hcju/proofwikips huggingface dataset:
Definition:Class Mapping
Let $S$ and $T$ be classes. A '''class mapping $f$ from $S$ to $T$''', denoted $f: S \to T$, is a class relation $f = \mathcal R \subseteq S \times T$ such that: :$\forall x \in S: \forall y_1, y_2 \in T: \left({x, y_1}\right) \in f \land \left({x, y_2}\right) \in f \implies y_1 = y_2$ and :$\forall x \in S: \exists y \in T: \left({x, y}\right) \in f$
Definition:Class Injection
Let $A$ and $B$ be classes. Let $f: A \to B$ be a class mapping from $A$ to $B$. Then $f$ is said to be a '''class injection''' if and only if: :$\forall x_1, x_2 \in A: f \left({x_1}\right) = f \left({x_2}\right) \implies x_1 = x_2$
Definition:Class Surjection
Let $A$ and $B$ be classes. Let $f: A \to B$ be a class mapping from $A$ to $B$. Then $f$ is said to be a '''class surjection''' if and only if: :$\forall y \in B: \exists x \in A: f \left({x}\right) = y$
Definition:Class Bijection
Let $A$ and $B$ be classes. Let $f: A \to B$ be a class mapping from $A$ to $B$. Then $f$ is said to be a '''class bijection''' if and only if: :$f$ is both a class injection and a class surjection.
Definition:Greatest Common Divisor/Polynomial Ring over Field
Let $F$ be a field. Let $P, Q, R \in F \left[{X}\right]$ be polynomials. Then $R$ is '''the greatest common divisor''' of $P$ and $Q$  it is a monic greatest common divisor. This is denoted $\gcd \left({P, Q}\right) = R$.
Definition:Euclidean Number Theory
'''Euclidean number theory''' concerns aspects of number theory and real analysis as presented in . Particular attention is paid to certain properties of the irrational numbers.
Definition:Göbel's Sequence
'''Göbel's sequence''' is the sequence defined recursively as: :$x_n = \begin{cases} 1 & : n = 0 \\ \displaystyle \left({1 + \sum_{k \mathop = 0}^{n - 1} {x_k}^2}\right) / n & : n > 0 \end{cases}$
Definition:Göbel's Sequence/Sequence
'''Göbel's sequence''' begins: :$1, 2, 3, 5, 10, 28, 154, 3520, 1 \, 551 \, 880, 267 \, 593 \, 772 \, 160, \ldots$
Definition:Göbel's Sequence/Historical Note
Some sources link Göbel's sequence with the name of {{AuthorRef|Michael Somos}}, but it appears that the latter has a different sequence named for him.
Definition:Göbel's Sequence/General/Examples/3
The '''$3$-Göbel sequence''' begins: :$1, 2, 5, 45, 22 \, 815, 2 \, 375 \, 152 \, 056 \, 927, \ldots$
Definition:Class Relation
Let $S$ and $T$ be classes. Let $S \times T$ be the Cartesian Product of $S$ and $T$. Then a '''relation''' on $S \times T$ is defined as a subclass of $S \times T$: :$\mathcal R \subseteq S \times T$
Definition:Conjugate Point/Dependent on N Functions
Let $K$ be a functional such that: :$\ds K \sqbrk h = \int_a^b \paren {\mathbf h'\mathbf P \mathbf h' + \mathbf h \mathbf Q \mathbf h} \rd x$ Consider Euler's equation related to the functional $K$: :$-\map {\dfrac \d {\d x} } {\mathbf P \mathbf h'} + \mathbf Q \mathbf h = 0$ where $\mathbf P$ and $\mathbf Q$ are symmetric matrices. Let the general solution to this equation be: :$\set {\mathbf h^{\paren i} = \paren {\sequence {h_{ij} } }: i,j \in \N_{\le N} }$ Let: :$\exists j: \forall k \ne j: \paren {\map {\mathbf h^{\paren j} } a = 0} \land \paren {\map {h_{j j}'} a = 1, h'_{j k} = 0}$ Let the determinant, built from $h_{ij}$, be such that: :$\size {h_{i j} } \paren {\tilde a} = 0$ Here $i$ denotes rows, and $j$ denotes columns. Then $\tilde a$ is said to be '''conjugate''' to point $a$  the functional $K$.
