hcju/proofwikips huggingface dataset:
Value of Multiplicative Function is Product of Values of Prime Power Factors
Let $f: \N \to \C$ be a multiplicative function. Let $n = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$ be the prime decomposition of $n$. Then: :$\map f n = \map f {p_1^{k_1} } \, \map f {p_2^{k_2} } \dotsm \map f {p_r^{k_r} }$
-1^n by -n choose k-1 equals -1^k by -k choose n-1
:$\paren {-1}^n \dbinom {-n} {k - 1} = \paren {-1}^k \dbinom {-k} {n - 1}$
One Choose n
:$\dbinom 1 n = \begin{cases} 1 & : n \in \left\{ {0, 1}\right\} \\ 0 & : \text {otherwise} \end{cases}$
Summations of Products of Binomial Coefficients
This page gathers together some identities concerning summations of products of binomial coefficients. In the following, unless otherwise specified: :$k, m, n \in \Z$ :$r, s, t \in \R$.
Separable Discrete Space is Countable
Let $T = \struct {S, \tau}$ be a discrete topological space. Let $T$ be separable. Then $S$ is countable.
Sum over k of r-tk Choose k by s-t(n-k) Choose n-k by r over r-tk/Proof 1/Basis for the Induction
Let $r, s, t \in \R, n \in \Z$. Consider the equation: :$\displaystyle (1): \quad \sum_{k \mathop \ge 0} \binom {r - t k} k \binom {s - t \paren {n - k} } {n - k} \frac r {r - t k} = \binom {r + s - t n} n$ where $\dbinom {r - t k} k$ etc. are binomial coefficients. Then equation $(1)$ holds for the special case where $s = n - 1 - r + n t$.
Continuous Lattice iff Auxiliary Approximating Relation is Superset of Way Below Relation
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $\mathit{App}\left({L}\right)$ be the set of all auxiliary approximating relation on $S$. Then :$L$ is continuous  :$\forall \mathcal R \in \mathit{App}\left({L}\right): \ll \subseteq \mathcal R$ and $\ll$ is an approximating relation
Completely Additive Function is Additive
Let $f: \N \to \C$ be a completely additive function. Then $f$ is also additive.
Real Logarithm is Completely Additive
Let $\log_b: \R_{>0} \to \R$ be the real logarithm to base $b$. Then $\log_b$ is completely additive.
Sum over k of r-tk Choose k by s-t(n-k) Choose n-k by r over r-tk/Proof 1/Lemma
Let this hold for $\tuple {r, s, t, n}$: :$\displaystyle \sum_{k \mathop \ge 0} \binom {r - t k} k \binom {s - t \paren {n - k} } {n - k} \frac r {r - t k} = \binom {r + s - t n} n$ and also for $\tuple {r, s - t, t, n - 1}$. Then it also holds for $\tuple {r, s + 1, t, n}$.
Sum over k of r-kt choose k by r over r-kt by z^k
Let $n \in \Z_{\ge 0}$ be a positive integer. Let $A_n \left({x, t}\right)$ be the polynomial of degree $n$ defined as: :$A_n \left({x, t}\right) := \dbinom {x - n t} n \dfrac x {x - n t}$ for $x \ne n t$. Let $z = x^{t + 1} - x^t$. Then: :$\displaystyle \sum_k A_k \left({r, t}\right) z^k = x^r$ for sufficiently small $z$.
Not Preceding implies Approximating Relation and not Preceding
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $x, y \in S$ such that :$x \npreceq y$ Let $\mathcal R$ be an approximating relation on $S$. Then :$\exists u \in S: \left({u, x}\right) \in \mathcal R \land u \npreceq y$
Auxiliary Approximating Relation has Quasi Interpolation Property
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $x, z \in S$. Let $\mathcal R$ be an auxiliary approximating relation on $S$ such that :$\left({x, z}\right) \in \mathcal R \land x \ne z$ Then :$\exists y \in S: x \preceq y \land \left({y, z}\right) \in \mathcal R \land x \ne y$
Preceding iff Join equals Larger Operand
Let $\left({S, \preceq}\right)$ be a join semilattice. Let $x, y \in S$. Then :$x \preceq y$  $x \vee y = y$
Sum over k of -1^k by n choose k by r-kt choose n by r over r-kt
:$\displaystyle \sum_k \left({-1}\right)^k \dbinom n k \dbinom {r - k t} n \dfrac r {r - k t} = \delta_{n 0}$ where $\delta_{n 0}$ is the Kronecker delta.
