hcju/proofwikips huggingface dataset:
Definition:Dedekind Cut
Let $\struct {S, \preceq}$ be a totally ordered set. === Definition 1 === {{:Definition:Dedekind Cut/Definition 1}} === Definition 2 === {{:Definition:Dedekind Cut/Definition 2}}
Definition:Complex Number
{{:Definition:Complex Number/Definition 1}}
Definition:Determinant/Matrix
Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. That is, let: :$\mathbf A = \begin {bmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\  \vdots &  \vdots & \ddots &  \vdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n} \\ \end {bmatrix}$ === Definition 1 === {{:Definition:Determinant/Matrix/Definition 1}} === Definition 2 === {{:Definition:Determinant/Matrix/Definition 2}}
Definition:Cofactor
Let $R$ be a commutative ring with unity. Let $\mathbf A \in R^{n \times n}$ be a square matrix of order $n$.  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 $n$. === Cofactor of an Element === {{:Definition:Cofactor/Element}} === Cofactor of a Minor === {{:Definition:Cofactor/Minor}}
Definition:Discriminant of Polynomial
Let $k$ be a field. Let $\map f X \in k \sqbrk X$ be a polynomial of degree $n$. Let $\overline k$ be an algebraic closure of $k$. Let the roots of $f$ in $\overline k$ be $\alpha_1, \alpha_2, \ldots, \alpha_n$. Then the '''discriminant''' $\map \Delta f$ of $f$ is defined as: :$\displaystyle \map \Delta f := \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {\alpha_i - \alpha_j}^2$ === Quadratic Equation === The concept is usually encountered in the context of a quadratic equation $a x^2 + b x + c$: {{:Definition:Discriminant of Polynomial/Quadratic Equation}} === Cubic Equation === In the context of a cubic equation $a x^3 + b x^2 + c x + d$: {{Definition:Discriminant of Polynomial/Cubic Equation}}
Definition:Minor of Determinant
Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Consider the order $k$ square submatrix $\mathbf B$ obtained by deleting $n - k$ rows and $n - k$ columns from $\mathbf A$. Let $\map \det {\mathbf B}$ denote the determinant of $\mathbf B$. Then $\map \det {\mathbf B}$ is an '''order-$k$ minor''' of $\map \det {\mathbf A}$. Thus a '''minor''' is a determinant formed from the elements (in the same relative order) of $k$ specified rows and columns.
Definition:Diagonal Matrix
Let $\mathbf A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \\ \end{bmatrix}$ be a square matrix of order $n$. Then $\mathbf A$ is a '''diagonal matrix'''  all elements of $\mathbf A$ are zero except for possibly its diagonal elements.
Definition:Triangular Matrix
Let $\mathbf T = \begin {bmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\  \vdots &  \vdots & \ddots &  \vdots \\ a_{m 1} & a_{m 2} & \cdots & a_{m n} \\ \end {bmatrix}$ be a matrix of order $m \times n$. Then $\mathbf T$ is a '''triangular matrix'''  all the elements either above or below the diagonal are zero. === Upper Triangular Matrix === {{:Definition:Triangular Matrix/Upper Triangular Matrix}} === Lower Triangular Matrix === {{:Definition:Triangular Matrix/Lower Triangular Matrix}}
Definition:Leading Coefficient of Matrix
Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix. The '''leading coefficient''' of each row of $\mathbf A$ is the leftmost non-zero element of that row. A zero row has no '''leading coefficient'''.
Definition:Echelon Matrix
Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix. === Echelon Form === {{:Definition:Echelon Matrix/Echelon Form}} === Reduced Echelon Form === {{:Definition:Echelon Matrix/Reduced Echelon Form}}
Definition:Complex Conjugate
Let $z = a + i b$ be a complex number. Then the '''(complex) conjugate''' of $z$ is denoted $\overline z$ and is defined as: :$\overline z := a - i b$
Definition:Root (Analysis)
Let $x, y \in \R_{\ge 0}$ be positive real numbers. Let $n \in \Z$ be an integer such that $n \ne 0$. Then $y$ is the '''positive $n$th root of $x$''' : :$y^n = x$ and we write: :$y = \sqrt[n] x$ Using the power notation, this can also be written: :$y = x^{1/n}$
Definition:Power (Algebra)
=== Natural Numbers === {{:Definition:Power (Algebra)/Natural Number}} === Integers === {{:Definition:Power (Algebra)/Integer}} === Rational Numbers === {{:Definition:Power (Algebra)/Rational Number}} === Real Numbers === {{:Definition:Power (Algebra)/Real Number/Definition 1}} === Complex Numbers === {{Definition:Power (Algebra)/Complex Number}}
Definition:Exponential Function
The '''exponential function''' is denoted $\exp$ and can be defined in several ways, as described below.
