hcju/proofwikips huggingface dataset:
Tarski-Vaught Test
Let $\mathcal M, \mathcal N$ be $\mathcal L$-structures such that $\mathcal M$ is a substructure of $\mathcal N$. {{wtd|The page Definition:Structure is a disambiguation page, in which the form in which it is used here may not be included. The level of clarity in this page generally needs improving. Hence the invocation of the Disambiguate template.}} {{Disambiguate|Definition:Structure}} Then $\mathcal M$ is an elementary substructure of $\mathcal N$ : :for every $\mathcal L$-formula $\phi \left({x, \bar v}\right)$ and for every $\bar a$ in $\mathcal M$: ::if there exists an $n$ in $\mathcal N$ such that $\mathcal N \models \phi \left({n, \bar a}\right)$ ::then there exists an $m$ in $\mathcal M$ such that $\mathcal N \models \phi \left({m, \bar a}\right)$. {{wtd|Before sense can be made of this page, Definition:Substructure and  Definition:Elementary Substructure need to be written.}} {{Disambiguate|Definition:Logical Formula}} The condition on the right side of the  statement above can be rephrased as: :Every existential statement with parameters from $\mathcal M$ which is satisfied in $\mathcal N$ can be witnessed by an element from the substructure $\mathcal M$.
Conditional and Inverse are not Equivalent
A conditional statement: :$p \implies q$ is not logically equivalent to its inverse: :$\lnot p \implies \lnot q$
Weight of Discrete Topology equals Cardinality of Space
Let $T = \left({S, \tau}\right)$ be a discrete topological space. Then: :$w \left({T}\right) = \left\vert{S}\right\vert$ where  :$w \left({T}\right)$ denotes the weight of $T$ :$\left\vert{S}\right\vert$ denotes the cardinality of $S$.
Primitive of Reciprocal of a squared minus x squared/Logarithm Form 2
:$\displaystyle \int \frac {\d x} {a^2 - x^2} = \dfrac 1 {2 a} \ln \size {\dfrac {a + x} {a - x} } + C$
Open Extension Topology is Topology
Let $T = \struct {S, \tau}$ be a topological space. Let $\tau^*_{\bar p}$ be the open extension topology of $\tau$. Then $\tau^*_{\bar p}$ is a topology on $S^*_p = S \cup \set p$.
Symmetric Difference with Intersection forms Boolean Ring
Let $S$ be a set. Then $\struct {\powerset S, *, \cap}$ is a Boolean ring.
1-Seminorm on Continuous on Closed Interval Real-Valued Functions is Norm
Let $\CC \closedint a b$ be the space of continuous on closed interval real-valued functions. Let $x \in \CC \closedint a b$ be a continuous real valued function. Let $\displaystyle \norm x_1 := \int_a^b \size {\map x t} \rd t$ be the 1-seminorm. Then $\norm {\, \cdot \,}_1$ is a norm on $\CC \closedint a b$.
Perpendicularity is Symmetric Relation
Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \in S$, let $l_1 \perp l_2$ denote that $l_1$ is perpendicular to $l_2$. Then $\perp$ is a symmetric relation on $S$.
Perpendicularity is Antitransitive Relation
Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \in S$, let $l_1 \perp l_2$ denote that $l_1$ is perpendicular to $l_2$. Then $\perp$ is an antitransitive relation on $S$.
Multiplication using Parabola
:500pxrightthumb Let the parabola $P$ defined as $y = x^2$ be plotted on the Cartesian plane. Let $A = \tuple {x_a, y_a}$ and $B = \tuple {x_b, y_b}$ be points on the curve $\map f x$ so that $x_a < x_b$. Then the line segment joining $A B$ will cross the $y$-axis at $-x_a x_b$. Thus $P$ can be used as a nomogram to calculate the product of two numbers $x_a$ and $x^b$, as follows: :$(1) \quad$ Find the points $-x_a$ and $x_b$ on the $x$-axis. :$(2) \quad$ Find the points $A$ and $B$ where the lines $x = -x_a$ and $x = x_b$ cut $P$. :$(3) \quad$ Lay a straightedge on the straight line joining $A$ and $B$ and locate its $y$-intercept $c$. Then $x_a x_b$ can be read off from the $y$-axis as the position of $c$.
