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-22400 | Category of Semilattices is Category | Let $\mathbf{SLat}$ denote the category of semilattices.
Then:
:$\mathbf{SLat}$ is a metacategory | Let us verify the axioms $(\text C 1)$ up to $(\text C 3)$ for a metacategory.
For any two semilattice homomorphisms their composition (in the usual set theoretic sense) is again a semilattice homomorphism by Composite Semilattice Homomorphisms is Semilattice Homomorphism.
For any semilattice $\struct{S, \circ}$, we ha... | Let $\mathbf{SLat}$ denote the [[Definition:Category of Semilattices|category of semilattices]].
Then:
:$\mathbf{SLat}$ is a [[Definition:Metacategory|metacategory]] | Let us verify the axioms $(\text C 1)$ up to $(\text C 3)$ for a [[Definition:Metacategory|metacategory]].
For any two [[Definition:Semilattice Homomorphism|semilattice homomorphisms]] their [[Definition:Composition of Mappings|composition]] (in the usual [[Definition:Set Theory|set theoretic]] sense) is again a [[De... | Category of Semilattices is Category | https://proofwiki.org/wiki/Category_of_Semilattices_is_Category | https://proofwiki.org/wiki/Category_of_Semilattices_is_Category | [
"Category of Semilattices"
] | [
"Definition:Category of Semilattices",
"Definition:Metacategory"
] | [
"Definition:Metacategory",
"Definition:Semilattice Homomorphism",
"Definition:Composition of Mappings",
"Definition:Set Theory",
"Definition:Semilattice Homomorphism",
"Composite Semilattice Homomorphisms is Semilattice Homomorphism",
"Definition:Semilattice",
"Definition:Identity Mapping",
"Identit... |
proofwiki-22401 | Composite Semilattice Homomorphisms is Semilattice Homomorphism | Let $S_1 = \struct{A_1, \circ_1}$, $S_2 = \struct{A_2, \circ_2}$ and $S_3 = \struct{A_3, \circ_3}$ be semilattices.
Let $\phi_1: S_1 \to S_2$ and $\phi_2: S_2 \to S_3$ be semilattice homomorphisms.
Let $\phi_2 \circ \phi_1 : A_1 \to A_3$ be the composite mapping of $\phi_1$ and $\phi_2$
Then:
:$\phi_2 \circ \phi_1$ i... | Follows immediately from Composite of Homomorphisms is Homomorphism
{{qed}}
Category:Semilattice Homomorphisms
1et9giqbcjvdauvmpsn69utj7xxx85j | Let $S_1 = \struct{A_1, \circ_1}$, $S_2 = \struct{A_2, \circ_2}$ and $S_3 = \struct{A_3, \circ_3}$ be [[Definition:Semilattice|semilattices]].
Let $\phi_1: S_1 \to S_2$ and $\phi_2: S_2 \to S_3$ be [[Definition:Semilattice Homomorphism|semilattice homomorphisms]].
Let $\phi_2 \circ \phi_1 : A_1 \to A_3$ be the [[Def... | Follows immediately from [[Composite of Homomorphisms is Homomorphism]]
{{qed}}
[[Category:Semilattice Homomorphisms]]
1et9giqbcjvdauvmpsn69utj7xxx85j | Composite Semilattice Homomorphisms is Semilattice Homomorphism | https://proofwiki.org/wiki/Composite_Semilattice_Homomorphisms_is_Semilattice_Homomorphism | https://proofwiki.org/wiki/Composite_Semilattice_Homomorphisms_is_Semilattice_Homomorphism | [
"Semilattice Homomorphisms"
] | [
"Definition:Semilattice",
"Definition:Semilattice Homomorphism",
"Definition:Composition of Mappings",
"Definition:Semilattice Homomorphism"
] | [
"Composite of Homomorphisms is Homomorphism",
"Category:Semilattice Homomorphisms"
] |
proofwiki-22402 | Identity Mapping is Semilattice Homomorphism | Let $S = \struct{A, \circ}$ be a semilattice.
Let $\operatorname{id}_A$ denote the identity mapping on $A$.
Then:
:$\operatorname{id}_A$ is a semilattice homomorphism of $S$ to $S$ | Follows immediately from Identity Mapping is Automorphism.
{{qed}}
Category:Semilattice Homomorphisms
npgqu71tccimkqe5gc43a4q5ky38tmm | Let $S = \struct{A, \circ}$ be a [[Definition:Semilattice|semilattice]].
Let $\operatorname{id}_A$ denote the [[Definition:Identity Mapping|identity mapping]] on $A$.
Then:
:$\operatorname{id}_A$ is a [[Definition:Semilattice Homomorphism|semilattice homomorphism]] of $S$ to $S$ | Follows immediately from [[Identity Mapping is Automorphism]].
{{qed}}
[[Category:Semilattice Homomorphisms]]
npgqu71tccimkqe5gc43a4q5ky38tmm | Identity Mapping is Semilattice Homomorphism | https://proofwiki.org/wiki/Identity_Mapping_is_Semilattice_Homomorphism | https://proofwiki.org/wiki/Identity_Mapping_is_Semilattice_Homomorphism | [
"Semilattice Homomorphisms"
] | [
"Definition:Semilattice",
"Definition:Identity Mapping",
"Definition:Semilattice Homomorphism"
] | [
"Identity Mapping is Automorphism",
"Category:Semilattice Homomorphisms"
] |
proofwiki-22403 | Scalene Triangle Tessellates the Plane | Let $T$ be a triangle, which can be any shape at all, even scalene.
Then $T$ can tessellate the plane. | By placing $2$ copies of $T$ together so their corresponding sides coincide, it is possible to form a quadrilateral.
The result follows from Quadrilateral Tessellates the Plane.
{{qed}} | Let $T$ be a [[Definition:Triangle (Geometry)|triangle]], which can be any shape at all, even [[Definition:Scalene Triangle|scalene]].
Then $T$ can [[Definition:Tessellation|tessellate]] [[Definition:The Plane|the plane]]. | By placing $2$ copies of $T$ together so their corresponding [[Definition:Side of Polygon|sides]] coincide, it is possible to form a [[Definition:Quadrilateral|quadrilateral]].
The result follows from [[Quadrilateral Tessellates the Plane]].
{{qed}} | Scalene Triangle Tessellates the Plane | https://proofwiki.org/wiki/Scalene_Triangle_Tessellates_the_Plane | https://proofwiki.org/wiki/Scalene_Triangle_Tessellates_the_Plane | [
"Scalene Triangles",
"Tessellations"
] | [
"Definition:Triangle (Geometry)",
"Definition:Triangle (Geometry)/Scalene",
"Definition:Tessellation",
"Definition:Plane Surface/The Plane"
] | [
"Definition:Polygon/Side",
"Definition:Quadrilateral",
"Quadrilateral Tessellates the Plane"
] |
proofwiki-22404 | Surface Area of Torus/Formulation 1 | Let $\TT$ be a torus.
Let $r$ be the radius of the generating circle of $\TT$.
Let $R$ be the distance of the center of the generating circle from the axis of revolution of $\TT$.
Then the area $\AA$ of $\TT$ is given by:
:$\AA = 4 \pi^2 r R$ | Recall Pappus's Centroid Theorem for Surface Area:
{{:Pappus's Centroid Theorem for Surface Area}}
In this context:
:$C$ is the generating circle of $\TT$, which has radius $r$
:the centroid of $C$ is the center of $C$, which is at a distance $R$ from the axis of revolution.
From Perimeter of Circle, the perimeter of $... | Let $\TT$ be a [[Definition:Torus (Geometry)|torus]].
Let $r$ be the [[Definition:Radius of Circle|radius]] of the [[Definition:Generating Curve of Surface of Revolution|generating]] [[Definition:Circle|circle]] of $\TT$.
Let $R$ be the [[Definition:Perpendicular Distance between Point and Straight Line|distance]] of... | Recall [[Pappus's Centroid Theorem for Surface Area]]:
{{:Pappus's Centroid Theorem for Surface Area}}
In this context:
:$C$ is the [[Definition:Generating Curve of Surface of Revolution|generating]] [[Definition:Circle|circle]] of $\TT$, which has [[Definition:Radius of Circle|radius $r$]]
:the [[Definition:Centroi... | Surface Area of Torus/Formulation 1 | https://proofwiki.org/wiki/Surface_Area_of_Torus/Formulation_1 | https://proofwiki.org/wiki/Surface_Area_of_Torus/Formulation_1 | [
"Surface Area of Torus"
] | [
"Definition:Torus (Geometry)",
"Definition:Circle/Radius",
"Definition:Generating Curve of Surface of Revolution",
"Definition:Circle",
"Definition:Perpendicular Distance between Point and Straight Line",
"Definition:Circle/Center",
"Definition:Generating Curve of Surface of Revolution",
"Definition:C... | [
"Second Pappus-Guldinus Theorem",
"Definition:Generating Curve of Surface of Revolution",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Centroid",
"Definition:Circle/Center",
"Definition:Perpendicular Distance between Point and Straight Line",
"Definition:Axis of Revolution",
"Perimet... |
proofwiki-22405 | Volume of Torus/Formulation 1 | Let $\TT$ be a torus.
Let $r$ be the radius of the generating circle of $\TT$.
Let $R$ be the distance of the center of the generating circle from the axis of revolution of $\TT$.
Then the volume $\VV$ enclosed by $\TT$ is given by:
:$\VV = 2 \pi^2 r^2 R$ | Recall Pappus's Centroid Theorem for Volume:
{{:Pappus's Centroid Theorem for Volume}}
In this context:
:$C$ is the generating circle of $\TT$, which has radius $r$
:the centroid of $C$ is the center of $C$, which is at a distance $R$ from the axis of revolution.
From Area of Circle, the area of $C$ is $\pi r^2$.
From ... | Let $\TT$ be a [[Definition:Torus (Geometry)|torus]].
Let $r$ be the [[Definition:Radius of Circle|radius]] of the [[Definition:Generating Curve of Surface of Revolution|generating]] [[Definition:Circle|circle]] of $\TT$.
Let $R$ be the [[Definition:Perpendicular Distance between Point and Straight Line|distance]] of... | Recall [[Pappus's Centroid Theorem for Volume]]:
{{:Pappus's Centroid Theorem for Volume}}
In this context:
:$C$ is the [[Definition:Generating Curve of Surface of Revolution|generating]] [[Definition:Circle|circle]] of $\TT$, which has [[Definition:Radius of Circle|radius $r$]]
:the [[Definition:Centroid|centroid]]... | Volume of Torus/Formulation 1 | https://proofwiki.org/wiki/Volume_of_Torus/Formulation_1 | https://proofwiki.org/wiki/Volume_of_Torus/Formulation_1 | [
"Volume of Torus"
] | [
"Definition:Torus (Geometry)",
"Definition:Circle/Radius",
"Definition:Generating Curve of Surface of Revolution",
"Definition:Circle",
"Definition:Perpendicular Distance between Point and Straight Line",
"Definition:Circle/Center",
"Definition:Generating Curve of Surface of Revolution",
"Definition:C... | [
"First Pappus-Guldinus Theorem",
"Definition:Generating Curve of Surface of Revolution",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Centroid",
"Definition:Circle/Center",
"Definition:Perpendicular Distance between Point and Straight Line",
"Definition:Axis of Revolution",
"Area of ... |
proofwiki-22406 | Equivalence of Definitions of Embedding of Categories | Let $\mathbf C$ and $\mathbf D$ be categories.
Let $F : \mathbf C \to \mathbf D$ be a functor.
{{TFAE|def=Embedding of Categories}}
=== Definition 1 ===
{{:Definition:Embedding of Categories/Definition 1}}
=== Definition 2 ===
{{:Definition:Embedding of Categories/Definition 2}}
=== Definition 3 ===
{{:Definition:Embed... | {{ProofWanted}}
Category:Embeddings of Categories
masz0f4irqjpkki8dub7whs14e559da | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Category|categories]].
Let $F : \mathbf C \to \mathbf D$ be a [[Definition:Functor|functor]].
{{TFAE|def=Embedding of Categories}}
=== [[Definition:Embedding of Categories/Definition 1|Definition 1]] ===
{{:Definition:Embedding of Categories/Definition 1}}
=== [[Defin... | {{ProofWanted}}
[[Category:Embeddings of Categories]]
masz0f4irqjpkki8dub7whs14e559da | Equivalence of Definitions of Embedding of Categories | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Embedding_of_Categories | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Embedding_of_Categories | [
"Embeddings of Categories"
] | [
"Definition:Category",
"Definition:Functor",
"Definition:Embedding of Categories/Definition 1",
"Definition:Embedding of Categories/Definition 2",
"Definition:Embedding of Categories/Definition 3"
] | [
"Category:Embeddings of Categories"
] |
proofwiki-22407 | Transpose of Column Matrix is Row Matrix | Let $\mathbf x = \sqbrk x_{1 n} = \begin {bmatrix} x_1 & x_2 & \cdots & x_n \end {bmatrix}$ be a column matrix.
Then $\mathbf x^\intercal$, the transpose of $\mathbf x$, is a row matrix:
:$\begin {bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}^\intercal = \begin {bmatrix} x_1 & x_2 & \cdots & x_n \end {bmatrix}$ | Self-evident.
{{Qed}} | Let $\mathbf x = \sqbrk x_{1 n} = \begin {bmatrix} x_1 & x_2 & \cdots & x_n \end {bmatrix}$ be a [[Definition:Column Matrix|column matrix]].
Then $\mathbf x^\intercal$, the [[Definition:Transpose of Matrix|transpose]] of $\mathbf x$, is a [[Definition:Row Matrix|row matrix]]:
:$\begin {bmatrix} x_1 \\ x_2 \\ \vdots ... | Self-evident.
{{Qed}} | Transpose of Column Matrix is Row Matrix | https://proofwiki.org/wiki/Transpose_of_Column_Matrix_is_Row_Matrix | https://proofwiki.org/wiki/Transpose_of_Column_Matrix_is_Row_Matrix | [
"Transposes of Matrices",
"column Matrices",
"Row Matrices"
] | [
"Definition:Column Matrix",
"Definition:Transpose of Matrix",
"Definition:Row Matrix"
] | [] |
proofwiki-22408 | Cotangent is Cosecant divided by Secant | Let $\theta$ be an angle such that $\sin \theta \ne 0$.
Then:
:$\cot \theta = \dfrac {\cosec \theta} {\sec \theta}$
where $\cot$, $\cosec$ and $\sec$ mean cotangent, cosecant and secant respectively. | {{begin-eqn}}
{{eqn | l = \cot \theta
| r = \dfrac {\cos \theta} {\sin \theta}
| c = Cotangent is Cosine divided by Sine, which holds when $\sin \theta \ne 0$
}}
{{eqn | r = \dfrac {1 / \sec \theta} {1 / \cosec \theta}
| c = Secant is Reciprocal of Cosine, Cosecant is Reciprocal of Sine
}}
{{eqn | r =... | Let $\theta$ be an [[Definition:Angle|angle]] such that $\sin \theta \ne 0$.
Then:
:$\cot \theta = \dfrac {\cosec \theta} {\sec \theta}$
where $\cot$, $\cosec$ and $\sec$ mean [[Definition:Cotangent of Angle|cotangent]], [[Definition:Cosecant of Angle|cosecant]] and [[Definition:Secant of Angle|secant]] respectively. | {{begin-eqn}}
{{eqn | l = \cot \theta
| r = \dfrac {\cos \theta} {\sin \theta}
| c = [[Cotangent is Cosine divided by Sine]], which holds when $\sin \theta \ne 0$
}}
{{eqn | r = \dfrac {1 / \sec \theta} {1 / \cosec \theta}
| c = [[Secant is Reciprocal of Cosine]], [[Cosecant is Reciprocal of Sine]]
}}... | Cotangent is Cosecant divided by Secant | https://proofwiki.org/wiki/Cotangent_is_Cosecant_divided_by_Secant | https://proofwiki.org/wiki/Cotangent_is_Cosecant_divided_by_Secant | [
"Secant Function",
"Cosecant Function",
"Cotangent Function"
] | [
"Definition:Angle",
"Definition:Cotangent/Definition from Triangle",
"Definition:Cosecant/Definition from Triangle",
"Definition:Secant Function/Definition from Triangle"
] | [
"Cotangent is Cosine divided by Sine",
"Secant is Reciprocal of Cosine",
"Cosecant is Reciprocal of Sine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-22409 | Sine of Obtuse Angle is Positive | Let $\theta$ be an obtuse angle.
Then:
:$\sin \theta > 0$
where $\sin$ denotes the sine function. | {{ProofWanted|trivial but I don't want to be distracted.}} | Let $\theta$ be an [[Definition:Obtuse Angle|obtuse angle]].
Then:
:$\sin \theta > 0$
where $\sin$ denotes the [[Definition:Sine Function|sine function]]. | {{ProofWanted|trivial but I don't want to be distracted.}} | Sine of Obtuse Angle is Positive | https://proofwiki.org/wiki/Sine_of_Obtuse_Angle_is_Positive | https://proofwiki.org/wiki/Sine_of_Obtuse_Angle_is_Positive | [
"Sine Function",
"Obtuse Angles"
] | [
"Definition:Obtuse Angle",
"Definition:Sine"
] | [] |
proofwiki-22410 | Cosine of Obtuse Angle is Negative | Let $\theta$ be an obtuse angle.
Then:
:$\cos \theta < 0$
where $\cos$ denotes the cosine function. | Let $\theta$ be an obtuse angle.
By definition of obtuse angle:
:$90 \degrees < \theta < 180 \degrees$
Let us align $\theta$ in a Cartesian plane such that:
:$\theta$ itself is at the origin
:one arm of $\theta$ is aligned with the positive $x$-axis.
Then the other arm of $\theta$ is in either the second quadrant or th... | Let $\theta$ be an [[Definition:Obtuse Angle|obtuse angle]].
Then:
:$\cos \theta < 0$
where $\cos$ denotes the [[Definition:Cosine Function|cosine function]]. | Let $\theta$ be an [[Definition:Obtuse Angle|obtuse angle]].
By definition of [[Definition:Obtuse Angle|obtuse angle]]:
:$90 \degrees < \theta < 180 \degrees$
Let us align $\theta$ in a [[Definition:Cartesian Plane|Cartesian plane]] such that:
:$\theta$ itself is at the [[Definition:Origin|origin]]
:one [[Definition:... | Cosine of Obtuse Angle is Negative | https://proofwiki.org/wiki/Cosine_of_Obtuse_Angle_is_Negative | https://proofwiki.org/wiki/Cosine_of_Obtuse_Angle_is_Negative | [
"Cosine Function",
"Obtuse Angles"
] | [
"Definition:Obtuse Angle",
"Definition:Cosine"
] | [
"Definition:Obtuse Angle",
"Definition:Obtuse Angle",
"Definition:Cartesian Plane",
"Definition:Coordinate System/Origin",
"Definition:Angle/Arm",
"Definition:Axis/Positive Direction",
"Definition:Axis/X-Axis",
"Definition:Angle/Arm",
"Definition:Cartesian Plane/Quadrants/Second",
"Definition:Cart... |
proofwiki-22411 | Tangent of Obtuse Angle is Negative | Let $\theta$ be an obtuse angle.
Then:
:$\tan \theta < 0$
where $\tan$ denotes the tangent function. | {{ProofWanted|trivial but I don't want to be distracted.}} | Let $\theta$ be an [[Definition:Obtuse Angle|obtuse angle]].
Then:
:$\tan \theta < 0$
where $\tan$ denotes the [[Definition:Tangent Function|tangent function]]. | {{ProofWanted|trivial but I don't want to be distracted.}} | Tangent of Obtuse Angle is Negative | https://proofwiki.org/wiki/Tangent_of_Obtuse_Angle_is_Negative | https://proofwiki.org/wiki/Tangent_of_Obtuse_Angle_is_Negative | [
"Tangent Function",
"Obtuse Angles"
] | [
"Definition:Obtuse Angle",
"Definition:Tangent Function"
] | [] |
proofwiki-22412 | Scalar Triple Product as Product of Magnitudes | Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be vectors in a Cartesian $3$-space:
{{begin-eqn}}
{{eqn | l = \mathbf a
| r = a_i \mathbf i + a_j \mathbf j + a_k \mathbf k
}}
{{eqn | l = \mathbf b
| r = b_i \mathbf i + b_j \mathbf j + b_k \mathbf k
}}
{{eqn | l = \mathbf c
| r = c_i \mathbf i + c_j \mat... | We have:
{{begin-eqn}}
{{eqn | l = \sqbrk {\mathbf a, \mathbf b, \mathbf c}
| r = \mathbf a \cdot \paren {\mathbf b \times \mathbf c}
| c = {{Defof|Scalar Triple Product}}
}}
{{eqn | r = \size {\mathbf a} \size {\mathbf b \times \mathbf c} \cos \alpha
| c = {{Defof|Dot Product}} in real Euclidean spac... | Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be [[Definition:Vector Quantity|vectors]] in a [[Definition:Cartesian 3-Space|Cartesian $3$-space]]:
{{begin-eqn}}
{{eqn | l = \mathbf a
| r = a_i \mathbf i + a_j \mathbf j + a_k \mathbf k
}}
{{eqn | l = \mathbf b
| r = b_i \mathbf i + b_j \mathbf j + b_k \mathb... | We have:
{{begin-eqn}}
{{eqn | l = \sqbrk {\mathbf a, \mathbf b, \mathbf c}
| r = \mathbf a \cdot \paren {\mathbf b \times \mathbf c}
| c = {{Defof|Scalar Triple Product}}
}}
{{eqn | r = \size {\mathbf a} \size {\mathbf b \times \mathbf c} \cos \alpha
| c = {{Defof|Dot Product}} in [[Definition:Real E... | Scalar Triple Product as Product of Magnitudes | https://proofwiki.org/wiki/Scalar_Triple_Product_as_Product_of_Magnitudes | https://proofwiki.org/wiki/Scalar_Triple_Product_as_Product_of_Magnitudes | [
"Scalar Triple Product"
] | [
"Definition:Vector Quantity",
"Definition:Cartesian 3-Space",
"Definition:Standard Ordered Basis/Vector Space",
"Definition:Scalar Triple Product",
"Definition:Angle",
"Definition:Angle"
] | [
"Definition:Euclidean Space/Real"
] |
proofwiki-22413 | Velocity of Point in Straight Line | Let $\mathbf v$ be the velocity of a particle $P$ in space.
Let $P$ be moving along a straight line $\LL$ whose positive direction has been established.
Then the motion of $P$ can be defined by:
:$\mathbf v = \dfrac {\d s} {\d t} \mathbf i$
where $\mathbf i$ denotes the unit vector in the positive direction of $\LL$. | We have {{hypothesis}} that $P$ moves along a straight line $\LL$.
Then the rate of change of displacement perpendicular to $\LL$ is zero.
Let the $\LL$ be embedded in a Cartesian space $\CC$.
From the components of velocity vector, we have:
:$\mathbf v = \dfrac {\d \mathbf r} {\d t}$
where $\mathbf r$ is the displacem... | Let $\mathbf v$ be the [[Definition:Velocity|velocity]] of a [[Definition:Particle|particle]] $P$ in [[Definition:Ordinary Space|space]].
Let $P$ be [[Definition:Motion|moving]] along a [[Definition:Straight Line|straight line]] $\LL$ whose [[Definition:Positive Direction|positive direction]] has been established.
T... | We have {{hypothesis}} that $P$ [[Definition:Motion|moves]] along a [[Definition:Straight Line|straight line]] $\LL$.
Then the [[Definition:Rate of Change|rate of change]] of [[Definition:Displacement|displacement]] [[Definition:Perpendicular|perpendicular]] to $\LL$ is [[Definition:Zero Vector|zero]].
Let the $\LL$... | Velocity of Point in Straight Line | https://proofwiki.org/wiki/Velocity_of_Point_in_Straight_Line | https://proofwiki.org/wiki/Velocity_of_Point_in_Straight_Line | [
"Velocity"
] | [
"Definition:Velocity",
"Definition:Particle",
"Definition:Ordinary Space",
"Definition:Motion",
"Definition:Line/Straight Line",
"Definition:Axis/Positive Direction",
"Definition:Motion",
"Definition:Unit Vector",
"Definition:Axis/Positive Direction"
] | [
"Definition:Motion",
"Definition:Line/Straight Line",
"Definition:Rate of Change",
"Definition:Displacement",
"Definition:Right Angle/Perpendicular",
"Definition:Zero Vector",
"Definition:Cartesian Product/Cartesian Space",
"Component of Vector/Examples/Velocity",
"Definition:Displacement",
"Defin... |
proofwiki-22414 | Draft:Union of Set of Sets is Greatest Element under Subset Relation | Let $M$ be a set.
