id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-11300 | Lattice is Complete iff it Admits All Suprema | Let $\struct {S, \preceq}$ be an ordered set.
Then $\struct {S, \preceq}$ is a complete lattice {{iff}}
:$\forall X \subseteq S: X$ admits a supremum. | === Sufficient Condition ===
Let $\struct {S, \preceq}$ be a complete lattice.
Thus by definition of complete lattice:
:$\forall X \subseteq S: X$ admits a supremum.
{{qed|lemma}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Then $\struct {S, \preceq}$ is a [[Definition:Complete Lattice|complete lattice]] {{iff}}
:$\forall X \subseteq S: X$ admits a [[Definition:Supremum of Set|supremum]]. | === Sufficient Condition ===
Let $\struct {S, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Thus by definition of [[Definition:Complete Lattice|complete lattice]]:
:$\forall X \subseteq S: X$ admits a [[Definition:Supremum of Set|supremum]].
{{qed|lemma}} | Lattice is Complete iff it Admits All Suprema | https://proofwiki.org/wiki/Lattice_is_Complete_iff_it_Admits_All_Suprema | https://proofwiki.org/wiki/Lattice_is_Complete_iff_it_Admits_All_Suprema | [
"Complete Lattices"
] | [
"Definition:Ordered Set",
"Definition:Complete Lattice",
"Definition:Supremum of Set"
] | [
"Definition:Complete Lattice",
"Definition:Complete Lattice",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Definition:Complete Latti... |
proofwiki-11301 | Gauss-Bonnet Theorem | Let $M$ be a compact $2$-dimensional Riemannian manifold with boundary $\partial M$.
Let $\kappa$ be the Gaussian curvature of $M$.
Let $k_g$ be the geodesic curvature of $\partial M$.
Then:
:$\ds \int_M \kappa \rd A + \int_{\partial M} k_g \rd s = 2 \pi \map \chi M$
where:
:$\d A$ is the element of area of the surface... | {{ProofWanted}}
{{Namedfor|Carl Friedrich Gauss|name2 = Pierre Ossian Bonnet|cat = Gauss|cat2 = Bonnet}} | Let $M$ be a [[Definition:Compact Topological Space|compact]] [[Definition:Dimension (Geometry)|$2$-dimensional]] [[Definition:Riemannian Manifold|Riemannian manifold]] with [[Definition:Boundary (Topology)|boundary]] $\partial M$.
Let $\kappa$ be the [[Definition:Gaussian Curvature|Gaussian curvature]] of $M$.
Let $... | {{ProofWanted}}
{{Namedfor|Carl Friedrich Gauss|name2 = Pierre Ossian Bonnet|cat = Gauss|cat2 = Bonnet}} | Gauss-Bonnet Theorem | https://proofwiki.org/wiki/Gauss-Bonnet_Theorem | https://proofwiki.org/wiki/Gauss-Bonnet_Theorem | [
"Differential Geometry"
] | [
"Definition:Compact Topological Space",
"Definition:Dimension (Geometry)",
"Definition:Riemannian Manifold",
"Definition:Boundary (Topology)",
"Definition:Gaussian Curvature",
"Definition:Geodesic Curvature",
"Definition:Area Element",
"Definition:Surface",
"Definition:Line Element",
"Definition:E... | [] |
proofwiki-11302 | Unique Factorization Theorem for Gaussian Integers | The ring of Gaussian integers:
:$\struct {\Z \sqbrk i, +, \times}$
forms a unique factorization domain.
That is, every Gaussian integer can be expressed as the product of one or more Gaussian primes, uniquely up to the order in which they appear. | We have that:
:Gaussian Integers form Euclidean Domain
:Euclidean Domain is Principal Ideal Domain
:Principal Ideal Domain is Unique Factorization Domain.
So $\struct {\Z \sqbrk i, +, \times}$ forms a unique factorization domain.
Hence the result, by definition of unique factorization domain.
{{qed}} | The [[Definition:Ring of Gaussian Integers|ring of Gaussian integers]]:
:$\struct {\Z \sqbrk i, +, \times}$
forms a [[Definition:Unique Factorization Domain|unique factorization domain]].
That is, every [[Definition:Gaussian Integer|Gaussian integer]] can be expressed as the [[Definition:Complex Multiplication|produc... | We have that:
:[[Gaussian Integers form Euclidean Domain]]
:[[Euclidean Domain is Principal Ideal Domain]]
:[[Principal Ideal Domain is Unique Factorization Domain]].
So $\struct {\Z \sqbrk i, +, \times}$ forms a [[Definition:Unique Factorization Domain|unique factorization domain]].
Hence the result, by definition ... | Unique Factorization Theorem for Gaussian Integers | https://proofwiki.org/wiki/Unique_Factorization_Theorem_for_Gaussian_Integers | https://proofwiki.org/wiki/Unique_Factorization_Theorem_for_Gaussian_Integers | [
"Gaussian Integers",
"Unique Factorization Domains",
"Gaussian Primes",
"Algebraic Number Theory",
"Named Theorems"
] | [
"Definition:Ring of Gaussian Integers",
"Definition:Unique Factorization Domain",
"Definition:Gaussian Integer",
"Definition:Multiplication/Complex Numbers",
"Definition:Gaussian Prime"
] | [
"Gaussian Integers form Euclidean Domain",
"Euclidean Domain is Principal Ideal Domain",
"Principal Ideal Domain is Unique Factorization Domain",
"Definition:Unique Factorization Domain",
"Definition:Unique Factorization Domain"
] |
proofwiki-11303 | Galois Connection is Expressed by Minimum | Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets.
Let $g: S \to T$, $d: T \to S$ be mappings.
Then $\tuple {g, d}$ is a Galois connection {{iff}}:
:$g$ is an increasing mapping and
::$\forall t \in T: \map d t = \map \min {g^{-1} \sqbrk {t^\succsim} }$
where
:$\min$ denotes the minimum
:$g^{-1} \sqbr... | === Sufficient Condition ===
Let $\tuple {g, d}$ be a Galois connection.
Thus by definition of Galois connection:
:$g$ is an increasing mapping
Let $t \in T$.
By definition of reflexivity:
:$\map d t \preceq \map d t$
By definition of Galois connection:
:$t \precsim \map g {\map d t}$
By definition of upper closure:
:$... | Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be [[Definition:Ordered Set|ordered sets]].
Let $g: S \to T$, $d: T \to S$ be [[Definition:Mapping|mappings]].
Then $\tuple {g, d}$ is a [[Definition:Galois Connection|Galois connection]] {{iff}}:
:$g$ is an [[Definition:Increasing Mapping|increasing mapping]] and
... | === Sufficient Condition ===
Let $\tuple {g, d}$ be a [[Definition:Galois Connection|Galois connection]].
Thus by definition of [[Definition:Galois Connection|Galois connection]]:
:$g$ is an [[Definition:Increasing Mapping|increasing mapping]]
Let $t \in T$.
By definition of [[Definition:Reflexivity|reflexivity]]:
... | Galois Connection is Expressed by Minimum | https://proofwiki.org/wiki/Galois_Connection_is_Expressed_by_Minimum | https://proofwiki.org/wiki/Galois_Connection_is_Expressed_by_Minimum | [
"Galois Connections"
] | [
"Definition:Ordered Set",
"Definition:Mapping",
"Definition:Galois Connection",
"Definition:Increasing/Mapping",
"Definition:Smallest Element/Subset",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Upper Closure/Element"
] | [
"Definition:Galois Connection",
"Definition:Galois Connection",
"Definition:Increasing/Mapping",
"Definition:Reflexivity",
"Definition:Galois Connection",
"Definition:Upper Closure/Element",
"Definition:Image (Set Theory)/Relation/Subset",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of ... |
proofwiki-11304 | Closed Form for Pentagonal Numbers | The closed-form expression for the $n$th pentagonal number is:
:$P_n = \dfrac {n \paren {3 n - 1} } 2$ | Pentagonal numbers are $k$-gonal numbers where $k = 5$.
From Closed Form for Polygonal Numbers we have that:
:$\map P {k, n} = \dfrac n 2 \paren {\paren {k - 2} n - k + 4}$
Hence:
{{begin-eqn}}
{{eqn | l = P_n
| r = \frac n 2 \paren {\paren {5 - 2} n - 5 + 4}
| c = Closed Form for Polygonal Numbers
}}
{{eqn... | The [[Definition:Closed-Form Expression|closed-form expression]] for the $n$th [[Definition:Pentagonal Number|pentagonal number]] is:
:$P_n = \dfrac {n \paren {3 n - 1} } 2$ | [[Definition:Pentagonal Number|Pentagonal numbers]] are [[Definition:Polygonal Number|$k$-gonal numbers]] where $k = 5$.
From [[Closed Form for Polygonal Numbers]] we have that:
:$\map P {k, n} = \dfrac n 2 \paren {\paren {k - 2} n - k + 4}$
Hence:
{{begin-eqn}}
{{eqn | l = P_n
| r = \frac n 2 \paren {\paren {... | Closed Form for Pentagonal Numbers | https://proofwiki.org/wiki/Closed_Form_for_Pentagonal_Numbers | https://proofwiki.org/wiki/Closed_Form_for_Pentagonal_Numbers | [
"Pentagonal Numbers",
"Closed Forms"
] | [
"Definition:Closed Form Expression",
"Definition:Pentagonal Number"
] | [
"Definition:Pentagonal Number",
"Definition:Polygonal Number",
"Closed Form for Polygonal Numbers",
"Closed Form for Polygonal Numbers"
] |
proofwiki-11305 | Closed Form for Hexagonal Numbers | The closed-form expression for the $n$th hexagonal number is:
:$H_n = n \paren {2 n - 1}$ | Hexagonal numbers are $k$-gonal numbers where $k = 6$.
From Closed Form for Polygonal Numbers we have that:
:$\map P {k, n} = \dfrac n 2 \paren {\paren {k - 2} n - k + 4}$
Hence:
{{begin-eqn}}
{{eqn | l = H_n
| r = \frac n 2 \paren {\paren {6 - 2} n - 6 + 4}
| c = Closed Form for Polygonal Numbers
}}
{{eqn ... | The [[Definition:Closed-Form Expression|closed-form expression]] for the $n$th [[Definition:Hexagonal Number|hexagonal number]] is:
:$H_n = n \paren {2 n - 1}$ | [[Definition:Hexagonal Number|Hexagonal numbers]] are [[Definition:Polygonal Number|$k$-gonal numbers]] where $k = 6$.
From [[Closed Form for Polygonal Numbers]] we have that:
:$\map P {k, n} = \dfrac n 2 \paren {\paren {k - 2} n - k + 4}$
Hence:
{{begin-eqn}}
{{eqn | l = H_n
| r = \frac n 2 \paren {\paren {6 ... | Closed Form for Hexagonal Numbers | https://proofwiki.org/wiki/Closed_Form_for_Hexagonal_Numbers | https://proofwiki.org/wiki/Closed_Form_for_Hexagonal_Numbers | [
"Hexagonal Numbers",
"Closed Forms"
] | [
"Definition:Closed Form Expression",
"Definition:Hexagonal Number"
] | [
"Definition:Hexagonal Number",
"Definition:Polygonal Number",
"Closed Form for Polygonal Numbers",
"Closed Form for Polygonal Numbers"
] |
proofwiki-11306 | Sum of Even Integers is Even | The sum of any finite number of even integers is itself even. | Proof by induction:
For all $n \in \N$, let $\map P n$ be the proposition:
:The sum of $n$ even integers is an even integer.
$\map P 1$ is trivially true, as this just says:
:The sum of $1$ even integer is an even integer.
The sum of $0$ even integers is understood, from the definition of a vacuous summation, to be $0$... | The [[Definition:Integer Addition|sum]] of any [[Definition:Finite Set|finite number]] of [[Definition:Even Integer|even integers]] is itself [[Definition:Even Integer|even]]. | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:The sum of $n$ [[Definition:Even Integer|even integers]] is an [[Definition:Even Integer|even integer]].
$\map P 1$ is trivially true, as this just says:
:The sum of $1$ [[De... | Sum of Even Integers is Even/Proof 1 | https://proofwiki.org/wiki/Sum_of_Even_Integers_is_Even | https://proofwiki.org/wiki/Sum_of_Even_Integers_is_Even/Proof_1 | [
"Even Integers",
"Euclidean Number Theory",
"Sum of Even Integers is Even"
] | [
"Definition:Addition/Integers",
"Definition:Finite Set",
"Definition:Even Integer",
"Definition:Even Integer"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Even Integer",
"Definition:Even Integer",
"Definition:Even Integer",
"Definition:Even Integer",
"Definition:Even Integer",
"Definition:Summation/Vacuous Summation",
"Definition:Even Integer",
"Definition:Addition/Integers... |
proofwiki-11307 | Sum of Even Integers is Even | The sum of any finite number of even integers is itself even. | Let $S = \set {r_1, r_2, \ldots, r_n}$ be a set of $n$ even numbers.
By definition of even number, this can be expressed as:
:$S = \set {2 s_1, 2 s_2, \ldots, 2 s_n}$
where:
:$\forall k \in \closedint 1 n: r_k = 2 s_k$
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n r_k
| r = \sum_{k \mathop = 1}^n 2 s_k
... | The [[Definition:Integer Addition|sum]] of any [[Definition:Finite Set|finite number]] of [[Definition:Even Integer|even integers]] is itself [[Definition:Even Integer|even]]. | Let $S = \set {r_1, r_2, \ldots, r_n}$ be a [[Definition:Set|set]] of $n$ [[Definition:Even Integer|even numbers]].
By definition of [[Definition:Even Integer|even number]], this can be expressed as:
:$S = \set {2 s_1, 2 s_2, \ldots, 2 s_n}$
where:
:$\forall k \in \closedint 1 n: r_k = 2 s_k$
Then:
{{begin-eqn}}
{{eq... | Sum of Even Integers is Even/Proof 2 | https://proofwiki.org/wiki/Sum_of_Even_Integers_is_Even | https://proofwiki.org/wiki/Sum_of_Even_Integers_is_Even/Proof_2 | [
"Even Integers",
"Euclidean Number Theory",
"Sum of Even Integers is Even"
] | [
"Definition:Addition/Integers",
"Definition:Finite Set",
"Definition:Even Integer",
"Definition:Even Integer"
] | [
"Definition:Set",
"Definition:Even Integer",
"Definition:Even Integer",
"Definition:Even Integer"
] |
proofwiki-11308 | Galois Connection is Expressed by Maximum | Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets.
Let $g: S \to T$, $d: T \to S$ be mappings.
Then $\tuple {g, d}$ is a Galois connection {{iff}}
:$d$ is an increasing mapping and:
::$\forall s \in S: \map g s = \map \max {d^{-1} \sqbrk {s^\preceq} }$
where:
:$\max$ denotes the maximum
:$d^{-1} \sqbr... | === Sufficient Condition ===
Let $\tuple {g, d}$ be a Galois connection.
Thus by definition of Galois connection:
:$d$ is an increasing mapping.
Let $s \in S$.
By definition of reflexivity:
:$\map g s \preceq \map g s$
By definition of Galois connection:
:$\map d {\map g s} \preceq s$
By definition of lower closure:
:$... | Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be [[Definition:Ordered Set|ordered sets]].
Let $g: S \to T$, $d: T \to S$ be [[Definition:Mapping|mappings]].
Then $\tuple {g, d}$ is a [[Definition:Galois Connection|Galois connection]] {{iff}}
:$d$ is an [[Definition:Increasing Mapping|increasing mapping]] and:
... | === Sufficient Condition ===
Let $\tuple {g, d}$ be a [[Definition:Galois Connection|Galois connection]].
Thus by definition of [[Definition:Galois Connection|Galois connection]]:
:$d$ is an [[Definition:Increasing Mapping|increasing mapping]].
Let $s \in S$.
By definition of [[Definition:Reflexivity|reflexivity]]:... | Galois Connection is Expressed by Maximum | https://proofwiki.org/wiki/Galois_Connection_is_Expressed_by_Maximum | https://proofwiki.org/wiki/Galois_Connection_is_Expressed_by_Maximum | [
"Galois Connections"
] | [
"Definition:Ordered Set",
"Definition:Mapping",
"Definition:Galois Connection",
"Definition:Increasing/Mapping",
"Definition:Greatest Element/Subset",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Lower Closure/Element"
] | [
"Definition:Galois Connection",
"Definition:Galois Connection",
"Definition:Increasing/Mapping",
"Definition:Reflexivity",
"Definition:Galois Connection",
"Definition:Lower Closure/Element",
"Definition:Image (Set Theory)/Relation/Subset",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of ... |
proofwiki-11309 | Compass and Straightedge Construction for Regular Heptagon does not exist | There exists no compass and straightedge construction for the regular heptagon. | Construction of a regular heptagon is the equivalent of constructing the point $\tuple {\cos \dfrac {2 \pi} 7, \sin \dfrac {2 \pi} 7}$ from the points $\tuple {0, 0}$ and $\tuple {1, 0}$
Let $\epsilon = \map \exp {\dfrac {2 \pi} 7}$.
Then $\epsilon$ is a root of $x^7 - 1$.
We have:
{{begin-eqn}}
{{eqn | l = x^7 - 1
... | There exists no [[Definition:Compass and Straightedge Construction|compass and straightedge construction]] for the [[Definition:Regular Heptagon|regular heptagon]]. | Construction of a [[Definition:Regular Heptagon|regular heptagon]] is the equivalent of constructing the [[Definition:Point|point]] $\tuple {\cos \dfrac {2 \pi} 7, \sin \dfrac {2 \pi} 7}$ from the [[Definition:Point|points]] $\tuple {0, 0}$ and $\tuple {1, 0}$
Let $\epsilon = \map \exp {\dfrac {2 \pi} 7}$.
Then $\eps... | Compass and Straightedge Construction for Regular Heptagon does not exist/Proof 2 | https://proofwiki.org/wiki/Compass_and_Straightedge_Construction_for_Regular_Heptagon_does_not_exist | https://proofwiki.org/wiki/Compass_and_Straightedge_Construction_for_Regular_Heptagon_does_not_exist/Proof_2 | [
"Compass and Straightedge Construction for Regular Heptagon does not exist",
"Compass and Straightedge Constructions",
"Regular Polygons",
"7"
] | [
"Definition:Compass and Straightedge Construction",
"Definition:Heptagon/Regular"
] | [
"Definition:Heptagon/Regular",
"Definition:Point",
"Definition:Point",
"Definition:Root of Polynomial",
"Definition:Root of Polynomial",
"Definition:Polynomial",
"Irreducible Polynomial/Examples/x^3 + x^2 - 2 x - 1 in Rationals",
"Definition:Irreducible Polynomial",
"Algebraic Element of Degree 3 is... |
proofwiki-11310 | Integer to Rational Power is Irrational iff not Integer or Reciprocal | Let $m$ be a rational number.
Let $n$ be a positive integer.
Then $n^m$ is an irrational number {{iff}} $n^{\size m}$ is not an integer. | === Necessary Condition ===
Let $n^{\size m} \notin \Z$.
We have that:
:$n^m = n^{u/v}$ for some $u \in\Z$ and $v \in \Z_{\ne 0}$
Then it follows from the definition of a rational power and the existence of a real $v$th root of $n^u$ that:
:$n^m = n^{u/v} = \paren {n^u}^{1/v} \in \R$
{{AimForCont}} $n^m \in \Q$.
We hav... | Let $m$ be a [[Definition:Rational Number|rational number]].
Let $n$ be a [[Definition:Positive Integer|positive integer]].
Then $n^m$ is an [[Definition:Irrational Number|irrational number]] {{iff}} $n^{\size m}$ is not an [[Definition:Integer|integer]]. | === Necessary Condition ===
Let $n^{\size m} \notin \Z$.
We have that:
:$n^m = n^{u/v}$ for some $u \in\Z$ and $v \in \Z_{\ne 0}$
Then it follows from the definition of a [[Definition:Rational Power|rational power]] and the [[Existence of Positive Root of Positive Real Number|existence]] of a [[Definition:Real Numbe... | Integer to Rational Power is Irrational iff not Integer or Reciprocal | https://proofwiki.org/wiki/Integer_to_Rational_Power_is_Irrational_iff_not_Integer_or_Reciprocal | https://proofwiki.org/wiki/Integer_to_Rational_Power_is_Irrational_iff_not_Integer_or_Reciprocal | [
"Number Theory",
"Irrational Numbers"
] | [
"Definition:Rational Number",
"Definition:Positive/Integer",
"Definition:Irrational Number",
"Definition:Integer"
] | [
"Definition:Power (Algebra)/Rational Number",
"Existence of Positive Root of Positive Real Number",
"Definition:Real Number",
"Definition:Root of Number",
"Definition:Common Divisor/Integers",
"Definition:Prime Factor",
"Definition:Divisor (Algebra)/Integer",
"Fundamental Theorem of Arithmetic",
"De... |
proofwiki-11311 | Area Enclosed by First Turn of Archimedean Spiral | Let $S$ be the Archimedean spiral defined by the equation:
:$r = a \theta$
The area $\AA$ enclosed by the first turn of $S$ and the polar axis is given by:
:$\AA = \dfrac {4 \pi^3 a^2} 3$
:500px | {{begin-eqn}}
{{eqn | l = \AA
| r = \int_0^{2 \pi} \frac {\paren {a \theta}^2} 2 \rd \theta
| c = Area between Radii and Curve in Polar Coordinates
}}
{{eqn | r = \intlimits {\frac {a^2 \theta^3} 6} 0 {2 \pi}
| c = Primitive of Power
}}
{{eqn | r = \frac {\paren {2 \pi}^3 a^2} 6
| c =
}}
{{eqn ... | Let $S$ be the [[Definition:Archimedean Spiral|Archimedean spiral]] defined by the equation:
:$r = a \theta$
The [[Definition:Area|area]] $\AA$ enclosed by the first turn of $S$ and the [[Definition:Polar Axis (Polar Coordinates)|polar axis]] is given by:
:$\AA = \dfrac {4 \pi^3 a^2} 3$
:[[File:ArchimedeanSpiralAre... | {{begin-eqn}}
{{eqn | l = \AA
| r = \int_0^{2 \pi} \frac {\paren {a \theta}^2} 2 \rd \theta
| c = [[Area between Radii and Curve in Polar Coordinates]]
}}
{{eqn | r = \intlimits {\frac {a^2 \theta^3} 6} 0 {2 \pi}
| c = [[Primitive of Power]]
}}
{{eqn | r = \frac {\paren {2 \pi}^3 a^2} 6
| c =
}... | Area Enclosed by First Turn of Archimedean Spiral | https://proofwiki.org/wiki/Area_Enclosed_by_First_Turn_of_Archimedean_Spiral | https://proofwiki.org/wiki/Area_Enclosed_by_First_Turn_of_Archimedean_Spiral | [
"Archimedean Spiral"
] | [
"Definition:Archimedean Spiral",
"Definition:Area",
"Definition:Polar Coordinates/Polar Axis",
"File:ArchimedeanSpiralArea.png"
] | [
"Area between Radii and Curve in Polar Coordinates",
"Primitive of Power"
] |
proofwiki-11312 | Trisecting the Angle/Archimedean Spiral | Let $\alpha$ be an angle which is to be trisected.
This can be achieved by means of an Archimedean spiral. | Let the equation of the Archimedean spiral be $r = a \theta$.
Then:
{{begin-eqn}}
{{eqn | l = \angle DOB
| r = \frac {OD} a
}}
{{eqn | r = \frac {OA/3} a
}}
{{eqn | r = \frac 1 3 \angle AOB
}}
{{end-eqn}}
{{qed}} | Let $\alpha$ be an [[Definition:Angle|angle]] which is to be [[Definition:Trisection|trisected]].
This can be achieved by means of an [[Definition:Archimedean Spiral|Archimedean spiral]]. | Let the equation of the [[Definition:Archimedean Spiral|Archimedean spiral]] be $r = a \theta$.
