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-21300 | Real Addition is Computably Uniformly Continuous | Let $f : \R^2 \to \R$ be defined as:
:$\map f {x, y} = x + y$
Then $f$ is computably uniformly continuous. | We will define $d : \N \to \N$ as:
:$\map d n = 2 n + 1$
which is primitive recursive by:
* Addition is Primitive Recursive
* Constant Function is Primitive Recursive
and thus total recursive by:
* Primitive Recursive Function is Total Recursive Function
Let $x_1, y_1, x_2, y_2 \in \R$ and $n \in \N$ be arbitrary, and ... | Let $f : \R^2 \to \R$ be defined as:
:$\map f {x, y} = x + y$
Then $f$ is [[Definition:Computably Uniformly Continuous Real-Valued Function|computably uniformly continuous]]. | We will define $d : \N \to \N$ as:
:$\map d n = 2 n + 1$
which is [[Definition:Primitive Recursive Function|primitive recursive]] by:
* [[Addition is Primitive Recursive]]
* [[Constant Function is Primitive Recursive]]
and thus [[Definition:Total Recursive Function|total recursive]] by:
* [[Primitive Recursive Function... | Real Addition is Computably Uniformly Continuous | https://proofwiki.org/wiki/Real_Addition_is_Computably_Uniformly_Continuous | https://proofwiki.org/wiki/Real_Addition_is_Computably_Uniformly_Continuous | [
"Computability Theory"
] | [
"Definition:Computably Uniformly Continuous Real-Valued Function"
] | [
"Definition:Primitive Recursive/Function",
"Addition is Primitive Recursive",
"Constant Function is Primitive Recursive",
"Definition:Total Recursive Function",
"Primitive Recursive Function is Total Recursive Function",
"Triangle Inequality/Vectors in Euclidean Space",
"Triangle Inequality/Real Numbers... |
proofwiki-21301 | Real Addition is Computable | Let $f : \R^2 \to \R$ be defined as:
:$\map f {x, y} = x + y$
Then $f$ is computable. | Follows immediately from:
* Real Addition is Sequentially Computable
* Real Addition is Computably Uniformly Continuous
{{qed}}
Category:Computability Theory
q5v2d6lq0aqoiq11qx58a6tfb88roqr | Let $f : \R^2 \to \R$ be defined as:
:$\map f {x, y} = x + y$
Then $f$ is [[Definition:Computable Real-Valued Function|computable]]. | Follows immediately from:
* [[Real Addition is Sequentially Computable]]
* [[Real Addition is Computably Uniformly Continuous]]
{{qed}}
[[Category:Computability Theory]]
q5v2d6lq0aqoiq11qx58a6tfb88roqr | Real Addition is Computable | https://proofwiki.org/wiki/Real_Addition_is_Computable | https://proofwiki.org/wiki/Real_Addition_is_Computable | [
"Computability Theory",
"Computability Theory"
] | [
"Definition:Computable Real-Valued Function"
] | [
"Real Addition is Sequentially Computable",
"Real Addition is Computably Uniformly Continuous",
"Category:Computability Theory"
] |
proofwiki-21302 | Sum of Computable Rational Sequences is Computable | Let $\sequence {x_n}$ and $\sequence {y_n}$ be computable rational sequences.
Then, $\sequence {x_n + y_n}$ is a computable rational sequence. | By definition of computable rational sequence, there exist total recursive $f_x, g_x, f_y, g_y : \N \to \N$ such that:
:$x_n = \dfrac {k_n} {\map {g_x} n + 1}$
:$y_n = \dfrac {\ell_n} {\map {g_y} n + 1}$
where:
:$\map {f_x} n$ codes the integer $k_n$
:$\map {f_y} n$ codes the integer $\ell_n$
We define:
:$\map f n = \p... | Let $\sequence {x_n}$ and $\sequence {y_n}$ be [[Definition:Computable Rational Sequence|computable rational sequences]].
Then, $\sequence {x_n + y_n}$ is a [[Definition:Computable Rational Sequence|computable rational sequence]]. | By definition of [[Definition:Computable Rational Sequence|computable rational sequence]], there exist [[Definition:Total Recursive Function|total recursive]] $f_x, g_x, f_y, g_y : \N \to \N$ such that:
:$x_n = \dfrac {k_n} {\map {g_x} n + 1}$
:$y_n = \dfrac {\ell_n} {\map {g_y} n + 1}$
where:
:$\map {f_x} n$ [[Definit... | Sum of Computable Rational Sequences is Computable | https://proofwiki.org/wiki/Sum_of_Computable_Rational_Sequences_is_Computable | https://proofwiki.org/wiki/Sum_of_Computable_Rational_Sequences_is_Computable | [
"Computability Theory"
] | [
"Definition:Computable Rational Sequence",
"Definition:Computable Rational Sequence"
] | [
"Definition:Computable Rational Sequence",
"Definition:Total Recursive Function",
"Definition:Code Number for Integer",
"Definition:Code Number for Integer",
"Definition:Total Recursive Function",
"Code Number for Non-Negative Integer is Primitive Recursive",
"Multiplication of Integers is Primitive Rec... |
proofwiki-21303 | Product of Computable Rational Sequences is Computable | Let $\sequence {x_n}$ and $\sequence {y_n}$ be computable rational sequences.
Then, $\sequence {x_n y_n}$ is a computable rational sequence. | By definition of computable rational sequence, there exist total recursive $f_x, g_x, f_y, g_y : \N \to \N$ such that:
:$x_n = \dfrac {k_n} {\map {g_x} n + 1}$
:$y_n = \dfrac {\ell_n} {\map {g_y} n + 1}$
where:
:$\map {f_x} n$ codes the integer $k_n$
:$\map {f_y} n$ codes the integer $\ell_n$
We define:
:$\map f n = \m... | Let $\sequence {x_n}$ and $\sequence {y_n}$ be [[Definition:Computable Rational Sequence|computable rational sequences]].
Then, $\sequence {x_n y_n}$ is a [[Definition:Computable Rational Sequence|computable rational sequence]]. | By definition of [[Definition:Computable Rational Sequence|computable rational sequence]], there exist [[Definition:Total Recursive Function|total recursive]] $f_x, g_x, f_y, g_y : \N \to \N$ such that:
:$x_n = \dfrac {k_n} {\map {g_x} n + 1}$
:$y_n = \dfrac {\ell_n} {\map {g_y} n + 1}$
where:
:$\map {f_x} n$ [[Definit... | Product of Computable Rational Sequences is Computable | https://proofwiki.org/wiki/Product_of_Computable_Rational_Sequences_is_Computable | https://proofwiki.org/wiki/Product_of_Computable_Rational_Sequences_is_Computable | [
"Computability Theory"
] | [
"Definition:Computable Rational Sequence",
"Definition:Computable Rational Sequence"
] | [
"Definition:Computable Rational Sequence",
"Definition:Total Recursive Function",
"Definition:Code Number for Integer",
"Definition:Code Number for Integer",
"Definition:Total Recursive Function",
"Multiplication of Integers is Primitive Recursive",
"Addition is Primitive Recursive",
"Multiplication i... |
proofwiki-21304 | Condition for Closure of Open Ball to be Closed Ball of Same Radius | Let $\struct {M, d}$ be a metric space.
{{TFAE}}
:$(1) \quad$ for all $x \in X$ and $r > 0$ we have $\map \cl {\map {B_r} x} = \map { {B_r}^-} x$, where $\cl$ denotes closure
:$(2) \quad$ for each $\epsilon > 0$ and $x, y \in X$ with $x \ne y$, there exists $z \in X$ with $\map d {z, y} < \epsilon$ and $\map d {x, z} ... | === $(1)$ implies $(2)$ ===
Suppose that $(1)$ holds.
Let $\epsilon > 0$.
Let $x, y \in X$ have $x \ne y$.
We aim to show that there exists $z \in X$ with $\map d {z, y} < \epsilon$ and $\map d {x, z} < \map d {x, y}$.
Let $r = \map d {x, y}$.
From $(1)$, we have $\map \cl {\map {B_r} x} = \map { {B_r}^-} x$.
So $y ... | Let $\struct {M, d}$ be a [[Definition:Metric Space|metric space]].
{{TFAE}}
:$(1) \quad$ for all $x \in X$ and $r > 0$ we have $\map \cl {\map {B_r} x} = \map { {B_r}^-} x$, where $\cl$ denotes [[Definition:Closure (Metric Space)|closure]]
:$(2) \quad$ for each $\epsilon > 0$ and $x, y \in X$ with $x \ne y$, there ... | === $(1)$ implies $(2)$ ===
Suppose that $(1)$ holds.
Let $\epsilon > 0$.
Let $x, y \in X$ have $x \ne y$.
We aim to show that there exists $z \in X$ with $\map d {z, y} < \epsilon$ and $\map d {x, z} < \map d {x, y}$.
Let $r = \map d {x, y}$.
From $(1)$, we have $\map \cl {\map {B_r} x} = \map { {B_r}^-} x$.
... | Condition for Closure of Open Ball to be Closed Ball of Same Radius | https://proofwiki.org/wiki/Condition_for_Closure_of_Open_Ball_to_be_Closed_Ball_of_Same_Radius | https://proofwiki.org/wiki/Condition_for_Closure_of_Open_Ball_to_be_Closed_Ball_of_Same_Radius | [
"Closed Balls",
"Open Balls"
] | [
"Definition:Metric Space",
"Definition:Closure (Topology)/Metric Space"
] | [
"Definition:Closure (Topology)",
"Definition:Closure (Topology)"
] |
proofwiki-21305 | Computable Real Sequence iff Limits of Computable Rational Sequences | Let $\sequence {x_m}$ be a sequence of real numbers.
Then, $\sequence {x_m}$ is computable {{iff}} there exist:
* A computable rational sequence $\sequence {a_k}$
* Total recursive functions $\phi, \psi : \N^2 \to \N$
such that:
:$\forall m, p \in \N: \forall n \ge \map \psi {m, p}: \size {a_{\map \phi {m, n}} - x_m} <... | === Necessary Condition ===
Suppose $\sequence {x_m}$ is computable.
Then, there exists a total recursive $f : \N^2 \to \N$ such that, for all $m, p \in \N$:
:$\dfrac {c_{m,p} - 1} {p + 1} < x_m < \dfrac {c_{m,p} + 1} {p + 1}$
where $\map f {m, p}$ codes the integer $c_{m,p}$.
To produce the computable rational sequenc... | Let $\sequence {x_m}$ be a [[Definition:Infinite Sequence|sequence]] of [[Definition:Real Number|real numbers]].
Then, $\sequence {x_m}$ is [[Definition:Computable Real Sequence|computable]] {{iff}} there exist:
* A [[Definition:Computable Rational Sequence|computable rational sequence]] $\sequence {a_k}$
* [[Definiti... | === Necessary Condition ===
Suppose $\sequence {x_m}$ is [[Definition:Computable Real Sequence|computable]].
Then, there exists a [[Definition:Total Recursive Function|total recursive]] $f : \N^2 \to \N$ such that, for all $m, p \in \N$:
:$\dfrac {c_{m,p} - 1} {p + 1} < x_m < \dfrac {c_{m,p} + 1} {p + 1}$
where $\map... | Computable Real Sequence iff Limits of Computable Rational Sequences | https://proofwiki.org/wiki/Computable_Real_Sequence_iff_Limits_of_Computable_Rational_Sequences | https://proofwiki.org/wiki/Computable_Real_Sequence_iff_Limits_of_Computable_Rational_Sequences | [
"Computability Theory"
] | [
"Definition:Sequence/Infinite Sequence",
"Definition:Real Number",
"Definition:Computable Real Sequence",
"Definition:Computable Rational Sequence",
"Definition:Total Recursive Function"
] | [
"Definition:Computable Real Sequence",
"Definition:Total Recursive Function",
"Definition:Code Number for Integer",
"Definition:Computable Rational Sequence",
"Inverse of Cantor Pairing Function",
"Definition:Code Number for Integer",
"Definition:Cantor Pairing Function",
"Definition:Total Recursive F... |
proofwiki-21306 | Computable Rational Sequence is Computable Real Sequence | Let $\sequence {x_n}$ be a computable rational sequence.
Then, $\sequence {x_n}$ is a computable real sequence. | By Computable Real Sequence iff Limits of Computable Rational Sequences, it suffices to show that there exist:
:A computable rational sequence $\sequence {a_k}$
:Total recursive functions $\phi, \psi : \N^2 \to \N$
such that:
:$\forall m, p \in \N: \forall n \ge \map \psi {m, p}: \size {a_{\map \phi {m, n}} - x_m} < \d... | Let $\sequence {x_n}$ be a [[Definition:Computable Rational Sequence|computable rational sequence]].
Then, $\sequence {x_n}$ is a [[Definition:Computable Real Sequence|computable real sequence]]. | By [[Computable Real Sequence iff Limits of Computable Rational Sequences]], it suffices to show that there exist:
:A [[Definition:Computable Rational Sequence|computable rational sequence]] $\sequence {a_k}$
:[[Definition:Total Recursive Function|Total recursive functions]] $\phi, \psi : \N^2 \to \N$
such that:
:$\for... | Computable Rational Sequence is Computable Real Sequence | https://proofwiki.org/wiki/Computable_Rational_Sequence_is_Computable_Real_Sequence | https://proofwiki.org/wiki/Computable_Rational_Sequence_is_Computable_Real_Sequence | [
"Computability Theory"
] | [
"Definition:Computable Rational Sequence",
"Definition:Computable Real Sequence"
] | [
"Computable Real Sequence iff Limits of Computable Rational Sequences",
"Definition:Computable Rational Sequence",
"Definition:Total Recursive Function",
"Definition:Primitive Recursive/Function",
"Definition:Total Recursive Function",
"Primitive Recursive Function is Total Recursive Function",
"Definit... |
proofwiki-21307 | Constant Sequence of Rational Number is Computable | Let $r \in \Q$ be a rational number.
Let $\sequence {x_n}$ be defined as:
:$x_n = r$
Then, $\sequence {x_n}$ is a computable rational sequence. | By Existence of Canonical Form of Rational Number, let:
:$r = \dfrac p q$
where:
:$p \in \Z$
:$q \in \Z_{>0}$
Let $m$ be the code number for the integer $p$.
As $q \ge 1$, it follows that:
:$q - 1 \in \N$
We define $N, D : \N \to \N$ as:
:$\map N n = m$
:$\map D n = q - 1$
which are total recursive by:
* Constant Funct... | Let $r \in \Q$ be a [[Definition:Rational Number|rational number]].
Let $\sequence {x_n}$ be defined as:
:$x_n = r$
Then, $\sequence {x_n}$ is a [[Definition:Computable Rational Sequence|computable rational sequence]]. | By [[Existence of Canonical Form of Rational Number]], let:
:$r = \dfrac p q$
where:
:$p \in \Z$
:$q \in \Z_{>0}$
Let $m$ be the [[Definition:Code Number for Integer|code number for the integer]] $p$.
As $q \ge 1$, it follows that:
:$q - 1 \in \N$
We define $N, D : \N \to \N$ as:
:$\map N n = m$
:$\map D n = q - 1$... | Constant Sequence of Rational Number is Computable | https://proofwiki.org/wiki/Constant_Sequence_of_Rational_Number_is_Computable | https://proofwiki.org/wiki/Constant_Sequence_of_Rational_Number_is_Computable | [
"Computability Theory"
] | [
"Definition:Rational Number",
"Definition:Computable Rational Sequence"
] | [
"Existence of Canonical Form of Rational Number",
"Definition:Code Number for Integer",
"Definition:Total Recursive Function",
"Constant Function is Primitive Recursive",
"Primitive Recursive Function is Total Recursive Function",
"Definition:Code Number for Integer",
"Definition:Computable Rational Seq... |
proofwiki-21308 | Rational Number is Computable Real Number | Let $r \in \Q$ be a rational number.
Then, $r$ is a computable real number. | Let $\sequence {x_n}$ be defined as:
:$x_n = r$
By Constant Sequence of Rational Number is Computable:
:$\sequence {x_n}$ is a computable rational sequence.
By Computable Rational Sequence is Computable Real Sequence:
:$\sequence {x_n}$ is a computable real sequence.
By Term of Computable Real Sequence is Computable:
:... | Let $r \in \Q$ be a [[Definition:Rational Number|rational number]].
Then, $r$ is a [[Definition:Computable Real Number|computable real number]]. | Let $\sequence {x_n}$ be defined as:
:$x_n = r$
By [[Constant Sequence of Rational Number is Computable]]:
:$\sequence {x_n}$ is a [[Definition:Computable Rational Sequence|computable rational sequence]].
By [[Computable Rational Sequence is Computable Real Sequence]]:
:$\sequence {x_n}$ is a [[Definition:Computable ... | Rational Number is Computable Real Number | https://proofwiki.org/wiki/Rational_Number_is_Computable_Real_Number | https://proofwiki.org/wiki/Rational_Number_is_Computable_Real_Number | [
"Computability Theory"
] | [
"Definition:Rational Number",
"Definition:Computable Real Number"
] | [
"Constant Sequence of Rational Number is Computable",
"Definition:Computable Rational Sequence",
"Computable Rational Sequence is Computable Real Sequence",
"Definition:Computable Real Sequence",
"Term of Computable Real Sequence is Computable",
"Definition:Computable Real Number",
"Category:Computabili... |
proofwiki-21309 | Computable Subsequence of Computable Rational Sequence is Computable | Let $\sequence {x_n}_{n \in \N}$ be a computable rational sequence.
Let $\phi : \N \to \N$ be a total recursive function.
Then:
:$\sequence {x_{\map \phi n}}_{n \in \N}$
is a computable rational sequence. | By definition of computable rational sequence, there exist total recursive $N, D : \N \to \N$ such that, for every $n \in \N$:
:$x_n = \dfrac {k_n} {\map D n + 1}$
where $\map N n$ codes the integer $k_n$.
We define total recursive $N', D' : \N \to \N$ as:
:$\map {N'} n = \map N {\map \phi n}$
:$\map {D'} n = \map N {\... | Let $\sequence {x_n}_{n \in \N}$ be a [[Definition:Computable Rational Sequence|computable rational sequence]].
Let $\phi : \N \to \N$ be a [[Definition:Total Recursive Function|total recursive function]].
Then:
:$\sequence {x_{\map \phi n}}_{n \in \N}$
is a [[Definition:Computable Rational Sequence|computable ration... | By definition of [[Definition:Computable Rational Sequence|computable rational sequence]], there exist [[Definition:Total Recursive Function|total recursive]] $N, D : \N \to \N$ such that, for every $n \in \N$:
:$x_n = \dfrac {k_n} {\map D n + 1}$
where $\map N n$ [[Definition:Code Number for Integer|codes the integer]... | Computable Subsequence of Computable Rational Sequence is Computable | https://proofwiki.org/wiki/Computable_Subsequence_of_Computable_Rational_Sequence_is_Computable | https://proofwiki.org/wiki/Computable_Subsequence_of_Computable_Rational_Sequence_is_Computable | [
"Computability Theory"
] | [
"Definition:Computable Rational Sequence",
"Definition:Total Recursive Function",
"Definition:Computable Rational Sequence"
] | [
"Definition:Computable Rational Sequence",
"Definition:Total Recursive Function",
"Definition:Code Number for Integer",
"Definition:Total Recursive Function",
"Definition:Code Number for Integer",
"Category:Computability Theory"
] |
proofwiki-21310 | Computable Subsequence of Computable Rational Sequence is Computable/Corollary | Let $\sequence {x_k}$ be a computable rational sequence.
Let $\phi : \N^2 \to \N$ be a total recursive function.
Then, there exists a computable rational sequence $\sequence {y_k}$ such that, for all $n, m \in \N$:
:$y_{\map \pi {n, m}} = x_{\map \phi {n, m}}$
where $\pi : \N^2 \to \N$ is the Cantor pairing function. | Let $\psi : \N^2 \to \N$ be defined as:
:$\map \psi k = \map \phi {\map {\pi_1} k, \map {\pi_2} k}$
where $\pi_1, \pi_2 : \N \to \N$ are the projections on the Cantor pairing function.
By Inverse of Cantor Pairing Function is Primitive Recursive, we have that $\psi$ is total recursive.
Therefore, by Computable Subseque... | Let $\sequence {x_k}$ be a [[Definition:Computable Rational Sequence|computable rational sequence]].
Let $\phi : \N^2 \to \N$ be a [[Definition:Total Recursive Function|total recursive function]].
Then, there exists a [[Definition:Computable Rational Sequence|computable rational sequence]] $\sequence {y_k}$ such tha... | Let $\psi : \N^2 \to \N$ be defined as:
:$\map \psi k = \map \phi {\map {\pi_1} k, \map {\pi_2} k}$
where $\pi_1, \pi_2 : \N \to \N$ are the [[Inverse of Cantor Pairing Function|projections on the Cantor pairing function]].
By [[Inverse of Cantor Pairing Function is Primitive Recursive]], we have that $\psi$ is [[Defi... | Computable Subsequence of Computable Rational Sequence is Computable/Corollary | https://proofwiki.org/wiki/Computable_Subsequence_of_Computable_Rational_Sequence_is_Computable/Corollary | https://proofwiki.org/wiki/Computable_Subsequence_of_Computable_Rational_Sequence_is_Computable/Corollary | [
"Computability Theory"
] | [
"Definition:Computable Rational Sequence",
"Definition:Total Recursive Function",
"Definition:Computable Rational Sequence",
"Definition:Cantor Pairing Function"
] | [
"Inverse of Cantor Pairing Function",
"Inverse of Cantor Pairing Function is Primitive Recursive",
"Definition:Total Recursive Function",
"Computable Subsequence of Computable Rational Sequence is Computable",
"Definition:Computable Rational Sequence",
"Inverse of Cantor Pairing Function",
"Definition:P... |
proofwiki-21311 | Product of Computable Real Sequences is Computable | Let $\sequence {x_m}$ and $\sequence {y_m}$ be computable real sequences.
Then $\sequence {x_m y_m}$ is a computable real sequence. | By definition of computable real sequence, there exist total recursive $f, g : \N^2 \to \N$ such that, for all $m, n \in \N$:
:$\dfrac {k_{m,n} - 1} {n + 1} < x_m < \dfrac {k_{m,n} + 1} {n + 1}$
:$\dfrac {\ell_{m,n} - 1} {n + 1} < y_m < \dfrac {\ell_{m,n} + 1} {n + 1}$
By Computable Real Sequence iff Limits of Computab... | Let $\sequence {x_m}$ and $\sequence {y_m}$ be [[Definition:Computable Real Sequence|computable real sequences]].
Then $\sequence {x_m y_m}$ is a [[Definition:Computable Real Sequence|computable real sequence]]. | By definition of [[Definition:Computable Real Sequence|computable real sequence]], there exist [[Definition:Total Recursive Function|total recursive]] $f, g : \N^2 \to \N$ such that, for all $m, n \in \N$:
:$\dfrac {k_{m,n} - 1} {n + 1} < x_m < \dfrac {k_{m,n} + 1} {n + 1}$
:$\dfrac {\ell_{m,n} - 1} {n + 1} < y_m < \df... | Product of Computable Real Sequences is Computable | https://proofwiki.org/wiki/Product_of_Computable_Real_Sequences_is_Computable | https://proofwiki.org/wiki/Product_of_Computable_Real_Sequences_is_Computable | [
"Computability Theory"
] | [
"Definition:Computable Real Sequence",
"Definition:Computable Real Sequence"
] | [
"Definition:Computable Real Sequence",
"Definition:Total Recursive Function",
"Computable Real Sequence iff Limits of Computable Rational Sequences",
"Definition:Computable Rational Sequence",
"Definition:Total Recursive Function",
"Computable Subsequence of Computable Rational Sequence is Computable/Coro... |
proofwiki-21312 | Real Multiplication is Sequentially Computable | Let $f : \R^2 \to \R$ be defined as:
:$\map f {x, y} = x y$
Then $f$ is sequentially computable. | Follows immediately from Product of Computable Real Sequences is Computable.
{{qed}}
Category:Computability Theory
buyid8unyfwvt87jsb37szhzr7ondsm | Let $f : \R^2 \to \R$ be defined as:
:$\map f {x, y} = x y$
Then $f$ is [[Definition:Sequentially Computable Real-Valued Function|sequentially computable]]. | Follows immediately from [[Product of Computable Real Sequences is Computable]].
{{qed}}
[[Category:Computability Theory]]
buyid8unyfwvt87jsb37szhzr7ondsm | Real Multiplication is Sequentially Computable | https://proofwiki.org/wiki/Real_Multiplication_is_Sequentially_Computable | https://proofwiki.org/wiki/Real_Multiplication_is_Sequentially_Computable | [
"Computability Theory",
"Computability Theory"
] | [
"Definition:Sequentially Computable Real-Valued Function"
] | [
"Product of Computable Real Sequences is Computable",
"Category:Computability Theory"
] |
proofwiki-21313 | Reciprocal of Computable Rational Sequence is Computable | Let $\sequence {x_n}$ be a computable rational sequence.
Suppose that, for all $n \in \N$:
:$x_n \ne 0$
Then:
:$\sequence {\dfrac 1 {x_n}}$ is a computable rational sequence. | By definition of computable rational sequence, there exist total recursive $f, g : \N \to \N$ such that:
:$x_n = \dfrac {k_n} {\map g n + 1}$
where $\map f n$ codes the integer $k_n$.
We will define $f', g' : \N \to \N$ as:
:$\map {f'} n = \map {\sgn_\Z} {k_n} \times_\Z \paren {\map g n + 1}_\Z$
:$\map {g'} n = \map {\... | Let $\sequence {x_n}$ be a [[Definition:Computable Rational Sequence|computable rational sequence]].