Definition:Square Root
A '''square root''' of a number $n$ is a number $z$ such that $z$ squared equals $n$.
Definition:Arithmetic Mean
Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \ldots, x_n$ is defined as: :$\displaystyle A_n := \dfrac 1 n \sum_{k \mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a set of numbers, add them all up and divide by how many there are.
Definition:Geometric Mean
Let $x_1, x_2, \ldots, x_n \in \R_{>0}$ be (strictly) positive real numbers. The '''geometric mean''' of $x_1, x_2, \ldots, x_n$ is defined as: :$\displaystyle G_n := \paren {\prod_{k \mathop = 1}^n x_k}^{1/n}$
Definition:Harmonic Mean
Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers which are all positive. The '''harmonic mean''' of $x_1, x_2, \ldots, x_n$ is defined as: :$\displaystyle H_n := \paren {\frac 1 n \paren {\sum_{k \mathop = 1}^n \frac 1 {x_k} } }^{-1}$ That is, to find the '''harmonic mean''' of a set of $n$ numbers, take the reciprocal of the arithmetic mean of their reciprocals.
Definition:Reciprocal
Let $x \in \R$ be a real number such that $x \ne 0$. Then $\dfrac 1 x$ is called the '''reciprocal of $x$'''.
Definition:Euler's Number
=== As the Limit of a Sequence === {{:Definition:Euler's Number/Limit of Sequence}} === As the Limit of a Series === {{:Definition:Euler's Number/Limit of Series}} === As the Base of the Natural Logarithm === As the base of the Natural Logarithm: {{:Definition:Euler's Number/Base of Logarithm}} === In Terms of the Exponential Function === In terms of the exponential function: {{:Definition:Euler's Number/Exponential Function}} === As the Base of the Exponential with Derivative One at Zero === {{:Definition:Euler's Number/Base of Exponential}}
Definition:Field Extension
Let $F$ be a field. A '''field extension over $F$''' is a field $E$ where $F \subseteq E$. That is, such that $F$ is a subfield of $E$.
Definition:Unbounded Divergent Sequence
=== Real Sequence === {{:Definition:Unbounded Divergent Sequence/Real Sequence}} === Complex Sequence === As the Complex Numbers cannot be Ordered Compatibly with Ring Structure, there is no concept of $-\infty$ in discussions relating to $\C$. So we can use only the following definition: {{:Definition:Unbounded Divergent Sequence/Complex Sequence}}
Definition:Oscillating Sequence
Let $S$ be one of the standard number fields $\Q$, $\R$ or $\C$. Let $\sequence {x_n}$ be a sequence in $S$. Let $\sequence {x_n}$ be divergent. Suppose $\sequence {x_n}$ is ''not'' unbounded. That is, let: :$\neg x_n \to \infty$ as $n \to \infty$ Then $\sequence {x_n}$ is said to '''oscillate'''.
Definition:Field Adjoined Element
{{delete|think it fits more naturally here}} Let $E/F$ be a field extension, $\alpha \in E$. Then: :$F[\alpha] $ denotes the smallest subring of $E$ containing $F \cup \alpha$. :$F(\alpha) $ denotes the smallest subfield of $E$ containing $F \cup \alpha$.  We say this as '''$F$ adjoined with $\alpha$'''.
Definition:Simple Field Extension
Let $E / F$ be a field extension. Then $E$ is a '''simple extension over $F$''' : : $\exists \alpha \in E: E = F \sqbrk \alpha$ where $F \sqbrk \alpha$ is the field extension generated by $\alpha$.
Definition:Subsequence
Let $\sequence {x_n}$ be a sequence in a set $S$. Let $\sequence {n_r}$ be a strictly increasing sequence in $\N$. Then the composition $\sequence {x_{n_r} }$ is called a '''subsequence of $\sequence {x_n}$'''.
