hcju/proofwikips huggingface dataset:
Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom
Let $S$ be a finite set. Let $\mathscr B$ be a non-empty set of subsets of $S$. {{TFAE|def=Base Axiom (Matroid)|view = Matroid Base Axiom}}
Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Lemma
Let $B_1, B_2 \subseteq S$. Let $x \in B_1 \setminus B_2$. Let $y \in B_2 \setminus B_1$. Then: :$\paren{B_1 \setminus \set x} \cup \set y = \paren{B_1 \cup \set y} \setminus \set x$
Number of Parameters of Moving Average 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 $\tilde z_t$ be the deviation 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 '''moving average model''' on $S$ of order $q$: :$\tilde z_t = a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$ Then $M$ has $q + 2$ parameters.
Characteristic of Field by Annihilator/Characteristic Zero
Suppose that: :$\map {\mathrm {Ann} } F = \set 0$ That is, the annihilator of $F$ consists of the zero only. Then: :$\Char F = 0$ That is, the characteristic of $F$ is zero.
Characteristic of Field by Annihilator/Prime Characteristic
Suppose that: :$\exists n \in \map {\mathrm {Ann} } F: n \ne 0$ That is, there exists (at least one) non-zero integer in the annihilator of $F$. If this is the case, then the characteristic of $F$ is non-zero: :$\Char F = p \ne 0$ and the annihilator of $F$ consists of the set of integer multiples of $p$: :$\map {\mathrm {Ann} } F = p \Z$ where $p$ is a prime number.
Necessary Condition for Autoregressive Process to be Stationary
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 $p$: :$\map \phi B \tilde z_t = a_t$ where $\map \phi B := 1 - \phi_1 B - \phi_2 B^2 - \dotsb - \phi_p B^p$ is the autoregressive operator of order $p$. Consider the polynomial equation in $B$ of degree $p$: :$(1): \quad \map \phi B = 0$ Let $\map R \phi \subseteq \C$ denote the set of roots of $(1)$, considered as a polynomial of degree $p$. It is noted that the elements of $\map R \phi$ may be real or complex. For $S$ modelled by $M$ to be a stationary process, it is necessary that the elements of $\map R \phi$ have a complex modulus greater than $1$: :$\forall z \in \map R \phi: \size z > 1$
ARIMA Model subsumes ARMA Model
Let $S$ be a stochastic process based on an equispaced time series. Let $M$ be an ARMA model for $S$. Then $M$ is also an implementation of an ARIMA model.
ARIMA Model subsumes Autoregressive Model
Let $S$ be a stochastic process based on an equispaced time series. Let $M$ be an autoregressive model for $S$. Then $M$ is also an implementation of an ARIMA model.
Leigh.Samphier/Sandbox/Matroid Base Axiom Implies Sets Have Same Cardinality
Let $S$ be a finite set. Let $\mathscr B$ be a non-empty set of subsets of $S$. Let $\mathscr B$ satisfy the base axiom: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1}} Then: :$\forall B_1, B_2 \in \mathscr B : \card{B_1} = \card{B_2}$ where $\card{B_1}$ and $\card{B_2}$ denote the cardinality of the sets $B_1$ and $B_2$ respectively.
Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 4 Iff Definition 5/Lemma
:$\forall B_1, B_2 \in \mathscr B : \card{B_1} = \card{B_2}$
Negative of Field Negative
Let $\struct {F, +, \times}$ be a field whose zero is $0_F$. Let $a \in F$ and let $-a$ be the field negative of $a$. Then: :$-\paren {-a} = a$
Field Product with Zero
Let $\struct {F, +, \times}$ be a field whose zero is $0_F$. Let $a \in F$. Then: :$a \times 0_F = 0_F$
Product with Field Negative
Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $a, b \in F$. Then: :$-\paren {a \times b} = a \times \paren {-b} = \paren {-a} \times b$
Product with Field Negative/Corollary
:$\paren {-1_F} \times a = \paren {-a}$
Condition for Division by Field Elements to be Unity
Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $a, b \in F$. Then: :$\dfrac a b = 1_F$  :$a = b$ where $\dfrac a b$ denotes division.
