atomic masses. | Atomic mass units | 0.813891 |
16,217 | Perrin estimated the Avogadro number by a variety of methods, at the turn of the 20th century. He was awarded the 1926 Nobel Prize in Physics, largely for this work. | Atomic mass units | 0.813891 |
16,218 | A dry lab is a laboratory where the nature of the experiments does not involve significant risk. This is in contrast to a wet lab where it is necessary to handle various types of chemicals and biological hazards. An example of a dry lab is one where computational or applied mathematical analyses are done on a computer-... | Dry lab | 0.813888 |
16,219 | As computing power has grown exponentially this approach to research, often referred to as in silico (as opposed to in vitro and in vivo), has amassed more attention, especially in the area of bioinformatics. More specifically, within bioinformatics, is the study of proteins or proteomics, which is the elucidation of t... | Dry lab | 0.813888 |
16,220 | Once we can see how a protein folds then we can see how it works as a catalyst, or in intracellular communication, e.g. neuroreceptor-neurotransmitter interaction. How certain compounds may be used to enhance or prevent the function of these proteins and how an elucidated protein overall plays a role in disease can als... | Dry lab | 0.813888 |
16,221 | Phycocyanin can be used in many practices, it is particularly used medicine and foods applications. It can also be used in genetics, where it acts a tracer due to its natural fluorescence. | Phycocyanin | 0.813882 |
16,222 | Any intersection of any family of closed sets is closed (this includes intersections of infinitely many closed sets) The union of finitely many closed sets is closed. The empty set is closed. The whole set is closed.In fact, if given a set X {\displaystyle X} and a collection F ≠ ∅ {\displaystyle \mathbb {F} \neq \varn... | Closed set | 0.813882 |
16,223 | Bacterial colony optimization is an algorithm used in evolutionary computing. The algorithm is based on a lifecycle model that simulates some typical behaviors of E. coli bacteria during their whole lifecycle, including chemotaxis, communication, elimination, reproduction, and migration. | Microbial intelligence | 0.813881 |
16,224 | These behaviors occur in single species populations, or mixed species populations. Examples are colonies or swarms of myxobacteria, quorum sensing, and biofilms.It has been suggested that a bacterial colony loosely mimics a biological neural network. The bacteria can take inputs in form of chemical signals, process the... | Microbial intelligence | 0.813881 |
16,225 | Micropredators that inhabit soil, including social predatory bacteria, have significant implications for its ecology. Soil biodiversity, managed in part by these micropredators, is of significant importance for carbon cycling and ecosystem functioning.The complicated interaction of microbes in the soil has been propose... | Microbial intelligence | 0.813881 |
16,226 | Protozoan ciliate Tetrahymena has the capacity to 'memorize' the geometry of its swimming area. Cells that were separated and confined in a droplet of water, recapitulated circular swimming trajectories upon release. This may result mainly from a rise in intracellular calcium. | Microbial intelligence | 0.813881 |
16,227 | Linnik's theorem in analytic number theory answers a natural question after Dirichlet's theorem on arithmetic progressions. It asserts that there exist positive c and L such that, if we denote p(a,d) the least prime in the arithmetic progression a + n d , {\displaystyle a+nd,\ } where n runs through the positive intege... | Linnik's constant | 0.813878 |
16,228 | Both definitions give the same result for a general static spherically symmetric metric.Theorem: Subject to an energy condition, a black hole in any spherically symmetric spacetime must be surrounded by a photon sphere. Conversely, subject to an energy condition, any photon sphere must cover more than a certain amount ... | Photon surface | 0.813878 |
16,229 | For a single polynomial equation, root-finding algorithms can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of algebraic geometry, a difficult branch ... | Nonlinear system | 0.813878 |
16,230 | Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials (of degree greater than one) to zero. For example, x 2 + x − 1 = 0 . {\displaystyle x^{2}+x-1=0\,.} | Nonlinear system | 0.813878 |