Let $\bigcup M \in M$.
Let $(M, \subseteq)$ be the ordered set formed on $M$ by the subset relation (see Subset Relation is Ordering).
Then $\bigcup M$ is the greatest set by set inclusion ($M$ corresponds to $\TT$) of $(M, \subseteq)$. | By Set is Subset of Union:
:$\forall \paren {N \in M}: N \subseteq \bigcup M$.
Therefore, $\bigcup M$ is the greatest element under the subset relation.
{{qed}}
Category:Set Union
t8lxntmh3eyvirjlfrkqmh01w7u7mtb | Let $M$ be a [[Definition:Set|set]].
Let $\bigcup M \in M$.
Let $(M, \subseteq)$ be the [[Definition:Ordered Set|ordered set]] formed on $M$ by the [[Definition:Subset Relation|subset relation]] (see [[Subset Relation is Ordering]]).
Then $\bigcup M$ is the [[Definition:Greatest Set by Set Inclusion|greatest set by... | By [[Set is Subset of Union/Set of Sets|Set is Subset of Union]]:
:$\forall \paren {N \in M}: N \subseteq \bigcup M$.
Therefore, $\bigcup M$ is the [[Definition:Greatest Element|greatest element]] under the subset relation.
{{qed}}
[[Category:Set Union]]
t8lxntmh3eyvirjlfrkqmh01w7u7mtb | Draft:Union of Set of Sets is Greatest Element under Subset Relation | https://proofwiki.org/wiki/Draft:Union_of_Set_of_Sets_is_Greatest_Element_under_Subset_Relation | https://proofwiki.org/wiki/Draft:Union_of_Set_of_Sets_is_Greatest_Element_under_Subset_Relation | [
"Set Union"
] | [
"Definition:Set",
"Definition:Ordered Set",
"Definition:Subset Relation",
"Subset Relation is Ordering",
"Definition:Greatest Set by Set Inclusion"
] | [
"Set is Subset of Union/Set of Sets",
"Definition:Greatest Element",
"Category:Set Union"
] |
proofwiki-22415 | Binomial Coefficient over Power Not Greater than Reciprocal of Factorial | Let $r > 0$.
Let $k \in \N$ such that $k \le 2r + 1$.
Then:
:$\dfrac {\dbinom r k} {r^k} \le \dfrac 1 {k!}$ | {{begin-eqn}}
{{eqn | l = \frac {\binom r k} {r^k}
| r = \frac 1 {k !} \cdot \frac {r^{\underline k} } {r^k}
| c = {{Defof|Binomial Coefficient/Real Numbers|Binomial Coefficient}}
}}
{{eqn | r = \frac 1 {k !} \prod_{j \mathop = 0}^{k - 1} \frac {r - j} r
| c = {{Defof|Falling Factorial}}, {{Defof|Inte... | Let $r > 0$.
Let $k \in \N$ such that $k \le 2r + 1$.
Then:
:$\dfrac {\dbinom r k} {r^k} \le \dfrac 1 {k!}$ | {{begin-eqn}}
{{eqn | l = \frac {\binom r k} {r^k}
| r = \frac 1 {k !} \cdot \frac {r^{\underline k} } {r^k}
| c = {{Defof|Binomial Coefficient/Real Numbers|Binomial Coefficient}}
}}
{{eqn | r = \frac 1 {k !} \prod_{j \mathop = 0}^{k - 1} \frac {r - j} r
| c = {{Defof|Falling Factorial}}, {{Defof|Inte... | Binomial Coefficient over Power Not Greater than Reciprocal of Factorial | https://proofwiki.org/wiki/Binomial_Coefficient_over_Power_Not_Greater_than_Reciprocal_of_Factorial | https://proofwiki.org/wiki/Binomial_Coefficient_over_Power_Not_Greater_than_Reciprocal_of_Factorial | [
"Binomial Coefficients",
"Factorials"
] | [] | [
"Absolute Value Function is Completely Multiplicative",
"Negative of Absolute Value",
"Category:Binomial Coefficients",
"Category:Factorials"
] |
proofwiki-22416 | Exponential Sequence Converges Compactly | For each $n \in \N$, let $f_n : \C \to \C$ be defined as:
:$\map {f_n} z = \paren {1 + \dfrac z n}^n$
Then, $\sequence {f_n}$ converges compactly to the complex exponential function. | Let $D$ be a compact subset of $\C$.
By definition, $D$ is bounded, so there is some $M \in \R$ such that:
:$\forall z \in D: \size z \le M$
Let $\epsilon > 0$ be arbitrary.
By definition of the exponential function, there is some $N \in \N$ such that:
:$\forall n \ge N: \map \exp M - \paren {1 + \dfrac M n}^n < \epsil... | For each $n \in \N$, let $f_n : \C \to \C$ be defined as:
:$\map {f_n} z = \paren {1 + \dfrac z n}^n$
Then, $\sequence {f_n}$ [[Definition:Compact Convergence|converges compactly]] to the [[Definition:Exponential Function/Complex|complex exponential function]]. | Let $D$ be a [[Definition:Compact Subset of Complex Plane|compact subset]] of $\C$.
By definition, $D$ is [[Definition:Bounded Subset of Complex Plane|bounded]], so there is some $M \in \R$ such that:
:$\forall z \in D: \size z \le M$
Let $\epsilon > 0$ be arbitrary.
By definition of the [[Definition:Real Exponenti... | Exponential Sequence Converges Compactly | https://proofwiki.org/wiki/Exponential_Sequence_Converges_Compactly | https://proofwiki.org/wiki/Exponential_Sequence_Converges_Compactly | [
"Compact Convergence",
"Exponential Function"
] | [
"Definition:Compact Convergence",
"Definition:Exponential Function/Complex"
] | [
"Definition:Compact Space/Metric Space/Complex",
"Definition:Bounded Metric Space/Complex",
"Definition:Exponential Function/Real",
"Binomial Theorem/Integral Index",
"Triangle Inequality/Complex Numbers",
"Binomial Coefficient over Power Not Greater than Reciprocal of Factorial",
"Binomial Theorem/Inte... |
proofwiki-22417 | Wald's Equation | Let $\sequence {X_n}_{n \ge 0}$ be a sequence of i.i.d. real-valued random variables.
Let $\sequence {\FF_n}_{n \ge 0}$ be the filtration generated by $\sequence {X_i}_{n \ge 0}$, that is:
:$\FF_n := \map \sigma {X_0, \ldots, X_n}$
the $\sigma$-algebra generated by $X_0, \ldots ,X_n$.
Let $T$ be a stopping time with re... | Let $M,N > 0$.
Let $\mathbb I = \closedint {-M} N$.
Let $\map {\phi_\mathbb{I} }x := \begin{cases} x & : x \in \mathbb I \\ 0 & : x \notin \mathbb I \end{cases}$.
For $n\ge 0$, let:
:$X^\mathbb{I}_n := \map {\phi_\mathbb{I} } {X_n}$
:$S^\mathbb{I}_n := X^\mathbb{I}_0 + \cdots + X^\mathbb{I}_n$
:$M^\mathbb{I}_n := S^\ma... | Let $\sequence {X_n}_{n \ge 0}$ be a [[Definition:Sequence|sequence]] of [[Definition:Independent and Identically Distributed|i.i.d.]] [[Definition:Real-Valued Random Variable|real-valued random variables]].
Let $\sequence {\FF_n}_{n \ge 0}$ be the [[Definition:Filtration of Sigma-Algebra/Discrete Time|filtration]] ge... | Let $M,N > 0$.
Let $\mathbb I = \closedint {-M} N$.
Let $\map {\phi_\mathbb{I} }x := \begin{cases} x & : x \in \mathbb I \\ 0 & : x \notin \mathbb I \end{cases}$.
For $n\ge 0$, let:
:$X^\mathbb{I}_n := \map {\phi_\mathbb{I} } {X_n}$
:$S^\mathbb{I}_n := X^\mathbb{I}_0 + \cdots + X^\mathbb{I}_n$
:$M^\mathbb{I}_n := S^... | Wald's Equation | https://proofwiki.org/wiki/Wald's_Equation | https://proofwiki.org/wiki/Wald's_Equation | [
"Stopping Times"
] | [
"Definition:Sequence",
"Definition:Random Sample (Probability Theory)",
"Definition:Random Variable/Real-Valued",
"Definition:Filtration of Sigma-Algebra/Discrete Time",
"Definition:Sigma-Algebra Generated by Collection of Random Variables",
"Definition:Stopping Time/Discrete Time"
] | [
"Definition:Martingale",
"Doob's Optional Stopping Theorem/Discrete Time/Martingale",
"Monotone Convergence Theorem (Measure Theory)"
] |
proofwiki-22418 | Convergent Sequences Characterize Metrizable Topology | Let $X$ be a set.
Let $\tau_1$ and $\tau_2$ be metrizable topologies induced by metrics $d_1$ and $d_2$ respectively.
Suppose that:
:a sequence $\sequence {x_n}_{n \in \N} \subseteq X$ converges to $x \in X$ in $\struct {X, d_1}$ {{iff}} it converges to $x$ in $\struct {X, d_2}$.
Then $\tau_1 = \tau_2$. | Let $U$ be open in $\struct {X, d_1}$.
Then $X \setminus U$ is closed in $\struct {X, d_1}$.
That is, from the definition of closedness in a metric space, for each $x \in X \setminus U$ there exists a sequence $\sequence {x_n}_{n \in \N}$ in $X \setminus U$ converging to $x$ in $\struct {X, d_1}$.
By hypothesis, each ... | Let $X$ be a [[Definition:Set|set]].
Let $\tau_1$ and $\tau_2$ be [[Definition:Metrizable Topology|metrizable topologies]] [[Definition:Topology Induced by Metric|induced by]] [[Definition:Metric|metrics]] $d_1$ and $d_2$ respectively.
Suppose that:
:a [[Definition:Sequence|sequence]] $\sequence {x_n}_{n \in \N} \su... | Let $U$ be [[Definition:Open Set (Topology)|open]] in $\struct {X, d_1}$.
Then $X \setminus U$ is [[Definition:Closed Set (Topology)|closed]] in $\struct {X, d_1}$.
That is, from the definition of [[Definition:Closed Set (Metric Space)|closedness in a metric space]], for each $x \in X \setminus U$ there exists a [[D... | Convergent Sequences Characterize Metrizable Topology | https://proofwiki.org/wiki/Convergent_Sequences_Characterize_Metrizable_Topology | https://proofwiki.org/wiki/Convergent_Sequences_Characterize_Metrizable_Topology | [
"Convergent Sequences (Metric Space)",
"Metric Spaces",
"Convergent Sequences (Metric Space)"
] | [
"Definition:Set",
"Definition:Metrizable Space",
"Definition:Topology Induced by Metric",
"Definition:Metric Space/Metric",
"Definition:Sequence",
"Definition:Convergent Sequence/Metric Space",
"Definition:Convergent Sequence/Metric Space"
] | [
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Metric Space",
"Definition:Sequence",
"Definition:Convergent Sequence",
"Definition:Convergent Sequence",
"Definition:Closed Set/Metric Space",
"Definition:Open Set/Topology",
"Category:Metric Spaces",
"Catego... |
proofwiki-22419 | Open Subspace of Polish Space is Polish Space | Let $\struct {X, \tau}$ be a Polish space.
Let $d$ be a metric on $X$ that induces $\tau$ and is such that $\struct {X, d}$ is a complete metric space.
Let $U \subseteq X$ be open.
Let $\tau_U$ be the subspace topology on $U$ induced by $\tau$.
Then $\struct {U, \tau_U}$ is a Polish space. | From Subspace of Separable Metric Space is Separable, $\struct {U, \tau_U}$ is separable.
Next, we have that $X \setminus U$ is closed in $\struct {X, \tau}$.
Define $f : U \to \hointr 0 \infty$ by:
:$\map f x = \map d {x, X \setminus U}$
for each $x \in U$.
From Point at Distance Zero from Closed Set is Element, we h... | Let $\struct {X, \tau}$ be a [[Definition:Polish Space|Polish space]].
Let $d$ be a [[Definition:Metric Space|metric]] on $X$ that [[Definition:Topology Induced by Metric|induces]] $\tau$ and is such that $\struct {X, d}$ is a [[Definition:Complete Metric Space|complete metric space]].
Let $U \subseteq X$ be [[Defini... | From [[Subspace of Separable Metric Space is Separable]], $\struct {U, \tau_U}$ is [[Definition:Separable Space|separable]].
Next, we have that $X \setminus U$ is [[Definition:Closed Set (Topology)|closed]] in $\struct {X, \tau}$.
Define $f : U \to \hointr 0 \infty$ by:
:$\map f x = \map d {x, X \setminus U}$
for ea... | Open Subspace of Polish Space is Polish Space | https://proofwiki.org/wiki/Open_Subspace_of_Polish_Space_is_Polish_Space | https://proofwiki.org/wiki/Open_Subspace_of_Polish_Space_is_Polish_Space | [
"Polish Spaces"
] | [
"Definition:Polish Space",
"Definition:Metric Space",
"Definition:Topology Induced by Metric",
"Definition:Complete Metric Space",
"Definition:Open Set/Topology",
"Definition:Topological Subspace",
"Definition:Polish Space"
] | [
"Subspace of Separable Metric Space is Separable",
"Definition:Separable Space",
"Definition:Closed Set/Topology",
"Point at Distance Zero from Closed Set is Element",
"Definition:Metric Space/Metric",
"Definition:Complete Metric Space",
"Definition:Topology Induced by Metric",
"Definition:Complete Me... |
proofwiki-22420 | Primitive of Arccosine of a x | :$\ds \int \arccos a x \rd x = x \arccos a x - \dfrac 1 a \sqrt {1 - a^2 x^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \arccos x \rd x
| r = x \arccos x - \sqrt {1 - x^2} + C
| c = Primitive of $\arccos x$
}}
{{eqn | ll= \leadsto
| l = \int \arccos a x \rd x
| r = \dfrac 1 a \paren {\paren {a x} \arccos a x - \sqrt {1 - \paren {a x}^2} } + C
| c = Primitive of Function of Con... | :$\ds \int \arccos a x \rd x = x \arccos a x - \dfrac 1 a \sqrt {1 - a^2 x^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \arccos x \rd x
| r = x \arccos x - \sqrt {1 - x^2} + C
| c = [[Primitive of Arccosine Function|Primitive of $\arccos x$]]
}}
{{eqn | ll= \leadsto
| l = \int \arccos a x \rd x
| r = \dfrac 1 a \paren {\paren {a x} \arccos a x - \sqrt {1 - \paren {a x}^2} } + C
... | Primitive of Arccosine of a x | https://proofwiki.org/wiki/Primitive_of_Arccosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Arccosine_of_a_x | [
"Primitives involving Inverse Cosine Function"
] | [] | [
"Primitive of Arccosine Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-22421 | Sequence of Mappings Converges Pointwise iff Converges at Each Point | Let $D$ be a set.
Let $T$ be a topological space.
For each $n \in \N$, let $f_n : D \to T$ be a mapping.
Let $f : D \to T$ also be a mapping.
Then:
:$\sequence {f_n}$ converges pointwise to $f$
{{iff}}
:for every $x \in D$, $\sequence {\map {f_n} x}$ converges to $\map f x$ | Follows from Sequence on Product Space Converges to Point iff Projections Converge to Projections of Point, with:
:$I = D$
:$\forall i \in D: T_i = T$
{{qed}} | Let $D$ be a [[Definition:Set|set]].
Let $T$ be a [[Definition:Topological Space|topological space]].
For each $n \in \N$, let $f_n : D \to T$ be a [[Definition:Mapping|mapping]].
Let $f : D \to T$ also be a [[Definition:Mapping|mapping]].
Then:
:$\sequence {f_n}$ [[Definition:Pointwise Convergence/Topology|conver... | Follows from [[Sequence on Product Space Converges to Point iff Projections Converge to Projections of Point]], with:
:$I = D$
:$\forall i \in D: T_i = T$
{{qed}} | Sequence of Mappings Converges Pointwise iff Converges at Each Point | https://proofwiki.org/wiki/Sequence_of_Mappings_Converges_Pointwise_iff_Converges_at_Each_Point | https://proofwiki.org/wiki/Sequence_of_Mappings_Converges_Pointwise_iff_Converges_at_Each_Point | [
"Convergence"
] | [
"Definition:Set",
"Definition:Topological Space",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Pointwise Convergence/Topology",
"Definition:Convergent Sequence/Topology"
] | [
"Sequence on Product Space Converges to Point iff Projections Converge to Projections of Point"
] |
proofwiki-22422 | Center of Mass of Uniform Solid Hemisphere | Let $\BB$ be a solid hemisphere of radius $r$ of uniform density.
Then the center of mass of $\BB$ is the point $\dfrac {3 r} 8$ from the center of $\BB$ along the radius of $\BB$ perpendicular to the base of $\BB$. | Let $V$ be the volume of $\BB$.
Let the density $\map \rho {\mathbf r}$ of $\BB$ be a constant $\rho$.
Let $M = \rho V$ be the total mass of $\BB$.
Let $\mathbf r$ be a position within $V$.
Let the base of the hemisphere lie in the $x y$ plane centered at the origin with all of $\BB$ in the $z \geq 0$ region.
Let $\d V... | Let $\BB$ be a [[Definition:Solid Figure|solid]] [[Definition:Hemisphere|hemisphere]] of [[Definition:Radius of Hemisphere|radius]] $r$ of [[Definition:Uniform Density|uniform density]].
Then the [[Definition:Center of Mass|center of mass]] of $\BB$ is the [[Definition:Point|point]] $\dfrac {3 r} 8$ from the [[Definit... | Let $V$ be the [[Definition:Volume|volume]] of $\BB$.
Let the [[Definition:Mass Density|density]] $\map \rho {\mathbf r}$ of $\BB$ be a [[Definition:Constant|constant]] $\rho$.
Let $M = \rho V$ be the [[Definition:Mass|total mass]] of $\BB$.
Let $\mathbf r$ be a [[Definition:Position Vector|position]] within $V$.
L... | Center of Mass of Uniform Solid Hemisphere/Proof 1 | https://proofwiki.org/wiki/Center_of_Mass_of_Uniform_Solid_Hemisphere | https://proofwiki.org/wiki/Center_of_Mass_of_Uniform_Solid_Hemisphere/Proof_1 | [
"Center of Mass of Uniform Solid Hemisphere",
"Centers of Mass",
"Hemispheres",
"Uniform Density"
] | [
"Definition:Geometric Figure/Three-Dimensional Figure",
"Definition:Hemisphere",
"Definition:Hemisphere/Radius",
"Definition:Uniform Density",
"Definition:Center of Mass",
"Definition:Point",
"Definition:Hemisphere/Center",
"Definition:Hemisphere/Radius",
"Definition:Right Angle/Perpendicular/Plane"... | [
"Definition:Volume",
"Definition:Mass Density",
"Definition:Constant",
"Definition:Mass",
"Definition:Position Vector",
"Definition:Hemisphere/Base",
"Definition:Hemisphere",
"Definition:Cartesian Plane",
"Definition:Hemisphere/Center",
"Definition:Coordinate System/Origin",
"Definition:Region",... |
proofwiki-22423 | Equivalence of Definitions of Lattice Isomorphism | Let $L_1 = \struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ be lattices.
{{TFAE|def=Lattice Isomorphism}}
=== Definition 1 ===
{{:Definition:Lattice Isomorphism/Definition 1}}
=== Definition 2 ===
{{:Definition:Lattice Isomorphism/Definition 2}} | === Definition 1 implies Definition 2 ===
Let $\phi : L_1 \to L_2$ be a bijective lattice homomorphism.
{{:Equivalence of Definitions of Lattice Isomorphism/Definition 1 Implies Definition 2}}{{qed|lemma}} | Let $L_1 = \struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ be [[Definition:Lattice (Order Theory)|lattices]].
{{TFAE|def=Lattice Isomorphism}}
=== [[Definition:Lattice Isomorphism/Definition 1|Definition 1]] ===
{{:Definition:Lattice Isomorphism/Definition 1}}
=== ... | === [[Equivalence of Definitions of Lattice Isomorphism/Definition 1 Implies Definition 2|Definition 1 implies Definition 2]] ===
Let $\phi : L_1 \to L_2$ be a [[Definition:Bijection|bijective]] [[Definition:Lattice Homomorphism|lattice homomorphism]].
{{:Equivalence of Definitions of Lattice Isomorphism/Definition 1... | Equivalence of Definitions of Lattice Isomorphism | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Lattice_Isomorphism | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Lattice_Isomorphism | [
"Lattice Isomorphisms",
"Equivalence of Definitions of Lattice Isomorphism"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Lattice Isomorphism/Definition 1",
"Definition:Lattice Isomorphism/Definition 2"
] | [
"Equivalence of Definitions of Lattice Isomorphism/Definition 1 Implies Definition 2",
"Definition:Bijection",
"Definition:Lattice Homomorphism"
] |
proofwiki-22424 | Center of Mass of Uniform Circular Arc | Let $\WW$ be a wire of uniform density.
Let $\WW$ be bent into the shape of the arc of a circle $\CC$ of radius $r$ subtending an angle of $2 \alpha$ from the center of $\CC$.
Then the center of mass of $\WW$ is the point $\dfrac {r \sin \alpha} \alpha$ from the center of $\CC$. | Let $\lambda$ be the linear mass density of the wire.
From definition of radian, the total length of the wire is:
:$L = 2 \alpha r$
Therefore, the total mass of the wire is:
:$M = L \lambda = 2 \alpha r \lambda$
Arrange a coordinate system so that:
:the wire is within the $x$-$y$ plane
:the center of the circle contain... | Let $\WW$ be a [[Definition:Wire|wire]] of [[Definition:Uniform Density|uniform]] [[Definition:Linear Mass Density|density]].
Let $\WW$ be bent into the shape of the [[Definition:Arc of Circle|arc]] of a [[Definition:Circle|circle]] $\CC$ of [[Definition:Radius of Circle|radius]] $r$ [[Definition:Angle Subtended by Ar... | Let $\lambda$ be the [[Definition:Linear Mass Density|linear mass density]] of the [[Definition:Wire|wire]].
From definition of [[Definition:Radian|radian]], the total [[Definition:Arc Length|length]] of the [[Definition:Wire|wire]] is:
:$L = 2 \alpha r$
Therefore, the total [[Definition:Mass|mass]] of the [[Definiti... | Center of Mass of Uniform Circular Arc | https://proofwiki.org/wiki/Center_of_Mass_of_Uniform_Circular_Arc | https://proofwiki.org/wiki/Center_of_Mass_of_Uniform_Circular_Arc | [
"Centers of Mass",
"Circles",
"Wires",
"Uniform Density"
] | [
"Definition:Wire",
"Definition:Uniform Density",
"Definition:Mass Density/Linear",
"Definition:Circle/Arc",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Arc/Subtend",
"Definition:Angle",
"Definition:Circle/Center",
"Definition:Center of Mass",
"Definition:Point",
"Defini... | [
"Definition:Mass Density/Linear",
"Definition:Wire",
"Definition:Angular Measure/Radian",
"Definition:Arc Length",
"Definition:Wire",
"Definition:Mass",
"Definition:Wire",
"Definition:Coordinate System",
"Definition:Wire",
"Definition:Cartesian Plane",
"Definition:Circle/Center",
"Definition:C... |
proofwiki-22425 | Center of Mass of Uniform Circular Sector | Let $\WW$ be a uniform lamina in the shape of the sector of a circle $\CC$ of radius $r$ of an angle of $2 \alpha$.
Then the center of mass of $\PP$ is the point $\dfrac {2 r \sin \alpha} {3 \alpha}$ from the center of $\CC$. | Let the sector have a constant area mass density of $\sigma$.
Let $\CC$ lie in the $x$-$y$ plane of a Cartesian coordinate system.
Let the center of $\CC$ coincide with the origin.
Let the $x$-axis bisect the sector.
We have the center of mass equation:
:$\ds M \bar {\mathbf r} = \int_V \map \rho {\mathbf r} \mathbf r ... | Let $\WW$ be a [[Definition:Uniform Lamina|uniform lamina]] in the shape of the [[Definition:Sector of Circle|sector]] of a [[Definition:Circle|circle]] $\CC$ of [[Definition:Radius of Circle|radius]] $r$ of an [[Definition:Angle of Sector|angle]] of $2 \alpha$.
