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- 05:13, 23 April 2025 Basis of Vector Space/Examples/2-Dimensional Space (hist | edit) [1,768 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Basis of Vector Space == <onlyinclude> Consider a vector space $V$ spanning the plane. Then $2$ arbitrary non-zero non-parallel vectors of $V$ can serve as a basis for $V$. </onlyinclude> == Proof == Let $\mathbf u$ and $\mathb...")
- 04:58, 23 April 2025 Basis of Vector Space/Examples/3-Dimensional Space (hist | edit) [1,911 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Basis of Vector Space == <onlyinclude> Consider a vector space $V$ spanning ordinary space of $3$ dimensions. Then $3$ arbitrary non-zero non-coplanar vectors of $V$ can serve as a basis f...")
- 21:04, 22 April 2025 Basis of Vector Space/Examples (hist | edit) [437 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Bases of Vector Spaces == <onlyinclude> === $-Dimensional Space === {{:Basis of Vector Space/Examples/3-Dimensional Space}} === $-Dimensional Space === {{:Basis of Vector Space/Examples/2-Dimensional Space}}</onlyinclude> Category:Examples of Bases of Vector Spaces")
- 21:50, 18 April 2025 Index Number/Examples/Arbitrary Example 2 (hist | edit) [898 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Index Number == <onlyinclude> Suppose an '''index number''' is assigned $100$ in the base year. Suppose prices have risen $17 \%$. Then the '''index number''' would be $117$. </onlyinclude> == Sources == * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|...")
- 06:53, 18 April 2025 Ideal and Zero Locus are Order Reversing (hist | edit) [2,804 bytes] Aidan.epperly (talk | contribs) (I saw that this was missing from the nullstellensatz.)
- 19:07, 17 April 2025 Four Fours/64/Solutions/7 (hist | edit) [193 bytes] Prime.mover (talk | contribs) (Created page with "== Puzzle: Four Fours: $64$ == {{:Four Fours/64}} == Solution == <onlyinclude> :$64 = \paren {4 + 4} \uparrow \paren {4 - \sqrt 4}$ </onlyinclude> Category:Four Fours/64")
- 19:06, 17 April 2025 Four Fours/64/Solutions/6 (hist | edit) [205 bytes] Prime.mover (talk | contribs) (Created page with "== Puzzle: Four Fours: $ == {{:Four Fours/64}} == Solution == <onlyinclude> :$64 = \sqrt 4 \uparrow {\paren {\sqrt 4 + \sqrt 4 + \sqrt 4} }$ </onlyinclude> Category:Four Fours/64")
- 19:04, 17 April 2025 Four Fours/64/Solutions/5 (hist | edit) [186 bytes] Prime.mover (talk | contribs) (Created page with "== Puzzle: Four Fours: $ == {{:Four Fours/64}} == Solution == <onlyinclude> :$64 = \sqrt 4 \uparrow {\paren {\sqrt 4 + \sqrt 4 + \sqrt 4} }$ </onlyinclude> Category:Four Fours/64")
- 05:21, 17 April 2025 Number Base/Examples/8/4703 (hist | edit) [1,114 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Number expressed in Base $ Notation == <onlyinclude> The integer expressed in decimal as $4703$ is expressed in base $ as $11137_8$. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = 4703 | r = 1 \times 8^4 + 1 \times 8^3 + 1 \times 8^2 + 3 \times 8^1 + 7 \times 8^0 | c = }} {{eqn | r = 4096 + 512 + 64 + 24 + 7 | c = }} {...")
- 05:03, 17 April 2025 Number Base/Examples/10/4703 (hist | edit) [869 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Number expressed in Base $5$ Notation == <onlyinclude> The integer expressed in decimal as $4703$ is: :$4 \times 10^3 + 7 \times 10^2 + 0 \times 10^1 + 3 \times 10^0$ </onlyinclude> == Sources == * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Base of Pyramid|n...")
