hcju/proofwikips huggingface dataset:
Resonance Frequency is less than Natural Frequency
Consider a physical system $S$ whose behaviour is defined by the second order ODE: :$(1): \quad \dfrac {\d^2 y} {\d x^2} + 2 b \dfrac {\d y} {\d x} + a^2 x = K \cos \omega x$ where: :$K \in \R: k > 0$ :$a, b \in \R_{>0}: b < a$ Then the resonance frequency of $S$ is smaller than the natural frequency of the associated second order ODE: :$(2): \quad \dfrac {\d^2 y} {\d x^2} + 2 b \dfrac {\d y} {\d x} + a^2 x = 0$
Circle is Bisected by Diameter
A circle is bisected by a diameter.
Power to Characteristic Power of Field is Monomorphism
Let $F$ be a field whose characteristic is $p$ where $p \ne 0$. Let $n \in \Z_{\ge 0}$ be any positive integer. Let $\phi_n: F \to F$ be the mapping on $F$ defined as: :$\forall x \in F: \map {\phi_n} x = x^{p^n}$ Then $\phi_n$ is a (field) monomorphism.
Logarithmic Spiral is Equiangular
The logarithmic spiral is '''equiangular''', in the following sense: Let $P = \left\langle{r, \theta}\right\rangle$ be a point on a logarithmic spiral $S$ expressed in polar coordinates as: :$r = a e^{b \theta}$ Then the angle $\psi$ that the tangent makes to the radius vector of $S$ is constant.
Area Enclosed by First Turn of Archimedean Spiral
Let $S$ be the Archimedean spiral defined by the equation: :$r = a \theta$ The area $\mathcal A$ enclosed by the first turn of $S$ and the polar axis is given by: :$\mathcal A = \dfrac {4 \pi^3 a^2} 3$ :500px
Trisecting the Angle/Archimedean Spiral
Let $\alpha$ be an angle which is to be trisected. This can be achieved by means of an Archimedean spiral.
Upper Adjoint Preserves All Infima
Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets. Let $g: S \to T$ be an upper adjoint of Galois connection. Then $g$ preserves all infima.
Trisecting the Angle/Parabola
Let $\alpha$ be an angle which is to be trisected. This can be achieved by means of a parabola.
Cardinality of Set of Induced Equivalence Classes of Injection
Let $f: S \to T$ be a mapping. Let $\mathcal R_f \subseteq S \times S$ be the relation induced by $f$: :$\tuple {s_1, s_2} \in \mathcal R_f \iff \map f {s_1} = \map f {s_2}$ Let $f$ be an injection. Then there are $\card S$ different $\mathcal R_f$-classes.
Multiplicative Group of Positive Rationals is Non-Cyclic
Let $\struct {\Q_{>0}, \times}$ be the multiplicative group of positive rational numbers. Then $\struct {\Q_{>0}, \times}$ is not a cyclic group.
Top equals to Relative Pseudocomplement in Brouwerian Lattice
Let $\struct {S, \vee, \wedge, \preceq}$ be a Brouwerian lattice with greatest element $\top$. Let $a, b \in S$. Then :$\top = a \to b$  $a \preceq b$
Approximate Value of Nth Prime Number
The $n$th prime number is approximately $n \ln n$.
Fortissimo Space is not Compact
Let $T = \struct {S, \tau_p}$ be a Fortissimo space. Then $T$ is not a compact space.
Fortissimo Space is not Sequentially Compact
Let $T = \struct {S, \tau_p}$ be a Fortissimo space. Then $T$ is not sequentially compact.
Rational Number is Algebraic of Degree 1
Let $r \in \Q$ be a rational number. Then $r$ is an algebraic number of degree $1$.
Double Pointed Fortissimo Space is Lindelöf
Let $T = \struct {S, \tau}$ be a Fortissimo space. Let $T \times D$ be the double pointed topology on $T$. Then $T \times D$ is a Lindelöf space.
