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A quadric curve in P 2 {\displaystyle \mathbf {P} ^{2}} is called a conic. A split conic over k is isomorphic to the projective line P 1 {\displaystyle \mathbf {P} ^{1}} over k, embedded in P 2 {\displaystyle \mathbf {P} ^{2}} by the 2nd Veronese embedding. (For example, ellipses, parabolas and hyperbolas are different... | Wikipedia - Quadric (algebraic geometry) | null | null | null |
)A split quadric surface X is isomorphic to P 1 × P 1 {\displaystyle \mathbf {P} ^{1}\times \mathbf {P} ^{1}} , embedded in P 3 {\displaystyle \mathbf {P} ^{3}} by the Segre embedding. The space of lines in the quadric surface X has two connected components, each isomorphic to P 1 {\displaystyle \mathbf {P} ^{1}} .A sp... | Wikipedia - Quadric (algebraic geometry) | null | null | null |
Namely, given a 4-dimensional vector space V with a symplectic form, the quadric 3-fold X can be identified with the space LGr(2,4) of 2-planes in V on which the form restricts to zero. Furthermore, the space of lines in the quadric 3-fold X is isomorphic to P 3 {\displaystyle \mathbf {P} ^{3}} .A split quadric 4-fold ... | Wikipedia - Quadric (algebraic geometry) | null | null | null |
The space of 2-planes in the quadric 4-fold X has two connected components, each isomorphic to P 3 {\displaystyle \mathbf {P} ^{3}} .The space of 2-planes in a split quadric 5-fold is isomorphic to a split quadric 6-fold. Likewise, both components of the space of 3-planes in a split quadric 6-fold are isomorphic to a s... | Wikipedia - Quadric (algebraic geometry) | null | null | null |
Let X be a topological space and P X = { γ: → X } {\displaystyle PX=\{\gamma :\,\to \,X\}} be the space of all continuous paths in X. Define the projection π: P X → X × X {\displaystyle \pi :PX\to \,X\times X} by π ( γ ) = ( γ ( 0 ) , γ ( 1 ) ) {\displaystyle \pi (\gamma )=(\gamma (0),\gamma (1))} . The topological co... | Wikipedia - Topological complexity | null | null | null |
Let X be a topological space. The Borel space associated to X is the pair (X,B), where B is the σ-algebra of Borel sets of X. George Mackey defined a Borel space somewhat differently, writing that it is "a set together with a distinguished σ-field of subsets called its Borel sets." However, modern usage is to call the ... | Wikipedia - Borel set | null | null | null |
There exist measurable spaces that are not Borel spaces, for any choice of topology on the underlying space.Measurable spaces form a category in which the morphisms are measurable functions between measurable spaces. A function f: X → Y {\displaystyle f:X\rightarrow Y} is measurable if it pulls back measurable sets, i.... | Wikipedia - Borel set | null | null | null |
Then X as a Borel space is isomorphic to one of R, Z, a finite space. (This result is reminiscent of Maharam's theorem.) Considered as Borel spaces, the real line R, the union of R with a countable set, and Rn are isomorphic. | Wikipedia - Borel set | null | null | null |
A standard Borel space is the Borel space associated to a Polish space. A standard Borel space is characterized up to isomorphism by its cardinality, and any uncountable standard Borel space has the cardinality of the continuum. For subsets of Polish spaces, Borel sets can be characterized as those sets that are the ra... | Wikipedia - Borel set | null | null | null |
Note however, that the range of a continuous noninjective map may fail to be Borel. See analytic set. Every probability measure on a standard Borel space turns it into a standard probability space. | Wikipedia - Borel set | null | null | null |
Let X be an affine algebraic variety embedded into the affine space k n {\displaystyle k^{n}} , with defining ideal I ⊂ k {\displaystyle I\subset k} . For any polynomial f, let in ( f ) {\displaystyle \operatorname {in} (f)} be the homogeneous component of f of the lowest degree, the initial term of f, and let in ... | Wikipedia - Tangent cone | null | null | null |
(The tangent cone at a point of k n {\displaystyle k^{n}} that is not contained in X is empty.) For example, the nodal curve C: y 2 = x 3 + x 2 {\displaystyle C:y^{2}=x^{3}+x^{2}} is singular at the origin, because both partial derivatives of f(x, y) = y2 − x3 − x2 vanish at (0, 0). Thus the Zariski tangent space to C ... | Wikipedia - Tangent cone | null | null | null |
