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23,900 | The method of lines (MOL, NMOL, NUMOL) is a technique for solving partial differential equations (PDEs) in which all but one dimension is discretized. By reducing a PDE to a single continuous dimension, the method of lines allows solutions to be computed via methods and software developed for the numerical integration ... | Method of lines | 0.808544 |
23,901 | MOL requires that the PDE problem is well-posed as an initial value (Cauchy) problem in at least one dimension, because ODE and DAE integrators are initial value problem (IVP) solvers. Thus it cannot be used directly on purely elliptic partial differential equations, such as Laplace's equation. However, MOL has been us... | Method of lines | 0.808544 |
23,902 | The measurements were made with a multigridded planar retarding potential analyzer very similar in concept and geometry to the instruments carried on the Atmospheric Explorer satellites. The retarding potential was variable in the range from approximately +32 to 0 volts. The details of this voltage trace, and whether i... | Dynamics Explorer 2 | 0.808542 |
23,903 | This investigation yielded information on: (1) the ion convection (electric field) pattern in the auroral and polar ionosphere; (2) the flow of plasma along magnetic field lines within the plasmasphere, which determines whether this motion was simply a breathing of the protonosphere, a refilling of this region after a ... | Dynamics Explorer 2 | 0.808542 |
23,904 | The Rank Prize for Optoelectronics supports, encourages, and rewards researchers working at the cutting edge of optoelectronics research, initially (from 1976) awarded annually, now a biennial prize worth £100,000. Optoelectronics relates to the interface between optics and electronics, and related phenomena.The Commit... | Rank Prizes | 0.808542 |
23,905 | van Lenteren, Marcel Dicke, and Louise E.M. Vet for "fundamental studies of plant-pest-natural enemy interactions and the development of practical methods of pest control". 2010 − Peter E. Hartmann and Robyn Owens for their "research on human lactation, including methods for the non-invasive measurement of the rate of ... | Rank Prizes | 0.808542 |
23,906 | Disadvantages include a tendency for 'wind-up' (a torsional spring rate) in the low torque region. Strain wave gearing is commonly used in robotics and aerospace. It can provide gear reduction but may also be used to increase rotational speed, or for differential gearing. | Strain wave gearing | 0.80854 |
23,907 | A locally Noetherian scheme is a locally Noetherian formal scheme in the canonical way: the formal completion along itself. In other words, the category of locally Noetherian formal schemes contains all locally Noetherian schemes. Formal schemes were motivated by and generalize Zariski's theory of formal holomorphic fu... | Formal algebraic geometry | 0.808535 |
23,908 | In mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. Unlike an ordinary scheme, a formal scheme includes infinitesimal data that, in effect, points in a direction off of the scheme. For this reason, formal schemes frequently appear in topics ... | Formal algebraic geometry | 0.808535 |
23,909 | For any ideal I and ring A we can define the I-adic topology on A, defined by its basis consisting of sets of the form a+In. This is preadmissible, and admissible if A is I-adically complete. In this case Spf A is the topological space Spec A/I with sheaf of rings lim n O Spec A / I n = lim n A / I n ~ {\displaystyle {... | Formal algebraic geometry | 0.808535 |
23,910 | Engineering professions emerge when new technologies, new problems or new opportunities arise. This was the case when safety engineering grew in the early 1900s to combat the high workplace injury and fatality rates. In the 1960s, Environmental engineering emerged as a discipline to reduce industrial pollution and miti... | Transition engineering | 0.808535 |
23,911 | Transition Engineers apply the science of climate change, petroleum geology, ecology, hydrology etc. to describe the "forward operating environment". Determining the probability of each future outcome creates a future operating environment envelope. This gives engineers and decision makers constraints with varying leve... | Transition engineering | 0.808535 |
