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The installation examined, questioned, and expanded the newly established public sphere of connection and interchange through the possibilities of voice. Arab Voices built upon a range of her previous works that promote audience participation, some of which include WAY and Voices and Morals, Ethics, Values. Studio Pape... | Wikipedia - Betty Beaumont - Works 1984-Present |
Shredded and altered studio papers spanning those forty years- project research, descriptions, proposals and notes- were shredded and combined with musical instruments. Beaumont has said that the installation was informed by a dream she had the first week moving to NYC. | Wikipedia - Betty Beaumont - Works 1984-Present |
"My possessions were on fire, yet were not destroyed, and were burning—a surrealist image of energy." In early 2016, an exhibition at the DiMattio Gallery in Monmouth, New Jersey presented installation studies for a project addressing language attrition. These conceptual studies include site-specific arrangements of wo... | Wikipedia - Betty Beaumont - Works 1984-Present |
In the following years, polymer chemists started studying the characteristics of this polymer and worked on enhancing its thermal stability and mechanical properties. In particular, Moore and coworkers conducted rigorous mechanistic studies on poly(phthalaldehyde) by modifying the type of catalyst used, as well as the ... | Wikipedia - Poly(phthalaldehyde) - Current trends |
In the following years, the burning was mentioned regularly in the press, with Drummond and Cauty often relegated to a cultural status of "the men who burnt a million quid". A February 2000 article in The Observer newspaper again insisted that the duo really had burnt one million pounds. "It wasn't a stunt. | Wikipedia - K Foundation Burn a Million Quid - Later reaction |
They really did it. If you want to rile Bill Drummond, you call him a hoaxer. 'I knew it was real,' a long-time friend and associate of his group The KLF tells me, 'because afterwards, Jimmy and Bill looked so harrowed and haunted. | Wikipedia - K Foundation Burn a Million Quid - Later reaction |
And to be honest, they've never really been the same since'".A 2004 listener poll by BBC Radio 6 Music saw The KLF/K Foundation placed second after The Who in a list of "rock excesses".Drummond's former protégé Julian Cope was unimpressed, claiming that Drummond still owed him money. "He burned a million pounds which w... | Wikipedia - K Foundation Burn a Million Quid - Later reaction |
In the following years, the knowledge concerning cytochalasin B was broadened. As the more general knowledge had been elucidated, more detailed analysis of e.g. the mechanism of action took place. | Wikipedia - Cytochalasin B - From 1980 |
In the following, # represents the Caml prompt. | Wikipedia - Categorical Abstract Machine Language - Examples |
In the following, ( Ω , F , F ∗ , P ) {\displaystyle (\Omega ,F,F_{*},\mathbf {P} )} will be a filtered probability space where F ∗ = ( F t ) t ≥ 0 {\displaystyle F_{*}=(F_{t})_{t\geq 0}} , and N: . {\displaystyle N_{s}\geq \operatorname {E} {\big }.} | Wikipedia - Lévy's martingale convergence theorem - Statements for the general case |
In the following, Exponentiation stands for repeated application of the group operation Juxtaposition stands for multiplication on the set of congruence classes or application of the group operation (as applicable) Subtraction stands for subtraction on the set of congruence classes M ∈ { 0 , 1 } ∗ {\displaystyle M\in \... | Wikipedia - Schnorr signature - Notation |
In the following, Marvin Minsky defines the numbers to be computed in a manner similar to those defined by Alan Turing in 1936; i.e., as "sequences of digits interpreted as decimal fractions" between 0 and 1: A computable number one for which there is a Turing machine which, given n on its initial tape, terminates wit... | Wikipedia - Uncomputable number - Informal definition using a Turing machine as example |
In the following, R {\displaystyle \mathbb {R} } represents the real numbers with their usual topology. The subspace topology of the natural numbers, as a subspace of R {\displaystyle \mathbb {R} } , is the discrete topology. The rational numbers Q {\displaystyle \mathbb {Q} } considered as a subspace of R {\displaysty... | Wikipedia - Relative topology - Examples |
