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-20600 | Reciprocal of 103 | :$\dfrac 1 {103} = 0 \cdotp \dot 00970 \, 87378 \, 64077 \, 66990 \, 29126 \, 21359 \, 223 \dot 3$ | Performing the calculation using long division:
<pre>
0.0097087378640776699029126213592233009...
--------------------------------------------
103)1.0000000000000000000000000000000000000...
927 618 927 103 309
--- --- --- --- ---
730 420 300 370 1000... | :$\dfrac 1 {103} = 0 \cdotp \dot 00970 \, 87378 \, 64077 \, 66990 \, 29126 \, 21359 \, 223 \dot 3$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.0097087378640776699029126213592233009...
--------------------------------------------
103)1.0000000000000000000000000000000000000...
927 618 927 103 309
--- --- --- --- ---
730 ... | Reciprocal of 103 | https://proofwiki.org/wiki/Reciprocal_of_103 | https://proofwiki.org/wiki/Reciprocal_of_103 | [
"103",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:103",
"Category:Examples of Reciprocals"
] |
proofwiki-20601 | Reciprocal of 23 | :$\dfrac 1 {23} = 0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3$ | Performing the calculation using long division:
<pre>
0.043478260869565217391304...
-----------------------------
23)1.00000000000000000000000000
92 184 23
-- --- --
80 160 170
69 138 161
-- --- ---
110 220 90
92 207 69
... | :$\dfrac 1 {23} = 0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.043478260869565217391304...
-----------------------------
23)1.00000000000000000000000000
92 184 23
-- --- --
80 160 170
69 138 161
-- --- ---
110 220 ... | Reciprocal of 23 | https://proofwiki.org/wiki/Reciprocal_of_23 | https://proofwiki.org/wiki/Reciprocal_of_23 | [
"23",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:23",
"Category:Examples of Reciprocals"
] |
proofwiki-20602 | Reciprocal of 29 | :$\dfrac 1 {29} = 0 \cdotp \dot 03448 \, 27586 \, 20689 \, 65517 \, 24137 \, 93 \dot 1$ | Performing the calculation using long division:
<pre>
0.034482758620689655172413793103...
------------------------------------
29)1.000000000000000000000000000000...
87
--
130 180 150 230
116 174 145 203
--- --- --- ---
140 60 50 270
1... | :$\dfrac 1 {29} = 0 \cdotp \dot 03448 \, 27586 \, 20689 \, 65517 \, 24137 \, 93 \dot 1$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.034482758620689655172413793103...
------------------------------------
29)1.000000000000000000000000000000...
87
--
130 180 150 230
116 174 145 203
--- --- --- ---
140... | Reciprocal of 29 | https://proofwiki.org/wiki/Reciprocal_of_29 | https://proofwiki.org/wiki/Reciprocal_of_29 | [
"29",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:29",
"Category:Examples of Reciprocals"
] |
proofwiki-20603 | Reciprocal of 31 | :$\dfrac 1 {31} = 0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$ | Performing the calculation using long division:
<pre>
0.03225806451612903...
-----------------------
31)1.00000000000000000000
93 155
-- ---
70 50
62 31
-- --
80 190
62 186
-- ---
180 40
155 31
--- ... | :$\dfrac 1 {31} = 0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.03225806451612903...
-----------------------
31)1.00000000000000000000
93 155
-- ---
70 50
62 31
-- --
80 190
62 186
-- ---
180 40
... | Reciprocal of 31 | https://proofwiki.org/wiki/Reciprocal_of_31 | https://proofwiki.org/wiki/Reciprocal_of_31 | [
"31",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:31",
"Category:Examples of Reciprocals"
] |
proofwiki-20604 | Reciprocal of 37 | :$\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$ | Performing the calculation using long division:
<pre>
0.027...
--------
37)1.00000
74
--
260
259
---
100
74
---
...
</pre>
It is to be noted that this is because $999 = 27 \times 37$.
{{qed}}
Category:37
Category:Examples of Reciprocals
b8x8urhl335q0n7odqvo8rn... | :$\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.027...
--------
37)1.00000
74
--
260
259
---
100
74
---
...
</pre>
It is to be noted that this is because $999 = 27 \times 37$.
{{qed}}
[[Category:37]]
[[Category:Examples ... | Reciprocal of 37 | https://proofwiki.org/wiki/Reciprocal_of_37 | https://proofwiki.org/wiki/Reciprocal_of_37 | [
"37",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:37",
"Category:Examples of Reciprocals"
] |
proofwiki-20605 | Reciprocal of 41 | :$\dfrac 1 {41} = 0 \cdotp \dot 0243 \dot 9$ | Performing the calculation using long division:
<pre>
0.0243902...
-------------
41)1.0000000...
82
----
180
164
---
160
123
---
370
369
---
100
82
---
...
</pre>
{{qed}}
Category:41
Category:Examples o... | :$\dfrac 1 {41} = 0 \cdotp \dot 0243 \dot 9$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.0243902...
-------------
41)1.0000000...
82
----
180
164
---
160
123
---
370
369
---
100
82
---
...
</pre>
{{qed}}
... | Reciprocal of 41 | https://proofwiki.org/wiki/Reciprocal_of_41 | https://proofwiki.org/wiki/Reciprocal_of_41 | [
"41",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:41",
"Category:Examples of Reciprocals"
] |
proofwiki-20606 | Composition of Mapping with Mapping Restricted to Image | Let $A, B, C$ be sets.
Let $f : A \to B, g: B \to C$ be mappings.
Let $g \circ f : A \to C$ denote the composite mapping of $f$ with $g$.
Let $f \sqbrk A$ denote the image of $A$ under $f$.
Let $f \restriction_{A \times f \sqbrk A} : A \to f \sqbrk A$ denote the restriction of $f$ to $A \times f \sqbrk A$.
Let $g \rest... | From Restriction of Mapping is Mapping:
:$f \restriction_{A \times f \sqbrk A} : A \to f \sqbrk A$ is a mapping
and
:$g \restriction_{f \sqbrk A} : f \sqbrk A \to C$ is a mapping
By definition of composite mapping:
:$g \restriction_{f \sqbrk A} \circ f \restriction_{A \mathop \times f \sqbrk A} \mathop : A \to C$ is a... | Let $A, B, C$ be [[Definition:Set|sets]].
Let $f : A \to B, g: B \to C$ be [[Definition:Mapping|mappings]].
Let $g \circ f : A \to C$ denote the [[Definition:Composite Mapping|composite mapping]] of $f$ with $g$.
Let $f \sqbrk A$ denote the [[Definition:Image of Subset under Mapping|image]] of $A$ under $f$.
Let ... | From [[Restriction of Mapping is Mapping (General Result)|Restriction of Mapping is Mapping]]:
:$f \restriction_{A \times f \sqbrk A} : A \to f \sqbrk A$ is a [[Definition:Mapping|mapping]]
and
:$g \restriction_{f \sqbrk A} : f \sqbrk A \to C$ is a [[Definition:Mapping|mapping]]
By definition of [[Definition:Composit... | Composition of Mapping with Mapping Restricted to Image | https://proofwiki.org/wiki/Composition_of_Mapping_with_Mapping_Restricted_to_Image | https://proofwiki.org/wiki/Composition_of_Mapping_with_Mapping_Restricted_to_Image | [
"Mapping Theory",
"Composite Mappings",
"Restrictions"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Composition of Mappings",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Restriction/Mapping",
"Definition:Restriction/Mapping",
"Definition:Composition of Mappings"
] | [
"Restriction of Mapping is Mapping/General Result",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Composition of Mappings",
"Definition:Well-Defined",
"Definition:Mapping",
"Equality of Mappings",
"Category:Mapping Theory",
"Category:Composite Mappings",
"Category:Restrictions"
] |
proofwiki-20607 | Reciprocal of 43 | :$\dfrac 1 {43} = 0 \cdotp \dot 02325 \, 58139 \, 53488 \, 37209 \, \dot 3$ | Performing the calculation using long division:
<pre>
0.02325581395348837209302...
-----------------------------
43)1.00000000000000000000000...
86
----
140 410 310
129 387 301
--- --- ---
110 230 90
86 215 86
--- --- --
... | :$\dfrac 1 {43} = 0 \cdotp \dot 02325 \, 58139 \, 53488 \, 37209 \, \dot 3$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.02325581395348837209302...
-----------------------------
43)1.00000000000000000000000...
86
----
140 410 310
129 387 301
--- --- ---
110 230 90
86 215 86... | Reciprocal of 43 | https://proofwiki.org/wiki/Reciprocal_of_43 | https://proofwiki.org/wiki/Reciprocal_of_43 | [
"43",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:43",
"Category:Examples of Reciprocals"
] |
proofwiki-20608 | Reciprocal of 53 | :$\dfrac 1 {53} = 0 \cdotp \dot 01886 \, 79245 \, 28 \dot 3$ | Performing the calculation using long division:
<pre>
0.018867924528301...
---------------------
53)1.000000000000000000
53 212
-- ---
470 280
424 265
--- ---
460 150
424 106
--- ---
360 440
318 424
--- ---
... | :$\dfrac 1 {53} = 0 \cdotp \dot 01886 \, 79245 \, 28 \dot 3$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.018867924528301...
---------------------
53)1.000000000000000000
53 212
-- ---
470 280
424 265
--- ---
460 150
424 106
--- ---
360 440
... | Reciprocal of 53 | https://proofwiki.org/wiki/Reciprocal_of_53 | https://proofwiki.org/wiki/Reciprocal_of_53 | [
"53",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:53",
"Category:Examples of Reciprocals"
] |
proofwiki-20609 | Reciprocal of 49 | :$\dfrac 1 {49} = 0 \cdotp \dot 02040 \, 81632 \, 65306 \, 12244 \, 89795 \, 91836 \, 73469 \, 38775 \, 5 \dot 1$ | Performing the calculation using long division:
<pre>
0.02040816326530612244897959183673469387755102...
---------------------------------------------
49)1.00000000000000000000000000000000000000000000000
98 294 196 441 196 245
-- --- --- --- --- ---
200 260 ... | :$\dfrac 1 {49} = 0 \cdotp \dot 02040 \, 81632 \, 65306 \, 12244 \, 89795 \, 91836 \, 73469 \, 38775 \, 5 \dot 1$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.02040816326530612244897959183673469387755102...
---------------------------------------------
49)1.00000000000000000000000000000000000000000000000
98 294 196 441 196 245
-- --- --- --- ... | Reciprocal of 49 | https://proofwiki.org/wiki/Reciprocal_of_49 | https://proofwiki.org/wiki/Reciprocal_of_49 | [
"49",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:49",
"Category:Examples of Reciprocals"
] |
proofwiki-20610 | Long Period Prime/Examples/61 | The prime number $61$ is a long period prime:
:$\dfrac 1 {61} = 0 \cdotp \dot 01639 \, 34426 \, 22950 \, 81967 \, 21311 \, 47540 \, 98360 \, 65573 \, 77049 \, 18032 \, 78688 \, 5245 \dot 9$ | From Reciprocal of $61$:
{{:Reciprocal of 61}}
Counting the digits, it is seen that this has a period of recurrence of $60$.
Inspecting the expansion and counting the digits, we find that each one appears exactly $6$ times.
Hence the result.
{{qed}} | The [[Definition:Prime Number|prime number]] $61$ is a [[Definition:Long Period Prime|long period prime]]:
:$\dfrac 1 {61} = 0 \cdotp \dot 01639 \, 34426 \, 22950 \, 81967 \, 21311 \, 47540 \, 98360 \, 65573 \, 77049 \, 18032 \, 78688 \, 5245 \dot 9$ | From [[Reciprocal of 61|Reciprocal of $61$]]:
{{:Reciprocal of 61}}
Counting the digits, it is seen that this has a [[Definition:Period of Recurrence|period of recurrence]] of $60$.
Inspecting the expansion and counting the [[Definition:Digit|digits]], we find that each one appears exactly $6$ times.
Hence the result... | Long Period Prime/Examples/61 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/61 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/61 | [
"61",
"Examples of Long Period Primes"
] | [
"Definition:Prime Number",
"Definition:Long Period Prime"
] | [
"Reciprocal of 61",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Digit"
] |
proofwiki-20611 | Operator Norm of Quotient Mapping in Quotient Normed Vector Space is 1 | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ be a normed vector space over $\Bbb F$.
Let $N$ be a closed linear subspace of $X$.
Let $\struct {X/N, \norm {\, \cdot \,}_{X/N} }$ be the normed quotient vector space associated with quotient vector space $X/N$.
Let $\pi : X \to X/N$ be the quotient mapping associated with $X/N... | Let $B_X$ be the unit open ball in $\struct {X, \norm {\, \cdot \,} }$.
Let $B_{X/N}$ be the unit open ball in $\struct {X/N, \norm {\, \cdot \,} }$.
From Quotient Mapping is Bounded in Normed Quotient Vector Space, $\pi$ is bounded.
We have:
{{begin-eqn}}
{{eqn | l = \norm \pi_{\map B {X, X/N} }
| r = \sup_{x \in B... | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ be a [[Definition:Normed Vector Space|normed vector space]] over $\Bbb F$.
Let $N$ be a [[Definition:Closed Linear Subspace|closed linear subspace]] of $X$.
Let $\struct {X/N, \norm {\, \cdot \,}_{X/N} }$ be the [[Definition:Normed Quotient Vector Space|normed quotient vector... | Let $B_X$ be the [[Definition:Unit Open Ball|unit open ball]] in $\struct {X, \norm {\, \cdot \,} }$.
Let $B_{X/N}$ be the [[Definition:Unit Open Ball|unit open ball]] in $\struct {X/N, \norm {\, \cdot \,} }$.
From [[Quotient Mapping is Bounded in Normed Quotient Vector Space]], $\pi$ is [[Definition:Bounded Linear T... | Operator Norm of Quotient Mapping in Quotient Normed Vector Space is 1 | https://proofwiki.org/wiki/Operator_Norm_of_Quotient_Mapping_in_Quotient_Normed_Vector_Space_is_1 | https://proofwiki.org/wiki/Operator_Norm_of_Quotient_Mapping_in_Quotient_Normed_Vector_Space_is_1 | [
"Quotient Mappings",
"Normed Quotient Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Closed Linear Subspace",
"Definition:Normed Quotient Vector Space",
"Definition:Quotient Vector Space",
"Definition:Quotient Mapping",
"Definition:Norm/Bounded Linear Transformation"
] | [
"Definition:Unit Open Ball",
"Definition:Unit Open Ball",
"Quotient Mapping is Bounded in Normed Quotient Vector Space",
"Definition:Bounded Linear Transformation",
"Quotient Mapping Maps Unit Open Ball in Normed Vector Space to Unit Open Ball in Normed Quotient Vector Space",
"Category:Quotient Mappings"... |
proofwiki-20612 | Reciprocal of 239 | :$\dfrac 1 {239} = 0 \cdotp \dot 00418 \, 4 \dot 1$ | Performing the calculation using long division:
<pre>
0.0041841004...
-------------
239)1.0000000000...
956
----
440
239
---
2010
1912
----
980
956
---
240
239
---
1000
... | :$\dfrac 1 {239} = 0 \cdotp \dot 00418 \, 4 \dot 1$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.0041841004...
-------------
239)1.0000000000...
956
----
440
239
---
2010
1912
----
980
956
---
240
239
... | Reciprocal of 239 | https://proofwiki.org/wiki/Reciprocal_of_239 | https://proofwiki.org/wiki/Reciprocal_of_239 | [
"239",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:239",
"Category:Examples of Reciprocals"
] |
proofwiki-20613 | Reciprocal of 73 | :$\dfrac 1 {73} = 0 \cdotp \dot 01369 \, 86 \dot 3$ | Performing the calculation using long division:
<pre>
0.0136986301...
-------------
73)1.0000000000...
73
----
270
219
---
510
438
---
720
657
---
630
584
---
460
438
---
220
219
... | :$\dfrac 1 {73} = 0 \cdotp \dot 01369 \, 86 \dot 3$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.0136986301...
-------------
73)1.0000000000...
73
----
270
219
---
510
438
---
720
657
---
630
584
---
460
438
---
... | Reciprocal of 73 | https://proofwiki.org/wiki/Reciprocal_of_73 | https://proofwiki.org/wiki/Reciprocal_of_73 | [
"73",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:73",
"Category:Examples of Reciprocals"
] |
proofwiki-20614 | Reciprocal of 79 | :$\dfrac 1 {79} = 0 \cdotp \dot 01265 \, 82278 \, 48 \dot 1$ | Performing the calculation using long division:
<pre>
0.012658227848101...
---------------------
79)1.000000000000000...
79
----
210 670
158 632
--- ---
520 380
474 316
--- ---
460 640
395 632
--- ---
650 80
... | :$\dfrac 1 {79} = 0 \cdotp \dot 01265 \, 82278 \, 48 \dot 1$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.012658227848101...
---------------------
79)1.000000000000000...
79
----
210 670
158 632
--- ---
520 380
474 316
--- ---
460 640
395 632
--- ... | Reciprocal of 79 | https://proofwiki.org/wiki/Reciprocal_of_79 | https://proofwiki.org/wiki/Reciprocal_of_79 | [
"79",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:79",
"Category:Examples of Reciprocals"
] |
proofwiki-20615 | Reciprocal of 59 | :$\dfrac 1 {59} = 0 \cdotp \dot 01694 \, 91525 \, 42372 \, 88135 \, 59322 \, 03389 \, 83050 \, 84745 \, 76271 \, 18644 \, 06779 \, 66 \dot 1$ | Performing the calculation using long division:
<pre>
0.016949152542372881355932203389830508474576271186440677966101...
------------------------------------------------------------------
59)1.000000000000000000000000000000000000000000000000000000000000...
59
----
410 150 170 350 230 500 ... | :$\dfrac 1 {59} = 0 \cdotp \dot 01694 \, 91525 \, 42372 \, 88135 \, 59322 \, 03389 \, 83050 \, 84745 \, 76271 \, 18644 \, 06779 \, 66 \dot 1$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.016949152542372881355932203389830508474576271186440677966101...
------------------------------------------------------------------
59)1.000000000000000000000000000000000000000000000000000000000000...
59
----
410 15... | Reciprocal of 59 | https://proofwiki.org/wiki/Reciprocal_of_59 | https://proofwiki.org/wiki/Reciprocal_of_59 | [
"59",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:59",
"Category:Examples of Reciprocals"
] |
proofwiki-20616 | Reciprocal of 67 | :$\dfrac 1 {67} = 0 \cdotp \dot 01492 \, 53731 \, 34328 \, 35820 \, 89552 \, 23880 \, 59 \dot 7$ | Performing the calculation using long division:
<pre>
0.01492537313432835820895522388059701...
-----------------------------------------
67)1.00000000000000000000000000000000000...
67
----
330 210 560 640 590
268 201 536 603 536
--- --- --- --- ---
620 90 ... | :$\dfrac 1 {67} = 0 \cdotp \dot 01492 \, 53731 \, 34328 \, 35820 \, 89552 \, 23880 \, 59 \dot 7$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.01492537313432835820895522388059701...
-----------------------------------------
67)1.00000000000000000000000000000000000...
67
----
330 210 560 640 590
268 201 536 603 536
--- --- ---... | Reciprocal of 67 | https://proofwiki.org/wiki/Reciprocal_of_67 | https://proofwiki.org/wiki/Reciprocal_of_67 | [
"67",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:67",
"Category:Examples of Reciprocals"
] |
proofwiki-20617 | Reciprocal of 71 | :$\dfrac 1 {71} = 0 \cdotp \dot 01408 \, 45070 \, 42253 \, 52112 \, 67605 \, 63380 \, 2816 \dot 9$ | Performing the calculation using long division:
<pre>
0.0140845070422535211267605633802816901...
-------------------------------------------
71)1.0000000000000000000000000000000000000...
71
----
290 160 80 400 580
284 142 71 355 568
--- --- -- --- --... | :$\dfrac 1 {71} = 0 \cdotp \dot 01408 \, 45070 \, 42253 \, 52112 \, 67605 \, 63380 \, 2816 \dot 9$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.0140845070422535211267605633802816901...
-------------------------------------------
71)1.0000000000000000000000000000000000000...
71
----
290 160 80 400 580
284 142 71 355 568
-... | Reciprocal of 71 | https://proofwiki.org/wiki/Reciprocal_of_71 | https://proofwiki.org/wiki/Reciprocal_of_71 | [
"71",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:71",
"Category:Examples of Reciprocals"
] |
proofwiki-20618 | Reciprocal of 83 | :$\dfrac 1 {83} = 0 \cdotp \dot 01204 \, 81927 \, 71084 \, 33734 \, 93975 \, 90361 \, 44578 \, 31325 \, \dot 3$ | Performing the calculation using long division:
<pre>
0.0120481927710843373493975903614457831325301...
-------------------------------------------------
83)1.0000000000000000000000000000000000000000000...
83
----
170 640 310 810 120 260 100
166 581 249 747 83 249 8... | :$\dfrac 1 {83} = 0 \cdotp \dot 01204 \, 81927 \, 71084 \, 33734 \, 93975 \, 90361 \, 44578 \, 31325 \, \dot 3$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.0120481927710843373493975903614457831325301...
-------------------------------------------------
83)1.0000000000000000000000000000000000000000000...
83
----
170 640 310 810 120 260 100
166 581 ... | Reciprocal of 83 | https://proofwiki.org/wiki/Reciprocal_of_83 | https://proofwiki.org/wiki/Reciprocal_of_83 | [
"83",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:83",
"Category:Examples of Reciprocals"
] |
proofwiki-20619 | Long Period Prime/Examples/59 | The prime number $59$ is a long period prime:
:$\dfrac 1 {59} = 0 \cdotp \dot 01694 \, 91525 \, 42372 \, 88135 \, 59322 \, 03389 \, 83050 \, 84745 \, 76271 \, 18644 \, 06779 \, 66 \dot 1$ | From Reciprocal of $59$:
{{:Reciprocal of 59}}
Counting the digits, it is seen that this has a period of recurrence of $58$.
Hence the result.
{{qed}}
Category:59
Category:Examples of Long Period Primes
e79e661a2xglmq24dw1daqzlozridby | The [[Definition:Prime Number|prime number]] $59$ is a [[Definition:Long Period Prime|long period prime]]:
:$\dfrac 1 {59} = 0 \cdotp \dot 01694 \, 91525 \, 42372 \, 88135 \, 59322 \, 03389 \, 83050 \, 84745 \, 76271 \, 18644 \, 06779 \, 66 \dot 1$ | From [[Reciprocal of 59|Reciprocal of $59$]]:
{{:Reciprocal of 59}}
Counting the digits, it is seen that this has a [[Definition:Period of Recurrence|period of recurrence]] of $58$.
Hence the result.