Then the [[Definition:Center of Mass|center of mass]] of... | Let the [[Definition:Sector of Circle|sector]] have a [[Definition:Constant|constant]] [[Definition:Area Mass Density|area mass density]] of $\sigma$.
Let $\CC$ lie in the [[Definition:XY Plane|$x$-$y$ plane]] of a [[Definition:Cartesian Coordinate System|Cartesian coordinate system]].
Let the [[Definition:Center of ... | Center of Mass of Uniform Circular Sector | https://proofwiki.org/wiki/Center_of_Mass_of_Uniform_Circular_Sector | https://proofwiki.org/wiki/Center_of_Mass_of_Uniform_Circular_Sector | [
"Centers of Mass",
"Sectors of Circles",
"Uniform Laminae"
] | [
"Definition:Lamina/Uniform",
"Definition:Sector of Circle",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Sector of Circle/Angle",
"Definition:Center of Mass",
"Definition:Point",
"Definition:Circle/Center"
] | [
"Definition:Sector of Circle",
"Definition:Constant",
"Definition:Mass Density/Area",
"Definition:Cartesian Plane",
"Definition:Cartesian Coordinate System",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Axis/X-Axis",
"Definition:Angle Bisector",
"Definition:Sector... |
proofwiki-22426 | Moment of Inertia of Uniform Rectangular Lamina through Midpoint about Perpendicular Axis | Let $\LL$ be a uniform lamina of mass $M$ in the shape of a rectangle whose sides are of length $2 a$ and $2 b$.
Let $\AA$ be the straight line through the centroid of $\LL$ perpendicular to $\LL$.
Then the moment of inertia $\II$ of $\LL$ about $\AA$ is given by:
:$\II = \dfrac {M \paren {a^2 + b^2} } 3$ | Let $\AA_a$ be the straight line:
:through the centroid of $\LL$
:in the plane of $\LL$
:perpendicular to the side of $\LL$ of length $2 a$.
Let $\AA_b$ be the straight line:
:through the centroid of $\LL$
:in the plane of $\LL$
:perpendicular to the side of $\LL$ of length $2 b$.
Let $\II_a$ be the moment of inertia o... | Let $\LL$ be a [[Definition:Uniform Lamina|uniform lamina]] of [[Definition:Mass|mass]] $M$ in the shape of a [[Definition:Rectangle|rectangle]] whose [[Definition:Side of Polygon|sides]] are of [[Definition:Length (Linear Measure)|length]] $2 a$ and $2 b$.
Let $\AA$ be the [[Definition:Straight Line|straight line]] t... | Let $\AA_a$ be the [[Definition:Straight Line|straight line]]:
:through the [[Definition:Centroid of Surface|centroid]] of $\LL$
:in the [[Definition:Plane|plane]] of $\LL$
:[[Definition:Line Perpendicular to Plane|perpendicular]] to the [[Definition:Side of Polygon|side]] of $\LL$ of [[Definition:Length (Linear Measur... | Moment of Inertia of Uniform Rectangular Lamina through Midpoint about Perpendicular Axis | https://proofwiki.org/wiki/Moment_of_Inertia_of_Uniform_Rectangular_Lamina_through_Midpoint_about_Perpendicular_Axis | https://proofwiki.org/wiki/Moment_of_Inertia_of_Uniform_Rectangular_Lamina_through_Midpoint_about_Perpendicular_Axis | [
"Moments of Inertia",
"Uniform Laminae",
"Rectangles"
] | [
"Definition:Lamina/Uniform",
"Definition:Mass",
"Definition:Quadrilateral/Rectangle",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Line/Straight Line",
"Definition:Centroid/Surface",
"Definition:Right Angle/Perpendicular/Plane",
"Definition:Moment of Inertia"
] | [
"Definition:Line/Straight Line",
"Definition:Centroid/Surface",
"Definition:Plane Surface",
"Definition:Right Angle/Perpendicular/Plane",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Line/Straight Line",
"Definition:Centroid/Surface",
"Definition:Plane Surface",
"Defi... |
proofwiki-22427 | Moment of Inertia of Uniform Elliptical Lamina through Midpoint about Perpendicular Axis | Let $\LL$ be a uniform lamina of mass $M$ in the shape of a ellipse whose axes are of length $2 a$ and $2 b$.
Let $\AA$ be the straight line through the centroid of $\LL$ perpendicular to $\LL$.
Then the moment of inertia $\II$ of $\LL$ about $\AA$ is given by:
:$\II = \dfrac {M \paren {a^2 + b^2} } 4$ | Let $\AA_a$ be the straight line coinciding with the axis of $\LL$ of length $2 a$.
Let $\AA_b$ be the straight line coinciding with the axis of $\LL$ of length $2 b$.
Let $\II_a$ be the moment of inertia of $\LL$ about $\AA_a$.
Let $\II_b$ be the moment of inertia of $\LL$ about $\AA_b$.
We note that $\AA$ is perpendi... | Let $\LL$ be a [[Definition:Uniform Lamina|uniform lamina]] of [[Definition:Mass|mass]] $M$ in the shape of a [[Definition:Ellipse|ellipse]] whose [[Definition:Axis of Ellipse|axes]] are of [[Definition:Length (Linear Measure)|length]] $2 a$ and $2 b$.
Let $\AA$ be the [[Definition:Straight Line|straight line]] throug... | Let $\AA_a$ be the [[Definition:Straight Line|straight line]] coinciding with the [[Definition:Axis of Ellipse|axis]] of $\LL$ of [[Definition:Length (Linear Measure)|length]] $2 a$.
Let $\AA_b$ be the [[Definition:Straight Line|straight line]] coinciding with the [[Definition:Axis of Ellipse|axis]] of $\LL$ of [[Defi... | Moment of Inertia of Uniform Elliptical Lamina through Midpoint about Perpendicular Axis | https://proofwiki.org/wiki/Moment_of_Inertia_of_Uniform_Elliptical_Lamina_through_Midpoint_about_Perpendicular_Axis | https://proofwiki.org/wiki/Moment_of_Inertia_of_Uniform_Elliptical_Lamina_through_Midpoint_about_Perpendicular_Axis | [
"Moments of Inertia",
"Uniform Laminae",
"Ellipses"
] | [
"Definition:Lamina/Uniform",
"Definition:Mass",
"Definition:Ellipse",
"Definition:Ellipse/Axis",
"Definition:Linear Measure/Length",
"Definition:Line/Straight Line",
"Definition:Centroid/Surface",
"Definition:Right Angle/Perpendicular/Plane",
"Definition:Moment of Inertia"
] | [
"Definition:Line/Straight Line",
"Definition:Ellipse/Axis",
"Definition:Linear Measure/Length",
"Definition:Line/Straight Line",
"Definition:Ellipse/Axis",
"Definition:Linear Measure/Length",
"Definition:Moment of Inertia",
"Definition:Moment of Inertia",
"Definition:Right Angle/Perpendicular",
"M... |
proofwiki-22428 | Category of Lattices is Category | Let $\mathbf{Lat}$ denote the category of lattices.
Then:
:$\mathbf{Lat}$ is a metacategory | Let us verify the axioms $(\text C 1)$ up to $(\text C 3)$ for a metacategory.
For any two lattice homomorphisms their composition (in the usual set theoretic sense) is again a lattice homomorphism by Composite Lattice Homomorphisms is Lattice Homomorphism.
For any lattice $\struct{L, \vee, \wedge, \preceq}$, we have t... | Let $\mathbf{Lat}$ denote the [[Definition:Category of Lattices|category of lattices]].
Then:
:$\mathbf{Lat}$ is a [[Definition:Metacategory|metacategory]] | Let us verify the axioms $(\text C 1)$ up to $(\text C 3)$ for a [[Definition:Metacategory|metacategory]].
For any two [[Definition:Lattice Homomorphism|lattice homomorphisms]] their [[Definition:Composition of Mappings|composition]] (in the usual [[Definition:Set Theory|set theoretic]] sense) is again a [[Definition... | Category of Lattices is Category | https://proofwiki.org/wiki/Category_of_Lattices_is_Category | https://proofwiki.org/wiki/Category_of_Lattices_is_Category | [
"Category of Lattices"
] | [
"Definition:Category of Lattices",
"Definition:Metacategory"
] | [
"Definition:Metacategory",
"Definition:Lattice Homomorphism",
"Definition:Composition of Mappings",
"Definition:Set Theory",
"Definition:Lattice Homomorphism",
"Composite Lattice Homomorphisms is Lattice Homomorphism",
"Definition:Lattice (Order Theory)",
"Definition:Identity Mapping",
"Identity Map... |
proofwiki-22429 | Composite Lattice Homomorphisms is Lattice Homomorphism | Let $L_1 = \struct{A_1, \vee_1, \wedge_1, \preceq_1}$, $L_2 = \struct{A_2, \vee_2, \wedge_2, \preceq_2}$ and $L_3 = \struct{A_3, \vee_3, \wedge_3, \preceq_3}$ be lattices.
Let $\phi_1: L_1 \to L_2$ and $\phi_2: L_2 \to L_3$ be lattice homomorphisms.
Let $\phi_2 \circ \phi_1 : A_1 \to A_3$ be the composite mapping of $\... | Follows immediately from Composite of Homomorphisms is Homomorphism
{{qed}}
Category:Lattice Homomorphisms
om3nw5mtzzsvjroe9k0fllt2g207brh | Let $L_1 = \struct{A_1, \vee_1, \wedge_1, \preceq_1}$, $L_2 = \struct{A_2, \vee_2, \wedge_2, \preceq_2}$ and $L_3 = \struct{A_3, \vee_3, \wedge_3, \preceq_3}$ be [[Definition:Lattice (Order Theory)|lattices]].
Let $\phi_1: L_1 \to L_2$ and $\phi_2: L_2 \to L_3$ be [[Definition:Lattice Homomorphism|lattice homomorphis... | Follows immediately from [[Composite of Homomorphisms is Homomorphism]]
{{qed}}
[[Category:Lattice Homomorphisms]]
om3nw5mtzzsvjroe9k0fllt2g207brh | Composite Lattice Homomorphisms is Lattice Homomorphism | https://proofwiki.org/wiki/Composite_Lattice_Homomorphisms_is_Lattice_Homomorphism | https://proofwiki.org/wiki/Composite_Lattice_Homomorphisms_is_Lattice_Homomorphism | [
"Lattice Homomorphisms"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Lattice Homomorphism",
"Definition:Composition of Mappings",
"Definition:Lattice Homomorphism"
] | [
"Composite of Homomorphisms is Homomorphism",
"Category:Lattice Homomorphisms"
] |
proofwiki-22430 | Identity Mapping is Lattice Homomorphism | Let $L = \struct{A, \vee, \wedge, \preceq}$ be a lattice.
Let $\operatorname{id}_A$ denote the identity mapping on $A$.
Then:
:$\operatorname{id}_A$ is a lattice homomorphism of $L$ to $L$ | Follows from Identity Mapping is Automorphism applied to $\struct{A, \vee}$ and $\struct{A, \wedge}$.
{{qed}}
Category:Lattice Homomorphisms
o3t1i1hguvpqxsl9jh1hima14sabg0l | Let $L = \struct{A, \vee, \wedge, \preceq}$ be a [[Definition:Lattice|lattice]].
Let $\operatorname{id}_A$ denote the [[Definition:Identity Mapping|identity mapping]] on $A$.
Then:
:$\operatorname{id}_A$ is a [[Definition:Lattice Homomorphism|lattice homomorphism]] of $L$ to $L$ | Follows from [[Identity Mapping is Automorphism]] applied to $\struct{A, \vee}$ and $\struct{A, \wedge}$.
{{qed}}
[[Category:Lattice Homomorphisms]]
o3t1i1hguvpqxsl9jh1hima14sabg0l | Identity Mapping is Lattice Homomorphism | https://proofwiki.org/wiki/Identity_Mapping_is_Lattice_Homomorphism | https://proofwiki.org/wiki/Identity_Mapping_is_Lattice_Homomorphism | [
"Lattice Homomorphisms"
] | [
"Definition:Lattice",
"Definition:Identity Mapping",
"Definition:Lattice Homomorphism"
] | [
"Identity Mapping is Automorphism",
"Category:Lattice Homomorphisms"
] |
proofwiki-22431 | Equivalence of Definitions of Lattice Isomorphism/Definition 1 Implies Definition 2 | Let $L_1 = \struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ be lattices.
Let $\phi : L_1 \to L_2$ be a bijective lattice homomorphism.
Then:
:$\phi : \struct{A_1, \preceq_1} \to \struct{A_2, \preceq_2}$ is an order isomorphism by definition. | From Inverse of Bijective Lattice Homomorphism is Bijective Lattice Homomorphism:
:$\phi^{-1}: L_2 \to L_1$ is a bijective lattice homomorphism.
From Lattice Homomorphism is Order-Preserving:
:$\phi : \struct{A_1, \preceq_1} \to \struct{A_2, \preceq_2}$ and $\phi^{-1} : \struct{A_2, \preceq_2} \to \struct{A_1, \preceq_... | Let $L_1 = \struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ be [[Definition:Lattice (Order Theory)|lattices]].
Let $\phi : L_1 \to L_2$ be a [[Definition:Bijection|bijective]] [[Definition:Lattice Homomorphism|lattice homomorphism]].
Then:
:$\phi : \struct{A_1, \pre... | From [[Inverse of Bijective Lattice Homomorphism is Bijective Lattice Homomorphism]]:
:$\phi^{-1}: L_2 \to L_1$ is a [[Definition:Bijection|bijective]] [[Definition:Lattice Homomorphism|lattice homomorphism]].
From [[Lattice Homomorphism is Order-Preserving]]:
:$\phi : \struct{A_1, \preceq_1} \to \struct{A_2, \preceq... | Equivalence of Definitions of Lattice Isomorphism/Definition 1 Implies Definition 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Lattice_Isomorphism/Definition_1_Implies_Definition_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Lattice_Isomorphism/Definition_1_Implies_Definition_2 | [
"Equivalence of Definitions of Lattice Isomorphism"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Bijection",
"Definition:Lattice Homomorphism",
"Definition:Order Isomorphism"
] | [
"Inverse of Bijective Lattice Homomorphism is Bijective Lattice Homomorphism",
"Definition:Bijection",
"Definition:Lattice Homomorphism",
"Lattice Homomorphism is Order-Preserving",
"Definition:Increasing",
"Definition:Order Isomorphism"
] |
proofwiki-22432 | Equivalence of Definitions of Lattice Isomorphism/Definition 2 Implies Definition 1 | Let $L_1 = \struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ be lattices.
Let $\phi : \struct{A_1, \preceq_1} \to \struct{A_2, \preceq_2}$ be an order isomorphism.
Then:
:$\phi$ is a bijective lattice homomorphism | By definition of order isomorphism:
:$\phi : \struct{A_1, \preceq_1} \to \struct{A_2, \preceq_2}$ is an order-preserving bijection
Let $\phi^{-1} : A_2 \to A_1$ be the inverse of $\phi : A_1 \to A_2$.
From Inverse of Order Isomorphism is Order Isomorphism:
:$\phi^{-1} : \struct{A_2, \preceq_2} \to \struct{A_1, \preceq... | Let $L_1 = \struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ be [[Definition:Lattice (Order Theory)|lattices]].
Let $\phi : \struct{A_1, \preceq_1} \to \struct{A_2, \preceq_2}$ be an [[Definition:Order Isomorphism|order isomorphism]].
Then:
:$\phi$ is a [[Definition:... | By definition of [[Definition:Order Isomorphism|order isomorphism]]:
:$\phi : \struct{A_1, \preceq_1} \to \struct{A_2, \preceq_2}$ is an [[Definition:Order-preserving|order-preserving]] [[Definition:Bijection|bijection]]
Let $\phi^{-1} : A_2 \to A_1$ be the [[Definition:Inverse Mapping|inverse]] of $\phi : A_1 \to A_... | Equivalence of Definitions of Lattice Isomorphism/Definition 2 Implies Definition 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Lattice_Isomorphism/Definition_2_Implies_Definition_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Lattice_Isomorphism/Definition_2_Implies_Definition_1 | [
"Equivalence of Definitions of Lattice Isomorphism"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Order Isomorphism",
"Definition:Bijection",
"Definition:Lattice Homomorphism"
] | [
"Definition:Order Isomorphism",
"Definition:Increasing",
"Definition:Bijection",
"Definition:Inverse Mapping",
"Inverse of Order Isomorphism is Order Isomorphism",
"Definition:Order Isomorphism",
"Definition:Morphism Property",
"Definition:Dual Statement (Order Theory)",
"Dual Pairs (Order Theory)",... |
proofwiki-22433 | Carnot's Theorem | All reversible heat engines operating between the same temperatures are equally efficient. | {{ProofWanted}}
{{Namedfor|Nicolas Léonard Sadi Carnot|cat = Carnot, Sadi}} | All [[Definition:Reversible Heat Engine|reversible heat engines]] operating between the same [[Definition:Temperature|temperatures]] are equally [[Definition:Efficiency|efficient]]. | {{ProofWanted}}
{{Namedfor|Nicolas Léonard Sadi Carnot|cat = Carnot, Sadi}} | Carnot's Theorem | https://proofwiki.org/wiki/Carnot's_Theorem | https://proofwiki.org/wiki/Carnot's_Theorem | [
"Thermodynamics"
] | [
"Definition:Reversible Heat Engine",
"Definition:Temperature",
"Definition:Efficiency"
] | [] |
proofwiki-22434 | Inverse of Bijective Lattice Homomorphism is Bijective Lattice Homomorphism | Let $L_1 = \struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ be lattices.
Let $\phi: L_1 \to L_2$ be a bijective lattice homomorphism.
Let $\phi^{-1} : A_2 \to A_1$ be the inverse of $\phi : A_1 \to A_2$.
Then:
:$\phi^{-1} : L_2 \to L_1$ is a bijective lattice homomorphi... | From Inverse of Bijection is Bijection:
:$\phi^{-1}$ is a bijection
It remains to show that $\phi^{-1}$ is a lattice homomorphism. | Let $L_1 = \struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ be [[Definition:Lattice (Order Theory)|lattices]].
Let $\phi: L_1 \to L_2$ be a [[Definition:Bijection|bijective]] [[Definition:Lattice Homomorphism|lattice homomorphism]].
Let $\phi^{-1} : A_2 \to A_1$ be t... | From [[Inverse of Bijection is Bijection]]:
:$\phi^{-1}$ is a [[Definition:Bijection|bijection]]
It remains to show that $\phi^{-1}$ is a [[Definition:Lattice Homomorphism|lattice homomorphism]]. | Inverse of Bijective Lattice Homomorphism is Bijective Lattice Homomorphism | https://proofwiki.org/wiki/Inverse_of_Bijective_Lattice_Homomorphism_is_Bijective_Lattice_Homomorphism | https://proofwiki.org/wiki/Inverse_of_Bijective_Lattice_Homomorphism_is_Bijective_Lattice_Homomorphism | [
"Lattice Homomorphisms",
"Bijections"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Bijection",
"Definition:Lattice Homomorphism",
"Definition:Inverse Mapping",
"Definition:Bijection",
"Definition:Lattice Homomorphism"
] | [
"Inverse of Bijection is Bijection",
"Definition:Bijection",
"Definition:Lattice Homomorphism",
"Definition:Lattice Homomorphism"
] |
proofwiki-22435 | Number of Eigenvalues of Square Complex Matrix | Let $\mathbf A$ be a square matrix of order $n$ over the complex numbers $\C$.
Then $\mathbf A$ has $n$ eigenvalues. | Let $p_A$ be the characteristic polynomial of $\mathbf A$.
By definition, an eigenvalue of $\mathbf A$ is a root of $p_A$.
Since Degree of Characteristic Polynomial of Matrix equals Order of Matrix, the degree of $p_A$ is $n$.
From the Number of Roots of Polynomial With Complex Coefficients Equals Degree of Polynomial,... | Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order]] $n$ over the [[Definition:Complex Number|complex numbers]] $\C$.
Then $\mathbf A$ has $n$ [[Definition:Eigenvalue of Square Matrix|eigenvalues]]. | Let $p_A$ be the [[Definition:Characteristic Polynomial of Matrix|characteristic polynomial]] of $\mathbf A$.
By definition, an [[Definition:Eigenvalue of Square Matrix|eigenvalue]] of $\mathbf A$ is a [[Definition:Root of Polynomial|root]] of $p_A$.
Since [[Degree of Characteristic Polynomial of Matrix equals Order ... | Number of Eigenvalues of Square Complex Matrix | https://proofwiki.org/wiki/Number_of_Eigenvalues_of_Square_Complex_Matrix | https://proofwiki.org/wiki/Number_of_Eigenvalues_of_Square_Complex_Matrix | [
"Eigenvalues of Square Matrices",
"Square Matrices",
"Complex Matrices"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Complex Number",
"Definition:Eigenvalue/Square Matrix"
] | [
"Definition:Characteristic Polynomial of Matrix",
"Definition:Eigenvalue/Square Matrix",
"Definition:Root of Polynomial",
"Degree of Characteristic Polynomial of Matrix equals Order of Matrix",
"Definition:Degree of Polynomial",
"Number of Roots of Polynomial With Complex Coefficients Equals Degree of Pol... |
proofwiki-22436 | Number of Linearly Independent Eigenvectors of Square Complex Matrix | Let $\mathbf A$ be a square matrix of order $n$ over the complex numbers $\C$.
Then $\mathbf A$ has no more than $n$ linearly independent eigenvectors. | By the definition of eigenvalue, it is a root of the characteristic polynomial of $\mathbf A$.
By Degree of Characteristic Polynomial of Matrix equals Order of Matrix, the degree of the characteristic polynomial is $n$.
By Number of Roots of Polynomial With Complex Coefficients Equals Degree of Polynomial, the number o... | Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order]] $n$ over the [[Definition:Complex Number|complex numbers]] $\C$.
Then $\mathbf A$ has no more than $n$ [[Definition:Linearly Independent Set of Complex Vectors|linearly independent]] [[Definition:Eigenvector ... | By the [[Definition:Eigenvalue of Square Matrix|definition of eigenvalue]], it is a root of the [[Definition:Characteristic Polynomial of Matrix|characteristic polynomial]] of $\mathbf A$.
By [[Degree of Characteristic Polynomial of Matrix equals Order of Matrix]], the [[Definition:Degree of Polynomial|degree]] of the... | Number of Linearly Independent Eigenvectors of Square Complex Matrix | https://proofwiki.org/wiki/Number_of_Linearly_Independent_Eigenvectors_of_Square_Complex_Matrix | https://proofwiki.org/wiki/Number_of_Linearly_Independent_Eigenvectors_of_Square_Complex_Matrix | [
"Eigenvectors of Square Matrices",
"Linear Independence",
"Square Matrices",
"Complex Matrices"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Complex Number",
"Definition:Linearly Independent/Set/Complex Vector Space",
"Definition:Eigenvector/Square Matrix"
] | [
"Definition:Eigenvalue/Square Matrix",
"Definition:Characteristic Polynomial of Matrix",
"Degree of Characteristic Polynomial of Matrix equals Order of Matrix",
"Definition:Degree of Polynomial",
"Definition:Characteristic Polynomial of Matrix",
"Number of Roots of Polynomial With Complex Coefficients Equ... |
proofwiki-22437 | Velocity from Integration of Equation of Motion | Let $P$ be a particle of constant mass $m$ moving under a force $\mathbf F$ as a function of time $t$ according to the '''equation of motion''':
:$m \dfrac {\d^2 \mathbf r} {\d t^2} = \map {\mathbf F} {\mathbf r}$
where $\mathbf r$ is the position vector of $P$.