The set as a subspace of R {\displaystyle \mathbb {R} } is both open and closed, whereas as a subset of R {\displaystyle \mathbb {R} } it is only closed. As a subspace of R {\displaystyle \mathbb {R} } , ∪ is composed of two disjoint open subsets (which happen also to be closed), and is therefore a disconnected spac... | Wikipedia - Relative topology - Examples |
Then [0, 1⁄2) is open in S but not in R {\displaystyle \mathbb {R} } . Likewise [1⁄2, 1) is closed in S but not in R {\displaystyle \mathbb {R} } . S is both open and closed as a subset of itself but not as a subset of R {\displaystyle \mathbb {R} } . | Wikipedia - Relative topology - Examples |
In the following, X , Y , Z , W {\displaystyle X,Y,Z,W} are Banach spaces, B ( X , Y ) {\displaystyle B(X,Y)} is the space of bounded operators X → Y {\displaystyle X\to Y} under the operator norm, and K ( X , Y ) {\displaystyle K(X,Y)} denotes the space of compact operators X → Y {\displaystyle X\to Y} . Id X {\displa... | Wikipedia - Compact operator - Important properties |
Conversely, if X , Y {\displaystyle X,Y} are Hilbert spaces, then every compact operator from X → Y {\displaystyle X\to Y} is the limit of finite rank operators. Notably, this "approximation property" is false for general Banach spaces X and Y. B ( Y , Z ) ∘ K ( X , Y ) ∘ B ( W , X ) ⊆ K ( W , Z ) . | Wikipedia - Compact operator - Important properties |
{\displaystyle B(Y,Z)\circ K(X,Y)\circ B(W,X)\subseteq K(W,Z).} In particular, K ( X ) {\displaystyle K(X)} forms a two-sided ideal in B ( X ) {\displaystyle B(X)} . Any compact operator is strictly singular, but not vice versa. | Wikipedia - Compact operator - Important properties |
A bounded linear operator between Banach spaces is compact if and only if its adjoint is compact (Schauder's theorem). If T: X → Y {\displaystyle T:X\to Y} is bounded and compact, then: the closure of the range of T {\displaystyle T} is separable. if the range of T {\displaystyle T} is closed in Y, then the range of T ... | Wikipedia - Compact operator - Important properties |
If X {\displaystyle X} is a Banach space and there exists an invertible bounded compact operator T: X → X {\displaystyle T:X\to X} then X {\displaystyle X} is necessarily finite-dimensional.Now suppose that X {\displaystyle X} is a Banach space and T: X → X {\displaystyle T:X\to X} is a compact linear operator, and T ∗... | Wikipedia - Compact operator - Important properties |
One can notice the similarity between this property and the fact that, if M {\displaystyle M} and N {\displaystyle N} are subspaces of X {\displaystyle X} where M {\displaystyle M} is closed and N {\displaystyle N} is finite-dimensional, then M + N {\displaystyle M+N} is also closed. If S: X → X {\displaystyle S:X\to X... | Wikipedia - Compact operator - Important properties |
If λ ≠ 0 {\displaystyle \lambda \neq 0} then the following are finite and equal: dim ker ( T − λ Id X ) = dim X / Im ( T − λ Id X ) = dim ker ( T ∗ − λ Id X ∗ ) = dim X ∗ / Im ( T ∗ − λ Id X ∗ ) {\displaystyle \dim \ker \left(T-\lambda \operatorname {Id} _{X}\right)=\dim X/\operatorname {Im} \left(T-\la... | Wikipedia - Compact operator - Important properties |
In the following, a few typical special triangle conics are discussed. In the descriptions, the standard notations are used: The reference triangle is always denoted by ABC. The angles at the vertices A, B, C are denoted by A, B, C and the lengths of the sides opposite to the vertices A, B, C are respectively a. b, c. ... | Wikipedia - Triangle conic - Special triangle conics |
In the following, a short introduction to input-output analysis and its environmental extension for the calculation of material footprints or RME indicators is provided. The inter-industry flows within an economy form an n×n matrix Z and the total output of each industry forms an n×1 vector x. By dividing each flow int... | Wikipedia - Environmentally extended input–output analysis - Input-Output Analysis for EEIOA |
Next to the inter-industry flows recorded in Z, each industry requires additional inputs (e.g. energy, materials, capital, labour) and outputs (e.g. emissions) which can be introduced into the calculation with the help of an environmental extension. This commonly takes the shape of an m×n matrix M of total factor input... | Wikipedia - Environmentally extended input–output analysis - Input-Output Analysis for EEIOA |
