Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 3 0 88538 475394 474883 2020-06-22T16:20:15Z Prime.mover 59 wikitext text/x-wiki == Corollary to Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations == <onlyinclude> Let $S$ be a system of homogeneous simultaneous linear equations: :$\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$ If $m < n$, then $S$ has at least one non-trivial solution. </onlyinclude> == Proof == Let $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ denote the augmented matrix of $S$. Because $S$ is homogeneous, we have that $\mathbf b = \mathbf 0$, and so its augmented matrix is $\begin {pmatrix} \mathbf A & \mathbf 0 \end {pmatrix}$. Let $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$ be a reduced echelon matrix derived from $\begin {pmatrix} \mathbf A & \mathbf 0 \end {pmatrix}$. Let $S'$ be the system of simultaneous linear equations: :$\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \rho_{i j} x_j = 0$ whose augmented matrix is $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$. By Corollary 1, $S$ and $S'$ are equivalent. Hence any every solution to $S'$ is also a solution to $S$. Consider the structure of $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$. Suppose the leading coefficients appear in columns which we name $j_1, j_2, \ldots, j_l$. Let the remaining columns be named $j_{l + 1}, j_{l + 2}, \ldots, j_n$. Then we have that $S'$ can be expressed as: {{begin-eqn}} {{eqn | l = x_{j_1} + \sum_{k \mathop = l + 1}^n \rho_{i j_k} x_{j_k}       | r = 0 }} {{eqn | l = x_{j_2} + \sum_{k \mathop = l + 1}^n \rho_{2 j_k} x_{j_k}       | r = 0 }} {{eqn | o = \cdots }} {{eqn | l = x_{j_l} + \sum_{k \mathop = l + 1}^n \rho_{2 j_k} x_{j_k}       | r = 0 }} {{end-eqn}} where $l + 1 \le m$. Setting arbitrary values to $x_{j_k}$ for $l < k \le n$ gives us a non-trivial solution for $S$. {{qed}} == Sources == * {{BookReference|Linear Algebra: An Introduction|1982|A.O. Morris|ed = 2nd|edpage = Second Edition|prev = Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 2|next = Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 4}}: Chapter $1$: Linear Equations and Matrices: $1.3$ Applications to Linear Equations: Corollary $3$ Category:Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations 8ifa6ehw6jyhti1mfzf2hng29hhikxb


Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 4 0 88539 474887 2020-06-20T12:54:48Z Prime.mover 59 Created page with "== Corollary to Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations == <onlyinclude> Let $S$ be a system of Definiti..." wikitext text/x-wiki == Corollary to Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations == <onlyinclude> Let $S$ be a system of homogeneous simultaneous linear equations: :$\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$ Let $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$ be a reduced echelon matrix derived from $\begin {pmatrix} \mathbf A & \mathbf 0 \end {pmatrix}$. Let the number of non-zero rows of $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$ be $l$. If $l = n$, then the only solution to $S$ is the trivial solution. </onlyinclude> == Proof == Consider the structure of $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$. Suppose the leading coefficients appear in columns which we name $j_1, j_2, \ldots, j_n$. As there are $n$ columns as well as $n$ non-zero rows: :each row has exactly one $1$ in it :each column has exactly one $1$ in it. Thus $S'$ can be expressed as: {{begin-eqn}} {{eqn | l = x_{j_1}       | r = 0 }} {{eqn | l = x_{j_2}       | r = 0 }} {{eqn | o = \cdots }} {{eqn | l = x_{j_n}       | r = 0 }} {{end-eqn}} and the result follows. {{qed}} == Sources == * {{BookReference|Linear Algebra: An Introduction|1982|A.O. Morris|ed = 2nd|edpage = Second Edition|prev = Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 3|next = Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 5}}: Chapter $1$: Linear Equations and Matrices: $1.3$ Applications to Linear Equations: Corollary $3$ Category:Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations 6gal5yyvtfrmls1qx2ti5kldjih0to5


Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse 0 88767 476442 476440 2020-06-30T09:23:35Z Prime.mover 59 Replaced content with "== Theorem == <onlyinclude> Let $e$ be an elementary row operation.  Let $\mathbf E$ be the Definition:Elementary Row Matrix|elem..." wikitext text/x-wiki == Theorem == <onlyinclude> Let $e$ be an elementary row operation. Let $\mathbf E$ be the elementary row matrix corresponding to $e$. Let $e'$ be the inverse of $e$. Then the elementary row matrix corresponding to $e'$ is the inverse of $\mathbf E$. </onlyinclude> == Proof 1 == {{:Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse/Proof 1}} == Proof 2 == {{:Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse/Proof 2}} == Also see == * Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse Category:Elementary Row Operations Category:Elementary Matrices Category:Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse b7onpxovwpfws6crk4vyoqqino8jaqy


Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse 0 88788 476460 2020-06-30T10:39:05Z Prime.mover 59 Created page with "== Theorem == <onlyinclude> Let $e$ be an elementary column operation.  Let $\mathbf E$ be the Definition:Elementary Column Matrix..." wikitext text/x-wiki == Theorem == <onlyinclude> Let $e$ be an elementary column operation. Let $\mathbf E$ be the elementary column matrix corresponding to $e$. Let $e'$ be the inverse of $e$. Then the elementary column matrix corresponding to $e'$ is the inverse of $\mathbf E$. </onlyinclude> == Proof 1 == {{:Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse/Proof 1}} == Proof 2 == {{:Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse/Proof 2}} == Also see == * Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse Category:Elementary Column Operations Category:Elementary Matrices Category:Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse hcex4o5gt7xef84ftxuu3z9n252qmd2


Elementary Column Operations as Matrix Multiplications/Corollary 0 88796 476469 2020-06-30T13:33:38Z Prime.mover 59 Created page with "== Theorem == <onlyinclude> Let $\mathbf X$ and $\mathbf Y$ be two $m \times n$ matrices that differ by exactly one Definition:Elementary Column Operat..." wikitext text/x-wiki == Theorem == <onlyinclude> Let $\mathbf X$ and $\mathbf Y$ be two $m \times n$ matrices that differ by exactly one elementary column operation. Then there exists an elementary column matrix $\mathbf E$ of order $n$ such that: :$\mathbf X \mathbf E = \mathbf Y$ </onlyinclude> == Proof == Let $e$ be the elementary column operation such that $e \paren {\mathbf X} = \mathbf Y$. Then this result follows immediately from Elementary Column Operations as Matrix Multiplications: :$e \paren {\mathbf X} = \mathbf X \mathbf E = \mathbf Y$ where $\mathbf E = e \paren {\mathbf I}$. {{qed}} Category:Conventional Matrix Multiplication Category:Elementary Column Operations ntseqkowo0j4dpokncrjvwc10rc0lfv


Fibonacci's Greedy Algorithm

Fibonacci's Greedy Algorithm 0 89051 478027 477999 2020-07-12T15:37:32Z Prime.mover 59 wikitext text/x-wiki == Algorithm == Let $\dfrac p q$ denote a proper fraction expressed in canonical form. '''Fibonacci's greedy algorithm''' is a greedy algorithm which calculates a sequence of distinct unit fractions which together sum to $\dfrac p q$: {{begin-eqn}} {{eqn | l = \dfrac p q       | r = \sum_{\substack {1 \mathop \le k \mathop \le m \\ n_j \mathop \le n_{j + 1} } } \dfrac 1 {n_k}       | c =  }} {{eqn | r = \dfrac 1 {n_1} + \dfrac 1 {n_2} + \dotsb + \dfrac 1 {n_m}       | c =  }} {{end-eqn}} <onlyinclude> '''Fibonacci's Greedy Algorithm''' is as follows: :$(1) \quad$ Let $p = x_0$ and $q = y_0$ and set $k = 0$. :$(2) \quad$ Is $x_k = 1$? If so, the algorithm has finished. :$(3) \quad$ Find the largest unit fraction $\dfrac 1 {m_k}$ less than $\dfrac {x_k} {y_k}$.  :$(4) \quad$ Calculate $\dfrac {x_{k + 1} } {y_{k + 1} } = \dfrac {x_k} {y_k} - \dfrac 1 {m_k}$ expressed in canonical form. :$(5) \quad$ Go to step $(2)$. </onlyinclude> == Also see == * Proper Fraction can be Expressed as Finite Sum of Unit Fractions/Fibonacci's Greedy Algorithm, which proves that Fibonacci's Greedy Algorithm works as expected {{Namedfor|Leonardo Fibonacci|cat = Fibonacci}} == Sources == * {{BookReference|Liber Abaci|1202|Leonardo Fibonacci}} * {{BookReference|Curious and Interesting Puzzles|1992|David Wells|prev = Proper Fraction can be Expressed as Finite Sum of Unit Fractions|next = Proper Fraction can be Expressed as Finite Sum of Unit Fractions/Fibonacci's Greedy Algorithm}}: Egyptian Fractions Category:Greedy Algorithms Category:Fibonacci's Greedy Algorithm o09ig10g3tj3495ctsekudc30kafuya


Largest Number not Expressible as Sum of Multiples of Coprime Integers/Generalization 0 89295 479146 2020-07-22T04:02:15Z RandomUndergrad 3904 Created page with "== Theorem ==  <onlyinclude> Let $a, b$ be integers greater than $1$.  Let $d = \gcd \set {a, b}$.  Then the largest multiple of $d$ not expressible as..." wikitext text/x-wiki == Theorem == <onlyinclude> Let $a, b$ be integers greater than $1$. Let $d = \gcd \set {a, b}$. Then the largest multiple of $d$ not expressible as a sum of multiples of $a$ and $b$ is the number: :$\dfrac {a b} d - a - b$ </onlyinclude> == Proof == By Integers Divided by GCD are Coprime: :$\dfrac a d \perp \dfrac b d$ By Largest Number not Expressible as Sum of Multiples of Coprime Integers, the largest number not expressible as a sum of multiples of $\dfrac a d$ and $\dfrac b d$ is the number: :$\dfrac {a b} {d^2} - \dfrac a d - \dfrac b d$ Let $k d$ be a multiple of $d$ expressible as a sum of multiples of $a$ and $b$: :$\exists s, t \in \N: s a + t b = k d$ Then: :$s \dfrac a d + t \dfrac b d = k$ showing that $k$ is a sum of multiples of $\dfrac a d$ and $\dfrac b d$. This argument reverses. Hence the largest multiple of $d$ not expressible as a sum of multiples of $a$ and $b$ is the number: :$d \paren {\dfrac {a b} {d^2} - \dfrac a d - \dfrac b d} = \dfrac {a b} d - a - b$ {{qed}} Category:Largest Number not Expressible as Sum of Multiples of Coprime Integers Category:Integer Combinations 9j77t0y7ednibo94kkth5ti58vhi8i0


Inscribing Equilateral Triangle inside Square with a Coincident Vertex/Construction 4 0 89408 479724 479717 2020-07-25T12:59:29Z Prime.mover 59 wikitext text/x-wiki == Construction for Inscribing Equilateral Triangle inside Square with a Coincident Vertex == {{:Inscribing Equilateral Triangle inside Square with a Coincident Vertex}} == Construction == <onlyinclude> :300px By Construction of Equilateral Triangle, let an equilateral triangle $\triangle ABN$ be constructed on $AB$ such that $N$ is inside $\Box ABCD$. Let $DN$ be produced to cut $BC$ at $H$. Construct $H$ on $BC$ such that $DH = DG$. Then $DGH$ is the required equilateral triangle. </onlyinclude> == Proof == First a lemma: === Lemma === {{:Inscribing Equilateral Triangle inside Square with a Coincident Vertex/Lemma}}{{qed|lemma}} Because $\triangle ABN$ is equilateral: :$AB = AN$ :$\angle BAN = 60 \degrees$ Thus $\triangle ADN$ is isosceles with apex at $A$. Then $\angle DAN = 90 \degrees - 60 \degrees = 30 \degrees$. From Sum of Angles of Triangle equals Two Right Angles: :$\angle ADN + \angle AND = 180 \degrees - 30 \degrees = 150 \degrees$ From Isosceles Triangle has Two Equal Angles: :$\angle ADN = \angle AND = \dfrac {150 \degrees} 2 = 75 \degrees$ Thus: :$\angle CDH = 15 \degrees$ The result follows from the lemma. {{qed}} Category:Inscribing Equilateral Triangle inside Square with a Coincident Vertex qskjduxorus4x54p5ch2u4abeqe9wnc


Inscribing Equilateral Triangle inside Square with a Coincident Vertex/Lemma 0 89410 479728 479723 2020-07-25T13:15:46Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $\Box ABCD$ be a square. Let $\triangle DGH$ be an isosceles triangle inscribed within $\Box ABCD$ such that the apex $D$ of $\triangle DGH$ coincides with vertex $D$ of $\Box ABCD$. :300px Then: :$\triangle DGH$ is equilateral triangle {{iff}}: :$\angle ADG = 15 \degrees \text { or } \angle CDH = 15 \degrees$ (and in fact both are the case). </onlyinclude> == Proof == First note that $\triangle DGH$ is isosceles. First we note that: {{begin-eqn}} {{eqn | l = CD       | r = AD       | c = as they are the sides of a square }} {{eqn | l = DH       | r = DG       | c = as $\triangle DGH$ is isosceles }} {{eqn | l = \angle DCH       | r = \angle DAG = 90 \degrees       | c = as $\Box ABCD$ is a square }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = CH^2       | r = DH^2 - CD^2       | c = Pythagoras's Theorem }} {{eqn | r = DG^2 - AD^2       | c =  }} {{eqn | r = AG^2       | c =  }} {{eqn | ll= \leadsto       | l = CH       | r = AG       | c =  }} {{end-eqn}} So by Triangle Side-Side-Side Equality: :$\triangle GAD = \triangle CDH$ and in particular: :$\angle GAD = \angle CDH$ === Necessary Condition === Let $\angle GAD = \angle CDH = 15 \degrees$. Then: :$\angle GDH = 90 \degrees - 2 \times 15 \degrees = 60 \degrees$ We have that $\triangle DGH$ is isosceles. Hence from Isosceles Triangle has Two Equal Angles: :$\angle DGH = \angle DHG$ From Sum of Angles of Triangle equals Two Right Angles it follows that: :$\angle DGH + \angle DHG = 180 \degrees - 60 \degrees = 120 \degrees$ from which: :$\angle DGH = \angle DHG = 60 \degrees$ Hence all the vertices of $\triangle DGH$ equal $60 \degrees$. It follows from Equiangular Triangle is Equilateral that $\triangle DGH$ is equilateral. {{qed|lemma}} === Sufficient Condition === Let $\triangle DGH$ be equilateral. From Internal Angle of Equilateral Triangle: :$\angle GDH = 60 \degrees$ Because $\Box ABCD$ is a square: :$\angle ADC = 90 \degrees$ Thus: {{begin-eqn}} {{eqn | l = \angle GAD + \angle GDH + \angle CDH       | r = 90 \degrees       | c =  }} {{eqn | ll= \leadsto       | l = \angle GAD + \angle CDH       | r = 90 \degrees - \angle GDH       | c =  }} {{eqn | r = 90 \degrees - 60 \degrees       | c =  }} {{eqn | r = 30 \degrees       | c =  }} {{end-eqn}} But we have that: :$\angle GAD = \angle CDH$ and so: :$\angle GAD = \angle CDH = 15 \degrees$ {{qed}} Category:Inscribing Equilateral Triangle inside Square with a Coincident Vertex lnchnr9pz5v2faxis0mmygg664pqlz9


