BRIGHT: A Realistic and Challenging Benchmark for Reasoning-Intensive Retrieval
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0 | \begin{definition}[Definition:Addition]
'''Addition''' is the basic operation $+$ everyone is familiar with.
For example:
:$2 + 3 = 5$
:$47 \cdotp 3 + 191\cdotp 4 = 238 \cdotp 7$
\end{definition} | |
1 | \begin{definition}[Definition:Addition/Complex Numbers]
The '''addition operation''' in the domain of complex numbers $\C$ is written $+$.
Let $z = a + i b, w = c + i d$ where $a, b, c, d \in \R, i^2 = -1$.
Then $z + w$ is defined as:
:$\paren {a + i b} + \paren {c + i d} = \paren {a + c} + i \paren {b + d}$
\end{defin... | |
2 | \begin{definition}[Definition:Addition/Integers]
The '''addition operation''' in the domain of integers $\Z$ is written $+$.
We have that the set of integers is the Inverse Completion of Natural Numbers.
Thus it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times... | |
3 | \begin{definition}[Definition:Addition/Multiindices]
The addition operation in the domain of multiindices $Z$ is written $+$.
Let $k = \sequence {k_j}_{j \mathop \in J}$ and $l = \sequence {l_j}_{j \mathop \in J}$ be multiindices.
Then $k + l$ is defined as:
:$\paren {\forall j \in J} \paren {\paren {k + l}_j = k_j + l... | |
4 | \begin{definition}[Definition:Addition/Natural Numbers]
Every attempt to describe the natural numbers via suitable axioms should reproduce the intuitive behaviour of $+$.
The same holds for any construction of $\N$ in an ambient theory.
Let $\N$ be the natural numbers.
\end{definition} | |
5 | \begin{definition}[Definition:Addition/Naturally Ordered Semigroup]
Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.
The operation $\circ$ in $\struct {S, \circ, \preceq}$ is called '''addition'''.
Category:Definitions/Naturally Ordered Semigroup
\end{definition} | |
6 | \begin{definition}[Definition:Addition/Peano Structure]
Let $\struct {P, 0, s}$ be a Peano structure.
The binary operation $+$ is defined on $P$ as follows:
:$\forall m, n \in P: \begin{cases}
m + 0 & = m \\
m + \map s n & = \map s {m + n}
\end{cases}$
This operation is called '''addition'''.
The definition can equival... | |
7 | \begin{definition}[Definition:Addition/Rational Numbers]
The addition operation in the domain of rational numbers $\Q$ is written $+$.
Let:
:$a = \dfrac p q, b = \dfrac r s$
where:
:$p, q \in \Z$
:$r, s \in \Z_{\ne 0}$
Then $a + b$ is defined as:
:$\dfrac p q + \dfrac r s = \dfrac {p s + r q} {q s}$
This definition fol... | |
8 | \begin{definition}[Definition:Addition/Real Modulo Addition]
Let $z \in \R$.
Let $\R_z$ be the set of residue classes modulo $z$ of $\R$.
The addition operation is defined on $\R_z$ as follows:
:$\eqclass a z +_z \eqclass b z = \eqclass {a + b} z$
This operation is called '''addition modulo $z$'''.
\end{definition} | |
9 | \begin{definition}[Definition:Addition/Real Numbers]
The addition operation in the domain of real numbers $\R$ is written $+$.
From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.
Let $x = \eqclass {\sequence {x_n} } {}, y... | |
10 | \begin{definition}[Definition:Addition/Sum]
Let $a + b$ denote the operation of addition on two objects $a$ and $b$.
Then the result $a + b$ is referred to as the '''sum''' of $a$ and $b$.
Note that the nature of $a$ and $b$ has deliberately been left unspecified.
They could be, for example, numbers, matrices or more c... | |
11 | \begin{definition}[Definition:Addition/Summand]
Let $a + b$ denote the operation of addition on two objects.
The objects $a$ and $b$ are known as the '''summands''' of $a + b$.
Note that the nature of $a$ and $b$ has deliberately been left unspecified.
They could be, for example, numbers, matrices or more complex expre... | |
12 | \begin{definition}[Definition:Addition (Affine Geometry)]
Let $\tuple {\EE, +, -}$ be an affine space.
Then the mapping $+$ is called '''affine addition'''.