Sum over k of r-kt choose k by z^k
Let $n \in \Z_{\ge 0}$ be a non-negative integer. Then: :$\displaystyle \sum_k \dbinom {r - t k} k z^k = \frac {x^{r + 1} } {\left({t + 1}\right)x - t}$ where $\dbinom {r - t k} k$ denotes a binomial coefficient.
Auxiliary Approximating Relation has Interpolation Property
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $x, z \in S$ such that :$x \ll z \land x \ne z$ Let $\mathcal R$ be an auxiliary approximating relation on $S$. Then :$\exists y \in S: \left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \land x \ne y$
Way Below has Strong Interpolation Property
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below continuous lattice. Let $x, z \in S$ such that :$x \ll z \land x \ne z$ Then :$\exists y \in S: x \ll y \land y \ll z \land x \ne y$
Way Below has Interpolation Property
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below continuous lattice. Let $x, z \in S$ such that :$x \ll z$ Then :$\exists y \in S: x \ll y \land y \ll z$
Left Coset Equals Subgroup iff Element in Subgroup
:$x H = H \iff x \in H$
Right Coset Equals Subgroup iff Element in Subgroup
:$H x = H \iff x \in H$
Elements in Same Left Coset iff Product with Inverse in Subgroup
: $x, y$ are in the same left coset of $H$  $x^{-1} y \in H$.
Elements in Same Right Coset iff Product with Inverse in Subgroup
: $x, y$ are in the same right coset of $H$  $x y^{-1} \in H$
Sum over k of m+r+s Choose k by n+r-s Choose n-k by r+k Choose m+n
Let $m, n \in \Z_{\ge 0}$. Then: :$\displaystyle \sum_k \binom {m - r + s} k \binom {n + r - s} {n - k} \binom {r + k} {m - n} = \binom r m \binom s n$
Sum over k of Unsigned Stirling Numbers of First Kind by x^k
:$\displaystyle \sum_k \left[{n \atop k}\right] x^k = x^{\overline n}$ where: :$\displaystyle \left[{n \atop k}\right]$ denotes an unsigned Stirling number of the first kind :$x^{\overline n}$ denotes $x$ to the $n$ rising.
Capelli's Sum
:$\displaystyle \left({x + y}\right)^{\overline n} = \sum_k \binom n k x^{\overline k} y^{\overline {n - k} }$ where: :$\dbinom n k$ denotes a binomial coefficient :$x^{\overline k}$ denotes $x$ to the $k$ rising.
Way Below iff Second Operand Preceding Supremum of Directed Set There Exists Element of Directed Set First Operand Way Below Element
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below continuous lattice. Let $x, y$ be elements of $S$. Then :$x \ll y$  :for every directed subset $D$ of $S$ such that $y \preceq \sup D$ ::there exists an element $d$ of $D$: $x \ll d$
Continuous iff Way Below Closure is Ideal and Element Precedes Supremum
Let $L = \left({S, \wedge, \preceq}\right)$ be an up-complete meet semilattice. Then :$L$ is continuous  :$\forall x \in S: x^\ll$ is an ideal in $L$ and $x \preceq \sup \left({x^\ll}\right)$ and ::for every ideal $I$ in $L$: $x \preceq \sup I \implies x^\ll \subseteq I$ where $x^\ll$ denotes the way below closure of $x$.
Characterization of Euclidean Borel Sigma-Algebra/Open equals Closed
Let $\mathcal O^n$, $\mathcal C^n$ and $\mathcal K^n$ be the collections of open and closed subsets of the Euclidean space $\left({\R^n, \tau}\right)$, respectively. Then: :$\sigma \left({\mathcal O^n}\right) = \sigma \left({\mathcal C^n}\right)$ where $\sigma$ denotes generated $\sigma$-algebra.
Characterization of Euclidean Borel Sigma-Algebra/Closed equals Compact
Let $\mathcal C^n$ and $\mathcal K^n$ be the collections of closed and compact subsets of the Euclidean space $\left({\R^n, \tau}\right)$, respectively. Then: :$\sigma \left({\mathcal C^n}\right) = \sigma \left({\mathcal K^n}\right)$ where $\sigma$ denotes generated $\sigma$-algebra.
