Berger loves synopsis, and he generalizes in ways that are at times outright misleading.
|
|
On the other hand, rehabilitation relies on the idea that motor skill generalizes across different tasks.
|
|
Of course, it's not clear if it generalizes to adults, but I would assume it does.
|
|
If they start to do a bunch of them, he tells me, that learning generalizes to other situations.
|
|
"[Frustrated silence] Dr. Macpherson: "This woman takes one remark and generalizes it to all communication in their relationship.
|
|
The ruling generalizes to other government social media accounts, and it's resulted in more lawsuits — including several against Rep.
|
|
The similarity here between the different perturbations means that the same basic idea generalizes well across the various classification networks.
|
|
In time that response generalizes to what we call systemic inflammation, or an overactive, overly excited immune response across the body.
|
|
EyeEm says its image recognition model generalizes well with a few samples, and can be trained in near real-time using GPUs.
|
|
Source: RecruitLoop Source: RecruitLoop This framework generalizes the recruitment value chain and ignores thousands of startups and innovations that haven't had widespread impact.
|
|
It generalizes a heterogeneous mix of people, it pits minorities against one another, and brushes aside the discrimination Asians do experience every day.
|
|
So that means it would take around nine days of poor sleep to gain a pound, if the study generalizes to the real world.
|
|
In fact, most times what the public calls bipolar disorder or generalizes to be "mental illness" could actually qualify as antisocial personality disorder or some variant.
|
|
That means the growth in the Hispanic population is causing an anxiety that generalizes into a broad sense that white people's status needs to be reinforced.
|
|
Others may consider bringing in more senses (smell is an obvious one) or seeing how a model produced from one dog (or many) generalizes to other dogs.
|
|
I think because of his stature and his money, [Trump] feels extremely powerful, and generalizes his [importance] because of that—even when his perception may be wrong.
|
|
It's "a case study about a given set of rules on a relatively small system, so it's maybe a bit early to say whether it generalizes," Lässig said.
|
|
I'm not sure how well my experience generalizes, though, because the things I find innovative about the View have confused most of the colleagues I've tested it on.
|
|
I think that's part of the issue that we're all grappling for ways to cohere what we're feeling into language in some sense, that generalizes to another person.
|
|
"I want to make clear that our decision is based solely on the part of your post that generalizes and advances stereotypes about women versus men," Google's talking points stated.
|
|
A leader or a government draws a circle around a group of people and generalizes them and demonizes them, making it a matter of necessity to hate them as a whole.
|
|
Although "You" gets some facts about serial killers correct, like their their ability to charm unsuspecting victims, the show also generalizes serial killer traits and actions through its depiction of Goldberg.
|
|
If the purpose of the machine-learning algorithm built into the router is to maximize bandwidth, it might stumble upon this solution by accident, which it then generalizes across the entire suite of router products.
|
|
This post generalizes the requirements of enterprise software investors in the intelligence era in the hope that it helps founders of enterprise software companies think about how to sequence their fundraising, product development and data strategy.
|
|
Letter To the Editor: " 'A New Sheriff' Enjoys Loyalty at Small Firms" (front page, June 3) grossly generalizes the views of American small-business owners toward President Trump's decision to withdraw from the Paris climate agreement.
|
|
Mr. Takano blames much of the problem on the myth of the "model minority," which over-generalizes Asians as diligent and high-achieving, and the idea that Asian-Americans do not have any of the same challenges as blacks or Latinos.
|
|
Like so many on the right—and some on the left, like New York magazine's Jonathan Chait—Kristof generalizes about liberal intolerance on campus and cites extreme examples, such as Oberlin students' protest of a local bakery accused (wrongly, it seems) of racial profiling.
|
|
Fridging is far from the only trope that generalizes women to an extreme; in fact, often in film, even when women are presented as complex people who have actual personalities, those personalities are all presented in exactly the same way: You only have to be these three things. pic.twitter.
|
|
Although we understand the critique that Roberson's talk was a little too focused on women waiting around for men and generalizes these relationship struggles between only straight men and straight women, it's not too difficult for anyone — no matter their gender or sexual orientation — to take what she said an apply it to their own life.
|
|
His opinions on those programs and advice for Google regarding them are certainly protected, she found, and a document prepared by an HR manager ahead of speaking to Damore (not an email to him as I previously had put here) emphasizes this (brackets NLRB's): I want to make clear that our decision is based solely on the part of your post that generalizes and advances stereotypes about women versus men.
|
|
In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes. A morphism of algebraic stacks generalizes a morphism of schemes.
|
|
In mathematics, a partial cyclic order is a ternary relation that generalizes a cyclic order in the same way that a partial order generalizes a linear order.
|
|
The Bunyakovsky conjecture generalizes Dirichlet's theorem to higher-degree polynomials. Whether or not even simple quadratic polynomials such as (known from Landau's fourth problem) attain infinitely many prime values is an important open problem. The Dickson's conjecture generalizes Dirichlet's theorem to more than one polynomial. The Schinzel's hypothesis H generalizes these two conjectures, i.e.
|
|
The multiplicative version of the Jordan-Chevalley decomposition generalizes to a decomposition in a linear algebraic group, and the additive version of the decomposition generalizes to a decomposition in a Lie algebra.
|
|
Taylor's theorem also generalizes to multivariate and vector valued functions.
|
|
The operation of tensor contraction generalizes the trace to arbitrary tensors.
|
|
The equation generalizes to constraint forces that do not obey D'Alembert's principle.
|
|
The theory of characteristic classes generalizes the idea of obstructions to our extensions.
|
|
Finally, the contraction criterion generalizes immediately from the bipartite to the multipartite case.
|
|
Non‑convex phenomena in economics have been studied with nonsmooth analysis, which generalizes convex analysis.
|
|
The notion generalizes to higher-order symmetric derivatives and also to n-dimensional Euclidean spaces.
|
|
The Carathéodory–Jacobi–Lie theorem is a theorem in symplectic geometry which generalizes Darboux's theorem.
|
|
In geometry, the Gram–Euler theorem generalizes the internal angle sum formula to higher-dimensional polytopes.
|
|
The Poincaré series generalizes the theta series to automorphic forms with respect to arbitrary Fuchsian groups.
|
|
In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom.
|
|
Thus, the concept of the Hausdorff measure generalizes the lebesgue measure and its notions of counting, length, and area. It also generalizes volume. In fact, there are d-dimensional Hausdorff measures for any d ≥ 0, which is not necessarily an integer. These measures are fundamental in geometric measure theory.
|
|
In mathematics, the fractional Laplacian is an operator which generalizes the notion of spatial derivatives to fractional powers.
|
|
In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives.
|
|
Instead of a single central point, one can ask for multiple points such that the variation from these points is minimized. This leads to cluster analysis, where each point in the data set is clustered with the nearest "center". Most commonly, using the 2-norm generalizes the mean to k-means clustering, while using the 1-norm generalizes the (geometric) median to k-medians clustering. Using the 0-norm simply generalizes the mode (most common value) to using the k most common values as centers.
|
|
In mathematical physics and harmonic analysis, the quadratic Fourier transform is an integral transform that generalizes the fractional Fourier transform, which in turn generalizes the Fourier transform. Roughly speaking, the Fourier transform corresponds to a change of variables from time to frequency (in the context of harmonic analysis) or from position to momentum (in the context of quantum mechanics). In phase space, this is a 90 degree rotation. The fractional Fourier transform generalizes this to any angle rotation, giving a smooth mixture of time and frequency, or of position and momentum.
|
|
The result by Dudley et al. generalizes this result to the setting of Gaussian measures on a general topological vector space.
|
|
The Stone–von Neumann theorem generalizes Stone's theorem to a pair of self- adjoint operators, (P,Q), satisfying the canonical commutation relation, and shows that these are all unitarily equivalent to the position operator and momentum operator on L^2(\R). The Hille–Yosida theorem generalizes Stone's theorem to strongly continuous one-parameter semigroups of contractions on Banach spaces.
|
|
Ivo Herzog showed in 1997 how to define the Ziegler spectrum of a locally coherent Grothendieck category, which generalizes the construction above.
|
|
The dual concept to a subobject is a '. This generalizes concepts such as quotient sets, quotient groups, quotient spaces, quotient graphs, etc.
|
|
The group algebra generalizes to the monoid ring and thence to the category algebra, of which another example is the incidence algebra.
|
|
Allowing inequality constraints, the KKT approach to nonlinear programming generalizes the method of Lagrange multipliers. It can be applied under differentiability and convexity.
|
|
In mathematics, Muirhead's inequality, named after Robert Franklin Muirhead, also known as the "bunching" method, generalizes the inequality of arithmetic and geometric means.
|
|
This is one of the steps used in the proof of the Weil conjectures. Behrend's trace formula generalizes the formula to algebraic stacks.
|
|
In particular, an atomic Schreier domain is a unique factorization domain; this generalizes the fact that an atomic GCD domain is a UFD.
|
|
In mathematics, the discrete Fourier transform over an arbitrary ring generalizes the discrete Fourier transform of a function whose values are complex numbers.
|
|
In geometry, the polar sine generalizes the sine function of angle to the vertex angle of a polytope. It is denoted by psin.
|
|
The solution for the hidden subgroup problem in the abelian case generalizes to finding the normal core in case of subgroups of arbitrary groups.
|
|
The Poisson random measure generalizes to the Poisson-type random measures, where members of the PT family are invariant under restriction to a subspace.
|
|
The original AL procedure crucially relies on the assumption that the item rankings are strict. generalizes this procedure to general rankings, with possible indifferences.
|
|
In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra.
|
|
This adds the H-box as a generator, that generalizes the Hadamard gate from the ZX-calculus. It can naturally describe quantum circuits involving Toffoli gates.
|
|
In metric- affine () gravity, one generalizes things even further, treating both the metric and connection independently, and assuming the matter Lagrangian depends on the connection as well.
|
|
This result can be proven numerous different ways; the dimension counting argument is most direct, and generalizes to higher degree, while other proofs are special to conics.
|
|
A partial cyclic order is a ternary relation that generalizes a (total) cyclic order in the same way that a partial order generalizes a total order. It is cyclic, asymmetric, and transitive, but it need not be total. An order variety is a partial cyclic order that satisfies an additional spreading axiom . Replacing the asymmetry axiom with a complementary version results in the definition of a co-cyclic order.
|
|
This is the product of Frobenius polynomials, and thus generalizes to arbitrary fields. The Prouhet–Thue–Morse constant was shown to be transcendental by Kurt Mahler in 1929.
|
|
Monoids, groups, rings, and algebras can all be viewed as categories with a single object. The construction of the opposite category generalizes the opposite group, opposite ring, etc.
|
|
In signal processing, the polynomial Wigner–Ville distribution is a quasiprobability distribution that generalizes the Wigner distribution function. It was proposed by Boualem Boashash and Peter O'Shea in 1994.
|
|
In metric geometry, the space of directions at a point describes the directions of curves that start at the point. It generalizes the tangent space in a differentiable manifold.
|
|
Main areas of Northeast Caucasian languages Avarian is a collective term for the Avar, Andi and Tsez (Dido) peoples and generalizes to various ethnic groups native to the foothills.
|
|
In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology.
|
|
In mathematics, in the field of sheaf theory and especially in algebraic geometry, the direct image functor generalizes the notion of a section of a sheaf to the relative case.
|
|
Another application of this Fourier series is to solve the Basel problem by using Parseval's theorem. The example generalizes and one may compute ζ(2n), for any positive integer n.
|
|
In mathematics, particularly in functional analysis and convex analysis, the Ursescu theorem is a theorem that generalizes the closed graph theorem, the open mapping theorem, and the uniform boundedness principle.
|
|
In mathematics, a Kleene algebra ( ; named after Stephen Cole Kleene) is an idempotent (and thus partially ordered) semiring endowed with a closure operator. It generalizes the operations known from regular expressions.
|
|
Although there is no 3-dimensional analogue of the complex numbers, the relationship between the positions of the centers can be re-expressed as a matrix equation, which also generalizes to dimensions.
|
|
A set T of non-constant polynomials is called a triangular set if all polynomials in T have distinct main variables. This generalizes triangular systems of linear equations in a natural way.
|
|
The algebraization of many-sorted logic is explained in an article by Caleiro and Gonçalves, which generalizes abstract algebraic logic to the many-sorted case, but can also be used as introductory material.
|
|
In algebraic geometry, a derived stack is, roughly, a stack together with a sheaf of commutative ring spectra. It generalizes a derived scheme. Derived stacks are the "spaces" studied in derived algebraic geometry.
|
|
A probabilistic and decision theoretic extension of affect control theoryHoey, Alhothali, and Schroeder (2013). generalizes the original theory in order to allow for uncertainty about identities, changing identities, and explicit non-affective goals.
|
|
Body essence is an entity invariant to interface reflection, and has two degrees of freedom. The Gaussian coefficient generalizes a conventional simple thresholding scheme, and it provides detailed use of body color similarity.
|
|
Bretschneider's formula generalizes Brahmagupta's formula for the area of a cyclic quadrilateral, which in turn generalizes Heron's formula for the area of a triangle. The trigonometric adjustment in Bretschneider's formula for non-cyclicality of the quadrilateral can be rewritten non-trigonometrically in terms of the sides and the diagonals and to giveJ. L. Coolidge, "A historically interesting formula for the area of a quadrilateral", American Mathematical Monthly, 46 (1939) 345–347. (JSTOR)E. W. Hobson: A Treatise on Plane Trigonometry.
|
|
The idea of describing algebraic structures with finite-automata can be generalized from groups to other structures. For instance, it generalizes naturally to automatic semigroups., Section 6.1, "Semigroups and Specialized Axioms", pp. 114–116.
|
|
In mathematics, Vitale's random Brunn–Minkowski inequality is a theorem due to Richard Vitale that generalizes the classical Brunn–Minkowski inequality for compact subsets of n-dimensional Euclidean space Rn to random compact sets.
|
|
This expression generalizes definition and can be taken as the definition. Then using invariance of the interval, equation follows from it in the same way follows from , except that here arbitrary coordinate changes are allowed.
|
|
In category theory and related fields of mathematics, a refinement is a construction that generalizes the operations of "interior enrichment", like bornologification or saturation of a locally convex space. A dual construction is called envelope.
|
|
In mathematics, specifically measure theory, a complex measure generalizes the concept of measure by letting it have complex values. In other words, one allows for sets whose size (length, area, volume) is a complex number.
|
|
The 1996 paper also defined a nested variant called NMAC. FIPS PUB 198 generalizes and standardizes the use of HMACs. HMAC is used within the IPsec, SSH and TLS protocols and for JSON Web Tokens.
|
|
In statistics, the generalized canonical correlation analysis (gCCA), is a way of making sense of cross-correlation matrices between the sets of random variables when there are more than two sets. While a conventional CCA generalizes principal component analysis (PCA) to two sets of random variables, a gCCA generalizes PCA to more than two sets of random variables. The canonical variables represent those common factors that can be found by a large PCA of all of the transformed random variables after each set underwent its own PCA.
|
|
In the finite projective space PG() with , a set of points such that no points lie in a common hyperplane is called a (spatial) -arc. This definition generalizes the definition of a -arc in a plane (where ).
|
|
The following generalizes Cayley's formula to labelled forests: Let Tn,k be the number of labelled forests on n vertices with k connected components, such that vertices 1, 2, ..., k all belong to different connected components. Then .
|
|
12, 2015. His collaborations with Michael Atiyah to figure out a geometric model for matter. Starting from the idea of an ultimate granular structure of the space-time, which generalizes the continuum and the discontinuum aspects of the space-time, he developed a formalism which generalizes the differential equations and difference equations theories to treat such spaces that appear continuous at low energies and exhibit a dual continuum and discontinuum aspects at high energy. Apart from his deep-routed interests in foundational sciences, he is also an aficionado of classical music.
|
|
299-336, 1999. () Instead of a complete classification, a set of aspects are enumerated below which generalizes the existing taxonomies by allowing classification along multiple viewpoints.Egri, P.: Coordination in production networks. PhD Thesis, Eötvös Loránd University, Budapest, 2008.
|
|
For a discussion on choosing between the t-test and nonparametric alternatives, see Lumley, et al. (2002). One-way analysis of variance (ANOVA) generalizes the two-sample t-test when the data belong to more than two groups.
|
|
The proof of the Korovkin version follows closely the version on , which however generalizes it to some extent by considering admissible functionals instead of non-negative measures and inequalities \leq and \geq respectively in conditions 1 and 2.
|
|
ISR generalizes the concept of a system of distinct representatives (SDR, also known as transversal). Every transversal is an ISR where in the underlying graph, all and only copies of the same vertex from different sets are connected.
|
|
The principle generalizes beyond the aggregation via majority rule to any reasonable aggregation rule, demonstrating that the aggregation of individual preferences into a social welfare function is fraught with severe difficulties (see Arrow's impossibility theorem and social choice theory).
|
|
The Gram–Euler theorem similarly generalizes the alternating sum of internal angles \sum \varphi for convex polyhedra to higher-dimensional polytopes:M. A. Perles and G. C. Shephard. 1967. "Angle sums of convex polytopes". Math. Scandinavica, Vol 21, No 2.
|
|
In valuation theory, the ramification theory of valuations studies the set of extensions of a valuation of a field K to an extension field of K. This generalizes the notions in algebraic number theory, local fields, and Dedekind domains.
|
|
In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a projective object is that of an injective object.
|
|
Itô's theorem is a result in the mathematical discipline of representation theory due to Noboru Itô. It generalizes the well-known result that the dimension of an irreducible representation of a group must divide the order of that group.
|
|
A definition of truth for IF sentences can be given, alternatively, by means of a translation into existential second-order logic. The translation generalizes the Skolemization procedure of first-order logic. Falsity is defined by a dual procedure called Kreiselization.
|
|
As another corollary from the Second Fundamental Theorem, one can obtain that : T(r,f')\leq 2 T(r,f)+S(r,f),\, which generalizes the fact that a rational function of degree d has 2d − 2 < 2d critical points.
|
|
In this theory, among other things, the stress tensor considered is no longer the Cauchy tensor or any similar, double symmetric tensor, but an asymmetric tensor that generalizes it: and this gives to the theory many further important structural peculiarities.
|
|
The $25,000,000,000 eigenvector: The linear algebra behind Google. SIAM Review, 48(3):569–581, 2006. Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions. Matrices are used in economics to describe systems of economic relationships.
|
|
In the infinite-dimensional case, this generalizes to a relationship between two minimal Stinespring representations. It is a consequence of Stinespring's theorem that all quantum operations can be implemented by unitary evolution after coupling a suitable ancilla to the original system.
|
|
Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite. A special case of this is a Lorentzian manifold, which is the mathematical basis of Einstein's general relativity theory of gravity.
|
|
In physics and engineering, the divergence theorem is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to integration by parts. In two dimensions, it is equivalent to Green's theorem.
|
|
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré.
|
|
In probability and statistics, a Markov renewal process (MRP) is a random process that generalizes the notion of Markov jump processes. Other random processes like Markov chains, Poisson processes and renewal processes can be derived as special cases of MRP's.
|
|
In mathematics, the generalized polygamma function or balanced negapolygamma function is a function introduced by Olivier Espinosa Aldunate and Victor H. Moll. It generalizes the polygamma function to negative and fractional order, but remains equal to it for integer positive orders.
|
|
If K is a commutative ring instead of a field, then everything that has been said above about linear combinations generalizes to this case without change. The only difference is that we call spaces like this V modules instead of vector spaces. If K is a noncommutative ring, then the concept still generalizes, with one caveat: Since modules over noncommutative rings come in left and right versions, our linear combinations may also come in either of these versions, whatever is appropriate for the given module. This is simply a matter of doing scalar multiplication on the correct side.
|
|
Meyer, pp 386+387 Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object.
|
|
This is easily generalizes to bottom topography with oscillations around a mean depth. Ardhuin, Fabrice. "Large scale forces under surface gravity waves at a wavy bottom: a mechanism for the generation of primary microseisms." Geophys. Res. Lett. 45 (2018), doi: 10.1029/2018GL078855.
|
|
An algebraic character is a formal expression attached to a module in representation theory of semisimple Lie algebras that generalizes the character of a finite-dimensional representation and is analogous to the Harish-Chandra character of the representations of semisimple Lie groups.
|
|
If one generalizes from these three examples, the active membership is probably less than 10% of the total claimed. Although the Spirit of Jesus Church continues to report thousands of baptisms, these claims are not accompanied by a serious increase in "active" members.
|
|
In the song's hook, E-40 generalizes that every person has choices to make, and that he is getting money. In the verses he presents scenarios and/or questions and then gives his own answer, with it being either "yup" or "nope".
|
|
486.) is an abstract data type that generalizes a queue, for which elements can be added to or removed from either the front (head) or back (tail).Donald Knuth. The Art of Computer Programming, Volume 1: Fundamental Algorithms, Third Edition. Addison-Wesley, 1997. .
|
|
In algebraic geometry, the Ramanujam vanishing theorem is an extension of the Kodaira vanishing theorem due to , that in particular gives conditions for the vanishing of first cohomology groups of coherent sheaves on a surface. The Kawamata–Viehweg vanishing theorem generalizes it.
|
|
On such deformations is the right invariant metric of Computational Anatomy which generalizes the metric of non-compressible Eulerian flows but to include the Sobolev norm ensuring smoothness of the flows, metrics have now been defined associated to Hamiltonian controls of diffeomorphic flows.
|
|
One such deformation is the right invariant metric of computational anatomy which generalizes the metric of non-compressible Eulerian flows to include the Sobolev norm, ensuring smoothness of the flows. Metrics have also been defined that are associated to Hamiltonian controls of diffeomorphic flows.
|
|
But saying "user" or "client" over-generalizes; in reality, the interaction takes place through a personal computer, an Internet protocol (IP) phone, or similar network device. Each of these must run supplicant software that initiates or reacts to IEEE 802.1X authentication requests for association.
|
|
In mathematics, the metric derivative is a notion of derivative appropriate to parametrized paths in metric spaces. It generalizes the notion of "speed" or "absolute velocity" to spaces which have a notion of distance (i.e. metric spaces) but not direction (such as vector spaces).
|
|
They also produced a new version in 2014. Kressner et al. tested a speech corpus different from the dataset used to develop HASQI and showed that the index generalizes well for listeners without a hearing loss with a performance comparable to PESQ. Kendrick et al.
|
|
Wedderburn's little theorem states that every finite domain is a field. In other words, for finite rings, there is no distinction between domains, skew- fields and fields. The Artin–Zorn theorem generalizes the theorem to alternative rings: every finite simple alternative ring is a field.
|
|
In algebraic geometry, a group-stack is an algebraic stack whose categories of points have group structures or even groupoid structures in a compatible way. It generalizes a group scheme, which is a scheme whose sets of points have group structures in a compatible way.
|
|
In mathematics and theoretical computer science, a k-regular sequence is a sequence satisfying linear recurrence equations that reflect the base-k representations of the integers. The class of k-regular sequences generalizes the class of k-automatic sequences to alphabets of infinite size.
|
|
The forward rate of chemical reactions is the reciprocal of the narrow escape time, which generalizes the classical Smoluchowski formula for Brownian particles located in an infinite medium. A Markov description can be used to estimate the binding and unbinding to a small number of sites.
|
|
In mathematics, an idempotent binary relation is a binary relation R on a set X (a subset of Cartesian product X × X) for which the composition of relations R ∘ R is the same as R. Here:p.3 This notion generalizes that of an idempotent function to relations.
|
|
Riemann also developed Riemannian geometry, which unifies and vastly generalizes the three types of geometry. The 19th century saw the beginning of a great deal of abstract algebra. Hermann Grassmann in Germany gave a first version of vector spaces, William Rowan Hamilton in Ireland developed noncommutative algebra.
|
|
Part 3 generalizes parts 1 and 2 to deal with imaginary and complex datatypes and arithmetic and elementary functions on such values. Much of the specifications in LIA-3 are inspired by the specifications for imaginary and complex datatypes and operations in C, Ada and Common Lisp.
|
|
The association found was relatively weak considering the large sample size used in the study. Overall, the study provided evidence that the BFLPE generalizes broadly over different levels of SES and is not moderated by SES. In other words, the BFLPE affects students from all economic backgrounds.
|
|
In mathematics, Wedderburn's little theorem states that every finite domain is a field. In other words, for finite rings, there is no distinction between domains, skew-fields and fields. The Artin–Zorn theorem generalizes the theorem to alternative rings: every finite alternative division ring is a field.
|
|
A cycle basis of the graph is a set of simple cycles that forms a basis of the cycle space.. Using ideas from algebraic topology, the binary cycle space generalizes to vector spaces or modules over other rings such as the integers, rational or real numbers, etc..
|
|
The same study went on to determine whether or not the difference generalizes to an auditory description of a person. The impaired subjects made entirely normal perceptions in this task. Overall, the research suggests that the amygdala is important for the making and retrieval of social judgements.
|
|
The Massey product is an algebraic generalization of the phenomenon of Borromean rings. In algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product. The Massey product was created by William S. Massey, an American algebraic topologist.
|
|
In mathematics, Verdier duality is a duality in sheaf theory that generalizes Poincaré duality for manifolds. Verdier duality was introduced by as an analog for locally compact spaces of the coherent duality for schemes due to Alexander Grothendieck. It is commonly encountered when studying constructible or perverse sheaves.
|
|
A matroid is a mathematical structure that generalizes the notion of linear independence from vector spaces to arbitrary sets. If an optimization problem has the structure of a matroid, then the appropriate greedy algorithm will solve it optimally.Papadimitriou, Christos H., and Kenneth Steiglitz. Combinatorial optimization: algorithms and complexity.
|
|
In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the action of the special linear group SL2(R) on the time–frequency plane (domain). The LCT generalizes the Fourier, fractional Fourier, Laplace, Gauss–Weierstrass, Bargmann and the Fresnel transforms as particular cases. The name "linear canonical transformation" is from canonical transformation, a map that preserves the symplectic structure, as SL2(R) can also be interpreted as the symplectic group Sp2, and thus LCTs are the linear maps of the time–frequency domain which preserve the symplectic form.
|
|
Pure Appl. Math. 19 (1966),261-286. Large deviations theory formalizes the heuristic ideas of concentration of measures and widely generalizes the notion of convergence of probability measures. Roughly speaking, large deviations theory concerns itself with the exponential decline of the probability measures of certain kinds of extreme or tail events.
|
|
The statement of the Adian–Rabin theorem generalizes a similar earlier result for semigroups by Andrey Markov, Jr.,A. Markov, "Невозможность алгорифмов распознавания некоторых свойств ассоциативных систем" [The impossibility of algorithms for the recognition of certain properties of associative systems]. Doklady Akademii Nauk SSSR vol. 77, (1951), pp. 953–956.
|
|
Infinitary logic allows infinitely long sentences. For example, one may allow a conjunction or disjunction of infinitely many formulas, or quantification over infinitely many variables. Infinitely long sentences arise in areas of mathematics including topology and model theory. Infinitary logic generalizes first-order logic to allow formulas of infinite length.
|
|
In mathematics, particularly q-analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant form when written in terms of the Ramanujan theta. The function is named after Srinivasa Ramanujan.
|
|
In algebra, a module spectrum is a spectrum with an action of a ring spectrum; it generalizes a module in abstract algebra. The ∞-category of (say right) module spectra is stable; hence, it can be considered as either analog or generalization of the derived category of modules over a ring.
|
|
A nonassociative ring is an algebraic structure that satisfies all of the ring axioms except the associative property and the existence of a multiplicative identity. A notable example is a Lie algebra. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.
|
|
This notion of inverse also readily generalizes to categories. An inverse category is simply a category in which every morphism f:X→Y has a generalized inverse g:Y→X such that fgf = f and gfg = g. An inverse category is selfdual. The category of sets and partial bijections is the prime example.
|
|
In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after the German mathematician Heinz Prüfer.
|
|
The Borel functional calculus is more general than the continuous functional calculus, and has a different focus from the holomorphic functional calculus. More precisely, the Borel functional calculus allows us to apply an arbitrary Borel function to a self-adjoint operator, in a way which generalizes applying a polynomial function.
|
|
Anti- functionalist generalizes post-Durkheimian point, which opposes any form of essentialist thinking about society. Therefore, in order to "grasp the continuing power of Durkheim's idea, we must discard the functionalist framework which shaped his work and think the question of social order (and its construction) from a new perspective".
|
|
An association scheme is a collection of binary relations satisfying certain compatibility conditions. Association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory. In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups.
|
|
The generalized variance is a scalar value which generalizes variance for multivariate random variables. It was introduced by Samuel S. Wilks. The generalized variance is defined as the determinant of the covariance matrix, \det(\Sigma). It can be shown to be related to the multidimensional scatter of points around their mean.
|
|
In mathematics, an orbital integral is an integral transform that generalizes the spherical mean operator to homogeneous spaces. Instead of integrating over spheres, one integrates over generalized spheres: for a homogeneous space X = G/H, a generalized sphere centered at a point x0 is an orbit of the isotropy group of x0.
|
|
A Montesinos link. In this example, e=-3 , \alpha_1 /\beta_1=-3/2 and \alpha_2 /\beta_2=5/2 . A Montesinos link is a special kind of link that generalizes pretzel links (a pretzel link can also be described as a Montesinos link with integer tangles). A Montesinos link which is also a knot (i.e.