Returning Explorer Puzzle

Returning Explorer Puzzle 0 90240 482712 482685 2020-08-23T09:36:27Z Prime.mover 59 wikitext text/x-wiki == Puzzle == An explorer walks: :one mile due south,  :one mile due east, :and one mile due north. He finds himself back where he started. He shoots a {{WP|Bear|bear}}. What colour is that {{WP|Bear|bear}}? === Variant === {{:Returning Explorer Puzzle/Variant}} == Solution == White. == Proof == The explorer starts from the North Pole. Every direction from the North Pole is due south. Walking due east keeps him the same distance (one mile) from the North Pole. Walking due north takes him back to the North Pole again. The bear, as a consequence, must be a {{WP|Polar_bear|polar bear}}. {{qed}} == Sources == * {{BookReference|Mathematical Puzzles and Diversions|1965|Martin Gardner|prev = |next = Returning Explorer Puzzle/Variant}}: Nine Problems: $1$ * {{BookReference|Hexaflexagons and Other Mathematical Diversions|1988|Martin Gardner|prev = |next = Returning Explorer Puzzle/Variant}}: Nine Problems: $1$ Category:Geography Puzzles 9udj5kjwnpqk3j4x5bwj5pxk5aku2a4


Rationals are Everywhere Dense in Reals/Topology 0 90798 486361 2020-09-08T09:33:47Z Julius 3095 Created page with "== Theorem == <onlyinclude> Let $\struct {\R, \tau_d}$ denote the real number line with the usual (Euclidean) topology...." wikitext text/x-wiki == Theorem == <onlyinclude> Let $\struct {\R, \tau_d}$ denote the real number line with the usual (Euclidean) topology. Let $\Q$ be the set of rational numbers. Then $\Q$ is everywhere dense in $\struct {\R, \tau_d}$. </onlyinclude> == Proof == Let $x \in \R$. Let $U \subseteq \R$ be an open set of $\struct {\R, \tau_d}$ such that $x \in U$. From Basis for Euclidean Topology on Real Number Line, there exists an open interval $V = \openint {x - \epsilon} {x + \epsilon} \subseteq U$ for some $\epsilon > 0$ such that $x \in V$. Now consider the open interval $\openint x {x + \epsilon} \subseteq V$. By Subset Relation is Transitive it follows that $\openint x {x + \epsilon} \subseteq U$. Note that $x \notin \openint x {x + \epsilon}$. From Between two Real Numbers exists Rational Number, there exists $y \in \Q: y \in \openint x {x + \epsilon}$. As $x \notin \openint x {x + \epsilon}$, it must be the case that $x \ne y$. That is, $V$ is an open set of $\struct {\R, \tau_d}$ containing $x$ which also contains an element of $\Q$ other than $x$. As $V$ is arbitrary, it follows that every open set of $\struct {\R, \tau_d}$ containing $x$ also contains an element of $\Q$ other than $x$. That is, $x$ is by definition a limit point of $\Q$. As $x$ is arbitrary, it follows that all elements of $\R$ are limit points of $\Q$. The result follows from the definition of everywhere dense. {{qed}} Category:Real Analysis Category:Rational Number Space Category:Real Number Line with Euclidean Topology Category:Denseness gqg5ugj8zepiyrt2c4szbkgnh6767q7


Convergent Sequence is Cauchy Sequence/Metric Space 0 91017 487795 2020-09-14T11:31:55Z Julius 3095 Created page with "== Theorem == <onlyinclude> Let $M = \struct {A, d}$ be a metric space.   Every Definition:Convergent Sequence in Metric Space|convergent sequenc..." wikitext text/x-wiki == Theorem == <onlyinclude> Let $M = \struct {A, d}$ be a metric space. Every convergent sequence in $M$ is a Cauchy sequence. </onlyinclude> == Proof == Let $\sequence {x_n}$ be a sequence in $A$ that converges to the limit $l \in A$. Let $\epsilon > 0$.  Then also $\dfrac \epsilon 2 > 0$. Because $\sequence {x_n}$ converges to $l$, we have: :$\exists N: \forall n > N: \map d {x_n, l} < \dfrac \epsilon 2$ So if $m > N$ and $n > N$, then: {{begin-eqn}} {{eqn | l = \map d {x_n, x_m}       | o = \le       | r = \map d {x_n, l} + \map d {l, x_m}       | c = Triangle Inequality }} {{eqn | o = <       | r = \frac \epsilon 2 + \frac \epsilon 2       | c = (by choice of $N$) }} {{eqn | r = \epsilon       | c =  }} {{end-eqn}} Thus $\sequence {x_n}$ is a Cauchy sequence. {{qed}} == Also see == * Definition:Complete Metric Space, where the converse is true. * Real Convergent Sequence is Cauchy Sequence * Complex Sequence is Cauchy iff Convergent == Sources == * {{BookReference|Principles of Mathematical Analysis|1953|Walter Rudin|next = Compact Metric Space is Complete}}: $3.11a$ * {{BookReference|Introduction to Metric and Topological Spaces|1975|W.A. Sutherland|prev = Definition:Cauchy Sequence (Metric Space)|next = Subsequence of Sequence in Metric Space with Limit}}: $7.2$: Sequential compactness: Proposition $7.2.4$ * {{BookReference|Counterexamples in Topology|1978|Lynn Arthur Steen|author2 = J. Arthur Seebach, Jr.|ed = 2nd|edpage = Second Edition|prev = Definition:Cauchy Sequence (Metric Space)|next = Cauchy Sequence in Metric Space is not necessarily Convergent}}: Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Complete Metric Spaces Category:Cauchy Sequences 3fp1bcpai3vu42ov3pmvs4jpzk155qx


Cauchy Sequence is Bounded/Metric Space

Cauchy Sequence is Bounded/Metric Space 0 91058 488083 2020-09-15T11:26:19Z Julius 3095 Created page with "== Theorem == <onlyinclude> Let $M = \struct {A, d}$ be a metric space.   Then every Cauchy sequence..." wikitext text/x-wiki == Theorem == <onlyinclude> Let $M = \struct {A, d}$ be a metric space. Then every Cauchy sequence in $M$ is bounded. </onlyinclude> == Proof == Let $\sequence {x_n}$ be a Cauchy sequence in $M$. By definition: :$\forall \epsilon > 0: \exists N \in \N: \forall m, n > N: \map d {x_n, x_m} < \epsilon$ Particularly, setting $\epsilon = 1$: :$\exists N_1: \forall m, n > N_1: \map d {x_n, x_m} < 1$ Note that since $N_1 \ge N_1$, this means that: :$\forall n \ge N_1: \map d {x_n, x_{N_1} } < 1$ To show $\sequence {x_n}$ is bounded, we need to show that there exists $a \in A$ and $K \in \R$ such that $\map d {x_n, a} \le K$ for all $x_n \in \sequence {x_n}$. Let $K'$ be the maximum distance from $x_{N_1}$ to any of the earlier terms in the sequence. That is, $K' = \max \set {\map d {x_{N_1}, x_1}, \map d {x_{N_1},  x_2}, \ldots, \map d {x_{N_1}, x_{N_1 - 1} } }$ Then: :Each $x_n$ for $n \ge N_1$ satisfies $\map d {x_{N_1}, x_n} \le 1$ by choice of $N_1$ as mentioned above :Each $x_n$ for $n < N_1$ satisfies $\map d {x_{N_1}, x_n} \le K'$ by choice of $K'$. Thus, taking $a = x_{N_1}$ and $K = \max \set {K', 1}$, we have shown that each $x_n$ satisfies $\map d {a, x_n} \le K$. So, $\sequence {x_n}$ is bounded. {{qed}} == Sources == * {{BookReference|Mathematical Analysis|1957|Tom M. Apostol|prev = Real Convergent Sequence is Cauchy Sequence|next = Unbounded Sequence is Divergent}}: $\S 12$-$2$: Convergent and divergent sequences Category:Cauchy Sequences s8kcwka7sx73viuxqvaryz87ba99gi3


Autocorrelation of Strictly Stationary Stochastic Process 0 91323 493070 489826 2020-10-07T15:20:48Z Prime.mover 59 wikitext text/x-wiki == Example of Strictly Stationary Stochastic Process == Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. <onlyinclude> It is necessary that: :The autocorrelation between every two observations $z_t, z_{t + k}$ separated by a given lag $k$  is the same as: :the autocorrelation between every other two observations $z_{t + m}, z_{t + m + k}$ separated by a given lag $k$ </onlyinclude> For such a strictly stationary stochastic process: :$\rho_k = \dfrac {\gamma_k} {\gamma_0}$ where $\gamma_k$ denotes the autocovariance of $S$. == Proof == The autocorrelation is defined as: :$\rho_k := \dfrac {\expect {\paren {z_t - \mu} \paren {z_{t + k} - \mu} } } {\sqrt {\expect {\paren {z_t - \mu}^2} \expect {\paren {z_{t + k} - \mu}^2} } }$ The autocovariance is defined as: :$\gamma_k := \expect {\paren {z_t - \mu} \paren {z_{t - k} - \mu} }$ Hence: :$\rho_k := \dfrac {\gamma_k} {\sqrt {\expect {\paren {z_t - \mu}^2} \expect {\paren {z_{t + k} - \mu}^2} } }$ Then we have that for a strictly stationary stochastic process: :$\expect {\paren {z_t - \mu}^2} = \sigma_z^2$ where $\sigma_z^2$ is the variance of $S$ and, for a strictly stationary stochastic process, is constant. Thus: :$\rho_k := \dfrac {\gamma_k} {\sigma_t^2}$ Then from Autocovariance at Zero Lag for Strictly Stationary Stochastic Process is Variance: :$\sigma_z^2 = \gamma_0$ Hence: :$\rho_k = \dfrac {\gamma_k} {\gamma_0}$ {{qed}} == Sources == * {{BookReference|Time Series Analysis: Forecasting and Control|1994|George E.P. Box|author2 = Gwilym M. Jenkins|author3 = Gregory C. Reinsel|ed = 3rd|edpage = Third Edition|prev = Definition:Autocorrelation|next = Autocovariance at Zero Lag for Strictly Stationary Stochastic Process is Variance}}: ::Part $\text {I}$: Stochastic Models and their Forecasting: :::$2$: Autocorrelation Function and Spectrum of Stationary Processes: ::::$2.1$ Autocorrelation Properties of Stationary Models: :::::$2.1.2$ Stationary Stochastic Processes: Autocovariance and autocorrelation coefficients: $(2.1.6)$ Category:Autocorrelation Category:Stationary Stochastic Processes oxk3s6u49i9odf3w3jttrcnsq6frhb7


Autocovariance at Zero Lag for Strictly Stationary Stochastic Process is Variance 0 91332 489817 489808 2020-09-21T15:36:53Z Prime.mover 59 wikitext text/x-wiki == Example of Strictly Stationary Stochastic Process == Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Then the autocovariance at zero lag is given by: :$\gamma_0 = \sigma_z^2$ where $\sigma_z^2$ is the variance of $S$. == Proof == By definition, the '''autocovariance''' of $S$ at lag $k$ is defined as: :$\gamma_k := \cov {z_t, z_{t + k} } = \expect {\paren {z_t - \mu} \paren {z_{t - k} - \mu} }$ where: :$z_t$ is the observation at time $t$ :$\mu$ is the mean of $S$ :$\expect \cdot$ is the expectation. For a strictly stationary stochastic process: :$\expect {\paren {z_t - \mu}^2} = \sigma_z^2$ where: :$\mu$ is the constant mean level of $S$ :$\expect {\paren {z_t - \mu}^2}$ is the expectation of $\paren {z_t - \mu}^2$ :$\sigma_z^2$ is the variance of $S$ and, for a strictly stationary stochastic process, is constant. Hence we have that: {{begin-eqn}} {{eqn | l = \gamma_0       | r = \expect {\paren {z_t - \mu} \paren {z_{t + 0} - \mu} }       | c =  }} {{eqn | r = \expect {\paren {z_t - \mu}^2}       | c =  }} {{eqn | r = \sigma_z^2       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Time Series Analysis: Forecasting and Control|1994|George E.P. Box|author2 = Gwilym M. Jenkins|author3 = Gregory C. Reinsel|ed = 3rd|edpage = Third Edition|prev = Autocorrelation of Strictly Stationary Stochastic Process|next = Autocorrelation at Zero Lag for Strictly Stationary Stochastic Process is 1}}: ::Part $\text {I}$: Stochastic Models and their Forecasting: :::$2$: Autocorrelation Function and Spectrum of Stationary Processes: ::::$2.1$ Autocorrelation Properties of Stationary Models: :::::$2.1.2$ Stationary Stochastic Processes: Autocovariance and autocorrelation coefficients Category:Stationary Stochastic Processes Category:Autocovariance csnx3gtbincn8tjz9921ydy7z84u086