- 04:59, 17 April 2025 Number Base/Examples/10 (hist | edit) [249 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Numbers expressed in Base $ Notation == <onlyinclude> ==== 03$ in Base $ ==== {{:Number Base/Examples/10/4703}}</onlyinclude> Category:Examples of Number Bases")
- 04:59, 17 April 2025 Number Base/Examples/8 (hist | edit) [254 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Numbers expressed in Base $ Notation == <onlyinclude> ==== 03_{10}$ expressed in Base $ ==== {{:Number Base/Examples/8/4703}}</onlyinclude> Category:Examples of Number Bases")
- 23:07, 16 April 2025 Four Fours/88/Solutions/3 (hist | edit) [155 bytes] Medians (talk | contribs) (Created page with "== Puzzle: Four Fours: $88$ == {{:Four Fours/88}} == Solution == <onlyinclude> :$88 = 44 + 44$ </onlyinclude> Category:Four Fours/88")
- 22:08, 16 April 2025 Barycentric Coordinates of Centroid of Triangle (hist | edit) [889 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be the position vectors of the $3$ vertices of a triangle $T$ in the plane. Let $p$ be the centroid of $T$. Let $\alpha$, $\beta$ and $\gamma$ be the barycentric coordinates of $p$ {{WRT}} $T$ Then: :...")
- 22:05, 16 April 2025 Barycentric Coordinates of Point inside Triangle (hist | edit) [865 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be the position vectors of the $3$ vertices of a triangle $T$ in the plane. Let $p$ be an arbitrary point inside $T$. Then the barycentric coordinates of $p$ {{WRT}} $T$ are all Definition:Positive Real Number|posi...")
- 20:04, 14 April 2025 Randomized Block Design/Examples/Repeated Measures Design (hist | edit) [842 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Randomized Block Design == <onlyinclude> A '''repeated measures design''' is an example of a '''randomized block design'''. </onlyinclude> == Sources == * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Block Design/Also define...")
- 08:36, 14 April 2025 Interior of Topological n-Manifold with Boundary is Topological n-Manifold without Boundary (hist | edit) [1,595 bytes] Austrodata (talk | contribs) (Created page with "== Theorem == Let $M$ be a topological $n$-manifold with boundary. <onlyinclude> The '''interior''' of $M$, denoted $\Int M$, is a topological $n$-manifold (without boundary). </onlyinclude> == Proof == {{Recall|Locally Euclidean Space|locally Euclidean space}} {{:Definition:Locally Euclidean Space}} {{Recall|Topological Ma...")
- 05:16, 14 April 2025 Randomized Block Design/Examples (hist | edit) [318 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Randomized Block Designs == <onlyinclude> === Repeated Measures Design === {{:Randomized Block Design/Examples/Repeated Measures Design}}</onlyinclude> Category:Examples of Randomized Block Designs")
- 22:28, 13 April 2025 Balanced Set/Examples/Sphere at Origin (hist | edit) [936 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Balanced Set == <onlyinclude> A sphere embedded in Cartesian space with its center at the origin is an example of a '''balanced set'''. </onlyinclude> == Sources == * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|e...")
- 22:23, 13 April 2025 Euclidean Space is Second-Countable (hist | edit) [1,287 bytes] Austrodata (talk | contribs) (Created page with "== Theorem == Let $\R^n$ be an $n$-dimensional real vector space. Let $\struct {\R^n, \tau_d}$ $n$-dimensional real Euclidean space with the usual topology. Then, $\struct {\R^n, \tau_d}$ is second-countable. == Proof == Let $\struct {\R, \tau}$ be the Definition:Real Number...")
- 21:43, 13 April 2025 Balanced Set/Examples (hist | edit) [239 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Balanced Set == <onlyinclude> === Sphere at Origin === {{:Balanced Set/Examples/Sphere at Origin}}</onlyinclude> Category:Examples of Balanced Sets")
- 20:54, 13 April 2025 Baire Category Theorem/Historical Note (hist | edit) [890 bytes] Prime.mover (talk | contribs) (Created page with "== Historical Note on Baire Category Theorem == <onlyinclude> The '''Baire Category Theorem''' has important uses in functional analysis. One of its uses is implying that an algebraic or Hamel basis of an infinite dimensional Banach space must be Definition:Uncountable Set|unco...")
