hcju/proofwikips huggingface dataset:
Definition:Chebyshev-Sylvester Constant
The '''Chebyshev-Sylvester constant''' is a mathematical constant $\alpha$ which was shown by {{AuthorRef|Pafnuty Lvovich Chebyshev}} and {{AuthorRef|James Joseph Sylvester}} that for sufficiently large $x$, there exists at least one prime number $p$ satisfying: :$x < p < \paren {1 + \alpha} x$ The number $\alpha$ was demonstrated to be $0 \cdotp 092 \ldots$ In $1896$, {{AuthorRef|Jacques Salomon Hadamard}} and {{AuthorRef|Charles de la Vallée Poussin}} independently proved the Prime Number Theorem. This demonstrated that the above inequality is true for all $\alpha > 0$ for sufficiently large $x$,. Hence this constant is now only of historical interest. {{NamedforDef|Pafnuty Lvovich Chebyshev|name2 = James Joseph Sylvester|cat = Chebyshev|cat2 = Sylvester}}
Definition:Du Bois-Reymond Constants
The '''du Bois-Reymond constants''' are the constants $C_n$ where: :$C_n = \displaystyle \int_0^\infty \size {\map {\dfrac \d {\d t} } {\dfrac {\sin t} t}^n} \rd t - 1$
Definition:Copeland-Erdős Constant
The '''Copeland-Erdős constant''' is the real number whose decimal expansion is formed by concatenating the prime numbers in ascending order: :$C_{10} = 0 \cdotp 23571 \, 11317 \, 19232 \, 93137 \, 4143 \ldots$ {{OEIS|A033308}}
Definition:Koebe Constant
The Koebe Quarter Theorem states: {{:Koebe Quarter Theorem}} That constant $\dfrac 1 4$ by which $\size {\map {f'} 0}$ is bounded below can be found referred to as the '''Koebe constant'''. {{NamedforDef|Paul Koebe|cat = Koebe}}
Definition:Meissel-Mertens Constant
Consider the expression: :$\displaystyle M = \map {\lim_{n \mathop \to \infty} } {\sum_{\substack {p \mathop \le n \\ \text {$p$ prime} } } \dfrac 1 p - \ln \ln p}$ Then: :$M \approx 0 \cdotp 26149 \, 72128 \, 47642 \, 78375 \, 54268 \, 38608 \, 69585 \, 90516 \ldots$ {{OEIS|A077761}} The constant $M$ is known as the '''Meissel-Mertens Constant'''.
Definition:Matroid Theory
'''Matroid Theory''' is the branch of mathematics which concerns the role which matroids play in disparate branches of combinatorial theory and algebra such as graph theory, lattice theory, combinatorial optimization, and linear algebra.
Definition:Kakeya's Constant
'''Kakeya's constant''' is defined as the area of the smallest simple convex domain in which one can put a line segment of length $1$ which will coincide with itself when rotated $180 \degrees$: :$K = \dfrac {\paren {5 - 2 \sqrt 2} \pi} {24} \approx 0 \cdotp 28425 \, 82246 \ldots$ {{OEIS|A093823}} {{expand|Needs considerable work done here by someone who understands exactly what is going on here. The case of the equilateral triangle is well known; so is the case of the Perron tree; I also remember a piece by Martin Gardner on the subject which demonstrates that a star-shaped area derived from the deltoid can be made arbitrarily small; and so on. Exactly what is meant here by simple convex domain needs rigorous clarification.}}
Definition:Matroid
Let $M = \struct{S,\mathscr I}$ be an independence system.
Definition:Matroid Axioms
 Let $S$ be a finite set. Let $\mathscr I$ be a set of subsets of $S$. The '''matroid axioms''' are the conditions on $S$ and $\mathscr I$ in order for the ordered pair $\struct {S, \mathscr I}$ to be a matroid:  === Axioms 1 === {{:Definition:Matroid Axioms/Axioms 1}} === Axioms 2 === {{:Definition:Matroid Axioms/Axioms 2}} === Axioms 3 === {{:Definition:Matroid Axioms/Axioms 3}} === Axioms 4 === {{:Definition:Matroid Axioms/Axioms 4}}
Definition:Matroid/Independent Set
An element of $\mathscr I$ is called an '''independent set''' of $M$.
