nyuuzyou/wikis huggingface dataset:
Geodesic Ball as Metric Ball in Riemannian Manifold

== Theorem ==
Let $\struct {M, g}$ be a connected Riemannian manifold.
Let $U^c$ and $U^o$ be closed and open geodesic balls with radii $R_c$ and $R_o$ respectively in $M$.
Then $U^c$ is a closed metric ball with radius $\epsilon_c = R_c$, and $U^o$ is an open metric ball with radius $\epsilon_o = R_o$.
== Proof ==
== Sources ==
* : $\S 6$: Geodesics and Distance. Geodesics Are Locally Minimizing
Category:Riemannian Manifolds
Geodesic Equation/2d Surface Embedded in 3d Euclidean Space

== Theorem ==
Let $\sigma: U \subset \R^2 \to V \subset \R^3$ be a smooth surface specified by a vector-valued function:
:$\mathbf r = \map {\mathbf r} {u, v}$
Then a geodesic of $\sigma$ satisfies the following system of differential equations:
:$\dfrac {E_u u'^2 + 2 F_u u' v' + G_u v'^2} {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } - \dfrac \d {\d t} \dfrac {2 \paren {E u' + F v'} } {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } = 0$
:$\dfrac {E_v u'^2 + 2 F_v u' v' + G_v v'^2} {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } - \dfrac \d {\d t} \dfrac {2 \paren {F u' + G v'} } {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } = 0$
where $E, F, G$ are the functions of the first fundamental form:
:$E = {\mathbf r}_u \cdot {\mathbf r}_u, F = {\mathbf r}_u \cdot {\mathbf r}_v, G = {\mathbf r}_v \cdot {\mathbf r}_v$
== Proof ==
A curve on the surface $\mathbf r$ can be specified as $u = \map u t$, $v = \map v t$
The arc length between the points corresponding to $t_0$ and $t_1$ equals:
:$\ds J \sqbrk {u, v} = \int_{t_0}^{t_1} \sqrt {E u'^2 + 2 F u'v' + G v'^2} \rd t$
The following derivatives will appear in Euler's Equations:
Analogous relations hold for derivatives $v$ and $v'$.
Euler's Equation together with the results above yield the geodesic equations.
== Sources ==
* : $\S 2.9$: The Fixed End Point Problem for n Unknown Functions
Category:Calculus of Variations
Geodesic Equation/2d Surface Embedded in 3d Euclidean Space/Cylinder

== Theorem ==
Let $\sigma$ be the surface of a cylinder.
Let $\sigma$ be embedded in 3-dimensional Euclidean space.
Let $\sigma$ be parameterised by $\tuple {\phi, z}$ as
:$\mathbf r = \tuple {a \cos \phi, a \sin \phi, z}$
where
:$a > 0$
and
:$z, \phi \in \R$
Then geodesics on $\sigma$ are of the following form:
:$z = C_1 \phi + C_2$
where $C_1, C_2$ are real arbitrary constants.
== Proof ==
From the given parametrization it follows that:
where $E, F, G$ are the functions of the first fundamental form.
Furthermore, all derivatives of $E, F, G$ $\phi$ and $z$ vanish.
Then geodesic equations read:
:$\dfrac \d {\d t} \dfrac {a^2 \phi'} {\sqrt {a^2 \phi'^2 + z'^2} } = 0$
:$\dfrac \d {\d t} \dfrac {z'} {\sqrt {a^2 \phi'^2 + z'^2} } = 0$
Integrate these differential equations once:
:$\dfrac {a^2 \phi'} {\sqrt {a^2 \phi'^2 + z'^2} } = b_1$
:$\dfrac {z'} {\sqrt {a^2 \phi'^2 + z'^2} } = b_2$
where $b_1, b_2$ are real arbitrary constants.
Divide the first equation by the second one:
:$\dfrac {a^2 \phi'} {z'} = \dfrac {b_1} {b_2}$
To solve this in terms of $z$ as a function of $\phi$, define:
:$C_1 = \dfrac {a^2 b_2} {b_1}$ 
and use the chain rule:
:$\dfrac {\d z} {\d \phi} = C_1$
Integration $\phi$ yields the desired result.
In other words, geodesics are helical lines.
== Sources ==
* : $\S 2.9$: The Fixed End Point Problem for n Unknown Functions
Category:Calculus of Variations
Geodesic Sphere as Metric Sphere in Riemannian Manifold

