hcju/proofwikips huggingface dataset:
Definite Integral from 0 to Pi of Sec x by Logarithm of One plus b Cosine x over One plus a Cosine x
:$\displaystyle \int_0^{\pi/2} \sec x \map \ln {\frac {1 + b \cos x} {1 + a \cos x} } \rd x = \frac 1 2 \paren {\paren {\arccos a}^2 - \paren {\arccos b}^2}$
Definite Integral to Infinity of Cosine p x minus Cosine q x over x Squared
:$\displaystyle \int_0^\infty \frac {\cos p x - \cos q x} {x^2} \rd x = \frac {\pi \paren {\size q - \size p} } 2$
Definite Integral to Infinity of Exponential of -a x minus Exponential of -b x over x by Cosecant of p x
:$\displaystyle \int_0^\infty \frac {e^{-a x} - e^{-b x} } {x \csc p x} \rd x = \arctan \frac b p - \arctan \frac a p$
Definite Integral of Periodic Function
Let $f$ be a Darboux integrable periodic function with period $L$. Let $\alpha \in \R$ and $n \in \Z$. Then: :$\displaystyle \int_\alpha^{\alpha + n L} \map f x \d x = n \int_0^L \map f x \d x$
Independent Subset is Contained in Maximal Independent Subset
Let $M = \struct{S, \mathscr I}$ be a matroid. Let $A \subseteq S$. Let $X \in \mathscr I$ such that $X \subseteq A$. Then: :$\exists Y \in \mathscr I : X \subseteq Y \subseteq A : \size Y = \map \rho A$ where $\rho$ is the rank function on $M$.
Automorphic Numbers in Base 10
If leading zeroes are allowed, there are exactly $4$ $n$-digit automorphic numbers in base $10$: :$00 \dots 00$ :$00 \dots 01$ :$5^{2^{n - 1} } \pmod {10^n}$ :$6^{5^{n - 1} } \pmod {10^n}$
Seventeen Horses/General Problem 1
A man dies, leaving $n$ indivisible and indistinguishable objects to be divided among $3$ heirs. They are to be distributed in the ratio $\dfrac 1 a : \dfrac 1 b : \dfrac 1 c$. Let $\dfrac 1 a + \dfrac 1 b + \dfrac 1 c < 1$. Then there are $7$ possible values of $\tuple {n, a, b, c}$ such that the required shares are: :$\dfrac {n + 1} a, \dfrac {n + 1} b, \dfrac {n + 1} c$ These values are: :$\tuple {7, 2, 4, 8}, \tuple {11, 2, 4, 6}, \tuple {11, 2, 3, 12}, \tuple {17, 2, 3, 9}, \tuple {19, 2, 4, 5}, \tuple {23, 2, 3, 8}, \tuple {41, 2, 3, 7}$ leading to shares, respectively, of: :$\tuple {4, 2, 1}, \tuple {6, 3, 2}, \tuple {6, 4, 1}, \tuple {9, 6, 2}, \tuple {10, 5, 4}, \tuple {12, 8, 3}, \tuple {21, 14, 6}$
Seventeen Horses/General Problem 2
A man dies, leaving $n$ indivisible and indistinguishable objects to be divided among $m$ heirs. They are to be distributed in the ratio $\dfrac 1 {a_1} : \dfrac 1 {a_2} : \cdots : \dfrac 1 {a_m}$. Let $t = \dfrac q r = \displaystyle \sum_{k \mathop = 1}^m \dfrac 1 {a_k}$ expressed in canonical form. Let $t \ne 1$. Then it is possible to achieve the required share by adding $s$ objects to the existing $n$ such that: :$s + q = r$ when $q = n$. This still works whether $q$ is positive or negative.
