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4,500 | Cronin and Fitch received the Nobel Prize in Physics for this discovery in 1980. It turns out that although the KL and KS are weak eigenstates (because they have definite lifetimes for decay by way of the weak force), they are not quite CP eigenstates. Instead, for small ε (and up to normalization), KL = K2 + εK1and si... | Theta-tau puzzle | 0.827961 |
4,501 | While trying to verify Adair's results, J. Christenson, James Cronin, Val Fitch and Rene Turlay of Princeton University found decays of KL into two pions (CP = +1) in an experiment performed in 1964 at the Alternating Gradient Synchrotron at the Brookhaven laboratory. As explained in an earlier section, this required t... | Theta-tau puzzle | 0.827961 |
4,502 | In particle physics, a kaon (), also called a K meson and denoted K, is any of a group of four mesons distinguished by a quantum number called strangeness. In the quark model they are understood to be bound states of a strange quark (or antiquark) and an up or down antiquark (or quark). Kaons have proved to be a copiou... | Theta-tau puzzle | 0.827961 |
4,503 | Other emerging polarizable forcefields which have been applied to condensed phase systems are; the Sum of Interactions between Fragments ab initio computed (SIBFA) and the Quantum Chemical Topology Force Field (QCTFF). Polarizable water models are also being produced. The so-called charge on spring (COS) model gives wa... | Solvent model | 0.827955 |
4,504 | These models are commonly geometrically constrained with aspects of the geometry fixed such as the bond length or angles.Advancements around 2010 onwards in explicit solvent modelling see the use of a new generation of polarizable force fields, which are currently being created. These force fields are able to account f... | Solvent model | 0.827955 |
4,505 | Typically the known data comes from experimental measurement, although there is no reason why similar methods can not be used to correlate descriptor(s) with theoretical or predicted values. It is currently debated whether if more accurate experimental data was used to train these models whether the prediction from suc... | Solvent model | 0.827955 |
4,506 | Quantitative Structure–Activity Relationships (QSAR)/Quantitative Structure–Property Relationships (QSPR), whilst unable to directly model the physical process occurring in a condensed solvent phase, can provide useful predictions of solvent and solvation properties and activities; such as the solubility of a solute. T... | Solvent model | 0.827955 |
4,507 | In addition to these basic inequalities, one has Harnack's inequality, which states that positive harmonic functions on bounded domains are roughly constant. One important use of these inequalities is to prove convergence of families of harmonic functions or sub-harmonic functions, see Harnack's theorem. These converge... | Potential Theory | 0.82794 |
4,508 | A fruitful approach to the study of harmonic functions is the consideration of inequalities they satisfy. Perhaps the most basic such inequality, from which most other inequalities may be derived, is the maximum principle. Another important result is Liouville's theorem, which states the only bounded harmonic functions... | Potential Theory | 0.82794 |
4,509 | Knockdown may also be achieved by RNA interference (RNAi). Alternatively, genes may be substituted into an organism's genome (also known as a transgene) to create a gene knock-in and result in a gain of function by the host. Although these techniques have some inherent bias regarding the decision to link a phenotype to... | Molecular Genetics | 0.827934 |
4,510 | Reverse genetics is the term for molecular genetics techniques used to determine the phenotype resulting from an intentional mutation in a gene of interest. The phenotype is used to deduce the function of the un-mutated version of the gene. Mutations may be random or intentional changes to the gene of interest. Mutatio... | Molecular Genetics | 0.827934 |
4,511 | Replication of DNA and transcription from DNA to mRNA occurs in the nucleus while translation from RNA to proteins occurs in the ribosome. The genetic code is made of four base pairs: adenine, cytosine, uracil, and guanine and is redundant meaning multiple combinations of these base pairs (which are read in triplicate)... | Molecular Genetics | 0.827934 |
