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21,600 | That is, if it is equal to its non-singular locus. By the implicit function theorem for holomorphic functions, every complex manifold is in particular a non-singular complex analytic variety, but is not in general affine or projective. By Serre's GAGA theorem, every projective complex analytic variety is actually a pro... | Complex geometry | 0.810079 |
21,601 | Since the definition of a singular point is local, the definition given for an affine analytic/algebraic variety applies to the points of any complex analytic or algebraic variety. The set of points of a variety X {\displaystyle X} which are singular is called the singular locus, denoted X s i n g {\displaystyle X^{sin... | Complex geometry | 0.810079 |
21,602 | In the analytic case, one typically allows X {\displaystyle X} to have a topology that is locally equivalent to the subspace topology due to the identification with open subsets of C n {\displaystyle \mathbb {C} ^{n}} , whereas in the algebraic case X {\displaystyle X} is often equipped with a Zariski topology. Again w... | Complex geometry | 0.810079 |
21,603 | To define a projective complex algebraic variety, one requires the subset X ⊆ C P n {\displaystyle X\subseteq \mathbb {CP} ^{n}} to locally be given by the zero set of finitely many homogeneous polynomials. In order to define a general complex algebraic or complex analytic variety, one requires the notion of a locally ... | Complex geometry | 0.810079 |
21,604 | Complex geometry is concerned not only with complex spaces, but other holomorphic objects attached to them. The classification of holomorphic line bundles on a complex variety X {\displaystyle X} is given by the Picard variety Pic ( X ) {\displaystyle \operatorname {Pic} (X)} of X {\displaystyle X} . The picard varie... | Complex geometry | 0.810079 |
21,605 | As a result of this, one can readily study singular spaces in complex geometry, such as singular complex analytic varieties or singular complex algebraic varieties, whereas in differential geometry the study of singular spaces is often avoided. In practice, complex geometry sits in the intersection of differential geom... | Complex geometry | 0.810079 |
21,606 | This equivalence indicates that complex geometry is in some sense closer to algebraic geometry than to differential geometry. Another example of this which links back to the nature of the complex plane is that, in complex analysis of a single variable, singularities of meromorphic functions are readily describable. In ... | Complex geometry | 0.810079 |
21,607 | This is because the complex plane C {\displaystyle \mathbb {C} } is, after forgetting its complex structure, isomorphic to the real plane R 2 {\displaystyle \mathbb {R} ^{2}} . However, complex geometry is not typically seen as a particular sub-field of differential geometry, the study of smooth manifolds. In particula... | Complex geometry | 0.810079 |
21,608 | One major theme in complex geometry is classification. Due to the rigid nature of complex manifolds and varieties, the problem of classifying these spaces is often tractable. Classification in complex and algebraic geometry often occurs through the study of moduli spaces, which themselves are complex manifolds or varie... | Complex geometry | 0.810079 |
21,609 | Additionally, the extra structure of complex geometry allows, especially in the compact setting, for global analytic results to be proven with great success, including Shing-Tung Yau's proof of the Calabi conjecture, the Hitchin–Kobayashi correspondence, the nonabelian Hodge correspondence, and existence results for Kä... | Complex geometry | 0.810079 |
21,610 | Because of the blend of techniques and ideas from various areas, problems in complex geometry are often more tractable or concrete than in general. For example, the classification of complex manifolds and complex algebraic varieties through the minimal model program and the construction of moduli spaces sets the field ... | Complex geometry | 0.810079 |
21,611 | In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holom... | Complex geometry | 0.810079 |
21,612 | These equations are not microscopic: they do not describe the equation of motion of individual nuclear magnetic moments. These are governed and described by laws of quantum mechanics. Bloch equations are macroscopic: they describe the equations of motion of macroscopic nuclear magnetization that can be obtained by summ... | Bloch equations | 0.810079 |