Sum of Sequence of Products of 3 Consecutive Reciprocals
:$\displaystyle \sum_{j \mathop = 1}^n \frac 1 {j \paren {j + 1} \paren {j + 2} } = \frac {n \paren {n + 3} } {4 \paren {n + 1} \paren {n + 2} }$
Sum of Sequence of Products of 3 Consecutive Reciprocals/Corollary
:$\displaystyle \sum_{j \mathop = 1}^\infty \frac 1 {j \paren {j + 1} \paren {j + 2} } = \frac 1 4$
Heronian Triangle whose Altitude and Sides are Consecutive Integers
There exists exactly one Heronian triangle one of whose altitudes and its sides are all consecutive integers. This is the Heronian triangle whose sides are $\tuple {13, 14, 15}$ and which has an altitude $12$.
Integer Heronian Triangle can be Scaled so Area equals Perimeter
Let $T_1$ be an integer Heronian triangle whose sides are $a$, $b$ and $c$. Then there exists a rational number $k$ such that the Heronian triangle $T_2$ whose sides are $k a$, $k b$ and $k c$ such that the perimeter of $T$ is equal to the area of $T$.
Heronian Triangle is Similar to Integer Heronian Triangle
Let $\triangle {ABC}$ be a Heronian triangle. Then there exists an integer Heronian triangle $\triangle {A'B'C'}$ such that $\triangle {ABC}$ and $\triangle {A'B'C'}$ are similar.
Triple with Sum and Product Equal
For $a, b, c \in \Z$, $a \le b \le c$, the solutions to the equation: :$a + b + c = a b c$ are: :$\tuple {1, 2, 3}$ :$\tuple {-3, -2, -1}$ and the trivial solution set: :$\set {\tuple {-z, 0, z}: z \in \N}$
Triple with Product Quadruple the Sum
Let $a, b, c \in \N$ such that $a \le b \le c$. Then the solutions to: :$a b c = 4 \paren {a + b + c}$ are: :$\tuple {0, 0, 0}, \tuple {1, 5, 24}, \tuple {1, 6, 14}, \tuple {1, 8, 9}, \tuple {2, 3, 10}, \tuple {2, 4, 6}$
Proper Integer Heronian Triangle whose Area is 24
There exists exactly one proper integer Heronian triangle whose area equals $24$. That is, the obtuse triangle whose sides are of length $4$, $13$ and $15$.
Semiperimeter of Integer Heronian Triangle is Composite
The semiperimeter of an integer Heronian triangle is always a composite number.
Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Necessary Condition
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\mathscr B$ be the set of bases of the matroid on $M$. Then $\mathscr B$ satisfies the base axiom: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1}}
Independent Subset is Base if Cardinality Equals Rank of Matroid
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\rho: \powerset S \to \Z$ be the rank function of $M$. Let $B \in \mathscr I$ such that: :$\size B = \map \rho S$ Then: :$B$ is a base of $M$.
Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition
Let $S$ be a finite set. Let $\mathscr B$ be a non-empty set of subsets of $S$ satisfying the base axiom: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1}} Then $\mathscr B$ is the set of bases of a matroid on $S$.
Morley's Trisector Theorem
Let $\triangle ABC$ be a triangle. Let the internal angles of $\triangle ABC$ be trisected. Let the points where these angle trisectors first intersect be $D$, $E$ and $F$. :500px Then $\triangle EDF$ is equilateral.
Mean Number of Elements Fixed by Self-Map
Let $n \in \Z_{>0}$ be a strictly positive integer. Let $S$ be a set of cardinality $n$. Let $S^S$ be the set of all mappings from $S$ to itself. Let $\map \mu n$ denote the arithmetic mean of the number of fixed points of all the mappings in $S^S$. Then: :$\map \mu n = 1$
Condition for 3 over n producing 3 Egyptian Fractions using Greedy Algorithm when 2 Sufficient
Consider proper fractions of the form $\dfrac 3 n$ expressed in canonical form. Let Fibonacci's Greedy Algorithm be used to generate a sequence $S$ of Egyptian fractions for $\dfrac 3 n$. Then $S$ consists of $3$ terms, where $2$ would be sufficient  the following conditions hold: :$n \equiv 1 \pmod 6$ :$\exists d: d \divides n$ and $d \equiv 2 \pmod 3$
P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences form Vector Space
Let $\ell^p$ be the p-sequence space. Let $\struct {\R, +_\R, \times_\R}$ be the field of real numbers. Let $\paren +$ be the pointwise addition on the ring of sequences. Let $\paren {\, \cdot \,}$ be the pointwise multiplication on the ring of sequences. Then $\struct {\ell^p, +, \, \cdot \,}_\R$ is a vector space.