\end{definition} | |
13 | \begin{definition}[Definition:Addition for Natural Numbers in Real Numbers]
Let $\struct {\R, +, \times, \le}$ be the field of real numbers.
Let $\N$ be the natural numbers in $\R$.
Then the restriction of $+$ to $\N$ is called '''addition'''.
Category:Definitions/Natural Numbers
\end{definition} | |
14 | \begin{definition}[Definition:Addition in Minimal Infinite Successor Set]
Let $\omega$ be the minimal infinite successor set.
The binary operation $+$ is defined on $\omega$ as follows:
:$\forall m,n \in \omega: \begin{cases} m + 0 &= m \\ m + n^+ &= \left({m + n}\right)^+\end{cases}$
where $m^+$ is the successor set o... | |
15 | \begin{definition}[Definition:Addition of Codewords in Linear Code]
Let $\map V {n, p}$ denote the linear $\tuple {n, n}$-code modulo $p$.
The operation of '''addition''' on $\map V {n, p}$ is defined as follows.
Let $A$ and $B$ be elements of $\map V {n, p}$, that is, sequences of length $n$ of residue classes modulo ... | |
16 | \begin{definition}[Definition:Addition of Cuts]
Let $\alpha$ and $\beta$ be cuts.
Let the operation of '''addition''' be defined on $\alpha$ and $\beta$ as:
:$\gamma := \alpha + \beta$
where $\gamma$ is the set of all rational numbers $r$ such that:
:$\exists p \in \alpha, q \in \beta: r = p + q$
In this context, $\gam... | |
17 | \begin{definition}[Definition:Addition of Order Types]
Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.
Let $\alpha := \map \ot {S_1, \preccurlyeq_1}$ and $\beta := \map \ot {S_2, \preccurlyeq_2}$ denote the order types of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \precc... | |
18 | \begin{definition}[Definition:Addition of Polynomials]
Let $\struct {R, +, \circ}$ be a ring.
Let $\struct {S, +, \circ}$ be a subring of $R$.
For arbitrary $x \in R$, let $S \sqbrk x$ be the set of polynomials in $x$ over $S$.
Let $p, q \in S \sqbrk x$ be polynomials in $x$ over $S$:
:$\ds p = \sum_{k \mathop = 0}^m ... | |
19 | \begin{definition}[Definition:Addition of Polynomials/Polynomial Forms]
Let:
:$\ds f = \sum_{k \mathop \in Z} a_k \mathbf X^k$
:$\ds g = \sum_{k \mathop \in Z} b_k \mathbf X^k$
be polynomials in the indeterminates $\set {X_j: j \in J}$ over $R$.
{{explain|What is $Z$ in the above? Presumably the integers, in which case... | |
20 | \begin{definition}[Definition:Addition of Polynomials/Sequence]
Let:
:$f = \sequence {a_k} = \tuple {a_0, a_1, a_2, \ldots}$
and:
:$g = \sequence {b_k} = \tuple {b_0, b_1, b_2, \ldots}$
be polynomials over a field $F$.
Then the operation of '''(polynomial) addition''' is defined as:
:$f + g := \tuple {a_0 + b_0, a_1 + ... | |
21 | \begin{definition}[Definition:Addition on 1-Based Natural Numbers]
Let $\N_{>0}$ be the $1$-based natural numbers, axiomatized by:
{{:Axiom:Axiomatization of 1-Based Natural Numbers}}
The operation $+$ in this axiomatizaton is called '''addition'''.
Category:Definitions/Natural Numbers
\end{definition} | |
22 | \begin{definition}[Definition:Additive Arithmetic Function]
Let $f : \N \to \C$ be an arithmetic function.
Then $f$ is '''additive''' {{iff}}:
:$m \perp n \implies \map f {m n} = \map f m + \map f n$
\end{definition} | |
23 | \begin{definition}[Definition:Additive Category]
Let $A$ be a preadditive category.
\end{definition} | |
24 | \begin{definition}[Definition:Additive Function (Conventional)]
Let $f: S \to S$ be a mapping on an algebraic structure $\struct {S, +}$.
Then $f$ is an '''additive function''' {{iff}} it preserves the addition operation:
:$\forall x, y \in S: \map f {x + y} = \map f x + \map f y$
\end{definition} | |
25 | \begin{definition}[Definition:Additive Function (Measure Theory)]
Let $\SS$ be an algebra of sets.