Excluded Point Space is Locally Path-Connected
Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $T$ is locally path-connected.
Occurrence in Polish Notation has Unique Scope
Let $\mathcal F$ be a formal language in Polish notation. Let $\mathbf A$ be a well-formed formula of $\mathcal F$. Let $a$ be an occurrence in $\mathbf A$. Then $a$ has a unique scope.
Exclusive Or as Conjunction of Disjunctions
: $p \oplus q \dashv \vdash \left({p \lor q}\right) \land \left({\neg p \lor \neg q}\right)$
Space of Almost-Zero Sequences is Everywhere Dense in 2-Sequence Space
Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the 2-sequence space equipped with Euclidean norm. Let $c_{00}$ be the space of almost-zero sequences. Then $c_{00}$ is everywhere dense in $\struct {\ell^2, \norm {\, \cdot \,}_2}$
Ordering of Integers is Reversed by Negation
Let $x, y \in \Z$ such that $x > y$. Then: :$-x < -y$
Excluded Point Space is Sequentially Compact
Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $T$ is a sequentially compact space.
Sufficient Conditions for Weak Stationarity of Order 2
Let $S$ be a stochastic process giving rise to a time series $T$. Let the mean of $S$ be fixed. Let the autocovariance matrix of $S$ be of the form: :$\boldsymbol \Gamma_n = \begin {pmatrix} \gamma_0 & \gamma_1 & \gamma_2 & \cdots & \gamma_{n - 1} \\ \gamma_1 & \gamma_0 & \gamma_1 & \cdots & \gamma_{n - 2} \\ \gamma_2 & \gamma_1 & \gamma_0 & \cdots & \gamma_{n - 3} \\ \vdots   & \vdots   & \vdots   & \ddots & \vdots \\ \gamma_{n - 1} & \gamma_{n - 2} & \gamma_{n - 3} & \cdots & \gamma_0 \end {pmatrix} = \sigma_z^2 \mathbf P_n = \begin {pmatrix} 1 & \rho_1 & \rho_2 & \cdots & \rho_{n - 1} \\ \rho_1 & 1 & \rho_1 & \cdots & \rho_{n - 2} \\ \rho_2 & \rho_1 & 1 & \cdots & \rho_{n - 3} \\ \vdots   & \vdots   & \vdots   & \ddots & \vdots \\ \rho_{n - 1} & \rho_{n - 2} & \rho_{n - 3} & \cdots & 1 \end {pmatrix}$ Then $S$ is weakly stationary of order $2$.
Strict Ordering on Integers is Trichotomy
Let $\eqclass {a, b} {}$ and $\eqclass {c, d} {}$ be integers, as defined by the formal definition of integers. Then exactly one of the following holds: {{begin-eqn}} {{eqn | l = \eqclass {a, b} {}       | o = <       | r = \eqclass {c, d} {}       | c =  }} {{eqn | l = \eqclass {a, b} {}       | o = =       | r = \eqclass {c, d} {}       | c =  }} {{eqn | l = \eqclass {a, b} {}       | o = >       | r = \eqclass {c, d} {}       | c =  }} {{end-eqn}} That is, strict ordering is a trichotomy.
Excluded Set Topology is not T0
Let $T = \struct {S, \tau_{\bar H} }$ be an excluded set space where $H$ has at least two distinct points. Then $T$ is not a $T_0$ (Kolmogorov) space.