Orthogonal Latin Squares of Order 6 do not Exist
Two orthogonal Latin squares of order $6$ do not exist.
Divisor of Product
Let $a, b, c \in \Z$ be integers. Let the symbol $\divides$ denote the divisibility relation. Let $a \divides b c$. Then there exist integers $r, s$ such that: :$a = r s$, where $r \divides b$ and $s \divides c$.
Equivalence of Definitions of Matroid
Let $M = \struct {S, \mathscr I}$ be an independence system. {{TFAE|def=Matroid}}
P-adic Norm is Well Defined
P-adic norm $\norm {\, \cdot \,}_p$ is well defined.
Equivalence of Definitions of Matroid/Definition 1 implies Definition 2
Let $M = \struct {S, \mathscr I}$ be an independence system. Let $M$ also satisfy: {{begin-axiom}} {{axiom | n = \text I 3         | q = \forall U, V \in \mathscr I         | mr= \size V < \size U \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I }} {{end-axiom}} Then $M$ satisfies: {{begin-axiom}} {{axiom | n = \text I 3'         | q = \forall U, V \in \mathscr I         | mr= \size U = \size V + 1 \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I }} {{end-axiom}}
Equivalence of Definitions of Matroid/Definition 2 implies Definition 3
Let $M = \struct {S, \mathscr I}$ be an independence system. Let $M$ also satisfy: {{begin-axiom}} {{axiom | n = \text I 3'         | q = \forall U, V \in \mathscr I         | mr= \size U = \size V + 1 \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I }} {{end-axiom}} Then $M$ satisfies: {{begin-axiom}} {{axiom | n = \text I 3''         | q = \forall U, V \in \mathscr I         | mr= \size V < \size U \implies \exists Z \subseteq U \setminus V : \paren{V \cup Z \in \mathscr I} \land \paren{ \size {V \cup Z} = \size U} }} {{end-axiom}}
Equivalence of Definitions of Matroid/Definition 3 implies Definition 1
Let $M = \struct {S, \mathscr I}$ be an independence system. Let $M$ also satisfy: {{begin-axiom}} {{axiom | n = \text I 3''         | q = \forall U, V \in \mathscr I         | mr= \size V < \size U \implies \exists Z \subseteq U \setminus V : \paren {V \cup Z \in \mathscr I} \land \paren {\size {V \cup Z} = \size U} }} {{end-axiom}} Then $M$ satisfies: {{begin-axiom}} {{axiom | n = \text I 3         | q = \forall U, V \in \mathscr I         | mr= \size V < \size U \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I }} {{end-axiom}}
Equivalence of Definitions of Matroid/Definition 1 implies Definition 4
Let $M = \struct {S, \mathscr I}$ be an independence system. Let $M$ also satisfy: {{begin-axiom}} {{axiom | n = \text I 3         | q = \forall U, V \in \mathscr I         | mr= \size V < \size U \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I }} {{end-axiom}} Then $M$ satisfies: {{begin-axiom}} {{axiom | n = \text I 3'''         | q = \forall A \subseteq S         | mr= \text{ all maximal subsets } Y \subseteq A \text{ with } Y \in \mathscr I \text{ have the same cardinality} }} {{end-axiom}}
Equivalence of Definitions of Matroid/Definition 4 implies Definition 1
Let $M = \struct {S, \mathscr I}$ be an independence system. Let $M$ also satisfy: {{begin-axiom}} {{axiom | n = \text I 3'''         | q = \forall A \subseteq S         | mr= \text{ all maximal subsets } Y \subseteq A \text{ with } Y \in \mathscr I \text{ have the same cardinality} }} {{end-axiom}} Then $M$ satisfies: {{begin-axiom}} {{axiom | n = \text I 3         | q = \forall U, V \in \mathscr I         | mr= \size V < \size U \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I }} {{end-axiom}}
Cardinality of Set Difference
Let $S$ and $T$ be sets such that $T$ is finite. Then: :$\card {S \setminus T} = \card S - \card {S \cap T}$ where $\card S$ denotes the cardinality of $S$.