21,613 | In physics and chemistry, specifically in nuclear magnetic resonance (NMR), magnetic resonance imaging (MRI), and electron spin resonance (ESR), the Bloch equations are a set of macroscopic equations that are used to calculate the nuclear magnetization M = (Mx, My, Mz) as a function of time when relaxation times T1 and... | Bloch equations | 0.810079 |
21,614 | Toric varieties are complex algebraic varieties of dimension n {\displaystyle n} containing an open dense subset biholomorphic to ( C ∗ ) n {\displaystyle (\mathbb {C} ^{*})^{n}} , equipped with an action of ( C ∗ ) n {\displaystyle (\mathbb {C} ^{*})^{n}} which extends the action on the open dense subset. A toric vari... | Complex geometry | 0.810079 |
21,615 | Since sheaf cohomology measures obstructions in complex geometry, one technique that is used is to prove vanishing theorems. Examples of vanishing theorems in complex geometry include the Kodaira vanishing theorem for the cohomology of line bundles on compact Kähler manifolds, and Cartan's theorems A and B for the coho... | Complex geometry | 0.810079 |
21,616 | For example, famous problems in the analysis of several complex variables preceding the introduction of modern definitions are the Cousin problems, asking precisely when local meromorphic data may be glued to obtain a global meromorphic function. These old problems can be simply solved after the introduction of sheaves... | Complex geometry | 0.810079 |
21,617 | Due to the rigidity of holomorphic functions and complex manifolds, the techniques typically used to study complex manifolds and complex varieties differ from those used in regular differential geometry, and are closer to techniques used in algebraic geometry. For example, in differential geometry, many problems are ap... | Complex geometry | 0.810079 |
21,618 | A point p ∈ X {\displaystyle p\in X} is singular if the Jacobian matrix of the vector of holomorphic functions ( f 1 , … , f k ) {\displaystyle (f_{1},\dots ,f_{k})} does not have full rank at p {\displaystyle p} , and non-singular otherwise. A projective complex analytic variety is a subset X ⊆ C P n {\displaystyle X\... | Complex geometry | 0.810079 |
21,619 | In contrast to complex manifolds which are always smooth, complex geometry is also concerned with possibly singular spaces. An affine complex analytic variety is a subset X ⊆ C n {\displaystyle X\subseteq \mathbb {C} ^{n}} such that about each point p ∈ X {\displaystyle p\in X} , there is an open neighbourhood U {\disp... | Complex geometry | 0.810079 |
21,620 | Complex geometry is concerned with the study of complex manifolds, and complex algebraic and complex analytic varieties. In this section, these types of spaces are defined and the relationships between them presented. A complex manifold is a topological space X {\displaystyle X} such that: X {\displaystyle X} is Hausdo... | Complex geometry | 0.810079 |
21,621 | In some sense, the novelty of complex geometry may be traced back to this fundamental observation. It is true that every complex manifold is in particular a real smooth manifold. | Complex geometry | 0.810079 |
21,622 | For example, whereas smooth manifolds admit partitions of unity, collections of smooth functions which can be identically equal to one on some open set, and identically zero elsewhere, complex manifolds admit no such collections of holomorphic functions. Indeed, this is the manifestation of the identity theorem, a typi... | Complex geometry | 0.810079 |
21,623 | Broadly, complex geometry is concerned with spaces and geometric objects which are modelled, in some sense, on the complex plane. Features of the complex plane and complex analysis of a single variable, such as an intrinsic notion of orientability (that is, being able to consistently rotate 90 degrees counterclockwise ... | Complex geometry | 0.810079 |
21,624 | A complex Fano variety is a complex algebraic variety with ample anti-canonical line bundle (that is, K X ∗ {\displaystyle K_{X}^{*}} is ample). Fano varieties are of considerable interest in complex algebraic geometry, and in particular birational geometry, where they often arise in the minimal model program. Fundamen... | Complex geometry | 0.810079 |