Determinant of Autocorrelation Matrix is Strictly Positive/Examples/Order 2 0 91368 490102 2020-09-22T10:34:28Z Prime.mover 59 Created page with "== Example of Use of Determinant of Autocorrelation Matrix is Strictly Positive == <onlyinclude> Let $\rho_1$ be the autocorrelation of a ..." wikitext text/x-wiki == Example of Use of Determinant of Autocorrelation Matrix is Strictly Positive == <onlyinclude> Let $\rho_1$ be the autocorrelation of a strictly stationary stochastic process $S$ at lag $1$. Then: :$-1 < \rho_1 < 1$ </onlyinclude> == Proof == Consider the autocorrelation matrix of order $2$: {{begin-eqn}} {{eqn | l = \map \det {\mathbf P_2}       | o = >       | r = 0       | c = Determinant of Autocorrelation Matrix is Strictly Positive }} {{eqn | l = \begin {vmatrix} 1 & \rho_1 \\ \rho_1 & 1 \end {vmatrix}       | o = >       | r = 0       | c = {{Defof|Autocorrelation Matrix}} }} {{eqn | l = 1 - \rho_1^2       | o = >       | r = 0       | c = {{Defof|Determinant}} }} {{eqn | ll= \leadsto       | l = \rho_1^2       | o = <       | r = 1       | c =  }} {{eqn | ll= \leadsto       | l = \size {\rho_1}       | o = <       | r = 1       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Time Series Analysis: Forecasting and Control|1994|George E.P. Box|author2 = Gwilym M. Jenkins|author3 = Gregory C. Reinsel|ed = 3rd|edpage = Third Edition|prev = Determinant of Autocorrelation Matrix is Strictly Positive|next = Determinant of Autocorrelation Matrix is Strictly Positive/Examples/Order 3}}: ::Part $\text {I}$: Stochastic Models and their Forecasting: :::$2$: Autocorrelation Function and Spectrum of Stationary Processes: ::::$2.1$ Autocorrelation Properties of Stationary Models: :::::$2.1.3$ Positive Definiteness and the Autocovariance Matrix: Conditions satisfied by the autocorrelations of a stationary process Category:Determinant of Autocorrelation Matrix is Strictly Positive nsyymouh42ctehv27y9wj35pcqp9uyw


Negative of Logarithm of x plus Root x squared plus a squared/Corollary 0 91523 490958 2020-09-26T13:22:45Z Prime.mover 59 Created page with "== Theorem ==  Let $x \in \R: \size x > 1$.  Let $x > 1$.  Then: <onlyinclude> :$-\map \ln {x + \sqrt {x^2 + a^2} } = \ln \size {x - \sqrt {x^2 + a^2} } - \map \ln {a^2}$ </on..." wikitext text/x-wiki == Theorem == Let $x \in \R: \size x > 1$. Let $x > 1$. Then: <onlyinclude> :$-\map \ln {x + \sqrt {x^2 + a^2} } = \ln \size {x - \sqrt {x^2 + a^2} } - \map \ln {a^2}$ </onlyinclude> == Proof == We have that $\sqrt {x^2 + a^2} > x$ for all $x$. Hence for all $x$: :$-x + \sqrt {x^2 + a^2} > 0$ and so: :$x - \sqrt {x^2 + a^2} < 0$ Hence: :$-x + \sqrt {x^2 + a^2} = \size {x - \sqrt {x^2 + a^2} }$ Then we have: {{begin-eqn}} {{eqn | l = -\map \ln {x + \sqrt {x^2 - a^2} }       | r = \map \ln {-x + \sqrt {x^2 + a^2} } - \map \ln {a^2}       | c = Negative of Logarithm of x plus Root x squared plus a squared }} {{eqn | ll= \leadsto       | l = -\map \ln {x + \sqrt {x^2 - a^2} }       | r = \ln \size {x - \sqrt {x^2 + a^2} } - \map \ln {a^2}       | c =  }} {{end-eqn}} {{qed}} == Also see == * Negative of Logarithm of x plus Root x squared plus a squared Category:Logarithms bq1gb4ofmsp608jt6gjqt15zeqxcjlc


Primitive of Reciprocal of a squared minus x squared/Logarithm Form/Lemma 0 91571 491217 2020-09-27T16:21:54Z Prime.mover 59 Created page with "== Lemma == <onlyinclude> Let $a \in \R_{>0}$ be a strictly positive real constant.  Then: :$\map \ln {\df..." wikitext text/x-wiki == Lemma == <onlyinclude> Let $a \in \R_{>0}$ be a strictly positive real constant. Then: :$\map \ln {\dfrac {a + x} {a - x} }$ is defined {{iff}} $\size x < a$ :$\map \ln {\dfrac {x + a} {x - a} }$ is defined {{iff}} $\size x > a$ </onlyinclude> == Proof == We have that the real natural logarithm is defined only on the strictly positive real numbers. Hence: :$\map \ln {\dfrac {a + x} {a - x} }$ is defined {{iff}} $\dfrac {a + x} {a - x} > 0$ :$\map \ln {\dfrac {x + a} {x - a} }$ is defined {{iff}} $\dfrac {x + a} {x - a} > 0$ First we note that if $\size x = a$, then either the numerator or denominator of the arguments of the logarithm functions in question are either $0$ or undefined. Hence the expressions have no meaning unless $\size x \ne a$. The following table indicates whether each of $x + a$, $a - x$ and $x - a$ are positive $(+)$ or negative $(-)$ on the domains in question. $\begin {array} {c|ccc|cc} & x + a & a - x & x - a & \dfrac {a + x} {a - x} & \dfrac {x + a} {x - a} \\ \hline  a < x      & + & - & + & + & - \\ 0 < x < a  & + & + & - & - & + \\ -a < x < 0 & + & + & - & - & + \\ x < -a     & - & + & - & + & - \\ \end {array}$ Hence: :$\map \ln {\dfrac {a + x} {a - x} }$ is defined {{iff}} $-a < x < a$ :$\map \ln {\dfrac {x + a} {x - a} }$ is defined {{iff}} $x > a$ or $x < -a$ as we were required to show. {{qed}} Category:Logarithms Category:Primitive of Reciprocal of a squared minus x squared czlzx7khcp4z5tjwrrh1qrxjlihs7d6


Primitive of Reciprocal of x squared minus a squared/Logarithm Form/Lemma 0 91592 491313 2020-09-27T22:16:21Z Prime.mover 59 Created page with "== Lemma == <onlyinclude> Let $a \in \R_{>0}$ be a strictly positive real constant.  Then: :$\map \ln {\df..." wikitext text/x-wiki == Lemma == <onlyinclude> Let $a \in \R_{>0}$ be a strictly positive real constant. Then: :$\map \ln {\dfrac {x - a} {x + a} }$ is defined {{iff}} $\size x > a$ :$\map \ln {\dfrac {a - x} {a + x} }$ is defined {{iff}} $\size x < a$ </onlyinclude> == Proof == We have that the real natural logarithm is defined only on the strictly positive real numbers. Hence: :$\map \ln {\dfrac {x - a} {x + a} }$ is defined {{iff}} $\dfrac {x - a} {x + a} > 0$ :$\map \ln {\dfrac {a - x} {a + x} }$ is defined {{iff}} $\dfrac {a - x} {a + x} > 0$ First we note that if $\size x = a$, then either the numerator or denominator of the arguments of the logarithm functions in question are either $0$ or undefined. Hence the expressions have no meaning unless $\size x \ne a$. The following table indicates whether each of $x + a$, $a - x$ and $x - a$ are positive $(+)$ or negative $(-)$ on the domains in question. $\begin {array} {c|ccc|cc} & x + a & a - x & x - a & \dfrac {x - a} {x + a} & \dfrac {a - x} {a + x} \\ \hline  a < x      & + & - & + & + & - \\ 0 < x < a  & + & + & - & - & + \\ -a < x < 0 & + & + & - & - & + \\ x < -a     & - & + & - & + & - \\ \end {array}$ Hence: :$\map \ln {\dfrac {x - a} {x + a} }$ is defined {{iff}} $x > a$ or $x < -a$ :$\map \ln {\dfrac {a - x} {a + x} }$ is defined {{iff}} $-a < x < a$ as we were required to show. {{qed}} Category:Logarithms Category:Primitive of Reciprocal of x squared minus a squared 2du9lctlzc521fqaiwvjzyok1zbdl2o


Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Tangent Form/Proof 0 91596 491322 2020-09-27T22:30:09Z Prime.mover 59 Created page with "== Theorem == {{:Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Tangent Form}}  == Proof == <onlyinclude> Let $\size x < a$.  Let:  {{begin-eqn}} {{eq..." wikitext text/x-wiki == Theorem == {{:Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Tangent Form}} == Proof == <onlyinclude> Let $\size x < a$. Let: {{begin-eqn}} {{eqn | l = u       | r = \tanh^{-1} {\frac x a}       | c = {{Defof|Real Inverse Hyperbolic Tangent}}, which is defined where $\size {\dfrac x a} < 1$ }} {{eqn | ll= \leadsto       | l = x       | r = a \tanh u       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d x} {\d u}       | r = a \sech^2 u       | c = Derivative of Hyperbolic Cotangent }} {{eqn | ll= \leadsto       | l = \int \frac 1 {x^2 - a^2} \rd x       | r = \int \frac {a \sech^2 u} {a^2 \tanh^2 u - a^2} \rd u       | c = Integration by Substitution }} {{eqn | r = \frac a {a^2} \int \frac {\sech^2 u} {-\paren {1 - \tanh^2 u} } \rd u       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 a \int \frac {\sech^2 u} {-\sech^2 u} \rd u       | c = Sum of Squares of Hyperbolic Secant and Tangent }} {{eqn | r = -\frac 1 a \int \rd u }} {{eqn | r = -\frac 1 a u + C       | c = Integral of Constant }} {{eqn | r = -\frac 1 a \tanh^{-1} {\frac x a} + C       | c = Definition of $u$ }} {{end-eqn}} </onlyinclude>{{qed}} Category:Primitive of Reciprocal of x squared minus a squared cum8etbjs33ubz9sorsuh7ts10svvrq


Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma/Lemma 1 0 91645 491748 491696 2020-09-30T06:34:44Z Leigh.Samphier 3031 wikitext text/x-wiki {{Proofread}} == Theoren == Let $U, V$ be finite sets. Let $\card V < \card U$. Then: <onlyinclude> :$\card{V \setminus U} < \card{U \setminus V}$ </onlyinclude> == Proof == We have: {{begin-eqn}} {{eqn | l = \card{U \setminus V}       | r = \card U - \card{U \cap V}       | c = Cardinality of Set Difference }} {{eqn | o = >        | r = \card V - \card{U \cap V}       | c = As $\card V < \card U$ }} {{eqn | r = \card{V \setminus U}       | c = Cardinality of Set Difference }} {{end-eqn}} {{qed}} Category:Matroid Satisfies Base Axiom ncp1vmjeal1kfh62yi04bort1blni90


Transitivity of Integrality/Lemma

Transitivity of Integrality/Lemma 0 92667 497829 2020-11-05T14:11:07Z Prime.mover 59 Created page with "== Lemma for Transitivity of Integrality == <onlyinclude> Let $A \subseteq B$ be a ring extension.  Let $x_1, \dotsc, x_n \in B$ be Definit..." wikitext text/x-wiki == Lemma for Transitivity of Integrality == <onlyinclude> Let $A \subseteq B$ be a ring extension. Let $x_1, \dotsc, x_n \in B$ be integral over $A$. Let $A \sqbrk {x_1, \dotsc, x_n}$ be the subring of $B$ generated by $A \cup \set {x_1, \dotsc, x_n}$ over $A$. Then $A \sqbrk {x_1, \dotsc, x_n}$ is integral over $A$. </onlyinclude> == Proof == Let $C$ be the integral closure of $A$ in $B$. Since the $x_i$ are integral over $A$, they lie in $C$. So by Integral Closure is Subring, all sums of the form: :$\ds \sum_{\text {finite} } r x_1^{\alpha_1} \dotsm x_n^{\alpha_n}, \quad r \in A,\ \alpha_j \in \N \cup \set 0$ lie in $C$. That is, they are integral over $A$.  But the set of such sums is precisely $A \sqbrk {x_1, \dotsc, x_n}$. {{qed}} == Linguistic Note == {{:Transitivity of Integrality/Linguistic Note}} Category:Transitivity of Integrality qnem75iriv50mo5gee7s0cgqervax5e


Schur-Zassenhaus Theorem

Let $G$ be a finite group and $N$ be a normal subgroup in $G$. Let $N$ be a Hall subgroup of $G$. Then there exists $H$, a complement of $N$, such that $G$ is the semidirect product of $N$ and $H$.


Non-Zero Integer has Unique Positive Integer Associate

Let $a \in \Z$ be an integer such that $a \ne 0$. Then $a$ has a unique associate $b \in \Z_{>0}$.


Element of Cyclic Group is not necessarily Generator

Let $\gen g = G$ be a cyclic group. Let $a \in G$ Then it is not necessarily the case that $a$ is also a generator of $G$.


Equivalent Statements for Congruence Modulo Subgroup/Left

Let $x \equiv^l y \pmod H$ denote that $x$ is left congruent modulo $H$ to $y$. Then the following statements are equivalent: {{begin-eqn}} {{eqn | n = 1       | l = x       | o = \equiv^l       | r = y \pmod H }} {{eqn | n = 2       | l = x^{-1} y       | o = \in       | r = H }} {{eqn | n = 3       | l = \exists h \in H: x^{-1} y       | r = h }} {{eqn | n = 4       | l = \exists h \in H: y       | r = x h }} {{end-eqn}}


Element of Group is in Unique Coset of Subgroup/Left

There exists a exactly one left coset of $H$ containing $x$, that is: $x H$


Order of Product of Commuting Group Elements of Coprime Order is Product of Orders

Let $G$ be a group. Let $g_1, g_2 \in G$ be commuting elements such that: {{begin-eqn}} {{eqn | l = \order {g_1}       | r = n_1 }} {{eqn | l = \order {g_1}       | r = n_2 }} {{end-eqn}} where $\order {g_1}$ denotes the order of $g_1$ in $G$. Let $n_1$ and $n_2$ be coprime. Then: :$\order {g_1 g_2} = n_1 n_2$


Trivial Norm on Division Ring is Norm

Let $\struct {R, +, \circ}$ be a division ring, and denote its ring zero by $0_R$. Then the trivial norm $\norm {\, \cdot \,}: R \to \R_{\ge 0}$, which is given by: :$\norm x = \begin{cases}   0 & \text { if } x = 0_R\\   1 & \text { otherwise} \end{cases}$ defines a norm on $R$.


