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6,400 | In topology, a quotient space is a topological space formed on the set of equivalence classes of an equivalence relation on a topological space, using the original space's topology to create the topology on the set of equivalence classes. In abstract algebra, congruence relations on the underlying set of an algebra all... | Equivalence classes | 0.824281 |
6,401 | Joint compatibility branch and bound (JCBB) is an algorithm in computer vision and robotics commonly used for data association in simultaneous localization and mapping. JCBB measures the joint compatibility of a set of pairings that successfully rejects spurious matchings and is hence known to be robust in complex envi... | Joint compatibility branch and bound | 0.824281 |
6,402 | A measure on a set is a non-negative real-valued functional on a σ-algebra of subsets. Lp spaces on sets with a measure are defined from aforementioned real-valued measurable functions, although they are actually quotient spaces. More precisely, whereas a function satisfying an appropriate summability condition defines... | Numerical function | 0.824273 |
6,403 | The space of continuous functions on a compact Hausdorff space has a particular importance. Convergent sequences also can be considered as real-valued continuous functions on a special topological space. Continuous functions also form a vector space and an algebra as explained above in § Algebraic structure, and are a ... | Numerical function | 0.824273 |
6,404 | Real numbers form a topological space and a complete metric space. Continuous real-valued functions (which implies that X is a topological space) are important in theories of topological spaces and of metric spaces. The extreme value theorem states that for any real continuous function on a compact space its global max... | Numerical function | 0.824273 |
6,405 | in 1931, 1934 and 1939; and by The Literary Guild in 1934. Doubleday also issued a four-volume limited edition of the work in 1931, limited to 750 sets, with the first volume autographed on the limitation page by the three authors. Three of the Cassell spin-off books were also published by Doubleday in 1932: Evolution,... | The Science of Life | 0.824273 |
6,406 | VIII Man's mind and behaviour. IX Biology and the human race. In New York, it was published by Doubleday, Doran & Co. | The Science of Life | 0.824273 |
6,407 | The Science of Life is a book written by H. G. Wells, Julian Huxley and G. P. Wells, published in three volumes by The Waverley Publishing Company Ltd in 1929–30, giving a popular account of all major aspects of biology as known in the 1920s. It has been called "the first modern textbook of biology" and "the best popul... | The Science of Life | 0.824273 |
6,408 | Conduct 1390 Borderland Science and the Question of Personal Survival 1411 Book Nine—Biology of the Human Race Peculiarities of the Species Homo Sapiens 1436 The Present Phase of Human Association 1454 Index 1481–1515The pagination is that of the 1934 Literary Guild edition. | The Science of Life | 0.824273 |
6,409 | Introduction The Range, Nature, and Study of Living Things 1 Book One—The Living Body The Body is a Machine 24 The Complex Body-Machine and How It Works 32 The Harmony and Direction of the Body-Machine 97 The Wearing Out of the Machine and Its Reproduction 140 Book Two—The Chief Patterns of Life The First Great Phylum:... | The Science of Life | 0.824273 |
6,410 | "In 1975 van der Waerden described the sources he drew upon to write the book.In 1997 Saunders Mac Lane recollected the book's influence: Upon its publication it was soon clear that this was the way that algebra should be presented. Its simple but austere style set the pattern for mathematical texts in other subjects, ... | Modern Algebra | 0.824271 |
6,411 | Moderne Algebra is a two-volume German textbook on graduate abstract algebra by Bartel Leendert van der Waerden (1930, 1931), originally based on lectures given by Emil Artin in 1926 and by Emmy Noether (1929) from 1924 to 1928. The English translation of 1949–1950 had the title Modern algebra, though a later, extensiv... | Modern Algebra | 0.824271 |
6,412 | A Master of Chemistry (or MChem) degree is a specific master's degree for courses in the field of Chemistry. | Master of Chemistry | 0.824267 |