Boubaker's Theorem
Let $\left({R, +, \circ}\right)$ be a commutative ring. Let $\left({D, +, \circ}\right)$ be an integral subdomain of $R$ whose zero is $0_D$ and whose unity is $1_D$. Let $X \in R$ be transcendental over $D$. Let $D \left[{X}\right]$ be the ring of polynomial forms in $X$ over $D$. Finally, consider the following properties: {{begin-eqn}} {{eqn | n = 1       | l = \sum_{k \mathop = 1}^N {p_n \left({0}\right)}       | r = -2N }} {{eqn | n = 2       | l = \sum_{k \mathop = 1}^N {p_n \left({\alpha_k}\right)}       | r = 0 }} {{eqn | n = 3       | l = \left.{\sum_{k \mathop = 1}^N \frac {\mathrm d p_x \left({x}\right)} {\mathrm d x} }\right\vert_{x \mathop = 0}       | r = 0 }} {{eqn | n = 4       | l = \left.{\sum_{k \mathop = 1}^N \frac {\mathrm d {p_n}^2 \left({x}\right)} {\mathrm d x^2} }\right\vert_{x \mathop = 0}       | r = \frac 8 3 N \left({N^2 - 1}\right) }} {{end-eqn}} where, for a given positive integer $n$, $p_n \in D \left[{X}\right]$ is a non-null polynomial such that $p_n$ has $N$ roots $\alpha_k$ in $F$. Then the subsequence $\left \langle {B_{4 n} \left({x}\right)}\right \rangle$ of the Boubaker polynomials is the unique polynomial sequence of $D \left[{X}\right]$ which verifies simultaneously the four properties $(1) - (4)$.
Either-Or Topology is T0
Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is a $T_0$ (Kolmogorov) space.
Either-Or Topology is not T1
Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is not a $T_1$ (Fréchet) space.
Cardinality of Set of Self-Mappings on Finite Set
Let $S$ be a finite set. Let the cardinality of $S$ be $n$. The cardinality of the set of all mappings from $S$ to itself (that is, the total number of self-maps on $S$) is: :$\card {S^S} = n^n$
Cover of Interval By Closed Intervals is not Pairwise Disjoint
Let $\closedint a b$ be a closed interval in $\R$. {{explain|Title mentions only "interval"; this does not affect truth of statement so may "closed" above line be removed as superfluous?}} Let $\JJ$ be a set of two or more closed intervals contained in $\closedint a b$ such that $\displaystyle \bigcup \JJ = \closedint a b$. Then the intervals in $\JJ$ are not pairwise disjoint.
Factorization of Natural Numbers within 4 n + 1 not Unique
Let: :$S = \set {4 n + 1: n \in \N} = \set {1, 5, 9, 13, 17, \ldots}$ be the set of natural numbers of the form $4 n + 1$. Then not all elements of $S$ have a complete factorization by other elements of $S$ which is unique.
Either-Or Topology is not T3
Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is not a $T_3$ space.
Definite Integral to Infinity of Reciprocal of x Squared plus a Squared/Corollary
:$\ds \int_0^\infty \dfrac {\d x} {1 + x^2} = \frac \pi 2$ for $a \ne 0$.
Either-Or Topology is First-Countable
Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is a first-countable space.
Construction of Regular Heptadecagon
It is possible to construct a regular hepadecagon (that is, a regular polygon with $17$ sides) using a compass and straightedge construction.
Countable Stability implies Stability for All Infinite Cardinalities
Let $T$ be a complete $\mathcal L$-theory whose language $\mathcal L$ is countable. If $T$ is $\omega$-stable, then $T$ is $\kappa$-stable for all infinite $\kappa$.
Finite Complement Topology is Topology
Let $T = \struct {S, \tau}$ be a finite complement space. Then $\tau$ is a topology on $T$.
Event Space of Experiment with Final Sample Space has Even Cardinality
Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\Omega$ be a finite set. Then the event space $\Sigma$ consists of an even number of subsets of $\Omega$.
Set of Elementary Events belonging to k Events is Event
Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $A_1, A_2, \ldots, A_m$ be events in the event space $\Sigma$ of $\EE$. Let $S$ denote the set of all elementary events of $\EE$ which are elements of exactly $k$ of the events $A_1, A_2, \ldots, A_m$. Then $S$ is an event of $\Sigma$.