On the other hand, the tangent cone is the union of the tangent lines to the two branches of C at the origin, x = y , x = − y . {\displaystyle x=y,\quad x=-y.} Its defining ideal is the principal ideal of k generated by the initial term of f, namely y2 − x2 = 0. | Wikipedia - Tangent cone | null | null | null |
Let X be an equivariant algebraic scheme. | Wikipedia - Equivariant algebraic K-theory | null | null | null |
Let X be an ordered vector space over the reals that is finite-dimensional. Then the order of X is Archimedean if and only if the positive cone of X is closed for the unique topology under which X is a Hausdorff TVS. Let X be an ordered vector space over the reals with positive cone C. Then the following are equivalent... | Wikipedia - Ordered topological vector space | null | null | null |
Let X be the complex projective plane with its standard symplectic form (corresponding to the Fubini–Study metric) and complex structure. Let ℓ ∈ H 2 ( X ) {\displaystyle \ell \in H^{2}(X)} be the Poincaré dual of a line L. Then H ∗ ( X ) ≅ Z / ℓ 3 . {\displaystyle H^{*}(X)\cong \mathbf {Z} /\ell ^{3}.} The only nonze... | Wikipedia - Quantum cup product | null | null | null |
Therefore, ℓ ∗ ℓ = ℓ 2 e 0 + 0 e L = ℓ 2 , {\displaystyle \ell *\ell =\ell ^{2}e^{0}+0e^{L}=\ell ^{2},} ℓ ∗ ℓ 2 = 0 e 0 + 1 e L = e L . {\displaystyle \ell *\ell ^{2}=0e^{0}+1e^{L}=e^{L}.} | Wikipedia - Quantum cup product | null | null | null |
In this case it is convenient to rename e L {\displaystyle e^{L}} as q and use the simpler coefficient ring Z. This q is of degree 6 = 2 c 1 ( L ) {\displaystyle 6=2c_{1}(L)} . Then Q H ∗ ( X , Z ) ≅ Z / ( ℓ 3 = q ) . {\displaystyle QH^{*}(X,\mathbf {Z} )\cong \mathbf {Z} /(\ell ^{3}=q).} | Wikipedia - Quantum cup product | null | null | null |
Let X n i = x ∈ { 0 , 1 } {\displaystyle X_{ni}=x\in \{0,1\}} be a dichotomous random variable where, for example, x = 1 {\displaystyle x=1} denotes a correct response and x = 0 {\displaystyle x=0} an incorrect response to a given assessment item. In the Rasch model for dichotomous data, the probability of the outcome ... | Wikipedia - Rasch model | null | null | null |
Given two examinees with different ability parameters β 1 {\displaystyle \beta _{1}} and β 2 {\displaystyle \beta _{2}} and an arbitrary item with difficulty δ i {\displaystyle \delta _{i}} , compute the difference in logits for these two examinees by ( β 1 − δ i ) − ( β 2 − δ i ) {\displaystyle (\beta _{1}-\delta _{i}... | Wikipedia - Rasch model | null | null | null |
For example, l o g - o d d s { X n 1 = 1 ∣ r n = 1 } = δ 2 − δ 1 , {\displaystyle \operatorname {log-odds} \{X_{n1}=1\mid \ r_{n}=1\}=\delta _{2}-\delta _{1},\,} where r n {\displaystyle r_{n}} is the total score of person n over the two items, which implies a correct response to one or other of the items. Hence, the... | Wikipedia - Rasch model | null | null | null |
More generally, a number of item parameters can be estimated iteratively through application of a process such as Conditional Maximum Likelihood estimation (see Rasch model estimation). While more involved, the same fundamental principle applies in such estimations. The ICC of the Rasch model for dichotomous data is sh... | Wikipedia - Rasch model | null | null | null |
The grey line maps the probability of the discrete outcome X n i = 1 {\displaystyle X_{ni}=1} (that is, correctly answering the question) for persons with different locations on the latent continuum (that is, their level of abilities). The location of an item is, by definition, that location at which the probability th... | Wikipedia - Rasch model | null | null | null |
For example, in the case of an assessment item used in the context of educational psychology, these could represent the proportions of persons who answered the item correctly. Persons are ordered by the estimates of their locations on the latent continuum and classified into Class Intervals on this basis in order to gr... | Wikipedia - Rasch model | null | null | null |