Definition:Derangement/Historical Note
The number of a derangements of a finite set was first investigated by {{AuthorRef|Nicolaus I Bernoulli}} and {{AuthorRef|Pierre Raymond de Montmort}} between about $1708$ and $1713$. They solved it at around the same time. The question is often couched in the idea of counting the number of ways of placing letters at random in envelopes such that nobody receives the correct letter.
Definition:Kaprekar Triple
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Suppose that $n^3$, when expressed in number base $b$, can be split into three parts that add up to $n$. Then $n$ is a '''Kaprekar triple''' for base $b$.
Definition:Kaprekar Triple/Sequence
The sequence of Kaprekar triples begins: :$1, 8, 45, 297, 2322, 2728, 4445, 4544, 4949, 5049, 5455, 5554, \ldots$
Definition:Hexamorphic Number
A '''hexamorphic number''' is a hexagonal number $H_n$ whose decimal representation ends in $n$.
Definition:Hexamorphic Number/Sequence
The sequence of hexamorphic numbers, for $n \in \Z_{\ge 0}$, begins: {{begin-eqn}} {{eqn | l = H_1       | r = 1 }} {{eqn | l = H_5       | r = 45 }} {{eqn | l = H_6       | r = 66 }} {{eqn | l = H_{25}       | r = 1225 }} {{eqn | l = H_{26}       | r = 1326 }} {{eqn | l = H_{50}       | r = 4950 }} {{end-eqn}}
Definition:Uniform Absolute Convergence
Let $S$ be a set. Let $\left({V, \left\lVert{\, \cdot \,}\right\rVert}\right)$ be a normed vector space. Let $\left \langle {f_n} \right \rangle$ be a sequence of mappings $f_n: S \to V$. Then the series $\displaystyle \sum_{n \mathop = 1}^\infty f_n$ '''converges uniformly absolutely'''  the series $\displaystyle \sum_{n \mathop = 1}^\infty \left\lVert{f_n}\right\rVert$ converges uniformly.
Definition:Locally Uniform Convergence
=== General Definition === {{:Definition:Locally Uniform Convergence/General Definition}} === Complex Functions === {{:Definition:Locally Uniform Convergence/Complex Functions}} === Series === {{:Definition:Locally Uniform Convergence/Series}}
Definition:Locally Uniform Convergence/General Definition
Let $X$ be a topological space. Let $M$ be a metric space. Let $\left\langle{f_n}\right\rangle$ be a sequence of mappings $f_n: X \to M$. Then $f_n$ '''converges locally uniformly''' to $f: X \to M$ if every point of $X$ has a neighborhood on which $f_n$ converges uniformly to $f$.
Definition:Compact Convergence
Let $X$ be a topological space. Let $M$ be a metric space. Let $\left\langle{f_n}\right\rangle$ be a sequence of mappings $f_n : X \to M$. Let $f: X \to M$ be a mapping. Then $f_n$ '''converges compactly''' to $f$  $f_n$ converges uniformly to $f$ on every compact subset of $X$.
Definition:Amicable Pair
Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be (strictly) positive integers. === Definition 1 === {{:Definition:Amicable Pair/Definition 1}} === Definition 2 === {{:Definition:Amicable Pair/Definition 2}} === Definition 3 === {{:Definition:Amicable Pair/Definition 3}}
Definition:Quasiamicable Numbers
Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be (strictly) positive integers. === Definition 1 === {{:Definition:Quasiamicable Numbers/Definition 1}} === Definition 2 === {{:Definition:Quasiamicable Numbers/Definition 2}}
Definition:Uniform Convergence of Product/Compact Space
Let $X$ be a compact topological space. Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a complete valued field. Let $\sequence{f_n}$ be a sequence of continuous mappings $f_n: X \to \mathbb K$. The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ '''converges uniformly'''  there exists $n_0 \in \N$ such that the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty f_n$ converges uniformly and is nonzero.
Definition:Locally Uniform Absolute Convergence
Let $X$ be a topological space. Let $\left({V, \left\lVert{\, \cdot \,}\right\rVert}\right)$ be a normed vector space. Let $\left \langle {f_n} \right \rangle$ be a sequence of mappings $f_n: X \to V$. Then the series $\displaystyle \sum_{n \mathop = 1}^\infty f_n$ '''converges locally uniformly absolutely'''  the series $\displaystyle \sum_{n \mathop = 1}^\infty \left\lVert{f_n}\right\rVert$ converges locally uniformly.