16,231 | In mathematics and statistics, random projection is a technique used to reduce the dimensionality of a set of points which lie in Euclidean space. Random projection methods are known for their power, simplicity, and low error rates when compared to other methods. According to experimental results, random projection pre... | Random projections | 0.813877 |
16,232 | Dimensionality reduction, as the name suggests, is reducing the number of random variables using various mathematical methods from statistics and machine learning. Dimensionality reduction is often used to reduce the problem of managing and manipulating large data sets. Dimensionality reduction techniques generally use... | Random projections | 0.813877 |
16,233 | The Johnson-Lindenstrauss lemma states that large sets of vectors in a high-dimensional space can be linearly mapped in a space of much lower (but still high) dimension n with approximate preservation of distances. One of the explanations of this effect is the exponentially high quasiorthogonal dimension of n-dimension... | Random projections | 0.813877 |
16,234 | In solar physics, a transequatorial loop is a structure present in the solar corona that connects two different regions of opposite magnetic polarity in opposite hemispheres of the Sun. These connected regions are not limited to active regions, but are most commonly found during the times of maximum solar activity, the... | Transequatorial loop | 0.813874 |
16,235 | Nor can they be extended immediately to the case where the inquiry is not for potential surfaces but, say, for surfaces of least area, or surfaces of constant positive gaussian curvature, which are to pass through a prescribed twisted curve or to stretch over a given ring surface. It is my conviction that it will be po... | Hilbert's twentieth problem | 0.813872 |
16,236 | The original problem statement in its entirety is as follows: An important problem closely connected with the foregoing is the question concerning the existence of solutions of partial differential equations when the values on the boundary of the region are prescribed. This problem is solved in the main by the keen me... | Hilbert's twentieth problem | 0.813872 |
16,237 | Hilbert noted that there existed methods for solving partial differential equations where the function's values were given at the boundary, but the problem asked for methods for solving partial differential equations with more complicated conditions on the boundary (e.g., involving derivatives of the function), or for ... | Hilbert's twentieth problem | 0.813872 |
16,238 | In the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. To be useful in appli... | Hilbert's twentieth problem | 0.813872 |
16,239 | Discovered as a radio source in 1983, the blazar has since been observed across the entire electromagnetic spectrum. TXS 0506+056 is the first known source of high energy astrophysical neutrinos, identified following the IceCube-170922A neutrino event in an early example of multi-messenger astronomy. The only astronomi... | TXS 0506+056 | 0.813868 |
16,240 | The detection of both neutrinos and light from the same object was an early example of multi-messenger astronomy.A search of archived neutrino data from IceCube found evidence for an earlier flare of lower-energy neutrinos in 2014-2015 (a form of precovery), which supports identification of the blazar as a source of ne... | TXS 0506+056 | 0.813868 |
16,241 | This example demonstrates that the topological boundary of an open ball of radius r > 0 {\displaystyle r>0} is not necessarily equal to the corresponding sphere of radius r {\displaystyle r} (centered at the same point); it also shows that the closure of an open ball of radius r > 0 {\displaystyle r>0} is not necessari... | Boundary component | 0.813862 |
16,242 | In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set. Notatio... | Boundary component | 0.813862 |
16,243 | {\displaystyle \partial \Omega =\left\{(x,y):x^{2}+y^{2}=1\right\}.} If the disk is viewed as a set in R 3 {\displaystyle \mathbb {R} ^{3}} with its own usual topology, that is, Ω = { ( x , y , 0 ): x 2 + y 2 ≤ 1 } , {\displaystyle \Omega =\left\{(x,y,0):x^{2}+y^{2}\leq 1\right\},} then the boundary of the disk is the ... | Boundary component | 0.813862 |
16,244 | In the space of rational numbers with the usual topology (the subspace topology of R {\displaystyle \mathbb {R} } ), the boundary of ( − ∞ , a ) , {\displaystyle (-\infty ,a),} where a {\displaystyle a} is irrational, is empty. The boundary of a set is a topological notion and may change if one changes the topology. Fo... | Boundary component | 0.813862 |