Field Product with Non-Zero Element yields Unique Solution
Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $a, b, x \in F$ such that $b \ne 0_F$. Let: :$b \times x = a$ Then: :$x = a b^{-1}$ That is: :$x = \dfrac a b$ where $\dfrac a b$ denotes division.
Field Unity Divided by Element equals Multiplicative Inverse
Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $a \in F$. Then: :$\dfrac {1_F} a = a^{-1}$ where $\dfrac {1_F} a$ denotes division.
Set Difference of Larger Set with Smaller is Not Empty
Let $S$ and $T$ be finite sets. Let $\card S > \card T$. Then: :$S \setminus T \ne \O$
Autocovariance Matrix for Stationary Process is Variance by Autocorrelation Matrix
Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$ Let $\boldsymbol \Gamma_n$ denote the '''autocovariance matrix''' associated with $S$ for $\sequence {s_n}$. Let $\mathbf P_n$ denote the '''autocorrelation matrix''' associated with $S$ for $\sequence {s_n}$. Then: :$\boldsymbol \Gamma_n = \sigma_z^2 \mathbf P_n$ where $\sigma_z^2$ denotes the variance of $S$.
Variance of Linear Function of Observations of Stationary Process
Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$ Let $L_t$ be a linear function of $\sequence {s_n}$: :$L_t = l_1 z_t + l_2 z_{t - 1} + \dotsb + l_n z_{t - n + 1}$ Then the variance of $L_t$ is given by: :$\var {L_t} = \displaystyle \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n l_i l_j \gamma {\size {j - i} }$ where $\gamma_k$ is the autocovariance of $S$ at lag $k$.
Autocorrelation Matrix is Positive Definite
Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$ Let $\mathbf P_n$ denote the '''autocorrelation matrix''' associated with $S$ for $\sequence {s_n}$. Then $\mathbf P_n$ is a positive definite matrix.
Determinant of Autocorrelation Matrix is Strictly Positive
Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$ Let $\mathbf P_n$ denote the '''autocorrelation matrix''' associated with $S$ for $\sequence {s_n}$. The determinant of $\mathbf P_n$ is strictly positive.
Sum of Wholly Real Numbers is Wholly Real
Let $x = \tuple {a, 0}$ and $y = \tuple {b, 0}$ be wholly real complex numbers. Then $x + y$ is also wholly real.
Product of Wholly Real Numbers is Wholly Real
Let $x = \tuple {a, 0}$ and $y = \tuple {b, 0}$ be wholly real complex numbers. Then $x y$ is also wholly real.
Product of Imaginary Unit with Itself
Let $\tuple {0, 1}$ denote the imaginary unit. Then: :$\tuple {0, 1} \times \tuple {0, 1} = \tuple {-1, 0}$ where $\times$ denotes complex multiplication.
Element of Matroid Base and Circuit has a Substitute
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $B \subseteq S$ be a base of $M$. Let $C \subseteq S$ be a circuit of $M$. Let $x \in B \cap C$. Then: :$\exists y \in C \setminus B : \paren{B \setminus \set x} \cup \set y$ is a base of $M$
Set Difference with Proper Subset is Proper Subset
Let $S$ be a set. Let $T \subsetneq S$ be a proper subset of $S$.   Let $S \setminus T$ denote the set difference between $S$ and $T$. Then: :$S \setminus T$ is a proper subset of $S$
Element of Matroid Base and Circuit has a Substitute/Lemma 1
:$C \setminus \set x$ is an independent proper subset of $C$
All Bases of Matroid have same Cardinality/Corollary
Let $X \subseteq S$ be any independent subset of $M$. Then: :$\card X \le \card B$
Independent Set can be Augmented by Larger Independent Set/Corollary
Let $B \subseteq S$ be a base of $M$. Then: :$\exists Z \subseteq B \setminus X : \card{X \cup Z} = \card B  : X \cup Z$ is a base of $M$
Independent Subset is Base if Cardinality Equals Rank of Matroid/Corollary
Let $B \subseteq S$ be a base of $M$. Let $X \subseteq S$ be any independent subset of $M$. Let $\card X = \card B$. Then: :$X$ is a base of $M$.