Definition:Limit Superior
Let $\sequence {x_n}$ be a bounded sequence in $\R$. === Definition 1 === {{:Definition:Limit Superior/Definition 1}} === Definition 2 === {{:Definition:Limit Superior/Definition 2}}
Definition:Limit Inferior
Let $\sequence {x_n}$ be a bounded sequence in $\R$. === Definition 1 === {{:Definition:Limit Inferior/Definition 1}} === Definition 2 === {{:Definition:Limit Inferior/Definition 2}}
Definition:Series
=== General Definition === {{:Definition:Series/General}} === Series in a Standard Number Field === The usual context for the definition of a series occurs when $S$ is one of the standard number fields $\Q, \R, \C$. {{:Definition:Series/Number Field}} === Finite === {{:Definition:Series/Finite}}
Definition:Cauchy Criterion
The '''Cauchy criterion''' is the necessary and sufficient condition for an infinite sequence in a complete metric space to converge that the absolute difference between successive terms with sufficiently large indices tends to zero.
Definition:Absolutely Convergent Series
=== General Definition === {{:Definition:Absolutely Convergent Series/General}} === Real Numbers === {{:Definition:Absolutely Convergent Series/Real Numbers}} === Complex Numbers === {{:Definition:Absolutely Convergent Series/Complex Numbers}}
Definition:Rational Function
Let $F$ be a field. Let $P: F \to F$ and $Q: F \to F$ be polynomial functions on $F$. Let $S$ be the set $F$ from which all the roots of $Q$ have been removed. That is: :$S = F \setminus \set {x \in F: \map Q x = 0}$ Then the equation $y = \dfrac {\map P x} {\map Q x}$ defines a mapping from $S$ to $F$. Such a mapping is called a '''rational function'''.
Definition:Semiperimeter
The '''semiperimeter''' of a plane figure is defined to be one half of the perimeter.
Definition:Circumradius
=== Polygon === {{:Definition:Circumradius/Polygon}} === Polyhedron === {{:Definition:Circumradius/Polyhedron}}
Definition:Incircle of Triangle
The '''incircle''' of a triangle is the circle tangent to each side of the triangle. This circle is said to be inscribed in the triangle. :410px === Incenter === {{:Definition:Incircle of Triangle/Incenter}} === Inradius === {{:Definition:Incircle of Triangle/Inradius}}
Definition:Excircle of Triangle
Given a triangle, extend two sides in the direction opposite their common vertex. The circle tangent to both of these lines and to the third side of the triangle is called an '''excircle'''. :500px There are three '''excircles''' for every triangle. === Excenter === {{:Definition:Excircle of Triangle/Excenter}} === Exradius === {{:Definition:Excircle of Triangle/Exradius}}
Definition:Circumcircle of Triangle
The '''circumcircle of a triangle''' is the circle that passes through all three of the triangle's vertices. This circle is said to circumscribe the triangle. :320px === Circumcenter === {{:Definition:Circumcircle of Triangle/Circumcenter}} === Circumradius === {{:Definition:Circumcircle of Triangle/Circumradius}}
Definition:Triangle (Geometry)
:300px A '''triangle''' is a polygon with exactly three sides.
Definition:Altitude of Triangle
Let $\triangle ABC$ be a triangle. Let a perpendicular be dropped from $\angle A$ to its opposite side $a$: :400px The line $h_a$ so constructed is called the '''altitude''' of $\angle A$. === Foot of Altitude === {{:Definition:Altitude of Triangle/Foot}}
Definition:Polygon
A '''polygon''' is a closed plane figure made up of an unspecified number of non-crossing straight line segments that join in pairs at their endpoints. For example: :400px
Definition:Quadrilateral
A '''quadrilateral''' is a polygon with exactly four sides.
Definition:Perimeter
The '''perimeter''' of a plane figure is the length of its boundary.
Definition:Bisection
To '''bisect''' a finite geometrical object is to cut it into two equal parts.