Suppose that, for all $n \in \N$:
:$x_n \ne 0$
Then:
:$\sequence {\dfrac 1 {x_n}}$ is a [[Definition:Computable Rational Sequence|computable rational sequence]]. | By definition of [[Definition:Computable Rational Sequence|computable rational sequence]], there exist [[Definition:Total Recursive Function|total recursive]] $f, g : \N \to \N$ such that:
:$x_n = \dfrac {k_n} {\map g n + 1}$
where $\map f n$ [[Definition:Code Number for Integer|codes the integer]] $k_n$.
We will def... | Reciprocal of Computable Rational Sequence is Computable | https://proofwiki.org/wiki/Reciprocal_of_Computable_Rational_Sequence_is_Computable | https://proofwiki.org/wiki/Reciprocal_of_Computable_Rational_Sequence_is_Computable | [
"Computability Theory"
] | [
"Definition:Computable Rational Sequence",
"Definition:Computable Rational Sequence"
] | [
"Definition:Computable Rational Sequence",
"Definition:Total Recursive Function",
"Definition:Code Number for Integer",
"Definition:Total Recursive Function",
"Signum Function on Integers is Primitive Recursive",
"Multiplication of Integers is Primitive Recursive",
"Code Number for Non-Negative Integer ... |
proofwiki-21314 | Seminorm on Vector Space induces Norm on Quotient | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $p$ be a seminorm on $X$.
Let:
:$N = \set {x \in X : \map p x = 0}$
From Set of Points for which Seminorm is Zero is Vector Subspace, $N$ is a vector subspace.
Let $X/N$ be the quotient vector space of $X$ modulo $N$.
Let $\pi : X \to X/N$ be the qu... | We first want to show that if $\map \pi x = \map \pi y$ for $x, y \in X$, then $\map p x = \map p y$.
From Quotient Mapping is Linear Transformation and Kernel of Quotient Mapping, it is enough to show that:
:if $y - x \in N$ then $\map p x = \map p y$.
It is therefore enough to show that if $x \in X$ and $z \in N$, t... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $p$ be a [[Definition:Seminorm|seminorm]] on $X$.
Let:
:$N = \set {x \in X : \map p x = 0}$
From [[Set of Points for which Seminorm is Zero is Vector Subspace]], $N$ is a [[Definition:Vector Subspace|vector subspace]]... | We first want to show that if $\map \pi x = \map \pi y$ for $x, y \in X$, then $\map p x = \map p y$.
From [[Quotient Mapping is Linear Transformation]] and [[Kernel of Quotient Mapping]], it is enough to show that:
:if $y - x \in N$ then $\map p x = \map p y$.
It is therefore enough to show that if $x \in X$ and $z... | Seminorm on Vector Space induces Norm on Quotient | https://proofwiki.org/wiki/Seminorm_on_Vector_Space_induces_Norm_on_Quotient | https://proofwiki.org/wiki/Seminorm_on_Vector_Space_induces_Norm_on_Quotient | [
"Seminorms",
"Quotient Vector Spaces"
] | [
"Definition:Vector Space",
"Definition:Seminorm",
"Set of Points for which Seminorm is Zero is Vector Subspace",
"Definition:Vector Subspace",
"Definition:Quotient Vector Space",
"Definition:Quotient Mapping",
"Definition:Mapping",
"Definition:Norm/Vector Space"
] | [
"Quotient Mapping is Linear Transformation",
"Kernel of Quotient Mapping",
"Reverse Triangle Inequality/Seminormed Vector Space",
"Axiom:Norm Axioms",
"Kernel of Quotient Mapping"
] |
proofwiki-21315 | Completion Theorem (Normed Vector Space) | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Then there exists a Banach space $\struct {\widetilde X, \widetilde {\norm {\, \cdot \,} } }$ and a linear isometry $\phi : X \to \widetilde X$ such that $\phi \sqbrk X$ is dense in $\widetilde X$.
Further, the Banach space ... | === Proof of Existence ===
Let $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\ast \ast} } }$ be the second normed dual of $\struct {X, \norm {\, \cdot \,} }$.
From Normed Dual Space is Banach Space, $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\ast \ast} } }$ is a Banach space.
Let $\phi : X \to X^{\ast \ast}$ ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Then there exists a [[Definition:Banach Space|Banach space]] $\struct {\widetilde X, \widetilde {\norm {\, \cdot \,} } }$ and a [[Definition:Linear Isometry|linear isometry]] $\phi : X \... | === Proof of Existence ===
Let $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\ast \ast} } }$ be the [[Definition:Second Normed Dual|second normed dual]] of $\struct {X, \norm {\, \cdot \,} }$.
From [[Normed Dual Space is Banach Space]], $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\ast \ast} } }$ is a [[Defini... | Completion Theorem (Normed Vector Space) | https://proofwiki.org/wiki/Completion_Theorem_(Normed_Vector_Space) | https://proofwiki.org/wiki/Completion_Theorem_(Normed_Vector_Space) | [
"Completion Theorem (Normed Vector Space)",
"Banach Spaces",
"Completion Theorem"
] | [
"Definition:Normed Vector Space",
"Definition:Banach Space",
"Definition:Linear Isometry",
"Definition:Everywhere Dense",
"Definition:Banach Space",
"Definition:Isometric Isomorphism"
] | [
"Definition:Second Normed Dual",
"Normed Dual Space is Banach Space",
"Definition:Banach Space",
"Definition:Evaluation Linear Transformation/Normed Vector Space",
"Evaluation Linear Transformation on Normed Vector Space is Linear Isometry",
"Definition:Linear Isometry",
"Definition:Closure (Topology)",... |
proofwiki-21316 | Composition of Linear Isometries is Linear Isometry | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot_X}$, $\struct {Y, \norm \cdot_Y}$ and $\struct {Z, \norm \cdot_Z}$ be normed vector spaces over $\GF$.
Let $T : X \to Y$ and $S : Y \to Z$ be linear isometries.
Then $S T$ is a linear isometry. | From Composition of Linear Transformations is Linear Transformation, $S T$ is a linear transformation.
For $x \in X$, we have:
{{begin-eqn}}
{{eqn | l = \norm {S T x}_Z
| r = \norm {T x}_Y
| c = $S$ is a linear isometry
}}
{{eqn | r = \norm x_X
| c = $T$ is a linear isometry
}}
{{end-eqn}}
So $S T$ is a linear i... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot_X}$, $\struct {Y, \norm \cdot_Y}$ and $\struct {Z, \norm \cdot_Z}$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $T : X \to Y$ and $S : Y \to Z$ be [[Definition:Linear Isometry|linear isometries]].
Then $S T$ is a [[Definition:L... | From [[Composition of Linear Transformations is Linear Transformation]], $S T$ is a [[Definition:Linear Transformation|linear transformation]].
For $x \in X$, we have:
{{begin-eqn}}
{{eqn | l = \norm {S T x}_Z
| r = \norm {T x}_Y
| c = $S$ is a [[Definition:Linear Isometry|linear isometry]]
}}
{{eqn | r = \norm x... | Composition of Linear Isometries is Linear Isometry | https://proofwiki.org/wiki/Composition_of_Linear_Isometries_is_Linear_Isometry | https://proofwiki.org/wiki/Composition_of_Linear_Isometries_is_Linear_Isometry | [
"Linear Isometries"
] | [
"Definition:Normed Vector Space",
"Definition:Linear Isometry",
"Definition:Linear Isometry"
] | [
"Composition of Linear Transformations is Linear Transformation",
"Definition:Linear Transformation",
"Definition:Linear Isometry",
"Definition:Linear Isometry",
"Definition:Linear Isometry",
"Category:Linear Isometries"
] |
proofwiki-21317 | Reciprocal of Computable Real Sequence is Computable | Let $\sequence {x_m}$ be a computable real sequence.
Suppose that, for all $m \in \N$:
:$x_m \ne 0$
Then:
:$\sequence {\dfrac 1 {x_m}}$
is a computable real sequence. | === Lemma ===
{{:Reciprocal of Computable Real Sequence is Computable/Lemma}}{{qed|lemma}}
{{WIP}}
q5p794tqe40tdq8trij5g61hmx3n9ql | Let $\sequence {x_m}$ be a [[Definition:Computable Real Sequence|computable real sequence]].
Suppose that, for all $m \in \N$:
:$x_m \ne 0$
Then:
:$\sequence {\dfrac 1 {x_m}}$
is a [[Definition:Computable Real Sequence|computable real sequence]]. | === [[Reciprocal of Computable Real Sequence is Computable/Lemma|Lemma]] ===
{{:Reciprocal of Computable Real Sequence is Computable/Lemma}}{{qed|lemma}}
{{WIP}}
q5p794tqe40tdq8trij5g61hmx3n9ql | Reciprocal of Computable Real Sequence is Computable | https://proofwiki.org/wiki/Reciprocal_of_Computable_Real_Sequence_is_Computable | https://proofwiki.org/wiki/Reciprocal_of_Computable_Real_Sequence_is_Computable | [] | [
"Definition:Computable Real Sequence",
"Definition:Computable Real Sequence"
] | [
"Reciprocal of Computable Real Sequence is Computable/Lemma"
] |
proofwiki-21318 | Center of Mass in Barycentric Coordinates | Let $p_0, p_1, p_2, p_3$ be fixed non-coplanar points, such that $p_i = \tuple {x_i, y_1, z_i}$.
Let $p$ be a point in ordinary space expressed in barycentric coordinates {{WRT}} $\set {p_0, p_1, p_2, p_3}$:
:$p = \lambda_0 p_0 + \lambda_1 p_1 + \lambda_2 p_2 + \lambda_3 p_3$
such that:
:$\lambda_0 + \lambda_1 + \lambd... | From the definition of center of mass with discrete masses:
:$\ds Mp = \sum_{i \mathop = 0}^3 \lambda_i p_i$
From the restriction on barycentric coordinates:
:$\ds M = \sum_{i \mathop = 0}^3 \lambda_i = 1$
Thus:
:$\ds p = \sum_{i \mathop = 0}^3 \lambda_i p_i = \lambda_0 p_0 + \lambda_1 p_1 + \lambda_2 p_2 + \lambda_3 p... | Let $p_0, p_1, p_2, p_3$ be fixed non-[[Definition:Coplanar Points|coplanar]] [[Definition:Point|points]], such that $p_i = \tuple {x_i, y_1, z_i}$.
Let $p$ be a [[Definition:Point|point]] in [[Definition:Ordinary Space|ordinary space]] expressed in [[Definition:Barycentric Coordinates|barycentric coordinates]] {{WRT}... | From the definition of [[Definition:Center of Mass (Discrete)|center of mass with discrete masses]]:
:$\ds Mp = \sum_{i \mathop = 0}^3 \lambda_i p_i$
From the restriction on [[Definition:Barycentric Coordinates|barycentric coordinates]]:
:$\ds M = \sum_{i \mathop = 0}^3 \lambda_i = 1$
Thus:
:$\ds p = \sum_{i \mathop ... | Center of Mass in Barycentric Coordinates | https://proofwiki.org/wiki/Center_of_Mass_in_Barycentric_Coordinates | https://proofwiki.org/wiki/Center_of_Mass_in_Barycentric_Coordinates | [
"Centers of Mass",
"Barycentric Coordinates"
] | [
"Definition:Coplanar/Points",
"Definition:Point",
"Definition:Point",
"Definition:Ordinary Space",
"Definition:Barycentric Coordinates",
"Definition:Particle",
"Definition:Mass",
"Definition:Center of Mass"
] | [
"Definition:Center of Mass/Discrete",
"Definition:Barycentric Coordinates"
] |
proofwiki-21319 | Representation of Number Base in that Base | Let $b \in \Z$ be an integer such that $b > 1$.
Then $b$ is expressed in base $b$ as $10$. | By the Basis Representation Theorem, $b$ can be expressed uniquely in the form:
:$\ds b = \sum_{j \mathop = 0}^m r_j b^j$
where:
:$m$ is such that $b^m \le n < b^{m + 1}$
:all the $r_j$ are such that $0 \le r_j < b$.
As $b = b^1$, we have that:
:$b = 1 \times b^1 + 0 \times b^0$
That is, by definition of base $b$:
:$b ... | Let $b \in \Z$ be an [[Definition:Integer|integer]] such that $b > 1$.
Then $b$ is expressed in [[Definition:Number Base|base $b$]] as $10$. | By the [[Basis Representation Theorem]], $b$ can be expressed [[Definition:Unique|uniquely]] in the form:
:$\ds b = \sum_{j \mathop = 0}^m r_j b^j$
where:
:$m$ is such that $b^m \le n < b^{m + 1}$
:all the $r_j$ are such that $0 \le r_j < b$.
As $b = b^1$, we have that:
:$b = 1 \times b^1 + 0 \times b^0$
That is, by... | Representation of Number Base in that Base | https://proofwiki.org/wiki/Representation_of_Number_Base_in_that_Base | https://proofwiki.org/wiki/Representation_of_Number_Base_in_that_Base | [
"Representation of Number Base in that Base",
"Basis Representations",
"Number Bases"
] | [
"Definition:Integer",
"Definition:Number Base"
] | [
"Basis Representation Theorem",
"Definition:Unique",
"Definition:Number Base"
] |
proofwiki-21320 | Frequency of Beats | Let $W_1$ and $W_2$ be harmonic waves whose frequencies are $f_1$ and $f_2$.
Let the superpositon of $W_1$ onto $W_2$ exhibit the phenomenon of beats.
The frequency $f_b$ of those beats is:
:$f_b = \size {f_2 - f_1}$ | Let $\omega_1$ and $\omega_2$ denote the angular frequency of $W_1$ and $W_2$ respectively.
Let us consider the harmonic waves that are $W_1$ and $W_2$ as they disturb the medium at $x = 0$.
{{WLOG}}, therefore, let $W_1$ and $W_2$ be be expressed as:
{{begin-eqn}}
{{eqn | l = \map {\phi_1} t
| r = \sin \omega_1 ... | Let $W_1$ and $W_2$ be [[Definition:Harmonic Wave|harmonic waves]] whose [[Definition:Frequency of Harmonic Wave|frequencies]] are $f_1$ and $f_2$.
Let the superpositon of $W_1$ onto $W_2$ exhibit the phenomenon of [[Definition:Beats|beats]].
The [[Definition:Frequency of Harmonic Wave|frequency]] $f_b$ of those [[De... | Let $\omega_1$ and $\omega_2$ denote the [[Definition:Angular Frequency|angular frequency]] of $W_1$ and $W_2$ respectively.
Let us consider the [[Definition:Harmonic Wave|harmonic waves]] that are $W_1$ and $W_2$ as they disturb the medium at $x = 0$.
{{WLOG}}, therefore, let $W_1$ and $W_2$ be be expressed as:
{{b... | Frequency of Beats | https://proofwiki.org/wiki/Frequency_of_Beats | https://proofwiki.org/wiki/Frequency_of_Beats | [
"Beats"
] | [
"Definition:Harmonic Wave",
"Definition:Harmonic Wave/Frequency",
"Definition:Beats",
"Definition:Harmonic Wave/Frequency",
"Definition:Beats"
] | [
"Definition:Angular Frequency",
"Definition:Harmonic Wave",
"Definition:Harmonic Wave",
"Prosthaphaeresis Formulas/Sine plus Sine",
"Cosine Function is Even",
"Definition:Harmonic Wave",
"Definition:Multiplication",
"Definition:Harmonic Wave",
"Definition:Harmonic Wave/Frequency",
"File:Beats-with... |
proofwiki-21321 | Amplitude of Beats | Let $W_1$ and $W_2$ be harmonic waves whose frequencies are $f_1$ and $f_2$.
Let the amplitude of $W_1$ and $W_2$ both be $a$.
Let the superpositon of $W_1$ onto $W_2$ exhibit the phenomenon of beats.
The amplitude $A_b$ of those beats at time $t$ is:
:$A_b = 2 a \map \cos {\pi \size {f_1 - f_2} t - \dfrac \epsilon 2}$ | Let $\omega_1$ and $\omega_2$ denote the angular frequency of $W_1$ and $W_2$ respectively.
Let us consider the harmonic waves that are $W_1$ and $W_2$ as they disturb the medium at $x = 0$.
{{WLOG}}, therefore, let $W_1$ and $W_2$ be be expressed as:
{{begin-eqn}}
{{eqn | l = \map {\phi_1} t
| r = a \sin \omega_... | Let $W_1$ and $W_2$ be [[Definition:Harmonic Wave|harmonic waves]] whose [[Definition:Frequency of Harmonic Wave|frequencies]] are $f_1$ and $f_2$.
Let the [[Definition:Amplitude of Harmonic Wave|amplitude]] of $W_1$ and $W_2$ both be $a$.
Let the superpositon of $W_1$ onto $W_2$ exhibit the phenomenon of [[Definitio... | Let $\omega_1$ and $\omega_2$ denote the [[Definition:Angular Frequency|angular frequency]] of $W_1$ and $W_2$ respectively.
Let us consider the [[Definition:Harmonic Wave|harmonic waves]] that are $W_1$ and $W_2$ as they disturb the medium at $x = 0$.
{{WLOG}}, therefore, let $W_1$ and $W_2$ be be expressed as:
{{b... | Amplitude of Beats | https://proofwiki.org/wiki/Amplitude_of_Beats | https://proofwiki.org/wiki/Amplitude_of_Beats | [
"Beats"
] | [
"Definition:Harmonic Wave",
"Definition:Harmonic Wave/Frequency",
"Definition:Harmonic Wave/Amplitude",
"Definition:Beats",
"Definition:Harmonic Wave/Amplitude",
"Definition:Beats"
] | [
"Definition:Angular Frequency",
"Definition:Harmonic Wave",
"Definition:Harmonic Wave/Phase",
"Definition:Harmonic Wave",
"Prosthaphaeresis Formulas/Sine plus Sine",
"Cosine Function is Even",
"Definition:Harmonic Wave/Amplitude",
"Definition:Angular Frequency"
] |
proofwiki-21322 | Bounded Summation of Integers is Primitive Recursive | Let the function $f : \N^{k + 1} \to \N$ be primitive recursive.
Let the function $g : \N^{k + 1} \to \N$ be defined as:
:$\ds \map g {n_1, \dotsc, n_k, z} = \paren {\sum_{y = 0}^{z - 1} \ell_y}_\Z$
where $\map f {n_1, \dotsc, n_k, y}$ codes the integer $\ell_y$.
Then, $g$ is primitive recursive. | We can equivalently write $g$ as:
:$\map g {n_1, \dotsc, n_k, z} = \begin{cases} 0 & : z = 0 \\ \map g {n_1, \dotsc, n_k, z - 1} +_\Z \map f {n_1, \dotsc, n_k, z - 1} & : z > 0 \end{cases}$
which is clearly obtained by primitive recursion from:
* Addition of Integers is Primitive Recursive
{{qed}}
Category:Primitive Re... | Let the function $f : \N^{k + 1} \to \N$ be [[Definition:Primitive Recursive Function|primitive recursive]].
Let the function $g : \N^{k + 1} \to \N$ be defined as:
:$\ds \map g {n_1, \dotsc, n_k, z} = \paren {\sum_{y = 0}^{z - 1} \ell_y}_\Z$
where $\map f {n_1, \dotsc, n_k, y}$ [[Definition:Code Number for Integer|co... | We can equivalently write $g$ as:
:$\map g {n_1, \dotsc, n_k, z} = \begin{cases} 0 & : z = 0 \\ \map g {n_1, \dotsc, n_k, z - 1} +_\Z \map f {n_1, \dotsc, n_k, z - 1} & : z > 0 \end{cases}$
which is clearly obtained by [[Definition:Primitive Recursion/Several Variables|primitive recursion]] from:
* [[Addition of Intege... | Bounded Summation of Integers is Primitive Recursive | https://proofwiki.org/wiki/Bounded_Summation_of_Integers_is_Primitive_Recursive | https://proofwiki.org/wiki/Bounded_Summation_of_Integers_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Primitive Recursive/Function",
"Definition:Code Number for Integer",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Primitive Recursion/Several Variables",
"Addition of Integers is Primitive Recursive",
"Category:Primitive Recursive Functions"
] |
proofwiki-21323 | Generalized Sum over Union of Disjoint Index Sets | Let $\struct {G, +}$ be a commutative topological semigroup.
Let $I$ and $J$ be disjoint indexing sets.
Let $K = I \cup J$.
Let $\family{g_k}_{k \mathop \in K}$ be an indexed family of elements of $G$.
Let the generalized sums $\ds \paren{\sum_{i \mathop \in I} g_i}$ and $\ds \paren{\sum_{j \mathop \in J} g_j}$ converg... | Let $0_G$ be the identity of the semigroup $\struct {G, +}$.
Let $\family{f_k}_{k \mathop \in K}$ be an indexed family of elements of $G$ defined by:
:$\forall k \in K : f_k = \begin{cases}
g_k & : k \in I \\
0_G & : k \in J
\end{cases}$
Let $\family{h_k}_{k \mathop \in K}$ be an indexed family of elements of $G$ defi... | Let $\struct {G, +}$ be a [[Definition:Commutative Semigroup|commutative]] [[Definition:Topological Semigroup|topological semigroup]].
Let $I$ and $J$ be [[Definition:Disjoint Sets|disjoint]] [[Definition:Indexing Set|indexing sets]].
Let $K = I \cup J$.
Let $\family{g_k}_{k \mathop \in K}$ be an [[Definition:Index... | Let $0_G$ be the [[Definition:Identity Element|identity]] of the [[Definition:Semigroup|semigroup]] $\struct {G, +}$.
Let $\family{f_k}_{k \mathop \in K}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Element|elements]] of $G$ defined by:
:$\forall k \in K : f_k = \begin{cases}
g_k & : k \in I \\... | Generalized Sum over Union of Disjoint Index Sets | https://proofwiki.org/wiki/Generalized_Sum_over_Union_of_Disjoint_Index_Sets | https://proofwiki.org/wiki/Generalized_Sum_over_Union_of_Disjoint_Index_Sets | [
"Generalized Sums"
] | [
"Definition:Commutative Semigroup",
"Definition:Topological Semigroup",
"Definition:Disjoint Sets",
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Element",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:Generalized Sum",
"Definition:Convergent Net... | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Semigroup",
"Definition:Indexing Set/Family",
"Definition:Element",
"Definition:Indexing Set/Family",
"Definition:Element",
"Generalized Sum Restricted to Non-zero Summands",
"Sum Rule for Convergent Generalized Sums",
"Categor... |
proofwiki-21324 | Sum Rule for Convergent Nets | Let $\struct {G, +}$ be a commutative topological semigroup.
Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\family {g_\lambda}_{\lambda \mathop \in \Lambda}$ and $\family {h_\lambda}_{\lambda \mathop \in \Lambda}$ be an indexed family of elements in $G$.
Let $\family {g_\lambda}_{\lambda \mathop \in \Lambda}... | Let $U$ be an open neighborhood of $a + b$.
By definition of topological semigroup:
:the binary operation $+ : G \times G \to G$ is continuous.
By definition of continuous mapping:
:$\exists W, V$ open neighborhoods of $a$ and $b$ respectively:
::$+ \sqbrk {W \times V} \subseteq U$
By definition of convergence:
:$\exis... | Let $\struct {G, +}$ be a [[Definition:Commutative Semigroup|commutative]] [[Definition:Topological Semigroup|topological semigroup]].
Let $\struct {\Lambda, \preceq}$ be a [[Definition:Directed Set|directed set]].
Let $\family {g_\lambda}_{\lambda \mathop \in \Lambda}$ and $\family {h_\lambda}_{\lambda \mathop \in ... | Let $U$ be an [[Definition:Open Neighborhood|open neighborhood]] of $a + b$.
By definition of [[Definition:Topological Semigroup|topological semigroup]]:
:the [[Definition:Binary Operation|binary operation]] $+ : G \times G \to G$ is [[Definition:Continuous Mapping (Topology)|continuous]].
By definition of [[Defini... | Sum Rule for Convergent Nets | https://proofwiki.org/wiki/Sum_Rule_for_Convergent_Nets | https://proofwiki.org/wiki/Sum_Rule_for_Convergent_Nets | [
"Nets (Set Theory)"
] | [
"Definition:Commutative Semigroup",
"Definition:Topological Semigroup",
"Definition:Directed Preordering",
"Definition:Indexing Set/Family",
"Definition:Element",
"Definition:Convergent Net",
"Definition:Limit of Net",
"Definition:Indexing Set/Family",
"Definition:Convergent Net",
"Definition:Limi... | [
"Definition:Open Neighborhood",
"Definition:Topological Semigroup",
"Definition:Operation/Binary Operation",
"Definition:Continuous Mapping (Topology)",
"Definition:Continuous Mapping (Topology)",
"Definition:Open Neighborhood",
"Definition:Convergent Net",
"Definition:Directed Preordering",
"Defini... |
proofwiki-21325 | Sum Rule for Convergent Generalized Sums | Let $\struct {G, +}$ be a commutative topological semigroup.
Let $\family {g_i}_{i \mathop \in I}$ and $\family {h_i}_{i \mathop \in I}$ be an indexed family of elements in $G$.
Let the generalized sums $\ds \sum_{i \mathop \in I} g_i$ and $\ds \sum_{i \mathop \in I} h_i$ be convergent to the following limits:
:$\ds \s... | Consider the set $\FF$ of finite subsets of $I$.
By definition of Definition:Generalized Sum:
:$\ds \sum_{i \mathop \in I} g_i$ is the net $\ds \family{\sum_{i \mathop \in F} g_i}_{F \mathop \in \FF}$
and
:$\ds \sum_{i \mathop \in I} h_i$ is the net $\ds \family{\sum_{i \mathop \in F} h_i}_{F \mathop \in \FF}$
By defin... | Let $\struct {G, +}$ be a [[Definition:Commutative Semigroup|commutative]] [[Definition:Topological Semigroup|topological semigroup]].