Definition:Compact Absolute Convergence
Let $X$ be a topological space. Let $\left({V, \left\lVert{\, \cdot \,}\right\rVert}\right)$ be a normed vector space. Let $\left \langle {f_n} \right \rangle$ be a sequence of mappings $f_n: X \to V$. Then the series $\displaystyle \sum_{n \mathop = 1}^\infty f_n$ '''converges compactly (uniformly) absolutely'''  the series $\displaystyle \sum_{n \mathop = 1}^\infty \left\lVert{f_n}\right\rVert$ converges compactly.
Definition:Local Uniform Absolute Convergence of Product
Let $X$ be a topological space. Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field. Let $\sequence {f_n}$ be a sequence of mappings $f_n: X \to \mathbb K$. Then the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ '''converges locally uniformly absolutely'''  every point of $X$ has a neighborhood on which it converges uniformly absolutely.
Definition:Absolute Convergence of Product/General Definition
Let $\struct {\mathbb K, \norm{\,\cdot\,} }$ be a valued field. Let $\sequence {a_n}$ be a sequence in $\mathbb K$. === Definition 1 === {{:Definition:Absolute Convergence of Product/General Definition/Definition 1}} === Definition 2 === {{:Definition:Absolute Convergence of Product/General Definition/Definition 2}}
Definition:Absolute Convergence of Product/General Definition/Definition 2
The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \paren{1 + a_n}$ is '''absolutely convergent'''  the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is absolutely convergent.
Definition:Quasiamicable Numbers/Definition 1
$m$ and $n$ are '''quasiamicable numbers''' : :the sum of the proper divisors of $m$ is equal to $n$ and: :the sum of the proper divisors of $n$ is equal to $m$.
Definition:Quasiamicable Numbers/Definition 2
$m$ and $n$ are '''quasiamicable numbers''' : :$\sigma \left({m}\right) = \sigma \left({n}\right) = m + n + 1$ where $\sigma \left({m}\right)$ denotes the $\sigma$ function.
Definition:Absolute Convergence of Product/Complex Numbers/Definition 1
The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + a_n}\right)$ is '''absolutely convergent'''  $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + \left\vert{a_n}\right\vert}\right)$ is convergent.
Definition:Absolute Convergence of Product/Complex Numbers/Definition 2
The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + a_n}\right)$ is '''absolutely convergent'''  the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is absolutely convergent.
Definition:Absolute Convergence of Product/Complex Numbers/Definition 3
The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ is '''absolutely convergent'''  there exists $n_0 \in \N$ such that: :$a_n \ne -1$ for $n > n_0$ :The series $\displaystyle \sum_{n \mathop = n_0 + 1}^\infty \log \paren {1 + a_n}$ is absolutely convergent where $\log$ denotes the complex logarithm.
Definition:Trimorphic Number
An '''trimorphic number''' is a positive integer whose cube ends in that number.
Definition:Divergent Product/Divergence to Zero
If either: :there exist infinitely many $n \in \N$ with $a_n = 0$ :there exists $n_0 \in \N$ with $a_n \ne 0$ for all $n > n_0$ and the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0 + 1}^\infty a_n$ converges to $0$ the product '''diverges to $0$''', and we assign the value: :$\displaystyle \prod_{n \mathop = 1}^\infty a_n = 0$
Definition:Cauchy's Criterion for Products
Let $\struct {\mathbb K, \norm {\,\cdot\,}}$ be a valued field. Let $\sequence {a_n}$ be a sequence of elements of $\mathbb K$. The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ satisfies '''Cauchy's criterion''' : :$\displaystyle \forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m\geq n \ge N: \norm {\prod_{k \mathop = n}^m a_n - 1} < \epsilon$
Definition:Cauchy Sequence/Cauchy Criterion
The '''Cauchy criterion''' is the condition: :For any (strictly) positive real number $\epsilon \in \R_{>0}$, for a sufficiently large natural number $N \in \N$, the difference between the $m$th and $n$th terms of a Cauchy sequence, where $m, n \ge N$, will be less than $\epsilon$. Informally: :For any number you care to pick (however small), if you go out far enough into the sequence, past a certain point, the difference between any two terms in the sequence is less than the number you picked. Or to put it another way, the terms get arbitrarily close together the farther out you go.