In case of lacking data, expert opinions or additional modelling may be required to estimate the extension. Once completed, M can be transformed into a direct factor requirements matrix per unit of useful output F, and the calculation is analogous to determination of the monetary direct multipliers matrix A (see first ... | Wikipedia - Environmentally extended input–output analysis - Input-Output Analysis for EEIOA |
It allows us to allocate economy-wide material requirements to specific industries. With the help of the coefficients contained in the Leontief inverse (I−A)−1, the material requirements can be allocated to domestic or foreign (exports) final demand. In order to consider variations in production structures across diffe... | Wikipedia - Environmentally extended input–output analysis - Input-Output Analysis for EEIOA |
In the following, assume that the characteristic is not 2.Clifford algebras are Z2-graded algebras (also known as superalgebras). Indeed, the linear map on V defined by v ↦ −v (reflection through the origin) preserves the quadratic form Q and so by the universal property of Clifford algebras extends to an algebra autom... | Wikipedia - Clifford algebras - Grading |
The subspace Cl(V, Q) is called the odd part of Cl(V, Q) (it is not a subalgebra). This Z2-grading plays an important role in the analysis and application of Clifford algebras. | Wikipedia - Clifford algebras - Grading |
The automorphism α is called the main involution or grade involution. Elements that are pure in this Z2-grading are simply said to be even or odd. Remark. | Wikipedia - Clifford algebras - Grading |
The Clifford algebra is not a Z-graded algebra, but is Z-filtered, where CL ≤ i(V, Q) is the subspace spanned by all products of at most i elements of V. The degree of a Clifford number usually refers to the degree in the N-grading. The even subalgebra Cl(V, Q) of a Clifford algebra is itself isomorphic to a Clifford a... | Wikipedia - Clifford algebras - Grading |
In the following, colons and square brackets are used to denote homogeneous vectors. The rotation about axis r is a classical application of quaternions to space mapping. In terms of a homography, the rotation is expressed ( u 0 0 u ) = ∼ , {\displaystyle {\begin{pmatrix}u&0\\0&u\end{pmatrix}}=\thicksim ,} where u =... | Wikipedia - Quaternionic analysis - Homographies |
{\displaystyle {\begin{pmatrix}1&0\\p&1\end{pmatrix}}=.} Rotation and translation xr along the axis of rotation is given by ( u 0 u x r u ) = ∼ . {\displaystyle {\begin{pmatrix}u&0\\uxr&u\end{pmatrix}}=\thicksim .} | Wikipedia - Quaternionic analysis - Homographies |
Such a mapping is called a screw displacement. In classical kinematics, Chasles' theorem states that any rigid body motion can be displayed as a screw displacement. | Wikipedia - Quaternionic analysis - Homographies |
Just as the representation of a Euclidean plane isometry as a rotation is a matter of complex number arithmetic, so Chasles' theorem, and the screw axis required, is a matter of quaternion arithmetic with homographies: Let s be a right versor, or square root of minus one, perpendicular to r, with t = rs. Consider the a... | Wikipedia - Quaternionic analysis - Homographies |
Now in the (s,t)-plane the parameter θ traces out a circle u − 1 z = u − 1 ( 2 t sin θ ) = 2 sin θ ( t cos θ − s sin θ ) {\displaystyle u^{-1}z=u^{-1}(2t\sin \theta )=2\sin \theta (t\cos \theta -s\sin \theta )} in the half-plane { w t + x s: x > 0 } . {\displaystyle \lbrace wt+xs:x>0\rbrace .} Any p in this hal... | Wikipedia - Quaternionic analysis - Homographies |
In the following, expose ( v ) {\displaystyle {\text{expose}}(v)} extracts the left child l {\displaystyle l} , key k {\displaystyle k} , and right child r {\displaystyle r} of node v {\displaystyle v} into a tuple ( l , k , r ) {\displaystyle (l,k,r)} . Node ( l , k , r ) {\displaystyle {\text{Node}}(l,k,r)} creates a... | Wikipedia - Join-based tree algorithms - Join-based algorithms |
In the following, f k {\displaystyle f_{k}} denotes the value of the function f: { 0 , 1 } n → { 0 , 1 } {\displaystyle f:\{0,1\}^{n}\rightarrow \{0,1\}} when applied to an input vector of weight k {\displaystyle k} . | Wikipedia - Symmetric Boolean function - Properties |