Let $f: \SS \to \overline \R$ be a function, where $\overline \R$ denotes the set of extended real numbers.
Then $f$ is defined to be '''additive''' {{iff}}:
:$\forall S, T \in \SS: S \cap T = \O \implies \map f {S \cup T... | |
26 | \begin{definition}[Definition:Additive Function on UFD]
Let $R$ be a unique factorization domain.
Let $f : R \to \C$ be a complex-valued function.
Then $f$ is '''additive''' {{iff}}:
:For all coprime $x,y\in R$, $f(xy) = f(x) + f(y)$
\end{definition} | |
27 | \begin{definition}[Definition:Additive Group of Complex Numbers]
The '''additive group of complex numbers''' $\struct {\C, +}$ is the set of complex numbers under the operation of addition.
\end{definition} | |
28 | \begin{definition}[Definition:Additive Group of Field]
Let $\struct {F, +, \times}$ be a field.
The group $\struct {F, +}$ is known as the '''additive group of $F$'''.
\end{definition} | |
29 | \begin{definition}[Definition:Additive Group of Integer Multiples]
Let $n \in \Z_{>0}$.
The '''additive group $\left({n \Z, +}\right)$ of integer multiples of $n$''' is the set of integer multiples of $n$ under the operation of addition.
\end{definition} | |
30 | \begin{definition}[Definition:Additive Group of Integers Modulo 4]
The '''additive group of integers modulo $4$''' $\struct {\Z_4, +_4}$ is the set of integers modulo $4$ under the operation of addition modulo $4$.
\end{definition} | |
31 | \begin{definition}[Definition:Additive Group of Integers Modulo m]
Let $m \in \Z$ such that $m > 1$.
The '''additive group of integers modulo $m$''', denoted $\struct {\Z_m, +_m}$, is the set of integers modulo $m$ under the operation of addition modulo $m$.
\end{definition} | |
32 | \begin{definition}[Definition:Additive Group of Rational Numbers]
The '''additive group of rational numbers''' $\struct {\Q, +}$ is the set of rational numbers under the operation of addition.
\end{definition} | |
33 | \begin{definition}[Definition:Additive Group of Real Numbers]
The '''additive group of real numbers''' $\struct {\R, +}$ is the set of real numbers under the operation of addition.
\end{definition} | |
34 | \begin{definition}[Definition:Additive Group of Ring]
Let $\struct {R, +, \circ}$ be a ring.
The group $\struct {R, +}$ is known as the '''additive group of $R$'''.
\end{definition} | |
35 | \begin{definition}[Definition:Additive Inverse/Field]
Let $\struct {F, +, \times}$ be a field whose addition operation is $+$.
Let $a \in R$ be any arbitrary element of $F$.
The '''additive inverse''' of $a$ is its inverse under addition, denoted $-a$:
:$a + \paren {-a} = 0_F$
where $0_F$ is the zero of $R$.
\end{defin... | |
36 | \begin{definition}[Definition:Additive Inverse/Number]
Let $\Bbb F$ be one of the standard number systems: $\N$, $\Z$, $\Q$, $\R$, $\C$.
Let $a \in \Bbb F$ be any arbitrary number.
The '''additive inverse''' of $a$ is its inverse under addition, denoted $-a$:
:$a + \paren {-a} = 0$
\end{definition} | |
37 | \begin{definition}[Definition:Additive Inverse/Ring]
Let $\struct {R, +, \circ}$ be a ring whose ring addition operation is $+$.
Let $a \in R$ be any arbitrary element of $R$.
The '''additive inverse''' of $a$ is its inverse under ring addition, denoted $-a$:
:$a + \paren {-a} = 0_R$
where $0_R$ is the zero of $R$.
\en... | |
38 | \begin{definition}[Definition:Additive Monoid of Natural Numbers]
The '''additive monoid of natural numbers''' $\left({\N, +}\right)$ is the set of natural numbers under the operation of addition.
\end{definition} | |
39 | \begin{definition}[Definition:Additive Notation]
'''Additive notation''' is a convention often used for representing a commutative binary operation of an algebraic structure.
The symbol used for the operation is $+$.
Let $\struct {S, +}$ be such an algebraic structure, and let $x, y \in S$.
:$x + y$ is used to indicate... | |
40 | \begin{definition}[Definition:Additive Semiring]
An '''additive semiring''' is a semiring with a commutative distributand.