- 16:04, 13 April 2025 Baire Category Theorem/Hausdorff Space (hist | edit) [1,175 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $T = \struct {S, \tau}$ be a Hausdorff space. Let $T$ be locally compact. Then $T = \struct {S, \tau}$ is also a Baire space. </onlyinclude> == Proof == {{ProofWanted}} {{Namedfor|René-Louis Baire|cat = Baire}} == Historical Note == {{:Baire Category Theorem/Historical Note}} == Sourc...")
- 15:20, 13 April 2025 Real Number Line is Non-Meager/Proof 2 (hist | edit) [601 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Real Number Line is Non-Meager}} == Proof == <onlyinclude> This proof does not use the {{Axiom-link|Dependent Choice}}, as it uses intrinsic properties of the real numbers that do not necessarily hold for the general complete metric space. {{ProofWanted}} </onlyinclude> Category:Real Number Line is Non-Meager")
- 14:48, 13 April 2025 Real Number Line is Non-Meager/Proof 1 (hist | edit) [770 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Real Number Line is Non-Meager}} == Proof == <onlyinclude> We have that the Real Number Line is Complete Metric Space. From the Baire Category Theorem, a complete metric space is also a Baire space. The result follows from Baire Space is Non-Meager. {{qed}} </onlyinclude> == Sources == * {{BookReference|Counterexamples in Topology|1978|Lynn Arthur Steen|author2 =...")
- 19:58, 12 April 2025 Empty Topological Space is Locally Euclidean Space of any Dimension (hist | edit) [586 bytes] Austrodata (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $T = \struct {\O, \set \O}$ be the empty topological space. Then $T$ is locally euclidean space of any dimension. </onlyinclude> == Proof == {{ProofWanted}} {{qed}} Category:Empty Topological Space")
- 19:47, 12 April 2025 Empty Topological Space is Second Countable (hist | edit) [327 bytes] Austrodata (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $T = \struct {\O, \set \O}$ be the empty topological space. Then $T$ is second-countable. </onlyinclude> == Proof == Follows from Finite Space is Second-Countable. {{qed}} Category:Empty Topological Space")
- 19:26, 12 April 2025 Empty Topological Space is Hausdorff (hist | edit) [460 bytes] Austrodata (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $T = \struct {\O, \set \O}$ be the empty topological space. Then $T$ is Hausdorff. </onlyinclude> == Proof == By definition, a topological space $\struct {S, \tau}$ is Hausdorff {{iff}}: :any two distinct elements $x, y \in S$ t...")
- 17:27, 11 April 2025 Autonomous Differential Equation/Examples/v' = 3t^2 + 4t - 3 (hist | edit) [973 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Autonomous Differential Equation == <onlyinclude> The differential equation: :$\sfrac {\d v} {\d t} = 3 t^2 + 4 t - 3$ is ''not'' an '''autonomous differential equation'''. </onlyinclude> == Sources == * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|e...")
- 17:26, 11 April 2025 Autonomous Differential Equation/Examples/v' = -k v (hist | edit) [1,000 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Autonomous Differential Equation == <onlyinclude> The differential equation: :$\dfrac {\d v} {\d t} = -k v$ is an '''autonomous differential equation'''. </onlyinclude> == Sources == * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Editi...")
- 16:51, 11 April 2025 Autonomous Differential Equation/Examples (hist | edit) [535 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Autonomous Differential Equations == <onlyinclude> === Example: $\frac {\d v} {\d t} = -k v$ === {{:Autonomous Differential Equation/Examples/v' = -k v}} === Example: $\frac {\d v} {\d t} = 3 t^2 + 4 t - 3$ === {{:Autonomous Differential Equation/Examples/v' = -k v}}</onlyinclude> Categ...")
- 05:18, 11 April 2025 Complex Conjugation is not Complex Vector Space Automorphism (hist | edit) [804 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == The operation of complex conjugation: :$\forall z \in \C: z \mapsto \overline z$ is ''not'' a vector space automorphism over the complex vector space $\C^n$. == Proof == {{ProofWanted|first define exactly what the above means -- it gets a one-liner in the given source}} == Sources == * {{BookReference|The Concise Oxford Dictionary of Mathemat...")
