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-16500 | Reciprocal of 7 | :$\dfrac 1 7 = 0 \cdotp \dot 14285 \, \dot 7$ | Performing the calculation using long division:
<pre>
0.142857...
------------
7)1.000000000
7
--
30
28
--
20
14
--
60
56
--
40
35
--
50
49
--
10
7
--
..
</pr... | :$\dfrac 1 7 = 0 \cdotp \dot 14285 \, \dot 7$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.142857...
------------
7)1.000000000
7
--
30
28
--
20
14
--
60
56
--
40
35
--
50
49
--
10
7
... | Reciprocal of 7 | https://proofwiki.org/wiki/Reciprocal_of_7 | https://proofwiki.org/wiki/Reciprocal_of_7 | [
"7",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:7",
"Category:Examples of Reciprocals"
] |
proofwiki-16501 | Spherical Triangle is Polar Triangle of its Polar Triangle | Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Let $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$.
Then $\triangle ABC$ is... | :400px
Let $BC$ be produced to meet $A'B'$ and $A'C'$ at $L$ and $M$ respectively.
Because $A'$ is the pole of the great circle $LBCM$, the spherical angle $A'$ equals the side of the spherical triangle $ALM$.
By construction we have that $B'$ is the pole of $AC$.
Thus the length of the arc of the great circle from $B$... | Let $\triangle ABC$ be a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is $O$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a, b, c$ of $\triangle ABC$ be measured by the [[Definition:Subtend|ang... | :[[File:Polar-triangle.png|400px]]
Let $BC$ be [[Definition:Production|produced]] to meet $A'B'$ and $A'C'$ at $L$ and $M$ respectively.
Because $A'$ is the [[Definition:Pole of Circle|pole]] of the [[Definition:Great Circle|great circle]] $LBCM$, the [[Definition:Spherical Angle|spherical angle]] $A'$ equals the [[D... | Spherical Triangle is Polar Triangle of its Polar Triangle | https://proofwiki.org/wiki/Spherical_Triangle_is_Polar_Triangle_of_its_Polar_Triangle | https://proofwiki.org/wiki/Spherical_Triangle_is_Polar_Triangle_of_its_Polar_Triangle | [
"Polar Triangles"
] | [
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Spherical Triangle/Side",
"Definition:Subtend",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polar Triangle",
"Definition:Polar Triangle"
] | [
"File:Polar-triangle.png",
"Definition:Production",
"Definition:Pole of Circle",
"Definition:Great Circle",
"Definition:Spherical Angle",
"Definition:Spherical Triangle/Side",
"Definition:Spherical Triangle",
"Definition:Pole of Arc",
"Definition:Length of Arc of Great Circle",
"Definition:Circle/... |
proofwiki-16502 | Side of Spherical Triangle is Supplement of Angle of Polar Triangle | Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Let $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$.
Then $A'$ is the supple... | :400px
Let $BC$ be produced to meet $A'B'$ and $A'C'$ at $L$ and $M$ respectively.
Because $A'$ is the pole of the great circle $LBCM$, the spherical angle $A'$ equals the side of the spherical triangle $A'LM$.
That is:
:$(1): \quad \sphericalangle A' = LM$
From Spherical Triangle is Polar Triangle of its Polar Triangl... | Let $\triangle ABC$ be a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is $O$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a, b, c$ of $\triangle ABC$ be measured by the [[Definition:Subtend|ang... | :[[File:Polar-triangle.png|400px]]
Let $BC$ be [[Definition:Production|produced]] to meet $A'B'$ and $A'C'$ at $L$ and $M$ respectively.
Because $A'$ is the [[Definition:Pole of Circle|pole]] of the [[Definition:Great Circle|great circle]] $LBCM$, the [[Definition:Spherical Angle|spherical angle]] $A'$ equals the [[D... | Side of Spherical Triangle is Supplement of Angle of Polar Triangle | https://proofwiki.org/wiki/Side_of_Spherical_Triangle_is_Supplement_of_Angle_of_Polar_Triangle | https://proofwiki.org/wiki/Side_of_Spherical_Triangle_is_Supplement_of_Angle_of_Polar_Triangle | [
"Polar Triangles",
"Supplementary Angles"
] | [
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Spherical Triangle/Side",
"Definition:Subtend",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polar Triangle",
"Definition:Supplementary Angles"
] | [
"File:Polar-triangle.png",
"Definition:Production",
"Definition:Pole of Circle",
"Definition:Great Circle",
"Definition:Spherical Angle",
"Definition:Spherical Triangle/Side",
"Definition:Spherical Triangle",
"Spherical Triangle is Polar Triangle of its Polar Triangle",
"Definition:Polar Triangle",
... |
proofwiki-16503 | Integers are Dense in P-adic Integers/Unit Ball | The integers $\Z$ are dense in the closed ball $\map {B^-_1} 0$. | We have:
{{begin-eqn}}
{{eqn | l = \map {B^-_1} 0
| r = \map {B^-_{p^0} } 0
| c = Zeroth Power of Real Number equals One
}}
{{eqn | r = 0 + p^0 \Z_p
| c = Closed Balls of P-adic Number
}}
{{eqn | r = 0 + 1 \cdot \Z_p
}}
{{eqn | r = 0 + \Z_p
| c = {{Defof|Multiplicative Identity}}
}}
{{eqn | ... | The [[Definition:Integer|integers]] $\Z$ are [[Definition:Everywhere Dense|dense]] in the [[Definition:Closed Ball in P-adic Numbers|closed ball]] $\map {B^-_1} 0$. | We have:
{{begin-eqn}}
{{eqn | l = \map {B^-_1} 0
| r = \map {B^-_{p^0} } 0
| c = [[Zeroth Power of Real Number equals One]]
}}
{{eqn | r = 0 + p^0 \Z_p
| c = [[Closed Balls of P-adic Number]]
}}
{{eqn | r = 0 + 1 \cdot \Z_p
}}
{{eqn | r = 0 + \Z_p
| c = {{Defof|Multiplicative Identity}}
}}
... | Integers are Dense in P-adic Integers/Unit Ball | https://proofwiki.org/wiki/Integers_are_Dense_in_P-adic_Integers/Unit_Ball | https://proofwiki.org/wiki/Integers_are_Dense_in_P-adic_Integers/Unit_Ball | [
"Integers are Dense in P-adic Integers"
] | [
"Definition:Integer",
"Definition:Everywhere Dense",
"Definition:Closed Ball/P-adic Numbers"
] | [
"Zeroth Power of Real Number equals One",
"Closed Ball of P-adic Number",
"Integers are Dense in P-adic Integers",
"Definition:Integer",
"Definition:Everywhere Dense",
"Definition:Closed Ball/P-adic Numbers",
"Category:Integers are Dense in P-adic Integers"
] |
proofwiki-16504 | Analogue Formula for Spherical Law of Cosines/Corollary | {{begin-eqn}}
{{eqn | l = \sin A \cos b
| r = \cos B \sin C + \sin B \cos C \cos a
}}
{{eqn | l = \sin A \cos c
| r = \cos C \sin B + \sin C \cos B \cos a
}}
{{end-eqn}} | Let $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$.
Let the sides $a', b', c'$ of $\triangle A'B'C'$ be opposite $A', B', C'$ respectively.
From Spherical Triangle is Polar Triangle of its Polar Triangle we have that:
:not only is $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$
:but also $\tria... | {{begin-eqn}}
{{eqn | l = \sin A \cos b
| r = \cos B \sin C + \sin B \cos C \cos a
}}
{{eqn | l = \sin A \cos c
| r = \cos C \sin B + \sin C \cos B \cos a
}}
{{end-eqn}} | Let $\triangle A'B'C'$ be the [[Definition:Polar Triangle|polar triangle]] of $\triangle ABC$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a', b', c'$ of $\triangle A'B'C'$ be [[Definition:Opposite (in Triangle)|opposite]] $A', B', C'$ respectively.
From [[Spherical Triangle is Polar Triangle of its Pol... | Analogue Formula for Spherical Law of Cosines/Corollary | https://proofwiki.org/wiki/Analogue_Formula_for_Spherical_Law_of_Cosines/Corollary | https://proofwiki.org/wiki/Analogue_Formula_for_Spherical_Law_of_Cosines/Corollary | [
"Analogue Formula for Spherical Law of Cosines"
] | [] | [
"Definition:Polar Triangle",
"Definition:Spherical Triangle/Side",
"Definition:Triangle (Geometry)/Opposite",
"Spherical Triangle is Polar Triangle of its Polar Triangle",
"Definition:Polar Triangle",
"Definition:Polar Triangle",
"Analogue Formula for Spherical Law of Cosines",
"Side of Spherical Tria... |
proofwiki-16505 | Four-Parts Formula/Corollary | :$\cos A \cos c = \sin A \cot B - \sin c \cot b$ | Let $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$.
Let the sides $a', b', c'$ of $\triangle A'B'C'$ be opposite $A', B', C'$ respectively.
From Spherical Triangle is Polar Triangle of its Polar Triangle we have that:
:not only is $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$
:but also $\tria... | :$\cos A \cos c = \sin A \cot B - \sin c \cot b$ | Let $\triangle A'B'C'$ be the [[Definition:Polar Triangle|polar triangle]] of $\triangle ABC$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a', b', c'$ of $\triangle A'B'C'$ be [[Definition:Opposite (in Triangle)|opposite]] $A', B', C'$ respectively.
From [[Spherical Triangle is Polar Triangle of its Pol... | Four-Parts Formula/Corollary | https://proofwiki.org/wiki/Four-Parts_Formula/Corollary | https://proofwiki.org/wiki/Four-Parts_Formula/Corollary | [
"Analogue Formula for Spherical Law of Cosines"
] | [] | [
"Definition:Polar Triangle",
"Definition:Spherical Triangle/Side",
"Definition:Triangle (Geometry)/Opposite",
"Spherical Triangle is Polar Triangle of its Polar Triangle",
"Definition:Polar Triangle",
"Definition:Polar Triangle",
"Four-Parts Formula",
"Side of Spherical Triangle is Supplement of Angle... |
proofwiki-16506 | Equation of Witch of Agnesi/Cartesian | The equation of the Witch of Agnesi is given in cartesian coordinates as:
:$y = \dfrac {8 a^3} {x^2 + 4 a^2}$ | Let $P = \tuple {x, y}$ and $A = \tuple {d, y}$.
We have that:
:$\dfrac {OM} {MN} = \dfrac {2 a} x = \dfrac y d$.
Also, by Pythagoras's Theorem:
:$\paren {a - y}^2 + d^2 = a^2 \implies y \paren {2 a - y} = d^2$
Eliminating $d$ gives us:
:$\dfrac {y^2} {y \paren {2 a - y} } = \dfrac {\paren {2 a}^2} {x^2}$
Hence:
:$\dfr... | The equation of the [[Definition:Witch of Agnesi|Witch of Agnesi]] is given in [[Definition:Cartesian Coordinate System|cartesian coordinates]] as:
:$y = \dfrac {8 a^3} {x^2 + 4 a^2}$ | Let $P = \tuple {x, y}$ and $A = \tuple {d, y}$.
We have that:
:$\dfrac {OM} {MN} = \dfrac {2 a} x = \dfrac y d$.
Also, by [[Pythagoras's Theorem]]:
:$\paren {a - y}^2 + d^2 = a^2 \implies y \paren {2 a - y} = d^2$
Eliminating $d$ gives us:
:$\dfrac {y^2} {y \paren {2 a - y} } = \dfrac {\paren {2 a}^2} {x^2}$
Hence... | Equation of Witch of Agnesi/Cartesian | https://proofwiki.org/wiki/Equation_of_Witch_of_Agnesi/Cartesian | https://proofwiki.org/wiki/Equation_of_Witch_of_Agnesi/Cartesian | [
"Witch of Agnesi"
] | [
"Definition:Witch of Agnesi",
"Definition:Cartesian Coordinate System"
] | [
"Pythagoras's Theorem"
] |
proofwiki-16507 | Cosine in Terms of Haversine | :$\cos \theta = 1 - 2 \hav \theta$
where $\cos$ denotes cosine and $\hav$ denotes haversine. | {{begin-eqn}}
{{eqn | l = \hav \theta
| r = \dfrac 1 2 \paren {1 - \cos \theta}
| c = {{Defof|Haversine}}
}}
{{eqn | ll= \leadsto
| l = 2 \hav \theta
| r = 1 - \cos \theta
| c =
}}
{{eqn | ll= \leadsto
| l = \cos \theta
| r = 1 - 2 \hav \theta
| c =
}}
{{end-eqn}}
{{qed... | :$\cos \theta = 1 - 2 \hav \theta$
where $\cos$ denotes [[Definition:Cosine|cosine]] and $\hav$ denotes [[Definition:Haversine|haversine]]. | {{begin-eqn}}
{{eqn | l = \hav \theta
| r = \dfrac 1 2 \paren {1 - \cos \theta}
| c = {{Defof|Haversine}}
}}
{{eqn | ll= \leadsto
| l = 2 \hav \theta
| r = 1 - \cos \theta
| c =
}}
{{eqn | ll= \leadsto
| l = \cos \theta
| r = 1 - 2 \hav \theta
| c =
}}
{{end-eqn}}
{{qed... | Cosine in Terms of Haversine | https://proofwiki.org/wiki/Cosine_in_Terms_of_Haversine | https://proofwiki.org/wiki/Cosine_in_Terms_of_Haversine | [
"Cosine Function",
"Haversines"
] | [
"Definition:Cosine",
"Definition:Haversine"
] | [] |
proofwiki-16508 | Spherical Law of Haversines | Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Then:
:$\hav a = \map \hav {b - c} + \sin b \sin c \hav A$
where $\hav$ denotes haversin... | {{begin-eqn}}
{{eqn | l = \cos a
| r = \cos b \cos c + \sin b \sin c \cos A
| c = Spherical Law of Cosines
}}
{{eqn | ll= \leadsto
| l = 1 - 2 \hav a
| r = \cos b \cos c + \sin b \sin c \paren {1 - 2 \hav A}
| c = Cosine in Terms of Haversine
}}
{{eqn | r = \map \cos {b - c} - 2 \sin b \si... | Let $\triangle ABC$ be a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is $O$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a, b, c$ of $\triangle ABC$ be measured by the [[Definition:Subtend|ang... | {{begin-eqn}}
{{eqn | l = \cos a
| r = \cos b \cos c + \sin b \sin c \cos A
| c = [[Spherical Law of Cosines]]
}}
{{eqn | ll= \leadsto
| l = 1 - 2 \hav a
| r = \cos b \cos c + \sin b \sin c \paren {1 - 2 \hav A}
| c = [[Cosine in Terms of Haversine]]
}}
{{eqn | r = \map \cos {b - c} - 2 \s... | Spherical Law of Haversines | https://proofwiki.org/wiki/Spherical_Law_of_Haversines | https://proofwiki.org/wiki/Spherical_Law_of_Haversines | [
"Haversines",
"Spherical Trigonometry"
] | [
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Spherical Triangle/Side",
"Definition:Subtend",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Haversine"
] | [
"Spherical Law of Cosines",
"Cosine in Terms of Haversine",
"Cosine of Difference",
"Cosine in Terms of Haversine"
] |
proofwiki-16509 | Sign of Haversine | The haversine is non-negative for all $\theta \in \R$. | The haversine is conventionally defined on the real numbers only.
We have that:
:$\forall \theta \in \R: -1 < \cos \theta < 1$
and so:
:$\forall \theta \in \R: 0 < 1 - \cos \theta < 2$
from which the result follows by definition of haversine.
{{qed}} | The [[Definition:Haversine|haversine]] is [[Definition:Non-Negative Real Number|non-negative]] for all $\theta \in \R$. | The [[Definition:Haversine|haversine]] is conventionally defined on the [[Definition:Real Numbers|real numbers]] only.
We have that:
:$\forall \theta \in \R: -1 < \cos \theta < 1$
and so:
:$\forall \theta \in \R: 0 < 1 - \cos \theta < 2$
from which the result follows by definition of [[Definition:Haversine|haversine... | Sign of Haversine | https://proofwiki.org/wiki/Sign_of_Haversine | https://proofwiki.org/wiki/Sign_of_Haversine | [
"Haversines"
] | [
"Definition:Haversine",
"Definition:Positive/Real Number"
] | [
"Definition:Haversine",
"Definition:Real Number",
"Definition:Haversine"
] |
proofwiki-16510 | Haversine Function is Even | The haversine is an even function:
:$\forall \theta \in \R: \map \hav {-\theta} = \hav \theta$ | {{begin-eqn}}
{{eqn | l = \map \hav {-\theta}
| r = \dfrac 1 2 \paren {1 - \map \cos {-\theta} }
| c = {{Defof|Haversine}}
}}
{{eqn | r = \dfrac 1 2 \paren {1 - \cos \theta}
| c = Cosine Function is Even
}}
{{eqn | r = \hav \theta
| c = {{Defof|Haversine}}
}}
{{end-eqn}}
{{qed}} | The [[Definition:Haversine|haversine]] is an [[Definition:Even Function|even function]]:
:$\forall \theta \in \R: \map \hav {-\theta} = \hav \theta$ | {{begin-eqn}}
{{eqn | l = \map \hav {-\theta}
| r = \dfrac 1 2 \paren {1 - \map \cos {-\theta} }
| c = {{Defof|Haversine}}
}}
{{eqn | r = \dfrac 1 2 \paren {1 - \cos \theta}
| c = [[Cosine Function is Even]]
}}
{{eqn | r = \hav \theta
| c = {{Defof|Haversine}}
}}
{{end-eqn}}
{{qed}} | Haversine Function is Even | https://proofwiki.org/wiki/Haversine_Function_is_Even | https://proofwiki.org/wiki/Haversine_Function_is_Even | [
"Haversines",
"Examples of Even Functions"
] | [
"Definition:Haversine",
"Definition:Even Function"
] | [
"Cosine Function is Even"
] |
proofwiki-16511 | Sine of Half Side for Spherical Triangles | :$\sin \dfrac a 2 = \sqrt {\dfrac {-\cos S \, \map \cos {S - A} } {\sin B \sin C} }$
where $S = \dfrac {A + B + C} 2$. | Let $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$.
Let the sides $a', b', c'$ of $\triangle A'B'C'$ be opposite $A', B', C'$ respectively.
From Spherical Triangle is Polar Triangle of its Polar Triangle we have that:
:not only is $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$
:but also $\tria... | :$\sin \dfrac a 2 = \sqrt {\dfrac {-\cos S \, \map \cos {S - A} } {\sin B \sin C} }$
where $S = \dfrac {A + B + C} 2$. | Let $\triangle A'B'C'$ be the [[Definition:Polar Triangle|polar triangle]] of $\triangle ABC$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a', b', c'$ of $\triangle A'B'C'$ be [[Definition:Opposite (in Triangle)|opposite]] $A', B', C'$ respectively.
From [[Spherical Triangle is Polar Triangle of its Pol... | Sine of Half Side for Spherical Triangles | https://proofwiki.org/wiki/Sine_of_Half_Side_for_Spherical_Triangles | https://proofwiki.org/wiki/Sine_of_Half_Side_for_Spherical_Triangles | [
"Half Side Formulas for Spherical Triangles"
] | [] | [
"Definition:Polar Triangle",
"Definition:Spherical Triangle/Side",
"Definition:Triangle (Geometry)/Opposite",
"Spherical Triangle is Polar Triangle of its Polar Triangle",
"Definition:Polar Triangle",
"Definition:Polar Triangle",
"Cosine of Half Angle for Spherical Triangles",
"Side of Spherical Trian... |
proofwiki-16512 | Tangent of Half Side for Spherical Triangles | :$\tan \dfrac a 2 = \sqrt {\dfrac {-\cos S \, \map \cos {S - A} } {\map \cos {S - B} \, \map \cos {S - C} } }$
where $S = \dfrac {a + b + c} 2$. | {{begin-eqn}}
{{eqn | l = \tan \dfrac a 2
| r = \dfrac {\sqrt {\dfrac {-\cos S \, \map \cos {S - A} } {\sin B \sin C} } } {\sqrt {\dfrac {\map \cos {S - B} \, \map \cos {S - C} } {\sin B \sin C} } }
| c = Sine of Half Angle for Spherical Triangles, Cosine of Half Angle for Spherical Triangles
}}
{{eqn | r =... | :$\tan \dfrac a 2 = \sqrt {\dfrac {-\cos S \, \map \cos {S - A} } {\map \cos {S - B} \, \map \cos {S - C} } }$
where $S = \dfrac {a + b + c} 2$. | {{begin-eqn}}
{{eqn | l = \tan \dfrac a 2
| r = \dfrac {\sqrt {\dfrac {-\cos S \, \map \cos {S - A} } {\sin B \sin C} } } {\sqrt {\dfrac {\map \cos {S - B} \, \map \cos {S - C} } {\sin B \sin C} } }
| c = [[Sine of Half Angle for Spherical Triangles]], [[Cosine of Half Angle for Spherical Triangles]]
}}
{{e... | Tangent of Half Side for Spherical Triangles | https://proofwiki.org/wiki/Tangent_of_Half_Side_for_Spherical_Triangles | https://proofwiki.org/wiki/Tangent_of_Half_Side_for_Spherical_Triangles | [
"Half Side Formulas for Spherical Triangles"
] | [] | [
"Sine of Half Angle for Spherical Triangles",
"Cosine of Half Angle for Spherical Triangles"
] |
proofwiki-16513 | Zenith Distance is Complement of Celestial Altitude | Let $X$ be the position of a star (or other celestial body) on the celestial sphere.
The zenith distance $z$ of $X$ is the complement of the altitude $a$ of $X$:
:$z = 90 \degrees - a$ | The vertical circle through $X$ is defined as the great circle that passes through $Z$.
By definition, the angle of the arc from $Z$ to the horizon is a right angle.
Hence $z + a = 90 \degrees$.
The result follows.
{{qed}} | Let $X$ be the position of a [[Definition:Star (Physics)|star]] (or other [[Definition:Celestial Body|celestial body]]) on the [[Definition:Celestial Sphere|celestial sphere]].
The [[Definition:Zenith Distance|zenith distance]] $z$ of $X$ is the [[Definition:Complement of Angle|complement]] of the [[Definition:Celesti... | The [[Definition:Vertical Circle|vertical circle]] through $X$ is defined as the [[Definition:Great Circle|great circle]] that passes through $Z$.
By definition, the [[Definition:Spherical Angle|angle]] of the [[Definition:Arc of Circle|arc]] from $Z$ to the [[Definition:Celestial Horizon|horizon]] is a [[Definition:R... | Zenith Distance is Complement of Celestial Altitude | https://proofwiki.org/wiki/Zenith_Distance_is_Complement_of_Celestial_Altitude | https://proofwiki.org/wiki/Zenith_Distance_is_Complement_of_Celestial_Altitude | [
"Spherical Astronomy",
"Complementary Angles"
] | [
"Definition:Star (Physics)",
"Definition:Celestial Body",
"Definition:Celestial Sphere",
"Definition:Zenith Distance",
"Definition:Complementary Angles",
"Definition:Celestial Altitude"
] | [
"Definition:Vertical Circle",
"Definition:Great Circle",
"Definition:Spherical Angle",
"Definition:Circle/Arc",
"Definition:Celestial Horizon",
"Definition:Right Angle"
] |
proofwiki-16514 | P-adic Metric on P-adic Numbers is Non-Archimedean Metric/Corollary 1 | Then:
:$\forall x, y, z \in R: \norm {x - y}_p \le \max \set {\norm {x - z}_p, \norm {y - z}_p}$ | Let $d_p$ be the $p$-adic metric on $\Q_p$:
:$\forall x, y \in \Q_p: \map {d_p} {x, y} = \norm {x - y}_p$
From P-adic Metric on P-adic Numbers is Non-Archimedean Metric, $d_p$ is a non-Archimedean norm.