21,625 | The divergence of a vector field on Euclidean space is a function (or scalar field). In three-dimensions, the divergence is defined by with the obvious generalization to arbitrary dimensions. The divergence at a point represents the degree to which a small volume around the point is a source or a sink for the vector fl... | Complete vector field | 0.810076 |
21,626 | Then the components of the vector V in the new coordinates are required to satisfy the transformation law Such a transformation law is called contravariant. A similar transformation law characterizes vector fields in physics: specifically, a vector field is a specification of n functions in each coordinate system subje... | Complete vector field | 0.810076 |
21,627 | In physics, a vector is additionally distinguished by how its coordinates change when one measures the same vector with respect to a different background coordinate system. The transformation properties of vectors distinguish a vector as a geometrically distinct entity from a simple list of scalars, or from a covector.... | Complete vector field | 0.810076 |
21,628 | {\displaystyle \mathbb {R} .} In this topology, a set U {\displaystyle U} is a neighborhood of + ∞ {\displaystyle +\infty } if and only if it contains a set { x: x > a } {\displaystyle \{x:x>a\}} for some real number a . {\displaystyle a.} The notion of the neighborhood of − ∞ {\displaystyle -\infty } can be defined si... | Extended reals | 0.810076 |
21,629 | {\displaystyle .} Thus the topology is metrizable, corresponding (for a given homeomorphism) to the ordinary metric on this interval. There is no metric, however, that is an extension of the ordinary metric on R . | Extended reals | 0.810076 |
21,630 | The affinely extended real number system R ¯ {\displaystyle {\overline {\mathbb {R} }}} , defined as {\displaystyle } or R ∪ { − ∞ , + ∞ } {\displaystyle \mathbb {R} \cup \left\{-\infty ,+\infty \right\}} , can be turned into a totally ordered set by defining − ∞ ≤ a ≤ + ∞ {\displaystyle -\infty \leq a\leq +\infty } f... | Extended reals | 0.810076 |
21,631 | The octahedral numbers have a generating function z ( z + 1 ) 2 ( z − 1 ) 4 = ∑ n = 1 ∞ O n z n = z + 6 z 2 + 19 z 3 + ⋯ . {\displaystyle {\frac {z(z+1)^{2}}{(z-1)^{4}}}=\sum _{n=1}^{\infty }O_{n}z^{n}=z+6z^{2}+19z^{3}+\cdots .} Sir Frederick Pollock conjectured in 1850 that every positive integer is the sum of at most... | Octahedral number | 0.810074 |
21,632 | In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres. The n {\displaystyle n} th octahedral number O n {\displaystyle O_{n}} can be obtained by the formula: O n = n ( 2 n 2 + 1 ) 3 . {\displaystyle O_{n}={n(2n^{2}+1) \over 3}... | Octahedral number | 0.810074 |
21,633 | A physical data model (or database design) is a representation of a data design as implemented, or intended to be implemented, in a database management system. In the lifecycle of a project it typically derives from a logical data model, though it may be reverse-engineered from a given database implementation. A comple... | Physical schema | 0.810074 |
21,634 | Thus, the two dsRNA pathways, exogenous and endogenous, converge at the RISC.Exogenous dsRNA initiates RNAi by activating the ribonuclease protein Dicer, which binds and cleaves dsRNAs in plants, or short hairpin RNAs (shRNAs) in humans, to produce double-stranded fragments of 20–25 base pairs with a 2-nucleotide overh... | Post transcriptional gene silencing | 0.81007 |
21,635 | With the ability to generate a large amount of data per experiment, genome-scale RNAi screening has led to an explosion of data generation rates. Exploiting such large data sets is a fundamental challenge, requiring suitable statistics/bioinformatics methods. The basic process of cell-based RNAi screening includes the ... | Post transcriptional gene silencing | 0.810069 |
21,636 | Plant interactions with microorganisms are also mediated by chemistry. Both constitutive and induced secondary metabolites are involved in plant defense against pathogens and chemical signals are also important in the establishment and maintenance of resource mutualisms. For example, both rhizobia and mycorrhizae depen... | Chemical ecology | 0.810068 |