Characterization of Euclidean Borel Sigma-Algebra/Open equals Rectangle
Let $\mathcal O^n$ be the collection of open subsets of the Euclidean space $\left({\R^n, \tau}\right)$. Let $\mathcal J_{ho}^n$ be the collection of half-open rectangles in $\R^n$. Then: :$\sigma \left({\mathcal O^n}\right) = \sigma \left({\mathcal J_{ho}^n}\right)$ where $\sigma$ denotes generated $\sigma$-algebra.
Characterization of Euclidean Borel Sigma-Algebra/Rectangle equals Rational Rectangle
Let $\mathcal J_{ho}^n$ be the collection of half-open rectangles in $\R^n$. Let $\mathcal J^n_{ho, \text{rat}}$ be the collection of half-open rectangles in $\R^n$ with rational endpoints. Then: :$\sigma \left({\mathcal J_{ho}^n}\right) = \sigma \left({\mathcal J^n_{ho, \text{rat}}}\right)$ where $\sigma$ denotes generated $\sigma$-algebra.
Rising Factorial as Factorial by Binomial Coefficient
Let $x \in \R$ be a real number. Let $n \in \Z_{\ge 0}$ be a positive integer. :$x^{\overline n} = n! \dbinom {x + n - 1} n$ where: :$x^{\overline n}$ denotes $x$ to the $n$ rising :$n!$ denotes the factorial of $n$ :$\dbinom {x + n - 1} n$ denotes a binomial coefficient.
Ramus's Identity
Let $k, m, n \in \Z_{\ge 0}$ be positive integers such that $0 \le k < m$. Then: {{begin-eqn}} {{eqn | l = \sum_{j \mathop \ge 0} \binom n {j m + k}       | r = \dbinom n k + \dbinom n {m + k} +  \dbinom n {2 m + k} + \cdots        | c =  }} {{eqn | r = \dfrac 1 m \sum_{0 \mathop \le j \mathop < m} \paren {2 \cos \dfrac {j \pi} m}^n \cos \dfrac {j \paren {n - 2 k} \pi} m       | c =  }} {{end-eqn}}
Way Below Closure is Lower Set
Let $L = \struct {S, \vee, \preceq}$ be an ordered set. Let $x \in S$. Then :$x^\ll$ is a lower set.
Continuous iff For Every Element There Exists Ideal Element Precedes Supremum
Let $L = \left({S, \wedge, \preceq}\right)$ be an up-complete meet semilattice. Then :$L$ is continuous  :for every element $x$ of $S$ there exists ideal $I$ in $L$: ::$x \preceq \sup I$ and for every ideal $J$ in $L: x \preceq \sup J \implies I \subseteq J$
Beta Function of x+1 with y plus Beta Function of x with y+1
:$\Beta \left({x + 1, y}\right) + \Beta \left({x, y + 1}\right) = \Beta \left({x, y}\right)$
Beta Function of x with y+1 by x+y over y
:$\map \Beta {x, y} = \dfrac {x + y} y \map \Beta {x, y + 1}$
Partial Gamma Function expressed as Integral
Let $m \in \Z_{\ge 1}$. Let $\map {\Gamma_m} x$ denote the partial Gamma function, defined as: :$\map {\Gamma_m} x := \dfrac {m^x m!} {x \paren {x + 1} \paren {x + 2} \cdots \paren {x + m} }$ Then: :$\displaystyle \map {\Gamma_m} x = \int_0^m \paren {1 - \frac t m}^m t^{x - 1} \rd t$ for $x > 0$.
Partial Gamma Function expressed as Integral/Lemma
:$(1): \quad \displaystyle \int_0^m \paren {1 - \frac t m}^m t^{x - 1} \rd t = m^x \int_0^1 \paren {1 - t}^m t^{x - 1} \rd t$ for $x > 0$.