By definition of a non-Archimedean norm:
:$\forall x, y, z \in R: \norm {x - y}_p \le \max \set {\norm {x - z}_p, \no... | Then:
:$\forall x, y, z \in R: \norm {x - y}_p \le \max \set {\norm {x - z}_p, \norm {y - z}_p}$ | Let $d_p$ be the [[Definition:P-adic Metric on P-adic Numbers|$p$-adic metric]] on $\Q_p$:
:$\forall x, y \in \Q_p: \map {d_p} {x, y} = \norm {x - y}_p$
From [[P-adic Metric on P-adic Numbers is Non-Archimedean Metric]], $d_p$ is a [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean norm]].
By definition ... | P-adic Metric on P-adic Numbers is Non-Archimedean Metric/Corollary 1 | https://proofwiki.org/wiki/P-adic_Metric_on_P-adic_Numbers_is_Non-Archimedean_Metric/Corollary_1 | https://proofwiki.org/wiki/P-adic_Metric_on_P-adic_Numbers_is_Non-Archimedean_Metric/Corollary_1 | [
"Normed Division Rings",
"Non-Archimedean Norms"
] | [] | [
"Definition:P-adic Metric/P-adic Numbers",
"P-adic Metric on P-adic Numbers is Non-Archimedean Metric",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Category:Normed Division Rings",
"Category:Non-Archimedean Norms"
] |
proofwiki-16515 | Zenith Distance of North Celestial Pole equals Colatitude of Observer | Let $O$ be an observer of the celestial sphere.
Let $P$ be the position of the north celestial pole with respect to $O$.
Let $z$ denote the zenith distance of $P$.
Let $\psi$ denote the (terrestrial) colatitude of $O$.
Then:
:$z = \psi$ | Let $Z$ denote the zenith.
The zenith distance $z$ of $P$ is by definition the length of the arc $PZ$ of the prime vertical.
This in turn is defined as the angle $\angle POZ$ subtended by $PZ$ at $O$.
This is equivalent to the angle between the radius of Earth through $O$ and Earth's axis.
This is by definition the (te... | Let $O$ be an [[Definition:Celestial Observer|observer]] of the [[Definition:Celestial Sphere|celestial sphere]].
Let $P$ be the position of the [[Definition:North Celestial Pole|north celestial pole]] with respect to $O$.
Let $z$ denote the [[Definition:Zenith Distance|zenith distance]] of $P$.
Let $\psi$ denote th... | Let $Z$ denote the [[Definition:Zenith|zenith]].
The [[Definition:Zenith Distance|zenith distance]] $z$ of $P$ is by definition the [[Definition:Length of Arc of Great Circle|length]] of the [[Definition:Arc of Circle|arc]] $PZ$ of the [[Definition:Prime Vertical|prime vertical]].
This in turn is defined as the [[Def... | Zenith Distance of North Celestial Pole equals Colatitude of Observer | https://proofwiki.org/wiki/Zenith_Distance_of_North_Celestial_Pole_equals_Colatitude_of_Observer | https://proofwiki.org/wiki/Zenith_Distance_of_North_Celestial_Pole_equals_Colatitude_of_Observer | [
"Spherical Astronomy"
] | [
"Definition:Celestial Sphere/Observer",
"Definition:Celestial Sphere",
"Definition:North Celestial Pole",
"Definition:Zenith Distance",
"Definition:Colatitude/Terrestrial"
] | [
"Definition:Zenith",
"Definition:Zenith Distance",
"Definition:Length of Arc of Great Circle",
"Definition:Circle/Arc",
"Definition:Prime Vertical",
"Definition:Circle/Arc/Subtend",
"Definition:Spherical Angle",
"Definition:Sphere/Geometry/Radius",
"Definition:Earth",
"Definition:Earth's Axis",
... |
proofwiki-16516 | Altitude of North Celestial Pole equals Latitude of Observer | Let $O$ be an observer of the celestial sphere.
Let $P$ be the position of the north celestial pole with respect to $O$.
Let $a$ denote the altitude of $P$.
Let $\phi$ denote the (terrestrial) latitude of $O$.
Then:
:$a = \phi$ | Let $z$ denote the zenith distance of $P$.
Let $\psi$ denote the (terrestrial) colatitude of $O$.
By definition we have:
:$a = 90 \degrees - z$
:$\phi - 90 \degrees - \psi$
Then:
{{begin-eqn}}
{{eqn | l = z
| r = \psi
| c = Zenith Distance of North Celestial Pole equals Colatitude of Observer
}}
{{eqn | ll=... | Let $O$ be an [[Definition:Celestial Observer|observer]] of the [[Definition:Celestial Sphere|celestial sphere]].
Let $P$ be the position of the [[Definition:North Celestial Pole|north celestial pole]] with respect to $O$.
Let $a$ denote the [[Definition:Celestial Altitude|altitude]] of $P$.
Let $\phi$ denote the [[... | Let $z$ denote the [[Definition:Zenith Distance|zenith distance]] of $P$.
Let $\psi$ denote the [[Definition:Terrestrial Colatitude|(terrestrial) colatitude]] of $O$.
By definition we have:
:$a = 90 \degrees - z$
:$\phi - 90 \degrees - \psi$
Then:
{{begin-eqn}}
{{eqn | l = z
| r = \psi
| c = [[Zenith D... | Altitude of North Celestial Pole equals Latitude of Observer | https://proofwiki.org/wiki/Altitude_of_North_Celestial_Pole_equals_Latitude_of_Observer | https://proofwiki.org/wiki/Altitude_of_North_Celestial_Pole_equals_Latitude_of_Observer | [
"Spherical Astronomy"
] | [
"Definition:Celestial Sphere/Observer",
"Definition:Celestial Sphere",
"Definition:North Celestial Pole",
"Definition:Celestial Altitude",
"Definition:Latitude/Terrestrial"
] | [
"Definition:Zenith Distance",
"Definition:Colatitude/Terrestrial",
"Zenith Distance of North Celestial Pole equals Colatitude of Observer"
] |
proofwiki-16517 | Celestial Equator is Parallel to Geographical Equator | Consider the celestial sphere with observer $O$.
The plane of the celestial equator is parallel to the plane of the geographical equator. | Let $P$ denote the north celestial pole.
The celestial equator is by definition perpendicular to $OP$.
Also by definition, $OP$ is parallel to Earth's axis.
Also by definition, Earth's axis is perpendicular to the geographical equator.
Hence $OP$ is also perpendicular to the geographical equator.
The result follows fro... | Consider the [[Definition:Celestial Sphere|celestial sphere]] with [[Definition:Celestial Observer|observer]] $O$.
The [[Definition:Plane|plane]] of the [[Definition:Celestial Equator|celestial equator]] is [[Definition:Parallel Planes|parallel]] to the [[Definition:Plane|plane]] of the [[Definition:Geographical Equa... | Let $P$ denote the [[Definition:North Celestial Pole|north celestial pole]].
The [[Definition:Celestial Equator|celestial equator]] is by definition [[Definition:Line Perpendicular to Plane|perpendicular]] to $OP$.
Also by definition, $OP$ is [[Definition:Parallel Lines|parallel]] to [[Definition:Earth's Axis|Earth's... | Celestial Equator is Parallel to Geographical Equator | https://proofwiki.org/wiki/Celestial_Equator_is_Parallel_to_Geographical_Equator | https://proofwiki.org/wiki/Celestial_Equator_is_Parallel_to_Geographical_Equator | [
"Spherical Astronomy"
] | [
"Definition:Celestial Sphere",
"Definition:Celestial Sphere/Observer",
"Definition:Plane Surface",
"Definition:Celestial Equator",
"Definition:Parallel (Geometry)/Planes",
"Definition:Plane Surface",
"Definition:Geographical Equator"
] | [
"Definition:North Celestial Pole",
"Definition:Celestial Equator",
"Definition:Right Angle/Perpendicular/Plane",
"Definition:Parallel (Geometry)/Lines",
"Definition:Earth's Axis",
"Definition:Earth's Axis",
"Definition:Right Angle/Perpendicular/Plane",
"Definition:Geographical Equator",
"Definition:... |
proofwiki-16518 | Celestial Body moves along Parallel of Declination | Consider the celestial sphere $C$ whose observer is $O$.
Let $B$ be a celestial body on $C$.
Let $\delta$ be the declination of $B$.
Let $P_\delta$ be the parallel of declination whose declination is $\delta$.
Then $B$ appears to move along the path of $P_\delta$, in the direction from the north to east to south to wes... | The apparent movement of $C$ is caused by the rotation of Earth about its axis.
This rotation takes $1$ sidereal day, by definition, for the principal vertical circle to be in the same position again.
As the Earth rotates, the observer $O$ moves in an easterly direction.
Hence the celestial sphere $C$ appears to move i... | Consider the [[Definition:Celestial Sphere|celestial sphere]] $C$ whose [[Definition:Celestial Observer|observer]] is $O$.
Let $B$ be a [[Definition:Celestial Body|celestial body]] on $C$.
Let $\delta$ be the [[Definition:Declination|declination]] of $B$.
Let $P_\delta$ be the [[Definition:Parallel of Declination|pa... | The apparent movement of $C$ is caused by the [[Definition:Space Rotation|rotation]] of [[Definition:Earth|Earth]] about its [[Definition:Axis|axis]].
This rotation takes $1$ [[Definition:Sidereal Day|sidereal day]], by definition, for the [[Definition:Principal Vertical Circle|principal vertical circle]] to be in the... | Celestial Body moves along Parallel of Declination | https://proofwiki.org/wiki/Celestial_Body_moves_along_Parallel_of_Declination | https://proofwiki.org/wiki/Celestial_Body_moves_along_Parallel_of_Declination | [
"Spherical Astronomy"
] | [
"Definition:Celestial Sphere",
"Definition:Celestial Sphere/Observer",
"Definition:Celestial Body",
"Definition:Declination",
"Definition:Parallel of Declination",
"Definition:Declination",
"Definition:North Point of Horizon",
"Definition:East Point of Horizon",
"Definition:South Point of Horizon",
... | [
"Definition:Rotation (Geometry)/Space",
"Definition:Earth",
"Definition:Axis",
"Definition:Sidereal Day",
"Definition:Principal Vertical Circle",
"Definition:Earth",
"Definition:Celestial Sphere/Observer",
"Definition:East",
"Definition:Celestial Sphere",
"Definition:North Point of Horizon",
"De... |
proofwiki-16519 | Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
For all $\epsilon \in \R_{>0}$, let $\map { {B_\epsilon}^-} a$ denote the closed $\epsilon$-ball of $a$.
Let $n, m \in Z$, such that $n < m$.
Then:
:$(1) \quad \map {B^-_{p^{-n} } } a = \ds \bigcup_{i \matho... | === Condition $(1)$ ===
{{:Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Union of Closed Balls}}{{qed|lemma}} | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a \in \Q_p$.
For all $\epsilon \in \R_{>0}$, let $\map { {B_\epsilon}^-} a$ denote the [[Definition:Closed Ball in P-adic Numbers|closed $\epsil... | === [[Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Union of Closed Balls|Condition $(1)$]] ===
{{:Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Union of Closed Balls}}{{qed|lemma}} | Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers | https://proofwiki.org/wiki/Closed_Ball_is_Disjoint_Union_of_Smaller_Closed_Balls_in_P-adic_Numbers | https://proofwiki.org/wiki/Closed_Ball_is_Disjoint_Union_of_Smaller_Closed_Balls_in_P-adic_Numbers | [
"Topology of P-adic Numbers",
"Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Closed Ball/P-adic Numbers",
"Definition:Set",
"Definition:Pairwise Disjoint",
"Definition:Closed Ball/P-adic Numbers"
] | [
"Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Union of Closed Balls"
] |
proofwiki-16520 | Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Union of Closed Balls | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
For all $\epsilon \in \R_{>0}$, let $\map { {B_\epsilon}^-} a$ denote the closed $\epsilon$-ball of $a$.
Let $n, m \in Z$, such that $n < m$.
Then:
:$\map {B^-_{p^{-n} } } a = \ds \bigcup_{i \mathop = 0}^{p^... | ==== Lemma ====
{{:Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1}}{{qed|lemma}}
Let $0 \le i \le p^{\paren{m - n}} - 1$.
Let $x \in \map {B^{\,-}_{p^{-m} } } {a + i p^{-n} }$
By definition of a closed ball:
:$\norm {x - a - i p^{-n} } \le p^{-m}$
From Lemma:
:$\norm {x - a}_p \le p^{-n... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a \in \Q_p$.
For all $\epsilon \in \R_{>0}$, let $\map { {B_\epsilon}^-} a$ denote the [[Definition:Closed Ball in P-adic Numbers|closed $\epsil... | ==== [[Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1|Lemma]] ====
{{:Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1}}{{qed|lemma}}
Let $0 \le i \le p^{\paren{m - n}} - 1$.
Let $x \in \map {B^{\,-}_{p^{-m} } } {a + i p^{-n} }$
By definition of a [[De... | Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Union of Closed Balls | https://proofwiki.org/wiki/Closed_Ball_is_Disjoint_Union_of_Smaller_Closed_Balls_in_P-adic_Numbers/Union_of_Closed_Balls | https://proofwiki.org/wiki/Closed_Ball_is_Disjoint_Union_of_Smaller_Closed_Balls_in_P-adic_Numbers/Union_of_Closed_Balls | [
"Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Closed Ball/P-adic Numbers"
] | [
"Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1",
"Definition:Closed Ball/P-adic Numbers",
"Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1",
"Definition:Closed Ball/P-adic Numbers",
"Definition:Closed Ball/P-adic Numbers",
"Closed Ball is D... |
proofwiki-16521 | Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Disjoint Closed Balls | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
For all $\epsilon \in \R_{>0}$, let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$.
Then:
:$\forall n \in Z : \set{\map {B^-_{p^{-m} } } {a + i p^n} : i = 0, \dotsc, p^{\paren {m - n}} - 1}$ is... | Let $0 \le i, j \le p^{\paren {m - n}} - 1$.
Let $x \in \map {B^-_{p^{-m} } } {a + i p^n} \cap \map {B^-_{p^{-m} } } {a + j p^n}$
From Characterization of Open Ball in $p$-adic Numbers:
:$\norm {\paren {x -a} - i p^n}_p \le p^{-m}$
and:
:$\norm {\paren {x -a} - j p^n}_p \le p^{-m}$
We have that $p$-adic Norm satisfies ... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a \in \Q_p$.
For all $\epsilon \in \R_{>0}$, let $\map {B_\epsilon} a$ denote the [[Definition:Open Ball in P-adic Numbers|open $\epsilon$-ball]... | Let $0 \le i, j \le p^{\paren {m - n}} - 1$.
Let $x \in \map {B^-_{p^{-m} } } {a + i p^n} \cap \map {B^-_{p^{-m} } } {a + j p^n}$
From [[Characterization of Open Ball in P-adic Numbers|Characterization of Open Ball in $p$-adic Numbers]]:
:$\norm {\paren {x -a} - i p^n}_p \le p^{-m}$
and:
:$\norm {\paren {x -a} - j p^... | Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Disjoint Closed Balls | https://proofwiki.org/wiki/Closed_Ball_is_Disjoint_Union_of_Smaller_Closed_Balls_in_P-adic_Numbers/Disjoint_Closed_Balls | https://proofwiki.org/wiki/Closed_Ball_is_Disjoint_Union_of_Smaller_Closed_Balls_in_P-adic_Numbers/Disjoint_Closed_Balls | [
"Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Open Ball/P-adic Numbers",
"Definition:Set",
"Definition:Pairwise Disjoint",
"Definition:Open Ball/Normed Division Ring"
] | [
"Characterization of Open Ball in P-adic Numbers",
"P-adic Norm satisfies Non-Archimedean Norm Axioms",
"Integer is Congruent to Integer less than Modulus"
] |
proofwiki-16522 | Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1 | :$\forall y \in \Q_p: \norm y_p \le p^{-n}$ {{iff}} there exists $i \in \Z$ such that:
::$(1)\quad 0 \le i \le p^{\paren {m - n}} - 1$
::$(2)\quad \norm {y - i p^n}_p \le p^{-m}$ | === Necessary Condition ===
Let $y \in \Q_p$.
Let $\norm y_p \le p^{-n}$.
{{:Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1/Necessary Condition}}{{qed|lemma}} | :$\forall y \in \Q_p: \norm y_p \le p^{-n}$ {{iff}} there exists $i \in \Z$ such that:
::$(1)\quad 0 \le i \le p^{\paren {m - n}} - 1$
::$(2)\quad \norm {y - i p^n}_p \le p^{-m}$ | === [[Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1/Necessary Condition|Necessary Condition]] ===
Let $y \in \Q_p$.
Let $\norm y_p \le p^{-n}$.
{{:Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1/Necessary Condition}}{{qed|lemma}} | Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1 | https://proofwiki.org/wiki/Closed_Ball_is_Disjoint_Union_of_Smaller_Closed_Balls_in_P-adic_Numbers/Lemma_1 | https://proofwiki.org/wiki/Closed_Ball_is_Disjoint_Union_of_Smaller_Closed_Balls_in_P-adic_Numbers/Lemma_1 | [
"Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers"
] | [] | [
"Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1/Necessary Condition"
] |
proofwiki-16523 | Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1/Necessary Condition | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $n, m \in Z$, such that $n < m$.
Let $y \in \Q_p$.
Let $\norm{y}_p \le p^{-n}$.
Then there exists $i \in \Z$ such that:
{{begin-itemize}}
{{item|(1):|$0 \le i \le p^\paren {m - n} - 1$}}
{{item|(2):|$\norm {y - i p^n}_p \l... | We have that $p$-adic Norm satisfies Non-Archimedean Norm Axioms.
Hence:
{{begin-eqn}}
{{eqn | l = \norm y_p
| o = \le
| r = p^{-n}
}}
{{eqn | ll= \leadsto
| l = p^n \norm{y}_p
| o = \le
| r = 1
| c = multiplying both sides by $p^n$
}}
{{eqn | ll= \leadsto
| l = \norm{p^{-n}... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $n, m \in Z$, such that $n < m$.
Let $y \in \Q_p$.
Let $\norm{y}_p \le p^{-n}$.
Then there exists $i \in \Z$ such that:
{{begin-itemize}}
{{it... | We have that [[P-adic Norm satisfies Non-Archimedean Norm Axioms|$p$-adic Norm satisfies Non-Archimedean Norm Axioms]].
Hence:
{{begin-eqn}}
{{eqn | l = \norm y_p
| o = \le
| r = p^{-n}
}}
{{eqn | ll= \leadsto
| l = p^n \norm{y}_p
| o = \le
| r = 1
| c = multiplying both sides b... | Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1/Necessary Condition | https://proofwiki.org/wiki/Closed_Ball_is_Disjoint_Union_of_Smaller_Closed_Balls_in_P-adic_Numbers/Lemma_1/Necessary_Condition | https://proofwiki.org/wiki/Closed_Ball_is_Disjoint_Union_of_Smaller_Closed_Balls_in_P-adic_Numbers/Lemma_1/Necessary_Condition | [
"Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers"
] | [
"P-adic Norm satisfies Non-Archimedean Norm Axioms",
"Characterization of Closed Ball in P-adic Numbers",
"Integers are Dense in P-adic Integers/Unit Ball",
"Residue Classes form Partition of Integers",
"Definition:P-adic Norm"
] |
proofwiki-16524 | Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1/Sufficient Condition | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $y \in \Q_p$
Let $n, m \in Z$, such that $n < m$.
Let there exist $i \in \Z$:
:$(1): \quad 0 \le i \le p^\paren {m - n} - 1$
:$(2): \quad \norm {y - i p^n}_p \le p^{-m}$
Then:
:$\norm y_p \le p^{-n}$ | We have that P-adic Norm satisfies Non-Archimedean Norm Axioms:.
Hence:
{{begin-eqn}}
{{eqn | l = \norm y_p
| r = \norm {y - i p^n + i p^n}_p
}}
{{eqn | o = \le
| r = \max \set {\norm {y - i p^n}_p, \norm {i p^n}_p}
| c = {{NormAxiomNonArch|4}}
}}
{{end-eqn}}
By assumption:
:$\norm {y - i p^n} \le p^{... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $y \in \Q_p$
Let $n, m \in Z$, such that $n < m$.
Let there exist $i \in \Z$:
:$(1): \quad 0 \le i \le p^\paren {m - n} - 1$
:$(2): \quad \norm ... | We have that [[P-adic Norm satisfies Non-Archimedean Norm Axioms]]:.
Hence:
{{begin-eqn}}
{{eqn | l = \norm y_p
| r = \norm {y - i p^n + i p^n}_p
}}
{{eqn | o = \le
| r = \max \set {\norm {y - i p^n}_p, \norm {i p^n}_p}
| c = {{NormAxiomNonArch|4}}
}}
{{end-eqn}}
By assumption:
:$\norm {y - i p^n} ... | Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1/Sufficient Condition | https://proofwiki.org/wiki/Closed_Ball_is_Disjoint_Union_of_Smaller_Closed_Balls_in_P-adic_Numbers/Lemma_1/Sufficient_Condition | https://proofwiki.org/wiki/Closed_Ball_is_Disjoint_Union_of_Smaller_Closed_Balls_in_P-adic_Numbers/Lemma_1/Sufficient_Condition | [
"Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers"
] | [
"P-adic Norm satisfies Non-Archimedean Norm Axioms"
] |
proofwiki-16525 | Open and Closed Balls in P-adic Numbers are Totally Bounded | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
Let $n \in \Z$.
Then the open ball $\map {B_{p^{-n} } } a$ and closed ball $\map {B^-_{p^{-n} } } a$ are totally bounded subspaces. | We begin by proving the theorem for the closed ball $\map {B^-_{p^{-n} } } a$.
From Open Ball in P-adic Numbers is Closed Ball then the theorem will be proved.
Let $d$ denote the subspace metric induced by the norm $\norm {\,\cdot\,}_p$ on $\map {B^-_{p^{-n} } } a$.
That is, $d: \map {B^-_{p^{-n} } } a \times \map {B^-... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a \in \Q_p$.
Let $n \in \Z$.
Then the [[Definition:Open Ball in P-adic Numbers|open ball]] $\map {B_{p^{-n} } } a$ and [[Definition:Closed Bal... | We begin by proving the [[Definition:Theorem|theorem]] for the [[Definition:Closed Ball in P-adic Numbers|closed ball]] $\map {B^-_{p^{-n} } } a$.
From [[Open Ball in P-adic Numbers is Closed Ball]] then the [[Definition:Theorem|theorem]] will be proved.
Let $d$ denote the [[Definition:Metric Subspace|subspace metri... | Open and Closed Balls in P-adic Numbers are Totally Bounded | https://proofwiki.org/wiki/Open_and_Closed_Balls_in_P-adic_Numbers_are_Totally_Bounded | https://proofwiki.org/wiki/Open_and_Closed_Balls_in_P-adic_Numbers_are_Totally_Bounded | [
"Topology of P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Open Ball/P-adic Numbers",
"Definition:Closed Ball/P-adic Numbers",
"Definition:Totally Bounded Metric Space",
"Definition:Subspace"
] | [
"Definition:Theorem",
"Definition:Closed Ball/P-adic Numbers",
"Open Ball in P-adic Numbers is Closed Ball",
"Definition:Theorem",
"Definition:Metric Subspace",
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Norm/Division Ring",
"Definition:Metric Space",
"Definition:Totally Bound... |
proofwiki-16526 | Cofactor Sum Identity | Let $J_n$ be the $n \times n$ matrix of all ones.
Let $A$ be a square matrix of order $n$.