Definition:Differentiable Mapping
=== Real Function === {{:Definition:Differentiable Mapping/Real Function}} === Complex Function === {{:Definition:Differentiable Mapping/Complex Function}} === Real-Valued Function === {{:Definition:Differentiable Mapping/Real-Valued Function}} === Vector-Valued Function === {{:Definition:Differentiable Mapping/Vector-Valued Function}} === Between Differentiable Manifolds === {{Definition:Differentiable Mapping between Manifolds}}
Definition:Compact Space
=== Euclidean Space === {{:Definition:Compact Space/Euclidean Space}} === Topology === {{:Definition:Compact Space/Topology}} === Metric Space === {{:Definition:Compact Space/Metric Space}} === Normed Vector Space === {{:Definition:Compact Space/Normed Vector Space}}
Definition:Open Cover
Let $T = \struct {S, \tau}$ be a topological space. Let $\CC$ be a cover for $S$. Then $\CC$ is an '''open cover (of $T$)''' : :$\CC \subseteq \tau$ That is,  all the elements of $\CC$ are open sets. === Open Cover of Subset === {{:Definition:Open Cover/Subset}}
Definition:Closed Set
=== Topology === Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. {{:Definition:Closed Set/Topology}} === Metric Space === In the context of metric spaces, the same definition applies: {{:Definition:Closed Set/Metric Space}} === Normed Vector Space === {{:Definition:Closed Set/Normed Vector Space}} === Complex Analysis === {{:Definition:Closed Set/Complex Analysis}} === Real Analysis === {{:Definition:Closed Set/Real Analysis/Real Numbers}}
Definition:Submanifold
A '''submanifold''' is a manifold which is completely contained in some other manifold.
Definition:Codimension
Let $X$ be a manifold. Let $Z$ be a submanifold of $X$. The '''codimension of $Z$ in $X$''' is the dimension of $X$ minus the dimension of $Z$.
Definition:Critical Point (Topology)
Let $f: X \to Y$ be a smooth map of manifolds. A point $x \in X$ is called a '''critical point''' of $f$  $\d f_x: T_x \sqbrk X \to T_y \sqbrk Y$ is not surjective at $x$. {{explain|What are $\d f_x$, $T_x$ and $T_y$ in this context?}}
Definition:Homeomorphism
=== Topological Spaces === Let $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ be topological spaces. Let $f: T_\alpha \to T_\beta$ be a bijection. {{:Definition:Homeomorphism/Topological Spaces/Definition 1}} === Metric Spaces === The same definition applies to metric spaces: {{:Definition:Homeomorphism/Metric Spaces/Definition 1}} === Manifolds === The same definition applies to manifolds: {{:Definition:Homeomorphism/Manifolds}}
Definition:Hausdorff Space
{{:Definition:Hausdorff Space/Definition 1}}
Definition:Tychonoff Separation Axioms
The '''Tychonoff separation axioms''' are a classification system for topological spaces. They are not axiomatic as such, but conditions that may or may not apply to general or specific topological spaces.
Definition:Differential Form
Let $X$ be a smooth manifold. Let $\map {T_x} X$ denotes the tangent space of $X$. A '''$p$-form''' on $X$ is a function $\omega: \map {T_x} X^p \to \R$ defined at each point of $X$ which takes $p$ vectors as inputs, and outputs a real number.
Definition:Euclidean Space
Let $S$ be one of the standard number fields $\Q$, $\R$, $\C$. Let $S^n$ be a cartesian space for $n \in \N_{\ge 1}$. Let $d: S^n \times S^n \to \R$ be the usual (Euclidean) metric on $S^n$. Then $\tuple {S^n, d}$ is a '''Euclidean space'''.
Definition:Tangent Vector
Let $M$ be a smooth manifold. Let $m \in M$ be a point.  Let $V$ be an open neighborhood of $m$.  Let $C^\infty \left({V, \R}\right)$ be defined as the set of all smooth mappings $f: V \to \R$. === Definition 1 === {{:Definition:Tangent Vector/Definition 1}} === Definition 2 === {{:Definition:Tangent Vector/Definition 2}}
Definition:Tangent Bundle
Let $X$ be a differentiable manifold. The '''tangent bundle''' of $X$ is the disjoint union of all the tangent spaces of $X$ at $x$ as the base point $x$ ranges over the entire manifold: :$\displaystyle \map T X = \coprod_{x \mathop \in X} \map {T_x} X$ {{explain|probably also need to mention the natural differential structure on the tangent bundle.}}
Definition:P-adic Norm
=== Definition 1 === {{:Definition:P-adic Norm/Definition 1}} === Definition 2 === {{:Definition:P-adic Norm/Definition 2}} === $p$-adic Metric === {{:Definition:P-adic Norm/P-adic Metric}} === $p$-adic Numbers === Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. {{:Definition:P-adic Norm/P-adic Numbers}}
Definition:Homotopy
=== Free Homotopy === {{:Definition:Homotopy/Free}} === Relative Homotopy === {{:Definition:Homotopy/Relative}} === Path-Homotopy === {{:Definition:Homotopy/Path}}
Definition:Homotopy Group
Let $X$ be a topological space, and $x_0 \in X$. Let $\mathbb S^n \subseteq \R^{n+1}$ be the $n$-sphere, and $a \in \mathbb S^n$. Let $\pi_n \left({X, x_0}\right)$ be the set of homotopy classes relative to $a$ of continuous mappings $c: \mathbb S^n \to X$ such that $c(a) = x_0$. Let $* : \pi_n \left({X, x_0}\right) \times \pi_n \left({X, x_0}\right) \to \pi_n \left({X, x_0}\right)$ denote the concatenation of homotopy classes of paths. That is, if $\overline{c_1}, \overline{c_2}$ are two elements of $\pi_n \left({X, x_0}\right)$, then: :$\overline{c_1} * \overline{c_2} = \overline{c_1 \cdot c_2}$ where $\cdot$ denotes the usual concatenation of paths. Then $\left({\pi_n \left({X, x_0}\right), *}\right)$ is the '''$n$th fundamental group''' of $X$. The first homotopy group is usually called the '''fundamental group''' when higher homotopy groups are not in sight. For a path-connected manifold, by Fundamental Group is Independent of Base Point for Path-Connected Space, the isomorphism class of $\pi_1 \left({X, x_0}\right)$ does not depend on $x_0$ and we just write $\pi_1 \left({X}\right)$.