Set Difference and Intersection are Disjoint
Let $S$ and $T$ be sets. Then:  :$S \setminus T$ and $S \cap T$ are disjoint  where $S \setminus T$ denotes set difference and $S \cap T$ denotes set intersection.
Straight Line has Zero Curvature
A straight lines has zero curvature.
Partial Differential Equation of Spheres in 3-Space
The set of spheres in real Cartesian $3$-dimensional space can be described by the system of partial differential equations: :$\dfrac {1 + z_x^2} {z_{xx} } = \dfrac {z_x z_x} {z_{xy} } = \dfrac {1 + z_y^2} {z_{yy} }$ and if the spheres are expected to be real: :$z_{xx} z_{yy} > z_{xy}$
Fermat's Right Triangle Theorem
$x^4 + y^4 = z^2$ has no solutions in the (strictly) positive integers.
Taylor's Theorem/One Variable with Two Functions
Let $f$ and $g$ be real functions satisfying following conditions: :$(1): \quad f$ is $n + 1$ times differentiable on the open interval $\openint a x$ :$(2): \quad f$ is of differentiability class $C^n$ on the closed interval $\closedint a x$ :$(3): \quad g$ is $k + 1$ times differentiable on the open interval $\openint a x$ :$(4): \quad g$ is of differentiability class $C^k$ on the closed interval $\closedint a x$ :$(5): \quad \map {g^{\paren {k + 1}}} t \ne 0$ for any $t \in \openint a x$ Then the following equation holds for some real number $\xi \in \openint a x$: :$\dfrac {\map {f^{\paren {n + 1} } } \xi /n!} {\map {g^{\paren {k + 1} } } \xi /k!} \paren {x - \xi}^{n - k} = \dfrac {\map f x - \map f a - \map {f'} a \paren {x - a} - \dfrac {\map {f''} a} {2!} \paren {x - a}^2 - \dotsb - \dfrac {\map {f^{\paren n} } a} {n!} \paren {x - a}^n} {\map g x - \map g a - \map {g'} a \paren {x - a} - \dfrac {\map {g''} a} {2!} \paren {x - a}^2 - \dotsb - \dfrac {\map {g^{\paren k} } a} {k!} \paren {x - a}^k}$ or equivalently: {{begin-eqn}} {{eqn | l = \map f x       | r = \map f a + \map {f'} a \paren {x - a} + \dfrac {\map {f''} a} {2!} \paren {x - a}^2 + \dotsb + \dfrac {\map {f^{\paren n} } a} {n!} \paren {x - a}^n + R_n }} {{eqn | l = R_n       | r = \dfrac {\map {f^{\paren {n + 1} } } \xi / n!} {\map {g^{\paren {k + 1} } } \xi / k!} \paren {x - \xi}^{n - k} \paren {\map g x - \map g a - \map {g'} a \paren {x - a} - \dfrac {\map {g''} a} {2!} \paren {x - a}^2 - \dotsb - \dfrac {\map {g^{\paren k} } a} {k!} \paren {x - a}^k} }} {{end-eqn}}
Set of Even Integers is Countably Infinite
Let $\Bbb E$ be the set of even integers. Then $\Bbb E$ is countably infinite.
Basis Expansion of Rational Number
Let $x$ be a rational number. {{WIP|Time has caught up with me, I'll continue this later. I'm late for work.}} q2llusuyprk7567f7o3l3jnktxipxcq
Integration by Parts/Definite Integral
:$\displaystyle \int_a^b \map f t \map G t \rd t = \bigintlimits {\map F t \map G t} a b - \int_a^b \map F t \map g t \rd t$
Integration by Parts/Primitive
:$\ds \int \map f t \map G t \rd t = \map F t \map G t - \int \map F t \map g t \rd t$ on $\closedint a b$.