Let $A_{ij}$ denote the cofactor of element $\tuple {i, j}$ in $\map \det A$, $1 \le i, j \le n$.
Then:
:$\ds \map \det {A - J_n} = \map \det A - \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n A_{i j} $ | Let $P_j$ equal matrix $A$ with column $j$ replaced by ones, $1\le j \le n$.
Then by the Laplace Expansion Theorem for Determinants applied to column $j$ of $P_j$:
:$\ds \sum_{j \mathop = 1}^n \map \det {P_j} = \sum_{j \mathop = 1}^n \sum_{i \mathop = 1}^n A_{ij}$
To complete the proof, it suffices to prove the equival... | Let $J_n$ be the $n \times n$ [[Definition:Ones Matrix|matrix of all ones]].
Let $A$ be a [[Definition:Square Matrix|square matrix of order $n$]].
Let $A_{ij}$ denote the [[Definition:Cofactor|cofactor]] of element $\tuple {i, j}$ in $\map \det A$, $1 \le i, j \le n$.
Then:
:$\ds \map \det {A - J_n} = \map \det A ... | Let $P_j$ equal [[Definition:Matrix|matrix]] $A$ with column $j$ replaced by ones, $1\le j \le n$.
Then by the [[Laplace Expansion Theorem for Determinants]] applied to column $j$ of $P_j$:
:$\ds \sum_{j \mathop = 1}^n \map \det {P_j} = \sum_{j \mathop = 1}^n \sum_{i \mathop = 1}^n A_{ij}$
To complete the proof, it s... | Cofactor Sum Identity | https://proofwiki.org/wiki/Cofactor_Sum_Identity | https://proofwiki.org/wiki/Cofactor_Sum_Identity | [
"Cofactors",
"Determinants"
] | [
"Definition:Ones Matrix",
"Definition:Matrix/Square Matrix",
"Definition:Cofactor"
] | [
"Definition:Matrix",
"Laplace Expansion Theorem for Determinants",
"Definition:Ones Matrix",
"Definition:Matrix Addition",
"Definition:Negative Matrix",
"Determinant as Sum of Determinants",
"Multiple of Row Added to Row of Determinant",
"Determinant as Sum of Determinants",
"Definition:Matrix/Squar... |
proofwiki-16527 | Sum of Elements of Nonsingular Matrix | Let $\mathbf J_n$ be the $n \times n$ square ones matrix.
Let $\mathbf B$ be an $n \times n$ nonsingular matrix with entries $b_{i j}$, $1 \le i, j \le n$.
Then:
:$\ds \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n b_{i j} = 1 - \map \det {\mathbf B} \map \det {\mathbf B^{-1} - \mathbf J_n}$ | === Lemma ===
Let$\mathbf J_n$ be the $n \times n$ square ones matrix.
Let $\mathbf A$ be an $n \times n$ matrix.
Let $\mathbf A_{ij}$ denote the cofactor of element $a_{ij}$ in $\map \det {\mathbf A}$, $1 \le i, j \le n$.
Then:
{{begin-eqn}}
{{eqn | l = \map \det {\mathbf A - \mathbf J_n}
| r = \map \det {\mathb... | Let $\mathbf J_n$ be the $n \times n$ [[Definition:Square Ones Matrix|square ones matrix]].
Let $\mathbf B$ be an $n \times n$ [[Definition:Nonsingular Matrix|nonsingular matrix]] with entries $b_{i j}$, $1 \le i, j \le n$.
Then:
:$\ds \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n b_{i j} = 1 - \map \det {\mathbf B... | === Lemma ===
Let$\mathbf J_n$ be the $n \times n$ [[Definition:Square Ones Matrix|square ones matrix]].
Let $\mathbf A$ be an $n \times n$ matrix.
Let $\mathbf A_{ij}$ denote the [[Definition:Cofactor|cofactor]] of element $a_{ij}$ in $\map \det {\mathbf A}$, $1 \le i, j \le n$.
Then:
{{begin-eqn}}
{{eqn | l = \m... | Sum of Elements of Nonsingular Matrix | https://proofwiki.org/wiki/Sum_of_Elements_of_Nonsingular_Matrix | https://proofwiki.org/wiki/Sum_of_Elements_of_Nonsingular_Matrix | [
"Nonsingular Matrices"
] | [
"Definition:Ones Matrix/Square",
"Definition:Nonsingular Matrix"
] | [
"Definition:Ones Matrix/Square",
"Definition:Cofactor",
"Cofactor Sum Identity",
"Matrix Product with Adjugate Matrix",
"Determinant of Matrix Product",
"Determinant of Matrix Product",
"Category:Nonsingular Matrices"
] |
proofwiki-16528 | Open and Closed Balls in P-adic Numbers are Clopen in P-adic Metric | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
Let $n \in \Z$.
Then the open ball $\map {B_{p^{-n}}} a$ and closed ball $\map {B^-_{p^{-n}}} a$ are clopen in the $p$-adic metric. | We begin by proving the theorem for the closed ball $\map {B^-_{p^{-n}}} a$.
From Open Ball in P-adic Numbers is Closed Ball then the theorem will be proved.
From P-adic Numbers form Non-Archimedean Valued Field::
:$\norm {\,\cdot\,}_p$ is a non-Archimedean division ring norm.
By definition the $p$-adic closed ball:
:$... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a \in \Q_p$.
Let $n \in \Z$.
Then the [[Definition:Open Ball in P-adic Numbers|open ball]] $\map {B_{p^{-n}}} a$ and [[Definition:Closed Ball ... | We begin by proving the [[Definition:Theorem|theorem]] for the [[Definition:Closed Ball in P-adic Numbers|closed ball]] $\map {B^-_{p^{-n}}} a$.
From [[Open Ball in P-adic Numbers is Closed Ball]] then the [[Definition:Theorem|theorem]] will be proved.
From [[P-adic Numbers form Non-Archimedean Valued Field]]::
:$\n... | Open and Closed Balls in P-adic Numbers are Clopen in P-adic Metric | https://proofwiki.org/wiki/Open_and_Closed_Balls_in_P-adic_Numbers_are_Clopen_in_P-adic_Metric | https://proofwiki.org/wiki/Open_and_Closed_Balls_in_P-adic_Numbers_are_Clopen_in_P-adic_Metric | [
"Topology of P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Open Ball/P-adic Numbers",
"Definition:Closed Ball/P-adic Numbers",
"Definition:Clopen Set",
"Definition:P-adic Metric"
] | [
"Definition:Theorem",
"Definition:Closed Ball/P-adic Numbers",
"Open Ball in P-adic Numbers is Closed Ball",
"Definition:Theorem",
"P-adic Norm forms Non-Archimedean Valued Field/P-adic Numbers",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Closed Ball/P-adic Numbers",
"Definition:Cl... |
proofwiki-16529 | Closed Balls Centered on P-adic Number is Countable | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
Then the set of all closed balls centered on $a$ is the countable set:
:$\BB^- = \set {\map {B^-_{p^{-n} } } a: n \in \Z}$ | Let $\epsilon \in \R_{>0}$. | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a \in \Q_p$.
Then the [[Definition:Set|set]] of all [[Definition:Closed Ball in P-adic Numbers|closed balls]] [[Definition:Center of Closed Bal... | Let $\epsilon \in \R_{>0}$. | Closed Balls Centered on P-adic Number is Countable | https://proofwiki.org/wiki/Closed_Balls_Centered_on_P-adic_Number_is_Countable | https://proofwiki.org/wiki/Closed_Balls_Centered_on_P-adic_Number_is_Countable | [
"Topology of P-adic Numbers",
"Closed Balls Centered on P-adic Number is Countable"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Set",
"Definition:Closed Ball/P-adic Numbers",
"Definition:Closed Ball/P-adic Numbers/Center",
"Definition:Countable Set"
] | [] |
proofwiki-16530 | Closed Balls Centered on P-adic Number is Countable/Open Balls | Then the set of all open balls centered on $a$ is the countable set:
:$\BB = \set {\map {B_{p^{-n} } } a : n \in \Z}$ | Let $\epsilon \in \R_{\ge 0}$. | Then the [[Definition:Set|set]] of all [[Definition:Open Ball in P-adic Numbers|open balls]] [[Definition:Center of Open Ball in P-adic Numbers|centered]] on $a$ is the [[Definition:Countable|countable set]]:
:$\BB = \set {\map {B_{p^{-n} } } a : n \in \Z}$ | Let $\epsilon \in \R_{\ge 0}$. | Closed Balls Centered on P-adic Number is Countable/Open Balls | https://proofwiki.org/wiki/Closed_Balls_Centered_on_P-adic_Number_is_Countable/Open_Balls | https://proofwiki.org/wiki/Closed_Balls_Centered_on_P-adic_Number_is_Countable/Open_Balls | [
"Topology of P-adic Numbers",
"Closed Balls Centered on P-adic Number is Countable"
] | [
"Definition:Set",
"Definition:Open Ball/P-adic Numbers",
"Definition:Open Ball/P-adic Numbers/Center",
"Definition:Countable Set"
] | [] |
proofwiki-16531 | Closed Balls Centered on P-adic Number is Countable/Lemma | :$\exists n \in \Z : p^{-n} \le \epsilon < p^{-\paren {n - 1} }$ | From Sequence of Powers of Reciprocals is Null Sequence:
:$\exists n_1 \in \N: \forall k \ge n_1 : p^{-k} < \epsilon$
Similarly:
:$\exists n_2 \in \N: \forall k \ge n_2 : p^{-k} < \dfrac 1 \epsilon$
Hence:
:$p^{-n_1} < \epsilon$ and $ p^{-n_2} < \dfrac 1 \epsilon$
That is:
:$p^{-n_1} < \epsilon < p^{n_2}$
From Power F... | :$\exists n \in \Z : p^{-n} \le \epsilon < p^{-\paren {n - 1} }$ | From [[Sequence of Powers of Reciprocals is Null Sequence]]:
:$\exists n_1 \in \N: \forall k \ge n_1 : p^{-k} < \epsilon$
Similarly:
:$\exists n_2 \in \N: \forall k \ge n_2 : p^{-k} < \dfrac 1 \epsilon$
Hence:
:$p^{-n_1} < \epsilon$ and $ p^{-n_2} < \dfrac 1 \epsilon$
That is:
:$p^{-n_1} < \epsilon < p^{n_2}$
From... | Closed Balls Centered on P-adic Number is Countable/Lemma | https://proofwiki.org/wiki/Closed_Balls_Centered_on_P-adic_Number_is_Countable/Lemma | https://proofwiki.org/wiki/Closed_Balls_Centered_on_P-adic_Number_is_Countable/Lemma | [
"Closed Balls Centered on P-adic Number is Countable"
] | [] | [
"Sequence of Powers of Reciprocals is Null Sequence",
"Power Function on Base between Zero and One is Strictly Decreasing/Integer",
"Power Function on Base between Zero and One is Strictly Decreasing/Integer",
"Definition:Contradiction",
"Category:Closed Balls Centered on P-adic Number is Countable"
] |
proofwiki-16532 | Closed Balls Centered on P-adic Number is Countable/Open Balls/Lemma | :$\exists n \in \Z : p^{-\paren {n + 1} } < \epsilon \le p^{-n}$ | From Lemma for Closed Balls:
:$\exists m \in \Z : p^{-m} \le \epsilon < p^{-\paren {m - 1} }$
Suppose $\epsilon \ne p^{-m}$.
Then:
:$p^{-m} < \epsilon < p^{-\paren {m - 1} }$
and the theorem is proved with $n = m - 1$.
Now suppose $\epsilon = p^{-m}$.
From Power Function on Integer between Zero and One is Strictly Decr... | :$\exists n \in \Z : p^{-\paren {n + 1} } < \epsilon \le p^{-n}$ | From [[Closed Balls Centered on P-adic Number is Countable/Lemma|Lemma for Closed Balls]]:
:$\exists m \in \Z : p^{-m} \le \epsilon < p^{-\paren {m - 1} }$
Suppose $\epsilon \ne p^{-m}$.
Then:
:$p^{-m} < \epsilon < p^{-\paren {m - 1} }$
and the theorem is proved with $n = m - 1$.
Now suppose $\epsilon = p^{-m}$.
... | Closed Balls Centered on P-adic Number is Countable/Open Balls/Lemma | https://proofwiki.org/wiki/Closed_Balls_Centered_on_P-adic_Number_is_Countable/Open_Balls/Lemma | https://proofwiki.org/wiki/Closed_Balls_Centered_on_P-adic_Number_is_Countable/Open_Balls/Lemma | [
"Closed Balls Centered on P-adic Number is Countable"
] | [] | [
"Closed Balls Centered on P-adic Number is Countable/Lemma",
"Power Function on Base between Zero and One is Strictly Decreasing/Integer",
"Category:Closed Balls Centered on P-adic Number is Countable"
] |
proofwiki-16533 | Open Ball contains Smaller Open Ball | Let $M = \struct{A, d}$ be a metric space.
Let $a \in A$.
Let $\epsilon, \delta \in \R_{>0}$ such that $\epsilon \le \delta$.
Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball on $a$.
Let $\map {B_\delta} a$ be the open $\delta$-ball on $a$.
Then:
:$\map {B_\epsilon} a \subseteq \map {B_\delta} a$ | {{begin-eqn}}
{{eqn | l = x \in \map {B_\epsilon} a
| o = \leadstoandfrom
| r = \map d {x, a} < \epsilon
| c = {{Defof|Open Ball of Metric Space}}
}}
{{eqn | o = \leadsto
| r = \map d {x, a} < \delta
| c = As $\epsilon \le \delta$
}}
{{eqn | o = \leadstoandfrom
| r = x \in \map {B_\... | Let $M = \struct{A, d}$ be a [[Definition:Metric Space|metric space]].
Let $a \in A$.
Let $\epsilon, \delta \in \R_{>0}$ such that $\epsilon \le \delta$.
Let $\map {B_\epsilon} a$ be the [[Definition:Open Ball of Metric Space|open $\epsilon$-ball]] on $a$.
Let $\map {B_\delta} a$ be the [[Definition:Open Ball of Me... | {{begin-eqn}}
{{eqn | l = x \in \map {B_\epsilon} a
| o = \leadstoandfrom
| r = \map d {x, a} < \epsilon
| c = {{Defof|Open Ball of Metric Space}}
}}
{{eqn | o = \leadsto
| r = \map d {x, a} < \delta
| c = As $\epsilon \le \delta$
}}
{{eqn | o = \leadstoandfrom
| r = x \in \map {B_\... | Open Ball contains Smaller Open Ball | https://proofwiki.org/wiki/Open_Ball_contains_Smaller_Open_Ball | https://proofwiki.org/wiki/Open_Ball_contains_Smaller_Open_Ball | [
"Open Balls"
] | [
"Definition:Metric Space",
"Definition:Open Ball",
"Definition:Open Ball"
] | [
"Definition:Subset",
"Category:Open Balls"
] |
proofwiki-16534 | Closed Ball contains Smaller Closed Ball | Let $M = \struct {A, d}$ be a metric space.
Let $a \in A$.
Let $\epsilon, \delta \in \R_{> 0}$ such that $\epsilon \le \delta$.
Let $\map {B^-_\epsilon} a$ be the closed $\epsilon$-ball on $a$.
Let $\map {B^-_\delta} a$ be the closed $\delta$-ball on $a$.
Then:
:$\map {B^-_\epsilon} a \subseteq \map {B^-_\delta} a$ | {{begin-eqn}}
{{eqn | l = x \in \map {B^-_\epsilon} a
| o = \leadsto
| r = \map d {x, a} \le \epsilon
| c = {{Defof|Closed Ball}}
}}
{{eqn | o = \leadsto
| r = \map d {x, a} \le \delta
| c = As $\epsilon \le \delta$
}}
{{eqn | o = \leadsto
| r = x \in \map {B^-_\delta} a
| c ... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $a \in A$.
Let $\epsilon, \delta \in \R_{> 0}$ such that $\epsilon \le \delta$.
Let $\map {B^-_\epsilon} a$ be the [[Definition:Closed Ball|closed $\epsilon$-ball]] on $a$.
Let $\map {B^-_\delta} a$ be the [[Definition:Closed Ball|closed $\... | {{begin-eqn}}
{{eqn | l = x \in \map {B^-_\epsilon} a
| o = \leadsto
| r = \map d {x, a} \le \epsilon
| c = {{Defof|Closed Ball}}
}}
{{eqn | o = \leadsto
| r = \map d {x, a} \le \delta
| c = As $\epsilon \le \delta$
}}
{{eqn | o = \leadsto
| r = x \in \map {B^-_\delta} a
| c ... | Closed Ball contains Smaller Closed Ball | https://proofwiki.org/wiki/Closed_Ball_contains_Smaller_Closed_Ball | https://proofwiki.org/wiki/Closed_Ball_contains_Smaller_Closed_Ball | [
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Closed Ball",
"Definition:Closed Ball"
] | [
"Definition:Subset",
"Category:Metric Spaces"
] |
proofwiki-16535 | Open Ball contains Strictly Smaller Closed Ball | Let $M = \struct {A, d}$ be a metric space.
Let $a \in A$.
Let $\epsilon, \delta \in \R_{>0}$ such that $\epsilon < \delta$.
Let $\map {B^-_\epsilon} a$ be the closed $\epsilon$-ball on $a$.
Let $\map {B_\delta} a$ be the open $\delta$-ball on $a$.
Then:
:$\map {B^-_\epsilon} a \subseteq \map {B_\delta} a$ | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \map {B^-_\epsilon} a
| c =
}}
{{eqn | ll= \leadsto
| l = x
| o = \in
| r = \map \cl {\map {B_\epsilon} a}
| c = {{Defof|Closed Ball of Metric Space}}
}}
{{eqn | ll= \leadsto
| l = \map d {x, a}
| o = \le
| r = \e... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $a \in A$.
Let $\epsilon, \delta \in \R_{>0}$ such that $\epsilon < \delta$.
Let $\map {B^-_\epsilon} a$ be the [[Definition:Closed Ball of Metric Space|closed $\epsilon$-ball]] on $a$.
Let $\map {B_\delta} a$ be the [[Definition:Open Ball ... | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \map {B^-_\epsilon} a
| c =
}}
{{eqn | ll= \leadsto
| l = x
| o = \in
| r = \map \cl {\map {B_\epsilon} a}
| c = {{Defof|Closed Ball of Metric Space}}
}}
{{eqn | ll= \leadsto
| l = \map d {x, a}
| o = \le
| r = \e... | Open Ball contains Strictly Smaller Closed Ball | https://proofwiki.org/wiki/Open_Ball_contains_Strictly_Smaller_Closed_Ball | https://proofwiki.org/wiki/Open_Ball_contains_Strictly_Smaller_Closed_Ball | [
"Open Balls",
"Closed Balls"
] | [
"Definition:Metric Space",
"Definition:Closed Ball/Metric Space",
"Definition:Open Ball"
] | [
"Closure of Open Ball in Metric Space",
"Definition:Subset",
"Category:Open Balls",
"Category:Closed Balls"
] |
proofwiki-16536 | Closed Ball contains Smaller Open Ball | Let $M = \struct {A, d}$ be a metric space.
Let $a \in A$.
Let $\epsilon, \delta \in \R_{> 0}$ such that $\epsilon \le \delta$.
Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball on $a$.
Let $\map {B^-_\delta} a$ be the closed $\delta$-ball on $a$.
Then:
:$\map {B_\epsilon} a \subseteq \map {B^-_\delta} a$ | {{begin-eqn}}
{{eqn | l = x \in \map {B_\epsilon} a
| o = \leadsto
| r = \map d {x, a} < \epsilon
| c = {{Defof|Open Ball of Metric Space}}
}}
{{eqn | o = \leadsto
| r = \map d {x, a} < \delta
| c = As $\epsilon \le \delta$
}}
{{eqn | o = \leadsto
| r = \map d {x, a} \le \delta
... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $a \in A$.
Let $\epsilon, \delta \in \R_{> 0}$ such that $\epsilon \le \delta$.
Let $\map {B_\epsilon} a$ be the [[Definition:Open Ball of Metric Space|open $\epsilon$-ball]] on $a$.
Let $\map {B^-_\delta} a$ be the [[Definition:Closed Ball... | {{begin-eqn}}
{{eqn | l = x \in \map {B_\epsilon} a
| o = \leadsto
| r = \map d {x, a} < \epsilon
| c = {{Defof|Open Ball of Metric Space}}
}}
{{eqn | o = \leadsto
| r = \map d {x, a} < \delta
| c = As $\epsilon \le \delta$
}}
{{eqn | o = \leadsto
| r = \map d {x, a} \le \delta
... | Closed Ball contains Smaller Open Ball | https://proofwiki.org/wiki/Closed_Ball_contains_Smaller_Open_Ball | https://proofwiki.org/wiki/Closed_Ball_contains_Smaller_Open_Ball | [
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Open Ball",
"Definition:Closed Ball/Metric Space"
] | [
"Definition:Subset",
"Category:Metric Spaces"
] |
proofwiki-16537 | Sample Matrix Independence Test | Let $V$ be a vector space of real or complex-valued functions on a set $J$.
Let $f_1, \ldots, f_n$ be functions in $V$.
Let $x_1, \ldots, x_n$ from $J$ be given.
Define the '''sample matrix''':
:<nowiki>$S = \begin{bmatrix}
\map {f_1} {x_1} & \cdots & \map {f_n} {x_1} \\
\vdots & \ddots & \vdots \\
\map {f_1}... | The definition of linear independence is applied.
Assume a linear combination of the functions $f_1, \ldots, f_n$ is the zero function:
{{begin-eqn}}
{{eqn | n = 1
| l = \sum_{i \mathop = 1}^n c_i \map {f_i} x
| r = 0
| c = for all $x$
}}
{{end-eqn}}
Let $\vec c$ have components $c_1, \ldots, c_n$.
F... | Let $V$ be a [[Definition:Vector Space|vector space]] of real or complex-valued functions on a [[Definition:Set|set]] $J$.
Let $f_1, \ldots, f_n$ be functions in $V$.
Let $x_1, \ldots, x_n$ from $J$ be [[Definition:Given|given]].
Define the '''[[Definition:Sample MJatrix|sample matrix]]''':
:<nowiki>$S = \begin{bma... | The definition of [[Definition:Linearly Independent|linear independence]] is applied.
Assume a [[Definition:Linear Combination|linear combination]] of the functions $f_1, \ldots, f_n$ is the [[Definition:Zero Function|zero function]]:
{{begin-eqn}}
{{eqn | n = 1
| l = \sum_{i \mathop = 1}^n c_i \map {f_i} x
... | Sample Matrix Independence Test | https://proofwiki.org/wiki/Sample_Matrix_Independence_Test | https://proofwiki.org/wiki/Sample_Matrix_Independence_Test | [
"Sample Matrix Independence Test",
"Sample Matrices",
"Linear Independence"
] | [
"Definition:Vector Space",
"Definition:Set",
"Definition:Given",
"Definition:Sample MJatrix",
"Definition:Nonsingular Matrix",
"Definition:Linearly Independent"
] | [
"Definition:Linearly Independent",
"Definition:Linear Combination",
"Definition:Basic Primitive Recursive Function/Zero Function",
"Definition:Homogeneous Simultaneous Linear Equations",
"Definition:Nonsingular Matrix",
"Definition:Linearly Independent"
] |
proofwiki-16538 | Open and Closed Balls in P-adic Numbers are Compact Subspaces/P-adic Integers | Then the set of $p$-adic integers $\Z_p$ is compact. | By definition the $p$-adic integers $\Z_p$ is the closed ball $\map {B^-_1} 0$.
From Open and Closed Balls in P-adic Numbers are Compact Subspaces, $\map {B^-_1} 0$ is compact.