== Theorem ==
Let $\struct {M, g}$ be a connected Riemannian manifold.
Let $U$ be geodesic sphere with radius $R$ in $M$.
Then $U$ is a metric sphere with radius $\epsilon = R$.
== Proof ==
== Sources ==
* : $\S 6$: Geodesics and Distance. Geodesics Are Locally Minimizing
Category:Riemannian Manifolds
Geodesic in Normal Neighborhood

== Theorem ==
Let $\struct {M, g}$ be an $n$-dimensional Riemannian manifold.
Let $U_p$ be the normal neighborhood for $p \in M$.
Let $\struct {U_p, \tuple {x^i}}$ be a normal coordinate chart.
Let $T_p M$ be the tangent space of $M$ at $p \in M$.
Let $v = v^i \valueat{\dfrac \partial {\partial x^i}} p \in T_p M$.
Let $I \subseteq \R$ be a real interval.
Let $\map {\gamma_v} t : I \to M$ be the geodesic such that:
:$\map {\gamma_v} 0 = p$
:$\map {\gamma_v'} 0 = v$
where $\gamma_v'$ denotes the velocity of $\gamma_v$.
Suppose:
:$t \in I : 0 \in I : \map {\gamma_v} I \subseteq U_p$
Then in normal coordinates:
:$\map {\gamma_v} t = \tuple {t v^1, \ldots, t v^n}$
== Proof ==
== Sources ==
* : $\S 5$: The Levi-Civita Connection. Normal Neighborhoods and Normal Coordinates
Category:Normal Neighborhoods
Category:Geodesics
Geometric Congruence is Equivalence Relation

== Theorem ==
Let $S$ be the set of geometric figures.
For $F_1, F_2 \in S$, let $F_1 \cong F_2$ denote that $F_1$ is congruent to $F_2$.
Then $\cong$ is an equivalence relation on $S$.
== Proof ==
Checking in turn each of the criteria for equivalence:
=== Reflexivity ===
=== Symmetry ===
=== Transitivity ===
$\cong$ has been shown to be reflexive, symmetric and transitive.
Hence by definition it is an equivalence relation.
== Sources ==
* 
* 
Category:Congruence (Geometry)
Category:Examples of Equivalence Relations
Geometric Distribution Gives Rise to Probability Mass Function

== Theorem ==
Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.
Let $X$ have the geometric distribution with parameter $p$ (where $0 < p < 1$).
Then $X$ gives rise to a probability mass function.
=== Shifted Geometric Distribution ===
The same result applies to the shifted geometric distribution.
== Proof ==
By definition:
:$\map \Omega X = \N = \set {0, 1, 2, \ldots}$
:$\map \Pr {X = k} = p^k \paren {1 - p}$
Then:
The above result is valid, because $0 < p < 1$.
So $X$ satisfies $\map \Pr \Omega = 1$, and hence the result.
Category:Geometric Distribution
Geometric Distribution Gives Rise to Probability Mass Function/Shifted

== Theorem ==
Let $Y$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.
Let $Y$ have the shifted geometric distribution with parameter $p$ (where $0 < p < 1$).
Then $Y$ gives rise to a probability mass function.
== Proof ==
By definition:
:$\map \Omega Y = \N_{>0} = \set {1, 2, 3, \ldots}$
:$\map \Pr {Y = k} = p \paren {1 - p}^{k - 1}$
Then:
So $Y$ satisfies $\map \Pr \Omega = 1$, and hence the result.
== Sources ==
* : $\S 2.2$: Examples
Category:Shifted Geometric Distribution
Category:Probability Mass Functions
Geometric Mean of Reciprocals is Reciprocal of Geometric Mean