Double Pointed Fortissimo Space is Weakly Countably Compact
Let $T = \struct {S, \tau_p}$ be a Fortissimo space. Let $T \times D$ be the double pointed topology on $T$. Then $T \times D$ is weakly countably compact.
Double Pointed Fortissimo Space is not Pseudocompact
Let $T = \struct {S, \tau}$ be a Fortissimo space. Let $T \times D$ be the double pointed topology on $T$. Then $T \times D$ is not pseudocompact.
Modified Fort Space is not Locally Connected
Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Then $T$ is not locally connected.
Clopen Sets in Modified Fort Space
Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Let $A \subseteq S$ be both closed and open in $T$. If $a \in A$, then $b \in A$ as well. That is, any clopen set of $T$ must contain '''both''' or '''neither''' of $a$ and $b$.
Maximum Rate of Change of Y Coordinate of Cycloid
Let a circle $C$ of radius $a$ roll without slipping along the x-axis of a cartesian plane at a constant speed such that the center moves with a velocity $\mathbf v_0$ in the direction of increasing $x$. Consider a point $P$ on the circumference of this circle. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The maximum rate of change of $y$ is $\mathbf v_0$, which happens when $\theta = \dfrac \pi 2 + 2 n \pi$ where $n \in \Z$.
Vertices in Locally Finite Graph
Let $G$ be a locally finite graph. Then if $G$ is infinite, it contains an infinite number of vertices.
Relation between Equations for Hypocycloid and Epicycloid
Consider the hypocycloid defined by the equations: :$x = \paren {a - b} \cos \theta + b \map \cos {\paren {\dfrac {a - b} b} \theta}$ :$y = \paren {a - b} \sin \theta - b \map \sin {\paren {\dfrac {a - b} b} \theta}$ By replacing $b$ with $-b$, this converts to the equations which define an epicycloid: :$x = \paren {a + b} \cos \theta - b \map \cos {\paren {\dfrac {a + b} b} \theta}$ :$y = \paren {a + b} \sin \theta - b \map \sin {\paren {\dfrac {a + b} b} \theta}$
Derivative of Hyperbolic Sine Function
:$\map {\dfrac \d {\d x} } {\sinh u} = \cosh u \dfrac {\d u} {\d x}$
Radius of Curvature in Cartesian Form
Let $C$ be a curve defined by a real function which is twice differentiable. Let $C$ be embedded in a cartesian plane. The '''radius of curvature''' $\rho$ of $C$ at a point $P = \tuple {x, y}$ is given by: :$\rho = \dfrac {\paren {1 + y'^2}^{3/2} } {\size {y''} }$ where: :$y' = \dfrac {\d y} {\d x}$ is the derivative of $y$ {{WRT|Differentiation}} $x$ at $P$ :$y'' = \dfrac {\d^2 y} {\d x^2}$ is the second derivative of $y$ {{WRT|Differentiation}} $x$ at $P$.
Derivative of Hyperbolic Tangent Function
:$\map {\dfrac \d {\d x} } {\tanh u} = \sech^2 u \dfrac {\d u} {\d x}$
Quaternion Multplication is not Commutative
The operation of multplication on the quaternions $H$ is not commutative.
Vectors in Three Dimensional Space with Cross Product forms Lie Algebra
Let $S$ be the set of vectors in $3$ dimensional Euclidean space. Let $\times$ denote the vector cross product on $S$. Then $\struct {S, \times}$ is a Lie algebra.
Tangent Line to Convex Graph
Let $f$ be a real function that is continuous on some closed interval $\left[{a \,.\,.\, b}\right]$ and differentiable and convex on the open interval $\left({a \,.\,.\, b}\right)$.  Then all the tangent lines to $f$ are below the graph of $f$. {{explain|"below"}}
Characterization of Lower Semicontinuity
Let $f: S \to \overline \R$ be an extended real valued function. Let $S$ be endowed with a topology $\tau$. The following are equivalent: :$(1): \quad$ $f$ is lower semicontinuous (LSC) on $S$. :$(2): \quad$ The epigraph $\map {\operatorname{epi}} f$ of $f$ is a closed set in $S \times \R$ with the product topology. :$(3): \quad$ All lower level sets of $f$ are closed in $S$.