Coset of Subgroup of Subgroup

Let $G$ be a group. Let $H, K \le G$ be subgroups of $G$. Let $K \subseteq H$. Let $x \in G$. Then either: :$x K \subseteq H$ or: :$x K \cap H = \O$ where $x K$ denotes the left coset of $K$ by $x$.


Equivalence of Definitions of Characteristic of Ring

Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. {{TFAE|def = Characteristic of Ring}}


Non-Cyclic Group of Order p^2 has p+3 Subgroups

Let $p$ be a prime number. Let $G$ be a non-cyclic group whose order is $p^2$. Then $G$ has exactly $p + 3$ subgroups.


General Morphism Property for Groups

Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups. Let $\phi: G \to H$ be a homomorphism. Then: :$\forall g_k \in H: \map \phi {g_1 \circ g_2 \circ \cdots \circ g_n} = \map \phi {g_1} * \map \phi {g_2} * \cdots * \map \phi {g_n}$


Set Difference is Anticommutative

Set difference is an anticommutative operation: :$S = T \iff S \setminus T = T \setminus S = \varnothing$


Integers under Addition form Monoid

The set of integers under addition $\struct {\Z, +}$ forms a monoid.


Klein Four-Group is Group

The Klein $4$-group $K_4$ is a group.


Dihedral Group is Non-Abelian

Let $n \in \N$ be a natural number such that $n > 2$. Let $D_n$ denote the dihedral group of order $2 n$. Then $D_n$ is not abelian.


Summation Formula (Complex Analysis)

:$\displaystyle \sum_{n \in \Z \setminus X} \map f n = - \sum_{z_0 \mathop \in X} \Res {\pi \cot \paren {\pi z} \map f z} {z_0}$


Set with Complement forms Partition

Let $\varnothing \subset S \subset \mathbb U$. Then $S$ and its complement $\complement \left({S}\right)$ form a partition of the universal set $\mathbb U$.


Subgroups of Additive Group of Integers Modulo m

Let $n \in \Z_{> 0}$ be a (strictly) positive integer. Let $\struct {\Z_m, +_m}$ denote the additive group of integers modulo $m$. The subgroups of $\struct {\Z_m, +_m}$ are the additive groups of integers modulo $k$ where: :$k \divides m$


Finite Monoid with Left Cancellable Operation is Group

Let $\left({S, \circ}\right)$ be a finite monoid. Let $\circ$ be a left cancellable operation. Then $\left({S, \circ}\right)$ is a group.


Cartesian Product is Anticommutative

Let $S, T \ne \O$. Then: :$S \times T = T \times S \implies S = T$


Groups of Order 6

There exist exactly $2$ groups of order $6$, up to isomorphism: :$C_6$, the cyclic group of order $6$ :$S_3$, the symmetric group on $3$ letters.


Normalizer of Rotation in Dihedral Group

Let $n \in \N$ be a natural number such that $n \ge 3$. Let $D_n$ be the dihedral group of order $2 n$, given by: :$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$ Let $\map {N_{D_n} } {\set \alpha}$ denote the normalizer of the singleton containing the rotation element $\alpha$. Then: :$\map {N_{D_n} } {\set \alpha} = \gen \alpha$ where $\gen \alpha$ is the subgroup generated by $\alpha$.


Möbius Strip has Euler Characteristic Zero

Let $M$ be a Möbius Strip. Then: :$\map \chi M = 0$ where $\map \chi M$ denotes the Euler characteristic of the graph $M$.


Group Action of Symmetric Group Acts Transitively

Let $S$ be a set. Let $\struct {\map \Gamma S, \circ}$ be the symmetric group on $S$. Let $*: \map \Gamma S \times S \to S$ be the group action defined as: :$\forall \pi \in \map \Gamma S, \forall s \in S: \pi * s = \map \pi s$ Then $*$ is a transitive group action. In other words, $\struct {\map \Gamma S, \circ}$ acts transitively on $S$ by $*$.


Orbit of Trivial Group Action is Singleton

Let $\left({G, \circ}\right)$ be a group whose identity is $e$. Let $S$ be a set. Let $*: G \times S \to S$ be the trivial group action: :$\forall \left({g, s}\right) \in G \times S: g * s = s$ Let $s \in S$. Then the orbit of $s$ under $*$ is $\left\{{s}\right\}$.


Length of Orbit of Subgroup Action on Left Coset Space

Let $G$ be a group. Let $H$ and $K$ be subgroups of $G$. Let $K$ act on the left coset space $G / H^l$ by: :$\forall \tuple {k, g H} \in K \times G / H^l: k * g H := \paren {k g} H$ The length of the orbit of $g H$ is $\index K {K \cap H^g}$.


Conjugacy Action on Abelian Group is Trivial

Let $\struct {G, \circ}$ be an abelian group whose identity is $e$. Let $*: G \times G \to G$ be the conjugacy group action: : $\forall g, h \in G: g * h = g \circ h \circ g^{-1}$ Then $*$ is a trivial group action.


Relation Symmetry

Every non-null relation has exactly one of these properties: it is either: :symmetric, :asymmetric or :non-symmetric.


Stabilizer of Cartesian Product of Group Actions

Let $\struct {G, \circ}$ be a group. Let $S$ and $T$ be sets. Let $*_S: G \times S \to S$ and $*_T: G \times T \to T$ be group actions. Let the group action $*: G \times \paren {S \times T} \to S \times T$ be defined as: :$\forall \tuple {g, \tuple {s, t} } \in G \times \paren {S \times T}: g * \tuple {s, t} = \tuple {g *_S s, g *_T t}$ Then the stabilizer of $\tuple {s, t} \in S \times T$ is given by: :$\Stab {s, t} = \Stab s \cap \Stab t$ where $\Stab s$ and $\Stab t$ are the stabilizers of $s$ and $t$ under $*_S$ and $*_T$ respectively.


Divisibility by 12

Let $N \in \N$ be expressed as: :$N = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ Then $N$ is divisible by $12$ {{iff}} $a_0 - 2 a_1 + 4 \paren {\displaystyle \sum_{r \mathop = 2}^n a_r}$ is  divisible by $12$.


Composite Relation with Inverse is Symmetric

Let $\mathcal R \subseteq S \times T$ be a relation. Then the composition of $\mathcal R$ with its inverse $\mathcal R^{-1}$ is symmetric: * $\mathcal R^{-1} \circ \mathcal R$ is a symmetric relation on $S$ * $\mathcal R \circ \mathcal R^{-1}$ is a symmetric relation on $T$.


Minimum Rule for Real Sequences

:$\displaystyle \lim_{n \mathop \to \infty} \min \set {x_n, y_n} = \min \set {l, m}$


Union of Equivalences

The union of two equivalence relations is '''not''' necessarily an equivalence relation itself.


Topological Properties of Non-Archimedean Division Rings/Intersection of Open Balls

:$\map {B_r} x \cap \map {B_s} y \ne \O \iff \map {B_r} x \subseteq \map {B_s} y$ or $\map {B_s} y \subseteq \map {B_r} x$


Universal Negative implies Particular Negative iff First Predicate is not Vacuous

Consider the categorical statements: :$\map {\mathbf E} {S, P}: \quad$ The universal negative: $\forall x: \map S x \implies \neg \map P x$ :$\map {\mathbf O} {S, P}: \quad$ The particular negative: $\exists x: \map S x \land \neg \map P x$ Then: :$\map {\mathbf E} {S, P} \implies \map {\mathbf O} {S, P}$ {{iff}}: :$\exists x: \map S x$ Using the symbology of predicate logic: :$\exists x: \map S x \iff \paren {\paren {\forall x: \map S x \implies \neg \map P x} \implies \paren {\exists x: \map S x \land \neg \map P x} }$


Addition on Numbers has no Zero Element

On all the number systems: * natural numbers $\N$ * integers $\Z$ * rational numbers $\Q$ * real numbers $\R$ * complex numbers $\C$ there exists no zero element for addition.


Group has Latin Square Property/Additive Notation

Let $\struct {G, +}$ be a group. Then $G$ satisfies the Latin square property. That is, for all $a, b \in G$, there exists a unique $g \in G$ such that $a + g = b$. Similarly, there exists a unique $h \in G$ such that $h + a = b$.


Normed Division Ring Operations are Continuous/Inversion

:$\iota : \struct {R^* ,d^*} \to \struct {R, d} : \map \iota x = x^{-1}$  is continuous.


Intersection Operation on Supersets of Subset is Closed

Let $S$ be a set. Let $T \subseteq S$ be a given subset of $S$. Let $\powerset S$ denote the power set of $S$ Let $\mathscr S$ be the subset of $\powerset S$ defined as: :$\mathscr S = \set {Y \in \powerset S: T \subseteq Y}$ Then the algebraic structure $\struct {\mathscr S, \cap}$ is closed.


Composite of Quotient Mappings

Let $S$ be a set. Let $\mathcal R_1$ be an equivalence on $S$, and $\mathcal R_2$ be an equivalence on the quotient set $S / \mathcal R_1$. We can find an equivalence $\mathcal R_3$ on $S$ such that $\paren {S / \mathcal R_1} / \mathcal R_2$ is in one-to-one correspondence with $S / \mathcal R_3$ under the mapping: :$\phi: \paren {S / \mathcal R_1} / \mathcal R_2 \to S / \mathcal R_3: \eqclass {\eqclass x {\mathcal R_1} } {\mathcal R_2} \mapsto \eqclass x {\mathcal R_3}$.


Subgroup Generated by Commuting Elements is Abelian

Let $\struct {G, \circ}$ be a group. Let $S \subseteq G$ such that: :$\forall x, y \in S: x \circ y = y \circ x$ Then the subgroup generated by $S$ is abelian.


Stabilizer is Normal iff Stabilizer of Each Element of Orbit

Let $\struct {G, \circ}$ be a group. Let $S$ be a set. Let $*: G \times S \to S$ be a group action. Let $x \in S$. Let $\Stab x$ denote the stabilizer of $x$ under $*$. Let $\Orb x$ denote the orbit of $x$ under $*$. Then $\Stab x$ is normal in $G$ {{iff}} $\Stab x$ is also the stabilizer of every element in $\Orb x$.


Additive Group of Integers is Normal Subgroup of Rationals

Let $\struct {\Z, +}$ be the additive group of integers. Let $\struct {\Q, +}$ be the additive group of rational numbers. Then $\struct {\Z, +}$ is a normal subgroup of $\struct {\Q, +}$.


Empty Set and Set form Algebra of Sets

Let $S$ be any non-empty set. Then $\left\{{S, \varnothing}\right\}$ is (trivially) an algebra of sets, where $S$ is the unit.


Closure of Union and Complement imply Closure of Set Difference

Let $\RR$ be a system of sets on a universe $\mathbb U$ such that for all $A, B \in \RR$: :$(1): \quad A \cup B \in \RR$ :$(2): \quad \map \complement A \in \RR$ where $\cup$ denotes set union and $\complement$ denotes complement (relative to $\mathbb U$). Then: :$\forall A, B \in \RR: A \setminus B \in \RR$ where $\setminus$ denotes set difference.


Order of Alternating Group

Let $n \in \Z$ be an integer such that $n > 1$. Let $A_n$ be the alternating group on $n$ letters. Then: :$\order {A_n} = \dfrac {n!} 2$ where $\order {A_n}$ denotes the order of $A_n$.


Plane contains Infinite Number of Lines

A plane contains an infinite number of distinct lines.


Three Non-Collinear Planes have One Point in Common

Three planes which are not collinear have exactly one point in all three planes.


Union of Mappings which Agree is Mapping

Let $A, B, Y$ be sets. Let $f: A \to Y$ and $g: B \to Y$ be mappings. Let $X = A \cup B$. Let $f$ and $g$ agree on $A \cap B$. Then $f \cup g: X \to Y$ is a mapping.


Right Operation is Idempotent

The right operation is idempotent: :$\forall x: x \rightarrow x = x$


Index of Intersection of Subgroups/Corollary

Let $H$ be a subgroup of $G$. Let $K$ be a subgroup of finite index of $G$. Then: :$\index H {H \cap K} \le \index G K$


Centralizer of Self-Inverse Element of Non-Abelian Finite Simple Group is not That Group

Let $G$ be a non-abelian finite simple group. Let $t \in G$ be a self-inverse element of $G$. Then: :$\map {C_G} t \ne G$ where $\map {C_G} t$ denotes the centralizer of $t$ in $G$.


Sum of Sequence of Squares of Fibonacci Numbers

:$\forall n \ge 1: \displaystyle \sum_{j \mathop = 1}^n {F_j}^2 = F_n F_{n + 1}$ That is: :${F_1}^2 + {F_2}^2 + {F_3}^2 + \cdots + {F_n}^2 = F_n F_{n + 1}$


Properties of Norm on Division Ring/Norm of Integer

For all $n \in \N_{\gt 0}$, let $n \cdot 1_R$ denote the sum of $1_R$ with itself $n$-times. That is: :$n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{n \, times}$ Then: :$\norm {n \cdot 1_R} \le n$.


Equivalence Relation is Congruence for Constant Operation

Every equivalence relation is a congruence relation for the constant operation.


Quotient Structure on Subset Product

Let $\left({S, \circ}\right)$ be an algebraic structure. Let $\mathcal R$ be a congruence for $\circ$ on $S$. Then: :$\forall X, Y \in S / \mathcal R: X \circ_\mathcal P Y \subseteq X \circ_\mathcal R Y$ where: : $S / \mathcal R$ is the quotient of $S$ by $\mathcal R$ : $\circ_\mathcal P$ is the operation induced on $\mathcal P \left({S}\right)$ by $\circ$ : $\circ_\mathcal R$ is the operation induced on $S / \mathcal R$ by $\circ$


Even Power of 3 as Sum of Consecutive Positive Integers

Take the positive integers and group them in sets such that the $n$th set contains the next $3^n$ positive integers: :$\set 1, \set {2, 3, 4}, \set {5, 6, \ldots, 13}, \set {14, 15, \cdots, 40}, \ldots$ Let the $n$th such set be denoted $S_{n - 1}$, that is, letting $S_0 := \set 1$ be considered as the zeroth. Then the sum of all the elements of $S_n$ is $3^{2 n}$.