Then:
{{begin-eqn}}
{{eqn | l = \angle DOB
| r = \frac {OD} a
}}
{{eqn | r = \frac {OA/3} a
}}
{{eqn | r = \frac 1 3 \angle AOB
}}
{{end-eqn}}
{{qed}} | Trisecting the Angle/Archimedean Spiral | https://proofwiki.org/wiki/Trisecting_the_Angle/Archimedean_Spiral | https://proofwiki.org/wiki/Trisecting_the_Angle/Archimedean_Spiral | [
"Trisecting the Angle",
"Archimedean Spiral"
] | [
"Definition:Angle",
"Definition:Trisection",
"Definition:Archimedean Spiral"
] | [
"Definition:Archimedean Spiral"
] |
proofwiki-11313 | Archimedes' Limits to Value of Pi | The value of $\pi$ lies between $3 \frac {10} {71}$ and $3 \frac 1 7$:
:$3 \dfrac {10} {71} < \pi < 3 \dfrac 1 7$ | Let $O$ be an arbitrary circle.
Let $AB$ be the diameter of $O$.
Let $Q$ be the circumference of $O$.
By the definition of $\pi$:
:$\dfrac Q {AB} = \pi$
The bounds on $\pi$ that are to be demonstrated are:
:$\dfrac {223} {71} < \pi < \dfrac {22} 7$
:400px
;Upper bound
Let $AC$ be the tangent at $A$.
Let $\angle \theta ... | The value of [[Definition:Pi|$\pi$]] lies between $3 \frac {10} {71}$ and $3 \frac 1 7$:
:$3 \dfrac {10} {71} < \pi < 3 \dfrac 1 7$ | Let $O$ be an arbitrary [[Definition:Circle|circle]].
Let $AB$ be the [[Definition:Diameter of Circle|diameter]] of $O$.
Let $Q$ be the [[Definition:Circumference of Circle|circumference]] of $O$.
By the definition of [[Definition:Pi|$\pi$]]:
:$\dfrac Q {AB} = \pi$
The bounds on [[Definition:Pi|$\pi$]] that are to ... | Archimedes' Limits to Value of Pi/Archimedes' Iterative Proof | https://proofwiki.org/wiki/Archimedes'_Limits_to_Value_of_Pi | https://proofwiki.org/wiki/Archimedes'_Limits_to_Value_of_Pi/Archimedes'_Iterative_Proof | [
"Archimedes' Limits to Value of Pi",
"Pi",
"Circles"
] | [
"Definition:Pi"
] | [
"Definition:Circle",
"Definition:Circle/Diameter",
"Definition:Circle/Circumference",
"Definition:Pi",
"Definition:Pi",
"File:Api Upper2.png",
"Definition:Tangent Line",
"Definition:Right Angle",
"Dissection of an Equilateral Triangle",
"Pythagoras's Theorem",
"Definition:Polygon/Side",
"Defin... |
proofwiki-11314 | Archimedes' Limits to Value of Pi | The value of $\pi$ lies between $3 \frac {10} {71}$ and $3 \frac 1 7$:
:$3 \dfrac {10} {71} < \pi < 3 \dfrac 1 7$ | Let $O$ be a circle with diameter $AB = 1$
Let $Q$ be the circumference of $O$.
By the definition of $\pi$:
:$\dfrac Q {AB} = \pi$
Since $AB = 1$ then $Q$ is exactly $\pi$.
Using regular polygons inscribed and circumscribed about the circle $O$, we aim to demonstrate that the bounds on $\pi$ are:
:$\dfrac {223} {71} < ... | The value of [[Definition:Pi|$\pi$]] lies between $3 \frac {10} {71}$ and $3 \frac 1 7$:
:$3 \dfrac {10} {71} < \pi < 3 \dfrac 1 7$ | Let $O$ be a [[Definition:Circle|circle]] with [[Definition:Diameter of Circle|diameter]] $AB = 1$
Let $Q$ be the [[Definition:Circumference of Circle|circumference]] of $O$.
By the definition of [[Definition:Pi/Definition 1|$\pi$]]:
:$\dfrac Q {AB} = \pi$
Since $AB = 1$ then $Q$ is exactly [[Definition:Pi|$\pi$]].
... | Archimedes' Limits to Value of Pi/Trigonometric Proof | https://proofwiki.org/wiki/Archimedes'_Limits_to_Value_of_Pi | https://proofwiki.org/wiki/Archimedes'_Limits_to_Value_of_Pi/Trigonometric_Proof | [
"Archimedes' Limits to Value of Pi",
"Pi",
"Circles"
] | [
"Definition:Pi"
] | [
"Definition:Circle",
"Definition:Circle/Diameter",
"Definition:Circle/Circumference",
"Definition:Pi/Definition 1",
"Definition:Pi",
"Definition:Polygon/Regular",
"Definition:Inscribe",
"Definition:Circumscribe",
"Definition:Circle",
"Definition:Pi",
"Archimedes' Limits to Value of Pi/Lemma 1",
... |
proofwiki-11315 | Upper Adjoint Preserves All Infima | Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets.
Let $g: S \to T$ be an upper adjoint of Galois connection.
Then $g$ preserves all infima. | By definition of upper adjoint
:$\exists d: T \to S: \left({g, d}\right)$ is a Galois connection
Let $X$ be a subset of $S$ such that
:$X$ admits an infimum.
We will prove as lemma 1 that
:$\forall t \in T: t$ is lower bound for $g^\to\left({X}\right) \implies t \precsim g\left({\inf X}\right)$
Let $t \in T$ such that
... | Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be [[Definition:Ordered Set|ordered sets]].
Let $g: S \to T$ be an [[Definition:Galois Connection|upper adjoint of Galois connection]].
Then $g$ [[Definition:Mapping Preserves Infimum/All|preserves all infima]]. | By definition of [[Definition:Galois Connection|upper adjoint]]
:$\exists d: T \to S: \left({g, d}\right)$ is a [[Definition:Galois Connection|Galois connection]]
Let $X$ be a [[Definition:Subset|subset]] of $S$ such that
:$X$ admits an [[Definition:Infimum of Set|infimum]].
We will prove as lemma 1 that
:$\forall t ... | Upper Adjoint Preserves All Infima | https://proofwiki.org/wiki/Upper_Adjoint_Preserves_All_Infima | https://proofwiki.org/wiki/Upper_Adjoint_Preserves_All_Infima | [
"Galois Connections"
] | [
"Definition:Ordered Set",
"Definition:Galois Connection",
"Definition:Mapping Preserves Infimum/All"
] | [
"Definition:Galois Connection",
"Definition:Galois Connection",
"Definition:Subset",
"Definition:Infimum of Set",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Lower Bound of Set",
"Def... |
proofwiki-11316 | Lower Adjoint Preserves All Suprema | Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets.
Let $d: T \to S$ be an lower adjoint of Galois connection.
Then $d$ preserves all suprema. | By definition of lower adjoint
:$\exists g: S \to T: \left({g, d}\right)$ is a Galois connection
Let $X$ be a subset of $T$ such that
:$X$ admits a supremum.
We will prove as lemma 1 that
:$\forall s \in S: s$ is upper bound for $d^\to\left({X}\right) \implies d\left({\sup X}\right) \preceq s$
Let $s \in S$ such that
:... | Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be [[Definition:Ordered Set|ordered sets]].
Let $d: T \to S$ be an [[Definition:Galois Connection|lower adjoint of Galois connection]].
Then $d$ [[Definition:Mapping Preserves Supremum/All|preserves all suprema]]. | By definition of [[Definition:Galois Connection|lower adjoint]]
:$\exists g: S \to T: \left({g, d}\right)$ is a [[Definition:Galois Connection|Galois connection]]
Let $X$ be a [[Definition:Subset|subset]] of $T$ such that
:$X$ admits a [[Definition:Supremum of Set|supremum]].
We will prove as lemma 1 that
:$\forall s... | Lower Adjoint Preserves All Suprema | https://proofwiki.org/wiki/Lower_Adjoint_Preserves_All_Suprema | https://proofwiki.org/wiki/Lower_Adjoint_Preserves_All_Suprema | [
"Galois Connections"
] | [
"Definition:Ordered Set",
"Definition:Galois Connection",
"Definition:Mapping Preserves Supremum/All"
] | [
"Definition:Galois Connection",
"Definition:Galois Connection",
"Definition:Subset",
"Definition:Supremum of Set",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Upper Bound of Set",
"De... |
proofwiki-11317 | Oscillation at Point (Infimum) equals Oscillation at Point (Limit) | Let $f: D \to \R$ be a real function where $D \subseteq \R$.
Let $x$ be a point in $D$.
Let $N_x$ be the set of neighborhoods of $x$.
Let $\map {\omega_f} x$ be the oscillation of $f$ at $x$ as defined by:
:$\map {\omega_f} x = \inf \set {\map {\omega_f} {I \cap D}: I \in N_x}$
where $\map {\omega_f} {I \cap D}$ is the... | === Lemma ===
{{:Oscillation at Point (Infimum) equals Oscillation at Point (Limit)/Lemma}} {{qed|lemma}} | Let $f: D \to \R$ be a [[Definition:Real Function|real function]] where $D \subseteq \R$.
Let $x$ be a [[Definition:Element|point]] in $D$.
Let $N_x$ be the [[Definition:Set|set]] of [[Definition:Neighborhood of Real Number|neighborhoods]] of $x$.
Let $\map {\omega_f} x$ be the [[Definition:Oscillation/Real Space/Os... | === [[Oscillation at Point (Infimum) equals Oscillation at Point (Limit)/Lemma|Lemma]] ===
{{:Oscillation at Point (Infimum) equals Oscillation at Point (Limit)/Lemma}} {{qed|lemma}} | Oscillation at Point (Infimum) equals Oscillation at Point (Limit) | https://proofwiki.org/wiki/Oscillation_at_Point_(Infimum)_equals_Oscillation_at_Point_(Limit) | https://proofwiki.org/wiki/Oscillation_at_Point_(Infimum)_equals_Oscillation_at_Point_(Limit) | [
"Real Analysis"
] | [
"Definition:Real Function",
"Definition:Element",
"Definition:Set",
"Definition:Neighborhood (Real Analysis)/Open Subset",
"Definition:Oscillation/Real Space/Oscillation at Point/Infimum",
"Definition:Oscillation/Real Space/Oscillation on Set",
"Definition:Real Number",
"Definition:Set",
"Definition... | [
"Oscillation at Point (Infimum) equals Oscillation at Point (Limit)/Lemma",
"Oscillation at Point (Infimum) equals Oscillation at Point (Limit)/Lemma"
] |
proofwiki-11318 | Trisecting the Angle/Parabola | Let $\alpha$ be an angle which is to be trisected.
This can be achieved by means of a parabola. | First, notice that because $A$ lies on $\CC_1$:
:$A = \tuple {\cos \angle POQ, \sin \angle POQ}$
This means:
:$B = \tuple {0, \sin \angle POQ}$
Because $C$ is the midpoint of $AB$:
:$C = \tuple {\dfrac {\cos \angle POQ} 2, \sin \angle POQ}$
Because $D$ lies on $\CC_1$:
:$D = \tuple {0, 1}$
and so:
:$E = \tuple {\dfrac ... | Let $\alpha$ be an [[Definition:Angle|angle]] which is to be [[Definition:Trisection|trisected]].
This can be achieved by means of a [[Definition:Parabola|parabola]]. | First, notice that because $A$ lies on $\CC_1$:
:$A = \tuple {\cos \angle POQ, \sin \angle POQ}$
This means:
:$B = \tuple {0, \sin \angle POQ}$
Because $C$ is the [[Definition:Midpoint of Line|midpoint]] of $AB$:
:$C = \tuple {\dfrac {\cos \angle POQ} 2, \sin \angle POQ}$
Because $D$ lies on $\CC_1$:
:$D = \tuple {0... | Trisecting the Angle/Parabola | https://proofwiki.org/wiki/Trisecting_the_Angle/Parabola | https://proofwiki.org/wiki/Trisecting_the_Angle/Parabola | [
"Trisecting the Angle",
"Parabolas"
] | [
"Definition:Angle",
"Definition:Trisection",
"Definition:Parabola"
] | [
"Definition:Line/Midpoint",
"Equation of Circle",
"Definition:Cartesian Coordinate System/X Coordinate",
"Definition:Intersection (Geometry)",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane/Quadrants/First",
"Triple Angle Formulas/Cosine",
"Definition:Vertical",
"Definition:Unit ... |
proofwiki-11319 | All Infima Preserving Mapping is Upper Adjoint of Galois Connection | Let $\struct {S, \preceq}$ be a complete lattice.
Let $\struct {T, \precsim}$ be an ordered set.
Let $g: S \to T$ be an all infima preserving mapping.
Then there exists a mapping $d: T \to S$ such that $\struct {g, d}$ is Galois connection and:
:$\forall t \in T: \map d t = \map \min {g^{-1} \sqbrk {t^\succsim} }$
wher... | === Lemma 1 ===
{{:All Infima Preserving Mapping is Upper Adjoint of Galois Connection/Lemma 1}}{{qed|lemma}}
Let us define a mapping $d: T \to S$ as:
:$\forall t \in T: \map d t := \map \inf {g^{-1} \sqbrk {t^\succsim} }$ | Let $\struct {S, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $\struct {T, \precsim}$ be an [[Definition:Ordered Set|ordered set]].
Let $g: S \to T$ be an [[Definition:All Infima Preserving Mapping|all infima preserving mapping]].
Then there exists a [[Definition:Mapping|mapping]] $d: T \to ... | === [[All Infima Preserving Mapping is Upper Adjoint of Galois Connection/Lemma 1|Lemma 1]] ===
{{:All Infima Preserving Mapping is Upper Adjoint of Galois Connection/Lemma 1}}{{qed|lemma}}
Let us define a [[Definition:Mapping|mapping]] $d: T \to S$ as:
:$\forall t \in T: \map d t := \map \inf {g^{-1} \sqbrk {t^\succ... | All Infima Preserving Mapping is Upper Adjoint of Galois Connection | https://proofwiki.org/wiki/All_Infima_Preserving_Mapping_is_Upper_Adjoint_of_Galois_Connection | https://proofwiki.org/wiki/All_Infima_Preserving_Mapping_is_Upper_Adjoint_of_Galois_Connection | [
"Galois Connections",
"All Infima Preserving Mapping is Upper Adjoint of Galois Connection"
] | [
"Definition:Complete Lattice",
"Definition:Ordered Set",
"Definition:Mapping Preserves Infimum/All",
"Definition:Mapping",
"Definition:Galois Connection",
"Definition:Smallest Element/Subset",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Upper Closure/Element"
] | [
"All Infima Preserving Mapping is Upper Adjoint of Galois Connection/Lemma 1",
"Definition:Mapping"
] |
proofwiki-11320 | Image of Inverse Image | Let $S, T$ be sets.
Let $f: S \to T$ be a mapping.
Let $X$ be a subset of $T$.
Then:
:$f \sqbrk {f^{-1} \sqbrk X} \subseteq X$
where:
:$f^{-1} \sqbrk X$ denotes the image of $X$ under the relation $f^{-1}$. | Let $x \in f \sqbrk {f^{-1} \sqbrk X}$.
By definition of image of set:
:$\exists y \in S: y \in f^{-1} \sqbrk X \land x = \map f y$
By definition of image of set under relation:
:$\map f y \in X$
Thus $x \in X$
{{qed}} | Let $S, T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $X$ be a [[Definition:Subset|subset]] of $T$.
Then:
:$f \sqbrk {f^{-1} \sqbrk X} \subseteq X$
where:
:$f^{-1} \sqbrk X$ denotes the [[Definition:Image of Subset under Relation|image]] of $X$ under the [[Definition:Rela... | Let $x \in f \sqbrk {f^{-1} \sqbrk X}$.
By definition of [[Definition:Image of Subset under Mapping|image of set]]:
:$\exists y \in S: y \in f^{-1} \sqbrk X \land x = \map f y$
By definition of [[Definition:Image of Subset under Relation|image of set under relation]]:
:$\map f y \in X$
Thus $x \in X$
{{qed}} | Image of Inverse Image | https://proofwiki.org/wiki/Image_of_Inverse_Image | https://proofwiki.org/wiki/Image_of_Inverse_Image | [
"Images",
"Inverse Mappings"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Subset",
"Definition:Image (Set Theory)/Relation/Subset",
"Definition:Relation"
] | [
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Image (Set Theory)/Relation/Subset"
] |
proofwiki-11321 | All Suprema Preserving Mapping is Lower Adjoint of Galois Connection | Let $\struct {S, \preceq}$ be an ordered set.
Let $\struct {T, \precsim}$ be a complete lattice.
Let $d: T \to S$ be all suprema preserving mapping.
Then there exists a mapping $g: S \to T$ such that $\tuple {g, d}$ is Galois connection and:
:$\forall s \in S: \map g s = \map \max {d^{-1} \sqbrk {s^\preceq} }$
where
:$... | Define a mapping $d: T \to S$:
:$\forall s \in S: \map g s := \map \sup {d^{-1} \sqbrk {s^\preceq} }$
We will prove as lemma 1 that:
:$d$ is an increasing mapping.
Let $x, y \in T$ such that
:$x \precsim y$
By Lower Closure is Increasing:
:$x^\precsim \subseteq y^\precsim$
By Supremum of Lower Closure of Element:
:$\ma... | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $\struct {T, \precsim}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $d: T \to S$ be [[Definition:Mapping Preserves Supremum/All|all suprema preserving mapping]].
Then there exists a [[Definition:Mapping|mapping]] $g: S \to T... | Define a [[Definition:Mapping|mapping]] $d: T \to S$:
:$\forall s \in S: \map g s := \map \sup {d^{-1} \sqbrk {s^\preceq} }$
We will prove as lemma 1 that:
:$d$ is an [[Definition:Increasing Mapping|increasing mapping]].
Let $x, y \in T$ such that
:$x \precsim y$
By [[Lower Closure is Increasing]]:
:$x^\precsim \sub... | All Suprema Preserving Mapping is Lower Adjoint of Galois Connection | https://proofwiki.org/wiki/All_Suprema_Preserving_Mapping_is_Lower_Adjoint_of_Galois_Connection | https://proofwiki.org/wiki/All_Suprema_Preserving_Mapping_is_Lower_Adjoint_of_Galois_Connection | [
"Galois Connections"
] | [
"Definition:Ordered Set",
"Definition:Complete Lattice",
"Definition:Mapping Preserves Supremum/All",
"Definition:Mapping",
"Definition:Galois Connection",
"Definition:Greatest Element/Subset",
"Definition:Image (Set Theory)/Relation/Subset",
"Definition:Lower Closure/Element"
] | [
"Definition:Mapping",
"Definition:Increasing/Mapping",
"Lower Closure is Increasing",
"Supremum of Lower Closure of Element",
"Definition:Mapping Preserves Supremum/All",
"Definition:Mapping Preserves Supremum/Subset",
"Definition:Mapping Preserves Supremum/Subset",
"Definition:Mapping Preserves Supre... |
proofwiki-11322 | Second Derivative at Point of Inflection | Let $f$ be a real function which is twice differentiable on the open interval $\openint a b$.
Let $f$ have a point of inflection at $\xi \in \openint a b$.
Then:
:$\map {f' '} \xi = 0$
where $\map {f' '} \xi$ denotes the second derivative of $f$ at $\xi$. | By definition of point of inflection, $f'$ has either a local maximum or a local minimum at $\xi$.
From the Interior Extremum Theorem, it follows that the derivative of $f'$ at $\xi$ is zero, that is:
:$\map {f' '} \xi = 0$
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Second Derivative|twice]] [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$.
Let $f$ have a [[Definition:Point of Inflection|point of inflection]] at $\xi \in \open... | By definition of [[Definition:Point of Inflection|point of inflection]], $f'$ has either a [[Definition:Local Maximum|local maximum]] or a [[Definition:Local Minimum|local minimum]] at $\xi$.
From the [[Interior Extremum Theorem]], it follows that the [[Definition:Derivative|derivative]] of $f'$ at $\xi$ is zero, that... | Second Derivative at Point of Inflection | https://proofwiki.org/wiki/Second_Derivative_at_Point_of_Inflection | https://proofwiki.org/wiki/Second_Derivative_at_Point_of_Inflection | [
"Points of Inflection",
"Differential Calculus"
] | [
"Definition:Real Function",
"Definition:Derivative/Higher Derivatives/Second Derivative",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open",
"Definition:Point of Inflection",
"Definition:Derivative/Higher Derivatives/Second Derivative"
] | [
"Definition:Point of Inflection",
"Definition:Maximum Value of Real Function/Local",
"Definition:Minimum Value of Real Function/Local",
"Interior Extremum Theorem",
"Definition:Derivative"
] |
proofwiki-11323 | Galois Connection Implies Order on Mappings | Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets.
Let $g: S \to T$ and $d: T \to S$ be mappings such that
:$\tuple {g, d}$ is Galois connection.
Then $d \circ g \preceq I_S$ and $I_T \precsim g \circ d$
where
:$\preceq, \precsim$ denote the orderings on mappings,
:$I_S$ denotes the identity mapping o... | Let $s \in S$.
By definition of reflexivity:
:$\map g s \precsim \map g s$
By definition of Galois connection:
:$\map d {\map g s} \preceq s$
By definition of composition:
:$\map {\paren {d \circ g} } s \preceq s$
By definition of identity mapping:
:$\map {\paren {d \circ g} } s \preceq \map {I_S} s$
Thus by definition... | Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be [[Definition:Ordered Set|ordered sets]].
Let $g: S \to T$ and $d: T \to S$ be [[Definition:Mapping|mappings]] such that
:$\tuple {g, d}$ is [[Definition:Galois Connection|Galois connection]].
Then $d \circ g \preceq I_S$ and $I_T \precsim g \circ d$
where
:$\pr... | Let $s \in S$.
By definition of [[Definition:Reflexivity|reflexivity]]:
:$\map g s \precsim \map g s$
By definition of [[Definition:Galois Connection|Galois connection]]:
:$\map d {\map g s} \preceq s$
By definition of [[Definition:Composition of Mappings|composition]]:
:$\map {\paren {d \circ g} } s \preceq s$
By ... | Galois Connection Implies Order on Mappings | https://proofwiki.org/wiki/Galois_Connection_Implies_Order_on_Mappings | https://proofwiki.org/wiki/Galois_Connection_Implies_Order_on_Mappings | [
"Galois Connections"
] | [
"Definition:Ordered Set",
"Definition:Mapping",
"Definition:Galois Connection",
"Definition:Ordering on Mappings",
"Definition:Identity Mapping"
] | [
"Definition:Reflexivity",
"Definition:Galois Connection",
"Definition:Composition of Mappings",
"Definition:Identity Mapping",
"Definition:Ordering on Mappings",
"Definition:Reflexivity",
"Definition:Galois Connection",
"Definition:Composition of Mappings",
"Definition:Identity Mapping",
"Definiti... |
proofwiki-11324 | Ordering on Mappings Implies Galois Connection | Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets.
Let $g: S \to T$ and $d: T \to S$ be mappings such that:
:$g$ and $d$ are increasing mappings
and
:$d \circ g \preceq I_S$ and $I_T \precsim g \circ d$
Then
:$\struct {g, d}$ is Galois connection.
where
:$\preceq, \precsim$ denote the orderings on ma... | We will prove that:
:$\forall s \in S, t \in T: t \precsim \map g s \iff \map d t \preceq s$
Let $s \in S, t \in T$.
First implication:
Let
:$t \precsim \map g s$
By definition of increasing mapping:
:$\map d t \preceq \map d {\map g s}$
By definition of ordering on mappings:
:$\map {\paren {d \circ g} } s \preceq \map... | Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be [[Definition:Ordered Set|ordered sets]].