== Theorem ==
Let $x_1, x_2, \ldots, x_n \in \R_{> 0}$ be strictly positive real numbers.
Let $G_n$ denote the geometric mean of $x_1, x_2, \ldots, x_n$.
Let ${G_n}'$ denote the geometric mean of their reciprocals $\dfrac 1 {x_1}, \dfrac 1 {x_2}, \ldots, \dfrac 1 {x_n}$.
Then:
:${G_n}' = \dfrac 1 {G_n}$
== Proof ==
Category:Geometric Mean
Category:Reciprocals
Geometric Mean of two Positive Real Numbers is Between them

== Theorem ==
Let $a, b \in \R$ be real numbers such that $0 < a < b$.
Let $\map G {a, b}$ denote the geometric mean of $a$ and $b$.
Then:
:$a < \map G {a, b} < b$
== Proof ==
By definition of geometric mean:
:$\map G {a, b} := \sqrt {a b}$
where $\sqrt {a b}$ specifically denotes the positive square root of $a$ and $b$.
Thus:
and:
Category:Geometric Mean
Geometric Sequence in Lowest Terms has Coprime Extremes/Proof 1

== Theorem ==
== Proof ==
Let $a_0, a_1, a_2, \ldots, a_n$ be natural numbers.
Let $\sequence {a_k}_{0 \mathop \le k \mathop \le n}$ be a geometric sequence with common ratio $r$.
Let $a_0, a_1, \ldots, a_n$ be the smallest such natural numbers.
From , let $d_0, d_1$ be the smallest natural numbers such that $d_1 = r d_0$.
From one can build a sequence of $3, 4, \ldots, n$ terms with the same property.
Let the geometric sequence so constructed with $n$ terms be $\sequence {b_k}_{0 \mathop \le k \mathop \le n}$.
From , $d_0$ and $d_1$ are coprime.
From , each of the extreme terms of the intermediate geometric sequences is a power of a natural number.
From , each of those extreme terms is coprime.
We have that $a_0, \ldots, a_n$ are the smallest such natural numbers such that $\sequence {a_k}_{0 \mathop \le k \mathop \le n}$ is a geometric sequence with common ratio $r$.
We have also constructed $b_0, \ldots, b_n$ to have the same property.
Thus:
: $a_0 = b_0, a_1 = b_1, \ldots, a_n = b_n$
and the result follows.
== Sources ==
* : Book $\text{VIII}$. Propositions
Category:Geometric Sequence in Lowest Terms has Coprime Extremes
Geometric Sequence in Lowest Terms has Coprime Extremes/Proof 2

== Theorem ==
== Proof ==
Let $P$ be a geometric sequence of natural numbers of length $n$.
Let the common ratio of $P$ be expressed in canonical form as $\dfrac p q$.
From Construction of Geometric Sequence in Lowest Terms:
:$P = \paren {q^n, p q^{n - 1}, p^2 q^{n - 2}, \ldots, p^{n - 1} q, p^n}$
By definition of canonical form:
:$p \perp q$
It follows from Powers of Coprime Numbers are Coprime that:
:$p^n \perp q^n$
Hence the result.
Category:Geometric Sequence in Lowest Terms has Coprime Extremes
Geometric Sequence of Integers with Integer Common Ratio

== Theorem ==
Let $P = \sequence {a_j}_{1 \mathop \le j \mathop \le n}$ be a geometric sequence of length $n$ consisting entirely of integers.
Let $r$ be the common ratio of $P$.
Then $r$ is an integer :
:$\forall i, j \in \set {1, 2, \ldots, n}, i \le j: a_i \divides a_j$
That is, terms of $P$ divide later terms of $P$ $r$ is an integer.
== Proof ==
=== Necessary Condition ===
Let $r$ be an integer.
By definition of geometric sequence, the terms of $P$ are in the form:
:$a_k = b r^{k - 1}$
where $b$ and $k$ are integers.
It follows from Integer Multiplication is Closed that $a_k$ is an integer.
=== Sufficient Condition ===
From Common Ratio in Integer Geometric Sequence is Rational, $r$ is a rational number.
Let $r$ be expressed in canonical form as:
:$r = \dfrac p q$
where, by definition of canonical form, $p \perp q$, that is, $p$ is coprime to $q$.
From Form of Geometric Sequence of Integers, the terms of $P$ are in the form:
:$(1): \quad a_j = k q^{j - 1} p^{n - j}$
Let $a_i \divides a_j$.
From Powers of Coprime Numbers are Coprime:
:$(2): \quad q^{j - i} \perp p^{j - i}$
Then:
But from $(2)$:
:$q^{j - i} \perp p^{j - i}$
Thus $q^{j - i} \divides p^{j - i}$ can happen only when $q^{j - 1} = q = 1$.
That is, when $r$ is an integer.
Category:Geometric Sequences of Integers
Geometric Sequence with Coprime Extremes is in Lowest Terms