|
|
More recent work further generalizes and extends these models. As regards asset pricing, developments in equilibrium-based pricing are discussed under "Portfolio theory" below, while "Derivative pricing" relates to risk-neutral, i.e. arbitrage-free, pricing. As regards the use of capital, "Corporate finance theory" relates, mainly, to the application of these models.
|
|
In other words, it is a topological invariant. This argument for monopoles is a restatement of the lasso argument for a pure U(1) theory. It generalizes to dimensions with in several ways. One way is to extend everything into the extra dimensions, so that U(1) monopoles become sheets of dimension .
|
|
Fréchet differentiability is a strictly stronger condition than Gateaux differentiability, even in finite dimensions. Between the two extremes is the quasi-derivative. In measure theory, the Radon–Nikodym derivative generalizes the Jacobian, used for changing variables, to measures. It expresses one measure μ in terms of another measure ν (under certain conditions).
|
|
Menard was known for his "tinny" voice and popular guitar strumming style. Musician and historian Ann Savoy generalizes Cajun guitar strumming to two styles: Old Time Style (Cléoma Falcon) and D. L. Menard Style.Savoy 1984, p. 8. It uses bass runs on chord changes and incorporates up-strokes along with down-strokes.
|
|
See "Circle and B-Splines clipping algorithms" under the subject Clipping (computer graphics) for an example of use. A convex hull is the smallest convex volume containing the object. If the object is the union of a finite set of points, its convex hull is a polytope. A ' ('DOP) generalizes the bounding box.
|
|
Newton's method assumes the function f to have a continuous derivative. Newton's method may not converge if started too far away from a root. However, when it does converge, it is faster than the bisection method, and is usually quadratic. Newton's method is also important because it readily generalizes to higher-dimensional problems.
|
|
The fundamental theorem of finitely generated abelian groups can be stated two ways, generalizing the two forms of the fundamental theorem of finite abelian groups. The theorem, in both forms, in turn generalizes to the structure theorem for finitely generated modules over a principal ideal domain, which in turn admits further generalizations.
|
|
If \Gamma is any commutative monoid, then the notion of a \Gamma-graded Lie algebra generalizes that of an ordinary (\Z-) graded Lie algebra so that the defining relations hold with the integers \Z replaced by \Gamma. In particular, any semisimple Lie algebra is graded by the root spaces of its adjoint representation.
|
|
It has a straightforward extension to modules stating that every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodules. This contains the case for rings as a special case, considering the ring as a module over itself, so that ideals are submodules. This also generalizes the primary decomposition form of the structure theorem for finitely generated modules over a principal ideal domain, and for the special case of polynomial rings over a field, it generalizes the decomposition of an algebraic set into a finite union of (irreducible) varieties. The first algorithm for computing primary decompositions for polynomial rings over a field of characteristic 0Primary decomposition requires testing irreducibility of polynomials, which is not always algorithmically possible in nonzero characteristic.
|
|
Elliptic geometry was developed later in the 19th century by the German mathematician Bernhard Riemann; here no parallel can be found and the angles in a triangle add up to more than 180°. Riemann also developed Riemannian geometry, which unifies and vastly generalizes the three types of geometry, and he defined the concept of a manifold, which generalizes the ideas of curves and surfaces. The 19th century saw the beginning of a great deal of abstract algebra. Hermann Grassmann in Germany gave a first version of vector spaces, William Rowan Hamilton in Ireland developed noncommutative algebra. The British mathematician George Boole devised an algebra that soon evolved into what is now called Boolean algebra, in which the only numbers were 0 and 1.
|
|
Szemerédi's theorem is a result in arithmetic combinatorics concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured. that every set of integers A with positive natural density contains a k term arithmetic progression for every k. This conjecture, which became Szemerédi's theorem, generalizes the statement of van der Waerden's theorem.
|
|
A simplicial polytope is a polytope whose facets are all simplices. For example, every polygon is a simplicial polytope. The Euler line associated to such a polytope is the line determined by its centroid and circumcenter of mass. This definition of an Euler line generalizes the ones above.. Suppose that P is a polygon.
|
|
In mathematics, a time dependent vector field is a construction in vector calculus which generalizes the concept of vector fields. It can be thought of as a vector field which moves as time passes. For every instant of time, it associates a vector to every point in a Euclidean space or in a manifold.
|
|
In mathematics, Artin–Verdier duality is a duality theorem for constructible abelian sheaves over the spectrum of a ring of algebraic numbers, introduced by , that generalizes Tate duality. It shows that, as far as etale (or flat) cohomology is concerned, the ring of integers in a number field behaves like a 3-dimensional mathematical object.
|
|
In mathematics, the Bogomolov conjecture is a conjecture, named after Fedor Bogomolov, in arithmetic geometry about algebraic curves that generalizes the Manin-Mumford conjecture in arithmetic geometry. The conjecture was proved by Emmanuel Ullmo and Shou-Wu Zhang in 1998. A further generalization to general abelian varieties was also proved by Zhang in 1998.
|
|
Szemerédi's theorem is a result in arithmetic combinatorics, concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured. that every set of integers A with positive natural density contains a k term arithmetic progression for every k. This conjecture, which became Szemerédi's theorem, generalizes the statement of van der Waerden's theorem.
|
|
The Borel functional calculus for self-adjoint operators is constructed using integrals with respect to PVMs. In quantum mechanics, PVMs are the mathematical description of projective measurements. They are generalized by positive operator valued measures (POVMs) in the same sense that a mixed state or density matrix generalizes the notion of a pure state.
|
|
In combinatorial game theory, and particularly in the theory of impartial games in misère play, an indistinguishability quotient is a commutative monoid that generalizes and localizes the Sprague–Grundy theorem for a specific game's rule set. In the specific case of misere-play impartial games, such commutative monoids have become known as misere quotients.
|
|
Equitable resolvable coverings. Journal of Combinatorial Designs, 11(2), 113-123. The Oberwolfach problem, of decomposing a complete graph into edge-disjoint copies of a given 2-regular graph, also generalizes Kirkman's schoolgirl problem. Kirkman's problem is the special case of the Oberwolfach problem in which the 2-regular graph consists of five disjoint triangles.
|
|
In mathematics, a Schwartz–Bruhat function, named after Laurent Schwartz and François Bruhat, is a complex valued function on a locally compact abelian group, such as the adeles, that generalizes a Schwartz function on a real vector space. A tempered distribution is defined as a continuous linear functional on the space of Schwartz–Bruhat functions.
|
|
Determinants of matrices in superrings (that is, Z2-graded rings) are known as Berezinians or superdeterminants. The permanent of a matrix is defined as the determinant, except that the factors sgn(σ) occurring in Leibniz's rule are omitted. The immanant generalizes both by introducing a character of the symmetric group Sn in Leibniz's rule.
|
|
In mathematics, the Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition or singular value decomposition of matrices. Its history can be traced to the 1880s work of Élie Cartan and Wilhelm Killing.
|
|
In a Cartesian closed category, the exponential operation can be used to raise an arbitrary object to the power of another object. This generalizes the Cartesian product in the category of sets. If 0 is an initial object in a Cartesian closed category, then the exponential object 00 is isomorphic to any terminal object 1.
|
|
This generalizes Fourier series to spaces of the type L^2(X), where X is a Riemannian manifold. The Fourier series converges in ways similar to the [-\pi,\pi] case. A typical example is to take X to be the sphere with the usual metric, in which case the Fourier basis consists of spherical harmonics.
|
|
The classical recurrence relation generalizes: the Catalan number of a Coxeter diagram is equal to the sum of the Catalan numbers of all its maximal proper sub-diagrams.Sergey Fomin and Nathan Reading, "Root systems and generalized associahedra", Geometric combinatorics, IAS/Park City Math. Ser. 13, American Mathematical Society, Providence, RI, 2007, pp 63–131.
|
|
In case of a spontaneous change at constant T and V without electrical work, the last term will thus be negative. In case there are other external parameters, the above relation further generalizes to :dF = -S\,dT - \sum_i X_i\,dx_i + \sum_j \mu_j\,dN_j. Here the x_i are the external variables, and the X_i the corresponding generalized forces.
|
|
The problem can be dealt with by shifting to field theory and considering the splitting field of a polynomial. Modern Galois theory generalizes the above type of Galois groups to field extensions and establishes—via the fundamental theorem of Galois theory—a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.
|
|
The research work of the professor generalizes the results of experimental, analytic and computer studies on the basis of which new theoretical problems and new applications of vibro-impact mechanisms and devices were solved. On the basis of the results of the investigations a number of original devices of vibro-impact type were created which were recognized as inventions.
|
|
The n-torsion of an abelian variety and the n-torsion of its dual are dual to each other when n is coprime to the characteristic of the base. In general - for all n - the n-torsion group schemes of dual abelian varieties are Cartier duals of each other. This generalizes the Weil pairing for elliptic curves.
|
|
Top and bottom envelope functions for a modulated sine wave. In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes. The envelope thus generalizes the concept of a constant amplitude into an instantaneous amplitude. The figure illustrates a modulated sine wave varying between an upper and a lower envelope.
|
|
This is possible for simple PDEs, which are called separable partial differential equations, and the domain is generally a rectangle (a product of intervals). Separable PDEs correspond to diagonal matrices – thinking of "the value for fixed " as a coordinate, each coordinate can be understood separately. This generalizes to the method of characteristics, and is also used in integral transforms.
|
|
In information theory, the Rényi entropy generalizes the Hartley entropy, the Shannon entropy, the collision entropy and the min-entropy. Entropies quantify the diversity, uncertainty, or randomness of a system. The entropy is named after Alfréd Rényi. In the context of fractal dimension estimation, the Rényi entropy forms the basis of the concept of generalized dimensions.
|
|
Ilija creates the anti-illusionistic, the abstract. He does not use perspective; he eliminates reminiscences; he generalizes and prefigures by using symbols. Two-headed beings denote the duplicity of everything presented. In Biblical scenes or Serbian myths, legends and epics, the dynamics and morale prevail, while in scenes from Iliad we find humor, irony and grotesque.
|
|
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth manifold, it generalizes the classical notion of Euler characteristic. It is named after Leonhard Euler because of this.
|
|
In the mathematical subject of geometric group theory, an acylindrically hyperbolic group is a group admitting a non-elementary 'acylindrical' isometric action on some geodesic hyperbolic metric space. This notion generalizes the notions of a hyperbolic group and of a relatively hyperbolic group and includes a significantly wider class of examples, such as mapping class groups and Out(Fn).
|
|
Binary search trees take more space than sorted arrays. Binary search trees lend themselves to fast searching in external memory stored in hard disks, as binary search trees can be efficiently structured in filesystems. The B-tree generalizes this method of tree organization. B-trees are frequently used to organize long-term storage such as databases and filesystems.
|
|
For S a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S (this point may be x itself). This definition generalizes to any subset S of a metric space X. Fully expressed, for X a metric space with metric d, x is a point of closure of S if for every r > 0, there is a y in S such that the distance d(x, y) < r. (Again, we may have x = y.) Another way to express this is to say that x is a point of closure of S if the distance d(x, S) := inf{d(x, s) : s in S} = 0. This definition generalizes to topological spaces by replacing "open ball" or "ball" with "neighbourhood".
|
|
234–235 The fundamental theorem for finitely presented abelian groups was proven by Henry John Stephen Smith in , as integer matrices correspond to finite presentations of abelian groups (this generalizes to finitely presented modules over a principal ideal domain), and Smith normal form corresponds to classifying finitely presented abelian groups. The fundamental theorem for finitely generated abelian groups was proven by Henri Poincaré in , using a matrix proof (which generalizes to principal ideal domains). This was done in the context of computing the homology of a complex, specifically the Betti number and torsion coefficients of a dimension of the complex, where the Betti number corresponds to the rank of the free part, and the torsion coefficients correspond to the torsion part. Kronecker's proof was generalized to finitely generated abelian groups by Emmy Noether in .
|
|
In computational complexity theory, an alternating Turing machine (ATM) is a non-deterministic Turing machine (NTM) with a rule for accepting computations that generalizes the rules used in the definition of the complexity classes NP and co-NP. The concept of an ATM was set forth by Chandra and Stockmeyer and independently by Kozen in 1976, with a joint journal publication in 1981.
|
|
Surprisingly, there is a notion of "distributivity" applicable to semilattices, even though distributivity conventionally requires the interaction of two binary operations. This notion requires but a single operation, and generalizes the distributivity condition for lattices. A join- semilattice is distributive if for all a, b, and x with there exist and such that x = a' ∨ b' . Distributive meet-semilattices are defined dually.
|
|
In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number δ) between points. The definition, introduced by Mikhael Gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. Hyperbolicity is a large-scale property, and is very useful to the study of certain infinite groups called (Gromov-)hyperbolic groups.
|
|
The correspondences listed here go much farther and deeper. For example, cartesian closed categories are generalized by closed monoidal categories. The internal language of these categories is the linear type system (corresponding to linear logic), which generalizes simply-typed lambda calculus as the internal language of cartesian closed categories. Moreover, these can be shown to correspond to cobordisms,John c.
|
|
Compared with binary categorization, multi-class categorization looks for common features that can be shared across the categories at the same time. They turn to be more generic edge like features. During learning, the detectors for each category can be trained jointly. Compared with training separately, it generalizes better, needs less training data, and requires fewer features to achieve the same performance.
|
|
New York: Routledge, 267–275.Ollis, D., (2004). I’m just a home economics teacher. Does discipline background impact on teachers’ ability to affirm and include gender and sexual diversity in secondary school health education programs? AARE Conference, Melbourne 2004 This observation generalizes to attitude evaluations in other areas besides sexual orientation and is one of the strengths of Riddle's study.
|
|
The change of variables , for a real parameter , brings Abel's equation into the celebrated Schröder's equation, . The further change into Böttcher's equation, . The Abel equation is a special case of (and easily generalizes to) the translation equation,Aczél, János, (1966): Lectures on Functional Equations and Their Applications, Academic Press, reprinted by Dover Publications, . :\omega( \omega(x,u),v)=\omega(x,u+v) ~, e.g.
|
|
In mathematics, specifically group theory, isoclinism is an equivalence relation on groups which generalizes isomorphism. Isoclinism was introduced by to help classify and understand p-groups, although it is applicable to all groups. Isoclinism also has consequences for the Schur multiplier and the associated aspects of character theory, as described in and . The word "isoclinism" comes from the Greek ισοκλινης meaning equal slope.
|
|
Massey products generalize cup product, allowing one to define "higher order linking numbers", the Milnor invariants. The cup product is a binary (2-ary) operation; one can define a ternary (3-ary) and higher order operation called the Massey product, which generalizes the cup product. This is a higher order cohomology operation, which is only partly defined (only defined for some triples).
|
|
Evdev and libevdev form a prominent part of the Linux API. Ergonomy requires the response time to be below a certain threshold. evdev (short for 'event device') is a generic input event interface in the Linux kernel and FreeBSD."Linux Input drivers v1.0" It generalizes raw input events from device drivers and makes them available through character devices in the `/dev/input/` directory.
|
|
In mathematics, in particular in partial differential equations and differential geometry, an elliptic complex generalizes the notion of an elliptic operator to sequences. Elliptic complexes isolate those features common to the de Rham complex and the Dolbeault complex which are essential for performing Hodge theory. They also arise in connection with the Atiyah- Singer index theorem and Atiyah-Bott fixed point theorem.
|
|
The RBF neural network is constructed by the conventional subset selection algorithms. The network structure is determined by combining both the stepwise forward network configuration and the continuous RBF parameter optimization. It is used to efficiently and effectively produce a parsimonious RBF neural network that generalizes well. It is achieved through simultaneous network structure determination and parameter optimization on the continuous parameter space.
|
|
Although topology is the field of mathematics that formalizes and generalizes the intuitive notion of "continuous deformation" of objects, it gives rise to many discrete topics; this can be attributed in part to the focus on topological invariants, which themselves usually take discrete values. See combinatorial topology, topological graph theory, topological combinatorics, computational topology, discrete topological space, finite topological space, topology (chemistry).
|
|
Named set theory is a branch of theoretical mathematics that studies the structures of names. The named set is a theoretical concept that generalizes the structure of a name described by Frege. Its generalization bridges the descriptivists theory of a name, and its triad structure (name, sensation and reference),Burgin (2011), p. 19 with mathematical structures that define mathematical names using triplets.
|
|
It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded \Z-algebra. The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to non-associative algebras as well; e.g.
|
|
During this training, the model is evaluated based on how well it predicts the observations contained in the training set. In general, however, the goal of a machine learning scheme is to produce a model that generalizes, that is, that predicts previously unseen observations. Overfitting occurs when a model fits the data in the training set well, while incurring larger generalization error.
|
|
Amplitude amplification is a technique in quantum computing which generalizes the idea behind the Grover's search algorithm, and gives rise to a family of quantum algorithms. It was discovered by Gilles Brassard and Peter Høyer in 1997, and independently rediscovered by Lov Grover in 1998. In a quantum computer, amplitude amplification can be used to obtain a quadratic speedup over several classical algorithms.
|
|
Every point in the complex Grassmannian manifold defines an -plane in -space. Fibering these planes over the Grassmannian one arrives at the vector bundle which generalizes the tautological bundle of a projective space. Similarly the -dimensional orthogonal complements of these planes yield an orthogonal vector bundle . The integral cohomology of the Grassmannians is generated, as a ring, by the Chern classes of .
|
|
In fact this generalizes to Rn whereby deleting a -dimensional subspace from Rn leaves a non-simply connected space). 4\. If A is a strong deformation retract of a topological space X, then the inclusion map from A to X induces an isomorphism between fundamental groups (so the fundamental group of X can be described using only loops in the subspace A).
|
|
233]-252 — an interpretation, which Arrow (1963, pp. 106-7) notes as in agreement with his own view of the constitution. The result generalizes and deepens the voting paradox to any voting rule satisfying the conditions, however complex or comprehensive. The 1963 edition includes an additional chapter with a simpler proof of Arrow's Theorem and corrects an earlier point noted by Blau.
|
|
In 1976, Daniel Z. Freedman codiscovered (with Sergio Ferrara and Peter van Nieuwenhuizen) supergravity. Freedman and van Nieuwenhuizen were on the faculty of the Stony Brook University. Supergravity generalizes Einstein's theory of general relativity by incorporating the then-new idea of supersymmetry. In the following decades it had implications for physics beyond the Standard Model, for superstring theory and for mathematics.
|
|
In mathematical analysis, Korn's inequality is an inequality concerning the gradient of a vector field that generalizes the following classical theorem: if the gradient of a vector field is skew-symmetric at every point, then the gradient must be equal to a constant skew-symmetric matrix. Korn's theorem is a quantitative version of this statement, which intuitively says that if the gradient of a vector field is on average not far from the space of skew- symmetric matrices, then the gradient must not be far from a particular skew- symmetric matrix. The statement that Korn's inequality generalizes thus arises as a special case of rigidity. In (linear) elasticity theory, the symmetric part of the gradient is a measure of the strain that an elastic body experiences when it is deformed by a given vector-valued function.
|
|
The eigenvectors are used as the basis when representing the linear transformation as Λ. Conversely, suppose a matrix A is diagonalizable. Let P be a non-singular square matrix such that P−1AP is some diagonal matrix D. Left multiplying both by P, AP = PD. Each column of P must therefore be an eigenvector of A whose eigenvalue is the corresponding diagonal element of D. Since the columns of P must be linearly independent for P to be invertible, there exist n linearly independent eigenvectors of A. It then follows that the eigenvectors of A form a basis if and only if A is diagonalizable. A matrix that is not diagonalizable is said to be defective. For defective matrices, the notion of eigenvectors generalizes to generalized eigenvectors and the diagonal matrix of eigenvalues generalizes to the Jordan normal form.
|
|
This generalizes the concept of an involution. An mth root of unity is a permutation σ so that σm = 1 under permutation composition. Now every time we apply σ we move one step in parallel along all of its cycles. A cycle of length d applied d times produces the identity permutation on d elements (d fixed points) and d is the smallest value to do so.
|
|
In mathematics, the Horrocks construction is a method for constructing vector bundles, especially over projective spaces, introduced by . His original construction gave an example of an indecomposable rank 2 vector bundle over 3-dimensional projective space, and generalizes to give examples of vector bundles of higher ranks over other projective spaces. The Horrocks construction is used in the ADHM construction to construct instantons over the 4-sphere.
|
|
He is well known for his early work on Hilbert's tenth problem and for developing the theory of motivic integration in a series of papers with François Loeser. He has also worked on computational number theory. Recently he proved a conjecture of Jean-Louis Colliot-Thélène which generalizes the Ax–Kochen theorem. In 2002 Denef was an Invited Speaker at the International Congresses of Mathematicians in Beijing.
|
|
Theatre, specifically Broadway musicals, are a component of another stereotype, the "show queen", which generalizes that gay men are involved with the performing arts, and are theatrical, overly dramatic, and camp. The bear subculture of the LGBT community is composed of generally large, hairy men, referred to as bears. They embrace their image, and some will shun more effeminate gay men, such as twinks, and vice versa.
|
|
Dual numbers find applications in physics, where they constitute one of the simplest non-trivial examples of a superspace. Equivalently, they are supernumbers with just one generator; supernumbers generalize the concept to distinct generators , each anti- commuting, possibly taking to infinity. Superspace generalizes supernumbers slightly, by allowing multiple commuting dimensions. The motivation for introducing dual numbers into physics follows from the Pauli exclusion principle for fermions.
|
|
The Manin–Mumford conjecture of Yuri Manin and David Mumford, proved by Michel Raynaud, states that a curve C in its Jacobian variety J can only contain a finite number of points that are of finite order (a torsion point) in J, unless C = J. There are other more general versions, such as the Bogomolov conjecture which generalizes the statement to non-torsion points.
|
|
In the mathematical branch of algebraic topology, specifically homotopy theory, n-connectedness (sometimes, n-simple connectedness) generalizes the concepts of path-connectedness and simple connectedness. To say that a space is n-connected is to say that its first n homotopy groups are trivial, and to say that a map is n-connected means that it is an isomorphism "up to dimension n, in homotopy".
|
|
CoL defines a computational problem as a game played by a machine against its environment. Such problem is computable if there is a machine that wins the game against every possible behavior of the environment. Such game-playing machine generalizes the Church-Turing thesis to the interactive level. The classical concept of truth turns out to be a special, zero-interactivity-degree case of computability.
|
|
PropBank differs from FrameNet, the resource to which it is most frequently compared, in several ways. PropBank is a verb- oriented resource, while FrameNet is centered on the more abstract notion of frames, which generalizes descriptions across similar verbs (e.g. "describe" and "characterize") as well as nouns and other words (e.g. "description"). PropBank does not annotate events or states of affairs described using nouns.
|
|
They contain emergent particles with non-Abelian statistics which generalizes the well known Bose and Fermi statistics. Non-Abelian particles may allow us to perform fault tolerant quantum computations. With Michael Levin, he found that string-net condensations can give rise to a large class of topological orders (2005). In particular, string-net condensation provides a unified origin of photons, electrons, and other elementary particles (2003).
|
|
It unifies two fundamental phenomena: gauge interactions and Fermi statistics. He pointed out that topological order is nothing but the pattern of long range entanglements. This led to a notion of symmetry protected topological (SPT) order (short-range entangled states with symmetry) and its description by group cohomology of the symmetry group (2011). The notion of SPT order generalizes the notion of topological insulator to interacting cases.
|
|
In complex analysis, a discipline within mathematics, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. From a geometrical perspective, it is a special case of the generalized Stokes' theorem.
|
|
A toroidal polyhedron with 6 × 4 = 24 quadrilateral faces Polyhedra with the topological type of a torus are called toroidal polyhedra, and have Euler characteristic V − E + F = 0. For any number of holes, the formula generalizes to V − E + F = 2 − 2N, where N is the number of holes. The term "toroidal polyhedron" is also used for higher-genus polyhedra and for immersions of toroidal polyhedra.
|
|
In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case of dimension 0). Simplicial homology arose as a way to study topological spaces whose building blocks are n-simplices, the n-dimensional analogs of triangles.
|
|
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions. Examples are the product of sets, groups (described below), rings, and other algebraic structures.
|
|
Let C be a set of n squares or circles of identical size. There is a polynomial-time approximation scheme for finding an MDS using a simple shifted-grid strategy. It finds a solution within (1 − e) of the maximum in time nO(1/e2) time and linear space. The strategy generalizes to any collection of fat objects of roughly the same size (i.e.
|
|
In addition to a resource with multiple components, this same model generalizes to a resource with multiple stages, each of which is composed of multiple resources, each of which can be removed independently of each other (i.e., with no additional cost). This model can be further generalized to the case where multiple components with additional costs can be removed in multiple stages of processing through recursion.
|
|
A framed knot is the extension of a tame knot to an embedding of the solid torus in . The framing of the knot is the linking number of the image of the ribbon with the knot. A framed knot can be seen as the embedded ribbon and the framing is the (signed) number of twists. This definition generalizes to an analogous one for framed links.
|
|
In abstract algebra, the total quotient ring,Matsumura (1980), p. 12 or total ring of fractions,Matsumura (1989), p. 21 is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings R that may have zero divisors. The construction embeds R in a larger ring, giving every non-zero-divisor of R an inverse in the larger ring.
|
|
Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric. This is a concept of distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point. Riemannian geometry generalizes Euclidean geometry to spaces that are not necessarily flat, although they still resemble the Euclidean space at each point infinitesimally, i.e. in the first order of approximation.
|
|
Over time, this memory retrieval style becomes negatively reinforced and generalizes to other memories that could potentially be connected to the original negative memory, leading to OGM. However, with additional research on memory and OGM, the theory of Functional Avoidance could not be upheld on its own.Conway, M. A., Pleydell-Pearce, C. W., (2000). The construction of autobiographical memories in the self-memory system.
|
|
In mathematics, the Freidlin–Wentzell theorem is a result in the large deviations theory of stochastic processes. Roughly speaking, the Freidlin–Wentzell theorem gives an estimate for the probability that a (scaled-down) sample path of an Itō diffusion will stray far from the mean path. This statement is made precise using rate functions. The Freidlin–Wentzell theorem generalizes Schilder's theorem for standard Brownian motion.
|
|
Visualization of distributive law for positive numbers In mathematics, the distributive property of binary operations generalizes the distributive law from Boolean algebra and elementary algebra. In propositional logic, distribution refers to two valid rules of replacement. The rules allow one to reformulate conjunctions and disjunctions within logical proofs. For example, in arithmetic: : 2 ⋅ (1 + 3) = (2 ⋅ 1) + (2 ⋅ 3), but 2 / (1 + 3) ≠ (2 / 1) + (2 / 3).
|
|
A Cauchy net generalizes the notion of Cauchy sequence to nets defined on uniform spaces.. A net (xα) is a Cauchy net if for every entourage V there exists γ such that for all α, β ≥ γ, (xα, xβ) is a member of V.. More generally, in a Cauchy space, a net (xα) is Cauchy if the filter generated by the net is a Cauchy filter.
|
|
He also proved several theorems concerning convergence of sequences of measurable and holomorphic functions. The Vitali convergence theorem generalizes Lebesgue's dominated convergence theorem. Another theorem bearing his name gives a sufficient condition for the uniform convergence of a sequence of holomorphic functions on an open domain. This result has been generalized to normal families of meromorphic functions, holomorphic functions of several complex variables, and so on.
|
|
It follows from the three defining coherence conditions that a large class of diagrams (i.e. diagrams whose morphisms are built using \alpha, \lambda, \rho, identities and tensor product) commute: this is Mac Lane's "coherence theorem". It is sometimes inaccurately stated that all such diagrams commute. There is a general notion of monoid object in a monoidal category, which generalizes the ordinary notion of monoid from abstract algebra.
|
|
Functional encryption generalizes several existing primitives including Identity-based encryption (IBE) and attribute-based encryption (ABE). In the IBE case, define F(k,x) to be equal to x when k corresponds to an identity that is allowed to decrypt, and \perp otherwise. Similarly, in the ABE case, define F(k, x) = x when k encodes attributes with permission to decrypt and \perp otherwise.
|
|
A 2-dimensional spring system. In engineering and physics, a spring system or spring network is a model of physics described as a graph with a position at each vertex and a spring of given stiffness and length along each edge. This generalizes Hooke's law to higher dimensions. This simple model can be used to solve the pose of static systems from crystal lattice to springs.
|
|
Originally algorithmic induction methods extrapolated ordered sequences of strings. Methods were needed for dealing with other kinds of data. A 1999 report,"Two Kinds of Probabilistic Induction," The Computer Journal, Vol 42, No. 4, 1999. (pdf version) generalizes the Universal Distribution and associated convergence theorems to unordered sets of strings and a 2008 report,"Three Kinds of Probabilistic Induction, Universal Distributions and Convergence Theorems" 2008.
|
|
In arithmetic geometry, a Frobenioid is a category with some extra structure that generalizes the theory of line bundles on models of finite extensions of global fields. Frobenioids were introduced by . The word "Frobenioid" is a portmanteau of Frobenius and monoid, as certain Frobenius morphisms between Frobenioids are analogues of the usual Frobenius morphism, and some of the simplest examples of Frobenioids are essentially monoids.
|
|
In string theory, it is conventional to study the Ginzburg–Landau functional for the manifold M being a Riemann surface, and taking n=1, i.e. a line bundle. The phenomenon of Abrikosov vortices persists in these general cases, including M=\R^2, where one can specify any finite set of points where \psi vanishes, including multiplicity. The proof generalizes to arbitrary Riemann surfaces and to Kähler manifolds.