Exterior of Union of Singleton Rationals is Empty

Let $B_\alpha$ be the singleton containing the rational number $\alpha$. Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology $\tau_d$. Then the exterior in $\struct {\R, \tau_d}$ of the union of all $B_\alpha$ is the empty set: :$\displaystyle \paren {\bigcup_{\alpha \mathop \in \Q} B_\alpha}^e = \O$


Smallest 18 Primes in Arithmetic Sequence

The smallest $18$ primes in arithmetic sequence are: :$107\,928\,278\,317 + 9\,922\,782\,870 n$ for $n = 0, 1, \ldots, 16$.


Irrational Number Space is Non-Meager

Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is non-meager.


Quotient Structure of Inverse Completion

Let $\left({T, \circ'}\right)$ be an inverse completion of a commutative semigroup $\left({S, \circ}\right)$, where $C$ is the set of cancellable elements of $S$. Let $f: S \times C: T$ be the mapping defined as: :$\forall x \in S, y \in C: f \left({x, y}\right) = x \circ' y^{-1}$ Then the mapping $g: \left({S \times C}\right) / \mathcal R_f \to T$ defined by $g \left({\left[\!\left[{x, y}\right]\!\right]_{\mathcal R_f}}\right) = x \circ' y^{-1}$, where $\left({S \times C}\right) / \mathcal R_f$ is a quotient structure, is an isomorphism.


Square Matrices with +1 or -1 Determinant under Multiplication forms Group

Let $n \in \Z_{> 0}$ be a strictly positive integer. Let $S$ be the set of square matrices of order $n$ of real numbers whose determinant is either $1$ or $-1$. Let $\struct {S, \times}$ denote the algebraic structure formed by $S$ whose operation is (conventional) matrix multiplication. Then $\struct {S, \times}$ is a group.


Rational Number Space is not Weakly Sigma-Locally Compact

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is not weakly $\sigma$-locally compact.


Group of Order 3 is Unique

There exists exactly $1$ group of order $3$, up to isomorphism: :$C_3$, the cyclic group of order $3$.


Integer Reciprocal Space is Topological Space

Let $\struct {\R, \tau_d}$ be the real number line $\R$ under the usual (Euclidean) topology $\tau_d$. Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set {\dfrac 1 n: n \in \Z_{>0} }$ Then the integer reciprocal space $\struct {A, \tau_d}$ is a topological space.


Integer Reciprocal Space with Zero is not Extremally Disconnected

Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology. Then $A$ is not extremally disconnected.


Components of Integer Reciprocal Space with Zero are Single Points

Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology. Then the components of $A$ are singletons.


Non-Homeomorphic Sets may be Homeomorphic to Subsets of Each Other

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $H_1 \subseteq S_1$ and $H_2 \subseteq S_2$. Then it is possible for: :$(1): \quad T_1$ to be homeomorphic to $H_2$ :$(2): \quad T_2$ to be homeomorphic to $H_1$ but: :$(3): \quad T_1$ and $T_2$ to not be homeomorphic.


Superspace of Homeomorphic Subspaces may not have Homeomorphism to Itself containing Subspace Homeomorphism

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $H_1 \subseteq S_1$ and $H_2 \subseteq S_2$. Let $H_1$ and $H_2$ be a homeomorphic. Then it may be the case that there does not exist a homeomorphism $g: T_1 \to T_2$ such that: :$g \restriction_{H_1} = f$ where: :$g \restriction_{H_1}$ is the restriction of $g$ to $H_1$ :$f: H_1 \to H_2$ is a homeomorphism.


Finite Group is p-Group iff Order is Power of p

Let $p$ be a prime number. Let $G$ be a finite group. Then $G$ is a $p$-group {{iff}} the order of $G$ is a power of $p$.


Trivial Ring from Abelian Group

Any abelian group $\struct {G, +}$ may be turned into a trivial ring by defining the ring product to be: :$\forall x, y \in G: x \circ y = e_G$


Product with Inverse on Homomorphic Image is Group Homomorphism

Let $G$ be a group. Let $H$ be an abelian group. Let $\theta: G \to H$ be a (group) homomorphism. Let $\phi: G \times G \to H$ be the mapping defined as: :$\forall \tuple {g_1, g_2} \in G \times G: \map \phi {g_1, g_2} = \map \theta {g_1} \map \theta {g_2}^{-1}$ Then $\phi$ is a homomorphism.


Divisor of Product may not be Divisor of Factors

Let $a, b, c \in \Z_{>0}$ be (strictly) positive integers. Let: :$c \divides a b$ where $\divides$ expresses the relation of divisibility. Then it is not necessarily the case that either $c \divides a$ or $c \divides b$.


Equality of Complex Numbers

Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be complex numbers. Then $z_1 = z_2$ {{iff}} $a_1 = a_2$ and $b_1 = b_2$.


Divisor of Unit is Unit

Let $\struct {D, +, \circ}$ be an integral domain whose unity is $1_D$. Let $\struct {U_D, \circ}$ be the group of units of $\struct {D, +, \circ}$. Then: :$x \in D, u \in U_D: x \divides u \implies x \in U_D$ That is, if $x$ is a divisor of a unit, $x$ must itself be a unit.


Associatehood is Equivalence Relation

Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $\sim$ be the relation defined on $D$ as: $\forall x, y \in D: x \sim y$ {{iff}} $x$ is an associate of $y$ Then $\sim$ is an equivalence relation.


Field is Subfield of Itself

Let $\struct {F, +, \circ}$ be a field. Then $\struct {F, +, \circ}$ is a subfield of $\struct {F, +, \circ}$.


Tangent Exponential Formulation/Formulation 1

:$\tan z = i \dfrac {1 - e^{2 i z} } {1 + e^{2 i z} }$


Tangent Exponential Formulation/Formulation 3

:$\tan z = -i \paren {\dfrac {e^{i z} - e^{-i z} } {e^{i z} + e^{-i z} } }$


Roots of Complex Number/Exponential Form

:$z^{1 / n} = \set {r^{1 / n} e^{i \paren {\theta + 2 \pi k} / n}: k \in \set {0, 1, 2, \ldots, n - 1} }$


Normalizer of Center is Group

Let $G$ be a group. Let $\map Z G$ be the center of $G$. Let $x \in G$. Let $\map {N_G} x$ be the normalizer of $x$ in $G$. Then: :$\map Z G = \set {x \in G: \map {N_G} x = G}$ That is, the center of a group $G$ is the set of elements $x$ of $G$ such that the normalizer of $x$ is the whole of $G$. Thus: :$x \in \map Z G \iff \map {N_G} x = G$ and so: :$\index G {\map {N_G} x} = 1$ where $\index G {\map {N_G} x}$ is the index of $\map {N_G} x$ in $G$.


Characterisation of Non-Archimedean Division Ring Norms/Corollary 4

Let $R$ have characteristic $p > 0$. Then $\norm {\,\cdot\,}$ is a non_Archimedean norm on $R$.


Relations with Combinations of Reflexivity, Symmetry and Transitivity Properties

Let $S$ be a set which has at least $3$ elements. Then it is possible to set up a relation $\circledcirc$ on $S$ which has any combination of the $3$ properties: :Reflexivity :Symmetry :Transitivity but this is not possible for a set which has fewer than $3$ elements.


Half Angle Formulas/Hyperbolic Tangent/Corollary 1

:$\tanh \dfrac x 2 = \dfrac {\sinh x} {\cosh x + 1}$


Even Integer Plus 5 is Odd

Let $x \in \Z$ be an even integer. Then $x + 5$ is odd.


Minimal Smooth Surface Spanned by Contour

Let $\map z {x, y}: \R^2 \to \R$ be a real-valued function. Let $\Gamma$ be a closed contour in $3$-dimensional Euclidean space. Suppose this surface is smooth for every $x$ and $y$. Then it has to satisfy the following Euler's equation: :$r \paren {1 + q^2} - 2 s p q + t \paren {1 + p^2} = 0$ where: {{begin-eqn}} {{eqn | l = p       | r = z_x }} {{eqn | l = q       | r = z_y }} {{eqn | l = r       | r = z_{xx} }} {{eqn | l = s       | r = z_{xy} }} {{eqn | l = t       | r = z_{yy} }} {{end-eqn}} with subscript denoting respective partial derivatives. In other words, its mean curvature has to vanish.


Principle of Induction applied to Interval of Naturally Ordered Semigroup

Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup. Let $\left[{p \,.\,.\, q}\right]$ be a closed interval of $\left({S, \circ, \preceq}\right)$. Let $T \subseteq \left[{p \,.\,.\, q}\right]$ such that the minimal element of $\left[{p \,.\,.\, q}\right]$ is in $T$. Let: : $x \in T: x \prec q \implies x \circ 1 \in T$ Then: : $T = \left[{p \,.\,.\, q}\right]$


Lowest Common Multiple of Integers with Common Divisor

Let $b, d \in \Z_{>0}$ be (strictly) positive integers Then: :$\lcm \set {a b, a d} = a \lcm \set {b, d}$ where: :$a \in \Z_{>0}$ :$\lcm \set {b, d}$ denotes the lowest common multiple of $m$ and $n$.


GCD of Sum and Difference of Integers

:$\gcd \set {a + b, a - b} \ge \gcd \set {a, b}$


Inverse Hyperbolic Cotangent is Odd Function

:$\map {\coth^{-1} } {-x} = -\coth^{-1} x$


Inverse Cosine of Imaginary Number

:$\cos^{-1} x = \pm \, i \cosh^{-1} x$


Inverse Tangent of Imaginary Number

:$\tan^{-1} \left({i x}\right) = i \tanh^{-1} x$


LCM of 3 Integers in terms of GCDs of Pairs of those Integers

Let $a, b, c \in \Z_{>0}$ be strictly positive integers. Then: :$\lcm \set {a, b, c} = \dfrac {a b c \gcd \set {a, b, c} } {d_1 d_2 d_3}$ where: :$\gcd$ denotes greatest common divisor :$\lcm$ denotes lowest common multiple :$d_1 = \gcd \set {a, b}$ :$d_2 = \gcd \set {b, c}$ :$d_3 = \gcd \set {a, c}$


Integer Multiplication is Well-Defined

Integer multiplication is well-defined.


Inverse Hyperbolic Secant of Imaginary Number

:$\sech^{-1} x = \pm \, i \sec^{-1} x$


Sum of Sequence of Product of Fibonacci Number with Binomial Coefficient

Let $F_n$ denote the $n$th Fibonacci number. Then: {{begin-eqn}} {{eqn | lo= \forall n \in \Z_{>0}:       | l = F_{2 n}       | r = \sum_{k \mathop = 1}^n \dbinom n k F_k       | c =  }} {{eqn | r = \dbinom n 1 F_1 + \dbinom n 2 F_2 + \dbinom n 3 F_3 + \dotsb + \dbinom n {n - 1} F_{n - 1} + \dbinom n n F_n       | c =  }} {{end-eqn}} where $\dbinom n k$ denotes a binomial coefficient.


Cardano's Formula/Real Coefficients

:$(1): \quad$ If $D > 0$, then one root is real and two are complex conjugates. :$(2): \quad$ If $D = 0$, then all roots are real, and at least two are equal. :$(3): \quad$ If $D < 0$, then all roots are real and unequal.


Cardano's Formula/Trigonometric Form

Let $a, b, c, d \in \R$. Let the discriminant $D < 0$, where $D := Q^3 + R^2$. Then the solutions of $P$ can be expressed as: :$x_1 = 2 \sqrt {-Q} \map \cos {\dfrac \theta 3} - \dfrac b {3 a}$ :$x_2 = 2 \sqrt {-Q} \map \cos {\dfrac \theta 3 + \dfrac {2 \pi} 3} - \dfrac b {3 a}$ :$x_3 = 2 \sqrt {-Q} \map \cos {\dfrac \theta 3 + \dfrac {4 \pi} 3} - \dfrac b {3 a}$ where: : $\cos \theta = \dfrac R {\sqrt{-Q^3} }$


Triangle Inequality/Complex Numbers/General Result

Let $z_1, z_2, \dotsc, z_n \in \C$ be complex numbers. Let $\cmod z$ be the modulus of $z$. Then: :$\cmod {z_1 + z_2 + \dotsb + z_n} \le \cmod {z_1} + \cmod {z_2} + \dotsb + \cmod {z_n}$


Substitution for Equivalent Subformula is Equivalent

Let $\mathbf B$ a WFF of propositional logic. Let $\mathbf A, \mathbf A'$ be equivalent WFFs. Let $\mathbf A$ be a subformula of $\mathbf B$. Let $\mathbf B' = \mathbf B \left({\mathbf A \,//\, \mathbf A'}\right)$ be the substitution of $\mathbf A'$ for $\mathbf A$ in $\mathbf B$. Then $\mathbf B$ and $\mathbf B'$ are equivalent.


Congruence Modulo Power of p as Linear Combination of Congruences Modulo p

Let $p$ be a prime number. Let $S = \set {a_1, a_2, \ldots, a_p}$ be a complete residue system modulo $p$. Then for all integers $n \in \Z$ and non-negative integer $s \in \Z_{\ge 0}$, there exists a congruence of the form: :$n \equiv \displaystyle \sum_{j \mathop = 0}^s b_j p^j \pmod {p^{s + 1} }$ where $b_j \in S$.


Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 62 Cards

Let $D$ be a deck of $62$ cards. Let $D$ be given a sequence of modified perfect faro shuffles. Then after $6$ such shuffles, the cards of $D$ will be in the same order they started in.


Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 12 Cards

Let $D$ be a deck of $12$ cards. Let $D$ be given a sequence of modified perfect faro shuffles. Then after $6$ such shuffles, the cards of $D$ will be in the same order they started in.


Exclusive Or with Tautology

:$p \oplus \top \dashv \vdash \neg p$


Difference of Two Even-Times Odd Powers

Let $\F$ be one of the standard number systems, that is $\Z, \Q, \R$ and so on. Let $n \in \Z_{> 0}$ be a (strictly) positive odd integer. Then: {{begin-eqn}} {{eqn | l = a^{2 n} - b^{2 n}       | r = \paren {a - b} \paren {a + b} \paren {\sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j} \paren {\sum_{j \mathop = 0}^{n - 1} \paren {-1}^j a^{n - j - 1} b^j}       | c =  }} {{eqn | r = \paren {a - b} \paren {a + b} \paren {a^{n - 1} + a^{n - 2} b + a^{n - 3} b^2 + \dotsb + a b^{n - 2} + b^{n - 1} } \paren {a^{n - 1} - a^{n - 2} b + a^{n - 3} b^2 - \dotsb - a b^{n - 2} + b^{n - 1} }       | c =  }} {{end-eqn}}


Existence of Real Polynomial with no Real Root

There exist polynomials in real numbers $\R$ which have no roots in $\R$.