Element of Matroid Base and Circuit has a Substitute/Lemma 2
:$x \notin \paren{ C \setminus \set x} \cup X$
Element of Matroid Base and Circuit has a Substitute/Lemma 3
:$\exists y \in \paren{\paren{C \setminus \set x} \cup X} \setminus \paren{B \setminus \set x} : \paren{B \setminus \set x} \cup \set y \in \mathscr I : \card{\paren{B \setminus \set x} \cup \set y} = \card {\paren{ C \setminus \set x} \cup X}$
Matroid Unique Circuit Property
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $X \subseteq S$ be an independent subset of $M$. Let $x \in S$ such that: :$X \cup \set x$ is a dependent subset of $M$. Then there exists a unique circuit $C$ such that: :$x \in C \subseteq X \cup \set x$
Absolute Value of Negative
Let $x \in \R$ be a real number. Then: :$\size x = \size {-x}$ where $\size x$ denotes the absolute value of $x$.
Dependent Subset Contains a Circuit
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\mathscr C$ denote the set of all circuits of $M$. Let $A$ be a dependent subset. Then: :$\exists C \in \mathscr C : C \subseteq A$
Condition for Linear Operation on Complex Numbers to be of Finite Order
Let $A$ be the operation on the complex numbers $\C$ defined as: :$\map A x = \alpha x + \beta$ Then $A$ is of finite order greater than $1$  $\alpha$ is a root of unity other than $1$.
Positive Rational Numbers under Division do not form Group
Let $\struct {\Q, /}$ denote the algebraic structure consisting of the set of rational numbers $\Q$ under the operation $/$ of division. We have that $\struct {\Q, /}$ is not a group.
Derivative of Inverse Hyperbolic Sine of x over a/Corollary 1
:$\map {\dfrac \d {\d x} } {\ln \size {x + \sqrt {x^2 + a^2} } } = \dfrac 1 {\sqrt {x^2 + a^2} }$
Negative of Logarithm of x plus Root x squared plus a squared
Let $x \in \R$ be a real number. Then: :$-\map \ln {x + \sqrt {x^2 + a^2} } = \map \ln {-x + \sqrt {x^2 + a^2} } - \map \ln {a^2}$
Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form/Corollary
:$\displaystyle \int \frac {\d x} {-\sqrt {x^2 + a^2} } = \ln \size {x - \sqrt {x^2 + a^2} } + C$
Derivative of Inverse Hyperbolic Cosine of x over a/Corollary 1
:$\map {\dfrac \d {\d x} } {\map \ln {x + \sqrt {x^2 - a^2} } } = \dfrac 1 {\sqrt {x^2 - a^2} }$ for $x > a$.
Derivative of Inverse Hyperbolic Cosine of x over a/Corollary 2
:$\map {\dfrac \d {\d x} } {\map \ln {x - \sqrt {x^2 - a^2} } } = -\dfrac 1 {\sqrt {x^2 - a^2} }$ for $x > a$.
Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form/Corollary
:$\displaystyle \int \frac {\d x} {-\sqrt {x^2 - a^2} } = \ln \size {x - \sqrt {x^2 - a^2} } + C$
Derivative of Inverse Hyperbolic Tangent of x over a/Corollary
:$\map {\dfrac \d {\d x} } {\dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } } = \dfrac 1 {a^2 - x^2}$ where $\size x < a$.
Derivative of Inverse Hyperbolic Cotangent of x over a/Corollary
:$\map {\dfrac \d {\d x} } {\dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } } = \dfrac 1 {a^2 - x^2}$ where $\size x > a$.
Leigh.Samphier/Sandbox/Set Difference of Matroid Circuit with Element is Independent
:$C \setminus \set x$ is an independent subset of $C$
Union of Matroid Base with Element of Complement is Dependent
:$B \cup \set x$ is a dependent superset of $B$
Primitive of Power of a x + b/Proof 2
{{:Primitive of Power of a x + b}} where $n \ne 1$.
Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form
:$\displaystyle \int \dfrac {\d x} {a^2 - x^2} = \begin {cases} \dfrac 1 a \tanh^{-1} \dfrac x a + C & : \size x < a \\ & \\ \dfrac 1 a \coth^{-1} \dfrac x a + C & : \size x > a \\ & \\ \text {undefined} & : x = a \end {cases}$
Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form
:$\displaystyle \int \frac {\d x} {a^2 - x^2} = \frac 1 a \coth^{-1} \frac x a + C$ where $\size x > a$.
Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a
Let $\size x < a$. Then: :$\displaystyle \int \frac {\d x} {a^2 - x^2} = \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C$
Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x greater than a
Let $\size x > a$. Then: :$\displaystyle \int \frac {\d x} {a^2 - x^2} = \dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } + C$
Primitive of Reciprocal of a squared minus x squared/Logarithm Form
Let $a \in \R_{>0}$ be a strictly positive real constant. Let $x \in \R$ such that $\size x \ne a$.
Primitive of Reciprocal of x squared minus a squared/Logarithm Form
Let $a \in \R_{>0}$ be a strictly positive real constant. Let $x \in \R$ such that $\size x \ne a$.
Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1
:$\displaystyle \int \frac {\d x} {x^2 - a^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C & : \size x < a\\ & \\ \dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C & : \size x > a \\ & \\ \text {undefined} & : \size x = a \end {cases}$
Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x less than a
Let $\size x < a$. Then: :$\displaystyle \int \frac {\d x} {x^2 - a^2} = \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C$
Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x greater than a
Let $\size x > a$. Then: :$\ds \int \frac {\d x} {x^2 - a^2} = \dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C$
Power Set is Closed under Complement
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Then: :$\forall A \in \powerset S: \relcomp S A \in \powerset S$ where $\relcomp S A$ denotes the complement of $A$ relative to $S$.
Parallelism is Reflexive Relation
Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \in S$, let $l_1 \parallel l_2$ denote that $l_1$ is parallel to $l_2$. Then $\parallel$ is a reflexive relation on $S$.
Parallelism is Symmetric Relation
Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \in S$, let $l_1 \parallel l_2$ denote that $l_1$ is parallel to $l_2$. Then $\parallel$ is a symmetric relation on $S$.
Parallelism is Equivalence Relation/Transitivity
Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \in S$, let $l_1 \parallel l_2$ denote that $l_1$ is parallel to $l_2$. Then $\parallel$ is a transitive relation on $S$.
Approximation/Examples/22 over 7
$\dfrac {22} 7$ is a convenient approximation to $\pi$: :$\dfrac {22} 7 = 3 \cdotp \dot 14285 \dot 7$
Perpendicularity is Antireflexive Relation
Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \in S$, let $l_1 \perp l_2$ denote that $l_1$ is perpendicular to $l_2$. Then $\perp$ is an antireflexive relation on $S$.
Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma
:$\exists B_3 \in \mathscr B$: :::$V \subseteq B_3$ :::$\card{B_1 \cap B_3} > \card{B_1 \cap B_2}$
Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma/Lemma 2
{{begin-eqn}} {{eqn | l = \card {B_1}       | r = \card{B_1 \cap B_2} + \card{B_1 \setminus B_2} }} {{eqn | l = \card {B_2}       | r = \card{B_2 \cap B_1} + \card{V \setminus B_1} + \card{\paren{B_2 \setminus B_1} \setminus V} }} {{end-eqn}}
Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma/Lemma 3
:$\card{B_1 \cap B_3} = \card{B_1 \cap B_2} + 1$
Equivalence of Versions of Axiom of Choice/Formulation 2 implies Formulation 1
The following formulation of the Axiom of Choice:
Equivalence of Versions of Axiom of Choice/Formulation 1 implies Formulation 3
The following formulation of the Axiom of Choice:
Equivalence of Versions of Axiom of Choice/Formulation 3 implies Formulation 1
The following formulation of the Axiom of Choice:
C^k Norm is Norm
Let $I = \closedint a b$ be a closed real interval. Let $\struct {\map {\CC^k} I, +, \, \cdot \,}_\R$ be the vector space of real-valued functions, k-times differentiable on $I$. Let $x \in \map {\CC^k} I$ be a real-valued function of differentiability class $k$. Let $\norm {\, \cdot \,}_{\map {C^k} I}$ be the $C^k$ norm on $I$. Then $\norm {\, \cdot \,}_{\map {C^k} I}$ is a norm on $\struct {\map {\CC^k} I, +, \, \cdot \,}_\R$.