== Theorem ==
Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be a geometric sequence of integers.
Let:
:$a_0 \perp a_n$
where $\perp$ denotes coprimality.
Then $G_n$ is in its lowest terms.
== Proof 1 ==
== Proof 2 ==
Category:Geometric Sequences of Integers
Category:Geometric Sequence with Coprime Extremes is in Lowest Terms
Geometric Sequence with Coprime Extremes is in Lowest Terms/Proof 1

== Theorem ==
== Proof ==
Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be natural numbers in geometric sequence such that $a_0 \perp a_n$.
 there were to exist another set of natural numbers in geometric sequence:
:$G\,'_n = \sequence {b_0, b_1, \cdots, b_n}$
with the same common ratio where:
:$\forall k \in \N_{\le n}: a_k > b_k$
From :
:$a_0 : a_n = b_0 : b_n$
But by hypothesis:
:$a_0 \perp a_n$
and so from:
:
and:
:
it follows that:
:$a_0 \divides b_0$
However, this contradicts the assumption that $b_0 < a_0$.
Therefore $a_0, a_1, \cdots, a_n$ are the least of those with the same common ratio.
== Sources ==
* : Book $\text{VIII}$. Propositions
Category:Geometric Sequence with Coprime Extremes is in Lowest Terms
Geometric Sequence with Coprime Extremes is in Lowest Terms/Proof 2

== Theorem ==
== Proof ==
Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be natural numbers in geometric sequence such that $a_0 \perp a_n$.
 $G\,'_n = \sequence {b_0, b_1, \cdots, b_n}$ be another set of natural numbers in geometric sequence with the same common ratio where:
:$\forall k \in \N_{\le n}: a_k > b_k$
By definition of geometric sequence:
:$a_0 = r^n a_n$
:$b_0 = r^n b_n$
Hence:
:$\dfrac {a_0} {a_n} = \dfrac {b_0} {b_n}$
By hypothesis:
: $a_0 \perp a_n$
Thus $\dfrac {a_0} {a_n}$ is in canonical form.
From Canonical Form of Rational Number is Unique it follows that $a_0$ and $a_n$ are the only two integers fulfilling these conditions.
Thus:
:$\forall p, q: \dfrac p q = \dfrac {a_0} {a_n}: a_0 \le p, a_n \le q$
But it was supposed that:
:$b_0 < a_0$
:$b_n < a_n$
From this contradiction it follows that there can be no such $b_0, \ldots, b_n$.
Hence the result.
Category:Geometric Sequence with Coprime Extremes is in Lowest Terms
Geometric Sequences in Proportion have Same Number of Elements

== Theorem ==
Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers of length $n$.
Let $r$ be the common ratio of $P$.
Let $Q = \sequence {b_j}_{0 \mathop \le j \mathop \le m}$ be a geometric sequence of integers of length $m$.
Let $r$ be the common ratio of $Q$.
Let $b_0$ and $b_m$ be such that $\dfrac {b_0} {b_m} = \dfrac {a_0} {a_n}$.
Then $m = n$.
== Proof ==
Let $S = \sequence {c_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers of length $n$ such that:
:$r$ is the common ratio of $S$
:$S$ is in its lowest terms.
From Geometric Sequence in Lowest Terms has Coprime Extremes, $c_0$ is coprime to $c_n$.
Then:
:$\dfrac {c_0} {c_n} = \dfrac {a_0} {a_n} = \dfrac {b_0} {b_m}$
But:
:$c_n = r^n c_0$
and so:
:$b_m = r^n b_0$
The result follows.
== Sources ==
* : Book $\text{VIII}$. Propositions
Category:Geometric Sequences of Integers
Geometrical Interpretation of Complex Addition