{{qed}}
[[Category:59]]
[[Category:Examples of Long Period Primes]]
e79e661a2xglmq24dw1daqzlozridby | Long Period Prime/Examples/59 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/59 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/59 | [
"59",
"Examples of Long Period Primes"
] | [
"Definition:Prime Number",
"Definition:Long Period Prime"
] | [
"Reciprocal of 59",
"Definition:Basis Expansion/Recurrence/Period",
"Category:59",
"Category:Examples of Long Period Primes"
] |
proofwiki-20620 | Long Period Prime/Examples/97 | The prime number $97$ is a long period prime:
:$\dfrac 1 {97} = 0 \cdotp \dot 01030 \, 92783 \, 50515 \, 46391 \, 75257 \, 73195 \, 87628 \, 86597 \, 93814 \, 43298 \, 96907 \, 21649 \, 48453 \, 60824 \, 74226 \, 80412 \, 37113 \, 40206 \, 18556 \, \dot 7$ | From Reciprocal of $97$:
{{:Reciprocal of 97}}
Counting the digits, it is seen that this has a period of recurrence of $96$.
Hence the result.
{{qed}} | The [[Definition:Prime Number|prime number]] $97$ is a [[Definition:Long Period Prime|long period prime]]:
:$\dfrac 1 {97} = 0 \cdotp \dot 01030 \, 92783 \, 50515 \, 46391 \, 75257 \, 73195 \, 87628 \, 86597 \, 93814 \, 43298 \, 96907 \, 21649 \, 48453 \, 60824 \, 74226 \, 80412 \, 37113 \, 40206 \, 18556 \, \dot 7$ | From [[Reciprocal of 97|Reciprocal of $97$]]:
{{:Reciprocal of 97}}
Counting the digits, it is seen that this has a [[Definition:Period of Recurrence|period of recurrence]] of $96$.
Hence the result.
{{qed}} | Long Period Prime/Examples/97 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/97 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/97 | [
"97",
"Examples of Long Period Primes"
] | [
"Definition:Prime Number",
"Definition:Long Period Prime"
] | [
"Reciprocal of 97",
"Definition:Basis Expansion/Recurrence/Period"
] |
proofwiki-20621 | Long Period Prime/Examples/131 | The prime number $131$ is a long period prime:
:$\dfrac 1 {131} = 0 \cdotp \dot 00763 \, 35877 \, 86259 \, 54198 \, 47328 \, 24427 \, 48091 \, 60305 \, 34351 \, 14503 \, 81679 \, 38931 \, 29770 \, 99236 \, 64122 \, 13740 \, 45801 \, 52671 \, 75572 \, 51908 \, 39694 \, 65648 \, 85496 \, 18320 \, 61068 \, 7022 \dot 9$
It... | From Reciprocal of $131$:
{{:Reciprocal of 131}}
Counting the digits, it is seen that this has a period of recurrence of $130$.
Inspecting the expansion and counting the digits, we find that each one appears exactly $13$ times.
Hence the result.
{{qed}} | The [[Definition:Prime Number|prime number]] $131$ is a [[Definition:Long Period Prime|long period prime]]:
:$\dfrac 1 {131} = 0 \cdotp \dot 00763 \, 35877 \, 86259 \, 54198 \, 47328 \, 24427 \, 48091 \, 60305 \, 34351 \, 14503 \, 81679 \, 38931 \, 29770 \, 99236 \, 64122 \, 13740 \, 45801 \, 52671 \, 75572 \, 51908 \,... | From [[Reciprocal of 131|Reciprocal of $131$]]:
{{:Reciprocal of 131}}
Counting the digits, it is seen that this has a [[Definition:Period of Recurrence|period of recurrence]] of $130$.
Inspecting the expansion and counting the [[Definition:Digit|digits]], we find that each one appears exactly $13$ times.
Hence the r... | Long Period Prime/Examples/131 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/131 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/131 | [
"131",
"Examples of Long Period Primes"
] | [
"Definition:Prime Number",
"Definition:Long Period Prime",
"Definition:Digit"
] | [
"Reciprocal of 131",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Digit"
] |
proofwiki-20622 | String is Substring of Itself | Let $S$ be a string.
Then $S$ is a substring of itself. | By definition, a string $T$ is a substring of $S$ in $\AA$ {{iff}}:
:$S = S_1 T S_2$
where:
:$S_1$ and $S_2$ are strings in $\AA$ (possibly null)
:$S_1 T S_2$ is the concatenation of $S_1$, $T$ and $S_2$.
Let $S_1$ and $S_2$ both be the null string.
Then it follows that:
:$S = T$
Hence the result.
{{qed}}
Category:Subs... | Let $S$ be a [[Definition:String|string]].
Then $S$ is a [[Definition:Substring|substring]] of itself. | By definition, a [[Definition:String|string]] $T$ is a [[Definition:Substring|substring]] of $S$ in $\AA$ {{iff}}:
:$S = S_1 T S_2$
where:
:$S_1$ and $S_2$ are [[Definition:String|strings]] in $\AA$ (possibly [[Definition:Null String|null]])
:$S_1 T S_2$ is the [[Definition:Concatenation (Formal Systems)|concatenation]... | String is Substring of Itself | https://proofwiki.org/wiki/String_is_Substring_of_Itself | https://proofwiki.org/wiki/String_is_Substring_of_Itself | [
"Substrings"
] | [
"Definition:String",
"Definition:Substring"
] | [
"Definition:String",
"Definition:Substring",
"Definition:String",
"Definition:Null String",
"Definition:Concatenation (Formal Systems)",
"Definition:Null String",
"Category:Substrings"
] |
proofwiki-20623 | Restriction of Mapping is Mapping/General Result | Let $f: S \to T$ be a mapping.
Let $X \subseteq S$.
Let $f \sqbrk X \subseteq Y \subseteq T$.
Let $f \restriction_{X \times Y}$ be the restriction of $f$ to $X \times Y$.
Then $f \restriction_{X \times Y}: X \to Y$ is a mapping:
: whose domain is $X$
: whose preimage is $X$
: whose codomain is $Y$. | === $f \restriction_{X \times Y}$ is Many-to-one ===
We have:
{{begin-eqn}}
{{eqn | q = \forall x \in X
| l = \tuple {x, y_1}, \tuple {x, y_2} \in f \restriction_{X \times Y}
| o = \leadsto
| r = \tuple {x, y_1}, \tuple {x, y_2} \in f \cap \paren{X \times Y}
| c = {{Defof|Restriction of Mapping}... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $X \subseteq S$.
Let $f \sqbrk X \subseteq Y \subseteq T$.
Let $f \restriction_{X \times Y}$ be the [[Definition:Restriction of Mapping|restriction of $f$ to $X \times Y$]].
Then $f \restriction_{X \times Y}: X \to Y$ is a [[Definition:Mapping|mapping]]:
: ... | === $f \restriction_{X \times Y}$ is Many-to-one ===
We have:
{{begin-eqn}}
{{eqn | q = \forall x \in X
| l = \tuple {x, y_1}, \tuple {x, y_2} \in f \restriction_{X \times Y}
| o = \leadsto
| r = \tuple {x, y_1}, \tuple {x, y_2} \in f \cap \paren{X \times Y}
| c = {{Defof|Restriction of Mapping}... | Restriction of Mapping is Mapping/General Result | https://proofwiki.org/wiki/Restriction_of_Mapping_is_Mapping/General_Result | https://proofwiki.org/wiki/Restriction_of_Mapping_is_Mapping/General_Result | [
"Restrictions"
] | [
"Definition:Mapping",
"Definition:Restriction/Mapping",
"Definition:Mapping",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Preimage/Mapping/Mapping",
"Definition:Codomain (Set Theory)/Mapping"
] | [
"Intersection is Subset"
] |
proofwiki-20624 | Preimage of Open Sets forms Basis if Continuous Mappings Separate Points from Closed Sets | Let $X$ be a topological space.
Let $\family {Y_i}_{i \mathop \in I}$ be an indexed family of topological 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.
Let $\family {f_i}_{i \mathop \in I}$ separate points from closed sets.
Let $\BB = \s... | By definition of continuous:
:$\forall B \in \BB : B$ is open in $X$
Let $U \subseteq X$ be open in $X$.
Let $x \in U$.
By definition of closed subset:
:$X \setminus U$ is closed in $X$
By definition of mappings separating points from closed sets:
:$\exists i \in I : \map {f_i} x \notin f_i \sqbrk {X \setminus U}^-$
He... | 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:Topological Space|topological spaces]] for some [[Definition:Indexing Set|indexing set]] $I$.
Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be ... | By definition of [[Definition:Continuous Mapping (Topology)|continuous]]:
:$\forall B \in \BB : B$ is [[Definition:Open Set (Topology)|open]] in $X$
Let $U \subseteq X$ be [[Definition:Open Set (Topology)|open]] in $X$.
Let $x \in U$.
By definition of [[Definition:Closed Set (Topology)|closed subset]]:
:$X \setmin... | Preimage of Open Sets forms Basis if Continuous Mappings Separate Points from Closed Sets | https://proofwiki.org/wiki/Preimage_of_Open_Sets_forms_Basis_if_Continuous_Mappings_Separate_Points_from_Closed_Sets | https://proofwiki.org/wiki/Preimage_of_Open_Sets_forms_Basis_if_Continuous_Mappings_Separate_Points_from_Closed_Sets | [
"Continuous Mappings"
] | [
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Continuous Mapping (Topology)",
"Definition:Mappings Separating Points from Closed Sets",
"Definition:Basis (Topology)/Analytic Bas... | [
"Definition:Continuous Mapping (Topology)",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Mappings Separating Points from Closed Sets",
"Definition:Preimage/Mapping/Mapping",
"Topological Closure is Close... |
proofwiki-20625 | Group of Units in Unital Banach Algebra is Open | Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra.
Let $\map G A$ be the group of units of $A$.
Then $\map G A$ is open in $A$. | Let $x \in \map G A$.
We find an open neighborhood of $x$ contained in $\map G A$.
Clearly $x^{-1} \ne \mathbf 0_A$, so $\norm {x^{-1} } > 0$.
We have, for $y \in A$:
{{begin-eqn}}
{{eqn | l = \norm {1 - x^{-1} y}
| r = \norm {x^{-1} \paren {x - y} }
}}
{{eqn | o = \le
| r = \norm {x^{-1} } \norm {x - y}
}}
{{end... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital Banach algebra]].
Let $\map G A$ be the [[Definition:Group of Units|group of units]] of $A$.
Then $\map G A$ is [[Definition:Open Set in Normed Vector Space|open]] in $A$. | Let $x \in \map G A$.
We find an [[Definition:Open Neighborhood|open neighborhood]] of $x$ contained in $\map G A$.
Clearly $x^{-1} \ne \mathbf 0_A$, so $\norm {x^{-1} } > 0$.
We have, for $y \in A$:
{{begin-eqn}}
{{eqn | l = \norm {1 - x^{-1} y}
| r = \norm {x^{-1} \paren {x - y} }
}}
{{eqn | o = \le
| r = \... | Group of Units in Unital Banach Algebra is Open | https://proofwiki.org/wiki/Group_of_Units_in_Unital_Banach_Algebra_is_Open | https://proofwiki.org/wiki/Group_of_Units_in_Unital_Banach_Algebra_is_Open | [
"Unital Banach Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:Group of Units",
"Definition:Open Set/Normed Vector Space"
] | [
"Definition:Open Neighborhood",
"Element of Unital Banach Algebra Close to Identity is Invertible",
"Definition:Invertible Element",
"Definition:Group",
"Group of Units is Group",
"Definition:Open Ball",
"Definition:Open Ball/Center",
"Definition:Open Ball/Radius",
"Definition:Open Set/Normed Vector... |
proofwiki-20626 | Necessary and Sufficient Condition for Diagonal Operator to be Invertible | Let $\mathbb F \in \set {\R, \C}$.
Let $\sequence {\lambda_n}_{n \mathop \in \N_{> 0} }$ be a bounded sequence in $\mathbb F$.
Let $\ell^2$ be the $2$-sequence space.
Let $\map {CL} {\ell^2} := \map {CL} {\ell^2, \ell^2}$ be the continuous linear transformation space on $\ell^2$.
Let $\Lambda \in \map {CL} {\ell^2}$ be... | === Necessary Condition ===
Suppose $\ds \inf_{n \mathop \in \N_{> 0} } \sequence {\size {\lambda_n} } > 0$.
Then:
{{begin-eqn}}
{{eqn | q = \forall k \in \N_{> 0}
| l = \size {\lambda_k}
| o = \ge
| r = \inf_{n \mathop \in \N_{> 0} } \sequence {\size {\lambda_n} }
}}
{{eqn | o = >
| r = 0
}}
{{... | Let $\mathbb F \in \set {\R, \C}$.
Let $\sequence {\lambda_n}_{n \mathop \in \N_{> 0} }$ be a [[Definition:Bounded Complex Sequence|bounded sequence]] in $\mathbb F$.
Let $\ell^2$ be the [[Definition:P-Sequence Space|$2$-sequence space]].
Let $\map {CL} {\ell^2} := \map {CL} {\ell^2, \ell^2}$ be the [[Definition:Con... | === Necessary Condition ===
Suppose $\ds \inf_{n \mathop \in \N_{> 0} } \sequence {\size {\lambda_n} } > 0$.
Then:
{{begin-eqn}}
{{eqn | q = \forall k \in \N_{> 0}
| l = \size {\lambda_k}
| o = \ge
| r = \inf_{n \mathop \in \N_{> 0} } \sequence {\size {\lambda_n} }
}}
{{eqn | o = >
| r = 0
}}... | Necessary and Sufficient Condition for Diagonal Operator to be Invertible | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_Diagonal_Operator_to_be_Invertible | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_Diagonal_Operator_to_be_Invertible | [
"Diagonal Operator",
"Inverse Mappings"
] | [
"Definition:Bounded Sequence/Complex",
"Definition:P-Sequence Space",
"Definition:Continuous Linear Transformation Space",
"Definition:Diagonal Operator",
"Definition:Invertible Continuous Linear Operator",
"Definition:Infimum of Real Sequence"
] | [
"Definition:Mapping",
"Definition:Invertible Continuous Linear Operator",
"Invertible Continuous Linear Operator has Unique Inverse",
"Definition:Inverse of Continuous Linear Operator",
"Definition:Invertible Continuous Linear Operator",
"Invertible Continuous Linear Operator has Unique Inverse",
"Defin... |
proofwiki-20627 | Banach Space Valued Function is Analytic iff Weakly Analytic | Let $U$ be an open subset of $\C$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a Banach space over $\C$.
Let $f : U \to X$ be a function.
Then $f$ is analytic {{iff}} it is weakly analytic. | === Necessary Condition ===
Suppose that $f$ is analytic.
Define $f' : U \to X$ by:
:$\ds \map {f'} z = \lim_{w \mathop \to z} \frac {\map f w - \map f z} {w - z}$
for each $z \in U$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,} }$.
Let $\phi \in ... | Let $U$ be an [[Definition:Open Set (Complex Analysis)|open subset]] of $\C$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Banach Space|Banach space]] over $\C$.
Let $f : U \to X$ be a [[Definition:Function|function]].
Then $f$ is [[Definition:Analytic Function/Banach Space Valued Function|analytic]] {... | === Necessary Condition ===
Suppose that $f$ is [[Definition:Analytic Function/Banach Space Valued Function|analytic]].
Define $f' : U \to X$ by:
:$\ds \map {f'} z = \lim_{w \mathop \to z} \frac {\map f w - \map f z} {w - z}$
for each $z \in U$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Def... | Banach Space Valued Function is Analytic iff Weakly Analytic | https://proofwiki.org/wiki/Banach_Space_Valued_Function_is_Analytic_iff_Weakly_Analytic | https://proofwiki.org/wiki/Banach_Space_Valued_Function_is_Analytic_iff_Weakly_Analytic | [
"Analytic Functions (Banach Spaces)"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Banach Space",
"Definition:Function",
"Definition:Analytic Function/Banach Space Valued Function",
"Definition:Weakly Analytic Function"
] | [
"Definition:Analytic Function/Banach Space Valued Function",
"Definition:Normed Dual Space",
"Definition:Linear Functional",
"Continuity of Linear Functionals",
"Definition:Continuous Mapping (Normed Vector Space)",
"Definition:Analytic Function/Banach Space Valued Function",
"Definition:Weakly Analytic... |
proofwiki-20628 | Liouville's Theorem (Complex Analysis)/Banach Space | Let $\struct {X, \norm {\, \cdot \,} }$ be a Banach space over $\C$.
Let $f : \C \to X$ be an analytic function that is bounded.
Then $f$ is constant. | Take $M \ge 0$ such that:
:$\norm {\map f x} \le M$
for each $x \in X$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,} }$.
From Banach Space Valued Function is Analytic iff Weakly Analytic, $f$ is weakly analytic.
So for each $\phi \in X^\ast$, $\ph... | Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Banach Space|Banach space]] over $\C$.
Let $f : \C \to X$ be an [[Definition:Analytic Function/Banach Space Valued Function|analytic function]] that is [[Definition:Bounded Mapping on Normed Vector Space|bounded]].
Then $f$ is [[Definition:Constant Mapping|co... | Take $M \ge 0$ such that:
:$\norm {\map f x} \le M$
for each $x \in X$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {\, \cdot \,} }$.
From [[Banach Space Valued Function is Analytic iff Weakly Analytic]], $f$ is [[Definition... | Liouville's Theorem (Complex Analysis)/Banach Space | https://proofwiki.org/wiki/Liouville's_Theorem_(Complex_Analysis)/Banach_Space | https://proofwiki.org/wiki/Liouville's_Theorem_(Complex_Analysis)/Banach_Space | [
"Liouville's Theorem (Complex Analysis)",
"Banach Spaces"
] | [
"Definition:Banach Space",
"Definition:Analytic Function/Banach Space Valued Function",
"Definition:Bounded Mapping/Normed Vector Space",
"Definition:Constant Mapping"
] | [
"Definition:Normed Dual Space",
"Banach Space Valued Function is Analytic iff Weakly Analytic",
"Definition:Weakly Analytic Function",
"Definition:Analytic Function/Complex Plane",
"Definition:Bounded Mapping",
"Liouville's Theorem (Complex Analysis)",
"Definition:Constant Mapping",
"Definition:Linear... |
proofwiki-20629 | Null Ring is Ring with Unity | Let $R$ be the null ring.
Then $R$ is a ring with unity. | We have that $R$ is the null ring.
That is, by definition it has a single element, which can be denoted $0_R$, such that:
:$R := \struct {\set {0_R}, +, \circ}$
where ring addition and the ring product are defined as:
{{begin-eqn}}
{{eqn | l = 0_R + 0_R
| r = 0_R
}}
{{eqn | l = 0_R \circ 0_R
| r = 0_R
}}
{{... | Let $R$ be the [[Definition:Null Ring|null ring]].
Then $R$ is a [[Definition:Ring with Unity|ring with unity]]. | We have that $R$ is the [[Definition:Null Ring|null ring]].
That is, by definition it has a single [[Definition:Element|element]], which can be denoted $0_R$, such that:
:$R := \struct {\set {0_R}, +, \circ}$
where [[Definition:Ring Addition|ring addition]] and the [[Definition:Ring Product|ring product]] are defined ... | Null Ring is Ring with Unity | https://proofwiki.org/wiki/Null_Ring_is_Ring_with_Unity | https://proofwiki.org/wiki/Null_Ring_is_Ring_with_Unity | [
"Rings with Unity",
"Null Ring"
] | [
"Definition:Null Ring",
"Definition:Ring with Unity"
] | [
"Definition:Null Ring",
"Definition:Element",
"Definition:Ring (Abstract Algebra)/Addition",
"Definition:Ring (Abstract Algebra)/Product",
"Definition:Algebraic Structure/One Operation",
"Definition:Trivial Group",
"Definition:Unity (Abstract Algebra)/Ring",
"Category:Rings with Unity",
"Category:Nu... |
proofwiki-20630 | Multitape Turing Machine Reduces to Turing Machine | Let $T$ be a multitape Turing machine.
Then, there exists a Turing machine $T'$ that accepts precisely the same language as $T$, and halts on exactly the inputs that $T$ does. | {{Proofread|In particular, make sure the construction is correct. I keep spotting mistakes in it and fixing them; who knows how many more are left. The correctness proof is more or less secondary in this case, although if you want to make that more rigorous, go ahead.}}
Let $k$ be the number of tapes in $T$.
Let:
:$\Ga... | Let $T$ be a [[Definition:Multitape Turing Machine|multitape Turing machine]].
Then, there exists a [[Definition:Turing Machine|Turing machine]] $T'$ that accepts precisely the same language as $T$, and halts on exactly the inputs that $T$ does. | {{Proofread|In particular, make sure the construction is correct. I keep spotting mistakes in it and fixing them; who knows how many more are left. The correctness proof is more or less secondary in this case, although if you want to make that more rigorous, go ahead.}}
Let $k$ be the number of tapes in $T$.
Let:
:$\... | Multitape Turing Machine Reduces to Turing Machine | https://proofwiki.org/wiki/Multitape_Turing_Machine_Reduces_to_Turing_Machine | https://proofwiki.org/wiki/Multitape_Turing_Machine_Reduces_to_Turing_Machine | [
"Turing Machines"
] | [
"Definition:Multitape Turing Machine",
"Definition:Turing Machine"
] | [] |
proofwiki-20631 | Unary Representation of Natural Number | Let $x \in \N$ be a natural number.
Then:
:$x = \map s {\map s {\dots \map s 0} }$
where there are $x$ applications of the successor mapping to the constant $0$. | We shall proceed by induction. | Let $x \in \N$ be a [[Definition:Natural Number|natural number]].
Then:
:$x = \map s {\map s {\dots \map s 0} }$
where there are $x$ applications of the [[Definition:Successor Mapping on Natural Numbers|successor mapping]] to the [[Definition:Constant|constant]] $0$. | We shall proceed by [[Definition:Mathematical Induction|induction]]. | Unary Representation of Natural Number | https://proofwiki.org/wiki/Unary_Representation_of_Natural_Number | https://proofwiki.org/wiki/Unary_Representation_of_Natural_Number | [
"Natural Numbers",
"Proofs by Induction"
] | [
"Definition:Natural Numbers",
"Definition:Successor Mapping on Natural Numbers",
"Definition:Constant"
] | [
"Definition:Mathematical Induction"
] |
proofwiki-20632 | Addition of Natural Numbers is Provable | Let $x, y \in \N$ be natural numbers.
Then there exists formal proof of:
:$\sqbrk x + \sqbrk y = \sqbrk {x + y}$
in minimal arithmetic, where $\sqbrk a$ is the unary representation of $a$. | By Unary Representation of Natural Number, let $\sqbrk a$ denote the term $\map s {\dots \map s 0}$, where there are $a$ applications of the successor mapping to the constant $0$.
Proceed by induction on $y$. | Let $x, y \in \N$ be [[Definition:Natural Number|natural numbers]].
Then there exists [[Definition:Formal Proof|formal proof]] of:
:$\sqbrk x + \sqbrk y = \sqbrk {x + y}$
in [[Definition:Minimal Arithmetic|minimal arithmetic]], where $\sqbrk a$ is the [[Unary Representation of Natural Number|unary representation]] of ... | By [[Unary Representation of Natural Number]], let $\sqbrk a$ denote the [[Definition:Term (Predicate Logic)|term]] $\map s {\dots \map s 0}$, where there are $a$ applications of the [[Definition:Successor Mapping on Natural Numbers|successor mapping]] to the [[Definition:Constant|constant]] $0$.