Let $g: S \to T$ and $d: T \to S$ be [[Definition:Mapping|mappings]] such that:
:$g$ and $d$ are [[Definition:Increasing Mapping|increasing mappings]]
and
:$d \circ g \preceq I_S$ and $I_T \precsim g \circ d$
Then
:$\str... | We will prove that:
:$\forall s \in S, t \in T: t \precsim \map g s \iff \map d t \preceq s$
Let $s \in S, t \in T$.
First implication:
Let
:$t \precsim \map g s$
By definition of [[Definition:Increasing Mapping|increasing mapping]]:
:$\map d t \preceq \map d {\map g s}$
By definition of [[Definition:Ordering on M... | Ordering on Mappings Implies Galois Connection | https://proofwiki.org/wiki/Ordering_on_Mappings_Implies_Galois_Connection | https://proofwiki.org/wiki/Ordering_on_Mappings_Implies_Galois_Connection | [
"Galois Connections"
] | [
"Definition:Ordered Set",
"Definition:Mapping",
"Definition:Increasing/Mapping",
"Definition:Galois Connection",
"Definition:Ordering on Mappings",
"Definition:Identity Mapping",
"Definition:Composition of Mappings"
] | [
"Definition:Increasing/Mapping",
"Definition:Ordering on Mappings",
"Definition:Composition of Mappings",
"Definition:Identity Mapping",
"Definition:Transitivity (Relation Theory)",
"Definition:Increasing/Mapping",
"Definition:Ordering on Mappings",
"Definition:Composition of Mappings",
"Definition:... |
proofwiki-11325 | Liouville's Theorem (Hamiltonian Mechanics) | Consider a Hamiltonian system characterised by the Hamiltonian $\map \HH {t, \mathbf q, \mathbf p}$, where the state of the system, $\mathbf r$, is described by generalized coordinates $\mathbf q$ and $\mathbf p$, which correspond to $N$ generalized positions and $N$ generalized momenta, respectively.
Let the traject... | {{tidy}}
{{MissingLinks}}
{{refactor|level = advanced|There are a number of definitions and constructions in the below which really need to be put onto their own pages, where they can be evoked as need be by means of referencing}}
In this proof we first describe the measure space for the Hamiltonian system.
This formal... | Consider a [[Definition:Hamiltonian|Hamiltonian]] system characterised by the [[Definition:Hamiltonian|Hamiltonian]] $\map \HH {t, \mathbf q, \mathbf p}$, where the state of the system, $\mathbf r$, is described by generalized coordinates $\mathbf q$ and $\mathbf p$, which correspond to $N$ generalized positions and ... | {{tidy}}
{{MissingLinks}}
{{refactor|level = advanced|There are a number of definitions and constructions in the below which really need to be put onto their own pages, where they can be evoked as need be by means of referencing}}
In this proof we first describe the measure space for the [[Definition:Hamiltonian|Hamil... | Liouville's Theorem (Hamiltonian Mechanics) | https://proofwiki.org/wiki/Liouville's_Theorem_(Hamiltonian_Mechanics) | https://proofwiki.org/wiki/Liouville's_Theorem_(Hamiltonian_Mechanics) | [
"Hamiltonian Mechanics",
"Liouville's Theorem"
] | [
"Definition:Hamiltonian",
"Definition:Hamiltonian",
"Definition:Hamilton-Jacobi Equation",
"Definition:Hamiltonian",
"Definition:Joint Cumulative Distribution Function",
"Definition:Neighborhood (Metric Space)",
"Definition:Phase Space",
"Definition:Neighborhood (Metric Space)",
"Definition:Phase Sp... | [
"Definition:Hamiltonian",
"Definition:Continuity Equation",
"Definition:Continuity Equation",
"Definition:Hamiltonian",
"Definition:Hamiltonian",
"Definition:Hamiltonian",
"Definition:Phase Space",
"Definition:Joint Cumulative Distribution Function",
"Definition:Phase Space",
"Definition:Neighborh... |
proofwiki-11326 | Pi is Transcendental | $\pi$ (pi) is transcendental. | Proof by Contradiction:
{{AimForCont}} $\pi$ is not transcendental.
Hence by definition, $\pi$ is algebraic.
Let $\pi$ be the root of a non-zero polynomial with rational coefficients, namely $\map f x$.
Then, $\map g x := \map f {i x} \map f {-i x}$ is also a non-zero polynomial with rational coefficients such that:
:$... | [[Definition:Pi|$\pi$ (pi)]] is [[Definition:Transcendental Number|transcendental]]. | [[Proof by Contradiction]]:
{{AimForCont}} $\pi$ is not [[Definition:Transcendental Number|transcendental]].
Hence by definition, $\pi$ is [[Definition:Algebraic Number|algebraic]].
Let $\pi$ be the root of a non-zero [[Definition:Polynomial (Analysis)|polynomial]] with [[Definition:Rational Number|rational]] coeffi... | Pi is Transcendental | https://proofwiki.org/wiki/Pi_is_Transcendental | https://proofwiki.org/wiki/Pi_is_Transcendental | [
"Transcendental Number Theory",
"Pi"
] | [
"Definition:Pi",
"Definition:Transcendental Number"
] | [
"Proof by Contradiction",
"Definition:Transcendental Number",
"Definition:Algebraic Number",
"Definition:Polynomial",
"Definition:Rational Number",
"Definition:Polynomial",
"Definition:Rational Number",
"Definition:Algebraic Number",
"Hermite-Lindemann-Weierstrass Theorem/Weaker",
"Definition:Tran... |
proofwiki-11327 | Gelfond's Constant is Transcendental | '''Gelfond's constant''':
:$e^\pi$
is transcendental. | From the Gelfond-Schneider Theorem:
If:
:$\alpha$ and $\beta$ are algebraic numbers such that $\alpha \notin \set {0, 1}$
:$\beta$ is either irrational or not wholly real
then $\alpha^\beta$ is transcendental.
We have that:
{{begin-eqn}}
{{eqn | l = i^{-2 i}
| r = \paren {e^{i \pi / 2} }^{- 2 i}
| c =
}}
... | '''[[Definition:Gelfond's Constant|Gelfond's constant]]''':
:$e^\pi$
is [[Definition:Transcendental|transcendental]]. | From the [[Gelfond-Schneider Theorem]]:
If:
:$\alpha$ and $\beta$ are [[Definition:Algebraic Number|algebraic numbers]] such that $\alpha \notin \set {0, 1}$
:$\beta$ is either [[Definition:Irrational Number|irrational]] or not [[Definition:Wholly Real|wholly real]]
then $\alpha^\beta$ is [[Definition:Transcendental ... | Gelfond's Constant is Transcendental | https://proofwiki.org/wiki/Gelfond's_Constant_is_Transcendental | https://proofwiki.org/wiki/Gelfond's_Constant_is_Transcendental | [
"Exponential Function",
"Gelfond's Constant"
] | [
"Definition:Gelfond's Constant",
"Definition:Transcendental"
] | [
"Gelfond-Schneider Theorem",
"Definition:Algebraic Number",
"Definition:Irrational Number",
"Definition:Complex Number/Wholly Real",
"Definition:Transcendental Number",
"Definition:Algebraic Number",
"Definition:Algebraic Number",
"Definition:Complex Number/Wholly Real",
"Gelfond-Schneider Theorem"
... |
proofwiki-11328 | Riemann Surface is Path-Connected | A Riemann surface is path-connected. | By definition, a Riemann surface is a complex manifold.
Hence it is connected and locally path-connected.
{{explain|It needs to be established that a complex manifold indeed has both of these properties.}}
{{ProofWanted|A link needed to a page which derives the fact of path-connectedness from connectedness and local pa... | A [[Definition:Riemann Surface|Riemann surface]] is [[Definition:Path-Connected Space|path-connected]]. | By definition, a [[Definition:Riemann Surface|Riemann surface]] is a [[Definition:Complex Manifold|complex manifold]].
Hence it is [[Definition:Connected Topological Space|connected]] and [[Definition:Locally Path-Connected Space|locally path-connected]].
{{explain|It needs to be established that a complex manifold i... | Riemann Surface is Path-Connected | https://proofwiki.org/wiki/Riemann_Surface_is_Path-Connected | https://proofwiki.org/wiki/Riemann_Surface_is_Path-Connected | [
"Riemann Surfaces"
] | [
"Definition:Riemann Surface",
"Definition:Path-Connected/Topological Space"
] | [
"Definition:Riemann Surface",
"Definition:Topological Manifold/Complex Manifold",
"Definition:Connected Topological Space",
"Definition:Locally Path-Connected Space",
"Category:Riemann Surfaces"
] |
proofwiki-11329 | Radó's Theorem (Riemann Surfaces) | A Riemann surface is second countable. | {{ProofWanted}}
{{Namedfor|Tibor Radó|cat = Radó T}} | A [[Definition:Riemann Surface|Riemann surface]] is [[Definition:Second-Countable Space|second countable]]. | {{ProofWanted}}
{{Namedfor|Tibor Radó|cat = Radó T}} | Radó's Theorem (Riemann Surfaces) | https://proofwiki.org/wiki/Radó's_Theorem_(Riemann_Surfaces) | https://proofwiki.org/wiki/Radó's_Theorem_(Riemann_Surfaces) | [
"Riemann Surfaces"
] | [
"Definition:Riemann Surface",
"Definition:Second-Countable Space"
] | [] |
proofwiki-11330 | Conformal Isomorphism of Universal Cover of Riemann Surface | The universal cover of a Riemann surface is conformally isomorphic to either:
:the Riemann sphere
:the complex plane
or
:the unit disk. | By the Riemann Uniformization Theorem.
{{ProofWanted|Expand into a proof.}}
Category:Riemann Surfaces
h3pzxxsv4n5hnxjbbi8d4y8v7d91ypu | The [[Definition:Universal Cover|universal cover]] of a [[Definition:Riemann Surface|Riemann surface]] is [[Definition:Conformal Isomorphism|conformally isomorphic]] to either:
:the [[Definition:Riemann Sphere|Riemann sphere]]
:the [[Definition:Complex Plane|complex plane]]
or
:the [[Definition:Unit Disk|unit disk]]. | By the [[Riemann Uniformization Theorem]].
{{ProofWanted|Expand into a proof.}}
[[Category:Riemann Surfaces]]
h3pzxxsv4n5hnxjbbi8d4y8v7d91ypu | Conformal Isomorphism of Universal Cover of Riemann Surface | https://proofwiki.org/wiki/Conformal_Isomorphism_of_Universal_Cover_of_Riemann_Surface | https://proofwiki.org/wiki/Conformal_Isomorphism_of_Universal_Cover_of_Riemann_Surface | [
"Riemann Surfaces"
] | [
"Definition:Universal Cover",
"Definition:Riemann Surface",
"Definition:Conformal Isomorphism",
"Definition:Riemann Sphere",
"Definition:Complex Number/Complex Plane",
"Definition:Unit Disk"
] | [
"Riemann Uniformization Theorem",
"Category:Riemann Surfaces"
] |
proofwiki-11331 | Riemann Surface is Metrizable | A Riemann surface is metrizable. | Follows {{apriori}} from Riemann Surface admits Metric of Constant Curvature.
{{qed}}
Category:Riemann Surfaces
le3230wb1hxrfa2c8y8dlsqyxoqh0fv | A [[Definition:Riemann Surface|Riemann surface]] is [[Definition:Metrizable Space|metrizable]]. | Follows {{apriori}} from [[Riemann Surface admits Metric of Constant Curvature]].
{{qed}}
[[Category:Riemann Surfaces]]
le3230wb1hxrfa2c8y8dlsqyxoqh0fv | Riemann Surface is Metrizable | https://proofwiki.org/wiki/Riemann_Surface_is_Metrizable | https://proofwiki.org/wiki/Riemann_Surface_is_Metrizable | [
"Riemann Surfaces"
] | [
"Definition:Riemann Surface",
"Definition:Metrizable Space"
] | [
"Riemann Surface admits Metric of Constant Curvature",
"Category:Riemann Surfaces"
] |
proofwiki-11332 | Riemann Surface admits Metric of Constant Curvature | A Riemann surface admits a metric of constant curvature. | {{ProofWanted}}
Category:Riemann Surfaces
m5kylsa3axmq372rmkcc91uk2txeovp | A [[Definition:Riemann Surface|Riemann surface]] admits a [[Definition:Metric|metric]] of [[Definition:Constant Curvature Metric|constant curvature]]. | {{ProofWanted}}
[[Category:Riemann Surfaces]]
m5kylsa3axmq372rmkcc91uk2txeovp | Riemann Surface admits Metric of Constant Curvature | https://proofwiki.org/wiki/Riemann_Surface_admits_Metric_of_Constant_Curvature | https://proofwiki.org/wiki/Riemann_Surface_admits_Metric_of_Constant_Curvature | [
"Riemann Surfaces"
] | [
"Definition:Riemann Surface",
"Definition:Metric Space/Metric",
"Definition:Constant Curvature Metric"
] | [
"Category:Riemann Surfaces"
] |
proofwiki-11333 | Riemann Sphere is only Elliptic Riemann Surface | The Riemann sphere is the only elliptic Riemann surface (up to conformal isomorphism). | {{ProofWanted}}
Category:Riemann Surfaces
cdq7310y4pazt8ulnwuf5vy8g31pgq1 | The [[Definition:Riemann Sphere|Riemann sphere]] is the only [[Definition:Elliptic Riemann Surface|elliptic Riemann surface]] (up to [[Definition:Conformal Isomorphism|conformal isomorphism]]). | {{ProofWanted}}
[[Category:Riemann Surfaces]]
cdq7310y4pazt8ulnwuf5vy8g31pgq1 | Riemann Sphere is only Elliptic Riemann Surface | https://proofwiki.org/wiki/Riemann_Sphere_is_only_Elliptic_Riemann_Surface | https://proofwiki.org/wiki/Riemann_Sphere_is_only_Elliptic_Riemann_Surface | [
"Riemann Surfaces"
] | [
"Definition:Riemann Sphere",
"Definition:Riemann Surface/Elliptic",
"Definition:Conformal Isomorphism"
] | [
"Category:Riemann Surfaces"
] |
proofwiki-11334 | Parabolic Riemann Surface is Plane, Punctured Plane or Torus | A parabolic Riemann surface is conformally isomorphic to either:
:the complex plane
:the punctured complex plane $\C \setminus \set 0$
or:
:a torus. | {{ProofWanted}}
Category:Riemann Surfaces
a1rl0eigrhpgy6ha6rrzg3u3wsrszus | A [[Definition:Parabolic Riemann Surface|parabolic Riemann surface]] is [[Definition:Conformal Isomorphism|conformally isomorphic]] to either:
:the [[Definition:Complex Plane|complex plane]]
:the [[Definition:Punctured Complex Plane|punctured complex plane]] $\C \setminus \set 0$
or:
:a [[Definition:Torus (Topology)|to... | {{ProofWanted}}
[[Category:Riemann Surfaces]]
a1rl0eigrhpgy6ha6rrzg3u3wsrszus | Parabolic Riemann Surface is Plane, Punctured Plane or Torus | https://proofwiki.org/wiki/Parabolic_Riemann_Surface_is_Plane,_Punctured_Plane_or_Torus | https://proofwiki.org/wiki/Parabolic_Riemann_Surface_is_Plane,_Punctured_Plane_or_Torus | [
"Riemann Surfaces"
] | [
"Definition:Riemann Surface/Parabolic",
"Definition:Conformal Isomorphism",
"Definition:Complex Number/Complex Plane",
"Definition:Punctured Complex Plane",
"Definition:Torus (Topology)"
] | [
"Category:Riemann Surfaces"
] |
proofwiki-11335 | Riemann's Rearrangement Theorem | Let $\sequence {a_n}_{n \mathop \in \N}$ be a sequence in $\R$ such that:
:$\ds \sum_{n \mathop = 1}^\infty a_n$ is conditionally convergent.
Let $\alpha, \beta \in \overline \R$ be such that:
:$-\infty \le \alpha \le \beta \le \infty$
Then there exists a permutation $\pi : \N \to \N$ such that, when we define:
:$\ds S... | Define the positive and negative parts:
:$a_n^+ = \max \set {0, a_n}$
and:
:$a_n^- = -\min \set {0, a_n}$
From Difference of Positive and Negative Parts, we have:
:$a_n = a_n^+ - a_n^-$
From Sum of Positive and Negative Parts, we have:
:$\size {a_n} = a_n^+ + a_n^-$
From Linear Combination of Convergent Series, we hav... | Let $\sequence {a_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $\R$ such that:
:$\ds \sum_{n \mathop = 1}^\infty a_n$ is [[Definition:Conditionally Convergent Series|conditionally convergent]].
Let $\alpha, \beta \in \overline \R$ be such that:
:$-\infty \le \alpha \le \beta \le \infty$
Then there... | Define the [[Definition:Positive Part|positive]] and [[Definition:Negative Part|negative parts]]:
:$a_n^+ = \max \set {0, a_n}$
and:
:$a_n^- = -\min \set {0, a_n}$
From [[Difference of Positive and Negative Parts]], we have:
:$a_n = a_n^+ - a_n^-$
From [[Sum of Positive and Negative Parts]], we have:
:$\size {a_n} = ... | Riemann's Rearrangement Theorem | https://proofwiki.org/wiki/Riemann's_Rearrangement_Theorem | https://proofwiki.org/wiki/Riemann's_Rearrangement_Theorem | [
"Riemann's Rearrangement Theorem",
"Series"
] | [
"Definition:Sequence",
"Definition:Conditionally Convergent Series",
"Definition:Permutation",
"Definition:Series/Real",
"Definition:Conditionally Convergent Series",
"Definition:Term of Sequence",
"Definition:Permutation",
"Definition:Series/Number Field",
"Definition:Conditionally Convergent Serie... | [
"Definition:Positive Part",
"Definition:Negative Part",
"Difference of Positive and Negative Parts",
"Sum of Positive and Negative Parts",
"Linear Combination of Convergent Series",
"Definition:Conditionally Convergent Series",
"Definition:Conditionally Convergent Series",
"Definition:Strictly Increas... |
proofwiki-11336 | Particle on Curved Surface under no Force moves along Geodesic | Consider a particle $P$ which is constrained to move on a curved surface $C$.
Let $P$ be such that no force acts upon it.
Then $P$ moves along a geodesic. | {{ProofWanted|A consequence of Hamilton's Principle in the calculus of variations.}} | Consider a [[Definition:Particle|particle]] $P$ which is constrained to move on a [[Definition:Curved Surface|curved surface]] $C$.
Let $P$ be such that no [[Definition:Force|force]] acts upon it.
Then $P$ moves along a [[Definition:Geodesic Curve|geodesic]]. | {{ProofWanted|A consequence of [[Hamilton's Principle]] in the [[Definition:Calculus of Variations|calculus of variations]].}} | Particle on Curved Surface under no Force moves along Geodesic | https://proofwiki.org/wiki/Particle_on_Curved_Surface_under_no_Force_moves_along_Geodesic | https://proofwiki.org/wiki/Particle_on_Curved_Surface_under_no_Force_moves_along_Geodesic | [
"Calculus of Variations"
] | [
"Definition:Particle",
"Definition:Curved Surface",
"Definition:Force",
"Definition:Geodesic"
] | [
"Hamilton's Principle",
"Definition:Calculus of Variations"
] |
proofwiki-11337 | Sum of Reciprocals of Powers as Euler Product | Let $\zeta$ be the Riemann zeta function.
Let $s \in \C$ be a complex number with real part $\sigma > 1$.
Then:
:$\ds \map \zeta s = \prod_{\text {$p$ prime} } \frac 1 {1 - p^{-s} }$
where the infinite product runs over the prime numbers. | By definition of Euler product:
:$\ds \sum_{n \mathop = 1}^\infty a_n n^{-z} = \prod_p \frac 1 {1 - a_p p^{-z} }$
{{iff}} $\ds \sum_{n \mathop = 1}^\infty a_n n^{-z}$ is absolutely convergent.
{{explain|I believe only "if" is true as shown in Product Form of Sum on Completely Multiplicative Function. Or is there a proo... | Let $\zeta$ be the [[Definition:Riemann Zeta Function|Riemann zeta function]].
Let $s \in \C$ be a [[Definition:Complex Number|complex number]] with [[Definition:Real Part|real part]] $\sigma > 1$.
Then:
:$\ds \map \zeta s = \prod_{\text {$p$ prime} } \frac 1 {1 - p^{-s} }$
where the [[Definition:Infinite Product|i... | By definition of [[Definition:Euler Product|Euler product]]:
:$\ds \sum_{n \mathop = 1}^\infty a_n n^{-z} = \prod_p \frac 1 {1 - a_p p^{-z} }$
{{iff}} $\ds \sum_{n \mathop = 1}^\infty a_n n^{-z}$ is [[Definition:Absolutely Convergent Series|absolutely convergent]].
{{explain|I believe only "if" is true as shown in [[P... | Sum of Reciprocals of Powers as Euler Product/Proof 1 | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Powers_as_Euler_Product | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Powers_as_Euler_Product/Proof_1 | [
"Riemann Zeta Function",
"Sum of Reciprocals of Powers as Euler Product"
] | [
"Definition:Riemann Zeta Function",
"Definition:Complex Number",
"Definition:Complex Number/Real Part",
"Definition:Continued Product/Infinite",
"Definition:Prime Number"
] | [
"Definition:Euler Product",
"Definition:Absolutely Convergent Series",
"Product Form of Sum on Completely Multiplicative Function",
"Convergence of P-Series",
"Definition:Absolutely Convergent Series"
] |
proofwiki-11338 | Sum of Reciprocals of Powers as Euler Product | Let $\zeta$ be the Riemann zeta function.
Let $s \in \C$ be a complex number with real part $\sigma > 1$.
Then:
:$\ds \map \zeta s = \prod_{\text {$p$ prime} } \frac 1 {1 - p^{-s} }$
where the infinite product runs over the prime numbers. | {{Recall|Riemann Zeta Function}}
{{begin-eqn}}
{{eqn | l = \map \zeta s
| r = \sum_{n \mathop = 1}^\infty \frac 1 {n^s}
| c =
}}
{{eqn | r = 1 + \dfrac 1 {2^z} + \dfrac 1 {3^z} + \dfrac 1 {2^{2 z} } + \dfrac 1 {5^z} + \dfrac 1 {2^z 3^z} + \dfrac 1 {7^z} + \dfrac 1 {2^{3 z} } + \dfrac 1 {3^{2 z} } + \cdots
... | Let $\zeta$ be the [[Definition:Riemann Zeta Function|Riemann zeta function]].
Let $s \in \C$ be a [[Definition:Complex Number|complex number]] with [[Definition:Real Part|real part]] $\sigma > 1$.
Then:
:$\ds \map \zeta s = \prod_{\text {$p$ prime} } \frac 1 {1 - p^{-s} }$
where the [[Definition:Infinite Product|i... | {{Recall|Riemann Zeta Function}}
{{begin-eqn}}
{{eqn | l = \map \zeta s
| r = \sum_{n \mathop = 1}^\infty \frac 1 {n^s}
| c =
}}
{{eqn | r = 1 + \dfrac 1 {2^z} + \dfrac 1 {3^z} + \dfrac 1 {2^{2 z} } + \dfrac 1 {5^z} + \dfrac 1 {2^z 3^z} + \dfrac 1 {7^z} + \dfrac 1 {2^{3 z} } + \dfrac 1 {3^{2 z} } + \cdots
... | Sum of Reciprocals of Powers as Euler Product/Proof 2 | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Powers_as_Euler_Product | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Powers_as_Euler_Product/Proof_2 | [
"Riemann Zeta Function",
"Sum of Reciprocals of Powers as Euler Product"
] | [
"Definition:Riemann Zeta Function",
"Definition:Complex Number",
"Definition:Complex Number/Real Part",
"Definition:Continued Product/Infinite",
"Definition:Prime Number"
] | [
"Sum of Infinite Geometric Sequence",
"Convergence of P-Series",
"Definition:Absolutely Convergent Series",
"Fundamental Theorem of Arithmetic",
"Definition:Integer",
"Definition:Multiplication/Integers",
"Definition:Prime Number"
] |
proofwiki-11339 | Shift Mapping is Lower Adjoint iff Appropriate Maxima Exist | Let $\struct {S, \preceq}$ be a meet semilattice.