Power is Well-Defined/Rational
Let $x \in \R_{> 0}$ be a (strictly) positive real number. Let $q$ be a rational number. Then $x^q$ is  well-defined.
Integral of Arcsine Function
:$\displaystyle \int \arcsin x \rd x = x \arcsin x + \sqrt {1 - x^2} + C$ for $x \in \closedint {-1} 1$.
Ordinal is Subset of Ordinal Class
Suppose $A$ is an ordinal. Then: :$A \subseteq \On$ where $\On$ represents the class of all ordinals.
Power Function on Base Greater than One is Strictly Increasing/Integer
Let $a \in \R$ be a real number such that $a > 1$. Let $f: \Z \to \R$ be the real-valued function defined as: :$\map f k = a^k$ where $a^k$ denotes $a$ to the power of $k$. Then $f$ is strictly decreasing.
Subset Products of Normal Subgroup with Normal Subgroup of Subgroup
Let $G$ be a group. Let: :$(1): \quad H$ be a subgroup of $G$ :$(2): \quad K$ be a normal subgroup of $H$ :$(3): \quad N$ be a normal subgroup of $G$ Then: :$N K \lhd N H$ where: : $N K$ and $N H$ denote subset product : $\lhd$ denotes the relation of being a normal subgroup.
Intersection with Normal Subgroup is Normal/Examples/Subset Product of Normal Subgroup with Intersection
Let $\struct G$ be a group whose identity is $e$. Let $H_1, H_2$ be subgroups of $G$. Let: : $N_1 \lhd H_1$ : $N_2 \lhd H_2$ where $\lhd$ denotes the relation of being a normal subgroup. Then: :$N_1 \paren {H_1 \cap N_2} \lhd N_1 \paren {H_1 \cap H_2}$
Quotient Group of Direct Products
Let $G$ and $G'$ be groups. Let: :$H \lhd G$ :$H' \lhd G'$ where $\lhd$ denotes the relation of being a normal subgroup. Then: :$\paren {G \times G'} / \paren {H \times H'}$ is isomorphic to $\paren {G / H} \times \paren {G' / H'}$ where: :$G \times G'$ denotes the group direct product of $H$ and $H'$ :$G / H$ denotes the quotient group of $G$ by $H$.
Set of Non-Negative Real Numbers is not Well-Ordered by Usual Ordering
The set of non-negative real numbers $\R_{\ge 0}$ is not well-ordered under the usual ordering $\le$.
Characteristic Subgroup is Transitive
Let $G$ be a group. Let $H$ be a characteristic subgroup of $G$. Let $K$ be a characteristic subgroup of $H$. Then $K$ is a characteristic subgroup of $G$.
Integral Resulting in Arcsecant
:$\displaystyle \int \frac 1 {x \sqrt{x^2 - a^2} }\ \mathrm dx = \begin{cases} \dfrac 1 {\left\vert{a}\right\vert} \operatorname {arcsec} \dfrac x {\left\vert{a}\right\vert} + C & : x > \left\vert{a}\right\vert \\ -\dfrac 1 {\left\vert{a}\right\vert} \operatorname {arcsec} \dfrac x {\left\vert{a}\right\vert} + C & : x < -\left\vert{a}\right\vert \end{cases}$ where $a$ is a constant.
Upper Bound of Natural Logarithm/Corollary
:$\forall s \in \R_{>0}: \ln x \le \dfrac {x^s} s$
Finite Abelian Group is Solvable
Let $G$ be a finite abelian group. Then $G$ is solvable.
Powers Drown Logarithms/Corollary
:$\displaystyle \lim_{y \mathop \to 0_+} y^r \ln y = 0$
Logarithm of Logarithm in terms of Natural Logarithms
Let $b, x \in \R_{>0}$ be (strictly) positive real numbers. Then: :$\map {\log_b} {\log_b x} = \dfrac {\map \ln {\ln x} - \map \ln {\ln b} } {\ln b}$ where $\ln x$ denotes the natural logarithm of $x$.