|
|
If harmonic Maass forms are interpreted as harmonic sections of the line bundle of modular forms of weight k equipped with the Petersson metric over the modular curve, then this differential operator can be viewed as a composition of the Hodge star operator and the antiholomorphic differential. The notion of harmonic Maass forms naturally generalizes to arbitrary congruence subgroups and (scalar and vector valued) multiplier systems.
|
|
The above definition of the characteristic polynomial of a matrix A \in M_n(F) with entries in a field F generalizes without any changes to the case when F is just a commutative ring. defines the characteristic polynomial for elements of an arbitrary finite-dimensional (associative, but not necessarily commutative) algebra over a field F and proves the standard properties of the characteristic polynomial in this generality.
|
|
The work culminated in what Arrow called the "General Possibility Theorem," better known thereafter as Arrow's (impossibility) theorem. The theorem states that, absent restrictions on either individual preferences or neutrality of the constitution to feasible alternatives, there exists no social choice rule that satisfies a set of plausible requirements. The result generalizes the voting paradox, which shows that majority voting may fail to yield a stable outcome.
|
|
In 1976, Sergio Ferrara, Daniel Z. Freedman, and Peter van Nieuwenhuizen discovered Supergravity at Stony Brook University in New York. It was initially proposed as a four-dimensional theory. The theory of supergravity generalizes Einstein's general theory of relativity by incorporating the principles of supersymmetry. In 2019 the three were awarded a special Breakthrough Prize in Fundamental Physics of $3 million for the discovery.
|
|
A trophic function was first introduced in the differential equations of the Kolmogorov predator–prey model. It generalizes the linear case of predator–prey interaction firstly described by Volterra and Lotka in the Lotka–Volterra equation. A trophic function represents the consumption of prey assuming a given number of predators. The trophic function (also referred to as the functional response) was widely applied in chemical kinetics, biophysics, mathematical physics and economics.
|
|
In mathematics, duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices via Priestley spaces, spectral spaces, and pairwise Stone spaces. This duality, which is originally also due to Marshall H. Stone Stone (1938), generalizes the well-known Stone duality between Stone spaces and Boolean algebras. Let be a bounded distributive lattice, and let denote the set of prime filters of . For each , let }.
|
|
Table of height and weight for boys over time. The growth curve model (also known as GMANOVA) is used to analyze data such as this, where multiple observations are made on collections of individuals over time. The growth curve model in statistics is a specific multivariate linear model, also known as GMANOVA (Generalized Multivariate Analysis-Of-Variance). It generalizes MANOVA by allowing post-matrices, as seen in the definition.
|
|
An open subset of a locally path-connected space is connected if and only if it is path-connected. This generalizes the earlier statement about Rn and Cn, each of which is locally path-connected. More generally, any topological manifold is locally path-connected. The topologist's sine curve is connected, but it is not locally connected Locally connected does not imply connected, nor does locally path-connected imply path connected.
|
|
In mathematics, the Gross–Koblitz formula, introduced by expresses a Gauss sum using a product of values of the p-adic gamma function. It is an analog of the Chowla–Selberg formula for the usual gamma function. It implies the Hasse–Davenport relation and generalizes the Stickelberger theorem. gave another proof of the Gross–Koblitz formula (Boyarski being a pseudonym of Bernard Dwork), and gave an elementary proof.
|
|
The concept of a chirality center generalizes the concept of an asymmetric carbon atom (a carbon atom bonded to four different entities) such that an interchanging of any two groups gives rise to an enantiomer. In organic chemistry, a chirality center usually refers to a carbon, phosphorus, or sulfur atom, though it is also possible for other atoms to be chirality centers, especially in areas of organometallic and inorganic chemistry.
|
|
In planar graphs, colorings with k distinct colors are dual to nowhere zero flows over the ring \Z_k of integers modulo k. In this duality, the difference between the colors of two adjacent regions is represented by a flow value across the edge separating the regions. In particular, the existence of nowhere zero 4-flows is equivalent to the four color theorem. The snark theorem generalizes this result to nonplanar graphs.
|
|
The first of these generalizes chip-firing from Laplacian matrices of graphs to M-matrices, connecting this generalization to root systems and representation theory. The second considers chip-firing on abstract simplicial complexes instead of graphs. The third uses chip-firing to study graph-theoretic analogues of divisor theory and the Riemann–Roch theorem. And the fourth applies methods from commutative algebra to the study of chip-firing.
|
|
Example of geometric median (in yellow) of a series of points. In blue the Center of mass. The geometric median of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distances for one-dimensional data, and provides a central tendency in higher dimensions.
|
|
Random Search replaces the exhaustive enumeration of all combinations by selecting them randomly. This can be simply applied to the discrete setting described above, but also generalizes to continuous and mixed spaces. It can outperform Grid search, especially when only a small number of hyperparameters affects the final performance of the machine learning algorithm. In this case, the optimization problem is said to have a low intrinsic dimensionality.
|
|
Some selected inputs might consist of a negative number, zero, and a positive number. When using these numbers to test software in this way, the developer generalizes the set of all reals into three numbers. This is a more efficient and manageable method, but more prone to failure. Generalizing test cases is an example of just one technique to deal with failure—specifically, failure due to invalid user input.
|
|
Berliant, Thomson and Dunz introduced the criterion of group envy-freeness, which generalizes both Pareto-efficiency and envy-freeness. They proved the existence of group-envy-free allocations with additive utilities. Later, Berliant and Dunz studied some natural non-additive utility functions, motivated by the problem of land division. When utilities are not additive, a CEEI allocation is no longer guaranteed to exist, but it does exist under certain restrictions.
|
|
In mathematics, a pseudoreflection is an invertible linear transformation of a finite-dimensional vector space such that it is not the identity transformation, has a finite (multiplicative) order, and fixes a hyperplane. The concept of pseudoreflection generalizes the concepts of reflection and complex reflection, and is simply called reflection by some mathematicians. It plays an important role in Invariant theory of finite groups, including the Chevalley-Shephard-Todd theorem.
|
|
The Nijenhuis tensor of an almost complex structure J, is the Frölicher–Nijenhuis bracket of J with itself. An almost complex structure is a complex structure if and only if the Nijenhuis tensor is zero. With the Frölicher–Nijenhuis bracket it is possible to define the curvature and cocurvature of a vector-valued 1-form which is a projection. This generalizes the concept of the curvature of a connection.
|
|
Eugene Gendlin's original Focusing process, described in his 1978 book, is a process that he generalizes as having six steps: clearing a space, allowing a "felt sense" to form, finding a handle, resonating, asking, and receiving.Gendlin, Eugene. Focusing. Bantam Books, 1978. pp. 103–107. Inner Relationship Focusing, developed in the late 1980s through the late 1990s, is a more fluid process, and eschews or modifies certain aspects of Gendlin's.
|
|
ANOVA generalizes to the study of the effects of multiple factors. When the experiment includes observations at all combinations of levels of each factor, it is termed factorial. Factorial experiments are more efficient than a series of single factor experiments and the efficiency grows as the number of factors increases.Montgomery (2001, Section 5-2: Introduction to factorial designs; The advantages of factorials) Consequently, factorial designs are heavily used.
|
|
In mathematical physics, the Dirac equation in curved spacetime generalizes the original Dirac equation to curved space. It can be written by using vierbein fields and the gravitational spin connection. The vierbein defines a local rest frame, allowing the constant Dirac matrices to act at each spacetime point. In this way, Dirac's equation takes the following form in curved spacetime: :i\gamma^a e_a^\mu D_\mu \Psi - m \Psi = 0.
|
|
The dimension of a vector space is the number of vectors in any basis for the space, i.e. the number of coordinates necessary to specify any vector. This notion of dimension (the cardinality of a basis) is often referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension. For the non-free case, this generalizes to the notion of the length of a module.
|
|
The theory of association schemes arose in statistics, in the theory of experimental design for the analysis of variance. In mathematics, association schemes belong to both algebra and combinatorics. Indeed, in algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory. In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups.
|
|
Applications whose goal is to create a system that generalizes well to unseen examples, face the possibility of over-training. This arises in convoluted or over-specified systems when the network capacity significantly exceeds the needed free parameters. Two approaches address over-training. The first is to use cross- validation and similar techniques to check for the presence of over-training and to select hyperparameters to minimize the generalization error.
|
|
In another version, if both disjoint convex sets are open, then there is a hyperplane in between them, but not necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, because the orthogonal projections of the convex bodies onto the axis are disjoint. The hyperplane separation theorem is due to Hermann Minkowski. The Hahn–Banach separation theorem generalizes the result to topological vector spaces.
|
|
In probability theory and statistics, the Conway–Maxwell–Poisson (CMP or COM–Poisson) distribution is a discrete probability distribution named after Richard W. Conway, William L. Maxwell, and Siméon Denis Poisson that generalizes the Poisson distribution by adding a parameter to model overdispersion and underdispersion. It is a member of the exponential family, has the Poisson distribution and geometric distribution as special cases and the Bernoulli distribution as a limiting case.
|
|
The original Miller effect is implemented by capacitive impedance connected between the two nodes. Miller theorem generalizes Miller effect as it implies arbitrary impedance Z connected between the nodes. It is supposed also a constant coefficient K; then the expressions above are valid. But modifying properties of Miller theorem exist even when these requirements are violated and this arrangement can be generalized further by dynamizing the impedance and the coefficient.
|
|
In fact, this is another way to state the Lie–Kolchin theorem. Lie's theorem states that any nonzero representation of a solvable Lie algebra on a finite dimensional vector space over an algebraically closed field of characteristic 0 has a one-dimensional invariant subspace. The result for Lie algebras was proved by and for algebraic groups was proved by . The Borel fixed point theorem generalizes the Lie–Kolchin theorem.
|
|
In string theory and related theories such as supergravity theories, a brane is a physical object that generalizes the notion of a point particle to higher dimensions. For example, a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. It is also possible to consider higher-dimensional branes. In dimension , these are called -branes.
|
|
The convention of using colors originates from coloring the countries of a map, where each face is literally colored. This was generalized to coloring the faces of a graph embedded in the plane. By planar duality it became coloring the vertices, and in this form it generalizes to all graphs. In mathematical and computer representations, it is typical to use the first few positive or non-negative integers as the "colors".
|
|
The category of smooth manifolds with smooth maps lacks certain desirable properties, and people have tried to generalize smooth manifolds in order to rectify this. Diffeological spaces use a different notion of chart known as a "plot". Frölicher spaces and orbifolds are other attempts. A rectifiable set generalizes the idea of a piece-wise smooth or rectifiable curve to higher dimensions; however, rectifiable sets are not in general manifolds.
|
|
The bias-variance tradeoff is a central problem in supervised learning. Ideally, one wants to choose a model that both accurately captures the regularities in its training data, but also generalizes well to unseen data. Unfortunately, it is typically impossible to do both simultaneously. High-variance learning methods may be able to represent their training set well but are at risk of overfitting to noisy or unrepresentative training data.
|
|
The definition of a holomorphic function generalizes to several complex variables in a straightforward way. Let D denote an open subset of Cn, and let . The function f is analytic at a point p in D if there exists an open neighbourhood of p in which f is equal to a convergent power series in n complex variables.Gunning and Rossi, Analytic Functions of Several Complex Variables, p. 2.
|
|
A subset U of a metric space is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in M with y also belongs to U. Equivalently, U is open if every point in U has a neighborhood contained in U. This generalizes the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.
|
|
If is any function, then we have (where "∘" denotes function composition). In particular, is the identity element of the monoid of all functions from to . Since the identity element of a monoid is unique, one can alternately define the identity function on to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of need not be functions.
|
|
In mathematics, the Caristi fixed-point theorem (also known as the Caristi–Kirk fixed-point theorem) generalizes the Banach fixed-point theorem for maps of a complete metric space into itself. Caristi's fixed-point theorem modifies the ε-variational principle of Ekeland (1974, 1979). The conclusion of Caristi's theorem is equivalent to metric completeness, as proved by Weston (1977). The original result is due to the mathematicians James Caristi and William Arthur Kirk.
|
|
This property generalizes immediately to hyperdeterminants implying invariance when you add a multiple of one slice of a hypermatrix to another parallel slice. A hyperdeterminant is not the only polynomial algebraic invariant for the group acting on the hypermatrix. For example, other algebraic invariants can be formed by adding and multiplying hyperdeterminants. In general the invariants form a ring algebra and it follows from Hilbert's basis theorem that the ring is finitely generated.
|
|
Rudy Rucker's discussion of the CCM: CS.sjsu.edu For general families of holomorphic functions, the boundary of the Mandelbrot set generalizes to the bifurcation locus, which is a natural object to study even when the connectedness locus is not useful. The Multibrot set is obtained by varying the value of the exponent d. The article has a video that shows the development from d = 0 to 7, at which point there are 6 i.e.
|
|
Some scholars argue that feminism in some ways waters down an individual's cultural identity and generalizes women to a non-inclusive umbrella category, while Africana womanism allows one to maintain their cultural identity.Blackmon, Janiece L. "I Am Because We Are: Africana Womanism as a Vehicle of Empowerment and Influence". Thesis. Virginia Polytechnic Institute and State University, 2008. 1–58. Print. In terms of its distinction from womanism, Africana womanism is very ethnically specific.
|
|
Three main approaches have been proposed for the definition of the semantics of IF logic. The first two, based respectively on games of imperfect information and on Skolemization, are mainly used in the definition of IF sentences only. The former generalizes a similar approach, for first-order logic, which was based instead on games of perfect information. The third approach, team semantics, is a compositional semantics in the spirit of Tarskian semantics.
|
|
Hammond's FBI associates find Flyte, a British academic who theorizes the town has fallen victim to the Ancient Enemy, an entity he generalizes as "chaos in the flesh". It periodically wipes out civilizations including that of the Mayans and the Roanoke Island colonists. They are soon joined by an Army commando unit and a group of scientists led by General Copperfield who has come to Snowfield. They, along with Flyte, investigate the town.
|
|
In computer science, more precisely in automata theory, a rational set of a monoid is an element of the minimal class of subsets of this monoid that contains all finite subsets and is closed under union, product and Kleene star. Rational sets are useful in automata theory, formal languages and algebra. A rational set generalizes the notion of rational (regular) language (understood as defined by regular expressions) to monoids that are not necessarily free.
|
|
In mathematics, a Generalized Clifford algebra (GCA) is an associative algebra that generalizes the Clifford algebra, and goes back to the work of Hermann Weyl, who utilized and formalized these clock-and-shift operators introduced by J. J. Sylvester (1882),; ibid II (1883) 46; ibid III (1884) 7–9. Summarized in The Collected Mathematics Papers of James Joseph Sylvester (Cambridge University Press, 1909) v III . online and further. and organized by Cartan (1898) and Schwinger.
|
|
The definition in the previous section generalizes the notion of inverse in group relative to the notion of identity. It's also possible, albeit less obvious, to generalize the notion of an inverse by dropping the identity element but keeping associativity, i.e., in a semigroup. In a semigroup S an element x is called (von Neumann) regular if there exists some element z in S such that xzx = x; z is sometimes called a pseudoinverse.
|
|
One can then define functions from that type by induction on the way the elements of the type are generated. Induction-recursion generalizes this situation since one can simultaneously define the type and the function, because the rules for generating elements of the type are allowed to refer to the function. Induction-recursion can be used to define large types including various universe constructions. It increases the proof-theoretic strength of type theory substantially.
|
|
In finance, a barbell strategy is formed when a trader invests in long and short duration bonds, but does not invest in Intermediate duration bonds. This strategy is useful when interest rates are rising; as the short term maturities are rolled over they receive a higher interest rate, raising the value. Taleb generalizes the phenomenon and applies it to other domains. Essentially it is the transformation of anything from fragile to antifragile.
|
|
Many of the applications of Hilbert spaces exploit the fact that Hilbert spaces support generalizations of simple geometric concepts like projection and change of basis from their usual finite dimensional setting. In particular, the spectral theory of continuous self-adjoint linear operators on a Hilbert space generalizes the usual spectral decomposition of a matrix, and this often plays a major role in applications of the theory to other areas of mathematics and physics.
|
|
The binomial distribution generalizes this to the number of heads from performing n independent flips (Bernoulli trials) of the same coin. The multinomial distribution models the outcome of n experiments, where the outcome of each trial has a categorical distribution, such as rolling a k-sided dice n times. Let k be a fixed finite number. Mathematically, we have k possible mutually exclusive outcomes, with corresponding probabilities p1, ..., pk, and n independent trials.
|
|
In software engineering, the multiton pattern is a design pattern which generalizes the singleton pattern. Whereas the singleton allows only one instance of a class to be created, the multiton pattern allows for the controlled creation of multiple instances, which it manages through the use of a map. Rather than having a single instance per application (e.g. the object in the Java programming language) the multiton pattern instead ensures a single instance per key.
|
|
The partial derivative generalizes the notion of the derivative to higher dimensions. A partial derivative of a multivariable function is a derivative with respect to one variable with all other variables held constant. Partial derivatives may be combined in interesting ways to create more complicated expressions of the derivative. In vector calculus, the del operator ( abla) is used to define the concepts of gradient, divergence, and curl in terms of partial derivatives.
|
|
In probability theory, the central limit theorem says that, under certain conditions, the sum of many independent identically-distributed random variables, when scaled appropriately, converges in distribution to a standard normal distribution. The martingale central limit theorem generalizes this result for random variables to martingales, which are stochastic processes where the change in the value of the process from time t to time t + 1 has expectation zero, even conditioned on previous outcomes.
|
|
Differential games are related closely with optimal control problems. In an optimal control problem there is single control u(t) and a single criterion to be optimized; differential game theory generalizes this to two controls u_{1}(t),u_{2}(t) and two criteria, one for each player. Each player attempts to control the state of the system so as to achieve its goal; the system responds to the inputs of all players.
|
|
In terms of homogeneous coordinates, a point of the complex projective plane with coordinates (a,b,c) in the complex projective space for which there exists no complex number z such that za, zb, and zc are all real. This definition generalizes to complex projective spaces. The point with coordinates : (a_1,a_2,\ldots,a_n) is imaginary if there exists no complex number z such that : (za_1,za_2,\ldots,za_n) are all real coordinates..
|
|
In conformal field theory and representation theory, a W-algebra is an algebra that generalizes the Virasoro algebra. W-algebras were introduced by Alexander Zamolodchikov (), and the name "W-algebra" comes from the fact that Zamolodchikov used the letter W for one of the elements of one of his examples. There are at least three different but related notions called W-algebras: classical W-algebras, quantum W-algebras, and finite W-algebras.
|
|
The Hasse diagram of the set of all subsets of a three-element set {x, y, z}, ordered by inclusion. Distinct sets on the same horizontal level are incomparable with each other. Some other pairs, such as {x} and {y,z}, are also incomparable. In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.
|
|
An interval in a poset P is a subset of P with the property that, for any x and y in and any z in P, if x ≤ z ≤ y, then z is also in . (This definition generalizes the interval definition for real numbers.) For a ≤ b, the closed interval is the set of elements x satisfying a ≤ x ≤ b (i.e. a ≤ x and x ≤ b). It contains at least the elements a and b.
|
|
In optics two beams of light are said to interfere coherently, when the phase difference between their waves is constant; if this phase difference is random or changing the beams are incoherent. The coherent superposition of wave amplitudes is called first order interference. In analogy to that we have intensity or second order Hanbury Brown and Twiss (HBT) interference, which generalizes the interference between amplitudes to that between squares of amplitudes, i.e. between intensities.
|
|
In geometry, a ball is a region in space comprising all points within a fixed distance from a given point; that is, it is the region enclosed by a sphere or hypersphere. An -ball is a ball in -dimensional Euclidean space. The volume of a unit -ball is an important expression that occurs in formulas throughout mathematics; it generalizes the notion of the volume enclosed by a sphere in 3-dimensional space.
|
|
Because the class of ordinal numbers is well-ordered, there is a smallest infinite limit ordinal; denoted by ω (omega). The ordinal ω is also the smallest infinite ordinal (disregarding limit), as it is the least upper bound of the natural numbers. Hence ω represents the order type of the natural numbers. The next limit ordinal above the first is ω + ω = ω·2, which generalizes to ω·n for any natural number n.
|
|
Fulton and MacPherson extended the Chow ring to singular varieties by defining the "operational Chow ring" and more generally a bivariant theory associated to any morphism of schemes.Fulton, Intersection Theory, Chapter 17. A bivariant theory is a pair of covariant and contravariant functors that assign to a map a group and a ring respectively. It generalizes a cohomology theory, which is a contravariant functor that assigns to a space a ring, namely a cohomology ring.
|
|
Publisher: Taylor & Francis Books. Theory and Applications of Fractional Differential Equations, by Kilbas, A. A.; Srivastava, H. M.; and Trujillo, J. J. Amsterdam, Netherlands, Elsevier, February 2006. operator that generalizes the Riemann–Liouville fractional integral. Katugampola fractional derivative has been defined using the Katugampola fractional integral and as with any other fractional differential operator, it also extends the possibility of taking real number powers or complex number powers of the integral and differential operators.
|
|
The idea of scale invariance of a monomial generalizes in higher dimensions to the idea of a homogeneous polynomial, and more generally to a homogeneous function. Homogeneous functions are the natural denizens of projective space, and homogeneous polynomials are studied as projective varieties in projective geometry. Projective geometry is a particularly rich field of mathematics; in its most abstract forms, the geometry of schemes, it has connections to various topics in string theory.
|
|
However, in the presence of symmetry, even short range entangled states are nontrivial and can belong to different phases. Those phases are said to contain SPT order. SPT order generalizes the notion of topological insulator to interacting systems. Some suggest that topological order (or more precisely, string-net condensation) in local bosonic (spin) models have the potential to provide a unified origin for photons, electrons and other elementary particles in our universe.
|
|
In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator.
|
|
The Wiener filter as originally proposed by Norbert Wiener is a signal processing filter which uses knowledge of the statistical properties of both the signal and the noise to reconstruct an optimal estimate of the signal from a noisy one-dimensional time-ordered data stream. The generalized Wiener filter generalizes the same idea beyond the domain of one-dimensional time- ordered signal processing, with two-dimensional image processing being the most common application.
|
|
For example, given affine forms x,y for X and Y, one can obtain an affine form z for Z = X + Y simply by adding the forms -- that is, setting z_j \gets x_j + y_j for every j. Similarly, one can compute an affine form z for Z = \alphaX, where \alpha is a known constant, by setting z_j \gets \alpha x_j for every j. This generalizes to arbitrary affine operations like Z = \alphaX + \betaY + \gamma.
|
|
Schinzel is a professor at the Institute of Mathematics of the Polish Academy of Sciences (IM PAN). His principal interest is the theory of polynomials. His 1958 conjecture on the prime values of polynomials, known as Schinzel's hypothesis H, both extends the Bunyakovsky conjecture and broadly generalizes the twin prime conjecture. Schinzel is the author of over 200 research articles in various branches of number theory, including elementary, analytic and algebraic number theory.
|
|
The name, "Learning Classifier System (LCS)", is a bit misleading since there are many machine learning algorithms that 'learn to classify' (e.g. decision trees, artificial neural networks), but are not LCSs. The term 'rule-based machine learning (RBML)' is useful, as it more clearly captures the essential 'rule-based' component of these systems, but it also generalizes to methods that are not considered to be LCSs (e.g. association rule learning, or artificial immune systems).
|
|
Francis Bacon, articulating inductivism in England, is often falsely stereotyped as a naive inductivist. Crudely explained, the "Baconian model" advises to observe nature, propose a modest law that generalizes an observed pattern, confirm it by many observations, venture a modestly broader law, and confirm that, too, by many more observations, while discarding disconfirmed laws. Growing ever broader, the laws never quite exceed observations. Scientists, freed from preconceptions, thus gradually uncover nature's causal and material structure.
|
|
Inductivism infers from observations of similar effects to similar causes, and generalizes unrestrictedly—that is, by enumerative induction—to a universal law. Extending inductivism, Comtean positivism explicitly aims to oppose metaphysics, shuns imaginative theorizing, emphasizes observation, then making predictions, confirming them, and stating laws. Logical positivism, rather, would accept hypotheticodeductivsm in theory development, but nonetheless sought an inductive logic to objectively quantity a theory's confirmation by empirical evidence and, additionally, objectively compare rival theories.
|
|
In set theory, a mouse is a small model of (a fragment of) Zermelo–Fraenkel set theory with desirable properties. The exact definition depends on the context. In most cases, there is a technical definition of "premouse" and an added condition of iterability (referring to the existence of wellfounded iterated ultrapowers): a mouse is then an iterable premouse. The notion of mouse generalizes the concept of a level of Gödel's constructible hierarchy while being able to incorporate large cardinals.
|
|
That amounts to Σ13 correctness (in the usual sense) if M is x→x#. The core model can also be defined above a particular set of ordinals X: X belongs to K(X), but K(X) satisfies the usual properties of K above X. If there is no iterable inner model with ω Woodin cardinals, then for some X, K(X) exists. The above discussion of K and Kc generalizes to K(X) and Kc(X).
|
|
In mathematics, specifically in symplectic geometry, the momentum map (or moment mapMoment map is a misnomer and physically incorrect. It is an erroneous translation of the French notion application moment. See this mathoverflow question for the history of the name.) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum.
|
|
In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant. The directional derivative is a special case of the Gateaux derivative.
|
|
Klivans is the author of the book The Mathematics of Chip-Firing (CRC Press, 2018). Her research contributions include a disproof of a 50-year-old conjecture of Richard Stanley that every abstract simplicial complex whose face ring is a Cohen–Macaulay ring can be partitioned into disjoint intervals, each including a facet of the complex. Such a partition generalizes a shelling and (if it always existed) would have been helpful in understanding the -vectors of these complexes.
|
|
Blake generalizes here "about the spiritual history of mankind out the experience of his own spiritual history."Hirsch, p. 312 It can be also understood as a cycle history of relations between society and idea of liberty in the form of a male and female that grow older and younger in opposition to the other experiencing such changes. As a whole, the poem portrays conflicting ideas that make it difficult for the reader to attach to any given viewpoint.
|
|
The homotopy principle generalizes such results as Smale's proof of sphere eversion. In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs). The h-principle is good for underdetermined PDEs or PDRs, such as occur in the immersion problem, isometric immersion problem, fluid dynamics, and other areas. The theory was started by Yakov Eliashberg, Mikhail Gromov and Anthony V. Phillips.
|
|
The work of Bancal et al. generalizes Bell's result by proving that correlations achievable in quantum theory are also incompatible with a large class of superluminal hidden variable models. In this framework, faster-than-light signaling is precluded. However, the choice of settings of one party can influence hidden variables at another party's distant location, if there is enough time for a superluminal influence (of finite, but otherwise unknown speed) to propagate from one point to the other.
|
|
The localization theorem generalizes the localization theorem for abelian categories. Let C \subset D be exact categories. Then C is said to be cofinal in D if (i) it is closed under extension in D and if (ii) for each object M in D there is an N in D such that M \oplus N is in C. The prototypical example is when C is the category of free modules and D is the category of projective modules.
|
|
The structured support vector machine is a machine learning algorithm that generalizes the Support Vector Machine (SVM) classifier. Whereas the SVM classifier supports binary classification, multiclass classification and regression, the structured SVM allows training of a classifier for general structured output labels. As an example, a sample instance might be a natural language sentence, and the output label is an annotated parse tree. Training a classifier consists of showing pairs of correct sample and output label pairs.
|
|
Graph comparing leftThis principle generalizes to other classes of material. Naturally brittle materials, such as glass, are not difficult to toughen effectively. Most such techniques involve one of two mechanisms: to deflect or absorb the tip of a propagating crack or to create carefully controlled residual stresses so that cracks from certain predictable sources will be forced closed. The first principle is used in laminated glass where two sheets of glass are separated by an interlayer of polyvinyl butyral.
|
|
In mathematics, specifically linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so that the product is well-defined and square). It generalizes the statement that the determinant of a product of square matrices is equal to the product of their determinants. The formula is valid for matrices with the entries from any commutative ring.
|
|
The potential complexity of this setup is controlled by a set of conventional locations for common resources. The second idea means that processes can offer their services to other processes by providing virtual files that appear in the other processes' namespace. The client process's input/output on such a file becomes inter- process communication between the two processes. This way, Plan 9 generalizes the Unix notion of the filesystem as the central point of access to computing resources.
|
|
In string theory and related theories such as supergravity theories, a brane is a physical object that generalizes the notion of a point particle to higher dimensions. 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. Mathematically, branes can be represented within categories, and are studied in pure mathematics for insight into homological mirror symmetry and noncommutative geometry.
|
|
Uniform dimension generalizes some, but not all, aspects of the notion of the dimension of a vector space. Finite uniform dimension was a key assumption for several theorems by Goldie, including Goldie's theorem, which characterizes which rings are right orders in a semisimple ring. Modules of finite uniform dimension generalize both Artinian modules and Noetherian modules. In the literature, uniform dimension is also referred to as simply the dimension of a module or the rank of a module.
|
|
In mathematics and computer science, the probabilistic automaton (PA) is a generalization of the nondeterministic finite automaton; it includes the probability of a given transition into the transition function, turning it into a transition matrix. Thus, the probabilistic automaton also generalizes the concepts of a Markov chain and of a subshift of finite type. The languages recognized by probabilistic automata are called stochastic languages; these include the regular languages as a subset. The number of stochastic languages is uncountable.