{{qed}} | Then the [[Definition:Set|set]] of [[Definition:P-adic Integer|$p$-adic integers]] $\Z_p$ is [[Definition:Compact Metric Space|compact]]. | By definition the [[Definition:P-adic Integer|$p$-adic integers]] $\Z_p$ is the [[Definition:Closed Ball in P-adic Numbers|closed ball]] $\map {B^-_1} 0$.
From [[Open and Closed Balls in P-adic Numbers are Compact Subspaces]], $\map {B^-_1} 0$ is [[Definition:Compact Metric Space|compact]].
{{qed}} | Open and Closed Balls in P-adic Numbers are Compact Subspaces/P-adic Integers | https://proofwiki.org/wiki/Open_and_Closed_Balls_in_P-adic_Numbers_are_Compact_Subspaces/P-adic_Integers | https://proofwiki.org/wiki/Open_and_Closed_Balls_in_P-adic_Numbers_are_Compact_Subspaces/P-adic_Integers | [
"Topology of P-adic Numbers"
] | [
"Definition:Set",
"Definition:P-adic Integer",
"Definition:Compact Space/Metric Space"
] | [
"Definition:P-adic Integer",
"Definition:Closed Ball/P-adic Numbers",
"Open and Closed Balls in P-adic Numbers are Compact Subspaces",
"Definition:Compact Space/Metric Space"
] |
proofwiki-16539 | Open Balls form Local Basis for Point of Metric Space | Let $M = \struct {A, d}$ be a metric space.
Let $x \in A$.
Let $\BB_x$ be the set of all open balls of $M$ centered on $x$.
That is:
:$\BB_x = \set {\map {B_\epsilon} x : \epsilon \in \R_{>0}}$
Then $\BB$ is a local basis of $x$. | Let $U$ be an open set of $M$ which has $x$ as an element.
Then by definition of an open set:
:$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} x \subseteq U$
From Open Ball of Metric Space is Open Set, $\BB_x$ is a set of open set which have $x$ as an element.
By definition of a local basis, $\BB_x$ is a local basis o... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $x \in A$.
Let $\BB_x$ be the [[Definition:Set|set]] of all [[Definition:Open Ball of Metric Space|open balls]] of $M$ [[Definition:Center of Open Ball|centered]] on $x$.
That is:
:$\BB_x = \set {\map {B_\epsilon} x : \epsilon \in \R_{>0}}$
... | Let $U$ be an [[Definition:Open Set (Metric Space)|open set]] of $M$ which has $x$ as an [[Definition:Element|element]].
Then by definition of an [[Definition:Open Set (Metric Space)|open set]]:
:$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} x \subseteq U$
From [[Open Ball of Metric Space is Open Set]], $\BB_x$ is... | Open Balls form Local Basis for Point of Metric Space | https://proofwiki.org/wiki/Open_Balls_form_Local_Basis_for_Point_of_Metric_Space | https://proofwiki.org/wiki/Open_Balls_form_Local_Basis_for_Point_of_Metric_Space | [
"Open Balls"
] | [
"Definition:Metric Space",
"Definition:Set",
"Definition:Open Ball",
"Definition:Open Ball/Center",
"Definition:Local Basis"
] | [
"Definition:Open Set/Metric Space",
"Definition:Element",
"Definition:Open Set/Metric Space",
"Open Ball is Open Set/Pseudometric Space",
"Definition:Set",
"Definition:Open Set/Metric Space",
"Definition:Element",
"Definition:Local Basis",
"Definition:Local Basis",
"Category:Open Balls"
] |
proofwiki-16540 | Local Basis of P-adic Number/Cosets | Let $\Z_p$ be the $p$-adic integers.
Then the set $\set {a + p^n \Z_p: n \in Z}$ is a local basis of $a$ consisting of clopen sets. | From Local Basis of P-adic Number the set $\set { \map {B_{p^{-n} } } a : n \in \Z}$ is a local basis of clopen sets.
From Open Balls of P-adic Number:
:$\set {\map {B_{p^{-n} } } a : n \in \Z} = \set {a + p^{n + 1} \Z_p : n \in \Z} = \set {a + p^n \Z_p : n \in \Z}$
The result follows.
{{qed}} | Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]].
Then the [[Definition:Set|set]] $\set {a + p^n \Z_p: n \in Z}$ is a [[Definition:Local Basis|local basis]] of $a$ consisting of [[Definition:Clopen Set|clopen sets]]. | From [[Local Basis of P-adic Number]] the [[Definition:Set|set]] $\set { \map {B_{p^{-n} } } a : n \in \Z}$ is a [[Definition:Local Basis|local basis]] of [[Definition:Clopen Set|clopen sets]].
From [[Open Balls of P-adic Number]]:
:$\set {\map {B_{p^{-n} } } a : n \in \Z} = \set {a + p^{n + 1} \Z_p : n \in \Z} = \set... | Local Basis of P-adic Number/Cosets | https://proofwiki.org/wiki/Local_Basis_of_P-adic_Number/Cosets | https://proofwiki.org/wiki/Local_Basis_of_P-adic_Number/Cosets | [
"Topology of P-adic Numbers"
] | [
"Definition:P-adic Integer",
"Definition:Set",
"Definition:Local Basis",
"Definition:Clopen Set"
] | [
"Local Basis of P-adic Number",
"Definition:Set",
"Definition:Local Basis",
"Definition:Clopen Set",
"Open Balls of P-adic Number"
] |
proofwiki-16541 | Local Basis of P-adic Number/Closed Balls | Then the set of closed balls $\set {\map {B^-_{p^{-n} } } a: n \in Z}$ is a local basis of $a$ consisting of clopen sets. | From Local Basis of P-adic Number the set $\set {\map {B_{p^{-n } } } a: n \in \Z}$ is a local basis of clopen sets.
From Open Ball in P-adic Numbers is Closed Ball:
:$\set {\map {B_{p^{-n} } } a: n \in \Z} = \set {\map {B^-_{p^{-\paren{n + 1} } } } a: n \in \Z} = \set {\map {B^-_{p^{-n} } } a : n \in \Z}$
The result f... | Then the [[Definition:Set|set]] of [[Definition:Closed Ball in P-adic Numbers|closed balls]] $\set {\map {B^-_{p^{-n} } } a: n \in Z}$ is a [[Definition:Local Basis|local basis]] of $a$ consisting of [[Definition:Clopen Set|clopen sets]]. | From [[Local Basis of P-adic Number]] the [[Definition:Set|set]] $\set {\map {B_{p^{-n } } } a: n \in \Z}$ is a [[Definition:Local Basis|local basis]] of [[Definition:Clopen Set|clopen sets]].
From [[Open Ball in P-adic Numbers is Closed Ball]]:
:$\set {\map {B_{p^{-n} } } a: n \in \Z} = \set {\map {B^-_{p^{-\paren{n ... | Local Basis of P-adic Number/Closed Balls | https://proofwiki.org/wiki/Local_Basis_of_P-adic_Number/Closed_Balls | https://proofwiki.org/wiki/Local_Basis_of_P-adic_Number/Closed_Balls | [
"Topology of P-adic Numbers"
] | [
"Definition:Set",
"Definition:Closed Ball/P-adic Numbers",
"Definition:Local Basis",
"Definition:Clopen Set"
] | [
"Local Basis of P-adic Number",
"Definition:Set",
"Definition:Local Basis",
"Definition:Clopen Set",
"Open Ball in P-adic Numbers is Closed Ball",
"Category:Topology of P-adic Numbers"
] |
proofwiki-16542 | Measurement of Terrestrial Latitude/Pole Star | An observer's (terrestrial) latitude can be identified by measuring the (celestial) altitude of the Pole Star. | From Altitude of North Celestial Pole equals Latitude of Observer, all the observer needs to do is to measure the (celestial) altitude of the north celestial pole.
The star Polaris is situated at practically exactly that point.
Hence the result.
{{qed}}
Category:Measurement of Terrestrial Latitude
9houihdiff95zu5sxgfzi... | An [[Definition:Celestial Observer|observer]]'s [[Definition:Terrestrial Latitude|(terrestrial) latitude]] can be identified by measuring the [[Definition:Celestial Altitude|(celestial) altitude]] of the [[Definition:Polaris|Pole Star]]. | From [[Altitude of North Celestial Pole equals Latitude of Observer]], all the [[Definition:Celestial Observer|observer]] needs to do is to measure the [[Definition:Celestial Altitude|(celestial) altitude]] of the [[Definition:North Celestial Pole|north celestial pole]].
The [[Definition:Star (Physics)|star]] [[Defini... | Measurement of Terrestrial Latitude/Pole Star | https://proofwiki.org/wiki/Measurement_of_Terrestrial_Latitude/Pole_Star | https://proofwiki.org/wiki/Measurement_of_Terrestrial_Latitude/Pole_Star | [
"Measurement of Terrestrial Latitude"
] | [
"Definition:Celestial Sphere/Observer",
"Definition:Latitude/Terrestrial",
"Definition:Celestial Altitude",
"Definition:Polaris"
] | [
"Altitude of North Celestial Pole equals Latitude of Observer",
"Definition:Celestial Sphere/Observer",
"Definition:Celestial Altitude",
"Definition:North Celestial Pole",
"Definition:Star (Physics)",
"Definition:Polaris",
"Definition:Point",
"Category:Measurement of Terrestrial Latitude"
] |
proofwiki-16543 | Measurement of Terrestrial Latitude/Sun at Noon | An observer's (terrestrial) latitude can be identified by measuring the (celestial) altitude of the Sun at noon. | {{ProofWanted|Needs to reference the fact that the time of year needs to be known, and then tables can be used.}}
{{qed}} | An [[Definition:Celestial Observer|observer]]'s [[Definition:Terrestrial Latitude|(terrestrial) latitude]] can be identified by measuring the [[Definition:Celestial Altitude|(celestial) altitude]] of the [[Definition:Sun|Sun]] at [[Definition:Noon|noon]]. | {{ProofWanted|Needs to reference the fact that the time of year needs to be known, and then tables can be used.}}
{{qed}} | Measurement of Terrestrial Latitude/Sun at Noon | https://proofwiki.org/wiki/Measurement_of_Terrestrial_Latitude/Sun_at_Noon | https://proofwiki.org/wiki/Measurement_of_Terrestrial_Latitude/Sun_at_Noon | [
"Measurement of Terrestrial Latitude"
] | [
"Definition:Celestial Sphere/Observer",
"Definition:Latitude/Terrestrial",
"Definition:Celestial Altitude",
"Definition:Sun",
"Definition:Twelve O'Clock/Noon"
] | [] |
proofwiki-16544 | Coherent Sequence is Partial Sum of P-adic Expansion | Let $p$ be a prime number.
Let $\sequence {\alpha_n}$ be a coherent sequence.
Then there exists a unique $p$-adic expansion of the form:
:$\ds \sum_{n \mathop = 0}^\infty d_n p^n$
such that:
:$\forall n \in \N: \alpha_n = \ds \sum_{i \mathop = 0}^n d_i p^i$ | By definition of a coherent sequence:
:$\forall n \in \N: 0 \le \alpha_n < p^{n + 1}$
From Zero Padded Basis Representation, for all $n \in \N$ there exists a sequence $\sequence {b_{j, n} }_{0 \le j \le n} :$
:$(1) \quad \ds \alpha_n = \sum_{j \mathop = 0}^n b_{j, n} p^j$
:$(2) \quad \forall j \in \closedint 0 n : 0 \... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\sequence {\alpha_n}$ be a [[Definition:P-adically Coherent Sequence|coherent sequence]].
Then there exists a [[Definition:Unique|unique]] [[Definition:P-adic Expansion|$p$-adic expansion]] of the form:
:$\ds \sum_{n \mathop = 0}^\infty d_n p^n$
such that:
... | By definition of a [[Definition:P-adically Coherent Sequence|coherent sequence]]:
:$\forall n \in \N: 0 \le \alpha_n < p^{n + 1}$
From [[Zero Padded Basis Representation]], for all $n \in \N$ there exists a [[Definition:Sequence|sequence]] $\sequence {b_{j, n} }_{0 \le j \le n} :$
:$(1) \quad \ds \alpha_n = \sum_{j \m... | Coherent Sequence is Partial Sum of P-adic Expansion | https://proofwiki.org/wiki/Coherent_Sequence_is_Partial_Sum_of_P-adic_Expansion | https://proofwiki.org/wiki/Coherent_Sequence_is_Partial_Sum_of_P-adic_Expansion | [
"P-adic Number Theory",
"Coherent Sequence is Partial Sum of P-adic Expansion"
] | [
"Definition:Prime Number",
"Definition:P-adically Coherent Sequence",
"Definition:Unique",
"Definition:P-adic Expansion"
] | [
"Definition:P-adically Coherent Sequence",
"Zero Padded Basis Representation",
"Definition:Sequence"
] |
proofwiki-16545 | Difference of Consecutive terms of Coherent Sequence | Let $p$ be a prime number.
Let $\sequence {\alpha_n}$ be a coherent sequence.
Then:
:for all $n \in \N_{>0}$ there exists $c_n \in \N$ such that:
::$0 \le c_n < p$
::$\alpha_n - \alpha_{n - 1} = c_n p^n$ | By definition of a coherent sequence:
:$\forall n \in \N_{>0}: \alpha_n \equiv \alpha_{n - 1} \pmod {p^n}$
That is:
:$\forall n \in \N_{>0}: \exists c_n \in \Z : \alpha_n - \alpha_{n - 1} = c_n p^n$
So it remains to show that:
:$\forall n \in \N_{>0} : 0 \le c_n < p$
{{AimForCont}} for some $N \in \N_{>0}$:
:$c_N \ge p... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\sequence {\alpha_n}$ be a [[Definition:P-adically Coherent Sequence|coherent sequence]].
Then:
:for all $n \in \N_{>0}$ there exists $c_n \in \N$ such that:
::$0 \le c_n < p$
::$\alpha_n - \alpha_{n - 1} = c_n p^n$ | By definition of a [[Definition:P-adically Coherent Sequence|coherent sequence]]:
:$\forall n \in \N_{>0}: \alpha_n \equiv \alpha_{n - 1} \pmod {p^n}$
That is:
:$\forall n \in \N_{>0}: \exists c_n \in \Z : \alpha_n - \alpha_{n - 1} = c_n p^n$
So it remains to show that:
:$\forall n \in \N_{>0} : 0 \le c_n < p$
{{Ai... | Difference of Consecutive terms of Coherent Sequence | https://proofwiki.org/wiki/Difference_of_Consecutive_terms_of_Coherent_Sequence | https://proofwiki.org/wiki/Difference_of_Consecutive_terms_of_Coherent_Sequence | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:P-adically Coherent Sequence"
] | [
"Definition:P-adically Coherent Sequence",
"Definition:P-adically Coherent Sequence",
"Definition:Contradiction",
"Definition:P-adically Coherent Sequence",
"Definition:Contradiction",
"Definition:P-adically Coherent Sequence",
"Category:P-adic Number Theory"
] |
proofwiki-16546 | Vandermonde Matrix Identity for Cauchy Matrix | Assume values $\set {x_1, \ldots, x_n, y_1, \ldots, y_n}$ are distinct in matrix
{{begin-eqn}}
{{eqn
| l = C
| r = <nowiki>\begin {pmatrix}
\dfrac 1 {x_1 - y_1} & \dfrac 1 {x_1 - y_2} & \cdots & \dfrac 1 {x_1 - y_n} \\
\dfrac 1 {x_2 - y_1} & \dfrac 1 {x_2 - y_2} & \cdots & \dfrac 1 {x_2 - y_n} \\
... | Matrices $P$ and $Q$ are invertible because all diagonal elements are nonzero.
For $1 \le i \le n$ express polynomial $p_i$ as:
:$\ds \map {p_i} x = \sum_{k \mathop = 1}^n a_{i k} x^{k - 1}$
Then:
{{begin-eqn}}
{{eqn | l = \paren {\map {p_i} {x_j} }
| r = \paren {a_{i j} } V_x
| c = {{Defof|Matrix Product (... | Assume values $\set {x_1, \ldots, x_n, y_1, \ldots, y_n}$ are distinct in matrix
{{begin-eqn}}
{{eqn
| l = C
| r = <nowiki>\begin {pmatrix}
\dfrac 1 {x_1 - y_1} & \dfrac 1 {x_1 - y_2} & \cdots & \dfrac 1 {x_1 - y_n} \\
\dfrac 1 {x_2 - y_1} & \dfrac 1 {x_2 - y_2} & \cdots & \dfrac 1 {x_2 - y_n} \\
... | Matrices $P$ and $Q$ are invertible because all diagonal elements are nonzero.
For $1 \le i \le n$ express polynomial $p_i$ as:
:$\ds \map {p_i} x = \sum_{k \mathop = 1}^n a_{i k} x^{k - 1}$
Then:
{{begin-eqn}}
{{eqn | l = \paren {\map {p_i} {x_j} }
| r = \paren {a_{i j} } V_x
| c = {{Defof|Matrix Produ... | Vandermonde Matrix Identity for Cauchy Matrix | https://proofwiki.org/wiki/Vandermonde_Matrix_Identity_for_Cauchy_Matrix | https://proofwiki.org/wiki/Vandermonde_Matrix_Identity_for_Cauchy_Matrix | [
"Vandermonde Matrix Identity",
"Cauchy Matrices"
] | [
"Definition:Cauchy Matrix",
"Definition:Vandermonde Matrix/Formulation 1",
"Definition:Cauchy Matrix"
] | [] |
proofwiki-16547 | Function that Satisfies Axioms of Uncertainty | Let $n \in \N$ be a natural number.
Let $p_1, p_2, \dotsc, p_n$ be real numbers such that:
:$\forall i \in \set {1, 2, \dotsc, n}: p_i \ge 0$
:$\ds \sum_{i \mathop = 1}^n p_i = 1$
Let $\map H {p_1, p_2, \ldots, p_n}$ be a real-valued function which satisfies the axioms of uncertainty.
Then:
:$\ds \map H {p_1, p_2, \ldo... | Let $g: \Z \to \R$ be the mapping defined as:
:$(1): \quad \map g n = \map H {\dfrac 1 n, \dfrac 1 n, \dotsc, \dfrac 1 n}$
Let $k \in \Z_{>0}$.
We have:
{{begin-eqn}}
{{eqn | l = \map g {n^k}
| r = \map g {n \times n^{k - 1} }
| c =
}}
{{eqn | r = \map g n + \map g {n^{k - 1} }
| c = Axiom 7
}}
{{eqn... | Let $n \in \N$ be a [[Definition:Natural Number|natural number]].
Let $p_1, p_2, \dotsc, p_n$ be [[Definition:Real Number|real numbers]] such that:
:$\forall i \in \set {1, 2, \dotsc, n}: p_i \ge 0$
:$\ds \sum_{i \mathop = 1}^n p_i = 1$
Let $\map H {p_1, p_2, \ldots, p_n}$ be a [[Definition:Real-Valued Function|real-... | Let $g: \Z \to \R$ be the [[Definition:Mapping|mapping]] defined as:
:$(1): \quad \map g n = \map H {\dfrac 1 n, \dfrac 1 n, \dotsc, \dfrac 1 n}$
Let $k \in \Z_{>0}$.
We have:
{{begin-eqn}}
{{eqn | l = \map g {n^k}
| r = \map g {n \times n^{k - 1} }
| c =
}}
{{eqn | r = \map g n + \map g {n^{k - 1} }
... | Function that Satisfies Axioms of Uncertainty/Proof | https://proofwiki.org/wiki/Function_that_Satisfies_Axioms_of_Uncertainty | https://proofwiki.org/wiki/Function_that_Satisfies_Axioms_of_Uncertainty/Proof | [
"Uncertainty",
"Function that Satisfies Axioms of Uncertainty"
] | [
"Definition:Natural Numbers",
"Definition:Real Number",
"Definition:Real-Valued Function",
"Axiom:Axioms of Uncertainty"
] | [
"Definition:Mapping",
"Axiom:Axioms of Uncertainty/Axiom 7",
"Axiom:Axioms of Uncertainty/Axiom 7",
"Axiom:Axioms of Uncertainty/Axiom 7",
"Axiom:Axioms of Uncertainty/Axiom 5",
"Definition:Natural Logarithm",
"Logarithm is Strictly Increasing",
"Definition:Positive/Integer",
"Definition:Constant",
... |
proofwiki-16548 | Permutation is Product of Transpositions | Let $S_n$ denote the symmetric group on $n$ letters.
Every element of $S_n$ can be expressed as a product of transpositions. | Let $\pi \in S_n$.
From Existence and Uniqueness of Cycle Decomposition, $\pi$ can be uniquely expressed as a cycle decomposition, up to the order of factors.
From K-Cycle can be Factored into Transpositions, each one of the cyclic permutations that compose this cycle decomposition can be expressed as a product of tran... | Let $S_n$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
Every [[Definition:Element|element]] of $S_n$ can be expressed as a product of [[Definition:Transposition|transpositions]]. | Let $\pi \in S_n$.
From [[Existence and Uniqueness of Cycle Decomposition]], $\pi$ can be [[Definition:Unique|uniquely expressed]] as a [[Definition:Cycle Decomposition|cycle decomposition]], up to the order of factors.
From [[K-Cycle can be Factored into Transpositions]], each one of the [[Definition:Cyclic Permutat... | Permutation is Product of Transpositions | https://proofwiki.org/wiki/Permutation_is_Product_of_Transpositions | https://proofwiki.org/wiki/Permutation_is_Product_of_Transpositions | [
"Permutations",
"Transpositions"
] | [
"Definition:Symmetric Group/n Letters",
"Definition:Element",
"Definition:Transposition"
] | [
"Existence and Uniqueness of Cycle Decomposition",
"Definition:Unique",
"Definition:Cycle Decomposition",
"K-Cycle can be Factored into Transpositions",
"Definition:Cyclic Permutation",
"Definition:Cycle Decomposition",
"Definition:Transposition"
] |
proofwiki-16549 | Vandermonde Matrix Identity for Hilbert Matrix | Define polynomial root sets $\set {1, 2, \ldots, n}$ and $\set {0, -1, \ldots, -n + 1}$ for Definition:Cauchy Matrix.
Let $H$ be the Hilbert matrix of order $n$:
:<nowiki>$H = \begin {pmatrix}
1 & \dfrac 1 2 & \cdots & \dfrac 1 n \\
\dfrac 1 2 & \dfrac 1 3 & \cdots & \dfrac 1 {n + 1} \\
\vdots... | Apply Vandermonde Matrix Identity for Cauchy Matrix and Hilbert Matrix is Cauchy Matrix.
Matrices $V_x$ and $V_y$ are invertible by Inverse of Vandermonde Matrix.
Matrices $P$ and $Q$ are invertible because all diagonal elements are nonzero.
{{qed}} | Define polynomial root sets $\set {1, 2, \ldots, n}$ and $\set {0, -1, \ldots, -n + 1}$ for [[Definition:Cauchy Matrix]].
Let $H$ be the [[Definition:Hilbert Matrix|Hilbert matrix]] of order $n$:
:<nowiki>$H = \begin {pmatrix}
1 & \dfrac 1 2 & \cdots & \dfrac 1 n \\
\dfrac 1 2 & \dfrac 1 3 & \cdots ... | Apply [[Vandermonde Matrix Identity for Cauchy Matrix]] and [[Hilbert Matrix is Cauchy Matrix]].
Matrices $V_x$ and $V_y$ are invertible by [[Inverse of Vandermonde Matrix]].
Matrices $P$ and $Q$ are invertible because all diagonal elements are nonzero.