Definition:Matroid/Dependent Set
A subset of $S$ that is not an element of $\mathscr I$ is called a '''dependent set''' of $M$.
Definition:Base of Matroid
A '''base''' of $M$ is a maximal independent subset of $S$.
Definition:Rank Function (Matroid)
The '''rank function''' of $M$ is the mapping $\rho : \powerset S \to \Z$ from the power set of $S$ into the integers defined by: :$\forall A \subseteq S : \map \rho A = \max \set {\size X : X \subseteq A \land X \in \mathscr I}$ where $\size A$ denotes the cardinality of $A$.
Definition:Rank (Matroid)
The '''rank''' of $M$, denoted by $\map \rho M$, is the image of $S$ under $\rho$, that is: :$\map \rho M = \map \rho S$.
Definition:Flat (Matroid)
A subset $A \subseteq S$ is a '''flat''' of $M$ : :$\forall x \in S \setminus A : \map \rho {A \cup \set x} = \map \rho A + 1$
Definition:Depends Relation (Matroid)
For $x \in S$ and $A \subseteq S$, $x$ is said to '''depend''' on $A$, denoted as $x \sim A$, : :$\map \rho {A \cup \set x} = \map \rho A$
Definition:Closure Operator (Matroid)
The '''closure operator''' of the matroid $M$ is the mapping $\sigma : \powerset S \to \powerset S$ defined by: :$\map \sigma A$ is the set of elements of $S$ which depend on $A$
Definition:Spanning Subset (Matroid)
A subset $X \subseteq S$ is '''spanning''' in $M$ : :there exists a base $B$ contained in $X$
Definition:Circuit (Matroid)
A '''circuit''' of $M$ is a minimal dependent subset of $S$.
Definition:Base Axiom (Matroid)
{{begin-axiom}} {{axiom | n = \text B 1         | q = \forall B_1, B_2 \in \mathscr B         | mr= x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_1 \cup \set y} \setminus \set x \in \mathscr B }} {{end-axiom}}
Definition:Rank Axioms (Matroid)/Definition 1
{{begin-axiom}} {{axiom | n = \text R 1         | mr= \map \rho \O = 0 }} {{axiom | n = \text R 2         | q = \forall X \in \powerset S \land y \in S         | mr= \map \rho X \le \map \rho {X \cup \set y} \le \map \rho X + 1 }} {{axiom | n = \text R 3         | q = \forall X \in \powerset S \land y, z \in S         | mr= \map \rho {X \cup \set y} = \map \rho {X \cup \set z} = \map \rho X \implies \map \rho {X \cup \set{y,z} } = \map \rho X }} {{end-axiom}}
Definition:Rank Axioms (Matroid)/Definition 2
{{begin-axiom}} {{axiom | n = \text R 1'         | q = \forall X \in \powerset S         | mr = 0 \le \map \rho X \le \size X }} {{axiom | n = \text R 2'         | q = \forall X, Y \in \powerset S         | mr = X \subseteq Y \implies \map \rho X \le \map \rho Y }} {{axiom | n = \text R 3'         | q = \forall X, Y \in \powerset S         | mr = \map \rho {X \cup Y} + \map \rho {X \cap Y} \le \map \rho X + \map \rho Y }} {{end-axiom}}
Definition:Rank Axioms (Matroid)
Let $S$ be a finite set. Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ into the integers === Definition 1 === $\rho$ is said to satisfy the '''rank axioms''' : {{:Definition:Rank Axioms (Matroid)/Definition 1}} === Definition 2 === $\rho$ is said to satisfy the '''rank axioms''' : {{:Definition:Rank Axioms (Matroid)/Definition 2}}
Definition:Isomorphism (Matroid)
Let $M_1 = \struct{S_1, \mathscr I_1}$ and $M_2 = \struct{S_2, \mathscr I_2}$ be matroids. Let $f : S_1 \to S_2$ be a bijection. Then $f$ is called an '''isomorphism''' if: :$\forall X \subseteq S : X \in \mathscr I_1 \iff \map f X \in \mathscr I_2$ If $f$ is an '''isomorphism''' then $M_1$ is said to be '''isomorphic''' to $M_2$.