|
|
It was investigated how geometric algebra could serve to model some of the quantum-like aspects and be used for an approach to holographic representations of memory. The role of quantum structures in economics was investigated, and more specifically a quantum model was worked out for the situation of the Ellsberg paradox in economics accounting for the deviations from classical probability due to ambiguity aversion, and it was analyzed how this quantum model generalizes the classical expected utility hypothesis.
|
|
Every graphic matroid (and every co-graphic matroid) is regular.. Conversely, every regular matroid may be constructed by combining graphic matroids, co-graphic matroids, and a certain ten-element matroid that is neither graphic nor co-graphic, using an operation for combining matroids that generalizes the clique-sum operation on graphs.. The number of bases in a regular matroid may be computed as the determinant of an associated matrix, generalizing Kirchhoff's matrix- tree theorem for graphic matroids..
|
|
Schreier was introduced to group theory by Kurt Reidemeister and first examined knot groups in 1924 following work by Max Dehn. His best-known work is his habilitation thesis on the subgroups of free groups, in which he generalizes the results of Reidemeister about normal subgroups. He proved that subgroups of free groups themselves are free, generalizing a theorem by Jakob Nielsen (1921). In 1927 he showed that the topological fundamental group of a classical Lie group is abelian.
|
|
The problem of motion estimation generalizes to binocular vision when we consider occlusion or motion perception at relatively large distances, where binocular disparity is a poor cue to depth. This fundamental difficulty is referred to as the inverse problem. Nonetheless, some humans do perceive motion in depth. There are indications that the brain uses various cues, in particular temporal changes in disparity as well as monocular velocity ratios, for producing a sensation of motion in depth.
|
|
A zipper is a technique of representing an aggregate data structure so that it is convenient for writing programs that traverse the structure arbitrarily and update its contents, especially in purely functional programming languages. The zipper was described by Gérard Huet in 1997. It includes and generalizes the gap buffer technique sometimes used with arrays. The zipper technique is general in the sense that it can be adapted to lists, trees, and other recursively defined data structures.
|
|
The notion of a regular language has been generalized to infinite words (see ω-automata) and to trees (see tree automaton). Rational set generalizes the notion (of regular/rational language) to monoids that are not necessarily free. Likewise, the notion of a recognizable language (by a finite automaton) has namesake as recognizable set over a monoid that is not necessarily free. Howard Straubing notes in relation to these facts that “The term "regular language" is a bit unfortunate.
|
|
There exist, however, typed lambda calculi that are not strongly normalizing. For example the dependently typed lambda calculus with a type of all types (Type : Type) is not normalizing due to Girard's paradox. This system is also the simplest pure type system, a formalism which generalizes the Lambda cube. Systems with explicit recursion combinators, such as Plotkin's "Programming language for Computable Functions" (PCF), are not normalizing, but they are not intended to be interpreted as a logic.
|
|
Two greedy colorings of the same graph using different vertex orders. The right example generalizes to 2-colorable graphs with n vertices, where the greedy algorithm expends n/2 colors. The greedy algorithm considers the vertices in a specific order v_1,…, v_n and assigns to v_i the smallest available color not used by v_i’s neighbours among v_1,…, v_{i-1}, adding a fresh color if needed. The quality of the resulting coloring depends on the chosen ordering.
|
|
Since many linear and nonlinear systems that oscillate are modeled as harmonic oscillators near their equilibria, this section begins with a derivation of the resonant frequency for a driven, damped harmonic oscillator. The section then uses an RLC circuit to illustrate connections between resonance and a system's transfer function, frequency response, poles, and zeroes. Building off the RLC circuit example, the section then generalizes these relationships for higher- order linear systems with multiple inputs and outputs.
|
|
The generalized Fermat equation generalizes the statement of Fermat's last theorem by considering positive integer solutions a, b, c, m, n, k satisfying In particular, the exponents m, n, k need not be equal, whereas Fermat's last theorem considers the case The Beal conjecture, also known as the Mauldin conjecture and the Tijdeman-Zagier conjecture, states that there are no solutions to the generalized Fermat equation in positive integers a, b, c, m, n, k with a, b, and c being pairwise coprime and all of m, n, k being greater than 2. The Fermat–Catalan conjecture generalizes Fermat's last theorem with the ideas of the Catalan conjecture. The conjecture states that the generalized Fermat equation has only finitely many solutions (a, b, c, m, n, k) with distinct triplets of values (am, bn, ck), where a, b, c are positive coprime integers and m, n, k are positive integers satisfying The statement is about the finiteness of the set of solutions because there are 10 known solutions.
|
|
However, the resulting average-case complexity depends heavily on the probability distribution that is chosen over the input. The actual inputs and distribution of inputs may be different in practice from the assumptions made during the analysis: a random input may be very unlike a typical input. Because of this choice of data model, a theoretical average- case result might say little about practical performance of the algorithm. Smoothed analysis generalizes both worst-case and average-case analysis and inherits strengths of both.
|
|
M–O similarity theory further generalizes the mixing length theory in non-neutral conditions by using so-called "universal functions" of dimensionless height to characterize vertical distributions of mean flow and temperature. The Obukhov length (L), a characteristic length scale of surface layer turbulence derived by Obukhov in 1946, is used for non-dimensional scaling of the actual height. M–O similarity theory marked a significant landmark of modern micrometeorology, providing a theoretical basis for micrometerological experiments and measurement techniques.
|
|
When subatomic particles interact, different outcomes are possible. The evolution of the various possibilities is called a "tree" and the probability of a given outcome is called its scattering amplitude. According to the principle of unitarity, the sum of the probabilities for every possible outcome is 1. The on-shell scattering process "tree" may be described by a positive Grassmannian, a structure in algebraic geometry analogous to a convex polytope, that generalizes the idea of a simplex in projective space.
|
|
In computer science, SUHA (Simple Uniform Hashing Assumption) is a basic assumption that facilitates the mathematical analysis of hash tables. The assumption states that a hypothetical hashing function will evenly distribute items into the slots of a hash table. Moreover, each item to be hashed has an equal probability of being placed into a slot, regardless of the other elements already placed. This assumption generalizes the details of the hash function and allows for certain assumptions about the stochastic system.
|
|
Complete group varieties are called abelian varieties. This generalizes to the notion of abelian scheme; a group scheme G over a base S is abelian if the structural morphism from G to S is proper and smooth with geometrically connected fibers They are automatically projective, and they have many applications, e.g., in geometric class field theory and throughout algebraic geometry. A complete group scheme over a field need not be commutative, however; for example, any finite group scheme is complete.
|
|
Another model, which generalizes Gilbert's random graph model, is the random dot-product model. A random dot-product graph associates with each vertex a real vector. The probability of an edge uv between any vertices u and v is some function of the dot product u • v of their respective vectors. The network probability matrix models random graphs through edge probabilities, which represent the probability p_{i,j} that a given edge e_{i,j} exists for a specified time period.
|
|
The lyrics begin by stating that even though he and Ono seem to have everything, they are still as lonely and isolated as everyone else. The second verse focuses on the couple's political activism, which many oppose generating even further isolation. The third verse generalizes the situation further. Lennon acknowledges that the people who have caused his pain can't be blamed, since we are all part of the same irrational world, and thus we are all victims of the world's insanity.
|
|
It is therefore sometimes said that the expansion is bi-orthogonal since the random coefficients are orthogonal in the probability space while the deterministic functions are orthogonal in the time domain. The general case of a process that is not centered can be brought back to the case of a centered process by considering which is a centered process. Moreover, if the process is Gaussian, then the random variables are Gaussian and stochastically independent. This result generalizes the Karhunen–Loève transform.
|
|
Firms combine labour and capital, and can achieve far greater economies of scale (when the average cost per unit declines as more units are produced) than individual market trading. In perfectly competitive markets studied in the theory of supply and demand, there are many producers, none of which significantly influence price. Industrial organization generalizes from that special case to study the strategic behaviour of firms that do have significant control of price. It considers the structure of such markets and their interactions.
|
|
Their work generalizes algebraic geometry in a purely algebraic direction: instead of studying the prime ideals in a polynomial ring, one can study the prime ideals in any commutative ring. For example, Krull defined the dimension of any commutative ring in terms of prime ideals. At least when the ring is Noetherian, he proved many of the properties one would want from the geometric notion of dimension. Noether and Krull's commutative algebra can be viewed as an algebraic approach to affine algebraic varieties.
|
|
Amicable numbers (m, n) satisfy \sigma(m)-m=n and \sigma(n)-n=m which can be written together as \sigma(m)=\sigma(n)=m+n. This can be generalized to larger tuples, say (n_1,n_2,\ldots,n_k), where we require :\sigma(n_1)=\sigma(n_2)= \dots =\sigma(n_k) = n_1+n_2+ \dots +n_k For example, (1980, 2016, 2556) is an amicable triple , and (3270960, 3361680, 3461040, 3834000) is an amicable quadruple . Amicable multisets are defined analogously and generalizes this a bit further .
|
|
In mathematics, the Segre class is a characteristic class used in the study of cones, a generalization of vector bundles. For vector bundles the total Segre class is inverse to the total Chern class, and thus provides equivalent information; the advantage of the Segre class is that it generalizes to more general cones, while the Chern class does not. The Segre class was introduced in the non-singular case by . In the modern treatment of intersection theory in algebraic geometry, as developed e.g.
|
|
The American polyconic projection can be thought of as "rolling" a cone tangent to the Earth at all parallels of latitude. This generalizes the concept of a conic projection, which uses a single cone to project the globe onto. By using this continuously varying cone, each parallel becomes a circular arc having true scale, contrasting with a conic projection, which can only have one or two parallels at true scale. The scale is also true on the central meridian of the projection.
|
|
One of the interesting properties of the Fourier transform which we have mentioned, is that it carries convolutions to pointwise products. If that is the property which we seek to preserve, one can produce Fourier series on any compact group. Typical examples include those classical groups that are compact. This generalizes the Fourier transform to all spaces of the form L2(G), where G is a compact group, in such a way that the Fourier transform carries convolutions to pointwise products.
|
|
Thus if the number of sides n is odd, a tangential polygon is equilateral if and only if it is regular.. Viviani's theorem generalizes to equilateral polygons:. The sum of the perpendicular distances from an interior point to the sides of an equilateral polygon is independent of the location of the interior point. The principal diagonals of a hexagon each divide the hexagon into quadrilaterals. In any convex equilateral hexagon with common side a, there existsInequalities proposed in “Crux Mathematicorum”, .
|
|
There is an operad for which each P(n) is given by the symmetric group S_n. The composite \sigma \circ (\tau_1, \dots, \tau_n) permutes its inputs in blocks according to \sigma, and within blocks according to the appropriate \tau_i. Similarly, there is a non-\Sigma operad for which each P(n) is given by the Artin braid group B_n. Moreover, this non-\Sigma operad has the structure of a braided operad, which generalizes the notion of an operad from symmetric to braid groups.
|
|
Degenerate conics follow by continuity (the theorem is true for non-degenerate conics, and thus holds in the limit of degenerate conic). A short elementary proof of Pascal's theorem in the case of a circle was found by , based on the proof in . This proof proves the theorem for circle and then generalizes it to conics. A short elementary computational proof in the case of the real projective plane was found by We can infer the proof from existence of isogonal conjugate too.
|
|
In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational force, is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic. In general relativity, gravity can be regarded as not a force but a consequence of a curved spacetime geometry where the source of curvature is the stress–energy tensor (representing matter, for instance).
|
|
General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. It is the basis of current cosmological models of the universe. General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present.
|
|
The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology. Certain special classes of manifolds also have additional algebraic structure; they may behave like groups, for instance. In that case, they are called Lie Groups. Alternatively, they may be described by polynomial equations, in which case they are called algebraic varieties, and if they additionally carry a group structure, they are called algebraic groups.
|
|
The best vertex degree characterization of Hamiltonian graphs was provided in 1972 by the Bondy–Chvátal theorem, which generalizes earlier results by G. A. Dirac (1952) and Øystein Ore. Both Dirac's and Ore's theorems can also be derived from Pósa's theorem (1962). Hamiltonicity has been widely studied with relation to various parameters such as graph density, toughness, forbidden subgraphs and distance among other parameters. Dirac and Ore's theorems basically state that a graph is Hamiltonian if it has enough edges.
|
|
In probability and statistics, the generalized beta distributionMcDonald, James B. & Xu, Yexiao J. (1995) "A generalization of the beta distribution with applications," Journal of Econometrics, 66(1–2), 133–152 is a continuous probability distribution with five parameters, including more than thirty named distributions as limiting or special cases. It has been used in the modeling of income distribution, stock returns, as well as in regression analysis. The exponential generalized beta (EGB) distribution follows directly from the GB and generalizes other common distributions.
|
|
In computer science, a B-tree is a self-balancing tree data structure that maintains sorted data and allows searches, sequential access, insertions, and deletions in logarithmic time. The B-tree generalizes the binary search tree, allowing for nodes with more than two children. Unlike other self-balancing binary search trees, the B-tree is well suited for storage systems that read and write relatively large blocks of data, such as disks. It is commonly used in databases and file systems.
|
|
By Tychonoff's theorem this topological space is compact. For each finite subgraph of , let be the subset of consisting of assignments of colors that validly color . Then the system of sets is a family of closed sets with the finite intersection property, so by compactness it has a nonempty intersection. Every member of this intersection is a valid coloring of .. Gottschalk states his proof more generally as a proof of the theorem of that generalizes the De Bruijn–Erdős theorem.
|
|
Some of the basic concepts of general relativity can be outlined outside the relativistic domain. In particular, the idea that mass/energy generates curvature in space and that curvature affects the motion of masses can be illustrated in a Newtonian setting. General relativity generalizes the geodesic equation and the field equation to the relativistic realm in which trajectories in space are replaced with Fermi–Walker transport along world lines in spacetime. The equations are also generalized to more complicated curvatures.
|
|
Since its definition in 2004, the Decision Linear assumption has seen a variety of other applications. These include the construction of a pseudorandom function that generalizes the Naor-Reingold construction, Allison Bishop Lewko, Brent Waters: Efficient pseudorandom functions from the decisional linear assumption and weaker variants. CCS 2009: 112-120 an attribute-based encryption scheme, Lucas Kowalczyk, Allison Bishop Lewko: Bilinear Entropy Expansion from the Decisional Linear Assumption. CRYPTO 2015: 524-541 and a special class of non-interactive zero-knowledge proofs.
|
|
Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under elementary group theory. For example, repeated applications of the associativity axiom show that the unambiguity of :a ⋅ b ⋅ c = (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) generalizes to more than three factors. Because this implies that parentheses can be inserted anywhere within such a series of terms, parentheses are usually omitted. The axioms may be weakened to assert only the existence of a left identity and left inverses.
|
|
A classic result in semigroup theory due to D. B. McAlister states that every inverse semigroup has an E-unitary cover; besides being surjective, the homomorphism in this case is also idempotent separating, meaning that in its kernel an idempotent and non-idempotent never belong to the same equivalence class.; something slightly stronger has actually be shown for inverse semigroups: every inverse semigroup admits an F-inverse cover.Lawson p. 230 McAlister's covering theorem generalizes to orthodox semigroups: every orthodox semigroup has a unitary cover.
|
|
A continuous game is a mathematical concept, used in game theory, that generalizes the idea of an ordinary game like tic-tac-toe (noughts and crosses) or checkers (draughts). In other words, it extends the notion of a discrete game, where the players choose from a finite set of pure strategies. The continuous game concepts allows games to include more general sets of pure strategies, which may be uncountably infinite. In general, a game with uncountably infinite strategy sets will not necessarily have a Nash equilibrium solution.
|
|
In theoretical physics, S-duality (short for strong–weak duality) is an equivalence of two physical theories, which may be either quantum field theories or string theories. S-duality is useful for doing calculations in theoretical physics because it relates a theory in which calculations are difficult to a theory in which they are easier.Frenkel 2009, p.2 In quantum field theory, S-duality generalizes a well established fact from classical electrodynamics, namely the invariance of Maxwell's equations under the interchange of electric and magnetic fields.
|
|
In "Zero Tolerance," Jen gets a phone call from who is known to be her husband Olivier, who Jen wants a divorce from. In the same episode, Jen confronts Kelly about her sending a recommendation letter in which she characterized Jen as a narcissist with no moral compass. Kelly then warns Ryan about Jen's devious personality by telling him that Jen is a "compulsive liar, and practically a sociopath." Ryan however pays no attention to this, and generalizes the statement as jealousy from Kelly's part.
|
|
Occamism questions the physical and Aristotelian metaphysics and, in particular, insists the only reality of individuals accessible to knowledge intuitive. The universals, which exist only in the mind, have no correspondence with reality and are mere signs that symbolize a multiplicity of individuals. The further one goes from experience and generalizes, the more one imagines the constitution of the universal expressed by names. It is therefore necessary to revise the logical structures of discourse and language, taking care to separate the sign from the signified thing.
|
|
In his work with his coauthors Navin Aswal and Shurojit Chatterji, he provides a comprehensive description of environments where GS theorem holds. In his works and with coauthors Shurojit Chatterji, Huaxia Zeng, and Remzi Sanver, he identifies environments where GS theorem does not hold, i.e., well- behaved voting rules exist. In his work with coauthors Shurojit Chatterji and Huaxia Zeng, he has identified environments where the GS theorem type result continues to hold even if the voting rule allows for randomization (which generalizes Gibbard's theorem).
|
|
The argument above involving the unit circle generalizes to show that no branch of log z exists on an open set U containing a closed curve that winds around 0. To foil this argument, U is typically chosen as the complement of a ray or curve in the complex plane going from 0 (inclusive) to infinity in some direction. In this case, the curve is known as a branch cut. For example, the principal branch has a branch cut along the negative real axis.
|
|
This argument generalizes. The compact dimension imposes specific boundary conditions on all fields, for example periodic boundary conditions in the case of a periodic dimension, and typically Neumann or Dirichlet boundary conditions in other cases. Now suppose the size of the compact dimension is L; then the possible eigenvalues under gradient along this dimension are integer or half-integer multiples of 1/L (depending on the precise boundary conditions). In quantum mechanics this eigenvalue is the momentum of the field, and is therefore related to its energy.
|
|
Multi-objective parametric query optimization generalizes parametric and multi-objective query optimization. Plans are compared according to multiple cost metrics and plan costs may depend on parameters whose values are unknown at optimization time. The cost of a query plan is therefore modeled as a function from a multi-dimensional parameter space to a multi-dimensional cost space. The goal of optimization is to generate the set of query plans that can be optimal for each possible combination of parameter values and user preferences.
|
|
In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field K, a K-derivation is a K-linear map that satisfies Leibniz's law: : D(ab) = a D(b) + D(a) b. More generally, if M is an A-bimodule, a K-linear map that satisfies the Leibniz law is also called a derivation. The collection of all K-derivations of A to itself is denoted by DerK(A).
|
|
It is mathematically oriented (e.g. rotation of vectors by complex multiplication), and uses the simplex method and deferred drawing to solve overall size constraint issues between fixed-sized objects (labels and arrowheads) and objects that should scale with figure size. Asymptote fully generalizes MetaPost path construction algorithms to three dimensions,The 3D Asymptote Generalization of MetaPost Bézier Interpolation, J. C. Bowman, Proceedings in Applied Mathematics and Mechanics, 7:1, 2010021-2010022 (2007). and compiles commands into virtual machine code for speed without sacrificing portability.
|
|
A triangulated torus Another triangulation of the torus A triangulated dolphin shape In mathematics, topology generalizes the notion of triangulation in a natural way as follows: :A triangulation of a topological space X is a simplicial complex K, homeomorphic to X, together with a homeomorphism h: K → X. Triangulation is useful in determining the properties of a topological space. For example, one can compute homology and cohomology groups of a triangulated space using simplicial homology and cohomology theories instead of more complicated homology and cohomology theories.
|
|
The joint quantum entropy generalizes the classical joint entropy to the context of quantum information theory. Intuitively, given two quantum states \rho and \sigma, represented as density operators that are subparts of a quantum system, the joint quantum entropy is a measure of the total uncertainty or entropy of the joint system. It is written S(\rho,\sigma) or H(\rho,\sigma), depending on the notation being used for the von Neumann entropy. Like other entropies, the joint quantum entropy is measured in bits, i.e.
|
|
The noncentral t-distribution generalizes Student's t-distribution using a noncentrality parameter. Whereas the central probability distribution describes how a test statistic t is distributed when the difference tested is null, the noncentral distribution describes how t is distributed when the null is false. This leads to its use in statistics, especially calculating statistical power. The noncentral t-distribution is also known as the singly noncentral t-distribution, and in addition to its primary use in statistical inference, is also used in robust modeling for data.
|
|
The Hasse diagram of a partially ordered set (left) and its Dedekind–MacNeille completion (right). In mathematics, specifically order theory, the Dedekind–MacNeille completion of a partially ordered set (also called the completion by cuts or normal completion); . is the smallest complete lattice that contains it. It is named after Holbrook Mann MacNeille whose 1937 paper first defined and constructed it, and after Richard Dedekind because its construction generalizes the Dedekind cuts used by Dedekind to construct the real numbers from the rational numbers.
|
|
Girls who are displeased with their parents' selection may violently protest against the marriage by kicking and screaming and running away at the end of the ceremony. After she has run away, this may result in the dissolution of the arrangement. Half of all first-time marriages end in divorce, but because it is common, the divorce process is not long. Anthropologist Marjorie Shostak generalizes that, "Everyone in the village expresses a point of view" on the marriage and if the couple should be divorced or not.
|
|
The Feynman path integral in Euclidean space generalizes this to other problems of interest to statistical mechanics. Any probability distribution which obeys a condition on correlation functions called reflection positivity leads to a local quantum field theory after Wick rotation to Minkowski spacetime ( see Osterwalder- Schrader axioms ). The operation of renormalization is a specified set of mappings from the space of probability distributions to itself. A quantum field theory is called renormalizable if this mapping has a fixed point which gives a quantum field theory.
|
|
In algebra, a hypoalgebra is a generalization of a subalgebra of a Lie algebra introduced by . The relation between an algebra and a hypoalgebra is called a subjoining , which generalizes the notion of an inclusion of subalgebras. There is also a notion of restriction of a representation of a Lie algebra to a subjoined hypoalgebra, with branching rules similar to those for restriction to subalgebras except that some of the multiplicities in the branching rule may be negative. calculated many of these branching rules for hypoalgebras.
|
|
The notion of one-to-one correspondence generalizes to partial functions, where they are called partial bijections, although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partial function is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverse to be a total function, i.e. defined everywhere on its domain. The set of all partial bijections on a given base set is called the symmetric inverse semigroup.
|
|
In mathematics, Katugampola fractional operators are integral operators that generalize the Riemann–Liouville and the Hadamard fractional operators into a unique form.Katugampola, Udita N. (2011). On Generalized Fractional Integrals and Derivatives, Ph.D. Dissertation, Southern Illinois University, Carbondale, August, 2011. The Katugampola fractional integral generalizes both the Riemann–Liouville fractional integral and the Hadamard fractional integral into a single form and It is also closely related to the Erdelyi–Kober Fractional Integrals and Derivatives: Theory and Applications, by Samko, S.; Kilbas, A.A.; and Marichev, O. Hardcover: 1006 pages.
|
|
In cryptography, the Generalized DES Scheme (GDES or G-DES) is a variant of the DES symmetric-key block cipher designed with the intention of speeding up the encryption process while improving its security. The scheme was proposed by Ingrid Schaumuller-Bichl in 1981. In 1990, Eli Biham and Adi Shamir showed that GDES was vulnerable to differential cryptanalysis, and that any GDES variant faster than DES is also less secure than DES. GDES generalizes the Feistel network structure of DES to larger block sizes.
|
|
In algebraic geometry, a formal holomorphic function along a subvariety V of an algebraic variety W is an algebraic analog of a holomorphic function defined in a neighborhood of V. They are sometimes just called holomorphic functions when no confusion can arise. They were introduced by . The theory of formal holomorphic functions has largely been replaced by the theory of formal schemes which generalizes it: a formal holomorphic function on a variety is essentially just a section of the structure sheaf of a related formal scheme.
|
|
The consistent histories interpretation generalizes the conventional Copenhagen interpretation and attempts to provide a natural interpretation of quantum cosmology. The theory is based on a consistency criterion that allows the history of a system to be described so that the probabilities for each history obey the additive rules of classical probability. It is claimed to be consistent with the Schrödinger equation. According to this interpretation, the purpose of a quantum-mechanical theory is to predict the relative probabilities of various alternative histories (for example, of a particle).
|
|
Open strings attached to a pair of D-branes In string theory and related theories in physics, a brane is a physical object that generalizes the notion of a point particle to higher dimensions. For example, a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. It is also possible to consider higher-dimensional branes. The word brane comes from the word "membrane" which refers to a two-dimensional brane.
|
|
If is a subset of a Euclidean space, then is an interior point of if there exists an open ball centered at which is completely contained in . (This is illustrated in the introductory section to this article.) This definition generalizes to any subset of a metric space with metric : is an interior point of if there exists , such that is in whenever the distance . This definition generalises to topological spaces by replacing "open ball" with "open set". Let be a subset of a topological space .
|
|
Intense parts of a story were also accompanied by increased brain activity in a network of regions known to be involved in the processing of fear, including the amygdala. Because it rests on psychological principles, a reader-response approach readily generalizes to other arts: cinema (David Bordwell), music, or visual art (E. H. Gombrich), and even to history (Hayden White). In stressing the activity of the reader, reader- response theory may be employed to justify upsettings of traditional interpretations like deconstruction or cultural criticism.
|
|
In probability theory, a branching random walk is a stochastic process that generalizes both the concept of a random walk and of a branching process. At every generation (a point of discrete time), a branching random walk's value is a set of elements that are located in some linear space, such as the real line. Each element of a given generation can have several descendants in the next generation. The location of any descendant is the sum of its parent's location and a random variable.
|
|
Related statistics such as Yule's Y and Yule's Q normalize this to the correlation-like range . The odds ratio is generalized by the logistic model to model cases where the dependent variables are discrete and there may be one or more independent variables. The correlation ratio, entropy-based mutual information, total correlation, dual total correlation and polychoric correlation are all also capable of detecting more general dependencies, as is consideration of the copula between them, while the coefficient of determination generalizes the correlation coefficient to multiple regression.
|
|
These and other examples of the general idea of jet bundles play a significant role in the study of differential operators on manifolds. The notion of a frame also generalizes to the case of higher-order jets. Define a k-th order frame to be the k-jet of a diffeomorphism from Rn to M.See S. Kobayashi (1972). The collection of all k-th order frames, Fk(M), is a principal Gk bundle over M, where Gk is the group of k-jets; i.e.
|
|
One commenter stated that men dressed in clone style usually possessed a more self-assured attitude about themselves and their sexual orientation. Men could take parts of the appearance that they found attractive and that worked for them. For many men, the look was an outward sign of their freedom from social dicta and a celebration of their personal masculinity. Some fetishize the style while others find the appearance a sign of liberation, countering the homophobic stereotype that generalizes all gay men as effeminate.
|
|
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits. Limits and colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize.
|
|
That is, the inertial frame is the one where the fictitious forces vanish. So much for fictitious forces due to rotation. However, for linear acceleration, Newton expressed the idea of undetectability of straight-line accelerations held in common: This principle generalizes the notion of an inertial frame. For example, an observer confined in a free- falling lift will assert that he himself is a valid inertial frame, even if he is accelerating under gravity, so long as he has no knowledge about anything outside the lift.
|
|
If each edge has a distinct weight then there will be only one, unique minimum spanning tree. This is true in many realistic situations, such as the telecommunications company example above, where it's unlikely any two paths have exactly the same cost. This generalizes to spanning forests as well. Proof: # Assume the contrary, that there are two different MSTs A and B. # Since A and B differ despite containing the same nodes, there is at least one edge that belongs to one but not the other.
|
|
Many of the preceding results remain valid when the field of definition of E is a number field K, that is to say, a finite field extension of Q. In particular, the group E(K) of K-rational points of an elliptic curve E defined over K is finitely generated, which generalizes the Mordell–Weil theorem above. A theorem due to Loïc Merel shows that for a given integer d, there are (up to isomorphism) only finitely many groups that can occur as the torsion groups of E(K) for an elliptic curve defined over a number field K of degree d. More precisely, there is a number B(d) such that for any elliptic curve E defined over a number field K of degree d, any torsion point of E(K) is of order less than B(d). The theorem is effective: for d > 1, if a torsion point is of order p, with p prime, then :p < d^{3d^2} As for the integral points, Siegel's theorem generalizes to the following: Let E be an elliptic curve defined over a number field K, x and y the Weierstrass coordinates.
|
|
To allow for spontaneous processes at constant T and V, one needs to enlarge the thermodynamical state space of the system. In case of a chemical reaction, one must allow for changes in the numbers Nj of particles of each type j. The differential of the free energy then generalizes to :dF = -S\,dT - P\,dV + \sum_j \mu_j\,dN_j, where the N_j are the numbers of particles of type j, and the \mu_j are the corresponding chemical potentials. This equation is then again valid for both reversible and non-reversible uPT changes.
|
|
In group theory, two subgroups Γ1 and Γ2 of a group G are said to be commensurable if the intersection Γ1 ∩ Γ2 is of finite index in both Γ1 and Γ2. Example: Let a and b be nonzero real numbers. Then the subgroup of the real numbers R generated by a is commensurable with the subgroup generated by b if and only if the real numbers a and b are commensurable, in the sense that a/b is rational. Thus the group-theoretic notion of commensurability generalizes the concept for real numbers.