Integration by Substitution/Primitive
The primitive of $f$ can be evaluated by: :$\ds \int \map f x \rd x = \int \map f {\map \phi u} \dfrac \d {\d u} \map \phi u \rd u$ where $x = \map \phi u$.
Product Formula for Norms on Non-zero Rationals/Lemma
Let $z \in \Z_{\ne 0}$. Then the following infinite product converges: :$\size z \times \displaystyle\prod_{p \mathop \in \Bbb P}^{} \norm z_p = 1$
P-adic Open Ball is Instance of Open Ball of a Norm
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $B \subseteq \Q_p$. Then:  :$B$ is an open ball in $p$-adic numbers with radius $\epsilon$ and centre $a$ : :$B$ is an open ball of the normed division ring $\struct {\Q_p, \norm {\,\cdot\,}_p}$ with radius $\epsilon$ and centre $a$ . That is, the definition of an open ball in $p$-adic numbers is a specific instance of the general definition of an open ball in a normed division ring.
P-adic Closed Ball is Instance of Closed Ball of a Norm
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $B \subseteq \Q_p$. Then:  :$B$ is a closed ball in $p$-adic numbers with radius $\epsilon$ and centre $a$ : :$B$ is a closed ball of the normed division ring $\struct {\Q_p, \norm {\,\cdot\,}_p}$ with radius $\epsilon$ and centre $a$  That is, the definition of a closed ball in $p$-adic numbers is a specific instance of the general definition of a closed ball in a normed division ring.
P-adic Sphere is Instance of Sphere of a Norm
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $S \subseteq \Q_p$. Then:  :$S$ is a sphere in $p$-adic numbers with radius $\epsilon$ and centre $a$ : :$S$ is a sphere of the normed division ring $\struct {\Q_p, \norm {\,\cdot\,}_p}$ with radius $\epsilon$ and centre $a$  That is, the definition of a closed ball in $p$-adic numbers is a specific instance of the general definition of a sphere in a normed division ring.
Sphere is Set Difference of Closed Ball with Open Ball/P-adic Numbers
Let $p$ be a prime number. Let $\Q_p$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map {{B_\epsilon}^-} a$ denote the $\epsilon$-closed ball of $a$ in $\Q_p$. Let $\map {B_\epsilon} a$ denote the $\epsilon$-open ball of $a$ in $\Q_p$. Let $\map {S_\epsilon} a$ denote the $\epsilon$-sphere of $a$ in $\Q_p$. Then: :$\map {S_\epsilon} a = \map { {B_\epsilon}^-} a \setminus \map {B_\epsilon} a$
Set is Closed in Metric Space iff Closed in Induced Topological Space
Let $M = \struct {A, d}$ be a metric space. Let $\tau$ be the topology induced by the metric $d$. Let $F$ be a subset of $M$. Then: :$F$ is closed in $M$  $F$ is closed in $\struct {A, \tau}$
Primitive of Reciprocal of Root of a squared minus x squared/Arccosine Form
:$\displaystyle \int \frac 1 {\sqrt {a^2 - x^2} } \rd x = -\arccos \frac x a + C$
Negative of Logarithm of x plus Root x squared minus a squared
Let $x \in \R: \size x > 1$. Let $x > 1$. Then: :$-\map \ln {x + \sqrt {x^2 - a^2} } = \map \ln {x - \sqrt {x^2 - a^2} } - \map \ln {a^2}$
Supremum Norm on Vector Space of Real Matrices is Norm
Supremum Norm forms a norm on the vector space of real matrices.
Arccotangent Logarithmic Formulation
For any real number $x$: :$\arccot x = \dfrac 1 2 i \, \map \ln {\dfrac {1 + i x} {1 - i x} }$ where $\arccot x$ is the arccotangent and $i^2 = -1$.