23,912 | Thus they obey Boltzmann statistics and form a Boltzmann distribution. All species' number densities n i {\displaystyle n_{i}} are altered from their bulk (overall average) values n i 0 {\displaystyle n_{i}^{0}} by the corresponding Boltzmann factor e − z i q φ k B T {\displaystyle e^{-{\frac {z_{i}q\varphi }{k_{\text{... | Debye–Hückel equation | 0.808534 |
23,913 | D&H note that the Guldberg–Waage formula for electrolyte species in chemical reaction equilibrium in classical form is: 221 where ∏ {\textstyle \prod } is a notation for multiplication, i {\displaystyle i} is a dummy variable indicating the species, s {\displaystyle s} is the number of species participating in the reac... | Debye–Hückel equation | 0.808534 |
23,914 | A number of approaches have been proposed to extend the validity of the law to concentration ranges as commonly encountered in chemistry One such extended Debye–Hückel equation is given by: where γ {\displaystyle \gamma } as its common logarithm is the activity coefficient, z {\displaystyle z} is the integer charge of ... | Debye–Hückel equation | 0.808534 |
23,915 | The differential equation is ready for solution (as stated above, the equation only holds for low concentrations): Using the Buckingham π theorem on this problem results in the following dimensionless groups: Φ {\displaystyle \Phi } is called the reduced scalar electric potential field. R {\displaystyle R} is called th... | Debye–Hückel equation | 0.808534 |
23,916 | In the same year they first published this article, Debye and Hückel, hereinafter D&H, also released an article that covered their initial characterization of solutions under the influence of electric fields called "On the Theory of Electrolytes. II. Limiting Law for Electric Conductivity", but that subsequent article ... | Debye–Hückel equation | 0.808534 |
23,917 | In mathematical physics, the wave maps equation is a geometric wave equation that solves D α ∂ α u = 0 {\displaystyle D^{\alpha }\partial _{\alpha }u=0} where D {\displaystyle D} is a connection.It can be considered a natural extension of the wave equation for Riemannian manifolds. == References == | Wave maps equation | 0.808534 |
23,918 | The Cadiot–Chodkiewicz coupling in organic chemistry is a coupling reaction between a terminal alkyne and a haloalkyne catalyzed by a copper(I) salt such as copper(I) bromide and an amine base. The reaction product is a 1,3-diyne or di-alkyne. The reaction mechanism involves deprotonation by base of the terminal alkyne... | Cadiot–Chodkiewicz coupling | 0.808533 |
23,919 | Southern blot is a method used for detection and quantification of a specific DNA sequence in DNA samples. This method is used in molecular biology. Briefly, purified DNA from a biological sample (such as blood or tissue) is digested with restriction enzymes, and the resulting DNA fragments are separated by using an el... | Southern hybridization | 0.808533 |
23,920 | The number of instructions to perform the above basic "loop" (Fetch/Execute/calculate new address) depends on hardware but it could be accomplished on IBM S/360/370/390/ES9000 range of machines in around 12 or 13 instructions for many instruction types. Checking for valid memory locations or for conditional "pause"s ad... | Instruction Set Simulator | 0.80853 |
23,921 | Use of a monitoring program can detect these even without a storage protect key. Illegal operations: some operating systems (or hardware) require the application program to be in the correct "mode" for certain calls to the Operating system. Instruction simulation can detect these conditions before execution. Hot spot a... | Instruction Set Simulator | 0.80853 |
23,922 | This is because in intuitionistic logic, ¬ P {\displaystyle \neg P} is just a shorthand for P → ⊥ {\displaystyle P\rightarrow \bot } , and we also have P → ¬ ¬ P {\displaystyle P\rightarrow \neg \neg P} . Composing that last implication with triple negation ¬ ¬ P → ⊥ {\displaystyle \neg \neg P\rightarrow \bot } implies... | Logical negation | 0.808527 |
23,923 | This marks one important difference between classical and intuitionistic negation. Algebraically, classical negation is called an involution of period two. However, in intuitionistic logic, the weaker equivalence ¬ ¬ ¬ P ≡ ¬ P {\displaystyle \neg \neg \neg P\equiv \neg P} does hold. | Logical negation | 0.808527 |
23,924 | Let ⊕ {\displaystyle \oplus } denote the logical xor operation. In Boolean algebra, a linear function is one such that: If there exists a 0 , a 1 , … , a n ∈ { 0 , 1 } {\displaystyle a_{0},a_{1},\dots ,a_{n}\in \{0,1\}} , f ( b 1 , b 2 , … , b n ) = a 0 ⊕ ( a 1 ∧ b 1 ) ⊕ ⋯ ⊕ ( a n ∧ b n ) {\displaystyle f(b_{1},b_{2},\... | Logical negation | 0.808527 |