That is, an '''additive semiring''' is a ringoid $\left({S, *, \circ}\right)$ in which:
: $(1): \quad \left({S, *}\right)$ forms a commutative semigroup
: $(2): \quad \left({S, \circ}\right)$ forms... | |
41 | \begin{definition}[Definition:Additive Semiring/Axioms]
An additive semiring is an algebraic structure $\struct {R, *, \circ}$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions:
{{begin-axiom}}
{{axiom | n = \text A 0
| q = \forall a, b \in S
| m = a * b... | |
42 | \begin{definition}[Definition:Additive Subgroup]
Let $\struct {R, +, \times}$ be a ring.
Let $\struct {R, +}$ be the additive group of $\struct {R, +, \times}$.
Let $\struct {S, +}$ be a subgroup of $\struct {R, +}$
Then $\struct {S, +}$ is an '''additive subgroup of ''' $\struct {R, +, \times}$.
\end{definition} | |
43 | \begin{definition}[Definition:Adherent Point]
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$.
\end{definition} | |
44 | \begin{definition}[Definition:Adherent Point/Definition 1]
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$.
A point $x \in S$ is an '''adherent point of $H$''' {{iff}} every open neighborhood $U$ of $x$ satisfies:
:$H \cap U \ne \O$
\end{definition} | |
45 | \begin{definition}[Definition:Adherent Point/Definition 2]
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$.
A point $x \in S$ is an '''adherent point of $H$''' {{iff}} $x$ is an element of the closure of $H$.
\end{definition} | |
46 | \begin{definition}[Definition:Adherent Point/Definition 3]
Let $T = \struct{S, \tau}$ be a topological space.
Let $H \subseteq S$.
A point $x \in S$ is an '''adherent point of $H$''' {{iff}} every neighborhood $N$ of $x$ satisfies:
:$H \cap N \ne \O$
\end{definition} | |
47 | \begin{definition}[Definition:Adjacency Matrix]
An '''adjacency matrix''' is a matrix which describes a graph by representing which vertices are adjacent to which other vertices.
If $G$ is a graph of order $n$, then its '''adjacency matrix''' is a square matrix of order $n$, where each row and column corresponds to a v... | |
48 | \begin{definition}[Definition:Adjacent (Graph Theory)/Edges/Digraph]
Let $G = \struct {V, E}$ be a digraph.
Two arcs $e_1, e_2 \in E$ of $G$ '''adjacent''' if there exists a vertex $v \in V$ to which they are both incident.
Otherwise they are '''non-adjacent'''.
\end{definition} | |
49 | \begin{definition}[Definition:Adjacent (Graph Theory)/Edges/Non-Adjacent]
Let $G = \struct {V, E}$ be a graph.
Two edges $u, v \in V$ of $G$ are '''non-adjacent''' if they are not adjacent.
\end{definition} | |
50 | \begin{definition}[Definition:Adjacent (Graph Theory)/Edges/Undirected Graph]
Let $G = \struct {V, E}$ be an undirected graph.
Two edges $e_1, e_2 \in E$ of $G$ '''adjacent''' if there exists a vertex $v \in V$ to which they are both incident.
Otherwise they are '''non-adjacent'''.
\end{definition} | |
51 | \begin{definition}[Definition:Adjacent (Graph Theory)/Faces]
Let $G = \left({V, E}\right)$ be a planar graph.
Two faces of $G$ are '''adjacent''' {{iff}} they are both incident to the same edge (or edges).
Note that faces which are both incident to the same vertex are ''not'' considered adjacent unless they are also bo... | |
52 | \begin{definition}[Definition:Adjacent (Graph Theory)/Vertices/Digraph]
Let $G = \struct {V, E}$ be a digraph.
Two vertices $u, v \in V$ of $G$ are '''adjacent''' if there exists an arc $e = \left({u, v}\right) \in E$ of $G$ to which they are both incident.
Otherwise they are '''non-adjacent'''.
\end{definition} | |
53 | \begin{definition}[Definition:Adjacent (Graph Theory)/Vertices/Non-Adjacent]
Let $G = \struct {V, E}$ be a graph.
Two vertices $u, v \in V$ of $G$ are '''non-adjacent''' if they are not adjacent.
\end{definition} | |
54 | \begin{definition}[Definition:Adjacent (Graph Theory)/Vertices/Undirected Graph]
Let $G = \struct {V, E}$ be an undirected graph.