Primitive of Reciprocal of x squared plus a squared/Arccotangent Form
:$\ds \int \frac {\d x} {x^2 + a^2} = -\frac 1 a \arccot \frac x a + C$
Primitive of Reciprocal of x by Root of x squared plus a squared/Reciprocal Logarithm Form
:$\displaystyle \int \frac {\d x} {x \sqrt {x^2 + a^2} } = \frac 1 a \map \ln {\frac x {a + \sqrt {x^2 + a^2} } } + C$
Derivative of Hyperbolic Sine
:$\map {\dfrac \d {\d x} } {\sinh x} = \cosh x$
Open Ball is Open Set/Metric Space
Let $M = \left({A, d}\right)$ be a metric space. Let $x \in A$. Let $\epsilon \in \R_{>0}$. Let $B_\epsilon \left({x}\right)$ be an open $\epsilon$-ball of $x$ in $M$. Then $B_\epsilon \left({x}\right)$ is an open set of $M$.
Derivative of Hyperbolic Cosine
:$\map {\dfrac \d {\d x} } {\cosh x} = \sinh x$
Derivative of Hyperbolic Tangent
:$\map {\dfrac \d {\d x} } {\tanh x} = \sech^2 x$
Derivative of Hyperbolic Cotangent
:$\map {\dfrac \d {\d x} } {\coth x} = -\csch^2 x$
Derivative of Hyperbolic Secant
:$\map {\dfrac \d {\d x} } {\sech x} = -\sech x \tanh x$
Derivative of Hyperbolic Cosecant
:$\map {\dfrac \d {\d x} } {\csch x} = -\csch x \coth x$
Derivative of Inverse Hyperbolic Sine Function
:$\map {\dfrac \d {\d x} } {\sinh^{-1} u} = \dfrac 1 {\sqrt {1 + u^2} } \dfrac {\d u} {\d x}$
Derivative of Inverse Hyperbolic Cosine Function
:$\map {\dfrac \d {\d x} } {\cosh^{-1} u} = \dfrac 1 {\sqrt {u^2 - 1} } \dfrac {\d u} {\d x}$  where $u > 1$
Derivative of Inverse Hyperbolic Tangent Function
:$\map {\dfrac \d {\d x} } {\tanh^{-1} u} = \dfrac 1 {1 - u^2} \dfrac {\d u} {\d x}$  where $\size u < 1$
Derivative of Inverse Hyperbolic Cosecant Function
:$\map {\dfrac \d {\d x} } {\csch^{-1} u} = \dfrac {-1} {\size u \sqrt {1 + u^2} } \dfrac {\d u} {\d x}$
Intersection of Closed Sets is Closed/Topology
Let $T = \struct {S, \tau}$ be a topological space. Then the intersection of an arbitrary number of closed sets of $T$ (either finitely or infinitely many) is itself closed.
Finite Union of Closed Sets is Closed/Topology
Let $T = \struct {S, \tau}$ be a topological space. Then the union of finitely many closed sets of $T$ is itself closed.
Closed Ball is Closed/Metric Space
Let $M = \struct {A, d}$ be a metric space. Let $x \in A$. Let $\epsilon \in \R_{>0}$. Let $\map {B_\epsilon^-} x$ be the closed $\epsilon$-ball of $x$ in $M$. Then $\map {B_\epsilon^-} x$ is a closed set of $M$.