4,512 | This image shows an example of the central dogma using a DNA strand being transcribed then translated and showing important enzymes used in the processes The Central Dogma is the basis of all genetics and plays a key role in the study of molecular genetics. The Central Dogma states that DNA replicates itself, DNA is tr... | Molecular Genetics | 0.827934 |
4,513 | The mutant genes are then characterized as dominant (resulting in a gain of function), recessive (showing a loss of function), or epistatic (the mutant gene masks the phenotype of another gene). Finally, the location and specific nature of the mutation is mapped via sequencing. Forward genetics is an unbiased approach ... | Molecular Genetics | 0.827934 |
4,514 | Forward genetics is a molecular genetics technique used to identify genes or genetic mutations that produce a certain phenotype. In a genetic screen, random mutations are generated with mutagens (chemicals or radiation) or transposons and individuals are screened for the specific phenotype. Often, a secondary assay in ... | Molecular Genetics | 0.827934 |
4,515 | The culmination of all of those discoveries was a new field called genomics that links the molecular structure of a gene to the protein or RNA encoded by that segment of DNA and the functional expression of that protein within an organism. Today, through the application of molecular genetic techniques, genomics is bein... | Molecular Genetics | 0.827934 |
4,516 | Watson and Crick (in conjunction with Franklin and Wilkins) figured out the structure of DNA, a cornerstone for molecular genetics. The isolation of a restriction endonuclease in E. coli by Arber and Linn in 1969 opened the field of genetic engineering. Restriction enzymes were used to linearize DNA for separation by e... | Molecular Genetics | 0.827934 |
4,517 | Bacterial transformation is often induced by conditions of stress, and the function of transformation appears to be repair of genomic damage.The phage group was an informal network of biologists centered on Max Delbrück that contributed substantially to molecular genetics and the origins of molecular biology during the... | Molecular Genetics | 0.827934 |
4,518 | For molecular genetics to develop as a discipline, several scientific discoveries were necessary. The discovery of DNA as a means to transfer the genetic code of life from one cell to another and between generations was essential for identifying the molecule responsible for heredity. Molecular genetics arose initially ... | Molecular Genetics | 0.827934 |
4,519 | Molecular genetics is a sub-field of biology that addresses how differences in the structures or expression of DNA molecules manifests as variation among organisms. Molecular genetics often applies an "investigative approach" to determine the structure and/or function of genes in an organism's genome using genetic scre... | Molecular Genetics | 0.827934 |
4,520 | Like other curricula designed and developed during the 1990s to implement the NCTM Standards, Connected Math was criticized by supporters of more traditional curricula. Critics made the following claims: Reform curricula like CMP pay too little attention to the development of basic computational skills in number and al... | Connected Mathematics | 0.827934 |
4,521 | In the third edition, the collection of units was expanded to cover Common Core Standards for both grade eight and Algebra I. Each CMP grade level course aims to advance student understanding, skills, and problem solving in every content strand with increasing sophistication and challenge over the middle school grades.... | Connected Mathematics | 0.827934 |
4,522 | The first edition of Connected Mathematics, developed with financial support from the National Science Foundation, was designed to provide instructional materials for middle grades mathematics based on the 1989 Curriculum and Evaluation Standards and the 1991 Professional Standards for Teaching Mathematics from the Nat... | Connected Mathematics | 0.827934 |
4,523 | Studies have actually demonstrated that the two wavelengths together have a synergistic effect on the photosynthetic activity, rather than an additive one.Each photosystem has two parts: a reaction center, where the photochemistry occurs, and an antenna complex, which surrounds the reaction center. The antenna complex ... | Photosystem | 0.82792 |