Definition:Bloch's Constant
Recall Bloch's Theorem: {{:Bloch's Theorem}} The lower bound of $B$ is known as '''Bloch's constant'''.
Definition:Landau's Constant
Recall Landau's Theorem: {{:Landau's Theorem}} The constant of $L$ is known as '''Landau's constant'''.
Definition:Uniform Matroid
Let $S$ be a finite set of cardinality $n$. Let $\mathscr I_{k,n}$ be the set of all subsets of $S$ of cardinality less than or equal to $k$.  Then the ordered pair $\struct {S, \mathscr I_{k, n} }$ is called the '''uniform matroid of rank $k$''' and is denoted $U_{k,n}$.
Definition:Free Matroid
Let $S$ be a finite set. Let $\mathscr I = \powerset S$ be the power set of $S$. That is, let $\mathscr I$ be the set of all subsets of $S$: :$\mathscr I := \set {X: X \subseteq S}$ Then the ordered pair $\struct{S, \mathscr I}$ is called the '''free matroid of $S$'''.
Definition:Matroid Induced by Linear Independence/Vector Space
Let $V$ be a vector space. Let $S$ be a finite subset of $V$. Let $\mathscr I$ be the set of linearly independent subsets of $S$. Then the ordered pair $\struct{S, \mathscr I}$ is called a '''matroid induced on $S$ by linear independence in $V$'''.
Definition:Cycle Matroid
Let $G$ be a graph. Let $E$ be the edge set of $G$. Let $\mathscr I$ be the set of edge sets of subgraphs of $G$ that contain no cycles. Then the ordered pair $\struct{E, \mathscr I}$ is called the '''cycle matroid''' of the graph $G$.
Definition:Riemann Zeta Function/Zero
The '''zeroes''' of the Riemann zeta function are, according to the conventional definition of the zero of a (complex) function, the points $s \in \C$ such that: :$\map \zeta s = 0$ === Trivial Zeroes === {{:Definition:Riemann Zeta Function/Zero/Trivial}} === Nontrivial Zeroes === {{:Definition:Riemann Zeta Function/Zero/Nontrivial}}
Definition:Riemann Zeta Function/Zero/Trivial
The '''trivial zeroes''' of the Riemann zeta function $\zeta$ are the strictly negative even integers : :$\set {n \in \Z: n = -2 \times k: k \in \N_{\ne 0} } = \set {-2, -4, -6, \ldots}$
Definition:Brocard Points
:700px Let $\triangle ABC$ be a triangle such that $A \to B \to C \to A$ travels anticlockwise around $\triangle ABC$. === First Brocard Point === {{:Definition:Brocard Points/First}} === Second Brocard Point === {{:Definition:Brocard Points/Second}}
Definition:Brocard Points/First
Let the point $P$ be constructed such that: :$\angle PAB = \angle PBC = \angle PCA$ Then $P$ is the '''first Brocard point''' of $\triangle ABC$.
Definition:Brocard Points/Second
Let the point $Q$ be constructed such that: :$\angle QAC = \angle QCB = \angle QBA$ Then $Q$ is the '''second Brocard point''' of $\triangle ABC$.
Definition:Brocard Angle
:700px Let $\triangle ABC$ be a triangle such that $A \to B \to C \to A$ travels anticlockwise around $\triangle ABC$. Let $P$ and $Q$ be the first and second Brocard points of $\triangle ABC$ respectively. Let: {{begin-eqn}} {{eqn | l = \omega       | r = \angle PAB = \angle PBC = \angle PCA }} {{eqn | l = \omega'       | r = \angle QAC = \angle QCB = \angle QBA }} {{end-eqn}} From Brocard Angle is Unique: :$\omega = \omega'$ This angle $\omega = \omega'$ is the '''Brocard angle''' of $\triangle ABC$.