Let X {\displaystyle X\,\!} be a state variable which can take state values x ∈ X {\displaystyle x\in \mathbb {X} \,\!} . A multinomial opinion over X {\displaystyle X\,\!} | Wikipedia - Subjective logic | null | null | null |
is the composite tuple ω X = ( b X , u X , a X ) {\displaystyle \omega _{X}=(b_{X},u_{X},a_{X})\,\!} , where b X {\displaystyle b_{X}\,\!} is a belief mass distribution over the possible state values of X {\displaystyle X\,\!} | Wikipedia - Subjective logic | null | null | null |
, u X {\displaystyle u_{X}\,\!} is the uncertainty mass, and a X {\displaystyle a_{X}\,\!} is the prior (base rate) probability distribution over the possible state values of X {\displaystyle X\,\!} | Wikipedia - Subjective logic | null | null | null |
. These parameters satisfy u X + ∑ b X ( x ) = 1 {\displaystyle u_{X}+\sum b_{X}(x)=1\,\!} | Wikipedia - Subjective logic | null | null | null |
and ∑ a X ( x ) = 1 {\displaystyle \sum a_{X}(x)=1\,\!} as well as b X ( x ) , u X , a X ( x ) ∈ {\displaystyle b_{X}(x),u_{X},a_{X}(x)\in \,\!} . | Wikipedia - Subjective logic | null | null | null |
Trinomial opinions can be simply visualised as points inside a tetrahedron, but opinions with dimensions larger than trinomial do not lend themselves to simple visualisation. Dirichlet PDFs are normally denoted as D i r ( p X ; α X ) {\displaystyle \mathrm {Dir} (p_{X};\alpha _{X})\,\!} where p X {\displaystyle p_{X}\,... | Wikipedia - Subjective logic | null | null | null |
is a probability distribution over the state values of X {\displaystyle X} , and α X {\displaystyle \alpha _{X}\,\!} are the strength parameters. The Dirichlet PDF of a multinomial opinion ω X = ( b X , u X , a X ) {\displaystyle \omega _{X}=(b_{X},u_{X},a_{X})\,\!} is the function D i r ( p X ; α X ) {\displaystyle \m... | Wikipedia - Subjective logic | null | null | null |
Let X {\displaystyle X} and Y {\displaystyle Y} be TVSs, D ⊆ X , {\displaystyle D\subseteq X,} and f: D → Y {\displaystyle f:D\to Y} be a map. Then f: D → Y {\displaystyle f:D\to Y} is uniformly continuous if for every neighborhood U {\displaystyle U} of the origin in X , {\displaystyle X,} there exists a neighborhood ... | Wikipedia - Complete TVS | null | null | null |
If x ∙ = ( x i ) i ∈ I {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}} is a Cauchy net in D {\displaystyle D} then f ∘ x ∙ = ( f ( x i ) ) i ∈ I {\displaystyle f\circ x_{\bullet }=\left(f\left(x_{i}\right)\right)_{i\in I}} is a Cauchy net in Y . {\displaystyle Y.} If B {\displaystyle {\mathcal {B}}} is a Cauch... | Wikipedia - Complete TVS | null | null | null |
Let X {\displaystyle X} be a compact Riemann surface of genus g {\displaystyle g} and fix a canonical homology basis a 1 , … , a g , b 1 , … , b g {\displaystyle a_{1},\dots ,a_{g},b_{1},\dots ,b_{g}} of H 1 ( X , Z ) {\displaystyle H_{1}(X,\mathbf {Z} )} with intersection numbers a i ∘ a j = b i ∘ b j = 0 , a i ∘ b j ... | Wikipedia - Tau function (integrable systems) | null | null | null |
{\displaystyle \mathbf {S} _{g}=\left\{B\in \mathrm {Mat} _{g\times g}(\mathbf {C} )\ \colon \ B^{T}=B,\ {\text{Im}}(B){\text{ is positive definite}}\right\}.} The Riemann θ {\displaystyle \theta } function on C g {\displaystyle \mathbf {C} ^{g}} corresponding to the period matrix B {\displaystyle B} is defined to be θ... | Wikipedia - Tau function (integrable systems) | null | null | null |
{\displaystyle \theta (Z|B):=\sum _{N\in \mathbb {Z} ^{g}}e^{i\pi (N,BN)+2i\pi (N,Z)}.} Choose a point p ∞ ∈ X {\displaystyle p_{\infty }\in X} , a local parameter ζ {\displaystyle \zeta } in a neighbourhood of p ∞ {\displaystyle p_{\infty }} with ζ ( p ∞ ) = 0 {\displaystyle \zeta (p_{\infty })=0} and a positive divis... | Wikipedia - Tau function (integrable systems) | null | null | null |
For any positive integer k ∈ N + {\displaystyle k\in \mathbf {N} ^{+}} let Ω k {\displaystyle \Omega _{k}} be the unique meromorphic differential of the second kind characterized by the following conditions: The only singularity of Ω k {\displaystyle \Omega _{k}} is a pole of order k + 1 {\displaystyle k+1} at p = p ∞ ... | Wikipedia - Tau function (integrable systems) | null | null | null |