Let $\family {g_i}_{i \mathop \in I}$ and $\family {h_i}_{i \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Element|elements]] in $G$.
Let the [[D... | Consider the set $\FF$ of [[Definition:Finite Set|finite]] [[Definition:Subset|subsets]] of $I$.
By definition of [[Definition:Generalized Sum]]:
:$\ds \sum_{i \mathop \in I} g_i$ is the [[Definition:Net (Preordered Set)|net]] $\ds \family{\sum_{i \mathop \in F} g_i}_{F \mathop \in \FF}$
and
:$\ds \sum_{i \mathop \in... | Sum Rule for Convergent Generalized Sums | https://proofwiki.org/wiki/Sum_Rule_for_Convergent_Generalized_Sums | https://proofwiki.org/wiki/Sum_Rule_for_Convergent_Generalized_Sums | [
"Generalized Sums"
] | [
"Definition:Commutative Semigroup",
"Definition:Topological Semigroup",
"Definition:Indexing Set/Family",
"Definition:Element",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:Limit of Net",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:Limit of Net"... | [
"Definition:Finite Set",
"Definition:Subset",
"Definition:Generalized Sum",
"Definition:Net (Preordered Set)",
"Definition:Net (Preordered Set)",
"Definition:Convergent Net",
"Sum Rule for Convergent Nets",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:Limit of Net",
"Ca... |
proofwiki-21326 | Inequality Rule for Real Convergent Nets | Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ and $\family {y_\lambda}_{\lambda \mathop \in \Lambda}$ be indexed families of elements in $\R$.
Let $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ and $\family {y_\lambda}_{\lambda \mathop \in \Lambda}$ be ... | {{AimForCont}}
:$l > m$
Let $\epsilon = \dfrac {\paren{l - m}} 2$.
Hence:
:$\epsilon > 0$
and:
:$(1) \quad l - \epsilon = m + \epsilon$
By definition of convergence:
:$\exists \lambda_1 \in \Lambda : \forall \mu \in \Lambda : \lambda_1 \preceq \mu \implies \size{l - x_\mu} < \epsilon$
and:
:$\exists \lambda_2 \in \Lam... | Let $\struct {\Lambda, \preceq}$ be a [[Definition:Directed Set|directed set]].
Let $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ and $\family {y_\lambda}_{\lambda \mathop \in \Lambda}$ be [[Definition:Indexed Family|indexed families]] of [[Definition:Element|elements]] in $\R$.
Let $\family {x_\lambda}_{\lamb... | {{AimForCont}}
:$l > m$
Let $\epsilon = \dfrac {\paren{l - m}} 2$.
Hence:
:$\epsilon > 0$
and:
:$(1) \quad l - \epsilon = m + \epsilon$
By definition of [[Definition:Convergent Net|convergence]]:
:$\exists \lambda_1 \in \Lambda : \forall \mu \in \Lambda : \lambda_1 \preceq \mu \implies \size{l - x_\mu} < \epsilon... | Inequality Rule for Real Convergent Nets | https://proofwiki.org/wiki/Inequality_Rule_for_Real_Convergent_Nets | https://proofwiki.org/wiki/Inequality_Rule_for_Real_Convergent_Nets | [
"Nets (Set Theory)"
] | [
"Definition:Directed Preordering",
"Definition:Indexing Set/Family",
"Definition:Element",
"Definition:Convergent Net",
"Definition:Limit of Net"
] | [
"Definition:Convergent Net",
"Closed Interval Defined by Absolute Value",
"Definition:Directed Preordering",
"Definition:Contradiction",
"Definition:Hypothesis",
"Category:Nets (Set Theory)"
] |
proofwiki-21327 | Sequence of Partial Sums of Computable Real Sequence is Computable | Let $\sequence {x_m}_{m \in \N}$ be a computable real sequence.
Let $\sequence{y_N}_{N \in \N}$ be defined as:
:$\ds y_N = \sum_{m = 0}^{N - 1} x_m$
Then $\sequence{y_N}$ is a computable real sequence. | By definition of computable real sequence, there exists a total recursive function $f : \N^2 \to \N$ such that, for all $m, p \in \N$:
:$\dfrac {k_{m,p} - 1} {p + 1} < x_m < \dfrac {k_{m,p} + 1} {p + 1}$
where $\map f {m, p}$ codes the integer $k_{m,p}$
Define $g: \N^2 \to \N$ as:
:$\ds \map g {N, p} = \map {\operatorn... | Let $\sequence {x_m}_{m \in \N}$ be a [[Definition:Computable Real Sequence|computable real sequence]].
Let $\sequence{y_N}_{N \in \N}$ be defined as:
:$\ds y_N = \sum_{m = 0}^{N - 1} x_m$
Then $\sequence{y_N}$ is a [[Definition:Computable Real Sequence|computable real sequence]]. | By definition of [[Definition:Computable Real Sequence|computable real sequence]], there exists a [[Definition:Total Recursive Function|total recursive function]] $f : \N^2 \to \N$ such that, for all $m, p \in \N$:
:$\dfrac {k_{m,p} - 1} {p + 1} < x_m < \dfrac {k_{m,p} + 1} {p + 1}$
where $\map f {m, p}$ [[Definition:C... | Sequence of Partial Sums of Computable Real Sequence is Computable | https://proofwiki.org/wiki/Sequence_of_Partial_Sums_of_Computable_Real_Sequence_is_Computable | https://proofwiki.org/wiki/Sequence_of_Partial_Sums_of_Computable_Real_Sequence_is_Computable | [] | [
"Definition:Computable Real Sequence",
"Definition:Computable Real Sequence"
] | [
"Definition:Computable Real Sequence",
"Definition:Total Recursive Function",
"Definition:Code Number for Integer",
"Definition:Code Number for Integer",
"Definition:Total Recursive Function",
"Bounded Summation of Integers is Primitive Recursive",
"Predecessor Function is Primitive Recursive",
"Const... |
proofwiki-21328 | Generalized Sum over Subset of Absolutely Convergent Generalized Sum is Absolutely Convergent | Let $V$ be a Banach space.
Let $\family {v_i}_{i \mathop \in I}$ be an indexed family of elements of $V$.
Let the generalized sum $\ds \sum_{i \mathop \in I} v_i$ be absolutely net convergent.
Let $J \subseteq I$.
Then:
:the generalized sum $\ds \sum_{j \mathop \in J} v_j$ is absolutely net convergent. | By definition of absolute net convergence, let:
:$\ds \sum_{i \mathop \in I} \norm{v_i} = M$
Let $F \subseteq J$ be finite.
From Subset Relation is Transitive:
:$F \subseteq I$
From Absolutely Convergent Generalized Sum Converges to Supremum:
:$\ds \sum_{j \mathop \in F} \norm{v_j} \le M$
Since $F \subseteq J$ was arb... | Let $V$ be a [[Definition:Banach Space|Banach space]].
Let $\family {v_i}_{i \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Element|elements]] of $V$.
Let the [[Definition:Generalized Sum|generalized sum]] $\ds \sum_{i \mathop \in I} v_i$ be [[Definition:Absolute Net Convergence|a... | By definition of [[Definition:Absolute Net Convergence|absolute net convergence]], let:
:$\ds \sum_{i \mathop \in I} \norm{v_i} = M$
Let $F \subseteq J$ be [[Definition:Finite Set|finite]].
From [[Subset Relation is Transitive]]:
:$F \subseteq I$
From [[Absolutely Convergent Generalized Sum Converges to Supremum]]:... | Generalized Sum over Subset of Absolutely Convergent Generalized Sum is Absolutely Convergent | https://proofwiki.org/wiki/Generalized_Sum_over_Subset_of_Absolutely_Convergent_Generalized_Sum_is_Absolutely_Convergent | https://proofwiki.org/wiki/Generalized_Sum_over_Subset_of_Absolutely_Convergent_Generalized_Sum_is_Absolutely_Convergent | [
"Banach Spaces",
"Generalized Sums"
] | [
"Definition:Banach Space",
"Definition:Indexing Set/Family",
"Definition:Element",
"Definition:Generalized Sum",
"Definition:Generalized Sum/Absolute Net Convergence",
"Definition:Generalized Sum",
"Definition:Generalized Sum/Absolute Net Convergence"
] | [
"Definition:Generalized Sum/Absolute Net Convergence",
"Definition:Finite Set",
"Subset Relation is Transitive",
"Absolutely Convergent Generalized Sum Converges to Supremum",
"Definition:Finite Set",
"Bounded Generalized Sum is Absolutely Convergent",
"Definition:Generalized Sum/Absolute Net Convergenc... |
proofwiki-21329 | Inequality Rule for Absolutely Convergent Generalized Sums | Let $V$ be a Banach space.
Let $\family {v_i}_{i \mathop \in I}$ be an indexed family of elements of $V$.
Let the generalized sum $\ds \sum \set {v_i: i \in I}$ be absolutely net convergent.
Let $\family {w_i}_{i \mathop \in I}$ be an indexed family of elements of $V$:
:$\forall i \in I : \norm{w_i} \le \norm{v_i}$
The... | By definition of absolutely net convergence, let:
:$\ds \sum_{i \mathop \in I} \norm{v_i} = M$
Let $F \subseteq I$ be finite.
From Absolutely Convergent Generalized Sum Converges to Supremum:
:$\ds \sum_{i \mathop \in F} \norm{v_i} \le M$
So {{Hypothesis}}:
:$\ds \sum_{i \mathop \in F} \norm{w_i} \le \ds \sum_{i \mat... | Let $V$ be a [[Definition:Banach Space|Banach space]].
Let $\family {v_i}_{i \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Element|elements]] of $V$.
Let the [[Definition:Generalized Sum|generalized sum]] $\ds \sum \set {v_i: i \in I}$ be [[Definition:Absolute Net Convergence|abs... | By definition of [[Definition:Absolute Net Convergence|absolutely net convergence]], let:
:$\ds \sum_{i \mathop \in I} \norm{v_i} = M$
Let $F \subseteq I$ be [[Definition:Finite Set|finite]].
From [[Absolutely Convergent Generalized Sum Converges to Supremum]]:
:$\ds \sum_{i \mathop \in F} \norm{v_i} \le M$
So {{... | Inequality Rule for Absolutely Convergent Generalized Sums | https://proofwiki.org/wiki/Inequality_Rule_for_Absolutely_Convergent_Generalized_Sums | https://proofwiki.org/wiki/Inequality_Rule_for_Absolutely_Convergent_Generalized_Sums | [
"Banach Spaces",
"Generalized Sums"
] | [
"Definition:Banach Space",
"Definition:Indexing Set/Family",
"Definition:Element",
"Definition:Generalized Sum",
"Definition:Generalized Sum/Absolute Net Convergence",
"Definition:Indexing Set/Family",
"Definition:Element",
"Definition:Generalized Sum",
"Definition:Generalized Sum/Absolute Net Conve... | [
"Definition:Generalized Sum/Absolute Net Convergence",
"Definition:Finite Set",
"Absolutely Convergent Generalized Sum Converges to Supremum",
"Definition:Finite Set",
"Bounded Generalized Sum is Absolutely Convergent",
"Definition:Generalized Sum/Absolute Net Convergence",
"Inequality Rule for Real Con... |
proofwiki-21330 | Binomial Distribution Approximated by Normal Distribution | Let $X$ be a discrete random variable which has the binomial distribution $\Binomial n p$.
Then for large $n$ and such that both $n p$ and $n q$ are approximately $5$ or more:
:$\Binomial n p \approx \Gaussian {n p} {n p q}$
where $\Gaussian {n p} {n p q}$ denotes the normal distribution. | Let $Y_1, Y_2, \ldots, Y_n$ be independent random variables which have the Bernoulli Distribution $\Bernoulli p$.
By the Sum of Independent Bernoulli Random Variables is Binomial, we have that:
:$\ds \sum_{i \mathop = 1}^n Y_i \sim \Binomial n p$
{{MissingLinks|Search for theorem of Sum of Independent Bernoulli Random ... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] which has the [[Definition:Binomial Distribution|binomial distribution $\Binomial n p$]].
Then for large $n$ and such that both $n p$ and $n q$ are approximately $5$ or more:
:$\Binomial n p \approx \Gaussian {n p} {n p q}$
where $\Gaussian... | Let $Y_1, Y_2, \ldots, Y_n$ be [[Definition:Independent Random Variables|independent random variables]] which have the [[Definition:Bernoulli Distribution|Bernoulli Distribution $\Bernoulli p$]].
By the [[Sum of Independent Bernoulli Random Variables is Binomial]], we have that:
:$\ds \sum_{i \mathop = 1}^n Y_i \sim \... | Binomial Distribution Approximated by Normal Distribution | https://proofwiki.org/wiki/Binomial_Distribution_Approximated_by_Normal_Distribution | https://proofwiki.org/wiki/Binomial_Distribution_Approximated_by_Normal_Distribution | [
"Binomial Distribution",
"Normal Distribution"
] | [
"Definition:Random Variable/Discrete",
"Definition:Binomial Distribution",
"Definition:Normal Distribution"
] | [
"Definition:Independent Random Variables",
"Definition:Bernoulli Distribution",
"Sum of Independent Bernoulli Random Variables is Binomial",
"Central Limit Theorem",
"Linear Transformation of Normal Random Variable"
] |
proofwiki-21331 | Position Vector of Midpoint of Line | Let $\mathbf a$ and $\mathbf b$ be the position vectors of points $A$ and $B$.
The position vector $\mathbf r$ of the midpoint of the line segment $AB$ is given by:
:$\mathbf r = \dfrac {\mathbf a + \mathbf b} 2$ | From Point dividing Line Segment between Two Points in Given Ratio:
:the position vector $\mathbf r$ of a point $R$ on $AB$ which divides $AB$ in the ratio $m : n$ is given by:
:$\mathbf r = \dfrac {n \mathbf a + m \mathbf b} {m + n}$
In this case the ratio $m : n$ is $1 : 1$.
Hence when $\mathbf r$ is the position vec... | Let $\mathbf a$ and $\mathbf b$ be the [[Definition:Position Vector|position vectors]] of [[Definition:Point|points]] $A$ and $B$.
The [[Definition:Position Vector|position vector]] $\mathbf r$ of the [[Definition:Midpoint of Line|midpoint]] of the [[Definition:Line Segment|line segment]] $AB$ is given by:
:$\mathbf r... | From [[Point dividing Line Segment between Two Points in Given Ratio]]:
:the [[Definition:Position Vector|position vector]] $\mathbf r$ of a [[Definition:Point|point]] $R$ on $AB$ which divides $AB$ in the [[Definition:Ratio|ratio]] $m : n$ is given by:
:$\mathbf r = \dfrac {n \mathbf a + m \mathbf b} {m + n}$
In thi... | Position Vector of Midpoint of Line | https://proofwiki.org/wiki/Position_Vector_of_Midpoint_of_Line | https://proofwiki.org/wiki/Position_Vector_of_Midpoint_of_Line | [
"Bisection",
"Vector Algebra",
"Straight Lines"
] | [
"Definition:Position Vector",
"Definition:Point",
"Definition:Position Vector",
"Definition:Line/Midpoint",
"Definition:Line/Segment"
] | [
"Point dividing Line Segment between Two Points in Given Ratio",
"Definition:Position Vector",
"Definition:Point",
"Definition:Ratio",
"Definition:Ratio",
"Definition:Position Vector",
"Definition:Line/Midpoint"
] |
proofwiki-21332 | Absolutely Convergent Generalized Sum over Union of Disjoint Index Sets | Let $V$ be a Banach space.
Let $I$ and $J$ be disjoint indexing sets.
Let $K = I \cup J$.
Let $\family{v_k}_{k \mathop \in K}$ be an indexed family of elements of $V$.
Then:
:the generalized sum $\ds \sum_{k \mathop \in K} v_k$ converges absolutely
{{iff}}
:the generalized sums $\ds \paren{\sum_{i \mathop \in I} v_i}$ ... | === Necessary Condition ===
Let $\ds \sum_{k \mathop \in K} v_k$ converge absolutely.
By definition of absolute net convergence:
:$\ds \sum_{k \mathop \in K} \norm{v_k}$ converges.
From Generalized Sum over Subset of Absolutely Convergent Generalized Sum is Absolutely Convergent:
:$\ds \paren{\sum_{i \mathop \in I} \no... | Let $V$ be a [[Definition:Banach Space|Banach space]].
Let $I$ and $J$ be [[Definition:Disjoint Sets|disjoint]] [[Definition:Indexing Set|indexing sets]].
Let $K = I \cup J$.
Let $\family{v_k}_{k \mathop \in K}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Element|elements]] of $V$.
Then:
:... | === Necessary Condition ===
Let $\ds \sum_{k \mathop \in K} v_k$ [[Definition:Absolute Net Convergence|converge absolutely]].
By definition of [[Definition:Absolute Net Convergence|absolute net convergence]]:
:$\ds \sum_{k \mathop \in K} \norm{v_k}$ [[Definition:Convergent Net|converges]].
From [[Generalized Sum ove... | Absolutely Convergent Generalized Sum over Union of Disjoint Index Sets | https://proofwiki.org/wiki/Absolutely_Convergent_Generalized_Sum_over_Union_of_Disjoint_Index_Sets | https://proofwiki.org/wiki/Absolutely_Convergent_Generalized_Sum_over_Union_of_Disjoint_Index_Sets | [
"Generalized Sums"
] | [
"Definition:Banach Space",
"Definition:Disjoint Sets",
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Element",
"Definition:Generalized Sum",
"Definition:Generalized Sum/Absolute Net Convergence",
"Definition:Generalized Sum",
"Definition:Generalized Sum/Absolute Net Conver... | [
"Definition:Generalized Sum/Absolute Net Convergence",
"Definition:Generalized Sum/Absolute Net Convergence",
"Definition:Convergent Net",
"Generalized Sum over Subset of Absolutely Convergent Generalized Sum is Absolutely Convergent",
"Definition:Convergent Net",
"Definition:Generalized Sum/Absolute Net ... |
proofwiki-21333 | Product of Sequentially Computable Real-Valued Functions is Sequentially Computable | Let $D \subseteq \R^n$ be a subset of real cartesian $n$-space.
Let $f, g : D \to \R$ be sequentially computable.
Then, $h : D \to \R$ defined as:
:$\map h \bsx = \map f \bsx \map g \bsx$
is sequentially computable. | Follows immediately from:
* Real Multiplication is Sequentially Computable
* Composition of Sequentially Computable Real-Valued Functions is Sequentially Computable
{{qed}}
Category:Computability Theory
9cq5yuvkh4nb0jrwzner72b0uib2lhl | Let $D \subseteq \R^n$ be a [[Definition:Subset|subset]] of [[Definition:Real Cartesian Space|real cartesian $n$-space]].
Let $f, g : D \to \R$ be [[Definition:Sequentially Computable Real-Valued Function|sequentially computable]].
Then, $h : D \to \R$ defined as:
:$\map h \bsx = \map f \bsx \map g \bsx$
is [[Definit... | Follows immediately from:
* [[Real Multiplication is Sequentially Computable]]
* [[Composition of Sequentially Computable Real-Valued Functions is Sequentially Computable]]
{{qed}}
[[Category:Computability Theory]]
9cq5yuvkh4nb0jrwzner72b0uib2lhl | Product of Sequentially Computable Real-Valued Functions is Sequentially Computable | https://proofwiki.org/wiki/Product_of_Sequentially_Computable_Real-Valued_Functions_is_Sequentially_Computable | https://proofwiki.org/wiki/Product_of_Sequentially_Computable_Real-Valued_Functions_is_Sequentially_Computable | [
"Computability Theory"
] | [
"Definition:Subset",
"Definition:Cartesian Product/Cartesian Space/Real Cartesian Space",
"Definition:Sequentially Computable Real-Valued Function",
"Definition:Sequentially Computable Real-Valued Function"
] | [
"Real Multiplication is Sequentially Computable",
"Composition of Sequentially Computable Real-Valued Functions is Sequentially Computable",
"Category:Computability Theory"
] |
proofwiki-21334 | Sum of Sequentially Computable Real-Valued Functions is Sequentially Computable | Let $D \subseteq \R^n$ be a subset of real cartesian $n$-space.
Let $f, g : D \to \R$ be sequentially computable.
Then, $h : D \to \R$ defined as:
:$\map h \bsx = \map f \bsx + \map g \bsx$
is sequentially computable. | Follows immediately from:
* Real Addition is Sequentially Computable
* Composition of Sequentially Computable Real-Valued Functions is Sequentially Computable
{{qed}}
Category:Computability Theory
m0ylg14hcx42e8bv9sx9qjpfut2fz0x | Let $D \subseteq \R^n$ be a [[Definition:Subset|subset]] of [[Definition:Real Cartesian Space|real cartesian $n$-space]].
Let $f, g : D \to \R$ be [[Definition:Sequentially Computable Real-Valued Function|sequentially computable]].
Then, $h : D \to \R$ defined as:
:$\map h \bsx = \map f \bsx + \map g \bsx$
is [[Defin... | Follows immediately from:
* [[Real Addition is Sequentially Computable]]
* [[Composition of Sequentially Computable Real-Valued Functions is Sequentially Computable]]
{{qed}}
[[Category:Computability Theory]]
m0ylg14hcx42e8bv9sx9qjpfut2fz0x | Sum of Sequentially Computable Real-Valued Functions is Sequentially Computable | https://proofwiki.org/wiki/Sum_of_Sequentially_Computable_Real-Valued_Functions_is_Sequentially_Computable | https://proofwiki.org/wiki/Sum_of_Sequentially_Computable_Real-Valued_Functions_is_Sequentially_Computable | [
"Computability Theory"
] | [
"Definition:Subset",
"Definition:Cartesian Product/Cartesian Space/Real Cartesian Space",
"Definition:Sequentially Computable Real-Valued Function",
"Definition:Sequentially Computable Real-Valued Function"
] | [
"Real Addition is Sequentially Computable",
"Composition of Sequentially Computable Real-Valued Functions is Sequentially Computable",
"Category:Computability Theory"
] |
proofwiki-21335 | Constant Function of Computable Real Number is Sequentially Computable | Let $c \in \R$ be a computable real number.
Then, $f : \R^n \to \R$, defined as:
:$\map f \bsx = c$
is sequentially computable. | For any real sequence $\sequence {x_n}$, we have:
:$\sequence {\map f {x_n}} = \sequence c$
Thus, the result follows from Constant Sequence of Computable Real Number is Computable.
{{qed}}
Category:Computability Theory
pcvt3i05a6st2kb953vcu9sm5hn97a8 | Let $c \in \R$ be a [[Definition:Computable Real Number|computable real number]].
Then, $f : \R^n \to \R$, defined as:
:$\map f \bsx = c$
is [[Definition:Sequentially Computable Real-Valued Function|sequentially computable]]. | For any [[Definition:Real Sequence|real sequence]] $\sequence {x_n}$, we have:
:$\sequence {\map f {x_n}} = \sequence c$
Thus, the result follows from [[Constant Sequence of Computable Real Number is Computable]].
{{qed}}
[[Category:Computability Theory]]
pcvt3i05a6st2kb953vcu9sm5hn97a8 | Constant Function of Computable Real Number is Sequentially Computable | https://proofwiki.org/wiki/Constant_Function_of_Computable_Real_Number_is_Sequentially_Computable | https://proofwiki.org/wiki/Constant_Function_of_Computable_Real_Number_is_Sequentially_Computable | [
"Computability Theory"
] | [
"Definition:Computable Real Number",
"Definition:Sequentially Computable Real-Valued Function"
] | [
"Definition:Real Sequence",
"Constant Sequence of Computable Real Number is Computable",
"Category:Computability Theory"
] |
proofwiki-21336 | Condition for Limits of Computable Real Sequences to be Computable | Let $\sequence {x_k}_{k \in \N}$ be a computable real sequence.
Let $\sequence {y_n}_{n \in \N}$ be a real sequence.
If there exists total recursive function $\phi : \N^2 \to \N$ such that:
:$\forall n, p \in \N: \size {x_{\map \phi {n, p}} - y_n} < \dfrac 1 {p + 1}$
then:
:$\sequence {y_n}_{n \in \N}$
is a computable ... | By {{Corollary|Computable Real Sequence iff Limits of Computable Rational Sequences}}, there exists a computable rational sequence $\sequence {a_N}$ such that, for all $k, p \in \N$:
:$\size {a_{\map \pi {k, p} } - x_k} < \dfrac 1 {p + 1}$
where $\pi$ is the Cantor pairing function.
Let $\psi : \N^2 \to \N$ be defined ... | Let $\sequence {x_k}_{k \in \N}$ be a [[Definition:Computable Real Sequence|computable real sequence]].
Let $\sequence {y_n}_{n \in \N}$ be a [[Definition:Real Sequence|real sequence]].
If there exists [[Definition:Total Recursive Function|total recursive function]] $\phi : \N^2 \to \N$ such that:
:$\forall n, p \in ... | By {{Corollary|Computable Real Sequence iff Limits of Computable Rational Sequences}}, there exists a [[Definition:Computable Rational Sequence|computable rational sequence]] $\sequence {a_N}$ such that, for all $k, p \in \N$:
:$\size {a_{\map \pi {k, p} } - x_k} < \dfrac 1 {p + 1}$
where $\pi$ is the [[Definition:Cant... | Condition for Limits of Computable Real Sequences to be Computable | https://proofwiki.org/wiki/Condition_for_Limits_of_Computable_Real_Sequences_to_be_Computable | https://proofwiki.org/wiki/Condition_for_Limits_of_Computable_Real_Sequences_to_be_Computable | [
"Computability Theory"
] | [
"Definition:Computable Real Sequence",
"Definition:Real Sequence",
"Definition:Total Recursive Function",
"Definition:Computable Real Sequence"
] | [
"Definition:Computable Rational Sequence",
"Definition:Cantor Pairing Function",
"Definition:Total Recursive Function",
"Cantor Pairing Function is Primitive Recursive",
"Primitive Recursive Function is Total Recursive Function",
"Definition:Computable Rational Sequence",
"Triangle Inequality/Real Numbe... |
proofwiki-21337 | Algebra of Sets is Boolean Algebra | An algebra of sets is a Boolean algebra. | Let $\RR \subseteq \powerset S$ be a set $S$ upon which an algebra of sets has been constructed.