Proceed by [[Definiti... | Addition of Natural Numbers is Provable | https://proofwiki.org/wiki/Addition_of_Natural_Numbers_is_Provable | https://proofwiki.org/wiki/Addition_of_Natural_Numbers_is_Provable | [
"Natural Number Addition",
"Proofs by Induction"
] | [
"Definition:Natural Numbers",
"Definition:Proof System/Formal Proof",
"Definition:Minimal Arithmetic",
"Unary Representation of Natural Number"
] | [
"Unary Representation of Natural Number",
"Definition:Language of Predicate Logic/Formal Grammar/Term",
"Definition:Successor Mapping on Natural Numbers",
"Definition:Constant",
"Definition:Mathematical Induction",
"Definition:Successor Mapping on Natural Numbers",
"Definition:Constant",
"Unary Repres... |
proofwiki-20633 | Uniformly Continuous Semigroup Bounded on Compact Intervals | Let $\GF \in \set {\R, \C}$.
Let $X$ be a Banach space over $\GF$.
Let $\family {\map T t}_{t \ge 0}$ be a uniformly continuous semigroup.
Let $\struct {\map B X, \norm {\, \cdot \,}_{\map B X} }$ be the space of bounded linear transformations equipped with the canonical norm.
Let $T > 0$ be a real number.
Then ther... | Since $\family {\map T t}_{t \ge 0}$ is a uniformly continuous semigroup, we have:
:$\ds \lim_{t \mathop \to 0^+} \norm {\map T t - I}_{\map B X} = 0$
From Sequence in Normed Vector Space Convergent to Limit iff Norm of Sequence minus Limit is Null Sequence and Modulus of Limit: Normed Vector Space, we have:
:$\norm ... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\family {\map T t}_{t \ge 0}$ be a [[Definition:Uniformly Continuous Semigroup|uniformly continuous semigroup]].
Let $\struct {\map B X, \norm {\, \cdot \,}_{\map B X} }$ be the [[Definition:Space of Bounded Linear ... | Since $\family {\map T t}_{t \ge 0}$ is a [[Definition:Uniformly Continuous Semigroup|uniformly continuous semigroup]], we have:
:$\ds \lim_{t \mathop \to 0^+} \norm {\map T t - I}_{\map B X} = 0$
From [[Sequence in Normed Vector Space Convergent to Limit iff Norm of Sequence minus Limit is Null Sequence]] and [[Mo... | Uniformly Continuous Semigroup Bounded on Compact Intervals | https://proofwiki.org/wiki/Uniformly_Continuous_Semigroup_Bounded_on_Compact_Intervals | https://proofwiki.org/wiki/Uniformly_Continuous_Semigroup_Bounded_on_Compact_Intervals | [
"Uniformly Continuous Semigroups"
] | [
"Definition:Banach Space",
"Definition:Uniformly Continuous Semigroup",
"Definition:Space of Bounded Linear Transformations",
"Definition:Norm on Space of Bounded Linear Transformations",
"Definition:Real Number"
] | [
"Definition:Uniformly Continuous Semigroup",
"Sequence in Normed Vector Space Convergent to Limit iff Norm of Sequence minus Limit is Null Sequence",
"Modulus of Limit/Normed Vector Space",
"Identity Mapping on Normed Vector Space is Bounded Linear Operator",
"Norm on Bounded Linear Transformation is Submul... |
proofwiki-20634 | Semigroup of Bounded Linear Operators Uniformly Continuous iff Continuous as Map from Non-Negative Reals to Bounded Linear Operators | Let $\GF \in \set {\R, \C}$.
Let $X$ be a Banach space over $\GF$.
Let $\family {\map T t}_{t \ge 0}$ be a semigroup of bounded linear operators.
Let $\struct {\map B X, \norm {\, \cdot \,}_{\map B X} }$ be the space of bounded linear transformations equipped with the canonical norm.
Then $\family {\map T t}_{t \ge 0... | === Necessary Condition ===
If $T$ is continuous, then in particular it is continuous at $0$.
Since $\map T 0 = I$, we therefore have:
:$\ds \lim_{t \mathop \to 0^+} \norm {\map T t - I}_{\map B X} = 0$
So $\family {\map T t}_{t \ge 0}$ is uniformly continuous.
{{qed|lemma}} | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\family {\map T t}_{t \ge 0}$ be a [[Definition:Semigroup of Bounded Linear Operators|semigroup of bounded linear operators]].
Let $\struct {\map B X, \norm {\, \cdot \,}_{\map B X} }$ be the [[Definition:Space of B... | === Necessary Condition ===
If $T$ is [[Definition:Continuous Mapping (Normed Vector Space)|continuous]], then in particular it is [[Definition:Continuous at Point of Normed Vector Space|continuous at $0$]].
Since $\map T 0 = I$, we therefore have:
:$\ds \lim_{t \mathop \to 0^+} \norm {\map T t - I}_{\map B X} = 0$... | Semigroup of Bounded Linear Operators Uniformly Continuous iff Continuous as Map from Non-Negative Reals to Bounded Linear Operators | https://proofwiki.org/wiki/Semigroup_of_Bounded_Linear_Operators_Uniformly_Continuous_iff_Continuous_as_Map_from_Non-Negative_Reals_to_Bounded_Linear_Operators | https://proofwiki.org/wiki/Semigroup_of_Bounded_Linear_Operators_Uniformly_Continuous_iff_Continuous_as_Map_from_Non-Negative_Reals_to_Bounded_Linear_Operators | [
"Uniformly Continuous Semigroups",
"Semigroups of Bounded Linear Operators"
] | [
"Definition:Banach Space",
"Definition:Semigroup of Bounded Linear Operators",
"Definition:Space of Bounded Linear Transformations",
"Definition:Norm on Space of Bounded Linear Transformations",
"Definition:Uniformly Continuous Semigroup",
"Definition:Mapping",
"Definition:Continuous Mapping (Normed Vec... | [
"Definition:Continuous Mapping (Normed Vector Space)",
"Definition:Continuous Mapping (Normed Vector Space)/Point",
"Definition:Uniformly Continuous Semigroup",
"Definition:Uniformly Continuous Semigroup",
"Definition:Continuous Mapping (Normed Vector Space)/Point",
"Definition:Continuous Mapping (Normed ... |
proofwiki-20635 | Converting Decimal Expansion of Rational Number to Fraction | Let $x \in \Q$ be a rational number.
Let the decimal expansion of $x$ be:
:$0 \cdotp a_1 a_2 \ldots a_m \dot b_1 b_2 \ldots \dot b_n$
where $a_i: i \in \set {1, 2, \ldots, m}$ and $b_j: j \in \set {1, 2, \ldots, n}$ be the digits in the base $10$ expansion of $x$.
Then $x$ can be expressed as the following fraction:
:$... | First we note that:
{{begin-eqn}}
{{eqn | l = 0 \cdotp \dot b_1 b_2 \ldots \dot b_n \times 10^n
| r = b_1 b_2 \ldots b_n \cdotp \dot b_1 b_2 \ldots \dot b_n
| c =
}}
{{eqn | ll= \leadsto
| l = 0 \cdotp \dot b_1 b_2 \ldots \dot b_n \times 10^n - 0 \cdotp \dot b_1 b_2 \ldots \dot b_n
| r = b_1 b_... | Let $x \in \Q$ be a [[Definition:Rational Number|rational number]].
Let the [[Definition:Decimal Expansion|decimal expansion]] of $x$ be:
:$0 \cdotp a_1 a_2 \ldots a_m \dot b_1 b_2 \ldots \dot b_n$
where $a_i: i \in \set {1, 2, \ldots, m}$ and $b_j: j \in \set {1, 2, \ldots, n}$ be the [[Definition:Digit|digits]] in... | First we note that:
{{begin-eqn}}
{{eqn | l = 0 \cdotp \dot b_1 b_2 \ldots \dot b_n \times 10^n
| r = b_1 b_2 \ldots b_n \cdotp \dot b_1 b_2 \ldots \dot b_n
| c =
}}
{{eqn | ll= \leadsto
| l = 0 \cdotp \dot b_1 b_2 \ldots \dot b_n \times 10^n - 0 \cdotp \dot b_1 b_2 \ldots \dot b_n
| r = b_1 b_... | Converting Decimal Expansion of Rational Number to Fraction | https://proofwiki.org/wiki/Converting_Decimal_Expansion_of_Rational_Number_to_Fraction | https://proofwiki.org/wiki/Converting_Decimal_Expansion_of_Rational_Number_to_Fraction | [
"Converting Decimal Expansion of Rational Number to Fraction",
"Recurring Decimals",
"Rational Numbers",
"Decimal Notation",
"Proof Techniques"
] | [
"Definition:Rational Number",
"Definition:Decimal Expansion",
"Definition:Digit",
"Definition:Basis Expansion",
"Definition:Fraction"
] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-20636 | Bound on C0 Semigroup | Let $\GF \in \set {\R, \C}$.
Let $X$ be a Banach space over $\GF$.
Let $\family {\map T t}_{t \ge 0}$ be a $C_0$ semigroup.
Let $\struct {\map B X, \norm {\, \cdot \,}_{\map B X} }$ be the space of bounded linear transformations equipped with the canonical norm.
Then there exists $M \ge 1$ and $\omega \ge 0$ such tha... | We first show that $\norm {\map T t}_{\map B X}$ is bounded on $\closedint 0 \delta$ for some $\delta > 0$.
{{AimForCont}} $\norm {\map T t}_{\map B X}$ is unbounded on $\closedint 0 \delta$ for each $\delta > 0$.
Then for each $n \in \N$ we can pick $t_n \in \closedint 0 {\frac 1 n}$ such that $\norm {\map T {t_n} }_{... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\family {\map T t}_{t \ge 0}$ be a [[Definition:C0 Semigroup|$C_0$ semigroup]].
Let $\struct {\map B X, \norm {\, \cdot \,}_{\map B X} }$ be the [[Definition:Space of Bounded Linear Transformations|space of bounded ... | We first show that $\norm {\map T t}_{\map B X}$ is [[Definition:Bounded Real-Valued Function|bounded]] on $\closedint 0 \delta$ for some $\delta > 0$.
{{AimForCont}} $\norm {\map T t}_{\map B X}$ is [[Definition:Unbounded Real-Valued Function|unbounded]] on $\closedint 0 \delta$ for each $\delta > 0$.
Then for each ... | Bound on C0 Semigroup | https://proofwiki.org/wiki/Bound_on_C0_Semigroup | https://proofwiki.org/wiki/Bound_on_C0_Semigroup | [
"C0 Semigroups"
] | [
"Definition:Banach Space",
"Definition:C0 Semigroup",
"Definition:Space of Bounded Linear Transformations",
" Definition:Norm on Bounded Linear Transformation"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Definition:Bounded Mapping/Real-Valued/Unbounded",
"Definition:Bounded Linear Operator",
"Principle of Condensation of Singularities",
"Squeeze Theorem",
"Convergent Sequence in Normed Vector Space is Bounded",
"Definition:Bounded Mapping/Real-Valued",
"Norm ... |
proofwiki-20637 | Semigroup of Bounded Linear Operators is C0 iff Point Evaluations Continuous | Let $\GF \in \set {\R, \C}$.
Let $X$ be a Banach space over $\GF$.
Let $\family {\map T t}_{t \ge 0}$ be a semigroup of bounded linear operators.
For each $x \in X$, define $x^\wedge : \hointr 0 \infty \to X$ by:
:$\map {x^\wedge} t = \map T t x$
for each $t \in \hointr 0 \infty$.
Then $\family {\map T t}_{t \ge 0}... | === Sufficient Condition ===
Suppose that $x^\wedge$ is continuous for each $x \in X$.
In particular, for each $x \in X$ we have that $x^\wedge$ is continuous at $0$.
That is:
:$\ds \lim_{t \mathop \to 0^+} \map T t x = \map T 0 x = x$ for each $x \in X$.
So $\family {\map T t}_{t \ge 0}$ is a $C_0$ semigroup.
{{qed|... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\family {\map T t}_{t \ge 0}$ be a [[Definition:Semigroup of Bounded Linear Operators|semigroup of bounded linear operators]].
For each $x \in X$, define $x^\wedge : \hointr 0 \infty \to X$ by:
:$\map {x^\wedge} t... | === Sufficient Condition ===
Suppose that $x^\wedge$ is [[Definition:Continuous Mapping (Normed Vector Space)|continuous]] for each $x \in X$.
In particular, for each $x \in X$ we have that $x^\wedge$ is [[Definition:Continuous at Point of Normed Vector Space|continuous at $0$]].
That is:
:$\ds \lim_{t \mathop \t... | Semigroup of Bounded Linear Operators is C0 iff Point Evaluations Continuous | https://proofwiki.org/wiki/Semigroup_of_Bounded_Linear_Operators_is_C0_iff_Point_Evaluations_Continuous | https://proofwiki.org/wiki/Semigroup_of_Bounded_Linear_Operators_is_C0_iff_Point_Evaluations_Continuous | [
"C0 Semigroups",
"Semigroups of Bounded Linear Operators"
] | [
"Definition:Banach Space",
"Definition:Semigroup of Bounded Linear Operators",
"Definition:C0 Semigroup",
"Definition:Continuous Mapping (Normed Vector Space)"
] | [
"Definition:Continuous Mapping (Normed Vector Space)",
"Definition:Continuous Mapping (Normed Vector Space)/Point",
"Definition:C0 Semigroup",
"Definition:C0 Semigroup",
"Definition:Continuous Mapping (Normed Vector Space)",
"Definition:Continuous Mapping (Normed Vector Space)/Point",
"Definition:C0 Sem... |
proofwiki-20638 | Norm of Continuous Function is Continuous | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$.
Let $f : X \to Y$ be a continuous mapping.
Define $\norm f_Y : X \to \hointr 0 \infty$ by:
:$\map {\paren {\norm f_Y} } x = \norm {\map f x}_Y$
for each $x \in X$.
Then ... | Follows immediately from combining Norm on Vector Space is Continuous Function and Composite of Continuous Mappings is Continuous.
{{qed}}
Category:Normed Vector Spaces
o3874yevdkkhz0cwhupwzy2ewipn57n | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $f : X \to Y$ be a [[Definition:Continuous Mapping (Normed Vector Space)|continuous mapping]].
Define $\norm f_Y : X \to \hointr ... | Follows immediately from combining [[Norm on Vector Space is Continuous Function]] and [[Composite of Continuous Mappings is Continuous]].
{{qed}}
[[Category:Normed Vector Spaces]]
o3874yevdkkhz0cwhupwzy2ewipn57n | Norm of Continuous Function is Continuous | https://proofwiki.org/wiki/Norm_of_Continuous_Function_is_Continuous | https://proofwiki.org/wiki/Norm_of_Continuous_Function_is_Continuous | [
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Continuous Mapping (Normed Vector Space)",
"Definition:Continuous Mapping (Normed Vector Space)"
] | [
"Norm on Vector Space is Continuous Function",
"Composite of Continuous Mappings is Continuous",
"Category:Normed Vector Spaces"
] |
proofwiki-20639 | Substitution Property of Equality | Let $\map F z$ be an expression of the variable $z$.
Then:
:$\forall x, y: x = y \implies \map F x = \map F y$. | {{BeginTableau|\forall x, y: x {{=}} y \implies \map F x {{=}} \map F y}}
{{TableauLine|n = 1
|pool = 1
|f = x = y
|rlnk = Rule of Assumption
|rtxt = Assumption
}}
{{TableauLine|n = 2
|pool = 1
|f = \map F x = \map F x \iff \map F x = \map F ... | Let $\map F z$ be an [[Definition:Expression|expression]] of the [[Definition:Variable|variable]] $z$.
Then:
:$\forall x, y: x = y \implies \map F x = \map F y$. | {{BeginTableau|\forall x, y: x {{=}} y \implies \map F x {{=}} \map F y}}
{{TableauLine|n = 1
|pool = 1
|f = x = y
|rlnk = Rule of Assumption
|rtxt = Assumption
}}
{{TableauLine|n = 2
|pool = 1
|f = \map F x = \map F x \iff \map F x = \map F ... | Substitution Property of Equality | https://proofwiki.org/wiki/Substitution_Property_of_Equality | https://proofwiki.org/wiki/Substitution_Property_of_Equality | [
"Equality"
] | [
"Definition:Expression",
"Definition:Variable"
] | [
"Equality is Reflexive",
"Category:Equality"
] |
proofwiki-20640 | Multiplication of Natural Numbers is Provable | Let $x, y \in \N$ be natural numbers.
Then there exists a formal proof of:
:$\sqbrk x \times \sqbrk y = \sqbrk {x \times y}$
in minimal arithmetic, where $\sqbrk a$ is the unary representation of $a$. | By Unary Representation of Natural Number, let $\sqbrk a$ denote the term $\map s {\dots \map s 0}$, where there are $a$ applications of the successor mapping to the constant $0$.
Proceed by induction on $y$. | Let $x, y \in \N$ be [[Definition:Natural Number|natural numbers]].
Then there exists a [[Definition:Formal Proof|formal proof]] of:
:$\sqbrk x \times \sqbrk y = \sqbrk {x \times y}$
in [[Definition:Minimal Arithmetic|minimal arithmetic]], where $\sqbrk a$ is the [[Unary Representation of Natural Number|unary represen... | By [[Unary Representation of Natural Number]], let $\sqbrk a$ denote the [[Definition:Term (Predicate Logic)|term]] $\map s {\dots \map s 0}$, where there are $a$ applications of the [[Definition:Successor Mapping on Natural Numbers|successor mapping]] to the [[Definition:Constant|constant]] $0$.
Proceed by [[Definiti... | Multiplication of Natural Numbers is Provable | https://proofwiki.org/wiki/Multiplication_of_Natural_Numbers_is_Provable | https://proofwiki.org/wiki/Multiplication_of_Natural_Numbers_is_Provable | [
"Natural Number Multiplication",
"Proofs by Induction"
] | [
"Definition:Natural Numbers",
"Definition:Proof System/Formal Proof",
"Definition:Minimal Arithmetic",
"Unary Representation of Natural Number"
] | [
"Unary Representation of Natural Number",
"Definition:Language of Predicate Logic/Formal Grammar/Term",
"Definition:Successor Mapping on Natural Numbers",
"Definition:Constant",
"Definition:Mathematical Induction",
"Definition:Successor Mapping on Natural Numbers",
"Definition:Constant",
"Unary Repres... |
proofwiki-20641 | Subspace of Metrizable Space is Metrizable Space | Let $\struct {X, \tau}$ be a metrizable topological space.
Let $\struct {Y, \tau_Y}$ be a subspace of $\struct{X, \tau}$.
Then:
:$\struct {Y, \tau_Y}$ is a metrizable topological space | By definition of metrizable topological space:
:there exists a metric $d$ on $X$ such that the topology induced by $d$ is $\tau$
From Metric Subspace Induces Subspace Topology:
:$\tau_Y$ is the topology induced by the subspace metric $d_Y$ on $Y$
From Subspace of Metric Space is Metric Space:
:$\struct {Y, d_Y}$ is a m... | Let $\struct {X, \tau}$ be a [[Definition:Metrizable Space|metrizable topological space]].
Let $\struct {Y, \tau_Y}$ be a [[Definition:Topological Subspace|subspace]] of $\struct{X, \tau}$.
Then:
:$\struct {Y, \tau_Y}$ is a [[Definition:Metrizable Space|metrizable topological space]] | By definition of [[Definition:Metrizable Space|metrizable topological space]]:
:there exists a [[Definition:Metric|metric]] $d$ on $X$ such that the [[Definition:Topology Induced by Metric|topology induced by $d$]] is $\tau$
From [[Metric Subspace Induces Subspace Topology]]:
:$\tau_Y$ is the [[Definition:Topology In... | Subspace of Metrizable Space is Metrizable Space | https://proofwiki.org/wiki/Subspace_of_Metrizable_Space_is_Metrizable_Space | https://proofwiki.org/wiki/Subspace_of_Metrizable_Space_is_Metrizable_Space | [
"Metrizable Spaces",
"Topological Subspaces"
] | [
"Definition:Metrizable Space",
"Definition:Topological Subspace",
"Definition:Metrizable Space"
] | [
"Definition:Metrizable Space",
"Definition:Metric Space/Metric",
"Definition:Topology Induced by Metric",
"Metric Subspace Induces Subspace Topology",
"Definition:Topology Induced by Metric",
"Definition:Metric Subspace",
"Subspace of Metric Space is Metric Space",
"Definition:Metric Space",
"Defini... |
proofwiki-20642 | Electric Field Strength from Assemblage of Point Charges | Let $q_1, q_2, \ldots, q_n$ be point charges.
Let $\mathbf r_1, \mathbf r_2, \ldots, \mathbf r_n$ be the position vectors of $q_1, q_2, \ldots, q_n$ respectively.
Let $\map {\mathbf E} {\mathbf r}$ be the electric field strength at a point $P$ whose position vector is $\mathbf r$.
Then:
:$\ds \map {\mathbf E} {\mathbf ... | Let $q$ be a test charge in the vicinity of $q_1, q_2, \ldots, q_n$ at $\map P {\mathbf r}$.
For all $i$ in $\set {1, 2, \ldots, n}$, let $\mathbf F_i$ denote the force exerted on $q$ by $q_i$.
Let $\mathbf F$ be the force exerted on $q$ by the combined action of $q_1, q_2, \ldots, q_n$.
We have:
{{begin-eqn}}
{{eqn | ... | Let $q_1, q_2, \ldots, q_n$ be [[Definition:Point Charge|point charges]].
Let $\mathbf r_1, \mathbf r_2, \ldots, \mathbf r_n$ be the [[Definition:Position Vector|position vectors]] of $q_1, q_2, \ldots, q_n$ respectively.
Let $\map {\mathbf E} {\mathbf r}$ be the [[Definition:Electric Field Strength|electric field s... | Let $q$ be a [[Definition:Test Charge|test charge]] in the vicinity of $q_1, q_2, \ldots, q_n$ at $\map P {\mathbf r}$.
For all $i$ in $\set {1, 2, \ldots, n}$, let $\mathbf F_i$ denote the [[Definition:Force|force]] exerted on $q$ by $q_i$.
Let $\mathbf F$ be the [[Definition:Force|force]] exerted on $q$ by the com... | Electric Field Strength from Assemblage of Point Charges | https://proofwiki.org/wiki/Electric_Field_Strength_from_Assemblage_of_Point_Charges | https://proofwiki.org/wiki/Electric_Field_Strength_from_Assemblage_of_Point_Charges | [
"Electric Fields",
"Point Charges"
] | [
"Definition:Point Charge",
"Definition:Position Vector",
"Definition:Electric Field Strength",
"Definition:Point",
"Definition:Position Vector",
"Definition:Vacuum Permittivity"
] | [
"Definition:Test Charge",
"Definition:Force",
"Definition:Force",
"Total Force on Point Charge from Multiple Point Charges",
"Definition:Constant"
] |
proofwiki-20643 | Ampère's Force Law | Let $s_1$ and $s_2$ be wires in a vacuum carrying steady currents $I_1$ and $I_2$.