Definition:Lehmer's Constant
Let the sequence $\sequence {b_k}_{k \mathop \in \N}$ be defined by the recurrence relation: {{begin-eqn}} {{eqn | l = b_k       | r = \begin {cases} 0 & : k = 0 \\ {b_{k - 1} }^2 + b_{k - 1} + 1 & : \text {otherwise} \end{cases} }} {{eqn | r = \sequence {0, 1, 3, 13, 183, 33973 \ldots} }} {{end-eqn}} {{OEIS|A002065}} '''Lehmer's constant''' is the real number $\xi$ defined as: {{begin-eqn}} {{eqn | l = \xi       | r = \map \cot {\sum_{k \mathop = 0}^\infty \paren {-1}^k \arccot b_k} }} {{eqn | r = \map \cot {\arccot 0 - \arccot 1 + \arccot 3 - \arccot 13 + \arccot 183 - \arccot 33973 + \dotsb} }} {{eqn | o = \approx       | r = 0 \cdotp 59263 \, 27182 \ldots }} {{end-eqn}} {{OEIS|A030125}}
Definition:Vectorial Matroid
Let $V$ be a vector space. Let $S$ be a finite subset of $V$. Let $\struct{S, \mathscr I}$ be the matroid induced by linear independence in $V$ on $S$. From Matroid Induced by Linear Independence in Vector Space is Matroid, $\struct{S, \mathscr I}$ is a matroid. Then any matroid isomorphic to $\struct{S, \mathscr I}$ is called a '''vectorial matroid'''.
Definition:Matroid Induced by Algebraic Independence
Let $L / K$ be a field extension. Let $S \subseteq L$ be a finite subset of $L$. Let $\mathscr I$ be the set of algebraically independent subsets of $S$. Then $\struct {S, \mathscr I}$ is called the '''matroid induced by algebraic independence over $K$ on $S$'''.
Definition:Algebraic Matroid
Let $L / K$ be a field extension. Let $S \subseteq L$ be a finite subset of $L$. Let $\struct{S, \mathscr I}$ be the matroid induced by Algebraic Independence over $K$ on $S$. From Matroid Induced by Algebraic Independence is Matroid, $\struct{S, \mathscr I}$ is a matroid. Then any matroid isomorphic to $\struct{S, \mathscr I}$ is called an '''algebraic matroid'''.
Definition:Affinely Dependent
Let $\R^n$ be the $n$-dimensional real Euclidean space. Let $S = \set{x_1, \dots, x_r}$ be a finite subset of $\R^n$. An element $x \in \R^n$ is '''affinely dependent''' on $S$ if there exist real numbers $\set{\lambda_i: 1 \le i \le r}$ such that: :$(1) \quad x = \displaystyle \sum_{i = 1}^r \lambda_i x_i$  :$(2) \quad \displaystyle \sum_{i = 1}^r \lambda_i = 1$
Definition:Affinely Dependent/Independent
Let $\R^n$ be the $n$-dimensional real Euclidean space. Let $X = \set{x_1, \dots, x_r}$ be a finite subset of $\R^n$. The subset $X$ is '''affinely independent''' if no element $x \in X$ is affinely dependent on $X \setminus \set x$
Definition:Euler-Gompertz Constant
=== Integral Form === {{:Definition:Euler-Gompertz Constant/Integral Form}} === As a Continued Fraction === {{:Definition:Euler-Gompertz Constant/Continued Fraction}}
Definition:Torsion-Free Group
Let $G$ be a group. Then $G$ is a '''torsion-free group'''  the only torsion element of $G$ is the group identity element.
Definition:Matroid Induced by Affine Independence
Let $\R^n$ be the $n$-dimensional real Euclidean space. Let $S = \set{x_1, \dots, x_r}$ be a finite subset of $\R^n$. Let $\mathscr I$ be the set of affinely independent subsets of $S$. Then $\struct{S, \mathscr I}$ is called the '''matroid induced by affine independence on $S$'''.
Definition:Matroid Induced by Linear Independence/Abelian Group
Let $\struct{G, +}$ be a torsion-free Abelian group. Let $\struct{G, +, \times}$ be the $\Z$-module associated with $G$. Let $S$ be a finite subset of $G$. Let $\mathscr I$ be the set of linearly independent subsets of $S$. Then the ordered pair $\struct{S, \mathscr I}$ is called the '''matroid induced by linear independence in $G$ on $S$'''.