Integral Form of Gamma Function equivalent to Euler Form
{{TFAENocat|def = Gamma Function}} === Integral Form === {{:Definition:Integral Form of Gamma Function}} === Euler Form === {{:Definition:Euler Form of Gamma Function}}
Derivative of Composite Function/Second Derivative
:$D_x^2 w = D_u^2 w \paren {D_x^1 u}^2 + D_u^1 w D_x^2 u$
Faà di Bruno's Formula
Let $D_x^k u$ denote the $k$th derivative of a function $u$ {{WRT|Differentiation}} $x$. Then: :$\displaystyle D_x^n w = \sum_{j \mathop = 0}^n D_u^j w \sum_{\substack {\sum_{p \mathop \ge 1} k_p \mathop = j \\ \sum_{p \mathop \ge 1} p k_p \mathop = n \\ \forall p \mathop \ge 1: k_p \mathop \ge 0} } n! \prod_{m \mathop = 1}^n \dfrac {\left({D_x^m u}\right)^{k_m} } {k_m! \left({m!}\right)^{k_m} }$
Cardinality of Reduced Residue System
Let $n \ge 2$. Let $\Z'_n$ be the reduced residue system modulo $n$. Then: :$\card {\Z'_n} = \map \phi n$ where $\map \phi n$ is the Euler phi function.
Equivalence of Definitions of Norm of Linear Functional/Corollary
For all $h \in H$, the following inequality holds: :$\left|{Lh}\right| \le \left\|{L}\right\| \left\|{h}\right\|$
Faà di Bruno's Formula/Example/0/Proof
Consider Faà di Bruno's Formula: {{:Faà di Bruno's Formula}} When $n = 0$ we have: {{:Faà di Bruno's Formula/Example/0}}
Faà di Bruno's Formula/Example/1/Proof
Consider Faà di Bruno's Formula: {{:Faà di Bruno's Formula}} When $n = 1$ we have: {{:Faà di Bruno's Formula/Example/1}}
Faà di Bruno's Formula/Example/2/Proof
Consider Faà di Bruno's Formula: {{:Faà di Bruno's Formula}} When $n = 2$ we have: {{:Faà di Bruno's Formula/Example/2}}
Faà di Bruno's Formula/Lemma 1
:$D_x \left({\left({D_x^m u}\right)^{k_m} }\right) = k_m \left({D_x^m u}\right)^{k_m - 1} D_x^{m + 1} u$
Faà di Bruno's Formula/Lemma 2
:$\displaystyle D_x \left({\prod_{m \mathop = 1}^r \left({\dfrac {\left({D_x^m u}\right)^{k_m} } {k_m! \left({m!}\right)^{k_m} } }\right) }\right) = \prod_{m \mathop = 1}^r \left({\dfrac {\left({D_x^m u}\right)^{k_m} } {k_m! \left({m!}\right)^{k_m} } }\right) \sum_{m \mathop = 1}^r k_m \dfrac {D_x^{m + 1} u} {D_x^m u}$
Supremum of Ideals is Increasing
Let $L = \left({S, \preceq}\right)$ be an up-complete ordered set. Let $\mathit{Ids}\left({L}\right)$ be the set of all ideals in $L$. Let $P = \left({\mathit{Ids}\left({L}\right), \precsim}\right)$ be an ordered set where $\mathord \precsim = \subseteq\restriction_{\mathit{Ids}\left({L}\right)\times \mathit{Ids}\left({L}\right)}$ Let $f: \mathit{Ids}\left({L}\right) \to S$ be a mapping such that :$\forall I \in \mathit{Ids}\left({L}\right): f\left({I}\right) = \sup I$ Then $f$ is an increasing mapping.
Derivative of Composite Function/Third Derivative
:$D_x^3 w = D_u^3 w \paren {D_x^1 u}^3 + 3 D_u^2 w D_x^2 u D_x^1 u + D_u^1 w D_x^3 u$
Number of Set Partitions by Number of Components
Let $S$ be a (finite) set whose cardinality is $n$. Let $\map f {n, k}$ denote the number of different ways $S$ can be partitioned into $k$ components. Then: :$\displaystyle \map f {n, k} = {n \brace k}$ where $\displaystyle {n \brace k}$ denotes a Stirling number of the second kind.
Floor of Half of n+m plus Floor of Half of n-m+1
Let $n, m \in \Z$ be integers. :$\floor {\dfrac {n + m} 2} + \floor {\dfrac {n - m + 1} 2} = n$ where $\floor x$ denotes the floor of $x$.
Ceiling of Half of n+m plus Ceiling of Half of n-m+1
Let $n, m \in \Z$ be integers. :$\left\lceil{\dfrac {n + m} 2}\right\rceil + \left\lceil{\dfrac {n - m + 1} 2}\right\rceil = n + 1$ where $\left\lceil{x}\right\rceil$ denotes the ceiling of $x$.