4,524 | Photosystems are functional and structural units of protein complexes involved in photosynthesis. Together they carry out the primary photochemistry of photosynthesis: the absorption of light and the transfer of energy and electrons. Photosystems are found in the thylakoid membranes of plants, algae, and cyanobacteria.... | Photosystem | 0.82792 |
4,525 | In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to... | Distinct degree factorization | 0.82792 |
4,526 | A state transition function is a surjective function when every state has a predecessor (there can be no Garden of Eden). It is an injective function when no two states have the same successor. A surjunctive group is a group with the property that, when its elements are used as the cells of cellular automata, every inj... | Surjunctive group | 0.827898 |
4,527 | As a finite set, the alphabet has a discrete topology, and the collection of states can be given the product topology (called a prodiscrete topology because it is the product of discrete topologies). To be the transition function of a cellular automaton, a function from states to states must be a continuous function fo... | Surjunctive group | 0.827898 |
4,528 | The uses that have been derived from Vectorette PCR are many and have been useful to the science of biology. For example, it gives rise to methods that can help during the outbreaks of diseases by making it easier to subtype pathogens that are similar or closely related. It can also be used to help diagnose certain dis... | Vectorette PCR | 0.827896 |
4,529 | A lot of research has been conducted on Vectorette PCR and the applications it has in the field of biology. Scientists used Vectorette PCR to take the transgene flanking DNA and isolate it. They used this technique on the DNA belonging to mice that was next to transgene sections. From this the scientists were able to s... | Vectorette PCR | 0.827896 |
4,530 | The degree of a hypersurface F = 0 is the same as the total degree of the homogeneous polynomial F defining it (granted, in case F has repeated factors, that intersection theory is used to count intersections with multiplicity, as in Bézout's theorem). | Degree of an algebraic variety | 0.827896 |
4,531 | The optimization version of this problem is called max set splitting and requires finding the partition which maximizes the number of split elements of F. It is an APX-complete problem and hence in NPO. The set k-splitting problem is stated as follows: given S, F, and an integer k, does there exist a partition of S whi... | Set splitting problem | 0.827895 |
4,532 | Set splitting is special case of the not-all-equal satisfiability problem without negated variables. Additionally, Ek-set splitting equals non-monochromatic graph coloring of k-uniform hypergraphs. For k=2, the optimization variant reduces to the well-known maximum cut. == References == | Set splitting problem | 0.827895 |
4,533 | In mathematics, a bracket algebra is an algebraic system that connects the notion of a supersymmetry algebra with a symbolic representation of projective invariants. Given that L is a proper signed alphabet and Super is the supersymmetric algebra, the bracket algebra Bracket of dimension n over the field K is the quoti... | Bracket algebra | 0.827882 |
4,534 | The Sun is composed primarily of the chemical elements hydrogen and helium. At this time in the Sun's life, they account for 74.9% and 23.8%, respectively, of the mass of the Sun in the photosphere. All heavier elements, called metals in astronomy, account for less than 2% of the mass, with oxygen (roughly 1% of the Su... | Solar X-rays | 0.827877 |
4,535 | According to the virial theorem, half of this released gravitational energy goes into heating, which leads to a gradual increase in the rate at which fusion occurs and thus an increase in the luminosity. This process speeds up as the core gradually becomes denser. | Solar X-rays | 0.827877 |
4,536 | In organic chemistry, spiro compounds are compounds that have at least two molecular rings with only one common atom. The simplest spiro compounds are bicyclic (having just two rings), or have a bicyclic portion as part of the larger ring system, in either case with the two rings connected through the defining single c... | Spiro compound | 0.827862 |