== Theorem ==
Let $a, b \in \C$ be complex numbers expressed as vectors $\mathbf a$ and $\mathbf b$ respectively.
Let $OA$ and $OB$ be two adjacent sides of the parallelogram $OACB$ such that $OA$ corresponds to $\mathbf a$ and $OB$ corresponds to $\mathbf b$.
Then the diagonal $OC$ of $OACB$ corresponds to $\mathbf a + \mathbf b$, the sum of $a$ and $b$ expressed as a vector.
== Proof ==
:
Let $a = a_x + i a_y$ and $b = b_x + i b_y$.
Then by definition of complex addition:
:$a + b = \paren {a_x + b_x} + i \paren {a_y + b_y}$
Thus $\mathbf a + \mathbf b$ is the vector whose components are $a_x + b_x$ and $a_y + b_y$.
Similarly, we have:
:$b + a = \paren {b_x + a_x} + i \paren {b_y + a_y}$
Thus $\mathbf b + \mathbf a$ is the vector whose components are $b_x + a_x$ and $b_y + a_y$.
It follows that both $\mathbf a + \mathbf b$ and $\mathbf b + \mathbf a$ both correspond to the diagonal $OC$ of $OACB$.
== Sources ==
* : Chapter $\text I$: Functions of a Complex Variable: $\S 3$. Geometric Representation of Complex Numbers
* : $\S 2$. Geometrical Representations
Category:Complex Addition
Category:Geometry of Complex Plane
Geometrical Interpretation of Complex Modulus

== Theorem ==
Let $z \in \C$ be a complex number expressed in the complex plane.
Then the modulus of $z$ can be interpreted as the distance of $z$ from the origin.
== Proof ==
Let $z = x + i y$.
By definition of the complex plane, it can be represented by the point $\tuple {x, y}$.
By the Distance Formula, the distance $d$ of $z$ from the origin is:
which is precisely the modulus of $z$.
== Sources ==
* : Chapter $\text{II}$: Groups: Subgroups
Category:Complex Modulus
Category:Geometry of Complex Plane
Geometrical Interpretation of Complex Subtraction

== Theorem ==
Let $a, b \in \C$ be complex numbers expressed as vectors $\mathbf a$ and $\mathbf b$ respectively.
Let $OA$ and $OB$ be two adjacent sides of the parallelogram $OACB$ such that $OA$ corresponds to $\mathbf a$ and $OB$ corresponds to $\mathbf b$.
Then the diagonal $BA$ of $OACB$ corresponds to $\mathbf a - \mathbf b$, the difference of $a$ and $b$ expressed as a vector.
== Proof ==
:
By definition of vector addition:
:$OB + BA = OA$
That is:
:$\mathbf b + \vec {BA} = \mathbf a$
which leads directly to:
:$\vec {BA} = \mathbf a - \mathbf b$
== Sources ==
* : Chapter $\text I$: Functions of a Complex Variable: $\S 3$. Geometric Representation of Complex Numbers
* : $\S 2$. Geometrical Representations
* : $1$: Solved Problems: Graphical Representations of Complex Numbers. Vectors: $8 \ \text {(a)}$
Category:Complex Addition
Category:Geometry of Complex Plane
Gershgorin Circle Theorem

== Theorem ==
Let $n$ be a positive integer.
Let $A = \sqbrk {a_{i j} }$ be a complex square matrix of order $n$.
Let $\lambda$ be an eigenvalue of $A$.
Then there exists $i \in \set {1, 2, \ldots, n}$ such that:
:$\lambda \in \map {\mathbb D} {a_{i i}, R_i}$
where:
:$\ds R_i = \sum_{j \mathop \ne i} \cmod {a_{ i j} }$
:$\map {\mathbb D} {a, R}$ denotes the complex disk of center $a$ and radius $R$.
== Proof ==
== Sources ==
* 
* 
Category:Linear Algebra
Gibbs Phenomenon/Historical Note