In the following, fn is any of jn, yn, h(1)n, h(2)n for n = 0, ±1, ±2, ... | Wikipedia - Modified Bessel function of the second kind - Differential relations |
In the following, it is assumed that the system is 2-dimensional with z {\displaystyle z} as the invariant axis, i.e. ∂ ∂ z {\textstyle {\frac {\partial }{\partial z}}} produces 0 for any quantity. Then the magnetic field can be written in cartesian coordinates as or more compactly, where A ( x , y ) z ^ {\displaystyle... | Wikipedia - Grad–Shafranov equation - Derivation (in Cartesian coordinates) |
Two dimensional, stationary, magnetic structures are described by the balance of pressure forces and magnetic forces, i.e.: where p is the plasma pressure and j is the electric current. It is known that p is a constant along any field line, (again since ∇ p {\displaystyle \nabla p} is everywhere perpendicular to B). Ad... | Wikipedia - Grad–Shafranov equation - Derivation (in Cartesian coordinates) |
This means that j ⊥ × B ⊥ = 0 {\displaystyle \mathbf {j} _{\perp }\times \mathbf {B} _{\perp }=0} , i.e. j ⊥ {\displaystyle \mathbf {j} _{\perp }} is parallel to B ⊥ {\displaystyle \mathbf {B} _{\perp }} . The right hand side of the previous equation can be considered in two parts: where the ⊥ {\displaystyle \perp } su... | Wikipedia - Grad–Shafranov equation - Derivation (in Cartesian coordinates) |
In the following, it is assumed that triangulation is made on corresponding image points from two views generated by pinhole cameras. Generalization from these assumptions are discussed here. The image to the left illustrates the epipolar geometry of a pair of stereo cameras of pinhole model. A point x (3D point) in 3D... | Wikipedia - Triangulation (computer vision) - Introduction |
If y 1 {\displaystyle \mathbf {y} _{1}} and y 2 {\displaystyle \mathbf {y} _{2}} are given and the geometry of the two cameras are known, the two projection lines (green lines) can be determined and it must be the case that they intersect at point x (3D point). Using basic linear algebra that intersection point can be ... | Wikipedia - Triangulation (computer vision) - Introduction |
The position of the image points y 1 {\displaystyle \mathbf {y} _{1}} and y 2 {\displaystyle \mathbf {y} _{2}} cannot be measured exactly. The reason is a combination of factors such as Geometric distortion, for example lens distortion, which means that the 3D to 2D mapping of the camera deviates from the pinhole camer... | Wikipedia - Triangulation (computer vision) - Introduction |
A single ray of light from x (3D point) is dispersed in the lens system of the cameras according to a point spread function. The recovery of the corresponding image point from measurements of the dispersed intensity function in the images gives errors. In a digital camera, the image intensity function is only measured ... | Wikipedia - Triangulation (computer vision) - Introduction |
Inexact interpolation of the discrete intensity function have to be used to recover the true one. The image points y1' and y2' used for triangulation are often found using various types of feature extractors, for example of corners or interest points in general. There is an inherent localization error for any type of f... | Wikipedia - Triangulation (computer vision) - Introduction |
However, their projection lines (blue) do not have to intersect in 3D space or come close to x. In fact, these lines intersect if and only if y 1 ′ {\displaystyle \mathbf {y} '_{1}} and y 2 ′ {\displaystyle \mathbf {y} '_{2}} satisfy the epipolar constraint defined by the fundamental matrix. Given the measurement noise... | Wikipedia - Triangulation (computer vision) - Introduction |
Which 3D point xest is the best estimate of x given y 1 ′ {\displaystyle \mathbf {y} '_{1}} and y 2 ′ {\displaystyle \mathbf {y} '_{2}} and the geometry of the cameras? The answer is often found by defining an error measure which depends on xest and then minimizing this error. In the following sections, some of the var... | Wikipedia - Triangulation (computer vision) - Introduction |
In the following, it is thought that the system is a two-body system and the orbiting object has a negligible mass compared to the larger (central) object. In real-world orbital mechanics, it is the system's barycenter, not the larger object, which is at the focus. Specific orbital energy, or total energy, is equal to ... | Wikipedia - Orbital speed - Radial trajectories |