|
|
The coloured de Bruijn graphs can be used to genotype any DNA sample at a known loci, even when the coverage is less than sufficient for variant assembly. The first step to this process is to construct a graph of the reference allele, known variants and data from the sample. The algorithm then calculates the likelihood of each genotype and accounts for the structure of the graph, both of the local and genome-wide sequence. This then generalizes to multiple allelic types and helps genotype complex and compound variants.
|
|
World line of a circular orbit about the Earth depicted in two spatial dimensions X and Y (the plane of the orbit) and a time dimension, usually put as the vertical axis. Note that the orbit about the Earth is (almost) a circle in space, but its worldline is a helix in spacetime. General relativity generalizes the geodesic equation and the field equation to the relativistic realm in which trajectories in space are replaced with world lines in spacetime. The equations are also generalized to more complicated curvatures.
|
|
Torelli's theorem states that a complex curve is determined by its Jacobian (with its polarization). The Schottky problem asks which principally polarized abelian varieties are the Jacobians of curves. The Picard variety, the Albanese variety, generalized Jacobian, and intermediate Jacobians are generalizations of the Jacobian for higher-dimensional varieties. For varieties of higher dimension the construction of the Jacobian variety as a quotient of the space of holomorphic 1-forms generalizes to give the Albanese variety, but in general this need not be isomorphic to the Picard variety.
|
|
He proved that every finite tournament contains an odd number of Hamiltonian paths. He gave several proofs of the theorem on quadratic reciprocity. He proved important results concerning the invariants of the class groups of quadratic number fields. Iyanaga's pamphlet discusses and generalizes one of Rédei's theorems; it gives a "necessary and sufficient condition for the existence of an ideal class (in the restricted sense) of order 4 in a quadratic field k() ..." In several cases, he determined if the ring of integers of the real quadratic field Q() is Euclidean or not.
|
|
Two ideals A and B in the commutative ring R are called coprime (or comaximal) if A + B = R. This generalizes Bézout's identity: with this definition, two principal ideals (a) and (b) in the ring of integers Z are coprime if and only if a and b are coprime. If the ideals A and B of R are coprime, then AB = A∩B; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem can be generalized to any commutative ring, using coprime ideals.
|
|
Another realization of S-duality in quantum field theory is Seiberg duality, first introduced by Nathan Seiberg around 1995.Seiberg 1995 Unlike Montonen–Olive duality, which relates two versions of the maximally supersymmetric gauge theory in four-dimensional spacetime, Seiberg duality relates less symmetric theories called N=1 supersymmetric gauge theories. The two N=1 theories appearing in Seiberg duality are not identical, but they give rise to the same physics at large distances. Like Montonen–Olive duality, Seiberg duality generalizes the symmetry of Maxwell's equations that interchanges electric and magnetic fields.
|
|
A semantic reasoner, reasoning engine, rules engine, or simply a reasoner, is a piece of software able to infer logical consequences from a set of asserted facts or axioms. The notion of a semantic reasoner generalizes that of an inference engine, by providing a richer set of mechanisms to work with. The inference rules are commonly specified by means of an ontology language, and often a description logic language. Many reasoners use first-order predicate logic to perform reasoning; inference commonly proceeds by forward chaining and backward chaining.
|
|
CoBoosting was an attempt by Collins and Singer to improve on previous attempts to leverage redundancy in features for training classifiers in a semi-supervised fashion. CoTraining, a seminal work by Blum and Mitchell, was shown to be a powerful framework for learning classifiers given a small number of seed examples by iteratively inducing rules in a decision list. The advantage of CoBoosting to CoTraining is that it generalizes the CoTraining pattern so that it could be used with any classifier. CoBoosting accomplishes this feat by borrowing concepts from AdaBoost.
|
|
The rotation group generalizes quite naturally to n-dimensional Euclidean space, \R^n with its standard Euclidean structure. The group of all proper and improper rotations in n dimensions is called the orthogonal group O(n), and the subgroup of proper rotations is called the special orthogonal group SO(n), which is a Lie group of dimension . In special relativity, one works in a 4-dimensional vector space, known as Minkowski space rather than 3-dimensional Euclidean space. Unlike Euclidean space, Minkowski space has an inner product with an indefinite signature.
|
|
In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety by a group: a quotient variety, say, would be a coarse approximation of a quotient stack. The notion is of fundamental importance in the study of stacks: a stack that arises in nature is often either a quotient stack itself or admits a stratification by quotient stacks (e.g., a Deligne–Mumford stack.) A quotient stack is also used to construct other stacks like classifying stacks.
|
|
Paul Montel first coined the term "normal family" in 1911.P. Montel, C. R. Acad. Sci. Paris 153 (1911), 996–998; Jahrbuch 42, page 426 Because the concept of a normal family has continually been very important to complex analysis, Montel's terminology is still used to this day, even though from a modern perspective, the phrase pre-compact subset might be preferred by some mathematicians. Note that though the notion of compact open topology generalizes and clarifies the concept, in many applications the original definition is more practical.
|
|
The state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space. The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions.
|
|
The Radon point of three points in a one-dimensional space is just their median. The geometric median of a set of points is the point minimizing the sum of distances to the points in the set; it generalizes the one-dimensional median and has been studied both from the point of view of facility location and robust statistics. For sets of four points in the plane, the geometric median coincides with the Radon point. Another generalization for partition into r sets was given by and is now known as Tverberg's theorem.
|
|
Asymptotic phenomena for the systole of surfaces of large genus have been shown to be related to interesting ergodic phenomena, and to properties of congruence subgroups of arithmetic groups. Gromov's 1983 inequality for the homotopy systole implies, in particular, a uniform lower bound for the area of an aspherical surface in terms of its systole. Such a bound generalizes the inequalities of Loewner and Pu, albeit in a non-optimal fashion. Gromov's seminal 1983 paper also contains asymptotic bounds relating the systole and the area, which improve the uniform bound (valid in all dimensions).
|
|
In category theory, a branch of mathematics, compact closed categories are a general context for treating dual objects. The idea of a dual object generalizes the more familiar concept of the dual of a finite-dimensional vector space. So, the motivating example of a compact closed category is FdVect, the category having finite-dimensional vector spaces as objects and linear maps as morphisms, with tensor product as the monoidal structure. Another example is Rel, the category having sets as objects and relations as morphisms, with Cartesian monoidal structure.
|
|
The dependent Dirichlet process (DDP) originally formulated by MacEachern led to the development of the DDP mixture model (DDPMM) which generalizes DPMM by including birth, death and transition processes for the clusters in the model. In addition, a low- variance approximations to DDPMM have been derived leading to a dynamic clustering algorithm T. Campbell, M. Liu, B. Kulis, J. P. How, and L. Carin, Dynamic clustering via asymptotics of the Dependent Dirichlet Process., Neural Information Processing Systems (NIPS), 2013. . Under time-varying setting, it is natural to introduce different DP priors for different time steps.
|
|
This assertion, which naturally generalizes the uniformization of Riemann surfaces to arbitrary dimensions, is completely correct, as is the broad outline of Yamabe's proof. However, Yamabe's argument contains a subtle analytic mistake arising form the failure of certain natural inclusions of Sobolev spaces to be compact. This mistake was only corrected in stages, on a case-by-case basis, first by Trudinger ("Remarks Concerning the Conformal Deformation of Metrics to Constant Scalar Curvature", Ann. Scuola Norm. Sup. Pisa 22 (1968) 265–274), then by Aubin (Équations Différentielles Non Linéaires et Problème de Yamabe, J. Math.
|
|
In algebra, the S-construction is a construction in algebraic K-theory that produces a model that can be used to define higher K-groups. It is due to Friedhelm Waldhausen and concerns a category with cofibrations and weak equivalences; such a category is called a Waldhausen category and generalizes Quillen's exact category. A cofibration can be thought of as analogous to a monomorphism, and a category with cofibrations is one in which, roughly speaking, monomorphisms are stable under pushouts. According to Waldhausen, the "S" was chosen to stand for Graeme B. Segal.
|
|
Einstein's equivalence principle generalizes this analogy, stating that an accelerating reference frame is locally indistinguishable from an inertial reference frame with a gravity force acting upon it. In this way, the Gravity Probe A was a test of the equivalence principle, matching the observations in the inertial reference frame (of special relativity) of the Earth's surface affected by gravity, with the predictions of special relativity for the same frame treated as being accelerating upwards with respect to free fall reference, which can thought of being inertial and gravity-less.
|
|
Graphlets were first introduced by Nataša Pržulj, when they were used as a basis for designing two new highly sensitive measures of network local structural similarities: the relative graphlet frequency distance (RGF-distance) and the graphlet degree distribution agreement (GDD-agreement). Additionally, Pržulj group developed a novel measure of network topological similarity that generalizes the degree of a node in the network to its graphlet degree vector (GDV) or graphlet degree signature.Tijana Milenković and Nataša Pržulj, Uncovering Biological Network Function via Graphlet Degree Signatures, Cancer Informatics 2008, 6:257–273.
|
|
In a series of published articles from 1974–1979, and then in his 1988 book Mind Children, computer scientist and futurist Hans Moravec generalizes Moore's law to make predictions about the future of artificial life. Moore's law describes an exponential growth pattern in the complexity of integrated semiconductor circuits. Moravec extends this to include technologies from long before the integrated circuit to future forms of technology. Moravec outlines a timeline and a scenario in which robots will evolve into a new series of artificial species, starting around 2030–2040.
|
|
In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for tangency. One needs a definition of intersection number in order to state results like Bézout's theorem. The intersection number is obvious in certain cases, such as the intersection of x- and y-axes which should be one. The complexity enters when calculating intersections at points of tangency and intersections along positive dimensional sets.
|
|
In mathematics, especially order theory, a prefix ordered set generalizes the intuitive concept of a tree by introducing the possibility of continuous progress and continuous branching. Natural prefix orders often occur when considering dynamical systems as a set of functions from time (a totally- ordered set) to some phase space. In this case, the elements of the set are usually referred to as executions of the system. The name prefix order stems from the prefix order on words, which is a special kind of substring relation and, because of its discrete character, a tree.
|
|
The activity of the Royal Chamber Orchestra (founded by D. João V), which had been in the previous century one of the most important chamber orchestras in Europe, declines irreversibly. However, in the turn of the 19th century, generalizes the tradition of amateur academies performing the contemporary instrumental music. The generalization of public concerts is due to João Domingos Bomtempo (1775–1842), the most prominent musical figure of the first half of the 19th century. Bomtempo, son of an Italian musician of the court Orchestra, studied with the Patriarchal masters.
|
|
When specifically dealing with network graphs, often graphs are without loops or multiple edges to maintain simple relationships (where edges represent connections between two people or vertices). In this case, using Brandes' algorithm will divide final centrality scores by 2 to account for each shortest path being counted twice. Another algorithm generalizes the Freeman's betweenness computed on geodesics and Newman's betweenness computed on all paths, by introducing a hyper-parameter controlling the trade-off between exploration and exploitation. The time complexity is the number of edges times the number of nodes in the graph.
|
|
Dorwin Cartwright & Frank Harary (1979) "Balance and clusterability: An overview", pages 25 to 50 in Perspectives in Social Network Research, editors: Paul W. Holland & Samuel Leinhardt, Academic Press The theorem was published by Harary in 1953. It generalizes the theorem that an ordinary (unsigned) graph is bipartite if and only if every cycle has even length. A simple proof uses the method of switching. To prove Harary's theorem, one shows by induction that Σ can be switched to be all positive if and only if it is balanced.
|
|
In mathematics, a colored matroid is a matroid whose elements are labeled from a set of colors, which can be any set that suits the purpose, for instance the set of the first n positive integers, or the sign set {+, −}. The interest in colored matroids is through their invariants, especially the colored Tutte polynomial,. which generalizes the Tutte polynomial of a signed graph of .. There has also been study of optimization problems on matroids where the objective function of the optimization depends on the set of colors chosen as part of a matroid basis..
|
|
Despite the great potential complexity and diversity of biological networks, all first-order network behavior generalizes to one of four possible input-output motifs: hyperbolic or Michaelis–Menten, ultra-sensitive, bistable, and bistable irreversible (a bistability where negative and therefore biologically impossible input is needed to return from a state of high output). Examples of each in biological contexts can be found on their respective pages. Ultrasensitive, bistable, and irreversibly bistable networks all show qualitative change in network behavior around certain parameter values – these are their bifurcation points.
|
|
In computer science, in particular in formal language theory, the pumping lemma for context-free languages, also known as the Bar-Hillel lemma, is a lemma that gives a property shared by all context-free languages and generalizes the pumping lemma for regular languages. The pumping lemma can be used to construct a proof by contradiction that a specific language is not context-free. Conversely, the pumping lemma does not suffice to guarantee that a language is context-free; there are other necessary conditions, such as Ogden's lemma, or the Interchange lemma.
|
|
In mathematics, the Ax–Grothendieck theorem is a result about injectivity and surjectivity of polynomials that was proved independently by James Ax and Alexander Grothendieck... The theorem is often given as this special case: If P is an injective polynomial function from an n-dimensional complex vector space to itself then P is bijective. That is, if P always maps distinct arguments to distinct values, then the values of P cover all of Cn. The full theorem generalizes to any algebraic variety over an algebraically closed field.Éléments de géométrie algébrique, IV3, Proposition 10.4.11.
|
|
Given a convex shape (light blue) and its set of extreme points (red), the convex hull of is . In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces (TVSs). This theorem generalizes to infinite- dimensional spaces and to arbitrary compact convex sets the following basic observation: a convex (i.e. "filled") triangle, including its perimeter and the area "inside of it", is equal to the convex hull of its three vertices, where these vertices are exactly the extreme points of this shape.
|
|
For instance, the subspace theorem proved by demonstrates that points of small height (i.e. small complexity) in projective space lie in a finite number of hyperplanes and generalizes Siegel's theorem on integral points and solution of the S-unit equation. Height functions were crucial to the proofs of the Mordell–Weil theorem and Faltings's theorem by and respectively. Several outstanding unsolved problems about the heights of rational points on algebraic varieties, such as the Manin conjecture and Vojta's conjecture, have far-reaching implications for problems in Diophantine approximation, Diophantine equations, arithmetic geometry, and mathematical logic.
|
|
The Jones model is a growth model developed in 1995 by economist Charles I. Jones. The model is essentially identical to the Romer model (1990), in particular it generalizes or modifies the description of how new technologies, ideas or design instructions arise. This should take into account the criticism made of the Romer model that the long-term growth rate depends positively on the size of the population (economies of scale). This is problematic in several respects: on the one hand larger countries do not necessarily grow faster.
|
|
The filter algorithm generalizes Peterson's algorithm to processes. Instead of a Boolean flag, it requires an integer variable per process, stored in a single writer/multiple reader (SWMR) atomic register, and additional variables in similar registers. The registers can be represented in pseudocode as arrays: level : array of N integers last_to_enter : array of N−1 integers The variables take on values up to , each representing a distinct "waiting room" before the critical section. Processes advance from one room to the next, finishing in room which is the critical section.
|
|
This generalizes the construction of a line graph, in which every edge of the multigraph is replaced by a vertex. Fuzzy linear interval graphs are constructed in the same way as fuzzy circular interval graphs, but on a line rather than on a circle. Chudnovsky and Seymour classify arbitrary connected claw-free graphs into one of the following: # Six specific subclasses of claw-free graphs. Three of these are line graphs, proper circular arc graphs, and the induced subgraphs of an icosahedron; the other three involve additional definitions.
|
|
Children have been shown to be sensitive to socio- economic cues and differences. Children also look at the informant's stance relative to a perceived group, whether that be the group involved in play or a larger social group that the child identifies with, and will show a bias towards information coming from a less conflicting stance with the majority. Overall children, while biased to trust adults, still apply rational judgements to new information introduced during pretend play that influences their tendency to believe how much that information generalizes to reality.
|
|
The LCF approach provides similar trustworthiness to systems that generate explicit proof certificates but without the need to store proof objects in memory. Theorem data type can be easily implemented to optionally store proof objects, depending on system's run-time configuration, so it generalizes the basic proof-generation approach. The design decision to use a general-purpose programming language for developing theorems means that, depending on the complexity of programs written, it is possible to use the same language to write step-by-step proofs, decision procedures, or theorem provers.
|
|
Many positive results were proven by Peter Linnell. For example, if the group acting is a free group, then the l^2-Betti numbers are integers. The most general question open as of late 2011 is whether l^2-Betti numbers are rational if there is a bound on the orders of finite subgroups of the group which acts. In fact, a precise relationship between possible denominators and the orders in question is conjectured; in the case of torsion-free groups, this statement generalizes the zero-divisors conjecture.
|
|
This file contains information on every individual within every family within every household selected for participation in the NHIS. Survey data about individuals who were either not available at the time of the interview or under 18 were provided by an available adult in the household. This person- level file contains information on health status and limitation of activity, health care access and utilization, health insurance, socio-demographics, and income and assets. Examining this file using the final weight variable (WTFA) generalizes to the entire noninstitutional, civilian United States population.
|
|
A matroid is a structure that captures and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid, the most significant being in terms of independent sets, bases, circuits, closed sets or flats, closure operators, and rank functions. Matroid theory borrows extensively from the terminology of linear algebra and graph theory, largely because it is the abstraction of various notions of central importance in these fields. Matroids have found applications in geometry, topology, combinatorial optimization, network theory and coding theory.
|
|
Thus CO (with its variants) is the only general technique that does not require the typically costly distribution of local concurrency control information (e.g., local precedence relations, locks, timestamps, or tickets). It generalizes the popular strong strict two-phase locking (SS2PL) property, which in conjunction with the two-phase commit protocol (2PC) is the de facto standard to achieve global serializability across (SS2PL based) database systems. As a result, CO compliant database systems (with any, different concurrency control types) can transparently join such SS2PL based solutions for global serializability.
|
|
In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant of a finite dimensional linear operator. It is defined for bounded operators on a Hilbert space which differ from the identity operator by a trace-class operator. The function is named after the mathematician Erik Ivar Fredholm. Fredholm determinants have had many applications in mathematical physics, the most celebrated example being Gábor Szegő's limit formula, proved in response to a question raised by Lars Onsager and C. N. Yang on the spontaneous magnetization of the Ising model.
|
|
The Borel–Tanner distribution generalizes the Borel distribution. Let k be a positive integer. If X1, X2, … Xk are independent and each has Borel distribution with parameter μ, then their sum W = X1 + X2 + … + Xk is said to have Borel–Tanner distribution with parameters μ and k. This gives the distribution of the total number of individuals in a Poisson–Galton–Watson process starting with k individuals in the first generation, or of the time taken for an M/D/1 queue to empty starting with k jobs in the queue.
|
|
As a result, functional annotation of a "center" of a cluster is likely to result in the most accurate predictions for the other proteins in the cluster. In evolutionary terms these "cluster centers" are closest to the evolutionary ancestor of all the proteins in the cluster. Active Learning generalizes this intuition principle to produce recommendations for additional experiments that are likely to either produce accurate predictions or identify proteins that are not annotated correctly. In addition to evolutionary analysis and Active Learning COMBREX also points to other criteria that might be considered in considering experiments.
|
|
Levi-Civita's parallelogramoid In the mathematical field of differential geometry, the Levi-Civita parallelogramoid is a quadrilateral in a curved space whose construction generalizes that of a parallelogram in the Euclidean plane. It is named for its discoverer, Tullio Levi-Civita. Like a parallelogram, two opposite sides AA′ and BB′ of a parallelogramoid are parallel (via parallel transport side AB) and the same length as each other, but the fourth side A′B′ will not in general be parallel to or the same length as the side AB, although it will be straight (a geodesic).
|
|
In the mathematical field of algebraic geometry, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety which is not singular is said to be regular. An algebraic variety which has no singular point is said to be non-singular or smooth.
|
|
Another formula that appears to be as fast as Ryser's (or perhaps even twice as fast) is to be found in the two Ph.D. theses; see , ; also . The methods to find the formula are quite different, being related to the combinatorics of the Muir algebra, and to finite difference theory respectively. Another way, connected with invariant theory is via the polarization identity for a symmetric tensor . The formula generalizes to infinitely many others, as found by all these authors, although it is not clear if they are any faster than the basic one.
|
|
In matroid theory, a mathematical discipline, the girth of a matroid is the size of its smallest circuit or dependent set. The cogirth of a matroid is the girth of its dual matroid. Matroid girth generalizes the notion of the shortest cycle in a graph, the edge connectivity of a graph, Hall sets in bipartite graphs, even sets in families of sets, and general position of point sets. It is hard to compute, but fixed-parameter tractable for linear matroids when parameterized both by the matroid rank and the field size of a linear representation.
|
|
However, the principle of special relativity generalizes the notion of inertial frame to include all physical laws, not simply Newton's first law. Newton viewed the first law as valid in any reference frame that is in uniform motion relative to the fixed stars;The question of "moving uniformly relative to what?" was answered by Newton as "relative to absolute space". As a practical matter, "absolute space" was considered to be the fixed stars. For a discussion of the role of fixed stars, see that is, neither rotating nor accelerating relative to the stars.
|
|
Mikhail Khovanov and Lev Rozansky have since defined cohomology theories associated to sln for all n. In 2003, Catharina Stroppel extended Khovanov homology to an invariant of tangles (a categorified version of Reshetikhin-Turaev invariants) which also generalizes to sln for all n. Paul Seidel and Ivan Smith have constructed a singly graded knot homology theory using Lagrangian intersection Floer homology, which they conjecture to be isomorphic to a singly graded version of Khovanov homology. Ciprian Manolescu has since simplified their construction and shown how to recover the Jones polynomial from the chain complex underlying his version of the Seidel- Smith invariant.
|
|
Regularization perspectives on support-vector machines provide a way of interpreting support-vector machines (SVMs) in the context of other machine- learning algorithms. SVM algorithms categorize multidimensional data, with the goal of fitting the training set data well, but also avoiding overfitting, so that the solution generalizes to new data points. Regularization algorithms also aim to fit training set data and avoid overfitting. They do this by choosing a fitting function that has low error on the training set, but also is not too complicated, where complicated functions are functions with high norms in some function space.
|
|
It generalizes several specialized agreement coefficients by accepting any number of observers, being applicable to nominal, ordinal, interval, and ratio levels of measurement, being able to handle missing data, and being corrected for small sample sizes. Alpha emerged in content analysis where textual units are categorized by trained coders and is used in counseling and survey research where experts code open-ended interview data into analyzable terms, in psychometrics where individual attributes are tested by multiple methods, in observational studies where unstructured happenings are recorded for subsequent analysis, and in computational linguistics where texts are annotated for various syntactic and semantic qualities.
|
|
His algorithms include: Baby-step giant-step algorithm for computing the discrete logarithm, which is useful in public-key cryptography; Shanks's square forms factorization, an integer factorization method that generalizes Fermat's factorization method; and the Tonelli–Shanks algorithm that finds square roots modulo a prime, which is useful for the quadratic sieve method of integer factorization. In 1974, Shanks and John Wrench did some of the first computer work on estimating the value of Brun's constant, the sum of the reciprocals of the twin primes, calculating it over the twin primes among the first two million primes.
|
|
The best way of dealing with arbitrary computation from a logical point of view is still an actively debated research question, but one popular approach is based on using monads to segregate provably terminating from potentially non-terminating code (an approach that also generalizes to much richer models of computation, and is itself related to modal logic by a natural extension of the Curry–Howard isomorphism). A more radical approach, advocated by total functional programming, is to eliminate unrestricted recursion (and forgo Turing completeness, although still retaining high computational complexity), using more controlled corecursion wherever non- terminating behavior is actually desired.
|
|
In the more abstract setting of incidence geometry, which is a set having a symmetric and reflexive relation called incidence defined on its elements, a flag is a set of elements that are mutually incident. This level of abstraction generalizes both the polyhedral concept given above as well as the related flag concept from linear algebra. A flag is maximal if it is not contained in a larger flag. An incidence geometry (Ω, ) has rank if Ω can be partitioned into sets Ω1, Ω2, ..., Ω, such that each maximal flag of the geometry intersects each of these sets in exactly one element.
|
|
Thus (D1) is both true and false. Either way, (D1) is both true and false – the same paradox as (A) above. The multi-sentence version of the liar paradox generalizes to any circular sequence of such statements (wherein the last statement asserts the truth/falsity of the first statement), provided there are an odd number of statements asserting the falsity of their successor; the following is a three-sentence version, with each statement asserting the falsity of its successor: Assume (E1) is true. Then (E2) is false, which means (E3) is true, and hence (E1) is false, leading to a contradiction.
|
|
In Euclidean geometry, linear separability is a property of two sets of points. This is most easily visualized in two dimensions (the Euclidean plane) by thinking of one set of points as being colored blue and the other set of points as being colored red. These two sets are linearly separable if there exists at least one line in the plane with all of the blue points on one side of the line and all the red points on the other side. This idea immediately generalizes to higher-dimensional Euclidean spaces if the line is replaced by a hyperplane.
|
|
A (non-degenerate) conic is completely determined by five points in general position (no three collinear) in a plane and the system of conics which pass through a fixed set of four points (again in a plane and no three collinear) is called a pencil of conics.. The four common points are called the base points of the pencil. Through any point other than a base point, there passes a single conic of the pencil. This concept generalizes a pencil of circles.Berger, M., Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry (Berlin/Heidelberg: Springer, 2010), p. 127.
|
|
The third way is when the verbal community provides reinforcement contingent on the overt behavior and the organism generalizes that to the private event that is occurring. Skinner refers to this as "metaphorical or metonymical extension". The final method that Skinner suggests may help form our verbal behavior is when the behavior is initially at a low level and then turns into a private event (Skinner, 1957, p. 134). This notion can be summarized by understanding that the verbal behavior of private events can be shaped through the verbal community by extending the language of tacts (Catania, 2007, p. 263).
|
|
In algebra and in particular in algebraic combinatorics, a quasisymmetric function is any element in the ring of quasisymmetric functions which is in turn a subring of the formal power series ring with a countable number of variables. This ring generalizes the ring of symmetric functions. This ring can be realized as a specific limit of the rings of quasisymmetric polynomials in n variables, as n goes to infinity. This ring serves as universal structure in which relations between quasisymmetric polynomials can be expressed in a way independent of the number n of variables (but its elements are neither polynomials nor functions).
|
|
Block diagram of the (forward) lifting scheme transform The generalized lifting scheme was developed by Joel Solé and Philippe Salembier and published in Solé's PhD dissertation.Ph.D. dissertation: Optimization and Generalization of Lifting Schemes: Application to Lossless Image Compression. It is based on the classical lifting scheme and generalizes it by breaking out a restriction hidden in the scheme structure. The classical lifting scheme has three kinds of operations: # A lazy wavelet transform splits signal f_j[n] in two new signals: the odd-samples signal denoted by f_j^o[n] and the even-samples signal denoted by f_j^e[n].
|
|
The TIR hypothesis assumes the original memory trace of the target remains and becomes temporarily inaccessible, rather than being permanently changed by verbalization. Verbalization leads cognitive processing to an inappropriate style, which stops retrieval of the non-verbal information needed for facial recognition. Verbal overshadowing comes solely because "verbalization indices inappropriate processing operations which area incommensurate with the processes required for successful recognition performance, that is, there is a transfer inappropriate processing shift". A shift is not tied to a particular item that has been previously coded, but rather generalizes to new stimuli that have not been encountered before.
|
|
In mathematics, a quasisymmetric homeomorphism between metric spaces is a map that generalizes bi-Lipschitz maps. While bi-Lipschitz maps shrink or expand the diameter of a set by no more than a multiplicative factor, quasisymmetric maps satisfy the weaker geometric property that they preserve the relative sizes of sets: if two sets A and B have diameters t and are no more than distance t apart, then the ratio of their sizes changes by no more than a multiplicative constant. These maps are also related to quasiconformal maps, since in many circumstances they are in fact equivalent.
|
|
Illusie's construction of the cotangent complex generalizes that of Michel André and Daniel Quillen to morphisms of ringed topoi. The generality of the framework makes it possible to apply the formalism to various first- order deformation problems: schemes, morphisms of schemes, group schemes and torsors under group schemes. Results concerning commutative group schemes in particular were the key tool in Grothendieck's proof of his existence and structure theorem for infinitesimal deformations of Barsotti–Tate groups, an ingredient in Gerd Faltings' proof of the Mordell conjecture. In Chapter VIII of the second volume of the thesis, Illusie introduces and studies derived de Rham complexes.
|
|
GDD- agreement generalizes the notion of the degree distribution to the spectrum of graphlet degree distributions (GDDs) in the following way. The degree distribution measures the number of nodes of degree k in graph G, i.e., the number of nodes "touching" k edges, for each value of k. Note that an edge is the only graphlet with two nodes. GDDs generalize the degree distribution to other graphlets: they measure for each 2-5-node graphlet Gi, i = 0, 1,..., 29, such as a triangle or a square, the number of nodes "touching" k graphlets Gi at a particular node.
|
|
In mathematics, specifically in the theory of generalized functions, the limit of a sequence of distributions is the distribution that sequence approaches. The distance, suitably quantified, to the limiting distribution can be made arbitrarily small by selecting a distribution sufficiently far along the sequence. This notion generalizes a limit of functions; a limit as a distribution may exist when a limit of functions does not. The notion is a part of distributional calculus, a generalized form of calculus that is based on the notion of distributions, as opposed to classical calculus, which is based on the narrower concept of functions.