23,925 | Finally, recognizing that the set of unitary transformations on C 2 {\displaystyle \mathbb {C} ^{2}} can be expressed as { e i γ ( α − β ¯ β α ¯ ): α , β ∈ C , | α | 2 + | β | 2 = 1 , γ ∈ } {\displaystyle \left\{{\rm {e}}^{i\gamma }{\begin{pmatrix}\alpha &-{\overline {\beta }}\\\beta &{\overline {\alpha }}\end{pmatrix... | Jones Calculus | 0.808527 |
23,926 | In physics, the Heisenberg picture or Heisenberg representation is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but the state vectors are time-independent, an arbitrary fixed basis rigidly underlying the t... | Heisenberg's equation | 0.808527 |
23,927 | In physical chemistry, a hot atom is an atom that has a high kinetic or internal energy.When molecule AB adsorbs on a surface dissociatively, both A and B adsorb on the surface, or only A adsorbs on the surface, and B desorbs from the surface.In case 2, B gains a high translational energy from the adsorption energy of ... | Hot atom | 0.808525 |
23,928 | In relativity, the COM frame exists for an isolated massive system. This is a consequence of Noether's theorem. In the COM frame the total energy of the system is the rest energy, and this quantity (when divided by the factor c2, where c is the speed of light) gives the rest mass (invariant mass) of the system: m 0 = E... | Center-of-mass frame | 0.808524 |
23,929 | In physics, the center-of-momentum frame (COM frame), also known as zero-momentum frame, is the inertial frame in which the total momentum of the system vanishes. It is unique up to velocity, but not origin. The center of momentum of a system is not a location, but a collection of relative momenta/velocities: a referen... | Center-of-mass frame | 0.808524 |
23,930 | {\displaystyle \exp(X)=c(1).\ } This is called the exponential map, and it maps the Lie algebra g {\displaystyle {\mathfrak {g}}} into the Lie group G {\displaystyle G} . It provides a diffeomorphism between a neighborhood of 0 in g {\displaystyle {\mathfrak {g}}} and a neighborhood of e {\displaystyle e} in G {\displa... | Lie group theory | 0.808523 |
23,931 | Every element of G {\displaystyle G} that is sufficiently close to the identity is the exponential of a matrix in the Lie algebra.The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its repre... | Lie group theory | 0.808523 |
23,932 | }}+{\frac {X^{3}}{3! }}+\cdots } for matrices X {\displaystyle X} . If G {\displaystyle G} is a closed subgroup of G L ( n ; C ) {\displaystyle \mathrm {GL} (n;\mathbb {C} )} , then the exponential map takes the Lie algebra of G {\displaystyle G} into G {\displaystyle G} ; thus, we have an exponential map for all matri... | Lie group theory | 0.808523 |
23,933 | The exponential map from the Lie algebra to the Lie group is not always onto, even if the group is connected (though it does map onto the Lie group for connected groups that are either compact or nilpotent). For example, the exponential map of SL(2, R) is not surjective. Also, the exponential map is neither surjective ... | Lie group theory | 0.808523 |
23,934 | In case X {\displaystyle X} and Y {\displaystyle Y} commute, this formula reduces to the familiar exponential law exp ( X ) exp ( Y ) = exp ( X + Y ) {\displaystyle \exp(X)\exp(Y)=\exp(X+Y)} The exponential map relates Lie group homomorphisms. That is, if ϕ: G → H {\displaystyle \phi :G\to H} is a Lie group homom... | Lie group theory | 0.808523 |
23,935 | Because the exponential map is surjective on some neighbourhood N {\displaystyle N} of e {\displaystyle e} , it is common to call elements of the Lie algebra infinitesimal generators of the group G {\displaystyle G} . The subgroup of G {\displaystyle G} generated by N {\displaystyle N} is the identity component of G {\... | Lie group theory | 0.808523 |
23,936 | The exponential map from the Lie algebra M ( n ; C ) {\displaystyle \mathrm {M} (n;\mathbb {C} )} of the general linear group G L ( n ; C ) {\displaystyle \mathrm {GL} (n;\mathbb {C} )} to G L ( n ; C ) {\displaystyle \mathrm {GL} (n;\mathbb {C} )} is defined by the matrix exponential, given by the usual power series: ... | Lie group theory | 0.808523 |
23,937 | If the manifold is a circle these are called loop groups, and have central extensions whose Lie algebras are (more or less) Kac–Moody algebras. There are infinite-dimensional analogues of general linear groups, orthogonal groups, and so on. One important aspect is that these may have simpler topological properties: see... | Lie group theory | 0.808523 |