{{qed}} | Vandermonde Matrix Identity for Hilbert Matrix | https://proofwiki.org/wiki/Vandermonde_Matrix_Identity_for_Hilbert_Matrix | https://proofwiki.org/wiki/Vandermonde_Matrix_Identity_for_Hilbert_Matrix | [
"Vandermonde Matrix Identity",
"Hilbert Matrices"
] | [
"Definition:Cauchy Matrix",
"Definition:Hilbert Matrix",
"Vandermonde Matrix Identity for Cauchy Matrix",
"Hilbert Matrix is Cauchy Matrix",
"Definition:Vandermonde Matrix/Formulation 1",
"Definition:Diagonal Matrix"
] | [
"Vandermonde Matrix Identity for Cauchy Matrix",
"Hilbert Matrix is Cauchy Matrix",
"Inverse of Vandermonde Matrix"
] |
proofwiki-16550 | Number of k-Cycles on Set of k Elements | Let $k \in \N$ be a natural number.
Let $S_k$ denote the symmetric group on $k$ letters.
The number of elements of $S_k$ which are $k$-cycles is $\paren {k - 1}!$. | Let $\N_k$ denote the set $\set {1, 2, \ldots, k}$.
Let $\pi$ be a $k$-cycle in $S_k$.
By definition, $\pi$ moves all $k$ elements of $\set {1, 2, \ldots, k}$.
Let $j \in \N_k$ be an arbitrary element of $\N_k$.
By definition of $k$-cycle, $k$ is the smallest integer such that $\map {\pi^k} j = j$.
Thus for each $r \in... | Let $k \in \N$ be a [[Definition:Natural Number|natural number]].
Let $S_k$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $k$ letters]].
The number of [[Definition:Element|elements]] of $S_k$ which are [[Definition:K-Cycle|$k$-cycles]] is $\paren {k - 1}!$. | Let $\N_k$ denote the [[Definition:Set|set]] $\set {1, 2, \ldots, k}$.
Let $\pi$ be a [[Definition:K-Cycle|$k$-cycle]] in $S_k$.
By definition, $\pi$ [[Definition:Moved Element under Permutation|moves]] all $k$ [[Definition:Element|elements]] of $\set {1, 2, \ldots, k}$.
Let $j \in \N_k$ be an arbitrary [[Definition... | Number of k-Cycles on Set of k Elements | https://proofwiki.org/wiki/Number_of_k-Cycles_on_Set_of_k_Elements | https://proofwiki.org/wiki/Number_of_k-Cycles_on_Set_of_k_Elements | [
"Cyclic Permutations"
] | [
"Definition:Natural Numbers",
"Definition:Symmetric Group/n Letters",
"Definition:Element",
"Definition:Cyclic Permutation"
] | [
"Definition:Set",
"Definition:Cyclic Permutation",
"Definition:Fixed Element under Permutation/Moved",
"Definition:Element",
"Definition:Element",
"Definition:Cyclic Permutation",
"Definition:Integer",
"Definition:Cyclic Permutation",
"Definition:Distinct",
"Definition:Ordered Tuple",
"Definitio... |
proofwiki-16551 | Zero Padded Basis Representation | Let $b \in \Z: b > 1$.
Let $m \in \Z_{> 0}$.
For every $n \in \Z_{\ge 0}$ such that $n < b^m$, there exists one and only one sequence $\sequence {r_j}_{0 \mathop \le j \mathop \le m - 1}$ such that:
{{begin-eqn}}
{{eqn | n = 1
| l = n
| r = \sum_{j \mathop = 0}^{m - 1} r_j b^j
}}
{{eqn | n = 2
| q = \... | === Case 1 ===
Let $n \in \Z_{> 0} : n < b^m$.
From Basis Representation Theorem there exists a sequence $\sequence {s_j}_{0 \mathop \le j \mathop \le k}$ such that:
:$(a): \quad \ds n = \sum_{j \mathop = 0}^k s_j b^j$
:$(b): \quad \forall j \in \closedint 0 k: s_j \in \N_b$
:$(c): \quad s_k \ne 0$
First it is shown t... | Let $b \in \Z: b > 1$.
Let $m \in \Z_{> 0}$.
For every $n \in \Z_{\ge 0}$ such that $n < b^m$, there exists [[Definition:Exactly One|one and only one]] [[Definition:Sequence|sequence]] $\sequence {r_j}_{0 \mathop \le j \mathop \le m - 1}$ such that:
{{begin-eqn}}
{{eqn | n = 1
| l = n
| r = \sum_{j \math... | === Case 1 ===
Let $n \in \Z_{> 0} : n < b^m$.
From [[Basis Representation Theorem]] there exists a [[Definition:Sequence|sequence]] $\sequence {s_j}_{0 \mathop \le j \mathop \le k}$ such that:
:$(a): \quad \ds n = \sum_{j \mathop = 0}^k s_j b^j$
:$(b): \quad \forall j \in \closedint 0 k: s_j \in \N_b$
:$(c): \quad... | Zero Padded Basis Representation/Proof | https://proofwiki.org/wiki/Zero_Padded_Basis_Representation | https://proofwiki.org/wiki/Zero_Padded_Basis_Representation/Proof | [
"Zero Padded Basis Representation",
"Basis Representations"
] | [
"Definition:Unique",
"Definition:Sequence"
] | [
"Basis Representation Theorem",
"Definition:Sequence",
"Power Function on Base between Zero and One is Strictly Decreasing/Integer",
"Definition:Contradiction",
"Definition:Premise",
"Definition:Sequence",
"Definition:Unique",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Basis Represe... |
proofwiki-16552 | Sum of Elements of Inverse of Matrix with Column of Ones | Let $\mathbf B = \sqbrk b_n$ denote the inverse of a square matrix $\mathbf A$ of order $n$.
Let $\mathbf A$ be such that it has a row or column of all ones.
Then the sum of elements in $\mathbf B$ is one:
:$\ds \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n b_{ij} = 1$ | If ones appear in a row of $\mathbf A$, then replace $\mathbf A$ by $\mathbf A^T$ and $\mathbf B$ by $\mathbf B^T$.
Assume $\mathbf A$ has a column of ones.
Apply Sum of Elements of Nonsingular Matrix to the inverse $\mathbf B = \mathbf A^{-1}$:
:$\ds \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n b_{i j} = 1 - \map \de... | Let $\mathbf B = \sqbrk b_n$ denote the [[Definition:Inverse Matrix|inverse]] of a [[Definition:Square Matrix|square matrix]] $\mathbf A$ of [[Definition:Order of Square Matrix|order]] $n$.
Let $\mathbf A$ be such that it has a row or column of all ones.
Then the sum of elements in $\mathbf B$ is one:
:$\ds \sum_{i ... | If ones appear in a row of $\mathbf A$, then replace $\mathbf A$ by $\mathbf A^T$ and $\mathbf B$ by $\mathbf B^T$.
Assume $\mathbf A$ has a column of ones.
Apply [[Sum of Elements of Nonsingular Matrix]] to the [[Definition:Inverse Matrix|inverse]] $\mathbf B = \mathbf A^{-1}$:
:$\ds \sum_{i \mathop = 1}^n \sum_{j ... | Sum of Elements of Inverse of Matrix with Column of Ones | https://proofwiki.org/wiki/Sum_of_Elements_of_Inverse_of_Matrix_with_Column_of_Ones | https://proofwiki.org/wiki/Sum_of_Elements_of_Inverse_of_Matrix_with_Column_of_Ones | [
"Inverse Matrices"
] | [
"Definition:Inverse Matrix",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order"
] | [
"Sum of Elements of Nonsingular Matrix",
"Definition:Inverse Matrix",
"Definition:Ones Matrix/Square",
"Definition:Matrix/Square Matrix/Order",
"Sum of Elements of Nonsingular Matrix"
] |
proofwiki-16553 | Coherent Sequence is Partial Sum of P-adic Expansion/Lemma | :$\forall n \in \N: \alpha_n = \ds \sum_{i \mathop = 0}^n b_{i, i} p^i$ | The theorem is proved by induction: | :$\forall n \in \N: \alpha_n = \ds \sum_{i \mathop = 0}^n b_{i, i} p^i$ | The [[Definition:Theorem|theorem]] is proved by [[Principle of Mathematical Induction|induction]]: | Coherent Sequence is Partial Sum of P-adic Expansion/Lemma | https://proofwiki.org/wiki/Coherent_Sequence_is_Partial_Sum_of_P-adic_Expansion/Lemma | https://proofwiki.org/wiki/Coherent_Sequence_is_Partial_Sum_of_P-adic_Expansion/Lemma | [
"Coherent Sequence is Partial Sum of P-adic Expansion"
] | [] | [
"Definition:Theorem",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-16554 | Numbers Equal to Sum of Squares of Digits | There are exactly $2$ integers which are equal to the sum of the squares of their digits when expressed in base $10$:
:$0 = 0^2$
:$1 = 1^2$ | We see the cases $0$ and $1$ above hold.
Suppose $N > 1$ is equal to the sum of the squares of their digits when expressed in base $10$.
Since $N^2 > N$, $N$ cannot be a $1$-digit integer.
Suppose $N$ is a $2$-digit integer.
Write $N = \sqbrk {a b} = 10 a + b$.
Then we have $a^2 + b^2 = 10 a + b$.
This can be reduced t... | There are exactly $2$ [[Definition:Integer|integers]] which are equal to the [[Definition:Integer Addition|sum]] of the [[Definition:Square|squares]] of their [[Definition:Digit|digits]] when expressed in [[Definition:Decimal Notation|base $10$]]:
:$0 = 0^2$
:$1 = 1^2$ | We see the cases $0$ and $1$ above hold.
Suppose $N > 1$ is equal to the [[Definition:Integer Addition|sum]] of the [[Definition:Square|squares]] of their [[Definition:Digit|digits]] when expressed in [[Definition:Decimal Notation|base $10$]].
Since $N^2 > N$, $N$ cannot be a $1$-[[Definition:Digit|digit]] [[Definit... | Numbers Equal to Sum of Squares of Digits | https://proofwiki.org/wiki/Numbers_Equal_to_Sum_of_Squares_of_Digits | https://proofwiki.org/wiki/Numbers_Equal_to_Sum_of_Squares_of_Digits | [
"Number Theory"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Square",
"Definition:Digit",
"Definition:Decimal Notation"
] | [
"Definition:Addition/Integers",
"Definition:Square",
"Definition:Digit",
"Definition:Decimal Notation",
"Definition:Digit",
"Definition:Integer",
"Definition:Digit",
"Definition:Integer",
"Definition:Even Integer",
"Definition:Even Integer",
"Definition:Digit",
"Definition:Integer",
"Definit... |
proofwiki-16555 | Convergent Sequence in P-adic Numbers has Unique Limit | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\sequence {x_n} $ be a sequence in $\Q_p$.
Then $\sequence {x_n}$ can have at most one limit. | From P-adic Metric on P-adic Numbers is Non-Archimedean Metric the $p$-adic metric is a metric on $\Q_p$.
By definition, the sequence $\sequence {x_n}$ converges in $\Q_p$ {{iff}}:
:$\sequence {x_n}$ converges in the $p$-adic metric.
The result then follows from Convergent Sequence in Metric Space has Unique Limit.
{{... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\sequence {x_n} $ be a [[Definition:Sequence|sequence]] in $\Q_p$.
Then $\sequence {x_n}$ can have at most one [[Definition:Limit of Real Seque... | From [[P-adic Metric on P-adic Numbers is Non-Archimedean Metric]] the [[Definition:P-adic Metric|$p$-adic metric]] is a [[Definition:Metric Space|metric]] on $\Q_p$.
By definition, the [[Definition:Sequence|sequence]] $\sequence {x_n}$ [[Definition:Convergent Sequence/P-adic Numbers/Definition 3|converges in $\Q_p$]]... | Convergent Sequence in P-adic Numbers has Unique Limit | https://proofwiki.org/wiki/Convergent_Sequence_in_P-adic_Numbers_has_Unique_Limit | https://proofwiki.org/wiki/Convergent_Sequence_in_P-adic_Numbers_has_Unique_Limit | [
"Sequences",
"Convergence",
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Sequence",
"Definition:Limit of Sequence/Real Numbers"
] | [
"P-adic Metric on P-adic Numbers is Non-Archimedean Metric",
"Definition:P-adic Metric",
"Definition:Metric Space",
"Definition:Sequence",
"Definition:Convergent Sequence/P-adic Numbers/Definition 3",
"Definition:Convergent Sequence/Metric Space",
"Definition:P-adic Metric",
"Convergent Sequence in Me... |
proofwiki-16556 | Limit to Infinity of Summation of Euler Phi Function over Square | :$\ds \lim_{n \mathop \to \infty} \dfrac {\map \Phi n} {n^2} = \dfrac 3 {\pi^2}$
where:
:$\map \Phi n = \ds \sum_{k \mathop = 1}^n \map \phi k$
:$\map \phi k$ is the Euler $\phi$ function of $k$.
Numerically, this evaluates to:
:$\dfrac 3 {\pi^2} \approx 0 \cdotp 30396 35509 \ldots$ | {{begin-eqn}}
{{eqn | l = \map \Phi n
| r = \sum_{k \mathop = 1}^n \map \phi k
}}
{{eqn | r = \sum_{k \mathop = 1}^n \paren {\sum_{d \mathop \divides k} \map \mu d \frac k d}
| c = Euler Phi Function in terms of Möbius Function
}}
{{eqn | r = \sum_{d d' \mathop \le n} d' \map \mu d
| c = taking $d d' ... | :$\ds \lim_{n \mathop \to \infty} \dfrac {\map \Phi n} {n^2} = \dfrac 3 {\pi^2}$
where:
:$\map \Phi n = \ds \sum_{k \mathop = 1}^n \map \phi k$
:$\map \phi k$ is the [[Definition:Euler Phi Function|Euler $\phi$ function]] of $k$.
Numerically, this evaluates to:
:$\dfrac 3 {\pi^2} \approx 0 \cdotp 30396 35509 \ldot... | {{begin-eqn}}
{{eqn | l = \map \Phi n
| r = \sum_{k \mathop = 1}^n \map \phi k
}}
{{eqn | r = \sum_{k \mathop = 1}^n \paren {\sum_{d \mathop \divides k} \map \mu d \frac k d}
| c = [[Euler Phi Function in terms of Möbius Function]]
}}
{{eqn | r = \sum_{d d' \mathop \le n} d' \map \mu d
| c = taking $d... | Limit to Infinity of Summation of Euler Phi Function over Square/Proof 1 | https://proofwiki.org/wiki/Limit_to_Infinity_of_Summation_of_Euler_Phi_Function_over_Square | https://proofwiki.org/wiki/Limit_to_Infinity_of_Summation_of_Euler_Phi_Function_over_Square/Proof_1 | [
"Euler Phi Function"
] | [
"Definition:Euler Phi Function"
] | [
"Euler Phi Function in terms of Möbius Function",
"Closed Form for Triangular Numbers",
"Approximate Size of Sum of Harmonic Series",
"Reciprocal of Riemann Zeta Function",
"Basel Problem",
"Powers Drown Logarithms"
] |
proofwiki-16557 | Complex Sine Function is Unbounded | Let $\sin: \C \to \C$ be the complex sine function.
Then $\sin$ is unbounded. | By Complex Sine Function is Entire, we have that $\sin$ is an entire function.
{{AimForCont}} that $\sin$ was a bounded function.
Then, by Liouville's Theorem, we would have that $\sin$ is a constant function.
However we have, for instance, by Sine of Zero is Zero:
:$\sin 0 = 0$
and by Sine of 90 Degrees:
:$\sin \df... | Let $\sin: \C \to \C$ be the [[Definition:Complex Sine Function|complex sine function]].
Then $\sin$ is [[Definition:Unbounded Complex-Valued Function|unbounded]]. | By [[Complex Sine Function is Entire]], we have that $\sin$ is an [[Definition:Entire Function|entire function]].
{{AimForCont}} that $\sin$ was a [[Definition:Bounded Complex-Valued Function|bounded function]].
Then, by [[Liouville's Theorem (Complex Analysis)|Liouville's Theorem]], we would have that $\sin$ is a [... | Complex Sine Function is Unbounded | https://proofwiki.org/wiki/Complex_Sine_Function_is_Unbounded | https://proofwiki.org/wiki/Complex_Sine_Function_is_Unbounded | [
"Sine Function"
] | [
"Definition:Sine/Complex Function",
"Definition:Bounded Mapping/Complex-Valued/Unbounded"
] | [
"Complex Sine Function is Entire",
"Definition:Entire Function",
"Definition:Bounded Mapping/Complex-Valued",
"Liouville's Theorem (Complex Analysis)",
"Definition:Constant Mapping",
"Sine of Zero is Zero",
"Sine of Right Angle",
"Definition:Constant Mapping",
"Proof by Contradiction",
"Definition... |
proofwiki-16558 | Complex Sine Function is Entire | Let $\sin: \C \to \C$ be the complex sine function.
Then $\sin$ is entire. | By the definition of the complex sine function, $\sin$ admits a power series expansion about $0$:
:$\ds \sin z = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!}$
By Complex Function is Entire iff it has Everywhere Convergent Power Series, to show that $\sin$ is entire it suffices to ... | Let $\sin: \C \to \C$ be the [[Definition:Complex Sine Function|complex sine function]].
Then $\sin$ is [[Definition:Entire Function|entire]]. | By the definition of the [[Definition:Complex Sine Function|complex sine function]], $\sin$ admits a [[Definition:Power Series|power series]] expansion about $0$:
:$\ds \sin z = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!}$
By [[Complex Function is Entire iff it has Everywhere C... | Complex Sine Function is Entire/Proof 1 | https://proofwiki.org/wiki/Complex_Sine_Function_is_Entire | https://proofwiki.org/wiki/Complex_Sine_Function_is_Entire/Proof_1 | [
"Complex Sine Function is Entire",
"Sine Function",
"Entire Functions"
] | [
"Definition:Sine/Complex Function",
"Definition:Entire Function"
] | [
"Definition:Sine/Complex Function",
"Definition:Power Series",
"Complex Function is Entire iff it has Everywhere Convergent Power Series",
"Definition:Everywhere Convergence",
"Radius of Convergence from Limit of Sequence/Complex Case"
] |
proofwiki-16559 | Complex Sine Function is Entire | Let $\sin: \C \to \C$ be the complex sine function.
Then $\sin$ is entire. | Let:
{{begin-eqn}}
{{eqn | l = \map f z
| r = \exp z
}}
{{eqn | l = \map g z
| r = i z
}}
{{eqn | l = \map h z
| r = -i z
}}
{{end-eqn}}
for all $z \in \C$.
By Complex Exponential Function is Entire, we have that $f$ is entire.
By Polynomial is Entire, we have that $g$ and $h$ are entire.
Therefor... | Let $\sin: \C \to \C$ be the [[Definition:Complex Sine Function|complex sine function]].
Then $\sin$ is [[Definition:Entire Function|entire]]. | Let:
{{begin-eqn}}
{{eqn | l = \map f z
| r = \exp z
}}
{{eqn | l = \map g z
| r = i z
}}
{{eqn | l = \map h z
| r = -i z
}}
{{end-eqn}}
for all $z \in \C$.
By [[Complex Exponential Function is Entire]], we have that $f$ is [[Definition:Entire Function|entire]].
By [[Polynomial is Entire]], we ... | Complex Sine Function is Entire/Proof 2 | https://proofwiki.org/wiki/Complex_Sine_Function_is_Entire | https://proofwiki.org/wiki/Complex_Sine_Function_is_Entire/Proof_2 | [
"Complex Sine Function is Entire",
"Sine Function",
"Entire Functions"
] | [
"Definition:Sine/Complex Function",
"Definition:Entire Function"
] | [
"Complex Exponential Function is Entire",
"Definition:Entire Function",
"Polynomial is Entire",
"Definition:Entire Function",
"Composition of Entire Functions is Entire",
"Definition:Entire Function",
"Linear Combination of Entire Functions is Entire",
"Definition:Entire Function",
"Euler's Sine Ide... |
proofwiki-16560 | Complex Exponential Function is Entire | Let $\exp: \C \to \C$ be the complex exponential function.
Then $\exp$ is entire. | By the definition of the complex exponential function, $\exp$ admits a power series expansion about $0$:
:$\ds \exp z = \sum_{n \mathop = 0}^\infty \frac {z^n} {n!}$
By Complex Function is Entire iff it has Everywhere Convergent Power Series, to show that $\exp$ is entire it suffices to show that this series is everyw... | Let $\exp: \C \to \C$ be the [[Definition:Exponential Function/Complex|complex exponential function]].
Then $\exp$ is [[Definition:Entire Function|entire]]. | By the definition of the [[Definition:Exponential Function/Complex|complex exponential function]], $\exp$ admits a [[Definition:Power Series|power series]] expansion about $0$:
:$\ds \exp z = \sum_{n \mathop = 0}^\infty \frac {z^n} {n!}$
By [[Complex Function is Entire iff it has Everywhere Convergent Power Series]]... | Complex Exponential Function is Entire | https://proofwiki.org/wiki/Complex_Exponential_Function_is_Entire | https://proofwiki.org/wiki/Complex_Exponential_Function_is_Entire | [
"Exponential Function",
"Entire Functions"
] | [
"Definition:Exponential Function/Complex",
"Definition:Entire Function"
] | [
"Definition:Exponential Function/Complex",
"Definition:Power Series",
"Complex Function is Entire iff it has Everywhere Convergent Power Series",
"Definition:Everywhere Convergence",
"Radius of Convergence of Power Series over Factorial/Complex Case",
"Definition:Everywhere Convergence"
] |
proofwiki-16561 | Complex Cosine Function is Entire | Let $\cos: \C \to \C$ be the complex cosine function.
Then $\cos$ is entire. | By the definition of the complex cosine function, $\cos$ admits a power series expansion about $0$:
:$\ds \cos z = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}$
By Complex Function is Entire iff it has Everywhere Convergent Power Series, to show that $\cos$ is entire it suffices to show t... | Let $\cos: \C \to \C$ be the [[Definition:Complex Cosine Function|complex cosine function]].
Then $\cos$ is [[Definition:Entire Function|entire]]. | By the definition of the [[Definition:Complex Cosine Function|complex cosine function]], $\cos$ admits a [[Definition:Power Series|power series]] expansion about $0$:
:$\ds \cos z = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}$
By [[Complex Function is Entire iff it has Everywhere Conve... | Complex Cosine Function is Entire/Proof 1 | https://proofwiki.org/wiki/Complex_Cosine_Function_is_Entire | https://proofwiki.org/wiki/Complex_Cosine_Function_is_Entire/Proof_1 | [
"Complex Cosine Function is Entire",
"Cosine Function",
"Entire Functions"
] | [
"Definition:Cosine/Complex Function",
"Definition:Entire Function"
] | [
"Definition:Cosine/Complex Function",
"Definition:Power Series",
"Complex Function is Entire iff it has Everywhere Convergent Power Series",
"Definition:Everywhere Convergence",
"Radius of Convergence from Limit of Sequence/Complex Case"
] |
proofwiki-16562 | Complex Cosine Function is Entire | Let $\cos: \C \to \C$ be the complex cosine function.
Then $\cos$ is entire. | Let:
{{begin-eqn}}
{{eqn | l = \map f z
| r = \exp z
}}
{{eqn | l = \map g z
| r = i z
}}
{{eqn | l = \map h z
| r = -i z
}}
{{end-eqn}}
for all $z \in \C$.
By Complex Exponential Function is Entire, we have that $f$ is entire.
By Polynomial is Entire, we have that $g$ and $h$ are entire.
Therefor... | Let $\cos: \C \to \C$ be the [[Definition:Complex Cosine Function|complex cosine function]].