We identify:
{{begin-axiom}}
{{axiom | lc= Set union:
| m = \cup
| rc= with join $\vee$
}}
{{axiom | lc= Set intersection:
| m = \cap
| rc= with meet $\vee$
}}
{{axiom | lc= Relative compleme... | An [[Definition:Algebra of Sets|algebra of sets]] is a [[Definition:Boolean Algebra|Boolean algebra]]. | Let $\RR \subseteq \powerset S$ be a [[Definition:Set|set]] $S$ upon which an [[Definition:Algebra of Sets|algebra of sets]] has been constructed.
We identify:
{{begin-axiom}}
{{axiom | lc= [[Definition:Set Union|Set union]]:
| m = \cup
| rc= with [[Definition:Join (Boolean Algebra)|join]] $\vee$
}}
{... | Algebra of Sets is Boolean Algebra | https://proofwiki.org/wiki/Algebra_of_Sets_is_Boolean_Algebra | https://proofwiki.org/wiki/Algebra_of_Sets_is_Boolean_Algebra | [
"Algebras of Sets",
"Boolean Algebras"
] | [
"Definition:Algebra of Sets",
"Definition:Boolean Algebra"
] | [
"Definition:Set",
"Definition:Algebra of Sets",
"Definition:Set Union",
"Definition:Boolean Algebra/Join",
"Definition:Set Intersection",
"Definition:Boolean Algebra/Meet",
"Definition:Relative Complement",
"Definition:Boolean Algebra/Complement",
"Axiom:Boolean Algebra/Axioms/Formulation 2",
"De ... |
proofwiki-21338 | L1 Mean Ergodic Theorem | Let $\struct {X, \BB, \mu, T}$ be a measure-preserving dynamical system.
Let $\map {L^1_\C} \mu$ be the complex-valued $L^1$ space of $\mu$.
Then for each $f \in \map {L^1_\C} \mu$ there is a $T$-invariant function $\tilde f \in \map {L^1_\C} \mu$ such that:
:$\ds \lim_{N \mathop \to \infty} \dfrac 1 N \sum_{n \mathop ... | For $c > 0$ let:
:$f_c = f \cdot \chi_{\set {\cmod f \mathop \le c} }$
where $\chi_B$ is the characteristic function of $B$.
By Lebesgue's Dominated Convergence Theorem:
{{begin-eqn}}
{{eqn | l = f - f_c
| r = f \cdot \paren {1 -\chi_{\set {\cmod f \mathop \le c} } }
}}
{{eqn | r = f \cdot \chi_{\set {\cmod f \ma... | Let $\struct {X, \BB, \mu, T}$ be a [[Definition:Measure-Preserving Dynamical System|measure-preserving dynamical system]].
Let $\map {L^1_\C} \mu$ be the [[Definition:Complex-Valued Function|complex-valued]] [[Definition:Lp Space|$L^1$ space]] of $\mu$.
Then for each $f \in \map {L^1_\C} \mu$ there is a $T$-[[Defin... | For $c > 0$ let:
:$f_c = f \cdot \chi_{\set {\cmod f \mathop \le c} }$
where $\chi_B$ is the [[Definition:Characteristic Function|characteristic function]] of $B$.
By [[Lebesgue's Dominated Convergence Theorem]]:
{{begin-eqn}}
{{eqn | l = f - f_c
| r = f \cdot \paren {1 -\chi_{\set {\cmod f \mathop \le c} } }
}}... | L1 Mean Ergodic Theorem | https://proofwiki.org/wiki/L1_Mean_Ergodic_Theorem | https://proofwiki.org/wiki/L1_Mean_Ergodic_Theorem | [
"Mean Ergodic Theorem",
"Lp Spaces"
] | [
"Definition:Measure-Preserving Dynamical System",
"Definition:Complex-Valued Function",
"Definition:Lp Space",
"Definition:Transformation Invariant Function",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Lp Norm"
] | [
"Definition:Characteristic Function",
"Lebesgue's Dominated Convergence Theorem",
"Mean Ergodic Theorem",
"Convergent Sequence is Cauchy Sequence",
"Cauchy-Bunyakovsky-Schwarz Inequality/Lebesgue 2-Space",
"Definition:Cauchy Sequence/Normed Vector Space",
"Riesz-Fischer Theorem"
] |
proofwiki-21339 | Topological Space Separated by Mappings is Hausdorff | Let $X$ be a topological space.
Let $\family {Y_i}_{i \mathop \in I}$ be an indexed family of Hausdorff spaces for some indexing set $I$.
Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be an indexed family of continuous mappings.
Suppose $\family {f_i : X \to Y_i}_{i \mathop \in I}$ separates the points of $X$.
Then... | Let $x \ne y$ be elements of $X$.
By definition of separating points, there exists some $i \in I$ such that:
:$\map {f_i} x \ne \map {f_i} y$
As $Y_i$ is Hausdorff, there exist open sets $U, V \subseteq Y_i$ such that:
:$\map {f_i} x \in U$
:$\map {f_i} y \in V$
:$U \cap V = \O$
By definition of continuous mapping, we ... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $\family {Y_i}_{i \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Hausdorff Space|Hausdorff spaces]] for some [[Definition:Indexing Set|indexing set]] $I$.
Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be an [[... | Let $x \ne y$ be [[Definition:Element|elements]] of $X$.
By definition of [[Definition:Mappings Separating Points|separating points]], there exists some $i \in I$ such that:
:$\map {f_i} x \ne \map {f_i} y$
As $Y_i$ is [[Definition:Hausdorff Space|Hausdorff]], there exist [[Definition:Open Set|open sets]] $U, V \subs... | Topological Space Separated by Mappings is Hausdorff | https://proofwiki.org/wiki/Topological_Space_Separated_by_Mappings_is_Hausdorff | https://proofwiki.org/wiki/Topological_Space_Separated_by_Mappings_is_Hausdorff | [
"Hausdorff Spaces"
] | [
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:T2 Space",
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Continuous Mapping (Topology)",
"Definition:Mappings Separating Points",
"Definition:T2 Space"
] | [
"Definition:Element",
"Definition:Mappings Separating Points",
"Definition:T2 Space",
"Definition:Open Set",
"Definition:Continuous Mapping (Topology)/Everywhere/Open Sets",
"Definition:Open Set",
"Definition:Preimage/Mapping/Subset",
"Definition:Disjoint Sets",
"Definition:Preimage/Mapping/Subset",... |
proofwiki-21340 | Boundary of Compact Convex Set with Nonempty Interior is Homeomorphic to Sphere | Let $n \in \N_{> 0}$.
Let $C \subseteq \R^n$ be a compact convex subset of real Euclidean $n$-space.
Suppose that the interior $C^\circ$ is non-empty.
Then, the boundary $\partial C$ is homeomorphic to $\Bbb S^{n - 1}$, the unit $n - 1$-sphere. | Let $\bsx_0 \in C^\circ$ be an element of $C^\circ$.
Define $\phi : \R^n \setminus \set {\bsx_0} \to \Bbb S^{n - 1}$ as:
:$\map \phi \bsx = \dfrac 1 {\norm {\bsx - \bsx_0}} \paren {\bsx - \bsx_0}$
As Normed Vector Space is Hausdorff Topological Vector Space, it follows that $\phi$ is continuous.
Define $\phi^* : \parti... | Let $n \in \N_{> 0}$.
Let $C \subseteq \R^n$ be a [[Definition:Compact Set (Topology)|compact]] [[Definition:Convex Set (Vector Space)|convex]] [[Definition:Subset|subset]] of [[Definition:Real Euclidean Space|real Euclidean $n$-space]].
Suppose that the [[Definition:Interior (Topology)|interior]] $C^\circ$ is [[Defi... | Let $\bsx_0 \in C^\circ$ be an [[Definition:Element|element]] of $C^\circ$.
Define $\phi : \R^n \setminus \set {\bsx_0} \to \Bbb S^{n - 1}$ as:
:$\map \phi \bsx = \dfrac 1 {\norm {\bsx - \bsx_0}} \paren {\bsx - \bsx_0}$
As [[Normed Vector Space is Hausdorff Topological Vector Space]], it follows that $\phi$ is [[Defi... | Boundary of Compact Convex Set with Nonempty Interior is Homeomorphic to Sphere | https://proofwiki.org/wiki/Boundary_of_Compact_Convex_Set_with_Nonempty_Interior_is_Homeomorphic_to_Sphere | https://proofwiki.org/wiki/Boundary_of_Compact_Convex_Set_with_Nonempty_Interior_is_Homeomorphic_to_Sphere | [
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Compact Topological Space/Subspace",
"Definition:Convex Set (Vector Space)",
"Definition:Subset",
"Definition:Euclidean Space/Real",
"Definition:Interior (Topology)",
"Definition:Non-Empty Set",
"Definition:Boundary (Topology)",
"Definition:Homeomorphism/Topological Spaces",
"Definition:... | [
"Definition:Element",
"Normed Vector Space is Hausdorff Topological Vector Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Well-Defined",
"Definition:Boundary (Topology)",
"Definition:Disjoint Sets",
"Definition:Interior (Topology)",
"Definition:Boundary (Topology)/Definitio... |
proofwiki-21341 | Similarity Dimension of Cantor Set | The Cantor set is a fractal with similarity dimension of $\dfrac {\ln 2} {\ln 3}$, which is $0 \cdotp 63$ to $2$ decimal places. | Let $C$ denote the Cantor set.
$C$ has the following self-similarities:
{{begin-eqn}}
{{eqn | l = x
| o = \mapsto
| r = \dfrac x 3
| c = with scale factor $r_1 = \dfrac 1 3$
}}
{{eqn | l = x
| o = \mapsto
| r = \dfrac 2 3 + \dfrac x 3
| c = with scale factor $r_2 = \dfrac 1 3$
}}
{{e... | The [[Definition:Cantor Set|Cantor set]] is a [[Definition:Fractal|fractal]] with [[Definition:Similarity Dimension|similarity dimension]] of $\dfrac {\ln 2} {\ln 3}$, which is $0 \cdotp 63$ [[Definition:Accurate to n Decimal Places|to $2$ decimal places]]. | Let $C$ denote the [[Definition:Cantor Set|Cantor set]].
$C$ has the following [[Definition:Self-Similarity|self-similarities]]:
{{begin-eqn}}
{{eqn | l = x
| o = \mapsto
| r = \dfrac x 3
| c = with [[Definition:Scale Factor|scale factor]] $r_1 = \dfrac 1 3$
}}
{{eqn | l = x
| o = \mapsto
... | Similarity Dimension of Cantor Set | https://proofwiki.org/wiki/Similarity_Dimension_of_Cantor_Set | https://proofwiki.org/wiki/Similarity_Dimension_of_Cantor_Set | [
"Cantor Set",
"Similarity Dimensions"
] | [
"Definition:Cantor Set",
"Definition:Fractal",
"Definition:Fractal Dimension/Similarity Dimension",
"Definition:Accuracy/Decimal Places"
] | [
"Definition:Cantor Set",
"Definition:Self-Similarity",
"Definition:Similarity Mapping/Scale Factor",
"Definition:Similarity Mapping/Scale Factor",
"Definition:Natural Logarithm"
] |
proofwiki-21342 | Chord Length for Regular Polygon | Let $P$ be a regular polygon of $n$ sides.
Let $P$ be inscribed into a circumcircle with radius $r$.
:400px
Let $CF$ be a chord of $P$, also a chord of the circumcircle.
$CF$ divides $P$ into two polygons containing $k$ and $n - k$ sides of $P$.
The length of the chord is $2 r \map \sin {\dfrac {k \pi} n}$.
{{improve|I... | Let $\theta = \angle BAC$ be an inscribed angle, subtending one of the sides of $P$, with its vertex coinciding in one of the other vertices of $P$.
Because $P$ is a regular $n$-gon, its sides are all the same length.
We have {{hypothesis}} that $P$ is inscribed into a circle.
Therefore, the central angles subtending a... | Let $P$ be a [[Definition:Regular Polygon|regular polygon]] of $n$ [[Definition:Side of Polygon|sides]].
Let $P$ be [[Definition:Polygon Inscribed in Circle|inscribed]] into a [[Definition:Circumcircle|circumcircle]] with [[Definition:Radius of Circle|radius]] $r$.
:[[File:Heptagon and chords.png|400px]]
Let $CF$ be... | Let $\theta = \angle BAC$ be an [[Definition:Angle Inscribed in Circle|inscribed angle]], [[Definition:Subtend|subtending]] one of the [[Definition:Side of Polygon|sides]] of $P$, with its [[Definition:Vertex of Angle|vertex]] coinciding in one of the other [[Definition:Vertex of Polygon|vertices]] of $P$.
Because $P$... | Chord Length for Regular Polygon | https://proofwiki.org/wiki/Chord_Length_for_Regular_Polygon | https://proofwiki.org/wiki/Chord_Length_for_Regular_Polygon | [
"Regular Polygons"
] | [
"Definition:Polygon/Regular",
"Definition:Polygon/Side",
"Definition:Inscribe/Polygon in Circle",
"Definition:Circumcircle",
"Definition:Circle/Radius",
"File:Heptagon and chords.png",
"Definition:Polygon/Chord",
"Definition:Chord",
"Definition:Circumcircle",
"Definition:Polygon",
"Definition:Po... | [
"Definition:Angle Inscribed in Circle",
"Definition:Subtend",
"Definition:Polygon/Side",
"Definition:Angle/Vertex",
"Definition:Polygon/Vertex",
"Definition:Polygon/Regular",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Inscribe/Polygon in Circle",
"Definition:Circle"... |
proofwiki-21343 | Complement of Bounded Set has Exactly One Unbounded Component | Let $n \in \N_{> 1}$
Let $A \subseteq \R^n$ be a bounded subspace of real Euclidean $n$-space.
Then, $\R^n \setminus A$ has exactly one unbounded component. | By definition of bounded, there exists some $\bsx_0 \in \R^n$ and $\epsilon > 0$ such that:
:$A \subseteq \map {B_\epsilon} {\bsx_0}$ | Let $n \in \N_{> 1}$
Let $A \subseteq \R^n$ be a [[Definition:Bounded Metric Space|bounded subspace]] of [[Definition:Real Euclidean Space|real Euclidean $n$-space]].
Then, $\R^n \setminus A$ has [[Definition:Exactly One|exactly one]] [[Definition:Unbounded Metric Space|unbounded]] [[Definition:Component (Topology)|... | By definition of [[Definition:Bounded Metric Space/Definition 3|bounded]], there exists some $\bsx_0 \in \R^n$ and $\epsilon > 0$ such that:
:$A \subseteq \map {B_\epsilon} {\bsx_0}$ | Complement of Bounded Set has Exactly One Unbounded Component | https://proofwiki.org/wiki/Complement_of_Bounded_Set_has_Exactly_One_Unbounded_Component | https://proofwiki.org/wiki/Complement_of_Bounded_Set_has_Exactly_One_Unbounded_Component | [] | [
"Definition:Bounded Metric Space",
"Definition:Euclidean Space/Real",
"Definition:Unique",
"Definition:Bounded Metric Space/Unbounded",
"Definition:Component (Topology)"
] | [
"Definition:Bounded Metric Space/Definition 3",
"Definition:Bounded Metric Space",
"Definition:Bounded Metric Space/Definition 3"
] |
proofwiki-21344 | Convex Cone is Convex Set | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $P \subseteq X$ be a convex cone in $X$.
Then $P$ is convex. | Let $x, y \in P$.
Let $t \in \closedint 0 1$ so that:
:$t \ge 0$ and $1 - t \ge 0$.
Since $P$ is a cone, we have:
:$t x \in P$ and $\paren {1 - t} y \in P$.
Since $P$ is a convex cone, we have:
:$t x + \paren {1 - t} y \in P$
So $P$ is convex.
{{qed}} | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $P \subseteq X$ be a [[Definition:Convex Cone|convex cone]] in $X$.
Then $P$ is [[Definition:Convex Set (Vector Space)|convex]]. | Let $x, y \in P$.
Let $t \in \closedint 0 1$ so that:
:$t \ge 0$ and $1 - t \ge 0$.
Since $P$ is a [[Definition:Cone (Vector Space)|cone]], we have:
:$t x \in P$ and $\paren {1 - t} y \in P$.
Since $P$ is a [[Definition:Convex Cone|convex cone]], we have:
:$t x + \paren {1 - t} y \in P$
So $P$ is [[Definition:Conv... | Convex Cone is Convex Set | https://proofwiki.org/wiki/Convex_Cone_is_Convex_Set | https://proofwiki.org/wiki/Convex_Cone_is_Convex_Set | [
"Convex Cones"
] | [
"Definition:Vector Space",
"Definition:Convex Cone",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:Cone (Vector Space)",
"Definition:Convex Cone",
"Definition:Convex Set (Vector Space)"
] |
proofwiki-21345 | Characterization of Preordered Vector Spaces | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $\succeq$ be a preordering on $X$.
Then $\struct {X, \succeq}$ is a preordered vector space {{iff}} there exists a convex cone $P \subseteq X$ such that $\succeq$ is the preordering on $X$ induced by $P$. | === Necessary Condition ===
Suppose that $\struct {X, \succeq}$ is a preordered vector space.
Let:
:$P = \set {x \in X : x \succeq 0}$
Let $\succeq^P$ be the preordering on $X$ induced by $P$.
We want to show that, for $x, y \in X$, we have $x \succeq y$ {{iff}} $x \succeq^P y$.
Let $x, y \in X$.
By the definition of t... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $\succeq$ be a [[Definition:Preordering|preordering]] on $X$.
Then $\struct {X, \succeq}$ is a [[Definition:Preordered Vector Space|preordered vector space]] {{iff}} there exists a [[Definition:Convex Cone|convex cone... | === Necessary Condition ===
Suppose that $\struct {X, \succeq}$ is a [[Definition:Preordered Vector Space|preordered vector space]].
Let:
:$P = \set {x \in X : x \succeq 0}$
Let $\succeq^P$ be the [[Definition:Preordering Induced by Convex Cone|preordering on $X$ induced by $P$]].
We want to show that, for $x, y \i... | Characterization of Preordered Vector Spaces | https://proofwiki.org/wiki/Characterization_of_Preordered_Vector_Spaces | https://proofwiki.org/wiki/Characterization_of_Preordered_Vector_Spaces | [
"Preordered Vector Spaces"
] | [
"Definition:Vector Space",
"Definition:Preordering",
"Definition:Preordered Vector Space",
"Definition:Convex Cone",
"Definition:Preordering Induced by Convex Cone"
] | [
"Definition:Preordered Vector Space",
"Definition:Preordering Induced by Convex Cone",
"Definition:Preordering Induced by Convex Cone",
"Definition:Preordered Vector Space",
"Definition:Preordering Induced by Convex Cone",
"Definition:Preordering Induced by Convex Cone",
"Definition:Preordered Vector Sp... |
proofwiki-21346 | Lagrange's Trigonometric Identities/Sine/Cosine Form | :$\ds \sum_{k \mathop = 0}^n \sin k \theta = \dfrac {\map \cos {\frac 1 2 \theta} - \map \cos {n \theta + \frac 1 2 \theta} } {2 \map \sin {\frac 1 2 \theta} }$ | {{begin-eqn}}
{{eqn | l = \map \sin {\alpha} \map \sin {\beta}
| r = \dfrac {\map \cos {\alpha - \beta} - \map \cos {\alpha + \beta} } 2
| c = Werner Formula for Sine by Sine
}}
{{eqn | l = 2 \map \sin {\beta} \map \sin {\alpha}
| r = \map \cos {\alpha - \beta} - \map \cos {\alpha + \beta}
| c =... | :$\ds \sum_{k \mathop = 0}^n \sin k \theta = \dfrac {\map \cos {\frac 1 2 \theta} - \map \cos {n \theta + \frac 1 2 \theta} } {2 \map \sin {\frac 1 2 \theta} }$ | {{begin-eqn}}
{{eqn | l = \map \sin {\alpha} \map \sin {\beta}
| r = \dfrac {\map \cos {\alpha - \beta} - \map \cos {\alpha + \beta} } 2
| c = [[Werner Formula for Sine by Sine]]
}}
{{eqn | l = 2 \map \sin {\beta} \map \sin {\alpha}
| r = \map \cos {\alpha - \beta} - \map \cos {\alpha + \beta}
|... | Lagrange's Trigonometric Identities/Sine/Cosine Form | https://proofwiki.org/wiki/Lagrange's_Trigonometric_Identities/Sine/Cosine_Form | https://proofwiki.org/wiki/Lagrange's_Trigonometric_Identities/Sine/Cosine_Form | [
"Cosine Form of Lagrange's Sine Identity",
"Lagrange's Sine Identity",
"Lagrange's Trigonometric Identities"
] | [] | [
"Werner Formulas/Sine by Sine",
"Definition:Telescoping Series",
"Cosine Function is Even"
] |
proofwiki-21347 | Product of nth Roots of Unity | Let $n \in \Z$ be an integer such that $n > 0$.
Let $z \in \C$ be a complex number such that $z^n = 1$.
Then:
:$U_n = \set {e^{2 i k \pi / n}: k \in \N_n}$
where $U_n$ is the set of $n$th roots of unity.
That is:
:$z \in \set {1, e^{2 i \pi / n}, e^{4 i \pi / n}, \ldots, e^{2 \paren {n - 1} i \pi / n} }$
Then the produ... | {{begin-eqn}}
{{eqn | l = \prod_{k \mathop = 0}^{n - 1} e^{2 i k \pi / n}
| r = e^{2 i \paren {0} \pi / n} e^{2 i \paren {1} \pi / n} e^{2 i \paren {2} \pi / n} \dotsm e^{2 i \paren {n - 1} \pi / n}
| c =
}}
{{eqn | r = e^{\paren {2 i \pi / n} \paren {0 + 1 + \dotsm + \paren {n - 1} } }
| c = Product... | Let $n \in \Z$ be an [[Definition:Integer|integer]] such that $n > 0$.
Let $z \in \C$ be a [[Definition:Complex Number|complex number]] such that $z^n = 1$.
Then:
:$U_n = \set {e^{2 i k \pi / n}: k \in \N_n}$
where $U_n$ is the [[Definition:Complex Roots of Unity|set of $n$th roots of unity]].
That is:
:$z \in \set ... | {{begin-eqn}}
{{eqn | l = \prod_{k \mathop = 0}^{n - 1} e^{2 i k \pi / n}
| r = e^{2 i \paren {0} \pi / n} e^{2 i \paren {1} \pi / n} e^{2 i \paren {2} \pi / n} \dotsm e^{2 i \paren {n - 1} \pi / n}
| c =
}}
{{eqn | r = e^{\paren {2 i \pi / n} \paren {0 + 1 + \dotsm + \paren {n - 1} } }
| c = [[Produ... | Product of nth Roots of Unity | https://proofwiki.org/wiki/Product_of_nth_Roots_of_Unity | https://proofwiki.org/wiki/Product_of_nth_Roots_of_Unity | [
"Complex Roots of Unity"
] | [
"Definition:Integer",
"Definition:Complex Number",
"Definition:Root of Unity/Complex"
] | [
"Exponent Combination Laws/Product of Powers",
"Closed Form for Triangular Numbers",
"Euler's Formula/Examples/e^i pi",
"Category:Complex Roots of Unity"
] |
proofwiki-21348 | Equiangular Right Triangles are Similar | Equiangular right triangles are similar. | This is an instance of the theorem Equiangular Triangles are Similar.
{{qed}} | [[Definition:Equiangular Geometric Figures|Equiangular]] [[Definition:Right Triangle|right triangles]] are [[Definition:Similar Figures|similar]]. | This is an instance of the theorem [[Equiangular Triangles are Similar]].
{{qed}} | Equiangular Right Triangles are Similar/Proof 1 | https://proofwiki.org/wiki/Equiangular_Right_Triangles_are_Similar | https://proofwiki.org/wiki/Equiangular_Right_Triangles_are_Similar/Proof_1 | [
"Equiangular Right Triangles are Similar",
"Right Triangles",
"Similar Triangles"
] | [
"Definition:Equiangular Geometric Figures",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Similar Figures"
] | [
"Equiangular Triangles are Similar"
] |
proofwiki-21349 | Equiangular Right Triangles are Similar | Equiangular right triangles are similar. | Let $ABC$ be an arbitrary right triangle with $\angle ABC$ a right angle.
Construct a straight line from $A$ parallel to $BC$.
{{:Euclid:Proposition/I/31}}
Construct a second straight line from $C$ parallel to $AB$, meeting the first straight line at $D$.
By Quadrilateral is Parallelogram iff Both Pairs of Opposite Sid... | [[Definition:Equiangular Geometric Figures|Equiangular]] [[Definition:Right Triangle|right triangles]] are [[Definition:Similar Figures|similar]]. | Let $ABC$ be an arbitrary [[Definition:Right Triangle|right triangle]] with $\angle ABC$ a [[Definition:Right Angle|right angle]].
Construct a [[Definition:Straight Line|straight line]] from $A$ [[Definition:Parallel Lines|parallel]] to $BC$.