21,637 | The substances that serve in such roles are typically small, readily-diffusible organic molecules, but can also include larger molecules and small peptides. In practice, chemical ecology relies extensively on chromatographic techniques, such as thin-layer chromatography, high performance liquid chromatography, and gas ... | Chemical ecology | 0.810068 |
21,638 | Chemical ecology is the study of chemically mediated interactions between living organisms, and the effects of those interactions on the demography, behavior and ultimately evolution of the organisms involved. It is thus a vast and highly interdisciplinary field. Chemical ecologists seek to identify the specific molecu... | Chemical ecology | 0.810068 |
21,639 | Chemical ecology has been utilized in the development of sustainable pest control strategies. Semiochemicals (especially insect sex pheromones) are widely used in integrated pest management for surveillance, trapping and mating disruption of pest insects. Unlike conventional insecticides, pheromone-based methods of pes... | Chemical ecology | 0.810068 |
21,640 | Around this time, Lincoln Brower also published a series of important ecological studies on monarch sequestration of cardenolides. Brower has been credited with popularizing the term "ecological chemistry" which appeared in the title of a paper he published in Science in 1968 and again the following year in an article ... | Chemical ecology | 0.810068 |
21,641 | During this period, Thomas Eisner and his close collaborator Jerrold Meinwald published a series seminal papers on chemical defenses in plants and insects. A number of other scientists at Cornell were also working on topics related to chemical ecology during this period, including Paul Feeny, Wendell L. Roelofs, Robert... | Chemical ecology | 0.810068 |
21,642 | In 1959, Adolf Butenandt identified the first intraspecific chemical signal (bombykol) from the silk moth, Bombyx mori, with material obtained by grinding up 500,000 moths. The same year, Karlson and Lüscher proposed the term 'pheromone' to describe this type of signal. Also in 1959, Gottfried S. Fraenkel also publishe... | Chemical ecology | 0.810068 |
21,643 | The chemical ecology of plant-insect interaction is a significant subfield of chemical ecology. In particular, plants and insects are often involved in a chemical evolutionary arms race. As plants develop chemical defenses to herbivory, insects which feed on them evolve immunity to these poisons, and in some cases, rep... | Chemical ecology | 0.810068 |
21,644 | Plant chemical ecology focuses on the role of chemical cues and signals in mediating interactions of plants with their biotic environment (e.g. microorganisms, phytophagous insects, and pollinators). | Chemical ecology | 0.810068 |
21,645 | Therefore, adopting the full double negation principle ¬ ¬ B → B {\displaystyle \neg \neg B\to B} in minimal logic brings the calculus back to classical logic, also skipping all intermediate logics. There are also propositional logic statements that are unprovable in minimal logic, but do hold intuitionistically. With ... | Minimal logic | 0.810068 |
21,646 | The tactic of generalizing ¬ A {\displaystyle \neg A} to A → C {\displaystyle A\to C} does not work to prove all classically valid statements involving double negations. In particular, unsurprisingly, the naive generalization of the double negation elimination ¬ ¬ B → B {\displaystyle \neg \neg B\to B} cannot be provab... | Minimal logic | 0.810068 |
21,647 | Each formula of this restricted minimal logic corresponds to a type in the simply typed lambda calculus, see Curry–Howard correspondence. In that context, the phrase minimal logic is sometimes used to mean this restriction of minimal logic. This implicational fragment of minimal logic is the same as the positive, impli... | Minimal logic | 0.810068 |
21,648 | Functional programming calculi already foremostly depend on the implication connective, see e.g. the calculus of constructions for a predicate logic framework. In this section we mention the system obtained by restricting minimal logic to implication only, and describe it formally. It can be defined by the following se... | Minimal logic | 0.810068 |
21,649 | From B → ( ( B → C ) → C ) {\displaystyle B\to ((B\to C)\to C)} follows ( ( ( B → C ) → C ) → D ) → ( B → D ) {\displaystyle {\big (}((B\to C)\to C)\to D{\big )}\to (B\to D)} . So B → ¬ ¬ B {\displaystyle B\to \neg \neg B} and ¬ ¬ ¬ B ↔ ¬ B {\displaystyle \neg \neg \neg B\leftrightarrow \neg B} Related to the negation ... | Minimal logic | 0.810068 |