Then $\cos$ is [[Definition:Entire Function|entire]]. | Let:
{{begin-eqn}}
{{eqn | l = \map f z
| r = \exp z
}}
{{eqn | l = \map g z
| r = i z
}}
{{eqn | l = \map h z
| r = -i z
}}
{{end-eqn}}
for all $z \in \C$.
By [[Complex Exponential Function is Entire]], we have that $f$ is [[Definition:Entire Function|entire]].
By [[Polynomial is Entire]], we h... | Complex Cosine Function is Entire/Proof 2 | https://proofwiki.org/wiki/Complex_Cosine_Function_is_Entire | https://proofwiki.org/wiki/Complex_Cosine_Function_is_Entire/Proof_2 | [
"Complex Cosine Function is Entire",
"Cosine Function",
"Entire Functions"
] | [
"Definition:Cosine/Complex Function",
"Definition:Entire Function"
] | [
"Complex Exponential Function is Entire",
"Definition:Entire Function",
"Polynomial is Entire",
"Definition:Entire Function",
"Composition of Entire Functions is Entire",
"Definition:Entire Function",
"Linear Combination of Entire Functions is Entire",
"Definition:Entire Function",
"Euler's Cosine I... |
proofwiki-16563 | Arccosecant Logarithmic Formulation | Let $x$ be a real number.
Let $x \in \hointl {-\infty} {-1} \cup \hointr 1 {\infty}$.
Then:
:$\arccsc x = -i \map \Ln {\sqrt {1 - \dfrac 1 {x^2} } + \dfrac i x}$
where:
:$\arccsc$ is the arccosecant function
:$\Ln$ is the principal branch of the complex logarithm whose imaginary part lies in $\hointl {-\pi} \pi$. | {{begin-eqn}}
{{eqn | l = \arccsc x
| r = \map \arcsin {\frac 1 x}
| c = Arccosecant of Reciprocal equals Arcsine
}}
{{eqn | r = -i \map \Ln {\sqrt {1 - \paren {\frac 1 x}^2} + i \times \frac 1 x}
| c = Arcsine Logarithmic Formulation
}}
{{eqn | r = -i \map \Ln {\sqrt {1 - \frac 1 {x^2} } + \frac i x}... | Let $x$ be a [[Definition:Real Number|real number]].
Let $x \in \hointl {-\infty} {-1} \cup \hointr 1 {\infty}$.
Then:
:$\arccsc x = -i \map \Ln {\sqrt {1 - \dfrac 1 {x^2} } + \dfrac i x}$
where:
:$\arccsc$ is the [[Definition:Arccosecant|arccosecant function]]
:$\Ln$ is the [[Definition:Principal Branch of Comple... | {{begin-eqn}}
{{eqn | l = \arccsc x
| r = \map \arcsin {\frac 1 x}
| c = [[Arccosecant of Reciprocal equals Arcsine]]
}}
{{eqn | r = -i \map \Ln {\sqrt {1 - \paren {\frac 1 x}^2} + i \times \frac 1 x}
| c = [[Arcsine Logarithmic Formulation]]
}}
{{eqn | r = -i \map \Ln {\sqrt {1 - \frac 1 {x^2} } + \f... | Arccosecant Logarithmic Formulation | https://proofwiki.org/wiki/Arccosecant_Logarithmic_Formulation | https://proofwiki.org/wiki/Arccosecant_Logarithmic_Formulation | [
"Arccosecant Function"
] | [
"Definition:Real Number",
"Definition:Inverse Cosecant/Real/Arccosecant",
"Definition:Natural Logarithm/Complex/Principal Branch",
"Definition:Natural Logarithm/Complex"
] | [
"Arccosecant of Reciprocal equals Arcsine",
"Arcsine Logarithmic Formulation"
] |
proofwiki-16564 | Arcsecant Logarithmic Formulation | Let $x$ be a real number.
Let $x \in \hointl \gets {-1} \cup \hointr 1 \to$.
Then:
:$\ds \arcsec x = -i \map \Ln {i \sqrt {1 - \frac 1 {x^2} } + \frac 1 x}$
where:
:$\arcsec$ is the arcsecant function
:$\Ln$ is the principal branch of the complex logarithm whose imaginary part lies in $\hointl {-\pi} \pi$. | {{begin-eqn}}
{{eqn | l = \arcsec x
| r = \map \arccos {\frac 1 x}
| c = Arcsecant of Reciprocal equals Arccosine
}}
{{eqn | r = -i \map \Ln {i \sqrt {1 - \paren {\frac 1 x}^2} + \frac 1 x}
| c = Arccosine Logarithmic Formulation
}}
{{eqn | r = -i \map \Ln {i \sqrt {1 - \frac 1 {x^2} } + \frac 1 x}
}}
{{end-eqn}}
{... | Let $x$ be a [[Definition:Real Number|real number]].
Let $x \in \hointl \gets {-1} \cup \hointr 1 \to$.
Then:
:$\ds \arcsec x = -i \map \Ln {i \sqrt {1 - \frac 1 {x^2} } + \frac 1 x}$
where:
:$\arcsec$ is the [[Definition:Real Arcsecant|arcsecant function]]
:$\Ln$ is the [[Definition:Principal Branch of Complex ... | {{begin-eqn}}
{{eqn | l = \arcsec x
| r = \map \arccos {\frac 1 x}
| c = [[Arcsecant of Reciprocal equals Arccosine]]
}}
{{eqn | r = -i \map \Ln {i \sqrt {1 - \paren {\frac 1 x}^2} + \frac 1 x}
| c = [[Arccosine Logarithmic Formulation]]
}}
{{eqn | r = -i \map \Ln {i \sqrt {1 - \frac 1 {x^2} } + \frac 1 x}
}}
{{end... | Arcsecant Logarithmic Formulation | https://proofwiki.org/wiki/Arcsecant_Logarithmic_Formulation | https://proofwiki.org/wiki/Arcsecant_Logarithmic_Formulation | [
"Arcsecant Function"
] | [
"Definition:Real Number",
"Definition:Inverse Secant/Real/Arcsecant",
"Definition:Natural Logarithm/Complex/Principal Branch",
"Definition:Natural Logarithm/Complex"
] | [
"Arcsecant of Reciprocal equals Arccosine",
"Arccosine Logarithmic Formulation"
] |
proofwiki-16565 | Nontrivial Zeroes of Riemann Zeta Function are Symmetrical with respect to Critical Line | The nontrivial zeroes of the Riemann $\zeta$ function are distributed symmetrically {{WRT}} the critical line.
That is, suppose $s_1 = \sigma_1 + i t$ is a nontrivial zero of $\map \zeta s$.
Then there exists another nontrivial zero $s_2$ of $\map \zeta s$ such that:
:$s_2 = 1 - s_1$ | From Functional Equation for Riemann Zeta Function, we have:
:$\ds \pi^{-s/2} \map \Gamma {\dfrac s 2} \map \zeta s = \pi^{\paren {s/2 - 1/2 } } \map \Gamma {\dfrac {1 - s} 2} \map \zeta {1 - s}$
We suppose $s_1 = \sigma_1 + i t$ is a nontrivial zero of $\map \zeta s$.
Then we have:
{{begin-eqn}}
{{eqn | l = \map \zeta... | The [[Definition:Nontrivial Zero of Riemann Zeta Function|nontrivial zeroes]] of the [[Definition:Riemann Zeta Function|Riemann $\zeta$ function]] are distributed [[Definition:Bilateral Symmetry|symmetrically]] {{WRT}} the [[Definition:Critical Line|critical line]].
That is, suppose $s_1 = \sigma_1 + i t$ is a [[Defi... | From [[Functional Equation for Riemann Zeta Function]], we have:
:$\ds \pi^{-s/2} \map \Gamma {\dfrac s 2} \map \zeta s = \pi^{\paren {s/2 - 1/2 } } \map \Gamma {\dfrac {1 - s} 2} \map \zeta {1 - s}$
We suppose $s_1 = \sigma_1 + i t$ is a [[Definition:Nontrivial Zero of Riemann Zeta Function|nontrivial zero]] of $\m... | Nontrivial Zeroes of Riemann Zeta Function are Symmetrical with respect to Critical Line | https://proofwiki.org/wiki/Nontrivial_Zeroes_of_Riemann_Zeta_Function_are_Symmetrical_with_respect_to_Critical_Line | https://proofwiki.org/wiki/Nontrivial_Zeroes_of_Riemann_Zeta_Function_are_Symmetrical_with_respect_to_Critical_Line | [
"Riemann Zeta Function"
] | [
"Definition:Riemann Zeta Function/Zero/Nontrivial",
"Definition:Riemann Zeta Function",
"Definition:Bilateral Symmetry",
"Definition:Riemann Zeta Function/Critical Line",
"Definition:Riemann Zeta Function/Zero/Nontrivial",
"Definition:Riemann Zeta Function/Zero/Nontrivial"
] | [
"Functional Equation for Riemann Zeta Function",
"Definition:Riemann Zeta Function/Zero/Nontrivial",
"Definition:Zero (Number)/Complex",
"Definition:Exponential Function/Complex",
"Definition:Exponential Function/Complex",
"Definition:Zero (Number)/Complex",
"Zeroes of Gamma Function",
"Definition:Zer... |
proofwiki-16566 | Prime-Counting Function in terms of Eulerian Logarithmic Integral/Riemann Hypothesis Holds | If the Riemann Hypothesis holds, then:
:$\map \pi x = \map \Li x + \map \OO {\sqrt x \ln x}$ | {{tidy}}
{{MissingLinks}}
{{refactor|level = medium|Usual exercise}}
Let $x$ denote a real number.
Let $n$ denote a natural number.
Let $p$ denote a prime number.
Let $\map \li x$ denote the logarithmic integral.
{{refactor|level = medium|These definitions are to be removed from here and referenced to their definition... | If the [[Riemann Hypothesis]] holds, then:
:$\map \pi x = \map \Li x + \map \OO {\sqrt x \ln x}$ | {{tidy}}
{{MissingLinks}}
{{refactor|level = medium|Usual exercise}}
Let $x$ denote a [[Definition:Real Number|real number]].
Let $n$ denote a [[Definition:Natural Numbers|natural number]].
Let $p$ denote a [[Definition:Prime Number|prime number]].
Let $\map \li x$ denote the [[Definition:Logarithmic Integral | log... | Prime-Counting Function in terms of Eulerian Logarithmic Integral/Riemann Hypothesis Holds | https://proofwiki.org/wiki/Prime-Counting_Function_in_terms_of_Eulerian_Logarithmic_Integral/Riemann_Hypothesis_Holds | https://proofwiki.org/wiki/Prime-Counting_Function_in_terms_of_Eulerian_Logarithmic_Integral/Riemann_Hypothesis_Holds | [
"Analytic Number Theory"
] | [
"Riemann Hypothesis"
] | [
"Definition:Real Number",
"Definition:Natural Numbers",
"Definition:Prime Number",
"Definition:Logarithmic Integral ",
"Definition:Prime Number"
] |
proofwiki-16567 | Mertens' Third Theorem | :$\ds \lim_{x \mathop \to \infty} \ln x \prod_{\substack {p \mathop \le x \\ \text {$p$ prime} } } \paren {1 - \dfrac 1 p} = e^{-\gamma}$
where $\gamma$ denotes the Euler-Mascheroni constant. | {{ProofWanted}}
{{Namedfor|Franz Mertens|cat = Mertens}} | :$\ds \lim_{x \mathop \to \infty} \ln x \prod_{\substack {p \mathop \le x \\ \text {$p$ prime} } } \paren {1 - \dfrac 1 p} = e^{-\gamma}$
where $\gamma$ denotes the [[Definition:Euler-Mascheroni Constant|Euler-Mascheroni constant]]. | {{ProofWanted}}
{{Namedfor|Franz Mertens|cat = Mertens}} | Mertens' Third Theorem | https://proofwiki.org/wiki/Mertens'_Third_Theorem | https://proofwiki.org/wiki/Mertens'_Third_Theorem | [
"Euler-Mascheroni Constant"
] | [
"Definition:Euler-Mascheroni Constant"
] | [] |
proofwiki-16568 | Abi-Khuzam Inequality | Let $\triangle ABC$ be a triangle.
Then:
:$\sin A \cdot \sin B \cdot \sin C \le k A \cdot B \cdot C$
where:
:$A, B, C$ are measured in radians
:$k = \paren {\dfrac {3 \sqrt 3} {2 \pi} }^3 \approx 0 \cdotp 56559 \, 56245 \ldots$ | {{tidy}}
{{MissingLinks}}
For $0 < \alpha_i < \pi, \alpha_1 + \alpha_2 + \alpha_3 = \pi$ we consider the function:
:$\map F {\alpha_1, \alpha_2, \alpha_3} = \dfrac {\sin \alpha_1 \sin \alpha_2 \sin \alpha_3} {\alpha_1 \alpha_2 \alpha_3}$
$F$ is defined on the region $G$ in the $\tuple {\alpha_1, \alpha_2, \alpha_3}$-s... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Then:
:$\sin A \cdot \sin B \cdot \sin C \le k A \cdot B \cdot C$
where:
:$A, B, C$ are measured in [[Definition:Radian|radians]]
:$k = \paren {\dfrac {3 \sqrt 3} {2 \pi} }^3 \approx 0 \cdotp 56559 \, 56245 \ldots$ | {{tidy}}
{{MissingLinks}}
For $0 < \alpha_i < \pi, \alpha_1 + \alpha_2 + \alpha_3 = \pi$ we consider the function:
:$\map F {\alpha_1, \alpha_2, \alpha_3} = \dfrac {\sin \alpha_1 \sin \alpha_2 \sin \alpha_3} {\alpha_1 \alpha_2 \alpha_3}$
$F$ is defined on the region $G$ in the $\tuple {\alpha_1, \alpha_2, \alpha_3}$... | Abi-Khuzam Inequality | https://proofwiki.org/wiki/Abi-Khuzam_Inequality | https://proofwiki.org/wiki/Abi-Khuzam_Inequality | [
"Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Angular Measure/Radian"
] | [
"Definition:Boundary (Geometry)",
"Definition:Continuous Real Function",
"Definition:Differentiable Mapping/Real Function",
"Definition:Lagrange multiplier",
"Definition:Decreasing/Real Function",
"Definition:Boundary (Topology)"
] |
proofwiki-16569 | Yff's Conjecture | Let $\triangle ABC$ be a triangle.
Let $\omega$ be the Brocard angle of $\triangle ABC$.
Then:
:$8 \omega^3 < ABC$
where $A, B, C$ are measured in radians. | The Abi-Khuzam Inequality states that
:$\sin A \cdot \sin B \cdot \sin C \le \paren {\dfrac {3 \sqrt 3} {2 \pi} }^3 A \cdot B \cdot C$
The maximum value of $A B C - 8 \omega^3$ occurs when two of the angles are equal.
So taking $A = B$, and using $A + B + C = \pi$, the maximum occurs at the maximum of:
:$\map f A = A^... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $\omega$ be the [[Definition:Brocard Angle|Brocard angle]] of $\triangle ABC$.
Then:
:$8 \omega^3 < ABC$
where $A, B, C$ are measured in [[Definition:Radian|radians]]. | The [[Abi-Khuzam Inequality]] states that
:$\sin A \cdot \sin B \cdot \sin C \le \paren {\dfrac {3 \sqrt 3} {2 \pi} }^3 A \cdot B \cdot C$
The maximum value of $A B C - 8 \omega^3$ occurs when two of the [[Definition:Angle|angles]] are equal.
So taking $A = B$, and using $A + B + C = \pi$, the maximum occurs at th... | Yff's Conjecture | https://proofwiki.org/wiki/Yff's_Conjecture | https://proofwiki.org/wiki/Yff's_Conjecture | [
"Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Brocard Angle",
"Definition:Angular Measure/Radian"
] | [
"Abi-Khuzam Inequality",
"Definition:Angle"
] |
proofwiki-16570 | Limit to Infinity of x minus Gamma of Reciprocal of x | thumbright600px
:$\ds \lim_{x \mathop \to \infty} \paren {x - \map \Gamma {\dfrac 1 x} } = \gamma$
where:
:$\Gamma$ denotes the $\Gamma$ (Gamma) function
:$\gamma$ denotes the Euler-Mascheroni constant. | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to \infty} \paren {x - \map \Gamma {\frac 1 x} }
| r = \lim_{x \mathop \to 0} \paren {\frac 1 x - \map \Gamma x}
}}
{{eqn | r = \lim_{x \mathop \to 0} \paren {\frac 1 x - \frac {\map \Gamma {x + 1} } x}
| c = Gamma Difference Equation
}}
{{eqn | r = \lim_{x \mathop... | [[File:Limit-of-x-minus-gamma-reciprocal-x.png|thumb|right|600px]]
:$\ds \lim_{x \mathop \to \infty} \paren {x - \map \Gamma {\dfrac 1 x} } = \gamma$
where:
:$\Gamma$ denotes the [[Definition:Gamma Function|$\Gamma$ (Gamma) function]]
:$\gamma$ denotes the [[Definition:Euler-Mascheroni Constant|Euler-Mascheroni const... | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to \infty} \paren {x - \map \Gamma {\frac 1 x} }
| r = \lim_{x \mathop \to 0} \paren {\frac 1 x - \map \Gamma x}
}}
{{eqn | r = \lim_{x \mathop \to 0} \paren {\frac 1 x - \frac {\map \Gamma {x + 1} } x}
| c = [[Gamma Difference Equation]]
}}
{{eqn | r = \lim_{x \ma... | Limit to Infinity of x minus Gamma of Reciprocal of x | https://proofwiki.org/wiki/Limit_to_Infinity_of_x_minus_Gamma_of_Reciprocal_of_x | https://proofwiki.org/wiki/Limit_to_Infinity_of_x_minus_Gamma_of_Reciprocal_of_x | [
"Gamma Function",
"Euler-Mascheroni Constant"
] | [
"File:Limit-of-x-minus-gamma-reciprocal-x.png",
"Definition:Gamma Function",
"Definition:Euler-Mascheroni Constant"
] | [
"Gamma Difference Equation",
"Gamma Function Extends Factorial",
"Definition:Derivative/Real Function/Derivative at Point/Definition 2",
"Derivative of Gamma Function at 1"
] |
proofwiki-16571 | Jung's Theorem | Let $S \subseteq \R^n$ be a compact subspace of an $n$-dimensional (real) Euclidean space.
Let $d = \ds \max_{x, y \mathop \in S} \map d {x, y}$ be the diameter of $S$.
Then there exists a closed ball ${B_r}^-$ with radius $r$ such that:
:$r = d \sqrt {\dfrac n {2 \paren {n + 1} } }$
such that $S \subseteq {B_r}^-$. | {{ProofWanted}}
{{Namedfor|Heinrich Wilhelm Ewald Jung|cat = Jung}} | Let $S \subseteq \R^n$ be a [[Definition:Compact Subspace|compact subspace]] of an [[Definition:Dimension of Vector Space|$n$-dimensional]] [[Definition:Real Euclidean Space|(real) Euclidean space]].
Let $d = \ds \max_{x, y \mathop \in S} \map d {x, y}$ be the [[Definition:Diameter of Subset of Metric Space|diameter]]... | {{ProofWanted}}
{{Namedfor|Heinrich Wilhelm Ewald Jung|cat = Jung}} | Jung's Theorem | https://proofwiki.org/wiki/Jung's_Theorem | https://proofwiki.org/wiki/Jung's_Theorem | [
"Jung's Theorem",
"Compact Topological Spaces",
"Real Euclidean Spaces"
] | [
"Definition:Compact Topological Space/Subspace",
"Definition:Dimension of Vector Space",
"Definition:Euclidean Space/Real",
"Definition:Diameter of Subset of Metric Space",
"Definition:Closed Ball",
"Definition:Closed Ball/Metric Space/Radius"
] | [] |
proofwiki-16572 | Jung's Theorem in the Plane | Let $S \subseteq \R^2$ be a compact region in a Euclidean plane.
Let $d$ be the diameter of $S$.
Then there exists a circle $C$ with radius $r$ such that:
:$r = d \dfrac {\sqrt 3} 3$
such that $S \subseteq C$. | This is an instance of Jung's Theorem, setting $n = 2$.
{{qed}}
{{Namedfor|Heinrich Wilhelm Ewald Jung}} | Let $S \subseteq \R^2$ be a [[Definition:Compact Subspace|compact region]] in a [[Definition:Euclidean Plane|Euclidean plane]].
Let $d$ be the [[Definition:Diameter of Geometric Figure|diameter]] of $S$.
Then there exists a [[Definition:Circle|circle]] $C$ with [[Definition:Radius of Closed Ball|radius]] $r$ such th... | This is an instance of [[Jung's Theorem]], setting $n = 2$.
{{qed}}
{{Namedfor|Heinrich Wilhelm Ewald Jung}} | Jung's Theorem in the Plane | https://proofwiki.org/wiki/Jung's_Theorem_in_the_Plane | https://proofwiki.org/wiki/Jung's_Theorem_in_the_Plane | [
"Jung's Theorem",
"Compact Topological Spaces",
"Real Euclidean Spaces"
] | [
"Definition:Compact Topological Space/Subspace",
"Definition:Euclidean Plane",
"Definition:Geometric Figure/Diameter",
"Definition:Circle",
"Definition:Closed Ball/Metric Space/Radius"
] | [
"Jung's Theorem"
] |
proofwiki-16573 | Reciprocal of One minus x in terms of Gaussian Hypergeometric Function | :$\dfrac 1 {1 - x} = \map F {1, p; p; x}$ | {{begin-eqn}}
{{eqn | l = \map F {1, p; p; x}
| r = \sum_{n \mathop = 0}^\infty \frac {1^{\bar n} p^{\bar n} } {p^{\bar n} } \frac {x^n} {n!}
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty n! \frac {x^n} {n!}
| c = One to Integer Rising is Integer Factorial... | :$\dfrac 1 {1 - x} = \map F {1, p; p; x}$ | {{begin-eqn}}
{{eqn | l = \map F {1, p; p; x}
| r = \sum_{n \mathop = 0}^\infty \frac {1^{\bar n} p^{\bar n} } {p^{\bar n} } \frac {x^n} {n!}
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty n! \frac {x^n} {n!}
| c = [[One to Integer Rising is Integer Factori... | Reciprocal of One minus x in terms of Gaussian Hypergeometric Function | https://proofwiki.org/wiki/Reciprocal_of_One_minus_x_in_terms_of_Gaussian_Hypergeometric_Function | https://proofwiki.org/wiki/Reciprocal_of_One_minus_x_in_terms_of_Gaussian_Hypergeometric_Function | [
"Gaussian Hypergeometric Function",
"Hypergeometric Functions"
] | [] | [
"One to Integer Rising is Integer Factorial",
"Sum of Infinite Geometric Sequence"
] |
proofwiki-16574 | Arctangent Function in terms of Gaussian Hypergeometric Function | :$\arctan x = x \map F {\dfrac 1 2, 1; \dfrac 3 2; -x^2}$ | {{begin-eqn}}
{{eqn | l = x \map F {\frac 1 2, 1; \frac 3 2; -x^2}
| r = x \sum_{n \mathop = 0}^\infty \frac {\paren {\frac 1 2}^{\bar n} 1^{\bar n} } {\paren {\frac 3 2}^{\bar n} } \frac {\paren {-x^2}^n} {n!}
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = x \sum_{n \mathop = 0}^\infty \fr... | :$\arctan x = x \map F {\dfrac 1 2, 1; \dfrac 3 2; -x^2}$ | {{begin-eqn}}
{{eqn | l = x \map F {\frac 1 2, 1; \frac 3 2; -x^2}
| r = x \sum_{n \mathop = 0}^\infty \frac {\paren {\frac 1 2}^{\bar n} 1^{\bar n} } {\paren {\frac 3 2}^{\bar n} } \frac {\paren {-x^2}^n} {n!}
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = x \sum_{n \mathop = 0}^\infty \fr... | Arctangent Function in terms of Gaussian Hypergeometric Function | https://proofwiki.org/wiki/Arctangent_Function_in_terms_of_Gaussian_Hypergeometric_Function | https://proofwiki.org/wiki/Arctangent_Function_in_terms_of_Gaussian_Hypergeometric_Function | [
"Arctangent Function in terms of Gaussian Hypergeometric Function",
"Arctangent Function",
"Gaussian Hypergeometric Function",
"Hypergeometric Functions"
] | [] | [
"Rising Factorial as Quotient of Factorials",
"One to Integer Rising is Integer Factorial",
"Gamma Difference Equation",
"Power Series Expansion for Real Arctangent Function"
] |
proofwiki-16575 | Alternating Sum and Difference of Factorials to Infinity | According to {{AuthorRef|Leonhard Paul Euler}}:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 0}^\infty \paren {-1}^n n!
| r = \int_0^\infty \dfrac {e^{-u} } {1 + u} \rd u
| c =
}}
{{eqn | r = 0! - 1! + 2! - 3! + 4! - 5! + \cdots
| c =
}}
{{eqn | r = G
| c = the Euler-Gompertz constant
}}
{{eqn ... | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 0}^\infty \paren {-1}^n n!