Let X {\displaystyle X} be a non-trivial (i.e. X ≠ { 0 } {\displaystyle X\neq \{0\}} ) real or complex vector space and let d {\displaystyle d} be the translation-invariant trivial metric on X {\displaystyle X} defined by d ( x , x ) = 0 {\displaystyle d(x,x)=0} and d ( x , y ) = 1 for all x , y ∈ X {\displaystyle d(x,... | Wikipedia - Metrizable topological vector space | null | null | null |
What fails is that scalar multiplication isn't continuous on ( X , τ ) . {\displaystyle (X,\tau ).} This example shows that a translation-invariant (pseudo)metric is not enough to guarantee a vector topology, which leads us to define paranorms and F-seminorms. | Wikipedia - Metrizable topological vector space | null | null | null |
Let X {\displaystyle X} be a point, A → {\displaystyle {\vec {A}}} a vector located at X {\displaystyle X} , and B → {\displaystyle {\vec {B}}} a vector field. The idea of differentiating B → {\displaystyle {\vec {B}}} at X {\displaystyle X} along the direction of A → {\displaystyle {\vec {A}}} in a physically meaningf... | Wikipedia - Mathematics of general relativity | null | null | null |
Notions of parallel transport can then be defined similarly as for the case of vector fields. By definition, a covariant derivative of a scalar field is equal to the regular derivative of the field. | Wikipedia - Mathematics of general relativity | null | null | null |
In the literature, there are three common methods of denoting covariant differentiation: Many standard properties of regular partial derivatives also apply to covariant derivatives: In general relativity, one usually refers to "the" covariant derivative, which is the one associated with Levi-Civita affine connection. B... | Wikipedia - Mathematics of general relativity | null | null | null |
Let X {\displaystyle X} be a real or complex vector space and let K {\displaystyle K} be an absorbing disk in X . {\displaystyle X.} p K {\displaystyle p_{K}} is a seminorm on X . {\displaystyle X.} | Wikipedia - Minkowski gauge | null | null | null |
p K {\displaystyle p_{K}} is a norm on X {\displaystyle X} if and only if K {\displaystyle K} does not contain a non-trivial vector subspace. p s K = 1 | s | p K {\displaystyle p_{sK}={\frac {1}{|s|}}p_{K}} for any scalar s ≠ 0. | Wikipedia - Minkowski gauge | null | null | null |
{\displaystyle s\neq 0.} If J {\displaystyle J} is an absorbing disk in X {\displaystyle X} and J ⊆ K {\displaystyle J\subseteq K} then p K ≤ p J . {\displaystyle p_{K}\leq p_{J}.} | Wikipedia - Minkowski gauge | null | null | null |
If K {\displaystyle K} is a set satisfying { x ∈ X: p ( x ) < 1 } ⊆ K ⊆ { x ∈ X: p ( x ) ≤ 1 } {\displaystyle \{x\in X:p(x)<1\}\;\subseteq \;K\;\subseteq \;\{x\in X:p(x)\leq 1\}} then K {\displaystyle K} is absorbing in X {\displaystyle X} and p = p K , {\displaystyle p=p_{K},} where p K {\displaystyle p_{K}} is the Mi... | Wikipedia - Minkowski gauge | null | null | null |
{\displaystyle \{x\in X:q(x)<1\}\;\subseteq \;K\;\subseteq \;\{x\in X:q(x)\leq 1\}.} If x ∈ X {\displaystyle x\in X} satisfies p K ( x ) < 1 {\displaystyle p_{K}(x)<1} then x ∈ K . {\displaystyle x\in K.} | Wikipedia - Minkowski gauge | null | null | null |
Let X {\displaystyle X} be a separable Hilbert space. Let P p ( X ) {\displaystyle {\mathcal {P}}_{p}(X)} denote the collection of probability measures on X {\displaystyle X} that have finite p {\displaystyle p} -th moment; let P p r ( X ) {\displaystyle {\mathcal {P}}_{p}^{r}(X)} denote those elements μ ∈ P p ( X ) {\... | Wikipedia - Optimal transport | null | null | null |
{\displaystyle \kappa =(\mathrm {id} _{X}\times r)_{*}(\mu )\in \Gamma (\mu ,\nu ).} Moreover, if ν {\displaystyle \nu } has bounded support, then r ( x ) = x − | ∇ φ ( x ) | q − 2 ∇ φ ( x ) {\displaystyle r(x)=x-|\nabla \varphi (x)|^{q-2}\,\nabla \varphi (x)} for μ {\displaystyle \mu } -almost all x ∈ X {\displaystyle... | Wikipedia - Optimal transport | null | null | null |
Let X {\displaystyle X} be a set, and let S {\displaystyle {\mathcal {S}}} be a collection of subsets of X . {\displaystyle X.} A function ϕ {\displaystyle \phi } on S {\displaystyle {\mathcal {S}}} with values in an abelian semigroup R {\displaystyle R} is called a valuation if it satisfies whenever A , {\displaystyle... | Wikipedia - Valuation (geometry) | null | null | null |