{{:Euclid:Proposition/I/31}}
Construct a second [[Definition:Straight Line... | Equiangular Right Triangles are Similar/Proof 2 | https://proofwiki.org/wiki/Equiangular_Right_Triangles_are_Similar | https://proofwiki.org/wiki/Equiangular_Right_Triangles_are_Similar/Proof_2 | [
"Equiangular Right Triangles are Similar",
"Right Triangles",
"Similar Triangles"
] | [
"Definition:Equiangular Geometric Figures",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Similar Figures"
] | [
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Right Angle",
"Definition:Line/Straight Line",
"Definition:Parallel (Geometry)/Lines",
"Definition:Line/Straight Line",
"Definition:Parallel (Geometry)/Lines",
"Definition:Line/Straight Line",
"Quadrilateral is Parallelogram iff Both Pairs of ... |
proofwiki-21350 | Component of Complement of Jordan Curve has Curve as Boundary | Let $\phi : \closedint 0 1 \to \R^2$ be a Jordan curve.
Let $J = \phi \closedint 0 1$ be the image of $\phi$.
Suppose that $\R^2 \setminus J$ has at least two distinct components.
Then, for any component $U$ of $\R^2 \setminus J$:
:$\partial U = J$
where $\partial U$ denotes the boundary of $U$. | By:
* Closed Real Interval is Compact Space
* Continuous Image of Compact Space is Compact
it follows that $J$ is compact.
Therefore, by Compact Subspace of Hausdorff Space is Closed:
:$J$ is closed in $\R^2$
so by definition of closed:
:$\R^2 \setminus J$ is open in $\R^2$
By definition of locally connected:
:Each com... | Let $\phi : \closedint 0 1 \to \R^2$ be a [[Definition:Jordan Curve|Jordan curve]].
Let $J = \phi \closedint 0 1$ be the [[Definition:Image of Mapping|image]] of $\phi$.
Suppose that $\R^2 \setminus J$ has at least two distinct [[Definition:Component (Topology)|components]].
Then, for any [[Definition:Component (To... | By:
* [[Closed Real Interval is Compact Space]]
* [[Continuous Image of Compact Space is Compact]]
it follows that $J$ is [[Definition:Compact Topological Space|compact]].
Therefore, by [[Compact Subspace of Hausdorff Space is Closed]]:
:$J$ is [[Definition:Closed Set|closed]] in $\R^2$
so by definition of [[Definitio... | Component of Complement of Jordan Curve has Curve as Boundary | https://proofwiki.org/wiki/Component_of_Complement_of_Jordan_Curve_has_Curve_as_Boundary | https://proofwiki.org/wiki/Component_of_Complement_of_Jordan_Curve_has_Curve_as_Boundary | [
"Jordan Curves"
] | [
"Definition:Jordan Curve",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Component (Topology)",
"Definition:Component (Topology)",
"Definition:Boundary (Topology)"
] | [
"Closed Real Interval is Compact Space",
"Continuous Image of Compact Space is Compact",
"Definition:Compact Topological Space",
"Compact Subspace of Hausdorff Space is Closed",
"Definition:Closed Set",
"Definition:Closed Set (Topology)/Definition 1",
"Definition:Open Set",
"Definition:Locally Connect... |
proofwiki-21351 | Lebesgue's Number Lemma/Compact Space | Let $M = \struct {X, d}$ be a metric space.
Let $M$ be compact.
Then there exists a Lebesgue number for every open cover of $M$. | Let $\UU$ be an open cover of $M$.
By definition of compact, there exists a finite subcover $\set {A_i}_{1 \le i \le n} \subseteq \UU$.
First, suppose some $A_i = X$.
Then, let $\epsilon = 1$.
For any $x \in X$:
:$\map {B_\epsilon} x \subseteq X = A_i \in \UU$
Therefore, $\epsilon$ is a Lebesgue number for $\UU$.
Now, ... | Let $M = \struct {X, d}$ be a [[Definition:Metric Space|metric space]].
Let $M$ be [[Definition:Compact Topological Space|compact]].
Then there exists a [[Definition:Lebesgue Number|Lebesgue number]] for every [[Definition:Open Cover|open cover]] of $M$. | Let $\UU$ be an [[Definition:Open Cover|open cover]] of $M$.
By definition of [[Definition:Compact Topological Space/Definition 1|compact]], there exists a [[Definition:Finite Subcover|finite subcover]] $\set {A_i}_{1 \le i \le n} \subseteq \UU$.
First, suppose some $A_i = X$.
Then, let $\epsilon = 1$.
For any $x ... | Lebesgue's Number Lemma/Compact Space | https://proofwiki.org/wiki/Lebesgue's_Number_Lemma/Compact_Space | https://proofwiki.org/wiki/Lebesgue's_Number_Lemma/Compact_Space | [
"Lebesgue's Number Lemma",
"Compact Topological Spaces"
] | [
"Definition:Metric Space",
"Definition:Compact Topological Space",
"Definition:Lebesgue Number",
"Definition:Open Cover"
] | [
"Definition:Open Cover",
"Definition:Compact Topological Space/Definition 1",
"Definition:Subcover/Finite",
"Definition:Lebesgue Number",
"Definition:Non-Empty Set",
"Distance in Pseudometric is Non-Negative",
"Definition:Bounded Below Set/Real Numbers",
"Greatest Lower Bound Property",
"Definition:... |
proofwiki-21352 | Catenary is Symmetric about Y-Axis | Consider a '''catenary''' $\CC$.
Let a cartesian plane be arranged so that the $y$-axis passes through the lowest point of the catenary.
$\CC$ exhibits reflectional symmetry in that $y$-axis.
</onlyinclude> | From Cartesian Equation of Catenary: Formulation $2$, we have the equation of $\CC$:
:$y = \dfrac a 2 \paren {e^{x / a} + e^{-x / a} } = a \cosh \dfrac x a$
The result follows directly from Hyperbolic Cosine Function is Even.
{{qed}} | Consider a '''[[Definition:Catenary|catenary]]''' $\CC$.
Let a [[Definition:Cartesian Plane|cartesian plane]] be arranged so that the [[Definition:Y-Axis|$y$-axis]] passes through the lowest point of the [[Definition:Catenary|catenary]].
$\CC$ exhibits [[Definition:Reflectional Symmetry|reflectional symmetry]] in th... | From [[Equation of Catenary/Cartesian/Formulation 2|Cartesian Equation of Catenary: Formulation $2$]], we have the equation of $\CC$:
:$y = \dfrac a 2 \paren {e^{x / a} + e^{-x / a} } = a \cosh \dfrac x a$
The result follows directly from [[Hyperbolic Cosine Function is Even]].
{{qed}} | Catenary is Symmetric about Y-Axis | https://proofwiki.org/wiki/Catenary_is_Symmetric_about_Y-Axis | https://proofwiki.org/wiki/Catenary_is_Symmetric_about_Y-Axis | [
"Catenary"
] | [
"Definition:Catenary",
"Definition:Cartesian Plane",
"Definition:Axis/Y-Axis",
"Definition:Catenary",
"Definition:Reflectional Symmetry",
"Definition:Axis/Y-Axis"
] | [
"Equation of Catenary/Cartesian/Formulation 2",
"Hyperbolic Cosine Function is Even"
] |
proofwiki-21353 | Thales' Theorem/Converse | 400px
Let $O$ be a circle.
Let $AOB$ be a diameter of $O$.
Then $\angle APB$ is a right angle {{iff}} $P$ lies on the circle $O$. | === Necessary Condition ===
Draw $OC \parallel PB$.
By Parallelism implies Equal Corresponding Angles:
:$\angle ACO = \angle APB$ and both are right angles.
$\triangle ACO$ and $\triangle APB$ share $\angle OAC$.
$\triangle ACO$ and $\triangle APB$ are both right triangles.
{{begin-eqn}}
{{eqn | l = \triangle ACO
... | [[File:Thales' theorem converse.png|400px]]
Let $O$ be a [[Definition:Circle|circle]].
Let $AOB$ be a [[Definition:Diameter of Circle|diameter]] of $O$.
Then $\angle APB$ is a [[Definition:Right Angle|right angle]] {{iff}} $P$ lies on the [[Definition:Circle|circle]] $O$. | === Necessary Condition ===
Draw $OC \parallel PB$.
By [[Parallelism implies Equal Corresponding Angles]]:
:$\angle ACO = \angle APB$ and both are [[Definition:Right Angle|right angles]].
$\triangle ACO$ and $\triangle APB$ share $\angle OAC$.
$\triangle ACO$ and $\triangle APB$ are both [[Definition:Right Triangle... | Thales' Theorem/Converse | https://proofwiki.org/wiki/Thales'_Theorem/Converse | https://proofwiki.org/wiki/Thales'_Theorem/Converse | [
"Thales' Theorem",
"Circles",
"Right Angles"
] | [
"File:Thales' theorem converse.png",
"Definition:Circle",
"Definition:Circle/Diameter",
"Definition:Right Angle",
"Definition:Circle"
] | [
"Parallelism implies Equal Corresponding Angles",
"Definition:Right Angle",
"Definition:Triangle (Geometry)/Right-Angled",
"Triangles with Two Equal Angles are Similar",
"Definition:Right Angle",
"Triangle Side-Angle-Side Congruence",
"Definition:Circle/Radius",
"Definition:Circle",
"Definition:Circ... |
proofwiki-21354 | Cauchy's Convergence Criterion/General | Let $\sequence {x_n}$ be a sequence in $\R$ or $\C$.
Then $\sequence {x_n}$ is a Cauchy sequence {{iff}}:
:$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall r \in \N: r \ge N: \forall k > 0: \size {\ds \sum_{i \mathop = 1}^k x_{r + i} } < \epsilon$ | {{ProofWanted}}
{{Namedfor|Augustin Louis Cauchy}} | Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $\R$ or $\C$.
Then $\sequence {x_n}$ is a [[Definition:Cauchy Sequence|Cauchy sequence]] {{iff}}:
:$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall r \in \N: r \ge N: \forall k > 0: \size {\ds \sum_{i \mathop = 1}^k x_{r + i} } < \epsilon$ | {{ProofWanted}}
{{Namedfor|Augustin Louis Cauchy}} | Cauchy's Convergence Criterion/General | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/General | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/General | [
"Cauchy's Convergence Criterion"
] | [
"Definition:Sequence",
"Definition:Cauchy Sequence"
] | [] |
proofwiki-21355 | Nth Root Test/Weak Form | Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a series of (strictly) positive real numbers $\R$.
Let the sequence $\sequence {a_n}$ be such that the limit $\ds \lim_{n \mathop \to \infty} \size {a_n}^{1/n} = l$.
Then:
:If $l > 1$, the series $\ds \sum_{n \mathop = 1}^\infty a_n$ diverges.
:If $l < 1$, the series $\ds \s... | {{ProofWanted|in due course}} | Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a [[Definition:Series|series]] of [[Definition:Strictly Positive Real Number|(strictly) positive real numbers]] $\R$.
Let the [[Definition:Sequence|sequence]] $\sequence {a_n}$ be such that the [[Definition:Limit of Real Sequence|limit]] $\ds \lim_{n \mathop \to \infty} \si... | {{ProofWanted|in due course}} | Nth Root Test/Weak Form | https://proofwiki.org/wiki/Nth_Root_Test/Weak_Form | https://proofwiki.org/wiki/Nth_Root_Test/Weak_Form | [
"Mistakes/The Penguin Dictionary of Mathematics"
] | [
"Definition:Series",
"Definition:Strictly Positive/Real Number",
"Definition:Sequence",
"Definition:Limit of Sequence/Real Numbers",
"Definition:Divergent Series",
"Definition:Absolutely Convergent Series"
] | [] |
proofwiki-21356 | Complex Numbers form Preordered Vector Space | Consider the complex numbers $\C$ as a vector space over itself.
Define the relation $\ge^\C$ by:
:$z \ge^\C w$
{{iff}}:
:$z - w \in \hointr 0 \infty$
for each $z, w \in \C$.
Then $\struct {\C, \ge^\C}$ is a preordered vector space. | From Characterization of Preordered Vector Spaces, it is enough to show that $\hointr 0 \infty$ is a convex cone.
Let $x \in \hointr 0 \infty$ and $\alpha \in \R_{\ge 0}$.
Then $\alpha x \ge 0$, so $\alpha x \in \hointr 0 \infty$.
So $\hointr 0 \infty$ is a cone.
Now let $x, y \in \hointr 0 \infty$.
Then we have $x +... | Consider the [[Definition:Complex Number|complex numbers]] $\C$ as a [[Definition:Vector Space|vector space]] over itself.
Define the [[Definition:Relation|relation]] $\ge^\C$ by:
:$z \ge^\C w$
{{iff}}:
:$z - w \in \hointr 0 \infty$
for each $z, w \in \C$.
Then $\struct {\C, \ge^\C}$ is a [[Definition:Preordered Vec... | From [[Characterization of Preordered Vector Spaces]], it is enough to show that $\hointr 0 \infty$ is a [[Definition:Convex Cone|convex cone]].
Let $x \in \hointr 0 \infty$ and $\alpha \in \R_{\ge 0}$.
Then $\alpha x \ge 0$, so $\alpha x \in \hointr 0 \infty$.
So $\hointr 0 \infty$ is a [[Definition:Cone (Vector S... | Complex Numbers form Preordered Vector Space | https://proofwiki.org/wiki/Complex_Numbers_form_Preordered_Vector_Space | https://proofwiki.org/wiki/Complex_Numbers_form_Preordered_Vector_Space | [
"Preordered Vector Spaces"
] | [
"Definition:Complex Number",
"Definition:Vector Space",
"Definition:Relation",
"Definition:Preordered Vector Space"
] | [
"Characterization of Preordered Vector Spaces",
"Definition:Convex Cone",
"Definition:Cone (Vector Space)",
"Definition:Convex Cone",
"Category:Preordered Vector Spaces"
] |
proofwiki-21357 | Element of *-Algebra Uniquely Decomposes into Hermitian Elements | Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$.
Let $a \in A$.
Then there exists unique Hermitian elements $b, c \in A$ such that:
:$a = b + i c$
In particular, $b = \map \Re a$ and $c = \map \Im a$ where $b$ and $c$ are the real and imaginary parts of $a$ respectively. | === Proof of Existence ===
{{finish|fill in $\text C^\ast x$ with template}}
Let:
:$b = \map \Re a = \dfrac 1 2 \paren {a + a^\ast}$
and:
:$c = \map \Im a = \dfrac 1 {2 i} \paren {a - a^\ast}$
Then we have using $(\text C^\ast 2)$ and $(\text C^\ast 1)$:
:$b^\ast = \dfrac 1 2 \paren {a^\ast + a^{\ast \ast} } = \dfrac... | Let $\struct {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $a \in A$.
Then there exists unique [[Definition:Hermitian Element of *-Algebra|Hermitian elements]] $b, c \in A$ such that:
:$a = b + i c$
In particular, $b = \map \Re a$ and $c = \map \Im a$ where $b$ and $c$ are the [[Definition... | === Proof of Existence ===
{{finish|fill in $\text C^\ast x$ with template}}
Let:
:$b = \map \Re a = \dfrac 1 2 \paren {a + a^\ast}$
and:
:$c = \map \Im a = \dfrac 1 {2 i} \paren {a - a^\ast}$
Then we have using $(\text C^\ast 2)$ and $(\text C^\ast 1)$:
:$b^\ast = \dfrac 1 2 \paren {a^\ast + a^{\ast \ast} } = \dfra... | Element of *-Algebra Uniquely Decomposes into Hermitian Elements | https://proofwiki.org/wiki/Element_of_*-Algebra_Uniquely_Decomposes_into_Hermitian_Elements | https://proofwiki.org/wiki/Element_of_*-Algebra_Uniquely_Decomposes_into_Hermitian_Elements | [
"Hermitian Elements of *-Algebras"
] | [
"Definition:*-Algebra",
"Definition:Hermitian Element of *-Algebra",
"Definition:Real Part of Element of *-Algebra",
"Definition:Imaginary Part of Element of *-Algebra"
] | [] |
proofwiki-21358 | Product of Element in *-Star Algebra with its Star is Hermitian | Let $\struct {A, \ast}$ be a $\ast$-algebra.
Let $a \in A$.
Then $a^\ast a$ and $a a^\ast$ are Hermitian. | We have:
{{begin-eqn}}
{{eqn | l = \paren {a^\ast a}^\ast
| r = a^\ast \paren {a^\ast}^\ast
| c = $(\text C^\ast 3)$
}}
{{eqn | r = a^\ast a
| c = $(\text C^\ast 1)$
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = \paren {a a^\ast}^\ast
| r = \paren {a^\ast}^\ast a^\ast
| c = $(\text C^\ast 3)$
}}
{{eqn | r = ... | Let $\struct {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]].
Let $a \in A$.
Then $a^\ast a$ and $a a^\ast$ are [[Definition:Hermitian Element of *-Algebra|Hermitian]]. | We have:
{{begin-eqn}}
{{eqn | l = \paren {a^\ast a}^\ast
| r = a^\ast \paren {a^\ast}^\ast
| c = $(\text C^\ast 3)$
}}
{{eqn | r = a^\ast a
| c = $(\text C^\ast 1)$
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = \paren {a a^\ast}^\ast
| r = \paren {a^\ast}^\ast a^\ast
| c = $(\text C^\ast 3)$
}}
{{eqn | r = ... | Product of Element in *-Star Algebra with its Star is Hermitian | https://proofwiki.org/wiki/Product_of_Element_in_*-Star_Algebra_with_its_Star_is_Hermitian | https://proofwiki.org/wiki/Product_of_Element_in_*-Star_Algebra_with_its_Star_is_Hermitian | [
"Hermitian Elements of *-Algebras"
] | [
"Definition:*-Algebra",
"Definition:Hermitian Element of *-Algebra"
] | [
"Definition:Hermitian Element of *-Algebra"
] |
proofwiki-21359 | Median of Cauchy Distribution | Let $X$ be a continuous random variable with a '''Cauchy distribution''':
:$\map {f_X} x = \dfrac 1 {\pi \lambda \paren {1 + \paren {\frac {x - \gamma} \lambda }^2} }$
for:
:$\lambda \in \R_{>0}$
:$\gamma \in \R$
The median of $X$ is $\gamma$. | From the definition of the Cauchy distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac 1 {\pi \lambda \paren {1 + \paren {\frac {x - \gamma} \lambda }^2} }$
Note that $f_X$ is non-zero, sufficient to ensure a unique median.
By the definition of a median, to prove that $\gamma$ is the median of $X... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with a '''[[Definition:Cauchy Distribution|Cauchy distribution]]''':
:$\map {f_X} x = \dfrac 1 {\pi \lambda \paren {1 + \paren {\frac {x - \gamma} \lambda }^2} }$
for:
:$\lambda \in \R_{>0}$
:$\gamma \in \R$
The [[Definition:Median of ... | From the definition of the [[Definition:Cauchy Distribution|Cauchy distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac 1 {\pi \lambda \paren {1 + \paren {\frac {x - \gamma} \lambda }^2} }$
Note that $f_X$ is non-zero, sufficient to ensure a unique ... | Median of Cauchy Distribution | https://proofwiki.org/wiki/Median_of_Cauchy_Distribution | https://proofwiki.org/wiki/Median_of_Cauchy_Distribution | [
"Cauchy Distribution",
"Medians",
"Arctangent Function"
] | [
"Definition:Random Variable/Continuous",
"Definition:Cauchy Distribution",
"Definition:Median of Continuous Random Variable"
] | [
"Definition:Cauchy Distribution",
"Definition:Probability Density Function",
"Definition:Median of Continuous Random Variable",
"Definition:Median of Continuous Random Variable",
"Definition:Median of Continuous Random Variable",
"Integration by Substitution",
"Primitive of Reciprocal of 1 plus x square... |
proofwiki-21360 | Cauchy Distribution is Symmetric about Median | Let $X$ be a continuous random variable with a '''Cauchy distribution''':
:$\map {f_X} x = \dfrac 1 {\pi \lambda \paren {1 + \paren {\frac {x - \gamma} \lambda }^2} }$
for:
:$\lambda \in \R_{>0}$
:$\gamma \in \R$
$X$ is (reflectionally) symmetric about the vertical line through the median $\gamma$. | Recall from Median of Cauchy Distribution that $\gamma$ is indeed the median of $X$.
{{begin-eqn}}
{{eqn | l = \map {f_X} {2 \gamma - x}
| r = \frac 1 {\pi \lambda \paren {1 + \paren {\frac {(2 \gamma - x) - \gamma} \lambda}^2} }
}}
{{eqn | r = \frac 1 {\pi \lambda \paren {1 + \paren {\frac {\gamma - x} \lambda}^... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with a '''[[Definition:Cauchy Distribution|Cauchy distribution]]''':
:$\map {f_X} x = \dfrac 1 {\pi \lambda \paren {1 + \paren {\frac {x - \gamma} \lambda }^2} }$
for:
:$\lambda \in \R_{>0}$
:$\gamma \in \R$
$X$ is [[Definition:Reflect... | Recall from [[Median of Cauchy Distribution]] that $\gamma$ is indeed the [[Definition:Median of Continuous Random Variable|median]] of $X$.
{{begin-eqn}}
{{eqn | l = \map {f_X} {2 \gamma - x}
| r = \frac 1 {\pi \lambda \paren {1 + \paren {\frac {(2 \gamma - x) - \gamma} \lambda}^2} }
}}
{{eqn | r = \frac 1 {\pi... | Cauchy Distribution is Symmetric about Median | https://proofwiki.org/wiki/Cauchy_Distribution_is_Symmetric_about_Median | https://proofwiki.org/wiki/Cauchy_Distribution_is_Symmetric_about_Median | [
"Cauchy Distribution",
"Medians"
] | [
"Definition:Random Variable/Continuous",
"Definition:Cauchy Distribution",
"Definition:Reflectional Symmetry",
"Definition:Vertical Line",
"Definition:Median of Continuous Random Variable"
] | [
"Median of Cauchy Distribution",
"Definition:Median of Continuous Random Variable"
] |
proofwiki-21361 | Duplicated Triangle Forms a Kite or a Parallelogram | Let a triangle be copied either by rotation or reflection.
Then if you mung them together by corresponding sides what you get is either a parallelogram or a kite.
{{mistake|you might get a dart}} | {{tidy|including about using definitions with / in them}}
Let $\triangle ABC$ be any triangle. We can form a congruent triangle by reflection or translation. | Let a [[Definition:Triangle (Geometry)|triangle]] be copied either by [[Definition:Rotation (Geometry)|rotation]] or [[Definition:Reflection|reflection]].
Then if you mung them together by corresponding sides what you get is either a [[Definition:Parallelogram|parallelogram]] or a [[Definition:Kite|kite]].
{{mistake|... | {{tidy|including about using definitions with / in them}}
Let $\triangle ABC$ be any [[Definition:Triangle (Geometry)|triangle]]. We can form a [[Definition:Congruence (Geometry)|congruent]] [[Definition:Triangle (Geometry)|triangle]] by [[Definition:Reflection|reflection]] or [[Definition:Translation in Euclidean Sp... | Duplicated Triangle Forms a Kite or a Parallelogram | https://proofwiki.org/wiki/Duplicated_Triangle_Forms_a_Kite_or_a_Parallelogram | https://proofwiki.org/wiki/Duplicated_Triangle_Forms_a_Kite_or_a_Parallelogram | [
"Triangles",
"Quadrilaterals",
"Parallelograms",
"Kites"
] | [
"Definition:Triangle (Geometry)",
"Definition:Rotation (Geometry)",
"Definition:Reflection (Geometry)",
"Definition:Quadrilateral/Parallelogram",
"Definition:Quadrilateral/Kite"
] | [
"Definition:Triangle (Geometry)",
"Definition:Congruence (Geometry)",
"Definition:Triangle (Geometry)",
"Definition:Reflection (Geometry)",
"Definition:Translation Mapping/Euclidean Space",
"Definition:Reflection (Geometry)",
"Definition:Reflection (Geometry)",
"Definition:Triangle (Geometry)",
"Def... |
proofwiki-21362 | Quadrilateral is Cyclic iff Opposite Angles sum to Two Right Angles | 500px
Given $\Box ABCD$ with $A, B$ and $D$ on a circle.
Let $\angle ABC$ and $\angle ADC$ add to two right angles.
Then $C$ lies on the circle, and $\Box ABCD$ is a cyclic quadrilateral. | === Sufficient Condition ===
By Sum of Internal Angles of Polygon, since $\Box ABCD$ is a quadrilateral:
:$\angle ABC + \angle BCD + \angle BAD + \angle ADC = 360$
Therefore $\angle BAD$ and $\angle BCD$ are also supplementary angles.
Suppose $C$ does not lie on the circle, but lies internal.
Extend $DC$ to meet the ... | [[File:Converse to Cyclic Quadrilateral Proof.png|500px]]
Given $\Box ABCD$ with $A, B$ and $D$ on a [[Definition:Circle|circle]].
Let $\angle ABC$ and $\angle ADC$ add to two [[Definition:Right Angle|right angles]].
Then $C$ lies on the [[Definition:Circle|circle]], and $\Box ABCD$ is a [[Definition:Cyclic Quadrila... | === Sufficient Condition ===
By [[Sum of Internal Angles of Polygon]], since $\Box ABCD$ is a [[Definition:Quadrilateral|quadrilateral]]:
:$\angle ABC + \angle BCD + \angle BAD + \angle ADC = 360$
Therefore $\angle BAD$ and $\angle BCD$ are also [[Definition:Supplementary Angles|supplementary angles]].