Definition:Uniform Absolute Convergence of Product/General Definition
Let $X$ be a set. Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\sequence {f_n} $ be a sequence of bounded mappings $f_n: X \to \mathbb K$. === Definition 1 === {{:Definition:Uniform Absolute Convergence of Product/General Definition/Definition 1}} === Definition 2 === {{:Definition:Uniform Absolute Convergence of Product/General Definition/Definition 2}}
Definition:Uniform Absolute Convergence of Product/General Definition/Definition 1
The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + f_n}\right)$ '''converges uniformly absolutely'''  the sequence of partial products of $\displaystyle \prod_{n \mathop = 1}^\infty(1+ \norm{f_n})$ converges uniformly.
Definition:Uniform Absolute Convergence of Product/General Definition/Definition 2
The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + f_n}\right)$ '''converges uniformly absolutely'''  the series $\displaystyle \sum_{n \mathop = 1}^\infty f_n$ converges uniformly absolutely.
Definition:Uniform Absolute Convergence of Product/Compact Space
{{refactor|integrate this in the other lists of definitions}} Let $X$ be a set. Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\sequence {f_n} $ be a sequence of bounded mappings $f_n: X \to \mathbb K$. The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + f_n}\right)$ '''converges uniformly absolutely'''  $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + \norm{f_n}}\right)$ converges uniformly.
Definition:Uniform Absolute Convergence of Product/Complex Functions
Let $X$ be a set. Let $\left \langle {f_n} \right \rangle$ be a sequence of bounded mappings $f_n: X \to \C$. === Definition 1 === {{:Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 1}} === Definition 2 === {{:Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 2}} === Definition 3 === {{:Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 3}}
Definition:Locally Uniform Convergence of Product
Let $T = \left({S, \tau}\right)$ be a weakly locally compact topological space. Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\left \langle {f_n} \right \rangle$ be a sequence of locally bounded mappings $f_n: S \to \mathbb K$. === Definition 1 === {{:Definition:Locally Uniform Convergence of Product/Definition 1}} === Definition 2 === {{:Definition:Locally Uniform Convergence of Product/Definition 2}}
Definition:Locally Uniform Convergence of Product/Definition 1
The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ '''converges locally uniformly'''  every point of $T$ has a compact neighborhood on which it converges uniformly.
Definition:Locally Uniform Convergence of Product/Definition 2
The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ '''converges locally uniformly'''  it converges uniformly on every compact subspace of $T$.
Definition:Normed Division Ring
Let $\struct {R, +, \circ}$ be a division ring. Let $\norm {\,\cdot\,}$ be a norm on $R$. Then $\struct {R, \norm{\,\cdot\,} }$ is a '''normed division ring'''.
Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 1
The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + f_n}\right)$ '''converges uniformly absolutely'''  the sequence of partial products of $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + \left\vert{f_n}\right\vert}\right)$ converges uniformly.
Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 3
The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + f_n}\right)$ '''converges uniformly absolutely'''  there exists $n_0 \in \N$ such that: :$(1): \quad f_n \left({x}\right) \ne -1$ for $n \ge n_0$ and $x \in X$ and: :$(2): \quad$ The series $\displaystyle \sum_{n \mathop = n_0}^\infty \log \left({1 + f_n}\right)$ is uniformly absolutely convergent.
Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 2
The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + f_n}\right)$ '''converges uniformly absolutely'''  the series $\displaystyle \sum_{n \mathop = 1}^\infty f_n$ converges uniformly absolutely.
Definition:Locally Uniform Convergence/Series
Let $X$ be a topological space. Let $V$ be a normed vector space. Let $\left\langle{f_n}\right\rangle$ be a sequence of mappings $f_n:X\to V$. Then the series $\displaystyle \sum_{n \mathop = 1}^\infty f_n$ '''converges locally uniformly'''  the sequence of partial sums converges locally uniformly.
Definition:Order of Entire Function/Definition 2
Let $f$ be not identically zero. The '''order''' $\alpha \in \closedint 0 {+\infty}$ of $f$ is the infimum of the $\beta \ge 0$ for which: :$\displaystyle \map \ln {\max_{\size z \mathop \le R} \size {\map f z} } = \map \OO {R^\beta}$ or $\infty$ if no such $\beta$ exists, where $\OO$ denotes big-O notation The '''order''' of $0$ is $0$.