6,413 | The role that a keystone species plays in its ecosystem is analogous to the role of a keystone in an arch. While the keystone is under the least pressure of any of the stones in an arch, the arch still collapses without it. Similarly, an ecosystem may experience a dramatic shift if a keystone species is removed, even t... | Keystone organism | 0.824263 |
6,414 | The description of the key binary relation properties has been formulated with the calculus of relations. The univalence property of functions describes a relation R {\displaystyle R} that satisfies the formula R T R ⊆ I , {\displaystyle R^{T}R\subseteq I,} where I {\displaystyle I} is the identity relation on the rang... | Algebra of logic | 0.824262 |
6,415 | Stone's Algebra Theorem: Let A be an ordered algebra with unit e such that e is an order unit in A, let A* denote the algebraic dual of A, and let K be the σ ( A ∗ , A ) {\displaystyle \sigma \left(A^{*},A\right)} -compact set of all multiplicative positive linear forms satisfying f(e) = 1. Then under the evaluation ma... | Ordered algebra | 0.824254 |
6,416 | Let A be an ordered algebra with unit e and let C* denote the cone in A* (the algebraic dual of A) of all positive linear forms on A. If f is a linear form on A such that f(e) = 1 and f generates an extreme ray of C* then f is a multiplicative homomorphism. | Ordered algebra | 0.824254 |
6,417 | In mathematics, an ordered algebra is an algebra over the real numbers R {\displaystyle \mathbb {R} } with unit e together with an associated order such that e is positive (i.e. e ≥ 0), the product of any two positive elements is again positive, and when A is considered as a vector space over R {\displaystyle \mathbb {... | Ordered algebra | 0.824254 |
6,418 | Sedimentation equilibrium in a suspension of different particles, such as molecules, exists when the rate of transport of each material in any one direction due to sedimentation equals the rate of transport in the opposite direction due to diffusion. Sedimentation is due to an external force, such as gravity or centrif... | Sedimentation equilibrium | 0.824252 |
6,419 | After Calabi–Yau manifolds had entered physics as a way to compactify extra dimensions in string theory, many physicists began studying these manifolds. In the late 1980s, several physicists noticed that given such a compactification of string theory, it is not possible to reconstruct uniquely a corresponding Calabi–Ya... | Ten-dimensional space | 0.824248 |
6,420 | Penrose also claims that the technical difficulty of modern physics forces young scientists to rely on the preferences of established researchers, rather than forging new paths of their own. Lee Smolin expresses a slightly different position in his critique, claiming that string theory grew out of a tradition of partic... | Ten-dimensional space | 0.824248 |
6,421 | In his book criticizing string theory, Peter Woit views the status of string theory research as unhealthy and detrimental to the future of fundamental physics. He argues that the extreme popularity of string theory among theoretical physicists is partly a consequence of the financial structure of academia and the fierc... | Ten-dimensional space | 0.824248 |
6,422 | Since the superstring revolutions of the 1980s and 1990s, string theory has been one of the dominant paradigms of high energy theoretical physics. Some string theorists have expressed the view that there does not exist an equally successful alternative theory addressing the deep questions of fundamental physics. In an ... | Ten-dimensional space | 0.824248 |
6,423 | The letter "D" in D-brane refers to a certain mathematical condition on the system known as the Dirichlet boundary condition. The study of D-branes in string theory has led to important results such as the AdS/CFT correspondence, which has shed light on many problems in quantum field theory.Branes are frequently studie... | Ten-dimensional space | 0.824248 |
6,424 | In dimension p, these are called p-branes. The word brane comes from the word "membrane" which refers to a two-dimensional brane.Branes are dynamical objects which can propagate through spacetime according to the rules of quantum mechanics. They have mass and can have other attributes such as charge. | Ten-dimensional space | 0.824248 |
6,425 | In addition to being an idea of considerable theoretical interest, string theory provides a framework for constructing models of real-world physics that combine general relativity and particle physics. Phenomenology is the branch of theoretical physics in which physicists construct realistic models of nature from more ... | Ten-dimensional space | 0.824248 |