23,938 | Some of the examples that have been studied include: The group of diffeomorphisms of a manifold. Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra is (more or less) the Witt algebra, whose central extension the Virasoro algebra (see Virasoro algebra from Witt algebra for a derivatio... | Lie group theory | 0.808523 |
23,939 | For example, an infinite-dimensional Lie algebra may or may not have a corresponding Lie group. That is, there may be a group corresponding to the Lie algebra, but it might not be nice enough to be called a Lie group, or the connection between the group and the Lie algebra might not be nice enough (for example, failure... | Lie group theory | 0.808523 |
23,940 | In this case the relation between the Lie algebra and the Lie group becomes rather subtle, and several results about finite-dimensional Lie groups no longer hold. The literature is not entirely uniform in its terminology as to exactly which properties of infinite-dimensional groups qualify the group for the prefix Lie ... | Lie group theory | 0.808523 |
23,941 | In population ecology and economics, optimum sustainable yield is the level of effort (LOE) that maximizes the difference between total revenue and total cost. Or, where marginal revenue equals marginal cost. This level of effort maximizes the economic profit, or rent, of the resource being utilized. It usually corresp... | Optimum sustainable yield | 0.808522 |
23,942 | The Fourier transform is useful in applied mathematics, particularly physics and signal processing. It is another integral operator; it is useful mainly because it converts a function on one (temporal) domain to a function on another (frequency) domain, in a way effectively invertible. No information is lost, as there ... | Operator (mathematics) | 0.808522 |
23,943 | In geometry, additional structures on vector spaces are sometimes studied. Operators that map such vector spaces to themselves bijectively are very useful in these studies, they naturally form groups by composition. For example, bijective operators preserving the structure of a vector space are precisely the invertible... | Operator (mathematics) | 0.808522 |
23,944 | Let U and V be two vector spaces over the same ordered field (for example, R {\displaystyle \mathbb {R} } ), and they are equipped with norms. Then a linear operator from U to V is called bounded if there exists C > 0 such thatfor all x in U. Bounded operators form a vector space. On this vector space we can introduce ... | Operator (mathematics) | 0.808522 |
23,945 | This description gives the standard model of elliptic geometry. The complex projective plane CP2 arises when K is taken to be the complex numbers, C. It is a closed complex 2-manifold, and hence a closed, orientable real 4-manifold. It and projective planes over other fields (known as pappian planes) serve as fundament... | Finite projective plane | 0.808519 |
23,946 | The real projective plane RP2 arises when K is taken to be the real numbers, R. As a closed, non-orientable real 2-manifold, it serves as a fundamental example in topology.In this construction, consider the unit sphere centered at the origin in R3. Each of the R3 lines in this construction intersects the sphere at two ... | Finite projective plane | 0.808519 |
23,947 | Dualizing this theorem and the first two axioms in the definition of a projective plane shows that the plane dual structure C* is also a projective plane, called the dual plane of C. If C and C* are isomorphic, then C is called self-dual. The projective planes PG(2, K) for any division ring K are self-dual. However, th... | Finite projective plane | 0.808519 |
23,948 | In finite desarguesian planes, PG(2, q), Fano subplanes exist if and only if q is even (that is, a power of 2). The situation in non-desarguesian planes is unsettled. They could exist in any non-desarguesian plane of order greater than 6, and indeed, they have been found in all non-desarguesian planes in which they hav... | Finite projective plane | 0.808519 |
23,949 | For example, the embedding ( x 1 , x 2 ) → ( x 2 , 1 , x 1 ) , {\displaystyle (x_{1},x_{2})\to (x_{2},1,x_{1}),} has as its complement those points of the form (x0, 0, x2), which are then regarded as points at infinity. When an affine plane does not have the form of K2 with K a division ring, it can still be embedded i... | Finite projective plane | 0.808519 |