Floor of Non-Integer
Let $x \in \R$ be a real number. Let $x \notin \Z$. Then: :$\left\lfloor{x}\right\rfloor < x$ where $\left\lfloor{x}\right\rfloor$ denotes the floor of $x$.
Logarithm of Base
Let $b \in \R_{>0}$ such that $b \ne 1$. Then: :$\log_b b = 1$ where $\log_b$ denotes the logarithm to base $b$.
Floor of x+m over n
Let $m, n \in \Z$ such that $n > 0$. Let $x \in \R$. Then: :$\floor {\dfrac {x + m} n} = \floor {\dfrac {\floor x + m} n}$ where $\floor x$ denotes the floor of $x$.
McEliece's Theorem (Integer Functions)
Let $f: \R \to \R$ be a continuous, strictly increasing real function defined on an interval $A$. Let: :$\forall x \in A: \left \lfloor{x}\right \rfloor \in A, \left \lceil{x}\right \rceil \in A$ where: :$\left \lfloor{x}\right \rfloor$ denotes the floor of $x$ :$\left \lceil{x}\right \rceil$ denotes the ceiling of $x$ Then: :$\forall x \in A: \left \lfloor{f \left({x}\right)}\right \rfloor = \left \lfloor{f \left({\left \lfloor{x}\right \rfloor}\right)}\right \rfloor \iff \left \lceil{f \left({x}\right)}\right \rceil = \left \lceil{f \left({\left \lceil{x}\right \rceil}\right)}\right \rceil \iff \left(f \left({x}\right) \in \Z \implies x \in \Z)\right)$
Ceiling of Non-Integer
Let $x \in \R$ be a real number. Let $x \notin \Z$. Then: :$\left \lceil{x}\right \rceil > x$ where $\left \lceil{x}\right \rceil$ denotes the ceiling  of $x$.
Ceiling of x+m over n
Let $m, n \in \Z$ such that $n > 0$. Let $x \in \R$. Then: :$\ceiling {\dfrac {x + m} n} = \ceiling {\dfrac {\ceiling x + m} n}$ where $\ceiling x$ denotes the ceiling of $x$.
Summation over k of Floor of mk+x over n
Let $m, n \in \Z$ such that $n > 0$. Let $x \in \R$. Then: :$\displaystyle \sum_{0 \mathop \le k \mathop < n} \left \lfloor{\dfrac {m k + x} n}\right \rfloor = \dfrac {\left({m - 1}\right) \left({n - 1}\right)} 2 + \dfrac {d - 1} 2 + d \left \lfloor{\dfrac x d}\right \rfloor$ where: :$\left \lfloor{x}\right \rfloor$ denotes the floor of $x$ :$d$ is the greatest common divisor of $m$ and $n$.
Summation over k of Floor of x plus k over y
Let $x, y \in \R$ such that $y > 0$. Then: :$\displaystyle \sum_{0 \mathop \le k \mathop < y} \left \lfloor{x + \dfrac k y}\right \rfloor = \left \lfloor{x y + \left \lfloor{x + 1}\right \rfloor \left({\left \lceil{y}\right \rceil - y}\right)}\right \rfloor$
Semantic Consequence preserved in Supersignature
Let $\mathcal L, \mathcal L'$ be signatures for the language of predicate logic. Let $\mathcal L'$ be a supersignature of $\mathcal L$. Let $\mathbf A$ be an $\mathcal L$-sentence. Let $\Sigma$ be a set of $\mathcal L$-sentences. Then the following are equivalent: :$\mathcal A \models_{\mathrm{PL}} \mathbf A$ for all $\mathcal L$-structure $\mathcal A$ for which $\mathcal A \models_{\mathrm{PL}} \Sigma$ :$\mathcal A' \models_{\mathrm{PL}} \mathbf A$ for all $\mathcal L'$-structure $\mathcal A'$ for which $\mathcal A' \models_{\mathrm{PL}} \Sigma$ where $\models_{\mathrm{PL}}$ denotes the models relation. That is to say, the notion of semantic consequence is preserved in passing to a supersignature.