Let X {\displaystyle X} be a topological space. According to De Morgan's laws, the collection T {\displaystyle T} of closed sets satisfies the following properties: The empty set and X {\displaystyle X} are elements of T {\displaystyle T} The intersection of any collection of sets in T {\displaystyle T} is also in T . ... | Wikipedia - Neighborhood characterization of topological spaces | null | null | null |
{\displaystyle T.} Now suppose that X {\displaystyle X} is only a set. Given any collection T {\displaystyle T} of subsets of X {\displaystyle X} which satisfy the above axioms, the corresponding set { U: X ∖ U ∈ T } {\displaystyle \{U:X\setminus U\in T\}} is a topology on X , {\displaystyle X,} and it is the only topo... | Wikipedia - Neighborhood characterization of topological spaces | null | null | null |
This is to say that a topology can be defined by declaring the closed sets. As such, one can rephrase all definitions to be in terms of closed sets: Given a second topological space Y , {\displaystyle Y,} a function f: X → Y {\displaystyle f:X\to Y} is continuous if and only if for every closed subset U {\displaystyle ... | Wikipedia - Neighborhood characterization of topological spaces | null | null | null |
a subset C {\displaystyle C} of X {\displaystyle X} is open if and only if its complement X ∖ C {\displaystyle X\setminus C} is closed. given a subset A {\displaystyle A} of X , {\displaystyle X,} the closure is the intersection of all closed sets containing A . {\displaystyle A.} given a subset A {\displaystyle A} of ... | Wikipedia - Neighborhood characterization of topological spaces | null | null | null |
Let X {\displaystyle X} be a topological space. The Hurewicz game played on X {\displaystyle X} is a game with two players Alice and Bob. 1st round: Alice chooses an open cover U 1 {\displaystyle {\mathcal {U}}_{1}} of X {\displaystyle X} . Bob chooses a finite set F 1 ⊂ U 1 {\displaystyle {\mathcal {F}}_{1}\subset {\m... | Wikipedia - Hurewicz space | null | null | null |
2nd round: Alice chooses an open cover U 2 {\displaystyle {\mathcal {U}}_{2}} of X {\displaystyle X} . Bob chooses a finite set F 2 ⊂ U 2 {\displaystyle {\mathcal {F}}_{2}\subset {\mathcal {U}}_{2}} . | Wikipedia - Hurewicz space | null | null | null |
etc. If every point of the space X {\displaystyle X} belongs to all but finitely many sets ⋃ F 1 , ⋃ F 2 , … {\displaystyle \bigcup {\mathcal {F}}_{1},\bigcup {\mathcal {F}}_{2},\ldots } , then Bob wins the Hurewicz game. Otherwise, Alice wins. A player has a winning strategy if he knows how to play in order to win the... | Wikipedia - Hurewicz space | null | null | null |
Let X {\displaystyle X} be a topological space. The Menger game G fin ( O , O ) {\displaystyle {\text{G}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )} played on X {\displaystyle X} is a game for two players, Alice and Bob. It has an inning per each natural number n {\displaystyle n} . At the n t h {\displaystyle n^{th}} in... | Wikipedia - Selection principle | null | null | null |
If the family ⋃ n = 1 ∞ F n {\displaystyle \bigcup _{n=1}^{\infty }{\mathcal {F}}_{n}} is a cover of the space X {\displaystyle X} , then Bob wins the game. Otherwise, Alice wins. A strategy for a player is a function determining the move of the player, given the earlier moves of both players. | Wikipedia - Selection principle | null | null | null |
A strategy for a player is a winning strategy if each play where this player sticks to this strategy is won by this player. A topological space is S fin ( O , O ) {\displaystyle {\text{S}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )} if and only if Alice has no winning strategy in the game G fin ( O , O ) {\displaystyle {\... | Wikipedia - Selection principle | null | null | null |
Bob has a winning strategy in the game G fin ( O , O ) {\displaystyle {\text{G}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )} played on the space X {\displaystyle X} if and only if the space X {\displaystyle X} is σ {\displaystyle \sigma } -compact.Note that among Lindelöf spaces, metrizable is equivalent to regular and se... | Wikipedia - Selection principle | null | null | null |