Then the force on $s_1$ due to the magnetic field generated by $s_2$ is given by:
:$\ds \mathbf F_1 \propto I_1 I_2 \oint_{s_1} \oint_{s_2} \rd \mathbf l_1 \times \paren {\dfrac {\d \mathbf l_2 \times \paren {\mathbf r_1 - \mathbf r_2} }... | Let $\mathbf B_2$ be the magnetic field produced by $I_2$.
The magnetic force felt by $s_1$ is given by
{{begin-eqn}}
{{eqn | l = \mathbf F_1
| r = \oint_{s_1} \d \mathbf F_1
}}
{{eqn | r = \oint_{s_1} I_1 \d \mathbf l_1 \times \mathbf B_2
| c = Magnetic Force on Conductor carrying Steady Current
}}
{{eqn |... | Let $s_1$ and $s_2$ be [[Definition:Wire (Electricity)|wires]] in a [[Definition:Vacuum|vacuum]] carrying [[Definition:Steady Current|steady currents]] $I_1$ and $I_2$.
Then the [[Definition:Force|force]] on $s_1$ due to the [[Definition:Magnetic Field|magnetic field]] generated by $s_2$ is given by:
:$\ds \mathbf F_1... | Let $\mathbf B_2$ be the [[Definition:Magnetic Field|magnetic field]] produced by $I_2$.
The magnetic force felt by $s_1$ is given by
{{begin-eqn}}
{{eqn | l = \mathbf F_1
| r = \oint_{s_1} \d \mathbf F_1
}}
{{eqn | r = \oint_{s_1} I_1 \d \mathbf l_1 \times \mathbf B_2
| c = [[Magnetic Force on Conductor c... | Ampère's Force Law | https://proofwiki.org/wiki/Ampère's_Force_Law | https://proofwiki.org/wiki/Ampère's_Force_Law | [
"Ampère's Force Law",
"Electric Current",
"Electromagnetism",
"Magnetic Forces"
] | [
"Definition:Wire (Electricity)",
"Definition:Vacuum",
"Definition:Steady Current",
"Definition:Force",
"Definition:Magnetic Field",
"Definition:Infinitesimal",
"Definition:Vector Quantity",
"Definition:Position Vector"
] | [
"Definition:Magnetic Field",
"Magnetic Force on Conductor carrying Steady Current",
"Biot-Savart Law",
"Linear Combination of Integrals"
] |
proofwiki-20644 | Equality of Terms of Natural Numbers is Provable | Let $A$ and $B$ be terms in the language of arithmetic containing no variables.
Let $A = B$ when interpreted over the natural numbers $\N$.
Then there is a formal proof of $A = B$ in minimal arithmetic. | Let $k = A = B$.
By Unary Representation of Natural Number, there is a term $\sqbrk k$ consisting of only the successor mapping and the constant $0$ such that $\sqbrk k = k$.
We want to show that $A = \sqbrk k$ has a formal proof.
Proceed by induction on the structure of the term $A$.
By definition, a term consists of ... | Let $A$ and $B$ be [[Definition:Term (Predicate Logic)|terms]] in the [[Definition:Language of Arithmetic|language of arithmetic]] containing no [[Definition:Variable|variables]].
Let $A = B$ when interpreted over the [[Definition:Natural Number|natural numbers]] $\N$.
Then there is a [[Definition:Formal Proof|formal... | Let $k = A = B$.
By [[Unary Representation of Natural Number]], there is a [[Definition:Term (Predicate Logic)|term]] $\sqbrk k$ consisting of only the [[Definition:Successor Mapping on Natural Numbers|successor mapping]] and the [[Definition:Constant|constant]] $0$ such that $\sqbrk k = k$.
We want to show that $A ... | Equality of Terms of Natural Numbers is Provable | https://proofwiki.org/wiki/Equality_of_Terms_of_Natural_Numbers_is_Provable | https://proofwiki.org/wiki/Equality_of_Terms_of_Natural_Numbers_is_Provable | [
"Natural Numbers"
] | [
"Definition:Language of Predicate Logic/Formal Grammar/Term",
"Definition:Language of Arithmetic",
"Definition:Variable",
"Definition:Natural Numbers",
"Definition:Proof System/Formal Proof",
"Definition:Minimal Arithmetic"
] | [
"Unary Representation of Natural Number",
"Definition:Language of Predicate Logic/Formal Grammar/Term",
"Definition:Successor Mapping on Natural Numbers",
"Definition:Constant",
"Definition:Proof System/Formal Proof",
"Definition:Language of Predicate Logic/Formal Grammar/Term",
"Definition:Language of ... |
proofwiki-20645 | Value of Vacuum Permeability | The value of the '''vacuum permeability''' is calculated as:
:$\mu_0 = 1 \cdotp 25663 \, 70621 \, 2 (19) \times 10^{-6} \, \mathrm H \, \mathrm m^{-1}$ (henries per metre)
with a relative uncertainty of $1 \cdotp 5 \times 10^{-10}$. | The '''vacuum permeability''' is the physical constant denoted $\mu_0$ defined as:
:$\mu_0:= \dfrac {2 \alpha h} {e^2 c}$
where:
:$e$ is the elementary charge
:$\alpha$ is the fine-structure constant
:$h$ is Planck's constant
:$c$ is the speed of light defined in $\mathrm m \, \mathrm s^{-1}$
$e$ is defined precisely a... | The value of the '''[[Definition:Vacuum Permeability|vacuum permeability]]''' is calculated as:
:$\mu_0 = 1 \cdotp 25663 \, 70621 \, 2 (19) \times 10^{-6} \, \mathrm H \, \mathrm m^{-1}$ ([[Definition:Henry|henries]] per [[Definition:Metre|metre]])
with a [[Definition:Relative Uncertainty|relative uncertainty]] of $1... | The '''[[Definition:Vacuum Permeability|vacuum permeability]]''' is the [[Definition:Physical Constant|physical constant]] denoted $\mu_0$ defined as:
:$\mu_0:= \dfrac {2 \alpha h} {e^2 c}$
where:
:$e$ is the [[Definition:Elementary Charge|elementary charge]]
:$\alpha$ is the [[Definition:Fine-Structure Constant|fine-... | Value of Vacuum Permeability/Proof 1 | https://proofwiki.org/wiki/Value_of_Vacuum_Permeability | https://proofwiki.org/wiki/Value_of_Vacuum_Permeability/Proof_1 | [
"Value of Vacuum Permeability",
"Vacuum Permeability"
] | [
"Definition:Vacuum Permeability",
"Definition:Henry",
"Definition:Metric System/Length/Metre",
"Definition:Relative Uncertainty"
] | [
"Definition:Vacuum Permeability",
"Definition:Physical Constant",
"Definition:Electric Charge/Quantum",
"Definition:Fine-Structure Constant",
"Definition:Planck's Constant",
"Definition:Speed of Light",
"Definition:Coulomb",
"Definition:SI/Energy/Joule",
"Definition:Time/Unit/Second",
"Definition:... |
proofwiki-20646 | Value of Vacuum Permeability | The value of the '''vacuum permeability''' is calculated as:
:$\mu_0 = 1 \cdotp 25663 \, 70621 \, 2 (19) \times 10^{-6} \, \mathrm H \, \mathrm m^{-1}$ (henries per metre)
with a relative uncertainty of $1 \cdotp 5 \times 10^{-10}$. | The '''vacuum permeability''' is the physical constant denoted $\mu_0$ defined as:
:$\mu_0 := \dfrac 1 {\varepsilon_0c^2}$
where:
:$\varepsilon_0$ is the vacuum permittivity defined in $\mathrm F \, \mathrm m^{-1}$ (farads per metre)
:$c$ is the speed of light defined in $\mathrm m \, \mathrm s^{-1}$
$\varepsilon_0$ ha... | The value of the '''[[Definition:Vacuum Permeability|vacuum permeability]]''' is calculated as:
:$\mu_0 = 1 \cdotp 25663 \, 70621 \, 2 (19) \times 10^{-6} \, \mathrm H \, \mathrm m^{-1}$ ([[Definition:Henry|henries]] per [[Definition:Metre|metre]])
with a [[Definition:Relative Uncertainty|relative uncertainty]] of $1... | The '''[[Definition:Vacuum Permeability|vacuum permeability]]''' is the [[Definition:Physical Constant|physical constant]] denoted $\mu_0$ defined as:
:$\mu_0 := \dfrac 1 {\varepsilon_0c^2}$
where:
:$\varepsilon_0$ is the [[Definition:Vacuum Permittivity|vacuum permittivity]] defined in $\mathrm F \, \mathrm m^{-1}$ (... | Value of Vacuum Permeability/Proof 2 | https://proofwiki.org/wiki/Value_of_Vacuum_Permeability | https://proofwiki.org/wiki/Value_of_Vacuum_Permeability/Proof_2 | [
"Value of Vacuum Permeability",
"Vacuum Permeability"
] | [
"Definition:Vacuum Permeability",
"Definition:Henry",
"Definition:Metric System/Length/Metre",
"Definition:Relative Uncertainty"
] | [
"Definition:Vacuum Permeability",
"Definition:Physical Constant",
"Definition:Vacuum Permittivity",
"Definition:Farad",
"Definition:Metric System/Length/Metre",
"Definition:Speed of Light",
"Definition:Farad/Base Units",
"Definition:Henry/Base Units"
] |
proofwiki-20647 | Mean Ergodic Theorem (Hilbert Space) | Let $\GF \in \set {\R, \C}$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\mathbb F$.
Let $U : \HH \to \HH$ be a bounded linear operator such that:
:$\forall f \in \HH : \norm {\map U f} \le \norm f$
Then for each $f \in \HH$:
:$\ds \lim_{N \mathop \to \infty} \dfrac 1 N \sum_{n \mathop = 0}^{N ... | Note that $I$ is a closed linear subspace of $\HH$, since $U$ is bounded.
Especially, $P : \HH \to I$ is well-defined.
Moreover, by Direct Sum of Subspace and Orthocomplement:
:$\HH = I \oplus I^\perp$
Let $f \in \HH$.
We can write:
:$ f = \map P f + f^\perp$
where $f^\perp \in I^\perp$.
Then we have:
{{begin-eqn}}
{{e... | Let $\GF \in \set {\R, \C}$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\mathbb F$.
Let $U : \HH \to \HH$ be a [[Definition:Bounded Linear Operator/Inner Product Space|bounded linear operator]] such that:
:$\forall f \in \HH : \norm {\map U f} \le \norm f$
Then... | Note that $I$ is a [[Definition:Closed Linear Subspace|closed linear subspace]] of $\HH$, since $U$ is [[Definition:Bounded Linear Operator/Inner Product Space|bounded]].
Especially, $P : \HH \to I$ is [[Definition:Well-Defined|well-defined]].
Moreover, by [[Direct Sum of Subspace and Orthocomplement]]:
:$\HH = I \op... | Mean Ergodic Theorem (Hilbert Space) | https://proofwiki.org/wiki/Mean_Ergodic_Theorem_(Hilbert_Space) | https://proofwiki.org/wiki/Mean_Ergodic_Theorem_(Hilbert_Space) | [
"Mean Ergodic Theorem"
] | [
"Definition:Hilbert Space",
"Definition:Bounded Linear Operator/Inner Product Space",
"Definition:Composition of Mappings",
"Definition:Orthogonal Projection"
] | [
"Definition:Closed Linear Subspace",
"Definition:Bounded Linear Operator/Inner Product Space",
"Definition:Well-Defined",
"Direct Sum of Subspace and Orthocomplement"
] |
proofwiki-20648 | Lower Section of Natural Number is Provable | Let $x \in \N$ be a natural number.
Then the following WFF:
:$\forall y: y = 0 \lor y = \map s 0 \lor \dotso \lor y = \sqbrk {x - 1} \lor \neg \paren {y < \sqbrk x}$
is a theorem of minimal arithmetic. | Proceed by induction on $x$. | Let $x \in \N$ be a [[Definition:Natural Number|natural number]].
Then the following [[Definition:WFF of Predicate Logic|WFF]]:
:$\forall y: y = 0 \lor y = \map s 0 \lor \dotso \lor y = \sqbrk {x - 1} \lor \neg \paren {y < \sqbrk x}$
is a [[Definition:Theorem (Formal Systems)|theorem]] of [[Definition:Minimal Arithmet... | Proceed by [[Definition:Mathematical Induction|induction]] on $x$. | Lower Section of Natural Number is Provable | https://proofwiki.org/wiki/Lower_Section_of_Natural_Number_is_Provable | https://proofwiki.org/wiki/Lower_Section_of_Natural_Number_is_Provable | [
"Natural Numbers",
"Proofs by Induction"
] | [
"Definition:Natural Numbers",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Theorem/Formal System",
"Definition:Minimal Arithmetic"
] | [
"Definition:Mathematical Induction"
] |
proofwiki-20649 | Ampère-Maxwell Law | Let $\mathbf B$ be a magnetic field due to a steady current $I$ flowing through a wire.
Then:
:$\ds \oint \mathbf B \cdot \rd \mathbf l = \mu_0 I$
where:
:the line integral is taken around a closed path that surrounds the wire
:$\d \mathbf l$ is an infinitesimal vector associated with the closed path
:$\mu_0$ denotes t... | Let $R$ be a simply connected region of space.
Let $\partial S$ be a simple closed contour within $R$.
Let $S$ be an open orientable surface bounded by $\partial S$.
Then:
{{begin-eqn}}
{{eqn | l = \oint_{\partial S} \mathbf B \cdot \rd \mathbf l
| r = \iint_S \paren {\nabla \times \mathbf B} \cdot \mathbf n \rd ... | Let $\mathbf B$ be a [[Definition:Magnetic Field|magnetic field]] due to a [[Definition:Steady Current|steady current]] $I$ flowing through a [[Definition:Wire (Electricity)|wire]].
Then:
:$\ds \oint \mathbf B \cdot \rd \mathbf l = \mu_0 I$
where:
:the [[Definition:Line Integral|line integral]] is taken around a [[Def... | Let $R$ be a [[Definition:Simply Connected|simply connected]] [[Definition:Region|region]] of [[Definition:Ordinary Space|space]].
Let $\partial S$ be a [[Definition:Simple Contour|simple]] [[Definition:Closed Contour|closed contour]] within $R$.
Let $S$ be an [[Definition:Unclosed Surface|open]] [[Definition:Orienta... | Ampère-Maxwell Law | https://proofwiki.org/wiki/Ampère-Maxwell_Law | https://proofwiki.org/wiki/Ampère-Maxwell_Law | [
"Ampère-Maxwell Law",
"Electromagnetism",
"Magnetic Fields",
"Electric Current"
] | [
"Definition:Magnetic Field",
"Definition:Steady Current",
"Definition:Wire (Electricity)",
"Definition:Contour Integral",
"Definition:Cycle (Graph Theory)",
"Definition:Wire (Electricity)",
"Definition:Infinitesimal",
"Definition:Vector Quantity",
"Definition:Cycle (Graph Theory)",
"Definition:Vac... | [
"Definition:Simply Connected",
"Definition:Region",
"Definition:Ordinary Space",
"Definition:Contour/Simple",
"Definition:Contour/Closed",
"Definition:Unclosed Surface",
"Definition:Orientable Manifold",
"Definition:Boundary (Geometry)",
"Kelvin-Stokes Theorem",
"Ampère's Law with Maxwell's Additi... |
proofwiki-20650 | Characterization of Paracompactness in T3 Space | Let $T = \struct {X, \tau}$ be a $T_3$ space.
{{TFAE}}
:$(1): \quad T$ is paracompact
:$(2): \quad$ every open cover of $T$ has a locally finite refinement
:$(3): \quad$ every open cover of $T$ has a closed locally finite refinement
:$(4): \quad$ every open cover of $T$ is even
:$(5): \quad$ every open cover of $T$ has... | === Statement $(1)$ implies Statement $(2)$ ===
{{:Characterization of Paracompactness in T3 Space/Statement 1 implies Statement 2}}{{qed|lemma}} | Let $T = \struct {X, \tau}$ be a [[Definition:T3 Space|$T_3$ space]].
{{TFAE}}
:$(1): \quad T$ is [[Definition:Paracompact Space|paracompact]]
:$(2): \quad$ every [[Definition:Open Cover|open cover]] of $T$ has a [[Definition:Locally Finite Set of Subsets|locally finite]] [[Definition:Refinement of Cover|refinement]]... | === [[Characterization of Paracompactness in T3 Space/Statement 1 implies Statement 2|Statement $(1)$ implies Statement $(2)$]] ===
{{:Characterization of Paracompactness in T3 Space/Statement 1 implies Statement 2}}{{qed|lemma}} | Characterization of Paracompactness in T3 Space | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space | [
"Characterization of Paracompactness in T3 Space",
"T3 Spaces",
"Paracompact Spaces"
] | [
"Definition:T3 Space",
"Definition:Paracompact Space",
"Definition:Open Cover",
"Definition:Locally Finite Set of Subsets",
"Definition:Refinement of Cover",
"Definition:Open Cover",
"Definition:Closed Locally Finite Set of Subsets",
"Definition:Refinement of Cover",
"Definition:Open Cover",
"Defi... | [
"Characterization of Paracompactness in T3 Space/Statement 1 implies Statement 2"
] |
proofwiki-20651 | Characterization of Paracompactness in T3 Space/Statement 1 implies Statement 2 | Let $T = \struct{X, \tau}$ be a topological space.
If $T$ is paracompact then:
:every open cover of $T$ has a locally finite refinement | Let $T$ be paracompact.
By definition of paracompact:
:every open cover of $S$ has an open refinement which is locally finite.
By definition of open refinement:
:every open refinement of a cover is a refinement of the cover.
It follows that:
:every open cover of $T$ has a locally finite refinement. | Let $T = \struct{X, \tau}$ be a [[Definition:Topological Space|topological space]].
If $T$ is [[Definition:Paracompact Space|paracompact]] then:
:every [[Definition:Open Cover|open cover]] of $T$ has a [[Definition:Locally Finite Set of Subsets|locally finite]] [[Definition:Refinement of Cover|refinement]] | Let $T$ be [[Definition:Paracompact Space|paracompact]].
By definition of [[Definition:Paracompact Space|paracompact]]:
:every [[Definition:Open Cover|open cover]] of $S$ has an [[Definition:Open Refinement|open refinement]] which is [[Definition:Locally Finite Cover|locally finite]].
By definition of [[Definition:O... | Characterization of Paracompactness in T3 Space/Statement 1 implies Statement 2 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Statement_1_implies_Statement_2 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Statement_1_implies_Statement_2 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Topological Space",
"Definition:Paracompact Space",
"Definition:Open Cover",
"Definition:Locally Finite Set of Subsets",
"Definition:Refinement of Cover"
] | [
"Definition:Paracompact Space",
"Definition:Paracompact Space",
"Definition:Open Cover",
"Definition:Open Refinement",
"Definition:Locally Finite Cover",
"Definition:Open Refinement",
"Definition:Open Refinement",
"Definition:Cover of Set",
"Definition:Refinement of Cover",
"Definition:Cover of Se... |
proofwiki-20652 | Characterization of Paracompactness in T3 Space/Statement 2 implies Statement 3 | Let $T = \struct{X, \tau}$ be a $T_3$ space.
If every open cover of $T$ has a locally finite refinement then:
:every open cover of $T$ has a closed locally finite refinement | Let every open cover of $T$ have a locally finite refinement.
Let $\UU$ be an open cover of $T$.
Let $\VV = \set{V \in \tau : \exists U \in \UU : V^- \subseteq U}$ where $V^-$ denotes the closure of $V$ in $T$.
==== Lemma 1 ====
{{:Characterization of Paracompactness in T3 Space/Lemma 1}}{{qed|lemma}}
By assumption:
:t... | Let $T = \struct{X, \tau}$ be a [[Definition:T3 Space|$T_3$ space]].
If every [[Definition:Open Cover|open cover]] of $T$ has a [[Definition:Locally Finite Set of Subsets|locally finite]] [[Definition:Refinement of Cover|refinement]] then:
:every [[Definition:Open Cover|open cover]] of $T$ has a [[Definition:Closed L... | Let every [[Definition:Open Cover|open cover]] of $T$ have a [[Definition:Locally Finite Set of Subsets|locally finite]] [[Definition:Refinement of Cover|refinement]].
Let $\UU$ be an [[Definition:Open Cover|open cover]] of $T$.
Let $\VV = \set{V \in \tau : \exists U \in \UU : V^- \subseteq U}$ where $V^-$ denotes ... | Characterization of Paracompactness in T3 Space/Statement 2 implies Statement 3 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Statement_2_implies_Statement_3 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Statement_2_implies_Statement_3 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:T3 Space",
"Definition:Open Cover",
"Definition:Locally Finite Set of Subsets",
"Definition:Refinement of Cover",
"Definition:Open Cover",
"Definition:Closed Locally Finite Set of Subsets",
"Definition:Refinement of Cover"
] | [
"Definition:Open Cover",
"Definition:Locally Finite Set of Subsets",
"Definition:Refinement of Cover",
"Definition:Open Cover",
"Definition:Closure (Topology)",
"Characterization of Paracompactness in T3 Space/Lemma 1",
"Definition:Locally Finite Set of Subsets",
"Definition:Refinement of Cover",
"C... |
proofwiki-20653 | Characterization of Paracompactness in T3 Space/Statement 3 implies Statement 4 | Let $T = \struct{X, \tau}$ be a topological space.
If every open cover of $T$ has a closed locally finite refinement then:
:every open cover of $T$ is even | This follows immediately from Open Cover with Closed Locally Finite Refinement is Even Cover. | Let $T = \struct{X, \tau}$ be a [[Definition:Topological Space|topological space]].
If every [[Definition:Open Cover|open cover]] of $T$ has a [[Definition:Closed Locally Finite Set of Subsets|closed locally finite]] [[Definition:Refinement of Cover|refinement]] then:
:every [[Definition:Open Cover|open cover]] of $T... | This follows immediately from [[Open Cover with Closed Locally Finite Refinement is Even Cover]]. | Characterization of Paracompactness in T3 Space/Statement 3 implies Statement 4 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Statement_3_implies_Statement_4 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Statement_3_implies_Statement_4 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Topological Space",
"Definition:Open Cover",
"Definition:Closed Locally Finite Set of Subsets",
"Definition:Refinement of Cover",
"Definition:Open Cover",
"Definition:Even Cover"
] | [
"Open Cover with Closed Locally Finite Refinement is Even Cover"
] |
proofwiki-20654 | Characterization of Paracompactness in T3 Space/Statement 4 implies Statement 5 | Let $T = \struct{X, \tau}$ be a topological space.
If every open cover of $T$ is even then:
:every open cover of $T$ has an open $\sigma$-discrete refinement | Let every open cover of $T$ be even.
Let $\UU$ be an open cover of $T$.