Derivative of Sine of Function
:$\map {\dfrac \d {\d x} } {\sin u} = \cos u \dfrac {\d u} {\d x}$
Derivative of Secant of Function
:$\map {\dfrac \d {\d x} } {\sec u} = \sec u \tan u \dfrac {\d u} {\d x}$
Derivative of Cosecant of Function
:$\map {\dfrac \d {\d x} } {\csc u} = \csc u \cot u \dfrac {\d u} {\d x}$
Derivative of Natural Logarithm of Function
:$\map {\dfrac \d {\d x} } {\ln u} = \dfrac 1 u \dfrac {\d u} {\d x}$
Derivative of Constant to Power of Function
:$\map {\dfrac \d {\d x} } {a^u} = a^u \ln a \dfrac {\d u} {\d x}$
Derivative of Arcsine of Function
:$\map {\dfrac \d {\d x} } {\arcsin u} = \dfrac 1 {\sqrt {1 - u^2} } \dfrac {\d u} {\d x}$
Derivative of Arccosine of Function
:$\map {\dfrac \d {\d x} } {\arccos u} = -\dfrac 1 {\sqrt {1 - u^2} } \dfrac {\d u} {\d x}$
Derivative of Arctangent of Function
:$\map {\dfrac \d {\d x} } {\arctan u} = \dfrac 1 {1 + u^2} \dfrac {\d u} {\d x}$
Derivative of Arccotangent of Function
:$\map {\dfrac \d {\d x} } {\arccot u} = -\dfrac 1 {1 + u^2} \dfrac {\d u} {\d x}$
Derivative of Arcsecant of Function
:$\map {\dfrac \d {\d x} } {\arcsec u} = \dfrac 1 {\size u \sqrt {u^2 - 1} } \dfrac {\d u} {\d x}$
Derivative of Arccosecant of Function
:$\map {\dfrac \d {\d x} } {\arccsc u} = -\dfrac 1 {\size u \sqrt {u^2 - 1} } \dfrac {\d u} {\d x}$
Derivative of Odd Function is Even
Let $f$ be a differentiable real function such that $f$ is odd. Then its derivative $f'$ is an even function.
Form of Prime Sierpiński Number of the First Kind
Suppose $S_n = n^n + 1$ is a prime Sierpiński number of the first kind. Then: :$n = 2^{2^k}$ for some integer $k$.
Motion of Body Falling through Air
The motion of a body $B$ falling through air can be described using the following differential equation: :$m \dfrac {\d^2 y} {\d t^2} = m g - k \dfrac {d y} {d t}$ where: :$m$ denotes mass of $B$ :$y$ denotes the height of $B$ from an arbitrary reference :$t$ denotes time elapsed from an arbitrary reference :$g$ denotes the Acceleration Due to Gravity of $B$ :$k$ denotes the coefficient of resistive force exerted on $B$ by the air (assumed to be proportional to the speed of $B$)
Rationals are Everywhere Dense in Reals/Normed Vector Space
Let $\struct {\R, \size {\, \cdot \,}}$ be the normed vector space of real numbers. Let $\Q$ be the set of rational numbers. Then $\Q$ are everywhere dense in $\struct {\R, \size {\, \cdot \,}}$
Number of Parameters of Autoregressive Model
Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$ Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$: :$\tilde z_t = z_t - \mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$ Let $M$ be an '''autoregessive model''' on $S$ of order $p$: :$\tilde z_t = \phi_1 \tilde z_{t - 1} + \phi_2 \tilde z_{t - 2} + \dotsb + \phi_p \tilde z_{t - p} + a_t$ Then $M$ has $p + 2$ parameters.
Autoregressive Model is Special Case of Linear Filter Model
Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$ Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$: :$\tilde z_t = z_t - \mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$ Let $M$ be an '''autoregessive model''' on $S$ of order $p$: :$(1): \quad \tilde z_t = \phi_1 \tilde z_{t - 1} + \phi_2 \tilde z_{t - 2} + \dotsb + \phi_p \tilde z_{t - p} + a_t$ Then $M$ is a special case of a linear filter model.
Irrationals are Everywhere Dense in Reals/Topology
Let $T = \struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology. Let $\R \setminus \Q$ be the set of irrational numbers. Then $\R \setminus \Q$ is everywhere dense in $T$.
Irrationals are Everywhere Dense in Reals/Normed Vector Space
Let $\struct {\R, \size {\, \cdot \,}}$ be the normed vector space of real numbers. Let $\R \setminus \Q$ be the set of irrational numbers. Then $\R \setminus \Q$ are everywhere dense in $\struct {\R, \size {\, \cdot \,}}$
Real Number Subtracted from Itself leaves Zero
Let $x \in \R$ be a real number. Then: :$x - x = 0$ where $x - x$ denotes the operation of real subtraction.
Real Number Ordering is Compatible with Multiplication/Positive Factor/Corollary
:$\forall a, b, c, d \in \R: 0 < a < b \land 0 < c < d \implies a c < b d$