16,245 | Consider the real line R {\displaystyle \mathbb {R} } with the usual topology (that is, the topology whose basis sets are open intervals) and Q , {\displaystyle \mathbb {Q} ,} the subset of rational numbers (whose topological interior in R {\displaystyle \mathbb {R} } is empty). Then ∂ ( 0 , 5 ) = ∂ = ∂ = { 0 , 5 } {... | Boundary component | 0.813862 |
16,246 | Stress orientation: Increased subduction angle increases the arc parallel component. Rheology and anisotropy: Mechanical properties of the wedge: (coulomb vs plastic) influence the wedge geometry. Boundary conditions: The friction and geometry between the backstop and the wedge constitute the boundary conditions. | Strain partitioning | 0.813857 |
16,247 | In structural geology, strain partitioning is the distribution of the total strain experienced on a rock, area, or region, in terms of different strain intensity and strain type (i.e. pure shear, simple shear, dilatation). This process is observed on a range of scales spanning from the grain – crystal scale to the plat... | Strain partitioning | 0.813857 |
16,248 | Uniqueness can then be deduced from the Bruhat-Tits fixed point theorem. Indeed, any bounded closed set in a Hadamard space is contained in a unique smallest closed ball, the center of which is called its circumcenter. In particular a compact group acting by isometries must fix the circumcenter of each of its orbits. | Maximal compact subgroup | 0.813857 |
16,249 | For a real semisimple Lie group, Cartan's proof of the existence and uniqueness of a maximal compact subgroup can be found in Borel (1950) and Helgason (1978). Cartier (1955) and Hochschild (1965) discuss the extension to connected Lie groups and connected locally compact groups. For semisimple groups, existence is a c... | Maximal compact subgroup | 0.813857 |
16,250 | The algebraic topology of the Lie groups is also largely carried by a maximal compact subgroup K. To be precise, a connected Lie group is a topological product (though not a group theoretic product) of a maximal compact K and a Euclidean space – G = K × Rd – thus in particular K is a deformation retract of G, and is ho... | Maximal compact subgroup | 0.813857 |
16,251 | The Cartan-Iwasawa-Malcev theorem asserts that every connected Lie group (and indeed every connected locally compact group) admits maximal compact subgroups and that they are all conjugate to one another. For a semisimple Lie group uniqueness is a consequence of the Cartan fixed point theorem, which asserts that if a c... | Maximal compact subgroup | 0.813857 |
16,252 | Mostow (1955) also related the general problem for semisimple groups to the case of GL(n, R). The corresponding symmetric space is the space of positive symmetric matrices. A direct proof of uniqueness relying on elementary properties of this space is given in Hilgert & Neeb (2012). Let g {\displaystyle {\mathfrak {g}}... | Maximal compact subgroup | 0.813857 |
16,253 | In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups. Maximal compact subgroups play an important role in the classification of Lie groups and especially semi-simple Lie groups. Maximal compact subgrou... | Maximal compact subgroup | 0.813857 |
16,254 | The Helmholtz machine (named after Hermann von Helmholtz and his concept of Helmholtz free energy) is a type of artificial neural network that can account for the hidden structure of a set of data by being trained to create a generative model of the original set of data. The hope is that by learning economical represen... | Helmholtz machines | 0.813855 |
16,255 | The only plane curves of genus seven are singular, since seven is not a triangular number, and the minimum degree for such a curve is six. The butterfly curve has branching number and multiplicity two, and hence the singularity link has two components, pictured at right. The area of the algebraic butterfly curve is giv... | Butterfly curve (algebraic) | 0.813855 |
16,256 | In mathematics, the algebraic butterfly curve is a plane algebraic curve of degree six, given by the equation x 6 + y 6 = x 2 . {\displaystyle x^{6}+y^{6}=x^{2}.} The butterfly curve has a single singularity with delta invariant three, which means it is a curve of genus seven. | Butterfly curve (algebraic) | 0.813855 |
16,257 | Make a compensatory change in J 1 {\displaystyle J_{1}} by decreasing e 1 {\displaystyle e_{1}} such that S 1 {\displaystyle S_{1}} is restored to its original concentration ( δ S 1 = 0 {\displaystyle \delta S_{1}=0} ). Since e 1 {\displaystyle e_{1}} and S 1 {\displaystyle S_{1}} have not changed, δ J 1 = 0 {\displays... | Branched pathways | 0.813852 |