23,950 | The projective spaces in turn generalize to the Grassmannian spaces. It can be shown that if Desargues' theorem holds in a projective space of dimension greater than two, then it must also hold in all planes that are contained in that space. Since there are projective planes in which Desargues' theorem fails (non-Desar... | Finite projective plane | 0.808519 |
23,951 | As in the construction given earlier, the points of the d-dimensional projective space PG(d, K) are the lines through the origin in Kd+1 and a line in PG(d, K) corresponds to a plane through the origin in Kd+1. In fact, each i-dimensional object in PG(d, K), with i < d, is an (i + 1)-dimensional (algebraic) vector subs... | Finite projective plane | 0.808519 |
23,952 | Projective planes may be thought of as projective geometries of "geometric" dimension two. Higher-dimensional projective geometries can be defined in terms of incidence relations in a manner analogous to the definition of a projective plane. These turn out to be "tamer" than the projective planes since the extra degree... | Finite projective plane | 0.808519 |
23,953 | In a pilot study of 16 men from 1997, testis FNA was used to "map" the organ for mature sperm. The concept for mapping testis for sperm was inspired by the work of Gottschalk-Sabag and colleagues and modeled after the approach to prostate biology in which multiple prostate biopsies are used to detect foci of prostate c... | FNA Mapping | 0.808518 |
23,954 | It has become clear to reproductive urologists who regularly perform testis sperm retrieval in azoospermic men for IVF-ICSI that spermatogenesis varies geographically within the testis. This underscores the limitation and inadequacy of a single, localized testis biopsy or a single FNA specimen to accurately reflect the... | FNA Mapping | 0.808518 |
23,955 | In 1927, in the same year as Zhegalkin's paper, the American mathematician Eric Temple Bell published a sophisticated arithmetization of Boolean algebra based on Richard Dedekind's ideal theory and general modular arithmetic (as opposed to arithmetic mod 2). The much simpler arithmetic character of Zhegalkin polynomial... | Boolean polynomial | 0.808517 |
23,956 | Since − 1 ≡ 1 {\displaystyle -1\equiv 1} in the Zhegalkin algebra, the Möbius function collapses to being the constant 1. The set of divisors of a given number x is also the order ideal generated by that number: ⟨ x ⟩ {\displaystyle \langle x\rangle } . Since summation is modulo 2, the formula can be restated as g ( x ... | Boolean polynomial | 0.808517 |
23,957 | Prior to 1927, Boolean algebra had been considered a calculus of logical values with logical operations of conjunction, disjunction, negation, and so on. Zhegalkin showed that all Boolean operations could be written as ordinary numeric polynomials, representing the false and true values as 0 and 1, the integers mod 2. ... | Boolean polynomial | 0.808517 |
23,958 | Notable organisations that use the D programming language for projects include Facebook, eBay, and Netflix.D has been successfully used for AAA games, language interpreters, virtual machines, an operating system kernel, GPU programming, web development, numerical analysis, GUI applications, a passenger information syst... | D programming language | 0.808516 |
23,959 | An important special case of a metric connection is a Riemannian connection. This is a connection ∇ {\displaystyle \nabla } on the tangent bundle of a pseudo-Riemannian manifold (M, g) such that ∇ X g = 0 {\displaystyle \nabla _{X}g=0} for all vector fields X on M. Equivalently, ∇ {\displaystyle \nabla } is Riemannian ... | Metric connection | 0.808516 |
23,960 | The connection is skew-symmetric in the vector-space (fiber) indices; that is, for a given vector field X ∈ T M {\displaystyle X\in TM} , the matrix A ( X ) {\displaystyle A(X)} is skew-symmetric; equivalently, it is an element of the Lie algebra o ( n ) {\displaystyle {\mathfrak {o}}(n)} . This can be seen as follows.... | Metric connection | 0.808516 |
23,961 | Radiobiology (also known as radiation biology, and uncommonly as actinobiology) is a field of clinical and basic medical sciences that involves the study of the effects of ionizing radiation on living things, in particular health effects of radiation. Ionizing radiation is generally harmful and potentially lethal to li... | Radiation biology | 0.808516 |
23,962 | The Christoffel symbols find frequent use in Einstein's theory of general relativity, where spacetime is represented by a curved 4-dimensional Lorentz manifold with a Levi-Civita connection. The Einstein field equations—which determine the geometry of spacetime in the presence of matter—contain the Ricci tensor, and so... | Connection coefficients | 0.808514 |