Bob has a winning Markov strategy in the game G fin ( O , O ) {\displaystyle {\text{G}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )} played on the space X {\displaystyle X} if and only if the space X {\displaystyle X} is σ {\displaystyle \sigma } -compact. Let X {\displaystyle X} be a second-countable space. Bob has a winn... | Wikipedia - Selection principle | null | null | null |
In all these cases a topological space has a property from the Scheepers Diagram if and only if Alice has no winning strategy in the corresponding game. But this does not hold in general: Let K {\displaystyle \mathbf {K} } be the family of k-covers of a space. That is, such that every compact set in the space is covere... | Wikipedia - Selection principle | null | null | null |
Let X {\displaystyle X} be a topological space. The Rothberger game G 1 ( O , O ) {\displaystyle {\text{G}}_{1}(\mathbf {O} ,\mathbf {O} )} played on X {\displaystyle X} is a game with two players Alice and Bob. 1st round: Alice chooses an open cover U 1 {\displaystyle {\mathcal {U}}_{1}} of X {\displaystyle X} . Bob c... | Wikipedia - Rothberger space | null | null | null |
2nd round: Alice chooses an open cover U 2 {\displaystyle {\mathcal {U}}_{2}} of X {\displaystyle X} . Bob chooses a set U 2 ∈ U 2 {\displaystyle U_{2}\in {\mathcal {U}}_{2}} . | Wikipedia - Rothberger space | null | null | null |
etc. If the family { U n: n ∈ N } {\displaystyle \{U_{n}:n\in \mathbb {N} \}} is a cover of the space X {\displaystyle X} , then Bob wins the game G 1 ( O , O ) {\displaystyle {\text{G}}_{1}(\mathbf {O} ,\mathbf {O} )} . Otherwise, Alice wins. A player has a winning strategy if he knows how to play in order to win the ... | Wikipedia - Rothberger space | null | null | null |
A topological space is Rothberger iff Alice has no winning strategy in the game G 1 ( O , O ) {\displaystyle {\text{G}}_{1}(\mathbf {O} ,\mathbf {O} )} played on this space. Let X {\displaystyle X} be a metric space. Bob has a winning strategy in the game G 1 ( O , O ) {\displaystyle {\text{G}}_{1}(\mathbf {O} ,\mathbf... | Wikipedia - Rothberger space | null | null | null |
Let X {\displaystyle X} be a topological space. The γ {\displaystyle \gamma } -has a pseudo intersection if there is a set game played on X {\displaystyle X} is a game with two players Alice and Bob. 1st round: Alice chooses an open ω {\displaystyle \omega } -cover U 1 {\displaystyle {\mathcal {U}}_{1}} of X {\displays... | Wikipedia - Γ-space | null | null | null |
2nd round: Alice chooses an open ω {\displaystyle \omega } -cover U 2 {\displaystyle {\mathcal {U}}_{2}} of X {\displaystyle X} . Bob chooses a set U 2 ∈ U 2 {\displaystyle U_{2}\in {\mathcal {U}}_{2}} . | Wikipedia - Γ-space | null | null | null |
etc. If { U n: n ∈ N } {\displaystyle \{U_{n}:n\in \mathbb {N} \}} is a γ {\displaystyle \gamma } -cover of the space X {\displaystyle X} , then Bob wins the game. Otherwise, Alice wins. A player has a winning strategy if he knows how to play in order to win the game (formally, a winning strategy is a function). A topo... | Wikipedia - Γ-space | null | null | null |
Let X {\displaystyle X} be a topological vector space (TVS). | Wikipedia - Ultrabornological space | null | null | null |
Let X {\displaystyle X} be a vector space and let U ∙ = ( U i ) i = 1 ∞ {\displaystyle U_{\bullet }=\left(U_{i}\right)_{i=1}^{\infty }} be a sequence of subsets of X . {\displaystyle X.} Each set in the sequence U ∙ {\displaystyle U_{\bullet }} is called a knot of U ∙ {\displaystyle U_{\bullet }} and for every index i ... | Wikipedia - Linear topological space | null | null | null |
The set U 1 {\displaystyle U_{1}} is called the beginning of U ∙ . {\displaystyle U_{\bullet }.} The sequence U ∙ {\displaystyle U_{\bullet }} is/is a: Summative if U i + 1 + U i + 1 ⊆ U i {\displaystyle U_{i+1}+U_{i+1}\subseteq U_{i}} for every index i . | Wikipedia - Linear topological space | null | null | null |
{\displaystyle i.} Balanced (resp. absorbing, closed, convex, open, symmetric, barrelled, absolutely convex/disked, etc.) if this is true of every U i . | Wikipedia - Linear topological space | null | null | null |
{\displaystyle U_{i}.} String if U ∙ {\displaystyle U_{\bullet }} is summative, absorbing, and balanced. Topological string or a neighborhood string in a TVS X {\displaystyle X} if U ∙ {\displaystyle U_{\bullet }} is a string and each of its knots is a neighborhood of the origin in X . | Wikipedia - Linear topological space | null | null | null |