Definition:Order of Entire Function/Definition 3
Let $f$ be non-constant. The '''order''' $\alpha \in \closedint 0 {+\infty}$ of $f$ is the limit superior: :$\displaystyle \limsup_{R \mathop \to \infty} \frac {\displaystyle \ln \ln \max_{\cmod z \le R} \cmod f} {\ln R}$ The '''order''' of a constant function is $0$.
Definition:Order of Entire Function/Definition 1
The '''order''' $\alpha \in \closedint 0 {+\infty}$ of $f$ is the infimum of the $\beta \ge 0$ for which: :$\map f z = \map \OO {\map \exp {\size z^\beta} }$ or $\infty$ if no such $\beta$ exists, where $\OO$ denotes big-O notation.
Definition:Jacobi's Equation of Functional/Dependent on N Functions
Let: :$\ds \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ be a (real) functional, where $\map {\mathbf y} a = A$ and $\map {\mathbf y} b = B$. Let: :$\ds \int_a^b \paren {\mathbf h' \mathbf P \mathbf h' + \mathbf h \mathbf Q \mathbf h} \rd x$ be a quadratic functional, where: :$P_{ij} = \dfrac 1 2 F_{y_i'y_j'}$ :$Q_{ij} = \dfrac 1 2 \paren {F_{y_i y_j} - \dfrac \d {\d x} F_{y_i y_j'} }$ Then the Euler's equation of the latter functional: :$-\map {\dfrac \d {\d x} } {\mathbf P \mathbf h'} + \mathbf Q \mathbf h = \mathbf 0$ is called '''Jacobi's Equation''' of the former functional. {{NamedforDef|Carl Gustav Jacob Jacobi|cat = Jacobi}}
Definition:Exponent of Convergence
Let $\sequence {a_n}$ be a sequence of nonzero complex numbers. The '''exponent of convergence''' of $\sequence {a_n}$ is the infimum of $\tau \ge 0$ for which the series $\displaystyle \sum_{n \mathop = 1}^\infty \size {a_n}^{-\tau}$ converges. The '''exponent of convergence''' of a finite sequence is $0$.
Definition:Rank of Entire Function
Let $f: \C \to \C$ be an entire function. Let $\left\langle{a_n}\right\rangle$ be the sequence of nonzero zeroes of $f$, repeated according to multiplicity. The '''rank''' of $f$ is the smallest integer $p\geq0$ for which the series $\displaystyle\sum_{n\mathop=1}^\infty |a_n|^{-p-1}$ converges, or $\infty$ if there is no such integer. If $f$ has finitely many zeroes, its '''rank''' is $0$.
Definition:Hadamard's Canonical Factorization
Let $f: \C \to \C$ be a nonzero entire function of finite rank $p \in \N$. Let $0$ be a zero of $f$ of multiplicity $m\geq0$. Let $\left\langle{a_n}\right\rangle$ be the sequence of nonzero zeroes of $f$, repeated according to multiplicity. The '''canonical representation''' of $f$ is: :$\displaystyle f \left({z}\right) = z^m e^{g \left({z}\right)} \prod_{n \mathop = 1}^\infty E_p\left({\frac z {a_n} }\right)$ where: :$g: \C \to \C$ is an entire function :$E_p$ denotes the $p$th Weierstrass elementary factor. If $f$ has finitely many zeroes, the product is understood to be finite. {{NamedforDef|Jacques Salomon Hadamard|cat = Hadamard}}
Definition:Pentagonal Pyramidal Number
'''Pentagonal pyramidal numbers''' are those denumerating a collection of objects which can be arranged in the form of a pentagonal pyramid.
Definition:Hexagonal Pyramidal Number
'''Hexagonal pyramidal numbers''' are those denumerating a collection of objects which can be arranged in the form of a hexagonal pyramid.