|
|
Maya Cakmak, assistant professor of computer science and engineering at the University of Washington, is trying to create a robot that learns by imitating - a technique called "programming by demonstration". A researcher shows it a cleaning technique for the robot's vision system and it generalizes the cleaning motion from the human demonstration as well as identifying the "state of dirt" before and after cleaning. Similarly the Baxter industrial robot can be taught how to do something by grabbing its arm and showing it the desired movements. It can also use deep learning to teach itself to grasp an unknown object.
|
|
However, the definition of a cup product generalizes to complexes and topological manifolds. This is an advantage for mathematicians who are interested in complexes and topological manifolds (not only in PL and smooth manifolds). When the 4-manifold is smooth, then in de Rham cohomology, if a and b are represented by 2-forms \alpha and \beta, then the intersection form can be expressed by the integral : Q(a,b)= \int_M \alpha \wedge \beta where \wedge is the wedge product. The definition using cup product has a simpler analogue modulo 2 (which works for non-orientable manifolds).
|
|
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among group means in a sample. ANOVA was developed by the statistician Ronald Fisher. The ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the t-test beyond two means.
|
|
The systematic exploration of finite groups of Lie type started with Camille Jordan's theorem that the projective special linear group PSL(2, q) is simple for q ≠ 2, 3. This theorem generalizes to projective groups of higher dimensions and gives an important infinite family PSL(n, q) of finite simple groups. Other classical groups were studied by Leonard Dickson in the beginning of 20th century. In the 1950s Claude Chevalley realized that after an appropriate reformulation, many theorems about semisimple Lie groups admit analogues for algebraic groups over an arbitrary field k, leading to construction of what are now called Chevalley groups.
|
|
In more than two dimensions, the spin–statistics theorem states that any multiparticle state of indistinguishable particles has to obey either Bose–Einstein or Fermi–Dirac statistics. For any d > 2, the Lie groups SO(d,1) (which generalizes the Lorentz group) and Poincaré(d,1) have Z2 as their first homotopy group. Because the cyclic group Z2 is composed of two elements, only two possibilities remain. (The details are more involved than that, but this is the crucial point.) The situation changes in two dimensions. Here the first homotopy group of SO(2,1), and also Poincaré(2,1), is Z (infinite cyclic).
|
|
The psychologist George Boeree, in the tradition of J. J. Gibson, specifically assigns color to light, and extends the idea of color realism to all sensory experience, an approach he refers to as "quality realism". Jonathan Cohen (of UCSD) and Michael Tye (of UT Austin) have also written many essays on color vision. Cohen argues for the uncontroversial position of color relationalism with respect to semantics of color vision in Relationalist Manifesto. In The Red and the Real, Cohen argues for the position, with respect to color ontology that generalizes from his semantics to his metaphysics.
|
|
In mathematics, a diffeology on a set declares what the smooth parametrizations in the set are. In some sense a diffeology generalizes the concept of smooth charts in a differentiable manifold. The concept was first introduced by Jean-Marie Souriau in the 1980s and developed first by his students Paul Donato (homogeneous spaces and coverings) and Patrick Iglesias (diffeological fiber bundles, higher homotopy, etc.), later by other people. A related idea was introduced by Kuo-Tsaï Chen (陳國才, Chen Guocai) in the 1970s, using convex sets instead of open sets for the domains of the plots.
|
|
The concept of Cartesian coordinates generalizes to allow axes that are not perpendicular to each other, and/or different units along each axis. In that case, each coordinate is obtained by projecting the point onto one axis along a direction that is parallel to the other axis (or, in general, to the hyperplane defined by all the other axes). In such an oblique coordinate system the computations of distances and angles must be modified from that in standard Cartesian systems, and many standard formulas (such as the Pythagorean formula for the distance) do not hold (see affine plane).
|
|
The Nagell–Lutz theorem generalizes to arbitrary number fields and more general cubic equations.See, for example, Theorem VIII.7.1 of Joseph H. Silverman (1986), "The arithmetic of elliptic curves", Springer, . For curves over the rationals, the generalization says that, for a nonsingular cubic curve whose Weierstrass form :y^2 +a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6 has integer coefficients, any rational point P=(x,y) of finite order must have integer coordinates, or else have order 2 and coordinates of the form x=m/4, y=n/8, for m and n integers.
|
|
Compassion fatigue refers to a reduced capacity to help as a health care professional after being exposed to the suffering and distress of their patients. Secondary traumatic stress was later renamed compassion fatigue in 1995 by Charles Figley who described compassion fatigue as the natural emotions that arise as a result of learning about a significant others' experience with a traumatic event. Overall, compassion fatigue is often used interchangeably with secondary traumatic stress but the difference between the two is that STS is specific to individuals who treat traumatized populations whereas CF generalizes to individuals who treat an array of other populations.
|
|
The simplest example of a lattice is the integer lattice of all points with integer coefficients; its determinant is 1. For , the theorem claims that a convex figure in the Euclidean plane symmetric about the origin and with area greater than 4 encloses at least one lattice point in addition to the origin. The area bound is sharp: if is the interior of the square with vertices then is symmetric and convex, and has area 4, but the only lattice point it contains is the origin. This example, showing that the bound of the theorem is sharp, generalizes to hypercubes in every dimension .
|
|
The theorem as stated above is only valid for simple polygons, i.e., ones that consist of a single, non-self- intersecting boundary (and thus do not contain holes). For a general polygon, Pick's formula generalizes to :A = v - \frac 1 2 e_b + h - 1 where v is the number of vertices both in and on the boundary of the polygon, e_b is the number of lattice edges on the boundary of the polygon, and h is the number of holes in the polygon. As an example, consider the "polygon" made by connecting the points (0, 0), (2, 0).
|
|
Suppose V is a non-trivial variety of algebras, i.e. V contains algebras with more than one element. One can show that for every set S, the variety V contains a free algebra FS on S. This means that there is an injective set map i : S -> FS which satisfies the following universal property: given any algebra A in V and any map k : S -> A, there exists a unique V-homomorphism f : FS -> A such that f\circ i = k. This generalizes the notions of free group, free abelian group, free algebra, free module etc.
|
|
In mathematics, a GCD domain is an integral domain R with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique minimal principal ideal containing the ideal generated by two given elements. Equivalently, any two elements of R have a least common multiple (LCM). A GCD domain generalizes a unique factorization domain (UFD) to a non-Noetherian setting in the following sense: an integral domain is a UFD if and only if it is a GCD domain satisfying the ascending chain condition on principal ideals (and in particular if it is Noetherian).
|
|
In this view, many mutations are deleterious and so never observed, and most of the remainder are neutral, i.e. are not under selection. With the fate of each neutral mutation left to chance (genetic drift), the direction of evolutionary change is driven by which mutations occur, and so cannot be captured by models of change in the frequency of (existing) alleles alone. The origin-fixation view of population genetics generalizes this approach beyond strictly neutral mutations, and sees the rate at which a particular change happens as the product of the mutation rate and the fixation probability.
|
|
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field.
|
|
The Pólya enumeration theorem, also known as the Redfield–Pólya theorem and Pólya counting, is a theorem in combinatorics that both follows from and ultimately generalizes Burnside's lemma on the number of orbits of a group action on a set. The theorem was first published by J. Howard Redfield in 1927. In 1937 it was independently rediscovered by George Pólya, who then greatly popularized the result by applying it to many counting problems, in particular to the enumeration of chemical compounds. The Pólya enumeration theorem has been incorporated into symbolic combinatorics and the theory of combinatorial species.
|
|
Envy-freeness becomes easier to attain when it is assumed that agents' valuations are quasilinear in money, and thus transferable across agents. Demange, Gale and Sotomayor showed a natural ascending auction that achieves an envy-free allocation using monetary payments for unit demand bidders (where each bidder is interested in at most one item). Fair by Design is a general framework for optimization problems with envy-freeness guarantee that naturally extends fair item assignments using monetary payments. Cavallo generalizes the traditional binary criteria of envy-freeness, proportionality, and efficiency (welfare) to measures of degree that range between 0 and 1.
|
|
In statistics, multinomial logistic regression is a classification method that generalizes logistic regression to multiclass problems, i.e. with more than two possible discrete outcomes. That is, it is a model that is used to predict the probabilities of the different possible outcomes of a categorically distributed dependent variable, given a set of independent variables (which may be real-valued, binary-valued, categorical-valued, etc.). Multinomial logistic regression is known by a variety of other names, including polytomous LR, multiclass LR, softmax regression, multinomial logit (mlogit), the maximum entropy (MaxEnt) classifier, and the conditional maximum entropy model.
|
|
In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets, a finite matroid is equivalent to a geometric lattice. Matroid theory borrows extensively from the terminology of linear algebra and graph theory, largely because it is the abstraction of various notions of central importance in these fields.
|
|
In the late 1960s matroid theorists asked for a more general notion that shares the different aspects of finite matroids and generalizes their duality. Many notions of infinite matroids were defined in response to this challenge, but the question remained open. One of the approaches examined by D.A. Higgs became known as B-matroids and was studied by Higgs, Oxley and others in the 1960s and 1970s. According to a recent result by , it solves the problem: Arriving at the same notion independently, they provided five equivalent systems of axiom—in terms of independence, bases, circuits, closure and rank.
|
|
There are several theories that attempt to explain the use of force within an intimate relationship. Cultural spill-over effect posits that the more a culture supports the use of violence to achieve their objectives, the more likely individuals in that culture will legitimize violence and generalize those beliefs across multiple domains, which include those where the use of violence or aggression is not socially appropriate. Occupational stress spill- over theory posits that male-dominated, hypermasculine occupations may inadvertently emphasize control though the use of physical force, which generalizes across domains where the use of force is socially unacceptable.
|
|
Inspired by the work of on Morse theory, found another proof, using Deligne's l-adic Fourier transform, which allowed him to simplify Deligne's proof by avoiding the use of the method of Hadamard and de la Vallée Poussin. His proof generalizes the classical calculation of the absolute value of Gauss sums using the fact that the norm of a Fourier transform has a simple relation to the norm of the original function. used Laumon's proof as the basis for their exposition of Deligne's theorem. gave a further simplification of Laumon's proof, using monodromy in the spirit of Deligne's first proof.
|
|
In model theory, a branch of mathematical logic, the Hrushovski construction generalizes the Fraïssé limit by working with a notion of strong substructure \leq rather than \subseteq. It can be thought of as a kind of "model-theoretic forcing", where a (usually) stable structure is created, called the generic or rich Slides on Hrushovski construction from Frank Wagner model. The specifics of \leq determine various properties of the generic, with its geometric properties being of particular interest. It was initially used by Ehud Hrushovski to generate a stable structure with an "exotic" geometry, thereby refuting Zil'ber's Conjecture.
|
|
The space of oriented lines is a double cover of the space of unoriented lines. The Crofton formula is often stated in terms of the corresponding density in the latter space, in which the numerical factor is not 1/4 but 1/2. Since a convex curve intersects almost every line either twice or not at all, the unoriented Crofton formula for convex curves can be stated without numerical factors: the measure of the set of straight lines which intersect a convex curve is equal to its length. The Crofton formula generalizes to any Riemannian surface; the integral is then performed with the natural measure on the space of geodesics.
|
|
The aPC generalizes chaos expansion techniques towards arbitrary distributions with arbitrary probability measures, which can be either discrete, continuous, or discretized continuous and can be specified either analytically (as probability density/cumulative distribution functions), numerically as histogram or as raw data sets. The aPC at finite expansion order only demands the existence of a finite number of moments and does not require the complete knowledge or even existence of a probability density function. This avoids the necessity to assign parametric probability distributions that are not sufficiently supported by limited available data. Alternatively, it allows modellers to choose freely of technical constraints the shapes of their statistical assumptions.
|
|
In geometric group theory, groups are studied by their actions on metric spaces. A principle that generalizes the bilipschitz invariance of word metrics says that any finitely generated word metric on G is quasi-isometric to any proper, geodesic metric space on which G acts, properly discontinuously and cocompactly. Metric spaces on which G acts in this manner are called model spaces for G. It follows in turn that any quasi-isometrically invariant property satisfied by the word metric of G or by any model space of G is an isomorphism invariant of G. Modern geometric group theory is in large part the study of quasi-isometry invariants.
|
|
For a function of more than one variable, the second- derivative test generalizes to a test based on the eigenvalues of the function's Hessian matrix at the critical point. In particular, assuming that all second-order partial derivatives of f are continuous on a neighbourhood of a critical point x, then if the eigenvalues of the Hessian at x are all positive, then x is a local minimum. If the eigenvalues are all negative, then x is a local maximum, and if some are positive and some negative, then the point is a saddle point. If the Hessian matrix is singular, then the second- derivative test is inconclusive.
|
|
An intersection graph of disks, with at most k = 5 disks covering any point of the plane The circle separator method of generalizes to the intersection graphs of any system of d-dimensional balls with the property that any point in space is covered by at most some constant number k of balls, to k-nearest-neighbor graphs in d dimensions, and to the graphs arising from finite element meshes.. The sphere separators constructed in this way partition the input graph into subgraphs of at most vertices. The size of the separators for k-ply ball intersection graphs and for k-nearest- neighbor graphs is O(k1/dn1 − 1/d).
|
|
Finally, sequent calculus generalizes the form of a natural deduction judgment to : A_1, \ldots, A_n \vdash B_1, \ldots, B_k, a syntactic object called a sequent. The formulas on left-hand side of the turnstile are called the antecedent, and the formulas on right-hand side are called the succedent or consequent; together they are called cedents or sequents. Again, A_i and B_i are formulae, and n and k are nonnegative integers, that is, the left-hand-side or the right-hand-side (or neither or both) may be empty. As in natural deduction, theorems are those B where \vdash B is the conclusion of a valid proof.
|
|
The name is motivated by the importance of changes of coordinate in physics: the covariant derivative transforms covariantly under a general coordinate transformation, that is, linearly via the Jacobian matrix of the transformation. This article presents an introduction to the covariant derivative of a vector field with respect to a vector field, both in a coordinate free language and using a local coordinate system and the traditional index notation. The covariant derivative of a tensor field is presented as an extension of the same concept. The covariant derivative generalizes straightforwardly to a notion of differentiation associated to a connection on a vector bundle, also known as a Koszul connection.
|
|
It's stuffed with interesting nuggets. It's brightly written. And if you get away from the generational mumbo jumbo, it illuminates changes that really do seem to be taking place.” Further, Brooks wrote that the generations aren't treated equally: "Basically, it sounds as if America has two greatest generations at either end of the age scale and two crummiest in the middle". In 2001, reviewer Dina Gomez wrote in NEA Today that they make their case “convincingly,” with “intriguing analysis of popular culture” but conceded that it "over-generalizes". Gomez argued that it is “hard to resist its hopeful vision for our children and future.
|
|
The Kerr–Newman metric is the most general asymptotically flat, stationary solution of the Einstein–Maxwell equations in general relativity that describes the spacetime geometry in the region surrounding an electrically charged, rotating mass. It generalizes the Kerr metric by taking into account the field energy of an electromagnetic field, in addition to describing rotation. It is one of a large number of various different electrovacuum solutions, that is, of solutions to the Einstein–Maxwell equations which account for the field energy of an electromagnetic field. Such solutions do not include any electric charges other than that associated with the gravitational field, and are thus termed vacuum solutions.
|
|
The theorem generalizes the Jordan–Hölder decomposition for finite groups (in which the primes are the finite simple groups), to all finite transformation semigroups (for which the primes are again the finite simple groups plus all subsemigroups of the "flip-flop" (see above)). Both the group and more general finite automata decomposition require expanding the state-set of the general, but allow for the same number of input symbols. In the general case, these are embedded in a larger structure with a hierarchical "coordinate system". One must be careful in understanding the notion of "prime" as Krohn and Rhodes explicitly refer to their theorem as a "prime decomposition theorem" for automata.
|
|
In mathematics, a Malcev algebra (or Maltsev algebra or Moufang–Lie algebra) over a field is a nonassociative algebra that is antisymmetric, so that :xy = -yx and satisfies the Malcev identity :(xy)(xz) = ((xy)z)x + ((yz)x)x + ((zx)x)y. They were first defined by Anatoly Maltsev (1955). Malcev algebras play a role in the theory of Moufang loops that generalizes the role of Lie algebras in the theory of groups. Namely, just as the tangent space of the identity element of a Lie group forms a Lie algebra, the tangent space of the identity of a smooth Moufang loop forms a Malcev algebra.
|
|
The lower K-groups were discovered first, in the sense that adequate descriptions of these groups in terms of other algebraic structures were found. For example, if F is a field, then is isomorphic to the integers Z and is closely related to the notion of vector space dimension. For a commutative ring R, the group is related to the Picard group of R, and when R is the ring of integers in a number field, this generalizes the classical construction of the class group. The group K1(R) is closely related to the group of units , and if R is a field, it is exactly the group of units.
|
|
Any two isomorphic rings are Morita equivalent. The ring of n-by-n matrices with elements in R, denoted Mn(R), is Morita-equivalent to R for any n > 0. Notice that this generalizes the classification of simple artinian rings given by Artin–Wedderburn theory. To see the equivalence, notice that if X is a left R-module then Xn is an Mn(R)-module where the module structure is given by matrix multiplication on the left of column vectors from X. This allows the definition of a functor from the category of left R-modules to the category of left Mn(R)-modules.
|
|
A set can be regarded as an algebra having no nontrivial operations or defining equations. From this perspective, the idea of the power set of as the set of subsets of generalizes naturally to the subalgebras of an algebraic structure or algebra. The power set of a set, when ordered by inclusion, is always a complete atomic Boolean algebra, and every complete atomic Boolean algebra arises as the lattice of all subsets of some set. The generalization to arbitrary algebras is that the set of subalgebras of an algebra, again ordered by inclusion, is always an algebraic lattice, and every algebraic lattice arises as the lattice of subalgebras of some algebra.
|
|
Such movement often involves compliant motion, a process where movement requires maintaining physical contact with an object. Moravec's paradox generalizes that low-level sensorimotor skills that humans take for granted are, counterintuitively, difficult to program into a robot; the paradox is named after Hans Moravec, who stated in 1988 that "it is comparatively easy to make computers exhibit adult level performance on intelligence tests or playing checkers, and difficult or impossible to give them the skills of a one-year-old when it comes to perception and mobility". This is attributed to the fact that, unlike checkers, physical dexterity has been a direct target of natural selection for millions of years.
|
|
There are many mathematically equivalent formulations of quantum mechanics. One of the oldest and most common is the "transformation theory" proposed by Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics – matrix mechanics (invented by Werner Heisenberg) and wave mechanics (invented by Erwin Schrödinger). Especially since Heisenberg was awarded the Nobel Prize in Physics in 1932 for the creation of quantum mechanics, the role of Max Born in the development of QM was overlooked until the 1954 Nobel award. The role is noted in a 2005 biography of Born, which recounts his role in the matrix formulation and the use of probability amplitudes.
|
|
The fifth concerns graph enumeration and random graphs, the sixth generalizes from graphs to hypergraphs, and the seventh concerns infinite graphs. The book concludes with a chapter of stories about Erdős from one of his oldest friends, Andrew Vázsonyi. Each chapter begins with a survey of the history and major results in the subtopic of graph theory that it covers; Erdős himself figures prominently in the history of several of these subtopics. The individual history, motivation, known progress, and bibliographic references for each problem are included, along with (in some cases) prizes for a solution originally offered by Erdős and maintained by Chung and Graham.
|
|
In the second part of the article, he generalizes this argument against universals to address concepts as a whole. He points out that it is "facile" to treat concepts as if they were "an article of property". Such questions as "Do we possess such-and-such a concept" and "how do we come to possess such-and-such a concept" are meaningless, because concepts are not the sort of thing that one possesses. In the final part of the paper, Austin further extends the discussion to relations, presenting a series of arguments to reject the idea that there is some thing that is a relation.
|
|
Finite groups of Lie type were among the first groups to be considered in mathematics, after cyclic, symmetric and alternating groups, with the projective special linear groups over prime finite fields, PSL(2, p) being constructed by Évariste Galois in the 1830s. The systematic exploration of finite groups of Lie type started with Camille Jordan's theorem that the projective special linear group PSL(2, q) is simple for q ≠ 2, 3. This theorem generalizes to projective groups of higher dimensions and gives an important infinite family PSL(n, q) of finite simple groups. Other classical groups were studied by Leonard Dickson in the beginning of 20th century.
|
|
The index is defined for all points of an ROC curve, and the maximum value of the index may be used as a criterion for selecting the optimum cut-off point when a diagnostic test gives a numeric rather than a dichotomous result. The index is represented graphically as the height above the chance line, and it is also equivalent to the area under the curve subtended by a single operating point. Youden's index is also known as deltap and generalizes from the dichotomous to the multiclass case as informedness. The use of a single index is "not generally to be recommended",Everitt B.S. (2002) The Cambridge Dictionary of Statistics.
|
|
AC Time to failure LT 100 hours An alternative to graphing the probability that the failure time is less than or equal to 100 hours is to graph the probability that the failure time is greater than 100 hours. The probability that the failure time is greater than 100 hours must be 1 minus the probability that the failure time is less than or equal to 100 hours, because total probability must sum to 1. This gives P(failure time > 100 hours) = 1 - P(failure time < 100 hours) = 1 – 0.81 = 0.19. This relationship generalizes to all failure times: P(T > t) = 1 - P(T < t) = 1 – cumulative distribution function.
|
|
For the same figure, the other two relations are analogous: :a^2=b^2+c^2-2bc\cos\alpha, :b^2=a^2+c^2-2ac\cos\beta. The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle is a right angle (of measure 90 degrees, or radians), then , and thus the law of cosines reduces to the Pythagorean theorem: :c^2 = a^2 + b^2. The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known.
|
|
The exterior algebra has notable applications in differential geometry, where it is used to define differential forms. Differential forms are mathematical objects that evaluate the length of vectors, areas of parallelograms, and volumes of higher-dimensional bodies, so they can be integrated over curves, surfaces and higher dimensional manifolds in a way that generalizes the line integrals and surface integrals from calculus. A differential form at a point of a differentiable manifold is an alternating multilinear form on the tangent space at the point. Equivalently, a differential form of degree k is a linear functional on the k-th exterior power of the tangent space.
|
|
The Injury and Poisoning Episode file contains event-level information about any injury event experienced by an individual within the household in the three months leading up to the interview. For each reported injury or poisoning event, a number of questions examining the severity and effect of the event were asked, such as whether a physician was consulted and what kind of injury was experienced. This episode file can be used to generalize about the frequency of all injuries experienced by Americans over a defined window of time. The final weight (WTFA) generalizes to the number of injuries that occur among the U.S. population.
|
|
Kenneth Arrow (1963) generalizes the analysis. Along earlier lines, his version of a social welfare function, also called a 'constitution', maps a set of individual orderings (ordinal utility functions) for everyone in the society to a social ordering, a rule for ranking alternative social states (say passing an enforceable law or not, ceteris paribus). Arrow finds that nothing of behavioral significance is lost by dropping the requirement of social orderings that are real-valued (and thus cardinal) in favor of orderings, which are merely complete and transitive, such as a standard indifference curve map. The earlier analysis mapped any set of individual orderings to one social ordering, whatever it was.
|
|
Dynamic Tonality is a new paradigm for music which generalizes the special relationship between Just Intonation and the Harmonic Series to apply to a much wider set of pseudo-Just tunings and pseudo-Harmonic timbres. Alt URL Dynamic Tonality enables many new musical effects that could expand the frontiers of tonality, including polyphonic tuning bends, tuning modulations, new chord progressions, temperament modulations and progressions, and novel timbre effects such as dynamic changes to primeness, conicality, and richness. 50px The definitions of primeness, conicality, and richness were copied from this source, which is available under a Creative Commons Attribution- ShareAlike 3.0 Unported license and the GNU Free Documentation License.
|
|
It quickly became clear that QED was almost "magical" in its relative tractability, and that most of the ways that one might imagine extending it would not produce rational calculations. However, one class of field theories remained promising: gauge theories, in which the objects in the theory represent equivalence classes of physically indistinguishable field configurations, any two of which are related by a gauge transformation. This generalizes the QED idea of a local change of phase to a more complicated Lie group. QED itself is a gauge theory, as is general relativity, although the latter has proven resistant to quantization so far, for reasons related to renormalization.
|
|
By treating fields of sets on pre-orders as a category in its own right this deep connection can be formulated as a category theoretic duality that generalizes Stone representation without topology. R. Goldblatt had shown that with restrictions to appropriate homomorphisms such a duality can be formulated for arbitrary modal algebras and modal frames. Naturman showed that in the case of interior algebras this duality applies to more general topomorphisms and can be factored via a category theoretic functor through the duality with topological fields of sets. The latter represent the Lindenbaum–Tarski algebra using sets of points satisfying sentences of the S4 theory in the topological semantics.
|
|
Seib noted that the Al Jazeera effect can be seen as parallel to the CNN effect, which states that coverage of international events can force otherwise uninvolved governments to take action. Whereas the CNN effect is used in the context of mainstream, traditional media networks such as CNN, the Al Jazeera effect generalizes this to newer media such as citizen journalist blogs, internet radio, and satellite broadcasting. He also argues that new media strengthen the identity of and give voice to previously marginalized groups, which previously lacked their own media outlets; he cites the Kurdish people as an example. Many of the new media organizations are affiliated with such groups, social movements or similar organizations.
|
|
Gradient boosting is a machine learning technique for regression and classification problems, which produces a prediction model in the form of an ensemble of weak prediction models, typically decision trees. It builds the model in a stage-wise fashion like other boosting methods do, and it generalizes them by allowing optimization of an arbitrary differentiable loss function. The idea of gradient boosting originated in the observation by Leo Breiman that boosting can be interpreted as an optimization algorithm on a suitable cost function. Explicit regression gradient boosting algorithms were subsequently developed by Jerome H. Friedman, simultaneously with the more general functional gradient boosting perspective of Llew Mason, Jonathan Baxter, Peter Bartlett and Marcus Frean.
|
|
Modus ponens represents an instance of the Law of total probability which for a binary variable is expressed as: \Pr(Q)=\Pr(Q\mid P)\Pr(P)+\Pr(Q\mid \lnot P)\Pr(\lnot P)\,, where e.g. \Pr(Q) denotes the probability of Q and the conditional probability \Pr(Q\mid P) generalizes the logical implication P \to Q. Assume that \Pr(Q) = 1 is equivalent to Q being TRUE, and that \Pr(Q) = 0 is equivalent to Q being FALSE. It is then easy to see that \Pr(Q) = 1 when \Pr(Q\mid P) = 1 and \Pr(P) = 1. Hence, the law of total probability represents a generalization of modus ponens.
|
|
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude. This formulation has proven crucial to the subsequent development of theoretical physics, because manifest Lorentz covariance (time and space components of quantities enter equations in the same way) is easier to achieve than in the operator formalism of canonical quantization. Unlike previous methods, the path integral allows one to easily change coordinates between very different canonical descriptions of the same quantum system.
|
|
In group theory, the wreath product is a specialized product of two groups, based on a semidirect product. Wreath products are used in the classification of permutation groups and also provide a way of constructing interesting examples of groups. Given two groups A and H, there exist two variations of the wreath product: the unrestricted wreath product A Wr H (also written A≀H) and the restricted wreath product A wr H. Given a set Ω with an H-action there exists a generalization of the wreath product which is denoted by A WrΩ H or A wrΩ H respectively. The notion generalizes to semigroups and is a central construction in the Krohn–Rhodes structure theory of finite semigroups.
|
|
In statistics, the generalized linear model (GLM) is a flexible generalization of ordinary linear regression that allows for response variables that have error distribution models other than a normal distribution. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a link function and by allowing the magnitude of the variance of each measurement to be a function of its predicted value. Generalized linear models were formulated by John Nelder and Robert Wedderburn as a way of unifying various other statistical models, including linear regression, logistic regression and Poisson regression. They proposed an iteratively reweighted least squares method for maximum likelihood estimation of the model parameters.
|
|
Economists have increasingly studied non-convex sets with nonsmooth analysis, which generalizes convex analysis. Convex analysis centers on convex sets and convex functions, for which it provides powerful ideas and clear results, but it is not adequate for the analysis of non-convexities, such as increasing returns to scale.: "Non-convexities in [both] production and consumption ... required mathematical tools that went beyond convexity, and further development had to await the invention of non-smooth calculus": For example, Clarke's differential calculus for Lipschitz continuous functions, which uses Rademacher's theorem and which is described by and ,Chapter 8 "Applications to economics", especially Section 8.5.3 "Enter nonconvexity" (and the remainder of the chapter), particularly page 495: according to .
|
|
The starting point of the program may be seen as Emil Artin's reciprocity law, which generalizes quadratic reciprocity. The Artin reciprocity law applies to a Galois extension of an algebraic number field whose Galois group is abelian; it assigns L-functions to the one-dimensional representations of this Galois group, and states that these L-functions are identical to certain Dirichlet L-series or more general series (that is, certain analogues of the Riemann zeta function) constructed from Hecke characters. The precise correspondence between these different kinds of L-functions constitutes Artin's reciprocity law. For non-abelian Galois groups and higher-dimensional representations of them, one can still define L-functions in a natural way: Artin L-functions.
|
|
Another way of proving the chain rule is to measure the error in the linear approximation determined by the derivative. This proof has the advantage that it generalizes to several variables. It relies on the following equivalent definition of differentiability at a point: A function g is differentiable at a if there exists a real number g′(a) and a function ε(h) that tends to zero as h tends to zero, and furthermore :g(a + h) - g(a) = g'(a) h + \varepsilon(h) h. Here the left-hand side represents the true difference between the value of g at a and at , whereas the right-hand side represents the approximation determined by the derivative plus an error term.