- 05:17, 11 April 2025 Complex Conjugation is Real Vector Space Automorphism (hist | edit) [806 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == The operation of complex conjugation: :$\forall z \in \C: z \mapsto \overline z$ is a vector space automorphism over the real vector space $\R^n$. == Proof == {{ProofWanted|first define exactly what the above means -- it gets a one-liner in the given source}} == Sources == * {{BookReference|The Concise Oxford Dictionary of Mathematics|2021|Richa...")
- 07:00, 10 April 2025 Arzelà's Dominated Convergence Theorem (hist | edit) [885 bytes] CircuitCraft (talk | contribs) (Created page with "== Theorem == For each $n \in \N$, let $f_n : \closedint a b \to \R$ be a real function. Suppose that each $f_n$ is Darboux integrable on $\closedint a b$. Further suppose that there is a constant $M > 0$ such that: :$\size {\map {f_n} x} \le M$ for all $n \in \N$ and $x \in \closedint a b$. Let $\sequence {f_n}$ converge pointwise to a Definition:Real Func...") originally created as "Arzelà's Bounded Convergence Theorem"
- 06:59, 10 April 2025 Approximation of Lower Darboux Integral by Continuous Function (hist | edit) [6,938 bytes] CircuitCraft (talk | contribs) (Created page with "== Theorem == Let $f : \closedint a b \to \R$ be a bounded real function. Then, for each $\epsilon > 0$, there is a continuous real function $g : \closedint a b \to \R$ such that: :For all $x \in \closedint a b$, $\map g x \le \map f x$. :The infimums of $f$ and $g$ on $\closedint a b$ are the same. :$\ds \under...")
- 20:54, 8 April 2025 Condition for Fixed Point to be Attracting (hist | edit) [1,297 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $x_{n + 1} = \map \phi {x_n}$ be a direct iteration for the purpose of finding a solution to the equation: :$x = \map \phi x$ for some real function $\phi: \R \to \R$. Let $\alpha$ be such that $\alpha = \map \phi \alpha$. Let $\sequence {\map \phi {x_n} }$ converge to $\alpha$ for $\size {x_0 - \alpha}$ Definiti...")
- 20:39, 7 April 2025 Asymptotically Equal Real Functions/Examples/(x + 1)^3 (x + 2)^4 and x^7 (hist | edit) [801 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Asymptotically Equal Real Functions == <onlyinclude> Let $f: \R \to \R$ be the real function defined as: :$\forall x \in \R: \map f x = \paren {x + 1}^3 \paren {x + 2}^4$ Let $g: \R \to \R$ be the real function defined as: :$\forall x \in \R: \map g x = x^7$ Then: :$f \sim g$ as $x \to +\infty$. </onlyinclude> == Proof == {{ProofWanted}} == Sou...")
- 14:33, 7 April 2025 Asymptote/Examples/x^3 over x^2 + x + 1 (hist | edit) [1,179 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Asymptotes == :600px|right|thumb|Asymptote ({{Color|red}}) to $\dfrac {x^3} {x^2 + x + 1}$ ({{color|blue}}) <onlyinclude> Consider the curve whose equation in the Cartesian plane is given by: :$y = \dfrac {x^3} {x^2 + x + 1}$ This has $1$ asymptote, neither vertical nor Defi...")
- 14:27, 7 April 2025 Asymptote/Examples/3 x^2 over x^2 + x + 1 (hist | edit) [1,095 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Asymptotes == :600px|right|thumb|Asymptote ({{Color|red}}) to $\dfrac {3 x^2} {x^2 + x + 1}$ ({{color|blue}}) <onlyinclude> Consider the curve whose equation in the Cartesian plane is given by: :$y = \dfrac {3 x^2} {x^2 + x + 1}$ This has $1$ horizontal asymptote: :the Definition:Straight Line|s...")