Two vertices $u, v \in V$ of $G$ are '''adjacent''' if there exists an edge $e = \set {u, v} \in E$ of $G$ to which they are both incident.
Otherwise they are '''non-adjacent'''.
\end{defini... | |
55 | \begin{definition}[Definition:Adjoint Linear Transformation]
Let $\HH$ and $\KK$ be Hilbert spaces.
Let $\map \BB {\HH, \KK}$ be the set of bounded linear transformations from $\HH$ to $\KK$.
Let $A \in \map \BB {\HH, \KK}$ be a bounded linear transformation.
By Existence and Uniqueness of Adjoint, there exists a uniqu... | |
56 | \begin{definition}[Definition:Adjugate]
Let:
: $D = \begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn}\end{vmatrix}$
be a determinant of order $n$.
Let $a_{rs}$ be an element of $D$.
Let $A_{rs}$ be the co... | |
57 | \begin{definition}[Definition:Adjugate Matrix]
Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.
Let $\mathbf C$ be its cofactor matrix.
The '''adjugate matrix''' of $\mathbf A$ is the transpose of $\mathbf C$:
:$\adj {\mathbf A} = \mathbf C^\intercal$
\end{definition} | |
58 | \begin{definition}[Definition:Adjunction]
Let $\mathbf {Set}$ be the category of sets.
Let $\mathbf C$, $\mathbf D$ be locally small categories.
An '''adjunction''' between $\mathbf C$ and $\mathbf D$ is a triple $\tuple {F, G, \alpha}$, where
:$F : \mathbf D \to \mathbf C$ is a functor.
:$G : \mathbf C \to \mathbf D$ ... | |
59 | \begin{definition}[Definition:Admiralty Mile]
The '''admiralty mile''' is an obsolete British unit of length which was used for air and sea navigation.
It was defined as the average length measured along Earth's surface of one minute of latitude along a meridian, rounded to $4$ significant figures.
{{begin-eqn}}
{{eqn ... | |
60 | \begin{definition}[Definition:Admissible Subdivision for Admissible Curve]
Let $M$ be a smooth manifold.
Let $I := \closedint a b$ be a closed real interval.
Let $\gamma : I \to M$ be an admissible curve.
Let $\tuple {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ be a finite subdivision of $I$.
Then $\tuple {x_0, x_1, x_2, ... | |
61 | \begin{definition}[Definition:Aether/Historical Note]
The existence of the aether or ether, as a hitherto undetected entity in physics was posited as late as the $20$th century as a medium through which electromagnetic radiation was able to travel.
Its existence was disproved by the Michelson-Morley Experiment.
Categor... | |
62 | \begin{definition}[Definition:Affine Algebraic Set]
Let $K$ be a field.
Let $A = K \sqbrk {X_1, \ldots, X_n}$ be the ring of polynomial functions in $n$ variables over $K$.
Then a subset $X \subseteq K^n$ is an '''affine algebraic set''' {{iff}} it is the zero locus of some set $T \subseteq A$.
\end{definition} | |
63 | \begin{definition}[Definition:Affine Algebraic Variety]
Let $k$ be a field.
Then a subset $X \subseteq k^n$ is an '''affine algebraic variety''' if the following hold:
* $X$ is an affine algebraic set
* $X$ is irreducible
\end{definition} | |
64 | \begin{definition}[Definition:Affine Frame]
Let $\EE$ be an affine space with difference space $V$.
\end{definition} | |
65 | \begin{definition}[Definition:Affine Frame/Definition 1]
Let $\EE$ be an affine space with difference space $V$.
An '''affine frame''' in $\EE$ is an ordered tuple $\tuple {p_0, e_1, \ldots, e_n}$, where:
:$p_0$ is an element of $\EE$ called the '''origin'''
:$\tuple {e_1, \ldots, e_n}$ is an ordered basis for $V$.
Cat... | |
66 | \begin{definition}[Definition:Affine Frame/Definition 2]
Let $\EE$ be an affine space with difference space $V$.
An '''affine frame''' may be given by the set of $n + 1$ points:
:$\tuple {q_0, \ldots, q_n} = \tuple {p_0, p_0 + e_1, \ldots, p_0 + e_n}$
The frame $\tuple {p_0, e_1, \ldots, e_n}$ is then recovered by:
:$\... | |
67 | \begin{definition}[Definition:Affine Geometry]
'''Affine geometry''' is the study of the geometry of affine spaces.