23,963 | Computability in Analysis and Physics is a monograph on computable analysis by Marian Pour-El and J. Ian Richards. It was published by Springer-Verlag in their Perspectives in Mathematical Logic series in 1989, and reprinted by the Association for Symbolic Logic and Cambridge University Press in their Perspectives in L... | Computability in Analysis and Physics | 0.808512 |
23,964 | In contrast to the Russian school of computable analysis led by Andrey Markov Jr., it views computability as a distinguishing property of mathematical objects among others, rather than developing a theory that concerns only computable objects.After an initial section of the book, introducing computable analysis and lea... | Computability in Analysis and Physics | 0.808512 |
23,965 | The book concerns computable analysis, a branch of mathematical analysis founded by Alan Turing and concerned with the computability of constructions in analysis. This area is connected to, but distinct from, constructive analysis, reverse mathematics, and numerical analysis. The early development of the field was summ... | Computability in Analysis and Physics | 0.808512 |
23,966 | This chain of extensions canonically embeds the natural numbers in the other number systems. Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combi... | Counting numbers | 0.808505 |
23,967 | Magnetic fields are used throughout modern technology, particularly in electrical engineering and electromechanics. Rotating magnetic fields are used in both electric motors and generators. The interaction of magnetic fields in electric devices such as transformers is conceptualized and investigated as magnetic circuit... | Magnetic field strength | 0.808504 |
23,968 | Accumulation points of sets If Y ⊆ X {\displaystyle Y\subseteq X} does not contain p, Y has no accumulation point (because Y is closed in X and discrete in the subspace topology).If Y ⊆ X {\displaystyle Y\subseteq X} contains p, every point x ≠ p {\displaystyle x\neq p} is an accumulation point of Y, since { x , p } {\... | Particular point topology | 0.808503 |
23,969 | T0 X is T0 (since {x, p} is open for each x) but satisfies no higher separation axioms (because all non-empty open sets must contain p).Not regular Since every non-empty open set contains p, no closed set not containing p (such as X \ {p}) can be separated by neighbourhoods from {p}, and thus X is not regular. Since co... | Particular point topology | 0.808503 |
23,970 | The discovery of cosmic microwave background radiation constitutes a major development in modern physical cosmology. In 1964, US physicist Arno Allan Penzias and radio-astronomer Robert Woodrow Wilson discovered the CMB, estimating its temperature as 3.5 K, as they experimented with the Holmdel Horn Antenna. The new me... | Discovery of cosmic microwave background radiation | 0.808503 |
23,971 | In the first, Dicke and his associates outlined the importance of cosmic background radiation as substantiation of the Big Bang Theory. In a second note, jointly signed by Penzias and Wilson titled, "A Measurement of Excess Antenna Temperature at 4080 Megacycles per Second," they reported the existence of a 3.5 K resid... | Discovery of cosmic microwave background radiation | 0.808503 |
23,972 | With the proper instrumentation, this radiation should be detectable, albeit as microwaves, due to a massive redshift. When his friend Bernard F. Burke, a professor of physics at MIT, told Penzias about a preprint paper he had seen by Jim Peebles on the possibility of finding radiation left over from an explosion that ... | Discovery of cosmic microwave background radiation | 0.808503 |
23,973 | Other cryptocurrencies (e.g. Ethereum, NEO, STRATIS, ...) use proof of stake, in which nodes compete to append blocks and earn associated rewards in proportion to stake, or existing cryptocurrency allocated and locked or staked for some time period. One advantage of a 'proof of stake' over a 'proof of work' system, is ... | Consensus (computer science) | 0.808499 |
23,974 | In the most traditional single-value consensus protocols such as Paxos, cooperating nodes agree on a single value such as an integer, which may be of variable size so as to encode useful metadata such as a transaction committed to a database. A special case of the single-value consensus problem, called binary consensus... | Consensus (computer science) | 0.808499 |