Probability of Set Difference of Events
Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$. Let $A, B \in \Sigma$ be events of $\EE$. Let $\map \Pr A$ denote the probability of event $A$ occurring. Then: :$\map \pr {A \setminus B} = \map \Pr A - \map \Pr {A \cap B}$
Distance Moved by Body from Rest under Constant Acceleration
Let a body $B$ be stationary. Let $B$ be subject to a constant acceleration. Then the distance travelled by $B$ is proportional to the square of the length of time $B$ is under the acceleration.
Set of Endomorphisms of Non-Abelian Group is not Ring
Let $\struct {G, \oplus}$ be a group which is non-abelian. Let $\mathbb G$ be the set of all group endomorphisms of $\struct {G, \oplus}$. Let $*: \mathbb G \times \mathbb G \to \mathbb G$ be the operation defined as: :$\forall u, v \in \mathbb G: u * v = u \circ v$ where $u \circ v$ is defined as composition of mappings. Then the algebraic structure $\struct {\mathbb G, \oplus, *}$ is not a ring.
Density not greater than Weight
Let $T = \left({S, \tau}\right)$ be a topological space. Then :$d \left({T}\right) \leq w \left({T}\right)$ where :$d \left({T}\right)$ denotes the density of $T$, :$w \left({T}\right)$ denotes the weight of $T$.
Countable Complement Space is not Countably Compact
Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is not a countably compact space.
Closed Form for Number of Derangements on Finite Set
The number of derangements $D_n$ on a finite set $S$ of cardinality $n$ is: :$D_n = n! \paren {1 - \dfrac 1 {1!} + \dfrac 1 {2!} - \dfrac 1 {3!} + \cdots + \dfrac {\paren {-1}^n} {n!} }$
Characteristics of Birkhoff-James Orthogonality
Let $\struct {V, \norm {\,\cdot\,} }$ be a normed linear space. Let $x, y \in V$. Then $x$ and $y$ are '''Birkhoff-James orthogonal'''  either: :$(1): \quad x = 0$ or: :$(2): \quad$ there exists a continuous functional $ f$ on $\struct {V, \norm {\,\cdot\,} }$ such that: ::::$\norm f = 1$ ::::$\map f x = \norm x$ ::::$\map f y = 0$
Countable Complement Space is Pseudocompact
Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is pseudocompact.
Countable Complement Space is not Countably Metacompact
Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is not countably metacompact.
Direct Product of Banach Spaces is Banach Space
Let $\struct {X, \norm {\, \cdot \,}}$ and $\struct {Y, \norm {\, \cdot \,}}$ be normed vector spaces. Let $V = X \times Y$ be a direct product of vector spaces $X$ and $Y$ together with induced component-wise operations. Let $\norm {\tuple {x, y} }$ be the direct product norm. Suppose $X$ and $Y$ are Banach spaces. Then $V$ is a Banach space.
Rational Numbers are F-Sigma Set in Real Line
Let $\left({\R, \tau}\right)$ be the real number line considered asa  topological space with the usual (Euclidean) topology. Then: :$\Q$ is an $F_\sigma$ set in $\left({\R, \tau}\right)$.
Confocal Conics are Self-Orthogonal
The confocal conics defined by: :$\quad \dfrac {x^2} {a^2} + \dfrac {y^2} {a^2 - c^2} = 1$ forms a family of orthogonal trajectories which is self-orthogonal. :500px
Compact Complement Topology is Connected
Let $T = \struct {\R, \tau}$ be the compact complement topology. Then $T$ is a connected space.
Compact Complement Topology is not Ultraconnected
Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$. Then $T$ is not an ultraconnected space.