|
|
"The Age of Robots", Extro 1, Proceedings of the First Extropy Institute Conference on TransHumanist Thought (1994) pp. 84–100. June 1993 version available online. In a 2007 guest editorial in the journal Science on the topic of "Robot Ethics", SF author Robert J. Sawyer argues that since the U.S. military is a major source of funding for robotic research (and already uses armed unmanned aerial vehicles to kill enemies) it is unlikely such laws would be built into their designs. In a separate essay, Sawyer generalizes this argument to cover other industries stating: > The development of AI is a business, and businesses are notoriously > uninterested in fundamental safeguards — especially philosophic ones.
|
|
This space is said to parametrize the longitudinal circles on the torus. In this case, mirror symmetry is equivalent to T-duality acting on the longitudinal circles, changing their radii from R to \alpha'/R, with \alpha ' the inverse of the string tension. The SYZ conjecture generalizes this idea to the more complicated case of six-dimensional Calabi-Yau manifolds like the one illustrated above. As in the case of a torus, one can divide a six-dimensional Calabi-Yau manifold into simpler pieces, which in this case are 3-tori (three- dimensional objects which generalize the notion of a torus) parametrized by a 3-sphere (a three-dimensional generalization of a sphere).
|
|
More generally, if the action of G on X is ergodic (meaning that X cannot be reduced by invariant proper Borel sets of X) then any system of imprimitivity on X is homogeneous. We now discuss how the structure of homogeneous systems of imprimitivity can be expressed in a form which generalizes the Koopman representation given in the example above. In the following, we assume that μ is a σ-finite measure on a standard Borel G-space X such that the action of G respects the measure class of μ. This condition is weaker than invariance, but it suffices to construct a unitary translation operator similar to the Koopman operator in the example above.
|
|
In mathematics, the Daniell integral is a type of integration that generalizes the concept of more elementary versions such as the Riemann integral to which students are typically first introduced. One of the main difficulties with the traditional formulation of the Lebesgue integral is that it requires the initial development of a workable measure theory before any useful results for the integral can be obtained. However, an alternative approach is available, developed by that does not suffer from this deficiency, and has a few significant advantages over the traditional formulation, especially as the integral is generalized into higher-dimensional spaces and further generalizations such as the Stieltjes integral. The basic idea involves the axiomatization of the integral.
|
|
In algebraic topology, a branch of mathematics, the calculus of functors or Goodwillie calculus is a technique for studying functors by approximating them by a sequence of simpler functors; it generalizes the sheafification of a presheaf. This sequence of approximations is formally similar to the Taylor series of a smooth function, hence the term "calculus of functors". Many objects of central interest in algebraic topology can be seen as functors, which are difficult to analyze directly, so the idea is to replace them with simpler functors which are sufficiently good approximations for certain purposes. The calculus of functors was developed by Thomas Goodwillie in a series of three papers in the 1990s and 2000s,T.
|
|
The Gray graph can be constructed from the 27 points of a 3 × 3 × 3 grid and the 27 axis-parallel lines through these points. This collection of points and lines forms a projective configuration: each point has exactly three lines through it, and each line has exactly three points on it. The Gray graph is the Levi graph of this configuration; it has a vertex for every point and every line of the configuration, and an edge for every pair of a point and a line that touch each other. This construction generalizes (Bouwer 1972) to any dimension n ≥ 3, yielding an n-valent Levi graph with algebraic properties similar to those of the Gray graph.
|
|
In descriptive set theory, the Kleene–Brouwer order or Lusin–Sierpiński order is a linear order on finite sequences over some linearly ordered set (X, <), that differs from the more commonly used lexicographic order in how it handles the case when one sequence is a prefix of the other. In the Kleene–Brouwer order, the prefix is later than the longer sequence containing it, rather than earlier. The Kleene–Brouwer order generalizes the notion of a postorder traversal from finite trees to trees that are not necessarily finite. For trees over a well-ordered set, the Kleene–Brouwer order is itself a well- ordering if and only if the tree has no infinite branch.
|
|
Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative stack quotients). For example, noncommutative algebraic geometry is supposed to extend a notion of an algebraic scheme by suitable gluing of spectra of noncommutative rings; depending on how literally and how generally this aim (and a notion of spectrum) is understood in noncommutative setting, this has been achieved in various level of success. The noncommutative ring generalizes here a commutative ring of regular functions on a commutative scheme.
|
|
Multiplying electric parameters of both problems by arbitrary real constants produces a coherent interaction of light with matter which generalizes Einstein's theory which is now considered as founding theory of lasers: it is not necessary to study a large set of identical molecules to get coherent amplification in the mode obtained by arbitrary multiplications of advanced and retarded fields. To compute energy, it is necessary to use the absolute fields which includes the zero point field; otherwise, an error appears, for instance in photon counting. It is into important to take into account the zero point field discovered by Planck. It replaces Einstein's "A" coefficient and explains that the classical electron is stable on Rydberg's classical orbits.
|
|
The conjecture is studied in the more general context of graph homomorphisms, especially because of interesting relations to the category of graphs (with graphs as objects and homomorphisms as arrows). For any fixed graph K, one considers graphs G that admit a homomorphism to K, written G → K. These are also called K-colorable graphs. This generalizes the usual notion of graph coloring, since it follows from definitions that a k-coloring is the same as a Kk-coloring (a homomorphism into the complete graph on k vertices). A graph K is called multiplicative if for any graphs G, H, the fact that G × H → K holds implies that G → K or H → K holds.
|
|
The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), ... The fundamental theorem of arithmetic generalizes to the Lasker–Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers. The spectrum of a ring is a geometric space whose points are the prime ideals of the ring. Arithmetic geometry also benefits from this notion, and many concepts exist in both geometry and number theory. For example, factorization or ramification of prime ideals when lifted to an extension field, a basic problem of algebraic number theory, bears some resemblance with ramification in geometry.
|
|
Let A be an m-by-n matrix. Then # rank(A) = dim(rowsp(A)) = dim(colsp(A)), # rank(A) = number of pivots in any echelon form of A, # rank(A) = the maximum number of linearly independent rows or columns of A. If one considers the matrix as a linear transformation from Rn to Rm, then the column space of the matrix equals the image of this linear transformation. The column space of a matrix A is the set of all linear combinations of the columns in A. If A = [a1, ...., an], then colsp(A) = span {a1, ...., an}. The concept of row space generalizes to matrices over C, the field of complex numbers, or over any field.
|
|
In geometry, a normal is an object such as a line or vector that is perpendicular to a given object. For example, in the two-dimensional case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point. In the three-dimensional case a surface normal, or simply normal, to a surface at a point P is a vector that is perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality.
|
|
Our inferences depend on this presupposition,... This presupposition thus gives us a built-in advantage in understanding what the world is like, and thereby makes inductive understanding of the world a real possibility. : When a population [kind] is uniform with respect to some [generic] property, [inductive] inferences from small samples, and indeed, from a single case, are perfectly reliable. If I note that a [generic] sample of [universal] copper conducts electricity and straightaway conclude that all copper conducts electricity, then I will do just as well as someone ... checking a very large number of copper samples for their conductivity. Kornblith didn't explain how tedious modern induction accurately generalizes from a few generic traits to all of some universal kind.
|
|
The remainder of this article explains the details of these concepts with a much more rigorous and precise mathematical and physical description. People are ill-suited to visualizing things in five or more dimensions, but mathematical equations are not similarly challenged and can represent five-dimensional concepts in a way just as appropriate as the methods that mathematical equations use to describe easier to visualize three and four-dimensional concepts. There is a particularly important implication of the more precise mathematical description that differs from the analogy-based heuristic description of de Sitter space and anti-de Sitter space above. The mathematical description of anti-de Sitter space generalizes the idea of curvature.
|
|
As previously noted, it is believed that FA develops from a maladaptive coping style: the initial avoidance of the particular memory allows the individual to reduce emotional distress, though as this memory retrieval style is reinforced over time, it generalizes to other memories and becomes an impairment. The third pathway toward OGM involves Impairment in Executive Capacity (X). As described by Williams, autobiographical memory retrieval requires working memory capacity, the ability to maintain working memory, and inhibition of irrelevant information. In relation to testing for OGM, Sumner points out that executive control is necessary to even complete the autobiographical memory test, as the instructions must be kept in working memory as the person searches and retrieves a memory.
|
|
PLN begins with a term logic foundation, and then adds on elements of probabilistic and combinatory logic, as well as some aspects of predicate logic and autoepistemic logic, to form a complete inference system, tailored for easy integration with software components embodying other (not explicitly logical) aspects of intelligence. PLN represents truth values as intervals, but with different semantics than in Imprecise Probability Theory. In addition to the interpretation of truth in a probabilistic fashion, a truth value in PLN also has an associated amount of certainty. This generalizes the notion of truth values used in autoepistemic logic, where truth values are either known or unknown, and when known, are either true or false.
|
|
This approach is computationally expensive, but it yields the full posterior distributions for the regression parameters and allows expert knowledge to be incorporated through the use of informative priors. A classical GLM formulation for a CMP regression has been developed which generalizes Poisson regression and logistic regression.Sellers, K. S. and Shmueli, G. (2010), "A Flexible Regression Model for Count Data", Annals of Applied Statistics, 4 (2), 943–961 This takes advantage of the exponential family properties of the CMP distribution to obtain elegant model estimation (via maximum likelihood), inference, diagnostics, and interpretation. This approach requires substantially less computational time than the Bayesian approach, at the cost of not allowing expert knowledge to be incorporated into the model.
|
|
It is also NP-hard to find an optimal coloring of the graph, because (via line graphs) this problem generalizes the NP-hard problem of computing the chromatic index of a graph. For the same reason, it is NP-hard to find a coloring that achieves an approximation ratio better than 4/3. However, an approximation ratio of two can be achieved by a greedy coloring algorithm, because the chromatic number of a claw-free graph is greater than half its maximum degree. A generalization of the edge list coloring conjecture states that, for claw-free graphs, the list chromatic number equals the chromatic number; these two numbers can be far apart in other kinds of graphs.
|
|
A (non- degenerate) conic is completely determined by five points in general position (no three collinear) in a plane and the system of conics which pass through a fixed set of four points (again in a plane and no three collinear) is called a pencil of conics.. The four common points are called the base points of the pencil. Through any point other than a base point, there passes a single conic of the pencil. This concept generalizes a pencil of circles. In a projective plane defined over an algebraically closed field any two conics meet in four points (counted with multiplicity) and so, determine the pencil of conics based on these four points.
|
|
Residual topology is a descriptive stereochemical term to classify a number of intertwined and interlocked molecules, which cannot be disentangled in an experiment without breaking of covalent bonds, while the strict rules of mathematical topology allow such a disentanglement. Examples of such molecules are rotaxanes, catenanes with covalently linked rings (so-called pretzelanes), and open knots (pseudoknots) which are abundant in proteins. The term "residual topology" was suggested on account of a striking similarity of these compounds to the well-established topologically nontrivial species, such as catenanes and knotanes (molecular knots). The idea of residual topological isomerism introduces a handy scheme of modifying the molecular graphs and generalizes former efforts of systemization of mechanically bound and bridged molecules.
|
|
The above discussion assumed a static world in which policy actions and outcomes for only one moment in time were considered. However, the analysis generalizes to a context of multiple time periods in which both policy actions take place and target variable outcomes matter, and in which time lags in the effects of policy actions exist. In this dynamic stochastic control context with multiplier uncertainty, a key result is that the "certainty equivalence principle" does not apply: while in the absence of multiplier uncertainty (that is, with only additive uncertainty) the optimal policy with a quadratic loss function coincides with what would be decided if the uncertainty were ignored, this no longer holds in the presence of multiplier uncertainty.
|
|
Trivially, a graph with edges has intersection number at most , for each edge forms a clique and these cliques together cover all the edges.. It is also true that every graph with vertices has intersection number at most . More strongly, the edges of every -vertex graph can be partitioned into at most cliques, all of which are either single edges or triangles. This generalizes Mantel's theorem that a triangle-free graph has at most edges, for in a triangle-free graph the only optimal clique edge cover has one clique per edge and therefore the intersection number equals the number of edges. An even tighter bound is possible when the number of edges is strictly greater than .
|
|
The autoregressive fractionally integrated moving average (ARFIMA) model generalizes the former three. Extensions of these classes to deal with vector-valued data are available under the heading of multivariate time-series models and sometimes the preceding acronyms are extended by including an initial "V" for "vector", as in VAR for vector autoregression. An additional set of extensions of these models is available for use where the observed time-series is driven by some "forcing" time-series (which may not have a causal effect on the observed series): the distinction from the multivariate case is that the forcing series may be deterministic or under the experimenter's control. For these models, the acronyms are extended with a final "X" for "exogenous".
|
|
It may also be computed in polynomial time for graphs of bounded treewidth including series-parallel graphs, outerplanar graphs, and Halin graphs,; as well as for split graphs,; . for the complements of chordal graphs, credits this result to the 1993 Ph.D. thesis of Ton Kloks; Garbe's polynomial time algorithm for comparability graphs of interval orders generalizes this result, since any chordal graph must be a comparability graph of this type. for permutation graphs, for cographs, for circular-arc graphs,. for the comparability graphs of interval orders, and of course for interval graphs themselves, since in that case the pathwidth is just one less than the maximum number of intervals covering any point in an interval representation of the graph.
|
|
The torus can be made an abelian group isomorphic to the product of the circle group. This abelian group is a Klein four-group-module, where the group acts by reflection in each of the coordinate directions (here depicted by red and blue arrows intersecting at the identity element). In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation of G. Group (co)homology provides an important set of tools for studying general G-modules. The term G-module is also used for the more general notion of an R-module on which G acts linearly (i.e.
|
|
Much of Welzl's research has been in computational geometry. With David Haussler, he showed that machinery from computational learning theory including ε-nets and VC dimension could be useful in geometric problems such as the development of space-efficient range searching data structures.. He devised linear time randomized algorithms for the smallest circle problem. and for low-dimensional linear programming, and developed the combinatorial framework of LP-type problems that generalizes both of these problems.. Other highly cited research publications by Welzl and his co-authors describe algorithms for constructing visibility graphs and using them to find shortest paths among obstacles in the plane,. test whether two point sets can be mapped to each other by a combination of a geometric transformation and a small perturbation,.
|
|
In abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition. A ring is left Artinian if it satisfies the descending chain condition on left ideals, right Artinian if it satisfies the descending chain condition on right ideals, and Artinian or two-sided Artinian if it is both left and right Artinian.
|
|
Slow motion computer simulation of the black hole binary system GW150914 as seen by a nearby observer, during 0.33 s of its final inspiral, merge, and ringdown. The star field behind the black holes is being heavily distorted and appears to rotate and move, due to extreme gravitational lensing, as spacetime itself is distorted and dragged around by the rotating black holes. General relativity, also known as the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time or four-dimensional spacetime.
|
|
In three dimensions, a surface normal, or simply normal, to a surface at point P is a vector perpendicular to the tangent plane of the surface at P. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality (right angles). The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at point P is the set of vectors which are orthogonal to the tangent space at P. Normal vectors are of special interest in the case of smooth curves and smooth surfaces.
|
|
Descartes' theorem on total angular defect of a polyhedron is the polyhedral analog: it states that the sum of the defect at all the vertices of a polyhedron which is homeomorphic to the sphere is 4π. More generally, if the polyhedron has Euler characteristic \chi=2-2g (where g is the genus, meaning "number of holes"), then the sum of the defect is 2\pi \chi. This is the special case of Gauss–Bonnet, where the curvature is concentrated at discrete points (the vertices). Thinking of curvature as a measure, rather than as a function, Descartes' theorem is Gauss–Bonnet where the curvature is a discrete measure, and Gauss–Bonnet for measures generalizes both Gauss–Bonnet for smooth manifolds and Descartes' theorem.
|
|
Sunnafrank's (1986) study examined a potential initial acquaintance association between perceived similarity and attraction may be present, though undetected, in previous interpersonal goals research and finds support for interpersonal goals claims regarding the perceived similarity/attraction relationship. The findings extended Sunnafrank (1983, 1984, 1985) and Sunnafrank and Miller's (1981) claim that attitude similarity and attraction are unassociated in beginning communicative relationships. While their research supports this claim for attitude similarity revealed in pre-acquaintance, it also shows that this claim generalizes to more normally occurring perceptions of attitude similarity produced during initial relational stages. The results strongly suggested that traditional beliefs about, and theoretical interpretations of, the attitude similarity/attraction association are incorrect, at least regarding the initial acquaintance period.
|
|
Some authors restrict the phrase "toroidal polyhedra" to mean more specifically polyhedra topologically equivalent to the (genus 1) torus.. In this area, it is important to distinguish embedded toroidal polyhedra, whose faces are flat polygons in three-dimensional Euclidean space that do not cross themselves or each other, from abstract polyhedra, topological surfaces without any specified geometric realization.. Intermediate between these two extremes are polyhedra formed by geometric polygons or star polygons in Euclidean space that are allowed to cross each other. In all of these cases the toroidal nature of a polyhedron can be verified by its orientability and by its Euler characteristic being non-positive. The Euler characteristic generalizes to V − E + F = 2 − 2N, where N is the number of holes.
|
|
A rect function turns into a sinc function as the order of the fractional Fourier transform becomes 1 The usual interpretation of the Fourier transform is as a transformation of a time domain signal into a frequency domain signal. On the other hand, the interpretation of the inverse Fourier transform is as a transformation of a frequency domain signal into a time domain signal. Apparently, fractional Fourier transforms can transform a signal (either in the time domain or frequency domain) into the domain between time and frequency: it is a rotation in the time-frequency domain. This perspective is generalized by the linear canonical transformation, which generalizes the fractional Fourier transform and allows linear transforms of the time- frequency domain other than rotation.
|
|
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector fields, the theorem gives necessary and sufficient integrability conditions for the existence of a foliation by maximal integral manifolds whose tangent bundles are spanned by the given vector fields. The theorem generalizes the existence theorem for ordinary differential equations, which guarantees that a single vector field always gives rise to integral curves; Frobenius gives compatibility conditions under which the integral curves of r vector fields mesh into coordinate grids on r-dimensional integral manifolds. The theorem is foundational in differential topology and calculus on manifolds.
|
|
In measure-theoretic analysis and related branches of mathematics, Lebesgue–Stieltjes integration generalizes Riemann–Stieltjes and Lebesgue integration, preserving the many advantages of the former in a more general measure-theoretic framework. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be associated to any function of bounded variation on the real line. The Lebesgue–Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is of this kind. Lebesgue–Stieltjes integrals, named for Henri Leon Lebesgue and Thomas Joannes Stieltjes, are also known as Lebesgue–Radon integrals or just Radon integrals, after Johann Radon, to whom much of the theory is due.
|
|
In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold. A metric tensor is called positive-definite if it assigns a positive value to every nonzero vector .
|
|
These results remain true for the Henstock–Kurzweil integral, which allows a larger class of integrable functions . In higher dimensions Lebesgue's differentiation theorem generalizes the Fundamental theorem of calculus by stating that for almost every x, the average value of a function f over a ball of radius r centered at x tends to f(x) as r tends to 0. Part II of the theorem is true for any Lebesgue integrable function f, which has an antiderivative F (not all integrable functions do, though). In other words, if a real function F on admits a derivative f(x) at every point x of and if this derivative f is Lebesgue integrable on then :F(b) - F(a) = \int_a^b f(t) \, dt.
|
|
It provides a mathematical foundation of industrial organization, discussed above, to model different types of firm behaviour, for example in a solipsistic industry (few sellers), but equally applicable to wage negotiations, bargaining, contract design, and any situation where individual agents are few enough to have perceptible effects on each other. In behavioural economics, it has been used to model the strategies agents choose when interacting with others whose interests are at least partially adverse to their own.• • In this, it generalizes maximization approaches developed to analyse market actors such as in the supply and demand model and allows for incomplete information of actors. The field dates from the 1944 classic Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern.
|
|
In category theory, the concept of an element, or a point, generalizes the more usual set theoretic concept of an element of a set to an object of any category. This idea often allows restating of definitions or properties of morphisms (such as monomorphism or product) given by a universal property in more familiar terms, by stating their relation to elements. Some very general theorems, such as Yoneda's lemma and the Mitchell embedding theorem, are of great utility for this, by allowing one to work in a context where these translations are valid. This approach to category theory, in particular the use of the Yoneda lemma in this way, is due to Grothendieck, and is often called the method of the functor of points.
|
|
NURBS curve - polynomial curve defined in homogeneous coordinates (blue) and its projection on plane - rational curve (red) In computer aided design, computer aided manufacturing, and computer graphics, a powerful extension of B-splines is non-uniform rational B-splines (NURBS). NURBS are essentially B-splines in homogeneous coordinates. Like B-splines, they are defined by their order, and a knot vector, and a set of control points, but unlike simple B-splines, the control points each have a weight. When the weight is equal to 1, a NURBS is simply a B-spline and as such NURBS generalizes both B-splines and Bézier curves and surfaces, the primary difference being the weighting of the control points which makes NURBS curves "rational".
|
|
Observe that the above argument also gives the following corollary: if we let A be the set of all eight-digit numbers whose digits are all either 1, 2, 3 (thus A contains numbers such as 11333233), and we color A with two colors, then A contains at least one arithmetic progression of length three, all of whose elements are the same color. This is simply because all of the combinatorial lines appearing in the above proof of the Hales–Jewett theorem, also form arithmetic progressions in decimal notation. A more general formulation of this argument can be used to show that the Hales–Jewett theorem generalizes van der Waerden's theorem. Indeed the Hales–Jewett theorem is substantially a stronger theorem.
|
|
Cartesian trees were introduced and named by . The name is derived from the Cartesian coordinate system for the plane: in Vuillemin's version of this structure, as in the two-dimensional range searching application discussed above, a Cartesian tree for a point set has the sorted order of the points by their x-coordinates as its symmetric traversal order, and it has the heap property according to the y-coordinates of the points. and subsequent authors followed the definition here in which a Cartesian tree is defined from a sequence; this change generalizes the geometric setting of Vuillemin to allow sequences other than the sorted order of x-coordinates, and allows the Cartesian tree to be applied to non-geometric problems as well.
|
|
Bedford and Cooke show that any assignment of values in the open interval (−1, 1) to the edges in any partial correlation vine is consistent, the assignments are algebraically independent, and there is a one- to-one relation between all such assignments and the set of correlation matrices. In other words, partial correlation vines provide an algebraically independent parametrization of the set of correlation matrices, whose terms have an intuitive interpretation. Moreover, the determinant of the correlation matrix is the product over the edges of (1 − ρ2ik;D(ik)) where ρik;D(ik) is the partial correlation assigned to the edge with conditioned variables i,k and conditioning variables D(ik). A similar decomposition characterizes the mutual information, which generalizes the determinant of the correlation matrix.
|
|
In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests (sometimes called first-order necessary conditions) for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied. Allowing inequality constraints, the KKT approach to nonlinear programming generalizes the method of Lagrange multipliers, which allows only equality constraints. Similar to the Lagrange approach, the constrained maximization (minimization) problem is rewritten as a Lagrange function whose optimal point is a saddle point, i.e. a global maximum (minimum) over the domain of the choice variables and a global minimum (maximum) over the multipliers, which is why the Karush–Kuhn–Tucker theorem is sometimes referred to as the saddle-point theorem.
|
|
On the one hand σ-completeness is too weak to characterize inverse image maps (completeness is required), on the other hand it is too restrictive for a generalization. (Sikorski remarked on using non-σ- complete homomorphisms but included σ-completeness in his axioms for closure algebras.) Later J. Schmid defined a continuous homomorphism or continuous morphism for interior algebras as a Boolean homomorphism f between two interior algebras satisfying f(xC) ≤ f(x)C. This generalizes the forward image map of a continuous map - the image of a closure is contained in the closure of the image. This construction is covariant but not suitable for category theoretic applications as it only allows construction of continuous morphisms from continuous maps in the case of bijections. (C.
|
|
Although the discovery of the motion silencing illusion is relatively new, there has been some interesting research done looking to investigate the parameters of the effect. One study was conducted concerning whether silencing is exclusively caused by motion or whether it can be produced by other coherent visual changes such as in color or size. It was found that silencing can occur without motion or coherent changes. Another study sought to examine whether the motion silencing illusion generalizes to infants, specifically four-month-olds, to test the hypothesis that the mechanisms underlying the ability to integrate motion patterns of individual dots into coherent global motion to the extent that it hinders the perception of the dots’ colour changes would be developed by this early age.
|
|
Originally, Hilbert defined syzygies for ideals in polynomial rings, but the concept generalizes trivially to (left) modules over any ring. Given a generating set g_1, \ldots, g_k of a module over a ring , a relation or first syzygy between the generators is a -tuple (a_1, \ldots, a_k) of elements of such thatThe theory is presented for finitely generated modules, but extends easily to arbitrary modules. :a_1g_1 + \cdots + a_kg_k =0. Let L_0 be the free module with basis (G_1, \ldots, G_k), the relation (a_1, \ldots, a_k) may be identified with the element :a_1G_1 + \cdots + a_kG_k, and the relations form the kernel R_1 of the linear map L_0 \to M defined by G_i \mapsto g_i. In other words, one has an exact sequence :0 \to R_1 \to L_0 \to M \to 0.
|
|
The ergodicity of the geodesic flow on compact Riemann surfaces of variable negative curvature and on compact manifolds of constant negative curvature of any dimension was proved by Eberhard Hopf in 1939, although special cases had been studied earlier: see for example, Hadamard's billiards (1898) and Artin billiard (1924). The relation between geodesic flows on Riemann surfaces and one-parameter subgroups on SL(2, R) was described in 1952 by S. V. Fomin and I. M. Gelfand. The article on Anosov flows provides an example of ergodic flows on SL(2, R) and on Riemann surfaces of negative curvature. Much of the development described there generalizes to hyperbolic manifolds, since they can be viewed as quotients of the hyperbolic space by the action of a lattice in the semisimple Lie group SO(n,1).
|
|
The term \pi_2(B) is the second homotopy group of B, which is defined to be the set of homotopy classes of maps from S^2 to B, in direct analogy with the definition of \pi_1. If E happens to be path-connected and simply connected, this sequence reduces to an isomorphism :\pi_1(B) \cong \pi_0(F) which generalizes the above fact about the universal covering (which amounts to the case where the fiber F is also discrete). If instead F happens to be connected and simply connected, it reduces to an isomorphism :\pi_1(E) \cong \pi_1(B). What is more, the sequence can be continued at the left with the higher homotopy groups \pi_n of the three spaces, which gives some access to computing such groups in the same vein.
|
|
In construction grammar, the grammar of a language is made up of taxonomic networks of families of constructions, which are based on the same principles as those of the conceptual categories known from cognitive linguistics, such as inheritance, prototypicality, extensions, and multiple parenting. Four different models are proposed in relation to how information is stored in the taxonomies: #Full-entry model #:In the full-entry model information is stored redundantly at all relevant levels in the taxonomy, which means that it operates, if at all, with minimal generalization. #: #Usage-based model #:The usage-based model is based on inductive learning, meaning that linguistic knowledge is acquired in a bottom- up manner through use. It allows for redundancy and generalizations, because the language user generalizes over recurring experiences of use.
|
|
Repeated application of the separator theorem produces a separator hierarchy which may take the form of either a tree decomposition or a branch- decomposition of the graph. Separator hierarchies may be used to devise efficient divide and conquer algorithms for planar graphs, and dynamic programming on these hierarchies can be used to devise exponential time and fixed-parameter tractable algorithms for solving NP-hard optimization problems on these graphs. Separator hierarchies may also be used in nested dissection, an efficient variant of Gaussian elimination for solving sparse systems of linear equations arising from finite element methods. Bidimensionality theory of Demaine, Fomin, Hajiaghayi, and Thilikos generalizes and greatly expands the applicability of the separator theorem for a vast set of minimization problems on planar graphs and more generally graphs excluding a fixed minor.
|
|
The concept of duality can be extended to graph embeddings on two-dimensional manifolds other than the plane. The definition is the same: there is a dual vertex for each connected component of the complement of the graph in the manifold, and a dual edge for each graph edge connecting the two dual vertices on either side of the edge. In most applications of this concept, it is restricted to embeddings with the property that each face is a topological disk; this constraint generalizes the requirement for planar graphs that the graph be connected. With this constraint, the dual of any surface-embedded graph has a natural embedding on the same surface, such that the dual of the dual is isomorphic to and isomorphically embedded to the original graph.
|
|
In a more quantified version: for natural numbers k and m, if n = km + 1 objects are distributed among m sets, then the pigeonhole principle asserts that at least one of the sets will contain at least k+1 objects. For arbitrary n and m this generalizes to k + 1 = \lfloor(n - 1)/m \rfloor + 1 = \lceil n/m\rceil, where \lfloor\cdots\rfloor and \lceil\cdots\rceil denote the floor and ceiling functions, respectively. Though the most straightforward application is to finite sets (such as pigeons and boxes), it is also used with infinite sets that cannot be put into one-to-one correspondence. To do so requires the formal statement of the pigeonhole principle, which is "there does not exist an injective function whose codomain is smaller than its domain".