16,258 | This also means that in this situation, there can be more than one Rate-limiting step (biochemistry) in a pathway. Unlike a linear pathway, values for C e 3 J 2 {\displaystyle C_{e_{3}}^{J_{2}}} and C e 1 J 2 {\displaystyle C_{e_{1}}^{J_{2}}} are not bounded between zero and one. Depending on the values of the elastici... | Branched pathways | 0.813852 |
16,259 | Hence the flux through e 2 {\displaystyle e_{2}} is almost entirely governed by the activity of e 2 {\displaystyle e_{2}} . Because of the flux summation theorem and the fact that C e 2 J 2 = 1 {\displaystyle C_{e_{2}}^{J_{2}}=1} , it means that the remaining two coefficients must be equal and opposite in value. Since ... | Branched pathways | 0.813852 |
16,260 | Some important work was done in enumerative geometry in particular, by Schubert, that is now considered as anticipating the theory of Chern classes, taken as representing the algebraic topology of Grassmannians. Projective geometry later proved key to Paul Dirac's invention of quantum mechanics. At a foundational level... | Projective Geometry | 0.813848 |
16,261 | This period in geometry was overtaken by research on the general algebraic curve by Clebsch, Riemann, Max Noether and others, which stretched existing techniques, and then by invariant theory. Towards the end of the century, the Italian school of algebraic geometry (Enriques, Segre, Severi) broke out of the traditional... | Projective Geometry | 0.813848 |
16,262 | Techniques were supposed to be synthetic: in effect projective space as now understood was to be introduced axiomatically. As a result, reformulating early work in projective geometry so that it satisfies current standards of rigor can be somewhat difficult. Even in the case of the projective plane alone, the axiomatic... | Projective Geometry | 0.813848 |
16,263 | These transformations represent projectivities of the complex projective line. In the study of lines in space, Julius Plücker used homogeneous coordinates in his description, and the set of lines was viewed on the Klein quadric, one of the early contributions of projective geometry to a new field called algebraic geome... | Projective Geometry | 0.813848 |
16,264 | The works of Gaspard Monge at the end of 18th and beginning of 19th century were important for the subsequent development of projective geometry. The work of Desargues was ignored until Michel Chasles chanced upon a handwritten copy during 1845. Meanwhile, Jean-Victor Poncelet had published the foundational treatise on... | Projective Geometry | 0.813848 |
16,265 | He made Euclidean geometry, where parallel lines are truly parallel, into a special case of an all-encompassing geometric system. Desargues's study on conic sections drew the attention of 16-year-old Blaise Pascal and helped him formulate Pascal's theorem. | Projective Geometry | 0.813848 |
16,266 | The first geometrical properties of a projective nature were discovered during the 3rd century by Pappus of Alexandria. Filippo Brunelleschi (1404–1472) started investigating the geometry of perspective during 1425 (see the history of perspective for a more thorough discussion of the work in the fine arts that motivate... | Projective Geometry | 0.813848 |
16,267 | In projective geometry the intersection of lines formed by corresponding points of a projectivity in a plane are of particular interest. The set of such intersections is called a projective conic, and in acknowledgement of the work of Jakob Steiner, it is referred to as a Steiner conic. Suppose a projectivity is formed... | Projective Geometry | 0.813848 |
16,268 | Given three non-collinear points, there are three lines connecting them, but with four points, no three collinear, there are six connecting lines and three additional "diagonal points" determined by their intersections. The science of projective geometry captures this surplus determined by four points through a quatern... | Projective Geometry | 0.813848 |
16,269 | A projective space is of: (M1) at most dimension 0 if it has no more than 1 point, (M2) at most dimension 1 if it has no more than 1 line, (M3) at most dimension 2 if it has no more than 1 plane,and so on. It is a general theorem (a consequence of axiom (3)) that all coplanar lines intersect—the very principle Projecti... | Projective Geometry | 0.813848 |
16,270 | One can add further axioms restricting the dimension or the coordinate ring. For example, Coxeter's Projective Geometry, references Veblen in the three axioms above, together with a further 5 axioms that make the dimension 3 and the coordinate ring a commutative field of characteristic not 2. | Projective Geometry | 0.813848 |