It provides a modern axiomatic approach to the study of configurations of lines, planes and hypersurfaces.
In particular an affine space can be thought of as a finite dimensional vector space with no dist... | |
68 | \begin{definition}[Definition:Affine Group of One Dimension]
Let $S$ be the set of mappings $f_{a b}: \R \to \R$ defined as:
:$S := \set {f_{a b}: x \mapsto a x + b: a \in \R_{\ne 0}, b \in \R}$
The algebraic structure $\struct {S, \circ}$, where $\circ$ denotes composition of mappings, is called the '''$1$-dimensional... | |
69 | \begin{definition}[Definition:Affine Monoid]
An '''affine monoid''' is a monoid that is:
: finitely generated
and:
: isomorphic to a submonoid of a free abelian group $\Z^d$, for some $d \in \Z_{\ge 0}$.
\end{definition} | |
70 | \begin{definition}[Definition:Affine Open Subset]
Let $\struct {X, \OO_X}$ be a scheme.
Let $U \subset X$ be an open subset.
We say, that $U$ is an '''affine open subset''', if the restriction $\struct {U, \OO_X {\restriction U}}$ of $\struct {X, \OO_X}$ to $U$ is an affine scheme.
Category:Definitions/Algebraic Geomet... | |
71 | \begin{definition}[Definition:Affine Scheme]
An '''affine scheme''' is a ringed space which is isomorphic to the spectrum of a commutative ring with unity.
\end{definition} | |
72 | \begin{definition}[Definition:Affine Space/Associativity Axioms]
Let $K$ be a field.
Let $\struct {V, +_V, \circ}$ be a vector space over $K$.
Let $\EE$ be a set on which two mappings are defined:
:$+ : \EE \times V \to \EE$
:$- : \EE \times \EE \to V$
satisfying the following associativity conditions:
{{begin-axiom}}
... | |
73 | \begin{definition}[Definition:Affine Space/Group Action]
Let $K$ be a field.
Let $\struct {V, +_V, \circ}$ be a vector space over $K$.
Let $\EE$ be a set.
Let $\phi: \EE \times V \to \EE$ be a free and transitive group action of $\struct {V, +_V}$ on $\EE$.
Then the ordered pair $\struct {\EE, \phi}$ is an '''affine sp... | |
74 | \begin{definition}[Definition:Affine Space/Weyl's Axioms]
Let $K$ be a field.
Let $\struct{V, +_V, \circ}$ be a vector space over $K$.
Let $\EE$ be a set on which a mapping is defined:
:$- : \EE \times \EE \to V$
satisfying the following associativity conditions:
{{begin-axiom}}
{{axiom | n = \text W 1
| q = \f... | |
75 | \begin{definition}[Definition:Affine Subspace]
Let $\EE$ be an affine space with tangent space $E$.
Let $\FF \subseteq \EE$ be a subset of $\EE$.
Then $\FF$ is an '''affine subspace''' of $\EE$ {{iff}} there exists a point $p \in \EE$ such that:
:$F_p := \set {q - p: q \in \FF}$
is a vector subspace of the vector space... | |
76 | \begin{definition}[Definition:Affine Transformation]
Let $\EE$ and $\FF$ be affine spaces with difference spaces $E$ and $F$ respectively.
Let $\LL: \EE \to \FF$ be a mapping.
Then $\LL$ is an '''affine transformation''' or '''affine mapping''' if there exists a linear transformation $L: E \to F$ such that for every pa... | |
77 | \begin{definition}[Definition:Affinely Dependent]
Let $\R^n$ be the $n$-dimensional real Euclidean space.
Let $S = \set {x_1, \dots, x_r}$ be a finite subset of $\R^n$.
An element $x \in \R^n$ is '''affinely dependent''' on $S$ if there exist real numbers $\set {\lambda_i: 1 \le i \le r}$ such that:
:$(1): \quad x = \d... | |
78 | \begin{definition}[Definition:Affinely Dependent/Independent]
Let $\R^n$ be the $n$-dimensional real Euclidean space.
Let $X = \set {x_1, \dots, x_r}$ be a finite subset of $\R^n$.
The subset $X$ is '''affinely independent''' if no element $x \in X$ is affinely dependent on $X \setminus \set x$.