4,537 | Bicyclic ring structures in organic chemistry that have two fully carbocyclic (all carbon) rings connected through just one atom are present both in natural products, as well as in esoteric targets of chemical synthesis. The two carbocycles can be different in nature, or identical. In common targets derived from natura... | Spiro compound | 0.827862 |
4,538 | Spiroaromaticity in organic chemistry refers to a special case of aromaticity in which conjugation is interrupted by a single spiroatom. Although this spiro center disrupts the continuous overlap of p-orbitals, traditionally thought to be a requirement for aromaticity, considerable thermodynamic stability and many of t... | Spiro compound | 0.827862 |
4,539 | The solution of carbon dioxide in water is thought to have made the seas slightly acidic, with a pH of about 5.5. The Hadean atmosphere has been characterized as a "gigantic, productive outdoor chemical laboratory," similar to volcanic gases today which still support some abiotic chemistry.Oceans may have appeared as s... | Prebiotic chemistry | 0.827858 |
4,540 | For instance, a person listening to a sentence like, "In the summer it is hot, and in the winter it is..." would be highly likely to predict the sentence completion "cold" in advance of actually hearing it. A form of prediction is also thought to occur in some types of lexical priming, a phenomenon whereby a word becom... | Prediction in language comprehension | 0.827853 |
4,541 | In the history of thermodynamics, Thermodynamik chemischer Vorgänge (Chemical thermodynamic process) is a sequence of three papers (1882–1883) written by German physicist Hermann von Helmholtz. It is one of the founding papers in thermodynamics, along with Josiah Willard Gibbs's 1876 paper "On the Equilibrium of Hetero... | Thermodynamik chemischer Vorgänge | 0.827849 |
4,542 | A lattice constant or lattice parameter is one of the physical dimensions and angles that determine the geometry of the unit cells in a crystal lattice, and is proportional to the distance between atoms in the crystal. A simple cubic crystal has only one lattice constant, the distance between atoms, but in general latt... | Lattice constant | 0.827847 |
4,543 | The Institute for Solar Physics (Swedish: Institutet för solfysik) is a Swedish research institute. It is managed as an independent institute associated with Stockholm University through its Department of Astronomy. It is also a national research infrastructure under the Swedish Research Council. The institute was firs... | Institute for Solar Physics | 0.827846 |
4,544 | The normalizer of a subset S of a Lie algebra (or Lie ring) L {\displaystyle {\mathfrak {L}}} is given by N L ( S ) = { x ∈ L ∣ ∈ S for all s ∈ S } . {\displaystyle \mathrm {N} _{\mathfrak {L}}(S)=\{x\in {\mathfrak {L}}\mid \in S{\text{ for all }}s\in S\}.} While this is the standard usage of the term "normalizer" in ... | Self-normalizing subgroup | 0.827843 |
4,545 | If R is a ring or an algebra over a field, and S is a subset of R, then the centralizer of S is exactly as defined for groups, with R in the place of G. If L {\displaystyle {\mathfrak {L}}} is a Lie algebra (or Lie ring) with Lie product , then the centralizer of a subset S of L {\displaystyle {\mathfrak {L}}} is defin... | Self-normalizing subgroup | 0.827843 |
4,546 | Source: Centralizers in rings and in algebras over a field are subrings and subalgebras over a field, respectively; centralizers in Lie rings and in Lie algebras are Lie subrings and Lie subalgebras, respectively. The normalizer of S in a Lie ring contains the centralizer of S. CR(CR(S)) contains S but is not necessari... | Self-normalizing subgroup | 0.827843 |
4,547 | Upper bounds are usually easy to show in applications, and this is why people have worked more on proving lower bounds. Neglecting other forces, it is reasonable to assume that ordinary matter is composed of negative and positive non-relativistic charges (electrons and nuclei), interacting solely via the Coulomb force.... | Stability of matter | 0.827836 |
4,548 | In physics, stability of matter refers to the problem of showing rigorously that a large number of charged quantum particles can coexist and form macroscopic objects, like ordinary matter. The first proof was provided by Freeman Dyson and Andrew Lenard in 1967–1968, but a shorter and more conceptual proof was found lat... | Stability of matter | 0.827836 |