23,975 | Each player validates the change by applying the delta to their own game state and comparing the game state hashes. If the hashes do not agree then a vote is cast, and those players whose game state is in the minority are disconnected and removed from the game (known as a desync.) Another well-known approach is called ... | Consensus (computer science) | 0.808499 |
23,976 | The after-effects phenomena suggests that prior to the movement, the CNS generates an internal-model, a sort of internal-map that guides the body in the course of the movement, and adapt to environmental forces. This observation suggests that in programming the motor output to the muscles of the arm, the CNS uses an in... | Motor adaptation | 0.808498 |
23,977 | For example, the closed subscheme of the affine line defined by x2 = 0 is different from the subscheme defined by x = 0 (the origin). More generally, the fiber of a morphism of schemes X → Y at a point of Y may be non-reduced, even if X and Y are reduced. Geometrically, this says that fibers of good mappings may have n... | Algebraic varieties | 0.808495 |
23,978 | A more significant modification is to allow nilpotents in the sheaf of rings, that is, rings which are not reduced. This is one of several generalizations of classical algebraic geometry that are built into Grothendieck's theory of schemes. Allowing nilpotent elements in rings is related to keeping track of "multiplici... | Algebraic varieties | 0.808495 |
23,979 | These varieties have been called "varieties in the sense of Serre", since Serre's foundational paper FAC on sheaf cohomology was written for them. They remain typical objects to start studying in algebraic geometry, even if more general objects are also used in an auxiliary way. One way that leads to generalizations is... | Algebraic varieties | 0.808495 |
23,980 | A scheme is a locally ringed space such that every point has a neighbourhood that, as a locally ringed space, is isomorphic to a spectrum of a ring. Basically, a variety over k is a scheme whose structure sheaf is a sheaf of k-algebras with the property that the rings R that occur above are all integral domains and are... | Algebraic varieties | 0.808495 |
23,981 | The basic definitions and facts above enable one to do classical algebraic geometry. To be able to do more — for example, to deal with varieties over fields that are not algebraically closed — some foundational changes are required. The modern notion of a variety is considerably more abstract than the one above, though... | Algebraic varieties | 0.808495 |
23,982 | Molecular electronics operates in the quantum realm of distances less than 100 nanometers. The miniaturization down to single molecules brings the scale down to a regime where quantum mechanics effects are important. In conventional electronic components, electrons can be filled in or drawn out more or less like a cont... | Molecular circuit | 0.808493 |
23,983 | In mathematics and computer science, a history monoid is a way of representing the histories of concurrently running computer processes as a collection of strings, each string representing the individual history of a process. The history monoid provides a set of synchronization primitives (such as locks, mutexes or thr... | History monoid | 0.808484 |
23,984 | The Benjamin Franklin Award is an annual award for Open Access in the Life Sciences presented by Bioinformatics.org to an individual who has, in his or her practice, promoted free and open access to the materials and methods used in the life sciences. | Benjamin Franklin Award (Bioinformatics) | 0.808484 |
23,985 | Source: bioinformatics.org 2002 - Michael B. Eisen 2003 - Jim Kent 2004 - Lincoln D. Stein 2005 - Ewan Birney 2006 - Michael Ashburner 2007 - Sean Eddy 2008 - Robert Gentleman 2009 - Philip E. Bourne 2010 - Alex Bateman 2011 - Jonathan Eisen 2012 - Heng Li 2013 - Steven Salzberg 2014 - Helen M. Berman 2015 - Owen White... | Benjamin Franklin Award (Bioinformatics) | 0.808484 |
23,986 | With a diverse set of applications in mind, it was natural to try to put the whole subject on a uniform basis. There were several attempts before the subject settled down. An approximate history can be stated as follows: Cartan-Eilenberg: In their 1956 book "Homological Algebra", these authors used projective and injec... | Homological algebra | 0.808479 |