{{TFAE}}
:$(1): \quad \forall x \in S, f: S \to S: \paren {\forall s \in S: \map f s = x \wedge s} \implies f$ is lower adjoint
:$(2): \quad \forall x, t \in S: \max \set {s \in S: x \wedge s \preceq t}$ exists. | === $(1) \implies (2)$ ===
Assume that:
:$\forall x \in S, f: S \to S: \paren {\forall s \in S: \map f s = x \wedge s} \implies f$ is lower adjoint
Let $x, t \in S$.
Define $f: S \to S$:
:$\forall s \in S: \map f s = x \wedge s$
By assumption:
:$f$ is lower adjoint
By definition of lower adjoint:
:$\exists g: S \to S: ... | Let $\struct {S, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
{{TFAE}}
:$(1): \quad \forall x \in S, f: S \to S: \paren {\forall s \in S: \map f s = x \wedge s} \implies f$ is [[Definition:Galois Connection|lower adjoint]]
:$(2): \quad \forall x, t \in S: \max \set {s \in S: x \wedge s \preceq t... | === $(1) \implies (2)$ ===
Assume that:
:$\forall x \in S, f: S \to S: \paren {\forall s \in S: \map f s = x \wedge s} \implies f$ is [[Definition:Galois Connection|lower adjoint]]
Let $x, t \in S$.
Define $f: S \to S$:
:$\forall s \in S: \map f s = x \wedge s$
By assumption:
:$f$ is [[Definition:Galois Connection|... | Shift Mapping is Lower Adjoint iff Appropriate Maxima Exist | https://proofwiki.org/wiki/Shift_Mapping_is_Lower_Adjoint_iff_Appropriate_Maxima_Exist | https://proofwiki.org/wiki/Shift_Mapping_is_Lower_Adjoint_iff_Appropriate_Maxima_Exist | [
"Galois Connections"
] | [
"Definition:Meet Semilattice",
"Definition:Galois Connection"
] | [
"Definition:Galois Connection",
"Definition:Galois Connection",
"Definition:Galois Connection",
"Definition:Galois Connection",
"Galois Connection is Expressed by Maximum",
"Definition:Image (Set Theory)/Relation/Subset",
"Definition:Lower Closure/Element",
"Galois Connection is Expressed by Maximum",... |
proofwiki-11340 | Brouwerian Lattice iff Shift Mapping is Lower Adjoint | Let $\struct {S, \preceq}$ be a lattice.
Then $\struct {S, \preceq}$ is a Brouwerian lattice {{iff}}:
:$\forall x \in S, f: S \to S: \paren {\forall s \in S: \map f s = x \wedge s} \implies f$ is a lower adjoint | :$\struct {S, \preceq}$ is a Brouwerian lattice
{{iff}}:
:$\forall x, y \in S: x$ has relative pseudocomplement with respect to $y$ by definition of Brouwerian lattice
{{iff}}:
:$\forall x, y \in S: \max \set {s \in S: x \wedge s \preceq y}$ exists by definition of relative pseudocomplement
{{iff}}:
:$\forall x \in S, ... | Let $\struct {S, \preceq}$ be a [[Definition:Lattice (Order Theory)|lattice]].
Then $\struct {S, \preceq}$ is a [[Definition:Brouwerian Lattice|Brouwerian lattice]] {{iff}}:
:$\forall x \in S, f: S \to S: \paren {\forall s \in S: \map f s = x \wedge s} \implies f$ is a [[Definition:Galois Connection|lower adjoint]] | :$\struct {S, \preceq}$ is a [[Definition:Brouwerian Lattice|Brouwerian lattice]]
{{iff}}:
:$\forall x, y \in S: x$ has [[Definition:Relative Pseudocomplement|relative pseudocomplement]] with respect to $y$ by definition of [[Definition:Brouwerian Lattice|Brouwerian lattice]]
{{iff}}:
:$\forall x, y \in S: \max \set {s... | Brouwerian Lattice iff Shift Mapping is Lower Adjoint | https://proofwiki.org/wiki/Brouwerian_Lattice_iff_Shift_Mapping_is_Lower_Adjoint | https://proofwiki.org/wiki/Brouwerian_Lattice_iff_Shift_Mapping_is_Lower_Adjoint | [
"Galois Connections",
"Brouwerian Lattices"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Brouwerian Lattice",
"Definition:Galois Connection"
] | [
"Definition:Brouwerian Lattice",
"Definition:Relative Pseudocomplement",
"Definition:Brouwerian Lattice",
"Definition:Relative Pseudocomplement",
"Definition:Galois Connection",
"Shift Mapping is Lower Adjoint iff Appropriate Maxima Exist"
] |
proofwiki-11341 | Brouwerian Lattice is Distributive | Let $\struct {S, \preceq}$ be a Brouwerian lattice.
Then $\struct {S, \preceq}$ is a distributive lattice | Let $x, y, z \in S$.
By Brouwerian Lattice iff Shift Mapping is Lower Adjoint:
:$\forall x \in S, f: S \to S: \paren {\forall s \in S: \map f s = x \wedge s} \implies f$ is a lower adjoint
Define a mapping $f: S \to S$:
:$\forall s \in S: \map f s = x \wedge s$
Then:
:$f$ is a lower adjoint
By Lower Adjoint Preserves A... | Let $\struct {S, \preceq}$ be a [[Definition:Brouwerian Lattice|Brouwerian lattice]].
Then $\struct {S, \preceq}$ is a [[Definition:Distributive Lattice|distributive lattice]] | Let $x, y, z \in S$.
By [[Brouwerian Lattice iff Shift Mapping is Lower Adjoint]]:
:$\forall x \in S, f: S \to S: \paren {\forall s \in S: \map f s = x \wedge s} \implies f$ is a [[Definition:Galois Connection|lower adjoint]]
Define a [[Definition:Mapping|mapping]] $f: S \to S$:
:$\forall s \in S: \map f s = x \wedge... | Brouwerian Lattice is Distributive | https://proofwiki.org/wiki/Brouwerian_Lattice_is_Distributive | https://proofwiki.org/wiki/Brouwerian_Lattice_is_Distributive | [
"Distributive Lattices",
"Brouwerian Lattices"
] | [
"Definition:Brouwerian Lattice",
"Definition:Distributive Lattice"
] | [
"Brouwerian Lattice iff Shift Mapping is Lower Adjoint",
"Definition:Galois Connection",
"Definition:Mapping",
"Definition:Galois Connection",
"Lower Adjoint Preserves All Suprema",
"Definition:Mapping Preserves Supremum/All",
"Definition:Mapping Preserves Supremum/All",
"Definition:Mapping Preserves ... |
proofwiki-11342 | Definite Integral of Reciprocal of Root of a Squared minus x Squared | :$\ds \int_0^x \frac {\d t} {\sqrt{1 - t^2} } = \arcsin x$ | {{begin-eqn}}
{{eqn | l = \int_0^x \frac {\d t} {\sqrt{1 - t^2} }
| r = \intlimits {\arcsin \frac t 1} 0 x
| c = Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$, {{Defof|Definite Integral}}
}}
{{eqn | r = \arcsin x - \arcsin 0
| c =
}}
{{eqn | r = \arcsin x
}}
{{end-eqn}}
{{qed}} | :$\ds \int_0^x \frac {\d t} {\sqrt{1 - t^2} } = \arcsin x$ | {{begin-eqn}}
{{eqn | l = \int_0^x \frac {\d t} {\sqrt{1 - t^2} }
| r = \intlimits {\arcsin \frac t 1} 0 x
| c = [[Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form|Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$]], {{Defof|Definite Integral}}
}}
{{eqn | r = \arcsin x - \arcsin 0
... | Definite Integral of Reciprocal of Root of a Squared minus x Squared | https://proofwiki.org/wiki/Definite_Integral_of_Reciprocal_of_Root_of_a_Squared_minus_x_Squared | https://proofwiki.org/wiki/Definite_Integral_of_Reciprocal_of_Root_of_a_Squared_minus_x_Squared | [
"Definite Integrals",
"Arcsine Function",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form"
] |
proofwiki-11343 | Entire Function is Transcendental iff Power Series Expansion is Infinite | Let $f$ be an entire function.
Then $f$ is '''transcendental''' {{iff}} the power series expansion of $f$ has an infinite number of coefficients which are non-zero. | {{ProofWanted}}
Category:Entire Functions
miflckvdvsw7isipgspnhc4j3jjh42b | Let $f$ be an [[Definition:Entire Function|entire function]].
Then $f$ is '''[[Definition:Transcendental Entire Function|transcendental]]''' {{iff}} the [[Definition:Power Series|power series expansion]] of $f$ has an [[Definition:Infinite|infinite]] number of [[Definition:Polynomial Coefficient|coefficients]] which a... | {{ProofWanted}}
[[Category:Entire Functions]]
miflckvdvsw7isipgspnhc4j3jjh42b | Entire Function is Transcendental iff Power Series Expansion is Infinite | https://proofwiki.org/wiki/Entire_Function_is_Transcendental_iff_Power_Series_Expansion_is_Infinite | https://proofwiki.org/wiki/Entire_Function_is_Transcendental_iff_Power_Series_Expansion_is_Infinite | [
"Entire Functions"
] | [
"Definition:Entire Function",
"Definition:Entire Function/Transcendental",
"Definition:Power Series",
"Definition:Infinite",
"Definition:Coefficient of Polynomial"
] | [
"Category:Entire Functions"
] |
proofwiki-11344 | Entire Function is Transcendental iff not Complex Polynomial Function | Let $f$ be an entire function.
Then $f$ is '''transcendental''' {{iff}} it is not a complex polynomial function. | {{ProofWanted}}
Category:Entire Functions
gyuedjao26bgt0dgqyejduxfigo72tg | Let $f$ be an [[Definition:Entire Function|entire function]].
Then $f$ is '''[[Definition:Transcendental Entire Function|transcendental]]''' {{iff}} it is not a [[Definition:Complex Polynomial Function|complex polynomial function]]. | {{ProofWanted}}
[[Category:Entire Functions]]
gyuedjao26bgt0dgqyejduxfigo72tg | Entire Function is Transcendental iff not Complex Polynomial Function | https://proofwiki.org/wiki/Entire_Function_is_Transcendental_iff_not_Complex_Polynomial_Function | https://proofwiki.org/wiki/Entire_Function_is_Transcendental_iff_not_Complex_Polynomial_Function | [
"Entire Functions"
] | [
"Definition:Entire Function",
"Definition:Entire Function/Transcendental",
"Definition:Polynomial Function/Complex"
] | [
"Category:Entire Functions"
] |
proofwiki-11345 | Complex Function is Entire iff it has Everywhere Convergent Power Series | Let $f: \C \to \C$ be a complex function.
Then $f$ is an '''entire function''' {{iff}} $f$ can be given by an everywhere convergent power series:
:$\ds \map f z = \sum_{n \mathop = 0}^\infty a_n z^n; \quad \lim_{n \mathop \to \infty} \sqrt [n] {\size {a_n} } = 0$ | {{explain|This needs tightening.}} | Let $f: \C \to \C$ be a [[Definition:Complex Function|complex function]].
Then $f$ is an '''[[Definition:Entire Function|entire function]]''' {{iff}} $f$ can be given by an [[Definition:Everywhere Convergence|everywhere convergent]] [[Definition:Power Series|power series]]:
:$\ds \map f z = \sum_{n \mathop = 0}^\infty... | {{explain|This needs tightening.}} | Complex Function is Entire iff it has Everywhere Convergent Power Series | https://proofwiki.org/wiki/Complex_Function_is_Entire_iff_it_has_Everywhere_Convergent_Power_Series | https://proofwiki.org/wiki/Complex_Function_is_Entire_iff_it_has_Everywhere_Convergent_Power_Series | [
"Entire Functions"
] | [
"Definition:Complex Function",
"Definition:Entire Function",
"Definition:Everywhere Convergence",
"Definition:Power Series"
] | [] |
proofwiki-11346 | Brouwerian Lattice is Upper Bounded | Let $\struct {S, \vee, \wedge, \preceq}$ be a Brouwerian lattice.
Then $S$ is upper bounded. | By assumption:
:$S \ne \O$
By definition of non-empty set:
:$\exists s: s \in S$
By definition of Brouwerian lattice:
:$s$ has relative pseudocomplement with respect to $s$
By definition of relative pseudocomplement:
:$\max \set {x \in S: s \wedge x \preceq s}$ exists and equals $s \to s$
Let $x \in S$.
By Meet Precede... | Let $\struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Brouwerian Lattice|Brouwerian lattice]].
Then $S$ is [[Definition:Bounded Above Set|upper bounded]]. | By assumption:
:$S \ne \O$
By definition of [[Definition:Non-Empty Set|non-empty set]]:
:$\exists s: s \in S$
By definition of [[Definition:Brouwerian Lattice|Brouwerian lattice]]:
:$s$ has [[Definition:Relative Pseudocomplement|relative pseudocomplement]] with respect to $s$
By definition of [[Definition:Relative P... | Brouwerian Lattice is Upper Bounded | https://proofwiki.org/wiki/Brouwerian_Lattice_is_Upper_Bounded | https://proofwiki.org/wiki/Brouwerian_Lattice_is_Upper_Bounded | [
"Bounded Lattices",
"Brouwerian Lattices"
] | [
"Definition:Brouwerian Lattice",
"Definition:Bounded Above Set"
] | [
"Definition:Non-Empty Set",
"Definition:Brouwerian Lattice",
"Definition:Relative Pseudocomplement",
"Definition:Relative Pseudocomplement",
"Meet Precedes Operands",
"Definition:Greatest Element/Subset",
"Definition:Upper Bound of Set",
"Definition:Bounded Above Set"
] |
proofwiki-11347 | Weierstrass Extreme Value Theorem | Let $f$ be a real function which is continuous in a closed real interval $\closedint a b$.
Then:
:$\exists x_M: \forall x \in \closedint a b: \map f {x_M} \ge \map f x$
:$\exists x_m: \forall x \in \closedint a b: \map f {x_m} \le \map f x$ | {{tidy}}
We will prove the case for $x_M$, the maximum.
The case for $x_m$, the minimum, is similar.
First it is shown that $\map f x$ is bounded above in the closed real interval $\closedint a b$.
{{AimForCont}} $\map f x$ has no upper bound.
Then:
:$\forall N \in \N: \exists x_N \in \closedint a b: \map f {x_N} > N$
... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] in a [[Definition:Closed Real Interval|closed real interval]] $\closedint a b$.
Then:
:$\exists x_M: \forall x \in \closedint a b: \map f {x_M} \ge \map f x$
:$\exists x_m: \forall x \in \closedint a b: \... | {{tidy}}
We will prove the case for $x_M$, the [[Definition:Maximum Value of Real Function|maximum]].
The case for $x_m$, the [[Definition:Minimum Value of Real Function|minimum]], is similar.
First it is shown that $\map f x$ is [[Definition:Bounded Above Real-Valued Function|bounded above]] in the [[Definition:Cl... | Weierstrass Extreme Value Theorem | https://proofwiki.org/wiki/Weierstrass_Extreme_Value_Theorem | https://proofwiki.org/wiki/Weierstrass_Extreme_Value_Theorem | [
"Weierstrass Extreme Value Theorem",
"Continuous Real Functions"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed"
] | [
"Definition:Maximum Value of Real Function",
"Definition:Minimum Value of Real Function",
"Definition:Bounded Above Mapping/Real-Valued",
"Definition:Real Interval/Closed",
"Definition:Upper Bound of Mapping/Real-Valued",
"Definition:Sequence",
"Definition:Bounded Sequence/Real",
"Definition:Bounded A... |
proofwiki-11348 | Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice | Let $\struct {S, \vee, \wedge, \preceq}$ be a Brouwerian lattice.
Let $a$ be an element of $S$.
Let $g, d: S \to S$ be mappings such that:
:$\forall s \in S: \map g s = a \to s$
and
:$\forall s \in S: \map d s = a \wedge s$
Then $\struct {g, d}$ is a Galois connection. | By Brouwerian Lattice iff Shift Mapping is Lower Adjoint:
:$d$ is lower adjoint
By definition of lower adjoint:
:$\exists g': S \to S: \struct {g', d}$ is Galois connection
By Galois Connection is Expressed by Maximum:
:$\forall s \in S: \map {g'} s = \map \max {d^{-1} \sqbrk {s^\preceq} }$
By definition of image of se... | Let $\struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Brouwerian Lattice|Brouwerian lattice]].
Let $a$ be an [[Definition:Element|element]] of $S$.
Let $g, d: S \to S$ be [[Definition:Mapping|mappings]] such that:
:$\forall s \in S: \map g s = a \to s$
and
:$\forall s \in S: \map d s = a \wedge s$
Then $\struc... | By [[Brouwerian Lattice iff Shift Mapping is Lower Adjoint]]:
:$d$ is [[Definition:Galois Connection|lower adjoint]]
By definition of [[Definition:Galois Connection|lower adjoint]]:
:$\exists g': S \to S: \struct {g', d}$ is [[Definition:Galois Connection|Galois connection]]
By [[Galois Connection is Expressed by Max... | Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice | https://proofwiki.org/wiki/Relative_Pseudocomplement_and_Shift_Mapping_form_Galois_Connection_in_Brouwerian_Lattice | https://proofwiki.org/wiki/Relative_Pseudocomplement_and_Shift_Mapping_form_Galois_Connection_in_Brouwerian_Lattice | [
"Galois Connections",
"Brouwerian Lattices"
] | [
"Definition:Brouwerian Lattice",
"Definition:Element",
"Definition:Mapping",
"Definition:Galois Connection"
] | [
"Brouwerian Lattice iff Shift Mapping is Lower Adjoint",
"Definition:Galois Connection",
"Definition:Galois Connection",
"Definition:Galois Connection",
"Galois Connection is Expressed by Maximum",
"Definition:Image (Set Theory)/Relation/Subset",
"Definition:Lower Closure/Element",
"Definition:Relativ... |
proofwiki-11349 | Inequality with Meet Operation is Equivalent to Inequality with Relative Pseudocomplement in Brouwerian Lattice | Let $\struct {S, \vee, \wedge, \preceq}$ be s Brouwerian lattice.
Let $a, x, y \in S$.
Then
:$a \wedge x \preceq y$ {{iff}} $x \preceq a \to y$ | Define a mapping $d: S \to S$:
:$\forall s \in S: \map d s = a \wedge s$
Define a mapping $g: S \to S$:
:$\forall s \in S: \map g s = a \to s$
By Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice:
:$\tuple {g, d}$ is Galois connection.
By definition of Galois connection:
:$x \prec... | Let $\struct {S, \vee, \wedge, \preceq}$ be s [[Definition:Brouwerian Lattice|Brouwerian lattice]].
Let $a, x, y \in S$.
Then
:$a \wedge x \preceq y$ {{iff}} $x \preceq a \to y$ | Define a [[Definition:Mapping|mapping]] $d: S \to S$:
:$\forall s \in S: \map d s = a \wedge s$
Define a [[Definition:Mapping|mapping]] $g: S \to S$:
:$\forall s \in S: \map g s = a \to s$
By [[Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice]]:
:$\tuple {g, d}$ is [[Definition... | Inequality with Meet Operation is Equivalent to Inequality with Relative Pseudocomplement in Brouwerian Lattice | https://proofwiki.org/wiki/Inequality_with_Meet_Operation_is_Equivalent_to_Inequality_with_Relative_Pseudocomplement_in_Brouwerian_Lattice | https://proofwiki.org/wiki/Inequality_with_Meet_Operation_is_Equivalent_to_Inequality_with_Relative_Pseudocomplement_in_Brouwerian_Lattice | [
"Brouwerian Lattices"
] | [
"Definition:Brouwerian Lattice"
] | [
"Definition:Mapping",
"Definition:Mapping",
"Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice",
"Definition:Galois Connection",
"Definition:Galois Connection"
] |
proofwiki-11350 | Weierstrass Approximation Theorem | Let $f$ be a real function which is continuous on the closed interval $\Bbb I = \closedint a b$.
Then $f$ can be uniformly approximated on $\Bbb I$ by a polynomial function to any given degree of accuracy. | Let $\map f t: \Bbb I = \closedint a b \to \R$ be a continuous function.
Introduce $\map x t$ with a rescaled domain:
:$\map f t \mapsto \map x {a + t \paren {b - a} } : \closedint a b \to \closedint 0 1$
From now on we will work with $x: \closedint 0 1 \to \R$, which is also continuous.
Let $n \in \N$.
For $t \in \clo... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\Bbb I = \closedint a b$.
Then $f$ can be [[Definition:Uniform Approximation|uniformly approximated]] on $\Bbb I$ by a [[Definition:Real Polynom... | Let $\map f t: \Bbb I = \closedint a b \to \R$ be a [[Definition:Continuous Real Function|continuous function]].
Introduce $\map x t$ with a rescaled [[Definition:Domain of Mapping|domain]]:
:$\map f t \mapsto \map x {a + t \paren {b - a} } : \closedint a b \to \closedint 0 1$
From now on we will work with $x: \clos... | Weierstrass Approximation Theorem/Proof 1 | https://proofwiki.org/wiki/Weierstrass_Approximation_Theorem | https://proofwiki.org/wiki/Weierstrass_Approximation_Theorem/Proof_1 | [
"Weierstrass Approximation Theorem",
"Real Analysis"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Uniform Approximation",
"Definition:Polynomial Function/Real"
] | [
"Definition:Continuous Real Function",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Continuous Real Function",
"Definition:Bernstein Polynomial",
"Binomial Theorem/Integral Index",
"Weierstrass Approximation Theorem/Lemma 1",
"Weierstrass Approximation Theorem/Lemma 2",
"Definition:Summation"... |
proofwiki-11351 | Weierstrass Approximation Theorem | Let $f$ be a real function which is continuous on the closed interval $\Bbb I = \closedint a b$.
Then $f$ can be uniformly approximated on $\Bbb I$ by a polynomial function to any given degree of accuracy. | {{WLOG}}, assume $\Bbb I = \closedint 0 1$.
For each $n \in \N$, let:
:$\ds \map {P_n} x := \sum_{k \mathop = 0}^n \map f {\dfrac k n } \dbinom n k x^k \paren {1 - x}^{n - k}$
We shall show that $\ds \lim_{n \mathop \to \infty} \norm { P_n - f}_\infty = 0$.
Let $\epsilon \in \R_{>0}$.
By Heine-Cantor Theorem, there is ... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\Bbb I = \closedint a b$.
Then $f$ can be [[Definition:Uniform Approximation|uniformly approximated]] on $\Bbb I$ by a [[Definition:Real Polynom... | {{WLOG}}, assume $\Bbb I = \closedint 0 1$.
For each $n \in \N$, let:
:$\ds \map {P_n} x := \sum_{k \mathop = 0}^n \map f {\dfrac k n } \dbinom n k x^k \paren {1 - x}^{n - k}$
We shall show that $\ds \lim_{n \mathop \to \infty} \norm { P_n - f}_\infty = 0$.
Let $\epsilon \in \R_{>0}$.
By [[Heine-Cantor Theorem]], ... | Weierstrass Approximation Theorem/Proof 2 | https://proofwiki.org/wiki/Weierstrass_Approximation_Theorem | https://proofwiki.org/wiki/Weierstrass_Approximation_Theorem/Proof_2 | [
"Weierstrass Approximation Theorem",
"Real Analysis"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Uniform Approximation",
"Definition:Polynomial Function/Real"
] | [
"Heine-Cantor Theorem",
"Definition:Random Variable",
"Definition:Binomial Distribution",
"Bienaymé-Chebyshev Inequality",
"Expectation is Linear",
"Expectation of Binomial Distribution",
"Variance of Binomial Distribution",
"Cauchy's Mean Theorem",
"Expectation of Almost Surely Constant Random Vari... |
proofwiki-11352 | Weierstrass Approximation Theorem | Let $f$ be a real function which is continuous on the closed interval $\Bbb I = \closedint a b$.