The sign of the result may be positive, zero, or negative and the sign tells us something about the type of orbit: If the specific orbital energy is positive the orbit is unbound, or open, and will follow a hyperbola with the larger body the focus of the hyperbola. Objects in open orbits do not return; once past periap... | Wikipedia - Orbital speed - Radial trajectories |
See radial parabolic trajectory. Parabolic orbits are also open. If the total energy is negative, Ek − Ep < 0: The orbit is bound, or closed. | Wikipedia - Orbital speed - Radial trajectories |
The motion will be on an ellipse with one focus at the other body. See radial elliptic trajectory, free-fall time. Planets have bound orbits around the Sun. | Wikipedia - Orbital speed - Radial trajectories |
In the following, label is an optional identifier terminated by a colon, and block is a sequence of one of more Perl statements surrounded by braces. All looping constructs except for the C-style for-loop can have a continue block that is executed after each iteration of the loop body, before the loop condition is eval... | Wikipedia - Perl control structures - Loops |
The second expression is evaluated prior to each iteration and the loop is terminated if it evaluates to false. The third expression is evaluated after each iteration, prior to deciding whether to perform the next. This for loop is the only looping construct that can not have a continue block, but expr3 is functionally... | Wikipedia - Perl control structures - Loops |
label for var ( list ) block label for var ( list ) block continue block label foreach var ( list ) block label foreach var ( list ) block continue block In foreach, var is a scalar variable that defaults to $_ if omitted. For each element of list, var is aliased to the element, and the loop body is executed once. The ... | Wikipedia - Perl control structures - Loops |
label while ( expr ) block label while ( expr ) block continue block label until ( expr ) block label until ( expr ) block continue block The while loop repeatedly executes the loop body as long as the controlling expression is true. The condition is evaluated before the loop body. until is similar, but executes the lo... | Wikipedia - Perl control structures - Loops |
In the following, let G = ( V , E ) {\displaystyle G=(V,E)} be an arbitrary graph. | Wikipedia - Circular arc graph - Some subclasses |
In the following, let G = (V, E) denote an undirected Graph with a set of vertices V and a set of edges E ⊆ V × V. The set sizes are denoted by |V| = n and |E| = m. Additionally, if not noted otherwise, the metric space [0,1)d with the euclidean distance is considered, i.e. for any points x , y ∈ [ 0 , 1 ) d {\displays... | Wikipedia - Random geometric graphs - Definition |
In the following, let G = A ⋊ H {\displaystyle G=A\rtimes H} be a semidirect product such that the normal semidirect factor, A {\displaystyle A} , is abelian. The irreducible representations of such a group G , {\displaystyle G,} can be classified by showing that all irreducible representations of G {\displaystyle G} c... | Wikipedia - Complex representations of finite groups - Classification of representations of a semidirect product |
{\displaystyle \mathrm {X} ={\text{Hom}}(A,\mathbb {C} ^{\times }).} The group G {\displaystyle G} acts on X {\displaystyle \mathrm {X} } by ( s χ ) ( a ) = χ ( s − 1 a s ) {\displaystyle (s\chi )(a)=\chi (s^{-1}as)} for s ∈ G , χ ∈ X , a ∈ A . {\displaystyle s\in G,\chi \in \mathrm {X} ,a\in A.} | Wikipedia - Complex representations of finite groups - Classification of representations of a semidirect product |
Let ( χ j ) j ∈ X / H {\displaystyle (\chi _{j})_{j\in \mathrm {X} /H}} be a representative system of the orbit of H {\displaystyle H} in X . {\displaystyle \mathrm {X} .} For every j ∈ X / H {\displaystyle j\in \mathrm {X} /H} let H j = { t ∈ H: t χ j = χ j } . | Wikipedia - Complex representations of finite groups - Classification of representations of a semidirect product |
{\displaystyle H_{j}=\{t\in H:t\chi _{j}=\chi _{j}\}.} This is a subgroup of H . {\displaystyle H.} | Wikipedia - Complex representations of finite groups - Classification of representations of a semidirect product |