Intersection is Subset of Union of Intersections with Complements
Let $R, S, T$ be sets. Then: :$S \cap T \subseteq \paren {R \cap S} \cup \paren {\overline R \cap T}$ where $\overline R$ denotes the complement of $R$.
Adjoining Commutes with Inverting
Let $H$ be a Hilbert space. Let $A \in B \left({H, K}\right)$ be a bounded linear operator. Let $A^{-1} \in B \left({K, H}\right)$ be an inverse for $A$. Then the adjoint of $A$, $A^*$, is invertible. Furthermore, $\left({A^*}\right)^{-1} = \left({A^{-1}}\right)^*$.
Sum of Elements in Inverse of Combinatorial Matrix
Let $C_n$ be the combinatorial matrix of order $n$ given by: :$C_n = \begin{bmatrix} x + y & y & \cdots & y \\ y & x + y & \cdots & y \\ \vdots & \vdots & \ddots & \vdots \\ y & y & \cdots & x + y \end{bmatrix}$ Let $C_n^{-1}$ be its inverse, from Inverse of Combinatorial Matrix: :$b_{i j} = \dfrac {-y + \delta_{i j} \left({x + n y}\right)} {x \left({x + n y}\right)}$ where $\delta_{i j}$ is the Kronecker delta. The sum of all the elements of $C_n^{-1}$ is: :$\displaystyle \sum_{1 \mathop \le i, \ j \mathop \le n} b_{i j} = \dfrac n {x + n y}$
Elements of Inverse of Hilbert Matrix are Integers
Let $H_n$ be the Hilbert matrix of order $n$: :$\begin{bmatrix} a_{i j} \end{bmatrix} = \begin{bmatrix} \dfrac 1 {i + j - 1} \end{bmatrix}$ Consider its inverse $H_n^{-1}$. All the elements of $H_n^{-1}$ are integers.
Auxiliary Relation is Transitive
Let $\left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $\mathcal R$ be relation on $S$ satisfying conditions $(i)$ and $(ii)$ of auxiliary relation. Then :$\mathcal R$ is a transitive relation.
Ordered Set of Auxiliary Relations is Complete Lattice
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $\operatorname {Aux} \left({L}\right)$ be the set of all auxiliary relations on $S$. Let $P = \left({\operatorname {Aux} \left({L}\right), \precsim}\right)$ be an ordered set where $\precsim \mathop = \subseteq \restriction_{\operatorname {Aux} \left({L}\right) \times \operatorname {Aux} \left({L}\right)}$ Then :$P$ is a complete lattice.
Zorn's Lemma Implies Axiom of Choice
If Zorn's Lemma is true, then so must the Axiom of Choice be.
Number minus Modulo is Integer Multiple
Let $x, y \in \R$ be real numbers. Let $x \bmod y$ denote the modulo operation: :$x \bmod y := \begin{cases} x - y \left \lfloor {\dfrac x y}\right \rfloor & : y \ne 0 \\ x & : y = 0 \end{cases}$ where $\left \lfloor {\dfrac x y}\right \rfloor$ denotes the floor of $\dfrac x y$. Let $y < 0$. Then: :$x - \left({x \bmod y}\right)$ is an integer multiple of $y$.
Modulo Operation/Examples/5 mod 3
:$5 \bmod 3 = 2$
Modulo Multiplication is Well-Defined/Warning
Let $z \in \R$ be a real number. Let: :$a \equiv b \pmod z$ and: :$x \equiv y \pmod z$ where $a, b, x, y \in \R$. Then it does '''not''' necessarily hold that: : $a x \equiv b y \pmod z$
Constant to Power of Number of Distinct Prime Divisors is Multiplicative Function
Let $c \in \R$ be a constant. Let $f: \N \to \R$ denotes the mapping defined as: :$\forall n \in \N: f \left({n}\right) = c^k$ where $k$ is number of distinct primes that divide $n$. Then $f$ is multiplicative.