==== Lemma 8 ====
{{:Characterization of Paracompactness in T3 Space/Lemma 8}}{{qed|lemma}}
By definition of $\sigma$-discrete set of subsets:
:$\AA = \ds \bigcup_{n \in \N} \AA_n$ where $\AA_n$ is a discrete set of subsets for each $n \in \N$.
Let... | Let $T = \struct{X, \tau}$ be a [[Definition:Topological Space|topological space]].
If every [[Definition:Open Cover|open cover]] of $T$ is [[Definition:Even Cover|even]] then:
:every [[Definition:Open Cover|open cover]] of $T$ has an [[Definition:Open Sigma-Discrete Set of Subsets|open $\sigma$-discrete]] [[Definiti... | Let every [[Definition:Open Cover|open cover]] of $T$ be [[Definition:Even Cover|even]].
Let $\UU$ be an [[Definition:Open Cover|open cover]] of $T$.
==== [[Characterization of Paracompactness in T3 Space/Lemma 8|Lemma 8]] ====
{{:Characterization of Paracompactness in T3 Space/Lemma 8}}{{qed|lemma}}
By definition... | Characterization of Paracompactness in T3 Space/Statement 4 implies Statement 5 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Statement_4_implies_Statement_5 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Statement_4_implies_Statement_5 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Topological Space",
"Definition:Open Cover",
"Definition:Even Cover",
"Definition:Open Cover",
"Definition:Open Sigma-Discrete Set of Subsets",
"Definition:Refinement of Cover"
] | [
"Definition:Open Cover",
"Definition:Even Cover",
"Definition:Open Cover",
"Characterization of Paracompactness in T3 Space/Lemma 8",
"Definition:Sigma-Discrete Set of Subsets",
"Definition:Discrete Set of Subsets",
"Definition:Cartesian Product",
"Definition:Product Topology",
"Definition:Product S... |
proofwiki-20655 | Characterization of Paracompactness in T3 Space/Statement 5 implies Statement 6 | Let $T = \struct{X, \tau}$ be a topological space.
If every open cover of $T$ has an open $\sigma$-discrete refinement then:
:every open cover of $T$ has an open $\sigma$-locally finite refinement | Follows immediately from Sigma-Discrete Set of Subsets is Sigma-Locally Finite. | Let $T = \struct{X, \tau}$ be a [[Definition:Topological Space|topological space]].
If every [[Definition:Open Cover|open cover]] of $T$ has an [[Definition:Open Sigma-Discrete Set of Subsets|open $\sigma$-discrete]] [[Definition:Refinement of Cover|refinement]] then:
:every [[Definition:Open Cover|open cover]] of $... | Follows immediately from [[Sigma-Discrete Set of Subsets is Sigma-Locally Finite]]. | Characterization of Paracompactness in T3 Space/Statement 5 implies Statement 6 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Statement_5_implies_Statement_6 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Statement_5_implies_Statement_6 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Topological Space",
"Definition:Open Cover",
"Definition:Open Sigma-Discrete Set of Subsets",
"Definition:Refinement of Cover",
"Definition:Open Cover",
"Definition:Open Sigma-Locally Finite Set of Subsets",
"Definition:Refinement of Cover"
] | [
"Sigma-Discrete Set of Subsets is Sigma-Locally Finite"
] |
proofwiki-20656 | Characterization of Paracompactness in T3 Space/Statement 6 implies Statement 2 | Let $T = \struct{X, \tau}$ be a topological space.
If every open cover of $T$ has an open $\sigma$-locally finite refinement then:
:every open cover of $T$ has a locally finite refinement | Let every open cover of $T$ have an open $\sigma$-locally finite refinement.
Let $\UU$ be an open cover of $T$.
Let $\VV$ be an open $\sigma$-locally finite refinement of $\UU$ {{Hypothesis}}.
From Sigma-Locally Finite Cover has Locally Finite Refinement:
:there exists a locally finite refinement $\AA$ of $\VV$
From Re... | Let $T = \struct{X, \tau}$ be a [[Definition:Topological Space|topological space]].
If every [[Definition:Open Cover|open cover]] of $T$ has an [[Definition:Open Sigma-Locally Finite Set of Subsets|open $\sigma$-locally finite]] [[Definition:Refinement of Cover|refinement]] then:
:every [[Definition:Open Cover|open c... | Let every [[Definition:Open Cover|open cover]] of $T$ have an [[Definition:Open Sigma-Locally Finite Set of Subsets|open $\sigma$-locally finite]] [[Definition:Refinement of Cover|refinement]].
Let $\UU$ be an [[Definition:Open Cover|open cover]] of $T$.
Let $\VV$ be an [[Definition:Open Sigma-Locally Finite Set of... | Characterization of Paracompactness in T3 Space/Statement 6 implies Statement 2 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Statement_6_implies_Statement_2 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Statement_6_implies_Statement_2 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Topological Space",
"Definition:Open Cover",
"Definition:Open Sigma-Locally Finite Set of Subsets",
"Definition:Refinement of Cover",
"Definition:Open Cover",
"Definition:Locally Finite Set of Subsets",
"Definition:Refinement of Cover"
] | [
"Definition:Open Cover",
"Definition:Open Sigma-Locally Finite Set of Subsets",
"Definition:Refinement of Cover",
"Definition:Open Cover",
"Definition:Open Sigma-Locally Finite Set of Subsets",
"Definition:Refinement of Cover",
"Sigma-Locally Finite Cover has Locally Finite Refinement",
"Definition:Lo... |
proofwiki-20657 | Addition of Natural Numbers is Provable/General Form | Let $y \in \N$ be a natural number.
Let $s^a$ denote the application of the successor mapping $a$ times.
Let $\sqbrk a$ denote $\map {s^a} 0$.
Then there exists a formal proof of:
:$\forall x: x + \sqbrk y = \map {s^y} x$
in minimal arithmetic. | Proceed by induction on $y$. | Let $y \in \N$ be a [[Definition:Natural Number|natural number]].
Let $s^a$ denote the application of the [[Definition:Successor Mapping on Natural Numbers|successor mapping]] $a$ times.
Let $\sqbrk a$ denote $\map {s^a} 0$.
Then there exists a [[Definition:Formal Proof|formal proof]] of:
:$\forall x: x + \sqbrk y =... | Proceed by [[Definition:Mathematical Induction|induction]] on $y$. | Addition of Natural Numbers is Provable/General Form | https://proofwiki.org/wiki/Addition_of_Natural_Numbers_is_Provable/General_Form | https://proofwiki.org/wiki/Addition_of_Natural_Numbers_is_Provable/General_Form | [
"Natural Number Addition",
"Proofs by Induction"
] | [
"Definition:Natural Numbers",
"Definition:Successor Mapping on Natural Numbers",
"Definition:Proof System/Formal Proof",
"Definition:Minimal Arithmetic"
] | [
"Definition:Mathematical Induction"
] |
proofwiki-20658 | Electric Field caused by Point Charge | Let $q$ be a point charge.
Let $\mathbf r_q$ be the position vector of $q$.
Let $\map {\mathbf E} {\mathbf r}$ be the electric field strength due to $q$ at a point $P$ whose position vector is $\mathbf r$.
Then:
:$\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \epsilon_0} \dfrac {q \paren {\mathbf r - \mathbf r_q} ... | A specific application of Electric Field Strength from Assemblage of Point Charges for $1$ point charge.
{{qed}} | Let $q$ be a [[Definition:Point Charge|point charge]].
Let $\mathbf r_q$ be the [[Definition:Position Vector|position vector]] of $q$.
Let $\map {\mathbf E} {\mathbf r}$ be the [[Definition:Electric Field Strength|electric field strength]] due to $q$ at a [[Definition:Point|point]] $P$ whose [[Definition:Position Vec... | A specific application of [[Electric Field Strength from Assemblage of Point Charges]] for $1$ [[Definition:Point Charge|point charge]].
{{qed}} | Electric Field caused by Point Charge | https://proofwiki.org/wiki/Electric_Field_caused_by_Point_Charge | https://proofwiki.org/wiki/Electric_Field_caused_by_Point_Charge | [
"Electric Fields",
"Point Charges"
] | [
"Definition:Point Charge",
"Definition:Position Vector",
"Definition:Electric Field Strength",
"Definition:Point",
"Definition:Position Vector",
"Definition:Vacuum Permittivity"
] | [
"Electric Field Strength from Assemblage of Point Charges",
"Definition:Point Charge"
] |
proofwiki-20659 | Magnitude of Electric Field caused by Point Charge | Let $q$ be a point charge.
Let $\map {\mathbf E} {\mathbf r}$ be the electric field strength due to $q$ at a point $P$ whose position vector is $\mathbf r$.
The magnitude of the electric field strength due to $q$ at $P$ is given by:
Then:
:$\size {\map {\mathbf E} {\mathbf r} } = \dfrac {\size q} {4 \pi \epsilon_0 r^2}... | {{begin-eqn}}
{{eqn | l = \map {\mathbf E} {\mathbf r}
| r = \dfrac 1 {4 \pi \epsilon_0} \dfrac {q \paren {\mathbf r - \mathbf r_q} } {\size {\mathbf r - \mathbf r_q}^3}
| c = Electric Field caused by Point Charge
}}
{{eqn | ll= \leadsto
| l = \size {\map {\mathbf E} {\mathbf r} }
| r = \size {\... | Let $q$ be a [[Definition:Point Charge|point charge]].
Let $\map {\mathbf E} {\mathbf r}$ be the [[Definition:Electric Field Strength|electric field strength]] due to $q$ at a [[Definition:Point|point]] $P$ whose [[Definition:Position Vector|position vector]] is $\mathbf r$.
The [[Definition:Magnitude|magnitude]] of... | {{begin-eqn}}
{{eqn | l = \map {\mathbf E} {\mathbf r}
| r = \dfrac 1 {4 \pi \epsilon_0} \dfrac {q \paren {\mathbf r - \mathbf r_q} } {\size {\mathbf r - \mathbf r_q}^3}
| c = [[Electric Field caused by Point Charge]]
}}
{{eqn | ll= \leadsto
| l = \size {\map {\mathbf E} {\mathbf r} }
| r = \siz... | Magnitude of Electric Field caused by Point Charge | https://proofwiki.org/wiki/Magnitude_of_Electric_Field_caused_by_Point_Charge | https://proofwiki.org/wiki/Magnitude_of_Electric_Field_caused_by_Point_Charge | [
"Electric Fields",
"Point Charges"
] | [
"Definition:Point Charge",
"Definition:Electric Field Strength",
"Definition:Point",
"Definition:Position Vector",
"Definition:Magnitude",
"Definition:Electric Field Strength",
"Definition:Distance between Points",
"Definition:Vacuum Permittivity"
] | [
"Electric Field caused by Point Charge",
"Definition:Point",
"Definition:Magnitude",
"Definition:Circle",
"File:Field-lines-positive-charge.png"
] |
proofwiki-20660 | Invertibility of Identity Transformation Plus Product of Two Continuous Linear Transformations | Let $\struct {X, \norm{, \cdot ,} }$ be the normed vector space.
Let $I : X \to X$ be the identity mapping.
Let $\map {CL} X := \map {CL} {X, X}$ be the continuous linear transformation space on $X$.
Suppose $I + A \circ B$ is invertible, where $\circ$ denotes the composition of mappings.
Then $I + B \circ A$ is invert... | {{begin-eqn}}
{{eqn | l = \paren {I + B \circ A} \circ \paren {I - B \circ \paren {I + A \circ B}^{-1} \circ A}
| r = I - B \circ \paren {I + A \circ B}^{-1} \circ A + B \circ A - B \circ A \circ B \circ \paren {I + A \circ B}^{-1} \circ A
}}
{{eqn | r = I - B \circ \paren {I + A \circ B}^{-1} \circ A + B \circ A... | Let $\struct {X, \norm{, \cdot ,} }$ be the [[Definition:Normed Vector Space|normed vector space]].
Let $I : X \to X$ be the [[Definition:Identity Mapping|identity mapping]].
Let $\map {CL} X := \map {CL} {X, X}$ be the [[Definition:Continuous Linear Transformation Space|continuous linear transformation space]] on $X... | {{begin-eqn}}
{{eqn | l = \paren {I + B \circ A} \circ \paren {I - B \circ \paren {I + A \circ B}^{-1} \circ A}
| r = I - B \circ \paren {I + A \circ B}^{-1} \circ A + B \circ A - B \circ A \circ B \circ \paren {I + A \circ B}^{-1} \circ A
}}
{{eqn | r = I - B \circ \paren {I + A \circ B}^{-1} \circ A + B \circ A... | Invertibility of Identity Transformation Plus Product of Two Continuous Linear Transformations | https://proofwiki.org/wiki/Invertibility_of_Identity_Transformation_Plus_Product_of_Two_Continuous_Linear_Transformations | https://proofwiki.org/wiki/Invertibility_of_Identity_Transformation_Plus_Product_of_Two_Continuous_Linear_Transformations | [
"Continuous Linear Transformations",
"Inverse Mappings"
] | [
"Definition:Normed Vector Space",
"Definition:Identity Mapping",
"Definition:Continuous Linear Transformation Space",
"Definition:Invertible Continuous Linear Operator",
"Definition:Composition of Mappings",
"Definition:Invertible Continuous Linear Operator",
"Definition:Inverse of Continuous Linear Ope... | [] |
proofwiki-20661 | Direction of Electric Field caused by Point Charge | Let $q$ be a point charge.
Let $\map {\mathbf E} {\mathbf r}$ be the electric field strength due to $q$ at a point $P$ whose position vector is $\mathbf r$.
The direction of the electric field due to $q$ at $P$ is:
:for positive $q$, directly away from $q$
:for negative $q$, directly towards $q$. | From Electric Field caused by Point Charge:
{{begin-eqn}}
{{eqn | q =
| l = \map {\mathbf E} {\mathbf r}
| r = \dfrac 1 {4 \pi \epsilon_0} \dfrac {q \paren {\mathbf r - \mathbf r_q} } {\size {\mathbf r - \mathbf r_q}^3}
| c = Electric Field caused by Point Charge
}}
{{end-eqn}}
By definition of vecto... | Let $q$ be a [[Definition:Point Charge|point charge]].
Let $\map {\mathbf E} {\mathbf r}$ be the [[Definition:Electric Field Strength|electric field strength]] due to $q$ at a [[Definition:Point|point]] $P$ whose [[Definition:Position Vector|position vector]] is $\mathbf r$.
The [[Definition:Direction|direction]] of... | From [[Electric Field caused by Point Charge]]:
{{begin-eqn}}
{{eqn | q =
| l = \map {\mathbf E} {\mathbf r}
| r = \dfrac 1 {4 \pi \epsilon_0} \dfrac {q \paren {\mathbf r - \mathbf r_q} } {\size {\mathbf r - \mathbf r_q}^3}
| c = [[Electric Field caused by Point Charge]]
}}
{{end-eqn}}
By definitio... | Direction of Electric Field caused by Point Charge | https://proofwiki.org/wiki/Direction_of_Electric_Field_caused_by_Point_Charge | https://proofwiki.org/wiki/Direction_of_Electric_Field_caused_by_Point_Charge | [
"Electric Fields",
"Point Charges"
] | [
"Definition:Point Charge",
"Definition:Electric Field Strength",
"Definition:Point",
"Definition:Position Vector",
"Definition:Direction",
"Definition:Electric Field",
"Definition:Electric Charge/Polarity/Positive",
"Definition:Electric Charge/Polarity/Negative"
] | [
"Electric Field caused by Point Charge",
"Electric Field caused by Point Charge",
"Definition:Vector Subtraction",
"Definition:Vector Quantity",
"Definition:Electric Charge/Polarity/Positive",
"Definition:Direction",
"Definition:Electric Charge/Polarity/Positive",
"Definition:Direction",
"Definition... |
proofwiki-20662 | Direction of Line of Electric Force | Let $\mathbf E$ be an electric field.
Let $L$ be a '''line of force''' within $\mathbf E$.
Then $L$ has one of the following properties:
:$(1): \quad$ Begins on a positively charged body and ends on a negatively charged body
:$(2): \quad$ Begins on a positively charged body and goes to infinity without terminating
:$(3... | {{ProofWanted|Not sure if this is something you can prove or whether it's a physical law. Grant and Phillips state it without elaboration.}} | Let $\mathbf E$ be an [[Definition:Electric Field|electric field]].
Let $L$ be a '''[[Definition:Line of Electric Force|line of force]]''' within $\mathbf E$.
Then $L$ has one of the following [[Definition:Property|properties]]:
:$(1): \quad$ Begins on a [[Definition:Positive Electric Charge|positively charged]] [[D... | {{ProofWanted|Not sure if this is something you can prove or whether it's a physical law. Grant and Phillips state it without elaboration.}} | Direction of Line of Electric Force | https://proofwiki.org/wiki/Direction_of_Line_of_Electric_Force | https://proofwiki.org/wiki/Direction_of_Line_of_Electric_Force | [
"Lines of Electric Force"
] | [
"Definition:Electric Field",
"Definition:Line of Electric Force",
"Definition:Property",
"Definition:Electric Charge/Polarity/Positive",
"Definition:Body",
"Definition:Electric Charge/Polarity/Negative",
"Definition:Body",
"Definition:Electric Charge/Polarity/Positive",
"Definition:Body",
"Definit... | [] |
proofwiki-20663 | Line of Electric Force is Continuous | Let $\mathbf E$ be an electric field.
Let $L$ be a '''line of force''' within $\mathbf E$.
Then, except for the positive charge or negative charge at which $L$ may start or end, $L$ is continuous. | Follows directly from Vector Line is Continuous. | Let $\mathbf E$ be an [[Definition:Electric Field|electric field]].
Let $L$ be a '''[[Definition:Line of Electric Force|line of force]]''' within $\mathbf E$.
Then, except for the [[Definition:Positive Electric Charge|positive charge]] or [[Definition:Negative Electric Charge|negative charge]] at which $L$ may start... | Follows directly from [[Vector Line is Continuous]]. | Line of Electric Force is Continuous | https://proofwiki.org/wiki/Line_of_Electric_Force_is_Continuous | https://proofwiki.org/wiki/Line_of_Electric_Force_is_Continuous | [
"Lines of Electric Force"
] | [
"Definition:Electric Field",
"Definition:Line of Electric Force",
"Definition:Electric Charge/Polarity/Positive",
"Definition:Electric Charge/Polarity/Negative",
"Definition:Continuous Real Function"
] | [
"Vector Line is Continuous"
] |
proofwiki-20664 | Lines of Electric Force do not Cross | Let $\mathbf E$ be an electric field.
Let $L_1$ and $L_2$ be '''lines of force''' within $\mathbf E$.
Then $L_1$ and $L_2$ do not cross over each other. | Follows directly from Vector Lines do not Intersect. | Let $\mathbf E$ be an [[Definition:Electric Field|electric field]].
Let $L_1$ and $L_2$ be '''[[Definition:Line of Electric Force|lines of force]]''' within $\mathbf E$.
Then $L_1$ and $L_2$ do not [[Definition:Intersection (Geometry)|cross]] over each other. | Follows directly from [[Vector Lines do not Intersect]]. | Lines of Electric Force do not Cross | https://proofwiki.org/wiki/Lines_of_Electric_Force_do_not_Cross | https://proofwiki.org/wiki/Lines_of_Electric_Force_do_not_Cross | [
"Lines of Electric Force"
] | [
"Definition:Electric Field",
"Definition:Line of Electric Force",
"Definition:Intersection (Geometry)"
] | [
"Vector Lines do not Intersect"
] |
proofwiki-20665 | Recursive Set is Turing Computable | Let $f: S \to \set {0, 1}$, where $S \subseteq \N$, be a recursive function.
Let $\sqbrk x$ denote the reverse of the base-$2$ representation of $x$, possibly with additional $0$ at the end.
:That is, if the base-$2$ representation of $x$ is
::$\sqbrk {r_m r_{m - 1} \dotsm r_1 r_0}_2$
:then $\sqbrk x$ is:
::$r_0 r_1 \d... | {{Proofread}}
By Recursive Function is URM Computable, there is a URM program that computes $f$.
By Normalized URM Program, let $P$ be the normalization of it.
Let $k = \map \rho P$ be the highest register number used by $P$.
Construct the $k + 1$-tape Turing machine $T$ as follows, where $\map \rho P$ is the highest r... | Let $f: S \to \set {0, 1}$, where $S \subseteq \N$, be a [[Definition:Recursive Function|recursive function]].
Let $\sqbrk x$ denote the reverse of the [[Definition:Basis Representation|base-$2$ representation]] of $x$, possibly with additional $0$ at the end.
:That is, if the [[Definition:Basis Representation|base-$2... | {{Proofread}}
By [[Recursive Function is URM Computable]], there is a [[Definition:URM Program|URM program]] that computes $f$.
By [[Normalized URM Program]], let $P$ be the normalization of it.
Let $k = \map \rho P$ be the highest register number used by $P$.
Construct the $k + 1$-tape Turing machine $T$ as follows... | Recursive Set is Turing Computable | https://proofwiki.org/wiki/Recursive_Set_is_Turing_Computable | https://proofwiki.org/wiki/Recursive_Set_is_Turing_Computable | [
"Recursive Functions",
"Turing Machines"
] | [
"Definition:Recursive/Function",
"Definition:Basis Representation",
"Definition:Basis Representation",
"Definition:Turing Machine"
] | [
"Recursive Function is URM Computable",
"Definition:Unlimited Register Machine/Program",
"Normalized URM Program",
"Definition:Unlimited Register Machine/Program",
"Normalized URM Program"
] |
proofwiki-20666 | Signum Complement Function is Primitive Recursive | Let $\overline \sgn: \N \to \N$ by defined as the signum-bar function.
Then $\overline \sgn$ is primitive recursive. | From Signum Complement Function on Natural Numbers as Characteristic Function, $\map {\overline \sgn} n = \chi_{\set 0} n$.
From Set Containing Only Zero is Primitive Recursive, $\chi_{\set 0}$ is primitive recursive.
Hence the result.
{{qed}}
Category:Signum Function
Category:Primitive Recursive Functions
okzns6v9dhc9... | Let $\overline \sgn: \N \to \N$ by defined as the [[Definition:Signum Complement|signum-bar function]].
Then $\overline \sgn$ is [[Definition:Primitive Recursive Set|primitive recursive]]. | From [[Signum Complement Function on Natural Numbers as Characteristic Function]], $\map {\overline \sgn} n = \chi_{\set 0} n$.
From [[Set Containing Only Zero is Primitive Recursive]], $\chi_{\set 0}$ is [[Definition:Primitive Recursive Set|primitive recursive]].
Hence the result.