Let the velocity of $P$ at time $t = 0$ be $\mathbf v_0$.... | {{begin-eqn}}
{{eqn | l = m \dfrac {\d^2 \mathbf r} {\d t^2}
| r = \map {\mathbf F} {\mathbf r}
| c =
}}
{{eqn | r = \dfrac {\map \d {m \mathbf v} } {\d t}
| c = Newton's First Law of Motion
}}
{{eqn | ll= \leadsto
| l = m \dfrac {\d^2 \mathbf r} {\d t^2}
| r = m \dfrac {\d \mathbf v} {\d... | Let $P$ be a [[Definition:Particle|particle]] of [[Definition:Constant|constant]] [[Definition:Mass|mass]] $m$ moving under a [[Definition:Force|force]] $\mathbf F$ as a [[Definition:Real Function|function]] of [[Definition:Time|time]] $t$ according to the '''[[Definition:Equation of Motion|equation of motion]]''':
:$m... | {{begin-eqn}}
{{eqn | l = m \dfrac {\d^2 \mathbf r} {\d t^2}
| r = \map {\mathbf F} {\mathbf r}
| c =
}}
{{eqn | r = \dfrac {\map \d {m \mathbf v} } {\d t}
| c = [[Newton's First Law of Motion]]
}}
{{eqn | ll= \leadsto
| l = m \dfrac {\d^2 \mathbf r} {\d t^2}
| r = m \dfrac {\d \mathbf v}... | Velocity from Integration of Equation of Motion | https://proofwiki.org/wiki/Velocity_from_Integration_of_Equation_of_Motion | https://proofwiki.org/wiki/Velocity_from_Integration_of_Equation_of_Motion | [
"Velocity",
"Equations of Motion"
] | [
"Definition:Particle",
"Definition:Constant",
"Definition:Mass",
"Definition:Force",
"Definition:Real Function",
"Definition:Time",
"Definition:Equation of Motion",
"Definition:Position Vector",
"Definition:Velocity",
"Definition:Velocity"
] | [
"Newton's Laws of Motion/First Law",
"Definition:Mass",
"Definition:Constant",
"Definition:Mass",
"Definition:Constant",
"Definition:Primitive (Calculus)/Integration",
"Fundamental Theorem of Calculus"
] |
proofwiki-22438 | Lattice Isomorphism is Isomorphism in Category Lat | Let $\mathbf{Lat}$ denote the category of lattices.
Let $f : L_1 \to L_2$ be a morphism of $\mathbf{Lat}$.
Then:
:$f$ is an isomorphism of $\mathbf{Lat}$ {{iff}} $f$ is a lattice isomorphsm | Let $L_1$ and $L_2$ be the lattices $L_1 = \struct{A_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct{A_2, \vee_2, \wedge_2, \preceq_2}$ respectively.
By definition of category of lattices:
:$f$ is a lattice homomorphisms
By definition of an isomorphism:
:$f$ be an isomorphism of $\mathbf{Lat}$
{{iff}}:
:$(1):$ ther... | Let $\mathbf{Lat}$ denote the [[Definition:Category of Lattices|category of lattices]].
Let $f : L_1 \to L_2$ be a [[Definition:Morphism|morphism]] of $\mathbf{Lat}$.
Then:
:$f$ is an [[Definition:Isomorphism (Category Theory)|isomorphism]] of $\mathbf{Lat}$ {{iff}} $f$ is a [[Definition:Lattice Isomorphism|lattice... | Let $L_1$ and $L_2$ be the [[Definition:Lattice (Order Theory)|lattices]] $L_1 = \struct{A_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct{A_2, \vee_2, \wedge_2, \preceq_2}$ respectively.
By definition of [[Definition:Category of Lattices|category of lattices]]:
:$f$ is a [[Definition:Lattice Homomorphism|lattic... | Lattice Isomorphism is Isomorphism in Category Lat | https://proofwiki.org/wiki/Lattice_Isomorphism_is_Isomorphism_in_Category_Lat | https://proofwiki.org/wiki/Lattice_Isomorphism_is_Isomorphism_in_Category_Lat | [
"Lattice Isomorphisms",
"Category of Lattices"
] | [
"Definition:Category of Lattices",
"Definition:Morphism",
"Definition:Isomorphism (Category Theory)",
"Definition:Lattice Isomorphism"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Category of Lattices",
"Definition:Lattice Homomorphism",
"Definition:Isomorphism (Category Theory)",
"Definition:Isomorphism (Category Theory)",
"Definition:Morphism",
"Definition:Identity Morphism",
"Definition:Category of Lattices",
"Definition:Lat... |
proofwiki-22439 | Creation of Orthogonal Vector from Independent Vectors | Let $\mathbf V$ be a vector space.
Let $\mathbf a$ and $\mathbf b$ be vectors of $\mathbf V$ such that $\mathbf a$ and $\mathbf b$ form a linearly independent set.
Let $\mathbf a$ and $\mathbf b$ be expressed as column vectors.
Let $\mathbf b'$ be calculated as:
:$\mathbf b' = \mathbf b - \dfrac {\mathbf a^\intercal \m... | {{Recall|Orthogonal (Linear Algebra)|orthogonal}}:
{{:Definition:Orthogonal (Linear Algebra)}}
We will show that $\mathbf b'^\intercal \mathbf a = 0$.
{{explain|why the inner product of $\mathbf a$ and $\mathbf b$ is the operation $\mathbf b'^\intercal \mathbf a$}}
{{begin-eqn}}
{{eqn | l = \mathbf b'^\intercal \mathbf... | Let $\mathbf V$ be a [[Definition:Vector Space|vector space]].
Let $\mathbf a$ and $\mathbf b$ be [[Definition:Vector|vectors]] of $\mathbf V$ such that $\mathbf a$ and $\mathbf b$ form a [[Definition:Linearly Independent Set|linearly independent set]].
Let $\mathbf a$ and $\mathbf b$ be expressed as [[Definition:Col... | {{Recall|Orthogonal (Linear Algebra)|orthogonal}}:
{{:Definition:Orthogonal (Linear Algebra)}}
We will show that $\mathbf b'^\intercal \mathbf a = 0$.
{{explain|why the inner product of $\mathbf a$ and $\mathbf b$ is the operation $\mathbf b'^\intercal \mathbf a$}}
{{begin-eqn}}
{{eqn | l = \mathbf b'^\intercal \mat... | Creation of Orthogonal Vector from Independent Vectors | https://proofwiki.org/wiki/Creation_of_Orthogonal_Vector_from_Independent_Vectors | https://proofwiki.org/wiki/Creation_of_Orthogonal_Vector_from_Independent_Vectors | [
"Orthogonality (Linear Algebra)",
"Linear Independence"
] | [
"Definition:Vector Space",
"Definition:Vector",
"Definition:Linearly Independent/Set",
"Definition:Column Matrix",
"Definition:Transpose of Matrix",
"Definition:Orthogonal (Linear Algebra)"
] | [] |
proofwiki-22440 | Coset Space wrt Subring forms Ring iff Subring is Ideal | Let $\struct {R, +, \circ}$ be a ring.
Let $S$ be a subring of $R$.
For $a \in R$, let $\paren {a + S}$ denote the coset of $S$ by $a$.
Then:
:The coset space of $R$ {{WRT}} $S$ forms a ring
{{iff}}:
:$S$ is an ideal of $R$. | {{ProofWanted|bit more consolidation of background needed yet}} | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $S$ be a [[Definition:Subring|subring]] of $R$.
For $a \in R$, let $\paren {a + S}$ denote the [[Definition:Coset of Subring|coset of $S$]] by $a$.
Then:
:The [[Definition:Coset Space of Ring|coset space]] of $R$ {{WRT}} $S$ forms a [... | {{ProofWanted|bit more consolidation of background needed yet}} | Coset Space wrt Subring forms Ring iff Subring is Ideal | https://proofwiki.org/wiki/Coset_Space_wrt_Subring_forms_Ring_iff_Subring_is_Ideal | https://proofwiki.org/wiki/Coset_Space_wrt_Subring_forms_Ring_iff_Subring_is_Ideal | [
"Coset Spaces of Rings",
"Ideals of Rings"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Subring",
"Definition:Coset of Subring",
"Definition:Coset Space of Ring",
"Definition:Ring (Abstract Algebra)",
"Definition:Ideal of Ring"
] | [] |
proofwiki-22441 | Basis of Point Lattice is not Necessarily Unique | Let $\LL$ be a point lattice of dimension $n$.
Let $\BB = \set {\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n}$ be a basis of $\LL$.
Then it is not necessarily the case that $\BB$ is the '''only''' basis of $\LL$. | {{MissingLinks}}
Let $\LL$ be a point lattice of dimension $n$.
Let $\BB = \set {\mathbf v_1,\mathbf v_2,\dots,\mathbf v_n}$ be a basis of $\LL$.
Let $\BB' = \set {\mathbf v_1, \mathbf v_1 + \mathbf v_2, \dots, \mathbf v_n} = \set {\mathbf w_1, \mathbf w_2, \dots, \mathbf w_n}$.
We will show that $\BB'$ is a basis for ... | Let $\LL$ be a [[Definition:Point Lattice|point lattice]] of [[Definition:Dimension of Point Lattice|dimension]] $n$.
Let $\BB = \set {\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n}$ be a [[Definition:Basis of Point Lattice|basis]] of $\LL$.
Then it is not necessarily the case that $\BB$ is the '''[[Definition:Uniqu... | {{MissingLinks}}
Let $\LL$ be a point lattice of dimension $n$.
Let $\BB = \set {\mathbf v_1,\mathbf v_2,\dots,\mathbf v_n}$ be a basis of $\LL$.
Let $\BB' = \set {\mathbf v_1, \mathbf v_1 + \mathbf v_2, \dots, \mathbf v_n} = \set {\mathbf w_1, \mathbf w_2, \dots, \mathbf w_n}$.
We will show that $\BB'$ is a basis ... | Basis of Point Lattice is not Necessarily Unique | https://proofwiki.org/wiki/Basis_of_Point_Lattice_is_not_Necessarily_Unique | https://proofwiki.org/wiki/Basis_of_Point_Lattice_is_not_Necessarily_Unique | [
"Bases of Point Lattices",
"Point Lattices"
] | [
"Definition:Point Lattice",
"Definition:Point Lattice/Dimension",
"Definition:Point Lattice/Basis",
"Definition:Unique",
"Definition:Point Lattice/Basis"
] | [] |
proofwiki-22442 | Packing of Circles into Plane | The most space-efficient way to pack unit circles into the plane is to place their centers at the lattice points of the point lattice in $\R^n$ with a basis $\set {\tuple {2, 0}, \tuple {1, \sqrt 3} }$. | {{ProofWanted|see Point Lattice/Examples/2-Dimensional}} | The most space-efficient way to pack [[Definition:Unit Circle|unit circles]] into [[Definition:The Plane|the plane]] is to place their [[Definition:Center of Circle|centers]] at the [[Definition:Lattice Point|lattice points]] of the [[Definition:Point Lattice|point lattice]] in $\R^n$ with a [[Definition:Basis of Point... | {{ProofWanted|see [[Point Lattice/Examples/2-Dimensional]]}} | Packing of Circles into Plane | https://proofwiki.org/wiki/Packing_of_Circles_into_Plane | https://proofwiki.org/wiki/Packing_of_Circles_into_Plane | [
"Unit Circles",
"Point Lattices"
] | [
"Definition:Unit Circle",
"Definition:Plane Surface/The Plane",
"Definition:Circle/Center",
"Definition:Lattice Point",
"Definition:Point Lattice",
"Definition:Point Lattice/Basis"
] | [
"Point Lattice/Examples/2-Dimensional"
] |
proofwiki-22443 | Inverse of Frame Isomorphism is Frame Isomorphism | Let $L_1 = \struct {S_1, \preceq_1}$ and $L_2 = \struct {S_2, \preceq_2}$ be frames.
Let $\phi: L_1 \to L_2$ be a frame isomorphism.
Let $\phi^{-1} : S_2 \to S_1$ be the inverse of $\phi : S_1 \to S_2$.
Then:
:$\phi^{-1} : L_2 \to L_1$ is a frame isomorphism | By definition of frame isomorphism:
:$\phi: L_1 \to L_2$ is a complete lattice isomorphism
From Inverse of Complete Lattice Isomorphism is Complete Lattice Isomorphism:
:$\phi^{-1} : L_2 \to L_1$ is a complete lattice isomorphism
By definition of frame isomorphism:
:$\phi^{-1}: L_2 \to L_1$ is a frame isomorphism
{{qed... | Let $L_1 = \struct {S_1, \preceq_1}$ and $L_2 = \struct {S_2, \preceq_2}$ be [[Definition:Frame (Lattice Theory)|frames]].
Let $\phi: L_1 \to L_2$ be a [[Definition:Frame Isomorphism|frame isomorphism]].
Let $\phi^{-1} : S_2 \to S_1$ be the [[Definition:Inverse Mapping|inverse]] of $\phi : S_1 \to S_2$.
Then:
:$\p... | By definition of [[Definition:Frame Isomorphism|frame isomorphism]]:
:$\phi: L_1 \to L_2$ is a [[Definition:Complete Lattice Isomorphism|complete lattice isomorphism]]
From [[Inverse of Complete Lattice Isomorphism is Complete Lattice Isomorphism]]:
:$\phi^{-1} : L_2 \to L_1$ is a [[Definition:Complete Lattice Isomor... | Inverse of Frame Isomorphism is Frame Isomorphism | https://proofwiki.org/wiki/Inverse_of_Frame_Isomorphism_is_Frame_Isomorphism | https://proofwiki.org/wiki/Inverse_of_Frame_Isomorphism_is_Frame_Isomorphism | [
"Frame Isomorphisms"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Frame Isomorphism",
"Definition:Inverse Mapping",
"Definition:Frame Isomorphism"
] | [
"Definition:Frame Isomorphism",
"Definition:Complete Lattice Isomorphism",
"Inverse of Complete Lattice Isomorphism is Complete Lattice Isomorphism",
"Definition:Complete Lattice Isomorphism",
"Definition:Frame Isomorphism",
"Definition:Frame Isomorphism",
"Category:Frame Isomorphisms"
] |
proofwiki-22444 | Inverse of Lattice Isomorphism is Lattice Isomorphism | Let $L_1 = \struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ be lattices.
Let $\phi: L_1 \to L_2$ be a lattice isomorphism.
Let $\phi^{-1} : A_2 \to A_1$ be the inverse of $\phi : A_1 \to A_2$.
Then:
:$\phi^{-1} : L_2 \to L_1$ is a lattice isomorphism | By definition, a lattice isomorphism is a bijective lattice homomorphism.
The result follows from Inverse of Bijective Lattice Homomorphism is Bijective Lattice Homomorphism.
{{qed}}
Category:Lattice Isomorphisms
i95oqn5vwwxq6ea85bsheaycwjh75sh | Let $L_1 = \struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ be [[Definition:Lattice (Order Theory)|lattices]].
Let $\phi: L_1 \to L_2$ be a [[Definition:Lattice Isomorphism|lattice isomorphism]].
Let $\phi^{-1} : A_2 \to A_1$ be the [[Definition:Inverse Mapping|inver... | By definition, a [[Definition:Lattice Isomorphism|lattice isomorphism]] is a [[Definition:Bijection|bijective]] [[Definition:Lattice Homomorphism|lattice homomorphism]].
The result follows from [[Inverse of Bijective Lattice Homomorphism is Bijective Lattice Homomorphism]].
{{qed}}
[[Category:Lattice Isomorphisms]]
... | Inverse of Lattice Isomorphism is Lattice Isomorphism | https://proofwiki.org/wiki/Inverse_of_Lattice_Isomorphism_is_Lattice_Isomorphism | https://proofwiki.org/wiki/Inverse_of_Lattice_Isomorphism_is_Lattice_Isomorphism | [
"Lattice Isomorphisms"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Lattice Isomorphism",
"Definition:Inverse Mapping",
"Definition:Lattice Isomorphism"
] | [
"Definition:Lattice Isomorphism",
"Definition:Bijection",
"Definition:Lattice Homomorphism",
"Inverse of Bijective Lattice Homomorphism is Bijective Lattice Homomorphism",
"Category:Lattice Isomorphisms"
] |
proofwiki-22445 | Frame Homomorphism is Lattice Homomorphism | Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be frames.
Let $\phi: L_1 \to L_2$ be a frame homomorphism.
Then:
:$\phi: \struct{S_1, \vee_1, \wedge_1, \preceq_1} \to \struct{S_2, \vee_2, \wedge_2, \preceq_2}$ is a lattice homomorphism
where:
:$\wedge_1, \vee_1$ denote the meet and join on $L_... | By definition of frame:
:$L_1$ and $L_2$ are complete lattices
From Complete Lattice is Lattice:
:$\struct{S_1, \vee_1, \wedge_1, \preceq_1}$ and $\struct{S_2, \vee_2, \wedge_2, \preceq_2}$ are lattices
where:
:$\wedge_1, \vee_1$ denote the meet and join on $L_1$
:$\wedge_2, \vee_2$ denote the meet and join on $L_2$
... | Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be [[Definition:Frame (Lattice Theory)|frames]].
Let $\phi: L_1 \to L_2$ be a [[Definition:Frame Homomorphism|frame homomorphism]].
Then:
:$\phi: \struct{S_1, \vee_1, \wedge_1, \preceq_1} \to \struct{S_2, \vee_2, \wedge_2, \preceq_2}$ is a [[Def... | By definition of [[Definition:Frame (Lattice Theory)|frame]]:
:$L_1$ and $L_2$ are [[Definition:Complete Lattice|complete lattices]]
From [[Complete Lattice is Lattice]]:
:$\struct{S_1, \vee_1, \wedge_1, \preceq_1}$ and $\struct{S_2, \vee_2, \wedge_2, \preceq_2}$ are [[Definition:Lattice (Order Theory)|lattices]]
wher... | Frame Homomorphism is Lattice Homomorphism | https://proofwiki.org/wiki/Frame_Homomorphism_is_Lattice_Homomorphism | https://proofwiki.org/wiki/Frame_Homomorphism_is_Lattice_Homomorphism | [
"Frame Homomorphisms",
"Lattice Homomorphisms"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Frame Homomorphism",
"Definition:Lattice Homomorphism",
"Definition:Meet",
"Definition:Join",
"Definition:Meet",
"Definition:Join"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Complete Lattice",
"Complete Lattice is Lattice",
"Definition:Lattice (Order Theory)",
"Definition:Meet",
"Definition:Join",
"Definition:Meet",
"Definition:Join",
"Definition:Frame Homomorphism",
"Definition:Arbitrary Join Preserving Mapping",
"De... |
proofwiki-22446 | Ramanujan's Continued Fraction of One | Let $x \notin \Z_{<0}$.
Then:
:$1 = \cfrac {x + 1} {x + \cfrac {x + 2} {x + 1 + \cfrac {x + 3 } {x + 2 + \ddots} } }$ | We have:
{{begin-eqn}}
{{eqn | l = 1
| r = \frac {x + 1} {x + 1}
| c =
}}
{{eqn | r = \cfrac {x + 1} {x + \cfrac {x + 2} {x + 2} }
| c =
}}
{{eqn | r = \cfrac {x + 1} {x + \cfrac {x + 2} {x + 1 + 1} }
| c =
}}
{{eqn | r = \cfrac {x + 1} {x + \cfrac {x + 2} {x + 1 + \cfrac {x + 3} {x + 3} } }
... | Let $x \notin \Z_{<0}$.
Then:
:$1 = \cfrac {x + 1} {x + \cfrac {x + 2} {x + 1 + \cfrac {x + 3 } {x + 2 + \ddots} } }$ | We have:
{{begin-eqn}}
{{eqn | l = 1
| r = \frac {x + 1} {x + 1}
| c =
}}
{{eqn | r = \cfrac {x + 1} {x + \cfrac {x + 2} {x + 2} }
| c =
}}
{{eqn | r = \cfrac {x + 1} {x + \cfrac {x + 2} {x + 1 + 1} }
| c =
}}
{{eqn | r = \cfrac {x + 1} {x + \cfrac {x + 2} {x + 1 + \cfrac {x + 3} {x + 3} } }... | Ramanujan's Continued Fraction of One | https://proofwiki.org/wiki/Ramanujan's_Continued_Fraction_of_One | https://proofwiki.org/wiki/Ramanujan's_Continued_Fraction_of_One | [
"Ramanujan's Continued Fraction of One",
"Continued Fractions",
"Number Theory"
] | [] | [] |
proofwiki-22447 | Continued Fraction for Real Arctangent Function | :$\arctan x = \cfrac x {1 + \cfrac {x^2} {3 - x^2 + \cfrac {\paren {3 x}^2} {5 - 3 x^2 + \cfrac {\paren {5 x}^2} {7 - 5 x^2 + \cfrac {\paren {7 x}^2} {\ddots } } } } }$ | {{begin-eqn}}
{{eqn | l = \arctan x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {2 n + 1}
| c = Power Series Expansion for Real Arctangent Function for $-1 \le x \le 1$
}}
{{eqn | r = x - \dfrac {x^3} 3 + \dfrac {x^5} 5 - \dfrac {x^7} 7 + \cdots
| c =
}}
{{eqn | r = x + x \pa... | :$\arctan x = \cfrac x {1 + \cfrac {x^2} {3 - x^2 + \cfrac {\paren {3 x}^2} {5 - 3 x^2 + \cfrac {\paren {5 x}^2} {7 - 5 x^2 + \cfrac {\paren {7 x}^2} {\ddots } } } } }$ | {{begin-eqn}}
{{eqn | l = \arctan x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {2 n + 1}
| c = [[Power Series Expansion for Real Arctangent Function]] for $-1 \le x \le 1$
}}
{{eqn | r = x - \dfrac {x^3} 3 + \dfrac {x^5} 5 - \dfrac {x^7} 7 + \cdots
| c =
}}
{{eqn | r = x + x... | Continued Fraction for Real Arctangent Function | https://proofwiki.org/wiki/Continued_Fraction_for_Real_Arctangent_Function | https://proofwiki.org/wiki/Continued_Fraction_for_Real_Arctangent_Function | [
"Arctangent Function",
"Continued Fractions",
"Euler's Continued Fraction Formula",
"Examples of Euler's Continued Fraction Formula"
] | [] | [
"Power Series Expansion for Real Arctangent Function",
"Euler's Continued Fraction Formula"
] |
proofwiki-22448 | Probability Mass Function of Negative Binomial Distribution (Type 2)/Also defined as | Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.
Let $X$ have the '''type $2$ negative binomial distribution with parameters $r$ and $p$''', defined as the number of failures before $r$ successes have occurred.
Then the probability mass function of $X$ is given by:
:$\map \P... | {{Recall|Negative Binomial Distribution (Type 2)|subdef = Also defined as}}
{{:Definition:Negative Binomial Distribution (Type 2)/Also defined as}}
The number of Bernoulli trials may be as few as $0$, so the image is correct:
:$\Img X = \set {0, 1, 2, \ldots}$
If $X$ takes the value $k$, then there must have been $k + ... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] on a [[Definition:Probability Space|probability space]] $\struct {\Omega, \Sigma, \Pr}$.
Let $X$ have the '''[[Definition:Negative Binomial Distribution (Type 2)|type $2$ negative binomial distribution]] with parameters $r$ and $p$''', defin... | {{Recall|Negative Binomial Distribution (Type 2)|subdef = Also defined as}}
{{:Definition:Negative Binomial Distribution (Type 2)/Also defined as}}
The number of [[Definition:Bernoulli Trial|Bernoulli trials]] may be as few as $0$, so the [[Definition:Image of Mapping|image]] is correct:
:$\Img X = \set {0, 1, 2, \ld... | Probability Mass Function of Negative Binomial Distribution (Type 2)/Also defined as | https://proofwiki.org/wiki/Probability_Mass_Function_of_Negative_Binomial_Distribution_(Type_2)/Also_defined_as | https://proofwiki.org/wiki/Probability_Mass_Function_of_Negative_Binomial_Distribution_(Type_2)/Also_defined_as | [
"Probability Mass Function of Negative Binomial Distribution",
"Negative Binomial Distribution (Type 2)",
"Probability Mass Functions"
] | [
"Definition:Random Variable/Discrete",
"Definition:Probability Space",
"Definition:Negative Binomial Distribution/Type 2",
"Definition:Bernoulli Distribution",
"Definition:Bernoulli Distribution",
"Definition:Occurrence",
"Definition:Probability Mass Function"
] | [
"Definition:Bernoulli Trial",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Experiment",
"Definition:Experiment",
"Definition:Bernoulli Distribution",
"Definition:Experiment",
"Definition:Bernoulli Distribution",
"Definition:Probability",
"Definition:Event/Occurrence",
"Definition:E... |
proofwiki-22449 | Probability Mass Function of Negative Binomial Distribution (Type 1)/Also defined as | Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.
Let $X$ have the '''type $1$ negative binomial distribution with parameters $r$ and $p$''', defined as number of number of Bernoulli trials before the $r$th failure has occurred.