16,271 | The distance between points is given by a Cayley-Klein metric, known to be invariant under the translations since it depends on cross-ratio, a key projective invariant. The translations are described variously as isometries in metric space theory, as linear fractional transformations formally, and as projective linear ... | Projective Geometry | 0.813847 |
16,272 | Poncelet examined the projective properties of objects (those invariant under central projection) and, by basing his theory on the concrete pole and polar relation with respect to a circle, established a relationship between metric and projective properties. The non-Euclidean geometries discovered soon thereafter were ... | Projective Geometry | 0.813847 |
16,273 | In algebra, Solomon's descent algebra of a Coxeter group is a subalgebra of the integral group ring of the Coxeter group, introduced by Solomon (1976). | Descent algebra | 0.813847 |
16,274 | In the special case of the symmetric group Sn, the descent algebra is given by the elements of the group ring such that permutations with the same descent set have the same coefficients. (The descent set of a permutation σ consists of the indices i such that σ(i) > σ(i+1).) The descent algebra of the symmetric group Sn... | Descent algebra | 0.813847 |
16,275 | In 2004, ECCB was jointly held with the Intelligent Systems for Molecular Biology (ISMB) conference for the first time. It was also co-located with the Genes, Proteins & Computers conference. This meeting, held in Glasgow, UK, was the largest bioinformatics conference ever held, attended by 2,136 delegates, submitting ... | European Conference on Computational Biology | 0.813847 |
16,276 | ECCB was formed with the intent of providing a European conference focusing on advances in computational biology and their application to problems in molecular biology. The conference was initially to be held on a rotating basis, with the idea that previously successful regional conferences (for instance, the German Co... | European Conference on Computational Biology | 0.813847 |
16,277 | Since 2007, the conference has been held jointly with Intelligent Systems for Molecular Biology (ISMB) every second year. The conference also hosts the European ISCB Student Council Symposium. The proceedings of the conference are published by the journal Bioinformatics. | European Conference on Computational Biology | 0.813847 |
16,278 | The European Conference on Computational Biology (ECCB) is a scientific meeting on the subjects of bioinformatics and computational biology. It covers a wide spectrum of disciplines, including bioinformatics, computational biology, genomics, computational structural biology, and systems biology. ECCB is organized annua... | European Conference on Computational Biology | 0.813847 |
16,279 | Formally, a coalgebra over a field K is a vector space C over K together with K-linear maps Δ: C → C ⊗ C and ε: C → K such that ( i d C ⊗ Δ ) ∘ Δ = ( Δ ⊗ i d C ) ∘ Δ {\displaystyle (\mathrm {id} _{C}\otimes \Delta )\circ \Delta =(\Delta \otimes \mathrm {id} _{C})\circ \Delta } ( i d C ⊗ ε ) ∘ Δ = i d C = ( ε ⊗ i d C ) ... | Coalgebra | 0.813839 |
16,280 | The shuffle product is appropriate, because it preserves the order of the terms appearing in the product, as is needed by non-commutative algebras. The singular homology of a topological space forms a graded coalgebra whenever the Künneth isomorphism holds, e.g. if the coefficients are taken to be a field.If C is the K... | Coalgebra | 0.813839 |
16,281 | Examples of coalgebras include the tensor algebra, the exterior algebra, Hopf algebras and Lie bialgebras. Unlike the polynomial case above, none of these are commutative. Therefore, the coproduct becomes the shuffle product, rather than the divided power structure given above. | Coalgebra | 0.813839 |
16,282 | Take an arbitrary set S and form the K-vector space C = K(S) with basis S, as follows. The elements of this vector space C are those functions from S to K that map all but finitely many elements of S to zero; identify the element s of S with the function that maps s to 1 and all other elements of S to 0. Define Δ(s) = ... | Coalgebra | 0.813839 |
16,283 | The peculiar form of having the j {\displaystyle \mathbf {j} } appear only once in the coproduct, rather than (for example) defining j ↦ j ⊗ j {\displaystyle \mathbf {j} \mapsto \mathbf {j} \otimes \mathbf {j} } is in order to maintain linearity: for this example, (and for representation theory in general), the coprodu... | Coalgebra | 0.813839 |