Definition:Euler-Gompertz Constant/Integral Form
The '''Euler-Gompertz constant''' is the real number $G$ defined as: :$G = \displaystyle \int_0^\infty \dfrac {e^{-u} } {1 + u} \rd u$
Definition:Twin Primes Constant
The '''twin primes constant''' is the real number: {{begin-eqn}} {{eqn | l = \Pi_2       | o = :=       | r = \displaystyle \prod_{\substack {p \mathop \ge 3 \\ \text {$p$ prime} } } \paren {1 - \dfrac 1 {\paren {p - 1}^2} } }} {{eqn | o = \approx       | r = 0 \cdotp 66016 \, 18       | c =  }} {{end-eqn}} {{OEIS|A005597}}
Definition:Robbins Constant
The '''Robbins constant''' $R$ is defined as the mean distance $D$ between $2$ points chosen at random from the interior of a unit cube: {{begin-eqn}} {{eqn | l = R       | r = \frac {4 + 17 \sqrt 2 - 6 \sqrt3 - 7 \pi} {105} + \frac {\map \ln {1 + \sqrt 2 } } 5 + \frac {2 \, \map \ln {2 + \sqrt 3} } 5 }} {{eqn | o = \approx       | r = 0 \cdotp 66170 \, 71822 \, 67176 \, 23515 \, 582 \ldots       | c =  }} {{end-eqn}} {{OEIS|A073012}}
Definition:Matroid Induced by Linear Independence
=== Vector Space === {{:Definition:Matroid Induced by Linear Independence/Vector Space}} === Abelian Group === {{:Definition:Matroid Induced by Linear Independence/Abelian Group}}
Definition:Loop (Matroid)
A '''loop''' of $M$ is an element $x$ of $S$ such that $\set x$ is a dependent subset of $S$. That is, $x \in S$ is a '''loop'''  $\set x \not \in \mathscr I$.
Definition:Parallel (Matroid)
Two elements $x, y \in S$ are said to be '''parallel''' in $M$  they are not loops but $\set {x, y}$ is a dependent subset of $S$. That is, $x, y \in S$ are '''parallel''' : :$\set x, \set y \in \mathscr I$ and $\set {x, y} \notin \mathscr I$.
Definition:Central Binomial Coefficient
Let $n \in \N$ be a natural number. The binomial coefficient: :$\dbinom {2 n} n$ is known as a '''central binomial coefficient'''.
Definition:Landau-Ramanujan Constant
{{begin-eqn}} {{eqn | l = k       | r = \sqrt {\dfrac 1 2 \displaystyle \prod_{\substack {r \mathop = 4 n \mathop + 3 \\ \text {$r$ prime} } } \paren {1 - \dfrac 1 {r^2} }^{-1} } }} {{eqn | o = \approx       | r = 0 \cdotp 76422 \, 3653 \ldots }} {{end-eqn}}
Definition:Stolarsky-Harborth Constant
The '''Stolarsky-Harborth constant''' is the lower bound for the number $\beta$ defined as: :$\beta > \dfrac {P_n} {n^{\lg 3} }$ where: :$P_n$ is the number of odd elements in the first $n$ rows of Pascal's triangle :$\lg 3$ denotes the logarithm base $2$ of $3$. Its value is given by: :$\beta \approx 0 \cdotp 81255 \, 65590 \, 160063 \, 8769 \ldots$ {{OEIS|A077464}}
Definition:Lehmer's Polynomial
'''Lehmer's polynomial''' is the expression: :$x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$
Definition:Salem Number
A '''Salem number''' is a real algebraic integer $\alpha$ such that: :$\alpha > 1$ : its conjugate roots all have absolute value no greater than $1$, at least one of which has absolute value exactly $1$.