Suppose $C$ ... | Quadrilateral is Cyclic iff Opposite Angles sum to Two Right Angles | https://proofwiki.org/wiki/Quadrilateral_is_Cyclic_iff_Opposite_Angles_sum_to_Two_Right_Angles | https://proofwiki.org/wiki/Quadrilateral_is_Cyclic_iff_Opposite_Angles_sum_to_Two_Right_Angles | [
"Cyclic Quadrilaterals"
] | [
"File:Converse to Cyclic Quadrilateral Proof.png",
"Definition:Circle",
"Definition:Right Angle",
"Definition:Circle",
"Definition:Cyclic Quadrilateral"
] | [
"Sum of Internal Angles of Polygon",
"Definition:Quadrilateral",
"Definition:Supplementary Angles",
"Definition:Circle",
"Definition:Circle",
"Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles",
"Definition:Supplementary Angles",
"External Angle of Triangle equals Sum of other Internal A... |
proofwiki-21363 | Quotient of Normal Distributions has Cauchy Distribution/Corollary | Let $X$ and $Y$ be independent continuous random variables each with a '''normal distribution''' with:
:zero expectation
:the same variance $\sigma$:
{{begin-eqn}}
{{eqn | l = X
| o = \sim
| r = \Gaussian 0 {\sigma^2}
}}
{{eqn | l = Y
| o = \sim
| r = \Gaussian 0 {\sigma^2}
}}
{{end-eqn}}
Let $U... | From Quotient of Normal Distributions has Cauchy Distribution:
:$U \sim \Cauchy 0 \lambda$
where:
:$\lambda = \dfrac {\sigma_x} {\sigma_y}$
and such that:
:$\sigma_x = \sigma_y = \sigma$
Hence $\lambda = 1$ and the result follows.
Similarly we have:
:$\dfrac 1 U = \dfrac Y X$
and again the result follows.
{{qed}} | Let $X$ and $Y$ be [[Definition:Independent Random Variables|independent]] [[Definition:Continuous Random Variable|continuous random variables]] each with a '''[[Definition:Normal Distribution|normal distribution]]''' with:
:[[Definition:Zero (Number)|zero]] [[Definition:Expectation|expectation]]
:the same [[Definition... | From [[Quotient of Normal Distributions has Cauchy Distribution]]:
:$U \sim \Cauchy 0 \lambda$
where:
:$\lambda = \dfrac {\sigma_x} {\sigma_y}$
and such that:
:$\sigma_x = \sigma_y = \sigma$
Hence $\lambda = 1$ and the result follows.
Similarly we have:
:$\dfrac 1 U = \dfrac Y X$
and again the result follows.
{{qed... | Quotient of Normal Distributions has Cauchy Distribution/Corollary | https://proofwiki.org/wiki/Quotient_of_Normal_Distributions_has_Cauchy_Distribution/Corollary | https://proofwiki.org/wiki/Quotient_of_Normal_Distributions_has_Cauchy_Distribution/Corollary | [
"Quotient of Normal Distributions has Cauchy Distribution"
] | [
"Definition:Independent Random Variables",
"Definition:Random Variable/Continuous",
"Definition:Normal Distribution",
"Definition:Zero (Number)",
"Definition:Expectation",
"Definition:Variance",
"Definition:Random Variable/Continuous",
"Definition:Cauchy Distribution"
] | [
"Quotient of Normal Distributions has Cauchy Distribution"
] |
proofwiki-21364 | Tangent Points of Incircle in Terms of Semiperimeter | Let $\triangle ABC$ be a triangle whose sides are $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Let $s$ denote the semiperimeter of $\triangle ABC$.
Then the distance from $A$ to a point tangent to the incircle is equal to $s - a$. | 400px
Let $D$, $E$ and $F$ be the points where the incircle is tangent to the sides $AC$, $AB$ and $BC$ respectively.
Then:
{{begin-eqn}}
{{eqn | l = AD
| r = AE
| c = Tangents to Circle from Point are of Equal Length
}}
{{eqn | l = CD
| r = CF
| c = Tangents to Circle from Point are of Equal Le... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose [[Definition:Side of Polygon|sides]] are $a$, $b$ and $c$ [[Definition:Opposite (in Triangle)|opposite]] [[Definition:Vertex of Polygon|vertices]] $A$, $B$ and $C$ respectively.
Let $s$ denote the [[Definition:Semiperimeter|semiperimeter]] of $... | [[File:IncenterLengthProof.png|400px]]
Let $D$, $E$ and $F$ be the [[Definition:Point|points]] where the [[Definition:Incircle of Triangle|incircle]] is [[Definition:Tangent to Circle|tangent]] to the [[Definition:Side of Polygon|sides]] $AC$, $AB$ and $BC$ respectively.
Then:
{{begin-eqn}}
{{eqn | l = AD
| r... | Tangent Points of Incircle in Terms of Semiperimeter | https://proofwiki.org/wiki/Tangent_Points_of_Incircle_in_Terms_of_Semiperimeter | https://proofwiki.org/wiki/Tangent_Points_of_Incircle_in_Terms_of_Semiperimeter | [
"Incircles of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Vertex",
"Definition:Semiperimeter",
"Definition:Distance between Points",
"Definition:Point",
"Definition:Tangent Line/Circle",
"Definition:Incircle of Triangle"
] | [
"File:IncenterLengthProof.png",
"Definition:Point",
"Definition:Incircle of Triangle",
"Definition:Tangent Line/Circle",
"Definition:Polygon/Side",
"Tangents to Circle from Point are of Equal Length",
"Tangents to Circle from Point are of Equal Length",
"Tangents to Circle from Point are of Equal Leng... |
proofwiki-21365 | Length of Tangent to Excircle | The excircle on one side of a triangle is tangent to that side and divides it into the same two lengths as the tangent to the incircle, but in opposite order.
{{rename|because the existing name does not match what is being proved}} | <onlyinclude>
:700px
Let the sides opposite vertex $A, B, C$ be $a,b,c$.
Let the semiperimeter of $\triangle ABC$ be $s$.
Construct the incircle of $\triangle ABC$ with center $M$.
Let $\triangle ABC$ be tangent to the incircle at $H$.
Construct the excircle of $\triangle ABC$ on side $c$ with center $N$.
Let $\triangl... | The [[Definition:Excircle of Triangle|excircle]] on one side of a [[Definition:Triangle (Geometry)|triangle]] is [[Definition:Tangent to Circle|tangent]] to that side and divides it into the same two lengths as the [[Definition:Tangent to Circle|tangent]] to the [[Definition:Incircle of Triangle|incircle]], but in oppo... | <onlyinclude>
:[[File:Heron5.png|700px]]
Let the [[Definition:Side of Polygon|sides]] opposite [[Definition:Vertex|vertex]] $A, B, C$ be $a,b,c$.
Let the [[Definition:Semiperimeter|semiperimeter]] of $\triangle ABC$ be $s$.
Construct the [[Definition:Incircle of Triangle|incircle]] of $\triangle ABC$ with [[Definiti... | Length of Tangent to Excircle | https://proofwiki.org/wiki/Length_of_Tangent_to_Excircle | https://proofwiki.org/wiki/Length_of_Tangent_to_Excircle | [
"Incircles of Triangles",
"Excircles of Triangles"
] | [
"Definition:Excircle of Triangle",
"Definition:Triangle (Geometry)",
"Definition:Tangent Line/Circle",
"Definition:Tangent Line/Circle",
"Definition:Incircle of Triangle"
] | [
"File:Heron5.png",
"Definition:Polygon/Side",
"Definition:Vertex",
"Definition:Semiperimeter",
"Definition:Incircle of Triangle",
"Definition:Circle/Center",
"Definition:Tangent",
"Definition:Incircle of Triangle",
"Definition:Excircle of Triangle",
"Definition:Circle/Center",
"Definition:Tangen... |
proofwiki-21366 | Student's t-Distribution with One Degree of Freedom is Standard Cauchy Distribution | The Student's $t$-distribution with one degree of freedom is a special case of a standard Cauchy distribution. | Let $X$ be a continuous random variables with a Student's $t$-distribution with one degree of freedom.
Then $X$ has probability density function:
:$\map {f_X} x = \dfrac {\map \Gamma {\frac {k + 1} 2} } {\sqrt {\pi k} \map \Gamma {\frac k 2} } \paren {1 + \dfrac {x^2} k}^{-\frac {k + 1} 2}$
where $k = 1$.
Hence:
{{be... | The [[Definition:Student's t-Distribution|Student's $t$-distribution]] with one [[Definition:Degrees of Freedom of Student's t-Distribution|degree of freedom]] is a special case of a [[Definition:Standard Cauchy Distribution|standard Cauchy distribution]]. | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variables]] with a [[Definition:Student's t-Distribution|Student's $t$-distribution]] with one [[Definition:Degrees of Freedom of Student's t-Distribution|degree of freedom]].
Then $X$ has [[Definition:Probability Density Function|probability dens... | Student's t-Distribution with One Degree of Freedom is Standard Cauchy Distribution | https://proofwiki.org/wiki/Student's_t-Distribution_with_One_Degree_of_Freedom_is_Standard_Cauchy_Distribution | https://proofwiki.org/wiki/Student's_t-Distribution_with_One_Degree_of_Freedom_is_Standard_Cauchy_Distribution | [
"Student's t-Distribution",
"Standard Cauchy Distribution"
] | [
"Definition:Student's t-Distribution",
"Definition:Student's t-Distribution/Degrees of Freedom",
"Definition:Cauchy Distribution/Standard"
] | [
"Definition:Random Variable/Continuous",
"Definition:Student's t-Distribution",
"Definition:Student's t-Distribution/Degrees of Freedom",
"Definition:Probability Density Function",
"Gamma Function Extends Factorial",
"Gamma Function of One Half",
"Definition:Cauchy Distribution/Standard"
] |
proofwiki-21367 | Set of Invertible Continuous Transformations is Open Subset of Continuous Linear Transformations in Supremum Operator Norm Topology | Let $X$ be a Banach space.
Let $\map {CL} X$ be the continuous linear operator space on $X$.
Let $\map {GL} X$ denote the set of all invertible continuous linear operators on $X$.
Then $\map {GL} X \subseteq \map {CL} X$ in the supremum operator norm topology. | Let $T_0 \in \map {GL} X$.
By definition:
:$T_0^{-1} \in \map {CL} X$.
Let $T \in \map {B_\epsilon} {T_0}$ where $\map {B_\epsilon} x$ is an open ball in $\struct {\map {GL} X, \norm {\, \cdot \,} }$ topology.
By definition:
:$\norm {T - T_0} < \epsilon$
We also have that:
{{begin-eqn}}
{{eqn | l = \norm {\paren {T - T... | Let $X$ be a [[Definition:Banach Space|Banach space]].
Let $\map {CL} X$ be the [[Definition:Invertible Continuous Linear Operator|continuous linear operator space]] on $X$.
Let $\map {GL} X$ denote the set of all [[Definition:Invertible Continuous Linear Operator|invertible continuous linear operators]] on $X$.
Th... | Let $T_0 \in \map {GL} X$.
By [[Definition:Invertible Continuous Linear Operator|definition]]:
:$T_0^{-1} \in \map {CL} X$.
Let $T \in \map {B_\epsilon} {T_0}$ where $\map {B_\epsilon} x$ is an [[Definition:Open Ball in Normed Vector Space|open ball]] in $\struct {\map {GL} X, \norm {\, \cdot \,} }$ [[Definition:Top... | Set of Invertible Continuous Transformations is Open Subset of Continuous Linear Transformations in Supremum Operator Norm Topology | https://proofwiki.org/wiki/Set_of_Invertible_Continuous_Transformations_is_Open_Subset_of_Continuous_Linear_Transformations_in_Supremum_Operator_Norm_Topology | https://proofwiki.org/wiki/Set_of_Invertible_Continuous_Transformations_is_Open_Subset_of_Continuous_Linear_Transformations_in_Supremum_Operator_Norm_Topology | [
"Continuous Linear Transformations",
"Inverse Mappings",
"Banach Spaces",
"Topology"
] | [
"Definition:Banach Space",
"Definition:Invertible Continuous Linear Operator",
"Definition:Invertible Continuous Linear Operator",
"Definition:Supremum Operator Norm",
"Definition:Topology"
] | [
"Definition:Invertible Continuous Linear Operator",
"Definition:Open Ball/Normed Vector Space",
"Definition:Topology",
"Definition:Open Ball/Normed Vector Space",
"Supremum Operator Norm on Continuous Linear Transformation Space is Submultiplicative",
"Definition:Zero Operator",
"Definition:Invertible C... |
proofwiki-21368 | Quotient Space of Compact Space is Compact | Let $T = \struct {X, \tau}$ be a compact topological space.
Let $\RR \subseteq X \times X$ be an equivalence relation on $X$.
Then, the quotient space:
:$T / \RR$
is compact. | Let $\UU$ be an open cover of $T / \RR$.
By definition of quotient topology, for every $U \in \UU$:
:$q_\RR^{-1} \sqbrk U \in \tau$
where $q_\RR$ is the quotient mapping induced by $\RR$.
Therefore:
:$\VV = \set {q_\RR^{-1} \sqbrk U : U \in \UU}$
is a set of open sets of $T$.
Let $x \in X$ be arbitrary.
By definition o... | Let $T = \struct {X, \tau}$ be a [[Definition:Compact Topological Space|compact topological space]].
Let $\RR \subseteq X \times X$ be an [[Definition:Equivalence Relation|equivalence relation]] on $X$.
Then, the [[Definition:Quotient Space (Topology)|quotient space]]:
:$T / \RR$
is [[Definition:Compact Topological S... | Let $\UU$ be an [[Definition:Open Cover|open cover]] of $T / \RR$.
By definition of [[Definition:Quotient Topology|quotient topology]], for every $U \in \UU$:
:$q_\RR^{-1} \sqbrk U \in \tau$
where $q_\RR$ is the [[Definition:Quotient Mapping|quotient mapping]] induced by $\RR$.
Therefore:
:$\VV = \set {q_\RR^{-1} \sq... | Quotient Space of Compact Space is Compact/Proof 1 | https://proofwiki.org/wiki/Quotient_Space_of_Compact_Space_is_Compact | https://proofwiki.org/wiki/Quotient_Space_of_Compact_Space_is_Compact/Proof_1 | [
"Quotient Space of Compact Space is Compact",
"Compact Topological Spaces",
"Quotient Spaces (Topology)"
] | [
"Definition:Compact Topological Space",
"Definition:Equivalence Relation",
"Definition:Quotient Topology/Quotient Space",
"Definition:Compact Topological Space"
] | [
"Definition:Open Cover",
"Definition:Quotient Topology",
"Definition:Quotient Mapping",
"Definition:Set",
"Definition:Open Set/Topology",
"Definition:Cover of Set",
"Definition:Cover of Set",
"Definition:Cover of Set",
"Definition:Open Set/Topology",
"Definition:Open Cover",
"Definition:Compact ... |
proofwiki-21369 | Quotient Space of Compact Space is Compact | Let $T = \struct {X, \tau}$ be a compact topological space.
Let $\RR \subseteq X \times X$ be an equivalence relation on $X$.
Then, the quotient space:
:$T / \RR$
is compact. | Let $q_\RR$ be the quotient mapping induced by $\RR$.
By the definition of the quotient topology, $q_\RR$ is continuous.
Further, $T / \RR = q_\RR \sqbrk T$.
From Continuous Image of Compact Space is Compact, $q_\RR \sqbrk T$ is compact.
Hence $T / \RR$ is compact.
{{qed}} | Let $T = \struct {X, \tau}$ be a [[Definition:Compact Topological Space|compact topological space]].
Let $\RR \subseteq X \times X$ be an [[Definition:Equivalence Relation|equivalence relation]] on $X$.
Then, the [[Definition:Quotient Space (Topology)|quotient space]]:
:$T / \RR$
is [[Definition:Compact Topological S... | Let $q_\RR$ be the [[Definition:Quotient Mapping|quotient mapping]] induced by $\RR$.
By the definition of the [[Definition:Quotient Topology|quotient topology]], $q_\RR$ is [[Definition:Continuous Mapping|continuous]].
Further, $T / \RR = q_\RR \sqbrk T$.
From [[Continuous Image of Compact Space is Compact]], $q_\... | Quotient Space of Compact Space is Compact/Proof 2 | https://proofwiki.org/wiki/Quotient_Space_of_Compact_Space_is_Compact | https://proofwiki.org/wiki/Quotient_Space_of_Compact_Space_is_Compact/Proof_2 | [
"Quotient Space of Compact Space is Compact",
"Compact Topological Spaces",
"Quotient Spaces (Topology)"
] | [
"Definition:Compact Topological Space",
"Definition:Equivalence Relation",
"Definition:Quotient Topology/Quotient Space",
"Definition:Compact Topological Space"
] | [
"Definition:Quotient Mapping",
"Definition:Quotient Topology",
"Definition:Continuous Mapping",
"Continuous Image of Compact Space is Compact",
"Definition:Compact Topological Space",
"Definition:Compact Topological Space"
] |
proofwiki-21370 | Maximal Inequality for Positive Operators | Let $\struct {X, \BB, \mu}$ be a probability space.
Let $\map {L^1} \mu$ be a real-valued $L^1$ space with respect to $\mu$.
Let $U : \map {L^1} \mu \to \map {L^1} \mu$ be a positive linear operator, that is:
:$\forall f \in \map {L^1} \mu : f \ge 0 \implies U f \ge 0$
Suppose:
:$\norm U \le 1$
where $\norm \cdot$ deno... | Let:
:$G_N := \max \set {f_n : 0 \le n \le N}$
where $f_0 := 0$.
For all $0 \le n \le N$:
:$U G_N + f \ge U f_n + f = f_{n+1}$
In particular:
:$\paren 1 :\quad U G_N + f \ge F_N$
Let $x \in X$.
If $\map {F_N} x > 0$, then:
{{begin-eqn}}
{{eqn | l = \map {G_N} x
| r = \max \set {\map {f_n} x : 0 \le n \le N}
}}
{{... | Let $\struct {X, \BB, \mu}$ be a [[Definition:Probability Space|probability space]].
Let $\map {L^1} \mu$ be a [[Definition:Real-Valued Function|real-valued]] [[Definition:Lp Space|$L^1$ space]] with respect to $\mu$.
Let $U : \map {L^1} \mu \to \map {L^1} \mu$ be a [[Definition:Positive Linear Operator|positive]] [[... | Let:
:$G_N := \max \set {f_n : 0 \le n \le N}$
where $f_0 := 0$.
For all $0 \le n \le N$:
:$U G_N + f \ge U f_n + f = f_{n+1}$
In particular:
:$\paren 1 :\quad U G_N + f \ge F_N$
Let $x \in X$.
If $\map {F_N} x > 0$, then:
{{begin-eqn}}
{{eqn | l = \map {G_N} x
| r = \max \set {\map {f_n} x : 0 \le n \le N... | Maximal Inequality for Positive Operators | https://proofwiki.org/wiki/Maximal_Inequality_for_Positive_Operators | https://proofwiki.org/wiki/Maximal_Inequality_for_Positive_Operators | [
"Ergodic Theory",
"Operator Theory"
] | [
"Definition:Probability Space",
"Definition:Real-Valued Function",
"Definition:Lp Space",
"Definition:Positive Linear Operator",
"Definition:Linear Operator",
"Definition:Norm/Bounded Linear Transformation"
] | [] |
proofwiki-21371 | Compact Convex Set with Nonempty Interior is Homeomorphic to Cone on Boundary | Let $n \in \N_{> 0}$.
Let $T \subseteq \R^n$ be a compact convex subset of real Euclidean $n$-space.
Suppose the interior of $T$ is non-empty.
Then, $T$ is homeomorphic to the cone on its boundary. | Let $\bsx_0 \in T^\circ$ be an interior point of $T$.
Let $C \partial T$ denote the cone on the boundary of $T$.
Define $\phi : C \partial T \to T$ as:
:$\map \phi {\eqclass {\tuple {e, \bsx, t}} \RR} = t \bsx + \paren {1 - t} \bsx_0$
where:
:$e$ is the unique element of the trivial topological space used in the constr... | Let $n \in \N_{> 0}$.
Let $T \subseteq \R^n$ be a [[Definition:Compact Set (Topology)|compact]] [[Definition:Convex Set (Vector Space)|convex]] [[Definition:Subset|subset]] of [[Definition:Real Euclidean Space|real Euclidean $n$-space]].
Suppose the [[Definition:Interior (Topology)|interior]] of $T$ is [[Definition:N... | Let $\bsx_0 \in T^\circ$ be an [[Definition:Interior Point (Topology)|interior point]] of $T$.
Let $C \partial T$ denote the [[Definition:Cone (Topology)|cone]] on the [[Definition:Boundary (Topology)|boundary]] of $T$.
Define $\phi : C \partial T \to T$ as:
:$\map \phi {\eqclass {\tuple {e, \bsx, t}} \RR} = t \bsx +... | Compact Convex Set with Nonempty Interior is Homeomorphic to Cone on Boundary | https://proofwiki.org/wiki/Compact_Convex_Set_with_Nonempty_Interior_is_Homeomorphic_to_Cone_on_Boundary | https://proofwiki.org/wiki/Compact_Convex_Set_with_Nonempty_Interior_is_Homeomorphic_to_Cone_on_Boundary | [
"Convex Sets (Vector Spaces)",
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Compact Topological Space/Subspace",
"Definition:Convex Set (Vector Space)",
"Definition:Subset",
"Definition:Euclidean Space/Real",
"Definition:Interior (Topology)",
"Definition:Non-Empty Set",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Cone (Topology)",
"Definition:Boun... | [
"Definition:Interior Point (Topology)",
"Definition:Cone (Topology)",
"Definition:Boundary (Topology)",
"Definition:Trivial Topological Space",
"Definition:Cone (Topology)",
"Definition:Equivalence Relation",
"Definition:Join (Topology)",
"Definition:Join (Topology)",
"Definition:Trivial Topological... |
proofwiki-21372 | Hurwitz's Theorem (Normed Division Algebras) | The only normed division algebras over the real numbers are:
:the real numbers $\R$ themselves
:the complex numbers $\C$
:the quaternions $\H$
:the octonions $\mathbb O$ | {{ProofWanted}}
{{Namedfor|Adolf Hurwitz|cat = Hurwitz}} | The only [[Definition:Normed Division Algebra|normed division algebras]] over the [[Definition:Real Number|real numbers]] are:
:the [[Definition:Real Number|real numbers]] $\R$ themselves
:the [[Definition:Complex Number|complex numbers]] $\C$
:the [[Definition:Quaternion|quaternions]] $\H$
:the [[Definition:Octonion|... | {{ProofWanted}}
{{Namedfor|Adolf Hurwitz|cat = Hurwitz}} | Hurwitz's Theorem (Normed Division Algebras) | https://proofwiki.org/wiki/Hurwitz's_Theorem_(Normed_Division_Algebras) | https://proofwiki.org/wiki/Hurwitz's_Theorem_(Normed_Division_Algebras) | [
"Hurwitz's Theorem",
"Normed Division Algebras"
] | [
"Definition:Normed Division Algebra",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Quaternion",
"Definition:Octonion"
] | [] |
proofwiki-21373 | Ray from Bounded Set Meets Boundary | Let $A \subseteq \R^n$ be a bounded subspace of real Euclidean $n$-space.
Let $\bsx_0 \in A$ be point of $A$.
Then, for every $\bsy \in \R^n \setminus \set \bszero$, there is some $t \in \R_{\ge 0}$ such that:
:$\bsx_0 + t \bsy \in \partial A$
where $\partial A$ denotes the boundary of $A$. | For every $t \in \R$, let:
:$\map \bsx t = \bsx_0 + t \bsy$
Define:
:$D = \set {t \in \R_{\ge 0} : \map \bsx t \in A}$
As $\map \bsx 0 = \bsx_0 \in A$:
:$0 \in D$
so:
:$D$ is non-empty
By definition of bounded, there exists some $K \in \R$ such that:
:$\forall \bsx \in A: \size {\bsx - \bsx_0} \le K$
Suppose for arbitr... | Let $A \subseteq \R^n$ be a [[Definition:Bounded Metric Space|bounded]] [[Definition:Topological Subspace|subspace]] of [[Definition:Real Euclidean Space|real Euclidean $n$-space]].
Let $\bsx_0 \in A$ be point of $A$.
Then, for every $\bsy \in \R^n \setminus \set \bszero$, there is some $t \in \R_{\ge 0}$ such that:
... | For every $t \in \R$, let:
:$\map \bsx t = \bsx_0 + t \bsy$
Define:
:$D = \set {t \in \R_{\ge 0} : \map \bsx t \in A}$
As $\map \bsx 0 = \bsx_0 \in A$:
:$0 \in D$
so:
:$D$ is [[Definition:Non-Empty Set|non-empty]]
By definition of [[Definition:Bounded Metric Space/Definition 4|bounded]], there exists some $K \in \R... | Ray from Bounded Set Meets Boundary | https://proofwiki.org/wiki/Ray_from_Bounded_Set_Meets_Boundary | https://proofwiki.org/wiki/Ray_from_Bounded_Set_Meets_Boundary | [
"Bounded Metric Spaces"
] | [
"Definition:Bounded Metric Space",
"Definition:Topological Subspace",
"Definition:Euclidean Space/Real",
"Definition:Boundary (Topology)"
] | [
"Definition:Non-Empty Set",
"Definition:Bounded Metric Space/Definition 4",
"Definition:Bounded Above Set/Real Numbers",
"Least Upper Bound Property",
"Definition:Neighborhood (Topology)/Point",
"Characterizing Property of Supremum of Subset of Real Numbers",
"Definition:Supremum of Set/Real Numbers",
... |
proofwiki-21374 | Ray from Interior of Compact Convex Set Meets Boundary Exactly Once | Let $A \subseteq \R^n$ be a compact convex subspace of real Euclidean $n$-space.
Let $\bsx_0 \in A^\circ$ be an interior point of $A$.