GCD from Congruence Modulo m

Let $a, b \in \Z, m \in \N$.   Let $a$ be congruent to $b$ modulo $m$. Then the GCD of $a$ and $m$ is equal to the GCD of $b$ and $m$. That is: :$a \equiv b \pmod m \implies \gcd \set {a, m} = \gcd \set {b, m}$


Equivalence of Definitions of Closed Set in Metric Space

{{TFAE|def = Closed Set (Metric Space)|view = Closed Set|context = Metric Space|contextview = Metric Spaces}} Let $M = \left({A, d}\right)$ be a metric space. Let $H \subseteq A$.


Bolzano-Weierstrass Theorem/General Form

Every infinite bounded space in a real Euclidean space has at least one limit point.


Sum of Euler Phi Function over Divisors

Let $n \in \Z_{>0}$ be a strictly positive integer. Then $\displaystyle \sum_{d \mathop \divides n} \map \phi d = n$ where: :$\displaystyle \sum_{d \mathop \divides n}$ denotes the sum over all of the divisors of $n$ :$\map \phi d$ is the Euler $\phi$ function, the number of integers less than $d$ that are prime to $d$. That is, the total of all the totients of all divisors of a number equals that number.


Valuation Ideal is Maximal Ideal of Induced Valuation Ring/Corollary 1

:$\OO$ is a local ring.


Bessel's Correction

Let $X_1, X_2, \ldots, X_n$ form a random sample from a population with mean $\mu$ and variance $\sigma^2$. Let:  :$\displaystyle \bar X = \frac 1 n \sum_{i \mathop = 1}^n X_i$ Then: :$\displaystyle \hat {\sigma^2} = \frac 1 {n - 1} \sum_{i \mathop = 1}^n \paren {X_i - \bar X}^2$ is an unbiased estimator of $\sigma^2$.


Sum of Cosines of Fractions of Pi

Let $n \in \Z$ such that $n > 1$. Then: :$\displaystyle \sum_{k \mathop = 1}^{n - 1} \cos \frac {2 k \pi} n = -1$


Area of Quadrilateral in Determinant Form

Let $A = \tuple {x_1, y_1}$, $B = \tuple {x_2, y_2}$, $C = \tuple {x_3, y_3}$ and $D = \tuple {x_4, y_4}$ be points in the Cartesian plane. Let $A$, $B$, $C$ and $D$ form the vertices of a quadrilateral. The area $\mathcal A$ of $\Box ABCD$ is given by: :$\mathcal A = \dfrac 1 2 \paren {\size {\paren {\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix} } } + \size {\paren {\begin{vmatrix} x_1 & y_1 & 1 \\ x_4 & y_4 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix} } } }$


Equation for Perpendicular Bisector of Two Points

Let $\tuple {x_1, y_1}$ and $\tuple {y_1, y_2}$ be two points in the cartesian plane. Let $L$ be the perpendicular bisector of the straight line through $z_1$ and $z_2$ in the complex plane. $L$ can be expressed by the equation: :$y - \dfrac {y_1 + y_2} 2 = \dfrac {x_1 - x_2} {y_2 - y_1} \paren {x - \dfrac {x_1 + x_2} 2}$


Finite Order Elements of Infinite Abelian Group form Normal Subgroup

Let $G$ be an infinite  abelian group. Let $H \subseteq G$ be the subset of $G$ defined as: :$H := \set {x \in G: x \text { is of finite order in } G}$ Then $H$ forms a normal subgroup of $G$.


Subgroup of Index 3 does not necessarily contain all Cubes of Group Elements

Let $G$ be a group. Let $H$ be a subgroup of $G$ whose index is $3$. Then it is not necessarily the case that: :$\forall x \in G: x^3 \in H$


Complex Cross Product Distributes over Addition

Let $z_1, z_2, z_3 \in \C$ be complex numbers. Then: :$z_1 \times \paren {z_2 + z_3} = z_1 \times z_2 + z_1 \times z_3$ where $\times$ denotes cross product.


Finite Union of Open Sets in Complex Plane is Open

Let $S_1, S_2, \ldots, S_n$ be open sets of $\C$. Then $\displaystyle \bigcup_{k \mathop = 1}^n S_k$ is an open set of $\C$.


Limit Point of Set in Complex Plane not Element is Boundary Point

Let $S \subseteq \C$ be a subset of the complex plane. Let $z \in \C$ be a limit point of $S$ such that $z \notin S$. Then $z$ is a boundary point of $S$.


Order of Isomorphic Image of Group Element

Let $G$ and $H$ be groups whose identities are $e_G$ and $e_H$. Let $\phi: G \to H$ be a group isomorphism. Then: :$a \in G \implies \order {\map \phi a} = \order a$


Ring is Subring of Itself

Let $R$ be a ring. Then $R$ is a subring of itself.


Cyclic Group Elements whose Powers equal Identity

Let $G$ be a cyclic group whose identity is $e$ and whose order is $n$. Let $d \divides n$. Then there exist exactly $d$ elements $x \in G$ satisfying the equation $x^d = e$. These are the elements of the group $G_d$ generated by $g^{n / d}$:  :$G_d = \gen {g^{n / d} }$


Conditional iff Biconditional of Antecedent with Conjunction

:$p \implies q \dashv \vdash p \iff \left({p \land q}\right)$


Element in Integral Domain is Unit iff Principal Ideal is Whole Domain

:$x \in U_D \iff \ideal x = D$


Conjunction iff Biconditional of Biconditional with Disjunction

:$p \land q \dashv \vdash \left({p \iff q}\right) \iff \left({p \lor q}\right)$


Group Generated by Normal Intersection is Normal

Let $I$ be an indexing set, and $\left\{{N_i: i \in I}\right\}$ be a set of normal subgroups of the group $G$. Then $\left \langle {N_i: i \in I} \right \rangle$ is a normal subgroup of $G$.


Abelian Quotient Group

Let $G$ be a group. Let $H$ be a normal subgroup of $G$. Let $G / H$ denote the quotient group of $G$ by $H$. Then $G / H$ is abelian {{iff}} $H$ contains every element of $G$ of the form $a b a^{-1} b^{-1}$ where $a, b \in G$.


Generator of Quotient Groups

Let $N \lhd G$ be a normal subgroup of $G$. Let: :$N \le A \le G$ :$N \le B \le G$ For a subgroup $H$ of $G$, let $\alpha$ be the bijection defined as: :$\map \alpha H = \set {h N: h \in H}$ Then: :$\map \alpha {\gen {A, B} } = \gen {\map \alpha A, \map \alpha B}$ where $\gen {A, B}$ denotes the subgroup of $G$ generated by $\set {A, B}$.


Inner Automorphism Group is Isomorphic to Quotient Group with Center

Let $G$ be a group. Let $\Inn G$ be the inner automorphism group of $G$. Let $\map Z G$ be the center of $G$. Let $G / \map Z G$ be the quotient group of $G$ by $\map Z G$. Then $G / \map Z G \cong \Inn G$.


Real Natural Logarithm is Restriction of Complex Natural Logarithm

Let $\ln: \C_{\ne 0} \to \C$ be the complex natural logarithm. Let $\ln': \R_{>0} \to \R$ be the real natural logarithm. Then: :$\ln' = \ln \restriction_{\R_{>0} \times \R}$ That is, the real natural logarithm is the restriction of the complex natural logarithm.


Inverse of Central Unit of Ring is in Center

Let $R$ be a ring. Let $\map Z R$ denote the center of $R$. Let $u \in R$ be a unit of $R$. Then: :$u \in \map Z R \implies u^{-1} \in \map Z R$


Commutativity of Group Direct Product

Let $\struct {G, \circ_g}$ and $\struct {H, \circ_h}$ be groups. Let $\struct {G \times H, \circ}$ be the group direct product of $\struct {G, \circ_g}$ and $\struct {H, \circ_h}$, where the operation $\circ$ is defined as: :$\tuple {g_1, h_1} \circ \tuple {g_2, h_2} = \tuple {g_1 \circ_g g_2, h_1 \circ_h h_2}$ Let $\struct {H \times G, \star}$ be the group direct product of $\struct {H, \circ_h}$ and $\struct {G, \circ_g}$, where the operation $\star$ is defined as: :$\tuple {h_1, g_1} \star \tuple {h_2, g_2} = \tuple {h_1 \circ_h h_2, g_1 \circ_g g_2}$ The group direct product $\struct {G \times H, \circ}$ is isomorphic to the $\struct {H \times G, \star}$.


Associativity of Group Direct Product

The group direct product $G \times \paren {H \times K}$ is (group) isomorphic to $\paren {G \times H} \times K$.


Cyclic Group of Order 6

Let $C_n$ be the cyclic group of order $n$. Then: : $C_2 \times C_3 \cong C_6$ : $C_6$ is the internal group direct product of $C_2$ and $C_3$.


Direct Product of Central Subgroup with Inverse Isomorphism is Central Subgroup

Let $G$ and $H$ be groups. Let $\map Z G$ denote the center of $G$. Let $Z$ and $W$ be central subgroups of $G$ and $H$ respectively. Let: :$Z \cong W$ where $\cong$ denotes isomorphism. Let such a group isomorphism be $\theta: Z \to W$. Let $X$ be the set defined as: :$X = \set {\tuple {x, \map \theta x^{-1} }: x \in Z}$ Then $X$ is a central subgroup of $G \times H$.


Non-Commutative Ring with Unity and 2 Ideals not necessarily Division Ring

Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_F$ and whose unity is $1_F$. Let $\struct {R, +, \circ}$ specifically not be commutative. Let $\struct {R, +, \circ}$ be such that the only ideals of $\struct {R, +, \circ}$ are $\set {0_R}$ and $R$ itself. Then it is not necessarily the case that $\struct {R, +, \circ}$ is a division ring.


Equivalence of Definitions of Complex Inverse Hyperbolic Tangent

{{TFAE|def = Complex Inverse Hyperbolic Tangent}} Let $S$ be the subset of the complex plane: :$S = \C \setminus \left\{{-1 + 0 i, 1 + 0 i}\right\}$


Natural Numbers form Inductive Set

Let $\N$ denote the natural numbers as subset of the real numbers $\R$. Then $\N$ is an inductive set.


Additive Group of Reals is Normal Subgroup of Complex

Let $\struct {\R, +}$ be the additive group of real numbers. Let $\struct {\C, +}$ be the additive group of complex numbers. Then $\struct {\R, +}$ is a normal subgroup of $\struct {\C, +}$.


Generators of Additive Group of Integers

The only generators of the additive group of integers $\struct {\Z, +}$ are $1$ and $-1$.


Prime Power Mapping on Galois Field is Automorphism

Let $\GF$ be a Galois field whose zero is $0_\GF$ and whose characteristic is $p$. Let $\sigma: \GF \to \GF$ be defined as: :$\forall x \in \GF: \map \sigma x = x^p$ Then $\sigma$ is an automorphism of $\GF$.


Principal Ideal Domain cannot have Infinite Strictly Increasing Sequence of Ideals

Let $\struct {D, +, \circ}$ be a principal ideal domain. Then $D$ cannot have an infinite sequence of ideals $\sequence {j_n}_{n \mathop \in \N}$ such that: :$\forall n \in \N: J_n \subsetneq j_{n + 1}$


Field Norm of Complex Number is not Norm

Let $\C$ denote the set of complex numbers. Let $N: \C \to \R_{\ge 0}$ denote the field norm on complex numbers: :$\forall z \in \C: \map N z = \cmod z^2$ where $\cmod z$ denotes the complex modulus of $z$. Then $N$ is not a norm on $\C$.


Internal Angles of Square

The internal angles of a square are right angles.


Ideal of Ring of Polynomials over Field has Unique Monic Polynomial forming Principal Ideal

Let $F$ be a field. Let $F \sqbrk X$ be the ring of polynomials in $X$ over $F$. Let $J$ be a non-null ideal of $F \sqbrk X$. Then there exists exactly one monic polynomial $f \in F \sqbrk X$ such that: :$J = \ideal f$ where $\ideal f$ is the principal ideal generated by $f$ in $F \sqbrk X$.


Fixed Elements form 1-Cycles

Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi \in S_n$. Let $\Fix \pi$ be the set of elements fixed by $\pi$. For any $\pi \in S_n$, all the elements of $\Fix \pi$ form $1$-cycles.


Polynomial X^2 + 1 is Irreducible in Ring of Real Polynomials

Let $\R \sqbrk X$ be the ring of polynomials in $X$ over the real numbers $\R$. Then the polynomial $X^2 + 1$ is an irreducible element of $\R \sqbrk X$.


Equivalence of Definitions of Concave Real Function

Let $f$ be a real function which is defined on a real interval $I$. {{TFAE|def = Concave Real Function}}


Ring Subtraction equals Zero iff Elements are Equal

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$ Then: :$\forall a, b \in R: a - b = 0_R \iff a = b$ where $a - b$ denotes ring subtraction.


Powers of Disjoint Permutations

Let $S_n$ denote the symmetric group on $n$ letters. Let $\rho, \sigma$ be disjoint permutations. Then: : $\forall k \in \Z: \paren {\sigma \rho}^k = \sigma^k \rho^k$


Equivalence of Definitions of Strictly Concave Real Function

Let $f$ be a real function which is defined on a real interval $I$. {{TFAE|def = Strictly Concave Real Function}}


Equivalence of Definitions of Strictly Convex Real Function

Let $f$ be a real function which is defined on a real interval $I$. {{TFAE|def = Strictly Convex Real Function}}


Group Action on Subgroup of Symmetric Group

Let $S_n$ be the symmetric group of $n$ elements. Let $H$ be a subgroup of $S_n$. Let $X$ be any set with $n$ elements. Then $H$ acts on $X$ as a group of transformations on $X$.


Kernel of Group Action is Normal Subgroup

Let $G$ be a group whose identity is $e$. Let $X$ be a set. Let $\phi: G \times X \to X$ be a group action. Let $G_0$ denote the kernel of $\phi$. Then $G_0$ is a normal subgroup of $G$.