Clopen Sets in Finite Complement Topology
Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Then the only clopen sets of $T$ are $S$ and $\O$.
Number of Digits in Factorial
Let $n!$ denote the factorial of $n$. The number of digits in $n!$ is approximately: :$1 + \left\lfloor{\dfrac 1 2 \left({\log_{10} 2 + \log_{10} \pi}\right) + \dfrac 1 2 \log_{10} n + n \left({\log_{10} n - \log_{10} e}\right)}\right\rfloor$ when $n!$ is shown in decimal notation. This evaluates to: :$1 + \left\lfloor{\left({n + \dfrac 1 2}\right) \log_{10} n - 0.43429 \ 4481 \, n + 0.39908 \ 9934}\right\rfloor$
Transformation of Unit Matrix into Inverse
Let $\mathbf A$ be a square matrix of order $n$ of the matrix space $\map {\MM_\R} n$. Let $\mathbf I$ be the unit matrix of order $n$. Suppose there exists a sequence of elementary row operations that reduces $\mathbf A$ to $\mathbf I$. Then $\mathbf A$ is invertible. Futhermore, the same sequence, when performed on $\mathbf I$, results in the inverse of $\mathbf A$.
Sum over k of r-kt Choose k by r over r-kt by s-(n-k)t Choose n-k by s over s-(n-k)t
For $n \in \Z_{\ge 0}$: :$\displaystyle \sum_k A_k \left({r, t}\right) A_{n - k} \left({s, t}\right) = A_n \left({r + s, t}\right)$ where $A_n \left({x, t}\right)$ is the polynomial of degree $n$ defined as: :$A_n \left({x, t}\right) = \dbinom {x - n t} n \dfrac x {x - n t}$ where $x \ne n t$.
Integer Multiples Greater than Positive Integer Closed under Addition
Let $n \Z$ be the set of integer multiples of $n$. Let $p \in \Z: p \ge 0$ be a positive integer. Let $S \subseteq n \Z$  be defined as: :$S := \set {x \in n \Z: x > p}$ that is, the set of integer multiples of $n$ greater than $p$. Then the algebraic structure $\struct {S, +}$ is closed under addition.
Signed Stirling Number of the First Kind of n+1 with 0
:$\map s {n + 1, 0} = 0$
Subset of Natural Numbers under Max Operation is Monoid
Let $S \subseteq \N$ be a subset of the natural numbers $\N$. Let $\left({S, \max}\right)$ denote the algebraic structure formed from $S$ and the max operation. Then $\left({S, \max}\right)$ is a monoid. Its identity element is the smallest element of $S$.
Stirling Number of the Second Kind of n+1 with 2
:$\displaystyle {n + 1 \brace 2} = 2^n - 1$
Preceding is Approximating Relation
Let $\left({S, \preceq}\right)$ be an ordered set. Then $\preceq$ is an approximating relation on $S$.
Substitution Instance of WFF is WFF
Let $\mathbf A$ be a WFF of predicate logic. Let $\tau$ be a term of predicate logic. Let $x \in \mathrm{VAR}$ be a variable. Let $\mathbf A \left({x \gets \tau}\right)$ be the substitution instance of $\mathbf A$ substituting $\tau$ for $x$. Then $\mathbf A \left({x \gets \tau}\right)$ is a WFF.
Sequence Converges to Within Half Limit/Real Numbers
Let $\sequence {x_n}$ be a sequence in $\R$. Let $\sequence {x_n}$ be convergent to the limit $l$. That is, let $\displaystyle \lim_{n \mathop \to \infty} x_n = l$. Suppose $l > 0$. Then: : $\exists N: \forall n > N: x_n > \dfrac l 2$ Similarly, suppose $l < 0$. Then: : $\exists N: \forall n > N: x_n < \dfrac l 2$
Binomial Coefficient/Examples/Number of Bridge Hands
The total number $N$ of possible different hands for a game of [https://en.wikipedia.org/wiki/Contract_bridge bridge] is: :$N = \dfrac {52!} {13! \, 39!} = 635 \ 013 \ 559 \ 600$
Symmetric Difference is Subset of Union of Symmetric Differences
Let $R, S, T$ be sets. Then: :$R * S \subseteq \left({R * T}\right) \cup \left({S * T}\right)$ where $R * S$ denotes the symmetric difference between $R$ and $S$.