Satisfiability preserved in Supersignature
Let $\mathcal L, \mathcal L'$ be signatures for the language of predicate logic. Let $\mathcal L'$ be a supersignature of $\mathcal L$. Let $\Sigma$ be a set of $\mathcal L$-sentences. Then the following are equivalent: :$\mathcal A \models_{\mathrm{PL}} \Sigma$ for some $\mathcal L$-structure $\mathcal A$ :$\mathcal A' \models_{\mathrm{PL}} \Sigma$ for some $\mathcal L'$-structure $\mathcal A'$ where $\models_{\mathrm{PL}}$ is the models relation. That is to say, the notion of satisfiability is preserved in passing to a supersignature.
Subtract Half is Replicative Function
Let $f: \R \to \R$ be the real function defined as: :$\forall x \in \R: \map f x = x - \dfrac 1 2$ Then $f$ is a replicative function.
Membership of Set of Integers is Replicative Function
Let $f: \R \to \R$ be the real function defined as: :$\forall x \in \R: \map f x = \sqbrk {x \in \Z}$ where $\sqbrk {\cdots}$ is Iverson's convention. Then $f$ is a replicative function.
Logarithm of Absolute Value of 2 times Sine of pi x is Replicative Function
Let $f: \R \to \R$ be the real function defined as: :$\forall x \in \R: \map f x = \log \, \size {2 \sin \pi x}$ Then $f$ is a replicative function.
Constant Multiple of Replicative Function is Replicative
Let $f: \R \to \R$ be a real function. Let $f$ be a replicative function. Let $c \in \R$ be a constant. Let $g: \R \to \R$ be the real function defined as: :$g \left({x}\right) = c f \left({x}\right)$ Then $g$ is also a replicative function.
Replicative Function of x minus Floor of x is Replicative
Let $f: \R \to \R$ be a real function. Let $f$ be a replicative function. Let $g: \R \to \R$ be the real function defined as: :$g \left({x}\right) = f \left({x - \left \lfloor{x}\right \rfloor}\right)$ Then $g$ is also a replicative function.
Closed Form for Sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ...
Let $a_1, a_2, a_3, \ldots$ be the sequence: :$\left \langle{a_n}\right \rangle = 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \ldots$ Then: :$a_n = \left \lceil{\dfrac {\sqrt {1 + 8 n} - 1} 2}\right \rceil$
Sum of Sequence as Summation of Difference of Adjacent Terms
Let $n \in \Z_{> 0}$ be a strictly positive integer. Then: :$\displaystyle \sum_{k \mathop = 1}^n a_k = n a_n - \sum_{k \mathop = 1}^{n - 1} k \left({a_{k + 1} - a_k}\right)$
Continuous Lattice and Way Below implies Preceding implies Preceding
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a continuous complete lattice. Let $a, b \in S$. Let :$\forall c \in S: c \ll a \implies c \preceq b$ Then $a \preceq b$
Sum over k of Floor of Log base b of k
Let $n \in \Z_{> 0}$ be a strictly positive integer. Let $b \in \Z$ such that $b \ge 2$. Then: :$\displaystyle \sum_{k \mathop = 1}^n \left \lfloor{\log_b k}\right \rfloor = \left({n + 1}\right) \left \lfloor{\log_b n}\right \rfloor - \dfrac {b^{\left \lfloor{\log_b n}\right \rfloor + 1} - b} {b - 1}$
Sum over k of Floor of Root k
Let $n \in \Z_{> 0}$ be a strictly positive integer. Let $b \in \Z$ such that $b \ge 2$. Then: :$\displaystyle \sum_{k \mathop = 1}^n \left \lfloor{\sqrt k}\right \rfloor = \left \lfloor{\sqrt n}\right \rfloor \left({n - \dfrac {\left({2 \left \lfloor{\sqrt n}\right \rfloor + 5}\right) \left({\left \lfloor{\sqrt n}\right \rfloor - 1}\right)} 6 }\right)$
Top is Meet Irreducible
Let $\left({S, \wedge, \preceq}\right)$ be a bounded above meet semilattice. Then $\top$ is meet irreducible where $\top$ denotes the greatest element in $S$.