Partial Derivatives of x tan^-1 (x^2 + y)
Let: :$\map f {x, y} = x \map \arctan {x^2 + y}$ Then: {{begin-eqn}} {{eqn | l = \map {f_1} {1, 0}       | r = \dfrac \pi 4 + 1 }} {{eqn | l = \map {f_2} {x, y}       | r = \dfrac x {1 + \paren {x^2 + y}^2} }} {{end-eqn}}
Partial Derivatives of x ln y^2 + y e^z
Let: :$\map f {x, y, z} = x \ln y^2 + y e^z$ Then: {{begin-eqn}} {{eqn | l = \map {f_1} {1, -1, 0}       | r = 0 }} {{eqn | l = \map {f_2} {x, x y, y + z}       | r = \dfrac 2 y + e^{y + z} }} {{end-eqn}}
Partial Derivative/Examples/u - v + 2 w, 2 u + v + 2 w, u - v + w
Let: {{begin-eqn}} {{eqn | l = u - v + 2 w       | r = x + 2 z }} {{eqn | l = 2 u + v - 2 w       | r = 2 x - 2 z }} {{eqn | l = u - v + w       | r = z - y }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \dfrac {\partial u} {\partial y}       | r = 0 }} {{eqn | l = \dfrac {\partial v} {\partial y}       | r = 2 }} {{eqn | l = \dfrac {\partial w} {\partial y}       | r = 1 }} {{end-eqn}}
Compact Complement Topology is Sequentially Compact
Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$. Then $T$ is a sequentially compact space.
Theories with Infinite Models have Models with Order Indiscernibles
Let $T$ be an $\LL$-theory with infinite models. Let $\struct {I, <$ be an infinite strict linearly ordered set. There is a model $\MM \models T$ containing an order indiscernible set $\set {x_i : i \in I}$.
Fort Space is Sequentially Compact
Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is a sequentially compact space.
First Order ODE/x y' = y + 2 x exp (- y over x)
is a homogeneous differential equation with solution: :$e^{y / x} = \ln x^2 + C$
Fort Space is not Extremally Disconnected
Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is not an extremally disconnected space.
Non-Forking Types have Non-Forking Completions
Let $T$ be a complete $\mathcal L$-theory. Let $\mathfrak C$ be a monster model for $T$. Let $A\subseteq B$ be subsets of the universe of $\mathfrak C$. Let $\pi(\bar x)$ be an $n$-type over $B$. If $\pi$ does not fork over $A$, then there is a complete $n$-type $p$ over $B$ such that $\pi \subseteq p$ and $p$ does not fork over $A$.
Fortissimo Space is Completely Normal
Let $T = \struct {S, \tau_p}$ be a Fortissimo space. Then $T$ is a completely normal space. Consequently, $T$ satisfies all weaker separation axioms.
First Order ODE/-1 over y sine x over y dx + x over y^2 sine x over y dy
is an exact differential equation with solution: :$\dfrac x y = C$
First Order ODE/dx = (y over (1 - x^2 y^2)) dx + (x over (1 - x^2 y^2)) dy
is an exact differential equation with solution: :$\map \ln {\dfrac {1 + x y} {1 - x y} } - 2 x = C$
First-Countability is not Continuous Invariant
Let $T_A = \struct {A, \tau_A}$ and $T_B = \struct {B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a continuous mapping. If $T_A$ is a first-countable space, then it does not necessarily follow that $T_B$ is also first-countable.
First Order ODE/(x y - 1) dx + (x^2 - x y) dy = 0
The first order ODE: :$(1): \quad \paren {x y - 1} \rd x + \paren {x^2 - x y} \rd y = 0$ has the general solution: :$x y - \ln x - \dfrac {y^2} 2 + C$
First Order ODE/x dy = (x^5 + x^3 y^2 + y) dx
The first order ODE: :$(1): \quad x \rd y = \paren {x^5 + x^3 y^2 + y} \rd x$ has the general solution: :$\arctan \dfrac x y = -\dfrac {x^4} 4 + C$
Linear First Order ODE/y' + y = 2 x exp -x + x^2
The linear first order ODE: :$(1): \quad y' + y = 2 x e^{-x} + x^2$ has the general solution: :$y = x^2 e^{-x} + x^2 - 2 x + 2 + C e^{-x}$
First Order ODE/(exp y - 2 x y) y' = y^2
The first order ODE: :$(1): \quad \paren {e^y - 2 x y} y' = y^2$ has the general solution: :$x y^2 = e^y + C$
Differential Equation of Family of Linear Combination of Functions is Linear
Consider the one-parameter family of curves: :$(1): \quad y = C \map f x + \map g x$ The differential equation that describes $(1)$ is linear and of first order.