{\displaystyle X.} If U {\displaystyle U} is an absorbing disk in a vector space X {\displaystyle X} then the sequence defined by U i := 2 1 − i U {\displaystyle U_{i}:=2^{1-i}U} forms a string beginning with U 1 = U . {\displaystyle U_{1}=U.} | Wikipedia - Linear topological space | null | null | null |
This is called the natural string of U {\displaystyle U} Moreover, if a vector space X {\displaystyle X} has countable dimension then every string contains an absolutely convex string. Summative sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functi... | Wikipedia - Linear topological space | null | null | null |
A proof of the above theorem is given in the article on metrizable topological vector spaces. If U ∙ = ( U i ) i ∈ N {\displaystyle U_{\bullet }=\left(U_{i}\right)_{i\in \mathbb {N} }} and V ∙ = ( V i ) i ∈ N {\displaystyle V_{\bullet }=\left(V_{i}\right)_{i\in \mathbb {N} }} are two collections of subsets of a vector ... | Wikipedia - Linear topological space | null | null | null |
Set of knots: Knots U ∙ := { U i: i ∈ N } . {\displaystyle \ \operatorname {Knots} U_{\bullet }:=\left\{U_{i}:i\in \mathbb {N} \right\}.} Kernel: ker U ∙ := ⋂ i ∈ N U i . | Wikipedia - Linear topological space | null | null | null |
{\textstyle \ \ker U_{\bullet }:=\bigcap _{i\in \mathbb {N} }U_{i}.} Scalar multiple: s U ∙ := ( s U i ) i ∈ N . {\displaystyle \ sU_{\bullet }:=\left(sU_{i}\right)_{i\in \mathbb {N} }.} | Wikipedia - Linear topological space | null | null | null |
Sum: U ∙ + V ∙ := ( U i + V i ) i ∈ N . {\displaystyle \ U_{\bullet }+V_{\bullet }:=\left(U_{i}+V_{i}\right)_{i\in \mathbb {N} }.} Intersection: U ∙ ∩ V ∙ := ( U i ∩ V i ) i ∈ N . | Wikipedia - Linear topological space | null | null | null |
{\displaystyle \ U_{\bullet }\cap V_{\bullet }:=\left(U_{i}\cap V_{i}\right)_{i\in \mathbb {N} }.} If S {\displaystyle \mathbb {S} } is a collection sequences of subsets of X , {\displaystyle X,} then S {\displaystyle \mathbb {S} } is said to be directed (downwards) under inclusion or simply directed downward if S {\di... | Wikipedia - Linear topological space | null | null | null |
{\displaystyle \mathbb {S} .} Defining vector topologies using collections of strings is particularly useful for defining classes of TVSs that are not necessarily locally convex. If S {\displaystyle \mathbb {S} } is the set of all topological strings in a TVS ( X , τ ) {\displaystyle (X,\tau )} then τ S = τ . {\display... | Wikipedia - Linear topological space | null | null | null |
Let X {\displaystyle X} be a vector space over a field K {\displaystyle \mathbb {K} } where K {\displaystyle \mathbb {K} } has a bornology B K . {\displaystyle {\mathcal {B}}_{\mathbb {K} }.} A bornology B {\displaystyle {\mathcal {B}}} on X {\displaystyle X} is called a vector bornology on X {\displaystyle X} if it is... | Wikipedia - Mackey convergence | null | null | null |
A vector bornology B {\displaystyle {\mathcal {B}}} is called a convex vector bornology if it is stable under the formation of convex hulls (i.e. the convex hull of a bounded set is bounded) then B . {\displaystyle {\mathcal {B}}.} And a vector bornology B {\displaystyle {\mathcal {B}}} is called separated if the only ... | Wikipedia - Mackey convergence | null | null | null |
Let X {\displaystyle X} be an algebraic curve given by an affine equation F ( x , y ) = 0 {\displaystyle F(x,y)=0} over an algebraically closed field K {\displaystyle K} of characteristic zero, and consider a point p {\displaystyle p} on X {\displaystyle X} which we can assume to be ( 0 , 0 ) {\displaystyle (0,0)} . We... | Wikipedia - Puiseux series | null | null | null |
Let X {\displaystyle X} be an operator acting on an N {\displaystyle N} -dimensional Hilbert space H N {\displaystyle {\mathcal {H}}_{N}} . Let Λ ( X ) {\displaystyle \mathrm {\Lambda } (X)} denote its numerical range, i.e. the set of all λ {\displaystyle \lambda } such that there exists a normalized state | ψ ⟩ ∈ H N ... | Wikipedia - Product numerical range | null | null | null |