Continuous iff Meet-Continuous and There Exists Smallest Auxiliary Approximating Relation
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice. Then: :$L$ is continuous  :$L$ is meet-continuous and there exists the smallest auxiliary approximating relation on $S$ That is: :$L$ is continuous  :$L$ is meet-continuous and there exists an auxiliary approximating relation $\mathcal R$ on $S$ ::for every auxiliary approximating relation $\mathcal Q$ on $S$: $\mathcal R \subseteq \mathcal Q$
Sum over k of r+tk choose k by s-tk choose n-k
Let $n \in \Z_{\ge 0}$ be a non-negative integer. Then: :$\displaystyle \sum_k \dbinom {r + t k} k \dbinom {s - t k} {n - k} = \sum_{k \mathop \ge 0} \dbinom {r + s - k} {n - k} t^k$ where $\dbinom {r + t k} k$ etc. denotes a binomial coefficient.
Homogeneous System has Zero Vector as Solution
Every homogeneous system of linear equations has the zero vector as a solution.
Summation over Lower Index of Unsigned Stirling Numbers of the First Kind
Let $n \in \Z_{\ge 0}$ be a positive integer. Then: :$\displaystyle \sum_k \left[{n \atop k}\right] = n!$ where: :$\displaystyle \left[{n \atop k}\right]$ denotes an unsigned Stirling number of the first kind :$n!$ denotes the factorial of $n$.
Co-Countable Measure is Probability Measure
Let $X$ be an uncountable set. Let $\mathcal A$ be the $\sigma$-algebra of countable sets on $X$. Then the co-countable measure $\mu$ on $X$ is a probability measure.
Supremum of Ideals is Upper Adjoint
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below continuous join semilattice. 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 upper adjoint of Galois connection.
Summation of Summation over Divisors of Function of Two Variables
Let $c, d, n \in \Z$. Then: :$\displaystyle \sum_{d \mathop \divides n} \sum_{c \mathop \divides d} \map f {c, d} = \sum_{c \mathop \divides n} \sum_{d \mathop \divides \paren {n / c} } \map f {c, c d}$ where $c \divides d$ denotes that $c$ is a divisor of $d$.
Summation over k of Ceiling of k over 2
:$\displaystyle \sum_{k \mathop = 1}^n \left \lceil{\dfrac k 2}\right \rceil = \left \lceil{\dfrac {n \left({n + 2}\right)} 4}\right \rceil$
Open Rectangles Closed under Intersection
Let $\left(({\mathbf a \,.\,.\, \mathbf b}\right))$ and $\left(({\mathbf c \,.\,.\, \mathbf d}\right))$ be open $n$-rectangles. Then $\left(({\mathbf a \,.\,.\, \mathbf b}\right)) \cap \left(({\mathbf c \,.\,.\, \mathbf d}\right))$ is also an open $n$-rectangle.
Euclidean Borel Sigma-Algebra Closed under Scalar Multiplication
Let $\mathcal B \left({\R^n}\right)$ be the Borel $\sigma$-algebra on $\R^n$. Let $B \in \mathcal B$, and let $t \in \R_{>0}$. Then also $t \cdot B := \left\{{t \mathbf b: \mathbf b \in B}\right\} \in \mathcal B$.
Lebesgue Measure of Scalar Multiple
Let $\lambda^n$ be the $n$-dimensional Lebesgue measure on $\R^n$ equipped with the Borel $\sigma$-algebra $\mathcal B \left({\R^n}\right)$. Let $B \in \mathcal B$, and let $t \in \R_{>0}$. Then $\lambda^n \left({t \cdot B}\right) = t^n \lambda^n \left({B}\right)$, where $t \cdot B$ is the set $\left\{{t \mathbf b: \mathbf b \in B}\right\}$.