Then the probability mass function of $X$ is gi... | {{Recall|Negative Binomial Distribution (Type 1)|subdef = Also defined as}}
{{:Definition:Negative Binomial Distribution (Type 1)/Also defined as}}
First note that the number of Bernoulli trials has to be at least $r$, so the image is correct: $\Img X = \set {r, r + 1, r + 2, \ldots}$.
Now, note that if $X$ takes the v... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] on a [[Definition:Probability Space|probability space]] $\struct {\Omega, \Sigma, \Pr}$.
Let $X$ have the '''[[Definition:Negative Binomial Distribution (Type 1)|type $1$ negative binomial distribution]] with parameters $r$ and $p$''', defin... | {{Recall|Negative Binomial Distribution (Type 1)|subdef = Also defined as}}
{{:Definition:Negative Binomial Distribution (Type 1)/Also defined as}}
First note that the number of [[Definition:Bernoulli Trial|Bernoulli trials]] has to be at least $r$, so the [[Definition:Image of Mapping|image]] is correct: $\Img X = \s... | Probability Mass Function of Negative Binomial Distribution (Type 1)/Also defined as | https://proofwiki.org/wiki/Probability_Mass_Function_of_Negative_Binomial_Distribution_(Type_1)/Also_defined_as | https://proofwiki.org/wiki/Probability_Mass_Function_of_Negative_Binomial_Distribution_(Type_1)/Also_defined_as | [
"Probability Mass Function of Negative Binomial Distribution",
"Negative Binomial Distribution (Type 1)",
"Probability Mass Functions"
] | [
"Definition:Random Variable/Discrete",
"Definition:Probability Space",
"Definition:Negative Binomial Distribution/Type 1",
"Definition:Bernoulli Trial",
"Definition:Bernoulli Distribution",
"Definition:Occurrence",
"Definition:Probability Mass Function",
"Definition:Probability Mass Function"
] | [
"Definition:Bernoulli Trial",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Bernoulli Distribution",
"Definition:Bernoulli Distribution",
"Definition:Bernoulli Distribution",
"Definition:Experiment",
"Definition:Probability",
"Definition:Event/Occurrence",
"Definition:Event",
"Defin... |
proofwiki-22450 | Equivalence of Definitions of Frame Isomorphism/Definition 1 Implies Definition 2 | Let $L_1$ and $L_2$ be frames.
Let $\phi : L_1 \to L_2$ be a bijective frame homomorphism.
Then:
:$\phi : L_1 \to L_2$ is an order isomorphism. | From Frame Homomorphism is Lattice Homomorphism:
:$\phi : L_1 \to L_2$ is a bijective lattice homomorphism
By definition of lattice isomorphism:
:$\phi : L_1 \to L_2$ is a lattice isomorphism
From Inverse of Lattice Isomorphism is Lattice Isomorphism:
:$\phi^{-1} : L_2 \to L_1$ is a lattice isomorphism
By definition of... | Let $L_1$ and $L_2$ be [[Definition:Frame (Lattice Theory)|frames]].
Let $\phi : L_1 \to L_2$ be a [[Definition:Bijection|bijective]] [[Definition:Frame Homomorphism|frame homomorphism]].
Then:
:$\phi : L_1 \to L_2$ is an [[Definition:Order Isomorphism|order isomorphism]]. | From [[Frame Homomorphism is Lattice Homomorphism]]:
:$\phi : L_1 \to L_2$ is a [[Definition:Bijection|bijective]] [[Definition:Lattice Homomorphism|lattice homomorphism]]
By definition of [[Definition:Lattice Isomorphism|lattice isomorphism]]:
:$\phi : L_1 \to L_2$ is a [[Definition:Lattice Isomorphism|lattice isomo... | Equivalence of Definitions of Frame Isomorphism/Definition 1 Implies Definition 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Frame_Isomorphism/Definition_1_Implies_Definition_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Frame_Isomorphism/Definition_1_Implies_Definition_2 | [
"Equivalence of Definitions of Frame Isomorphism"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Bijection",
"Definition:Frame Homomorphism",
"Definition:Order Isomorphism"
] | [
"Frame Homomorphism is Lattice Homomorphism",
"Definition:Bijection",
"Definition:Lattice Homomorphism",
"Definition:Lattice Isomorphism",
"Definition:Lattice Isomorphism",
"Inverse of Lattice Isomorphism is Lattice Isomorphism",
"Definition:Lattice Isomorphism",
"Definition:Lattice Isomorphism",
"D... |
proofwiki-22451 | Expectation of Negative Binomial Distribution (Type 2)/Also defined as | Let $X$ be a discrete random variable with the type $2$ negative binomial distribution with parameters $r$ and $p$.
Let $X$ use the definition as the number of failures before $r$ successes have occurred.
Then the expectation of $X$ is given by:
:$\expect X = \dfrac {r q} p$
where $q = 1 - p$. | Let $Y$ be a discrete random variable with the type $2$ negative binomial distribution with parameters $r$ and $p$.
Let $Y$ use the definition as the number of successes before $r$ failures have occurred.
From Expectation of Negative Binomial Distribution (Type 2):
:$\expect X = \dfrac {r p} q$
By definition:
:$p$ is t... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] with the [[Definition:Negative Binomial Distribution (Type 2)|type $2$ negative binomial distribution]] with parameters $r$ and $p$.
Let $X$ use the definition as the number of [[Definition:Failure|failures]] before $r$ [[Definition:Success|... | Let $Y$ be a [[Definition:Discrete Random Variable|discrete random variable]] with the [[Definition:Negative Binomial Distribution (Type 2)|type $2$ negative binomial distribution]] with parameters $r$ and $p$.
Let $Y$ use the definition as the number of [[Definition:Success|successes]] before $r$ [[Definition:Failure... | Expectation of Negative Binomial Distribution (Type 2)/Also defined as | https://proofwiki.org/wiki/Expectation_of_Negative_Binomial_Distribution_(Type_2)/Also_defined_as | https://proofwiki.org/wiki/Expectation_of_Negative_Binomial_Distribution_(Type_2)/Also_defined_as | [
"Expectation of Negative Binomial Distribution",
"Negative Binomial Distribution (Type 2)",
"Expectation"
] | [
"Definition:Random Variable/Discrete",
"Definition:Negative Binomial Distribution/Type 2",
"Definition:Bernoulli Distribution",
"Definition:Bernoulli Distribution",
"Definition:Occurrence",
"Definition:Expectation"
] | [
"Definition:Random Variable/Discrete",
"Definition:Negative Binomial Distribution/Type 2",
"Definition:Bernoulli Distribution",
"Definition:Bernoulli Distribution",
"Definition:Occurrence",
"Expectation of Negative Binomial Distribution/Type 2",
"Definition:Probability",
"Definition:Bernoulli Distribu... |
proofwiki-22452 | Equivalence of Definitions of Frame Isomorphism/Definition 4 Implies Definition 1 | Let $L_1 = \struct {S_1, \preceq_1}$ and $L_2 = \struct {S_2, \preceq_2}$ be frames.
Let $\phi : L_1 \to L_2$ be a complete lattice isomorphism.
Then:
:$\phi$ is a bijective frame homomorphism | By definition of complete lattice isomorphism:
:$\phi : L_1 \to L_2$ is a bijective complete lattice homomorphism.
From Complete Lattice Homomorphism is Frame Homomorphism:
:$\phi : L_1 \to L_2$ is a bijective frame homomorphism. | Let $L_1 = \struct {S_1, \preceq_1}$ and $L_2 = \struct {S_2, \preceq_2}$ be [[Definition:Frame (Lattice Theory)|frames]].
Let $\phi : L_1 \to L_2$ be a [[Definition:Complete Lattice Isomorphism|complete lattice isomorphism]].
Then:
:$\phi$ is a [[Definition:Bijection|bijective]] [[Definition:Frame Homomorphism|fra... | By definition of [[Definition:Complete Lattice Isomorphism|complete lattice isomorphism]]:
:$\phi : L_1 \to L_2$ is a [[Definition:Bijection|bijective]] [[Definition:Complete Lattice Homomorphism|complete lattice homomorphism]].
From [[Complete Lattice Homomorphism is Frame Homomorphism]]:
:$\phi : L_1 \to L_2$ is a ... | Equivalence of Definitions of Frame Isomorphism/Definition 4 Implies Definition 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Frame_Isomorphism/Definition_4_Implies_Definition_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Frame_Isomorphism/Definition_4_Implies_Definition_1 | [
"Equivalence of Definitions of Frame Isomorphism"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Complete Lattice Isomorphism",
"Definition:Bijection",
"Definition:Frame Homomorphism"
] | [
"Definition:Complete Lattice Isomorphism",
"Definition:Bijection",
"Definition:Complete Lattice Homomorphism",
"Complete Lattice Homomorphism is Frame Homomorphism",
"Definition:Bijection",
"Definition:Frame Homomorphism"
] |
proofwiki-22453 | Continued Fraction for Exponential Function | :$e^x = \cfrac 1 {1 - \cfrac x {1 + x - \cfrac x {2 + x - \cfrac {2 x} {3 + x - \cfrac {3 x} {4 + x - \cfrac \ddots \ddots} } } } }$ | {{begin-eqn}}
{{eqn | l = e^x
| r = \sum_{n \mathop = 0}^\infty \frac {x^n } {n!}
| c = Power Series Expansion for Exponential Function
}}
{{eqn | r = 1 + x + \frac {x^2} {2!} + \frac {x^3} {3!} + \frac {x^4} {4!} + \cdots
| c =
}}
{{eqn | r = 1 + \paren 1 \paren x + \paren 1 \paren x \paren {\dfrac ... | :$e^x = \cfrac 1 {1 - \cfrac x {1 + x - \cfrac x {2 + x - \cfrac {2 x} {3 + x - \cfrac {3 x} {4 + x - \cfrac \ddots \ddots} } } } }$ | {{begin-eqn}}
{{eqn | l = e^x
| r = \sum_{n \mathop = 0}^\infty \frac {x^n } {n!}
| c = [[Power Series Expansion for Exponential Function]]
}}
{{eqn | r = 1 + x + \frac {x^2} {2!} + \frac {x^3} {3!} + \frac {x^4} {4!} + \cdots
| c =
}}
{{eqn | r = 1 + \paren 1 \paren x + \paren 1 \paren x \paren {\df... | Continued Fraction for Exponential Function | https://proofwiki.org/wiki/Continued_Fraction_for_Exponential_Function | https://proofwiki.org/wiki/Continued_Fraction_for_Exponential_Function | [
"Continued Fraction for Exponential Function",
"Euler's Continued Fraction Formula",
"Examples of Euler's Continued Fraction Formula",
"Examples of Continued Fractions",
"Exponential Function"
] | [] | [
"Power Series Expansion for Exponential Function",
"Euler's Continued Fraction Formula"
] |
proofwiki-22454 | One Represented With Infinite Twos | {{begin-eqn}}
{{eqn | l = 1
| r = \cfrac 2 {\cfrac 2 {\cfrac 2 {\cfrac 2 {\cfrac 2 {\ddots + \ddots} + \cfrac 2 {\ddots + \ddots} } + \cfrac 2 {\cfrac 2 {\ddots + \ddots} + \cfrac 2 {\ddots + \ddots} } } + \cfrac 2 {\cfrac 2 {\cfrac 2 {\ddots + \ddots} + \cfrac 2 {\ddots + \ddots} } + \cfrac 2 {\cfrac 2 {\ddots +... | We have:
{{begin-eqn}}
{{eqn | l = 1
| r = \cfrac 2 {1 + 1}
| c = One Layer Deep: $2^1$ ones and $\paren{2^1 - 1}$ twos
}}
{{eqn | r = \cfrac 2 {\cfrac 2 {1 + 1} + \cfrac 2 {1 + 1} }
| c = Two Layers Deep: $2^2$ ones and $\paren{2^2 - 1}$ twos
}}
{{eqn | r = \cfrac 2 {\cfrac 2 {\cfrac 2 {1 + 1} + \cfr... | {{begin-eqn}}
{{eqn | l = 1
| r = \cfrac 2 {\cfrac 2 {\cfrac 2 {\cfrac 2 {\cfrac 2 {\ddots + \ddots} + \cfrac 2 {\ddots + \ddots} } + \cfrac 2 {\cfrac 2 {\ddots + \ddots} + \cfrac 2 {\ddots + \ddots} } } + \cfrac 2 {\cfrac 2 {\cfrac 2 {\ddots + \ddots} + \cfrac 2 {\ddots + \ddots} } + \cfrac 2 {\cfrac 2 {\ddots +... | We have:
{{begin-eqn}}
{{eqn | l = 1
| r = \cfrac 2 {1 + 1}
| c = One Layer Deep: $2^1$ ones and $\paren{2^1 - 1}$ twos
}}
{{eqn | r = \cfrac 2 {\cfrac 2 {1 + 1} + \cfrac 2 {1 + 1} }
| c = Two Layers Deep: $2^2$ ones and $\paren{2^2 - 1}$ twos
}}
{{eqn | r = \cfrac 2 {\cfrac 2 {\cfrac 2 {1 + 1} + \... | One Represented With Infinite Twos | https://proofwiki.org/wiki/One_Represented_With_Infinite_Twos | https://proofwiki.org/wiki/One_Represented_With_Infinite_Twos | [
"Continued Fractions",
"Number Theory",
"Recreational Mathematics"
] | [] | [
"Category:Continued Fractions",
"Category:Number Theory",
"Category:Recreational Mathematics"
] |
proofwiki-22455 | Continued Fraction for Logarithm of 1 + x | :$\map \ln {1 + x} = \cfrac x {1 + \cfrac x {2 - x + \cfrac {2^2 x} {3 - 2 x + \cfrac {3^2 x} {4 - 3 x + \cfrac {\ddots} {\ddots} } } } }$ | {{begin-eqn}}
{{eqn | l = \map \ln {1 + x}
| r = \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n
| c = Power Series Expansion for Logarithm of 1 + x
}}
{{eqn | r = x - \frac {x^2} 2 + \frac {x^3} 3 - \frac {x^4} 4 + \cdots
| c =
}}
{{eqn | r = \paren x + \paren x \paren {-\dfrac x 2} + ... | :$\map \ln {1 + x} = \cfrac x {1 + \cfrac x {2 - x + \cfrac {2^2 x} {3 - 2 x + \cfrac {3^2 x} {4 - 3 x + \cfrac {\ddots} {\ddots} } } } }$ | {{begin-eqn}}
{{eqn | l = \map \ln {1 + x}
| r = \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n
| c = [[Power Series Expansion for Logarithm of 1 + x]]
}}
{{eqn | r = x - \frac {x^2} 2 + \frac {x^3} 3 - \frac {x^4} 4 + \cdots
| c =
}}
{{eqn | r = \paren x + \paren x \paren {-\dfrac x 2... | Continued Fraction for Logarithm of 1 + x | https://proofwiki.org/wiki/Continued_Fraction_for_Logarithm_of_1_+_x | https://proofwiki.org/wiki/Continued_Fraction_for_Logarithm_of_1_+_x | [
"Continued Fractions",
"Euler's Continued Fraction Formula",
"Examples of Euler's Continued Fraction Formula",
"Natural Logarithms"
] | [] | [
"Power Series Expansion for Logarithm of 1 + x",
"Euler's Continued Fraction Formula",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-22456 | Continued Fraction for Real Arcsine Function | :$\arcsin x = \cfrac x {1 - \cfrac {x^2} {2 \times 3 + x^2 - \cfrac {2 \times 3 \times \paren {3 x}^2} {4 \times 5 + \paren {3 x}^2 - \cfrac {4 \times 5 \times \paren {5 x}^2} {6 \times 7 + \paren {5 x}^2 - \cfrac {6 \times 7 \times \paren {7 x}^2} {\ddots } } } } }$ | {{begin-eqn}}
{{eqn | l = \arcsin x
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}
| c = Power Series Expansion for Real Arcsine Function for $-1 \le x \le 1$
}}
{{eqn | r = x + \frac {x^3} {2 \times 3} + \frac {\paren {1 \times 3} x^5} {2 \tim... | :$\arcsin x = \cfrac x {1 - \cfrac {x^2} {2 \times 3 + x^2 - \cfrac {2 \times 3 \times \paren {3 x}^2} {4 \times 5 + \paren {3 x}^2 - \cfrac {4 \times 5 \times \paren {5 x}^2} {6 \times 7 + \paren {5 x}^2 - \cfrac {6 \times 7 \times \paren {7 x}^2} {\ddots } } } } }$ | {{begin-eqn}}
{{eqn | l = \arcsin x
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}
| c = [[Power Series Expansion for Real Arcsine Function]] for $-1 \le x \le 1$
}}
{{eqn | r = x + \frac {x^3} {2 \times 3} + \frac {\paren {1 \times 3} x^5} {2 ... | Continued Fraction for Real Arcsine Function | https://proofwiki.org/wiki/Continued_Fraction_for_Real_Arcsine_Function | https://proofwiki.org/wiki/Continued_Fraction_for_Real_Arcsine_Function | [
"Arcsine Function",
"Continued Fractions",
"Examples of Euler's Continued Fraction Formula",
"Euler's Continued Fraction Formula"
] | [] | [
"Power Series Expansion for Real Arcsine Function",
"Euler's Continued Fraction Formula"
] |
proofwiki-22457 | Inverse of Bijective Complete Lattice Homomorphism is Bijective Complete Lattice Homomorphism | Let $L_1 = \struct {A_1, \preceq_1}$ and $L_2 = \struct {A_2, \preceq_2}$ be complete lattices.
Let $\phi: L_1 \to L_2$ be a bijective complete lattice homomorphism.
Let $\phi^{-1} : A_2 \to A_1$ be the inverse of $\phi : A_1 \to A_2$.
Then:
:$\phi^{-1} : L_2 \to L_1$ is a bijective complete lattice homomorphism. | From Inverse of Bijection is Bijection:
:$\phi^{-1}$ is a bijection
It remains to show that $\phi^{-1}$ is a complete lattice homomorphism. | Let $L_1 = \struct {A_1, \preceq_1}$ and $L_2 = \struct {A_2, \preceq_2}$ be [[Definition:Complete Lattice|complete lattices]].
Let $\phi: L_1 \to L_2$ be a [[Definition:Bijection|bijective]] [[Definition:Complete Lattice Homomorphism|complete lattice homomorphism]].
Let $\phi^{-1} : A_2 \to A_1$ be the [[Definition... | From [[Inverse of Bijection is Bijection]]:
:$\phi^{-1}$ is a [[Definition:Bijection|bijection]]
It remains to show that $\phi^{-1}$ is a [[Definition:Complete Lattice Homomorphism|complete lattice homomorphism]]. | Inverse of Bijective Complete Lattice Homomorphism is Bijective Complete Lattice Homomorphism | https://proofwiki.org/wiki/Inverse_of_Bijective_Complete_Lattice_Homomorphism_is_Bijective_Complete_Lattice_Homomorphism | https://proofwiki.org/wiki/Inverse_of_Bijective_Complete_Lattice_Homomorphism_is_Bijective_Complete_Lattice_Homomorphism | [
"Complete Lattice Homomorphisms",
"Bijections"
] | [
"Definition:Complete Lattice",
"Definition:Bijection",
"Definition:Complete Lattice Homomorphism",
"Definition:Inverse Mapping",
"Definition:Bijection",
"Definition:Complete Lattice Homomorphism"
] | [
"Inverse of Bijection is Bijection",
"Definition:Bijection",
"Definition:Complete Lattice Homomorphism",
"Definition:Complete Lattice Homomorphism"
] |
proofwiki-22458 | Converse of Fermat's Little Theorem does not hold | Let $n$ be a natural number.
Let:
:$n \divides a^n - a$
where:
:$a$ is a natural number such that $n$ is not a divisor of $a$.
:$\divides$ denotes divisibility.
Then it is not necessarily the case that $n$ is prime | ;Proof by Counterexample
We have that:
:$2^{341} \equiv 2 \pmod {341}$
despite the fact that $341$ is not prime:
:$341 = 11 \times 31$
{{qed}} | Let $n$ be a [[Definition:Natural Number|natural number]].
Let:
:$n \divides a^n - a$
where:
:$a$ is a [[Definition:Natural Number|natural number]] such that $n$ is not a [[Definition:Divisor of Integer|divisor]] of $a$.
:$\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
Then it is not necessarily t... | ;[[Proof by Counterexample]]
We have that:
:$2^{341} \equiv 2 \pmod {341}$
despite the fact that $341$ is not [[Definition:Prime Number|prime]]:
:$341 = 11 \times 31$
{{qed}} | Converse of Fermat's Little Theorem does not hold | https://proofwiki.org/wiki/Converse_of_Fermat's_Little_Theorem_does_not_hold | https://proofwiki.org/wiki/Converse_of_Fermat's_Little_Theorem_does_not_hold | [
"Fermat's Little Theorem",
"Fermat Pseudoprimes"
] | [
"Definition:Natural Numbers",
"Definition:Natural Numbers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Number"
] | [
"Proof by Counterexample",
"Definition:Prime Number"
] |
proofwiki-22459 | Category DLat is Full Subcategory of Lat | Let $\mathbf {DLat}$ denote the category of distributive lattices.
Let $\mathbf {Lat}$ denote the category of lattices.
Then:
:$\mathbf {DLat}$ is a full subcategory of $\mathbf {Lat}$ | By definition of distributive lattice:
:a distributive lattice is a lattice
Hence the objects of $\mathbf {DLat}$ is a collection of objects of $\mathbf {Lat}$
By definition of category of distributive lattices:
:every morphism of $\mathbf {DLat}$ is a morphism of $\mathbf {Lat}$
Hence the morphisms of $\mathbf {DLat}$... | Let $\mathbf {DLat}$ denote the [[Definition:Category of Distributive Lattices|category of distributive lattices]].
Let $\mathbf {Lat}$ denote the [[Definition:Category of Lattices|category of lattices]].
Then:
:$\mathbf {DLat}$ is a [[Definition:Full Subcategory|full subcategory]] of $\mathbf {Lat}$ | By definition of [[Definition:Distributive Lattice|distributive lattice]]:
:a [[Definition:Distributive Lattice|distributive lattice]] is a [[Definition:Lattice (Order Theory)|lattice]]
Hence the [[Definition:Object (Category Theory)|objects]] of $\mathbf {DLat}$ is a [[Definition:Collection|collection]] of [[Definiti... | Category DLat is Full Subcategory of Lat | https://proofwiki.org/wiki/Category_DLat_is_Full_Subcategory_of_Lat | https://proofwiki.org/wiki/Category_DLat_is_Full_Subcategory_of_Lat | [
"Category of Distributive Lattices"
] | [
"Definition:Category of Distributive Lattices",
"Definition:Category of Lattices",
"Definition:Full Subcategory"
] | [
"Definition:Distributive Lattice",
"Definition:Distributive Lattice",
"Definition:Lattice (Order Theory)",
"Definition:Object (Category Theory)",
"Definition:Collection",
"Definition:Object (Category Theory)",
"Definition:Category of Distributive Lattices",
"Definition:Morphism",
"Definition:Morphis... |
proofwiki-22460 | Category of Distributive Lattices is Category | Let $\mathbf {DLat}$ denote the category of distributive lattices.
Then:
:$\mathbf {DLat}$ is a metacategory | From Category DLat is Full Subcategory of Lat:
:$\mathbf {DLat}$ is a full subcategory of the category of lattices
By definition of subcategory:
:$\mathbf {DLat}$ is a metacategory
{{qed}}
Category:Category of Distributive Lattices
h6ioghc0zqpd1un1b97ltg2f80zeicz | Let $\mathbf {DLat}$ denote the [[Definition:Category of Distributive Lattices|category of distributive lattices]].