6,426 | This problem was solved by the nineteenth-century German mathematician Hermann Schubert, who found that there are exactly 2,875 such lines. In 1986, geometer Sheldon Katz proved that the number of curves, such as circles, that are defined by polynomials of degree two and lie entirely in the quintic is 609,250.By the ye... | Ten-dimensional space | 0.824248 |
6,427 | The Calabi–Yau manifolds used in string theory are of interest in pure mathematics, and mirror symmetry allows mathematicians to solve problems in enumerative geometry, a branch of mathematics concerned with counting the numbers of solutions to geometric questions.Enumerative geometry studies a class of geometric objec... | Ten-dimensional space | 0.824248 |
6,428 | A disk-covering method is a divide-and-conquer meta-technique for large-scale phylogenetic analysis which has been shown to improve the performance of both heuristics for NP-hard optimization problems and polynomial-time distance-based methods. Disk-covering methods are a meta-technique in that they have flexibility in... | Disk-covering method | 0.824243 |
6,429 | Efficient cake-cutting is a problem in economics and computer science. It involves a heterogeneous resource, such as a cake with different toppings or a land with different coverings, that is assumed to be divisible - it is possible to cut arbitrarily small pieces of it without destroying their value. The resource has ... | Efficient cake-cutting | 0.824241 |
6,430 | This does not follow directly from the above argument, since envy-freeness is not preserved by Pareto improvements. However, it is proved explicitly as Weller's theorem.Finding such allocations may be hard even with strictly-positive valuations, depending on the computational model: In the query model, no finite algori... | Efficient cake-cutting | 0.824241 |
6,431 | In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct types of groups: in the study of finite groups, a Z-group is a finite group whose Sylow subgroups are all cyclic. in the study of infinite groups, a Z-group is a group which possesses a very general ... | Z-group | 0.824239 |
6,432 | If G {\displaystyle G} is a Lie group then from an inner product g e {\displaystyle g_{e}} on the tangent space g {\displaystyle {\mathfrak {g}}} (the Lie algebra of G {\displaystyle G} ) one can construct a Riemannian metric on G {\displaystyle G} as follows: if v , w {\displaystyle v,w} belong to the tangent space at... | Lattice (group theory) | 0.824221 |
6,433 | A consequence of local rigidity and the Kazhdan-Margulis theorem is Wang's theorem: in a given group (with a fixed Haar measure), for any v>0 there are only finitely many (up to conjugation) lattices with covolume bounded by v. The Mostow rigidity theorem states that for lattices in simple Lie groups not locally isomor... | Lattice (group theory) | 0.824221 |
6,434 | Another group of phenomena concerning lattices in semisimple algebraic groups is collectively known as rigidity. Here are three classical examples of results in this category. Local rigidity results state that in most situations every subgroup which is sufficiently "close" to a lattice (in the intuitive sense, formalis... | Lattice (group theory) | 0.824221 |
6,435 | The criterion for nilpotent Lie groups to have a lattice given above does not apply to more general solvable Lie groups. It remains true that any lattice in a solvable Lie group is uniform and that lattices in solvable groups are finitely presented. Not all finitely generated solvable groups are lattices in a Lie group... | Lattice (group theory) | 0.824221 |
6,436 | More generally one can look at lattices in groups of the form G = ∏ p ∈ S G p {\displaystyle G=\prod _{p\in S}G_{p}} where G p {\displaystyle G_{p}} is a semisimple algebraic group over Q p {\displaystyle \mathbb {Q} _{p}} . Usually p = ∞ {\displaystyle p=\infty } is allowed, in which case G ∞ {\displaystyle G_{\infty ... | Lattice (group theory) | 0.824221 |
6,437 | More generally one can ask the following question: if H {\displaystyle H} is a closed subgroup of A u t ( T ) {\displaystyle \mathrm {Aut} (T)} , under which conditions does H {\displaystyle H} contain a lattice? The existence of a uniform lattice is equivalent to H {\displaystyle H} being unimodular and the quotient H... | Lattice (group theory) | 0.824221 |