Then $f$ can be uniformly approximated on $\Bbb I$ by a polynomial function to any given degree of accuracy. | Let $\AA \subseteq \map C {\Bbb I, \R}$ be the set of real polynomial functions.
$\AA$ is a subalgebra of $\map C {\Bbb I, \R}$ because polynomials over $\R$ form an algebra over $\R$.
Let $I$ denote the identity mapping on $\Bbb I$, i.e.:
:$\forall x \in \Bbb I : \map I x = x$
Then $I \in \AA$.
Thus $\AA$ separates th... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\Bbb I = \closedint a b$.
Then $f$ can be [[Definition:Uniform Approximation|uniformly approximated]] on $\Bbb I$ by a [[Definition:Real Polynom... | Let $\AA \subseteq \map C {\Bbb I, \R}$ be the [[Definition:Set|set]] of [[Definition:Real Polynomial Function|real polynomial functions]].
$\AA$ is a [[Definition:Subalgebra|subalgebra]] of $\map C {\Bbb I, \R}$ because [[Definition:Polynomial over Real Numbers|polynomials]] over $\R$ form an [[Definition:Algebra ove... | Weierstrass Approximation Theorem/Proof 3 | https://proofwiki.org/wiki/Weierstrass_Approximation_Theorem | https://proofwiki.org/wiki/Weierstrass_Approximation_Theorem/Proof_3 | [
"Weierstrass Approximation Theorem",
"Real Analysis"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Uniform Approximation",
"Definition:Polynomial Function/Real"
] | [
"Definition:Set",
"Definition:Polynomial Function/Real",
"Definition:Subalgebra",
"Definition:Polynomial/Real Numbers",
"Definition:Algebra over Ring",
"Definition:Identity Mapping",
"Stone-Weierstrass Theorem"
] |
proofwiki-11353 | Construction of Right Angle by Stretched Rope | Let a rope be knotted at regular intervals so that there are $12$ equal segments separated by knots.
Tie the rope in a loop consisting of those $12$ segments.
Fix one of the knots to the ground at the point you want the right angle to be placed.
Stretch the rope tightly in the direction of one of the legs of the right ... | The rope will be stretched in the form of a triangle with sides $3$, $4$ and $5$.
We have that:
:$3^2 + 4^2 = 9 + 16 = 25 = 5^2$
and so by Pythagoras's Theorem the triangle is right angled.
{{qed}} | Let a rope be knotted at regular intervals so that there are $12$ equal segments separated by knots.
Tie the rope in a loop consisting of those $12$ segments.
Fix one of the knots to the ground at the point you want the [[Definition:Right Angle|right angle]] to be placed.
Stretch the rope tightly in the direction of... | The rope will be stretched in the form of a [[Definition:Triangle (Geometry)|triangle]] with sides $3$, $4$ and $5$.
We have that:
:$3^2 + 4^2 = 9 + 16 = 25 = 5^2$
and so by [[Pythagoras's Theorem]] the [[Definition:Triangle (Geometry)|triangle]] is [[Definition:Right Triangle|right angled]].
{{qed}} | Construction of Right Angle by Stretched Rope | https://proofwiki.org/wiki/Construction_of_Right_Angle_by_Stretched_Rope | https://proofwiki.org/wiki/Construction_of_Right_Angle_by_Stretched_Rope | [
"Pythagoras's Theorem",
"Classic Problems"
] | [
"Definition:Right Angle",
"Definition:Right Angle",
"Definition:Right Angle",
"Definition:Right Angle",
"File:RightAngleRopeConstruction.png"
] | [
"Definition:Triangle (Geometry)",
"Pythagoras's Theorem",
"Definition:Triangle (Geometry)",
"Definition:Triangle (Geometry)/Right-Angled"
] |
proofwiki-11354 | Top equals to Relative Pseudocomplement in Brouwerian Lattice | Let $\struct {S, \vee, \wedge, \preceq}$ be a Brouwerian lattice with greatest element $\top$.
Let $a, b \in S$.
Then
:$\top = a \to b$ {{iff}} $a \preceq b$ | === Sufficient Condition ===
Let:
:$\top = a \to b$
By definition of reflexivity:
:$\top \preceq a \to b$
By Inequality with Meet Operation is Equivalent to Inequality with Relative Pseudocomplement in Brouwerian Lattice:
:$\top \wedge a \preceq b$
Thus
:$a \preceq b$
{{qed|lemma}} | Let $\struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Brouwerian Lattice|Brouwerian lattice]] with [[Definition:Greatest Element of Subset|greatest element]] $\top$.
Let $a, b \in S$.
Then
:$\top = a \to b$ {{iff}} $a \preceq b$ | === Sufficient Condition ===
Let:
:$\top = a \to b$
By definition of [[Definition:Reflexivity|reflexivity]]:
:$\top \preceq a \to b$
By [[Inequality with Meet Operation is Equivalent to Inequality with Relative Pseudocomplement in Brouwerian Lattice]]:
:$\top \wedge a \preceq b$
Thus
:$a \preceq b$
{{qed|lemma}} | Top equals to Relative Pseudocomplement in Brouwerian Lattice | https://proofwiki.org/wiki/Top_equals_to_Relative_Pseudocomplement_in_Brouwerian_Lattice | https://proofwiki.org/wiki/Top_equals_to_Relative_Pseudocomplement_in_Brouwerian_Lattice | [
"Brouwerian Lattices"
] | [
"Definition:Brouwerian Lattice",
"Definition:Greatest Element/Subset"
] | [
"Definition:Reflexivity",
"Inequality with Meet Operation is Equivalent to Inequality with Relative Pseudocomplement in Brouwerian Lattice",
"Inequality with Meet Operation is Equivalent to Inequality with Relative Pseudocomplement in Brouwerian Lattice"
] |
proofwiki-11355 | Up-Complete Product | Let $\struct {S, \preceq_1}$, $\struct {T, \preceq_2}$ be non-empty ordered sets.
Let $\struct {S \times T, \preceq}$ be the simple order product of $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$. | === Lemma 1 ===
{{:Up-Complete Product/Lemma 1}}{{qed|lemma}} | Let $\struct {S, \preceq_1}$, $\struct {T, \preceq_2}$ be [[Definition:Non-Empty Set|non-empty]] [[Definition:Ordered Set|ordered sets]].
Let $\struct {S \times T, \preceq}$ be the [[Definition:Simple Order Product|simple order product]] of $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$. | === [[Up-Complete Product/Lemma 1|Lemma 1]] ===
{{:Up-Complete Product/Lemma 1}}{{qed|lemma}} | Up-Complete Product | https://proofwiki.org/wiki/Up-Complete_Product | https://proofwiki.org/wiki/Up-Complete_Product | [
"Order Theory"
] | [
"Definition:Non-Empty Set",
"Definition:Ordered Set",
"Definition:Simple Order Product"
] | [
"Up-Complete Product/Lemma 1"
] |
proofwiki-11356 | Motion of Rocket in Outer Space | Let $B$ be a rocket travelling in outer space.
Let the velocity of $B$ at time $t$ be $\mathbf v$.
Let the mass of $B$ at time $t$ be $m$.
Let the exhaust velocity of $B$ be constant at $\mathbf b$.
Then the equation of motion of $B$ is given by:
:$m \dfrac {\d \mathbf v} {\d t} = - \mathbf b \dfrac {\d m} {\d t}$ | From Motion of Body with Variable Mass:
:$(1): \quad \mathbf w \dfrac {\d m} {\d t} + \mathbf F = m \dfrac {\d \mathbf v} {\d t}$
where:
:$\mathbf F$ is the external force being applied
:$\mathbf w$ is the velocity of the added mass relative to $B$.
In this scenario:
:there is no external force and so $\mathbf F = \mat... | Let $B$ be a [[Definition:Rocket|rocket]] travelling in outer space.
Let the [[Definition:Velocity|velocity]] of $B$ at [[Definition:Time|time]] $t$ be $\mathbf v$.
Let the [[Definition:Mass|mass]] of $B$ at [[Definition:Time|time]] $t$ be $m$.
Let the [[Definition:Exhaust Velocity|exhaust velocity]] of $B$ be const... | From [[Motion of Body with Variable Mass]]:
:$(1): \quad \mathbf w \dfrac {\d m} {\d t} + \mathbf F = m \dfrac {\d \mathbf v} {\d t}$
where:
:$\mathbf F$ is the external [[Definition:Force|force]] being applied
:$\mathbf w$ is the [[Definition:Velocity|velocity]] of the added [[Definition:Mass|mass]] relative to $B$.
... | Motion of Rocket in Outer Space/Proof 1 | https://proofwiki.org/wiki/Motion_of_Rocket_in_Outer_Space | https://proofwiki.org/wiki/Motion_of_Rocket_in_Outer_Space/Proof_1 | [
"Rocket Science"
] | [
"Definition:Rocket",
"Definition:Velocity",
"Definition:Time",
"Definition:Mass",
"Definition:Time",
"Definition:Exhaust Velocity"
] | [
"Motion of Body with Variable Mass",
"Definition:Force",
"Definition:Velocity",
"Definition:Mass",
"Definition:Force",
"Definition:Velocity",
"Definition:Mass"
] |
proofwiki-11357 | Motion of Rocket in Outer Space | Let $B$ be a rocket travelling in outer space.
Let the velocity of $B$ at time $t$ be $\mathbf v$.
Let the mass of $B$ at time $t$ be $m$.
Let the exhaust velocity of $B$ be constant at $\mathbf b$.
Then the equation of motion of $B$ is given by:
:$m \dfrac {\d \mathbf v} {\d t} = - \mathbf b \dfrac {\d m} {\d t}$ | From Newton's Second Law of Motion:
:$\mathbf F = \dfrac \d {\d t} \paren {m \mathbf v}$
At time $t + \Delta t$, let:
: the mass of $B$ be $m + \Delta m$
: the velocity of $B$ be $\mathbf v + \Delta \mathbf v$.
The fuel is being consumed, so the increase in mass of the fuel during time $\Delta t$ is $-\Delta m$.
Thus t... | Let $B$ be a [[Definition:Rocket|rocket]] travelling in outer space.
Let the [[Definition:Velocity|velocity]] of $B$ at [[Definition:Time|time]] $t$ be $\mathbf v$.
Let the [[Definition:Mass|mass]] of $B$ at [[Definition:Time|time]] $t$ be $m$.
Let the [[Definition:Exhaust Velocity|exhaust velocity]] of $B$ be const... | From [[Newton's Second Law of Motion]]:
:$\mathbf F = \dfrac \d {\d t} \paren {m \mathbf v}$
At time $t + \Delta t$, let:
: the [[Definition:Mass|mass]] of $B$ be $m + \Delta m$
: the [[Definition:Velocity|velocity]] of $B$ be $\mathbf v + \Delta \mathbf v$.
The fuel is being consumed, so the increase in [[Definition... | Motion of Rocket in Outer Space/Proof 2 | https://proofwiki.org/wiki/Motion_of_Rocket_in_Outer_Space | https://proofwiki.org/wiki/Motion_of_Rocket_in_Outer_Space/Proof_2 | [
"Rocket Science"
] | [
"Definition:Rocket",
"Definition:Velocity",
"Definition:Time",
"Definition:Mass",
"Definition:Time",
"Definition:Exhaust Velocity"
] | [
"Newton's Laws of Motion/Second Law",
"Definition:Mass",
"Definition:Velocity",
"Definition:Mass",
"Definition:Velocity",
"Definition:Velocity",
"Principle of Conservation of Linear Momentum",
"Definition:Linear Momentum",
"Definition:Constant"
] |
proofwiki-11358 | Velocity of Rocket in Outer Space | Let $B$ be a rocket travelling in outer space.
Let the velocity of $B$ at time $t$ be $\mathbf v$.
Let the mass of $B$ at time $t$ be $m$.
Let the exhaust velocity of $B$ be constant at $\mathbf b$.
Then the velocity of $B$ at time $t$ is given by:
:$\map {\mathbf v} t = \map {\mathbf v} 0 + \mathbf b \ln \dfrac {\map ... | From Motion of Rocket in Outer Space, the equation of motion of $B$ is given by:
:$m \dfrac {\d \mathbf v} {\d t} = -\mathbf b \dfrac {\d m} {\d t}$
Hence:
{{begin-eqn}}
{{eqn | l = \int_0^t \dfrac {\d \mathbf v} {\d t} \rd t
| r = -\int_0^t \mathbf b \frac 1 m \dfrac {\d m} {\d t} \rd t
| c =
}}
{{eqn | l... | Let $B$ be a [[Definition:Rocket|rocket]] travelling in outer space.
Let the [[Definition:Velocity|velocity]] of $B$ at [[Definition:Time|time]] $t$ be $\mathbf v$.
Let the [[Definition:Mass|mass]] of $B$ at [[Definition:Time|time]] $t$ be $m$.
Let the [[Definition:Exhaust Velocity|exhaust velocity]] of $B$ be const... | From [[Motion of Rocket in Outer Space]], the equation of motion of $B$ is given by:
:$m \dfrac {\d \mathbf v} {\d t} = -\mathbf b \dfrac {\d m} {\d t}$
Hence:
{{begin-eqn}}
{{eqn | l = \int_0^t \dfrac {\d \mathbf v} {\d t} \rd t
| r = -\int_0^t \mathbf b \frac 1 m \dfrac {\d m} {\d t} \rd t
| c =
}}
{{... | Velocity of Rocket in Outer Space | https://proofwiki.org/wiki/Velocity_of_Rocket_in_Outer_Space | https://proofwiki.org/wiki/Velocity_of_Rocket_in_Outer_Space | [
"Rocket Science"
] | [
"Definition:Rocket",
"Definition:Velocity",
"Definition:Time",
"Definition:Mass",
"Definition:Time",
"Definition:Exhaust Velocity",
"Definition:Velocity",
"Definition:Time",
"Definition:Velocity",
"Definition:Mass",
"Definition:Time"
] | [
"Motion of Rocket in Outer Space"
] |
proofwiki-11359 | Acceleration of Rocket in Outer Space | Let $B$ be a rocket travelling in outer space.
Let the velocity of $B$ at time $t$ be $\mathbf v$.
Let the mass of $B$ at time $t$ be $m$.
Let the exhaust velocity of $B$ be constant at $\mathbf b$.
Then the acceleration of $B$ at time $t$ is given by:
:$\mathbf a = \dfrac 1 m \paren {-\mathbf b \dfrac {\d m} {\d t} }$ | From Motion of Rocket in Outer Space, the equation of motion of $B$ is given by:
:$(1): \quad m \dfrac {\d \mathbf v} {\d t} = -\mathbf b \dfrac {\d m} {\d t}$
By definition, the acceleration of $B$ is its rate of change of velocity:
:$\mathbf a = \dfrac {\d \mathbf v} {\d t}$
The result follows by substituting $\mathb... | Let $B$ be a [[Definition:Rocket|rocket]] travelling in outer space.
Let the [[Definition:Velocity|velocity]] of $B$ at [[Definition:Time|time]] $t$ be $\mathbf v$.
Let the [[Definition:Mass|mass]] of $B$ at [[Definition:Time|time]] $t$ be $m$.
Let the [[Definition:Exhaust Velocity|exhaust velocity]] of $B$ be const... | From [[Motion of Rocket in Outer Space]], the equation of motion of $B$ is given by:
:$(1): \quad m \dfrac {\d \mathbf v} {\d t} = -\mathbf b \dfrac {\d m} {\d t}$
By definition, the [[Definition:Acceleration|acceleration]] of $B$ is its [[Definition:Rate of Change with respect to Time|rate of change]] of [[Definition... | Acceleration of Rocket in Outer Space | https://proofwiki.org/wiki/Acceleration_of_Rocket_in_Outer_Space | https://proofwiki.org/wiki/Acceleration_of_Rocket_in_Outer_Space | [
"Rocket Science"
] | [
"Definition:Rocket",
"Definition:Velocity",
"Definition:Time",
"Definition:Mass",
"Definition:Time",
"Definition:Exhaust Velocity",
"Definition:Acceleration",
"Definition:Time"
] | [
"Motion of Rocket in Outer Space",
"Definition:Acceleration",
"Definition:Rate of Change/Time",
"Definition:Velocity"
] |
proofwiki-11360 | Up-Complete Product/Lemma 1 | Let $X$ be a directed subset of $S$.
Let $Y$ be a directed subset of $T$.
Then $X \times Y$ is also a directed subset of $S \times T$. | Let $\tuple {s_1, t_1}, \tuple {s_2, t_2} \in X \times Y$.
By definition of Cartesian product:
:$s_1, s_2 \in X$ and $t_1, t_2 \in Y$
By definition of directed subset:
:$\exists h_1 \in X: s_1 \preceq_1 h_1 \land s_2 \preceq_1 h_1$
and
:$\exists h_2 \in X: t_1 \preceq_2 h_2 \land t_2 \preceq_2 h_2$
By definition of sim... | Let $X$ be a [[Definition:Directed Subset|directed subset]] of $S$.
Let $Y$ be a [[Definition:Directed Subset|directed subset]] of $T$.
Then $X \times Y$ is also a [[Definition:Directed Subset|directed subset]] of $S \times T$. | Let $\tuple {s_1, t_1}, \tuple {s_2, t_2} \in X \times Y$.
By definition of [[Definition:Cartesian Product|Cartesian product]]:
:$s_1, s_2 \in X$ and $t_1, t_2 \in Y$
By definition of [[Definition:Directed Subset|directed subset]]:
:$\exists h_1 \in X: s_1 \preceq_1 h_1 \land s_2 \preceq_1 h_1$
and
:$\exists h_2 \in ... | Up-Complete Product/Lemma 1 | https://proofwiki.org/wiki/Up-Complete_Product/Lemma_1 | https://proofwiki.org/wiki/Up-Complete_Product/Lemma_1 | [
"Order Theory"
] | [
"Definition:Directed Subset",
"Definition:Directed Subset",
"Definition:Directed Subset"
] | [
"Definition:Cartesian Product",
"Definition:Directed Subset",
"Definition:Simple Order Product",
"Definition:Directed Subset"
] |
proofwiki-11361 | Up-Complete Product/Lemma 2 | Let $X$ be a directed subset of $S \times T$.
Then
:$\map {\pr_1^\to} X$ and $\map {\pr_2^\to} X$ are directed
where
:$\pr_1$ denotes the first projection on $S \times T$
:$\pr_2$ denotes the second projection on $S \times T$
:$\map {\pr_1^\to} X$ denotes the image of $X$ under $\pr_1$ | Let $x, y \in \map {\pr_1^\to} X$.
By definitions of image of set and projections:
:$\exists x' \in T: \tuple {x, x'} \in X$
and
:$\exists y' \in T: \tuple {y, y'} \in X$
By definition of directed:
:$\exists \tuple {a, b} \in X: \tuple {x, x'} \preceq \tuple {a, b} \land \tuple {y, y'} \preceq \tuple {a, b}$
By definit... | Let $X$ be a [[Definition:Directed Subset|directed subset]] of $S \times T$.
Then
:$\map {\pr_1^\to} X$ and $\map {\pr_2^\to} X$ are [[Definition:Directed Subset|directed]]
where
:$\pr_1$ denotes the [[Definition:First Projection|first projection]] on $S \times T$
:$\pr_2$ denotes the [[Definition:Second Projection|s... | Let $x, y \in \map {\pr_1^\to} X$.
By definitions of [[Definition:Image of Subset under Mapping|image of set]] and [[Definition:Projection (Mapping Theory)|projections]]:
:$\exists x' \in T: \tuple {x, x'} \in X$
and
:$\exists y' \in T: \tuple {y, y'} \in X$
By definition of [[Definition:Directed Subset|directed]]:
:... | Up-Complete Product/Lemma 2 | https://proofwiki.org/wiki/Up-Complete_Product/Lemma_2 | https://proofwiki.org/wiki/Up-Complete_Product/Lemma_2 | [
"Order Theory"
] | [
"Definition:Directed Subset",
"Definition:Directed Subset",
"Definition:Projection (Mapping Theory)/First Projection",
"Definition:Projection (Mapping Theory)/Second Projection",
"Definition:Image (Set Theory)/Mapping/Subset"
] | [
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Projection (Mapping Theory)",
"Definition:Directed Subset",
"Definition:Simple Order Product",
"Definition:Directed Subset",
"Definition:Directed Subset"
] |
proofwiki-11362 | Lagrange's Trigonometric Identities/Cosine | {{begin-eqn}}
{{eqn | l = \frac 1 2 + \sum_{k \mathop = 1}^n \map \cos {k x}
| r = \frac 1 2 + \cos x + \cos 2 x + \cos 3 x + \cdots + \cos n x
| c =
}}
{{eqn| r = \frac {\map \sin {\paren {2 n + 1} x / 2} } {2 \map \sin {x / 2} }
| c =
}}
{{end-eqn}}
where $x$ is not an integer multiple of $2 \pi$. | By the Werner Formula for Cosine by Sine:
:$2 \cos \alpha \sin \beta = \map \sin {\alpha + \beta} - \map \sin {\alpha - \beta}$
Thus we establish the following sequence of identities:
{{begin-eqn}}
{{eqn | l = 2 \cdot \frac 1 2 \sin \frac x 2
| r = \sin \frac x 2
| c =
}}
{{eqn | l = 2 \cos x \sin \frac x ... | {{begin-eqn}}
{{eqn | l = \frac 1 2 + \sum_{k \mathop = 1}^n \map \cos {k x}
| r = \frac 1 2 + \cos x + \cos 2 x + \cos 3 x + \cdots + \cos n x
| c =
}}
{{eqn| r = \frac {\map \sin {\paren {2 n + 1} x / 2} } {2 \map \sin {x / 2} }
| c =
}}
{{end-eqn}}
where $x$ is not an [[Definition:Integer Multip... | By the [[Werner Formula for Cosine by Sine]]:
:$2 \cos \alpha \sin \beta = \map \sin {\alpha + \beta} - \map \sin {\alpha - \beta}$
Thus we establish the following sequence of identities:
{{begin-eqn}}
{{eqn | l = 2 \cdot \frac 1 2 \sin \frac x 2
| r = \sin \frac x 2
| c =
}}
{{eqn | l = 2 \cos x \sin \f... | Lagrange's Trigonometric Identities/Cosine | https://proofwiki.org/wiki/Lagrange's_Trigonometric_Identities/Cosine | https://proofwiki.org/wiki/Lagrange's_Trigonometric_Identities/Cosine | [
"Lagrange's Cosine Identity",
"Lagrange's Trigonometric Identities",
"Cosine Function",
"Telescoping Series"
] | [
"Definition:Integral Multiple/Real Numbers"
] | [
"Werner Formulas/Cosine by Sine",
"Definition:Telescoping Series"
] |
proofwiki-11363 | Eratosthenes' Measurement of Earth | The circumference of Earth was measured by {{AuthorRef|Eratosthenes of Cyrene}} to be of the order of $45 \, 000 \, \mathrm{km}$. | :700px
We have that the shape of the Earth is very close to being a sphere.
In the diagram, we have a great circle of Earth with center $O$ and radius $R$.
We have the cities of Alexandria, Egypt ($A$) and Syene, Egypt ($S$) lie on approximately the same meridian.
We have the distance $D$ between $A$ and $S$ is $5\,000... | The [[Definition:Circumference of Circle|circumference]] of [[Definition:Earth|Earth]] was measured by {{AuthorRef|Eratosthenes of Cyrene}} to be of the order of $45 \, 000 \, \mathrm{km}$. | :[[File:CircumferenceOfEarth.png|700px]]
We have that the [[Definition:Earth/Shape/First Approximation|shape]] of the [[Definition:Earth|Earth]] is [[Definition:Approximation|very close]] to being a [[Definition:Sphere|sphere]].