Then:
:$\mathbf {DLat}$ is a [[Definition:Metacategory|metacategory]] | From [[Category DLat is Full Subcategory of Lat]]:
:$\mathbf {DLat}$ is a [[Definition:Full Subcategory|full subcategory]] of the [[Definition:Category of Lattices|category of lattices]]
By definition of [[Definition:Subcategory|subcategory]]:
:$\mathbf {DLat}$ is a [[Definition:Metacategory|metacategory]]
{{qed}}
[... | Category of Distributive Lattices is Category | https://proofwiki.org/wiki/Category_of_Distributive_Lattices_is_Category | https://proofwiki.org/wiki/Category_of_Distributive_Lattices_is_Category | [
"Category of Distributive Lattices"
] | [
"Definition:Category of Distributive Lattices",
"Definition:Metacategory"
] | [
"Category DLat is Full Subcategory of Lat",
"Definition:Full Subcategory",
"Definition:Category of Lattices",
"Definition:Subcategory",
"Definition:Metacategory",
"Category:Category of Distributive Lattices"
] |
proofwiki-22461 | Tangent of Three Right Angles minus Angle | :$\map \tan {x - \dfrac {3 \pi} 2} = \cot x$ | {{begin-eqn}}
{{eqn | l = \map \tan {x - \frac {3 \pi} 2}
| r = \frac {\map \sin {x - \frac {3 \pi} 2} } {\map \cos {x - \frac {3 \pi} 2} }
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {-\cos x} {-\sin x}
| c = Sine of Angle minus Three Right Angles and Cosine of Angle minus Three Rig... | :$\map \tan {x - \dfrac {3 \pi} 2} = \cot x$ | {{begin-eqn}}
{{eqn | l = \map \tan {x - \frac {3 \pi} 2}
| r = \frac {\map \sin {x - \frac {3 \pi} 2} } {\map \cos {x - \frac {3 \pi} 2} }
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {-\cos x} {-\sin x}
| c = [[Sine of Angle minus Three Right Angles]] and [[Cosine of Angle minus... | Tangent of Three Right Angles minus Angle | https://proofwiki.org/wiki/Tangent_of_Three_Right_Angles_minus_Angle | https://proofwiki.org/wiki/Tangent_of_Three_Right_Angles_minus_Angle | [
"Tangent Function"
] | [] | [
"Tangent is Sine divided by Cosine",
"Sine of Angle minus Three Right Angles",
"Cosine of Angle minus Three Right Angles",
"Cotangent is Cosine divided by Sine"
] |
proofwiki-22462 | Gauss's Continued Fraction | :$\ds \frac {\map F {\alpha, \beta + 1; \gamma + 1; x} } {\map F {\alpha, \beta; \gamma; x} } = \cfrac 1 {1 - \cfrac {a x} {1 - \cfrac {b x} {1 - \cfrac {c x} {1 - \cfrac {d x} {1 - \cfrac {\ddots} {\ddots } } } } } }$
where:
:$\map F {a, b; c; x}$ is the Gaussian hypergeometric function of $x$: $\ds \sum_{k \mathop = ... | :First, we demonstate:
:$\ds \dfrac {\map F {\alpha, \beta + 1, \gamma + 1; x} } {\map F {\alpha, \beta, \gamma; x} } = \dfrac 1 {1 - \dfrac {\alpha \paren {\gamma - \beta} x } {\gamma \paren {\gamma + 1} } \dfrac {\map F {\alpha + 1, \beta + 1, \gamma + 2; x} } {\map F {\alpha, \beta + 1, \gamma + 1; x} } }$
{{begin-e... | :$\ds \frac {\map F {\alpha, \beta + 1; \gamma + 1; x} } {\map F {\alpha, \beta; \gamma; x} } = \cfrac 1 {1 - \cfrac {a x} {1 - \cfrac {b x} {1 - \cfrac {c x} {1 - \cfrac {d x} {1 - \cfrac {\ddots} {\ddots } } } } } }$
where:
:$\map F {a, b; c; x}$ is the [[Definition:Gaussian Hypergeometric Function|Gaussian hypergeo... | :First, we demonstate:
:$\ds \dfrac {\map F {\alpha, \beta + 1, \gamma + 1; x} } {\map F {\alpha, \beta, \gamma; x} } = \dfrac 1 {1 - \dfrac {\alpha \paren {\gamma - \beta} x } {\gamma \paren {\gamma + 1} } \dfrac {\map F {\alpha + 1, \beta + 1, \gamma + 2; x} } {\map F {\alpha, \beta + 1, \gamma + 1; x} } }$
{{begin-... | Gauss's Continued Fraction | https://proofwiki.org/wiki/Gauss's_Continued_Fraction | https://proofwiki.org/wiki/Gauss's_Continued_Fraction | [
"Continued Fractions",
"Gaussian Hypergeometric Function"
] | [
"Definition:Hypergeometric Function/Gaussian",
"Definition:Rising Factorial"
] | [
"Gaussian Hypergeometric Function Difference Equation/Formulation 1",
"Definition:Reciprocal",
"Gaussian Hypergeometric Function Difference Equation/Formulation 2",
"Definition:Reciprocal"
] |
proofwiki-22463 | Empty Mapping is Surjective iff Codomain is Empty | Let $T$ be a set.
Let $\O$ denote the empty set.
Let $e: \O \to T$ be the empty mapping.
Then $e$ is a surjection {{iff}} $T = \O$. | Let $T = \O$.
From Empty Mapping to Empty Set is Bijective, $e$ is a bijection.
Hence {{afortiori}} $e$ is a surjection.
{{qed|lemma}}
Let $T \ne \O$.
{{AimForCont}} $e$ is a surjection.
Let $t \in T$.
As $e$ is a surjection:
:$(1): \quad \exists s \in S: \map e s = t$
But by Null Relation is Mapping iff Domain is Empt... | Let $T$ be a [[Definition:Set|set]].
Let $\O$ denote the [[Definition:Empty Set|empty set]].
Let $e: \O \to T$ be the [[Definition:Empty Mapping|empty mapping]].
Then $e$ is a [[Definition:Surjection|surjection]] {{iff}} $T = \O$. | Let $T = \O$.
From [[Empty Mapping to Empty Set is Bijective]], $e$ is a [[Definition:Bijection|bijection]].
Hence {{afortiori}} $e$ is a [[Definition:Surjection|surjection]].
{{qed|lemma}}
Let $T \ne \O$.
{{AimForCont}} $e$ is a [[Definition:Surjection|surjection]].
Let $t \in T$.
As $e$ is a [[Definition:Surje... | Empty Mapping is Surjective iff Codomain is Empty | https://proofwiki.org/wiki/Empty_Mapping_is_Surjective_iff_Codomain_is_Empty | https://proofwiki.org/wiki/Empty_Mapping_is_Surjective_iff_Codomain_is_Empty | [
"Empty Mapping",
"Surjections"
] | [
"Definition:Set",
"Definition:Empty Set",
"Definition:Empty Mapping",
"Definition:Surjection"
] | [
"Empty Mapping to Empty Set is Bijective",
"Definition:Bijection",
"Definition:Surjection",
"Definition:Surjection",
"Definition:Surjection",
"Null Relation is Mapping iff Domain is Empty Set",
"Definition:Contradiction",
"Proof by Contradiction",
"Definition:Surjection",
"Definition:Surjection",
... |
proofwiki-22464 | Gaussian Hypergeometric Function Difference Equation/Formulation 1 | {{begin-eqn}}
{{eqn | l = \map F {a, b + 1, c + 1; x} - \map F {a, b, c; x}
| r = \frac {a \paren {c - b} x } {c \paren {c + 1} } \map F {a + 1, b + 1, c + 2; x}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map F {a, b + 1, c + 1; x}
| r = \sum_{n \mathop = 0}^\infty \frac {a^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^n
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac {a \paren {b + 1} } {\paren {c + 1} } x + \cdots + \fr... | {{begin-eqn}}
{{eqn | l = \map F {a, b + 1, c + 1; x} - \map F {a, b, c; x}
| r = \frac {a \paren {c - b} x } {c \paren {c + 1} } \map F {a + 1, b + 1, c + 2; x}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map F {a, b + 1, c + 1; x}
| r = \sum_{n \mathop = 0}^\infty \frac {a^{\overline n} \paren {b + 1}^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^n
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac {a \paren {b + 1} } {\paren {c + 1} } x + \cdots + \fr... | Gaussian Hypergeometric Function Difference Equation/Formulation 1 | https://proofwiki.org/wiki/Gaussian_Hypergeometric_Function_Difference_Equation/Formulation_1 | https://proofwiki.org/wiki/Gaussian_Hypergeometric_Function_Difference_Equation/Formulation_1 | [
"Gaussian Hypergeometric Function Difference Equation"
] | [] | [
"Gamma Difference Equation",
"Gamma Difference Equation",
"Gamma Difference Equation"
] |
proofwiki-22465 | Gaussian Hypergeometric Function Difference Equation/Formulation 2 | {{begin-eqn}}
{{eqn | l = \map F {a + 1, b, c + 1; x} - \map F {a, b, c; x}
| r = \frac {b \paren {c - a} x } {c \paren {c + 1} } \map F {a + 1, b + 1, c + 2; x}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map F {a + 1, b, c + 1; x}
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {a + 1}^{\overline n} b^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^n
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac {\paren {a + 1} b } {\paren {c + 1} } x + \cdots + \... | {{begin-eqn}}
{{eqn | l = \map F {a + 1, b, c + 1; x} - \map F {a, b, c; x}
| r = \frac {b \paren {c - a} x } {c \paren {c + 1} } \map F {a + 1, b + 1, c + 2; x}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map F {a + 1, b, c + 1; x}
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {a + 1}^{\overline n} b^{\overline n} } {\paren {c + 1}^{\overline n} n!} x^n
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac {\paren {a + 1} b } {\paren {c + 1} } x + \cdots + \... | Gaussian Hypergeometric Function Difference Equation/Formulation 2 | https://proofwiki.org/wiki/Gaussian_Hypergeometric_Function_Difference_Equation/Formulation_2 | https://proofwiki.org/wiki/Gaussian_Hypergeometric_Function_Difference_Equation/Formulation_2 | [
"Gaussian Hypergeometric Function Difference Equation"
] | [] | [
"Gamma Difference Equation",
"Gamma Difference Equation",
"Gamma Difference Equation"
] |
proofwiki-22466 | Classification of Stationary Points/Function of Two Variables | Let $\SS$ be a surface defined by the Cartesian equation $z = \map f {x, y}$.
Let $P$ be a stationary point on $\SS$.
Then $P$ is either:
:a local maximum
:a local minimum
:a saddle point. | {{ProofWanted|one direction immediate, the other not so much}} | Let $\SS$ be a [[Definition:Surface|surface]] defined by the [[Definition:Cartesian Coordinates|Cartesian equation]] $z = \map f {x, y}$.
Let $P$ be a [[Definition:Stationary Point of Function of Two Variables|stationary point]] on $\SS$.
Then $P$ is either:
:a [[Definition:Local Maximum|local maximum]]
:a [[Definit... | {{ProofWanted|one direction immediate, the other not so much}} | Classification of Stationary Points/Function of Two Variables | https://proofwiki.org/wiki/Classification_of_Stationary_Points/Function_of_Two_Variables | https://proofwiki.org/wiki/Classification_of_Stationary_Points/Function_of_Two_Variables | [
"Classification of Stationary Points"
] | [
"Definition:Surface",
"Definition:Cartesian Coordinate System",
"Definition:Stationary Point/Function of Two Variables",
"Definition:Maximum Value of Real Function/Local",
"Definition:Minimum Value of Real Function/Local",
"Definition:Saddle Point (Geometry)"
] | [] |
proofwiki-22467 | Gaussian Hypergeometric Function Difference Equation/Formulation 3 | {{begin-eqn}}
{{eqn | l = \map F {a, b + 1, c; x} - \map F {a, b, c; x}
| r = \frac {a x } c \map F {a + 1, b + 1, c + 1; x}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map F {a, b + 1, c; x}
| r = \sum_{n \mathop = 0}^\infty \frac {a^{\overline n} \paren {b + 1}^{\overline n} } {c^{\overline n} n!} x^n
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac {a \paren {b + 1} } c x + \cdots + \frac {\map \Gamma {a + n} \map \Gam... | {{begin-eqn}}
{{eqn | l = \map F {a, b + 1, c; x} - \map F {a, b, c; x}
| r = \frac {a x } c \map F {a + 1, b + 1, c + 1; x}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map F {a, b + 1, c; x}
| r = \sum_{n \mathop = 0}^\infty \frac {a^{\overline n} \paren {b + 1}^{\overline n} } {c^{\overline n} n!} x^n
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac {a \paren {b + 1} } c x + \cdots + \frac {\map \Gamma {a + n} \map \Gam... | Gaussian Hypergeometric Function Difference Equation/Formulation 3 | https://proofwiki.org/wiki/Gaussian_Hypergeometric_Function_Difference_Equation/Formulation_3 | https://proofwiki.org/wiki/Gaussian_Hypergeometric_Function_Difference_Equation/Formulation_3 | [
"Gaussian Hypergeometric Function Difference Equation"
] | [] | [
"Gamma Difference Equation",
"Gamma Difference Equation"
] |
proofwiki-22468 | Gaussian Hypergeometric Function Difference Equation/Formulation 4 | {{begin-eqn}}
{{eqn | l = \map F {a + 1, b, c; x} - \map F {a, b, c; x}
| r = \frac {b x } c \map F {a + 1, b + 1, c + 1; x}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map F {a + 1, b, c; x}
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {a + 1}^{\overline n} b^{\overline n} } {c^{\overline n} n!} x^n
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac {\paren {a + 1} b } c x + \cdots + \frac {\map \Gamma {a + 1 + n} \map ... | {{begin-eqn}}
{{eqn | l = \map F {a + 1, b, c; x} - \map F {a, b, c; x}
| r = \frac {b x } c \map F {a + 1, b + 1, c + 1; x}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map F {a + 1, b, c; x}
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {a + 1}^{\overline n} b^{\overline n} } {c^{\overline n} n!} x^n
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac {\paren {a + 1} b } c x + \cdots + \frac {\map \Gamma {a + 1 + n} \map ... | Gaussian Hypergeometric Function Difference Equation/Formulation 4 | https://proofwiki.org/wiki/Gaussian_Hypergeometric_Function_Difference_Equation/Formulation_4 | https://proofwiki.org/wiki/Gaussian_Hypergeometric_Function_Difference_Equation/Formulation_4 | [
"Gaussian Hypergeometric Function Difference Equation"
] | [] | [
"Gamma Difference Equation",
"Gamma Difference Equation"
] |
proofwiki-22469 | Hypergeometric Function Difference Equation/Formulation 1 | {{begin-eqn}}
{{eqn | l = \map { {}_0 \operatorname F_1} {a - 1; x} - \map { {}_0 \operatorname F_1} {a; x}
| r = \frac x {a \paren {a - 1} } \map { {}_0 \operatorname F_1} {a + 1; x}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map { {}_0 \operatorname F_1} {a - 1; x}
| r = \sum_{n \mathop = 0}^\infty \frac 1 {\paren {a - 1}^{\overline n} n!} x^n
| c = {{Defof|Hypergeometric Function/Generalized|Generalized Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac 1 {\paren {a - 1} } x + \cdots + \frac {\map \G... | {{begin-eqn}}
{{eqn | l = \map { {}_0 \operatorname F_1} {a - 1; x} - \map { {}_0 \operatorname F_1} {a; x}
| r = \frac x {a \paren {a - 1} } \map { {}_0 \operatorname F_1} {a + 1; x}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map { {}_0 \operatorname F_1} {a - 1; x}
| r = \sum_{n \mathop = 0}^\infty \frac 1 {\paren {a - 1}^{\overline n} n!} x^n
| c = {{Defof|Hypergeometric Function/Generalized|Generalized Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac 1 {\paren {a - 1} } x + \cdots + \frac {\map \G... | Hypergeometric Function Difference Equation/Formulation 1 | https://proofwiki.org/wiki/Hypergeometric_Function_Difference_Equation/Formulation_1 | https://proofwiki.org/wiki/Hypergeometric_Function_Difference_Equation/Formulation_1 | [
"Hypergeometric Function Difference Equation"
] | [] | [
"Gamma Difference Equation",
"Gamma Difference Equation"
] |
proofwiki-22470 | Hypergeometric Function Difference Equation/Formulation 2 | {{begin-eqn}}
{{eqn | l = \map { {}_1 \operatorname F_1} {a; b - 1; x} - \map { {}_1 \operatorname F_1} {a + 1; b; x}
| r = \frac {\paren {a - b + 1} x} {b \paren {b - 1} } \map { {}_1 \operatorname F_1} {a + 1; b + 1; x}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map { {}_1 \operatorname F_1} {a; b - 1; x}
| r = \sum_{n \mathop = 0}^\infty \frac {a^{\overline n} } {\paren {b - 1}^{\overline n} n!} x^n
| c = {{Defof|Hypergeometric Function/Generalized|Generalized Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac a {\paren {b - 1} } x + \cd... | {{begin-eqn}}
{{eqn | l = \map { {}_1 \operatorname F_1} {a; b - 1; x} - \map { {}_1 \operatorname F_1} {a + 1; b; x}
| r = \frac {\paren {a - b + 1} x} {b \paren {b - 1} } \map { {}_1 \operatorname F_1} {a + 1; b + 1; x}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map { {}_1 \operatorname F_1} {a; b - 1; x}
| r = \sum_{n \mathop = 0}^\infty \frac {a^{\overline n} } {\paren {b - 1}^{\overline n} n!} x^n
| c = {{Defof|Hypergeometric Function/Generalized|Generalized Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac a {\paren {b - 1} } x + \cd... | Hypergeometric Function Difference Equation/Formulation 2 | https://proofwiki.org/wiki/Hypergeometric_Function_Difference_Equation/Formulation_2 | https://proofwiki.org/wiki/Hypergeometric_Function_Difference_Equation/Formulation_2 | [
"Hypergeometric Function Difference Equation"
] | [] | [
"Gamma Difference Equation",
"Gamma Difference Equation",
"Gamma Difference Equation"
] |
proofwiki-22471 | Hypergeometric Function Difference Equation/Formulation 3 | {{begin-eqn}}
{{eqn | l = \map { {}_1 \operatorname F_1} {a; b - 1; x} - \map { {}_1 \operatorname F_1} {a; b; x}
| r = \frac {a x} {b \paren {b - 1} } \map { {}_1 \operatorname F_1} {a + 1; b + 1; x}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map { {}_1 \operatorname F_1} {a; b - 1; x}
| r = \sum_{n \mathop = 0}^\infty \frac {a^{\overline n} } {\paren {b - 1}^{\overline n} n!} x^n
| c = {{Defof|Hypergeometric Function/Generalized|Generalized Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac a {\paren {b - 1} } x + \cd... | {{begin-eqn}}
{{eqn | l = \map { {}_1 \operatorname F_1} {a; b - 1; x} - \map { {}_1 \operatorname F_1} {a; b; x}
| r = \frac {a x} {b \paren {b - 1} } \map { {}_1 \operatorname F_1} {a + 1; b + 1; x}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map { {}_1 \operatorname F_1} {a; b - 1; x}
| r = \sum_{n \mathop = 0}^\infty \frac {a^{\overline n} } {\paren {b - 1}^{\overline n} n!} x^n
| c = {{Defof|Hypergeometric Function/Generalized|Generalized Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac a {\paren {b - 1} } x + \cd... | Hypergeometric Function Difference Equation/Formulation 3 | https://proofwiki.org/wiki/Hypergeometric_Function_Difference_Equation/Formulation_3 | https://proofwiki.org/wiki/Hypergeometric_Function_Difference_Equation/Formulation_3 | [
"Hypergeometric Function Difference Equation"
] | [] | [
"Gamma Difference Equation",
"Gamma Difference Equation"
] |
proofwiki-22472 | Hyperbolic Sine Function in terms of Hypergeometric Function | :$\sinh x = x \paren {\map { {}_0 \operatorname F_1} {\dfrac 3 2; \dfrac {x^2} 4} }$ | {{begin-eqn}}
{{eqn | l = x \paren {\map { {}_0 \operatorname F_1} {\frac 3 2; \dfrac {x^2} 4} }
| r = x \sum_{n \mathop = 0}^\infty \frac 1 {\paren {\frac 3 2}^{\bar n} } \frac {\paren {\frac {x^2} 4 }^n} {n!}
| c = {{Defof|Generalized Hypergeometric Function}}
}}
{{eqn | r = x \sum_{n \mathop = 0}^\infty ... | :$\sinh x = x \paren {\map { {}_0 \operatorname F_1} {\dfrac 3 2; \dfrac {x^2} 4} }$ | {{begin-eqn}}
{{eqn | l = x \paren {\map { {}_0 \operatorname F_1} {\frac 3 2; \dfrac {x^2} 4} }
| r = x \sum_{n \mathop = 0}^\infty \frac 1 {\paren {\frac 3 2}^{\bar n} } \frac {\paren {\frac {x^2} 4 }^n} {n!}
| c = {{Defof|Generalized Hypergeometric Function}}
}}
{{eqn | r = x \sum_{n \mathop = 0}^\infty ... | Hyperbolic Sine Function in terms of Hypergeometric Function | https://proofwiki.org/wiki/Hyperbolic_Sine_Function_in_terms_of_Hypergeometric_Function | https://proofwiki.org/wiki/Hyperbolic_Sine_Function_in_terms_of_Hypergeometric_Function | [
"Hypergeometric Functions",
"Hyperbolic Sine Function"
] | [] | [
"Rising Factorial as Quotient of Factorials",
"Gamma Function Extends Factorial",
"Gamma Difference Equation",
"Exponent Combination Laws/Power of Power",
"Legendre's Duplication Formula",
"Gamma Function Extends Factorial",
"Power Series Expansion for Hyperbolic Sine Function"
] |
proofwiki-22473 | Area under Acceleration-Time Graph | Let $P$ be a particle moving in a straight line.
Let the acceleration of $P$ as a function of time be $\map a t$.
Let the velocity of $P$ as a function of time be $\map v t$.
Let the motion of $P$ be plotted on an acceleration-time-graph $G$.
Let $v_1$ and $v_2$ be the velocities of $P$ at times $t_1$ and $t_2$ respect... | Let $A$ be the area under $G$ between times $t_1$ and $t_2$.
Then
{{begin-eqn}}
{{eqn | l = A
| r = \int_{t_1}^{t_2} \map a t \rd t
| c = Area under Curve
}}
{{eqn | r = \int_{t_1}^{t_2} \map {\dfrac {\d v} {\d t} } t \rd t
| c = {{Defof|Acceleration}}
}}
{{eqn | r = \map v {t_2} - \map v {t_1}
... | Let $P$ be a [[Definition:Particle|particle]] [[Definition:Motion|moving]] in a [[Definition:Straight Line|straight line]].
Let the [[Definition:Acceleration|acceleration]] of $P$ as a [[Definition:Function|function]] of [[Definition:Time|time]] be $\map a t$.
Let the [[Definition:Velocity|velocity]] of $P$ as a [[De... | Let $A$ be the [[Definition:Area under Graph|area under $G$]] between [[Definition:Time|times]] $t_1$ and $t_2$.