Let G j = A ⋅ H j {\displaystyle G_{j}=A\cdot H_{j}} be the corresponding subgroup of G . {\displaystyle G.} We now extend the function χ j {\displaystyle \chi _{j}} onto G j {\displaystyle G_{j}} by χ j ( a t ) = χ j ( a ) {\displaystyle \chi _{j}(at)=\chi _{j}(a)} for a ∈ A , t ∈ H j . | Wikipedia - Complex representations of finite groups - Classification of representations of a semidirect product |
{\displaystyle a\in A,t\in H_{j}.} Thus, χ j {\displaystyle \chi _{j}} is a class function on G j . {\displaystyle G_{j}.} | Wikipedia - Complex representations of finite groups - Classification of representations of a semidirect product |
Moreover, since t χ j = χ j {\displaystyle t\chi _{j}=\chi _{j}} for all t ∈ H j , {\displaystyle t\in H_{j},} it can be shown that χ j {\displaystyle \chi _{j}} is a group homomorphism from G j {\displaystyle G_{j}} to C × . {\displaystyle \mathbb {C} ^{\times }.} | Wikipedia - Complex representations of finite groups - Classification of representations of a semidirect product |
Therefore, we have a representation of G j {\displaystyle G_{j}} of degree one which is equal to its own character. Let now ρ {\displaystyle \rho } be an irreducible representation of H j . {\displaystyle H_{j}.} | Wikipedia - Complex representations of finite groups - Classification of representations of a semidirect product |
Then we obtain an irreducible representation ρ ~ {\displaystyle {\tilde {\rho }}} of G j , {\displaystyle G_{j},} by combining ρ {\displaystyle \rho } with the canonical projection G j → H j . {\displaystyle G_{j}\to H_{j}.} Finally, we construct the tensor product of χ j {\displaystyle \chi _{j}} and ρ ~ . | Wikipedia - Complex representations of finite groups - Classification of representations of a semidirect product |
{\displaystyle {\tilde {\rho }}.} Thus, we obtain an irreducible representation χ j ⊗ ρ ~ {\displaystyle \chi _{j}\otimes {\tilde {\rho }}} of G j . {\displaystyle G_{j}.} | Wikipedia - Complex representations of finite groups - Classification of representations of a semidirect product |
To finally obtain the classification of the irreducible representations of G {\displaystyle G} we use the representation θ j , ρ {\displaystyle \theta _{j,\rho }} of G , {\displaystyle G,} which is induced by the tensor product χ j ⊗ ρ ~ . {\displaystyle \chi _{j}\otimes {\tilde {\rho }}.} Thus, we achieve the followin... | Wikipedia - Complex representations of finite groups - Classification of representations of a semidirect product |
θ j , ρ {\displaystyle \theta _{j,\rho }} is irreducible. If θ j , ρ {\displaystyle \theta _{j,\rho }} and θ j ′ , ρ ′ {\displaystyle \theta _{j',\rho '}} are isomorphic, then j = j ′ {\displaystyle j=j'} and additionally ρ {\displaystyle \rho } is isomorphic to ρ ′ . {\displaystyle \rho '.} | Wikipedia - Complex representations of finite groups - Classification of representations of a semidirect product |
Every irreducible representation of G {\displaystyle G} is isomorphic to one of the θ j , ρ . {\displaystyle \theta _{j,\rho }.} | Wikipedia - Complex representations of finite groups - Classification of representations of a semidirect product |
Amongst others, the criterion of Mackey and a conclusion based on the Frobenius reciprocity are needed for the proof of the proposition. Further details may be found in . In other words, we classified all irreducible representations of G = A ⋊ H . {\displaystyle G=A\rtimes H.} | Wikipedia - Complex representations of finite groups - Classification of representations of a semidirect product |
In the following, let M be a G-module. | Wikipedia - Crossed homomorphism - Properties |
In the following, let X {\displaystyle X} be a topological space, G , H {\displaystyle G,H} topological groups and a group homomorphism ϕ: H → G {\displaystyle \phi \colon H\to G} . | Wikipedia - Reduction of structure group - Definition |
In the following, let X {\displaystyle X} be our n {\displaystyle n} -dimensional input space. Let H {\displaystyle {\mathcal {H}}} be a class of functions that we wish to use in order to learn a { 0 , 1 } {\displaystyle \{0,1\}} -valued target function f {\displaystyle f} defined over X {\displaystyle X} . Let D {\dis... | Wikipedia - Error tolerance (PAC learning) - Notation and the Valiant learning model |