Equivalence of Definitions of Well-Ordered Integral Domain

{{TFAE|def = Well-Ordered Integral Domain}} Let $\struct {D, +, \times \le}$ be an ordered integral domain whose zero is $0_D$.


Second Principle of Finite Induction/Zero-Based

Let $S \subseteq \N$ be a subset of the natural numbers. Suppose that: :$(1): \quad 0 \in S$ :$(2): \quad \forall n \in \N: \paren {\forall k: 0 \le k \le n \implies k \in S} \implies n + 1 \in S$ Then: :$S = \N$


Integral of Exponent of Half Square over Reals

:$\displaystyle \int_{\mathop \to -\infty}^{\mathop \to +\infty} e^{- x^2 / 2} \rd x = \sqrt {2 \pi}$


Cayley's Representation Theorem

Let $S_n$ denote the symmetric group on $n$ letters. Every finite group is isomorphic to a subgroup of $S_n$ for some $n \in \Z$.


Coprimality Relation is not Antisymmetric

:$\perp$ is not antisymmetric.


Coprimality Relation is Non-Transitive

:$\perp$ is non-transitive.


Canonical Injection into Cartesian Product of Modules

Let $G$ be the cartesian product of a sequence $\sequence {G_n}$ of $R$-modules. Then for each $j \in \closedint 1 n$, the canonical injection $\inj_j$ from $G_j$ into $G$ is a monomorphism.


Gaussian Integers does not form Subfield of Complex Numbers

The ring of Gaussian integers: :$\struct {\Z \sqbrk i, +, \times}$ is not a subfield of $\C$.


Even Integers form Commutative Ring

Let $2 \Z$ be the set of even integers. Then $\struct {2 \Z, +, \times}$ is a commutative ring. However, $\struct {2 \Z, +, \times}$ is not an integral domain.


Leibniz's Rule/One Variable/Second Derivative

Let $f$ and $g$ be real functions defined on the open interval $I$. Let $x \in I$ be a point in $I$ at which both $f$ and $g$ are twice differentiable. Then: :$\paren {\map f x \, \map g x}'' = \map f x \, \map {g''} x + 2 \map {f'} x \, \map {g'} x + \map {f''} x \, \map g x$


Vector Space on Cartesian Product is Vector Space

Let $\struct {K, +, \circ}$ be a division ring. Let $n \in \N_{>0}$. Let $\struct {K^n, +, \times}_K$ be the '''$K$-vector space $K^n$'''. Then $\struct {K^n, +, \times}_K$ is a $K$-vector space.


Ring of Linear Operators

Let $\map {\LL_R} G$ be the set of all linear operators on $G$. {{explain|the precise nature of $G$}} Let $\phi \circ \psi$ denote the composition of the two linear operators $\phi$ and $\psi$. Then $\struct {\map {\LL_R} G, +, \circ}$ is a ring.


No Non-Trivial Norm on Rational Numbers is Complete

No non-trivial norm on the set of the rational numbers is complete.


Inverse Evaluation Isomorphism of Annihilator

Let $R$ be a commutative ring.  Let $G$ be a module over $R$ whose dimension is finite. Let $G^*$ be the algebraic dual of $G$. Let $G^{**}$ be the algebraic dual of $G^*$. Let $N$ be a submodule of $G^*$. Let $J$ be the evaluation isomorphism from $G$ onto $G^{**}$. Let $N^\circ$ be the annihilator of $N$. Then: :$J^{-1} \left({N^\circ}\right) = \left\{{x \in G: \forall t' \in N: t' \left({x}\right) = 0}\right\}$


Complex Numbers form Vector Space over Reals

Let $\R$ be the set of real numbers. Let $\C$ be the set of complex numbers. Then the $\R$-module $\C$ is a vector space.


Dimension of Vector Space on Cartesian Product

Let $\struct {K, +, \circ}$ be a division ring. Let $n \in \N_{>0}$. Let $\mathbf V := \struct {K^n, +, \times}_K$ be the '''$K$-vector space $K^n$'''. Then the dimension of $\mathbf V$ is $n$.


Finitely Generated Vector Space has Basis

Let $K$ be a division ring. Let $V$ be a finitely generated vector space over $K$. Then $V$ has a finite basis.


Primitive of Hyperbolic Cosecant Function/Logarithm Form

:$\displaystyle \int \csch x \rd x = -\ln \size {\csch x + \coth x} + C$ where $\csch x + \coth x \ne 0$.


Primitive of Hyperbolic Cosecant Function/Inverse Hyperbolic Cotangent Form

:$\displaystyle \int \csch x \rd x = -2 \map {\coth^{-1} } {e^x} + C$


Primitive of Product of Hyperbolic Secant and Tangent

:$\ds \int \sech x \tanh x \rd x = -\sech x + C$ where $C$ is an arbitrary constant.


Existence of Subgroup whose Index is Prime Power

Let $G$ be a finite group. Let $H$ be a normal subgroup of $G$ which has a finite index in $G$. Let: :$p^k \divides \index G H$ where: :$p$ is a prime number :$k \in \Z_{>0}$ is a (strictly) positive integer :$\divides$ denotes divisibility. Then $G$ contains a subgroup $K$ such that: :$\index K H = p^k$


Condition for Planes to be Parallel

Let $P: \alpha_1 x_1 + \alpha_2 x_2 + \alpha_3 x_3 = \gamma$ be a plane in $\R^3$. Then the plane $P'$ is parallel to $P$ iff there is a $\gamma' \in \R$ such that: :$P' = \left\{{ \left({x_1, x_2, x_3}\right) \in \R^3 : \alpha_1 x_1 + \alpha_2 x_2 + \alpha_3 x_3 = \gamma' }\right\}$


Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2

:$\ds \int \frac {\d x} {x^2 - a^2} = \frac 1 {2 a} \ln \size {\frac {x - a} {x + a} } + C$


Invertible Matrix Corresponds with Change of Basis

Let $R$ be a commutative ring with unity. Let $G$ be an $n$-dimensional unitary $R$-module. Let $\left \langle {a_n} \right \rangle$ be an ordered basis of $G$. Let $\mathbf P = \left[{\alpha}\right]_{n}$ be a square matrix of order $n$ over $R$. Let $\displaystyle \forall j \in \left[{1 \,.\,.\, n}\right]: b_j = \sum_{i \mathop = 1}^n \alpha_{i j} a_i$. Then $\left \langle {b_n} \right \rangle$ is an ordered basis of $G$ iff $\mathbf P$ is invertible.


Matrix Similarity is Equivalence Relation

Matrix similarity is an equivalence relation.


Similar Matrices are Equivalent

If two square matrices over a ring with unity $R$ are similar, then they are equivalent. It follows directly that every equivalence class for the relation of similarity on $\mathcal M_R \left({n}\right)$ is contained in an equivalence class for the relation of matrix equivalence. Here, $\mathcal M_R \left({n}\right)$ denotes the $n \times n$ matrix space over $R$.


Rank is Dimension of Subspace

Let $K$ be a field. Let $\mathbf A$ be an $m \times n$ matrix over $K$. Then the rank of $\mathbf A$ is the dimension of the subspace of $K^n$ generated by the rows of $\mathbf A$.


Primitive of Function of Nth Root of a x + b

:$\displaystyle \int F \left({\sqrt [n] {a x + b}}\right) \ \mathrm d x = \frac n a \int u^{n-1} F \left({u}\right) \ \mathrm d u$ where $u = \sqrt [n] {a x + b}$.


Primitive of Function of Root of a squared plus x squared

:$\displaystyle \int F \left({\sqrt {a^2 + x^2}}\right) \ \mathrm d x = a \int \sec^2 u \ F \left({a \sec u}\right) \ \mathrm d u$ where $x = a \tan u$.


Primitive of Function of Natural Logarithm

:$\displaystyle \int F \left({\ln x}\right) \rd x = \int F \left({u}\right) e^u \rd u$ where $u = \ln x$.


Primitive of Function of Arccosecant

:$\displaystyle \int F \left({\operatorname{arccsc} \frac x a}\right) \ \mathrm d x = -a \int F \left({u}\right) \left\vert{\csc u}\right\vert \cot u \ \mathrm d u$ where $u = \operatorname{arccsc} \dfrac x a$.


Subtraction of Subring is Subtraction of Ring

Let $\struct {R, +, \circ}$ be an ring. For each $x, y \in R$ let $x - y$ denote the subtraction of $x$ and $y$ in $R$. Let $\struct {S, + {\restriction_S}, \circ {\restriction_S}}$ be a subring of $R$. For each $x, y \in S$ let $x \sim y$ denote the subtraction of $x$ and $y$ in $S$. Then: :$\forall x, y \in S: x \sim y = x - y$


Area between Smooth Curve and Line is Maximized by Semicircle

Let $y$ be a smooth curve, embedded in $2$-dimensional Euclidean space. Let $y$ have a total length of $l$. Let it be contained in the upper half-plane with an exception of endpoints, which are on the $x$-axis. Suppose, $y$, together with a line segment connecting $y$'s endpoints, maximizes the enclosed area. Then $y$ is a semicircle.


Unique Representation in Polynomial Forms

Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\struct {D, +, \circ}$ be an integral subdomain of $R$. Let $X \in R$ be transcendental over $D$. Let $D \sqbrk X$ be the ring of polynomials in $X$ over $D$. Then each non-zero member of $D \left[{X}\right]$ can be expressed in just one way in the form: :$\ds f \in D \sqbrk X: f = \sum_{k \mathop = 0}^n {a_k \circ X^k}$


Rings of Polynomials in Ring Elements are Isomorphic

Let $R_1, R_2$ be commutative rings with unity. Let $D$ be an integral subdomain of both $R_1$ and $R_2$. Let $X_1, X_2 \in R$ be transcendental over $D$. Let $D \sqbrk {X_1}, D \sqbrk {X_2}$ be the rings of polynomials in $X_1$ and $X_2$ over $D$. Then $D \sqbrk {X_1}$ is isomorphic to $D \sqbrk {X_2}$.


Laplace Transform of Higher Order Derivatives

{{begin-eqn}} {{eqn | l = \laptrans {\map {f^{\paren n} } t}       | r = s^n \laptrans {\map f t} - \sum_{j \mathop = 1}^n s^{j - 1} \map {f^{\paren {n - j} } } 0 }} {{eqn | r = s^n \map F s - s^{n - 1} \, \map f 0 - s^{n - 2} \, \map {f'} 0 - s^{n - 3} \, \map {f''} 0 - \ldots - s \, \map {f^{\paren {n - 2} } } 0 - \map {f^{\paren {n - 1} } } 0 }} {{end-eqn}}


Laplace Transform of Error Function

:$\laptrans {\map \erf t} = \dfrac 1 s \, \map \exp {\dfrac {s^2} 4} \, \map \erfc {\dfrac s 2}$ where: :$\laptrans f$ denotes the Laplace transform of the function $f$ :$\erf$ denotes the error function :$\erfc$ denotes the complementary error function :$\exp$ denotes the exponential function.


Laplace Transform of Exponential Integral Function

:$\laptrans {\map \Ei t} = \dfrac {\map \ln {s + 1} } s$ where: :$\laptrans f$ denotes the Laplace transform of the function $f$ :$\Ei$ denotes the exponential integral function.


Laplace Transform of Shifted Dirac Delta Function

Let $\map \delta t$ denote the Dirac delta function. The Laplace transform of $\map \delta {t - a}$ is given by: :$\laptrans {\map \delta {t - a} } = e^{-a s}$


Primitive of x over Root of a x + b

:$\displaystyle \int \frac {x \ \mathrm d x} {\sqrt{a x + b} } = \frac {2 \left({a x - 2 b}\right) \sqrt{a x + b} } {3 a^2}$


Laplace Transform of t by Sine a t

Let $\sin$ denote the real sine function. Let $\laptrans f$ denote the Laplace transform of a real function $f$. Then: :$\laptrans {t \sin a t} = \dfrac {2 a s} {\paren {s^2 + a^2}^2}$


Laplace Transform of Sine of t over t/Corollary

:$\laptrans {\dfrac {\sin a t} t} = \arctan \dfrac a s$


Laplace Transform of Natural Logarithm

:$\laptrans {\ln t} = \dfrac {\map {\Gamma'} 1 - \ln s} s = -\dfrac {\gamma + \ln s} s$ where: :$\laptrans f$ denotes the Laplace transform of the function $f$ :$\Gamma$ denotes the Gamma function :$\gamma$ denotes the Euler-Mascheroni constant.


Convolution Theorem

Let $\mathbb F \in \set {\R, \C}$.  Let $f: \R \to \F$ and $g: \R \to \F$ be functions. Let their Laplace transforms $\laptrans {\map f t} = \map F s$ and $\laptrans {\map g t} = \map G s$ exist. Then: :$\map F s \map G s = \displaystyle \laptrans {\int_0^t \map f u \map g {t - u} \rd u}$


Primitive of p x + q over Root of a x + b

:$\displaystyle \int \frac {p x + q} {\sqrt {a x + b} } \rd x = \frac {2 \paren {a p x + 3 a q - 2 b p} } {3 a^2} \sqrt{a x + b}$


Negative of Supremum is Infimum of Negatives

Let $S$ be a subset of the real numbers $\R$. Let $S$ be bounded above. Then: :$(1): \quad \set {x \in \R: -x \in S}$ is bounded below :$(2): \quad \displaystyle -\sup_{x \mathop \in S} x = \map {\inf_{x \mathop \in S} } {-x}$ where $\sup$ and $\inf$ denote the supremum and infimum respectively.