== Historical Note on Gibbs Phenomenon ==
The Gibbs phenomenon was first reported in print in $1848$ by .
However, this went unnoticed at the time.
 published a short note in $1899$ on the subject of the Fourier series of a square wave, but failed to report on the phenomenon at that time.
Later that year he published a correction to that note in which the overshoot was described accurately.
In $1906$ gave a complete analysis of the mathematics behind the phenomenon, and called it the Gibbs Phenomenon.
 paper was later brought to light, but by that time it was generally attributed to .
In the words of ($1925$):
:We may still call this property of Fourier's series (and certain other series) Gibbs's phenomenon; but we must no longer claim that the property was first discovered by Gibbs.
== Sources ==
* 
* 
* 
* 
* 
* 
Category:Historical Notes
Gigantic Palindromic Prime

== Theorem ==
The integer defined as:
:$10^{11 \, 810} + 1 \, 465 \, 641 \times 10^{5902} + 1$
is a gigantic prime which is also palindromic.
That is:
:$1(0)_{5901}1465641(0)_{5901}1$
where $\left({a}\right)_b$ means $b$ instances of $a$ in a string.
== Proof ==
It is clear that this number is palindromic.
It is also noted that it has $1 + 5901 + 7 + 5901 + 1 = 11 \, 811$ digits, making it gigantic.
== Sources ==
* 
* : $10_{5901}14656410_{5901}1$
Category:Gigantic Primes
Category:Palindromic Primes
Giuga Number/Examples/1722

== Example of Giuga Number ==
$1722$ is a Giuga number:
:$\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 7 + \dfrac 1 {41} - \dfrac 1 {1722} = 1$
== Proof ==
We have that:
:$1722 = 2 \times 3 \times 7 \times 41$
Then:
Hence the result by definition of Giuga number.
== Sources ==
* 
Category:Giuga Numbers
Category:1722
Giuga Number/Examples/244,197,000,982,499,715,087,866,346

== Example of Giuga Number ==
$244 \, 197 \, 000 \, 982 \, 499 \, 715 \, 087 \, 866 \, 346$ is a Giuga number:
:$\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 {11} + \dfrac 1 {23} + \dfrac 1 {31} + \dfrac 1 {47 \, 137} + \dfrac 1 {28 \, 282 \, 147} + \dfrac 1 {3 \, 892 \, 535 \, 183} - \dfrac 1 {244 \, 197 \, 000 \, 982 \, 499 \, 715 \, 087 \, 866 \, 346} = 1$
== Proof ==
We have that:
:$244 \, 197 \, 000 \, 982 \, 499 \, 715 \, 087 \, 866 \, 346 = 2 \times 3 \times 11 \times 23 \times 31 \times 47 \, 137 \times 28 \, 282 \, 147 \times 3 \, 892 \, 535 \, 183$
Then we have:
Adding all these up together:
Dividing both sides by $244 \, 197 \, 000 \, 982 \, 499 \, 715 \, 087 \, 866 \, 347$ makes the result apparent by definition of Giuga number.
== Sources ==
* 
* : $244,197,000,982,499,715,087,866,346$
Category:Giuga Numbers
Category:244,197,000,982,499,715,087,866,346
Giuga Number/Examples/30

== Example of Giuga Number ==
$30$ is a Giuga number (the smallest):
:$\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 {30} = 1$
== Proof ==
We have that:
:$30 = 2 \times 3 \times 5$
Then:
Hence the result by definition of Giuga number.
== Sources ==
* 
* : $30$
Category:Giuga Numbers
Category:30
Giuga Number/Examples/858

== Example of Giuga Number ==
$858$ is a Giuga number:
:$\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 {11} + \dfrac 1 {13} - \dfrac 1 {858} = 1$
== Proof ==
We have that:
:$858 = 2 \times 3 \times 11 \times 13$
Then:
Hence the result by definition of Giuga number.
== Sources ==
* 
* : $30$
Category:Giuga Numbers
Category:858
Globally-Defined Symmetric Linear Operator is Bounded and Self-Adjoint