Definition:Wilbraham-Gibbs Constant
The '''Wilbraham-Gibbs constant''' is the real number defined as: {{begin-eqn}} {{eqn | l = G'       | r = \int_0^\pi \dfrac {\sin u} u \rd u       | c =  }} {{eqn | o = \approx       | r = 1 \cdotp 85193 \, 7052 \ldots       | c =  }} {{end-eqn}} {{OEIS|036792}}
Definition:Covertex
=== Covertex of Ellipse === Let $K$ be an ellipse. {{:Definition:Covertex of Ellipse}} === Covertex of Hyperbola === Let $K$ be a hyperbola. {{:Definition:Covertex of Hyperbola}}
Definition:Kepler's Equation
Consider a system consisting of two bodies $B_1$ and $B_2$ in elliptical orbit around each other. Let $e$ denote the eccentricity of that orbit. '''Kepler's equation''' is the equation that describes the relation between the mean anomaly and eccentric anomaly of $B_1$ with respect to $B_2$: :$M = E - e \sin E$ where: :$M$ is the mean anomaly :$E$ is the eccentric anomaly. {{NamedforDef|Johannes Kepler|cat = Kepler}}
Definition:Lemniscate of Bernoulli/Geometric Definition
Let $P_1$ and $P_2$ be points in the plane such that $P_1 P_2 = 2 a$ for some constant $a$. The '''lemniscate of Bernoulli''' is the locus of points $M$ in the plane such that: :$P_1 M \times P_2 M = a^2$
Definition:Lemniscate of Bernoulli/Cartesian Definition
The '''lemniscate of Bernoulli''' is the curve defined by the Cartesian equation: :$\paren {x^2 + y^2}^2 = 2 a^2 \paren {x^2 - y^2}$
Definition:Lemniscate of Bernoulli/Polar Definition
The '''lemniscate of Bernoulli''' is the curve defined by the polar equation: :$r^2 = 2 a^2 \cos 2 \theta$
Definition:Lemniscate of Bernoulli/Parametric Definition
The '''lemniscate of Bernoulli''' is the curve defined by the parametric equation: :$\begin{cases} x = \dfrac {a \sqrt 2 \cos t} {\sin^2 t + 1} \\ y = \dfrac {a \sqrt 2 \cos t \sin t} {\sin^2 t + 1} \end{cases}$
Definition:Lemniscate of Bernoulli/Cartesian Definition/Also defined as
Some sources define the Cartesian equation for the '''lemniscate of Bernoulli''' as: :$\paren {x^2 + y^2}^2 = a^2 \paren {x^2 - y^2}$ which is the same but for a scale factor: :620px
Definition:Lemniscate of Bernoulli/Polar Definition/Also defined as
Some sources define the polar equation for the '''lemniscate of Bernoulli''' as: :$r^2 = a^2 \cos 2 \theta$ which is the same but for a scale factor: :620px
Definition:Twin Primes Constant/Also defined as
{{AuthorRef|François Le Lionnais}} and {{AuthorRef|Jean Brette}}, in their {{BookLink|Les Nombres Remarquables|François Le Lionnais}} of $1983$, define the '''twin primes constant''' as: {{begin-eqn}} {{eqn | l = \Pi_2       | o = :=       | r = 2 \displaystyle \prod_{\substack {p \mathop \ge 3 \\ \text {$p$ prime} } } \paren {1 - \dfrac 1 {\paren {p - 1}^2} } }} {{eqn | o = \approx       | r = 1 \cdotp 32032 \, 36316 \ldots       | c =  }} {{end-eqn}}
Definition:Pisot-Vijayaraghavan Number
Let $\alpha$ be a real algebraic integer greater than $1$. Let $\alpha$ be such that the absolute values of its Galois conjugates are all less than $1$. Then $\alpha$ is a '''Pisot-Vijayaraghavan number'''.