In the diagram, we have a [[Definition:Great Circle|great circle]] of [[Definition:Earth... | Eratosthenes' Measurement of Earth | https://proofwiki.org/wiki/Eratosthenes'_Measurement_of_Earth | https://proofwiki.org/wiki/Eratosthenes'_Measurement_of_Earth | [
"Geodesy"
] | [
"Definition:Circle/Circumference",
"Definition:Earth"
] | [
"File:CircumferenceOfEarth.png",
"Definition:Earth/Shape/First Approximation",
"Definition:Earth",
"Definition:Approximation",
"Definition:Sphere",
"Definition:Great Circle",
"Definition:Earth",
"Definition:Sphere/Geometry/Center",
"Definition:Circle/Radius",
"Definition:Meridian/Terrestrial",
"... |
proofwiki-11364 | Existence of Prime-Free Sequence of Natural Numbers | Let $n$ be a natural number.
Then there exists a sequence of consecutive natural numbers of length $n$ which are all composite. | Consider the number:
:$N := \paren {n + 1}!$
where $!$ denotes the factorial.
Let $i \in I$ where:
:$I = \set {i \in \N: 2 \le i \le n + 1}$
We have that $\size I = n$.
Then for all $i \in I$:
{{begin-eqn}}
{{eqn | l = i
| o = \divides
| r = N
| c = where $\divides$ denotes divisibility
}}
{{eqn | ll=... | Let $n$ be a [[Definition:Natural Number|natural number]].
Then there exists a [[Definition:Sequence|sequence]] of consecutive [[Definition:Natural Number|natural numbers]] of length $n$ which are all [[Definition:Composite Number|composite]]. | Consider the number:
:$N := \paren {n + 1}!$
where $!$ denotes the [[Definition:Factorial|factorial]].
Let $i \in I$ where:
:$I = \set {i \in \N: 2 \le i \le n + 1}$
We have that $\size I = n$.
Then for all $i \in I$:
{{begin-eqn}}
{{eqn | l = i
| o = \divides
| r = N
| c = where $\divides$ denote... | Existence of Prime-Free Sequence of Natural Numbers | https://proofwiki.org/wiki/Existence_of_Prime-Free_Sequence_of_Natural_Numbers | https://proofwiki.org/wiki/Existence_of_Prime-Free_Sequence_of_Natural_Numbers | [
"Number Theory",
"Prime Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Sequence",
"Definition:Natural Numbers",
"Definition:Composite Number"
] | [
"Definition:Factorial",
"Definition:Divisor (Algebra)/Integer",
"Definition:Composite Number",
"Definition:Composite Number"
] |
proofwiki-11365 | Approximate Value of Nth Prime Number | The $n$th prime number is approximately $n \ln n$. | This will be demonstrated by showing that:
:$\ds \lim_{n \mathop \to \infty} \dfrac {p_n} {n \ln n} = 1$
where $p_n$ denotes the $n$th prime number.
By definition of prime-counting function:
:$\map \pi {p_n} = n$
The Prime Number Theorem gives:
:$\ds \lim_{x \mathop \to \infty} \dfrac {\map \pi x} {x / \ln x} = 1$
Thus... | The $n$th [[Definition:Prime Number|prime number]] is approximately $n \ln n$. | This will be demonstrated by showing that:
:$\ds \lim_{n \mathop \to \infty} \dfrac {p_n} {n \ln n} = 1$
where $p_n$ denotes the $n$th [[Definition:Prime Number|prime number]].
By definition of [[Definition:Prime-Counting Function|prime-counting function]]:
:$\map \pi {p_n} = n$
The [[Prime Number Theorem]] gives:
:... | Approximate Value of Nth Prime Number | https://proofwiki.org/wiki/Approximate_Value_of_Nth_Prime_Number | https://proofwiki.org/wiki/Approximate_Value_of_Nth_Prime_Number | [
"Prime Number Theorem",
"Prime Numbers",
"Analytic Number Theory"
] | [
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Prime-Counting Function",
"Prime Number Theorem",
"Prime Number Theorem",
"Definition:Logarithm"
] |
proofwiki-11366 | Prime Number Theorem in Eulerian Logarithmic Integral Form | The Prime Number Theorem is equivalent to:
:$\ds \lim_{x \mathop \to \infty} \frac {\map \pi x} {\map \Li x} = 1$
where:
:$\map \pi x$ is the prime-counting function
:$\map \Li x$ is the Eulerian logarithmic integral:
::$\ds \map \Li x := \int_2^x \dfrac {\d t} {\ln t}$ | Using Integration by Parts:
{{begin-eqn}}
{{eqn | l = \map \Li x
| r = \int_2^x \dfrac {\d t} {\ln t}
| c =
}}
{{eqn | n = 1
| r = \dfrac x {\ln x} - \dfrac 2 {\ln 2} + \int_2^x \dfrac {\d t} {\paren {\ln t}^2}
| c =
}}
{{end-eqn}}
We have that $\dfrac 1 {\paren {\ln t}^2}$ is positive and dec... | The [[Prime Number Theorem]] is equivalent to:
:$\ds \lim_{x \mathop \to \infty} \frac {\map \pi x} {\map \Li x} = 1$
where:
:$\map \pi x$ is the [[Definition:Prime-Counting Function|prime-counting function]]
:$\map \Li x$ is the [[Definition:Eulerian Logarithmic Integral|Eulerian logarithmic integral]]:
::$\ds \map \... | Using [[Integration by Parts]]:
{{begin-eqn}}
{{eqn | l = \map \Li x
| r = \int_2^x \dfrac {\d t} {\ln t}
| c =
}}
{{eqn | n = 1
| r = \dfrac x {\ln x} - \dfrac 2 {\ln 2} + \int_2^x \dfrac {\d t} {\paren {\ln t}^2}
| c =
}}
{{end-eqn}}
We have that $\dfrac 1 {\paren {\ln t}^2}$ is [[Definiti... | Prime Number Theorem in Eulerian Logarithmic Integral Form | https://proofwiki.org/wiki/Prime_Number_Theorem_in_Eulerian_Logarithmic_Integral_Form | https://proofwiki.org/wiki/Prime_Number_Theorem_in_Eulerian_Logarithmic_Integral_Form | [
"Prime Number Theorem",
"Analytic Number Theory"
] | [
"Prime Number Theorem",
"Definition:Prime-Counting Function",
"Definition:Logarithmic Integral/Eulerian"
] | [
"Integration by Parts",
"Definition:Positive Real Function",
"Definition:Decreasing/Real Function"
] |
proofwiki-11367 | Exponential of Rational Number is Irrational | Let $r$ be a rational number such that $r \ne 0$.
Then:
:$e^r$ is irrational
where $e$ is Euler's number. | Let $r = \dfrac p q$ be rational such that $r \ne 0$.
{{AimForCont}} $e^r$ is rational.
Then $\paren {e^r}^q = e^p$ is also rational.
Then if $e^{-p}$ is rational, it follows that $e^p$ is rational.
It is therefore sufficient to derive a contradiction from the supposition that $e^p$ is rational for every $p \in \Z_{>0}... | Let $r$ be a [[Definition:Rational Number|rational number]] such that $r \ne 0$.
Then:
:$e^r$ is [[Definition:Irrational Number|irrational]]
where $e$ is [[Definition:Euler's Number|Euler's number]]. | Let $r = \dfrac p q$ be [[Definition:Rational Number|rational]] such that $r \ne 0$.
{{AimForCont}} $e^r$ is [[Definition:Rational Number|rational]].
Then $\paren {e^r}^q = e^p$ is also [[Definition:Rational Number|rational]].
Then if $e^{-p}$ is [[Definition:Rational Number|rational]], it follows that $e^p$ is [[De... | Exponential of Rational Number is Irrational | https://proofwiki.org/wiki/Exponential_of_Rational_Number_is_Irrational | https://proofwiki.org/wiki/Exponential_of_Rational_Number_is_Irrational | [
"Exponential Function"
] | [
"Definition:Rational Number",
"Definition:Irrational Number",
"Definition:Euler's Number"
] | [
"Definition:Rational Number",
"Definition:Rational Number",
"Definition:Rational Number",
"Definition:Rational Number",
"Definition:Rational Number",
"Definition:Contradiction",
"Definition:Rational Number",
"Definition:Strictly Positive/Integer",
"Definition:Real Function",
"Definition:Integer",
... |
proofwiki-11368 | Rational Points on Graph of Exponential Function | Consider the graph $f$ of the exponential function in the real Cartesian plane $\R^2$:
:$f := \set {\tuple {x, y} \in \R^2: y = e^x}$
The only rational point of $f$ is $\tuple {0, 1}$. | From Exponential of Rational Number is Irrational:
:$r \in \Q_{\ne 0} \implies e^r \in \R - \Q$
Thus, apart from the point $\tuple {0, 1}$, when $x$ is rational, $e^x$ is not.
Hence the result.
{{qed}} | Consider the [[Definition:Graph of Mapping|graph]] $f$ of the [[Definition:Real Exponential Function|exponential function]] in the [[Definition:Cartesian Plane|real Cartesian plane]] $\R^2$:
:$f := \set {\tuple {x, y} \in \R^2: y = e^x}$
The only [[Definition:Rational Point in Plane|rational point]] of $f$ is $\tuple... | From [[Exponential of Rational Number is Irrational]]:
:$r \in \Q_{\ne 0} \implies e^r \in \R - \Q$
Thus, apart from the point $\tuple {0, 1}$, when $x$ is [[Definition:Rational Number|rational]], $e^x$ is not.
Hence the result.
{{qed}} | Rational Points on Graph of Exponential Function | https://proofwiki.org/wiki/Rational_Points_on_Graph_of_Exponential_Function | https://proofwiki.org/wiki/Rational_Points_on_Graph_of_Exponential_Function | [
"Exponential Function"
] | [
"Definition:Graph of Mapping",
"Definition:Exponential Function/Real",
"Definition:Cartesian Plane",
"Definition:Rational Point in Plane"
] | [
"Exponential of Rational Number is Irrational",
"Definition:Rational Number"
] |
proofwiki-11369 | Rational Points on Graph of Logarithm Function | Consider the graph of the logarithm function in the real Cartesian plane $\R^2$:
:$f := \set {\tuple {x, y} \in \R^2: y = \ln x}$
The only rational point of $f$ is $\tuple {1, 0}$. | Consider the graph of the exponential function in the real Cartesian plane $\R^2$:
:$g := \set {\tuple {x, y} \in \R^2: y = e^x}$
From Rational Points on Graph of Exponential Function, the only rational point of $g$ is $\tuple {0, 1}$.
By definition of the exponential function, $f$ and $g$ are inverses.
Thus:
:$\tuple ... | Consider the [[Definition:Graph of Mapping|graph]] of the [[Definition:Logarithm|logarithm function]] in the [[Definition:Cartesian Plane|real Cartesian plane]] $\R^2$:
:$f := \set {\tuple {x, y} \in \R^2: y = \ln x}$
The only [[Definition:Rational Point in Plane|rational point]] of $f$ is $\tuple {1, 0}$. | Consider the [[Definition:Graph of Mapping|graph]] of the [[Definition:Real Exponential Function|exponential function]] in the [[Definition:Cartesian Plane|real Cartesian plane]] $\R^2$:
:$g := \set {\tuple {x, y} \in \R^2: y = e^x}$
From [[Rational Points on Graph of Exponential Function]], the only [[Definition:Rati... | Rational Points on Graph of Logarithm Function | https://proofwiki.org/wiki/Rational_Points_on_Graph_of_Logarithm_Function | https://proofwiki.org/wiki/Rational_Points_on_Graph_of_Logarithm_Function | [
"Logarithms"
] | [
"Definition:Graph of Mapping",
"Definition:Logarithm",
"Definition:Cartesian Plane",
"Definition:Rational Point in Plane"
] | [
"Definition:Graph of Mapping",
"Definition:Exponential Function/Real",
"Definition:Cartesian Plane",
"Rational Points on Graph of Exponential Function",
"Definition:Rational Point in Plane",
"Definition:Exponential Function/Real/Inverse of Natural Logarithm",
"Definition:Inverse Mapping",
"Definition:... |
proofwiki-11370 | Meet-Continuous iff Ideal Supremum is Meet Preserving | Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be an up-complete lattice.
Let $f: \map {\it Ids} {\mathscr S} \to S$ be a mapping such that:
:$\forall I \in \map {\it Ids} {\mathscr S}: \map f I = \sup_{\mathscr S} I$
where
:$\map {\it Ids} {\mathscr S}$ denotes the set of all ideals in $\mathscr S$
Then
:$\mat... | === Sufficient Condition ===
Let $\mathscr S$ be meet-continuous.
We will prove that:
:for every directed subsets $D_1, D_2$ of $S$: $\paren {\sup D_1} \wedge \paren {\sup D_2} = \sup \set {d_1 \wedge d_2: d_1 \in D_1, d_2 \in D_2}$
Let $D_1, D_2$ be directed subsets of $S$.
we will prove as sublemma that:
:for every a... | Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Lattice (Order Theory)|lattice]].
Let $f: \map {\it Ids} {\mathscr S} \to S$ be a [[Definition:Mapping|mapping]] such that:
:$\forall I \in \map {\it Ids} {\mathscr S}: \map f I = \sup_{\mathscr S} I$
where ... | === Sufficient Condition ===
Let $\mathscr S$ be [[Definition:Meet-Continuous Lattice|meet-continuous]].
We will prove that:
:for every [[Definition:Directed Subset|directed subsets]] $D_1, D_2$ of $S$: $\paren {\sup D_1} \wedge \paren {\sup D_2} = \sup \set {d_1 \wedge d_2: d_1 \in D_1, d_2 \in D_2}$
Let $D_1, D_2$... | Meet-Continuous iff Ideal Supremum is Meet Preserving | https://proofwiki.org/wiki/Meet-Continuous_iff_Ideal_Supremum_is_Meet_Preserving | https://proofwiki.org/wiki/Meet-Continuous_iff_Ideal_Supremum_is_Meet_Preserving | [
"Meet-Continuous Lattices"
] | [
"Definition:Up-Complete",
"Definition:Lattice (Order Theory)",
"Definition:Mapping",
"Definition:Set of Sets",
"Definition:Ideal in Ordered Set",
"Definition:Meet-Continuous Lattice",
"Definition:Mapping Preserves Infimum/Meet",
"Definition:Mapping"
] | [
"Definition:Meet-Continuous Lattice",
"Definition:Directed Subset",
"Definition:Directed Subset",
"Definition:Element",
"Definition:Directed Subset",
"Definition:Directed Subset",
"Preceding iff Meet equals Less Operand",
"Definition:Reflexivity",
"Definition:Element",
"Definition:Directed Subset"... |
proofwiki-11371 | Rational Number is Algebraic | Let $r \in \Q$ be a rational number.
Then $r$ is also an algebraic number. | Let $r$ be expressed in the form:
:$r = \dfrac p q$
Consider the linear function in $x$:
:$q x - p = 0$
which has the solution:
:$x = \dfrac p q$
Hence the result, by definition of algebraic number.
{{qed}} | Let $r \in \Q$ be a [[Definition:Rational Number|rational number]].
Then $r$ is also an [[Definition:Algebraic Number|algebraic number]]. | Let $r$ be expressed in the form:
:$r = \dfrac p q$
Consider the [[Definition:Linear Function|linear function]] in $x$:
:$q x - p = 0$
which has the solution:
:$x = \dfrac p q$
Hence the result, by definition of [[Definition:Algebraic Number|algebraic number]].
{{qed}} | Rational Number is Algebraic | https://proofwiki.org/wiki/Rational_Number_is_Algebraic | https://proofwiki.org/wiki/Rational_Number_is_Algebraic | [
"Algebraic Numbers",
"Rational Numbers"
] | [
"Definition:Rational Number",
"Definition:Algebraic Number"
] | [
"Definition:Linear Function",
"Definition:Algebraic Number"
] |
proofwiki-11372 | Rational Number is Algebraic of Degree 1 | Let $r \in \Q$ be a rational number.
Then $r$ is an algebraic number of degree $1$. | Let $r$ be expressed in the form:
:$r = \dfrac p q$
By Rational Number is Algebraic, $r$ can be expressed as the root of the linear function:
:$q x - p = 0$
A polynomial of degree zero is a constant polynomial, and has no roots.
Hence the result by definition of degree of algebraic number.
{{qed}} | Let $r \in \Q$ be a [[Definition:Rational Number|rational number]].
Then $r$ is an [[Definition:Algebraic Number|algebraic number]] of [[Definition:Degree of Algebraic Number|degree]] $1$. | Let $r$ be expressed in the form:
:$r = \dfrac p q$
By [[Rational Number is Algebraic]], $r$ can be expressed as the [[Definition:Root of Polynomial|root]] of the [[Definition:Linear Function|linear function]]:
:$q x - p = 0$
A [[Definition:Polynomial (Analysis)|polynomial]] of [[Definition:Polynomial of Degree Zero|... | Rational Number is Algebraic of Degree 1 | https://proofwiki.org/wiki/Rational_Number_is_Algebraic_of_Degree_1 | https://proofwiki.org/wiki/Rational_Number_is_Algebraic_of_Degree_1 | [
"Algebraic Numbers",
"Rational Numbers"
] | [
"Definition:Rational Number",
"Definition:Algebraic Number",
"Definition:Algebraic Number/Degree"
] | [
"Rational Number is Algebraic",
"Definition:Root of Polynomial",
"Definition:Linear Function",
"Definition:Polynomial",
"Definition:Degree of Polynomial/Zero",
"Definition:Constant Polynomial",
"Definition:Root of Polynomial",
"Definition:Algebraic Number/Degree"
] |
proofwiki-11373 | Square Root of 2 is Algebraic of Degree 2 | The square root of $2$ is an algebraic number of degree $2$. | Suppose $\sqrt 2$ could be expressed as the root of the linear function:
:$a_1 x + a_0 = 0$
for some $a_0, a_1 \in \Q$.
Then:
:$\sqrt 2 = -\dfrac {a_0} {a_1}$
and would be rational.
But as Square Root of 2 is Irrational, this is not the case.
However, $\sqrt 2$ is a root of the polynomial of degree $2$:
:$x^2 - 2 = 0$
... | The [[Definition:Square Root|square root]] of $2$ is an [[Definition:Algebraic Number|algebraic number]] of [[Definition:Degree of Algebraic Number|degree]] $2$. | Suppose $\sqrt 2$ could be expressed as the [[Definition:Root of Polynomial|root]] of the [[Definition:Linear Function|linear function]]:
:$a_1 x + a_0 = 0$
for some $a_0, a_1 \in \Q$.
Then:
:$\sqrt 2 = -\dfrac {a_0} {a_1}$
and would be [[Definition:Rational Number|rational]].
But as [[Square Root of 2 is Irrational... | Square Root of 2 is Algebraic of Degree 2 | https://proofwiki.org/wiki/Square_Root_of_2_is_Algebraic_of_Degree_2 | https://proofwiki.org/wiki/Square_Root_of_2_is_Algebraic_of_Degree_2 | [
"Algebraic Numbers",
"Square Root of 2"
] | [
"Definition:Square Root",
"Definition:Algebraic Number",
"Definition:Algebraic Number/Degree"
] | [
"Definition:Root of Polynomial",
"Definition:Linear Function",
"Definition:Rational Number",
"Square Root of 2 is Irrational",
"Definition:Root of Polynomial",
"Definition:Polynomial",
"Definition:Degree of Polynomial",
"Definition:Algebraic Number/Degree"
] |
proofwiki-11374 | Gelfond-Schneider Constant is Transcendental | The Gelfond-Schneider constant:
:$2^{\sqrt 2}$
is transcendental. | From the Gelfond-Schneider Theorem:
If:
:$\alpha$ and $\beta$ are algebraic numbers such that $\alpha \notin \set {0, 1}$
:$\beta$ is either irrational or not wholly real
then $\alpha^\beta$ is transcendental.
From Rational Number is Algebraic:
:$2$ is algebraic.
From Square Root of 2 is Algebraic of Degree 2:
:$\sqrt ... | The [[Definition:Gelfond-Schneider Constant|Gelfond-Schneider constant]]:
:$2^{\sqrt 2}$
is [[Definition:Transcendental|transcendental]]. | From the [[Gelfond-Schneider Theorem]]:
If:
:$\alpha$ and $\beta$ are [[Definition:Algebraic Number|algebraic numbers]] such that $\alpha \notin \set {0, 1}$
:$\beta$ is either [[Definition:Irrational Number|irrational]] or not [[Definition:Wholly Real|wholly real]]
then $\alpha^\beta$ is [[Definition:Transcendental N... | Gelfond-Schneider Constant is Transcendental | https://proofwiki.org/wiki/Gelfond-Schneider_Constant_is_Transcendental | https://proofwiki.org/wiki/Gelfond-Schneider_Constant_is_Transcendental | [
"Real Analysis"
] | [
"Definition:Gelfond-Schneider Constant",
"Definition:Transcendental"
] | [
"Gelfond-Schneider Theorem",
"Definition:Algebraic Number",
"Definition:Irrational Number",
"Definition:Complex Number/Wholly Real",
"Definition:Transcendental Number",
"Rational Number is Algebraic",
"Definition:Algebraic Number",
"Square Root of 2 is Algebraic of Degree 2",
"Definition:Algebraic N... |
proofwiki-11375 | Sum of Bernoulli Numbers by Binomial Coefficients Vanishes | :$\forall n \in \Z_{>1}: \ds \sum_{k \mathop = 0}^{n - 1} \binom n k B_k = 0$
where $B_k$ denotes the $k$th Bernoulli number. | Take the definition of Bernoulli numbers:
:$\ds \frac x {e^x - 1} = \sum_{n \mathop = 0}^\infty \frac {B_n x^n} {n!}$
From the definition of the exponential function:
{{begin-eqn}}
{{eqn | l = e^x
| r = \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}
| c =
}}
{{eqn | r = 1 + \sum_{n \mathop = 1}^\infty \frac ... | :$\forall n \in \Z_{>1}: \ds \sum_{k \mathop = 0}^{n - 1} \binom n k B_k = 0$
where $B_k$ denotes the $k$th [[Definition:Bernoulli Numbers|Bernoulli number]]. | Take the definition of [[Definition:Bernoulli Numbers/Generating Function|Bernoulli numbers]]:
:$\ds \frac x {e^x - 1} = \sum_{n \mathop = 0}^\infty \frac {B_n x^n} {n!}$
From the definition of the [[Definition:Exponential Function/Real/Power Series Expansion|exponential function]]:
{{begin-eqn}}
{{eqn | l = e^x
... | Sum of Bernoulli Numbers by Binomial Coefficients Vanishes | https://proofwiki.org/wiki/Sum_of_Bernoulli_Numbers_by_Binomial_Coefficients_Vanishes | https://proofwiki.org/wiki/Sum_of_Bernoulli_Numbers_by_Binomial_Coefficients_Vanishes | [
"Bernoulli Numbers",
"Binomial Coefficients",
"Sum of Bernoulli Numbers by Binomial Coefficients Vanishes"
] | [
"Definition:Bernoulli Numbers"
] | [
"Definition:Bernoulli Numbers/Generating Function",
"Definition:Exponential Function/Real/Power Series Expansion",
"Definition:Series",
"Definition:Zero (Number)",
"Product of Absolutely Convergent Series",
"Definition:Subtraction",
"Definition:Binomial Coefficient"
] |
proofwiki-11376 | Bernoulli Numbers are Rational | The Bernoulli numbers are rational. | By the recurrence relation for the Bernoulli numbers:
:$B_n = \begin{cases} 1 & : n = 0 \\
\ds - \sum_{k \mathop = 0}^{n-1} \binom n k \frac {B_k} {n - k + 1} & : n > 0
\end{cases}$
:$B_0$ is rational.