6,438 | In this case, it implies that a = –b, so the equation should read 5 − 9 2 = − ( 4 − 9 2 ) {\displaystyle 5-{\frac {9}{2}}=-\left(4-{\frac {9}{2}}\right)} which, by adding 9/2 on both sides, correctly reduces to 5 = 5. Another example illustrating the danger of taking the square root of both sides of an equation involve... | Mathematical fallacies | 0.824217 |
6,439 | Pseudaria, an ancient lost book of false proofs, is attributed to Euclid.Mathematical fallacies exist in many branches of mathematics. In elementary algebra, typical examples may involve a step where division by zero is performed, where a root is incorrectly extracted or, more generally, where different values of a mul... | Mathematical fallacies | 0.824217 |
6,440 | The latter usually applies to a form of argument that does not comply with the valid inference rules of logic, whereas the problematic mathematical step is typically a correct rule applied with a tacit wrong assumption. Beyond pedagogy, the resolution of a fallacy can lead to deeper insights into a subject (e.g., the i... | Mathematical fallacies | 0.824217 |
6,441 | In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known e... | Mathematical fallacies | 0.824217 |
6,442 | In quantum mechanics, identical particles (also called indistinguishable or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, elementary particles (such as electrons), composite subatomic particles (su... | Identical particle | 0.824216 |
6,443 | Many examples of problems with checkable algorithms come from graph theory. For instance, a classical algorithm for testing whether a graph is bipartite would simply output a Boolean value: true if the graph is bipartite, false otherwise. In contrast, a certifying algorithm might output a 2-coloring of the graph in the... | Certifying algorithm | 0.824212 |
6,444 | In theoretical computer science, a certifying algorithm is an algorithm that outputs, together with a solution to the problem it solves, a proof that the solution is correct. A certifying algorithm is said to be efficient if the combined runtime of the algorithm and a proof checker is slower by at most a constant facto... | Certifying algorithm | 0.824212 |
6,445 | We now assume that the probability amplitudes are time-dependent, though the basis states are not. The Time-dependent Schrödinger equation states i ℏ ∂ t | ψ ⟩ = H | ψ ⟩ {\textstyle i\hbar \partial _{t}|\psi \rangle =H|\psi \rangle } , and proceeding as before (substituting for | ψ ⟩ {\displaystyle |\psi \rangle } and ... | Two-level quantum mechanical system | 0.824205 |
6,446 | As per the previous section, the solution to this equation has the Bloch vector precessing around ( ω 1 , 0 , ω 0 + ω r / 2 ) {\displaystyle (\omega _{1},0,\omega _{0}+\omega _{r}/2)} with a frequency that is twice the magnitude of the vector. If ω 0 {\displaystyle \omega _{0}} is sufficiently strong, some proportion o... | Two-level quantum mechanical system | 0.824205 |
6,447 | In physics or engineering education, a Fermi problem (or Fermi quiz, Fermi question, Fermi estimate), also known as a order-of-magnitude problem (or order-of-magnitude estimate, order estimation), is an estimation problem designed to teach dimensional analysis or approximation of extreme scientific calculations, and su... | Fermi Problem | 0.824205 |
6,448 | In physics, and in particular in biomechanics, the ground reaction force (GRF) is the force exerted by the ground on a body in contact with it. For example, a person standing motionless on the ground exerts a contact force on it (equal to the person's weight) and at the same time an equal and opposite ground reaction f... | Ground reaction force | 0.824201 |
6,449 | This book is aimed at beginning graduate students, although it could also be used by advanced undergraduates. Its many exercises make it suitable as a course textbook, and serve both to fill in details from the exposition in the book, and to provide pointers to additional topics. Although some mathematical sophisticati... | Introduction to Lattices and Order | 0.824195 |
6,450 | Introduction to Lattices and Order is a mathematical textbook on order theory by Brian A. Davey and Hilary Priestley. It was published by the Cambridge University Press in their Cambridge Mathematical Textbooks series in 1990, with a second edition in 2002. The second edition is significantly different in its topics an... | Introduction to Lattices and Order | 0.824195 |