Definition:Cubic Recurring Digital Invariant
A '''cubic recurring digital invariant''' is a recurring digital invariant of order $3$. === Sequence === {{:Definition:Cubic Recurring Digital Invariant/Sequence}}
Definition:Mutually Consistent Boundary Conditions
Let $\map {\mathbf y} x$, $\map {\boldsymbol \psi} {\mathbf y}$ be an N-dimensional vectors. Consider the system of differential equations: :$(1): \quad \mathbf y'' = \map {\mathbf f} {x, \mathbf y, \mathbf y'}$ Let derivatives of $\mathbf y$ satisfy: :$\bigintlimits {\mathbf y'} {x \mathop = x_1} {} = \bigintlimits {\map {\boldsymbol \psi^{\paren 1} } {\mathbf y} } {x \mathop = x_1} {}$ :$\bigintlimits {\mathbf y'} {x \mathop = x_2} {} = \bigintlimits {\map {\boldsymbol \psi^{\paren 2} } {\mathbf y} } {x \mathop = x_2} {}$ If every solution of $(1)$ satisfying conditions at $x = x_1$ automatically satisfies conditions at $x = x_2$ (or vice versa), then these boundary conditions are called '''mutually consistent'''.
Definition:Recurring Digital Invariant
Let $k \in \Z_{>0}$ be a positive integer. Let $f: \Z_{>0} \to \Z_{>0}$ be the mapping defined as: :$\forall m \in \Z_{>0}: f \left({m}\right) = $ the sum of the $k$th powers of the digits of $n$. Let $n_0 \in \Z_{>0}$ be a positive integer. Consider the sequence: :$s_n = \begin{cases} n_0 & : n = 0 \\ f \left({s_{n - 1} }\right) & : n > 0 \end{cases}$ If: :$\exists r \in \N_{>0}: s_r = n_0$ then the smallest of the terms $n_0, n_1, \ldots, n_r$ is a '''recurring digital invariant of order $k$'''.
Definition:Separated Sets/Definition 2
$A$ and $B$ are '''separated (in $T$)'''  there exist $U,V\in\tau$ with: :$A\subset U$ and $U\cap B = \varnothing$ :$B\subset V$ and $V\cap A = \varnothing$ where $\varnothing$ denotes the empty set.
Definition:Separated Sets/Definition 1
$A$ and $B$ are '''separated (in $T$)''' : :$A^- \cap B = A \cap B^- = \O$ where: :$A^-$ denotes the closure of $A$ in $T$ :$\O$ denotes the empty set.
Definition:Logarithmic Mean Value
Let $f: \N \to \C$ be an arithmetic function. The '''logarithmic mean value''' of $f$ is the limit: :$L \left({f}\right) = \displaystyle \lim_{x \mathop \to \infty} \frac 1 {\ln x} \sum_{n \mathop \le x} \frac {f \left({n}\right)} n$ if it exists.
Definition:Ordinary Mean Value
Let $f: \N \to \C$ be an arithmetic function. The '''(ordinary) mean value''' of $f$ is the limit: :$M \left({f}\right) = \displaystyle \lim_{x \mathop \to \infty} \frac 1 x \sum_{n \mathop \le x} f \left({n}\right)$ if it exists.
Definition:Summatory Function
Let $a: \N \to \C$ be an arithmetic function. Its '''summatory function''' is the mapping: :$A \left({x}\right) = \displaystyle\sum_{n \mathop \le x} a \left({n}\right)$ defined for $x \in \Z_{\ge 0}$.
Definition:Dirichlet Convolution/Definition 1
The '''Dirichlet convolution''' of $f$ and $g$ is the arithmetic function: :$\displaystyle \left({f * g}\right) \left({n}\right) := \sum_{d \mathop \backslash n} f \left({d}\right) g \left({\frac n d}\right)$ where the summation runs over the set of positive divisors $d$ of $n$.
Definition:Dirichlet Convolution/Definition 2
The '''Dirichlet convolution''' of $f$ and $g$ is the arithmetic function: :$\displaystyle \left({f * g}\right) \left({n}\right) := \sum_{a b \mathop = n} f \left({a}\right) g \left({b}\right)$ where the summation runs over all pairs of positive integers $\left({a, b}\right)$ with $a b = n$.
Definition:Pentatope Number
'''Pentatope numbers''' are those denumerating a collection of objects which can be arranged in $4$ dimensions in the form of a regular pentatope.
Definition:Polytope Number
'''Polytope numbers''' are those denumerating a collection of objects which can be arranged in $4$ dimensions in the form of a regular polytope.
Definition:Polytope Number/Examples
=== Pentatope Number === {{:Definition:Pentatope Number}}