4,549 | A chemical bonding model is a theoretical model used to explain atomic bonding structure, molecular geometry, properties, and reactivity of physical matter. This can refer to: VSEPR theory, a model of molecular geometry. Valence bond theory, which describes molecular electronic structure with localized bonds and lone p... | Chemical bonding model | 0.827823 |
4,550 | To be able to deliver the engineering degree, an École Master 's curriculum has to be validated by the Commission des titres d'ingénieur (Commission of the Engineering Title). It is important for the external observer to note that the system in France is extremely demanding in its entrance requirements (numerus clausus... | Engineering education | 0.827818 |
4,551 | After this period, some institutions offer specializations within the different fields of engineering (i.e. a student majoring in electrical engineering can choose to specialize in electronics or telecommunications) although most institutions balance their workload in order to give the students a consistent knowledge o... | Engineering education | 0.827818 |
4,552 | In Brazil, education in engineering is offered by both public and private institutions. A degree in engineering requires five to six years of studies, comprising the core courses, specific subjects, an internship and a Course Completion Paper. Due to the nature of college admissions in Brazil, most students have to dec... | Engineering education | 0.827818 |
4,553 | In Austria, similar to Germany, an engineering degree can be obtained from either universities or Fachhochschulen (universities of applied sciences). As in most of Europe, the education usually consists of a 3-year bachelor's degree and a 2-year master's degree. A lower engineering degree is offered by Höheren Technisc... | Engineering education | 0.827818 |
4,554 | Engineering education programs at universities in Argentina span a variety of disciplines and typically require 5–6 years of studies to complete. Most degree programs begin with foundational courses in mathematics, statistics, and the physical sciences during the first and second years, then move on to courses specific... | Engineering education | 0.827818 |
4,555 | The length of education, and training to qualify as a basic professional engineer, is typically 5 years, with 15–20 years for an engineer who takes responsibility for major projects. Science, technology, engineering, and mathematics (STEM) education in primary and secondary schools often serves as the foundation for en... | Engineering education | 0.827818 |
4,556 | An institution offering engineering education is called "teknisk högskola" (institute of technology). These schools primarily offers five-year programmes resulting in the civilingenjör degree (not to be confused with the narrower English term "civil engineer"), internationally corresponding to a Master of Science in En... | Engineering education | 0.827818 |
4,557 | Some of engineering specialties are completely traditional, such as machine building, computer and software engineering, automation, electrical engineering, electronics. Newer specialties are engineering design, mechatronics, aviation engineering, industrial engineering. The following technical universities prepare mai... | Engineering education | 0.827818 |
4,558 | The main subjects to be studied are mathematics, physics, chemistry, electrical engineering, etc. The degree received after completing with a state exam or defense of a thesis. Absolvents are awarded with the Ing. title always put in front of one's name. | Engineering education | 0.827818 |
4,559 | This article summarizes equations in the theory of quantum mechanics. | List of equations in quantum mechanics | 0.827812 |
4,560 | A fundamental physical constant occurring in quantum mechanics is the Planck constant, h. A common abbreviation is ħ = h/2π, also known as the reduced Planck constant or Dirac constant. The general form of wavefunction for a system of particles, each with position ri and z-component of spin sz i. Sums are over the disc... | List of equations in quantum mechanics | 0.827812 |
4,561 | In mathematics, an integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, the identity matrix, and the adjacency matrices used in graph theory, amongst many others. Integer matrices find frequent application in combinatorics. | Integer matrix | 0.827811 |