Trivial Topological Space is Non-Meager
Let $T = \struct {S, \tau}$ be a trivial topological space. Then $T$ is non-meager.
Hermitian Operators have Orthogonal Eigenvectors
The eigenvectors of a Hermitian operation are orthogonal.
First Order ODE/(exp x - 3 x^2 y^2) y' + y exp x = 2 x y^3
The first order ordinary differential equation: :$(1): \quad \paren {e^x - 3 x^2 y^2} y' + y e^x = 2 x y^3$ is an exact differential equation with solution: :$y e^x - x^2 y^3 = C$
First Order ODE/(y^2 exp x y + cosine x) dx + (exp x y + x y exp x y) dy = 0
The first order ordinary differential equation: :$(1): \quad \paren {y^2 e^{x y} + \cos x} \rd x + \paren {e^{x y} + x y e^{x y} } \rd y = 0$ is an exact differential equation with solution: :$y e^{x y} + \sin x = C$
First Order ODE/(y^2 - 3 x y - 2 x^2) dx = (x^2 - x y) dy
The first order ordinary differential equation: :$(1): \quad \paren {y^2 - 3 x y - 2 x^2} \rd x = \paren {x^2 - x y} \rd y$ is a homogeneous differential equation with solution: :$y^2 x^2 - 2 y x^3 + x^4 = C$
First Order ODE/exp x sine y dx + exp x cos y dy = y sine x y dx + x sine x y dy
The first order ordinary differential equation: :$(1): \quad e^x \sin y \rd x + e^x \cos y \rd y = y \sin x y \rd x + x \sin x y \rd y$ is an exact differential equation with solution: :$e^x \sin y + \cos x y = C$
Linear Second Order ODE/(1 + x^2) y'' + x y' = 0
The second order ODE: :$\paren {1 + x^2} y'' + x y' = 0$ has the general solution: :$y = C_1 \map \ln {x + \sqrt {x^2 + 1} } + C_2$
Circumscribing Circle about Regular Pentagon
About any given regular pentagon it is possible to circumscribe a circle. {{:Euclid:Proposition/IV/14}}
Pólya-Vinogradov Inequality
Let $p$ be a positive odd prime. Then: :$\forall m, n \in \N: \displaystyle \size {\sum_{k \mathop = m}^{m + n} \paren {\frac k p} } < \sqrt p \, \ln p$ where $\paren {\dfrac k p}$ is the Legendre symbol.
Multiplication of Real Numbers is Left Distributive over Subtraction
{{:Euclid:Proposition/V/5}} That is, for any numbers $a, b$ and for any integer $m$: : $m a - m b = m \paren {a - b}$
Linear Second Order ODE/y'' + y = 0/y(0) = 2, y'(0) = 3
The second order ODE: :$(1): \quad y'' + y = 0$ with initial conditions: :$\map y 0 = 2$ :$\map {y'} 0 = 3$ has the particular solution: :$y = 3 \sin x + 2 \cos x$
Legendre's Differential Equation/(1 - x^2) y'' - 2 x y' + 2 y = 0
The special case of Legendre's differential equation: :$(1): \quad \paren {1 - x^2} y'' - 2 x y' + 2 y = 0$ has the general solution: :$y = C_1 x + C_2 \paren {\dfrac x 2 \, \map \ln {\dfrac {1 + x} {1 - x} } - 1}$
Second Order ODE/y'' - f(x) y' + (f(x) - 1) y = 0
The second order ODE: :$(1): \quad y'' - \map f x y' + \paren {\map f x - 1} y = 0$ has the general solution: :$\displaystyle y = C_1 e^x + C_2 e^x \int e^{-2 x + \int \map f x \rd x} \rd x$
Linear Second Order ODE/2 x^2 y'' + 10 x y' + 8 y = 0
The second order ODE: :$(1): \quad 2 x^2 y'' + 10 x y' + 8 y = 0$ has the general solution: :$y = C_1 x^{-2} + C_2 x^{-2} \ln x$
Divisors obey Distributive Law
{{:Euclid:Proposition/VII/5}} In modern algebraic language: :$\displaystyle a = \frac 1 n b, c = \frac 1 n d \implies a + c = \frac 1 n \left({b + d}\right)$
Real Numbers under Addition Modulo 1 form Group
Let $S = \left\{{x \in \R: 0 \le x < 1}\right\}$. Let $\circ: S \times S \to S$ be the operation defined as: : $x \circ y = x + y - \left \lfloor {x + y} \right \rfloor$ That is, $\circ$ is defined as addition modulo $1$. Then $\left({S, \circ}\right)$ is a group.