\end{definition} | |
79 | \begin{definition}[Definition:Affirmative Categorical Statement]
An '''affirmative categorical statement''' is either one of:
\end{definition} | |
80 | \begin{definition}[Definition:Age (Model Theory)]
Let $\MM$ be an $\LL$-structure.
An '''age''' of $\MM$ is a class $K$ of $\LL$-structures such that:
* if $\AA$ is a finitely generated $\LL$-structure such that there is an $\LL$-embedding $\AA \to \MM$, then $\AA$ is isomorphic to some structure in $K$,
* no two struc... | |
81 | \begin{definition}[Definition:Age (Time)]
The '''age''' of a physical objectis defined as the period of time over which it has been in existence.
Category:Definitions/Applied Mathematics
\end{definition} | |
82 | \begin{definition}[Definition:Aggregate]
An '''aggregate''' is an archaic word for an '''infinite set''', as used by {{AuthorRef|Georg Cantor}}.
It is used in current parlance to mean a general accumulation of objects, but the concept is vague.
\end{definition} | |
83 | \begin{definition}[Definition:Agreement/Mappings]
Let:
:$(1): \quad f_1: S_1 \to T_1$ be a mapping from $S_1$ to $T_1$
:$(2): \quad f_2: S_2 \to T_2$ be a mapping from $S_2$ to $T_2$
:$(3): \quad X \subseteq S_1 \cap S_2$
Let:
:$\forall s \in X: \map {f_1} s = \map {f_2} s$
Then the mappings $f_1$ and $f_2$ are said to... | |
84 | \begin{definition}[Definition:Agreement/Relations]
Let:
:$(1): \quad \RR_1 \subseteq S_1 \times T_1$ be a relation on $S_1 \times T_1$
:$(2): \quad \RR_2 \subseteq S_2 \times T_2$ be a relation on $S_2 \times T_2$
:$(3): \quad X \subseteq S_1 \cap S_2$
Let:
:$\forall s \in X: \map {\RR_1} s = \map {\RR_2} s$
where $\ma... | |
85 | \begin{definition}[Definition:Airy's Equation]
'''Airy's equation''' is the second order ODE:
:$y'' = x y$
\end{definition} | |
86 | \begin{definition}[Definition:Aleph Mapping]
Let $\NN'$ denote the class of all infinite cardinal numbers.
Then $\aleph$ (that is: '''aleph''') is defined as the unique order isomorphism between the two ordered structures $\struct {\On, \in}$ and $\struct {\NN', \in}$
where $\On$ denotes the class of ordinal numbers.
{... | |
87 | \begin{definition}[Definition:Aleph Number]
The '''aleph numbers''' are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered.
The cardinality of the natural numbers is $\aleph_0$.
The next larger cardinality is $\aleph_1$.
Then $\aleph_2$ and so on.
\end{definition... | |
88 | \begin{definition}[Definition:Alexander Polynomial]
For a knot $K$ with Seifert matrix $V$, the '''Alexander polynomial''' of $K$ is defined as:
:$\map {\Delta_K} t = \map \det {V - t V^\intercal}$
{{NamedforDef|James Waddell Alexander II|cat=Alexander}}
Category:Definitions/Knot Theory
\end{definition} | |
89 | \begin{definition}[Definition:Alexandrian School]
The '''Alexandrian School''' was active in the city of {{WP|Alexandria|Alexandria}} for the approximate millennium leading up to the city's destruction in $641$ C.E.
It stood as the cultural hub of the Greek sphere of influence.
The {{WP|Musaeum|Alexandrian Musaeum}} st... | |
90 | \begin{definition}[Definition:Alexandroff Extension]
Let $T = \struct {S, \tau}$ be a non-empty topological space.
Let $p$ be a new element not in $S$.
Let $S^* := S \cup \set p$.
Let $\tau^*$ be the topology on $S^*$ defined such that $U \subseteq S^*$ is open {{iff}}:
:$U$ is an open set of $T$
or
:$U$ is the complem... | |
91 | \begin{definition}[Definition:Alexandroff Extension of Real Number Line]
The '''Alexandroff extension of the real number line''' $\R^*$ is defined as:
:$\R^* := \R \cup \set \infty$
that is, the set of real numbers together with an element $\infty$ which is not in $\R$.