21,650 | As seen above, the double negated excluded middle for any proposition is already provable in minimal logic. However, it is worth emphasizing that in the predicate calculus, the laws of minimal logic do not enable a proof of the double negation of an infinite conjunction of excluded middle statements. | Minimal logic | 0.810068 |
21,651 | The following Heyting arithmetic theorem allows for proofs of existence claims that cannot be proven, by means of this general result, without the explosion principle. The result is essentially a family of simple double negation elimination claims, ∃ {\displaystyle \exists } -sentences binding a computable predicate. L... | Minimal logic | 0.810068 |
21,652 | Here only theorems not already provable in the positive calculus are covered. | Minimal logic | 0.810068 |
21,653 | One possible scheme of extending the positive calculus to minimal logic is to treat ¬ B {\displaystyle \neg B} as an implication, in which case the theorems from the implication calculus of a logic carry over to negation statements. To this end, ⊥ {\displaystyle \bot } is introduced as a proposition, not provable unles... | Minimal logic | 0.810068 |
21,654 | Minimal logic, or minimal calculus, is a symbolic logic system originally developed by Ingebrigt Johansson. It is an intuitionistic and paraconsistent logic, that rejects both the law of the excluded middle as well as the principle of explosion (ex falso quodlibet), and therefore holding neither of the following two de... | Minimal logic | 0.810068 |
21,655 | RBF networks can be used to interpolate a function y: R n → R {\displaystyle y:\mathbb {R} ^{n}\to \mathbb {R} } when the values of that function are known on finite number of points: y ( x i ) = b i , i = 1 , … , N {\displaystyle y(\mathbf {x} _{i})=b_{i},i=1,\ldots ,N} . Taking the known points x i {\displaystyle \ma... | Radial basis function network | 0.810066 |
21,656 | There are occasions in which multiple objectives, such as smoothness as well as accuracy, must be optimized. In that case it is useful to optimize a regularized objective function such as H ( w ) = d e f K ( w ) + λ S ( w ) = d e f ∑ t = 1 T H t ( w ) {\displaystyle H(\mathbf {w} )\ {\stackrel {\mathrm {def} }{=}}\ K(\... | Radial basis function network | 0.810066 |
21,657 | Once the underlying geometry of the time series is estimated as in the previous examples, a prediction for the time series can be made by iteration: φ ( 0 ) = x ( 1 ) {\displaystyle \varphi (0)=x(1)} x ( t ) ≈ φ ( t − 1 ) {\displaystyle {x}(t)\approx \varphi (t-1)} x ( t + 1 ) ≈ φ ( t ) = φ {\displaystyle {x}(t+1)\app... | Radial basis function network | 0.810066 |
21,658 | In the field of mathematical modeling, a radial basis function network is an artificial neural network that uses radial basis functions as activation functions. The output of the network is a linear combination of radial basis functions of the inputs and neuron parameters. Radial basis function networks have many uses,... | Radial basis function network | 0.810066 |
21,659 | Elementary flows can be considered the basic building blocks (Fundamental solutions, local solutions and solitons) of the different types of equations derived from the Navier-Stokes. Some of the flows reflect specific cases constraints such as incompressible or irrotational flows, or both, as in the case of potential f... | Elementary flow | 0.810066 |
21,660 | In the larger context of Navier-Stokes Equations but especially in the context of potential theory Elementary flows are a collection of basic flows from which it is possible to construct more complex flows with different techniques. In this article the term flows is used interchangeably to the term solutions due to his... | Elementary flow | 0.810066 |
21,661 | As a summary, the results of the EPR thought experiment do not contradict the predictions of special relativity. Neither the EPR paradox nor any quantum experiment demonstrates that superluminal signaling is possible; however, the principle of locality appeals powerfully to physical intuition, and Einstein, Podolsky an... | EPR pairs | 0.810062 |