Existence of Minimal Polynomial for Square Matrix over Field
Let $K$ be a field. Let $n$ be a natural number. Let $K^{n \times n}$ be the set of $n \times n$ matrices over $K$.  Let $A \in K^{n \times n}$. Then the minimal polynomial of $A$ exists and has degree at most $n^2$.
Homeomorphic Topologies on Same Set may not be Identical
Let $S$ be a set. Let $\tau_1$ and $\tau_2$ both be topologies on $S$ such that the topological spaces $\struct {S, \tau_1}$ and $\struct {S, \tau_2}$ are homeomorphic. Then it is not necessarily the case that $\struct {S, \tau_1} = \struct {S, \tau_1}$.
Clopen Sets in Indiscrete Topology
The only subsets of $S$ which are both closed and open in $T$ are $S$ and $\O$.
Equivalence of Definitions of Connected Topological Space/No Separation iff No Clopen Sets
{{TFAE|def = Connected Topological Space}} Let $T = \struct {S, \tau}$ be a topological space.
Dependent Subset of Independent Set Union Singleton Contains Singleton
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $X$ be an independent subset of $M$. Let $x \in S$. Let $C$ be a dependent subset of $M$ such that: :$C \subseteq X \cup \set x$. Then: :$x \in C$
Matroid Unique Circuit Property/Corollary
Let $B$ be a base of $M$. Let $x \in S \setminus B$. Then there exists a unique circuit $C$ such that: :$x \in C \subseteq B \cup \set x$ That is, $C$ is the fundamental circuit of $x$ in $B$.
Leigh.Samphier/Sandbox/Matroid Base Substitution From Fundamental Circuit
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $B$ be a base of $M$. Let $y \in S \setminus B$. Let $\map C {y,B}$ denote the fundamental circuit of $y$ in $B$. Let $x \in B$. Then: :$\paren{B \setminus \set x} \cup \set y$ is a base of $M$  $x \in \map C {y,B}$
Socrates is Mortal/Variant
:$(1): \quad$ ''If {{AuthorRef|Socrates}} is a man then {{AuthorRef|Socrates}} is mortal.'' :$(2): \quad$ ''{{AuthorRef|Socrates}} is a man.'' :$(3): \quad$ ''Therefore {{AuthorRef|Socrates}} is mortal.''
Self-Distributive Law for Conditional/Forward Implication/Formulation 2
:$\paren {p \implies \paren {q \implies r} } \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }$
Self-Distributive Law for Conditional/Reverse Implication/Formulation 2
:$\vdash \paren {\paren {p \implies q} \implies \paren {p \implies r} } \implies \paren {p \implies \paren {q \implies r} }$
Odd Integers under Addition do not form Group
Let $S$ be the set of odd integers: :$S = \set {x \in \Z: \exists n \in \Z: x = 2 n + 1}$ Let $\struct {S, +}$ denote the algebraic structure formed by $S$ under the operation of addition. Then $\struct {S, +}$ is not a group.
Number of Digits in Number
Let $n \in \Z_{>0}$ be a strictly positive integer. Let $b \in \Z_{>1}$ be an integer greater than $1$. Let $n$ be expressed in base $b$. Then the number of digits $d$ in this expression for $n$ is: :$d = 1 + \floor {\log_b n}$ where: :$\floor {\, \cdot \,}$ denotes the floor function :$\log_b$ denotes the logarithm to base $b$.
Every Tenth Power of Two Minus Every Third Power of Ten is Divisible By Three
Let $x \in \Z_{\ge 0}$ be a non-negative integer. Then $2^{10 x} - 10^{3 x}$ is divisible by $3$. That is: :$2^{10 x} - 10^{3 x} \equiv 0 \pmod 3$
Gödel's Incompleteness Theorems/Second
Let $T$ be the set of theorems of some recursive set of sentences in the language of arithmetic such that $T$ contains minimal arithmetic. Let $\map {\mathrm {Cons} } T$ be the propositional function which states that $T$ is consistent. Then it is not possible to prove $\map {\mathrm {Cons} } T$ by means of formal statements within $T$ itself.