Sum over k of Sum over j of Floor of n + jb^k over b^k+1
Let $n, b \in \Z$ such that $n \ge 0$ and $b \ge 2$. Then: :$\displaystyle \sum_{k \mathop \ge 0} \sum_{1 \mathop \le j \mathop < b} \left \lfloor{\dfrac {n + j b^k} {b^{k + 1} } }\right \rfloor = n$
Equivalence of Definitions of Legendre Symbol
Let $p$ be an odd prime. Let $a \in \Z$. {{TFAE|def = Legendre Symbol}}
Upper Closure of Element without Element is Filter implies Element is Meet Irreducible
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a lattice. Let $x \in S$. Let : $x^\succeq \setminus \left\{ {x}\right\}$ be a filter in $L$. Then $x$ is meet irreducible.
Maximal Element of Complement of Filter is Meet Irreducible
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a lattice. Let $F$ be a filter in $L$. Let $p \in S$. Let $p = \max \complement_S\left({F}\right)$. Then $p$ is meet irreducible.
De Polignac's Formula/Technique
When calculating $\mu$, the easiest way to calculate the next term is simply to divide the previous term by $p$ and discard the remainder: :$\floor {\dfrac n {p^{k + 1} } } = \floor {\floor {\dfrac n {p^k} } / p}$
Not Preceding implies There Exists Meet Irreducible Element Not Preceding
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below continuous lattice. Let $x, y \in S$ such that :$y \npreceq x$ Then :$\exists p \in S: p$ is meet irreducible and $x \preceq p$ and $y \npreceq p$
Characteristics of Floor and Ceiling Function
Let $f: \R \to \Z$ be an integer-valued function which satisfies both of the following: :$(1): \quad \map f {x + 1} = \map f x + 1$ :$(2): \quad \forall n \in \Z_{> 0}: \map f x = \map f {\dfrac {\map f {n x} } n}$ Then either: :$\forall x \in \Q: \map f x = \floor x$ or: :$\forall x \in \Q: \map f x = \ceiling x$
Continuous Replicative Function
Let $f: \R \to \R$ be a real function. Let $f$ be continuous on $\R$. Let $f$ also be a replicative function. Then $f$ is of the form: :$f \left({x}\right) = \left({x - \dfrac 1 2}\right) a$ where $a \in \R$.
Sum over k to p over 2 of Floor of 2kq over p
Let $p \in \Z$ be an odd prime. Let $q \in \Z$ be an odd integer. Then: :$\displaystyle \sum_{0 \mathop \le k \mathop < p / 2} \floor {\dfrac {2 k q} p}r \equiv  \sum_{0 \mathop \le k \mathop < p / 2} \floor {\dfrac {k q} p} \pmod 2$
Way Below implies There Exists Way Below Open Filter Subset of Way Above Closure
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below continuous lattice. Let $x, y \in S$ such that :$x \ll y$ where $\ll$ denotes the way below relation. Then there exists a way below open filter in $L$: $y \in F \land F \subseteq x^\gg$ where $x^\gg$ denotes the way above closure of $x$.
Powers of Group Elements/Negative Index/Additive Notation
:$\forall n \in \Z: -\paren {n g} = \paren {-n} g = n \paren {-g}$
Powers of Group Elements/Product of Indices/Additive Notation
:$\forall m, n \in \Z: n \paren {m g} = \paren {m n} g = m \paren {n g}$
Power of Product in Abelian Group/Additive Notation
: $k \left({x + y}\right) = k x + k y$
Power of Idempotent Element
Let $\struct {S, \circ}$ be an algebraic structure. Let $s \in S$ be an idempotent element with respect to $\circ$. Then: :$\forall n \in \Z_{> 0}: s^n = s$ where $s^n$ is defined as: :$s^n = \begin{cases} s & : n = 1 \\ s^{n - 1} \circ s & : n > 1 \end{cases}$
Powers of Group Element Commute
Let $\struct {G, \circ}$ be a group. Let $g \in G$. Let $m, n \in \N_{>0}$. Then: :$\forall m, n \in \N_{>0}: g^n \circ g^m = g^m \circ g^n$
Strictly Positive Real Numbers under Multiplication form Subgroup of Non-Zero Real Numbers
Let $\R_{>0}$ be the set of strictly positive real numbers, that is: :$\R_{>0} = \set {x \in \R: x > 0}$ The structure $\struct {\R_{>0}, \times}$ forms a subgroup of $\struct {\R_{\ne 0}, \times}$, where $\R_{\ne 0}$ is the set of real numbers without zero, that is: :$\R_{\ne 0} = \R \setminus \set 0$