Consider first a bi–partite Hilbert space, H N = H K ⊗ H M , {\displaystyle {\mathcal {H}}_{N}={\mathcal {H}}_{K}\otimes {\mathcal {H}}_{M},} of a composite dimension N = K M {\displaystyle N=KM} . Let X {\displaystyle X} be an operator acting on the composite Hilbert space. We define the product numerical range Λ ⊗ ( ... | Wikipedia - Product numerical range | null | null | null |
Let X {\displaystyle X} be any set. The family consisting only of the empty set and the set X , {\displaystyle X,} called the minimal or trivial σ-algebra over X . {\displaystyle X.} The power set of X , {\displaystyle X,} called the discrete σ-algebra. | Wikipedia - Σ-algebra | null | null | null |
The collection { ∅ , A , X ∖ A , X } {\displaystyle \{\varnothing ,A,X\setminus A,X\}} is a simple σ-algebra generated by the subset A . {\displaystyle A.} The collection of subsets of X {\displaystyle X} which are countable or whose complements are countable is a σ-algebra (which is distinct from the power set of X {\... | Wikipedia - Σ-algebra | null | null | null |
This is the σ-algebra generated by the singletons of X . {\displaystyle X.} Note: "countable" includes finite or empty. The collection of all unions of sets in a countable partition of X {\displaystyle X} is a σ-algebra. | Wikipedia - Σ-algebra | null | null | null |
Let X {\displaystyle X} be some set, and let P ( X ) {\displaystyle P(X)} represent its power set. Then a subset Σ ⊆ P ( X ) {\displaystyle \Sigma \subseteq P(X)} is called a σ-algebra if and only if it satisfies the following three properties: X {\displaystyle X} is in Σ , {\displaystyle \Sigma ,} and X {\displaystyle... | Wikipedia - Product sigma-algebra | null | null | null |
Σ {\displaystyle \Sigma } is closed under countable unions: If A 1 , A 2 , A 3 , … {\displaystyle A_{1},A_{2},A_{3},\ldots } are in Σ , {\displaystyle \Sigma ,} then so is A = A 1 ∪ A 2 ∪ A 3 ∪ ⋯ . {\displaystyle A=A_{1}\cup A_{2}\cup A_{3}\cup \cdots .} From these properties, it follows that the σ-algebra is also clos... | Wikipedia - Product sigma-algebra | null | null | null |
It also follows that the empty set ∅ {\displaystyle \varnothing } is in Σ , {\displaystyle \Sigma ,} since by (1) X {\displaystyle X} is in Σ {\displaystyle \Sigma } and (2) asserts that its complement, the empty set, is also in Σ . {\displaystyle \Sigma .} Moreover, since { X , ∅ } {\displaystyle \{X,\varnothing \}} s... | Wikipedia - Product sigma-algebra | null | null | null |
{\displaystyle X.} The largest possible σ-algebra on X {\displaystyle X} is P ( X ) . {\displaystyle P(X).} | Wikipedia - Product sigma-algebra | null | null | null |
Elements of the σ-algebra are called measurable sets. An ordered pair ( X , Σ ) , {\displaystyle (X,\Sigma ),} where X {\displaystyle X} is a set and Σ {\displaystyle \Sigma } is a σ-algebra over X , {\displaystyle X,} is called a measurable space. A function between two measurable spaces is called a measurable functio... | Wikipedia - Product sigma-algebra | null | null | null |
The collection of measurable spaces forms a category, with the measurable functions as morphisms. Measures are defined as certain types of functions from a σ-algebra to . | Wikipedia - Product sigma-algebra | null | null | null |
{\displaystyle .} A σ-algebra is both a π-system and a Dynkin system (λ-system). The converse is true as well, by Dynkin's theorem (see below). | Wikipedia - Product sigma-algebra | null | null | null |
Let X {\displaystyle \ X\ } and Y {\displaystyle \ Y\ } be metric spaces with metrics (e.g., distances) d X {\displaystyle \ d_{X}\ } and d Y . {\displaystyle \ d_{Y}\ .} A map f: X → Y {\displaystyle \ f:X\to Y\ } is called an isometry or distance preserving if for any a , b ∈ X {\displaystyle \ a,b\in X\ } one has d ... | Wikipedia - Orthonormal transformation | null | null | null |
An isometry is automatically injective; otherwise two distinct points, a and b, could be mapped to the same point, thereby contradicting the coincidence axiom of the metric d. This proof is similar to the proof that an order embedding between partially ordered sets is injective. Clearly, every isometry between metric s... | Wikipedia - Orthonormal transformation | null | null | null |
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