Definition:Concatenation (Topology)
Let $T = \left({S, \tau}\right)$ be a topological space. Let $c_1, c_2: \left[{0  \,.\,.\, 1}\right]^n \to S$ be maps. Let $c_1$ and $c_2$ both satisfy the concatenation criterion $c \left({\partial \left[{0 \,.\,.\, 1}\right]^n}\right) = x_0$. <!-- For two maps $c_1, c_2: \left[{0 \,.\,.\, 1}\right]^n \to X$ which satisfy the concatenation criterion $c \left({\partial \left[{0 \,.\,.\, 1}\right]^n}\right) = x_0$--> Then the '''concatenation''' $c_1 * c_2$ is defined as: :$\left({c_1 * c_2}\right) \left({t_1, t_2, \ldots, t_n}\right) = \begin{cases} c_1 \left({2t_1, t_2, \ldots, t_n}\right) & : t_1 \in \left[{0 \,.\,.\, 1/2}\right] \\  c_2 \left({2t_1-1, t_2, \ldots, t_n}\right) & : t_1 \in \left[{1/2 \,.\,.\, 1}\right] \end{cases} $  where $\left({t_1, \ldots, t_n}\right)$ are coordinates in the $n$-cube. By Continuous Mapping on Finite Union of Closed Sets, $c_1*c_2$ is continuous. This resulting map is continuous, since: * $2 \left({\dfrac 1 2}\right) = 1$ and $2 \left({\dfrac 1 2}\right) - 1 = 0$; * anywhere any coordinate of $\hat t$ is either $1$ or $0$, $\left({c_1*c_2}\right) \left({\hat t}\right) = x_0$. The resulting map also clearly satisfies the concatenation criteria itself.
Definition:Fundamental Group
Let $(X, x_0)$ be a pointed topological space with base point $x_0$. The '''fundamental group''' $\pi_1 \left({X, x_0}\right)$ of $X$ at the base point $x_0$ is the set of homotopy classes of loops with base point $x_0$ with multiplication of homotopy classes of paths.
Definition:Maximum Value of Real Function/Local
Let $f$ be a real function defined on an open interval $\openint a b$. Let $\xi \in \openint a b$. Then $f$ has a '''local maximum at $\xi$''' : :$\exists \openint c d \subseteq \openint a b: \forall x \in \openint c d: \map f x \le \map f \xi$ That is,  there is some subinterval on which $f$ attains a maximum within that interval.
Definition:Minimum Value of Real Function/Local
Let $f$ be a real function defined on an open interval $\openint a b$. Let $\xi \in \openint a b$. Then $f$ has a '''local minimum at $\xi$''' : :$\exists \openint c d \subseteq \openint a b: \forall x \in \openint c d: \map f x \ge \map f \xi$ That is,  there is some subinterval on which $f$ attains a minimum within that interval.
Definition:Stationary Point
Let $f$ be a real function which is differentiable on the open interval $\openint a b$. Let $\exists \xi \in \openint a b: \map {f'} \xi = 0$, where $\map {f'} \xi$ is the derivative of $f$ at $\xi$. Then $\xi$ is known as a '''stationary point of $f$'''.