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \map \Gamma {n + 1}
| c = Gamma Function Extends Factorial
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \int_0^{\to \infty} u^{\paren {n + 1} - 1} e^{-u} \rd u
| c = {{Def... | According to {{AuthorRef|Leonhard Paul Euler}}:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 0}^\infty \paren {-1}^n n!
| r = \int_0^\infty \dfrac {e^{-u} } {1 + u} \rd u
| c =
}}
{{eqn | r = 0! - 1! + 2! - 3! + 4! - 5! + \cdots
| c =
}}
{{eqn | r = G
| c = the [[Definition:Euler-Gompertz Cons... | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 0}^\infty \paren {-1}^n n!
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \map \Gamma {n + 1}
| c = [[Gamma Function Extends Factorial]]
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \int_0^{\to \infty} u^{\paren {n + 1} - 1} e^{-u} \rd u
| c = {... | Alternating Sum and Difference of Factorials to Infinity | https://proofwiki.org/wiki/Alternating_Sum_and_Difference_of_Factorials_to_Infinity | https://proofwiki.org/wiki/Alternating_Sum_and_Difference_of_Factorials_to_Infinity | [
"Euler-Gompertz Constant",
"Factorials"
] | [
"Definition:Euler-Gompertz Constant"
] | [
"Gamma Function Extends Factorial",
"Fubini's Theorem",
"Sum of Infinite Geometric Sequence"
] |
proofwiki-16576 | Arcsine Function in terms of Gaussian Hypergeometric Function | :$\arcsin x = x \map F {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; x^2}$ | {{begin-eqn}}
{{eqn | l = x \map F {\frac 1 2, \frac 1 2; \frac 3 2; x^2}
| r = x \sum_{n \mathop = 0}^\infty \frac {\paren {\paren {\frac 1 2}^{\bar n} }^2} {\paren {\frac 3 2}^{\bar n} } \frac {x^{2 n} } {n!}
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac... | :$\arcsin x = x \map F {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; x^2}$ | {{begin-eqn}}
{{eqn | l = x \map F {\frac 1 2, \frac 1 2; \frac 3 2; x^2}
| r = x \sum_{n \mathop = 0}^\infty \frac {\paren {\paren {\frac 1 2}^{\bar n} }^2} {\paren {\frac 3 2}^{\bar n} } \frac {x^{2 n} } {n!}
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac... | Arcsine Function in terms of Gaussian Hypergeometric Function | https://proofwiki.org/wiki/Arcsine_Function_in_terms_of_Gaussian_Hypergeometric_Function | https://proofwiki.org/wiki/Arcsine_Function_in_terms_of_Gaussian_Hypergeometric_Function | [
"Arcsine Function in terms of Gaussian Hypergeometric Function",
"Arcsine Function",
"Gaussian Hypergeometric Function",
"Hypergeometric Functions"
] | [] | [
"Rising Factorial as Quotient of Factorials",
"Gamma Difference Equation",
"Gamma Function of One Half",
"Gamma Function of Positive Half-Integer",
"Power Series Expansion for Real Arcsine Function"
] |
proofwiki-16577 | Rational Number Expressible as Sum of Reciprocals of Distinct Squares | Let $x$ be a rational number such that $0 < x < \dfrac {\pi^2} 6 - 1$.
Then $x$ can be expressed as the sum of a finite number of reciprocals of distinct squares. | That no rational number such that $x \ge \dfrac {\pi^2} 6 - 1$ can be so expressed follows from Riemann Zeta Function of 2:
:$\ds \sum_{n \mathop = 1}^n \dfrac 1 {n^2} = 1 + \dfrac 1 {2^2} + \dfrac 1 {3^2} + \dotsb = \dfrac {\pi^2} 6$
That is, using ''all'' the reciprocals of distinct squares, you can never get as hig... | Let $x$ be a [[Definition:Rational Number|rational number]] such that $0 < x < \dfrac {\pi^2} 6 - 1$.
Then $x$ can be expressed as the [[Definition:Rational Addition|sum]] of a [[Definition:Finite Set|finite number]] of [[Definition:Reciprocal|reciprocals]] of [[Definition:Distinct Objects|distinct]] [[Definition:Squa... | That no [[Definition:Rational Number|rational number]] such that $x \ge \dfrac {\pi^2} 6 - 1$ can be so expressed follows from [[Riemann Zeta Function of 2]]:
:$\ds \sum_{n \mathop = 1}^n \dfrac 1 {n^2} = 1 + \dfrac 1 {2^2} + \dfrac 1 {3^2} + \dotsb = \dfrac {\pi^2} 6$
That is, using ''all'' the [[Definition:Recipro... | Rational Number Expressible as Sum of Reciprocals of Distinct Squares | https://proofwiki.org/wiki/Rational_Number_Expressible_as_Sum_of_Reciprocals_of_Distinct_Squares | https://proofwiki.org/wiki/Rational_Number_Expressible_as_Sum_of_Reciprocals_of_Distinct_Squares | [
"Square Numbers",
"Reciprocals",
"Unit Fractions",
"Rational Number Expressible as Sum of Reciprocals of Distinct Squares"
] | [
"Definition:Rational Number",
"Definition:Addition/Rational Numbers",
"Definition:Finite Set",
"Definition:Reciprocal",
"Definition:Distinct/Plural",
"Definition:Square Number"
] | [
"Definition:Rational Number",
"Riemann Zeta Function at Even Integers/Examples/2",
"Definition:Reciprocal",
"Definition:Distinct/Plural",
"Definition:Square Number",
"Definition:Rational Number",
"Definition:Subset"
] |
proofwiki-16578 | Mean Distance between Two Random Points in Cuboid | Let $B$ be a cuboid in the Cartesian $3$-space $\R^3$ as:
:$\size x \le a$, $\size y \le b$, $\size z \le c$
Let $E$ denote the mean distance $D$ between $2$ points chosen at random from the interior of $B$.
Then:
{{begin-eqn}}
{{eqn | l = E
| r = \dfrac {2 r} {15} - \dfrac 7 {45} \paren {\paren {r - r_1} \paren ... | Let $X_1$, $X_2$; $Y_1$, $Y_2$; $Z_1$, $Z_2$ be pairs of independent random variables with a continuous uniform distribution in $\closedint {-a} a$, $\closedint {-b} b$ and $\closedint {-c} c$.
The random variables $U = \size {X_1 - X_2}$, $V = \size {Y_1 - Y_2}$, $W = \size {Z_1 - Z_2}$ are likewise independent.
Thus ... | Let $B$ be a [[Definition:Cuboid|cuboid]] in the [[Definition:Cartesian Space|Cartesian $3$-space]] $\R^3$ as:
:$\size x \le a$, $\size y \le b$, $\size z \le c$
Let $E$ denote the [[Definition:Arithmetic Mean|mean]] [[Definition:Distance between Points|distance]] $D$ between $2$ [[Definition:Point|points]] chosen at ... | Let $X_1$, $X_2$; $Y_1$, $Y_2$; $Z_1$, $Z_2$ be pairs of [[Definition:Independent Random Variables|independent random variables]] with a [[Definition:Continuous Uniform Distribution|continuous uniform distribution]] in $\closedint {-a} a$, $\closedint {-b} b$ and $\closedint {-c} c$.
The [[Definition:Random Variable|r... | Mean Distance between Two Random Points in Cuboid | https://proofwiki.org/wiki/Mean_Distance_between_Two_Random_Points_in_Cuboid | https://proofwiki.org/wiki/Mean_Distance_between_Two_Random_Points_in_Cuboid | [
"Solid Geometry"
] | [
"Definition:Cuboid",
"Definition:Cartesian Product/Cartesian Space",
"Definition:Arithmetic Mean",
"Definition:Distance between Points",
"Definition:Point"
] | [
"Definition:Independent Random Variables",
"Definition:Uniform Distribution/Continuous",
"Definition:Random Variable",
"Definition:Independent Random Variables",
"Definition:Probability Density Function",
"Definition:Expectation",
"Definition:Pyramid",
"Definition:Plane Surface",
"Definition:Spheric... |
proofwiki-16579 | Mean Distance between Two Random Points in Unit Cube | The mean distance $R$ between $2$ points chosen at random from the interior of a unit cube is given by:
{{begin-eqn}}
{{eqn | l = R
| r = \frac {4 + 17 \sqrt 2 - 6 \sqrt3 - 7 \pi} {105} + \frac {\map \ln {1 + \sqrt 2 } } 5 + \frac {2 \, \map \ln {2 + \sqrt 3} } 5
}}
{{eqn | o = \approx
| r = 0 \cdotp 66170 ... | From Mean Distance between Two Random Points in Cuboid:
{{:Mean Distance between Two Random Points in Cuboid}}
The result follows by setting $a = b = c = \dfrac 1 2$.
Hence we have:
{{begin-eqn}}
{{eqn | l = r
| r = \sqrt {\dfrac 1 {2^2} + \dfrac 1 {2^2} + \dfrac 1 {2^2} }
| c =
}}
{{eqn | r = \sqrt {\dfra... | The [[Definition:Arithmetic Mean|mean]] [[Definition:Distance between Points|distance]] $R$ between $2$ [[Definition:Point|points]] chosen at random from the interior of a [[Definition:Unit Cube|unit cube]] is given by:
{{begin-eqn}}
{{eqn | l = R
| r = \frac {4 + 17 \sqrt 2 - 6 \sqrt3 - 7 \pi} {105} + \frac {\m... | From [[Mean Distance between Two Random Points in Cuboid]]:
{{:Mean Distance between Two Random Points in Cuboid}}
The result follows by setting $a = b = c = \dfrac 1 2$.
Hence we have:
{{begin-eqn}}
{{eqn | l = r
| r = \sqrt {\dfrac 1 {2^2} + \dfrac 1 {2^2} + \dfrac 1 {2^2} }
| c =
}}
{{eqn | r = \sqrt ... | Mean Distance between Two Random Points in Unit Cube | https://proofwiki.org/wiki/Mean_Distance_between_Two_Random_Points_in_Unit_Cube | https://proofwiki.org/wiki/Mean_Distance_between_Two_Random_Points_in_Unit_Cube | [
"Solid Geometry"
] | [
"Definition:Arithmetic Mean",
"Definition:Distance between Points",
"Definition:Point",
"Definition:Unit Cube",
"Definition:Robbins Constant"
] | [
"Mean Distance between Two Random Points in Cuboid",
"Inverse Hyperbolic Sine Logarithmic Formulation",
"Sine of 30 Degrees",
"Logarithm of Power"
] |
proofwiki-16580 | Recursion Property of Elementary Symmetric Function | Let $\set {z_1, z_2, \ldots, z_{n + 1} }$ be a set of $n + 1$ numbers, duplicate values permitted.
Then for $1 \le m \le n$:
:$\map {e_m} {\set {z_1, \ldots, z_n, z_{n + 1} } } = z_{n + 1} \map {e_{m - 1} } {\set {z_1, \ldots, z_n} } + \map {e_m} {\set {z_1, \ldots, z_n} }$ | Recall the definition of elementary symmetric function:
{{begin-eqn}}
{{eqn | l = \map {e_m} {\set {z_1, z_2, \ldots, z_n} }
| r = \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le n} \paren {\prod_{i \mathop = 1}^m z_{j_i} }
| c =
}}
{{eqn | r = \sum_{1 \mathop \le j... | Let $\set {z_1, z_2, \ldots, z_{n + 1} }$ be a set of $n + 1$ [[Definition:Number|numbers]], duplicate values permitted.
Then for $1 \le m \le n$:
:$\map {e_m} {\set {z_1, \ldots, z_n, z_{n + 1} } } = z_{n + 1} \map {e_{m - 1} } {\set {z_1, \ldots, z_n} } + \map {e_m} {\set {z_1, \ldots, z_n} }$ | Recall the definition of [[Definition:Elementary Symmetric Function|elementary symmetric function]]:
{{begin-eqn}}
{{eqn | l = \map {e_m} {\set {z_1, z_2, \ldots, z_n} }
| r = \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le n} \paren {\prod_{i \mathop = 1}^m z_{j_i} }
... | Recursion Property of Elementary Symmetric Function/Proof 2 | https://proofwiki.org/wiki/Recursion_Property_of_Elementary_Symmetric_Function | https://proofwiki.org/wiki/Recursion_Property_of_Elementary_Symmetric_Function/Proof_2 | [
"Recursion Property of Elementary Symmetric Function",
"Elementary Symmetric Functions"
] | [
"Definition:Number"
] | [
"Definition:Symmetric Function/Elementary",
"Definition:Summation/Summand",
"Definition:Summation/Summand",
"Definition:Summation/Summand",
"Definition:Summation/Summand",
"Definition:Summation/Summand"
] |
proofwiki-16581 | Uniform Matroid is Matroid | Let $S$ be a finite set of cardinality $n$.
Let $0 \le k \le n$.
Let $U_{k, n} = \struct{S, \mathscr I}$ be the uniform matroid of rank $k$.
Then $U_{k, n}$ is a matroid. | It needs to be shown that $\mathscr I$ satisfies the matroid axioms $(I1)$, $(I2)$ and $(I3)$. | Let $S$ be a [[Definition:Finite Set|finite set]] of [[Definition:Cardinality|cardinality]] $n$.
Let $0 \le k \le n$.
Let $U_{k, n} = \struct{S, \mathscr I}$ be the [[Definition:Uniform Matroid|uniform matroid of rank $k$]].
Then $U_{k, n}$ is a [[Definition:Matroid|matroid]]. | It needs to be shown that $\mathscr I$ satisfies the [[Axiom:Matroid Axioms|matroid axioms $(I1)$, $(I2)$ and $(I3)$]]. | Uniform Matroid is Matroid | https://proofwiki.org/wiki/Uniform_Matroid_is_Matroid | https://proofwiki.org/wiki/Uniform_Matroid_is_Matroid | [
"Matroid Theory"
] | [
"Definition:Finite Set",
"Definition:Cardinality",
"Definition:Uniform Matroid",
"Definition:Matroid"
] | [
"Axiom:Matroid Axioms",
"Axiom:Matroid Axioms",
"Axiom:Matroid Axioms",
"Axiom:Matroid Axioms"
] |
proofwiki-16582 | Free Matroid is Matroid | Let $S$ be a finite set.
Let $\struct {S, \powerset S}$ be the free matroid of $S$.
Then $\struct {S, \powerset S}$ is a matroid. | Let $S$ have cardinality $n$.
Let $\struct {S, \mathscr I_{n, n} }$ be the uniform matroid of rank $n$.
From Cardinality of Proper Subset of Finite Set, every subset of $S$ has cardinality less than or equal to $n$.
It follows that $\mathscr I_{n, n} = \powerset S$.
From Uniform Matroid is Matroid, then $\struct {S, \p... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\struct {S, \powerset S}$ be the [[Definition:Free Matroid|free matroid of $S$]].
Then $\struct {S, \powerset S}$ is a [[Definition:Matroid|matroid]]. | Let $S$ have [[Definition:Cardinality|cardinality]] $n$.
Let $\struct {S, \mathscr I_{n, n} }$ be the [[Definition:Uniform Matroid|uniform matroid of rank $n$]].
From [[Cardinality of Proper Subset of Finite Set]], every [[Definition:Subset|subset]] of $S$ has [[Definition:Cardinality|cardinality]] less than or equal... | Free Matroid is Matroid | https://proofwiki.org/wiki/Free_Matroid_is_Matroid | https://proofwiki.org/wiki/Free_Matroid_is_Matroid | [
"Matroid Theory"
] | [
"Definition:Finite Set",
"Definition:Free Matroid",
"Definition:Matroid"
] | [
"Definition:Cardinality",
"Definition:Uniform Matroid",
"Cardinality of Proper Subset of Finite Set",
"Definition:Subset",
"Definition:Cardinality",
"Uniform Matroid is Matroid",
"Definition:Matroid"
] |
proofwiki-16583 | Matroid Induced by Linear Independence in Vector Space is Matroid | Let $V$ be a vector space.
Let $S$ be a finite subset of $V$.
Let $\struct{S, \mathscr I}$ be the matroid induced on $S$ by linear independence in $V$.
That is, $\mathscr I$ is the set of linearly independent subsets of $S$.
Then $\struct{S, \mathscr I}$ is a matroid. | It needs to be shown that $\mathscr I$ satisfies the matroid axioms $(I1)$, $(I2)$ and $(I3)$. | Let $V$ be a [[Definition:Vector Space|vector space]].
Let $S$ be a [[Definition:Finite|finite]] [[Definition:Subset|subset]] of $V$.
Let $\struct{S, \mathscr I}$ be the [[Definition:Matroid Induced by Linear Independence (Vector Space)|matroid induced on $S$ by linear independence in $V$]].
That is, $\mathscr I$ is... | It needs to be shown that $\mathscr I$ satisfies the [[axiom:Matroid Axioms|matroid axioms $(I1)$, $(I2)$ and $(I3)$]]. | Matroid Induced by Linear Independence in Vector Space is Matroid | https://proofwiki.org/wiki/Matroid_Induced_by_Linear_Independence_in_Vector_Space_is_Matroid | https://proofwiki.org/wiki/Matroid_Induced_by_Linear_Independence_in_Vector_Space_is_Matroid | [
"Matroid Theory"
] | [
"Definition:Vector Space",
"Definition:Finite",
"Definition:Subset",
"Definition:Matroid Induced by Linear Independence/Vector Space",
"Definition:Set",
"Definition:Linearly Independent",
"Definition:Subset",
"Definition:Matroid"
] | [
"axiom:Matroid Axioms"
] |
proofwiki-16584 | Cycle Matroid is Matroid | Let $G = \struct {V, E}$ be a graph.
Let $\struct {E, \mathscr I}$ be the cycle matroid of $G$.
Then $\struct {E, \mathscr I}$ is a matroid. | It needs to be shown that $\mathscr I$ satisfies the matroid axioms $(\text I 1)$, $(\text I 2)$ and $(\text I 3)$. | Let $G = \struct {V, E}$ be a [[Definition:Graph (Graph Theory)|graph]].
Let $\struct {E, \mathscr I}$ be the [[Definition:Cycle Matroid|cycle matroid]] of $G$.
Then $\struct {E, \mathscr I}$ is a [[Definition:Matroid|matroid]]. | It needs to be shown that $\mathscr I$ satisfies the [[Axiom:Matroid Axioms|matroid axioms $(\text I 1)$, $(\text I 2)$ and $(\text I 3)$]]. | Cycle Matroid is Matroid | https://proofwiki.org/wiki/Cycle_Matroid_is_Matroid | https://proofwiki.org/wiki/Cycle_Matroid_is_Matroid | [
"Matroid Theory"
] | [
"Definition:Graph (Graph Theory)",
"Definition:Cycle Matroid",
"Definition:Matroid"
] | [
"Axiom:Matroid Axioms"
] |
proofwiki-16585 | Matroid Induced by Algebraic Independence is Matroid | Let $L / K$ be a field extension.
Let $S \subseteq L$ be a finite subset of $L$.
Let $\struct {S, \mathscr I}$ be the matroid induced by algebraic independence over $K$ on $S$.
That is, $\mathscr I$ is the set of algebraically independent subsets of $S$.
Then $\struct {S, \mathscr I}$ is a matroid. | It needs to be shown that $\mathscr I$ satisfies the matroid axioms $(\text I 1)$, $(\text I 2)$ and $(\text I 3)$. | Let $L / K$ be a [[Definition:Field Extension|field extension]].
Let $S \subseteq L$ be a [[Definition:Finite|finite]] [[Definition:Subset|subset]] of $L$.
Let $\struct {S, \mathscr I}$ be the [[Definition:Matroid Induced by Algebraic Independence|matroid induced by algebraic independence over $K$ on $S$]].
That is,... | It needs to be shown that $\mathscr I$ satisfies the [[Axiom:Matroid Axioms|matroid axioms $(\text I 1)$, $(\text I 2)$ and $(\text I 3)$]]. | Matroid Induced by Algebraic Independence is Matroid | https://proofwiki.org/wiki/Matroid_Induced_by_Algebraic_Independence_is_Matroid | https://proofwiki.org/wiki/Matroid_Induced_by_Algebraic_Independence_is_Matroid | [
"Matroid Theory"
] | [
"Definition:Field Extension",
"Definition:Finite",
"Definition:Subset",
"Definition:Matroid Induced by Algebraic Independence",
"Definition:Set",
"Definition:Algebraically Independent",
"Definition:Subset",
"Definition:Matroid"
] | [
"Axiom:Matroid Axioms"
] |
proofwiki-16586 | Matroid Induced by Affine Independence is Matroid | Let $\R^n$ be the $n$-dimensional real Euclidean space.
Let $S = \set {x_1, \dots, x_r}$ be a finite subset of $\R^n$.
Let $\struct {S, \mathscr I}$ be the matroid induced by affine independence on $S$.
That is, $\mathscr I$ is the set of affinely independent subsets of $S$.
Then $\struct{S, \mathscr I}$ is a matroid. | It needs to be shown that $\mathscr I$ satisfies the matroid axioms $(I1)$, $(I2)$ and $(I3)$. | Let $\R^n$ be the [[Definition:Dimension of Vector Space|$n$-dimensional]] [[Definition:Real Euclidean Space|real Euclidean space]].
Let $S = \set {x_1, \dots, x_r}$ be a [[Definition:Finite Set|finite]] [[Definition:Subset|subset]] of $\R^n$.
Let $\struct {S, \mathscr I}$ be the [[Definition:Matroid Induced by Affin... | It needs to be shown that $\mathscr I$ satisfies the [[Axiom:Matroid Axioms|matroid axioms $(I1)$, $(I2)$ and $(I3)$]]. | Matroid Induced by Affine Independence is Matroid | https://proofwiki.org/wiki/Matroid_Induced_by_Affine_Independence_is_Matroid | https://proofwiki.org/wiki/Matroid_Induced_by_Affine_Independence_is_Matroid | [
"Matroid Theory"
] | [
"Definition:Dimension of Vector Space",
"Definition:Euclidean Space/Real",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Matroid Induced by Affine Independence",
"Definition:Set",
"Definition:Affinely Dependent/Independent",
"Definition:Subset",
"Definition:Matroid"
] | [
"Axiom:Matroid Axioms"
] |
proofwiki-16587 | Matroid Induced by Linear Independence in Abelian Group is Matroid | Let $\struct{G, +}$ be a torsion-free Abelian group.