6,451 | The fourth of the introductory chapters concerns special classes of lattices, including modular lattices, distributive lattices, and Boolean lattices.In the second part of the book, chapter 5 concerns the theorem that every finite Boolean lattice is isomorphic to the lattice of subsets of a finite set, and (less trivia... | Introduction to Lattices and Order | 0.824195 |
6,452 | Both editions of the book have 11 chapters; in the second book they are organized with the first four providing a general reference for mathematicians and computer scientists, and the remaining seven focusing on more specialized material for logicians, topologists, and lattice theorists.The first chapter concerns parti... | Introduction to Lattices and Order | 0.824195 |
6,453 | In statistics, some Monte Carlo methods require independent observations in a sample to be drawn from a one-dimensional distribution in sorted order. In other words, all n order statistics are needed from the n observations in a sample. The naive method performs a sort and takes O(n log n) time. There are also O(n) alg... | Sampling in order | 0.824188 |
6,454 | In cosmology, the cosmological constant problem or vacuum catastrophe is the disagreement between the observed values of vacuum energy density (the small value of the cosmological constant) and theoretical large value of zero-point energy suggested by quantum field theory. Depending on the Planck energy cutoff and othe... | Cosmological constant problem | 0.824188 |
6,455 | In quantum mechanics, the vacuum itself should experience quantum fluctuations. In general relativity, those quantum fluctuations constitute energy that would add to the cosmological constant. However, this calculated vacuum energy density is many orders of magnitude bigger than the observed cosmological constant. Orig... | Cosmological constant problem | 0.824188 |
6,456 | 1987 Thibault and Naylor described how arbitrary polyhedra may be represented using a BSP tree as opposed to the traditional b-rep (boundary representation). This provided a solid representation vs. a surface based-representation. Set operations on polyhedra were described using a tool, enabling constructive solid geom... | Binary space partitioning | 0.824179 |
6,457 | In theoretical computer science the random-access stored-program (RASP) machine model is an abstract machine used for the purposes of algorithm development and algorithm complexity theory. The RASP is a random-access machine (RAM) model that, unlike the RAM, has its program in its "registers" together with its input. T... | Random-access stored-program machine | 0.824177 |
6,458 | A multilayer perceptron (MLP) is a misnomer for a modern feedforward artificial neural network, consisting of fully connected neurons with a nonlinear kind of activation function, organized in at least three layers, notable for being able to distinguish data that is not linearly separable. It is a misnomer because the ... | Multi-layer perceptron | 0.824172 |
6,459 | Radar systems measure the distance to a target by the time it takes a radio-wave pulse to return to the radar antenna after being reflected by the target: the distance to the target is half the round-trip transit time multiplied by the speed of light. A Global Positioning System (GPS) receiver measures its distance to ... | Velocity of light | 0.824171 |
6,460 | A continuous real function on the closed unit interval is uniformly continuous. A continuous real function on the closed unit interval is Riemann integrable. The Brouwer fixed point theorem (for continuous functions on a finite product of copies of the closed unit interval). | Reverse Mathematics | 0.824169 |
6,461 | The Heine–Borel theorem for complete totally bounded separable metric spaces (where covering is by a sequence of open balls). A continuous real function on the closed unit interval (or on any compact separable metric space, as above) is bounded (or: bounded and reaches its bounds). A continuous real function on the clo... | Reverse Mathematics | 0.824169 |
6,462 | WKL0 can prove a good number of classical mathematical results that do not follow from RCA0, however. These results are not expressible as first-order statements but can be expressed as second-order statements. The following results are equivalent to weak Kőnig's lemma and thus to WKL0 over RCA0: The Heine–Borel theore... | Reverse Mathematics | 0.824169 |