== Theorem ==
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\tuple {\HH, T}$ be a symmetric linear operator.
Then $\tuple {\HH, T}$ is bounded and self-adjoint.
== Proof ==
Let $\tuple {\map D {T^\ast}, T^\ast}$ be the adjoint of $\tuple {\HH, T}$. 
From Adjoint of Symmetric Densely-Defined Linear Operator Extends Operator, $\tuple {\map D {T^\ast}, T^\ast}$ extends $\tuple {\HH, T}$. 
However, $\map D {T^\ast} \subseteq \HH$. 
So we have $\map D {T^\ast} = \HH = \map D T$.
Hence $T^\ast x = T x$ for each $x \in \map D T$.
So $\tuple {\HH, T} = \tuple {\HH, T^\ast}$.
So $\tuple {\HH, T}$ is self-adjoint.
From Adjoint of Densely-Defined Linear Operator is Closed, $\tuple {\HH, T}$ is also closed.
Since $\HH$ is Banach, we have that:
:$T$ is bounded
from the Closed Graph Theorem.
== Sources ==
* : $13.11$: Theorem
Category:Symmetric Densely-Defined Linear Operators
Category:Self-Adjoint Densely-Defined Linear Operators
Golay Ternary Code Corrects 2 Errors

== Theorem ==
The Golay ternary code corrects $2$ transmission errors.
== Proof ==
We have that Golay Ternary Code has Minimum Distance 5.
The result follows from Error Correction Capability of Linear Code.
== Sources ==
* : Chapter $6$: Error-correcting codes: Example $6.15$
Category:Golay Ternary Code
Golay Ternary Code has Minimum Distance 5

== Theorem ==
The Golay ternary code has a minimum distance of $5$.
== Proof ==
Let $C$ denote the Golay ternary code.
By inspection of the standard generator matrix $G$ of $C$:
:$G := \begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 1 \\
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 \\
0 & 0 & 1 & 0 & 0 & 0 & 2 & 1 & 0 & 1 & 2 \\
0 & 0 & 0 & 1 & 0 & 0 & 2 & 2 & 1 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 
\end{pmatrix}$
it is seen that the smallest weight of all the codewords of $C$ that can be found in $G$ is $5$.
So it is immediately seen that the minimum distance of $C$ is at least $5$.
It remains to be shown that the minimum distance of $C$ is no more than $5$.
== Sources ==
* : Chapter $6$: Error-correcting codes: Example $6.15$
Category:Golay Ternary Code
Goldbach's Lesser Conjecture

== False Conjecture ==
Every positive odd integer $n$ can be expressed in the form:
:$n = 2 a^2 + p$
where:
:$a \in \Z_{\ge 0}$ is a non-negative integer
:$p$ is a prime number or $1$.
== Refutation ==
There are two known counterexamples:
:$5777$ and $5993$
as follows:
=== $5777$ is a Stern Number ===
=== $5993$ is a Stern Number ===
== Historical Note ==
== Sources ==
* : $5777$
* 
* 
* : $5777$
Category:Famous False Conjectures
Category:Prime Numbers
Category:Stern Numbers
Category:Polynomial Expressions for Primes
Category:Goldbach's Lesser Conjecture
Goldbach's Lesser Conjecture/5777

== Goldbach's Lesser Conjecture: $5777$ is a Stern Number ==
The number $5777$ cannot be represented in the form:
:$5777 = 2 a^2 + p$
where:
:$a \in \Z_{\ge 0}$ is a non-negative integer
:$p$ is a prime number.
== Proof ==
It will be shown that for all $a$ such that $2 a^2 \le 5777$, it is never the case that $5777 - 2 a^2$ is prime.
Thus:
That exhausts all $a$ such that $2 a^2 \le 5775$, as $2 \times 54^2 = 5832$.
Hence the result.
== Sources ==
* : $5777$
* : $5777$
Category:Goldbach's Lesser Conjecture
Category:5777
Goldbach's Lesser Conjecture/5993

== Goldbach's Lesser Conjecture: $5993$ is a Stern Number ==
The number $5993$ cannot be represented in the form:
:$5993 = 2 a^2 + p$
where:
:$a \in \Z_{\ge 0}$ is a non-negative integer
:$p$ is a prime number.
== Proof ==
It will be shown that for all $a$ such that $2 a^2 \le 5993$, it is never the case that $5993 - 2 a^2$ is prime.
Thus:
That exhausts all $a$ such that $2 a^2 \le 5993$, as $2 \times 55^2 = 6050$.
Hence the result.
== Sources ==
* : $5777$
* : $5777$
Category:Goldbach's Lesser Conjecture
Category:5993