Definition:Plastic Constant
The '''plastic constant''' is the real root of the cubic: :$x^3 - x - 1 = 0$ {{explain|There are $3$ roots of this equation -- needs to be established exactly which one we are talking about here.}} Its value is approximately: :$P = 1 \cdotp 32471 \, 795 \ldots$ {{OEIS|A060006}}
Definition:Hermite Constant
Let $n \in \N$ be a natural number. Let $L$ be an integer lattice in Euclidean space $R^n$ unit covolume. That is: :$\map {\operatorname {vol} } {\dfrac {R^n} L} = 1$ Let $\map {\lambda_1} L$ denote the least length of a nonzero element of $L$. Let $\sqrt {\gamma_n}$ be the maximum of $\map {\lambda_1} L$ over all such integer lattices $L$. The '''Hermite constant of dimension $n$''' is the constant $\gamma_n$. {{NamedforDef|Charles Hermite|cat = Hermite}}
Definition:Landau's Problems
'''Landau's problems''' are a set of $4$ (still) unsolved problems about prime numbers listed by {{AuthorRef|Edmund Georg Hermann Landau}} in an address at the $1912$ International Congress of Mathematicians. They are as follows: === Goldbach's Conjecture === {{:Goldbach's Conjecture}} === Twin Prime Conjecture === {{:Twin Prime Conjecture}} === Legendre's Conjecture === {{:Legendre's Conjecture}} === Is there an Infinite Number of Primes of Form $n^2 + 1$? === {{:Is there an Infinite Number of Primes of Form n^2 + 1?}}
Definition:Polar Equation
A '''polar equation''' is an equation defining the locus of a set of points in the polar coordinate plane. Such an equation is generally presented in terms of the variables: :$r$: the radial coordinate :$\theta$: the angular coordinate
Definition:Cartesian Equation
A '''Cartesian equation''' is an equation defining the locus of a set of points in a Cartesian space. Such an equation is generally presented in terms of the variables: :$x$ and $y$ for a $2$-dimensional space :$x$, $y$ and $z$ for a $3$-dimensional space :$x_1, x_2, \ldots, x_n$ for a general $n$-dimensional space
Definition:Lemniscate of Bernoulli/Focus
Each of the two points $P_1$ and $P_2$ can be referred to as a '''focus''' of the lemniscate.
Definition:Cardioid
A '''cardioid''' is an epicycloid with $1$ cusp. :500px
Definition:Trifolium/Cosine Form
The '''trifolium''' can be defined by the polar equation in the form: :$r = a \cos 3 \theta$ where: :$0 \le \theta \le \pi$ :400px
Definition:Trifolium/Sine Form
The '''trifolium''' can be defined by the polar equation in the form: :$r = a \sin 3 \theta$ where: :$0 \le \theta \le \pi$ :400px
Definition:Rhodonea Curve
A '''rhodonea curve''' is a curve defined by either one of the polar equations: {{begin-eqn}} {{eqn | l = r       | r = a \cos n \theta }} {{eqn | l = r       | r = a \sin n \theta }} {{end-eqn}}
Definition:Rhodonea Curve/Petal
Each of the loops of a '''rhodonea curve''' is referred to as a '''petal'''.
Definition:Quadrifolium/Cosine Form
The '''quadrifolium''' can be defined by the polar equation in the form: :$r = a \cos 2 \theta$ where: :$0 \le \theta \le 2 \pi$ :400px
Definition:Quadrifolium/Sine Form
The '''quadrifolium''' can be defined by the polar equation in the form: :$r = a \sin 2 \theta$ where: :$0 \le \theta \le 2 \pi$ :400px
Definition:Trochoid
Consider a circle $C$ rolling without slipping along a straight line. Consider a point $P$ on the line of a radius of $C$ at a distance $b$ from the center of $C$. The curve traced out by $P$ is called a '''trochoid'''.
Definition:Trochoid/Curtate
Let $b < a$. The curve traced out by $P$ is called a '''curtate trochoid'''. :500px
Definition:Trochoid/Prolate
Let $b > a$. The curve traced out by $P$ is called a '''prolate trochoid'''. :500px
Definition:Folium of Descartes/Cartesian Form
The '''folium of Descartes''' is the locus of the equation: :$x^3 + y^3 - 3 a x y = 0$
Definition:Omega Constant/Decimal Expansion
The decimal expansion of the omega constant $\Omega$ starts: :$0 \cdotp 56714 \, 32904 \, 097838 \, 72999 \, 96866 \, 22 \ldots$
Definition:Folium of Descartes/Parametric Form
The '''folium of Descartes''' is the locus of the equation given in parametric form as: :$\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$
Definition:Ovals of Cassini
Let $P_1$ and $P_2$ be points in the plane such that $P_1 P_2 = 2 a$ for some constant $a$. The '''ovals of Cassini''' are the loci of points $M$ in the plane such that: :$P_1 M \times P_2 M = b^2$ for a real constant $b$.