Propagation of Light in Inhomogeneous Medium

Let $v: \R^3 \to \R$ be a real function. Let $M$ be a 3-dimensional Euclidean space. Let $\gamma:t \in \R \to M$ be a smooth curve embedded in $M$, where $t$ is time. Denote its derivative {{WRT}} time by $v$. Suppose $M$ is filled with an optically inhomogeneous medium such that at each point speed of light is $v = \map v {x, y, z}$ Suppose $\map y x$ and $\map z x$ are real functions. Let the light move according to Fermat's principle. Then equations of motion have the following form: :$\dfrac {\partial v} {\partial y} \dfrac {\sqrt {1 + y'^2 + z'^2} } {v^2} + \dfrac \d {\d x} \dfrac {y'} {v \sqrt {1 + y'^2 + z'^2} } = 0$ :$\dfrac {\partial v} {\partial z} \dfrac {\sqrt {1 + y'^2 + z'^2} } {v^2} + \dfrac \d {\d x} \dfrac {z'} {v \sqrt {1 + y'^2 + z'^2} } = 0$


Primitive of Reciprocal of x by Power of x squared minus a squared

:$\displaystyle \int \frac {\mathrm d x} {x \left({x^2 - a^2}\right)^n} = \frac {-1} {2 \left({n - 1}\right) a^2 \left({x^2 - a^2}\right)^{n - 1} } - \frac 1 {a^2} \int \frac {\mathrm d x} {x \left({x^2 - a^2}\right)^{n - 1} }$ for $x^2 > a^2$.


Primitive of Reciprocal of Power of x by Power of x squared minus a squared

:$\displaystyle \int \frac {\d x} {x^m \paren {x^2 - a^2}^n} = \frac 1 {a^2} \int \frac {\d x} {x^{m - 2} \paren {x^2 - a^2}^n} - \frac 1 {a^2} \int \frac {\d x} {x^m \paren {x^2 - a^2}^{n - 1} }$ for $x^2 > a^2$.


Signum Function is Quotient of Number with Absolute Value

Let $x \in \R_{\ne 0}$ be a non-zero real number. Then: :$\operatorname{sgn} \left({x}\right) = \dfrac x {\left\vert{x}\right\vert} = \dfrac {\left\vert{x}\right\vert} x$ where: :$\operatorname{sgn} \left({x}\right)$ denotes the signum function of $x$ :$\left\vert{x}\right\vert$ denotes the absolute value of $x$.


Local Basis Generated from Neighborhood Basis

Let $T = \struct {S, \tau}$ be a topological space. Let $x$ be an element of $S$. Let $\BB$ be a neighborhood basis of $x$. For any subset $A \subseteq S$, let $A^\circ$ denote the interior of $A$. Then the set: :$\BB' = \set {H^\circ: H \in B}$ is a local basis of $x$.


Excess Kurtosis of Bernoulli Distribution

Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$. Then the excess kurtosis $\gamma_2$ of $X$ is given by:  :$\gamma_2 = \dfrac {1 - 6 p q} {p q}$ where $q = 1 - p$.


Skewness of Beta Distribution

Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname {Beta}$ denotes the Beta distribution.  Then the skewness $\gamma_1$ of $X$ is given by:  :$\gamma_1 = \dfrac {2 \paren {\beta - \alpha} \sqrt {\alpha + \beta + 1} } {\paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }$


Primitive of x over Power of a squared minus x squared

:$\displaystyle \int \frac {x \rd x} {\paren {a^2 - x^2}^n} = \frac 1 {2 \paren {n - 1} \paren {a^2 - x^2}^{n - 1} }$ for $x^2 < a^2$.


Primitive of Reciprocal of x by Power of a squared minus x squared

:$\displaystyle \int \frac {\mathrm d x} {x \left({a^2 - x^2}\right)^n} = \frac 1 {2 \left({n - 1}\right) a^2 \left({a^2 - x^2}\right)^{n - 1} } + \frac 1 {a^2} \int \frac {\mathrm d x} {x \left({a^2 - x^2}\right)^{n - 1} }$ for $x^2 < a^2$.


Primitive of x squared by Root of x squared plus a squared

:$\displaystyle \int x^2 \sqrt {x^2 + a^2} \rd x = \frac {x \paren {\sqrt {x^2 + a^2} }^3} 4 - \frac {a^2 x \sqrt {x^2 + a^2} } 8 - \frac {a^4} 8 \map \ln {x + \sqrt {x^2 + a^2} } + C$


Sum of Squares of Standard Gaussian Random Variables has Chi-Squared Distribution

Let $X_1, X_2, \ldots, X_n$ be independent random variables. Let $X_i \sim \Gaussian 0 1$ for $1 \le i \le n$ where $\Gaussian 0 1$ is the standard Gaussian Distribution. Then:  :$\displaystyle \sum_{i \mathop = 1}^n X^2_i \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.


Primitive of x squared over Root of x squared plus a squared cubed

:$\displaystyle \int \frac {x^2 \ \mathrm d x} {\left({\sqrt {x^2 + a^2} }\right)^3} = \frac {-x} {\sqrt {x^2 + a^2} } + \ln \left({x + \sqrt {x^2 + a^2} }\right) + C$


Primitive of Reciprocal of x squared by Root of x squared plus a squared cubed

:$\displaystyle \int \frac {\d x} {x^2 \paren {\sqrt {x^2 + a^2} }^3} = \frac {-\sqrt {x^2 + a^2} } {a^4 x} - \frac x {a^4 \sqrt {x^2 + a^2} } + C$


Variance of Student's t-Distribution

Let $k$ be a strictly positive integer.  Let $X \sim t_k$ where $t_k$ is the $t$-distribution with $k$ degrees of freedom. Then the variance of $X$ is given by:  :$\var X = \dfrac k {k - 2}$ for $k > 2$, and does not exist otherwise.


Primitive of Root of x squared plus a squared cubed over x

:$\ds \int \frac {\paren {\sqrt {x^2 + a^2} }^3} x \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^3} 3 + a^2 \sqrt {x^2 + a^2} - a^3 \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$


Primitive of Root of x squared plus a squared cubed over x squared

:$\displaystyle \int \frac{\left({\sqrt {x^2 + a^2} }\right)^3} {x^2} \ \mathrm d x = \frac {-\left({\sqrt {x^2 + a^2} }\right)^3} x + \frac{3 x \sqrt {x^2 + a^2} } 2 + \frac {3 a^2} 2 \ln \left({x + \sqrt {x^2 + a^2} }\right) + C$


Weak Whitney Immersion Theorem

Every $k$-dimensional manifold $X$ admits a one-to-one immersion in $\R^{2 k + 1}$. {{MissingLinks|$k$-dimensional}}


Primitive of Root of x squared plus a squared cubed over x cubed

:$\displaystyle \int \frac {\paren {\sqrt {x^2 + a^2} }^3} {x^3} \rd x = \frac {-\paren {\sqrt {x^2 + a^2} }^3} {2 x^2} + \frac {3 \sqrt {x^2 + a^2} } 2 - \frac {3 a} 2 \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$


Homotopy Group is Homeomorphism Invariant

Let $X$ and $Y$ be two topological spaces. Let $\phi: X \to Y$ be a homeomorphism. Let $x_0 \in X$, $y_0 \in Y$. Then for all $n \in \N$ the induced mapping: :$\phi_* : \pi_n \left({X, x_0}\right) \to \pi_n \left({Y, y_0}\right):$ ::$\left[{\!\left[{\, c \,}\right]\!}\right] \mapsto \left[{\!\left[{\, \phi \circ c \,}\right]\!}\right]$ is an isomorphism, where $\pi_n$ denotes the $n$th homotopy group.


Primitive of x cubed by Root of x squared minus a squared

:$\displaystyle \int x^3 \sqrt {x^2 - a^2} \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^5} 5 + \frac {a^2 \paren {\sqrt {x^2 - a^2} }^3} 3 + C$


Primitive of Reciprocal of x squared by Root of x squared minus a squared cubed

:$\displaystyle \int \frac {\d x} {x^2 \paren {\sqrt {x^2 - a^2} }^3} = \frac {-\sqrt {x^2 - a^2} } {a^4 x} - \frac x {a^4 \sqrt {x^2 - a^2} } + C$


Primitive of x squared by Root of x squared minus a squared cubed

:$\displaystyle \int x^2 \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {x \paren {\sqrt {x^2 - a^2} }^5} 6 + \frac {a^2 x \paren {\sqrt {x^2 - a^2} }^3} {24} - \frac {a^4 x \sqrt {x^2 - a^2} } {16} + \frac {a^6} {16} \ln \size {x + \sqrt {x^2 - a^2} } + C$


Binomial Coefficient 2 n Choose n is Divisible by All Primes between n and 2 n

Let $\dbinom {2 n} n$ denote a binomial coefficient. Then for all prime numbers $p$ such that $n < p < 2 n$: :$p \divides \dbinom {2 n} n$ where $\divides$ denotes divisibility.


Linear Combination of Gaussian Random Variables

Let $X_1, X_2, X_3, \ldots, X_n$ be independent random variables.  Let $\sequence {\alpha_i}_{1 \le i \le n}$ and $\sequence {\mu_i}_{1 \le i \le n}$ be sequences of real numbers. Let $\sequence {\sigma_i}_{1 \le i \le n}$ be a sequence of positive real numbers. Let $X_i \sim \Gaussian {\mu_i} {\sigma^2_i}$ for $1 \le i \le n$, where $\Gaussian {\mu_i} {\sigma^2_i}$ is the Gaussian distribution with parameters $\mu_i$ and $\sigma^2_i$. Then:  :$\displaystyle \sum_{i \mathop = 1}^n \alpha_i X_i \sim \Gaussian {\sum_{i \mathop = 1}^n \alpha_i \mu_i} {\sum_{i \mathop = 1}^n \alpha^2_i \sigma^2_i}$


Primitive of x squared by Root of a squared minus x squared

:$\displaystyle \int x^2 \sqrt {a^2 - x^2} \rd x = \frac {-x \paren {\sqrt {a^2 - x^2} }^3} 4 + \frac {a^2 x \sqrt {a^2 - x^2} } 8 + \frac {a^4} 8 \sinh^{-1} \frac x a + C$


Existence of Non-Measurable Subset of Real Numbers

There exists a subset of the real numbers which is not measurable.


Cube as Difference between Two Squares

A cube number can be expressed as the difference between two squares.


Factorial Greater than Cube for n Greater than 5

Let $n \in \Z$ be an integer such that $n > 5$. Then $n! > n^3$.


Existence of Euler-Mascheroni Constant

The real sequence: :$\displaystyle \sequence {\sum_{k \mathop = 1}^n \frac 1 k - \ln n}$ converges to a limit. This limit is known as the Euler-Mascheroni constant.


Square Product of Three Consecutive Triangular Numbers

Let $T_n$ denote the $n$th triangular number for $n \in \Z_{>0}$ a (strictly) positive integer. Let $T_n \times T_{n + 1} \times T_{n + 2}$ be a square number. Then at least one value of $n$ fulfils this condition: :$n = 3$


Cube Modulo 9

Let $x \in \Z$ be an integer. Then one of the following holds: {{begin-eqn}} {{eqn | l = x^3       | o = \equiv       | r = 0 \pmod 9       | c =  }} {{eqn | l = x^3       | o = \equiv       | r = 1 \pmod 9       | c =  }} {{eqn | l = x^3       | o = \equiv       | r = 8 \pmod 9       | c =  }} {{end-eqn}}


Acceleration of Particle moving in Circle

Let $P$ be a particle moving in a circular path $C$. Then the acceleration of $P$ is given as: :$\mathbf a = -\dfrac {\size {\mathbf v}^2 \mathbf r} {\size {\mathbf r}^2}$ where: :$\mathbf v$ is the instantaneous velocity of $P$ :$\mathbf r$ is the vector whose magnitude equals the length of the radius of $C$ and whose direction is from the center of $C$ to $P$ :$\size {\, \cdot \,}$ denotes the magnitude of a vector.


Primitive of Reciprocal of Root of a x squared plus b x plus c/a equal to 0

:$\displaystyle \int \frac {\mathrm d x} {\sqrt {a x^2 + b x + c} } = \frac {2 \sqrt {b x + c} } b + C$ when $a = 0$.


Primitive of x squared over Cube of Root of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {x^2 \ \mathrm d x} {\left({\sqrt {a x^2 + b x + c} }\right)^3} = \frac {\left({2 b^2 - 4 a c}\right) x + 2 b c} {a \left({4 a c - b^2}\right) \sqrt {a x^2 + b x + c} } + \frac 1 a \int \frac {\mathrm d x} {\sqrt {a x^2 + b x + c} }$


Primitive of Reciprocal of x squared by Cube of Root of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\d x} {x^2 \paren {\sqrt {a x^2 + b x + c} }^3} = -\frac {a x^2 + 2 b x + c} {c^2 x \sqrt {a x^2 + b x + c} } + \frac {b^2 - 2 a c} {2 c^2} \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3} - \frac {3 b} {2 c^2} \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }$


Dirichlet Beta Function at Odd Positive Integers

:$\map \beta {2 n + 1} = \paren {-1}^n \dfrac {E_{2 n} \pi^{2 n + 1} } {4^{n + 1} \paren {2 n}!}$


Primitive of Reciprocal of Half Integer Power of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\mathrm d x} {\left({a x^2 + b x + c}\right)^{n + \frac 1 2} } = \frac {2 \left({2 a x + b}\right)} {\left({2 n - 1}\right) \left({4 a c - b^2}\right) \left({a x^2 + b x + c}\right)^{n - \frac 1 2} } + \frac {8 a \left({n - 1}\right)} {\left({2 n - 1}\right) \left({4 a c - b^2}\right)} \int \frac {\mathrm d x} {\left({a x^2 + b x + c}\right)^{n - \frac 1 2} }$


Primitive of x squared over x cubed plus a cubed

:$\displaystyle \int \frac {x^2 \rd x} {x^3 + a^3} = \frac 1 3 \ln \size {x^3 + a^3} + C$


Primitive of x over x cubed plus a cubed squared

:$\displaystyle \int \frac {x \ \mathrm d x} {\left({x^3 + a^3}\right)^2} = \frac {x^2} {3 a^3 \left({x^3 + a^3}\right)} + \frac 1 {18 a^4} \ln \left({\frac {x^2 - a x + a^2} {\left({x + a}\right)^2} }\right) + \frac 1 {3 a^4 \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$


Compact First-Countable Space is Sequentially Compact

Let $T = \struct {S, \tau}$ be a compact first-countable topological space. Then every infinite sequence in $S$ has a convergent subsequence; that is, $T$ is sequentially compact.


Convergence in Indiscrete Space

Let $\left({S, \left\{{S, \varnothing}\right\}}\right)$ be an indiscrete space. Let $\left \langle {x_n} \right \rangle$ be any sequence in $S$. Then $\left \langle {x_n} \right \rangle$ converges to any point $x$ of $S$.