- 14:12, 7 April 2025 Asymptote/Examples/(x + 3) over (x + 2) (x - 1) (hist | edit) [1,267 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Asymptotes == :600px|right|thumb|Asymptotes ({{Color|red}}) to $\dfrac {x + 3} {\paren {x + 2} \paren {x - 1} }$ ({{color|blue}}) <onlyinclude> Consider the curve whose equation in the Cartesian plane is given by: :$y = \dfrac {x + 3} {\paren {x + 2} \paren {x - 1} }$ This has $2$ Definition:Vertical Asymptote|vertical asymp...")
- 13:45, 7 April 2025 Asymmetric Relation/Examples/Less Than (hist | edit) [1,007 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Asymmetric Relation == <onlyinclude> The relation $\RR$ defined on the set of real numbers $\R$: :$\forall x, y \in \R: x \mathop \RR y \iff x < y$ where: :$<$ denotes the conventional '''less than''' relation is an '''asymmetric relation'''. </onlyinclude> == Sources == * {{BookReference...")
- 11:31, 7 April 2025 Assumption/Examples/Arbitrary Example 1 (hist | edit) [487 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Assumption == <onlyinclude> The '''assumption''': :'''''Let $p$''' (be true) ...'' can be interpreted, in natural language, as: :'''''Let us assume''', for the sake of argument, that $p$ is true ...'' </onlyinclude> Category:Examples of A...")
- 11:28, 7 April 2025 Assumption/Examples (hist | edit) [236 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Assumptions == <onlyinclude> === Arbitrary Example === {{:Assumption/Examples/Arbitrary Example 1}}</onlyinclude> Category:Examples of Assumptions")
- 11:16, 7 April 2025 Assignment Problem/Examples/Workers to Jobs (hist | edit) [1,469 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Assignment Problem == <onlyinclude> Let an industrial firm have $n$ workers who are each to be assigned to one of $n$ jobs. Let it be possible to specify the value $v_{i j}$, measured in suitable units, of the $i$th worker when assigned to the $j$th job. The values could then be displayed in a ...")
- 10:58, 7 April 2025 Assignment Problem/Examples (hist | edit) [266 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Assignment Problems == <onlyinclude> === Workers to Jobs === {{:Assignment Problem/Examples/Workers to Jobs}}</onlyinclude> Category:Examples of Assignment Problems")
- 08:51, 7 April 2025 Arrow's Impossibility Theorem (hist | edit) [1,000 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let voters be given $3$ or more distinct choices to choose from. Then there is no ranking system which can aggregate the individual preferences of two or more individuals so that $4$ apparently reasonable conditions are met. </onlyinclude> {{Expand|Make this rigorous and provide examples}} == Proof == {{ProofWanted}} {{Namedfor|Kenneth Joseph Arrow|cat = Arrow}} == S...")
- 07:35, 7 April 2025 Array/Examples/Matrix (hist | edit) [888 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Array == <onlyinclude> A matrix is an example of an '''array''' with $2$ dimension: :$\begin {pmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m 1} & a_{m 2} & \cdots & a_{m n} \end {pmatrix}$ </onlyinclude> == Sources == * {{BookReference|The Concise Oxford Dictionary of Mathema...")
- 07:32, 7 April 2025 Array/Examples/Vector (hist | edit) [734 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Array == <onlyinclude> A vector is an example of an '''array''' with $1$ dimension: :$\begin {pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end {pmatrix}$ </onlyinclude> == Sources == * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Array|next = Array/...")
- 08:00, 6 April 2025 Area of Surface of Revolution/Polar Form (hist | edit) [1,542 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\SS$ be a surface of revolution such that: :$\SS$ is embedded in a cylindrical coordinate space $\polar {r, \theta, z}$ :the axis of revolution of $\SS$ is aligned with the polar axis :the curve $\CC$ being rotated to generate $\SS$ is the plane curve: ::$r...")
- 07:38, 6 April 2025 Area of Surface of Revolution/Parametric Form (hist | edit) [1,597 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\SS$ be a surface of revolution such that: :$\SS$ is embedded in a Cartesian $3$-space :the axis of revolution of $\SS$ is aligned with the $x$-axis :the curve $\CC$ being rotated to generate $\SS$ is the plane curve described by the Definition:Parametric Equation|p...")
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