Then, for every $\bsy \in \R^n \setminus \bszero$, there is a unique $t \in \R_{> 0}$ such that:
:$\bsx_0 + t \bsy \in \partial A$ | By Compact Subspace of Metric Space is Bounded:
:$A$ is bounded
Then, by Ray from Bounded Set Meets Boundary, there is some $t \in \R_{\ge 0}$ such that:
:$\bsx_0 + t \bsy \in \partial A$
But, if $t = 0$, then:
:$\bsx_0 \in \partial A$
contradicting the definition of boundary, since $\bsx_0 \in A^\circ$.
Thus, $t > 0$.... | Let $A \subseteq \R^n$ be a [[Definition:Compact Topological Space|compact]] [[Definition:Convex Set (Vector Space)|convex]] [[Definition:Topological Subspace|subspace]] of [[Definition:Real Euclidean Space|real Euclidean $n$-space]].
Let $\bsx_0 \in A^\circ$ be an [[Definition:Interior Point (Topology)|interior point... | By [[Compact Subspace of Metric Space is Bounded]]:
:$A$ is [[Definition:Bounded Metric Space|bounded]]
Then, by [[Ray from Bounded Set Meets Boundary]], there is some $t \in \R_{\ge 0}$ such that:
:$\bsx_0 + t \bsy \in \partial A$
But, if $t = 0$, then:
:$\bsx_0 \in \partial A$
contradicting the definition of [[Defi... | Ray from Interior of Compact Convex Set Meets Boundary Exactly Once | https://proofwiki.org/wiki/Ray_from_Interior_of_Compact_Convex_Set_Meets_Boundary_Exactly_Once | https://proofwiki.org/wiki/Ray_from_Interior_of_Compact_Convex_Set_Meets_Boundary_Exactly_Once | [
"Convex Sets (Vector Spaces)"
] | [
"Definition:Compact Topological Space",
"Definition:Convex Set (Vector Space)",
"Definition:Topological Subspace",
"Definition:Euclidean Space/Real",
"Definition:Interior Point (Topology)",
"Definition:Unique"
] | [
"Compact Subspace of Metric Space is Bounded",
"Definition:Bounded Metric Space",
"Ray from Bounded Set Meets Boundary",
"Definition:Boundary (Topology)/Definition 1",
"Definition:Interior Point (Topology)/Definition 2",
"Compact Subspace of Hausdorff Space is Closed",
"Definition:Convex Set (Vector Spa... |
proofwiki-21375 | Center of Mass of System of Particles in Cartesian Plane | Let $B$ be a system of $n$ discrete particles embedded in a cartesian plane, each with:
:mass $m_i$
:position $\tuple {x_i, y_i}$
where $i \in \set {1, 2, \ldots, n}$.
Then the coordinates $\tuple {\bar x, \bar y}$ of the center of mass of $B$ are given by:
{{begin-eqn}}
{{eqn | l = M \bar x
| r = \sum_{i \mathop... | {{ProofWanted|straightforward but utterly tedious}} | Let $B$ be a system of $n$ [[Definition:Discrete|discrete]] [[Definition:Particle|particles]] embedded in a [[Definition:Cartesian Plane|cartesian plane]], each with:
:[[Definition:Mass|mass]] $m_i$
:[[Definition:Position|position]] $\tuple {x_i, y_i}$
where $i \in \set {1, 2, \ldots, n}$.
Then the [[Definition:Carte... | {{ProofWanted|straightforward but utterly tedious}} | Center of Mass of System of Particles in Cartesian Plane | https://proofwiki.org/wiki/Center_of_Mass_of_System_of_Particles_in_Cartesian_Plane | https://proofwiki.org/wiki/Center_of_Mass_of_System_of_Particles_in_Cartesian_Plane | [
"Centers of Mass"
] | [
"Definition:Discrete",
"Definition:Particle",
"Definition:Cartesian Plane",
"Definition:Mass",
"Definition:Position",
"Definition:Cartesian Coordinate System",
"Definition:Center of Mass"
] | [] |
proofwiki-21376 | Center of Mass/Examples/Uniform Lamina | Let $\LL$ be a uniform lamina embedded in a cartesian plane in the shape of the area between the curve $\map f x$, the straight lines $x = a$ and $x = b$, and the $x$-axis.
Let the area of $\LL$ be $A$.
Then the coordinates $\tuple {\bar x, \bar y}$ of the center of mass of $B$ are given by:
{{begin-eqn}}
{{eqn | l = A... | {{ProofWanted|straightforward but I have other things I want to do}} | Let $\LL$ be a [[Definition:Uniform Lamina|uniform lamina]] embedded in a [[Definition:Cartesian Plane|cartesian plane]] in the shape of the [[Definition:Area|area]] between the [[Definition:Curve|curve]] $\map f x$, the [[Definition:Straight Line|straight lines]] $x = a$ and $x = b$, and the [[Definition:X-Axis|$x$-ax... | {{ProofWanted|straightforward but I have other things I want to do}} | Center of Mass/Examples/Uniform Lamina | https://proofwiki.org/wiki/Center_of_Mass/Examples/Uniform_Lamina | https://proofwiki.org/wiki/Center_of_Mass/Examples/Uniform_Lamina | [
"Centers of Mass"
] | [
"Definition:Lamina/Uniform",
"Definition:Cartesian Plane",
"Definition:Area",
"Definition:Line/Curve",
"Definition:Line/Straight Line",
"Definition:Axis/X-Axis",
"Definition:Area",
"Definition:Cartesian Coordinate System",
"Definition:Center of Mass"
] | [] |
proofwiki-21377 | Center of Gravity equals Center of Mass if it exists | Let $B$ be a body in a gravitational field $\mathbf G$.
Let $B$ have a center of gravity $P$.
Then the center of mass of $B$ is also $P$. | Let $Q$ denote the center of mass of $B$
First suppose that $\mathbf G$ is uniform.
From Center of Gravity in Uniform Gravitational Field is Center of Mass:
:$P = Q$
{{qed|lemma}}
Now suppose that $\mathbf G$ is non-uniform.
There are two possibilities:
:$(1): \quad$ $B$ is barycentric
:$(2): \quad$ $B$ is not barycent... | Let $B$ be a [[Definition:Body|body]] in a [[Definition:Gravitational Field|gravitational field]] $\mathbf G$.
Let $B$ have a [[Definition:Center of Gravity|center of gravity]] $P$.
Then the [[Definition:Center of Mass|center of mass]] of $B$ is also $P$. | Let $Q$ denote the [[Definition:Center of Mass|center of mass]] of $B$
First suppose that $\mathbf G$ is [[Definition:Uniform Field|uniform]].
From [[Center of Gravity in Uniform Gravitational Field is Center of Mass]]:
:$P = Q$
{{qed|lemma}}
Now suppose that $\mathbf G$ is non-[[Definition:Uniform Field|uniform]].... | Center of Gravity equals Center of Mass if it exists | https://proofwiki.org/wiki/Center_of_Gravity_equals_Center_of_Mass_if_it_exists | https://proofwiki.org/wiki/Center_of_Gravity_equals_Center_of_Mass_if_it_exists | [
"Centers of Mass",
"Centers of Gravity"
] | [
"Definition:Body",
"Definition:Gravitational Field",
"Definition:Center of Gravity",
"Definition:Center of Mass"
] | [
"Definition:Center of Mass",
"Definition:Uniform Field",
"Center of Gravity in Uniform Gravitational Field is Center of Mass",
"Definition:Uniform Field",
"Definition:Barycentric Body",
"Definition:Barycentric Body",
"Center of Gravity of Barycentric Body is Center of Mass",
"Center of Gravity in Non-... |
proofwiki-21378 | Centripetal Force on Body in Circular Path | Let $B$ be a body of mass $m$ constrained to move at constant speed $v$ in a circular path $C$ or radius $r$.
Let $\mathbf F$ denote the centripetal force on $B$.
Then:
:$(1): \quad$ The direction of $\mathbf F$ is towards the center of $C$
:$(2): \quad$ The magnitude of $\mathbf F$ is given by:
::::$\size {\mathbf F} ... | Let $\mathbf r$ be the position vector of $B$ {{WRT}} the center of $C$.
Let $\mathbf v$ be the velocity of $B$.
Let $\mathbf a$ be the acceleration of $B$.
We have:
{{begin-eqn}}
{{eqn | l = \mathbf F
| r = m \mathbf a
| c = Newton's Second Law
}}
{{eqn | r = m \paren {-\frac {\size {\mathbf v}^2 \mathbf r... | Let $B$ be a [[Definition:Body|body]] of [[Definition:Mass|mass]] $m$ constrained to move at constant [[Definition:Speed|speed]] $v$ in a [[Definition:Circle|circular]] path $C$ or [[Definition:Radius of Circle|radius]] $r$.
Let $\mathbf F$ denote the [[Definition:Centripetal Force|centripetal force]] on $B$.
Then:
... | Let $\mathbf r$ be the [[Definition:Position Vector|position vector]] of $B$ {{WRT}} the [[Definition:Center of Circle|center]] of $C$.
Let $\mathbf v$ be the [[Definition:Velocity|velocity]] of $B$.
Let $\mathbf a$ be the [[Definition:Acceleration|acceleration]] of $B$.
We have:
{{begin-eqn}}
{{eqn | l = \mathbf F... | Centripetal Force on Body in Circular Path | https://proofwiki.org/wiki/Centripetal_Force_on_Body_in_Circular_Path | https://proofwiki.org/wiki/Centripetal_Force_on_Body_in_Circular_Path | [
"Centripetal Force"
] | [
"Definition:Body",
"Definition:Mass",
"Definition:Speed",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Centripetal Force",
"Definition:Direction",
"Definition:Circle/Center",
"Definition:Magnitude",
"Definition:Angular Speed"
] | [
"Definition:Position Vector",
"Definition:Circle/Center",
"Definition:Velocity",
"Definition:Acceleration",
"Newton's Laws of Motion/Second Law",
"Acceleration of Particle moving in Circle at Constant Speed",
"Definition:Parenthesis",
"Definition:Positive/Number",
"Definition:Mass",
"Definition:Po... |
proofwiki-21379 | Center of Mass Operation is Associative | Let $S$ denote the set of massy particles in ordinary space.
Let $\circ$ denote the binary operation defined as:
:$\forall x, y \in S: x \circ y =$ the center of mass of $x$ and $y$
Then $\circ$ is an associative operation. | {{ProofWanted|Apparently it's equivalent to Ceva's Theorem.}} | Let $S$ denote the [[Definition:Set|set]] of [[Definition:Mass|massy]] [[Definition:Particle|particles]] in [[Definition:Ordinary Space|ordinary space]].
Let $\circ$ denote the [[Definition:Binary Operation|binary operation]] defined as:
:$\forall x, y \in S: x \circ y =$ the [[Definition:Center of Mass|center of mass... | {{ProofWanted|Apparently it's equivalent to [[Ceva's Theorem]].}} | Center of Mass Operation is Associative | https://proofwiki.org/wiki/Center_of_Mass_Operation_is_Associative | https://proofwiki.org/wiki/Center_of_Mass_Operation_is_Associative | [
"Centers of Mass",
"Examples of Associative Operations"
] | [
"Definition:Set",
"Definition:Mass",
"Definition:Particle",
"Definition:Ordinary Space",
"Definition:Operation/Binary Operation",
"Definition:Center of Mass",
"Definition:Associative Operation"
] | [
"Ceva's Theorem"
] |
proofwiki-21380 | Derivative of Composite Function/Jacobians | Let $k, m, n \in \N$.
Let $U$ be an open set of $\R^n$.
Let $V$ be an open set of $\R^m$.
Let $\mathbf f = \paren {f_1, f_2, \ldots, f_m}^\intercal: U \to V$ be a vector valued function, differentiable at $\mathbf x = \paren {x_1, x_2, \ldots, x_n}^\intercal \in U$.
Let $\mathbf g = \paren {g_1, g_2, \ldots, g_k}^\inte... | {{MissingLinks}}
The Jacobian matrix of $\mathbf g \circ \mathbf f$ at $\mathbf x$ is defined to be the $k \times n$ matrix:
:<nowiki>$\map {\mathbf J_{\mathbf g \circ \mathbf f} } {\mathbf x} = \begin{pmatrix}
\map {\dfrac {\map \partial {g_1 \circ \mathbf f} } {\partial x_1} } {\mathbf x} & \cdots & \map {\dfrac {\m... | Let $k, m, n \in \N$.
Let $U$ be an [[Definition:Open Set of Real Euclidean Space|open set]] of $\R^n$.
Let $V$ be an [[Definition:Open Set of Real Euclidean Space|open set]] of $\R^m$.
Let $\mathbf f = \paren {f_1, f_2, \ldots, f_m}^\intercal: U \to V$ be a [[Definition:Vector-Valued Function|vector valued function... | {{MissingLinks}}
The [[Definition:Jacobian Matrix|Jacobian matrix]] of $\mathbf g \circ \mathbf f$ at $\mathbf x$ is defined to be the $k \times n$ matrix:
:<nowiki>$\map {\mathbf J_{\mathbf g \circ \mathbf f} } {\mathbf x} = \begin{pmatrix}
\map {\dfrac {\map \partial {g_1 \circ \mathbf f} } {\partial x_1} } {\mathb... | Derivative of Composite Function/Jacobians | https://proofwiki.org/wiki/Derivative_of_Composite_Function/Jacobians | https://proofwiki.org/wiki/Derivative_of_Composite_Function/Jacobians | [
"Derivative of Composite Function"
] | [
"Definition:Open Set/Real Analysis/Real Euclidean Space",
"Definition:Open Set/Real Analysis/Real Euclidean Space",
"Definition:Vector-Valued Function",
"Definition:Differentiable Mapping/Vector-Valued Function",
"Definition:Vector-Valued Function",
"Definition:Differentiable Mapping/Vector-Valued Functio... | [
"Definition:Jacobian/Matrix",
"Definition:Jacobian/Matrix",
"Definition:Jacobian/Matrix",
"Derivative of Composite Function"
] |
proofwiki-21381 | Maximal Ergodic Theorem | Let $\struct {X, \BB, \mu, T}$ be a measure-preserving dynamical system.
Let $g : X \to \overline \R$ be a $\mu$-integrable function.
Let $\alpha \in \R$.
Let:
:$\ds E_\alpha := \set {x \in X : \sup_{n \ge 1} \frac 1 n \sum_{i \mathop = 0}^{n-1} \map g {T^i x} > \alpha }$
Then:
:$\ds \alpha \map \mu {E_\alpha} \le \int... | Let $\map {L^1} \mu$ be a real-valued $L^1$ space with respect to $\mu$.
Consider $U_T : \map {L^1} \mu \to \map {L^1} \mu$ defined by:
:$f \mapsto f \circ T$
Observe:
{{begin-eqn}}
{{eqn | l = E_\alpha
| r = \bigcup_{N \mathop =1}^\infty F_N
}}
{{end-eqn}}
where:
:$\ds F_N := \set {\max_{1 \le n \le N} \sum_{i \... | Let $\struct {X, \BB, \mu, T}$ be a [[Definition:Measure-Preserving Dynamical System|measure-preserving dynamical system]].
Let $g : X \to \overline \R$ be a [[Definition:Measure-Integrable Function|$\mu$-integrable function]].
Let $\alpha \in \R$.
Let:
:$\ds E_\alpha := \set {x \in X : \sup_{n \ge 1} \frac 1 n \sum... | Let $\map {L^1} \mu$ be a [[Definition:Real-Valued Function|real-valued]] [[Definition:Lp Space|$L^1$ space]] with respect to $\mu$.
Consider $U_T : \map {L^1} \mu \to \map {L^1} \mu$ defined by:
:$f \mapsto f \circ T$
Observe:
{{begin-eqn}}
{{eqn | l = E_\alpha
| r = \bigcup_{N \mathop =1}^\infty F_N
}}
{{end-... | Maximal Ergodic Theorem | https://proofwiki.org/wiki/Maximal_Ergodic_Theorem | https://proofwiki.org/wiki/Maximal_Ergodic_Theorem | [
"Ergodic Theory",
"Named Theorems"
] | [
"Definition:Measure-Preserving Dynamical System",
"Definition:Integrable Function/Measure Space"
] | [
"Definition:Real-Valued Function",
"Definition:Lp Space",
"Lebesgue's Dominated Convergence Theorem",
"Maximal Inequality for Positive Operators"
] |
proofwiki-21382 | Join of Compact Spaces is Compact | Let $A, B$ be compact topological spaces.
Then:
:$A \ast B$ is compact
where $A \ast B$ denotes the join of $A$ and $B$. | By Closed Real Interval is Compact Space:
:$\closedint 0 1$
is compact.
Thus, by Topological Product of Compact Spaces:
:$A \times B \times \closedint 0 1$
is compact.
Therefore, by Quotient Space of Compact Space is Compact:
:$A \ast B = \paren {A \times B \times \closedint 0 1} / \RR$
is compact.
{{qed}}
Category:Com... | Let $A, B$ be [[Definition:Compact Topological Space|compact topological spaces]].
Then:
:$A \ast B$ is [[Definition:Compact Topological Space|compact]]
where $A \ast B$ denotes the [[Definition:Join (Topology)|join]] of $A$ and $B$. | By [[Closed Real Interval is Compact Space]]:
:$\closedint 0 1$
is [[Definition:Compact Topological Space|compact]].
Thus, by [[Topological Product of Compact Spaces]]:
:$A \times B \times \closedint 0 1$
is [[Definition:Compact Topological Space|compact]].
Therefore, by [[Quotient Space of Compact Space is Compact]]... | Join of Compact Spaces is Compact | https://proofwiki.org/wiki/Join_of_Compact_Spaces_is_Compact | https://proofwiki.org/wiki/Join_of_Compact_Spaces_is_Compact | [
"Compact Topological Spaces",
"Quotient Spaces (Topology)"
] | [
"Definition:Compact Topological Space",
"Definition:Compact Topological Space",
"Definition:Join (Topology)"
] | [
"Closed Real Interval is Compact Space",
"Definition:Compact Topological Space",
"Topological Product of Compact Spaces",
"Definition:Compact Topological Space",
"Quotient Space of Compact Space is Compact",
"Definition:Compact Topological Space",
"Category:Compact Topological Spaces",
"Category:Quoti... |
proofwiki-21383 | Characteristic Function of Normal Distribution/Corollary | The characteristic function of the standard normal distribution is:
:$\map \phi t = e^{-\frac 1 2 t^2}$ | Recall Characteristic Function of Normal Distribution:
{{:Characteristic Function of Normal Distribution}}
The standard normal distribution is the normal distribution with $\mu = 0$ and $\sigma = 1$.
Hence:
{{begin-eqn}}
{{eqn | l = \map \phi t
| r = e^{i t \times 0 - \frac 1 2 t^2 \times 1^2}
| c =
}}
{{e... | The [[Definition:Characteristic Function of Random Variable|characteristic function]] of the [[Definition:Standard Normal Distribution|standard normal distribution]] is:
:$\map \phi t = e^{-\frac 1 2 t^2}$ | Recall [[Characteristic Function of Normal Distribution]]:
{{:Characteristic Function of Normal Distribution}}
The [[Definition:Standard Normal Distribution|standard normal distribution]] is the [[Definition:Normal Distribution|normal distribution]] with $\mu = 0$ and $\sigma = 1$.
Hence:
{{begin-eqn}}
{{eqn | l = \... | Characteristic Function of Normal Distribution/Corollary | https://proofwiki.org/wiki/Characteristic_Function_of_Normal_Distribution/Corollary | https://proofwiki.org/wiki/Characteristic_Function_of_Normal_Distribution/Corollary | [
"Characteristic Function of Normal Distribution"
] | [
"Definition:Characteristic Function of Random Variable",
"Definition:Standard Normal Distribution"
] | [
"Characteristic Function of Normal Distribution",
"Definition:Standard Normal Distribution",
"Definition:Normal Distribution"
] |
proofwiki-21384 | Median of Logistic Distribution | Let $X$ be a continuous random variable with a '''logistic distribution''':
:$\map {f_X} x = \dfrac {\map \exp {-\dfrac {\paren {x - \mu} } s} } {s \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2}$
for $\mu \in \R, s \in \R_{>0}$.
The median of $X$ is $\mu$. | From the definition of the logistic distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {\map \exp {-\dfrac {\paren {x - \mu} } s} } {s \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2}$
Note that $f_X$ is non-zero, sufficient to ensure a unique median.
{{explain|Why does being non-zero... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with a '''[[Definition:Logistic Distribution|logistic distribution]]''':
:$\map {f_X} x = \dfrac {\map \exp {-\dfrac {\paren {x - \mu} } s} } {s \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2}$
for $\mu \in \R, s \in \R_{>0}$... | From the definition of the [[Definition:Logistic Distribution|logistic distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac {\map \exp {-\dfrac {\paren {x - \mu} } s} } {s \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2}$
Note that $f_X$ i... | Median of Logistic Distribution | https://proofwiki.org/wiki/Median_of_Logistic_Distribution | https://proofwiki.org/wiki/Median_of_Logistic_Distribution | [
"Logistic Distribution",
"Medians"
] | [
"Definition:Random Variable/Continuous",
"Definition:Logistic Distribution",
"Definition:Median of Continuous Random Variable"
] | [
"Definition:Logistic Distribution",
"Definition:Probability Density Function",
"Definition:Median of Continuous Random Variable",
"Definition:Median of Continuous Random Variable",
"Definition:Median of Continuous Random Variable",
"Integration by Substitution",
"Power Rule for Derivatives",
"Derivati... |
proofwiki-21385 | Expansion of Characteristic Polynomial of Matrix | Let $R$ be a commutative ring with unity.
Let $R \sqbrk x$ be the polynomial ring in one variable over $R$.
Let $\mathbf A$ be a square matrix over $R$ of order $n > 0$.
Let $\map {p_{\mathbf A} } x$ be the characteristic polynomial of $\mathbf A$.
Then $\map {p_{\mathbf A} } x$ can be expressed as:
:$\map {p_{\mathbf ... | We have that the eigenvalues of $\mathbf A$ are the roots of $\map {p_{\mathbf A} } x$.
From Sum of Roots of Polynomial, the sum of the roots of $\map {p_{\mathbf A} } x$ is $-\dfrac {a_{n - 1} } {a_n}$
From Characteristic Polynomial of Matrix is Monic, $a_n = 1$.
Thus, from Trace of Matrix is Sum of Eigenvalues:
:$-a_... | Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $R \sqbrk x$ be the [[Definition:Polynomial Ring in One Variable|polynomial ring in one variable]] over $R$.
Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]] over $R$ of [[Definition:Order of Square Matrix|order]] ... | We have that the [[Definition:Eigenvalue of Square Matrix|eigenvalues]] of $\mathbf A$ are the [[Definition:Root of Polynomial|roots]] of $\map {p_{\mathbf A} } x$.
From [[Sum of Roots of Polynomial]], the [[Definition:Summation|sum]] of the [[Definition:Root of Polynomial|roots]] of $\map {p_{\mathbf A} } x$ is $-\d... | Expansion of Characteristic Polynomial of Matrix | https://proofwiki.org/wiki/Expansion_of_Characteristic_Polynomial_of_Matrix | https://proofwiki.org/wiki/Expansion_of_Characteristic_Polynomial_of_Matrix | [
"Characteristic Polynomial of Matrix",
"Traces of Matrices",
"Determinants",
"Eigenvalues of Square Matrices"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Polynomial Ring",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Characteristic Polynomial of Matrix",
"Definition:Trace (Linear Algebra)/Matrix",
"Definition:Determinant/Matrix"
] | [
"Definition:Eigenvalue/Square Matrix",
"Definition:Root of Polynomial",
"Sum of Roots of Polynomial",
"Definition:Summation",
"Definition:Root of Polynomial",
"Characteristic Polynomial of Matrix is Monic",
"Trace of Matrix is Sum of Eigenvalues",
"Product of Roots of Polynomial",
"Definition:Contin... |
proofwiki-21386 | Chinese Remainder Theorem/Construction of Solution | Recall the Chinese Remainder Theorem:
{{:Chinese Remainder Theorem}} | First we demonstrate that $x$ is a solution.
By construction:
{{begin-eqn}}
{{eqn | q = \forall j \in \set {1, 2, \ldots, r}
| l = n_j
| o = \divides
| r = y_i
| c =
}}
{{eqn | ll= \leadsto
| l = y_i
| o = \equiv
| r = 0
| rr= \pmod {n_i}
| c =
}}
{{eqn | n = 1
... | Recall the [[Chinese Remainder Theorem]]:
{{:Chinese Remainder Theorem}} | First we demonstrate that $x$ is a [[Definition:Solution of System of Simultaneous Congruences|solution]].
By construction:
{{begin-eqn}}
{{eqn | q = \forall j \in \set {1, 2, \ldots, r}
| l = n_j
| o = \divides
| r = y_i
| c =
}}
{{eqn | ll= \leadsto
| l = y_i
| o = \equiv
... | Chinese Remainder Theorem/Construction of Solution | https://proofwiki.org/wiki/Chinese_Remainder_Theorem/Construction_of_Solution | https://proofwiki.org/wiki/Chinese_Remainder_Theorem/Construction_of_Solution | [
"Chinese Remainder Theorem"
] | [
"Chinese Remainder Theorem"
] | [
"Definition:Simultaneous Congruences/Solution",
"Definition:Vanish",
"Definition:Simultaneous Congruences/Solution",
"Definition:Unique",
"Definition:Congruence (Number Theory)/Integers",
"Definition:Simultaneous Congruences/Solution",
"Definition:Pairwise Coprime/Integers",
"Category:Chinese Remainde... |
proofwiki-21387 | Maximal Radical implies Primary Ideal | Let $R$ be a commutative ring with unity.
Let $\mathfrak a$ be an ideal of $R$.
Let $\map \Rad {\mathfrak a}$ be the radical of $\mathfrak a$.