{{qed}}
[[Category:Signum Function... | Signum Complement Function is Primitive Recursive | https://proofwiki.org/wiki/Signum_Complement_Function_is_Primitive_Recursive | https://proofwiki.org/wiki/Signum_Complement_Function_is_Primitive_Recursive | [
"Signum Function",
"Primitive Recursive Functions"
] | [
"Definition:Signum Function/Signum Complement",
"Definition:Primitive Recursive/Set"
] | [
"Signum Complement Function on Natural Numbers as Characteristic Function",
"Set Containing Only Zero is Primitive Recursive",
"Definition:Primitive Recursive/Set",
"Category:Signum Function",
"Category:Primitive Recursive Functions"
] |
proofwiki-20667 | Basis Representation is Primitive Recursive | Let $\operatorname{basis} : \N^3 \to \N$ be defined as follows:
:$\map {\operatorname{basis} } {b, n, i} = \begin{cases}
1 & : b = 1 \land i < n \\
r_i & : b > 1 \land i \le m \\
0 & : \text {otherwise}
\end{cases}$
where $\sqbrk {r_m r_{m - 1} \dots r_1 r_0}_b$ is the base-$b$ representation of $n$. | Consider the following function:
:$\map f {b, n, i} = \begin{cases}
n & : i = 0 \\
\map {\operatorname{quot} } {\map f {b, n, i - 1}, b} & : i > 0
\end{cases}$
As $f$ is obtained by Primitive Recursion and:
* Constant Function is Primitive Recursive
* Quotient is Primitive Recursive
Thus, $f$ is primitive recursive.
Ob... | Let $\operatorname{basis} : \N^3 \to \N$ be defined as follows:
:$\map {\operatorname{basis} } {b, n, i} = \begin{cases}
1 & : b = 1 \land i < n \\
r_i & : b > 1 \land i \le m \\
0 & : \text {otherwise}
\end{cases}$
where $\sqbrk {r_m r_{m - 1} \dots r_1 r_0}_b$ is the [[Definition:Basis Representation|base-$b$ represe... | Consider the following function:
:$\map f {b, n, i} = \begin{cases}
n & : i = 0 \\
\map {\operatorname{quot} } {\map f {b, n, i - 1}, b} & : i > 0
\end{cases}$
As $f$ is obtained by [[Definition:Primitive Recursion|Primitive Recursion]] and:
* [[Constant Function is Primitive Recursive]]
* [[Quotient is Primitive Recu... | Basis Representation is Primitive Recursive | https://proofwiki.org/wiki/Basis_Representation_is_Primitive_Recursive | https://proofwiki.org/wiki/Basis_Representation_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Basis Representation"
] | [
"Definition:Primitive Recursion",
"Constant Function is Primitive Recursive",
"Quotient is Primitive Recursive",
"Definition:Primitive Recursive/Function",
"Basis Representation Theorem",
"Division Theorem",
"Basis Representation Theorem",
"Principle of Mathematical Induction/Zero-Based",
"Definitio... |
proofwiki-20668 | Nagata-Smirnov Metrization Theorem/Necessary Condition | A metrizable topological space $T = \struct {S, \tau}$ is regular and has a basis that is $\sigma$-locally finite. | ==== $T$ is Regular ====
We show that $T$ is regular.
{{Recall|Metrizable Space|metrizable space|index = 1}}
{{:Definition:Metrizable Space/Definition 1}}
From Metric Space is Regular Space:
:$T$ is a regular space.
{{qed|lemma}}
{{improve|We can replace the rest of this with Pseudometrizable Space has Sigma-Locally Fi... | A [[Definition:Metrizable Space|metrizable topological space]] $T = \struct {S, \tau}$ is [[Definition:Regular Space|regular]] and has a [[Definition:Basis (Topology)|basis]] that is [[Definition:Sigma-Locally Finite Basis|$\sigma$-locally finite]]. | ==== $T$ is Regular ====
We show that $T$ is [[Definition:Regular Space|regular]].
{{Recall|Metrizable Space|metrizable space|index = 1}}
{{:Definition:Metrizable Space/Definition 1}}
From [[Metric Space is Regular Space]]:
:$T$ is a [[Definition:Regular Space|regular space]].
{{qed|lemma}}
{{improve|We can replac... | Nagata-Smirnov Metrization Theorem/Necessary Condition | https://proofwiki.org/wiki/Nagata-Smirnov_Metrization_Theorem/Necessary_Condition | https://proofwiki.org/wiki/Nagata-Smirnov_Metrization_Theorem/Necessary_Condition | [
"Nagata-Smirnov Metrization Theorem"
] | [
"Definition:Metrizable Space",
"Definition:Regular Space",
"Definition:Basis (Topology)",
"Definition:Sigma-Locally Finite Basis"
] | [
"Definition:Regular Space",
"Metric Space is Regular",
"Definition:Regular Space",
"Pseudometrizable Space has Sigma-Locally Finite Basis",
"Definition:Sigma-Locally Finite Set of Subsets",
"Definition:Basis (Topology)",
"Definition:Open Ball",
"Definition:Open Ball/Radius",
"Open Balls of Same Radi... |
proofwiki-20669 | Nagata-Smirnov Metrization Theorem/Sufficient Condition | Let $T = \struct {S, \tau}$ be a regular topological space.
Let $T$ have a basis that is $\sigma$-locally finite.
Then $T$ is metrizable. | Let $\BB = \ds \bigcup_{n \mathop \in \N} \BB_n$ be a $\sigma$-locally finite basis where $\BB_n$ is locally finite set of subsets for each $n \in \N$.
From T3 Space with Sigma-Locally Finite Basis is Perfectly T4 Space:
:$T$ is a perfectly $T_4$ space
Let $I = \set{\tuple{B, n} : B \in \BB, B \in \BB_n}$.
By definitio... | Let $T = \struct {S, \tau}$ be a [[Definition:Regular Space|regular topological space]].
Let $T$ have a [[Definition:Basis (Topology)|basis]] that is [[Definition:Sigma-Locally Finite Basis|$\sigma$-locally finite]].
Then $T$ is [[Definition:Metrizable|metrizable]]. | Let $\BB = \ds \bigcup_{n \mathop \in \N} \BB_n$ be a [[Definition:Sigma-Locally Finite Basis|$\sigma$-locally finite basis]] where $\BB_n$ is [[Definition:Locally Finite Set of Subsets|locally finite set of subsets]] for each $n \in \N$.
From [[T3 Space with Sigma-Locally Finite Basis is Perfectly T4 Space]]:
:$T$ i... | Nagata-Smirnov Metrization Theorem/Sufficient Condition | https://proofwiki.org/wiki/Nagata-Smirnov_Metrization_Theorem/Sufficient_Condition | https://proofwiki.org/wiki/Nagata-Smirnov_Metrization_Theorem/Sufficient_Condition | [
"Nagata-Smirnov Metrization Theorem"
] | [
"Definition:Regular Space",
"Definition:Basis (Topology)",
"Definition:Sigma-Locally Finite Basis",
"Definition:Metrizable"
] | [
"Definition:Sigma-Locally Finite Basis",
"Definition:Locally Finite Set of Subsets",
"T3 Space with Sigma-Locally Finite Basis is Perfectly T4 Space",
"Definition:Perfectly T4 Space",
"Definition:Perfectly T4 Space",
"Definition:Continuous Mapping (Topology)",
"Definition:Generalized Hilbert Sequence Sp... |
proofwiki-20670 | Total Charge carried By Electron in Hydrogen Atom | Consider an atom of hydrogen $\mathrm H$.
Then:
:$\ds \int_V \map {\rho_{\mathrm {el} } } {\mathbf r} \rd \tau = -\E$
where:
:$\d \tau$ is an infinitesimal volume element
:$\mathbf r$ is the position vector of $\d \tau$
:$V$ is a volume large enough to completely enclose $\mathrm H$
:$\map {\rho_{\mathrm {el} } } {\mat... | The total electric charge on $\mathrm H$ carried by the electron is equal to the total charge on the electron.
By definition of the charge on the electron, this total is $-\E$.
The result follows.
{{qed}} | Consider an [[Definition:Atom (Physics)|atom]] of [[Definition:Hydrogen|hydrogen]] $\mathrm H$.
Then:
:$\ds \int_V \map {\rho_{\mathrm {el} } } {\mathbf r} \rd \tau = -\E$
where:
:$\d \tau$ is an [[Definition:Infinitesimal|infinitesimal]] [[Definition:Volume Element|volume element]]
:$\mathbf r$ is the [[Definition:Po... | The total [[Definition:Electric Charge|electric charge]] on $\mathrm H$ carried by the [[Definition:Electron|electron]] is equal to the total [[Definition:Electric Charge|charge]] on the [[Definition:Electron|electron]].
By definition of the [[Definition:Charge on Electron|charge on the electron]], this total is $-\E$... | Total Charge carried By Electron in Hydrogen Atom | https://proofwiki.org/wiki/Total_Charge_carried_By_Electron_in_Hydrogen_Atom | https://proofwiki.org/wiki/Total_Charge_carried_By_Electron_in_Hydrogen_Atom | [
"Total Charge carried By Electron in Hydrogen Atom",
"Hydrogen",
"Electrons"
] | [
"Definition:Atom (Physics)",
"Definition:Hydrogen",
"Definition:Infinitesimal",
"Definition:Volume Element",
"Definition:Position Vector",
"Definition:Volume",
"Definition:Electric Charge Density",
"Definition:Electric Charge",
"Definition:Electron",
"Definition:Electron Cloud",
"Definition:Elec... | [
"Definition:Electric Charge",
"Definition:Electron",
"Definition:Electric Charge",
"Definition:Electron",
"Definition:Electron/Charge"
] |
proofwiki-20671 | Total Charge carried By Electron in Hydrogen Atom/Cartesian Form | The total electric charge on $\mathrm H$ carried by the electron can be expressed in Cartesian coordinates as:
{{begin-eqn}}
{{eqn | l = \int_{\text {all space} } \map {\rho_{\mathrm {el} } } {\mathbf r} \rd \tau
| r =
| c =
}}
{{eqn | l = \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty ... | Follows directly.
{{qed}} | The total [[Definition:Electric Charge|electric charge]] on $\mathrm H$ carried by the [[Definition:Electron|electron]] can be expressed in [[Definition:Cartesian Coordinates|Cartesian coordinates]] as:
{{begin-eqn}}
{{eqn | l = \int_{\text {all space} } \map {\rho_{\mathrm {el} } } {\mathbf r} \rd \tau
| r =
... | Follows directly.
{{qed}} | Total Charge carried By Electron in Hydrogen Atom/Cartesian Form | https://proofwiki.org/wiki/Total_Charge_carried_By_Electron_in_Hydrogen_Atom/Cartesian_Form | https://proofwiki.org/wiki/Total_Charge_carried_By_Electron_in_Hydrogen_Atom/Cartesian_Form | [
"Total Charge carried By Electron in Hydrogen Atom"
] | [
"Definition:Electric Charge",
"Definition:Electron",
"Definition:Cartesian Coordinate System"
] | [] |
proofwiki-20672 | Binary Sequence Codes are Primitive Recursive | The following function is primitive recursive:
:$\map {\operatorname{bincode} } {n, i} = o_i$
where $o_i$ is the number of $1$ digits between the $i - 1$-th and $i$-th $0$ digit in the base-$2$ representation of $n$, starting from the least significant digits.
As a special case, $\map {\operatorname{bincode} } {n, 0}$ ... | By Basis Representation is Primitive Recursive, we have that $\map {\operatorname{basis} } {b, n, i}$ is primitive recursive.
Consider the function:
:$\map z {n, i} = \begin{cases}
\map {\mu j < n} {\map {\operatorname{basis} } {2, n, j} = 0} & : i = 0 \\
\map {\mu j < n} {j > \map z {n, i - 1} \land \map {\operatornam... | The following function is [[Definition:Primitive Recursive Function|primitive recursive]]:
:$\map {\operatorname{bincode} } {n, i} = o_i$
where $o_i$ is the number of $1$ digits between the $i - 1$-th and $i$-th $0$ digit in the [[Definition:Basis Representation|base-$2$ representation]] of $n$, starting from the least... | By [[Basis Representation is Primitive Recursive]], we have that $\map {\operatorname{basis} } {b, n, i}$ is [[Definition:Primitive Recursive Function|primitive recursive]].
Consider the function:
:$\map z {n, i} = \begin{cases}
\map {\mu j < n} {\map {\operatorname{basis} } {2, n, j} = 0} & : i = 0 \\
\map {\mu j < n... | Binary Sequence Codes are Primitive Recursive | https://proofwiki.org/wiki/Binary_Sequence_Codes_are_Primitive_Recursive | https://proofwiki.org/wiki/Binary_Sequence_Codes_are_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Primitive Recursive/Function",
"Definition:Basis Representation",
"Definition:Basis Representation",
"Definition:Length of String",
"Definition:String/Finite"
] | [
"Basis Representation is Primitive Recursive",
"Definition:Primitive Recursive/Function",
"Definition:Basis Representation",
"Definition:Basis Representation",
"Definition:Increasing/Mapping",
"Definition:Primitive Recursive/Function",
"Constant Function is Primitive Recursive",
"Equality Relation is ... |
proofwiki-20673 | Straight Line Segment is Shortest Path between Two Points | Let $A$ and $B$ be distinct points in a Euclidean space.
Let $\LL$ be the straight line segment lying on both $A$ and $B$.
Then $\LL$ is the shortest line that lies on both $A$ and $B$. | Let $A = \tuple {x_1, y_1}$ and $B = \tuple {x_2, y_2}$ be an arbitrary pair of distinct points embedded in the Cartesian $\R^2$-plane.
Let $\LL$ be the straight line segment from $A$ to $B$.
We are to demonstrate that there does not exist a differentiable real function $f: \R \to \R$ such that:
{{begin-eqn}}
{{eqn | l... | Let $A$ and $B$ be [[Definition:Distinct Elements|distinct]] [[Definition:Point|points]] in a [[Definition:Euclidean Space|Euclidean space]].
Let $\LL$ be the [[Definition:Straight Line Segment|straight line segment]] lying on both $A$ and $B$.
Then $\LL$ is the shortest [[Definition:Line|line]] that lies on both $A$... | Let $A = \tuple {x_1, y_1}$ and $B = \tuple {x_2, y_2}$ be an arbitrary [[Definition:Doubleton|pair]] of [[Definition:Distinct Elements|distinct]] [[Definition:Point|points]] embedded in the [[Definition:Cartesian Plane|Cartesian $\R^2$-plane]].
Let $\LL$ be the [[Definition:Straight Line Segment|straight line segment... | Straight Line Segment is Shortest Path between Two Points | https://proofwiki.org/wiki/Straight_Line_Segment_is_Shortest_Path_between_Two_Points | https://proofwiki.org/wiki/Straight_Line_Segment_is_Shortest_Path_between_Two_Points | [
"Straight Line Segment is Shortest Path between Two Points",
"Euclidean Geometry",
"Straight Lines"
] | [
"Definition:Distinct/Plural",
"Definition:Point",
"Definition:Euclidean Space",
"Definition:Line/Straight Line Segment",
"Definition:Line"
] | [
"Definition:Doubleton",
"Definition:Distinct/Plural",
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Line/Straight Line Segment",
"Definition:Differentiable Mapping/Real Function",
"Definition:Arc Length",
"Definition:Graph of Mapping",
"Definition:Linear Measure/Length",
"Fundament... |
proofwiki-20674 | Discrete Normal Subgroup of Connected Group is Contained in Center | Let $G$ be a connected topological group.
Let $\map Z G$ be the center of $G$.
Let $N$ be a discrete normal subgroup of $G$.
Then:
:$N \subseteq \map Z G$ | Let $h \in N$.
We shall show $h \in \map Z G$
Now, let:
:$A _h := \set {g \in G : g^{-1} h g = h}$
Then, we need to show:
:$A _h = G$
Let $e \in G$ denote the identity.
Since $e \in A _h$, we have $A _h \ne \O$.
Thus, it suffices to show that $A _h$ is clopen in view of {{Defof|Connected Topological Space|index = 4}}.
... | Let $G$ be a [[Definition:Connected Topological Space|connected]] [[Definition:Topological Group|topological group]].
Let $\map Z G$ be the [[Definition:Center of Group|center]] of $G$.
Let $N$ be a [[Definition:Discrete Subgroup|discrete]] [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Then:
:$N \subseteq ... | Let $h \in N$.
We shall show $h \in \map Z G$
Now, let:
:$A _h := \set {g \in G : g^{-1} h g = h}$
Then, we need to show:
:$A _h = G$
Let $e \in G$ denote the [[Definition:Identity of Group|identity]].
Since $e \in A _h$, we have $A _h \ne \O$.
Thus, it suffices to show that $A _h$ is [[Definition:Clopen Set|clo... | Discrete Normal Subgroup of Connected Group is Contained in Center | https://proofwiki.org/wiki/Discrete_Normal_Subgroup_of_Connected_Group_is_Contained_in_Center | https://proofwiki.org/wiki/Discrete_Normal_Subgroup_of_Connected_Group_is_Contained_in_Center | [
"Topological Groups"
] | [
"Definition:Connected Topological Space",
"Definition:Topological Group",
"Definition:Center (Abstract Algebra)/Group",
"Definition:Discrete Subgroup",
"Definition:Normal Subgroup"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Clopen Set",
"Definition:Discrete Subgroup",
"Definition:Open Set",
"Definition:Continuous Mapping",
"Definition:Open Set",
"Definition:Clopen Set",
"Category:Topological Groups"
] |
proofwiki-20675 | Length of Binary Sequence Code is Primitive Recursive | Let the function $\map {\operatorname{bincodelen} } n$ be defined as the least $m$ such that:
:For every $i \ge m$, $\map {\operatorname{bincode} } {n, i} = 0$
where $\operatorname{bincode}$ is defined as in Binary Sequence Codes are Primitive Recursive.
Then, $\operatorname{bincodelen}$ is primitive recursive. | The base-$2$ representation of $n$ is never longer than $n$ digits.
Therefore, there are at most $n$ digits that are $0$.
By definition of $\operatorname{bincode}$, there are thus at most $n$ nonzero values.
Thus, the following definition is correct:
:$\map {\operatorname{bincodelen} } n = \map {\mu m \le n} {\map {\mu... | Let the function $\map {\operatorname{bincodelen} } n$ be defined as the [[Definition:Least Element|least]] $m$ such that:
:For every $i \ge m$, $\map {\operatorname{bincode} } {n, i} = 0$
where $\operatorname{bincode}$ is defined as in [[Binary Sequence Codes are Primitive Recursive]].
Then, $\operatorname{bincodelen... | The [[Definition:Basis Representation|base-$2$ representation]] of $n$ is never longer than $n$ [[Definition:Digit|digits]].
Therefore, there are at most $n$ digits that are $0$.
By definition of $\operatorname{bincode}$, there are thus at most $n$ nonzero values.
Thus, the following definition is correct:
:$\map {\... | Length of Binary Sequence Code is Primitive Recursive | https://proofwiki.org/wiki/Length_of_Binary_Sequence_Code_is_Primitive_Recursive | https://proofwiki.org/wiki/Length_of_Binary_Sequence_Code_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Smallest Element",
"Binary Sequence Codes are Primitive Recursive",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Basis Representation",
"Definition:Digit",
"Definition:Bounded Minimization",
"Definition:Bounded Minimization",
"Binary Sequence Codes are Primitive Recursive",
"Set Operations on Primitive Recursive Relations",
"Ordering Relations are Primitive Recursive",
"Bounded Minimization is Primit... |
proofwiki-20676 | Recursive Relation is Turing Computable | Let $f: S \to \set {0, 1}$, where $S \subseteq \N^k$ be a recursive function.
Let the encoding of some $\tuple {x_1, x_2, \dotsc, x_k} \in \N^k$ be the string:
:$1^{x_1} 0 1^{x_2} 0 1^{x_3} 0 \dotsm 0 1^{x_k}$
Then, there exists a Turing machine such that:
* The input symbols of the machine are $\set {0, 1}$
* The acce... | Define the function $g: S' \to \set {0, 1}$, where $S \subseteq \N$, as follows:
:$\map g x = \map f {\map {\operatorname{bincode} } {x, 0}, \map {\operatorname{bincode} } {x, 1}, \dotsc, \map {\operatorname{bincode} } {x, k} }$
That is, assuming the input is properly formatted, $g$ takes the value of $f$ when applied ... | Let $f: S \to \set {0, 1}$, where $S \subseteq \N^k$ be a [[Definition:Recursive Function|recursive function]].
Let the encoding of some $\tuple {x_1, x_2, \dotsc, x_k} \in \N^k$ be the [[Definition:Finite String|string]]:
:$1^{x_1} 0 1^{x_2} 0 1^{x_3} 0 \dotsm 0 1^{x_k}$
Then, there exists a [[Definition:Turing Mach... | Define the function $g: S' \to \set {0, 1}$, where $S \subseteq \N$, as follows:
:$\map g x = \map f {\map {\operatorname{bincode} } {x, 0}, \map {\operatorname{bincode} } {x, 1}, \dotsc, \map {\operatorname{bincode} } {x, k} }$
That is, assuming the input is properly formatted, $g$ takes the value of $f$ when applied... | Recursive Relation is Turing Computable | https://proofwiki.org/wiki/Recursive_Relation_is_Turing_Computable | https://proofwiki.org/wiki/Recursive_Relation_is_Turing_Computable | [
"Recursive Functions",
"Turing Machines"
] | [
"Definition:Recursive/Function",
"Definition:String/Finite",
"Definition:Turing Machine",
"Definition:Set"
] | [
"Definition:Digit",
"Definition:Basis Representation",
"Constant Function is Primitive Recursive",
"Primitive Recursive Function is Total Recursive Function",
"Definition:Recursive/Function",
"Definition:Substitution (Mathematical Logic)",
"Definition:Recursive/Function",
"Recursive Set is Turing Comp... |
proofwiki-20677 | Recursive Relation is Turing Computable/Corollary | Let $f: S \to \set {0, 1}$, where $S \subseteq \N$ be a recursive function.
Let $x \in \N$ be encoded as:
:$1 1 \dotsm 1$
where $1$ is repeated $x$ times.
Then there exists a Turing machine such that:
* The input symbols of the machine are $\set 1$
* The accepted language are the encodings of $x \in \N$ such that $\map... | By Recursive Relation is Turing Computable, there is a Turing machine $T$ that satisfies the conditions.
The result follows from an identical machine, except that the input symbols are restricted to be only $\set 1$.
{{qed}}
Category:Recursive Functions
Category:Turing Machines
7cg3pk55p5mps73ne6aprqammb1023m | Let $f: S \to \set {0, 1}$, where $S \subseteq \N$ be a [[Definition:Recursive Function|recursive function]].
Let $x \in \N$ be encoded as:
:$1 1 \dotsm 1$
where $1$ is repeated $x$ times.
Then there exists a [[Definition:Turing Machine|Turing machine]] such that:
* The input symbols of the machine are $\set 1$
* The... | By [[Recursive Relation is Turing Computable]], there is a [[Definition:Turing Machine|Turing machine]] $T$ that satisfies the conditions.
The result follows from an identical machine, except that the input symbols are restricted to be only $\set 1$.
{{qed}}
[[Category:Recursive Functions]]
[[Category:Turing Machines... | Recursive Relation is Turing Computable/Corollary | https://proofwiki.org/wiki/Recursive_Relation_is_Turing_Computable/Corollary | https://proofwiki.org/wiki/Recursive_Relation_is_Turing_Computable/Corollary | [
"Recursive Functions",
"Turing Machines"
] | [
"Definition:Recursive/Function",
"Definition:Turing Machine"
] | [
"Recursive Relation is Turing Computable",
"Definition:Turing Machine",
"Category:Recursive Functions",
"Category:Turing Machines"
] |
proofwiki-20678 | Atomic Electric Field at Point within Body of Matter | Let $B$ be a body of matter.