Definition:Cauchy Equivalent Metrics
Let $X$ be a set upon which there are two metrics $d_1$ and $d_2$. That is, $\struct {X, d_1}$ and $\struct {X, d_2}$ are two different metric spaces on the same underlying set $X$. Then $d_1$ and $d_2$ are said to be '''Cauchy equivalent'''  for every sequence $\sequence {x_n}$ of points in $X$: :$\sequence {x_n}$ is a Cauchy sequence in $\struct {X, d_1} \iff \sequence {x_n}$ is a  is a Cauchy sequence in $\struct {X, d_2}$ {{NamedforDef|Augustin Louis Cauchy|cat = Cauchy}}
Definition:Homology Group
Let $X$ be a topological space. Let the standard n-simplex be denoted: :$\displaystyle \Delta^n := \left\{ {\left({x_0, \ldots, x_n}\right) \in {\R_{\ge 0} }^{n + 1}: \sum_{i \mathop = 1}^{n + 1} x_i = 1}\right\}$ Let $\mathcal C \left({\Delta^n, X}\right)$ be the set of continuous mappings from $\Delta^n$ to $X$.  For $n \ge 0$, let $C_n \left({X}\right)$ be the free abelian group generated by $\mathcal C \left({\Delta^n, X}\right)$.  Then there is a boundary map $\partial_n: C_n \left({X}\right) \to C_{n - 1} \left({X}\right)$ defined as follows. First, there are maps $s^i_n: \Delta^{n - 1} \to \Delta^n$, where $n > 0$ and $0 \le i \le n$, defined by: :$s^i_n \left({x_0, \ldots, x_{n - 1} }\right) = \left({x_0, \ldots, x_{i - 1}, 0, x_i, \ldots, x_{n-1} }\right)$ These can be considered as the inclusion of $\Delta^{n-1}$ as a 'face' of $\Delta^n$. For a continuous function $\phi: \Delta^n \to X$ we define: :$\displaystyle \partial_n \left({\phi}\right) = \sum_{i \mathop = 0}^n \left({-1}\right)^i \phi \circ s^i_n$ This definition, along with the requirement that $\partial_n$ be a group homomorphism, uniquely specifies $\partial_n$. In addition, one has $\partial_{n-1} \partial_n = 0$, meaning that the sequence of groups and morphisms: :$0 \gets C_0 \left({X}\right) \stackrel {\partial_1} {\longleftarrow} C_1 \left({X}\right) \stackrel {\partial_2} {\longleftarrow} C_2 \left({X}\right) \stackrel {\partial_3} {\longleftarrow} \cdots$ are a chain complex. This is demonstrated in Singular Chains form Chain Complex. Let $\partial_0$ denote the map $C_0 \left({X}\right) \to 0$. Thus the $n$th singular homology group of $X$ is defined as the $n$th homology group of this chain complex. Explicitly: Let $B_n \left({X}\right) \subset C_n \left({X}\right)$ denote the image of $\partial_{n+1}$. Let $Z_n \left({X}\right)$ denote the kernel of $\partial_n$. Since $\partial_n \partial_{n + 1} = 0$: :$B_n \left({X}\right) \subseteq Z_n \left({X}\right)$ Then define: :$H_n \left({X}\right) = \dfrac {Z_n \left({X}\right)} {B_n \left({X}\right)}$
Definition:Convex Real Function
Let $f$ be a real function which is defined on a real interval $I$. === Definition 1 === {{:Definition:Convex Real Function/Definition 1}} === Definition 2 === {{:Definition:Convex Real Function/Definition 2}} === Definition 3 === {{:Definition:Convex Real Function/Definition 3}}
Definition:Concave Real Function
Let $f$ be a real function which is defined on a real interval $I$. === Definition 1 === {{:Definition:Concave Real Function/Definition 1}} === Definition 2 === {{:Definition:Concave Real Function/Definition 2}} === Definition 3 === {{:Definition:Concave Real Function/Definition 3}}
Definition:Simply Connected
A path-connected topological space $T = \left({S, \tau}\right)$ is said to be '''simply connected''' if the fundamental group $\pi_1 \left({T}\right)$ is trivial. {{stub|Expand to include the context of regions in the plane.}}
Definition:Primitive (Calculus)
=== Primitive of Real Function === {{:Definition:Primitive (Calculus)/Real}}
Definition:Transcendental Number
A number (either real or complex) is '''transcendental'''  it is not algebraic.