Suppose that $\map \Rad {\mathfrak a}$ is a maximal ideal.
Then $\mathfrak a$ is a primary ideal. | Consider the quotient ring $R / \mathfrak a$.
By {{Defof|Nilradical of Ring|index=1}} and {{Defof|Radical of Ideal of Ring|index=1}}:
:$\Nil {R / \mathfrak a} = \map \Rad {\mathfrak a} / \mathfrak a$
On the other hand, {{hypothesis}}, $\map \Rad {\mathfrak a} / \mathfrak a$ is a maximal ideal of $R / \mathfrak a$.
Thus... | Let $R$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $\mathfrak a$ be an [[Definition:Ideal|ideal]] of $R$.
Let $\map \Rad {\mathfrak a}$ be the [[Definition:Radical of Ideal of Ring|radical]] of $\mathfrak a$.
Suppose that $\map \Rad {\mathfrak a}$ is a [[Definition:Maximal Idea... | Consider the [[Definition:Quotient Ring|quotient ring]] $R / \mathfrak a$.
By {{Defof|Nilradical of Ring|index=1}} and {{Defof|Radical of Ideal of Ring|index=1}}:
:$\Nil {R / \mathfrak a} = \map \Rad {\mathfrak a} / \mathfrak a$
On the other hand, {{hypothesis}}, $\map \Rad {\mathfrak a} / \mathfrak a$ is a [[Definit... | Maximal Radical implies Primary Ideal | https://proofwiki.org/wiki/Maximal_Radical_implies_Primary_Ideal | https://proofwiki.org/wiki/Maximal_Radical_implies_Primary_Ideal | [
"Radical of Ideals",
"Primary Ideals",
"Prime Ideals of Rings"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ideal",
"Definition:Radical of Ideal of Ring",
"Definition:Maximal Ideal of Ring",
"Definition:Primary Ideal"
] | [
"Definition:Quotient Ring",
"Definition:Maximal Ideal of Ring",
"Definition:Maximal Ideal of Ring",
"Proper Ideal of Ring is Contained in Maximal Ideal",
"Category:Radical of Ideals",
"Category:Primary Ideals",
"Category:Prime Ideals of Rings"
] |
proofwiki-21388 | Positive Definite Matrix has Cholesky Factorization | Let $\mathbf A$ be a positive definite matrix.
Then there exists a Cholesky factorization of $\mathbf A$. | {{tidy}}
{{Proofread}}
From the Spectral Theorem for Real Symmetric Matrices, there exists an orthogonal matrix $\Omega$ such that $\Omega^T \mathbf A \Omega$ is diagonal.
Let $\Omega^T \mathbf A \Omega = \map {\mathrm {diag} } {\lambda_1, \ldots, \lambda_n}$
Then, $\lambda_1, \cdots, \lambda_n$ are eigenvalues of $\ma... | Let $\mathbf A$ be a [[Definition:Positive Definite Matrix|positive definite matrix]].
Then there exists a [[Definition:Cholesky Factorization|Cholesky factorization]] of $\mathbf A$. | {{tidy}}
{{Proofread}}
From the [[Spectral Theorem for Real Symmetric Matrices]], there exists an [[Definition:Orthogonal_Matrix|orthogonal matrix]] $\Omega$ such that $\Omega^T \mathbf A \Omega$ is [[Definition:Diagonal Matrix|diagonal]].
Let $\Omega^T \mathbf A \Omega = \map {\mathrm {diag} } {\lambda_1, \ldots, \l... | Positive Definite Matrix has Cholesky Factorization | https://proofwiki.org/wiki/Positive_Definite_Matrix_has_Cholesky_Factorization | https://proofwiki.org/wiki/Positive_Definite_Matrix_has_Cholesky_Factorization | [
"Cholesky Factorizations",
"Positive Definite Matrices"
] | [
"Definition:Positive Definite Matrix",
"Definition:Cholesky Factorization"
] | [
"Spectral Theorem for Real Symmetric Matrices",
"Definition:Orthogonal_Matrix",
"Definition:Diagonal Matrix",
"Definition:Positive Definite Matrix",
"Definition:QR Factorization",
"Definition:Orthogonal_Matrix"
] |
proofwiki-21389 | Ring of Continuous Mappings is Subring of All Mappings | Let $\struct {S, \tau_{_S} }$ be a topological space.
Let $\struct {R, +, *, \tau_{_R} }$ be a topological ring.
Let $\struct {R^S, +, *}$ be the ring of mappings from $S$ to $R$.
Let $\struct{\map C {S, R}, +, *}$ be the ring of continuous mappings from $S$ to $R$.
Then:
:$\struct{\map C {S, R}, +, *}$ is a subring of... | From Structure Induced by Ring Operations is Ring:
:$\struct {R^S, +, *}$ is a ring.
From Structure Induced by Ring Operations is Ring:
:$\forall f \in R^S :$ the additive inverse of $f$ is the pointwise negation $-f$, defined by:
::$\forall s \in S: \map {\paren {-f} } s := - \map f s$
From the Subring Test:
:$\struct... | Let $\struct {S, \tau_{_S} }$ be a [[Definition:Topological Space|topological space]].
Let $\struct {R, +, *, \tau_{_R} }$ be a [[Definition:Topological Ring|topological ring]].
Let $\struct {R^S, +, *}$ be the [[Definition:Ring of Mappings|ring of mappings from $S$ to $R$]].
Let $\struct{\map C {S, R}, +, *}$ be th... | From [[Structure Induced by Ring Operations is Ring]]:
:$\struct {R^S, +, *}$ is a [[Definition:Ring (Abstract Algebra)|ring]].
From [[Structure Induced by Ring Operations is Ring]]:
:$\forall f \in R^S :$ the [[Definition:Additive Inverse in Ring|additive inverse]] of $f$ is the [[Definition:Pointwise Negation of Re... | Ring of Continuous Mappings is Subring of All Mappings | https://proofwiki.org/wiki/Ring_of_Continuous_Mappings_is_Subring_of_All_Mappings | https://proofwiki.org/wiki/Ring_of_Continuous_Mappings_is_Subring_of_All_Mappings | [
"Rings of Continuous Mappings"
] | [
"Definition:Topological Space",
"Definition:Topological Ring",
"Definition:Ring of Mappings",
"Definition:Ring of Continuous Mappings",
"Definition:Subring"
] | [
"Structure Induced by Ring Operations is Ring",
"Definition:Ring (Abstract Algebra)",
"Structure Induced by Ring Operations is Ring",
"Definition:Additive Inverse/Ring",
"Definition:Pointwise Negation of Real-Valued Function",
"Subring Test",
"Definition:Subring",
"Subring Test",
"Definition:Subring... |
proofwiki-21390 | Zero of Ring of Continuous Mappings | Let $\struct {S, \tau_{_S} }$ be a topological space.
Let $\struct {R, +, *, \tau_{_R} }$ be a topological ring with zero $0_R$.
Let $\struct{\map C {S, R}, +, *}$ be the ring of continuous mappings from $S$ to $R$.
Then:
:the zero of $\struct{\map C {S, R}, +, *}$ is the constant mapping $0_{R^S} : S \to R$ defined by... | Let $\struct {R^S, +, *}$ be the ring of mappings from $S$ to $R$.
From Ring of Continuous Mappings is Subring of All Mappings:
:$\struct{\map C {S, R}, +, *}$ is a subring of $\struct {R^S, +, *}$
From Induced Structure Identity:
:the zero of $\struct {R^S, +, *}$ is the constant mapping $0_{R^S} : S \to R$ defined by... | Let $\struct {S, \tau_{_S} }$ be a [[Definition:Topological Space|topological space]].
Let $\struct {R, +, *, \tau_{_R} }$ be a [[Definition:Topological Ring|topological ring]] with [[Definition:Ring Zero|zero]] $0_R$.
Let $\struct{\map C {S, R}, +, *}$ be the [[Definition:Ring of Continuous Mappings|ring of continuo... | Let $\struct {R^S, +, *}$ be the [[Definition:Ring of Mappings|ring of mappings from $S$ to $R$]].
From [[Ring of Continuous Mappings is Subring of All Mappings]]:
:$\struct{\map C {S, R}, +, *}$ is a [[Definition:Subring|subring]] of $\struct {R^S, +, *}$
From [[Induced Structure Identity]]:
:the [[Definition:Ring Z... | Zero of Ring of Continuous Mappings | https://proofwiki.org/wiki/Zero_of_Ring_of_Continuous_Mappings | https://proofwiki.org/wiki/Zero_of_Ring_of_Continuous_Mappings | [
"Rings of Continuous Mappings"
] | [
"Definition:Topological Space",
"Definition:Topological Ring",
"Definition:Ring Zero",
"Definition:Ring of Continuous Mappings",
"Definition:Ring Zero",
"Definition:Constant Mapping"
] | [
"Definition:Ring of Mappings",
"Ring of Continuous Mappings is Subring of All Mappings",
"Definition:Subring",
"Induced Structure Identity",
"Definition:Ring Zero",
"Definition:Constant Mapping",
"Zero of Subring is Zero of Ring"
] |
proofwiki-21391 | Church's Theorem | There exists no effective procedure for determining whether or not a given well-formed formula of the predicate calculus is a theorem.
That is, the decision problem for the predicate calculus is undecidable. | {{ProofWanted}}
{{Namedfor|Alonzo Church|cat = Church}} | There exists no [[Definition:Effective Procedure|effective procedure]] for determining whether or not a given [[Definition:WFF of Predicate Logic|well-formed formula]] of the [[Definition:Predicate Logic|predicate calculus]] is a [[Definition:Theorem|theorem]].
That is, the [[Definition:Decision Problem|decision probl... | {{ProofWanted}}
{{Namedfor|Alonzo Church|cat = Church}} | Church's Theorem | https://proofwiki.org/wiki/Church's_Theorem | https://proofwiki.org/wiki/Church's_Theorem | [
"Church's Theorem",
"Decision Problems",
"Mathematical Logic"
] | [
"Definition:Effective Procedure",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Predicate Logic",
"Definition:Theorem",
"Definition:Decision Problem",
"Definition:Predicate Logic",
"Definition:Undecidable"
] | [] |
proofwiki-21392 | Unity of Ring of Continuous Mappings | Let $\struct {S, \tau_{_S} }$ be a topological space.
Let $\struct {R, +, *, \tau_{_R} }$ be a topological ring with unity $1_R$.
Let $\struct{\map C {S, R}, +, *}$ be the ring of continuous mappings from $S$ to $R$.
Then:
:the unity of $\struct{\map C {S, R}, +, *}$ is the constant mapping $1_{R^S} : S \to R$ defined ... | Let $\struct {R^S, +, *}$ be the ring of mappings from $S$ to $R$.
From Ring of Continuous Mappings is Subring of All Mappings:
:$\struct{\map C {S, R}, +, *}$ is a subring of $\struct {R^S, +, *}$
From Induced Structure Identity:
:the unity of $\struct {R^S, +, *}$ is the constant mapping $1_{R^S} : S \to R$ defined b... | Let $\struct {S, \tau_{_S} }$ be a [[Definition:Topological Space|topological space]].
Let $\struct {R, +, *, \tau_{_R} }$ be a [[Definition:Topological Ring|topological ring]] with [[Definition:Unity of Ring|unity]] $1_R$.
Let $\struct{\map C {S, R}, +, *}$ be the [[Definition:Ring of Continuous Mappings|ring of con... | Let $\struct {R^S, +, *}$ be the [[Definition:Ring of Mappings|ring of mappings from $S$ to $R$]].
From [[Ring of Continuous Mappings is Subring of All Mappings]]:
:$\struct{\map C {S, R}, +, *}$ is a [[Definition:Subring|subring]] of $\struct {R^S, +, *}$
From [[Induced Structure Identity]]:
:the [[Definition:Unity ... | Unity of Ring of Continuous Mappings | https://proofwiki.org/wiki/Unity_of_Ring_of_Continuous_Mappings | https://proofwiki.org/wiki/Unity_of_Ring_of_Continuous_Mappings | [
"Rings of Continuous Mappings"
] | [
"Definition:Topological Space",
"Definition:Topological Ring",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Ring of Continuous Mappings",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Constant Mapping"
] | [
"Definition:Ring of Mappings",
"Ring of Continuous Mappings is Subring of All Mappings",
"Definition:Subring",
"Induced Structure Identity",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Constant Mapping",
"Constant Mapping is Continuous",
"Subring Containing Ring Unity has Unity",
"Defini... |
proofwiki-21393 | Commutativity of Ring of Continuous Mappings | Let $\struct {S, \tau_{_S} }$ be a topological space.
Let $\struct {R, +, *, \tau_{_R} }$ be a commutative topological ring.
Let $\struct{\map C {S, R}, +, *}$ be the ring of continuous mappings from $S$ to $R$.
Then:
:$\struct{\map C {S, R}, +, *}$ is a commutative ring | From Structure Induced by Commutative Operation is Commutative:
:the ring of mappings $\struct{R^S, +, *}$ is commutative
From Ring of Continuous Mappings is Subring of All Mappings:
:$\struct{\map C {S, R}, +, *}$ is a subring of $\struct{R^S, +, *}$
From Subring of Commutative Ring is Commutative:
:$\struct{\map C {S... | Let $\struct {S, \tau_{_S} }$ be a [[Definition:Topological Space|topological space]].
Let $\struct {R, +, *, \tau_{_R} }$ be a [[Definition:Commutative Ring|commutative]] [[Definition:Topological Ring|topological ring]].
Let $\struct{\map C {S, R}, +, *}$ be the [[Definition:Ring of Continuous Mappings|ring of conti... | From [[Structure Induced by Commutative Operation is Commutative]]:
:the [[Definition:Ring of Mappings|ring of mappings]] $\struct{R^S, +, *}$ is [[Definition:Commutative Ring|commutative]]
From [[Ring of Continuous Mappings is Subring of All Mappings]]:
:$\struct{\map C {S, R}, +, *}$ is a [[Definition:Subring|subrin... | Commutativity of Ring of Continuous Mappings | https://proofwiki.org/wiki/Commutativity_of_Ring_of_Continuous_Mappings | https://proofwiki.org/wiki/Commutativity_of_Ring_of_Continuous_Mappings | [
"Rings of Continuous Mappings"
] | [
"Definition:Topological Space",
"Definition:Commutative Ring",
"Definition:Topological Ring",
"Definition:Ring of Continuous Mappings",
"Definition:Commutative Ring"
] | [
"Structure Induced by Commutative Operation is Commutative",
"Definition:Ring of Mappings",
"Definition:Commutative Ring",
"Ring of Continuous Mappings is Subring of All Mappings",
"Definition:Subring",
"Subring of Commutative Ring is Commutative",
"Definition:Commutative Ring"
] |
proofwiki-21394 | Tangents to Circle from Point are of Equal Length | Let $\CC$ be a circle.
Let $P$ be a point in the exterior of $\CC$.
Let $PA$ and $PB$ be tangents to $\CC$ from $P$ touching $\CC$ at $A$ and $B$ respectively.
:400px
Then:
:$PA = PB$ | Let $O$ be the center of $\CC$.
Construct $OA$ and $OB$.
From Radius at Right Angle to Tangent:
:$PA \perp OA$ and $PB \perp OB$
and so $\angle OAP = \angle OBP$ which equals a right angle.
:400px
Consider the right triangles $\triangle OAP$ and $\triangle OBP$.
We have:
:$OA = OB$ by definition of radius of circle
:$\... | Let $\CC$ be a [[Definition:Circle|circle]].
Let $P$ be a [[Definition:Point|point]] in the [[Definition:Exterior|exterior]] of $\CC$.
Let $PA$ and $PB$ be [[Definition:Tangent to Circle|tangents]] to $\CC$ from $P$ touching $\CC$ at $A$ and $B$ respectively.
:[[File:Equal-Tangents-to-Circle.png|400px]]
Then:
:$P... | Let $O$ be the [[Definition:Center of Circle|center]] of $\CC$.
Construct $OA$ and $OB$.
From [[Radius at Right Angle to Tangent]]:
:$PA \perp OA$ and $PB \perp OB$
and so $\angle OAP = \angle OBP$ which equals a [[Definition:Right Angle|right angle]].
:[[File:Equal-Tangents-to-Circle-Proof.png|400px]]
Consider t... | Tangents to Circle from Point are of Equal Length | https://proofwiki.org/wiki/Tangents_to_Circle_from_Point_are_of_Equal_Length | https://proofwiki.org/wiki/Tangents_to_Circle_from_Point_are_of_Equal_Length | [
"Circles",
"Tangents to Circles"
] | [
"Definition:Circle",
"Definition:Point",
"Definition:Exterior",
"Definition:Tangent Line/Circle",
"File:Equal-Tangents-to-Circle.png"
] | [
"Definition:Circle/Center",
"Radius at Right Angle to Tangent",
"Definition:Right Angle",
"File:Equal-Tangents-to-Circle-Proof.png",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Circle/Radius",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Triangle Right-Angle-Hypotenuse-Sid... |
proofwiki-21395 | Tangents to Circle from Point subtend Equal Angles at Center | Let $\CC$ be a circle.
Let $P$ be a point in the exterior of $\CC$.
Let $PA$ and $PB$ be tangents to $\CC$ from $P$ touching $\CC$ at $A$ and $B$ respectively.
:400px
Then:
:$\angle OPA = \angle OPB$ | Let $O$ be the center of $\CC$.
Construct $OA$ and $OB$.
From Radius at Right Angle to Tangent:
:$PA \perp OA$ and $PB \perp OB$
and so $\angle OAP = \angle OBP$ which equals a right angle.
:400px
Consider the right triangles $\triangle OAP$ and $\triangle OBP$.
We have:
:$OA = OB$ by definition of radius of circle
:$\... | Let $\CC$ be a [[Definition:Circle|circle]].
Let $P$ be a [[Definition:Point|point]] in the [[Definition:Exterior|exterior]] of $\CC$.
Let $PA$ and $PB$ be [[Definition:Tangent to Circle|tangents]] to $\CC$ from $P$ touching $\CC$ at $A$ and $B$ respectively.
:[[File:Equal-Angles-Subtended-by-Tangents-to-Circle.png... | Let $O$ be the [[Definition:Center of Circle|center]] of $\CC$.
Construct $OA$ and $OB$.
From [[Radius at Right Angle to Tangent]]:
:$PA \perp OA$ and $PB \perp OB$
and so $\angle OAP = \angle OBP$ which equals a [[Definition:Right Angle|right angle]].
:[[File:Equal-Angles-Subtended-by-Tangents-to-Circle-Proof.png|... | Tangents to Circle from Point subtend Equal Angles at Center | https://proofwiki.org/wiki/Tangents_to_Circle_from_Point_subtend_Equal_Angles_at_Center | https://proofwiki.org/wiki/Tangents_to_Circle_from_Point_subtend_Equal_Angles_at_Center | [
"Circles",
"Tangents to Circles"
] | [
"Definition:Circle",
"Definition:Point",
"Definition:Exterior",
"Definition:Tangent Line/Circle",
"File:Equal-Angles-Subtended-by-Tangents-to-Circle.png"
] | [
"Definition:Circle/Center",
"Radius at Right Angle to Tangent",
"Definition:Right Angle",
"File:Equal-Angles-Subtended-by-Tangents-to-Circle-Proof.png",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Circle/Radius",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Triangle Right-... |
proofwiki-21396 | Tangents to Circle from Point subtend Equal Angles at Center/Corollary | Let $\CC$ be a circle.
Let $P$ be a point in the exterior of $\CC$.
Let $PA$ and $PB$ be tangents to $\CC$ from $P$ touching $\CC$ at $A$ and $B$ respectively.
:400px
Then $OP$ is a bisector of $\angle APB$. | From Tangents to Circle from Point subtend Equal Angles at Center:
:$\angle OPA = \angle OPB$
from which follows the result.
{{qed}} | Let $\CC$ be a [[Definition:Circle|circle]].
Let $P$ be a [[Definition:Point|point]] in the [[Definition:Exterior|exterior]] of $\CC$.
Let $PA$ and $PB$ be [[Definition:Tangent to Circle|tangents]] to $\CC$ from $P$ touching $\CC$ at $A$ and $B$ respectively.
:[[File:Equal-Angles-Subtended-by-Tangents-to-Circle.png... | From [[Tangents to Circle from Point subtend Equal Angles at Center]]:
:$\angle OPA = \angle OPB$
from which follows the result.
{{qed}} | Tangents to Circle from Point subtend Equal Angles at Center/Corollary | https://proofwiki.org/wiki/Tangents_to_Circle_from_Point_subtend_Equal_Angles_at_Center/Corollary | https://proofwiki.org/wiki/Tangents_to_Circle_from_Point_subtend_Equal_Angles_at_Center/Corollary | [
"Circles",
"Tangents to Circles"
] | [
"Definition:Circle",
"Definition:Point",
"Definition:Exterior",
"Definition:Tangent Line/Circle",
"File:Equal-Angles-Subtended-by-Tangents-to-Circle.png",
"Definition:Angle Bisector"
] | [
"Tangents to Circle from Point subtend Equal Angles at Center"
] |
proofwiki-21397 | Order of Convergence Implies Convergence | Let $\sequence {x_n}$ be an real sequence that converges to $\alpha$ with order $p$, where $p \ge 1$.
Then, $\sequence {x_n}$ converges to $\alpha$. | By definition of order of convergence, there exist a sequence $\sequence {\epsilon_n}$ and $c > 0$ such that:
:$\size {x_n - \alpha} \le \epsilon_n$
:$\ds \lim_{n \to \infty} \frac {\epsilon_{n + 1}} {\epsilon_n^p} = c$
:$p = 1 \implies c < 1$ | Let $\sequence {x_n}$ be an [[Definition:Real Sequence|real sequence]] that converges to $\alpha$ with [[Definition:Order of Convergence|order]] $p$, where $p \ge 1$.
Then, $\sequence {x_n}$ [[Definition:Convergent Real Sequence|converges]] to $\alpha$. | By definition of [[Definition:Order of Convergence|order of convergence]], there exist a [[Definition:Real Sequence|sequence]] $\sequence {\epsilon_n}$ and $c > 0$ such that:
:$\size {x_n - \alpha} \le \epsilon_n$
:$\ds \lim_{n \to \infty} \frac {\epsilon_{n + 1}} {\epsilon_n^p} = c$
:$p = 1 \implies c < 1$ | Order of Convergence Implies Convergence | https://proofwiki.org/wiki/Order_of_Convergence_Implies_Convergence | https://proofwiki.org/wiki/Order_of_Convergence_Implies_Convergence | [] | [
"Definition:Real Sequence",
"Definition:Order of Convergence",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Order of Convergence",
"Definition:Real Sequence"
] |
proofwiki-21398 | Equation of Circular Helix/Parametric Form | Let $\HH$ be a '''circular helix''' embedded in Cartesian $3$-space whose axis coincides with the $z$-axis.
$\HH$ can be described by the parametric equation:
:<nowiki>$\begin {cases}
x & = a \cos t \\
y & = a \sin t \\
z & = b t \\
\end {cases}$</nowiki>
where $t$ is the parameter. | {{ProofWanted|It needs to be proved that the tangent to $\HH$ is at a constant angle to the $z$-axis}} | Let $\HH$ be a '''[[Definition:Circular Helix|circular helix]]''' embedded in [[Definition:Cartesian 3-Space|Cartesian $3$-space]] whose [[Definition:Axis of Helix|axis]] coincides with the [[Definition:Z-Axis|$z$-axis]].
$\HH$ can be described by the [[Definition:Parametric Equation|parametric equation]]:
:<nowiki>$... | {{ProofWanted|It needs to be proved that the tangent to $\HH$ is at a constant [[Definition:Angle|angle]] to the [[Definition:Z-Axis|$z$-axis]]}} | Equation of Circular Helix/Parametric Form | https://proofwiki.org/wiki/Equation_of_Circular_Helix/Parametric_Form | https://proofwiki.org/wiki/Equation_of_Circular_Helix/Parametric_Form | [
"Circular Helices"
] | [
"Definition:Helix/Circular",
"Definition:Cartesian 3-Space",
"Definition:Helix/Axis",
"Definition:Axis/Z-Axis",
"Definition:Parametric Equation",
"Definition:Parameter"
] | [
"Definition:Angle",
"Definition:Axis/Z-Axis"
] |
proofwiki-21399 | Ring of Integers of Algebraic Number Field is UFD iff Class Number is 1 | Let $K$ be a field of algebraic numbers.
Let $\OO_K$ be the ring of integers of $K$.
Then $\OO_K$ is a unique factorization domain (UFD) {{iff}} the class number of $K$ is $1$. | {{ProofWanted|This may already exist in some form in {{ProofWiki}}. This will need to be ascertained by someone fluent in the language of algebraic number fields.}} | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]] of [[Definition:Algebraic Number|algebraic numbers]].
Let $\OO_K$ be the [[Definition:Ring of Integers of Number Field|ring of integers]] of $K$.
Then $\OO_K$ is a [[Definition:Unique Factorization Domain|unique factorization domain (UFD)]] {{iff}} the [[Def... | {{ProofWanted|This may already exist in some form in {{ProofWiki}}. This will need to be ascertained by someone fluent in the language of algebraic number fields.}} | Ring of Integers of Algebraic Number Field is UFD iff Class Number is 1 | https://proofwiki.org/wiki/Ring_of_Integers_of_Algebraic_Number_Field_is_UFD_iff_Class_Number_is_1 | https://proofwiki.org/wiki/Ring_of_Integers_of_Algebraic_Number_Field_is_UFD_iff_Class_Number_is_1 | [
"Class Groups"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Algebraic Number",
"Definition:Ring of Integers of Number Field",
"Definition:Unique Factorization Domain",
"Definition:Class Group/Class Number"
] | [] |
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