21,662 | Whichever axis she uses, she has a 50% probability of obtaining "+" and 50% probability of obtaining "−", completely at random; according to quantum mechanics, it is fundamentally impossible for her to influence what result she gets. Furthermore, Bob is able to perform his measurement only once: there is a fundamental ... | EPR pairs | 0.810062 |
21,663 | Locality has several different meanings in physics. EPR describe the principle of locality as asserting that physical processes occurring at one place should have no immediate effect on the elements of reality at another location. At first sight, this appears to be a reasonable assumption to make, as it seems to be a c... | EPR pairs | 0.810062 |
21,664 | A true quantum-mechanical wave would diffract from the inner hemisphere, leaving a diffraction pattern to be observed on the outer hemisphere. This is not really an objection, but rather an affirmation that a partial collapse of the wave function has occurred. If a diffraction pattern were not observed, one would be fo... | Renninger negative-result experiment | 0.810058 |
21,665 | In quantum mechanics, the Renninger negative-result experiment is a thought experiment that illustrates some of the difficulties of understanding the nature of wave function collapse and measurement in quantum mechanics. The statement is that a particle need not be detected in order for a quantum measurement to occur, ... | Renninger negative-result experiment | 0.810058 |
21,666 | The above formulation is inherently phrased in a non-relativistic language; and it is noted that elementary particles have relativistic decay products. This objection only serves to confuse the issue. The experiment can be reformulated so that the decay product is slow-moving. At any rate, special relativity is not in ... | Renninger negative-result experiment | 0.810058 |
21,667 | In this objection, it is noted that in real life, a decay product is either spin-1/2 (a fermion) or a photon (spin-1). This is taken to mean that the decay is not truly sphere symmetric, but rather has some other distribution, such as a p-wave. However, on closer examination, one sees this has no bearing on the spheric... | Renninger negative-result experiment | 0.810058 |
21,668 | In type theory, commonly used in programming languages, a tuple has a product type; this fixes not only the length, but also the underlying types of each component. Formally: ( x 1 , x 2 , … , x n ): T 1 × T 2 × … × T n {\displaystyle (x_{1},x_{2},\ldots ,x_{n}):{\mathsf {T}}_{1}\times {\mathsf {T}}_{2}\times \ldots \t... | Tuple | 0.810055 |
21,669 | As a consequence of the four-dimensional nature of space-time in relativity, relativistic quantum mechanics uses 4×4 matrices to describe spin operators and observables. | Spin-1/2 | 0.810051 |
21,670 | In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of 1/2. The spin number describes how many symmetrical facets a particle has in one full rotation; a spin of 1/2 means that the particle must be rotated by two ... | Spin-1/2 | 0.810051 |
21,671 | Spin-1/2 objects are all fermions (a fact explained by the spin–statistics theorem) and satisfy the Pauli exclusion principle. Spin-1/2 particles can have a permanent magnetic moment along the direction of their spin, and this magnetic moment gives rise to electromagnetic interactions that depend on the spin. One such ... | Spin-1/2 | 0.810051 |
21,672 | While NRQM defines spin 1/2 with 2 dimensions in Hilbert space with dynamics that are described in 3-dimensional space and time, relativistic quantum mechanics defines the spin with 4 dimensions in Hilbert space and dynamics described by 4-dimensional space-time. | Spin-1/2 | 0.810051 |
21,673 | Suppose a detector that can be rotated measures a particle in which the probabilities of detecting some state are affected by the rotation of the detector. When the system is rotated through 360°, the observed output and physics are the same as initially but the amplitudes are changed for a spin-1/2 particle by a facto... | Spin-1/2 | 0.810051 |
21,674 | In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetime that represents the relativistic counterpart of velocity, which is a three-dimensional vector in space. Physical events correspond to mathematical points in time and space, the set of a... | Four velocity | 0.810048 |