Let $P$ be a point inside $B$ whose position vector is $\mathbf r$.
The '''atomic electric field''' at $P$ is given by:
:$\ds \map {\mathbf E_{\text {atomic} } } {\mathbf r} = \dfrac 1 {4 \pi \varepsilon_0} \int_{\text {all space} } \dfrac {\paren {\mathbf r - \mathbf r'} \map {\rho_{\text ... | From Electric Field Strength from Assemblage of Point Charges, the electric field strength caused by an assemblage of point charges $q_1, q_2, \ldots, q_n$ is given by:
:$\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \epsilon_0} \sum_i \dfrac {\paren {\mathbf r - \mathbf r_i} q_i} {\size {\mathbf r - \mathbf r_i}^... | Let $B$ be a [[Definition:Body|body]] of [[Definition:Matter|matter]].
Let $P$ be a [[Definition:Point|point]] inside $B$ whose [[Definition:Position Vector|position vector]] is $\mathbf r$.
The '''[[Definition:Atomic Electric Field|atomic electric field]]''' at $P$ is given by:
:$\ds \map {\mathbf E_{\text {atomic... | From [[Electric Field Strength from Assemblage of Point Charges]], the [[Definition:Electric Field Strength|electric field strength]] caused by an assemblage of [[Definition:Point Charge|point charges]] $q_1, q_2, \ldots, q_n$ is given by:
:$\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \epsilon_0} \sum_i \dfrac {... | Atomic Electric Field at Point within Body of Matter | https://proofwiki.org/wiki/Atomic_Electric_Field_at_Point_within_Body_of_Matter | https://proofwiki.org/wiki/Atomic_Electric_Field_at_Point_within_Body_of_Matter | [
"Atomic Electric Fields"
] | [
"Definition:Body",
"Definition:Matter",
"Definition:Point",
"Definition:Position Vector",
"Definition:Atomic Electric Field",
"Definition:Infinitesimal",
"Definition:Volume Element",
"Definition:Position Vector",
"Definition:Atomic Charge Density",
"Definition:Electric Charge",
"Definition:Atom ... | [
"Electric Field Strength from Assemblage of Point Charges",
"Definition:Electric Field Strength",
"Definition:Point Charge",
"Definition:Position Vector",
"Definition:Atomic Charge Density",
"Definition:Summation",
"Definition:Definite Integral",
"Definition:Infinitesimal",
"Definition:Volume Elemen... |
proofwiki-20679 | Equivalence of Definitions of Cover of Set | Let $S$ be a set.
{{TFAE|def = Cover of Set}} | === Definition 1 Implies Definition 2 ===
Let $\CC$ be a set of sets such that:
:$\ds S \subseteq \bigcup \CC$
where $\bigcup \CC$ denotes the union of $\CC$.
By definition of subset:
:$\forall s \in S : s \in \ds \bigcup \CC$
By definition of set union:
:$\forall s \in S : \exists C \in \CC : s \in C$
{{qed|lemma}} | Let $S$ be a [[Definition:Set|set]].
{{TFAE|def = Cover of Set}} | === Definition 1 Implies Definition 2 ===
Let $\CC$ be a [[Definition:Set of Sets|set of sets]] such that:
:$\ds S \subseteq \bigcup \CC$
where $\bigcup \CC$ denotes the [[Definition:Union of Set of Sets|union]] of $\CC$.
By definition of [[Definition:Subset|subset]]:
:$\forall s \in S : s \in \ds \bigcup \CC$
By de... | Equivalence of Definitions of Cover of Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Cover_of_Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Cover_of_Set | [
"Covers"
] | [
"Definition:Set"
] | [
"Definition:Set of Sets",
"Definition:Set Union/Set of Sets",
"Definition:Subset",
"Definition:Set Union",
"Definition:Set of Sets",
"Definition:Subset",
"Definition:Subset"
] |
proofwiki-20680 | Unbounded Space Minus Bounded Space is Unbounded | Let $M$ be a metric space.
Let $A \subseteq M$ be unbounded in $M$.
Let $B \subseteq M$ be bounded in $M$.
Then $A \setminus B$ is unbounded in $M$. | {{AimForCont}} $A \setminus B$ is bounded.
Then by definition of bounded:
:$\exists K \in \R: \forall x, y \in A \setminus B: \map d {x, y} \le K$
By definition of bounded, choose some $a_b \in A$ and $d_b \in \R$ such that:
:$\forall x \in B: \map d {x, a_b} \le d_b$
By definition of unbounded, $A$ is not bounded is $... | Let $M$ be a [[Definition:Metric Space|metric space]].
Let $A \subseteq M$ be [[Definition:Unbounded Metric Space|unbounded]] in $M$.
Let $B \subseteq M$ be [[Definition:Bounded Metric Space|bounded]] in $M$.
Then $A \setminus B$ is [[Definition:Unbounded Metric Space|unbounded]] in $M$. | {{AimForCont}} $A \setminus B$ is [[Definition:Bounded Metric Space|bounded]].
Then by definition of [[Definition:Bounded Metric Space/Definition 2|bounded]]:
:$\exists K \in \R: \forall x, y \in A \setminus B: \map d {x, y} \le K$
By definition of [[Definition:Bounded Metric Space/Definition 1|bounded]], choose som... | Unbounded Space Minus Bounded Space is Unbounded/Proof 1 | https://proofwiki.org/wiki/Unbounded_Space_Minus_Bounded_Space_is_Unbounded | https://proofwiki.org/wiki/Unbounded_Space_Minus_Bounded_Space_is_Unbounded/Proof_1 | [
"Bounded Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Bounded Metric Space/Unbounded",
"Definition:Bounded Metric Space",
"Definition:Bounded Metric Space/Unbounded"
] | [
"Definition:Bounded Metric Space",
"Definition:Bounded Metric Space/Definition 2",
"Definition:Bounded Metric Space/Definition 1",
"Definition:Bounded Metric Space/Unbounded",
"Definition:Bounded Metric Space",
"Definition:Bounded Metric Space/Definition 4",
"Definition:Bounded Metric Space/Definition 4... |
proofwiki-20681 | Unbounded Space Minus Bounded Space is Unbounded | Let $M$ be a metric space.
Let $A \subseteq M$ be unbounded in $M$.
Let $B \subseteq M$ be bounded in $M$.
Then $A \setminus B$ is unbounded in $M$. | {{AimForCont}} $A \setminus B$ is bounded.
By Finite Union of Bounded Subsets:
:$\paren {A \setminus B} \cup B$
is bounded.
On the other hand, by {{Defof|Set Difference}}:
:$A \subseteq \paren {A \setminus B} \cup B$
Thus by Subset of Bounded Subset of Metric Space is Bounded, $A$ must be bounded, too.
This is a contra... | Let $M$ be a [[Definition:Metric Space|metric space]].
Let $A \subseteq M$ be [[Definition:Unbounded Metric Space|unbounded]] in $M$.
Let $B \subseteq M$ be [[Definition:Bounded Metric Space|bounded]] in $M$.
Then $A \setminus B$ is [[Definition:Unbounded Metric Space|unbounded]] in $M$. | {{AimForCont}} $A \setminus B$ is [[Definition:Bounded Metric Space|bounded]].
By [[Finite Union of Bounded Subsets]]:
:$\paren {A \setminus B} \cup B$
is [[Definition:Bounded Metric Space|bounded]].
On the other hand, by {{Defof|Set Difference}}:
:$A \subseteq \paren {A \setminus B} \cup B$
Thus by [[Subset of Boun... | Unbounded Space Minus Bounded Space is Unbounded/Proof 2 | https://proofwiki.org/wiki/Unbounded_Space_Minus_Bounded_Space_is_Unbounded | https://proofwiki.org/wiki/Unbounded_Space_Minus_Bounded_Space_is_Unbounded/Proof_2 | [
"Bounded Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Bounded Metric Space/Unbounded",
"Definition:Bounded Metric Space",
"Definition:Bounded Metric Space/Unbounded"
] | [
"Definition:Bounded Metric Space",
"Finite Union of Bounded Subsets",
"Definition:Bounded Metric Space",
"Subset of Bounded Subset of Metric Space is Bounded",
"Definition:Bounded Metric Space",
"Definition:Contradiction"
] |
proofwiki-20682 | Distribution of Macroscopic Electric Charge within Insulator | Let $B$ be a body made out of an electrically insulating substance.
Then it is possible for different volume elements of $B$ to have different (macroscopic) electric fields caused by intrinsic imbalance of the electric charges distributed throughout $B$. | In an electrical insulator, electric charges cannot flow, because of the nature of the substance.
Hence an electric charge in one point within $B$ cannot flow to another point within $B$ under a potential difference caused by the presence of those (macroscopic) electric fields.
{{qed}} | Let $B$ be a [[Definition:Body|body]] made out of an [[Definition:Electrical Insulator|electrically insulating]] [[Definition:Substance|substance]].
Then it is possible for different [[Definition:Volume Element|volume elements]] of $B$ to have different [[Definition:Macroscopic Electric Field|(macroscopic) electric fi... | In an [[Definition:Electrical Insulator|electrical insulator]], [[Definition:Electric Charge|electric charges]] cannot flow, because of the nature of the [[Definition:Substance|substance]].
Hence an [[Definition:Electric Charge|electric charge]] in one [[Definition:Point|point]] within $B$ cannot flow to another [[Def... | Distribution of Macroscopic Electric Charge within Insulator | https://proofwiki.org/wiki/Distribution_of_Macroscopic_Electric_Charge_within_Insulator | https://proofwiki.org/wiki/Distribution_of_Macroscopic_Electric_Charge_within_Insulator | [
"Electrical Insulators",
"Macroscopic Electric Fields"
] | [
"Definition:Body",
"Definition:Electrical Insulator",
"Definition:Substance",
"Definition:Volume Element",
"Definition:Macroscopic Electric Field",
"Definition:Electric Charge"
] | [
"Definition:Electrical Insulator",
"Definition:Electric Charge",
"Definition:Substance",
"Definition:Electric Charge",
"Definition:Point",
"Definition:Point",
"Definition:Potential/Electric",
"Definition:Macroscopic Electric Field"
] |
proofwiki-20683 | Distribution of Macroscopic Electric Charge within Conductor | Let $B$ be a body made out of an electrically conducting substance.
Then it is not possible for different volume elements of $B$ to have different (macroscopic) electric fields caused by intrinsic imbalance of the electric charges distributed throughout $B$. | In an electrical conductor, electric charges can flow, because of the action of conduction electrons within the substance.
Hence an electric charge in one point within $B$ will flow to another point within $B$ under a potential difference caused by differences in (macroscopic) electric fields within $B$.
Hence any imba... | Let $B$ be a [[Definition:Body|body]] made out of an [[Definition:Electrical Conductor|electrically conducting]] [[Definition:Substance|substance]].
Then it is not possible for different [[Definition:Volume Element|volume elements]] of $B$ to have different [[Definition:Macroscopic Electric Field|(macroscopic) electri... | In an [[Definition:Electrical Conductor|electrical conductor]], [[Definition:Electric Charge|electric charges]] can flow, because of the action of [[Definition:Conduction Electron|conduction electrons]] within the [[Definition:Substance|substance]].
Hence an [[Definition:Electric Charge|electric charge]] in one [[Defi... | Distribution of Macroscopic Electric Charge within Conductor | https://proofwiki.org/wiki/Distribution_of_Macroscopic_Electric_Charge_within_Conductor | https://proofwiki.org/wiki/Distribution_of_Macroscopic_Electric_Charge_within_Conductor | [
"Electrical Conductors",
"Macroscopic Electric Fields"
] | [
"Definition:Body",
"Definition:Electrical Conductor",
"Definition:Substance",
"Definition:Volume Element",
"Definition:Macroscopic Electric Field",
"Definition:Electric Charge"
] | [
"Definition:Electrical Conductor",
"Definition:Electric Charge",
"Definition:Conduction Electron",
"Definition:Substance",
"Definition:Electric Charge",
"Definition:Point",
"Definition:Point",
"Definition:Potential/Electric",
"Definition:Macroscopic Electric Field",
"Definition:Macroscopic Electri... |
proofwiki-20684 | T3 Space with Sigma-Locally Finite Basis is T4 Space | Let $T = \struct {S, \tau}$ be a $T_3$ topological space.
Let $\BB$ be a $\sigma$-locally finite basis.
Then:
:$T$ is a $T_4$ space | Let $A$ and $B$ be disjoint closed subsets of $T$.
=== Lemma 1 ===
{{:T3 Space with Sigma-Locally Finite Basis is T4 Space/Lemma 1}}{{qed|lemma}}
From {{Lemma|T3 Space with Sigma-Locally Finite Basis is T4 Space|1}}:
{{begin-itemize}}
{{item||there exists a countable open cover $\UU {{=}} \set {U_n : n \in \N}$ of $A$:... | Let $T = \struct {S, \tau}$ be a [[Definition:T3 Space|$T_3$ topological space]].
Let $\BB$ be a [[Definition:Sigma-Locally Finite Basis|$\sigma$-locally finite basis]].
Then:
:$T$ is a [[Definition:T4 Space|$T_4$ space]] | Let $A$ and $B$ be [[Definition:Disjoint Sets|disjoint]] [[Definition:Closed Subset|closed]] [[Definition:Subset|subsets]] of $T$.
=== [[T3 Space with Sigma-Locally Finite Basis is T4 Space/Lemma 1|Lemma 1]] ===
{{:T3 Space with Sigma-Locally Finite Basis is T4 Space/Lemma 1}}{{qed|lemma}}
From {{Lemma|T3 Space with... | T3 Space with Sigma-Locally Finite Basis is T4 Space/Proof 1 | https://proofwiki.org/wiki/T3_Space_with_Sigma-Locally_Finite_Basis_is_T4_Space | https://proofwiki.org/wiki/T3_Space_with_Sigma-Locally_Finite_Basis_is_T4_Space/Proof_1 | [
"T3 Spaces",
"T4 Spaces",
"T3 Space with Sigma-Locally Finite Basis is T4 Space"
] | [
"Definition:T3 Space",
"Definition:Sigma-Locally Finite Basis",
"Definition:T4 Space"
] | [
"Definition:Disjoint Sets",
"Definition:Closed Subset",
"Definition:Subset",
"T3 Space with Sigma-Locally Finite Basis is T4 Space/Lemma 1",
"Definition:Countable Set",
"Definition:Open Cover",
"Definition:Countable Set",
"Definition:Open Cover",
"Countable Open Covers Condition for Separated Sets",... |
proofwiki-20685 | T3 Space with Sigma-Locally Finite Basis is T4 Space | Let $T = \struct {S, \tau}$ be a $T_3$ topological space.
Let $\BB$ be a $\sigma$-locally finite basis.
Then:
:$T$ is a $T_4$ space | From T3 Space with Sigma-Locally Finite Basis is Paracompact:
:$T$ is a paracompact space.
From $T_3$ Space is Fully $T_4$ iff Paracompact:
:$T$ is a fully $T_4$ space.
From Fully $T_4$ Space is $T_4$:
:$T$ is a $T_4$ space.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:T3 Space|$T_3$ topological space]].
Let $\BB$ be a [[Definition:Sigma-Locally Finite Basis|$\sigma$-locally finite basis]].
Then:
:$T$ is a [[Definition:T4 Space|$T_4$ space]] | From [[T3 Space with Sigma-Locally Finite Basis is Paracompact]]:
:$T$ is a [[Definition:Paracompact Space|paracompact space]].
From [[T3 Space is Fully T4 iff Paracompact|$T_3$ Space is Fully $T_4$ iff Paracompact]]:
:$T$ is a [[Definition:Fully T4 Space|fully $T_4$ space]].
From [[Fully T4 Space is T4|Fully $T_4$ S... | T3 Space with Sigma-Locally Finite Basis is T4 Space/Proof 2 | https://proofwiki.org/wiki/T3_Space_with_Sigma-Locally_Finite_Basis_is_T4_Space | https://proofwiki.org/wiki/T3_Space_with_Sigma-Locally_Finite_Basis_is_T4_Space/Proof_2 | [
"T3 Spaces",
"T4 Spaces",
"T3 Space with Sigma-Locally Finite Basis is T4 Space"
] | [
"Definition:T3 Space",
"Definition:Sigma-Locally Finite Basis",
"Definition:T4 Space"
] | [
"T3 Space with Sigma-Locally Finite Basis is Paracompact",
"Definition:Paracompact Space",
"T3 Space is Fully T4 iff Paracompact",
"Definition:Fully T4 Space",
"Fully T4 Space is T4",
"Definition:T4 Space"
] |
proofwiki-20686 | Characterization of T3 Space | Let $T = \struct {S, \tau}$ be a topological space.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$T$ is a $T_3$ space}}
{{item|(2):|$\forall F \subseteq S : S \setminus F \in \tau, y \in S \setminus F : \exists V \in \tau : y \in V, V^- \cap F {{=}} \O$}}
{{item|(3):|$\forall U \in \tau, y \in U : \exists V \in \tau : y \in... | === Statement $(1)$ implies Statement $(2)$ ===
Let $T$ be a $T_3$ space.
Let $F \subseteq S : S \setminus F \in \tau$.
Let $y \in S \setminus F$.
By definition of $T_3$ space:
:$\exists V, W \in \tau : y \in V, F \subseteq W : V \cap W = \O$
From Subset of Set Difference iff Disjoint Set:
:$V \subseteq S \setminus W$
... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$T$ is a [[Definition:T3 Space|$T_3$ space]]}}
{{item|(2):|$\forall F \subseteq S : S \setminus F \in \tau, y \in S \setminus F : \exists V \in \tau : y \in V, V^- \cap F {{=}} \O$}}
{{item|(3)... | === Statement $(1)$ implies Statement $(2)$ ===
Let $T$ be a [[Definition:T3 Space|$T_3$ space]].
Let $F \subseteq S : S \setminus F \in \tau$.
Let $y \in S \setminus F$.
By definition of [[Definition:T3 Space|$T_3$ space]]:
:$\exists V, W \in \tau : y \in V, F \subseteq W : V \cap W = \O$
From [[Subset of Set D... | Characterization of T3 Space | https://proofwiki.org/wiki/Characterization_of_T3_Space | https://proofwiki.org/wiki/Characterization_of_T3_Space | [
"T3 Spaces"
] | [
"Definition:Topological Space",
"Definition:T3 Space",
"Definition:Closure (Topology)"
] | [
"Definition:T3 Space",
"Definition:T3 Space",
"Subset of Set Difference iff Disjoint Set",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Closure of Subset of Closed Set of Topological Space is Subset",
"Set Difference with Subset is Superset of Set Difference",
"Subset Relation ... |
proofwiki-20687 | Open Star Convex Set Minus Countable Set is Path-Connected in Real Euclidean Space | Let $k \ge 2$ be a natural number.
Let $A \subset \R^k$ be a star convex set, with star center $a \in A$.
Let $G \subset \R^k$ be finite or countable.
Suppose $a \notin G$.
Then $A \setminus G$ is path-connected. | Let $x \in A \setminus G$ be an arbitrary point.
By definition of dimension, there is some vector $\vec v \in \R^k$ that is linearly independent from $\vec {x a}$.
Now, consider the mapping $f: \R \to \R^k$:
:$\map {\vec f} k = \vec v + k \vec {x a}$
The vector $\vec {x a}$ is linearly independent from every vector in ... | Let $k \ge 2$ be a [[Definition:Natural Number|natural number]].
Let $A \subset \R^k$ be a [[Definition:Star Convex Set|star convex set]], with [[Definition:Star Center|star center]] $a \in A$.
Let $G \subset \R^k$ be [[Definition:Finite Set|finite]] or [[Definition:Countable Set|countable]].
Suppose $a \notin G$.
... | Let $x \in A \setminus G$ be an arbitrary [[Definition:Point of Set|point]].
By definition of [[Definition:Dimension of Vector Space|dimension]], there is some [[Definition:Vector|vector]] $\vec v \in \R^k$ that is [[Definition:Linearly Independent|linearly independent]] from $\vec {x a}$.
Now, consider the [[Definit... | Open Star Convex Set Minus Countable Set is Path-Connected in Real Euclidean Space | https://proofwiki.org/wiki/Open_Star_Convex_Set_Minus_Countable_Set_is_Path-Connected_in_Real_Euclidean_Space | https://proofwiki.org/wiki/Open_Star_Convex_Set_Minus_Countable_Set_is_Path-Connected_in_Real_Euclidean_Space | [
"Real Euclidean Spaces",
"Path-Connected Sets"
] | [
"Definition:Natural Numbers",
"Definition:Star Convex Set",
"Definition:Star Convex Set/Star Center",
"Definition:Finite Set",
"Definition:Countable Set",
"Definition:Path-Connected"
] | [
"Definition:Element",
"Definition:Dimension of Vector Space",
"Definition:Vector",
"Definition:Linearly Independent",
"Definition:Mapping",
"Definition:Vector",
"Definition:Linearly Independent",
"Definition:Vector",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Linearly Independent... |
proofwiki-20688 | Macroscopic Electric Field in Conductor in Static Equilibrium is Zero | Let $B$ be a body made out of an electrically conducting substance.
Let $B$ be in static equilibrium.
Then the macroscopic electric field within $B$ is everywhere zero. | By definition of static equilibrium there are no forces upon $B$.
That includes forces caused by an electric field.
From Distribution of Macroscopic Electric Charge within Conductor, the macroscopic electric field is constant throughout $B$.
An electric field arises through differences in electric charges.
As we have s... | Let $B$ be a [[Definition:Body|body]] made out of an [[Definition:Electrical Conductor|electrically conducting]] [[Definition:Substance|substance]].
Let $B$ be in [[Definition:Static Equilibrium|static equilibrium]].
Then the [[Definition:Macroscopic Electric Field|macroscopic electric field]] within $B$ is everywher... | By definition of [[Definition:Static Equilibrium|static equilibrium]] there are no [[Definition:Force|forces]] upon $B$.
That includes forces caused by an [[Definition:Electric Field|electric field]].
From [[Distribution of Macroscopic Electric Charge within Conductor]], the [[Definition:Macroscopic Electric Field|ma... | Macroscopic Electric Field in Conductor in Static Equilibrium is Zero | https://proofwiki.org/wiki/Macroscopic_Electric_Field_in_Conductor_in_Static_Equilibrium_is_Zero | https://proofwiki.org/wiki/Macroscopic_Electric_Field_in_Conductor_in_Static_Equilibrium_is_Zero | [
"Electrical Conductors",
"Macroscopic Electric Fields",
"Static Equilibrium"
] | [
"Definition:Body",
"Definition:Electrical Conductor",
"Definition:Substance",
"Definition:Static Equilibrium",
"Definition:Macroscopic Electric Field",
"Definition:Zero Vector"
] | [
"Definition:Static Equilibrium",
"Definition:Force",
"Definition:Electric Field",
"Distribution of Macroscopic Electric Charge within Conductor",
"Definition:Macroscopic Electric Field",
"Definition:Constant",
"Definition:Electric Field",
"Definition:Electric Charge"
] |
proofwiki-20689 | T3 Space with Sigma-Locally Finite Basis is Paracompact | Let $T = \struct {S, \tau}$ be a $T_3$ topological space.