Linear Function on Stationary Stochastic Model is Stationary
Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$ Let $L_t$ be a linear function of $\sequence {s_n}$: :$L_t = l_1 z_t + l_2 z_{t - 1} + \dotsb + l_n z_{t - n + 1}$ Then $L_t$ is itself stationary.
Strict Ordering on Integers is Well-Defined
Let $\eqclass {a, b} {}$ denote an integer, as defined by the formal definition of integers. Let: {{begin-eqn}} {{eqn | l = \eqclass {a, b} {}       | r = \eqclass {a', b'} {}       | c =  }} {{eqn | l = \eqclass {c, d} {}       | r = \eqclass {c', d'} {}       | c =  }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \eqclass {a, b} {}       | o = <       | r = \eqclass {c, d} {}       | c =  }} {{eqn | ll= \iff       | l = \eqclass {a', b'} {}       | o = <       | r = \eqclass {c', d'} {}       | c =  }} {{end-eqn}}
Definition:Gaussian Process
Let $S$ be a stochastic process giving rise to a time series $T$. Let the probability distribution of $T$ be a multivariate normal distribution. Then $S$ is called a '''Gaussian process'''.
Second Order Weakly Stationary Gaussian Stochastic Process is Strictly Stationary
Let $S$ be a Gaussian stochastic process giving rise to a time series $T$. Let $S$ be weakly stationary of order $2$. Then $S$ is strictly stationary of order $2$.
Ordering on Integers is Transitive
Let $\eqclass {a, b} {}$ denote an integer, as defined by the formal definition of integers. Then: {{begin-eqn}} {{eqn | l = \eqclass {a, b} {}       | o = \le       | r = \eqclass {c, d} {}       | c =  }} {{eqn | lo= \land       | l = \eqclass {c, d} {}       | o = \le       | r = \eqclass {e, f} {}       | c =  }} {{eqn | ll= \implies       | l = \eqclass {a, b} {}       | o = \le       | r = \eqclass {e, f} {}       | c =  }} {{end-eqn}} That is, ordering on the integers is transitive.
Strict Ordering on Integers is Asymmetric
Let $\eqclass {a, b} {}$ denote an integer, as defined by the formal definition of integers. Then: {{begin-eqn}} {{eqn | l = \eqclass {a, b} {}       | o = <       | r = \eqclass {c, d} {}       | c =  }} {{eqn | lo= \implies       | l = \eqclass {c, d} {}       | o = \not <       | r = \eqclass {a, b} {}       | c =  }} {{end-eqn}} That is, strict ordering on the integers is asymmetric.
Negative of Integer
Let $x \in \Z$ be an integer. Let $x = \eqclass {a, b} {}$ be defined from the formal definition of integers, where $\eqclass {a, b} {}$ is an equivalence class of ordered pairs of natural numbers. Then: :$-x = \eqclass {b, a} {}$
Symmetric Function Theorem
Let $f$ be a polynomial in $n$ variables. Let $f$ be of degree $r$ in each of its $n$ variables. Then $f$ is equal to a polynomial of total degree $r$ with integer coefficients in the elementary symmetric functions: :$ds \sum x_i \sim x_i x_j, \dotsc, \prod x_j$ and the coefficients of $f$.
Euler's Integral Theorem
:$\ds H_n = \ln n + \gamma + \map \OO {\dfrac 1 n}$ where: :$H_n$ denotes the $n$th harmonic number :$\gamma$ denotes the Euler-Mascheroni constant.
Ordering on Positive Integers is Equivalent to Ordering on Natural Numbers
Let $u, v \in \Z_{>0}$ be natural numbers. Consider the mapping $\phi: \N_{>0} \to \Z_{>0}$ defined as: :$\forall u \in \N_{>0}: \map \phi u = u'$ where $u' \in \Z$ denotes the (strictly) positive integer $\eqclass {b + u, b} {}$. Let $u', v' \in \Z_{>0}$ be strictly positive integers. Then: :$u > v \iff u' > v'$