16,284 | In this setting, the coproduct is the map Δ: J → J ⊗ J {\displaystyle \Delta :J\to J\otimes J} that takes Δ: j ↦ j ⊗ 1 + 1 ⊗ j {\displaystyle \Delta :\mathbf {j} \mapsto \mathbf {j} \otimes 1+1\otimes \mathbf {j} } For this example, J {\displaystyle J} can be taken to be one of the spin representations of the rotation ... | Coalgebra | 0.813839 |
16,285 | A tensor algebra comes with a tensor product (the internal one); it can also be equipped with a second tensor product, the "external" one, or the coproduct, having the form above. That they are two different products is emphasized by recalling that the internal tensor product of a vector and a scalar is just simple sca... | Coalgebra | 0.813839 |
16,286 | Given two systems A , B {\displaystyle A,B} with angular momenta j A {\displaystyle j_{A}} and j B {\displaystyle j_{B}} , a particularly important task is to find the total angular momentum j A + j B {\displaystyle j_{A}+j_{B}} given the combined state | A ⟩ ⊗ | B ⟩ {\displaystyle |A\rangle \otimes |B\rangle } . This ... | Coalgebra | 0.813839 |
16,287 | One frequently recurring example of coalgebras occurs in representation theory, and in particular, in the representation theory of the rotation group. A primary task, of practical use in physics, is to obtain combinations of systems with different states of angular momentum and spin. For this purpose, one uses the Cleb... | Coalgebra | 0.813839 |
16,288 | Some of its features were not incorporated into COBOL so that it would not look like IBM had dominated the design process, and Jean Sammet said in 1981 that there had been a "strong anti-IBM bias" from some committee members (herself included). In one case, after Roy Goldfinger, author of the COMTRAN manual and interme... | Named condition | 0.813836 |
16,289 | COBOL's data structures influenced subsequent programming languages. Its record and file structure influenced PL/I and Pascal, and the REDEFINES clause was a predecessor to Pascal's variant records. Explicit file structure definitions preceded the development of database management systems and aggregated data was a sig... | Named condition | 0.813836 |
16,290 | 9 (1st ed.). Butterworth-Heinemann. ISBN 978-0-7506-2636-1.Volume 9 builds on the original statistical physics book, with more applications to condensed matter theory. | Course of Theoretical Physics | 0.813836 |
16,291 | Lifshitz, Evgeny M.; Pitaevskii, Lev P. (1980). Statistical Physics, Part 2: Theory of the Condensed State. Vol. | Course of Theoretical Physics | 0.813836 |
16,292 | The presentation of material is advanced and typically considered suitable for graduate-level study. Despite this specialized character, it is estimated that a million volumes of the Course were sold by 2005.The series has been called "renowned" in Science and "celebrated" in American Scientist. A note in Mathematical ... | Course of Theoretical Physics | 0.813836 |
16,293 | ISBN 978-0-7506-3371-0.The original edition comprised two books, labelled part 1 and part 2. The first covered general aspects of relativistic quantum mechanics and relativistic quantum field theory, leading onto quantum electrodynamics. The second continued with quantum electrodynamics and what was then known about th... | Course of Theoretical Physics | 0.813836 |
16,294 | The Course of Theoretical Physics is a ten-volume series of books covering theoretical physics that was initiated by Lev Landau and written in collaboration with his student Evgeny Lifshitz starting in the late 1930s. It is said that Landau composed much of the series in his head while in an NKVD prison in 1938–1939. H... | Course of Theoretical Physics | 0.813836 |
16,295 | Vol. 5 (3rd ed.). Butterworth-Heinemann. ISBN 978-0-7506-3372-7.Volume 5 covers general statistical mechanics and thermodynamics and applications, including chemical reactions, phase transitions, and condensed matter physics. | Course of Theoretical Physics | 0.813836 |
16,296 | Landau, Lev D.; Lifshitz, Evgeny M. (1980). Statistical Physics. | Course of Theoretical Physics | 0.813836 |
16,297 | Clarendon Press. ASIN B00085BKZG. Statistical Physics. | Course of Theoretical Physics | 0.813836 |
16,298 | Early version: Landau, Lev D. (1938). Statistical Physics. | Course of Theoretical Physics | 0.813836 |
16,299 | Statistical Physics. Vol. 5 (1st ed.). 1951. | Course of Theoretical Physics | 0.813836 |
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