4,562 | Under the same assumption, there are also essentially no other properties that can be used to get polynomial time algorithms. This is formalized as follows: Theorem (Grohe): For a computable class of graphs G {\displaystyle {\mathcal {G}}} , the homomorphism problem for instances ( G , H ) {\displaystyle (G,H)} with G ... | Graph homomorphism | 0.827809 |
4,563 | CSP(Γ) is the constraint satisfaction problem where instances are only allowed to use constraints in Γ. Theorem (Feder, Vardi 1998): For every constraint language Γ, the problem CSP(Γ) is equivalent under polynomial-time reductions to some H-coloring problem, for some directed graph H.Intuitively, this means that every... | Graph homomorphism | 0.827809 |
4,564 | The homomorphism problem with a fixed graph H on the right side of each instance is also called the H-coloring problem. When H is the complete graph Kk, this is the graph k-coloring problem, which is solvable in polynomial time for k = 0, 1, 2, and NP-complete otherwise. In particular, K2-colorability of a graph G is e... | Graph homomorphism | 0.827809 |
4,565 | The order → for directed graphs is again a distributive lattice and a Heyting algebra, with join and meet operations defined as before. However, it is not dense. There is also a category with directed graphs as objects and homomorphisms as arrows, which is again a cartesian closed category. | Graph homomorphism | 0.827809 |
4,566 | Since these two operations are always defined, the category of graphs is a cartesian closed category. For the same reason, the lattice of equivalence classes of graphs under homomorphisms is in fact a Heyting algebra.For directed graphs the same definitions apply. In particular → is a partial order on equivalence class... | Graph homomorphism | 0.827809 |
4,567 | 107–166, doi:10.1007/978-94-015-8937-6_4 Godsil, C.; Royle, G. (2001), "6. Homomorphisms", Algebraic Graph Theory, Graduate Texts in Mathematics, vol. 207, Springer–Verlag New York, doi:10.1007/978-1-4613-0163-9, ISBN 978-1-4613-0163-9, S2CID 9661174 | Graph homomorphism | 0.827809 |
4,568 | Cameron, P. (2006), Graph Homomorphisms, Combinatorics Study Group Notes (PDF) Hell, Pavol; Nešetřil, Jaroslav (2004), Graphs and Homomorphisms, Oxford Lecture Series in Mathematics and Its Applications, vol. 28, Oxford University Press, ISBN 0-19-852817-5 Hahn, G.; Tardif, C. (1997), "Graph homomorphisms: structure an... | Graph homomorphism | 0.827809 |
4,569 | Bodirsky, M. (2007), Graph Homomorphisms and Universal Algebra, Course Notes (PDF) Hell, Pavol; Nešetřil, Jaroslav (2008), "Colouring, constraint satisfaction, and complexity" (PDF), Computer Science Review, 2 (3): 143–163, doi:10.1016/j.cosrev.2008.10.003 | Graph homomorphism | 0.827809 |
4,570 | Therefore, a graph G is k-colorable (has a homomorphism to Kk) if and only if some orientation of G has a homomorphism to T→k.A folklore theorem states that for all k, a directed graph G has a homomorphism to T→k if and only if it admits no homomorphism from the directed path P→k+1. Here P→n is the directed graph with ... | Graph homomorphism | 0.827809 |
4,571 | The fact that homomorphisms can be composed leads to rich algebraic structures: a preorder on graphs, a distributive lattice, and a category (one for undirected graphs and one for directed graphs). The computational complexity of finding a homomorphism between given graphs is prohibitive in general, but a lot is known ... | Graph homomorphism | 0.827809 |
4,572 | In the mathematical field of graph theory, a graph homomorphism is a mapping between two graphs that respects their structure. More concretely, it is a function between the vertex sets of two graphs that maps adjacent vertices to adjacent vertices. Homomorphisms generalize various notions of graph colorings and allow t... | Graph homomorphism | 0.827809 |
4,573 | This is often denoted as just G → H .The above definition is extended to directed graphs. Then, for a homomorphism f: G → H, (f(u),f(v)) is an arc (directed edge) of H whenever (u,v) is an arc of G. There is an injective homomorphism from G to H (i.e., one that maps distinct vertices in G to distinct vertices in H) if ... | Graph homomorphism | 0.827809 |