Membership of Set of Strictly Positive Integers is Replicative Function
Let $f: \R \to \R$ be the real function defined as: :$\forall x \in \R: \map f x = \sqbrk {x \in \Z_{> 0} }$ where $\sqbrk \cdots$ is Iverson's convention. Then $f$ is a replicative function.
Membership of Equivalence Class of m mod pi is Replicative Function
Let $f: \R \to \R$ be the real function defined as: :$\forall x \in \R: \map f x = \sqbrk {\exists r \in \Q, \exists m \in \Z: x = r \pi + m}$ where $\sqbrk {\cdots}$ is Iverson's convention. Then $f$ is a replicative function.
Sum of Replicative Functions is Replicative
Let $f: \R \to \R$ and $g: \R \to \R$ be real functions. Let $f$ and $g$ both be replicative functions. Then the pointwise sum of $f$ and $g$ is also a replicative function.
Meet Irreducible iff Finite Infimum equals Element
Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $x \in S$. Then :$x$ is meet irreducible  :for every non-empty finite subset $A$ of $S$: $x = \inf A \implies x \in A$
Sum over j of Function of Floor of mj over n
Let $f$ be a real function. Then: :$\displaystyle \sum_{0 \mathop \le j \mathop < n} \map f {\floor {\dfrac {m j} n} } = \sum_{0 \mathop \le r \mathop < m} \ceiling {\dfrac {r n} m} \paren {\map f {r - 1} - \map f r} + n \map f {m - 1}$
Isomorphism of External Direct Products/General Result
Let: : $(1): \quad \displaystyle \left({S, \circ}\right) = \prod_{k \mathop = 1}^n S_k = \left({S_1, \circ_1}\right) \times \left({S_2, \circ_2}\right) \times \cdots \times \left({S_n, \circ_n}\right)$ : $(2): \quad \displaystyle \left({T, \ast}\right) = \prod_{k \mathop = 1}^n T_k = \left({T_1, \ast_1}\right) \times \left({T_2, \ast_2}\right) \times \cdots \times \left({T_n, \ast_n}\right)$ be external direct products of algebraic structures. Let $\phi_k: \left({S_k, \circ_k}\right) \to \left({T_k, \ast_k}\right)$ be an isomorphism for each $k \in \left[{1 \,.\,.\, n}\right]$. Then: :$\phi: \left({s_1, \ldots, s_n}\right) \to \left({\phi_1 \left({s_1}\right), \ldots, \phi_n \left({s_n}\right)}\right)$ is an isomorphism from $\left({S, \circ}\right)$ to $\left({T, \ast}\right)$.
Cancellable Infinite Semigroup is not necessarily Group
Let $\struct {S, \circ}$ be a semigroup whose underlying set is infinite. Let $\struct {S, \circ}$ be such that all elements of $S$ are cancellable. Then it is not necessarily the case that $\struct {S, \circ}$ is a group.
Latin Square is not necessarily Cayley Table of Group
While it is true that the Cayley table of a (finite) group is in the form of a Latin square it is not necessarily the case that a Latin square is the Cayley table of a group.
Projection is Epimorphism/General Result
Let $\left({S, \circ}\right)$ be the external direct product of the algebraic structures $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_k, \circ_k}\right), \ldots, \left({S_n, \circ_n}\right)$. Then: :for each $j \in \left[{1 \,.\,.\, n}\right]$, $\operatorname{pr}_j$ is an epimorphism from $\left({S, \circ}\right)$ to $\left({S_j, \circ_j}\right)$ where $\operatorname{pr}_j: \left({S, \circ}\right) \to \left({S_j, \circ_j}\right)$ is the $j$th projection from $\left({S, \circ}\right)$ to $\left({S_j, \circ_j}\right)$.