Then
{{begin-eqn}}
{{eqn | l = A
| r = \int_{t_1}^{t_2} \map a t \rd t
| c = [[Area under Curve]]
}}
{{eqn | r = \int_{t_1}^{t_2} \map {\dfrac {\d v} {\d t} } t \rd t
| c = {{Defof|Acceler... | Area under Acceleration-Time Graph | https://proofwiki.org/wiki/Area_under_Acceleration-Time_Graph | https://proofwiki.org/wiki/Area_under_Acceleration-Time_Graph | [
"Acceleration-Time Graphs"
] | [
"Definition:Particle",
"Definition:Motion",
"Definition:Line/Straight Line",
"Definition:Acceleration",
"Definition:Function",
"Definition:Time",
"Definition:Velocity",
"Definition:Function",
"Definition:Time",
"Definition:Motion",
"Definition:Acceleration-Time Graph",
"Definition:Velocity",
... | [
"Definition:Darboux Integral/Geometric Interpretation",
"Definition:Time",
"Area under Curve",
"Fundamental Theorem of Calculus/Second Part"
] |
proofwiki-22474 | Hyperbolic Cosine Function in terms of Hypergeometric Function | :$\cosh x = \map { {}_0 \operatorname F_1} {\dfrac 1 2; \dfrac {x^2} 4}$ | {{begin-eqn}}
{{eqn | l = \map { {}_0 \operatorname F_1} {\frac 1 2; \dfrac {x^2} 4}
| r = \sum_{n \mathop = 0}^\infty \frac 1 {\paren {\frac 1 2}^{\bar n} } \frac {\paren {\frac {x^2} 4 }^n} {n!}
| c = {{Defof|Generalized Hypergeometric Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {\map \Gam... | :$\cosh x = \map { {}_0 \operatorname F_1} {\dfrac 1 2; \dfrac {x^2} 4}$ | {{begin-eqn}}
{{eqn | l = \map { {}_0 \operatorname F_1} {\frac 1 2; \dfrac {x^2} 4}
| r = \sum_{n \mathop = 0}^\infty \frac 1 {\paren {\frac 1 2}^{\bar n} } \frac {\paren {\frac {x^2} 4 }^n} {n!}
| c = {{Defof|Generalized Hypergeometric Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {\map \Gam... | Hyperbolic Cosine Function in terms of Hypergeometric Function | https://proofwiki.org/wiki/Hyperbolic_Cosine_Function_in_terms_of_Hypergeometric_Function | https://proofwiki.org/wiki/Hyperbolic_Cosine_Function_in_terms_of_Hypergeometric_Function | [
"Hypergeometric Functions",
"Hyperbolic Cosine Function"
] | [] | [
"Rising Factorial as Quotient of Factorials",
"Gamma Function Extends Factorial",
"Gamma Difference Equation",
"Exponent Combination Laws/Power of Power",
"Gamma Difference Equation",
"Legendre's Duplication Formula",
"Gamma Function Extends Factorial",
"Power Series Expansion for Hyperbolic Cosine Fu... |
proofwiki-22475 | Continued Fraction for Real Hyperbolic Tangent Function | :$\tanh x = \cfrac x {1 + \cfrac {x^2} {3 + \cfrac {x^2} {5 + \cfrac {x^2} {7 + \cfrac {x^2} {9 + \cfrac {x^2} \ddots} } } } }$ | {{begin-eqn}}
{{eqn | l = \tanh x
| r = \dfrac {\sinh x} {\cosh x}
| c = {{Defof|Hyperbolic Tangent}}
}}
{{eqn | r = \frac {x \map { {}_0 \operatorname F_1} {\dfrac 3 2; \dfrac {x^2} 4} } {\map { {}_0 \operatorname F_1} {\dfrac 1 2; \dfrac {x^2} 4} }
| c = Hyperbolic Sine Function in terms of Hypergeo... | :$\tanh x = \cfrac x {1 + \cfrac {x^2} {3 + \cfrac {x^2} {5 + \cfrac {x^2} {7 + \cfrac {x^2} {9 + \cfrac {x^2} \ddots} } } } }$ | {{begin-eqn}}
{{eqn | l = \tanh x
| r = \dfrac {\sinh x} {\cosh x}
| c = {{Defof|Hyperbolic Tangent}}
}}
{{eqn | r = \frac {x \map { {}_0 \operatorname F_1} {\dfrac 3 2; \dfrac {x^2} 4} } {\map { {}_0 \operatorname F_1} {\dfrac 1 2; \dfrac {x^2} 4} }
| c = [[Hyperbolic Sine Function in terms of Hyperg... | Continued Fraction for Real Hyperbolic Tangent Function | https://proofwiki.org/wiki/Continued_Fraction_for_Real_Hyperbolic_Tangent_Function | https://proofwiki.org/wiki/Continued_Fraction_for_Real_Hyperbolic_Tangent_Function | [
"Hyperbolic Tangent Function",
"Continued Fractions",
"Hypergeometric Functions"
] | [] | [
"Hyperbolic Sine Function in terms of Hypergeometric Function",
"Hyperbolic Cosine Function in terms of Hypergeometric Function",
"Hypergeometric Continued Fraction/Formulation 1",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-22476 | Cosine Function in terms of Hypergeometric Function | :$\cos x = \map { {}_0 \operatorname F_1} {\dfrac 1 2; \dfrac {-x^2} 4}$ | {{begin-eqn}}
{{eqn | l = \map { {}_0 \operatorname F_1} {\frac 1 2; \dfrac {-x^2} 4}
| r = \sum_{n \mathop = 0}^\infty \frac 1 {\paren {\frac 1 2}^{\bar n} } \frac {\paren {\frac {-x^2} 4 }^n} {n!}
| c = {{Defof|Generalized Hypergeometric Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {\map \G... | :$\cos x = \map { {}_0 \operatorname F_1} {\dfrac 1 2; \dfrac {-x^2} 4}$ | {{begin-eqn}}
{{eqn | l = \map { {}_0 \operatorname F_1} {\frac 1 2; \dfrac {-x^2} 4}
| r = \sum_{n \mathop = 0}^\infty \frac 1 {\paren {\frac 1 2}^{\bar n} } \frac {\paren {\frac {-x^2} 4 }^n} {n!}
| c = {{Defof|Generalized Hypergeometric Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {\map \G... | Cosine Function in terms of Hypergeometric Function | https://proofwiki.org/wiki/Cosine_Function_in_terms_of_Hypergeometric_Function | https://proofwiki.org/wiki/Cosine_Function_in_terms_of_Hypergeometric_Function | [
"Hypergeometric Functions",
"Cosine Function"
] | [] | [
"Rising Factorial as Quotient of Factorials",
"Gamma Function Extends Factorial",
"Gamma Difference Equation",
"Exponent Combination Laws/Power of Power",
"Gamma Difference Equation",
"Legendre's Duplication Formula",
"Gamma Function Extends Factorial",
"Power Series Expansion for Cosine Function"
] |
proofwiki-22477 | Sine Function in terms of Hypergeometric Function | :$\sin x = x \paren {\map { {}_0 \operatorname F_1} {\dfrac 3 2; \dfrac {-x^2} 4} }$ | {{begin-eqn}}
{{eqn | l = x \paren {\map { {}_0 \operatorname F_1} {\frac 3 2; \dfrac {-x^2} 4} }
| r = x \sum_{n \mathop = 0}^\infty \frac 1 {\paren {\frac 3 2}^{\bar n} } \frac {\paren {\frac {-x^2} 4 }^n} {n!}
| c = {{Defof|Generalized Hypergeometric Function}}
}}
{{eqn | r = x \sum_{n \mathop = 0}^\inft... | :$\sin x = x \paren {\map { {}_0 \operatorname F_1} {\dfrac 3 2; \dfrac {-x^2} 4} }$ | {{begin-eqn}}
{{eqn | l = x \paren {\map { {}_0 \operatorname F_1} {\frac 3 2; \dfrac {-x^2} 4} }
| r = x \sum_{n \mathop = 0}^\infty \frac 1 {\paren {\frac 3 2}^{\bar n} } \frac {\paren {\frac {-x^2} 4 }^n} {n!}
| c = {{Defof|Generalized Hypergeometric Function}}
}}
{{eqn | r = x \sum_{n \mathop = 0}^\inft... | Sine Function in terms of Hypergeometric Function | https://proofwiki.org/wiki/Sine_Function_in_terms_of_Hypergeometric_Function | https://proofwiki.org/wiki/Sine_Function_in_terms_of_Hypergeometric_Function | [
"Hypergeometric Functions",
"Sine Function"
] | [] | [
"Rising Factorial as Quotient of Factorials",
"Gamma Function Extends Factorial",
"Gamma Difference Equation",
"Exponent Combination Laws/Power of Power",
"Legendre's Duplication Formula",
"Gamma Function Extends Factorial",
"Power Series Expansion for Hyperbolic Sine Function"
] |
proofwiki-22478 | Continued Fraction for Tangent Function | :$\tan x = \cfrac x {1 - \cfrac {x^2} {3 - \cfrac {x^2} {5 - \cfrac {x^2} {7 - \cfrac {x^2} {9 - \cfrac {x^2} \ddots} } } } }$ | {{begin-eqn}}
{{eqn | l = \tan x
| r = \dfrac {\sin x} {\cos x}
| c = {{Defof|Tangent}}
}}
{{eqn | r = \frac {x \map { {}_0 \operatorname F_1} {\dfrac 3 2; \dfrac {-x^2} 4} } {\map { {}_0 \operatorname F_1} {\dfrac 1 2; \dfrac {-x^2} 4} }
| c = Sine Function in terms of Hypergeometric Function, Cosine... | :$\tan x = \cfrac x {1 - \cfrac {x^2} {3 - \cfrac {x^2} {5 - \cfrac {x^2} {7 - \cfrac {x^2} {9 - \cfrac {x^2} \ddots} } } } }$ | {{begin-eqn}}
{{eqn | l = \tan x
| r = \dfrac {\sin x} {\cos x}
| c = {{Defof|Tangent}}
}}
{{eqn | r = \frac {x \map { {}_0 \operatorname F_1} {\dfrac 3 2; \dfrac {-x^2} 4} } {\map { {}_0 \operatorname F_1} {\dfrac 1 2; \dfrac {-x^2} 4} }
| c = [[Sine Function in terms of Hypergeometric Function]], [[... | Continued Fraction for Tangent Function | https://proofwiki.org/wiki/Continued_Fraction_for_Tangent_Function | https://proofwiki.org/wiki/Continued_Fraction_for_Tangent_Function | [
"Tangent Function",
"Continued Fractions",
"Hypergeometric Functions"
] | [] | [
"Sine Function in terms of Hypergeometric Function",
"Cosine Function in terms of Hypergeometric Function",
"Hypergeometric Continued Fraction/Formulation 1",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-22479 | Hypergeometric Function Difference Equation/Formulation 4 | {{begin-eqn}}
{{eqn | l = \map { {}_1 \operatorname F_1} {a + 1; b; x} - \map { {}_1 \operatorname F_1} {a; b; x}
| r = \frac x b \map { {}_1 \operatorname F_1} {a + 1; b + 1; x}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map { {}_1 \operatorname F_1} {a + 1; b; x}
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {a + 1}^{\overline n} } {b^{\overline n} n!} x^n
| c = {{Defof|Hypergeometric Function/Generalized|Generalized Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac {\paren {a + 1} } b x + \cd... | {{begin-eqn}}
{{eqn | l = \map { {}_1 \operatorname F_1} {a + 1; b; x} - \map { {}_1 \operatorname F_1} {a; b; x}
| r = \frac x b \map { {}_1 \operatorname F_1} {a + 1; b + 1; x}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map { {}_1 \operatorname F_1} {a + 1; b; x}
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {a + 1}^{\overline n} } {b^{\overline n} n!} x^n
| c = {{Defof|Hypergeometric Function/Generalized|Generalized Hypergeometric Function}}
}}
{{eqn | r = 1 + \frac {\paren {a + 1} } b x + \cd... | Hypergeometric Function Difference Equation/Formulation 4 | https://proofwiki.org/wiki/Hypergeometric_Function_Difference_Equation/Formulation_4 | https://proofwiki.org/wiki/Hypergeometric_Function_Difference_Equation/Formulation_4 | [
"Hypergeometric Function Difference Equation"
] | [] | [
"Gamma Difference Equation",
"Gamma Difference Equation"
] |
proofwiki-22480 | Exponential Function in terms of Hypergeometric Function | :$e^x = \map { {}_1 \operatorname F_1} {p; p; x}$ | {{begin-eqn}}
{{eqn | l = \map { {}_1 \operatorname F_1} {p; p; x}
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {p}^{\bar n} } {\paren {p}^{\bar n} } \frac {x^n} {n!}
| c = {{Defof|Generalized Hypergeometric Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}
| c =
}}
{{eqn | r ... | :$e^x = \map { {}_1 \operatorname F_1} {p; p; x}$ | {{begin-eqn}}
{{eqn | l = \map { {}_1 \operatorname F_1} {p; p; x}
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {p}^{\bar n} } {\paren {p}^{\bar n} } \frac {x^n} {n!}
| c = {{Defof|Generalized Hypergeometric Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}
| c =
}}
{{eqn | r ... | Exponential Function in terms of Hypergeometric Function | https://proofwiki.org/wiki/Exponential_Function_in_terms_of_Hypergeometric_Function | https://proofwiki.org/wiki/Exponential_Function_in_terms_of_Hypergeometric_Function | [
"Hypergeometric Functions",
"Exponential Function"
] | [] | [
"Power Series Expansion for Exponential Function"
] |
proofwiki-22481 | Hypergeometric Continued Fraction/Formulation 2 | Let $a, b \in \C$.
Let $b \notin \Z_{\le 0}$.
:$\ds \frac {\map { {}_1 \operatorname F_1} {a + 1; b; x} } {\map { {}_1 \operatorname F_1} {a; b; x} } = \cfrac 1 {1 - \cfrac x {b + \cfrac {\paren {a + 1} x} {\paren {b + 1} + \cfrac {\paren {a - b } x} {\paren {b + 2} + \cfrac {\paren {a + 2} x} {\paren {b + 3} + \cfrac ... | :First, we demonstate:
:$\ds \frac {\map { {}_1 \operatorname F_1} {a + 1; b; x} } {\map { {}_1 \operatorname F_1} {a; b; x} } = \frac 1 {1 - \dfrac x b \dfrac {\map { {}_1 \operatorname F_1} {a + 1; b + 1; x} } {\map { {}_1 \operatorname F_1} {a + 1; b; x} } }$
{{begin-eqn}}
{{eqn | l = \map { {}_1 \operatorname F_1} ... | Let $a, b \in \C$.
Let $b \notin \Z_{\le 0}$.
:$\ds \frac {\map { {}_1 \operatorname F_1} {a + 1; b; x} } {\map { {}_1 \operatorname F_1} {a; b; x} } = \cfrac 1 {1 - \cfrac x {b + \cfrac {\paren {a + 1} x} {\paren {b + 1} + \cfrac {\paren {a - b } x} {\paren {b + 2} + \cfrac {\paren {a + 2} x} {\paren {b + 3} + \cfrac... | :First, we demonstate:
:$\ds \frac {\map { {}_1 \operatorname F_1} {a + 1; b; x} } {\map { {}_1 \operatorname F_1} {a; b; x} } = \frac 1 {1 - \dfrac x b \dfrac {\map { {}_1 \operatorname F_1} {a + 1; b + 1; x} } {\map { {}_1 \operatorname F_1} {a + 1; b; x} } }$
{{begin-eqn}}
{{eqn | l = \map { {}_1 \operatorname F_1}... | Hypergeometric Continued Fraction/Formulation 2 | https://proofwiki.org/wiki/Hypergeometric_Continued_Fraction/Formulation_2 | https://proofwiki.org/wiki/Hypergeometric_Continued_Fraction/Formulation_2 | [
"Continued Fractions",
"Hypergeometric Functions"
] | [] | [
"Hypergeometric Function Difference Equation/Formulation 4",
"Definition:Reciprocal",
"Hypergeometric Function Difference Equation/Formulation 3",
"Definition:Reciprocal",
"Hypergeometric Function Difference Equation/Formulation 2",
"Definition:Reciprocal",
"Category:Continued Fractions",
"Category:Hy... |
proofwiki-22482 | Exponential Function as Limit of Gaussian Hypergeometric Function | :$\ds e^x = \lim_{p \mathop \to \infty} \map F {1, p; 1; \dfrac x p}$ | {{begin-eqn}}
{{eqn | l = \lim_{p \mathop \to \infty} \map F {1, p; 1; \dfrac x p}
| r = \sum_{n \mathop = 0}^\infty \lim_{p \mathop \to \infty} \frac {\paren {p}^{\overline n} 1^{\overline n} } {1^{\overline n} } \frac {\paren {\dfrac x p}^n} {n!}
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn |... | :$\ds e^x = \lim_{p \mathop \to \infty} \map F {1, p; 1; \dfrac x p}$ | {{begin-eqn}}
{{eqn | l = \lim_{p \mathop \to \infty} \map F {1, p; 1; \dfrac x p}
| r = \sum_{n \mathop = 0}^\infty \lim_{p \mathop \to \infty} \frac {\paren {p}^{\overline n} 1^{\overline n} } {1^{\overline n} } \frac {\paren {\dfrac x p}^n} {n!}
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn |... | Exponential Function as Limit of Gaussian Hypergeometric Function | https://proofwiki.org/wiki/Exponential_Function_as_Limit_of_Gaussian_Hypergeometric_Function | https://proofwiki.org/wiki/Exponential_Function_as_Limit_of_Gaussian_Hypergeometric_Function | [
"Gaussian Hypergeometric Function",
"Hypergeometric Functions"
] | [] | [
"Rising Factorial as Quotient of Factorials",
"Exponent Combination Laws/Power of Quotient",
"Limit of Real Function/Examples/Reciprocal of x at Infinity",
"Power Series Expansion for Exponential Function"
] |
proofwiki-22483 | Group of Rationals Modulo One Induces Equivalence Relation | Define a relation $\sim$ on $\Q$ such that:
:$\forall p, q \in \Q: p \sim q \iff p - q \in \Z$
Then $\sim$ is an equivalence relation | {{proofread}}
From Integers under Addition form Abelian Group, $\struct {\Z, +}$ forms an abelian group.
Checking in turn each of the criteria for equivalence: | Define a [[Definition:Relation|relation]] $\sim$ on $\Q$ such that:
:$\forall p, q \in \Q: p \sim q \iff p - q \in \Z$
Then $\sim$ is an [[Definition:Equivalence Relation|equivalence relation]] | {{proofread}}
From [[Integers under Addition form Abelian Group]], $\struct {\Z, +}$ forms an [[Definition:Abelian Group|abelian group]].
Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]: | Group of Rationals Modulo One Induces Equivalence Relation | https://proofwiki.org/wiki/Group_of_Rationals_Modulo_One_Induces_Equivalence_Relation | https://proofwiki.org/wiki/Group_of_Rationals_Modulo_One_Induces_Equivalence_Relation | [
"Examples of Groups",
"Rational Numbers"
] | [
"Definition:Relation",
"Definition:Equivalence Relation"
] | [
"Integers under Addition form Abelian Group",
"Definition:Abelian Group",
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-22484 | Almost-Everywhere Equality Relation for Real-Valued Functions is Equivalence Relation | Let $\struct {X, \Sigma, \mu}$ be a measure space.
<onlyinclude>
Let $\map {\mathcal M} {X, \Sigma, \R}$ be the real-valued $\Sigma$-measurable functions on $X$.
Let $\sim_\mu$ be the $\mu$-almost-everywhere equality relation on $\map {\mathcal M} {X, \Sigma, \R}$ by:
:$f \sim_\mu g$ {{iff}} $\map f x = \map g x$ for... | Checking in turn each of the criteria for equivalence: | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
<onlyinclude>
Let $\map {\mathcal M} {X, \Sigma, \R}$ be the [[Definition:Space of Real-Valued Measurable Functions|real-valued $\Sigma$-measurable functions]] on $X$.
Let $\sim_\mu$ be the $\mu$-almost-everywhere equality relation on $\... | Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]: | Almost-Everywhere Equality Relation for Real-Valued Functions is Equivalence Relation | https://proofwiki.org/wiki/Almost-Everywhere_Equality_Relation_for_Real-Valued_Functions_is_Equivalence_Relation | https://proofwiki.org/wiki/Almost-Everywhere_Equality_Relation_for_Real-Valued_Functions_is_Equivalence_Relation | [
"Definitions/Almost-Everywhere Equality Relation"
] | [
"Definition:Measure Space",
"Definition:Space of Measurable Functions/Real-Valued",
"Definition:Almost Everywhere",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-22485 | Cosecant Function in terms of Gaussian Hypergeometric Function | :$\csc x = \dfrac 1 x \map F {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; \sin^2 x}$ | {{begin-eqn}}
{{eqn | l = \arcsin x
| r = x \map F {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; x^2}
| c = Arcsine Function in terms of Gaussian Hypergeometric Function
}}
{{eqn | ll = \leadsto
| l = \map \arcsin {\sin x}
| r = \sin x \map F {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; \sin^2 x}
| c = $x ... | :$\csc x = \dfrac 1 x \map F {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; \sin^2 x}$ | {{begin-eqn}}
{{eqn | l = \arcsin x
| r = x \map F {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; x^2}
| c = [[Arcsine Function in terms of Gaussian Hypergeometric Function]]
}}
{{eqn | ll = \leadsto
| l = \map \arcsin {\sin x}
| r = \sin x \map F {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; \sin^2 x}
| c =... | Cosecant Function in terms of Gaussian Hypergeometric Function | https://proofwiki.org/wiki/Cosecant_Function_in_terms_of_Gaussian_Hypergeometric_Function | https://proofwiki.org/wiki/Cosecant_Function_in_terms_of_Gaussian_Hypergeometric_Function | [
"Cosecant Function",
"Gaussian Hypergeometric Function",
"Hypergeometric Functions"
] | [] | [
"Arcsine Function in terms of Gaussian Hypergeometric Function",
"Sine is Reciprocal of Cosecant"
] |
proofwiki-22486 | Secant Function in terms of Gaussian Hypergeometric Function | :$\sec x = \map F {\dfrac 1 2, 1; 1; \sin^2 x}$ | {{begin-eqn}}
{{eqn | l = \paren {1 + x}^p
| r = \map F {-p, 1; 1; -x}
| c = Power of One plus x in terms of Gaussian Hypergeometric Function
}}
{{eqn | ll = \leadsto
| l = \frac 1 {\sqrt {\paren {1 - \sin^2 x} } }
| r = \map F {\frac 1 2, 1; 1; \sin^2 x}
| c = $p \gets -\dfrac 1 2$ and $-... | :$\sec x = \map F {\dfrac 1 2, 1; 1; \sin^2 x}$ | {{begin-eqn}}
{{eqn | l = \paren {1 + x}^p
| r = \map F {-p, 1; 1; -x}
| c = [[Power of One plus x in terms of Gaussian Hypergeometric Function]]
}}
{{eqn | ll = \leadsto
| l = \frac 1 {\sqrt {\paren {1 - \sin^2 x} } }
| r = \map F {\frac 1 2, 1; 1; \sin^2 x}
| c = $p \gets -\dfrac 1 2$ an... | Secant Function in terms of Gaussian Hypergeometric Function | https://proofwiki.org/wiki/Secant_Function_in_terms_of_Gaussian_Hypergeometric_Function | https://proofwiki.org/wiki/Secant_Function_in_terms_of_Gaussian_Hypergeometric_Function | [
"Secant Function",
"Gaussian Hypergeometric Function",
"Hypergeometric Functions"
] | [] | [
"Power of One plus x in terms of Gaussian Hypergeometric Function",
"Sum of Squares of Sine and Cosine",
"Cosine is Reciprocal of Secant"
] |
proofwiki-22487 | Almost-Everywhere Equality Relation for Lebesgue Space is Equivalence Relation | Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $p \in \closedint 1 \infty$.
Let $\map {\mathcal L^p} {X, \Sigma, \mu}$ be the Lebesgue $p$-space of $\struct {X, \Sigma, \mu}$.
We define the $\mu$-almost-everywhere equality relation $\sim_\mu$ on $\map {\mathcal L^p} {X, \Sigma, \mu}$ by:
:$\forall f, g \in... | Let $f, g, h \in \LL^p$.
Checking in turn each of the criteria for equivalence:
=== Reflexivity ===
From P-Seminorm of Function Zero iff A.E. Zero, we have:
:$\norm {f - f}_p = 0$
Therefore:
:$f \sim_\mu f$
Hence $\sim_\mu$ is a reflexive relation.
{{qed|lemma}}
=== Symmetry ===
Suppose:
:$f \sim_\mu g$
Then:
:$\norm {... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]], and let $p \in \closedint 1 \infty$.
Let $\map {\mathcal L^p} {X, \Sigma, \mu}$ be the [[Definition:Lebesgue Space|Lebesgue $p$-space]] of $\struct {X, \Sigma, \mu}$.
We define the [[Definition:Almost-Everywhere Equality Relation/Lebesgue... | Let $f, g, h \in \LL^p$.
Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]:
=== Reflexivity ===
From [[P-Seminorm of Function Zero iff A.E. Zero]], we have:
:$\norm {f - f}_p = 0$
Therefore:
:$f \sim_\mu f$
Hence $\sim_\mu$ is a [[Definition:Reflexive Relation|reflexive rel... | Almost-Everywhere Equality Relation for Lebesgue Space is Equivalence Relation/Proof 1 | https://proofwiki.org/wiki/Almost-Everywhere_Equality_Relation_for_Lebesgue_Space_is_Equivalence_Relation | https://proofwiki.org/wiki/Almost-Everywhere_Equality_Relation_for_Lebesgue_Space_is_Equivalence_Relation/Proof_1 | [
"Almost-Everywhere Equality Relation for Lebesgue Space is Equivalence Relation",
"Almost-Everywhere Equality Relation",
"Functional Analysis",
"Measure Theory",
"Lebesgue Spaces"
] | [
"Definition:Measure Space",
"Definition:Lebesgue Space",
"Definition:Almost-Everywhere Equality Relation/Lebesgue Space/Definition 1",
"Definition:P-Seminorm",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"P-Seminorm of Function Zero iff A.E. Zero",
"Definition:Reflexive Relation",
"Definition:Symmetric Relation",
"Definition:Transitive Relation",
"Definition:Equivalence Relation"
] |
proofwiki-22488 | Almost-Everywhere Equality Relation for Lebesgue Space is Equivalence Relation | Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $p \in \closedint 1 \infty$.