6,463 | To show that WKL0 is actually stronger than (not provable in) RCA0, it is sufficient to exhibit a theorem of WKL0 that implies that noncomputable sets exist. This is not difficult; WKL0 implies the existence of separating sets for effectively inseparable recursively enumerable sets. It turns out that RCA0 and WKL0 have... | Reverse Mathematics | 0.824169 |
6,464 | The separable Hahn–Banach theorem in the form: a bounded linear form on a subspace of a separable Banach space extends to a bounded linear form on the whole space. The Jordan curve theorem Gödel's completeness theorem (for a countable language). Determinacy for open (or even clopen) games on {0,1} of length ω. Every co... | Reverse Mathematics | 0.824169 |
6,465 | These theorems are, in a sense, below the reach of the reverse mathematics enterprise because they are already provable in the base system. The classical theorems provable in RCA0 include: Basic properties of the natural numbers, integers, and rational numbers (for example, that the latter form an ordered field). Basic... | Reverse Mathematics | 0.824169 |
6,466 | In particular, any set of natural numbers that can be proven to exist in RCA0 is computable, and thus any theorem that implies that noncomputable sets exist is not provable in RCA0. To this extent, RCA0 is a constructive system, although it does not meet the requirements of the program of constructivism because it is a... | Reverse Mathematics | 0.824169 |
6,467 | As another example, the "average time" between 11 PM and 1 AM is either midnight or noon, depending on whether the two times are part of a single night or part of a single calendar day. The circular mean is one of the simplest examples of directional statistics and of statistics of non-Euclidean spaces. This computatio... | Mean of circular quantities | 0.824163 |
6,468 | In mathematics and statistics, a circular mean or angular mean is a mean designed for angles and similar cyclic quantities, such as times of day, and fractional parts of real numbers. This is necessary since most of the usual means may not be appropriate on angle-like quantities. For example, the arithmetic mean of 0° ... | Mean of circular quantities | 0.824163 |
6,469 | Some examples of non-trophic interactions are habitat modification, mutualism and competition for space. It has been suggested recently that non-trophic interactions can indirectly affect food web topology and trophic dynamics by affecting the species in the network and the strength of trophic links. A number of recent... | Biological interactions | 0.824153 |
6,470 | Submodular and supermodular set functions are also studied in combinatorial optimization. Many of the results in (Shapley 1971) have analogues in (Edmonds 1970), where submodular functions were first presented as generalizations of matroids. In this context, the core of a convex cost game is called the base polyhedron,... | Cooperative game theory | 0.824152 |
6,471 | Unlike cognitive neuroscience, which historically has focused on the connection between human brain systems and behavior, the field of MCC has used model organisms, such as mice, to study how molecular (i.e. receptor, kinase activation, phosphatase regulation), intra-cellular (i.e. dendritic processes), and inter-cellu... | Molecular Cellular Cognition | 0.82415 |
6,472 | Molecular cellular cognition (MCC) is a branch of neuroscience that involves the study of cognitive processes with approaches that integrate molecular, cellular and behavioral mechanisms. Key goals of MCC studies include the derivation of molecular and cellular explanations of cognitive processes, as well as finding me... | Molecular Cellular Cognition | 0.82415 |
6,473 | The last form is to spot organic compounds in a homogenous solution without immobilization and drying effect, this platform is developed by Dr. Dhaval Gosalia and Dr. Scott Diamond and later commercialized as DiscoveryDot technology by Reaction Biology Corporation . Polymer Microarrays Polymer microarrays have been de... | Chemical Compound Microarray | 0.824145 |
6,474 | A chemical compound microarray is a collection of organic chemical compounds spotted on a solid surface, such as glass and plastic. This microarray format is very similar to DNA microarray, protein microarray and antibody microarray. In chemical genetics research, they are routinely used for searching proteins that bin... | Chemical Compound Microarray | 0.824145 |