Let $\BB$ be a $\sigma$-locally finite basis of $T$.
Then $T$ is a paracompact. | Let $\UU$ be an open cover of $T$.
Let $\VV = \set {B \in \BB : \exists U \in \UU : B \subseteq U}$.
Then:
:$\VV \subseteq \BB$
From Subset of Sigma-Locally Finite Set of Subsets is Sigma-Locally Finite:
:$\VV$ is $\sigma$-locally finite
Let $x \in S$.
By definition of open cover:
:$\exists U \in \UU : x \in U$
By defi... | Let $T = \struct {S, \tau}$ be a [[Definition:T3 Space|$T_3$ topological space]].
Let $\BB$ be a [[Definition:Sigma-Locally Finite Basis|$\sigma$-locally finite basis]] of $T$.
Then $T$ is a [[Definition:Paracompact Space|paracompact]]. | Let $\UU$ be an [[Definition:Open Cover|open cover]] of $T$.
Let $\VV = \set {B \in \BB : \exists U \in \UU : B \subseteq U}$.
Then:
:$\VV \subseteq \BB$
From [[Subset of Sigma-Locally Finite Set of Subsets is Sigma-Locally Finite]]:
:$\VV$ is [[Definition:Sigma-Locally Finite Set of Subsets|$\sigma$-locally finite... | T3 Space with Sigma-Locally Finite Basis is Paracompact | https://proofwiki.org/wiki/T3_Space_with_Sigma-Locally_Finite_Basis_is_Paracompact | https://proofwiki.org/wiki/T3_Space_with_Sigma-Locally_Finite_Basis_is_Paracompact | [
"T3 Spaces",
"Sigma-Locally Finite Bases",
"Paracompact Spaces"
] | [
"Definition:T3 Space",
"Definition:Sigma-Locally Finite Basis",
"Definition:Paracompact Space"
] | [
"Definition:Open Cover",
"Subset of Sigma-Locally Finite Set of Subsets is Sigma-Locally Finite",
"Definition:Sigma-Locally Finite Set of Subsets",
"Definition:Open Cover",
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Open Cover",
"Definition:Open Refinement",
"Definition:Open Cover",
... |
proofwiki-20690 | Nonexistence of Complex Matrices whose Commutator equals Identity | Let $d \in \N_{> 0}$ be a positive natural number.
Let $\mathbf A, \mathbf B \in \C^{d \times d}$ be complex matrices.
{{explain|Same applies to real matrices too, yes? I don't have the source to support that, but I don't see why not.<br>This theorem includes the result on the real case, because real matrices are espec... | We have that complex numbers form a commutative ring.
{{AimForCont}} there are $\mathbf A$, $\mathbf B$ such that $\mathbf A \mathbf B - \mathbf B \mathbf A = \mathbf I$.
Then:
{{begin-eqn}}
{{eqn | l = \map \tr {\mathbf I}
| r = d
| c = {{Defof|Trace of Matrix}}
}}
{{eqn | r = \map \tr {\mathbf A \mathbf ... | Let $d \in \N_{> 0}$ be a [[Definition:Positive Number|positive]] [[Definition:Natural Number|natural number]].
Let $\mathbf A, \mathbf B \in \C^{d \times d}$ be [[Definition:Complex Matrix|complex matrices]].
{{explain|Same applies to real matrices too, yes? I don't have the source to support that, but I don't see w... | We have that [[Properties of Complex Numbers|complex numbers form a commutative ring]].
{{AimForCont}} there are $\mathbf A$, $\mathbf B$ such that $\mathbf A \mathbf B - \mathbf B \mathbf A = \mathbf I$.
Then:
{{begin-eqn}}
{{eqn | l = \map \tr {\mathbf I}
| r = d
| c = {{Defof|Trace of Matrix}}
}}
{{e... | Nonexistence of Complex Matrices whose Commutator equals Identity | https://proofwiki.org/wiki/Nonexistence_of_Complex_Matrices_whose_Commutator_equals_Identity | https://proofwiki.org/wiki/Nonexistence_of_Complex_Matrices_whose_Commutator_equals_Identity | [
"Matrices",
"Commutativity"
] | [
"Definition:Positive/Number",
"Definition:Natural Numbers",
"Definition:Complex Matrix",
"Definition:Unit Matrix"
] | [
"Properties of Complex Numbers",
"Trace of Sum of Matrices is Sum of Traces",
"Trace of Product of Matrices",
"Definition:Contradiction"
] |
proofwiki-20691 | Cumulative Distribution Function of Logistic Distribution | Let $X$ be a continuous random variable with the logistic distribution.
Then the cumulative distribution function of $X$ is:
:$\map {F_X} x = \dfrac 1 {1 + \map \exp {- \dfrac {x - \mu} s} }$ | The derivative of $F_X$ is:
{{begin-eqn}}
{{eqn | l = \map {F'_X} x
| r = \paren {\frac 1 {1 + \map \exp {- \frac {x - \mu} s} } }'
}}
{{eqn | r = \frac {\paren 1' \paren {1 + \map \exp {- \frac {x - \mu} s} } - \paren 1 \paren {1 + \map \exp {- \frac {x - \mu} s} }'} {\paren {1 + \map \exp {- \frac {x - \mu} s} ... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with the [[Definition:Logistic Distribution|logistic distribution]].
Then the [[Definition:Cumulative Distribution Function|cumulative distribution function]] of $X$ is:
:$\map {F_X} x = \dfrac 1 {1 + \map \exp {- \dfrac {x - \mu} s} }$ | The derivative of $F_X$ is:
{{begin-eqn}}
{{eqn | l = \map {F'_X} x
| r = \paren {\frac 1 {1 + \map \exp {- \frac {x - \mu} s} } }'
}}
{{eqn | r = \frac {\paren 1' \paren {1 + \map \exp {- \frac {x - \mu} s} } - \paren 1 \paren {1 + \map \exp {- \frac {x - \mu} s} }'} {\paren {1 + \map \exp {- \frac {x - \mu} s} ... | Cumulative Distribution Function of Logistic Distribution | https://proofwiki.org/wiki/Cumulative_Distribution_Function_of_Logistic_Distribution | https://proofwiki.org/wiki/Cumulative_Distribution_Function_of_Logistic_Distribution | [
"Logistic Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Logistic Distribution",
"Definition:Cumulative Distribution Function"
] | [
"Quotient Rule for Derivatives",
"Derivative of Constant",
"Sum Rule for Derivatives",
"Derivative of Composite Function",
"Derivative of Exponential Function",
"Derivative of Constant Multiple",
"Derivative of Constant Multiple",
"Derivative of Identity Function",
"Derivative of Constant",
"Funda... |
proofwiki-20692 | Union of Turing Machines | Let $T_1, T_2$ be Turing machines.
Let $\Sigma_1, \Sigma_2$ be the input symbols of $T_1, T_2$, respectively.
Let $L_1, L_2$ be the languages accepted by $T_1, T_2$, respectively.
There exists a Turing machine $T$ such that:
* The input symbols of $T$ are $\Sigma_1 \cup \Sigma_2$.
* The language accepted by $T$ is $L_1... | Let:
:$T_1 = \tuple {Q_1, \Sigma_1, \Gamma_1, \delta_1, q_1, B, F_1}$
:$T_2 = \tuple {Q_2, \Sigma_2, \Gamma_2, \delta_2, q_2, B, F_2}$
As implied by the notation, the blank symbols of the two machines will be identified with each other, and denoted as $B$.
Construct the $2$-tape Turing machine $T' = \tuple{Q, \Sigma, \... | Let $T_1, T_2$ be [[Definition:Turing Machine|Turing machines]].
Let $\Sigma_1, \Sigma_2$ be the input symbols of $T_1, T_2$, respectively.
Let $L_1, L_2$ be the languages accepted by $T_1, T_2$, respectively.
There exists a [[Definition:Turing Machine|Turing machine]] $T$ such that:
* The input symbols of $T$ are $... | Let:
:$T_1 = \tuple {Q_1, \Sigma_1, \Gamma_1, \delta_1, q_1, B, F_1}$
:$T_2 = \tuple {Q_2, \Sigma_2, \Gamma_2, \delta_2, q_2, B, F_2}$
As implied by the notation, the blank symbols of the two machines will be identified with each other, and denoted as $B$.
Construct the [[Definition:Multitape Turing Machine|$2$-tape ... | Union of Turing Machines | https://proofwiki.org/wiki/Union_of_Turing_Machines | https://proofwiki.org/wiki/Union_of_Turing_Machines | [
"Turing Machines"
] | [
"Definition:Turing Machine",
"Definition:Turing Machine"
] | [
"Definition:Multitape Turing Machine",
"Multitape Turing Machine Reduces to Turing Machine",
"Category:Turing Machines"
] |
proofwiki-20693 | Finite State Machine is Turing Computable | Let $F = \tuple {S, A, I, \Sigma, T}$ be a finite state machine.
Then there exists a Turing machine $T$ that:
* Has input language $\Sigma$.
* Accepts exactly the same language as $F$.
* Halts on every input. | Define the Turing machine:
:$T = \tuple {S \cup \set {H}, \Sigma, \Sigma \cup \set B, \delta, I, B, \set H}$
where:
:$\map \delta {s, \sigma} = \tuple {\map T {s, \sigma}, \sigma, R}$ if $q \in S$ and $\sigma \ne B$
:$\map \delta {s, B} = \tuple {H, B, R}$ if $s \in A$
The machine behaves identically to $F$ while the i... | Let $F = \tuple {S, A, I, \Sigma, T}$ be a [[Definition:Finite State Machine|finite state machine]].
Then there exists a [[Definition:Turing Machine|Turing machine]] $T$ that:
* Has input language $\Sigma$.
* Accepts exactly the same language as $F$.
* Halts on every input. | Define the [[Definition:Turing Machine|Turing machine]]:
:$T = \tuple {S \cup \set {H}, \Sigma, \Sigma \cup \set B, \delta, I, B, \set H}$
where:
:$\map \delta {s, \sigma} = \tuple {\map T {s, \sigma}, \sigma, R}$ if $q \in S$ and $\sigma \ne B$
:$\map \delta {s, B} = \tuple {H, B, R}$ if $s \in A$
The machine behaves... | Finite State Machine is Turing Computable | https://proofwiki.org/wiki/Finite_State_Machine_is_Turing_Computable | https://proofwiki.org/wiki/Finite_State_Machine_is_Turing_Computable | [
"Turing Machines"
] | [
"Definition:Finite State Machine",
"Definition:Turing Machine"
] | [
"Definition:Turing Machine",
"Category:Turing Machines"
] |
proofwiki-20694 | Conjunction of Finite State Machines | Let $\Sigma$ be a finite set.
Let $F_1 = \tuple {S_1, A_1, I_1, \Sigma, T_1}$ and $F_2 = \tuple {S_2, A_2, I_2, \Sigma, T_2}$ be finite state machines.
Then there exists a finite state machine $F$ such that $\map L F = \map L {F_1} \cap \map L {F_2}$. | Construct $F = \tuple {S_1 \times S_2, A_1 \times A_2, \tuple {I_1, I_2}, \Sigma, T}$ where:
:$\map T {s_1, s_2, \Sigma} = \tuple {\map {T_1} {s_1}, \map {T_2} {s_2} }$
The machines are simultaneously emulated in the first and second components of the state tuple.
When the input word is exhausted, $F$ accepts {{iff}} b... | Let $\Sigma$ be a [[Definition:Finite Set|finite set]].
Let $F_1 = \tuple {S_1, A_1, I_1, \Sigma, T_1}$ and $F_2 = \tuple {S_2, A_2, I_2, \Sigma, T_2}$ be [[Definition:Finite State Machine|finite state machines]].
Then there exists a [[Definition:Finite State Machine|finite state machine]] $F$ such that $\map L F = \... | Construct $F = \tuple {S_1 \times S_2, A_1 \times A_2, \tuple {I_1, I_2}, \Sigma, T}$ where:
:$\map T {s_1, s_2, \Sigma} = \tuple {\map {T_1} {s_1}, \map {T_2} {s_2} }$
The machines are simultaneously emulated in the first and second components of the state tuple.
When the input word is exhausted, $F$ accepts {{iff}}... | Conjunction of Finite State Machines | https://proofwiki.org/wiki/Conjunction_of_Finite_State_Machines | https://proofwiki.org/wiki/Conjunction_of_Finite_State_Machines | [
"Abstract Machines"
] | [
"Definition:Finite Set",
"Definition:Finite State Machine",
"Definition:Finite State Machine"
] | [
"Category:Abstract Machines"
] |
proofwiki-20695 | Disjunction of Finite State Machines | Let $\Sigma$ be a finite set.
Let $F_1 = \tuple {S_1, A_1, I_1, \Sigma, T_1}$ and $F_2 = \tuple {S_2, A_2, I_2, \Sigma, T_2}$ be finite state machines.
Then there exists a finite state machine $F$ such that $\map L F = \map L {F_1} \cup \map L {F_2}$. | Construct $F = \tuple {S_1 \times S_2, A_1 \times S_2 \cup S_1 \times A_2, \tuple {I_1, I_2}, \Sigma, T}$ where:
:$\map T {s_1, s_2, \Sigma} = \tuple {\map {T_1} {s_1}, \map {T_2} {s_2} }$
The machines are simultaneously emulated in the first and second components of the state tuple.
When the input word is exhausted, $... | Let $\Sigma$ be a [[Definition:Finite Set|finite set]].
Let $F_1 = \tuple {S_1, A_1, I_1, \Sigma, T_1}$ and $F_2 = \tuple {S_2, A_2, I_2, \Sigma, T_2}$ be [[Definition:Finite State Machine|finite state machines]].
Then there exists a [[Definition:Finite State Machine|finite state machine]] $F$ such that $\map L F = \... | Construct $F = \tuple {S_1 \times S_2, A_1 \times S_2 \cup S_1 \times A_2, \tuple {I_1, I_2}, \Sigma, T}$ where:
:$\map T {s_1, s_2, \Sigma} = \tuple {\map {T_1} {s_1}, \map {T_2} {s_2} }$
The machines are simultaneously emulated in the first and second components of the state tuple.
When the input word is exhausted,... | Disjunction of Finite State Machines | https://proofwiki.org/wiki/Disjunction_of_Finite_State_Machines | https://proofwiki.org/wiki/Disjunction_of_Finite_State_Machines | [
"Abstract Machines"
] | [
"Definition:Finite Set",
"Definition:Finite State Machine",
"Definition:Finite State Machine"
] | [
"Category:Abstract Machines"
] |
proofwiki-20696 | Negation of Finite State Machine | Let $F = \tuple {S, A, I, \Sigma, T}$ be a finite state machine.
Then there exists a finite state machine $F'$ such that $\map L {F'} = \map \complement {\map L F}$. | Construct $F' = \tuple {S, S \setminus A, I, \Sigma, T}$.
The state transitions are identical to those of $F$.
However, when the input word is exhausted, the final state is an accepting state {{iff}} it is not an accepting state in $F$, as required.
{{qed}}
Category:Abstract Machines
fr1mwzhwds5tq73o9vka8feontorjrm | Let $F = \tuple {S, A, I, \Sigma, T}$ be a [[Definition:Finite State Machine|finite state machine]].
Then there exists a [[Definition:Finite State Machine|finite state machine]] $F'$ such that $\map L {F'} = \map \complement {\map L F}$. | Construct $F' = \tuple {S, S \setminus A, I, \Sigma, T}$.
The state transitions are identical to those of $F$.
However, when the input word is exhausted, the final state is an accepting state {{iff}} it is not an accepting state in $F$, as required.
{{qed}}
[[Category:Abstract Machines]]
fr1mwzhwds5tq73o9vka8feontor... | Negation of Finite State Machine | https://proofwiki.org/wiki/Negation_of_Finite_State_Machine | https://proofwiki.org/wiki/Negation_of_Finite_State_Machine | [
"Abstract Machines"
] | [
"Definition:Finite State Machine",
"Definition:Finite State Machine"
] | [
"Category:Abstract Machines"
] |
proofwiki-20697 | Symbol Count by Finite State Machine | Let $\Sigma$ be a finite set.
Let $\Pi \subseteq \Sigma$.
For a finite sequence $w = \sequence {a_1, a_2, \dotsc, a_n}$ of elements of $\Sigma$:
:Let $\map c w = \size {\set {i : a_i \in \Pi} }$
Let $k \in \N$.
Then, for each of the following, there exists a finite state machine whose accepted languages is all $w \in \... | === $\map c w < k$ ===
Define $F = \tuple {S, A, I, \Sigma, T}$ where:
:$S = \set {s_0, s_1, \dotsc, s_k}$
:$A = \set {s_0, s_1, \dotsc, s_{k - 1} }$
:$I = s_0$
:$\map T {s_i, \sigma} = \begin{cases}
s_{i + 1} & : \sigma \in \Pi \land i < k \\
s_i & : \text {otherwise}
\end{cases}$
The machine being in the state $s_i$ ... | Let $\Sigma$ be a [[Definition:Finite Set|finite set]].
Let $\Pi \subseteq \Sigma$.
For a [[Definition:Finite Sequence|finite sequence]] $w = \sequence {a_1, a_2, \dotsc, a_n}$ of [[Definition:Element of Set|elements]] of $\Sigma$:
:Let $\map c w = \size {\set {i : a_i \in \Pi} }$
Let $k \in \N$.
Then, for each of ... | === $\map c w < k$ ===
Define $F = \tuple {S, A, I, \Sigma, T}$ where:
:$S = \set {s_0, s_1, \dotsc, s_k}$
:$A = \set {s_0, s_1, \dotsc, s_{k - 1} }$
:$I = s_0$
:$\map T {s_i, \sigma} = \begin{cases}
s_{i + 1} & : \sigma \in \Pi \land i < k \\
s_i & : \text {otherwise}
\end{cases}$
The machine being in the state $s_i... | Symbol Count by Finite State Machine | https://proofwiki.org/wiki/Symbol_Count_by_Finite_State_Machine | https://proofwiki.org/wiki/Symbol_Count_by_Finite_State_Machine | [
"Abstract Machines"
] | [
"Definition:Finite Set",
"Definition:Finite Sequence",
"Definition:Element",
"Definition:Finite State Machine"
] | [] |
proofwiki-20698 | Intersection of Turing Machines | Let $T_1, T_2$ be Turing machines.
Let $\Sigma_1, \Sigma_2$ be the input symbols of $T_1, T_2$, respectively.
Let $L_1, L_2$ be the languages accepted by $T_1, T_2$, respectively.
There exists a Turing machine $T$ such that:
* The input symbols of $T$ are $\Sigma_1 \cap \Sigma_2$.
* The language accepted by $T$ is $L_1... | Let:
:$T_1 = \tuple {Q_1, \Sigma_1, \Gamma_1, \delta_1, q_1, B, F_1}$
:$T_2 = \tuple {Q_2, \Sigma_2, \Gamma_2, \delta_2, q_2, B, F_2}$
As implied by the notation, the blank symbols of the two machines will be identified with each other, and denoted as $B$.
Construct the $2$-tape Turing machine $T' = \tuple{Q, \Sigma, \... | Let $T_1, T_2$ be [[Definition:Turing Machine|Turing machines]].
Let $\Sigma_1, \Sigma_2$ be the input symbols of $T_1, T_2$, respectively.
Let $L_1, L_2$ be the languages accepted by $T_1, T_2$, respectively.
There exists a [[Definition:Turing Machine|Turing machine]] $T$ such that:
* The input symbols of $T$ are $... | Let:
:$T_1 = \tuple {Q_1, \Sigma_1, \Gamma_1, \delta_1, q_1, B, F_1}$
:$T_2 = \tuple {Q_2, \Sigma_2, \Gamma_2, \delta_2, q_2, B, F_2}$
As implied by the notation, the blank symbols of the two machines will be identified with each other, and denoted as $B$.
Construct the [[Definition:Multitape Turing Machine|$2$-tape ... | Intersection of Turing Machines | https://proofwiki.org/wiki/Intersection_of_Turing_Machines | https://proofwiki.org/wiki/Intersection_of_Turing_Machines | [
"Turing Machines"
] | [
"Definition:Turing Machine",
"Definition:Turing Machine"
] | [
"Definition:Multitape Turing Machine",
"Category:Turing Machines"
] |
proofwiki-20699 | Real and Imaginary Parts of Holomorphic Function are Harmonic | Let $D \subseteq \C$ be an open subset of the set of complex numbers $\C$.
Let $f: D \to \C$ be a holomorphic complex function on $D$.
Let $u, v: \set {\tuple {x, y} \in \R^2: x + i y = z \in D} \to \R$ be the two real-valued functions defined as:
{{begin-eqn}}
{{eqn | l = \map u {x, y}
| r = \map \Re {\map f {x ... | By Cauchy-Riemann Equations, $u$ and $v$ satisfy:
{{begin-eqn}}
{{eqn | n = 1
| l = \dfrac {\partial u} {\partial x}
| r = \dfrac {\partial v} {\partial y}
}}
{{eqn | n = 2
| l = \dfrac {\partial u} {\partial y}
| r = -\dfrac {\partial v} {\partial x}
}}
{{end-eqn}}
{{Explain|Need to justify som... | Let $D \subseteq \C$ be an [[Definition:Open Set (Complex Analysis)|open]] [[Definition:Subset|subset]] of the [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] $\C$.
Let $f: D \to \C$ be a [[Definition:Holomorphic Complex Function|holomorphic complex function]] on $D$.
Let $u, v: \set {\tuple {... | By [[Cauchy-Riemann Equations]], $u$ and $v$ satisfy:
{{begin-eqn}}
{{eqn | n = 1
| l = \dfrac {\partial u} {\partial x}
| r = \dfrac {\partial v} {\partial y}
}}
{{eqn | n = 2
| l = \dfrac {\partial u} {\partial y}
| r = -\dfrac {\partial v} {\partial x}
}}
{{end-eqn}}
{{Explain|Need to just... | Real and Imaginary Parts of Holomorphic Function are Harmonic | https://proofwiki.org/wiki/Real_and_Imaginary_Parts_of_Holomorphic_Function_are_Harmonic | https://proofwiki.org/wiki/Real_and_Imaginary_Parts_of_Holomorphic_Function_are_Harmonic | [
"Holomorphic Functions",
"Harmonic Functions"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Subset",
"Definition:Set",
"Definition:Complex Number",
"Definition:Holomorphic Function/Complex Plane",
"Definition:Real-Valued Function",
"Definition:Harmonic Function"
] | [
"Cauchy-Riemann Equations",
"Definition:Partial Derivative",
"Definition:Partial Derivative",
"Clairaut's Theorem",
"Definition:Harmonic Function",
"Definition:Partial Derivative",
"Definition:Partial Derivative",
"Clairaut's Theorem ",
"Definition:Harmonic Function"
] |
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