21,675 | In the first case, the isomorphism of ℓ 1 {\displaystyle \ell ^{1}} with c ∗ {\displaystyle c^{*}} is given as follows. If ( x 0 , x 1 , … ) ∈ ℓ 1 , {\displaystyle \left(x_{0},x_{1},\ldots \right)\in \ell ^{1},} then the pairing with an element ( y 0 , y 1 , … ) {\displaystyle \left(y_{0},y_{1},\ldots \right)} in c {\d... | C space | 0.810047 |
21,676 | Consequently, classical field theories are usually categorized as non-relativistic and relativistic. Modern field theories are usually expressed using the mathematics of tensor calculus. A more recent alternative mathematical formalism describes classical fields as sections of mathematical objects called fiber bundles. | Classical field theory | 0.810042 |
21,677 | Each vector represents the direction of the movement of air at that point, so the set of all wind vectors in an area at a given point in time constitutes a vector field. As the day progresses, the directions in which the vectors point change as the directions of the wind change. The first field theories, Newtonian grav... | Classical field theory | 0.810042 |
21,678 | A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories. In most contexts, 'classical field theory' is specifically... | Classical field theory | 0.810042 |
21,679 | In mathematics, the Samuelson–Berkowitz algorithm efficiently computes the characteristic polynomial of an n × n {\displaystyle n\times n} matrix whose entries may be elements of any unital commutative ring. Unlike the Faddeev–LeVerrier algorithm, it performs no divisions, so may be applied to a wider range of algebrai... | Samuelson–Berkowitz algorithm | 0.810039 |
21,680 | The { ϕ j ( x ) {\displaystyle \phi _{j}(x)\ } } satisfy orthonormality relations: ∫ a b ϕ i ∗ ( x ) ϕ j ( x ) d x = δ i j , {\displaystyle \int _{a}^{b}\phi _{i}^{*}(x)\phi _{j}(x)\,dx=\delta _{ij},} where δij is the Kronecker delta. Substituting function fn into these equations then leads to the n-dimensional Pythago... | Least-squares function approximation | 0.810037 |
21,681 | A generalization to approximation of a data set is the approximation of a function by a sum of other functions, usually an orthogonal set: f ( x ) ≈ f n ( x ) = a 1 ϕ 1 ( x ) + a 2 ϕ 2 ( x ) + ⋯ + a n ϕ n ( x ) , {\displaystyle f(x)\approx f_{n}(x)=a_{1}\phi _{1}(x)+a_{2}\phi _{2}(x)+\cdots +a_{n}\phi _{n}(x),\ } with ... | Least-squares function approximation | 0.810037 |
21,682 | In other words, the least squares approximation of f {\displaystyle f} is the function g ∈ subspace W {\displaystyle g\in {\text{ subspace }}W} closest to f {\displaystyle f} in terms of the inner product ⟨ f , g ⟩ {\displaystyle \left\langle f,g\right\rangle } . Furthermore, this can be applied with a theorem: Let f {... | Least-squares function approximation | 0.810037 |
21,683 | {\displaystyle a_{j}=\int _{a}^{b}\phi _{j}^{*}(x)f(x)\,dx.} The generalization of the n-dimensional Pythagorean theorem to infinite-dimensional real inner product spaces is known as Parseval's identity or Parseval's equation. Particular examples of such a representation of a function are the Fourier series and the gen... | Least-squares function approximation | 0.810036 |
21,684 | Microwave volumetric heating is a commercially available method of heating liquids, suspensions, or solids in a continuous flow on an industrial scale. Microwave volumetric heating has a greater penetration depth, of up to 42 millimetres (1.7 in), which is an even penetration through the entire volume of the flowing pr... | Dielectric heating | 0.810035 |
21,685 | An amidotransferase then converts the acid side chain of the glutamate to the amide, forming the correctly charged gln-tRNA-Gln. Interference with aminoacylation may be useful as an approach to treating some diseases: cancerous cells may be relatively vulnerable to disturbed aminoacylation compared to healthy cells. Th... | Transfer RNA | 0.810035 |
21,686 | The reasons why tRNA genes have been lost during evolution remains under debate but may relate improving resistance to viral infection. Because nucleotide triplets can present more combinations than there are amino acids and associated tRNAs, there is redundancy in the genetic code, and several different 3-nucleotide c... | Transfer RNA | 0.810035 |
21,687 | Organisms vary in the number of tRNA genes in their genome. For example, the nematode worm C. elegans, a commonly used model organism in genetics studies, has 29,647 genes in its nuclear genome, of which 620 code for tRNA. The budding yeast Saccharomyces cerevisiae has 275 tRNA genes in its genome. | Transfer RNA | 0.810035 |