6,475 | A presentation of a group determines a geometry, in the sense of geometric group theory: one has the Cayley graph, which has a metric, called the word metric. These are also two resulting orders, the weak order and the Bruhat order, and corresponding Hasse diagrams. An important example is in the Coxeter groups. Furthe... | Generators and relations | 0.824139 |
6,476 | The commuting probability can be defined for other algebraic structures such as finite rings. The commuting probability can be defined for infinite compact groups; the probability measure is then, after a renormalisation, the Haar measure. == References == | Commuting probability | 0.824128 |
6,477 | In mathematics and more precisely in group theory, the commuting probability (also called degree of commutativity or commutativity degree) of a finite group is the probability that two randomly chosen elements commute. It can be used to measure how close to abelian a finite group is. It can be generalized to infinite g... | Commuting probability | 0.824128 |
6,478 | The finite group G {\displaystyle G} is abelian if and only if p ( G ) = 1 {\displaystyle p(G)=1} . One has p ( G ) = k ( G ) # G {\displaystyle p(G)={\frac {k(G)}{\#G}}} where k ( G ) {\displaystyle k(G)} is the number of conjugacy classes of G {\displaystyle G} .If G {\displaystyle G} is not abelian then p ( G ) ≤ 5 ... | Commuting probability | 0.824128 |
6,479 | In terms of components, let A = A i d x i , {\displaystyle A=A_{i}dx^{i},} where d x i {\displaystyle dx^{i}} is the standard one-form coordinate bases on the cotangent bundle T*M. Inserting into the above, and expanding, one obtains (using the summation convention): F = 1 2 ( ∂ A j ∂ x i − ∂ A i ∂ x j + ) d x i ∧ d x... | Metric compatibility | 0.824126 |
6,480 | The degree of a map is a homotopy invariant; moreover for continuous maps from the sphere to itself it is a complete homotopy invariant, i.e. two maps f , g: S n → S n {\displaystyle f,g:S^{n}\to S^{n}\,} are homotopic if and only if deg ( f ) = deg ( g ) {\displaystyle \deg(f)=\deg(g)} . In other words, degree is ... | Degree of a mapping | 0.82412 |
6,481 | and one defines the degree of f to be r − s. This definition coincides with the algebraic topological definition above. The same definition works for compact manifolds with boundary but then f should send the boundary of X to the boundary of Y. One can also define degree modulo 2 (deg2(f)) the same way as before but ta... | Degree of a mapping | 0.82412 |
6,482 | In the language of differential topology, the degree of a smooth map can be defined as follows: If f is a smooth map whose domain is a compact manifold and p is a regular value of f, consider the finite set f − 1 ( p ) = { x 1 , x 2 , … , x n } . {\displaystyle f^{-1}(p)=\{x_{1},x_{2},\ldots ,x_{n}\}\,.} By p being a r... | Degree of a mapping | 0.82412 |
6,483 | It is broadly used in vitro as molecular biology research tool due to its capability of joining as sticky as blunt DNA ends. However, procedures for ligation without the use of standard DNA ligase are also popular. Human DNA ligases abnormalities have been linked to pathological disorders characterized by immunodeficie... | Ligation (molecular biology) | 0.824116 |
6,484 | A co-factor is generally involved in the reaction, and this is usually ATP or NAD+. Eukaryotic cells ligases belong to ATP type, and NAD+ - dependent are found in bacteria (e.g. E. coli).The discovery of DNA ligase dates back to 1967 and is an important event in the field of molecular biology. Ligation in the laborator... | Ligation (molecular biology) | 0.824116 |
6,485 | Seven dogs were missed (false negatives), and seven cats were correctly excluded (true negatives). The program's precision is then 5/8 (true positives / selected elements) while its recall is 5/12 (true positives / relevant elements). Adopting a hypothesis-testing approach from statistics, in which, in this case, the n... | Precision and recall | 0.824113 |
6,486 | In pattern recognition, information retrieval, object detection and classification (machine learning), precision and recall are performance metrics that apply to data retrieved from a collection, corpus or sample space. Precision (also called positive predictive value) is the fraction of relevant instances among the re... | Precision and recall | 0.824113 |