4,574 | It is common to model such a network with a set of coupled ordinary differential equations (ODEs) or SDEs, describing the reaction kinetics of the constituent parts. Suppose that our regulatory network has N {\displaystyle N} nodes, and let S 1 ( t ) , S 2 ( t ) , … , S N ( t ) {\displaystyle S_{1}(t),S_{2}(t),\ldots ,... | Genetic program | 0.827808 |
4,575 | {\displaystyle \lambda ^{n}(x+y)=\sum _{0}^{n}\lambda ^{i}(x)\lambda ^{n-i}(y).} For example, Z {\displaystyle \mathbb {Z} } is a λ-ring with λ n ( x ) = ( x n ) , {\displaystyle \lambda ^{n}(x)={\binom {x}{n}},} the binomial coefficients. The notion plays a central rule in the algebraic approach to the Riemann–Roch th... | Associative rings | 0.827798 |
4,576 | A ring R is a topological ring if its set of elements R is given a topology which makes the addition map ( +: R × R → R {\displaystyle +:R\times R\to R\,} ) and the multiplication map ⋅: R × R → R {\displaystyle \cdot :R\times R\to R} to be both continuous as maps between topological spaces (where X × X inherits the pr... | Associative rings | 0.827798 |
4,577 | The theorem may be illustrated by the following application to linear algebra. Let V be a finite-dimensional vector space over a field k and f: V → V a linear map with minimal polynomial q. Then, since k is a unique factorization domain, q factors into powers of distinct irreducible polynomials (that is, prime elements... | Associative rings | 0.827798 |
4,578 | An important class of integral domains that contain a PID is a unique factorization domain (UFD), an integral domain in which every nonunit element is a product of prime elements (an element is prime if it generates a prime ideal.) The fundamental question in algebraic number theory is on the extent to which the ring o... | Associative rings | 0.827798 |
4,579 | {\displaystyle \operatorname {End} _{R}(U)\simeq \prod _{i=1}^{r}\operatorname {M} _{m_{i}}(\operatorname {End} _{R}(U_{i})).} The Artin–Wedderburn theorem states any semisimple ring (cf. below) is of this form. A ring R and the matrix ring Mn(R) over it are Morita equivalent: the category of right modules of R is equi... | Associative rings | 0.827798 |
4,580 | As in linear algebra, a matrix ring may be canonically interpreted as an endomorphism ring: End R ( R n ) ≃ M n ( R ) . {\displaystyle \operatorname {End} _{R}(R^{n})\simeq \operatorname {M} _{n}(R).} This is a special case of the following fact: If f: ⊕ 1 n U → ⊕ 1 n U {\displaystyle f:\oplus _{1}^{n}U\to \oplus _... | Associative rings | 0.827798 |
4,581 | To any topological space X one can associate its integral cohomology ring H ∗ ( X , Z ) = ⨁ i = 0 ∞ H i ( X , Z ) , {\displaystyle H^{*}(X,\mathbb {Z} )=\bigoplus _{i=0}^{\infty }H^{i}(X,\mathbb {Z} ),} a graded ring. There are also homology groups H i ( X , Z ) {\displaystyle H_{i}(X,\mathbb {Z} )} of a space, and ind... | Associative rings | 0.827798 |
4,582 | A ring may be viewed as an abelian group (by using the addition operation), with extra structure: namely, ring multiplication. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example: An associative algebra is a ring that is also a vector space over a fie... | Associative rings | 0.827798 |
4,583 | Thus, if, say, k is algebraically closed, then all pi's are of the form t – λi and the above decomposition corresponds to the Jordan canonical form of f. In algebraic geometry, UFDs arise because of smoothness. More precisely, a point in a variety (over a perfect field) is smooth if the local ring at the point is a reg... | Associative rings | 0.827798 |
4,584 | This owns to the Cohen structure theorem, which says, roughly, that a complete local ring tends to look like a formal power series ring or a quotient of it. On the other hand, the interaction between the integral closure and completion has been among the most important aspects that distinguish modern commutative ring t... | Associative rings | 0.827798 |
4,585 | Let R be a commutative ring, and let I be an ideal of R. The completion of R at I is the projective limit R ^ = lim ← R / I n ; {\displaystyle {\hat {R}}=\varprojlim R/I^{n};} it is a commutative ring. The canonical homomorphisms from R to the quotients R / I n {\displaystyle R/I^{n}} induce a homomorphism R → R ^ . ... | Associative rings | 0.827798 |