Let $\struct{G, +, \times}$ be the $\Z$-module associated with $G$.
Let $S$ be a finite subset of $G$.
Let $\struct{S, \mathscr I}$ be the matroid induced by linear independence in $G$ on $S$.
That is, $\mathscr I$ is the set of linearly independent subsets of $S$.
T... | It needs to be shown that $\mathscr I$ satisfies the matroid axioms $(I1)$, $(I2)$ and $(I3)$. | Let $\struct{G, +}$ be a [[Definition:Torsion-Free Group|torsion-free]] [[Definition:Abelian Group|Abelian group]].
Let $\struct{G, +, \times}$ be the [[Definition:Z-Module Associated with Abelian Group|$\Z$-module associated]] with $G$.
Let $S$ be a [[Definition:Finite|finite]] [[Definition:Subset|subset]] of $G$.
... | It needs to be shown that $\mathscr I$ satisfies the [[Axiom:Matroid Axioms|matroid axioms $(I1)$, $(I2)$ and $(I3)$]]. | Matroid Induced by Linear Independence in Abelian Group is Matroid | https://proofwiki.org/wiki/Matroid_Induced_by_Linear_Independence_in_Abelian_Group_is_Matroid | https://proofwiki.org/wiki/Matroid_Induced_by_Linear_Independence_in_Abelian_Group_is_Matroid | [
"Matroid Theory"
] | [
"Definition:Torsion-Free Group",
"Definition:Abelian Group",
"Definition:Z-Module Associated with Abelian Group",
"Definition:Finite",
"Definition:Subset",
"Definition:Matroid Induced by Linear Independence/Abelian Group",
"Definition:Set",
"Definition:Linearly Independent",
"Definition:Subset",
"... | [
"Axiom:Matroid Axioms"
] |
proofwiki-16588 | Independent Set can be Augmented by Larger Independent Set | Let $M = \struct{S, \mathscr I}$ be a matroid.
Let $X, Y \in \mathscr I$ such that:
:$\size X < \size Y$
Then there exists non-empty $Z \subseteq Y \setminus X$ such that:
:$X \cup Z \in \mathscr I$
:$\size {X \cup Z} = \size Y$ | Let $\mathscr Z = \set {Z \subseteq Y \setminus X : X \cup Z \in \mathscr I}$
Note that $\O \in \mathscr Z $
So $\mathscr Z \ne \O$
Let $Z_0 \in \mathscr Z : \size {Z_0} = \max \set {\size Z : Z \in \mathscr Z}$
{{AimForCont}}:
:$\size {X \cup Z_0} < \size Y$
By matroid axiom $(\text I 3)$:
:$\exists y \in Y \setminus ... | Let $M = \struct{S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $X, Y \in \mathscr I$ such that:
:$\size X < \size Y$
Then there exists [[Definition:Non-Empty Set|non-empty]] $Z \subseteq Y \setminus X$ such that:
:$X \cup Z \in \mathscr I$
:$\size {X \cup Z} = \size Y$ | Let $\mathscr Z = \set {Z \subseteq Y \setminus X : X \cup Z \in \mathscr I}$
Note that $\O \in \mathscr Z $
So $\mathscr Z \ne \O$
Let $Z_0 \in \mathscr Z : \size {Z_0} = \max \set {\size Z : Z \in \mathscr Z}$
{{AimForCont}}:
:$\size {X \cup Z_0} < \size Y$
By [[Axiom:Matroid Axioms|matroid axiom $(\text I 3)$... | Independent Set can be Augmented by Larger Independent Set | https://proofwiki.org/wiki/Independent_Set_can_be_Augmented_by_Larger_Independent_Set | https://proofwiki.org/wiki/Independent_Set_can_be_Augmented_by_Larger_Independent_Set | [
"Matroid Independent Subsets"
] | [
"Definition:Matroid",
"Definition:Non-Empty Set"
] | [
"Axiom:Matroid Axioms",
"Cardinality of Singleton",
"Definition:Contradiction",
"Finite Set Contains Subset of Smaller Cardinality",
"Cardinality of Empty Set"
] |
proofwiki-16589 | All Bases of Matroid have same Cardinality | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $\rho: \powerset S \to \Z$ be the rank function of $M$.
Let $B$ be a base of $M$.
Then:
:$\card B = \map \rho S$
That is, all bases of $M$ have the same cardinality, which is the rank of $M$. | By definition of the rank function:
:$\map \rho S = \max \set {\card X: X \subseteq S, X \in \mathscr I}$
Let $B_1$ be an independent subset such that:
:$\card {B_1} = \map \rho S$
It is shown that $B_1$ is a base.
Let $X$ be an independent superset of $B_1$.
From Cardinality of Subset of Finite Set:
:$\card {B_1} \le ... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $\rho: \powerset S \to \Z$ be the [[Definition:Rank Function (Matroid)|rank function]] of $M$.
Let $B$ be a [[Definition:Base of Matroid|base]] of $M$.
Then:
:$\card B = \map \rho S$
That is, all [[Definition:Base of Matroid|bases]] of $M$... | By definition of the [[Definition:Rank Function (Matroid)|rank function]]:
:$\map \rho S = \max \set {\card X: X \subseteq S, X \in \mathscr I}$
Let $B_1$ be an [[Definition:Independent Subset (Matroid)|independent subset]] such that:
:$\card {B_1} = \map \rho S$
It is shown that $B_1$ is a [[Definition:Base of Matr... | All Bases of Matroid have same Cardinality | https://proofwiki.org/wiki/All_Bases_of_Matroid_have_same_Cardinality | https://proofwiki.org/wiki/All_Bases_of_Matroid_have_same_Cardinality | [
"Matroid Bases"
] | [
"Definition:Matroid",
"Definition:Rank Function (Matroid)",
"Definition:Base of Matroid",
"Definition:Base of Matroid",
"Definition:Cardinality",
"Definition:Rank (Matroid)"
] | [
"Definition:Rank Function (Matroid)",
"Definition:Matroid/Independent Set",
"Definition:Base of Matroid",
"Definition:Matroid/Independent Set",
"Definition:Subset/Superset",
"Cardinality of Subset of Finite Set",
"Cardinality of Proper Subset of Finite Set",
"Definition:Maximal",
"definition:Indepen... |
proofwiki-16590 | Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom | Let $S$ be a finite set.
Let $\mathscr B$ be a non-empty set of subsets of $S$.
Then $\mathscr B$ is the set of bases of a matroid on $S$ {{iff}} $\mathscr B$ satisfies formulation $1$ of base axiom:
{{:Axiom:Base Axiom (Matroid)/Formulation 1}} | === Necessary Condition ===
Let $\mathscr B$ be the set of bases of the matroid on $M = \struct{S, \mathscr I}$
{{:Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom/Necessary Condition}}{{qed|lemma}} | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\mathscr B$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$.
Then $\mathscr B$ is the set of [[Definition:Base of Matroid|bases]] of a [[Definition:Matroid|matroid]] on $S$ {{iff}} $\mathscr B$ satisfie... | === [[Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom/Necessary Condition|Necessary Condition]] ===
Let $\mathscr B$ be the set of [[Definition:Base of Matroid|bases]] of the [[Definition:Matroid|matroid]] on $M = \struct{S, \mathscr I}$
{{:Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axi... | Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom | https://proofwiki.org/wiki/Matroid_Bases_Iff_Satisfies_Formulation_1_of_Matroid_Base_Axiom | https://proofwiki.org/wiki/Matroid_Bases_Iff_Satisfies_Formulation_1_of_Matroid_Base_Axiom | [
"Matroid Bases",
"Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom"
] | [
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Subset",
"Definition:Base of Matroid",
"Definition:Matroid",
"Axiom:Base Axiom (Matroid)/Formulation 1"
] | [
"Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom/Necessary Condition",
"Definition:Base of Matroid",
"Definition:Matroid"
] |
proofwiki-16591 | Number of Integer Partitions into Sum of Consecutive Primes | Let $n$ be a natural number.
Let $\map f n$ denote the number of integer partitions of $n$ where the parts are consecutive prime numbers.
For example:
:$\map f {41} = 3$
because:
:$41 = 11 + 13 + 17 = 2 + 3 + 5 + 7 + 11 + 13$
Then:
:$\ds \lim_{x \mathop \to \infty} \dfrac 1 x \sum_{n \mathop = 1}^x \map f n = \ln 2$ | Let $\mathbb P$ denote the set of prime numbers.
Every set of consecutive primes whose sum is less than $x$ will contribute $1$ to the sum:
:$\map f 1 + \map f 2 + \dotsb + \map f x$
The number of such sets of $r$ primes is clearly at most:
:$\map \pi {x / r}$
and at least:
:$\map \pi {x / r} - r$
where $\pi$ is the pr... | Let $n$ be a [[Definition:Natural Number|natural number]].
Let $\map f n$ denote the number of [[Definition:Integer Partition|integer partitions]] of $n$ where the [[Definition:Part of Integer Partition|parts]] are consecutive [[Definition:Prime Number|prime numbers]].
For example:
:$\map f {41} = 3$
because:
:$41 = ... | Let $\mathbb P$ denote the set of [[Definition:Prime Number|prime numbers]].
Every set of consecutive primes whose sum is less than $x$ will contribute $1$ to the sum:
:$\map f 1 + \map f 2 + \dotsb + \map f x$
The number of such sets of $r$ primes is clearly at most:
:$\map \pi {x / r}$
and at least:
:$\map \pi {x ... | Number of Integer Partitions into Sum of Consecutive Primes | https://proofwiki.org/wiki/Number_of_Integer_Partitions_into_Sum_of_Consecutive_Primes | https://proofwiki.org/wiki/Number_of_Integer_Partitions_into_Sum_of_Consecutive_Primes | [
"Partition Theory",
"Prime Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Integer Partition",
"Definition:Integer Partition/Part",
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Prime-Counting Function",
"Definition:Positive/Real Number",
"Definition:Little-O Notation",
"Prime Number Theorem"
] |
proofwiki-16592 | Coherent Sequence Converges to P-adic Integer | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\sequence {\alpha_n}$ be a coherent sequence.
Let $\Z_p$ be the $p$-adic integers.
Then the sequence $\sequence {\alpha_n}$ converges to some $x \in \Z_p$.
That is, there exists $x \in \Z_p$ such that:
:$\ds \lim_{n \mat... | From Coherent Sequence is Partial Sum of P-adic Expansion there exists a unique $p$-adic expansion of the form:
:$\ds \sum_{n \mathop = 0}^\infty d_n p^n$
such that:
:$\forall n \in \N: \alpha_n = \ds \sum_{i \mathop = 0}^n d_i p^i$
From P-adic Expansion Converges to P-adic Number:
:$\exists x \in \Q_p : \ds \lim_{n \m... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\sequence {\alpha_n}$ be a [[Definition:P-adically Coherent Sequence|coherent sequence]].
Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adi... | From [[Coherent Sequence is Partial Sum of P-adic Expansion]] there exists a [[Definition:Unique|unique]] [[Definition:P-adic Expansion|$p$-adic expansion]] of the form:
:$\ds \sum_{n \mathop = 0}^\infty d_n p^n$
such that:
:$\forall n \in \N: \alpha_n = \ds \sum_{i \mathop = 0}^n d_i p^i$
From [[P-adic Expansion Conv... | Coherent Sequence Converges to P-adic Integer | https://proofwiki.org/wiki/Coherent_Sequence_Converges_to_P-adic_Integer | https://proofwiki.org/wiki/Coherent_Sequence_Converges_to_P-adic_Integer | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adically Coherent Sequence",
"Definition:P-adic Integer",
"Definition:Sequence",
"Definition:Convergent Sequence/P-adic Numbers"
] | [
"Coherent Sequence is Partial Sum of P-adic Expansion",
"Definition:Unique",
"Definition:P-adic Expansion",
"P-adic Expansion is a Cauchy Sequence in P-adic Norm/Converges to P-adic Number",
"Definition:P-adically Coherent Sequence",
"Definition:P-adic Integer",
"Definition:Closed Ball/P-adic Numbers",
... |
proofwiki-16593 | Dixon's Theorem (Group Theory) | Let $P_1$ and $P_2$ be distinct elements of the symmetric group on $n$ letters.
The probability that $\set {P_1, P_2}$ forms a generator of $S_n$ approaches $\dfrac 3 4$ as $n$ tends to infinity. | {{ProofWanted}}
{{Namedfor|John Douglas Dixon|cat = Dixon, J.D.}} | Let $P_1$ and $P_2$ be [[Definition:Distinct Elements|distinct]] [[Definition:Element|elements]] of the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
The [[Definition:Probability|probability]] that $\set {P_1, P_2}$ forms a [[Definition:Generator of Group|generator]] of $S_n$ approaches ... | {{ProofWanted}}
{{Namedfor|John Douglas Dixon|cat = Dixon, J.D.}} | Dixon's Theorem (Group Theory) | https://proofwiki.org/wiki/Dixon's_Theorem_(Group_Theory) | https://proofwiki.org/wiki/Dixon's_Theorem_(Group_Theory) | [
"Symmetric Groups",
"Generators of Groups"
] | [
"Definition:Distinct/Plural",
"Definition:Element",
"Definition:Symmetric Group/n Letters",
"Definition:Probability",
"Definition:Generator of Group",
"Definition:Infinity"
] | [] |
proofwiki-16594 | P-adic Integer is Limit of Unique P-adic Expansion | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ be the $p$-adic integers.
Let $x \in \Z_p$.
Then $x$ is the limit of a unique $p$-adic expansion of the form:
:$\ds \sum_{n \mathop = 0}^\infty d_n p^n$ | From P-adic Integer is Limit of Unique Coherent Sequence of Integers, there exists a unique coherent sequence $\sequence{\alpha_n}$ such that:
:$\ds \lim_{n \mathop \to \infty} \alpha_n = x$
From Coherent Sequence is Partial Sum of P-adic Expansion, there exists a unique $p$-adic expansion of the form:
:$\ds \sum_{n \m... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]].
Let $x \in \Z_p$.
Then $x$ is the [[Definition:Limit of P-adic Sequence|limit]] ... | From [[P-adic Integer is Limit of Unique Coherent Sequence of Integers]], there exists a [[Definition:Unique|unique]] [[Definition:P-adically Coherent Sequence|coherent sequence]] $\sequence{\alpha_n}$ such that:
:$\ds \lim_{n \mathop \to \infty} \alpha_n = x$
From [[Coherent Sequence is Partial Sum of P-adic Expansio... | P-adic Integer is Limit of Unique P-adic Expansion | https://proofwiki.org/wiki/P-adic_Integer_is_Limit_of_Unique_P-adic_Expansion | https://proofwiki.org/wiki/P-adic_Integer_is_Limit_of_Unique_P-adic_Expansion | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Integer",
"Definition:Limit of Sequence/P-adic Numbers",
"Definition:Unique",
"Definition:P-adic Expansion"
] | [
"P-adic Integer is Limit of Unique Coherent Sequence of Integers",
"Definition:Unique",
"Definition:P-adically Coherent Sequence",
"Coherent Sequence is Partial Sum of P-adic Expansion",
"Definition:Unique",
"Definition:P-adic Expansion"
] |
proofwiki-16595 | P-adic Number is Limit of Unique P-adic Expansion | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $x \in \Q_p$.
Then $x$ is the limit of a unique $p$-adic expansion. | Let $\Z_p$ be the $p$-adic integers.
From P-adic Number times Integer Power of p is P-adic Integer:
:$\exists m \in \Z_{\ge 0} : x p^m \in \Z_p$
From P-adic Integer is Limit of Unique P-adic Expansion, there exists a unique $p$-adic expansion of the form:
:$\ds \sum_{n \mathop = 0}^\infty d_n p^n$
such that:
:$\ds \lim... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $x \in \Q_p$.
Then $x$ is the [[Definition:Limit of P-adic Sequence|limit]] of a [[Definition:Unique|unique]] [[Definition:P-adic Expansion|$p$-... | Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]].
From [[P-adic Number times Integer Power of p is P-adic Integer]]:
:$\exists m \in \Z_{\ge 0} : x p^m \in \Z_p$
From [[P-adic Integer is Limit of Unique P-adic Expansion]], there exists a [[Definition:Unique|unique]] [[Definition:P-adic Expansion|$p$... | P-adic Number is Limit of Unique P-adic Expansion | https://proofwiki.org/wiki/P-adic_Number_is_Limit_of_Unique_P-adic_Expansion | https://proofwiki.org/wiki/P-adic_Number_is_Limit_of_Unique_P-adic_Expansion | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Limit of Sequence/P-adic Numbers",
"Definition:Unique",
"Definition:P-adic Expansion"
] | [
"Definition:P-adic Integer",
"P-adic Number times Integer Power of p is P-adic Integer",
"P-adic Integer is Limit of Unique P-adic Expansion",
"Definition:Unique",
"Definition:P-adic Expansion",
"Combination Theorem for Sequences/Normed Division Ring/Multiple Rule",
"Definition:P-adic Expansion",
"Def... |
proofwiki-16596 | Finite Set Contains Subset of Smaller Cardinality | Let $S$ be a finite sets.
Let
:$\size S = n$
where $\size {\, \cdot \,}$ denotes cardinality.
Let $0 \le m \le n$.
Then there exists a subset $X \subseteq S$ such that:
:$\size X = m$ | === Case 1 ===
Let $m = n$.
Then $X = S$ is a subset $X \subseteq S$ such that:
:$\size X = m$
{{qed|lemma}} | Let $S$ be a [[Definition:Finite Set|finite sets]].
Let
:$\size S = n$
where $\size {\, \cdot \,}$ denotes [[Definition:Cardinality|cardinality]].
Let $0 \le m \le n$.
Then there exists a [[Definition:Subset|subset]] $X \subseteq S$ such that:
:$\size X = m$ | === Case 1 ===
Let $m = n$.
Then $X = S$ is a [[Definition:Subset|subset]] $X \subseteq S$ such that:
:$\size X = m$
{{qed|lemma}} | Finite Set Contains Subset of Smaller Cardinality | https://proofwiki.org/wiki/Finite_Set_Contains_Subset_of_Smaller_Cardinality | https://proofwiki.org/wiki/Finite_Set_Contains_Subset_of_Smaller_Cardinality | [
"Set Theory",
"Subsets",
"Cardinality"
] | [
"Definition:Finite Set",
"Definition:Cardinality",
"Definition:Subset"
] | [
"Definition:Subset",
"Definition:Subset",
"Definition:Subset"
] |
proofwiki-16597 | Integral from 0 to 1 of Complete Elliptic Integral of First Kind | Let $G$ denote Catalan's constant.
Then:
:$2 G = \ds \int_0^1 \map K k \rd k$
where $\map K k$ denotes the complete elliptic integral of the first kind:
:$\map K k = \ds \int \limits_0^{\pi / 2} \dfrac {\d \phi} {\sqrt {1 - k^2 \sin^2 \phi} }$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \map K k \rd k
| r = \int_0^1 \int_0^{\pi / 2} \dfrac 1 {\sqrt {1 - k^2 \sin^2 \phi} } \rd \phi \rd k
| c = {{Defof|Complete Elliptic Integral of the First Kind}}
}}
{{eqn | r = \int_0^{\pi / 2} \int_0^1 \dfrac 1 {\sqrt {1 - k^2 \sin^2 \phi} } \rd k \rd \phi
| c = To... | Let $G$ denote [[Definition:Catalan's Constant|Catalan's constant]].
Then:
:$2 G = \ds \int_0^1 \map K k \rd k$
where $\map K k$ denotes the [[Definition:Complete Elliptic Integral of the First Kind|complete elliptic integral of the first kind]]:
:$\map K k = \ds \int \limits_0^{\pi / 2} \dfrac {\d \phi} {\sqrt {1 - ... | {{begin-eqn}}
{{eqn | l = \int_0^1 \map K k \rd k
| r = \int_0^1 \int_0^{\pi / 2} \dfrac 1 {\sqrt {1 - k^2 \sin^2 \phi} } \rd \phi \rd k
| c = {{Defof|Complete Elliptic Integral of the First Kind}}
}}
{{eqn | r = \int_0^{\pi / 2} \int_0^1 \dfrac 1 {\sqrt {1 - k^2 \sin^2 \phi} } \rd k \rd \phi
| c = [[... | Integral from 0 to 1 of Complete Elliptic Integral of First Kind | https://proofwiki.org/wiki/Integral_from_0_to_1_of_Complete_Elliptic_Integral_of_First_Kind | https://proofwiki.org/wiki/Integral_from_0_to_1_of_Complete_Elliptic_Integral_of_First_Kind | [
"Catalan's Constant",
"Complete Elliptic Integral of the First Kind"
] | [
"Definition:Catalan's Constant",
"Definition:Elliptic Integral of the First Kind/Complete"
] | [
"Tonelli's Theorem",
"Integration by Substitution",
"Arcsine as Integral",
"Integration by Substitution",
"Double Angle Formulas/Sine",
"Integration by Substitution",
"Power Series Expansion for Real Arctangent Function",
"Linear Combination of Integrals/Definite",
"Integral of Power"
] |
proofwiki-16598 | Trivial Group is Smallest Group | Let $G = \struct {\set e, \circ}$ be a trivial group.
Then $G$ is the smallest group possible, in that there exists no set with lower cardinality which is the underlying set of a group. | From Trivial Group is Group, we have that there does exist a group of cardinality $1$.
From Group is not Empty, there can be no group of smaller order.
{{qed}} | Let $G = \struct {\set e, \circ}$ be a [[Definition:Trivial Group|trivial group]].
Then $G$ is the smallest [[Definition:Group|group]] possible, in that there exists no [[Definition:Set|set]] with lower [[Definition:Cardinality|cardinality]] which is the [[Definition:Underlying Set of Structure|underlying set]] of a [... | From [[Trivial Group is Group]], we have that there does exist a [[Definition:Group|group]] of [[Definition:Cardinality|cardinality]] $1$.
From [[Group is not Empty]], there can be no [[Definition:Group|group]] of smaller [[Definition:Order of Group|order]].
{{qed}} | Trivial Group is Smallest Group | https://proofwiki.org/wiki/Trivial_Group_is_Smallest_Group | https://proofwiki.org/wiki/Trivial_Group_is_Smallest_Group | [
"Trivial Group"
] | [
"Definition:Trivial Group",
"Definition:Group",
"Definition:Set",
"Definition:Cardinality",
"Definition:Underlying Set/Abstract Algebra",
"Definition:Group"
] | [
"Trivial Group is Group",
"Definition:Group",
"Definition:Cardinality",
"Group is not Empty",
"Definition:Group",
"Definition:Order of Structure"
] |
proofwiki-16599 | Sum to Infinity of Reciprocal of n^4 by 2n Choose n | It is conjectured that:
:$\map \zeta 4 = \ds \dfrac {36} {17} \sum_{n \mathop = 1}^\infty \dfrac 1 {n^4 \dbinom {2 n} n}$ | {{ProofWanted|The significance of this result is not clear.}} | It is conjectured that:
:$\map \zeta 4 = \ds \dfrac {36} {17} \sum_{n \mathop = 1}^\infty \dfrac 1 {n^4 \dbinom {2 n} n}$ | {{ProofWanted|The significance of this result is not clear.}} | Sum to Infinity of Reciprocal of n^4 by 2n Choose n | https://proofwiki.org/wiki/Sum_to_Infinity_of_Reciprocal_of_n^4_by_2n_Choose_n | https://proofwiki.org/wiki/Sum_to_Infinity_of_Reciprocal_of_n^4_by_2n_Choose_n | [
"Central Binomial Coefficients",
"Riemann Zeta Function"
] | [] | [] |
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