|
|
Mirsky's theorem can be restated in terms of directed acyclic graphs (representing a partially ordered set by reachability of their vertices), as the statement that there exists a graph homomorphism from a given directed acyclic graph G to a k-vertex transitive tournament if and only if there does not exist a homomorphism from a (k + 1)-vertex path graph to G. For, the largest path graph that has a homomorphism to G gives the longest chain in the reachability ordering, and the sets of vertices with the same image in a homomorphism to a transitive tournament form a partition into antichains. This statement generalizes to the case that G is not acyclic, and is a form of the Gallai–Hasse–Roy–Vitaver theorem on graph colorings and orientations .
|
|
Bit complexity generalizes to any machine where the memory cells are of a fixed size that does not depend on the input; for this reason, algorithms that manipulate numbers much larger than the registers on ordinary PCs are typically analyzed in terms of their bit complexity. Put another way, the bit complexity is the complexity when the word size is a single bit, where word size is the size of each memory cell and register. Another commonly used model has words with log n bits, where n is a variable depending on the input. For example, in graph algorithms, it is typical to assume that the vertices are numbered 1 through n and that each memory cell can hold any of these values, so that they can refer to any vertex.
|
|
In terms of the completion of some of the underlying Grothendieck program of research, he defined absolute Hodge cycles, as a surrogate for the missing and still largely conjectural theory of motives. This idea allows one to get around the lack of knowledge of the Hodge conjecture, for some applications. The theory of mixed Hodge structures, a powerful tool in algebraic geometry that generalizes classical Hodge theory, was created by applying weight filtration, Hironaka's resolution of singularities and other methods, which he then used it to prove the Weil conjectures. He reworked the Tannakian category theory in his 1990 paper for the "Grothendieck Festschrift", employing Beck's theorem – the Tannakian category concept being the categorical expression of the linearity of the theory of motives as the ultimate Weil cohomology.
|
|
The context for the second stage of the Maimononidean Controversy was Hachmei Provence in southern France where Maimonides’ work became a platform on which the general conflict between philosophy and tradition could be contested. Maimonides’ work fell into a time of ideological formation of a Christian Europe, with the Crusades and the Spanish Reconquista. Mystical tendencies and kabbalistic circles were on the rise in Spain, and philosophy had enjoyed a great flourishing also of Jewish authors under Muslim rule in al-Andalus. Maimonides’ projects to combine Jewish tradition with Greco-Arabic Aristotelianism – a problem already in the Talmud addressed as “Greek wisdom” (hokhmah yevanit). Wolfson generalizes this to be an issue common to Latin, Arabic and Jewish traditions who all attempted “Philonic” structures to combine reason with revelation.
|
|
The definition works without any changes if instead of vector spaces over a field F, we use modules over a commutative ring R. It generalizes to n-ary functions, where the proper term is multilinear. For non-commutative rings R and S, a left R-module M and a right S-module N, a bilinear map is a map with T an -bimodule, and for which any n in N, is an R-module homomorphism, and for any m in M, is an S-module homomorphism. This satisfies :B(r ⋅ m, n) = r ⋅ B(m, n) :B(m, n ⋅ s) = B(m, n) ⋅ s for all m in M, n in N, r in R and s in S, as well as B being additive in each argument.
|
|
The notion of irreducible fraction generalizes to the field of fractions of any unique factorization domain: any element of such a field can be written as a fraction in which denominator and numerator are coprime, by dividing both by their greatest common divisor.. This applies notably to rational expressions over a field. The irreducible fraction for a given element is unique up to multiplication of denominator and numerator by the same invertible element. In the case of the rational numbers this means that any number has two irreducible fractions, related by a change of sign of both numerator and denominator; this ambiguity can be removed by requiring the denominator to be positive. In the case of rational functions the denominator could similarly be required to be a monic polynomial..
|
|
In an infinite distributed lag model, an infinite number of lag weights need to be estimated; clearly this can be done only if some structure is assumed for the relation between the various lag weights, with the entire infinitude of them expressible in terms of a finite number of assumed underlying parameters. In a finite distributed lag model, the parameters could be directly estimated by ordinary least squares (assuming the number of data points sufficiently exceeds the number of lag weights); nevertheless, such estimation may give very imprecise results due to extreme multicollinearity among the various lagged values of the independent variable, so again it may be necessary to assume some structure for the relation between the various lag weights. The concept of distributed lag models easily generalizes to the context of more than one right-side explanatory variable.
|
|
On her way to work, Liz Lemon (Tina Fey) is delayed by what she perceives to be the failure of other subway goers to observe basic social norms, which she generalizes to a breakdown of New York society. While she is complaining to Jack Donaghy (Alec Baldwin) about this in a call, his cell phone is stolen at knifepoint in a construction tunnel. Jack responds to his mugging by hiding in his office for days and organizing the wealthy to protect themselves from the apparently angry lower classes. Shortly, Liz comes down with a cold and dons an elderly woman costume from one of The Girlie Show with Tracy Jordan sketches, but discovers to her amazement that the getup causes people to avoid her on the subway, fearing that she is sick and potentially mentally ill.
|
|
Quicksort identifies a pivot element in the list and then partitions the list into two sublists, those elements less than the pivot and those greater than the pivot. Spreadsort generalizes this idea by partitioning the list into n/c partitions at each step, where n is the total number of elements in the list and c is a small constant (in practice usually between 4 and 8 when comparisons are slow, or much larger in situations where they are fast). It uses distribution-based techniques to accomplish this, first locating the minimum and maximum value in the list, and then dividing the region between them into n/c equal-sized bins. Where caching is an issue, it can help to have a maximum number of bins in each recursive division step, causing this division process to take multiple steps.
|
|
More generally, if S is a partially ordered set, a completion of S means a complete lattice L with an order-embedding of S into L. The notion of complete lattice generalizes the least-upper-bound property of the reals. One completion of S is the set of its downwardly closed subsets, ordered by inclusion. A related completion that preserves all existing sups and infs of S is obtained by the following construction: For each subset A of S, let Au denote the set of upper bounds of A, and let Al denote the set of lower bounds of A. (These operators form a Galois connection.) Then the Dedekind–MacNeille completion of S consists of all subsets A for which (Au)l = A; it is ordered by inclusion. The Dedekind-MacNeille completion is the smallest complete lattice with S embedded in it.
|
|
In the language of category theory, free Boolean algebras can be defined simply in terms of an adjunction between the category of sets and functions, Set, and the category of Boolean algebras and Boolean algebra homomorphisms, BA. In fact, this approach generalizes to any algebraic structure definable in the framework of universal algebra. Above, we said that a free Boolean algebra is a Boolean algebra with a set of generators that behave a certain way; alternatively, one might start with a set and ask which algebra it generates. Every set X generates a free Boolean algebra FX defined as the algebra such that for every algebra B and function f : X → B, there is a unique Boolean algebra homomorphism f′ : FX → B that extends f. Diagrammatically, center where iX is the inclusion, and the dashed arrow denotes uniqueness.
|
|
There has been much debate about how useful “brain games” really are. A 2011 study with over 11,000 participants found that participants improved on the tasks in which they were trained, but there was no transfer to tasks outside of the training tasks. This is a common finding in the cognitive training literature, and within the cognitive psychology literature in general. Studies that try to train specific cognitive abilities oftentimes only show task-specific improvements, and participants are unable to generalize their strategies to new tasks or problems. In 2016, there was some evidence that some of these programs improved performance on tasks in which users were trained, less evidence that improvements in performance generalize to related tasks, and almost no evidence that "brain training" generalizes to everyday cognitive performance; in addition most clinical studies were flawed.
|
|
In its original form by F. Riesz (1909) the theorem states that every continuous linear functional A[f] over the space C([0, 1]) of continuous functions in the interval [0,1] can be represented in the form :A[f] = \int_0^1 f(x)\,d\alpha(x), where α(x) is a function of bounded variation on the interval [0, 1], and the integral is a Riemann–Stieltjes integral. Since there is a one-to-one correspondence between Borel regular measures in the interval and functions of bounded variation (that assigns to each function of bounded variation the corresponding Lebesgue–Stieltjes measure, and the integral with respect to the Lebesgue–Stieltjes measure agrees with the Riemann–Stieltjes integral for continuous functions), the above stated theorem generalizes the original statement of F. Riesz. (See Gray(1984), for a historical discussion).
|
|
The construction of the real numbers from the rational numbers is an example of the Dedekind completion of a totally ordered set, and the Dedekind–MacNeille completion generalizes this concept from total orders to partial orders. If is an antichain (a set of elements no two of which are comparable) then the Dedekind–MacNeille completion of consists of itself together with two additional elements, a bottom element that is below every element in and a top element that is above every element in ., Example 7.44(2), p. 168. If is any finite set of objects, and is any finite set of unary attributes for the objects in , then one may form a partial order of height two in which the elements of the partial order are the objects and attributes, and in which when is an object that has attribute .
|
|
The first part consists of two introductory chapters defining partial words, compatibility and containment, and related concepts. The second part generalizes to partial words some standard results on repetitions in strings, and the third part studies the problem of characterizing and recognizing primitive partial words, the partial words that have no repetition. Part four concerns codes defined from sets of partial words, in the sense that no two distinct concatenations of partial words from the set can be compatible with each other. A final part includes three chapters on advanced topics including the construction of repetitions of given numbers of copies of partial words that are compatible with each other, enumeration of the possible patterns of repetitions of partial words, and sets of partial words with the property that every infinite string contains a substring matching the set.
|
|
The exhibition- characterization relation binds a refinee—the exhibitor—with one or more refineables, which shall identify features that characterize the exhibitor Graphically, a smaller black triangle inside a larger empty triangle with that larger triangle's apex connecting by a line to the exhibitor and the features connecting to the opposite (horizontal) base defines the exhibition- characterization relation link. ; Generalization-specialization and inheritance : These are structural relations which provide for abstracting any number of objects or process classes into superclasses, and assigning attributes of superclasses to subordinate classes. # Generalization- specialization link: The refinee—the general—generalizes the refineables, which are specializations of the general. Binds one or more specializations with the same persistence attribute value as the general, such that either the general and all its specializations are objects or the general and all its specializations are processes.
|
|
CO generalizes strong strict two phase locking (SS2PL), which in conjunction with the two-phase commit (2PC) protocol is the de facto standard for achieving global serializability across (SS2PL based) database systems. As a result, CO compliant database systems (with any, different concurrency control types) can transparently join existing SS2PL based solutions for global serializability. The same applies also to all other multiple (transactional) object systems that use atomic transactions and need global serializability for correctness (see examples above; nowadays such need is not smaller than with database systems, the origin of atomic transactions). The most significant aspects of CO that make it a uniquely effective general solution for global serializability are the following: #Seamless, low overhead integration with any concurrency control mechanism, with neither changing any transaction's operation scheduling or blocking it, nor adding any new operation.
|
|
The hidden linear function problem, is a search problem that generalizes the Bernstein–Vazirani problem. In the Bernstein–Vazirani problem, the hidden function is implicitly specified in an oracle; while in the 2D hidden linear function problem (2D HLF), the hidden function is explicitly specified by a matrix and a binary vector. 2D HLF can be solved exactly by a constant-depth quantum circuit restricted to a 2-dimensional grid of qubits using bounded fan-in gates but can't be solved by any sub-exponential size, constant-depth classical circuit using unbounded fan-in AND, OR, and NOT gates. While Bernstein–Vazirani's problem was designed to prove an oracle separation between complexity classes BQP and BPP, 2D HLF was designed to prove an explicit separation between the circuit classes QNC^{0} and NC^{0} (QNC^{0} subseteq NC^{0}).
|
|
All and only those numbers congruent to 0 or 1 (mod 4), except 4, are amenable. The first few amenable numbers are: 1, 5, 8, 9, 12, 13 ... A solution for integers of the form n = 4k + 1 could be given by a set of 2k (+1)s and 2k (-1)s and n itself. (This generalizes the example of 5 given above.) Although not obvious from the definition, the set of amenable numbers is closed under multiplication (the product of two amenable numbers is an amenable number). All composite numbers would be amenable if the multiset was allowed to be of any length, because, even if other solutions are available, one can always obtain a solution by taking the prime factorization (expressed with repeated factors rather than exponents) and add as many 1s as necessary to add up to n.
|
|
The terms "structure" and "organization" were focal for the Gestalt psychologists. Stimuli were said to have a certain structure, to be organized in a certain way, and that it is to this structural organization, rather than to individual sensory elements, that the organism responds. When an animal is conditioned, it does not simply respond to the absolute properties of a stimulus, but to its properties relative to its surroundings. To use a favorite example of Köhler's, if conditioned to respond in a certain way to the lighter of two gray cards, the animal generalizes the relation between the two stimuli rather than the absolute properties of the conditioned stimulus: it will respond to the lighter of two cards in subsequent trials even if the darker card in the test trial is of the same intensity as the lighter one in the original training trials.
|
|
The theory further generalizes to a theory of history, claiming to account for many salient events of the two-million-year course of the human lineage—from the evolution of the genus Homo to the inception of behavioral modernity to the neolithic revolution to the rise of the nation-state (Bingham, 1999 and 2000). He has presented his theory at The Stony Brook Human Evolution Symposium and Workshop, convened by Richard Leakey , and gave the keynote presentation at the NSF-sponsored symposium Strategies for Risk Communication in 2006. Most recently, Bingham joined Noam Chomsky, Marc Hauser, Ray Jackendoff, Philip Lieberman, Ian Tattersall and others to debate the issues surrounding the evolution of human speech at the Morris Symposium on language evolution . He and Souza presented their work on theories of human evolution, behavior and history at the 2009 meeting of the Cold Spring Harbor Symposium on Quantitative Biology.
|
|
When an animal is conditioned, it does not simply respond to the absolute properties of a stimulus, but to its properties relative to its surroundings. To use a favorite example of Köhler's, if conditioned to respond in a certain way to the lighter of two gray cards, the animal generalizes the relation between the two stimuli rather than the absolute properties of the conditioned stimulus: it will respond to the lighter of two cards in subsequent trials even if the darker card in the test trial is of the same intensity as the lighter one in the original training trials. In 1921 Koffka published a Gestalt-oriented text on developmental psychology, Growth of the Mind. With the help of American psychologist Robert Ogden, Koffka introduced the Gestalt point of view to an American audience in 1922 by way of a paper in Psychological Bulletin.
|
|
In mathematics, especially in the study of infinite groups, the Hirsch–Plotkin radical is a subgroup describing the normal nilpotent subgroups of the group. It was named by after Kurt Hirsch and Boris I. Plotkin, who proved that the product of locally nilpotent groups remains locally nilpotent; this fact is a key ingredient in its construction.... The Hirsch–Plotkin radical is defined as the subgroup generated by the union of the normal locally nilpotent subgroups (that is, those normal subgroups such that every finitely generated subgroup is nilpotent). The Hirsch–Plotkin radical is itself a locally nilpotent normal subgroup, so is the unique largest such.. The Hirsch–Plotkin radical generalizes the Fitting subgroup to infinite groups.. Unfortunately the subgroup generated by the union of infinitely many normal nilpotent subgroups need not itself be nilpotent,. so the Fitting subgroup must be modified in this case.. See p.
|
|
The van der Waals equation (or van der Waals equation of state; named after Dutch physicist Johannes Diderik van der Waals) is an equation of state that generalizes the ideal gas law based on plausible reasons that real gases do not act ideally. The ideal gas law treats gas molecules as point particles that interact with their containers but not each other, meaning they neither take up space nor change kinetic energy during collisions (i.e. all collisions are perfectly elastic). The ideal gas law states that volume (V) occupied by n moles of any gas has a pressure (P) at temperature (T) in kelvins given by the following relationship, where R is the gas constant: : PV = nRT To account for the volume that a real gas molecule takes up, the van der Waals equation replaces V in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume that is occupied by one mole of the molecules.
|
|
For relative convex hulls of simple polygons, an alternative but equivalent definition of convexity can be used. A simple polygon P within another simple polygon Q is relatively convex or Q-convex if every line segment contained in Q that connects two points of P lies within P. The relative convex hull of a simple polygon P within Q can be defined as the intersection of all Q-convex polygons that contain P, as the smallest Q-convex polygon that contains P, or as the minimum-perimeter simple polygon that contains P and is contained by Q. generalizes linear time algorithms for the convex hull of a simple polygon to the relative convex hull of one simple polygon within another. The resulting generalized algorithm is not linear time, however: its time complexity depends on the depth of nesting of certain features of one polygon within another. In this case, the relative convex hull is itself a simple polygon.
|
|
Bird with earthworm: Shepard gives example of bird using "generalization," based on experience with one previous worm, to decide if another worm is edible.The universal law of generalization is a theory of cognition stating that the probability of a response to one stimulus being generalized to another is a function of the “distance” between the two stimuli in a psychological space. It was introduced in 1987 by Roger N. Shepard, who began researching mechanisms of generalization while he was still a graduate student at Yale: > "I was now convinced that the problem of generalization was the most > fundamental problem confronting learning theory. Because we never encounter > exactly the same total situation twice, no theory of learning can be > complete without a law governing how what is learned in one situation > generalizes to another" Shepard’s 1987 paper gives a "generalization" example of a bird that has eaten one earthworm, and is presented with a slightly different-looking earthworm.
|
|
By 2016, companies offering products and services for cognitive training were marketing them as improving educational outcomes for children, and for adults as improving memory, processing speed, and problem-solving, and even as preventing dementia or Alzheimers. They often have supported their marketing with discussion about the educational or professional background of their founders, some discuss neuroscience that supports their approach—especially concepts of neuroplasticity and transfer of learning, and some cite evidence from clinical trials. The key claim made by these companies is that the specific training that they offer generalizes to other fields—academic or professional performance generally or everyday life. CogniFit was founded in 1999, Cogmed in 2001, Posit Science in 2002, and Brain Age was first released in 2005, all capitalizing on the growing interest within the public in neuroscience, along with heightened worries by parents about ADHD and other learning disabilities in their children, and concern about their own cognitive health as they aged.
|
|
For points in the plane or on an algebraic curve, the notion of general position is made algebraically precise by the notion of a regular divisor, and is measured by the vanishing of the higher sheaf cohomology groups of the associated line bundle (formally, invertible sheaf). As the terminology reflects, this is significantly more technical than the intuitive geometric picture, similar to how a formal definition of intersection number requires sophisticated algebra. This definition generalizes in higher dimensions to hypersurfaces (codimension 1 subvarieties), rather than to sets of points, and regular divisors are contrasted with superabundant divisors, as discussed in the Riemann–Roch theorem for surfaces. Note that not all points in general position are projectively equivalent, which is a much stronger condition; for example, any k distinct points in the line are in general position, but projective transformations are only 3-transitive, with the invariant of 4 points being the cross ratio.
|
|
Though the simple DC-nets protocol uses binary digits as its transmission alphabet, and uses the XOR operator to combine cipher texts, the basic protocol generalizes to any alphabet and combining operator suitable for one-time pad encryption. This flexibility arises naturally from the fact that the secrets shared between the many pairs of participants are, in effect, merely one-time pads combined together symmetrically within a single DC-net round. One useful alternate choice of DC- nets alphabet and combining operator is to use a finite group suitable for public-key cryptography as the alphabet—such as a Schnorr group or elliptic curve—and to use the associated group operator as the DC-net combining operator. Such a choice of alphabet and operator makes it possible for clients to use zero-knowledge proof techniques to prove correctness properties about the DC-net ciphertexts that they produce, such as that the participant is not "jamming" the transmission channel, without compromising the anonymity offered by the DC-net.
|
|
It is also possible to define a notion of branch-decomposition for matroids that generalizes branch-decompositions of graphs.. Section 12, "Tangles and Matroids", pp. 188–190. A branch- decomposition of a matroid is a hierarchical clustering of the matroid elements, represented as an unrooted binary tree with the elements of the matroid at its leaves. An e-separation may be defined in the same way as for graphs, and results in a partition of the set M of matroid elements into two subsets A and B. If ρ denotes the rank function of the matroid, then the width of an e-separation is defined as , and the width of the decomposition and the branchwidth of the matroid are defined analogously. The branchwidth of a graph and the branchwidth of the corresponding graphic matroid may differ: for instance, the three-edge path graph and the three-edge star have different branchwidths, 2 and 1 respectively, but they both induce the same graphic matroid with branchwidth 1.
|
|
The inversive distance has been used to define the concept of an inversive-distance circle packing: a collection of circles such that a specified subset of pairs of circles (corresponding to the edges of a planar graph ) have a given inversive distance with respect to each other. This concept generalizes the circle packings described by the circle packing theorem, in which specified pairs of circles are tangent to each other.. Although less is known about the existence of inversive distance circle packings than for tangent circle packings, it is known that, when they exist, they can be uniquely specified (up to Möbius transformations) by a given maximal planar graph and set of Euclidean or hyperbolic inversive distances. This rigidity property can be generalized broadly, to Euclidean or hyperbolic metrics on triangulated manifolds with angular defects at their vertices.. However, for manifolds with spherical geometry, these packings are no longer unique.. In turn, inversive-distance circle packings have been used to construct approximations to conformal mappings.
|
|
Eulerian matroids were defined by as a generalization of the Eulerian graphs, graphs in which every vertex has even degree. By Veblen's theorem the edges of every such graph may be partitioned into simple cycles, from which it follows that the graphic matroids of Eulerian graphs are examples of Eulerian matroids.. The definition of an Eulerian graph above allows graphs that are disconnected, so not every such graph has an Euler tour. observes that the graphs that have Euler tours can be characterized in an alternative way that generalizes to matroids: a graph G has an Euler tour if and only if it can be formed from some other graph H, and a cycle C in H, by contracting the edges of H that do not belong to C. In the contracted graph, C generally stops being a simple cycle and becomes instead an Euler tour. Analogously, Wilde considers the matroids that can be formed from a larger matroid by contracting the elements that do not belong to some particular circuit.
|
|
Given random variables X,Y,\ldots, that are defined on a probability space, the joint probability distribution for X,Y,\ldots is a probability distribution that gives the probability that each of X,Y,\ldots falls in any particular range or discrete set of values specified for that variable. In the case of only two random variables, this is called a bivariate distribution, but the concept generalizes to any number of random variables, giving a multivariate distribution. The joint probability distribution can be expressed either in terms of a joint cumulative distribution function or in terms of a joint probability density function (in the case of continuous variables) or joint probability mass function (in the case of discrete variables). These in turn can be used to find two other types of distributions: the marginal distribution giving the probabilities for any one of the variables with no reference to any specific ranges of values for the other variables, and the conditional probability distribution giving the probabilities for any subset of the variables conditional on particular values of the remaining variables.
|
|
GDP rate of Four Provinces of Pakistan. The NFC Award is enacted through mathematical formulation between federal government and provincial governments. The NFC generalizes the five kinds of taxation, including the income taxes, sales tax, wealth taxes, capital gains taxes, and custom duties taxes. The program is constituted under President of Pakistan who coordinated and supervise the studies and calculations conducted by financial specialists, economists, statisticians and mathematicians. In 1997, changes in NFC formulation was carried out by Prime Minister Nawaz Sharif after including the custom duties taxes with the award. Before 1991, the custom duties taxes revenue had been awarded to federal government while the revenue of Worker Welfare Fund (WWF) remain in the four provinces where they are collected. The principality of 1991 NFC Award specifies that 63.12% of collected revenue of taxes were directed to the federal government and 37% distributed to the four provinces. Prime Minister Sharif was widely given credited for declaring a consensus on formulating the 7th NFC Award with some positive recommendations.
|
|
A 2016 review of studies in the field found that there was some evidence that some of brain training programs improved performance on tasks in which users were trained, less evidence that improvements in performance generalize to related tasks, and almost no evidence that brain training generalizes to everyday cognitive performance; in addition most clinical studies were flawed. In 2017, a group of Australian scientists undertook a systematic review of what studies have been published of commercially available brain training programs in an attempt to give consumers and doctors credible information on which brain training programs are actually scientifically proved to work. Unfortunately, after reviewing close to 8,000 studies about brain training programs marketed to healthy older adults that were studied, most programs had no peer reviewed published evidence of their efficacy and of the seven brain training programs that did, only two of those had multiple studies, including at least one study of high quality: BrainHQ (from Posit Science) and CogniFit. To this date, these 2 online brain training programs are the only programs that were scientifically proven efficient.
|
|
In 1956–1957, working as a student of Alfred Tarski, Kallin helped simplify Tarski's axioms for the first-order theory of Euclidean geometry, by showing that several of the axioms originally presented by Tarski did not need to be stated as axioms, but could instead be proved as theorems from the other axioms... Kallin earned her Ph.D. in 1963 from Berkeley under the supervision of John L. Kelley. Her thesis, only 14 pages long, concerned function algebras, and a summary of its results was published in the Proceedings of the National Academy of Sciences.; One of its results, that not every topological algebra is localizable, has become a "well-known counterexample".. See in particular p. 89. In the study of complex vector spaces, a set S is said to be polynomially convex if, for every point x outside of S, there exists a polynomial whose complex absolute value at x is greater than at any point of S. This condition generalizes the ordinary notion of a convex set, which can be separated from any point outside the set by a linear function.
|
|
In addition, the importance of "sticking to one's role" in the particular play situation facilitates the play interaction, and allows fertile ground for the development of planning, self-regulation, impulse control, and perspective taking (Bodrova & Leong 2007). Researchers have cited numerous important developmental achievements generated by sociodramatic play (see summary in Bodrova & Leong 2007). They include: inhibition of impulses and self- regulation through adhering to playing a sociodramatic role; the overcoming of "cognitive egocentrism" by learning to take other points of view through playing various social roles; the development of imagination through voluntarily entering the imaginary situations involved in play; the ability to act on an internal mental plane; the integration of emotions and cognition; further development of object substitutions and symbolic thought; and development of the "learning motive" to continue to grow toward adulthood, which helps to propel children's next leading activity of learning in school (Karpov 2005). As one illustration of the benefits of play, dramatic role play encourages children to use language to regulate their own behavior and those of other children (to make sure everyone sticks to their dramatic role), and this use of language generalizes to other non-play tasks (Bodrova & Leong 2007).
|
|
He established the theorem of instability for the equations of a perturbed motion. Working on the perturbations of stable motions of Hamiltonian system he formulated and proved the theorem of the properties of the Poincaré variational equations that states: “If the unperturbed motion of a holonomic potential system is stable, then, first, the characteristic numbers of all solutions of the variational equations are equal to zero, second, these equations are regular in the sense of Lyapunov and are reduced to a system of equations with constant coefficients and have a quadratic integral of definite sign”. The Chetaev's theorem generalizes the Lagrange's theorem on an equilibrium and the Poincaré–Lyapunov theorem on a periodic motion. According to the theorem, for a stable unperturbed motion of a potential system, an infinitely near perturbed motion has an oscillatory, wave-like, character. # Chetaev’s method of constructing Lyapunov functions as a coupling (combination) of first integrals. The previous result gave rise to and substantiated the Chetaev’s concept of constructing Lyapunov functions using first integrals initially implemented in his famous book “Stability of Motion” as a coupling of first integrals in quadratic form .
|
|
Each quadruple consists of (i) a syntactic unit (or concatenation of units) of an idiolect system, (ii) a syntactic structure the unit or concatenation has in the system, (iii) an assignment of lexical meanings to the primitive constituents contained in the unit given the structure and the system (called a 'lexical interpretation'), and (iv) the system itself. The values of such grammatical functions are two-(or more)-place relations among constituents of the syntactic unit. (Grammatical functions are only one type of 'constituent functions,' which also include 'scope functions' like negation and qualification, and 'phoric functions' like antecedent; and there are other types of syntactic functions besides the constituent functions.) Syntactic functions play a central role, via their semantic content, in the composition process by which syntactic meanings of a syntactic unit are constructed from the lexical meanings of its primitive constituents. Incorporating features of Valency Grammar, Integrational Syntax construes subject and object functions as derived from more basic complement functions that simultaneously cover all complements of a single verbal nucleus; it generalizes the notion of valency to arbitrary lexical words, excluding purely auxiliary words.
|
|
The model flat geometry for the ambient construction is the future null cone in Minkowski space, with the origin deleted. The celestial sphere at infinity is the conformal manifold M, and the null rays in the cone determine a line bundle over M. Moreover, the null cone carries a metric which degenerates in the direction of the generators of the cone. The ambient construction in this flat model space then asks: if one is provided with such a line bundle, along with its degenerate metric, to what extent is it possible to extend the metric off the null cone in a canonical way, thus recovering the ambient Minkowski space? In formal terms, the degenerate metric supplies a Dirichlet boundary condition for the extension problem and, as it happens, the natural condition is for the extended metric to be Ricci flat (because of the normalization of the normal conformal connection.) The ambient construction generalizes this to the case when M is conformally curved, first by constructing a natural null line bundle N with a degenerate metric, and then solving the associated Dirichlet problem on N × (-1,1).
|
|
The "girth" terminology generalizes the use of girth in graph theory, meaning the length of the shortest cycle in a graph: the girth of a graphic matroid is the same as the girth of its underlying graph.. The girth of other classes of matroids also corresponds to important combinatorial problems. For instance, the girth of a co-graphic matroid (or the cogirth of a graphic matroid) equals the edge connectivity of the underlying graph, the number of edges in a minimum cut of the graph. The girth of a transversal matroid gives the cardinality of a minimum Hall set in a bipartite graph: this is a set of vertices on one side of the bipartition that does not form the set of endpoints of a matching in the graph.. Any set of points in Euclidean space gives rise to a real linear matroid by interpreting the Cartesian coordinates of the points as the vectors of a matroid representation. The girth of the resulting matroid equals one plus the dimension of the space when the underlying set of point is in general position, and is smaller otherwise.
|
|