6,487 | Inversion of control is not a new term in computer science. Martin Fowler traces the etymology of the phrase back to 1988, but it is closely related to the concept of program inversion described by Michael Jackson in his Jackson Structured Programming methodology in the 1970s. A bottom-up parser can be seen as an inver... | Inversion of Control | 0.824102 |
6,488 | In software engineering, inversion of control (IoC) is a design pattern in which custom-written portions of a computer program receive the flow of control from a generic framework. The term "inversion" is historical: a software architecture with this design "inverts" control as compared to procedural programming. In pr... | Inversion of Control | 0.824102 |
6,489 | In mathematics and statistics, a random number is either Pseudo-random or a number generated for, or part of, a set exhibiting statistical randomness. | Random number | 0.82409 |
6,490 | The universe of "possible worlds" depends upon the axioms and rules of the logical system in which one is working, but given some logical system, any logically consistent collection of statements is a possible world. The modal diamond operator ◊ {\displaystyle \lozenge } is used to express possibility: ◊ P {\displaysty... | Logical possibility | 0.824088 |
6,491 | That is, the truth-value of a sentence of the form "Mary believes that..." is not determined solely by the truth-value of its component sentence, and hence the (unary) connective (or simply operator since it is unary) is non-truth-functional. The class of classical logic connectives (e.g. &, →) used in the construction... | Truth-functional logic | 0.824074 |
6,492 | Some truth functions possess properties which may be expressed in the theorems containing the corresponding connective. Some of those properties that a binary truth function (or a corresponding logical connective) may have are: associativity: Within an expression containing two or more of the same associative connectiv... | Truth-functional logic | 0.824074 |
6,493 | Similar to torque and energy in physics; information-theoretic information and data storage size have the same dimensionality of units of measurement, but there is in general no meaning to adding, subtracting or otherwise combining the units mathematically, although one may act as a bound on the other. Units of informa... | Binary Digits | 0.82407 |
6,494 | In adiabatic dynamics, interatomic interactions are represented by single scalar potentials called potential energy surfaces. This is the Born–Oppenheimer approximation introduced by Born and Oppenheimer in 1927. Pioneering applications of this in chemistry were performed by Rice and Ramsperger in 1927 and Kassel in 19... | Quantum chemical | 0.824065 |
6,495 | Spectroscopic properties may also be predicted. Typically, such studies assume the electronic wave function is adiabatically parameterized by the nuclear positions (i.e., the Born–Oppenheimer approximation). A wide variety of approaches are used, including semi-empirical methods, density functional theory, Hartree-Fock... | Quantum chemical | 0.824065 |
6,496 | Common methods are infra-red (IR) spectroscopy, nuclear magnetic resonance (NMR) spectroscopy, and scanning probe microscopy. Quantum chemistry may be applied to the prediction and verification of spectroscopic data as well as other experimental data. Many quantum chemistry studies are focused on the electronic ground ... | Quantum chemical | 0.824065 |
6,497 | Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions to physical and chemical properties of molecules, materials, and solution... | Quantum chemical | 0.824065 |
6,498 | Multipole moments in mathematics and mathematical physics form an orthogonal basis for the decomposition of a function, based on the response of a field to point sources that are brought infinitely close to each other. These can be thought of as arranged in various geometrical shapes, or, in the sense of distribution t... | Multipole moment | 0.824063 |
6,499 | In mathematical logic and computer science, two-variable logic is the fragment of first-order logic where formulae can be written using only two different variables. This fragment is usually studied without function symbols. | Two-variable logic with counting | 0.824054 |
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