:$B_n$ is a finite sum of the products of a binomial coefficient with a Bernoulli number earlier in the sequence divid... | The [[Definition:Bernoulli Numbers|Bernoulli numbers]] are [[Definition:Rational Number|rational]]. | By the [[Definition:Bernoulli Numbers/Recurrence Relation|recurrence relation for the Bernoulli numbers]]:
:$B_n = \begin{cases} 1 & : n = 0 \\
\ds - \sum_{k \mathop = 0}^{n-1} \binom n k \frac {B_k} {n - k + 1} & : n > 0
\end{cases}$
:$B_0$ is [[Definition:Rational Number|rational]].
:$B_n$ is a finite sum of the ... | Bernoulli Numbers are Rational | https://proofwiki.org/wiki/Bernoulli_Numbers_are_Rational | https://proofwiki.org/wiki/Bernoulli_Numbers_are_Rational | [
"Bernoulli Numbers"
] | [
"Definition:Bernoulli Numbers",
"Definition:Rational Number"
] | [
"Definition:Bernoulli Numbers/Recurrence Relation",
"Definition:Rational Number",
"Definition:Binomial Coefficient",
"Definition:Bernoulli Numbers",
"Definition:Sequence",
"Definition:Integer",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Rational Number",
"Definiti... |
proofwiki-11377 | Power Series Expansion for Tangent Function/Proof of Convergence | The radius of convergence of the Power Series Expansion for Tangent Function:
:$\ds \tan x = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} } {\paren {2 n}!} x^{2 n - 1}$
where $B_{2 n}$ denotes the Bernoulli numbers, is given as:
:$\size x < \dfrac \pi 2$ | By Combination Theorem for Limits of Real Functions we can deduce the following.
{{begin-eqn}}
{{eqn | o =
| r = \lim_{n \mathop \to \infty} \size {\frac {\frac {\paren {-1}^n 2^{2 n + 2} \paren {2^{2 n + 2} - 1} B_{2 n + 2} } {\paren {2 n + 2}!} x^{2 n + 1} } {\frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} ... | The [[Definition:Radius of Convergence of Real Power Series|radius of convergence]] of the [[Power Series Expansion for Tangent Function]]:
:$\ds \tan x = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} } {\paren {2 n}!} x^{2 n - 1}$
where $B_{2 n}$ denotes the [[Definition:B... | By [[Combination Theorem for Limits of Real Functions]] we can deduce the following.
{{begin-eqn}}
{{eqn | o =
| r = \lim_{n \mathop \to \infty} \size {\frac {\frac {\paren {-1}^n 2^{2 n + 2} \paren {2^{2 n + 2} - 1} B_{2 n + 2} } {\paren {2 n + 2}!} x^{2 n + 1} } {\frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{... | Power Series Expansion for Tangent Function/Proof of Convergence | https://proofwiki.org/wiki/Power_Series_Expansion_for_Tangent_Function/Proof_of_Convergence | https://proofwiki.org/wiki/Power_Series_Expansion_for_Tangent_Function/Proof_of_Convergence | [
"Power Series Expansion for Tangent Function"
] | [
"Definition:Radius of Convergence/Real Domain",
"Power Series Expansion for Tangent Function",
"Definition:Bernoulli Numbers"
] | [
"Combination Theorem for Limits of Functions/Real",
"Asymptotic Formula for Bernoulli Numbers",
"Ratio Test",
"Definition:Convergent Series"
] |
proofwiki-11378 | Power Series Expansion for Tangent Function/Sequence | The Power Series Expansion for Tangent Function begins:
:$\tan x = x + \dfrac 1 3 x^3 + \dfrac 2 {15} x^5 + \dfrac {17} {315} x^7 + \dfrac {62} {2835} x^9 + \cdots$ | From Power Series Expansion for Tangent Function:
{{begin-eqn}}
{{eqn | l = \tan x
| r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} } {\paren {2 n}!} x^{2 n - 1}
| c =
}}
{{eqn | r = \frac {2^2 \paren {2^2 - 1} B_2} {2!} x - \frac {2^4 \paren {2^4 - 1} B_4} ... | The [[Power Series Expansion for Tangent Function]] begins:
:$\tan x = x + \dfrac 1 3 x^3 + \dfrac 2 {15} x^5 + \dfrac {17} {315} x^7 + \dfrac {62} {2835} x^9 + \cdots$ | From [[Power Series Expansion for Tangent Function]]:
{{begin-eqn}}
{{eqn | l = \tan x
| r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} } {\paren {2 n}!} x^{2 n - 1}
| c =
}}
{{eqn | r = \frac {2^2 \paren {2^2 - 1} B_2} {2!} x - \frac {2^4 \paren {2^4 - 1} ... | Power Series Expansion for Tangent Function/Sequence | https://proofwiki.org/wiki/Power_Series_Expansion_for_Tangent_Function/Sequence | https://proofwiki.org/wiki/Power_Series_Expansion_for_Tangent_Function/Sequence | [
"Power Series Expansion for Tangent Function"
] | [
"Power Series Expansion for Tangent Function"
] | [
"Power Series Expansion for Tangent Function",
"Definition:Bernoulli Numbers/Sequence"
] |
proofwiki-11379 | Tangent to Cycloid is Vertical at Cusps | The tangent to the cycloid whose locus is given by:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
is vertical at the cusps. | From Slope of Tangent to Cycloid, the slope of the tangent to $C$ at the point $\tuple {x, y}$ is given by:
:$\dfrac {\d y} {\d x} = \cot \dfrac \theta 2$
At the cusps, $\theta = 2 n \pi$ for $n \in \Z$.
Thus at the cusps, the slope of the tangent to $C$ is $\cot n \pi$.
From Shape of Cotangent Function:
:$\ds \lim_{\t... | The [[Definition:Tangent Line|tangent]] to the [[Definition:Cycloid|cycloid]] whose [[Definition:Locus|locus]] is given by:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
is [[Definition:Vertical Tangent Line|vertical]] at the [[Definition:Cusp of Cycloid|cusps]]. | From [[Slope of Tangent to Cycloid]], the [[Definition:Slope of Straight Line|slope]] of the [[Definition:Tangent Line|tangent]] to $C$ at the point $\tuple {x, y}$ is given by:
:$\dfrac {\d y} {\d x} = \cot \dfrac \theta 2$
At the [[Definition:Cusp of Cycloid|cusps]], $\theta = 2 n \pi$ for $n \in \Z$.
Thus at the [... | Tangent to Cycloid is Vertical at Cusps | https://proofwiki.org/wiki/Tangent_to_Cycloid_is_Vertical_at_Cusps | https://proofwiki.org/wiki/Tangent_to_Cycloid_is_Vertical_at_Cusps | [
"Cycloids"
] | [
"Definition:Tangent Line",
"Definition:Cycloid",
"Definition:Locus",
"Definition:Vertical Tangent Line",
"Definition:Cycloid/Cusp"
] | [
"Slope of Tangent to Cycloid",
"Definition:Slope/Straight Line",
"Definition:Tangent Line",
"Definition:Cycloid/Cusp",
"Definition:Cycloid/Cusp",
"Definition:Slope/Straight Line",
"Definition:Tangent Line",
"Shape of Cotangent Function",
"Definition:Vertical Tangent Line"
] |
proofwiki-11380 | Tangent to Cycloid passes through Top of Generating Circle | Let $C$ be a cycloid generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the tangent to $C$ at a point $P$ on $C$ passes through the top of the generating circle of $C$. | From Tangent to Cycloid, the equation for the tangent to $C$ at a point $P = \tuple {x, y}$ is given by:
:$(1): \quad y - a \paren {1 - \cos \theta} = \dfrac {\sin \theta} {1 - \cos \theta} \paren {x - a \theta + a \sin \theta}$
From Equation of Cycloid, the point at the top of the generating circle of $C$ has coordina... | Let $C$ be a [[Definition:Cycloid|cycloid]] generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the [[Definition:Tangent Line|tangent]] to $C$ at a point $P$ on $C$ passes through the top of the [[Definition:Generating Circle of Cycloid|generating circle]] of $C$. | From [[Tangent to Cycloid]], the equation for the [[Definition:Tangent Line|tangent]] to $C$ at a point $P = \tuple {x, y}$ is given by:
:$(1): \quad y - a \paren {1 - \cos \theta} = \dfrac {\sin \theta} {1 - \cos \theta} \paren {x - a \theta + a \sin \theta}$
From [[Equation of Cycloid]], the point at the top of the... | Tangent to Cycloid passes through Top of Generating Circle | https://proofwiki.org/wiki/Tangent_to_Cycloid_passes_through_Top_of_Generating_Circle | https://proofwiki.org/wiki/Tangent_to_Cycloid_passes_through_Top_of_Generating_Circle | [
"Cycloids"
] | [
"Definition:Cycloid",
"Definition:Tangent Line",
"Definition:Cycloid/Generating Circle"
] | [
"Tangent to Cycloid",
"Definition:Tangent Line",
"Equation of Cycloid",
"Definition:Cycloid/Generating Circle",
"Definition:Tangent Line"
] |
proofwiki-11381 | Tangent to Cycloid | Let $C$ be a cycloid generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the tangent to $C$ at a point $\tuple {x, y}$ on $C$ is given by the equation:
:$y - a \paren {1 - \cos \theta} = \dfrac {\sin \theta} {1 - \cos \theta} \paren {x - a \theta + a \sin \theta}$ | By Derivative of Curve at Point, the tangent to $C$ at the point $\tuple {x, y}$ is the derivative of its equation at that point.
Thus:
{{begin-eqn}}
{{eqn | l = \frac {\d x} {\d \theta}
| r = a \paren {1 - \cos \theta}
| c =
}}
{{eqn | l = \frac {\d y} {\d \theta}
| r = a \sin \theta
| c =
}}... | Let $C$ be a [[Definition:Cycloid|cycloid]] generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the [[Definition:Tangent Line|tangent]] to $C$ at a point $\tuple {x, y}$ on $C$ is given by the equation:
:$y - a \paren {1 - \cos \theta} = \dfrac {\sin \theta} {1 - ... | By [[Derivative of Curve at Point]], the [[Definition:Tangent Line|tangent]] to $C$ at the point $\tuple {x, y}$ is the [[Definition:Derivative of Real Function at Point|derivative]] of its equation at that point.
Thus:
{{begin-eqn}}
{{eqn | l = \frac {\d x} {\d \theta}
| r = a \paren {1 - \cos \theta}
| ... | Slope of Tangent to Cycloid/Proof 1 | https://proofwiki.org/wiki/Tangent_to_Cycloid | https://proofwiki.org/wiki/Slope_of_Tangent_to_Cycloid/Proof_1 | [
"Cycloids"
] | [
"Definition:Cycloid",
"Definition:Tangent Line"
] | [
"Derivative of Curve at Point",
"Definition:Tangent Line",
"Definition:Derivative/Real Function/Derivative at Point"
] |
proofwiki-11382 | Tangent to Cycloid | Let $C$ be a cycloid generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the tangent to $C$ at a point $\tuple {x, y}$ on $C$ is given by the equation:
:$y - a \paren {1 - \cos \theta} = \dfrac {\sin \theta} {1 - \cos \theta} \paren {x - a \theta + a \sin \theta}$ | Consider a polygon $ABCD$ being rolled along a straight line in the same way as the generating circle of $C$.
Let $A', B', C', D'$ be the points around which the $ABCD$ rotates while rolling.
:600px
The point $A$ traces out in succession several arcs of circles with centers $B', C', D'$.
The tangent to each of these ar... | Let $C$ be a [[Definition:Cycloid|cycloid]] generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the [[Definition:Tangent Line|tangent]] to $C$ at a point $\tuple {x, y}$ on $C$ is given by the equation:
:$y - a \paren {1 - \cos \theta} = \dfrac {\sin \theta} {1 - ... | Consider a [[Definition:Polygon|polygon]] $ABCD$ being rolled along a [[Definition:Straight Line|straight line]] in the same way as the [[Definition:Generating Circle of Cycloid|generating circle]] of $C$.
Let $A', B', C', D'$ be the [[Definition:Point|points]] around which the $ABCD$ rotates while rolling.
:[[File:... | Slope of Tangent to Cycloid/Proof 2 | https://proofwiki.org/wiki/Tangent_to_Cycloid | https://proofwiki.org/wiki/Slope_of_Tangent_to_Cycloid/Proof_2 | [
"Cycloids"
] | [
"Definition:Cycloid",
"Definition:Tangent Line"
] | [
"Definition:Polygon",
"Definition:Line/Straight Line",
"Definition:Cycloid/Generating Circle",
"Definition:Point",
"File:TangentToCycloid-construction.png",
"Definition:Circle/Arc",
"Definition:Circle/Center",
"Definition:Tangent Line",
"Definition:Circle/Arc",
"Definition:Right Angle/Perpendicula... |
proofwiki-11383 | Tangent to Cycloid | Let $C$ be a cycloid generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the tangent to $C$ at a point $\tuple {x, y}$ on $C$ is given by the equation:
:$y - a \paren {1 - \cos \theta} = \dfrac {\sin \theta} {1 - \cos \theta} \paren {x - a \theta + a \sin \theta}$ | From Slope of Tangent to Cycloid, the slope of the tangent to $C$ at the point $\tuple {x, y}$ is given by:
:$\dfrac {\d y} {\d x} = \cot \dfrac \theta 2$
This tangent to $C$ also passes through the point $\tuple {a \paren {\theta - \sin \theta}, a \paren {1 - \cos \theta} }$.
We have:
{{begin-eqn}}
{{eqn | l = \tan \... | Let $C$ be a [[Definition:Cycloid|cycloid]] generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the [[Definition:Tangent Line|tangent]] to $C$ at a point $\tuple {x, y}$ on $C$ is given by the equation:
:$y - a \paren {1 - \cos \theta} = \dfrac {\sin \theta} {1 - ... | From [[Slope of Tangent to Cycloid]], the [[Definition:Slope of Straight Line|slope]] of the [[Definition:Tangent Line|tangent]] to $C$ at the point $\tuple {x, y}$ is given by:
:$\dfrac {\d y} {\d x} = \cot \dfrac \theta 2$
This [[Definition:Tangent Line|tangent]] to $C$ also passes through the point $\tuple {a \pare... | Tangent to Cycloid | https://proofwiki.org/wiki/Tangent_to_Cycloid | https://proofwiki.org/wiki/Tangent_to_Cycloid | [
"Cycloids"
] | [
"Definition:Cycloid",
"Definition:Tangent Line"
] | [
"Slope of Tangent to Cycloid",
"Definition:Slope/Straight Line",
"Definition:Tangent Line",
"Definition:Tangent Line",
"Cotangent is Reciprocal of Tangent",
"Equation of Straight Line in Plane",
"Definition:Slope/Straight Line",
"Definition:Point",
"Definition:Line/Straight Line",
"Definition:Slop... |
proofwiki-11384 | Equation of Hypocycloid | Let a circle $C_1$ of radius $b$ roll without slipping around the inside of a circle $C_2$ of (larger) radius $a$.
Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin.
Let $P$ be a point on the circumference of $C_1$.
Let $C_1$ be initially positioned so that $P$ is its point of tangenc... | :500px
Let $C_1$ have rolled so that the line $OB$ through the radii of $C_1$ and $C_2$ is at angle $\theta$ to the $x$-axis.
Let $C_1$ have turned through an angle $\phi$ to reach that point.
By definition of sine and cosine, $P = \tuple {x, y}$ is defined by:
:$x = \paren {a - b} \cos \theta + b \map \cos {\phi - \th... | Let a [[Definition:Circle|circle]] $C_1$ of [[Definition:Radius of Circle|radius]] $b$ roll without slipping around the inside of a [[Definition:Circle|circle]] $C_2$ of (larger) [[Definition:Radius of Circle|radius]] $a$.
Let $C_2$ be embedded in a [[Definition:Cartesian Plane|cartesian plane]] with its [[Definition:... | :[[File:Hypocycloid.png|500px]]
Let $C_1$ have rolled so that the [[Definition:Straight Line|line]] $OB$ through the [[Definition:Radius of Circle|radii]] of $C_1$ and $C_2$ is at [[Definition:Angle|angle]] $\theta$ to the [[Definition:X-Axis|$x$-axis]].
Let $C_1$ have turned through an angle $\phi$ to reach that poi... | Equation of Hypocycloid | https://proofwiki.org/wiki/Equation_of_Hypocycloid | https://proofwiki.org/wiki/Equation_of_Hypocycloid | [
"Hypocycloids"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Cartesian Plane",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Point",
"Definition:Circle/Circumference",
"Definition:Tangent Circles",
"Definition:Poin... | [
"File:Hypocycloid.png",
"Definition:Line/Straight Line",
"Definition:Circle/Radius",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Sine",
"Definition:Cosine",
"Definition:Circle/Arc",
"Definition:Circle/Arc",
"Arc Length of Sector"
] |
proofwiki-11385 | Meet in Set of Ideals | Let $\mathscr S = \struct {S, \preceq}$ be a meet semilattice.
Let $\map {\it Ids} {\mathscr S}$ be the set of all ideals in $\mathscr S$.
Let $I_1, I_2$ be ideals in $\mathscr S$.
Then
:$I_1 \wedge I_2 = \set {i_1 \wedge i_2: i_1 \in I_1, i_2 \in I_2}$
where
:$I_1 \wedge I_2$ denotes the meet in $\struct {\map {\it Id... | By Meet is Intersection in Set of Ideals:
:$I_1 \wedge I_2 = I_1 \cap I_2$ | Let $\mathscr S = \struct {S, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $\map {\it Ids} {\mathscr S}$ be the [[Definition:Set of Sets|set]] of all [[Definition:Ideal in Ordered Set|ideals]] in $\mathscr S$.
Let $I_1, I_2$ be [[Definition:Ideal in Ordered Set|ideals]] in $\mathscr S$.
Then
... | By [[Meet is Intersection in Set of Ideals]]:
:$I_1 \wedge I_2 = I_1 \cap I_2$ | Meet in Set of Ideals | https://proofwiki.org/wiki/Meet_in_Set_of_Ideals | https://proofwiki.org/wiki/Meet_in_Set_of_Ideals | [
"Join and Meet Semilattices",
"Join and Meet"
] | [
"Definition:Meet Semilattice",
"Definition:Set of Sets",
"Definition:Ideal in Ordered Set",
"Definition:Ideal in Ordered Set",
"Definition:Meet (Order Theory)"
] | [
"Meet is Intersection in Set of Ideals"
] |
proofwiki-11386 | Intersection of Semilattice Ideals is Ideal | Let $\struct {S, \preceq}$ be a meet semilattice.
Let $I_1, I_2$ be ideals in $\struct {S, \preceq}$.
Then $I_1 \cap I_2$ is ideal in $\struct {S, \preceq}$ | === Directed Subset ===
Let $x, y \in I_1 \cap I_2$.
By definition of intersection:
:$x, y \in I_1$ and $x, y \in I_2$
By definition of directed subset:
:$\exists z_1 \in I_1: x \preceq z_1 \land y \preceq z_1$
and
:$\exists z_2 \in I_2: x \preceq z_2 \land y \preceq z_2$
By Meet Precedes Operands:
:$z_1 \wedge z_2 \pr... | Let $\struct {S, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $I_1, I_2$ be [[Definition:Ideal in Ordered Set|ideals]] in $\struct {S, \preceq}$.
Then $I_1 \cap I_2$ is [[Definition:Ideal in Ordered Set|ideal]] in $\struct {S, \preceq}$ | === Directed Subset ===
Let $x, y \in I_1 \cap I_2$.
By definition of [[Definition:Set Intersection|intersection]]:
:$x, y \in I_1$ and $x, y \in I_2$
By definition of [[Definition:Directed Subset|directed subset]]:
:$\exists z_1 \in I_1: x \preceq z_1 \land y \preceq z_1$
and
:$\exists z_2 \in I_2: x \preceq z_2 \l... | Intersection of Semilattice Ideals is Ideal | https://proofwiki.org/wiki/Intersection_of_Semilattice_Ideals_is_Ideal | https://proofwiki.org/wiki/Intersection_of_Semilattice_Ideals_is_Ideal | [
"Join and Meet Semilattices"
] | [
"Definition:Meet Semilattice",
"Definition:Ideal in Ordered Set",
"Definition:Ideal in Ordered Set"
] | [
"Definition:Set Intersection",
"Definition:Directed Subset",
"Meet Precedes Operands",
"Definition:Lower Section",
"Definition:Set Intersection",
"Definition:Meet (Order Theory)",
"Definition:Infimum of Set",
"Definition:Directed Subset",
"Definition:Set Intersection",
"Definition:Lower Section",
... |
proofwiki-11387 | Number of Cusps of Hypocycloid from Integral Ratio of Circle Radii | Let $H$ be a hypocycloid $H$ generated by a circle $C_1$ of radius $b$ rolling within a circle $C_2$ of (larger) radius $a$.
Let $a = n b$ where $n$ is an integer.
Then $H$ has $n$ cusps. | The length of the arc of $C_2$ between two adjacent cusps of $H$ is $2 \pi b$.
The total length of the circumference of $C_1$ is $2 \pi a$.
Thus the total number of cusps of $H$ is:
:$\dfrac {2 \pi a} {2 \pi b} = \dfrac {2 \pi n b} {2 \pi b} = n$
{{qed}} | Let $H$ be a [[Definition:Hypocycloid|hypocycloid]] $H$ generated by a [[Definition:Circle|circle]] $C_1$ of [[Definition:Radius of Circle|radius]] $b$ rolling within a [[Definition:Circle|circle]] $C_2$ of (larger) [[Definition:Radius of Circle|radius]] $a$.
Let $a = n b$ where $n$ is an [[Definition:Integer|integer]... | The [[Definition:Arc Length|length]] of the [[Definition:Arc of Circle|arc]] of $C_2$ between two adjacent [[Definition:Cusp of Hypocycloid|cusps]] of $H$ is $2 \pi b$.
The total length of the [[Definition:Circumference of Circle|circumference]] of $C_1$ is $2 \pi a$.
Thus the total number of [[Definition:Cusp of Hyp... | Number of Cusps of Hypocycloid from Integral Ratio of Circle Radii | https://proofwiki.org/wiki/Number_of_Cusps_of_Hypocycloid_from_Integral_Ratio_of_Circle_Radii | https://proofwiki.org/wiki/Number_of_Cusps_of_Hypocycloid_from_Integral_Ratio_of_Circle_Radii | [
"Hypocycloids"
] | [
"Definition:Hypocycloid",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Integer",
"Definition:Hypocycloid/Cusp"
] | [
"Definition:Arc Length",
"Definition:Circle/Arc",
"Definition:Hypocycloid/Cusp",
"Definition:Circle/Circumference",
"Definition:Hypocycloid/Cusp"
] |
proofwiki-11388 | Length of Tangent to Astroid between Axes equals Radius of Stator | Let $C_1$ be a circle of radius $b$ roll without slipping around the inside of a circle $C_2$ of radius $a = 4 b$.
Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin.
Let $P$ be a point on the circumference of $C_1$.
Let $C_1$ be initially positioned so that $P$ is its point of tangenc... | :400px
From Equation of Astroid, $H$ can be expressed as:
:$\begin {cases} x & = a \cos^3 \theta \\ y & = a \sin^3 \theta \end {cases}$
Thus the slope of the tangent to $H$ at $\tuple {x, y}$ is:
{{begin-eqn}}
{{eqn | l = \frac {\d y} {\d x}
| r = \frac {3 a \sin^2 \theta \cos \theta \rd \theta} {-3 a \cos^2 \the... | Let $C_1$ be a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $b$ roll without slipping around the inside of a [[Definition:Circle|circle]] $C_2$ of [[Definition:Radius of Circle|radius]] $a = 4 b$.