Let $\map {\mathcal L^p} {X, \Sigma, \mu}$ be the Lebesgue $p$-space of $\struct {X, \Sigma, \mu}$.
We define the $\mu$-almost-everywhere equality relation $\sim_\mu$ on $\map {\mathcal L^p} {X, \Sigma, \mu}$ by:
:$\forall f, g \in... | Let $f, g, h \in \LL^p$.
Checking in turn each of the criteria for equivalence:
=== Reflexivity ===
By Equality is Reflexive, we have:
:$f \sim_\mu f$ {{iff}} $\map f x = \map f x$ for $\mu$-almost all $x \in X$.
Therefore:
:$f \sim_\mu f$
Hence $\sim_\mu$ is a reflexive relation.
{{qed|lemma}}
=== Symmetry ===
Suppose... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]], and let $p \in \closedint 1 \infty$.
Let $\map {\mathcal L^p} {X, \Sigma, \mu}$ be the [[Definition:Lebesgue Space|Lebesgue $p$-space]] of $\struct {X, \Sigma, \mu}$.
We define the [[Definition:Almost-Everywhere Equality Relation/Lebesgue... | Let $f, g, h \in \LL^p$.
Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]:
=== Reflexivity ===
By [[Equality is Reflexive]], we have:
:$f \sim_\mu f$ {{iff}} $\map f x = \map f x$ for [[Definition:Almost All|$\mu$-almost all]] $x \in X$.
Therefore:
:$f \sim_\mu f$
Hence $\... | Almost-Everywhere Equality Relation for Lebesgue Space is Equivalence Relation/Proof 2 | https://proofwiki.org/wiki/Almost-Everywhere_Equality_Relation_for_Lebesgue_Space_is_Equivalence_Relation | https://proofwiki.org/wiki/Almost-Everywhere_Equality_Relation_for_Lebesgue_Space_is_Equivalence_Relation/Proof_2 | [
"Almost-Everywhere Equality Relation for Lebesgue Space is Equivalence Relation",
"Almost-Everywhere Equality Relation",
"Functional Analysis",
"Measure Theory",
"Lebesgue Spaces"
] | [
"Definition:Measure Space",
"Definition:Lebesgue Space",
"Definition:Almost-Everywhere Equality Relation/Lebesgue Space/Definition 1",
"Definition:P-Seminorm",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"Equality is Reflexive",
"Definition:Almost All",
"Definition:Reflexive Relation",
"Definition:Almost All",
"Equality is Symmetric",
"Definition:Almost All",
"Definition:Symmetric Relation",
"Definition:Almost All",
"Definition:Almost All",
"Equality is Transit... |
proofwiki-22489 | Cosine Function in terms of Gaussian Hypergeometric Function | :$\map \cos x = \map F {\dfrac 1 2, -\dfrac 1 2; \dfrac 1 2; \sin^2 x}$ | {{begin-eqn}}
{{eqn | l = \paren {1 + x}^p
| r = \map F {-p, 1; 1; -x}
| c = Power of One plus x in terms of Gaussian Hypergeometric Function
}}
{{eqn | ll = \leadsto
| l = \sqrt {\paren {1 - \sin^2 x} }
| r = \map F {-\frac 1 2, 1; 1; \sin^2 x}
| c = $p \to \frac 1 2$ and $-x \to \sin^2 x... | :$\map \cos x = \map F {\dfrac 1 2, -\dfrac 1 2; \dfrac 1 2; \sin^2 x}$ | {{begin-eqn}}
{{eqn | l = \paren {1 + x}^p
| r = \map F {-p, 1; 1; -x}
| c = [[Power of One plus x in terms of Gaussian Hypergeometric Function]]
}}
{{eqn | ll = \leadsto
| l = \sqrt {\paren {1 - \sin^2 x} }
| r = \map F {-\frac 1 2, 1; 1; \sin^2 x}
| c = $p \to \frac 1 2$ and $-x \to \sin... | Cosine Function in terms of Gaussian Hypergeometric Function | https://proofwiki.org/wiki/Cosine_Function_in_terms_of_Gaussian_Hypergeometric_Function | https://proofwiki.org/wiki/Cosine_Function_in_terms_of_Gaussian_Hypergeometric_Function | [
"Cosine Function",
"Gaussian Hypergeometric Function",
"Hypergeometric Functions"
] | [] | [
"Power of One plus x in terms of Gaussian Hypergeometric Function",
"Sum of Squares of Sine and Cosine",
"Real Multiplication is Commutative",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-22490 | Almost-Everywhere Equality Relation for Measurable Sets is Equivalence Relation | Let $\struct {X, \Sigma, \mu}$ be a measure space.
We define the $\mu$-almost-everywhere equality relation $\sim_\mu$ on $\Sigma$ by:
:$A \sim_\mu B$ {{iff}} $\map \mu {A \symdif B} = 0$
where $\symdif$ denotes set symmetric difference.
Then:
:$\sim_\mu$ is an equivalence relation. | Checking in turn each of the criteria for equivalence:
=== Reflexivity ===
By Symmetric Difference with Self is Empty Set, we have:
We have:
:$A \symdif A = \O$
By Measure of Empty Set is Zero:
:$\map \mu \O = 0$
So:
:$\map \mu {A \symdif A} = 0$
Therefore:
:$A \sim_\mu A$
Hence $\sim_\mu$ is a reflexive relation.
{{qe... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
We define the [[Definition:Almost-Everywhere Equality Relation/Measurable Sets|$\mu$-almost-everywhere equality relation]] $\sim_\mu$ on $\Sigma$ by:
:$A \sim_\mu B$ {{iff}} $\map \mu {A \symdif B} = 0$
where $\symdif$ denotes [[Definiti... | Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]:
=== Reflexivity ===
By [[Symmetric Difference with Self is Empty Set]], we have:
We have:
:$A \symdif A = \O$
By [[Measure of Empty Set is Zero]]:
:$\map \mu \O = 0$
So:
:$\map \mu {A \symdif A} = 0$
Therefore:
:$A \sim_\m... | Almost-Everywhere Equality Relation for Measurable Sets is Equivalence Relation/Proof | https://proofwiki.org/wiki/Almost-Everywhere_Equality_Relation_for_Measurable_Sets_is_Equivalence_Relation | https://proofwiki.org/wiki/Almost-Everywhere_Equality_Relation_for_Measurable_Sets_is_Equivalence_Relation/Proof | [
"Definitions/Almost-Everywhere Equality Relation"
] | [
"Definition:Measure Space",
"Definition:Almost-Everywhere Equality Relation/Measurable Sets",
"Definition:Symmetric Difference",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"Symmetric Difference with Self is Empty Set",
"Measure of Empty Set is Zero",
"Definition:Reflexive Relation",
"Definition:Symmetric Relation",
"Definition:Transitive Relation",
"Definition:Equivalence Relation"
] |
proofwiki-22491 | Space of Integrable Functions Under Pointwise Addition forms Abelian Group | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $\map {\LL^1} \mu$ be the space of real-valued $\mu$-integrable functions.
Then $\map {\LL^1} \mu$, endowed with pointwise addition, forms an abelian group over $\map {\LL^1} \mu$. | Suppose $f, g \in \map {\LL^1} \mu$.
By Pointwise Sum of Integrable Functions is Integrable Function:
:$f + g \in \map {\LL^1} \mu$
So $\map {\LL^1} \mu$ is closed under pointwise addition.
{{finish}}
Hence the result.
{{qed}} | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $\map {\LL^1} \mu$ be the [[Definition:Space of Integrable Functions|space of real-valued $\mu$-integrable functions]].
Then $\map {\LL^1} \mu$, endowed with [[Definition:Pointwise Addition of Real-Valued Functions|pointwise addition... | Suppose $f, g \in \map {\LL^1} \mu$.
By [[Pointwise Sum of Integrable Functions is Integrable Function]]:
:$f + g \in \map {\LL^1} \mu$
So $\map {\LL^1} \mu$ is [[Definition:Closed Operation|closed]] under [[Definition:Pointwise Addition of Real-Valued Functions|pointwise addition]].
{{finish}}
Hence the result.
{{q... | Space of Integrable Functions Under Pointwise Addition forms Abelian Group | https://proofwiki.org/wiki/Space_of_Integrable_Functions_Under_Pointwise_Addition_forms_Abelian_Group | https://proofwiki.org/wiki/Space_of_Integrable_Functions_Under_Pointwise_Addition_forms_Abelian_Group | [
"Measure-Integrable Functions",
"Examples of Abelian Groups"
] | [
"Definition:Measure Space",
"Definition:Space of Integrable Functions",
"Definition:Pointwise Addition of Real-Valued Functions",
"Definition:Abelian Group"
] | [
"Pointwise Sum of Integrable Functions is Integrable Function",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Pointwise Addition of Real-Valued Functions"
] |
proofwiki-22492 | Adjacency Matrix for Undirected Graph is Symmetric | Let $G$ be an undirected graph.
The adjacency matrix for $G$ is a symmetric matrix. | {{MissingLinks}}
{{tidy}}
Let $G = \struct {V, E}$ be an undirected graph and $M$ its adjacency matrix.
We will show that $M_{ij} = M_{ji}$.
Let $v_i, v_j \in V$ be vertices on $G$.
;Case 1: $v_i v_j \in E$
It follows from the definition of adjacency matrix that $M_{ij} = 1$.
It follows from the definition of undirect... | Let $G$ be an [[Definition:Undirected Graph|undirected graph]].
The [[Definition:Adjacency Matrix|adjacency matrix]] for $G$ is a [[Definition:Symmetric Matrix|symmetric matrix]]. | {{MissingLinks}}
{{tidy}}
Let $G = \struct {V, E}$ be an [[Definition:Undirected Graph|undirected graph]] and $M$ its [[Definition:Adjacency Matrix|adjacency matrix]].
We will show that $M_{ij} = M_{ji}$.
Let $v_i, v_j \in V$ be vertices on $G$.
;Case 1: $v_i v_j \in E$
It follows from the definition of [[Defin... | Adjacency Matrix for Undirected Graph is Symmetric | https://proofwiki.org/wiki/Adjacency_Matrix_for_Undirected_Graph_is_Symmetric | https://proofwiki.org/wiki/Adjacency_Matrix_for_Undirected_Graph_is_Symmetric | [
"Adjacency Matrices",
"Undirected Graphs",
"Symmetric Matrices"
] | [
"Definition:Undirected Graph",
"Definition:Adjacency Matrix",
"Definition:Symmetric Matrix"
] | [
"Definition:Undirected Graph",
"Definition:Adjacency Matrix",
"Definition:Adjacency Matrix",
"Definition:Undirected Graph",
"Definition:Adjacency Matrix",
"Definition:Undirected Graph",
"Definition:Contradiction",
"Definition:Adjacency Matrix"
] |
proofwiki-22493 | Logarithm of One plus x over One minus x in terms of Gaussian Hypergeometric Function | :$\map \ln {\dfrac {1 + x} {1 - x} } = 2 x \map F {\dfrac 1 2, 1; \dfrac 3 2; x^2}$ | {{begin-eqn}}
{{eqn | l = 2 x \map F {\dfrac 1 2, 1; \dfrac 3 2; x^2}
| r = 2 x \sum_{n \mathop = 0}^\infty \frac {\paren {\frac 1 2}^{\bar n} \paren {1}^{\bar n} } {\paren {\frac 3 2}^{\bar n} } \frac {\paren {x^2}^n } {n!}
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = 2 x \sum_{n \mathop... | :$\map \ln {\dfrac {1 + x} {1 - x} } = 2 x \map F {\dfrac 1 2, 1; \dfrac 3 2; x^2}$ | {{begin-eqn}}
{{eqn | l = 2 x \map F {\dfrac 1 2, 1; \dfrac 3 2; x^2}
| r = 2 x \sum_{n \mathop = 0}^\infty \frac {\paren {\frac 1 2}^{\bar n} \paren {1}^{\bar n} } {\paren {\frac 3 2}^{\bar n} } \frac {\paren {x^2}^n } {n!}
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = 2 x \sum_{n \mathop... | Logarithm of One plus x over One minus x in terms of Gaussian Hypergeometric Function | https://proofwiki.org/wiki/Logarithm_of_One_plus_x_over_One_minus_x_in_terms_of_Gaussian_Hypergeometric_Function | https://proofwiki.org/wiki/Logarithm_of_One_plus_x_over_One_minus_x_in_terms_of_Gaussian_Hypergeometric_Function | [
"Gaussian Hypergeometric Function",
"Hypergeometric Functions",
"Logarithms"
] | [] | [
"Rising Factorial as Quotient of Factorials",
"One to Integer Rising is Integer Factorial",
"Gamma Function Extends Factorial",
"Gamma Difference Equation",
"Exponent Combination Laws/Product of Powers",
"Power Series Expansion for Half Logarithm of 1 + x over 1 - x"
] |
proofwiki-22494 | Equivalence of Definitions of Affine Hull | Let $S$ be a set.
{{TFAE|def = Affine Hull}} | === Definition $(1)$ implies Definition $(2)$ ===
Let $\HH$ be the affine hull of $S$ by definition $1$.
{{Recall|Affine Hull|index = 1}}
{{:Definition:Affine Hull/Definition 1}}
{{finish}}
Thus $\HH$ is the affine hull of $S$ by definition $2$.
{{qed|lemma}} | Let $S$ be a [[Definition:Set|set]].
{{TFAE|def = Affine Hull}} | === Definition $(1)$ implies Definition $(2)$ ===
Let $\HH$ be the [[Definition:Affine Hull/Definition 1|affine hull of $S$ by definition $1$]].
{{Recall|Affine Hull|index = 1}}
{{:Definition:Affine Hull/Definition 1}}
{{finish}}
Thus $\HH$ is the [[Definition:Affine Hull/Definition 2|affine hull of $S$ by defini... | Equivalence of Definitions of Affine Hull | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Affine_Hull | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Affine_Hull | [
"Affine Hulls"
] | [
"Definition:Set"
] | [
"Definition:Affine Hull/Definition 1",
"Definition:Affine Hull/Definition 2",
"Definition:Affine Hull/Definition 2",
"Definition:Affine Hull/Definition 1"
] |
proofwiki-22495 | Arccosine Function in terms of Gaussian Hypergeometric Function | :$\arccos x = 2 \sqrt {\dfrac {1 - x} 2} \map F {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; \dfrac {1 - x} 2}$ | {{begin-eqn}}
{{eqn | l = u
| r = \map \arcsin {\sin u}
| c =
}}
{{eqn | r = \sin u \map F {\frac 1 2, \frac 1 2; \frac 3 2; \sin^2 u}
| c = Arcsine Function in terms of Gaussian Hypergeometric Function
}}
{{eqn | r = \sqrt {\dfrac {1 - \map \cos {2 u} } 2} \map F {\frac 1 2, \frac 1 2; \frac 3 2; \d... | :$\arccos x = 2 \sqrt {\dfrac {1 - x} 2} \map F {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; \dfrac {1 - x} 2}$ | {{begin-eqn}}
{{eqn | l = u
| r = \map \arcsin {\sin u}
| c =
}}
{{eqn | r = \sin u \map F {\frac 1 2, \frac 1 2; \frac 3 2; \sin^2 u}
| c = [[Arcsine Function in terms of Gaussian Hypergeometric Function]]
}}
{{eqn | r = \sqrt {\dfrac {1 - \map \cos {2 u} } 2} \map F {\frac 1 2, \frac 1 2; \frac 3 2... | Arccosine Function in terms of Gaussian Hypergeometric Function | https://proofwiki.org/wiki/Arccosine_Function_in_terms_of_Gaussian_Hypergeometric_Function | https://proofwiki.org/wiki/Arccosine_Function_in_terms_of_Gaussian_Hypergeometric_Function | [
"Arccosine Function",
"Gaussian Hypergeometric Function",
"Hypergeometric Functions"
] | [] | [
"Arcsine Function in terms of Gaussian Hypergeometric Function",
"Double Angle Formula for Cosine/Corollary 5"
] |
proofwiki-22496 | Area of Right Parabolic Segment | Let $ABC$ be a right parabolic segment where:
:$AB$ is the defining chord $\LL$ of $ABC$
:$C$ is the vertex of the defining parabola $\PP$ of $ABC$.
:480px
The area $\AA$ of $ABC$ is given by:
:$\AA = \dfrac {2 a b} 3$
where:
:$a$ is the length of the line segment $CF$, where $F$ is the point at which the axis of $\PP$... | Construct the triangle $\triangle ABC$:
:480px
From Quadrature of Parabola:
:$\AA = \dfrac 4 3 \triangle ABC$
From Area of Triangle in Terms of Side and Altitude, the area of $\triangle ABC$ equals $\dfrac {a b} 2$.
The result follows.
{{qed}} | Let $ABC$ be a [[Definition:Right Parabolic Segment|right parabolic segment]] where:
:$AB$ is the defining [[Definition:Chord of Parabola|chord]] $\LL$ of $ABC$
:$C$ is the [[Definition:Vertex of Parabola|vertex]] of the defining [[Definition:Parabola|parabola]] $\PP$ of $ABC$.
:[[File:Area-of-Right-Parabolic-Segment... | Construct the [[Definition:Triangle (Geometry)|triangle]] $\triangle ABC$:
:[[File:Area-of-Right-Parabolic-Segment-Proof.png|480px]]
From [[Quadrature of Parabola]]:
:$\AA = \dfrac 4 3 \triangle ABC$
From [[Area of Triangle in Terms of Side and Altitude]], the [[Definition:Area|area]] of $\triangle ABC$ equals $\d... | Area of Right Parabolic Segment | https://proofwiki.org/wiki/Area_of_Right_Parabolic_Segment | https://proofwiki.org/wiki/Area_of_Right_Parabolic_Segment | [
"Right Parabolic Segments",
"Area Formulas"
] | [
"Definition:Parabolic Segment/Right",
"Definition:Chord of Conic Section/Parabola",
"Definition:Parabola/Vertex",
"Definition:Parabola",
"File:Area-of-Right-Parabolic-Segment.png",
"Definition:Area",
"Definition:Linear Measure/Length",
"Definition:Line/Segment",
"Definition:Point",
"Definition:Par... | [
"Definition:Triangle (Geometry)",
"File:Area-of-Right-Parabolic-Segment-Proof.png",
"Quadrature of Parabola",
"Area of Triangle in Terms of Side and Altitude",
"Definition:Area"
] |
proofwiki-22497 | Logarithmic Approximation of Error Term of Stirling's Formula for Gamma Function/Lemma 1 | :If $\size {\arg z} \le \dfrac \pi 4 $, then $K_z = 1$ | {{begin-eqn}}
{{eqn | l = K_z
| r = \map \sup {\size {\dfrac {z^2} {u^2 + z^2} } }
| c =
}}
{{eqn | r = \map \sup {\size {\dfrac {\paren {\cmod z e^{i \size {\arg z} } }^2 } {u^2 + \paren {\cmod z e^{i \size {\arg z} } }^2} } }
| c =
}}
{{eqn | r = \map \sup {\size {\dfrac {\cmod z^2 e^{i 2 \size {\... | :If $\size {\arg z} \le \dfrac \pi 4 $, then $K_z = 1$ | {{begin-eqn}}
{{eqn | l = K_z
| r = \map \sup {\size {\dfrac {z^2} {u^2 + z^2} } }
| c =
}}
{{eqn | r = \map \sup {\size {\dfrac {\paren {\cmod z e^{i \size {\arg z} } }^2 } {u^2 + \paren {\cmod z e^{i \size {\arg z} } }^2} } }
| c =
}}
{{eqn | r = \map \sup {\size {\dfrac {\cmod z^2 e^{i 2 \size {\... | Logarithmic Approximation of Error Term of Stirling's Formula for Gamma Function/Lemma 1 | https://proofwiki.org/wiki/Logarithmic_Approximation_of_Error_Term_of_Stirling's_Formula_for_Gamma_Function/Lemma_1 | https://proofwiki.org/wiki/Logarithmic_Approximation_of_Error_Term_of_Stirling's_Formula_for_Gamma_Function/Lemma_1 | [
"Logarithmic Approximation of Error Term of Stirling's Formula for Gamma Function"
] | [] | [
"Definition:Complex Number/Real Part",
"Definition:Positive/Real Number",
"Category:Logarithmic Approximation of Error Term of Stirling's Formula for Gamma Function"
] |
proofwiki-22498 | Logarithmic Approximation of Error Term of Stirling's Formula for Gamma Function/Lemma 2 | :If $\dfrac \pi 4 < \size {\arg z} < \dfrac \pi 2 $, then $K_z = \map \csc {2 \size {\arg z} }$ | {{begin-eqn}}
{{eqn | l = K_z
| r = \map \sup {\size {\dfrac {z^2} {u^2 + z^2} } }
| c =
}}
{{eqn | r = \map \sup {\size {\dfrac {\cmod z^2 \paren {\map \cos {2 \size {\arg z} } + i \map \sin {2 \size {\arg z} } } } {u^2 + \cmod z^2 \paren {\map \cos {2 \size {\arg z} } + i \map \sin {2 \size {\arg z} } } ... | :If $\dfrac \pi 4 < \size {\arg z} < \dfrac \pi 2 $, then $K_z = \map \csc {2 \size {\arg z} }$ | {{begin-eqn}}
{{eqn | l = K_z
| r = \map \sup {\size {\dfrac {z^2} {u^2 + z^2} } }
| c =
}}
{{eqn | r = \map \sup {\size {\dfrac {\cmod z^2 \paren {\map \cos {2 \size {\arg z} } + i \map \sin {2 \size {\arg z} } } } {u^2 + \cmod z^2 \paren {\map \cos {2 \size {\arg z} } + i \map \sin {2 \size {\arg z} } } ... | Logarithmic Approximation of Error Term of Stirling's Formula for Gamma Function/Lemma 2 | https://proofwiki.org/wiki/Logarithmic_Approximation_of_Error_Term_of_Stirling's_Formula_for_Gamma_Function/Lemma_2 | https://proofwiki.org/wiki/Logarithmic_Approximation_of_Error_Term_of_Stirling's_Formula_for_Gamma_Function/Lemma_2 | [
"Logarithmic Approximation of Error Term of Stirling's Formula for Gamma Function"
] | [] | [
"Definition:Complex Number/Real Part",
"Definition:Negative/Real Number",
"Definition:Minimization",
"Definition:Fraction/Denominator",
"Definition:Maximum Value of Real Function",
"Definition:Ratio",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Category:Logarithmic Approxima... |
proofwiki-22499 | Volume of Circular Cylinder/Height | Let $\CC$ be a circular cylinder such that:
:the bases of $\CC$ are circles of radius $r$
:the height of $\CC$ is $h$.
The volume $\VV$ of $\CC$ is given by the formula:
:$\VV = \pi r^2 h$ | From Volume of Cylinder in terms of Height and Base Area:
:$\VV = \AA h$
where $\AA$ is the area of the base of $\CC$.
From Area of Circle:
:$\AA = \pi r^2$
The result follows.
{{qed}} | Let $\CC$ be a [[Definition:Circular Cylinder|circular cylinder]] such that:
:the [[Definition:Base of Cylinder|bases]] of $\CC$ are [[Definition:Circle|circles]] of [[Definition:Radius of Circle|radius]] $r$
:the [[Definition:Height of Cylinder|height]] of $\CC$ is $h$.
The [[Definition:Volume|volume]] $\VV$ of $\CC... | From [[Volume of Cylinder in terms of Height and Base Area]]:
:$\VV = \AA h$
where $\AA$ is the [[Definition:Area|area]] of the [[Definition:Base of Cylinder|base]] of $\CC$.
From [[Area of Circle]]:
:$\AA = \pi r^2$
The result follows.
{{qed}} | Volume of Circular Cylinder/Height | https://proofwiki.org/wiki/Volume_of_Circular_Cylinder/Height | https://proofwiki.org/wiki/Volume_of_Circular_Cylinder/Height | [
"Volume of Circular Cylinder"
] | [
"Definition:Circular Solid Figure/Cylinder",
"Definition:Cylinder/Base",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Cylinder/Height",
"Definition:Volume"
] | [
"Volume of Cylinder/Height and Base Area",
"Definition:Area",
"Definition:Cylinder/Base",
"Area of Circle"
] |
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