\end{definition} | |
92 | \begin{definition}[Definition:Alexandroff Plank]
:rightthumb600px
Let $\Omega$ be the first uncountable ordinal.
Let the closed ordinal space $\closedint 0 \Omega$ be given the interval topology.
Let the closed real interval $\closedint {-1} 1$ be given the interval topology.
Let $\struct {S, \tau}$ be the product spac... | |
93 | \begin{definition}[Definition:Algebra (Abstract Algebra)]
In the context of abstract algebra, in particular ring theory and linear algebra, the following varieties of '''algebra''' exist:
* Definition:Boolean Algebra
* Definition:Algebra over Ring: an $R$-module $G_R$ over a commutative ring $R$ with a bilinear mapping... | |
94 | \begin{definition}[Definition:Algebra (Mathematical Branch)]
'''Algebra''' is the branch of mathematics which studies the techniques of manipulation of objects and expressions.
\end{definition} | |
95 | \begin{definition}[Definition:Algebra Defined by Ring Homomorphism]
Let $R$ be a commutative ring.
Let $\struct {S, +, *}$ be a ring.
Let $f : R \to S$ be a ring homomorphism.
Let the image of $f$ be a subset of the center of $S$.
Let $S_R$ be the module defined by the ring homomorphism $f$.
The '''algebra defined by $... | |
96 | \begin{definition}[Definition:Algebra Homomorphism]
Let $R$ be a commutative ring.
Let $(A, *)$ and $(B, \times)$ be algebras over $R$.
An '''algebra homomorphism''' $f : A \to B$ is a module homomorphism such that:
:$\forall a, b \in A : f(a * b) = f(a) \times f(b)$
\end{definition} | |
97 | \begin{definition}[Definition:Algebra Loop]
An '''algebra loop''' $\struct {S, \circ}$ is a quasigroup with an identity element.
:$\exists e \in S: \forall x \in S: x \circ e = x = e \circ x$
\end{definition} | |
98 | \begin{definition}[Definition:Algebra of Sets/Definition 1]
Let $X$ be a set.
Let $\powerset X$ be the power set of $X$.
Let $\RR \subseteq \powerset X$ be a set of subsets of $X$.
Then $\RR$ is an '''algebra of sets over $X$''' {{iff}} the following conditions hold:
{{begin-axiom}}
{{axiom | n = \text {AS} 1
|... | |
99 | \begin{definition}[Definition:Algebra of Sets/Definition 2]
An '''algebra of sets''' is a ring of sets with a unit.
\end{definition} |
End of preview. Expand in Data Studio
BRIGHT: theoremqa_theorems
A format conversion of the standard passage-retrieval setting of
BRIGHT: A Realistic and Challenging Benchmark for Reasoning-Intensive Retrieval,
by the original BRIGHT authors. Source: xlangai/BRIGHT,
pinned to 3066d29c9651a576c8aba4832d249807b181ecae. This is a converted mirror, not an independently annotated dataset.
The original dataset is distributed under CC BY 4.0.
See SOURCE_README.md for the original dataset card and attribution.
| Table | Rows | Columns |
|---|---|---|
| corpus | 23,839 | id, title, text |
| queries | 76 | id, text |
| qrels | 151 | query-id, corpus-id, score |
| exclusions (optional) | 0 | query-id, corpus-id |
All published configs use the test split. corpus.text is exactly the original
documents.content; title is empty. Query IDs, document IDs and original examples.query
strings are preserved without normalization. qrels is the unique set of gold_ids pairs,
with binary score 1. Duplicate labels are collapsed; no relevance judgments are added.
Gold reasoning, gold answers, gold_ids_long, generated query rewrites and long_documents
are not part of the converted evaluation inputs.
This subset has no real excluded documents; the optional exclusions table is omitted.
The N/A exclusion sentinel is removed. Excluded IDs and positive IDs are validated against
the corpus and must be disjoint for each query. Keep ignore_identical_ids=false: matching
query/document ID strings alone do not imply a BRIGHT exclusion.
Evaluate each BRIGHT subset against its own complete corpus. Report nDCG@10 per subset and the unweighted mean across the 12 subsets. Record encoder context limits and truncation. The original evaluation code defines the protocol.
Converted with scripts/preprocess/dataset/convert_bright.py in the SPARSE project.
conversion.json records source and output SHA-256 hashes, row counts and validation checks.
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