Let $C_2$ be embedded in a [[Definition:Cartesian Plane|cartesian plane]] with its [[Definition:... | :[[File:AstroidTangent.png|400px]]
From [[Equation of Astroid]], $H$ can be expressed as:
:$\begin {cases} x & = a \cos^3 \theta \\ y & = a \sin^3 \theta \end {cases}$
Thus the [[Definition:Slope of Straight Line|slope]] of the [[Definition:Tangent Line|tangent]] to $H$ at $\tuple {x, y}$ is:
{{begin-eqn}}
{{eqn |... | Length of Tangent to Astroid between Axes equals Radius of Stator | https://proofwiki.org/wiki/Length_of_Tangent_to_Astroid_between_Axes_equals_Radius_of_Stator | https://proofwiki.org/wiki/Length_of_Tangent_to_Astroid_between_Axes_equals_Radius_of_Stator | [
"Astroids"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Cartesian Plane",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Point",
"Definition:Circle/Circumference",
"Definition:Tangent Circles",
"Definition:Poin... | [
"File:AstroidTangent.png",
"Equation of Astroid",
"Definition:Slope/Straight Line",
"Definition:Tangent Line",
"Definition:Tangent Line",
"Definition:Intercept/X-Intercept",
"Sum of Squares of Sine and Cosine",
"Definition:Intercept/Y-Intercept",
"Definition:Linear Measure/Length",
"Definition:Tan... |
proofwiki-11389 | Pendulum Contained by Cycloid moves along Cycloidal Path | Let a pendulum with a flexible rod be suspended from a point $P$.
Let the rod be contained by a pair of bodies shaped as the arcs of a cycloid such that $P$ is the cusp between those two arcs.
:400px
Then the bob is constrained to move such that its path traces the arc of a cycloid. | From Evolute of Cycloid is Cycloid, the evolute of a cycloid is another cycloid.
From Curve is Involute of Evolute, the involute of a cycloid is another cycloid as well.
But by the definition of involute, the path defined by the pendulum as described is the involute of the cycloid.
Hence the result.
{{qed}} | Let a [[Definition:Pendulum|pendulum]] with a [[Definition:Flexible Chain|flexible]] [[Definition:Rod|rod]] be suspended from a point $P$.
Let the [[Definition:Rod|rod]] be contained by a pair of bodies shaped as the [[Definition:Arc of Cycloid|arcs]] of a [[Definition:Cycloid|cycloid]] such that $P$ is the [[Definiti... | From [[Evolute of Cycloid is Cycloid]], the [[Definition:Evolute|evolute]] of a [[Definition:Cycloid|cycloid]] is another [[Definition:Cycloid|cycloid]].
From [[Curve is Involute of Evolute]], the [[Definition:Involute|involute]] of a [[Definition:Cycloid|cycloid]] is another [[Definition:Cycloid|cycloid]] as well.
B... | Pendulum Contained by Cycloid moves along Cycloidal Path | https://proofwiki.org/wiki/Pendulum_Contained_by_Cycloid_moves_along_Cycloidal_Path | https://proofwiki.org/wiki/Pendulum_Contained_by_Cycloid_moves_along_Cycloidal_Path | [
"Cycloids",
"Pendulums"
] | [
"Definition:Pendulum",
"Definition:Flexible Chain",
"Definition:Rod",
"Definition:Rod",
"Definition:Cycloid/Arc",
"Definition:Cycloid",
"Definition:Cycloid/Cusp",
"Definition:Cycloid/Arc",
"File:CycloidPendulum.png",
"Definition:Simple Pendulum/Bob",
"Definition:Cycloid/Arc",
"Definition:Cyclo... | [
"Evolute of Cycloid is Cycloid",
"Definition:Evolute",
"Definition:Cycloid",
"Definition:Cycloid",
"Curve is Involute of Evolute",
"Definition:Involute",
"Definition:Cycloid",
"Definition:Cycloid",
"Definition:Involute",
"Definition:Pendulum",
"Definition:Involute",
"Definition:Cycloid"
] |
proofwiki-11390 | Lower Closure of Meet of Lower Closures | Let $\struct {S, \preceq}$ be a meet semilattice.
Let $D_1, D_2$ be subsets of $S$.
Then:
:$\set {i_1 \wedge i_2: i_1 \in D_1^\preceq, i_2 \in D_2^\preceq}^\preceq = \set {i_1 \wedge i_2: i_1 \in D_1, i_2 \in D_2}^\preceq$
where
:$D_1^\preceq$ denotes the lower closure of $D_1$. | By Lower Closure is Closure Operator:
:$D_1 \subseteq D_1^\preceq$ and $D_2 \subseteq D_2^\preceq$
Then
:$\set {i_1 \wedge i_2: i_1 \in D_1, i_2 \in D_2} \subseteq \set {i_1 \wedge i_2: i_1 \in D_1^\preceq, i_2 \in D_2^\preceq}$
By Lower Closure is Closure Operator:
:$\set {i_1 \wedge i_2: i_1 \in D_1, i_2 \in D_2}^\pr... | Let $\struct {S, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $D_1, D_2$ be [[Definition:Subset|subsets]] of $S$.
Then:
:$\set {i_1 \wedge i_2: i_1 \in D_1^\preceq, i_2 \in D_2^\preceq}^\preceq = \set {i_1 \wedge i_2: i_1 \in D_1, i_2 \in D_2}^\preceq$
where
:$D_1^\preceq$ denotes the [[Defini... | By [[Lower Closure is Closure Operator]]:
:$D_1 \subseteq D_1^\preceq$ and $D_2 \subseteq D_2^\preceq$
Then
:$\set {i_1 \wedge i_2: i_1 \in D_1, i_2 \in D_2} \subseteq \set {i_1 \wedge i_2: i_1 \in D_1^\preceq, i_2 \in D_2^\preceq}$
By [[Lower Closure is Closure Operator]]:
:$\set {i_1 \wedge i_2: i_1 \in D_1, i_2 \i... | Lower Closure of Meet of Lower Closures | https://proofwiki.org/wiki/Lower_Closure_of_Meet_of_Lower_Closures | https://proofwiki.org/wiki/Lower_Closure_of_Meet_of_Lower_Closures | [
"Join and Meet Semilattices"
] | [
"Definition:Meet Semilattice",
"Definition:Subset",
"Definition:Lower Closure/Set"
] | [
"Lower Closure is Closure Operator",
"Lower Closure is Closure Operator",
"Definition:Set Equality",
"Definition:Lower Closure/Set",
"Definition:Lower Closure/Set",
"Meet Semilattice is Ordered Structure",
"Definition:Transitive",
"Definition:Lower Closure/Set"
] |
proofwiki-11391 | Equation of Cycloid in Cartesian Coordinates | Consider a circle of radius $a$ rolling without slipping along the $x$-axis of a cartesian plane.
Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis.
Consider the cycloid traced out by the point $P$.
Let $\tuple {x, y}$ be the coordinates of $P$ as it trav... | From Equation of Cycloid, the point $P = \tuple {x, y}$ is described by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Expressing $\theta$ and $\sin \theta$ in terms of $y$:
{{begin-eqn}}
{{eqn | l = \cos \theta
| r = 1 - \frac y a
| c =
}}
{{eqn | ll= \leadsto
... | Consider a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $a$ rolling without slipping along the [[Definition:X-Axis|$x$-axis]] of a [[Definition:Cartesian Plane|cartesian plane]].
Consider the [[Definition:Point|point]] $P$ on the [[Definition:Circumference of Circle|circumference]] of this [[... | From [[Equation of Cycloid]], the [[Definition:Point|point]] $P = \tuple {x, y}$ is described by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Expressing $\theta$ and $\sin \theta$ in terms of $y$:
{{begin-eqn}}
{{eqn | l = \cos \theta
| r = 1 - \frac y a
| c =
... | Equation of Cycloid in Cartesian Coordinates | https://proofwiki.org/wiki/Equation_of_Cycloid_in_Cartesian_Coordinates | https://proofwiki.org/wiki/Equation_of_Cycloid_in_Cartesian_Coordinates | [
"Cycloids"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Axis/X-Axis",
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Circle/Circumference",
"Definition:Circle",
"Definition:Coordinate System/Origin",
"Definition:Circle/Center",
"Definition:Axis/Y-Axis",
"Definition:Cycloid",
... | [
"Equation of Cycloid",
"Definition:Point"
] |
proofwiki-11392 | Second Derivative of Locus of Cycloid | Consider a circle of radius $a$ rolling without slipping along the x-axis of a cartesian plane.
Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis.
Consider the cycloid traced out by the point $P$.
Let $\tuple {x, y}$ be the coordinates of $P$ as it travel... | From Equation of Cycloid:
{{begin-eqn}}
{{eqn | l = x
| r = a \paren {\theta - \sin \theta}
}}
{{eqn | l = y
| r = a \paren {1 - \cos \theta}
}}
{{end-eqn}}
From Slope of Tangent to Cycloid:
{{begin-eqn}}
{{eqn | l = y'
| r = \cot \dfrac \theta 2
| c = Slope of Tangent to Cycloid
}}
{{eqn | ll= ... | Consider a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $a$ rolling without slipping along the [[Definition:X-Axis|x-axis]] of a [[Definition:Cartesian Plane|cartesian plane]].
Consider the [[Definition:Point|point]] $P$ on the [[Definition:Circumference of Circle|circumference]] of this [[De... | From [[Equation of Cycloid]]:
{{begin-eqn}}
{{eqn | l = x
| r = a \paren {\theta - \sin \theta}
}}
{{eqn | l = y
| r = a \paren {1 - \cos \theta}
}}
{{end-eqn}}
From [[Slope of Tangent to Cycloid]]:
{{begin-eqn}}
{{eqn | l = y'
| r = \cot \dfrac \theta 2
| c = [[Slope of Tangent to Cycloid]]
}... | Second Derivative of Locus of Cycloid | https://proofwiki.org/wiki/Second_Derivative_of_Locus_of_Cycloid | https://proofwiki.org/wiki/Second_Derivative_of_Locus_of_Cycloid | [
"Cycloids"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Axis/X-Axis",
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Circle/Circumference",
"Definition:Circle",
"Definition:Coordinate System/Origin",
"Definition:Circle/Center",
"Definition:Axis/Y-Axis",
"Definition:Cycloid",
... | [
"Equation of Cycloid",
"Slope of Tangent to Cycloid",
"Slope of Tangent to Cycloid",
"Derivative of Composite Function",
"Derivative of Cotangent Function",
"Derivative of Sine Function"
] |
proofwiki-11393 | Arc of Cycloid is Concave | Consider a circle of radius $a$ rolling without slipping along the x-axis of a cartesian plane.
Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis.
Consider the cycloid traced out by the point $P$.
Let $\tuple {x, y}$ be the coordinates of $P$ as it travel... | From Second Derivative of Locus of Cycloid:
:$y' ' = -\dfrac a {y^2}$
As $y \ge 0$ throughout, then $y' ' < 0$ wherever $y \ne 0$, which is at the cusps.
The result follows from Second Derivative of Concave Real Function is Non-Positive.
{{qed}} | Consider a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $a$ rolling without slipping along the [[Definition:X-Axis|x-axis]] of a [[Definition:Cartesian Plane|cartesian plane]].
Consider the [[Definition:Point|point]] $P$ on the [[Definition:Circumference of Circle|circumference]] of this [[De... | From [[Second Derivative of Locus of Cycloid]]:
:$y' ' = -\dfrac a {y^2}$
As $y \ge 0$ throughout, then $y' ' < 0$ wherever $y \ne 0$, which is at the [[Definition:Cusp of Cycloid|cusps]].
The result follows from [[Second Derivative of Concave Real Function is Non-Positive]].
{{qed}} | Arc of Cycloid is Concave | https://proofwiki.org/wiki/Arc_of_Cycloid_is_Concave | https://proofwiki.org/wiki/Arc_of_Cycloid_is_Concave | [
"Cycloids"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Axis/X-Axis",
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Circle/Circumference",
"Definition:Circle",
"Definition:Coordinate System/Origin",
"Definition:Circle/Center",
"Definition:Axis/Y-Axis",
"Definition:Cycloid",
... | [
"Second Derivative of Locus of Cycloid",
"Definition:Cycloid/Cusp",
"Second Derivative of Concave Real Function is Non-Positive"
] |
proofwiki-11394 | Normal to Cycloid passes through Bottom of Generating Circle | Let $C$ be a cycloid generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the normal to $C$ at a point $P$ on $C$ passes through the bottom of the generating circle of $C$. | From Normal to Cycloid, the equation for the normal to $C$ at a point $P = \tuple {x, y}$ is given by:
{{ProofWanted|We don't even have the definition of a normal to a curve posted up yet, so there's a long way to go here.}} | Let $C$ be a [[Definition:Cycloid|cycloid]] generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the [[Definition:Normal to Curve|normal]] to $C$ at a point $P$ on $C$ passes through the bottom of the [[Definition:Generating Circle of Cycloid|generating circle]] of ... | From [[Normal to Cycloid]], the equation for the [[Definition:Normal to Curve|normal]] to $C$ at a point $P = \tuple {x, y}$ is given by:
{{ProofWanted|We don't even have the definition of a normal to a curve posted up yet, so there's a long way to go here.}} | Normal to Cycloid passes through Bottom of Generating Circle/Proof 1 | https://proofwiki.org/wiki/Normal_to_Cycloid_passes_through_Bottom_of_Generating_Circle | https://proofwiki.org/wiki/Normal_to_Cycloid_passes_through_Bottom_of_Generating_Circle/Proof_1 | [
"Normal to Cycloid passes through Bottom of Generating Circle",
"Cycloids",
"Normals to Curves"
] | [
"Definition:Cycloid",
"Definition:Normal to Curve",
"Definition:Cycloid/Generating Circle"
] | [
"Normal to Cycloid",
"Definition:Normal to Curve"
] |
proofwiki-11395 | Normal to Cycloid passes through Bottom of Generating Circle | Let $C$ be a cycloid generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the normal to $C$ at a point $P$ on $C$ passes through the bottom of the generating circle of $C$. | From Tangent to Cycloid passes through Top of Generating Circle, the tangent to $C$ at a point $P = \tuple {x, y}$ passes through the top of the generating circle.
By definition, the normal to $C$ at $P$ is perpendicular to the tangent to $C$ at $P$.
From Thales' Theorem, the normal, the tangent and the diameter of the... | Let $C$ be a [[Definition:Cycloid|cycloid]] generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the [[Definition:Normal to Curve|normal]] to $C$ at a point $P$ on $C$ passes through the bottom of the [[Definition:Generating Circle of Cycloid|generating circle]] of ... | From [[Tangent to Cycloid passes through Top of Generating Circle]], the [[Definition:Tangent Line|tangent]] to $C$ at a point $P = \tuple {x, y}$ passes through the top of the [[Definition:Generating Circle of Cycloid|generating circle]].
By definition, the [[Definition:Normal to Curve|normal]] to $C$ at $P$ is [[Def... | Normal to Cycloid passes through Bottom of Generating Circle/Proof 2 | https://proofwiki.org/wiki/Normal_to_Cycloid_passes_through_Bottom_of_Generating_Circle | https://proofwiki.org/wiki/Normal_to_Cycloid_passes_through_Bottom_of_Generating_Circle/Proof_2 | [
"Normal to Cycloid passes through Bottom of Generating Circle",
"Cycloids",
"Normals to Curves"
] | [
"Definition:Cycloid",
"Definition:Normal to Curve",
"Definition:Cycloid/Generating Circle"
] | [
"Tangent to Cycloid passes through Top of Generating Circle",
"Definition:Tangent Line",
"Definition:Cycloid/Generating Circle",
"Definition:Normal to Curve",
"Definition:Right Angle/Perpendicular",
"Definition:Tangent Line",
"Thales' Theorem",
"Definition:Normal to Curve",
"Definition:Tangent Line"... |
proofwiki-11396 | Rate of Change of Cartesian Coordinates of Cycloid | Let a circle $C$ of radius $a$ roll without slipping along the x-axis of a cartesian plane at a constant speed such that the center moves with a velocity $\mathbf v_0$ in the direction of increasing $x$.
Consider a point $P$ on the circumference of this circle.
Let $\tuple {x, y}$ be the coordinates of $P$ as it travel... | Let the center of $C$ be $O$.
{{WLOG}}, let $P$ be at the origin at time $t = t_0$.
By definition, $P$ traces out a cycloid.
From Equation of Cycloid, the point $P = \tuple {x, y}$ is described by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Let $\tuple {x_c, y_c}$ be the coor... | Let a [[Definition:Circle|circle]] $C$ of [[Definition:Radius of Circle|radius]] $a$ roll without slipping along the [[Definition:X-Axis|x-axis]] of a [[Definition:Cartesian Plane|cartesian plane]] at a constant [[Definition:Speed|speed]] such that the [[Definition:Center of Circle|center]] moves with a [[Definition:Ve... | Let the [[Definition:Center of Circle|center]] of $C$ be $O$.
{{WLOG}}, let $P$ be at the [[Definition:Origin|origin]] at time $t = t_0$.
By definition, $P$ traces out a [[Definition:Cycloid|cycloid]].
From [[Equation of Cycloid]], the [[Definition:Point|point]] $P = \tuple {x, y}$ is described by the equations:
:$x... | Rate of Change of Cartesian Coordinates of Cycloid | https://proofwiki.org/wiki/Rate_of_Change_of_Cartesian_Coordinates_of_Cycloid | https://proofwiki.org/wiki/Rate_of_Change_of_Cartesian_Coordinates_of_Cycloid | [
"Cycloids"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Axis/X-Axis",
"Definition:Cartesian Plane",
"Definition:Speed",
"Definition:Circle/Center",
"Definition:Velocity",
"Definition:Point",
"Definition:Circle/Circumference",
"Definition:Circle",
"Definition:Cartesian Plane/Ordered Pair",
... | [
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cycloid",
"Equation of Cycloid",
"Definition:Point",
"Definition:Cartesian Plane/Ordered Pair",
"Definition:Constant",
"Equations of Motion with Constant Acceleration/Velocity after Time",
"Definition:Acceleration",
"De... |
proofwiki-11397 | Maximum Rate of Change of X Coordinate of Cycloid | Let a circle $C$ of radius $a$ roll without slipping along the x-axis of a cartesian plane at a constant speed such that the center moves with a velocity $\mathbf v_0$ in the direction of increasing $x$.
Consider a point $P$ on the circumference of this circle.
Let $\tuple {x, y}$ be the coordinates of $P$ as it travel... | From Rate of Change of Cartesian Coordinates of Cycloid, the rate of change of $x$ is given by:
:$\dfrac {\d x} {\d t} = \mathbf v_0 \paren {1 - \cos \theta}$
This is a maximum when $1 - \cos \theta$ is a maximum.
That happens when $\cos \theta$ is at a minimum.
That happens when $\cos \theta = -1$.
That happens when $... | Let a [[Definition:Circle|circle]] $C$ of [[Definition:Radius of Circle|radius]] $a$ roll without slipping along the [[Definition:X-Axis|x-axis]] of a [[Definition:Cartesian Plane|cartesian plane]] at a constant [[Definition:Speed|speed]] such that the [[Definition:Center of Circle|center]] moves with a [[Definition:Ve... | From [[Rate of Change of Cartesian Coordinates of Cycloid]], the [[Definition:Rate of Change with respect to Time|rate of change]] of $x$ is given by:
:$\dfrac {\d x} {\d t} = \mathbf v_0 \paren {1 - \cos \theta}$
This is a maximum when $1 - \cos \theta$ is a maximum.
That happens when $\cos \theta$ is at a minimum.
... | Maximum Rate of Change of X Coordinate of Cycloid | https://proofwiki.org/wiki/Maximum_Rate_of_Change_of_X_Coordinate_of_Cycloid | https://proofwiki.org/wiki/Maximum_Rate_of_Change_of_X_Coordinate_of_Cycloid | [
"Cycloids"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Axis/X-Axis",
"Definition:Cartesian Plane",
"Definition:Speed",
"Definition:Circle/Center",
"Definition:Velocity",
"Definition:Point",
"Definition:Circle/Circumference",
"Definition:Circle",
"Definition:Cartesian Plane/Ordered Pair",
... | [
"Rate of Change of Cartesian Coordinates of Cycloid",
"Definition:Rate of Change/Time",
"Definition:Circle"
] |
proofwiki-11398 | Maximum Rate of Change of Y Coordinate of Cycloid | Let a circle $C$ of radius $a$ roll without slipping along the x-axis of a cartesian plane at a constant speed such that the center moves with a velocity $\mathbf v_0$ in the direction of increasing $x$.
Consider a point $P$ on the circumference of this circle.
Let $\tuple {x, y}$ be the coordinates of $P$ as it travel... | From Rate of Change of Cartesian Coordinates of Cycloid, the rate of change of $y$ is given by:
:$\dfrac {\d y} {\d t} = \mathbf v_0 \sin \theta$.
This is a maximum when $\sin \theta$ is a maximum.
That happens when $\sin \theta = 1$.
That happens when $\theta = \dfrac \pi 2 + 2 n \pi$ where $n \in \Z$.
When $\sin \the... | Let a [[Definition:Circle|circle]] $C$ of [[Definition:Radius of Circle|radius]] $a$ roll without slipping along the [[Definition:X-Axis|x-axis]] of a [[Definition:Cartesian Plane|cartesian plane]] at a constant [[Definition:Speed|speed]] such that the [[Definition:Center of Circle|center]] moves with a [[Definition:Ve... | From [[Rate of Change of Cartesian Coordinates of Cycloid]], the [[Definition:Rate of Change with respect to Time|rate of change]] of $y$ is given by:
:$\dfrac {\d y} {\d t} = \mathbf v_0 \sin \theta$.
This is a maximum when $\sin \theta$ is a maximum.
That happens when $\sin \theta = 1$.
That happens when $\theta =... | Maximum Rate of Change of Y Coordinate of Cycloid | https://proofwiki.org/wiki/Maximum_Rate_of_Change_of_Y_Coordinate_of_Cycloid | https://proofwiki.org/wiki/Maximum_Rate_of_Change_of_Y_Coordinate_of_Cycloid | [
"Cycloids"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Axis/X-Axis",
"Definition:Cartesian Plane",
"Definition:Speed",
"Definition:Circle/Center",
"Definition:Velocity",
"Definition:Point",
"Definition:Circle/Circumference",
"Definition:Circle",
"Definition:Cartesian Plane/Ordered Pair",
... | [
"Rate of Change of Cartesian Coordinates of Cycloid",
"Definition:Rate of Change/Time"
] |
proofwiki-11399 | Lower Closure is Closure Operator | Let $\struct {S, \preceq}$ be an ordered set.
Then
:lower closure of set is a closure operator. | === Inflationary ===
Let $X$ be a subset of $S$.
Let $x \in X$.
By definition of reflexivity:
:$x \preceq x$
Thus by definition of lower closure of set:
:$x \in X^\preceq$
Thus by definition of subset:
:$X \subseteq X^\preceq$
{{qed|lemma}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Then
:[[Definition:Lower Closure of Subset|lower closure of set]] is a [[Definition:Closure Operator|closure operator]]. | === [[Definition:Inflationary Mapping|Inflationary]] ===
Let $X$ be a [[Definition:Subset|subset]] of $S$.
Let $x \in X$.
By definition of [[Definition:Reflexivity|reflexivity]]:
:$x \preceq x$
Thus by definition of [[Definition:Lower Closure of Subset|lower closure of set]]:
:$x \in X^\preceq$
Thus by definition ... | Lower Closure is Closure Operator | https://proofwiki.org/wiki/Lower_Closure_is_Closure_Operator | https://proofwiki.org/wiki/Lower_Closure_is_Closure_Operator | [
"Lower Closures",
"Closure Operators"
] | [
"Definition:Ordered Set",
"Definition:Lower Closure/Set",
"Definition:Closure Operator"
] | [
"Definition:Inflationary Mapping",
"Definition:Subset",
"Definition:Reflexivity",
"Definition:Lower Closure/Set",
"Definition:Subset",
"Definition:Subset",
"Definition:Lower Closure/Set",
"Definition:Subset",
"Definition:Lower Closure/Set",
"Definition:Subset",
"Definition:Subset",
"Definition... |
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