4,586 | The optimal "threading" of a protein sequence onto a known structure and the production of an optimal multiple sequence alignment have been shown to be NP-complete. However, this does not imply that the structural alignment problem is NP-complete. Strictly speaking, an optimal solution to the protein structure alignmen... | Structural alignment | 0.827785 |
4,587 | Indeed, evaluating such predictions often requires a structural alignment between the model and the true known structure to assess the model's quality. Structural alignments are especially useful in analyzing data from structural genomics and proteomics efforts, and they can be used as comparison points to evaluate ali... | Structural alignment | 0.827785 |
4,588 | Geomathematics (also: mathematical geosciences, mathematical geology, mathematical geophysics) is the application of mathematical methods to solve problems in geosciences, including geology and geophysics, and particularly geodynamics and seismology. | Geomathematics | 0.827755 |
4,589 | Crystallography is one of the traditional areas of geology that use mathematics. Crystallographers make use of linear algebra by using the Metrical Matrix. The Metrical Matrix uses the basis vectors of the unit cell dimensions to find the volume of a unit cell, d-spacings, the angle between two planes, the angle betwee... | Geomathematics | 0.827755 |
4,590 | Many geophysical data sets have spectra that follow a power law, meaning that the frequency of an observed magnitude varies as some power of the magnitude. An example is the distribution of earthquake magnitudes; small earthquakes are far more common than large earthquakes. This is often an indicator that the data sets... | Geomathematics | 0.827755 |
4,591 | Some statistical problems come under the heading of mathematical geophysics, including model validation and quantifying uncertainty. | Geomathematics | 0.827755 |
4,592 | In a weighted bipartite graph, the optimization problem is to find a maximum-weight matching; a dual problem is to find a minimum-weight matching. This problem is often called maximum weighted bipartite matching, or the assignment problem. The Hungarian algorithm solves the assignment problem and it was one of the begi... | Maximum matching problem | 0.827731 |
4,593 | Kőnig's theorem states that, in bipartite graphs, the maximum matching is equal in size to the minimum vertex cover. Via this result, the minimum vertex cover, maximum independent set, and maximum vertex biclique problems may be solved in polynomial time for bipartite graphs. Hall's marriage theorem provides a characte... | Maximum matching problem | 0.827731 |
4,594 | Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics and topological combinatorics. Topics in this area include: Discrete Laplace... | Combinatorial geometry | 0.82773 |
4,595 | Deep results of Borel, Harish-Chandra, Mostow, Tamagawa, M. S. Raghunathan, Margulis, Zimmer obtained from the 1950s through the 1970s provided examples and generalized much of the theory to the setting of nilpotent Lie groups and semisimple algebraic groups over a local field. In the 1990s, Bass and Lubotzky initiated... | Combinatorial geometry | 0.82773 |
4,596 | A lattice in a locally compact topological group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of Rn, this amounts to the usual geometric notion of a lattice, and both the algebraic structure of lattices and the geometry of the totality o... | Combinatorial geometry | 0.82773 |
4,597 | A discrete group is a group G equipped with the discrete topology. With this topology, G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one. For example, the integers, Z, form a discrete subgroup of the reals, R (with the standard metric... | Combinatorial geometry | 0.82773 |
4,598 | Structural rigidity is a combinatorial theory for predicting the flexibility of ensembles formed by rigid bodies connected by flexible linkages or hinges. Topics in this area include: Cauchy's theorem Flexible polyhedra | Combinatorial geometry | 0.82773 |
4,599 | Voltammetry is a category of electroanalytical methods used in analytical chemistry and various industrial processes. In voltammetry, information about an analyte is obtained by measuring the current as the potential is varied. The analytical data for a voltammetric experiment comes in the form of a voltammogram which ... | Voltammetry | 0.827729 |
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