23,987 | Cohomology theories have been defined for many different objects such as topological spaces, sheaves, groups, rings, Lie algebras, and C*-algebras. The study of modern algebraic geometry would be almost unthinkable without sheaf cohomology. Central to homological algebra is the notion of exact sequence; these can be us... | Homological algebra | 0.808479 |
23,988 | The class of Abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an Abelian category, or the category of functors from a small category to an Abelian category are Abelian as well. These stability properties make them inevitable in homological algebra and... | Homological algebra | 0.808479 |
23,989 | Homological algebra studies the relation, induced by the map f, between chain complexes associated with X and Y and their homology. This is generalized to the case of several objects and maps connecting them. Phrased in the language of category theory, homological algebra studies the functorial properties of various co... | Homological algebra | 0.808479 |
23,990 | On a technical level, homological algebra provides the tools for manipulating complexes and extracting this information. Here are two general illustrations. Two objects X and Y are connected by a map f between them. | Homological algebra | 0.808479 |
23,991 | For example, if X is a topological space then the singular chains Cn(X) are formal linear combinations of continuous maps from the standard n-simplex into X; if K is a simplicial complex then the simplicial chains Cn(K) are formal linear combinations of the n-simplices of K; if A = F/R is a presentation of an abelian g... | Homological algebra | 0.808479 |
23,992 | {\displaystyle H_{n}(C)=Z_{n}/B_{n}=\operatorname {Ker} \,d_{n}/\operatorname {Im} \,d_{n+1}.} A chain complex is called acyclic or an exact sequence if all its homology groups are zero. Chain complexes arise in abundance in algebra and algebraic topology. | Homological algebra | 0.808479 |
23,993 | The chain groups Cn may be endowed with extra structure; for example, they may be vector spaces or modules over a fixed ring R. The differentials must preserve the extra structure if it exists; for example, they must be linear maps or homomorphisms of R-modules. For notational convenience, restrict attention to abelian... | Homological algebra | 0.808479 |
23,994 | The notion of chain complex is central in homological algebra. An abstract chain complex is a sequence ( C ∙ , d ∙ ) {\displaystyle (C_{\bullet },d_{\bullet })} of abelian groups and group homomorphisms, with the property that the composition of any two consecutive maps is zero: C ∙: ⋯ ⟶ C n + 1 ⟶ d n + 1 C n ⟶ d n C n... | Homological algebra | 0.808479 |
23,995 | By definition, this means that fn is an injection, gn is a surjection, and Im fn = Ker gn. One of the most basic theorems of homological algebra, sometimes known as the zig-zag lemma, states that, in this case, there is a long exact sequence in homology ⋯ ⟶ H n ( L ) ⟶ H n ( f ) H n ( M ) ⟶ H n ( g ) H n ( N ) ⟶ δ n H ... | Homological algebra | 0.808479 |
23,996 | It follows that the homology groups H ∙ ( C ) {\displaystyle H_{\bullet }(C)} are functorial as well, so that morphisms between algebraic or topological objects give rise to compatible maps between their homology. The following definition arises from a typical situation in algebra and topology. A triple consisting of t... | Homological algebra | 0.808479 |
23,997 | The result was surprising, as many systems previously believed to be unrelated proved to be isomorphic. These include all finite stationary stochastic processes, subshifts of finite type, finite Markov chains, Anosov flows, and Sinai's billiards: these are all isomorphic to Bernoulli schemes. For the generalized case, ... | Bernoulli shift | 0.808475 |
23,998 | The Ornstein isomorphism theorem states that two Bernoulli schemes with the same entropy are isomorphic. The result is sharp, in that very similar, non-scheme systems, such as Kolmogorov automorphisms, do not have this property. The Ornstein isomorphism theorem is in fact considerably deeper: it provides a simple crite... | Bernoulli shift | 0.808475 |
23,999 | Manganese pentacarbonyl bromide is an organomanganese compound with the formula BrMn(CO)5. It is a bright orange solid that is a precursor to other manganese complexes. The compound is prepared by treatment of dimanganese decacarbonyl with bromine: Mn2(CO)10 + Br2 → 2 BrMn(CO)5The complex undergoes substitution by a va... | C5BrMnO5 | 0.808474 |
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