21,688 | Eq. 2 can then be used to calculate bond lengths which are found to lie within a few picometres of the observed bond lengths if no additional constraints are present. Additional constraints include electronic anisotropies (lone pairs and Jahn-Teller distortions) or steric constraints, (bonds stretched or compressed in ... | Bond valence method | 0.810034 |
21,689 | The bond valence model uses mostly classical physics, and with little more than a pocket calculator, it gives quantitative predictions of bond lengths and places limits on what structures can be formed. However, like all models, the bond valence model has its limitations. It is restricted to compounds with localized bo... | Bond valence method | 0.810034 |
21,690 | The association of the cation bonding electrons with the anion in the ionic model is purely formal. There is no change in physical locations of any electrons, and there is no change in the bond valence. The terms "anion" and "cation" in the bond valence model are defined in terms of the bond topology, not the chemical ... | Bond valence method | 0.810034 |
21,691 | Under these conditions, the bonds are all single bonds (or multiples of single bonds). Compounds can be constructed by linking carbon and hydrogen atoms with bonds that are all exactly equivalent. Under certain conditions, nitrogen can form three bonds and oxygen two, but since nitrogen and oxygen typically also form h... | Bond valence method | 0.810034 |
21,692 | Structures containing covalent bonds can be treated using the ionic model providing they satisfy the topological conditions given above, but a special situation applies to hydrocarbons which allows the bond valence model to be reduced to the traditional bond model of organic chemistry. If an atom has a valence, V, that... | Bond valence method | 0.810034 |
21,693 | It is this empirical relation that links the formal theorems of the bond valence model to the real world and allows the bond valence model to be used to predict the real structure, geometry and properties of a compound. If the structure of a compound is known, the empirical bond valence - bond length correlation of Eq.... | Bond valence method | 0.810034 |
21,694 | This definition contains a total of nine types of digital surfaces. The digital manifold was studied in the 1990s. A recursive definition of the digital k-manifold was proposed intuitively by Chen and Zhang in 1993. Many applications were found in image processing and computer vision. | Combinatorial manifold | 0.810032 |
21,695 | He also proposed (2008) a more general axiomatic theory of locally finite topological spaces and abstract cell complexes formerly suggested by Ernst Steinitz (1908). It is the Alexandrov topology. | Combinatorial manifold | 0.810032 |
21,696 | The labeling method for inferring a connected component was studied in the 1970s. Theodosios Pavlidis (1982) suggested the use of graph-theoretic algorithms such as the depth-first search method for finding connected components. Vladimir A. Kovalevsky (1989) extended the Alexandrov–Hopf 2D grid cell topology to three a... | Combinatorial manifold | 0.810032 |
21,697 | Digital topology was first studied in the late 1960s by the computer image analysis researcher Azriel Rosenfeld (1931–2004), whose publications on the subject played a major role in establishing and developing the field. The term "digital topology" was itself invented by Rosenfeld, who used it in a 1973 publication for... | Combinatorial manifold | 0.810032 |
21,698 | A digital manifold is a special kind of combinatorial manifold which is defined in digital space i.e. grid cell space. A digital form of the Gauss–Bonnet theorem is: Let M be a closed digital 2D manifold in direct adjacency (i.e., a (6,26)-surface in 3D). The formula for genus is g = 1 + ( M 5 + 2 M 6 − M 3 ) / 8 {\dis... | Combinatorial manifold | 0.810032 |
21,699 | The main differences between them are: (1) digital topology mainly studies digital objects that are formed by grid cells, and (2) digital topology also deals with non-Jordan manifolds. A combinatorial manifold is a kind of manifold which is a discretization of a manifold. It usually means a piecewise linear manifold ma... | Combinatorial manifold | 0.810032 |
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