Let us suppose we have a function s i z e ( f ) {\displaystyle size(f)} that can measure the complexity of f {\displaystyle f} . Let Oracle ( x ) {\displaystyle {\text{Oracle}}(x)} be an oracle that, whenever called, returns an example x {\displaystyle x} and its correct label f ( x ) {\displaystyle f(x)} . When no noi... | Wikipedia - Error tolerance (PAC learning) - Notation and the Valiant learning model |
In the following, let f: X → Y be a morphism of schemes. The composition of two proper morphisms is proper. Any base change of a proper morphism f: X → Y is proper. That is, if g: Z → Y is any morphism of schemes, then the resulting morphism X ×Y Z → Z is proper. | Wikipedia - Proper morphism - Properties and characterizations of proper morphisms |
Properness is a local property on the base (in the Zariski topology). That is, if Y is covered by some open subschemes Yi and the restriction of f to all f−1(Yi) is proper, then so is f. More strongly, properness is local on the base in the fpqc topology. For example, if X is a scheme over a field k and E is a field ex... | Wikipedia - Proper morphism - Properties and characterizations of proper morphisms |
More generally, finite morphisms are proper. This is a consequence of the going up theorem. | Wikipedia - Proper morphism - Properties and characterizations of proper morphisms |
By Deligne, a morphism of schemes is finite if and only if it is proper and quasi-finite. This had been shown by Grothendieck if the morphism f: X → Y is locally of finite presentation, which follows from the other assumptions if Y is noetherian. For X proper over a scheme S, and Y separated over S, the image of any mo... | Wikipedia - Proper morphism - Properties and characterizations of proper morphisms |
The Stein factorization theorem states that any proper morphism to a locally noetherian scheme can be factored as X → Z → Y, where X → Z is proper, surjective, and has geometrically connected fibers, and Z → Y is finite. Chow's lemma says that proper morphisms are closely related to projective morphisms. One version is... | Wikipedia - Proper morphism - Properties and characterizations of proper morphisms |
Nagata's compactification theorem, as generalized by Deligne, says that a separated morphism of finite type between quasi-compact and quasi-separated schemes factors as an open immersion followed by a proper morphism. Proper morphisms between locally noetherian schemes preserve coherent sheaves, in the sense that the h... | Wikipedia - Proper morphism - Properties and characterizations of proper morphisms |
As a very special case: the ring of regular functions on a proper scheme X over a field k has finite dimension as a k-vector space. By contrast, the ring of regular functions on the affine line over k is the polynomial ring k, which does not have finite dimension as a k-vector space. There is also a slightly stronger s... | Wikipedia - Proper morphism - Properties and characterizations of proper morphisms |
If the support of F is proper over S, then for each i ≥ 0 {\displaystyle i\geq 0} the higher direct image R i f ∗ F {\displaystyle R^{i}f_{*}F} is coherent. For a scheme X of finite type over the complex numbers, the set X(C) of complex points is a complex analytic space, using the classical (Euclidean) topology. For X... | Wikipedia - Proper morphism - Properties and characterizations of proper morphisms |
In the following, literary, epigraphical and documentary sources referring to watermills and other water-driven machines are listed. | Wikipedia - Roman watermill - Written sources |
In the following, matrices will be indicated by indexed variables. "Subject" indices will be indicated using letters a {\displaystyle a} , b {\displaystyle b} and c {\displaystyle c} , with values running from 1 {\displaystyle 1} to p {\displaystyle p} which is equal to 10 {\displaystyle 10} in the above example. "Fact... | Wikipedia - Factor loading - Mathematical model of the same example |
Thus, no generality is lost by assuming that the standard deviation of the factors for verbal intelligence is 1 {\displaystyle 1} . Likewise for mathematical intelligence. Moreover, for similar reasons, no generality is lost by assuming the two factors are uncorrelated with each other. | Wikipedia - Factor loading - Mathematical model of the same example |
In other words: ∑ i F p i F q i = δ p q {\displaystyle \sum _{i}F_{pi}F_{qi}=\delta _{pq}} where δ p q {\displaystyle \delta _{pq}} is the Kronecker delta ( 0 {\displaystyle 0} when p ≠ q {\displaystyle p\neq q} and 1 {\displaystyle 1} when p = q {\displaystyle p=q} ).The errors are assumed to be independent of the fac... | Wikipedia - Factor loading - Mathematical model of the same example |