Linear Second Order ODE/y'' + y = 2 cosine x
The second order ODE: :$(1): \quad y'' + y = 2 \cos x$ has the general solution: :$y = C_1 \sin x + C_2 \cos x + x \cos x$
Linear Second Order ODE/y'' + y = cosecant x
The second order ODE: :$(1): \quad y'' + y = \csc x$ has the general solution: :$y = C_1 \sin x + C_2 \cos x - x \cos x + \sin x \map \ln {\sin x}$
Quaternions Defined by Ordered Pairs
Consider the quaternions $\Bbb H$ as numbers in the form: : $a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ where: : $a, b, c, d$ are real numbers; : $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ are entities related to each other in the following way: {{begin-eqn}}      {{eqn | l = \mathbf i \mathbf j = - \mathbf j \mathbf i       | r = \mathbf k       | c =  }} {{eqn | l = \mathbf j \mathbf k = - \mathbf k \mathbf j       | r = \mathbf i       | c =  }} {{eqn | l = \mathbf k \mathbf i = - \mathbf i \mathbf k       | r = \mathbf j       | c =  }} {{eqn | l = \mathbf i^2 = \mathbf j^2 = \mathbf k^2 = \mathbf i \mathbf j \mathbf k       | r = - \mathbf 1       | c =  }} {{end-eqn}} Now consider the quaternions $\Bbb H$ defined as ordered pairs $\left({x, y}\right)$ where $x, y \in \C$ are complex numbers, on which the operation of multiplication is defined as follows: Let $w = a_1 + b_1 i, x = c_1 + d_1 i, y = a_2 + b_2 i, z = c_2 + d_2 i$ be complex numbers. Then $\left({w, x}\right) \left({y, z}\right)$ is defined as: :$\left({w, x}\right) \left({y, z}\right) := \left({w y - z \overline x, \overline w z + x y}\right)$ where $\overline w$ and $\overline x$ are the complex conjugates of $w$ and $x$ respectively. These two definitions are equivalent.
Linear Second Order ODE/y'' - 2 y' - 3 y = 64 x exp -x
The second order ODE: :$(1): \quad y'' - 2 y' - 3 y = 64 x e^{-x}$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-x} - e^{-x} \paren {8 x^2 + 4 x + 1}$
Linear Second Order ODE/(x^2 + x) y'' + (2 - x^2) y' - (2 + x) y = x (x + 1)^2
The second order ODE: :$(1): \quad \paren {x^2 + x} y'' + \paren {2 - x^2} y' - \paren {2 + x} y = x \paren {x + 1}^2$ has the general solution: :$y = C_1 e^x + \dfrac {C_2} x - x - 1 - \dfrac {x^2} 3$
Position of Cart attached to Wall by Spring under Damping/Critically Damped/x = x0 at t = 0
Let $C$ be pulled aside to $x = x_0$ and released from stationary at time $t = 0$. Then the horizontal position of $C$ at time $t$ can be expressed as: :$x = x_0 e^{-a t} \left({1 + a t}\right)$
Ring of Square Matrices over Real Numbers
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\struct {\map {\mathcal M_\R} n, +, \times}$ denote the ring of square matrices of order $n$ over $\R$. Then $\struct {\map {\mathcal M_\R} n, +, \times}$ is a ring with unity, but is not a commutative ring.