Even if they are uncorrelated, we cannot tell which factor corresponds to verbal intelligence and which corresponds to mathematical intelligence without an outside argument. The values of the loadings L {\displaystyle L} , the averages μ {\displaystyle \mu } , and the variances of the "errors" ε {\displaystyle \varepsi... | Wikipedia - Factor loading - Mathematical model of the same example |
The first term on the right is the "reduced correlation matrix" and will be equal to the correlation matrix except for its diagonal values which will be less than unity. These diagonal elements of the reduced correlation matrix are called "communalities" (which represent the fraction of the variance in the observed var... | Wikipedia - Factor loading - Mathematical model of the same example |
This is to be contrasted with principal component analysis which seeks to minimize the mean square error of all residuals. Before the advent of high-speed computers, considerable effort was devoted to finding approximate solutions to the problem, particularly in estimating the communalities by other means, which then s... | Wikipedia - Factor loading - Mathematical model of the same example |
With the advent of high-speed computers, the minimization problem can be solved iteratively with adequate speed, and the communalities are calculated in the process, rather than being needed beforehand. The MinRes algorithm is particularly suited to this problem, but is hardly the only iterative means of finding a solu... | Wikipedia - Factor loading - Mathematical model of the same example |
In the following, n {\displaystyle n} denotes the number of people in the initial circle, and k {\displaystyle k} denotes the count for each step, that is, k − 1 {\displaystyle k-1} people are skipped and the k {\displaystyle k} -th is executed. The people in the circle are numbered from 1 {\displaystyle 1} to n {\disp... | Wikipedia - Josephus permutation - Solution |
In the following, some cross sections which are of importance in a nuclear reactor are given. The thermal cross-section is averaged using a Maxwellian spectrum and the fast cross section is averaged using the uranium-235 fission spectrum. The cross sections are taken from the JEFF-3.1.1 library using JANIS software. * ... | Wikipedia - Neutron cross section - Typical cross sections |
In the following, technology is less emphasised. The boundaries between the below systems are often blurred. The example of transplantation is international but could be classified as a micro-economy, too. Mutualism in the sense of a (moneyless) economic theory. | Wikipedia - Non-monetary economy - Other moneyless systems |
People contribute to a community without payment not only to help but also because they expect to be helped by a member of the community when in need (a selfish interpretation of solidarity.) So, the term 'mutualism' is understood as aid by the community and not necessarily reciprocation. For example, if a transplantat... | Wikipedia - Non-monetary economy - Other moneyless systems |
Other examples are the anarchist communities during the Spanish Civil War, where quota and rations were used for distribution, and the Mink'a communal work. Debt system, as used in manorialism or with the aid of the tally stick. To continue the previous example, if a transplantation center donates a liver to another ce... | Wikipedia - Non-monetary economy - Other moneyless systems |
To save transportation cost, such obligations are passed from one creditor center to the other in the obvious way. The transplantation clearing house could further diminish transportation cost by deferring such alienation of the obligations. This would be similar to the fairs from which the banks and government-issued ... | Wikipedia - Non-monetary economy - Other moneyless systems |
The redistribution economy, which is a more authoritarian case of mutualism. For example, the Incas and possibly, also the empire of Majapahit. A combination of mutualism and redistribution: Uruganda and similar economies like Umuganda/Isarongo and Ubuntu. | Wikipedia - Non-monetary economy - Other moneyless systems |
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